2$ , then the rate $n^{-(p-2)/(2(p+1))}$ is sharp; if ${\\mathbb {E}}e^{t\\xi _1}<\\infty ,\\ |t|<\\delta , \\ \\delta >0,$ then the rate $\\ln n/\\sqrt{n}$ is sharp.","For details, see [7]; earlier works on the subject include [17], [18], [25].","A more general approach to investigating the rate of convergence is to find a bound on $\\sup _{\\varphi \\in \\mathcal {G}} |{\\mathbb {E}}\\varphi (S_n)-{\\mathbb {E}}\\varphi (W)|$ for a suitable class $\\mathcal {G}$ of functions $\\varphi : {\\mathcal {C}}(0,T)\\rightarrow {\\mathbb {R}}$ .","Barbour [3] used an infinite-dimensional version of Stein's method to establish the benchmark result $|{\\mathbb {E}}\\varphi (S_n)-{\\mathbb {E}}\\varphi (W)| \\le \\frac{C}{\\sqrt{n}}\\Big ( \\sqrt{\\ln n} + {\\mathbb {E}}|\\xi _1|^3\\Big )$ for a certain (rather restrictive) class $\\mathcal {G}$ ; the restrictive nature of this class ensures that there is no contradiction with [7] or [25].","For various other $\\mathcal {G}$ , there are bounds of the form $|{\\mathbb {E}}\\varphi (S_n)-{\\mathbb {E}}\\varphi (W)| \\le \\frac{C}{{n}^r},\\ 00: \\mu (A)\\le \\nu (A^{\\varepsilon })+\\varepsilon ,\\ A\\in \\mathcal {B}(E)\\rbrace ,$ where $A^{\\varepsilon }=\\lbrace x\\in E: \\inf \\limits _{y\\in A} \\rho (x,y)\\le \\varepsilon \\rbrace .$ We have $\\mathrm {d}_{_{BL}}(\\mu ,\\nu )\\le \\mathrm {d}_{{w}}(\\mu ,\\nu )$ (by definition), $\\mathrm {d}_{_{LP}}(\\mu ,\\nu )\\le \\sqrt{\\mathrm {d}_{_{BL}}(\\mu ,\\nu )}$ ([9]), and $\\mathrm {d}_{_{BL}}(\\mu ,\\nu )\\le 4 \\mathrm {d}_{_{LP}}(\\mu ,\\nu )$ ([9]).","In particular, convergence in the Wasserstein-1 metric implies weak convergence, that is, convergence in either Lévy-Prokhorov or bounded Lipschitz metric; the converse is not always true [9]; in fact, the diagram in [11] suggests that the Wasserstein-1 metric $\\mathrm {d}_{{w}}$ is the strongest possible for the CLT-type problems in function spaces.","Still, a sharp rate of convergence in one metric does not directly lead to a sharp rate in any other metric.","The invariance principle can hold if independence requirement for the random variables $\\xi _k$ is relaxed, for example, to a strictly stationary and ergodic martingale difference [20], or a stationary Markov process satisfying Doebling's condition [13] [where a bound of the type (REF ) is also established].","In continuous time, if $X=X(t),\\ t\\in {\\mathbb {R}},$ is a strictly stationary process with mean zero and covariance function $R(t)={\\mathbb {E}}\\big (X(t)X(0)\\big )$ satisfying $\\int _{-\\infty }^{+\\infty } R(t)\\, dt =1$ , then, under some additional conditions of weak dependence, the sequence of processes $S_n(t)=\\frac{1}{\\sqrt{n}}\\int _0^{nt} X(s)\\, ds,\\ n=1,2,\\ldots , t\\in [0,1],$ converges weakly to the standard Brownian motion; cf.","[20] or [14].","The objective of this paper is to show that if $X$ is a stationary Gauss-Markov process, in either discrete or continuous time, then the Wasserstein-1 distance between $S_n$ and $W$ in the space of continuous functions is of order $\\big (n^{-1}\\ln n\\big )^{1/2}$ .","In other words, if $S_n$ is Gaussian, then the convergence rate in Wasserstein-1 metric is slightly faster than the Lévy-Prokhorov rate $\\ln n/\\sqrt{n}$ .","This difference does not contradict the results from [7] and [25], and the discrepancy by a $\\sqrt{\\ln n}$ factor can be explained as follows: in the limit $\\sigma \\rightarrow 0+$ , the distance between a Gaussian distribution with mean zero and variance $\\sigma ^2$ and a point mass at zero is of order $\\sigma $ in the Wasserstein-1 metric, but it is of order $\\sigma \\sqrt{|\\ln \\sigma |}$ in the Lévy-Prokhorov metric.","Section discusses (the easier) continuous-time case (REF ).","Discrete-time case, a generalization of (REF ) for a stationary Gaussian sequence $\\lbrace \\xi _k,\\ k\\ge 0\\rbrace $ , is in Section .","In Section , the results are applied to weak approximation for some ordinary differential equations with additive noise.","Section is a summary.","Traditionally, models (REF ) and (REF ) are studied on a bounded time interval $[0,T]$ , and the index parameter $n=1,2,\\ldots $ is discrete.","In this paper, the time interval is $(0,+\\infty )$ and the index parameter $\\kappa \\ge 1$ is not necessarily an integer.","The following two properties of Gaussian processes will be used on several occasions: 1.","The Borell-TIS inequality [1]: If $X=X(t), \\ t\\in \\mathcal {T},$ is a zero-mean Gaussian process indexed by the set $\\mathcal {T}$ , and $\\mathbb {P}\\left( \\sup \\limits _{t\\in \\mathcal {T}} X(t) < \\infty \\right) =1$ , then ${\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} |X(t)| < \\infty $ and, with $X^*={\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} X(t),\\ \\ \\sigma _X^2=\\sup \\limits _{t\\in \\mathcal {T}}{\\mathbb {E}}X^2(t)$ , $\\mathbb {P}\\Big (\\sup \\limits _{t\\in \\mathcal {T}} X(t) - X^*>x\\Big )\\le e^{-x^2/(2\\sigma _X^2)},\\ x>0.$ 2.","The Fernique-Sudakov inequality [1]: If $X=X(t), \\ Y=Y(t)\\ t\\in \\mathcal {T},$ are zero-mean Gaussian processes indexed by the set $\\mathcal {T}$ , and, for all $t,s\\in \\mathcal {T}$ , ${\\mathbb {E}}|X(t)-X(s)|^2 \\le {\\mathbb {E}}|Y(t)-Y(s)|^2$ , then ${\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} X(t) \\le {\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} Y(t).$"],["Continuous Time","Let $X=X(t),\\ t\\in {\\mathbb {R}},$ be a (continuous version of a) stationary Gaussian process with mean zero and covariance ${\\mathbb {E}}X(t)X(s)=e^{-2|t-s|}.$ In particular, $X(t)$ is a standard Gaussian random variable for every $t$ .","Equivalent characterizations of $X$ are as follows: $X(t)&=e^{-t}W(e^{2t});\\\\X(t)&=2\\int _{-\\infty }^t e^{-2(t-s)}dW(s);\\\\dX(t)&=-2X(t)dt+2\\,dW(t),\\ t\\ge 0.$ In (REF ), (), and (), $W=W(t), \\ t\\ge 0,$ is a standard Brownian motion; in (), when $t<0$ , $W(t)=V(-t)$ for an independent copy $V$ of $W$ .","The initial condition $X(0)$ in () is a standard Gaussian random variable independent of $W$ .","Given a real number $\\kappa \\ge 1$ , we define $W^{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\int _0^{\\kappa t} X(s)\\, ds,\\ \\ t\\ge 0.$ Denote by ${\\mathcal {C}}_{(0)}$ the collection of continuous functions $f=f(t)$ on $[0,+\\infty )$ such that $f(0)=0,\\ \\lim _{t\\rightarrow +\\infty } \\frac{|f(t)|}{t}=0.$ Endowed with the norm $\\Vert f\\Vert _{(0)}=\\sup _{t>0}\\frac{|f(t)|}{1+t},$ ${\\mathcal {C}}_{(0)}$ becomes a separable Banach space; cf.","[8].","We have $\\mathbb {P}(W\\in {\\mathcal {C}}_{(0)})=1$ (either by the law of large numbers for square integrable martingales or using that $t\\mapsto tW(1/t)$ is a standard Brownian motion), and also $\\mathbb {P}(W^{\\kappa }\\in {\\mathcal {C}}_{(0)})=1$ , $\\kappa \\ge 1$ , because the ergodic theorem implies $\\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\int _0^t X(s)\\, ds = {\\mathbb {E}}X(0) = 0$ with probability one.","Proposition 2.1 There exists a constant $C_X$ such that, for every $\\kappa \\ge 1$ and every function $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfying $|\\varphi (f)-\\varphi (g)|\\le \\Vert f-g\\Vert _{(0)},$ we have $\\Big |{\\mathbb {E}}\\varphi \\big (W^{\\kappa }\\big )-{\\mathbb {E}}\\varphi (W)\\Big |\\le C_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Using (), $X(t)=X(0)e^{-2t}+2\\int _0^t e^{-2(t-s)}\\, dW(s).$ Changing the order of integration (stochastic Fubini theorem [23]), $W^{\\kappa }(t)=\\kappa ^{-1/2}W(\\kappa t)+\\frac{X(0)-X(\\kappa t)}{2\\sqrt{\\kappa }}.$ Define $V^{\\kappa }(t)=\\kappa ^{-1/2}W(\\kappa t),\\ \\ \\ X^{\\kappa }(t)=\\frac{X(\\kappa t)-X(0)}{2}.$ Because $t\\mapsto V^{\\kappa }(t)$ , $t\\ge 0$ , is standard Brownian motion for every $\\kappa >0$ , we have ${\\mathbb {E}}\\varphi \\big (W\\big )={\\mathbb {E}}\\varphi \\big (V^{\\kappa }\\big )$ and then, using (REF ), $|{\\mathbb {E}}\\varphi \\big (W^{\\kappa }\\big )-{\\mathbb {E}}\\big (\\varphi (W)\\big )\\Big |\\le \\frac{{\\mathbb {E}}\\Vert X^{\\kappa }\\Vert _{(0)}}{\\sqrt{\\kappa }}.$ Next, by [5], $\\lim _{T\\rightarrow +\\infty } \\frac{\\max _{0\\le t\\le T} X(t)}{\\sqrt{2\\ln T}}=1,$ with probability one.","Using the same arguments as in [19], we conclude from (REF ) that $\\limsup _{T\\rightarrow +\\infty } \\frac{\\max _{0\\le t\\le T} |X(t)|}{\\sqrt{2\\ln T}}\\le 2,$ and then continuity of $X$ implies that the random variable $\\zeta =\\sup _{t>0} \\frac{|X(t)-X(0)|}{2\\sqrt{\\ln (2+t)}}$ is finite with probability one.","Indeed, if $T^*=\\sup \\left\\lbrace t\\ge 0: \\frac{\\max _{0\\le t\\le T} |X(t)-X(0)|}{2\\sqrt{2\\ln (2+ T)}}>5\\right\\rbrace $ then, by (REF ), $\\mathbb {P}(T^*<\\infty )=1$ , so that $\\zeta \\le \\max _{0\\le t\\le T^*} \\frac{|X(t)-X(0)|}{2\\sqrt{\\ln (2+t)}} + 5 <\\infty .$ Moreover, because $t\\mapsto X(t)-X(0)$ is a Gaussian process with mean zero, the Borell-TIS inequality (REF ) implies ${\\mathbb {E}}\\zeta ^p <\\infty $ for all $p>0$ .","If $t>0$ and $\\kappa \\ge 1$ , then, by direct computation, $(2+\\kappa t)\\le (1+\\kappa )(1+t),\\ \\frac{1}{2}\\le \\sqrt{\\ln (1+\\kappa )}, \\ \\frac{\\sqrt{\\ln (1+t)}}{1+t}\\le \\frac{1}{2}.$ As a result, $\\frac{\\sqrt{\\ln (2+\\kappa t)}}{1+t}&\\le \\frac{\\sqrt{\\ln (1+t)}}{1+t} + \\sqrt{\\ln (1+\\kappa )}\\le 2\\sqrt{\\ln (1+\\kappa )},\\\\\\Vert X^{\\kappa }\\Vert _{(0)} &=\\sup _{t>0} \\frac{|X(\\kappa t)-X(0)|}{2(1+t)} \\\\&\\le \\left(\\sup _{t,\\kappa >0}\\frac{|X(\\kappa t)-X(0)|}{2\\sqrt{\\ln (2+\\kappa t)}}\\right)\\left( \\sup _{t>0} \\frac{\\sqrt{\\ln (2+\\kappa t)}}{1+t}\\right)\\le 2\\zeta \\sqrt{\\ln (1+\\kappa )},$ and (REF ) follows with $C_X={\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|X(t)-X(0)|}{\\sqrt{\\ln (2+t)}}\\right].$ $\\Box $ Denote by $\\mu _0$ and $\\mu _{\\kappa }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\kappa }$ .","The following is the main result of this section, showing that the convergence rate $\\kappa ^{-1/2}\\sqrt{\\ln \\kappa }$ is sharp for the Wasserstein-1 metric.","Theorem 2.2 There exist positive constants $C_X$ and $c_X$ such that, for every $\\kappa \\ge 1$ , $c_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ The upper bound in (REF ) follows from (REF ) and Proposition REF .","To establish the lower bound, we use (REF ) with a particular $\\varphi $ .","If $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ is a bounded linear functional, then (REF ) and (REF ) imply ${\\mathbb {E}}\\varphi (W^{\\kappa }) - {\\mathbb {E}}\\varphi (W)=\\kappa ^{-1/2}{\\mathbb {E}}\\varphi (X^{\\kappa }).$ For $f\\in {\\mathcal {C}}_{(0)},$ define $\\varphi _{\\kappa }: f \\mapsto \\frac{f(t^*_{\\kappa })}{2},$ where $t^*_{\\kappa } = \\arg \\max _{0\\le t \\le 1} X^{\\kappa }(t).$ In particular, $t^*_{\\kappa }\\in [0,1]$ .","Then $\\varphi _{\\kappa }$ is a bounded linear functional on ${\\mathcal {C}}_{(0)}$ : $|\\varphi _{\\kappa }(f)|\\le \\frac{\\max _{0\\le t\\le 1} |f(t)|}{2}\\le \\max _{0\\le t\\le 1}\\frac{|f(t)|}{1+t}\\le \\Vert f\\Vert _{(0)}.$ Therefore, by (REF ) and (REF ), $\\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0) \\ge \\kappa ^{-1/2}|{\\mathbb {E}}\\varphi _{\\kappa }(X^{\\kappa })| =\\frac{{\\mathbb {E}}\\max _{0\\le t \\le 1} X^{\\kappa }(t)}{2\\sqrt{\\kappa }}.$ Next, define $\\zeta _{\\kappa }=\\frac{\\max \\limits _{0\\le t \\le 1} X^{\\kappa }(t)}{\\sqrt{\\ln (1+\\kappa )}}=\\frac{\\max \\limits _{0\\le t \\le \\kappa }X(t)-X(0)}{2\\sqrt{\\ln (1+\\kappa )}}, \\ \\ \\kappa \\ge 1;$ the second equality follows from (REF ).","By (REF ), the family $\\lbrace \\zeta _{\\kappa },\\ \\kappa \\ge 1\\rbrace $ is uniformly integrable, so that (REF ) implies $\\lim _{\\kappa \\rightarrow \\infty } {\\mathbb {E}}\\zeta _{\\kappa } = \\frac{1}{\\sqrt{2}},$ which, in turn, means $\\inf _{\\kappa \\ge 1} {\\mathbb {E}}\\zeta _{\\kappa }>0.$ The lower bound in (REF ), with $c_X=\\frac{1}{2}\\inf _{\\kappa \\ge 1} \\frac{{\\mathbb {E}}\\left[ \\max \\limits _{0\\le t \\le \\kappa } X(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}},$ now follows from (REF ) and (REF ), because ${\\mathbb {E}}X(0)=0$ .","$\\Box $ To get a better idea about numerical values of $C_X$ and $c_X$ , we need Proposition 2.3 Let $X=X(t),\\ t\\in {\\mathbb {R}},$ be a stationary Gaussian process with mean zero and covariance $e^{-2|t-s|}$ .","Define the random variable $\\bar{\\eta }=\\max _{0\\le t\\le 1} X(t).$ Then $1.2& < {\\mathbb {E}}\\bar{\\eta } < 3.2;\\\\\\mathbb {P}(\\bar{\\eta }>x)&\\le e^{-(x-3.2)^2/2},\\ x\\ge 3.2.$ Let $B=B(t)$ , $0\\le t\\le 1$ , be the standard Brownian bridge and $W=W(t)$ , $0\\le t\\le 1$ , a standard Brownian motion.","Then ${\\mathbb {E}}|B(t)-B(s)|^2 &= |t-s|-|t-s|^2,\\ {\\mathbb {E}}|X(t)-X(s)|^2 = 2(1-e^{-2|t-s|}),\\\\& {\\mathbb {E}}|W(t)-W(s)|^2=|t-s|,$ and, because $x-x^2\\le 1-e^{-2x}\\le 2x,\\ x\\ge 0,$ inequality (REF ) implies $2\\,{\\mathbb {E}}\\max _{0\\le t\\le 1} B(t) \\le {\\mathbb {E}}\\bar{\\eta } \\le 4\\, {\\mathbb {E}}\\max _{0\\le t\\le 1} W(t).$ It is well known (e.g.","[9]) that $\\mathbb {P}\\left(\\max _{0\\le t\\le 1} B(t) >x\\right) = e^{-2x^2}, \\ \\ \\mathbb {P}\\left(\\max _{0\\le t\\le 1} W(t) >x\\right) = \\frac{2}{\\sqrt{2\\pi }}\\int _{x}^{+\\infty }e^{-t^2/2}dt.$ Then ${\\mathbb {E}}\\max _{0\\le t\\le 1} B(t) &= \\int _0^{+\\infty } e^{-2x^2}\\, dx = \\frac{\\sqrt{2\\pi }}{4}>0.6,\\\\{\\mathbb {E}}\\max _{0\\le t\\le 1} W(t) & =\\frac{2}{\\sqrt{2\\pi }}\\int _0^{+\\infty }xe^{-x^2/2}\\, dx = \\sqrt{\\frac{2}{\\pi }}<0.8,$ and (REF ) follows.","After that, () is a re-statement of the Borell-TIS inequality (REF ).","$\\Box $ We can now show that the number $C_X$ defined in (REF ) satisfies $1.4 < C_X < 14.$ For the lower bound, note that $C_X\\ge {\\mathbb {E}}\\left[\\sup _{t>0} \\frac{X(t)-X(0)}{\\sqrt{\\ln (2+t)}}\\right],$ whereas $\\sup _{t>0} \\frac{X(t)-X(0)}{\\sqrt{\\ln (2+t)}}\\ge \\limsup _{T\\rightarrow \\infty } \\frac{\\max \\limits _{0\\le t\\le T} \\big (X(t)-X(0)\\big )}{\\sqrt{\\ln (2+T)}},$ and it remains to apply (REF ).","For the upper bound in (REF ), start by writing $C_X\\le {\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|X(t)|}{\\sqrt{\\ln (2+t)}}\\right] + \\frac{{\\mathbb {E}}|X(0)|}{\\sqrt{\\ln 2}}; \\ \\ \\ \\frac{{\\mathbb {E}}|X(0)|}{\\sqrt{\\ln 2}}=\\sqrt{\\frac{2}{\\pi \\ln 2}}\\approx 0.96.$ Next, let $\\bar{X}$ be the process consisting of iid copies of $X(t),\\ t\\in [0,1),$ on each of the intervals $[k-1,k)$ , $k=1,2,\\ldots $ .","Then ${\\mathbb {E}}X^2(t)={\\mathbb {E}}\\bar{X}^2(t)$ , ${\\mathbb {E}}X(t)X(s)\\ge {\\mathbb {E}}\\bar{X}(t)\\bar{X}(s)$ , and so ${\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|X(t)|}{\\sqrt{\\ln (2+t)}}\\right] \\le 2{\\mathbb {E}}\\left[\\sup _{t>0} \\frac{X(t)}{\\sqrt{\\ln (2+t)}}\\right] \\le 2{\\mathbb {E}}\\left[\\sup _{t>0} \\frac{\\bar{X}(t)}{\\sqrt{\\ln (2+t)}}\\right],$ where the first inequality follows from [19], and the second, from (REF ).","On the other hand, if $\\bar{\\eta }_k$ , $k\\ge 1$ , are iid copies of the random variable $\\bar{\\eta }$ from (REF ), then ${\\mathbb {E}}\\left[\\sup _{t>0} \\frac{\\bar{X}(t)}{\\sqrt{\\ln (2+t)}}\\right]\\le {\\mathbb {E}}\\left[\\sup _{k\\ge 1} \\frac{\\bar{\\eta }_k}{\\sqrt{\\ln (1+k)}}\\right].$ Using () with $x>5$ , $\\mathbb {P}\\Big (\\sup _{k\\ge 1} \\frac{\\bar{\\eta }_k}{\\sqrt{\\ln (1+k)}} > x\\Big ) &\\le \\sum _{k\\ge 1} \\mathbb {P}\\Big (\\bar{\\eta }_k>x\\sqrt{\\ln (1+k)}\\Big )\\\\&\\le \\sum _{k\\ge 1}\\frac{1}{(1+k)^{-(x-3.2)^2/2}}\\le \\frac{2}{(x-3.2)^2-2},$ and therefore ${\\mathbb {E}}\\left[\\sup _{k\\ge 1} \\frac{\\bar{\\eta }_k}{\\sqrt{\\ln (1+k)}}\\right]\\le 5+2\\int _5^{+\\infty } \\frac{dx}{(x-3.2)^2-2} <6.5.$ Then (REF ) follows from (REF ) – (REF ).","Next, we will show that the number $c_X$ defined in (REF ) satisfies $0.2 < c_X < 0.8.$ Indeed, the upper bound follows immediately from (REF ).","For the lower bound, start by noting that, for $N\\le \\kappa < N+1$ , ${\\mathbb {E}}\\left[ \\max \\limits _{0\\le t \\le \\kappa } X(t)\\right] \\ge {\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} X(k)\\right]$ and ${\\mathbb {E}}|X(k)-X(m)|^2 = 2(1-e^{-2})> 1$ .","Now take iid Gaussian $Y_k$ , $k=1,\\ldots , N$ with mean zero and variance $1/2$ .","Then ${\\mathbb {E}}|Y(k)-Y(m)|^2=1$ and so ${\\mathbb {E}}|X(k)-X(m)|^2\\ge {\\mathbb {E}}|Y(k)-Y(m)|^2.$ By (REF ), ${\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} X(k)\\right] \\ge {\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} Y(k)\\right],$ and, by [19], ${\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} Y(k)\\right] \\ge 0.4\\sqrt{\\ln N},$ leading to the lower bound in (REF ).","To summarize, we can write (REF ) in a more explicit form $0.2\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le 14\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1.$ Theorem REF can be used to study convergence on a bounded interval.","For $T>0$ , let ${\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\mu _{0}^T$ and $\\mu _{\\kappa }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\kappa }$ .","Theorem 2.4 There exist positive constants $C_{X,T}$ and $c_{_{X,T}}$ such that, for every $\\kappa \\ge 1$ , $c_{_{X,T}}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T)\\le C_{X,T}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ If $f\\in {\\mathcal {C}}_{(0)}$ , then, for $00$ , let $\\lfloor x \\rfloor $ denote the largest integer that is less than or equal to $x$ .","Define the processes $W^{\\kappa }$ by $W^{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} X_{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ The second term on the right-hand side of (REF ) ensures that $W^{\\kappa }$ is a continuous function of $t$ .","The case $a=0$ , that is, the Gaussian version of the original Donsker theorem, is of special interest; the corresponding process $W^{\\kappa }$ will be denoted by $S_{\\kappa }$ : $S_{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } \\xi _n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} \\xi _{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ We have $\\mathbb {P}(W^{\\kappa }\\in {\\mathcal {C}}_{(0)})=1$ for every $\\kappa \\ge 1$ , and $a\\in (-1,1)$ , because the ergodic theorem implies $\\lim _{t\\rightarrow \\infty } \\frac{1}{\\lfloor \\kappa t \\rfloor }\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n = {\\mathbb {E}}X_0 = 0$ with probability one.","Let $W=W(t),\\ t\\ge 0$ , be a standard Brownian motion, and denote by $\\mu _0$ and $\\mu _{\\kappa }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\kappa }$ .","The following is the discrete-time analog of Theorem REF .","Theorem 3.1 There exist positive constants $C_a$ and $c_a$ such that, for every $\\kappa \\ge 1$ , $c_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2} \\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ The steps are the same as in the proof of Theorem REF .","Substituting (REF ) in (REF ) and changing the order of summation, $W^{\\kappa }(t)=S_{\\kappa }(t)+\\frac{X^{\\kappa }(t)}{\\sqrt{\\kappa }},$ where $S_{\\kappa }$ is from (REF ) and $X^{\\kappa }(t) = \\left( (a-a^{\\lfloor \\kappa t \\rfloor })\\sum _{n=-\\infty }^{0}a^{-n}\\xi _n -\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor }a^{\\lfloor \\kappa t \\rfloor -n}\\xi _n\\right).$ As a result, it is enough to establish (REF ) when $a=0$ : $c_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Then, similar to the continuous time case, we see that ${\\mathbb {E}}\\Vert X^{\\kappa }\\Vert _{(0)} \\le \\bar{C}_a \\sqrt{\\ln (1+\\kappa )},$ with a suitable constant $\\bar{C}_a$ , and then (REF ) follows from (REF ) with $C_a=C_0+\\bar{C}_a$ and $c_a=c_0$ .","To prove (REF ), we choose the random variables $\\xi _k$ in (REF ) as the increments of the Brownian motion $W$ : $\\frac{\\xi _n}{\\sqrt{\\kappa }} = W(n/\\kappa )- W((n-1)/\\kappa ).$ Then, for $(n-1)/\\kappa \\le t \\le n/\\kappa $ , the process $t\\mapsto S_{\\kappa }- W $ is a Brownian bridge, and, for every function $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfying (REF ), $|{\\mathbb {E}}\\varphi (S_{\\kappa })-{\\mathbb {E}}\\varphi (W)|\\le {\\mathbb {E}}\\Vert B^{\\kappa }\\Vert _{(0)},$ where $B^{\\kappa }$ is a collection of independent Brownian bridges on $[(n-1)/\\kappa , n/\\kappa ]$ , $n=1,2,\\ldots $ .","Direct computations show that, for $N=1,2,\\ldots $ , $& \\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)|\\le 8\\sqrt{{\\ln (N+1)}},\\\\&\\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } B^{\\kappa }(t) \\ge 0.3\\sqrt{{\\ln (N+1)}};$ the numbers $0.3$ and 8 do not necessarily provide optimal bounds.","Indeed, let $B=B(t),\\ t\\in [0,1],$ be the standard Brownian bridge, and let $\\eta =\\max _{0\\le t\\le 1} B(t).$ Then $\\sqrt{\\kappa } \\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)| = {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k|,$ where $\\eta _k,\\ k=1,\\ldots , N$ , are iid copies of $\\eta $ .","Also, ${\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k\\le {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k| \\le 2 {\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k.$ To derive (REF ), we repeat the arguments from the proof of Lemma 10.1 in [19] using (REF ) and conclude that ${\\mathbb {E}}\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\le 4 \\sqrt{\\ln (N+1)}$ .","Similarly, for (), we repeat the proof of Lemma 10.2 in [19].","Next, denote by $\\bar{B}$ the process $B^{\\kappa }$ corresponding to $\\kappa =1$ , that is, the collection of independent standard Brownian bridges on $[n-1,n]$ , $n\\ge 1$ .","Then we get the upper bound in (REF ), with $C_0={\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|\\bar{B}(t)|}{\\sqrt{\\ln (2+t)}}\\right]< 14,$ by combining (REF ) and (REF ); the upper bound on $C_0$ is from (REF ), because, by (REF ), $C_0\\le C_X$ .","The lower bound in (REF ), with $c_0=\\frac{1}{2}\\inf _{\\kappa \\ge 1} \\frac{{\\mathbb {E}}\\left[ \\max _{0\\le t \\le \\kappa } \\bar{B}(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}}> 0.2,$ follows from () after the same arguments as in the proof of Theorem REF ; the lower bound on $c_0$ follows from ().","$\\Box $ For $T>0$ , let ${\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\mu _{0}^T$ and $\\mu _{\\kappa }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\kappa }$ .","The discrete-time version of Theorem REF is obvious.","When $a=0$ , and there is no continuous-time analog, we also have the following result (cf.","[2]).","Proposition 3.2 If $\\mu _{\\kappa }^T$ is the measure on ${\\mathcal {C}}(0,T)$ generated by the process $S_{\\kappa }$ from (REF ), then $\\lim _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{\\kappa }{\\ln \\kappa }} \\,\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) = {\\sqrt{2}}.$ Using the random variables $\\eta _k$ from the proof of Theorem REF , $\\limsup _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }}\\; \\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\le \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|\\right]}{\\sqrt{\\ln N}},$ and $\\liminf _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }} \\;\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\ge \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\right]}{\\sqrt{\\ln N}}.$ By (REF ) and [24], $\\lim _{N\\rightarrow \\infty }\\frac{\\max \\limits _{k=1,\\ldots ,N}\\eta _k}{\\sqrt{\\ln N}}=\\lim _{N\\rightarrow \\infty } \\frac{\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|}{\\sqrt{\\ln N}}={\\sqrt{2}}$ with probability 1; then uniform integrability [22] implies (REF ).","$\\Box $ Applications Let $W$ be a standard Brownian motion and let $W^{\\kappa }$ be the process from (REF ) or (REF ).","Consider a continuous mapping $\\Psi : {\\mathcal {C}}_{(0)} \\rightarrow {\\mathcal {C}}_{(0)}$ .","Denote by $\\mu _{0,\\psi }$ and $\\mu _{\\kappa ,\\psi }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $\\Psi (W)$ and $\\Psi (W^{\\kappa })$ .","Proposition 4.1 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi } = \\mu _{0,\\psi }$ .","Moreover, if there exists a number $C_{\\psi }$ such that, for all $f,g\\in {\\mathcal {C}}_{(0)}$ , $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le C_{\\psi }\\Vert f-g\\Vert _{(0)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi },\\mu _{0,\\psi })\\le \\bar{C}_XC_{\\psi }\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.","Weak convergence follows by the continuous mapping theorem (e.g.","[4]).","To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfies (REF ), then $|\\varphi (\\Psi (f))-\\varphi (\\Psi (g))|\\le C_{\\psi }\\Vert f-g\\Vert _{(0)}.$ $\\Box $ Example 1.","Given ${\\alpha }>0$ , let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=-{\\alpha }\\int _0^t Y(s)\\, ds + W(t),\\ \\ t\\ge 0.$ Then $Y^{\\kappa }, Y\\in {\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\nu _{\\kappa }, \\nu $ on ${\\mathcal {C}}_{(0)}$ satisfy $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le 2\\bar{C}_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1;$ the constant $\\bar{C}_X$ is from Proposition REF .","Indeed, by direct computation, $Y^{\\kappa }(t)=\\Psi _{\\alpha }(W^{\\kappa })(t), \\ \\ Y(t)=\\Psi _{\\alpha }(W)(t),$ where $\\Psi _{\\alpha } : f(t)\\mapsto f(t)-{\\alpha }\\int _0^t e^{-{\\alpha }(t-s)} f(s) \\, ds,\\ \\ f \\in \\mathcal {C}_{(0)},$ is a linear operator.","To see that $\\Psi _{\\alpha }$ maps ${\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t}&\\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^t e^{-{\\alpha }(t-s)}|f(s)|\\, ds\\\\& \\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^T e^{-{\\alpha }(t-s)}|f(s)|\\, ds \\\\&+ \\frac{{\\alpha }e^{-{\\alpha }t}}{1+t}\\int _T^t (1+s)e^{{\\alpha }s}\\, \\frac{|f(s)|}{1+s}\\, ds.$ If $\\lim _{t\\rightarrow \\infty }|f(t)|/(1+t) = 0$ , then, for every $\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\varepsilon $ , $s>T$ .","As a result, keeping in mind that $\\frac{{\\alpha }}{1+t}\\int _T^t (1+s) e^{{\\alpha }s}\\, ds \\le {\\alpha }\\int _0^t e^{{\\alpha }s}\\, ds \\le e^{{\\alpha }t},$ we compute $\\limsup _{t\\rightarrow \\infty } \\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t} \\le \\varepsilon $ and conclude that $\\lim _{t\\rightarrow \\infty } |\\Psi _{\\alpha }(f)(t)|/(1+t)=0$ .","Similarly, $\\Vert \\Psi _{\\alpha }(f)\\Vert _{(0)} \\le \\Vert f\\Vert _{(0)} \\left(1+\\alpha \\int _0^{+\\infty }e^{-\\alpha t}\\, dt \\right) =2\\Vert f\\Vert _{(0)},$ so that (REF ) holds with $C_{\\psi }=2$ .","$\\Box $ The analog of Proposition REF on a bounded interval is as follows.","Let $\\Psi ^T$ be a continuous mapping of ${\\mathcal {C}}(0,T)$ to itself.","Denote by $\\mu _{0,\\psi }^T$ and $\\mu _{\\kappa ,\\psi }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $\\Psi ^T(W)$ and $\\Psi ^T(W^{\\kappa })$ .","Proposition 4.2 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi }^T = \\mu _{0,\\psi }^T$ .","Moreover, if there exists a number $C_{\\psi }^T$ such that, for all $f,g\\in {\\mathcal {C}}(0,T)$ , $\\Vert \\Psi ^T(f)-\\Psi ^T(g)\\Vert _{{\\mathcal {C}}(0,T)}\\le C_{\\psi }^T\\Vert f-g\\Vert _{{\\mathcal {C}}(0,T)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi }^T,\\mu _{0,\\psi }^T)\\le (1+T)\\bar{C}_{X}C_{\\psi }^T\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.","Example 2.","Let the function $b=b(x), \\ x\\in {\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\le K|x-y|,\\ x,y\\in {\\mathbb {R}},$ and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = \\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ 0\\le t\\le T.$ If $\\nu _{\\kappa }^T, \\nu ^T$ are the corresponding measures on ${\\mathcal {C}}(0,T)$ , then $\\mathrm {d}_{{w}}(\\nu _{\\kappa }^T,\\nu ^T)\\le (1+T)\\bar{C}_Xe^{KT}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}(0,T)$ , define $\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ 0\\le t\\le T.$ By direct computation (e.g.","[10]), we have (REF ) with $C_{\\psi }^T = e^{KT}$ .","$ $ $\\Box $ Example 3.","Let us combine Examples 1 and 2.","Take a positive number ${\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t)& = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds+\\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\\\Y(t)&= -{\\alpha }\\int _0^t Y(s)\\, ds+\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ t\\ge 0.$ If ${\\alpha }>K$ , then $Y^{\\kappa }, Y \\in {\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\nu _{\\kappa }, \\nu $ , $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le \\frac{2{\\alpha }\\bar{C}_X}{{\\alpha }-K}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}_{(0)}$ , define $\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\alpha }\\int _0^t y(s)\\, ds + \\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ t\\ge 0.$ Using variation of parameters formula and (REF ), $\\Psi (f)(t)=\\int _0^t e^{-{\\alpha }(t-s)} b\\big (\\Psi (f)(s)\\big )\\, ds + \\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\Psi $ maps ${\\mathcal {C}}_{(0)}$ to itself.","In particular, using (REF ) and (REF ), $\\Psi (f)(t)-\\Psi (g)(t) =\\int _0^t e^{-{\\alpha }(t-s)} \\Big ( b\\big (\\Psi (f)(s)\\big )-b\\big (\\Psi (g)(s)\\big )\\Big )\\, ds + \\Psi _{\\alpha }(f-g)(t)$ so that $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\le K \\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\int _0^{\\infty } e^{-as}\\, ds + 2\\Vert f-g\\Vert _{(0)}.$ As a result, if $\\alpha >K$ , then $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le \\frac{2{\\alpha }}{{\\alpha }-K}\\Vert f-g\\Vert _{(0)},$ and (REF ) follows from (REF ).","$\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .","Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.","When the underlying processes are Gaussian, some of the approximation errors are not present.","In continuous time, the first two steps are the equality (REF ).","There is no need for Skorokhod embedding because the martingale component is the Brownian motion.","In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.","For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.","Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.","For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].","Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.","For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."],["Discrete Time","Consider a stationary Gaussian sequence $X=\\lbrace X_n,\\ n\\ge 0\\rbrace ,$ with ${\\mathbb {E}}X_n=0$ and ${\\mathbb {E}}X_{k+n} X_k=(1-a)a^n/(1+a),\\ n,k\\ge 0$ , where $a\\in (-1,1)$ and $a=0$ corresponds to the sequence of iid standard Gaussian random variables.","The variance of $X_n$ is chosen so that, for all $a\\in (-1,1)$ , the covariance function $R(n)={\\mathbb {E}}X_{k+n} X_n$ of $X$ satisfies $R(0)+2\\sum _{n=1}^{\\infty } R(n)=1.$ Using a collection $\\xi _k,\\ k=0,\\pm 1,\\pm 2,\\ldots $ of iid standard normal random variables, we get the discrete-time analogs of () and (): $X_n&=(1-a)\\sum _{k=-\\infty }^n a^{n-k}\\xi _k,\\\\X_{n+1}&=aX_n+\\xi _{n+1};$ in (), the initial condition $X_0$ is independent of $\\xi _k,\\ k\\ge 1,$ and is a normal random variable with mean 0 and variance $(1-a)/(1+a)$ .","For $x>0$ , let $\\lfloor x \\rfloor $ denote the largest integer that is less than or equal to $x$ .","Define the processes $W^{\\kappa }$ by $W^{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} X_{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ The second term on the right-hand side of (REF ) ensures that $W^{\\kappa }$ is a continuous function of $t$ .","The case $a=0$ , that is, the Gaussian version of the original Donsker theorem, is of special interest; the corresponding process $W^{\\kappa }$ will be denoted by $S_{\\kappa }$ : $S_{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } \\xi _n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} \\xi _{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ We have $\\mathbb {P}(W^{\\kappa }\\in {\\mathcal {C}}_{(0)})=1$ for every $\\kappa \\ge 1$ , and $a\\in (-1,1)$ , because the ergodic theorem implies $\\lim _{t\\rightarrow \\infty } \\frac{1}{\\lfloor \\kappa t \\rfloor }\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n = {\\mathbb {E}}X_0 = 0$ with probability one.","Let $W=W(t),\\ t\\ge 0$ , be a standard Brownian motion, and denote by $\\mu _0$ and $\\mu _{\\kappa }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\kappa }$ .","The following is the discrete-time analog of Theorem REF .","Theorem 3.1 There exist positive constants $C_a$ and $c_a$ such that, for every $\\kappa \\ge 1$ , $c_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2} \\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ The steps are the same as in the proof of Theorem REF .","Substituting (REF ) in (REF ) and changing the order of summation, $W^{\\kappa }(t)=S_{\\kappa }(t)+\\frac{X^{\\kappa }(t)}{\\sqrt{\\kappa }},$ where $S_{\\kappa }$ is from (REF ) and $X^{\\kappa }(t) = \\left( (a-a^{\\lfloor \\kappa t \\rfloor })\\sum _{n=-\\infty }^{0}a^{-n}\\xi _n -\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor }a^{\\lfloor \\kappa t \\rfloor -n}\\xi _n\\right).$ As a result, it is enough to establish (REF ) when $a=0$ : $c_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Then, similar to the continuous time case, we see that ${\\mathbb {E}}\\Vert X^{\\kappa }\\Vert _{(0)} \\le \\bar{C}_a \\sqrt{\\ln (1+\\kappa )},$ with a suitable constant $\\bar{C}_a$ , and then (REF ) follows from (REF ) with $C_a=C_0+\\bar{C}_a$ and $c_a=c_0$ .","To prove (REF ), we choose the random variables $\\xi _k$ in (REF ) as the increments of the Brownian motion $W$ : $\\frac{\\xi _n}{\\sqrt{\\kappa }} = W(n/\\kappa )- W((n-1)/\\kappa ).$ Then, for $(n-1)/\\kappa \\le t \\le n/\\kappa $ , the process $t\\mapsto S_{\\kappa }- W $ is a Brownian bridge, and, for every function $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfying (REF ), $|{\\mathbb {E}}\\varphi (S_{\\kappa })-{\\mathbb {E}}\\varphi (W)|\\le {\\mathbb {E}}\\Vert B^{\\kappa }\\Vert _{(0)},$ where $B^{\\kappa }$ is a collection of independent Brownian bridges on $[(n-1)/\\kappa , n/\\kappa ]$ , $n=1,2,\\ldots $ .","Direct computations show that, for $N=1,2,\\ldots $ , $& \\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)|\\le 8\\sqrt{{\\ln (N+1)}},\\\\&\\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } B^{\\kappa }(t) \\ge 0.3\\sqrt{{\\ln (N+1)}};$ the numbers $0.3$ and 8 do not necessarily provide optimal bounds.","Indeed, let $B=B(t),\\ t\\in [0,1],$ be the standard Brownian bridge, and let $\\eta =\\max _{0\\le t\\le 1} B(t).$ Then $\\sqrt{\\kappa } \\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)| = {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k|,$ where $\\eta _k,\\ k=1,\\ldots , N$ , are iid copies of $\\eta $ .","Also, ${\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k\\le {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k| \\le 2 {\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k.$ To derive (REF ), we repeat the arguments from the proof of Lemma 10.1 in [19] using (REF ) and conclude that ${\\mathbb {E}}\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\le 4 \\sqrt{\\ln (N+1)}$ .","Similarly, for (), we repeat the proof of Lemma 10.2 in [19].","Next, denote by $\\bar{B}$ the process $B^{\\kappa }$ corresponding to $\\kappa =1$ , that is, the collection of independent standard Brownian bridges on $[n-1,n]$ , $n\\ge 1$ .","Then we get the upper bound in (REF ), with $C_0={\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|\\bar{B}(t)|}{\\sqrt{\\ln (2+t)}}\\right]< 14,$ by combining (REF ) and (REF ); the upper bound on $C_0$ is from (REF ), because, by (REF ), $C_0\\le C_X$ .","The lower bound in (REF ), with $c_0=\\frac{1}{2}\\inf _{\\kappa \\ge 1} \\frac{{\\mathbb {E}}\\left[ \\max _{0\\le t \\le \\kappa } \\bar{B}(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}}> 0.2,$ follows from () after the same arguments as in the proof of Theorem REF ; the lower bound on $c_0$ follows from ().","$\\Box $ For $T>0$ , let ${\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\mu _{0}^T$ and $\\mu _{\\kappa }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\kappa }$ .","The discrete-time version of Theorem REF is obvious.","When $a=0$ , and there is no continuous-time analog, we also have the following result (cf.","[2]).","Proposition 3.2 If $\\mu _{\\kappa }^T$ is the measure on ${\\mathcal {C}}(0,T)$ generated by the process $S_{\\kappa }$ from (REF ), then $\\lim _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{\\kappa }{\\ln \\kappa }} \\,\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) = {\\sqrt{2}}.$ Using the random variables $\\eta _k$ from the proof of Theorem REF , $\\limsup _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }}\\; \\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\le \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|\\right]}{\\sqrt{\\ln N}},$ and $\\liminf _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }} \\;\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\ge \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\right]}{\\sqrt{\\ln N}}.$ By (REF ) and [24], $\\lim _{N\\rightarrow \\infty }\\frac{\\max \\limits _{k=1,\\ldots ,N}\\eta _k}{\\sqrt{\\ln N}}=\\lim _{N\\rightarrow \\infty } \\frac{\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|}{\\sqrt{\\ln N}}={\\sqrt{2}}$ with probability 1; then uniform integrability [22] implies (REF ).","$\\Box $ Applications Let $W$ be a standard Brownian motion and let $W^{\\kappa }$ be the process from (REF ) or (REF ).","Consider a continuous mapping $\\Psi : {\\mathcal {C}}_{(0)} \\rightarrow {\\mathcal {C}}_{(0)}$ .","Denote by $\\mu _{0,\\psi }$ and $\\mu _{\\kappa ,\\psi }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $\\Psi (W)$ and $\\Psi (W^{\\kappa })$ .","Proposition 4.1 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi } = \\mu _{0,\\psi }$ .","Moreover, if there exists a number $C_{\\psi }$ such that, for all $f,g\\in {\\mathcal {C}}_{(0)}$ , $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le C_{\\psi }\\Vert f-g\\Vert _{(0)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi },\\mu _{0,\\psi })\\le \\bar{C}_XC_{\\psi }\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.","Weak convergence follows by the continuous mapping theorem (e.g.","[4]).","To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfies (REF ), then $|\\varphi (\\Psi (f))-\\varphi (\\Psi (g))|\\le C_{\\psi }\\Vert f-g\\Vert _{(0)}.$ $\\Box $ Example 1.","Given ${\\alpha }>0$ , let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=-{\\alpha }\\int _0^t Y(s)\\, ds + W(t),\\ \\ t\\ge 0.$ Then $Y^{\\kappa }, Y\\in {\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\nu _{\\kappa }, \\nu $ on ${\\mathcal {C}}_{(0)}$ satisfy $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le 2\\bar{C}_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1;$ the constant $\\bar{C}_X$ is from Proposition REF .","Indeed, by direct computation, $Y^{\\kappa }(t)=\\Psi _{\\alpha }(W^{\\kappa })(t), \\ \\ Y(t)=\\Psi _{\\alpha }(W)(t),$ where $\\Psi _{\\alpha } : f(t)\\mapsto f(t)-{\\alpha }\\int _0^t e^{-{\\alpha }(t-s)} f(s) \\, ds,\\ \\ f \\in \\mathcal {C}_{(0)},$ is a linear operator.","To see that $\\Psi _{\\alpha }$ maps ${\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t}&\\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^t e^{-{\\alpha }(t-s)}|f(s)|\\, ds\\\\& \\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^T e^{-{\\alpha }(t-s)}|f(s)|\\, ds \\\\&+ \\frac{{\\alpha }e^{-{\\alpha }t}}{1+t}\\int _T^t (1+s)e^{{\\alpha }s}\\, \\frac{|f(s)|}{1+s}\\, ds.$ If $\\lim _{t\\rightarrow \\infty }|f(t)|/(1+t) = 0$ , then, for every $\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\varepsilon $ , $s>T$ .","As a result, keeping in mind that $\\frac{{\\alpha }}{1+t}\\int _T^t (1+s) e^{{\\alpha }s}\\, ds \\le {\\alpha }\\int _0^t e^{{\\alpha }s}\\, ds \\le e^{{\\alpha }t},$ we compute $\\limsup _{t\\rightarrow \\infty } \\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t} \\le \\varepsilon $ and conclude that $\\lim _{t\\rightarrow \\infty } |\\Psi _{\\alpha }(f)(t)|/(1+t)=0$ .","Similarly, $\\Vert \\Psi _{\\alpha }(f)\\Vert _{(0)} \\le \\Vert f\\Vert _{(0)} \\left(1+\\alpha \\int _0^{+\\infty }e^{-\\alpha t}\\, dt \\right) =2\\Vert f\\Vert _{(0)},$ so that (REF ) holds with $C_{\\psi }=2$ .","$\\Box $ The analog of Proposition REF on a bounded interval is as follows.","Let $\\Psi ^T$ be a continuous mapping of ${\\mathcal {C}}(0,T)$ to itself.","Denote by $\\mu _{0,\\psi }^T$ and $\\mu _{\\kappa ,\\psi }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $\\Psi ^T(W)$ and $\\Psi ^T(W^{\\kappa })$ .","Proposition 4.2 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi }^T = \\mu _{0,\\psi }^T$ .","Moreover, if there exists a number $C_{\\psi }^T$ such that, for all $f,g\\in {\\mathcal {C}}(0,T)$ , $\\Vert \\Psi ^T(f)-\\Psi ^T(g)\\Vert _{{\\mathcal {C}}(0,T)}\\le C_{\\psi }^T\\Vert f-g\\Vert _{{\\mathcal {C}}(0,T)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi }^T,\\mu _{0,\\psi }^T)\\le (1+T)\\bar{C}_{X}C_{\\psi }^T\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.","Example 2.","Let the function $b=b(x), \\ x\\in {\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\le K|x-y|,\\ x,y\\in {\\mathbb {R}},$ and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = \\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ 0\\le t\\le T.$ If $\\nu _{\\kappa }^T, \\nu ^T$ are the corresponding measures on ${\\mathcal {C}}(0,T)$ , then $\\mathrm {d}_{{w}}(\\nu _{\\kappa }^T,\\nu ^T)\\le (1+T)\\bar{C}_Xe^{KT}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}(0,T)$ , define $\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ 0\\le t\\le T.$ By direct computation (e.g.","[10]), we have (REF ) with $C_{\\psi }^T = e^{KT}$ .","$ $ $\\Box $ Example 3.","Let us combine Examples 1 and 2.","Take a positive number ${\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t)& = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds+\\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\\\Y(t)&= -{\\alpha }\\int _0^t Y(s)\\, ds+\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ t\\ge 0.$ If ${\\alpha }>K$ , then $Y^{\\kappa }, Y \\in {\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\nu _{\\kappa }, \\nu $ , $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le \\frac{2{\\alpha }\\bar{C}_X}{{\\alpha }-K}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}_{(0)}$ , define $\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\alpha }\\int _0^t y(s)\\, ds + \\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ t\\ge 0.$ Using variation of parameters formula and (REF ), $\\Psi (f)(t)=\\int _0^t e^{-{\\alpha }(t-s)} b\\big (\\Psi (f)(s)\\big )\\, ds + \\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\Psi $ maps ${\\mathcal {C}}_{(0)}$ to itself.","In particular, using (REF ) and (REF ), $\\Psi (f)(t)-\\Psi (g)(t) =\\int _0^t e^{-{\\alpha }(t-s)} \\Big ( b\\big (\\Psi (f)(s)\\big )-b\\big (\\Psi (g)(s)\\big )\\Big )\\, ds + \\Psi _{\\alpha }(f-g)(t)$ so that $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\le K \\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\int _0^{\\infty } e^{-as}\\, ds + 2\\Vert f-g\\Vert _{(0)}.$ As a result, if $\\alpha >K$ , then $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le \\frac{2{\\alpha }}{{\\alpha }-K}\\Vert f-g\\Vert _{(0)},$ and (REF ) follows from (REF ).","$\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .","Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.","When the underlying processes are Gaussian, some of the approximation errors are not present.","In continuous time, the first two steps are the equality (REF ).","There is no need for Skorokhod embedding because the martingale component is the Brownian motion.","In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.","For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.","Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.","For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].","Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.","For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."],["Applications","Let $W$ be a standard Brownian motion and let $W^{\\kappa }$ be the process from (REF ) or (REF ).","Consider a continuous mapping $\\Psi : {\\mathcal {C}}_{(0)} \\rightarrow {\\mathcal {C}}_{(0)}$ .","Denote by $\\mu _{0,\\psi }$ and $\\mu _{\\kappa ,\\psi }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $\\Psi (W)$ and $\\Psi (W^{\\kappa })$ .","Proposition 4.1 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi } = \\mu _{0,\\psi }$ .","Moreover, if there exists a number $C_{\\psi }$ such that, for all $f,g\\in {\\mathcal {C}}_{(0)}$ , $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le C_{\\psi }\\Vert f-g\\Vert _{(0)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi },\\mu _{0,\\psi })\\le \\bar{C}_XC_{\\psi }\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.","Weak convergence follows by the continuous mapping theorem (e.g.","[4]).","To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfies (REF ), then $|\\varphi (\\Psi (f))-\\varphi (\\Psi (g))|\\le C_{\\psi }\\Vert f-g\\Vert _{(0)}.$ $\\Box $ Example 1.","Given ${\\alpha }>0$ , let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=-{\\alpha }\\int _0^t Y(s)\\, ds + W(t),\\ \\ t\\ge 0.$ Then $Y^{\\kappa }, Y\\in {\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\nu _{\\kappa }, \\nu $ on ${\\mathcal {C}}_{(0)}$ satisfy $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le 2\\bar{C}_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1;$ the constant $\\bar{C}_X$ is from Proposition REF .","Indeed, by direct computation, $Y^{\\kappa }(t)=\\Psi _{\\alpha }(W^{\\kappa })(t), \\ \\ Y(t)=\\Psi _{\\alpha }(W)(t),$ where $\\Psi _{\\alpha } : f(t)\\mapsto f(t)-{\\alpha }\\int _0^t e^{-{\\alpha }(t-s)} f(s) \\, ds,\\ \\ f \\in \\mathcal {C}_{(0)},$ is a linear operator.","To see that $\\Psi _{\\alpha }$ maps ${\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t}&\\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^t e^{-{\\alpha }(t-s)}|f(s)|\\, ds\\\\& \\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^T e^{-{\\alpha }(t-s)}|f(s)|\\, ds \\\\&+ \\frac{{\\alpha }e^{-{\\alpha }t}}{1+t}\\int _T^t (1+s)e^{{\\alpha }s}\\, \\frac{|f(s)|}{1+s}\\, ds.$ If $\\lim _{t\\rightarrow \\infty }|f(t)|/(1+t) = 0$ , then, for every $\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\varepsilon $ , $s>T$ .","As a result, keeping in mind that $\\frac{{\\alpha }}{1+t}\\int _T^t (1+s) e^{{\\alpha }s}\\, ds \\le {\\alpha }\\int _0^t e^{{\\alpha }s}\\, ds \\le e^{{\\alpha }t},$ we compute $\\limsup _{t\\rightarrow \\infty } \\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t} \\le \\varepsilon $ and conclude that $\\lim _{t\\rightarrow \\infty } |\\Psi _{\\alpha }(f)(t)|/(1+t)=0$ .","Similarly, $\\Vert \\Psi _{\\alpha }(f)\\Vert _{(0)} \\le \\Vert f\\Vert _{(0)} \\left(1+\\alpha \\int _0^{+\\infty }e^{-\\alpha t}\\, dt \\right) =2\\Vert f\\Vert _{(0)},$ so that (REF ) holds with $C_{\\psi }=2$ .","$\\Box $ The analog of Proposition REF on a bounded interval is as follows.","Let $\\Psi ^T$ be a continuous mapping of ${\\mathcal {C}}(0,T)$ to itself.","Denote by $\\mu _{0,\\psi }^T$ and $\\mu _{\\kappa ,\\psi }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $\\Psi ^T(W)$ and $\\Psi ^T(W^{\\kappa })$ .","Proposition 4.2 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi }^T = \\mu _{0,\\psi }^T$ .","Moreover, if there exists a number $C_{\\psi }^T$ such that, for all $f,g\\in {\\mathcal {C}}(0,T)$ , $\\Vert \\Psi ^T(f)-\\Psi ^T(g)\\Vert _{{\\mathcal {C}}(0,T)}\\le C_{\\psi }^T\\Vert f-g\\Vert _{{\\mathcal {C}}(0,T)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi }^T,\\mu _{0,\\psi }^T)\\le (1+T)\\bar{C}_{X}C_{\\psi }^T\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.","Example 2.","Let the function $b=b(x), \\ x\\in {\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\le K|x-y|,\\ x,y\\in {\\mathbb {R}},$ and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = \\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ 0\\le t\\le T.$ If $\\nu _{\\kappa }^T, \\nu ^T$ are the corresponding measures on ${\\mathcal {C}}(0,T)$ , then $\\mathrm {d}_{{w}}(\\nu _{\\kappa }^T,\\nu ^T)\\le (1+T)\\bar{C}_Xe^{KT}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}(0,T)$ , define $\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ 0\\le t\\le T.$ By direct computation (e.g.","[10]), we have (REF ) with $C_{\\psi }^T = e^{KT}$ .","$ $ $\\Box $ Example 3.","Let us combine Examples 1 and 2.","Take a positive number ${\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t)& = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds+\\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\\\Y(t)&= -{\\alpha }\\int _0^t Y(s)\\, ds+\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ t\\ge 0.$ If ${\\alpha }>K$ , then $Y^{\\kappa }, Y \\in {\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\nu _{\\kappa }, \\nu $ , $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le \\frac{2{\\alpha }\\bar{C}_X}{{\\alpha }-K}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}_{(0)}$ , define $\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\alpha }\\int _0^t y(s)\\, ds + \\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ t\\ge 0.$ Using variation of parameters formula and (REF ), $\\Psi (f)(t)=\\int _0^t e^{-{\\alpha }(t-s)} b\\big (\\Psi (f)(s)\\big )\\, ds + \\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\Psi $ maps ${\\mathcal {C}}_{(0)}$ to itself.","In particular, using (REF ) and (REF ), $\\Psi (f)(t)-\\Psi (g)(t) =\\int _0^t e^{-{\\alpha }(t-s)} \\Big ( b\\big (\\Psi (f)(s)\\big )-b\\big (\\Psi (g)(s)\\big )\\Big )\\, ds + \\Psi _{\\alpha }(f-g)(t)$ so that $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\le K \\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\int _0^{\\infty } e^{-as}\\, ds + 2\\Vert f-g\\Vert _{(0)}.$ As a result, if $\\alpha >K$ , then $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le \\frac{2{\\alpha }}{{\\alpha }-K}\\Vert f-g\\Vert _{(0)},$ and (REF ) follows from (REF ).","$\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .","Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.","When the underlying processes are Gaussian, some of the approximation errors are not present.","In continuous time, the first two steps are the equality (REF ).","There is no need for Skorokhod embedding because the martingale component is the Brownian motion.","In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.","For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.","Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.","For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].","Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.","For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."],["Concluding Remarks","A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .","Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.","When the underlying processes are Gaussian, some of the approximation errors are not present.","In continuous time, the first two steps are the equality (REF ).","There is no need for Skorokhod embedding because the martingale component is the Brownian motion.","In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.","For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.","Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.","For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].","Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.","For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."]],"string":"[\n [\n \"A Sharp Rate of Convergence in the Functional Central Limit Theorem with\\n Gaussian Input\"\n ],\n [\n \"Abstract When the underlying random variables are Gaussian, the classical Central Limit Theorem (CLT) is trivial, but the functional CLT is not.\",\n \"The objective of the paper is to investigate the functional CLT for stationary Gaussian processes in the Wasserstein-1 metric on the space of continuous functions.\",\n \"Matching upper and lower bounds are established, indicating that the convergence rate is slightly faster than in the L\\\\'{e}vy-Prokhorov metric.\"\n ],\n [\n \"Introduction\",\n \"By the Central Limit Theorem, given a collection $\\\\lbrace \\\\xi _k,\\\\ k\\\\ge 1\\\\rbrace $ of independent and identically distributed random variables, each with mean zero and variance one, the sequence $S_n=n^{-1/2}\\\\sum _{k=1}^n \\\\xi _k$ converges in distribution, as $n\\\\rightarrow \\\\infty $ , to the standard Gaussian random variable.\",\n \"The Berry-Esseen bound [16] gives the rate of convergence in the Kolmogorov metric: $\\\\sup _{x\\\\in {\\\\mathbb {R}}} |F_n(x)-\\\\Phi (x)|\\\\le \\\\frac{{\\\\mathbb {E}}|\\\\xi _1|^3}{\\\\sqrt{n}},$ where $F_n$ is the cumulative distribution function of $S_n$ and $\\\\Phi $ is the cumulative distribution function of the standard Gaussian random variable.\",\n \"While the rate $1/\\\\sqrt{n}$ is sharp in general, it can be improved by imposing additional conditions on the random variables $\\\\xi _k$ .\",\n \"For example, if ${\\\\mathbb {E}}\\\\xi _k^3=0$ and ${\\\\mathbb {E}}\\\\xi _k^4<\\\\infty $ , then the left-hand side of (REF ) is of order $1/n$ ; cf [21].\",\n \"Of course, if each $\\\\xi _k$ is standard normal, then the left-hand side of (REF ) is zero.\",\n \"The functional version of the Central Limit Theorem, also known as the Donsker invariance principle [16], establishes weak convergence of the sequence of processes $S_n(t)=n^{-1/2}\\\\sum _{k=1}^{\\\\lfloor n t \\\\rfloor } \\\\xi _k+ \\\\frac{nt-\\\\lfloor n t \\\\rfloor }{\\\\sqrt{n}} \\\\xi _{\\\\lfloor n t \\\\rfloor +1},\\\\ t\\\\ge 0,\\\\ n\\\\ge 1,$ to the standard Brownian motion $W$ .\",\n \"An analog of (REF ) becomes a bound on the distance between the distributions of $S_n$ and $W$ on the space of continuous functions ${\\\\mathcal {C}}(0,T)$ in the Lévy-Prokhorov metric.\",\n \"Compared to (REF ), the corresponding rate of convergence depends on integrability properties of $\\\\xi _k$ in a more complicated way: if ${\\\\mathbb {E}}|\\\\xi _1|^p<\\\\infty $ , $p>2$ , then the rate $n^{-(p-2)/(2(p+1))}$ is sharp; if ${\\\\mathbb {E}}e^{t\\\\xi _1}<\\\\infty ,\\\\ |t|<\\\\delta , \\\\ \\\\delta >0,$ then the rate $\\\\ln n/\\\\sqrt{n}$ is sharp.\",\n \"For details, see [7]; earlier works on the subject include [17], [18], [25].\",\n \"A more general approach to investigating the rate of convergence is to find a bound on $\\\\sup _{\\\\varphi \\\\in \\\\mathcal {G}} |{\\\\mathbb {E}}\\\\varphi (S_n)-{\\\\mathbb {E}}\\\\varphi (W)|$ for a suitable class $\\\\mathcal {G}$ of functions $\\\\varphi : {\\\\mathcal {C}}(0,T)\\\\rightarrow {\\\\mathbb {R}}$ .\",\n \"Barbour [3] used an infinite-dimensional version of Stein's method to establish the benchmark result $|{\\\\mathbb {E}}\\\\varphi (S_n)-{\\\\mathbb {E}}\\\\varphi (W)| \\\\le \\\\frac{C}{\\\\sqrt{n}}\\\\Big ( \\\\sqrt{\\\\ln n} + {\\\\mathbb {E}}|\\\\xi _1|^3\\\\Big )$ for a certain (rather restrictive) class $\\\\mathcal {G}$ ; the restrictive nature of this class ensures that there is no contradiction with [7] or [25].\",\n \"For various other $\\\\mathcal {G}$ , there are bounds of the form $|{\\\\mathbb {E}}\\\\varphi (S_n)-{\\\\mathbb {E}}\\\\varphi (W)| \\\\le \\\\frac{C}{{n}^r},\\\\ 00: \\\\mu (A)\\\\le \\\\nu (A^{\\\\varepsilon })+\\\\varepsilon ,\\\\ A\\\\in \\\\mathcal {B}(E)\\\\rbrace ,$ where $A^{\\\\varepsilon }=\\\\lbrace x\\\\in E: \\\\inf \\\\limits _{y\\\\in A} \\\\rho (x,y)\\\\le \\\\varepsilon \\\\rbrace .$ We have $\\\\mathrm {d}_{_{BL}}(\\\\mu ,\\\\nu )\\\\le \\\\mathrm {d}_{{w}}(\\\\mu ,\\\\nu )$ (by definition), $\\\\mathrm {d}_{_{LP}}(\\\\mu ,\\\\nu )\\\\le \\\\sqrt{\\\\mathrm {d}_{_{BL}}(\\\\mu ,\\\\nu )}$ ([9]), and $\\\\mathrm {d}_{_{BL}}(\\\\mu ,\\\\nu )\\\\le 4 \\\\mathrm {d}_{_{LP}}(\\\\mu ,\\\\nu )$ ([9]).\",\n \"In particular, convergence in the Wasserstein-1 metric implies weak convergence, that is, convergence in either Lévy-Prokhorov or bounded Lipschitz metric; the converse is not always true [9]; in fact, the diagram in [11] suggests that the Wasserstein-1 metric $\\\\mathrm {d}_{{w}}$ is the strongest possible for the CLT-type problems in function spaces.\",\n \"Still, a sharp rate of convergence in one metric does not directly lead to a sharp rate in any other metric.\",\n \"The invariance principle can hold if independence requirement for the random variables $\\\\xi _k$ is relaxed, for example, to a strictly stationary and ergodic martingale difference [20], or a stationary Markov process satisfying Doebling's condition [13] [where a bound of the type (REF ) is also established].\",\n \"In continuous time, if $X=X(t),\\\\ t\\\\in {\\\\mathbb {R}},$ is a strictly stationary process with mean zero and covariance function $R(t)={\\\\mathbb {E}}\\\\big (X(t)X(0)\\\\big )$ satisfying $\\\\int _{-\\\\infty }^{+\\\\infty } R(t)\\\\, dt =1$ , then, under some additional conditions of weak dependence, the sequence of processes $S_n(t)=\\\\frac{1}{\\\\sqrt{n}}\\\\int _0^{nt} X(s)\\\\, ds,\\\\ n=1,2,\\\\ldots , t\\\\in [0,1],$ converges weakly to the standard Brownian motion; cf.\",\n \"[20] or [14].\",\n \"The objective of this paper is to show that if $X$ is a stationary Gauss-Markov process, in either discrete or continuous time, then the Wasserstein-1 distance between $S_n$ and $W$ in the space of continuous functions is of order $\\\\big (n^{-1}\\\\ln n\\\\big )^{1/2}$ .\",\n \"In other words, if $S_n$ is Gaussian, then the convergence rate in Wasserstein-1 metric is slightly faster than the Lévy-Prokhorov rate $\\\\ln n/\\\\sqrt{n}$ .\",\n \"This difference does not contradict the results from [7] and [25], and the discrepancy by a $\\\\sqrt{\\\\ln n}$ factor can be explained as follows: in the limit $\\\\sigma \\\\rightarrow 0+$ , the distance between a Gaussian distribution with mean zero and variance $\\\\sigma ^2$ and a point mass at zero is of order $\\\\sigma $ in the Wasserstein-1 metric, but it is of order $\\\\sigma \\\\sqrt{|\\\\ln \\\\sigma |}$ in the Lévy-Prokhorov metric.\",\n \"Section discusses (the easier) continuous-time case (REF ).\",\n \"Discrete-time case, a generalization of (REF ) for a stationary Gaussian sequence $\\\\lbrace \\\\xi _k,\\\\ k\\\\ge 0\\\\rbrace $ , is in Section .\",\n \"In Section , the results are applied to weak approximation for some ordinary differential equations with additive noise.\",\n \"Section is a summary.\",\n \"Traditionally, models (REF ) and (REF ) are studied on a bounded time interval $[0,T]$ , and the index parameter $n=1,2,\\\\ldots $ is discrete.\",\n \"In this paper, the time interval is $(0,+\\\\infty )$ and the index parameter $\\\\kappa \\\\ge 1$ is not necessarily an integer.\",\n \"The following two properties of Gaussian processes will be used on several occasions: 1.\",\n \"The Borell-TIS inequality [1]: If $X=X(t), \\\\ t\\\\in \\\\mathcal {T},$ is a zero-mean Gaussian process indexed by the set $\\\\mathcal {T}$ , and $\\\\mathbb {P}\\\\left( \\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}} X(t) < \\\\infty \\\\right) =1$ , then ${\\\\mathbb {E}}\\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}} |X(t)| < \\\\infty $ and, with $X^*={\\\\mathbb {E}}\\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}} X(t),\\\\ \\\\ \\\\sigma _X^2=\\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}}{\\\\mathbb {E}}X^2(t)$ , $\\\\mathbb {P}\\\\Big (\\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}} X(t) - X^*>x\\\\Big )\\\\le e^{-x^2/(2\\\\sigma _X^2)},\\\\ x>0.$ 2.\",\n \"The Fernique-Sudakov inequality [1]: If $X=X(t), \\\\ Y=Y(t)\\\\ t\\\\in \\\\mathcal {T},$ are zero-mean Gaussian processes indexed by the set $\\\\mathcal {T}$ , and, for all $t,s\\\\in \\\\mathcal {T}$ , ${\\\\mathbb {E}}|X(t)-X(s)|^2 \\\\le {\\\\mathbb {E}}|Y(t)-Y(s)|^2$ , then ${\\\\mathbb {E}}\\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}} X(t) \\\\le {\\\\mathbb {E}}\\\\sup \\\\limits _{t\\\\in \\\\mathcal {T}} Y(t).$\"\n ],\n [\n \"Continuous Time\",\n \"Let $X=X(t),\\\\ t\\\\in {\\\\mathbb {R}},$ be a (continuous version of a) stationary Gaussian process with mean zero and covariance ${\\\\mathbb {E}}X(t)X(s)=e^{-2|t-s|}.$ In particular, $X(t)$ is a standard Gaussian random variable for every $t$ .\",\n \"Equivalent characterizations of $X$ are as follows: $X(t)&=e^{-t}W(e^{2t});\\\\\\\\X(t)&=2\\\\int _{-\\\\infty }^t e^{-2(t-s)}dW(s);\\\\\\\\dX(t)&=-2X(t)dt+2\\\\,dW(t),\\\\ t\\\\ge 0.$ In (REF ), (), and (), $W=W(t), \\\\ t\\\\ge 0,$ is a standard Brownian motion; in (), when $t<0$ , $W(t)=V(-t)$ for an independent copy $V$ of $W$ .\",\n \"The initial condition $X(0)$ in () is a standard Gaussian random variable independent of $W$ .\",\n \"Given a real number $\\\\kappa \\\\ge 1$ , we define $W^{\\\\kappa }(t)=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\int _0^{\\\\kappa t} X(s)\\\\, ds,\\\\ \\\\ t\\\\ge 0.$ Denote by ${\\\\mathcal {C}}_{(0)}$ the collection of continuous functions $f=f(t)$ on $[0,+\\\\infty )$ such that $f(0)=0,\\\\ \\\\lim _{t\\\\rightarrow +\\\\infty } \\\\frac{|f(t)|}{t}=0.$ Endowed with the norm $\\\\Vert f\\\\Vert _{(0)}=\\\\sup _{t>0}\\\\frac{|f(t)|}{1+t},$ ${\\\\mathcal {C}}_{(0)}$ becomes a separable Banach space; cf.\",\n \"[8].\",\n \"We have $\\\\mathbb {P}(W\\\\in {\\\\mathcal {C}}_{(0)})=1$ (either by the law of large numbers for square integrable martingales or using that $t\\\\mapsto tW(1/t)$ is a standard Brownian motion), and also $\\\\mathbb {P}(W^{\\\\kappa }\\\\in {\\\\mathcal {C}}_{(0)})=1$ , $\\\\kappa \\\\ge 1$ , because the ergodic theorem implies $\\\\lim _{t\\\\rightarrow \\\\infty } \\\\frac{1}{t} \\\\int _0^t X(s)\\\\, ds = {\\\\mathbb {E}}X(0) = 0$ with probability one.\",\n \"Proposition 2.1 There exists a constant $C_X$ such that, for every $\\\\kappa \\\\ge 1$ and every function $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ satisfying $|\\\\varphi (f)-\\\\varphi (g)|\\\\le \\\\Vert f-g\\\\Vert _{(0)},$ we have $\\\\Big |{\\\\mathbb {E}}\\\\varphi \\\\big (W^{\\\\kappa }\\\\big )-{\\\\mathbb {E}}\\\\varphi (W)\\\\Big |\\\\le C_X\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Using (), $X(t)=X(0)e^{-2t}+2\\\\int _0^t e^{-2(t-s)}\\\\, dW(s).$ Changing the order of integration (stochastic Fubini theorem [23]), $W^{\\\\kappa }(t)=\\\\kappa ^{-1/2}W(\\\\kappa t)+\\\\frac{X(0)-X(\\\\kappa t)}{2\\\\sqrt{\\\\kappa }}.$ Define $V^{\\\\kappa }(t)=\\\\kappa ^{-1/2}W(\\\\kappa t),\\\\ \\\\ \\\\ X^{\\\\kappa }(t)=\\\\frac{X(\\\\kappa t)-X(0)}{2}.$ Because $t\\\\mapsto V^{\\\\kappa }(t)$ , $t\\\\ge 0$ , is standard Brownian motion for every $\\\\kappa >0$ , we have ${\\\\mathbb {E}}\\\\varphi \\\\big (W\\\\big )={\\\\mathbb {E}}\\\\varphi \\\\big (V^{\\\\kappa }\\\\big )$ and then, using (REF ), $|{\\\\mathbb {E}}\\\\varphi \\\\big (W^{\\\\kappa }\\\\big )-{\\\\mathbb {E}}\\\\big (\\\\varphi (W)\\\\big )\\\\Big |\\\\le \\\\frac{{\\\\mathbb {E}}\\\\Vert X^{\\\\kappa }\\\\Vert _{(0)}}{\\\\sqrt{\\\\kappa }}.$ Next, by [5], $\\\\lim _{T\\\\rightarrow +\\\\infty } \\\\frac{\\\\max _{0\\\\le t\\\\le T} X(t)}{\\\\sqrt{2\\\\ln T}}=1,$ with probability one.\",\n \"Using the same arguments as in [19], we conclude from (REF ) that $\\\\limsup _{T\\\\rightarrow +\\\\infty } \\\\frac{\\\\max _{0\\\\le t\\\\le T} |X(t)|}{\\\\sqrt{2\\\\ln T}}\\\\le 2,$ and then continuity of $X$ implies that the random variable $\\\\zeta =\\\\sup _{t>0} \\\\frac{|X(t)-X(0)|}{2\\\\sqrt{\\\\ln (2+t)}}$ is finite with probability one.\",\n \"Indeed, if $T^*=\\\\sup \\\\left\\\\lbrace t\\\\ge 0: \\\\frac{\\\\max _{0\\\\le t\\\\le T} |X(t)-X(0)|}{2\\\\sqrt{2\\\\ln (2+ T)}}>5\\\\right\\\\rbrace $ then, by (REF ), $\\\\mathbb {P}(T^*<\\\\infty )=1$ , so that $\\\\zeta \\\\le \\\\max _{0\\\\le t\\\\le T^*} \\\\frac{|X(t)-X(0)|}{2\\\\sqrt{\\\\ln (2+t)}} + 5 <\\\\infty .$ Moreover, because $t\\\\mapsto X(t)-X(0)$ is a Gaussian process with mean zero, the Borell-TIS inequality (REF ) implies ${\\\\mathbb {E}}\\\\zeta ^p <\\\\infty $ for all $p>0$ .\",\n \"If $t>0$ and $\\\\kappa \\\\ge 1$ , then, by direct computation, $(2+\\\\kappa t)\\\\le (1+\\\\kappa )(1+t),\\\\ \\\\frac{1}{2}\\\\le \\\\sqrt{\\\\ln (1+\\\\kappa )}, \\\\ \\\\frac{\\\\sqrt{\\\\ln (1+t)}}{1+t}\\\\le \\\\frac{1}{2}.$ As a result, $\\\\frac{\\\\sqrt{\\\\ln (2+\\\\kappa t)}}{1+t}&\\\\le \\\\frac{\\\\sqrt{\\\\ln (1+t)}}{1+t} + \\\\sqrt{\\\\ln (1+\\\\kappa )}\\\\le 2\\\\sqrt{\\\\ln (1+\\\\kappa )},\\\\\\\\\\\\Vert X^{\\\\kappa }\\\\Vert _{(0)} &=\\\\sup _{t>0} \\\\frac{|X(\\\\kappa t)-X(0)|}{2(1+t)} \\\\\\\\&\\\\le \\\\left(\\\\sup _{t,\\\\kappa >0}\\\\frac{|X(\\\\kappa t)-X(0)|}{2\\\\sqrt{\\\\ln (2+\\\\kappa t)}}\\\\right)\\\\left( \\\\sup _{t>0} \\\\frac{\\\\sqrt{\\\\ln (2+\\\\kappa t)}}{1+t}\\\\right)\\\\le 2\\\\zeta \\\\sqrt{\\\\ln (1+\\\\kappa )},$ and (REF ) follows with $C_X={\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{|X(t)-X(0)|}{\\\\sqrt{\\\\ln (2+t)}}\\\\right].$ $\\\\Box $ Denote by $\\\\mu _0$ and $\\\\mu _{\\\\kappa }$ the measures on ${\\\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\\\kappa }$ .\",\n \"The following is the main result of this section, showing that the convergence rate $\\\\kappa ^{-1/2}\\\\sqrt{\\\\ln \\\\kappa }$ is sharp for the Wasserstein-1 metric.\",\n \"Theorem 2.2 There exist positive constants $C_X$ and $c_X$ such that, for every $\\\\kappa \\\\ge 1$ , $c_X\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}\\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0)\\\\le C_X\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ The upper bound in (REF ) follows from (REF ) and Proposition REF .\",\n \"To establish the lower bound, we use (REF ) with a particular $\\\\varphi $ .\",\n \"If $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ is a bounded linear functional, then (REF ) and (REF ) imply ${\\\\mathbb {E}}\\\\varphi (W^{\\\\kappa }) - {\\\\mathbb {E}}\\\\varphi (W)=\\\\kappa ^{-1/2}{\\\\mathbb {E}}\\\\varphi (X^{\\\\kappa }).$ For $f\\\\in {\\\\mathcal {C}}_{(0)},$ define $\\\\varphi _{\\\\kappa }: f \\\\mapsto \\\\frac{f(t^*_{\\\\kappa })}{2},$ where $t^*_{\\\\kappa } = \\\\arg \\\\max _{0\\\\le t \\\\le 1} X^{\\\\kappa }(t).$ In particular, $t^*_{\\\\kappa }\\\\in [0,1]$ .\",\n \"Then $\\\\varphi _{\\\\kappa }$ is a bounded linear functional on ${\\\\mathcal {C}}_{(0)}$ : $|\\\\varphi _{\\\\kappa }(f)|\\\\le \\\\frac{\\\\max _{0\\\\le t\\\\le 1} |f(t)|}{2}\\\\le \\\\max _{0\\\\le t\\\\le 1}\\\\frac{|f(t)|}{1+t}\\\\le \\\\Vert f\\\\Vert _{(0)}.$ Therefore, by (REF ) and (REF ), $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0) \\\\ge \\\\kappa ^{-1/2}|{\\\\mathbb {E}}\\\\varphi _{\\\\kappa }(X^{\\\\kappa })| =\\\\frac{{\\\\mathbb {E}}\\\\max _{0\\\\le t \\\\le 1} X^{\\\\kappa }(t)}{2\\\\sqrt{\\\\kappa }}.$ Next, define $\\\\zeta _{\\\\kappa }=\\\\frac{\\\\max \\\\limits _{0\\\\le t \\\\le 1} X^{\\\\kappa }(t)}{\\\\sqrt{\\\\ln (1+\\\\kappa )}}=\\\\frac{\\\\max \\\\limits _{0\\\\le t \\\\le \\\\kappa }X(t)-X(0)}{2\\\\sqrt{\\\\ln (1+\\\\kappa )}}, \\\\ \\\\ \\\\kappa \\\\ge 1;$ the second equality follows from (REF ).\",\n \"By (REF ), the family $\\\\lbrace \\\\zeta _{\\\\kappa },\\\\ \\\\kappa \\\\ge 1\\\\rbrace $ is uniformly integrable, so that (REF ) implies $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } {\\\\mathbb {E}}\\\\zeta _{\\\\kappa } = \\\\frac{1}{\\\\sqrt{2}},$ which, in turn, means $\\\\inf _{\\\\kappa \\\\ge 1} {\\\\mathbb {E}}\\\\zeta _{\\\\kappa }>0.$ The lower bound in (REF ), with $c_X=\\\\frac{1}{2}\\\\inf _{\\\\kappa \\\\ge 1} \\\\frac{{\\\\mathbb {E}}\\\\left[ \\\\max \\\\limits _{0\\\\le t \\\\le \\\\kappa } X(t)\\\\right]}{\\\\sqrt{\\\\ln (1+\\\\kappa )}},$ now follows from (REF ) and (REF ), because ${\\\\mathbb {E}}X(0)=0$ .\",\n \"$\\\\Box $ To get a better idea about numerical values of $C_X$ and $c_X$ , we need Proposition 2.3 Let $X=X(t),\\\\ t\\\\in {\\\\mathbb {R}},$ be a stationary Gaussian process with mean zero and covariance $e^{-2|t-s|}$ .\",\n \"Define the random variable $\\\\bar{\\\\eta }=\\\\max _{0\\\\le t\\\\le 1} X(t).$ Then $1.2& < {\\\\mathbb {E}}\\\\bar{\\\\eta } < 3.2;\\\\\\\\\\\\mathbb {P}(\\\\bar{\\\\eta }>x)&\\\\le e^{-(x-3.2)^2/2},\\\\ x\\\\ge 3.2.$ Let $B=B(t)$ , $0\\\\le t\\\\le 1$ , be the standard Brownian bridge and $W=W(t)$ , $0\\\\le t\\\\le 1$ , a standard Brownian motion.\",\n \"Then ${\\\\mathbb {E}}|B(t)-B(s)|^2 &= |t-s|-|t-s|^2,\\\\ {\\\\mathbb {E}}|X(t)-X(s)|^2 = 2(1-e^{-2|t-s|}),\\\\\\\\& {\\\\mathbb {E}}|W(t)-W(s)|^2=|t-s|,$ and, because $x-x^2\\\\le 1-e^{-2x}\\\\le 2x,\\\\ x\\\\ge 0,$ inequality (REF ) implies $2\\\\,{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le 1} B(t) \\\\le {\\\\mathbb {E}}\\\\bar{\\\\eta } \\\\le 4\\\\, {\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le 1} W(t).$ It is well known (e.g.\",\n \"[9]) that $\\\\mathbb {P}\\\\left(\\\\max _{0\\\\le t\\\\le 1} B(t) >x\\\\right) = e^{-2x^2}, \\\\ \\\\ \\\\mathbb {P}\\\\left(\\\\max _{0\\\\le t\\\\le 1} W(t) >x\\\\right) = \\\\frac{2}{\\\\sqrt{2\\\\pi }}\\\\int _{x}^{+\\\\infty }e^{-t^2/2}dt.$ Then ${\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le 1} B(t) &= \\\\int _0^{+\\\\infty } e^{-2x^2}\\\\, dx = \\\\frac{\\\\sqrt{2\\\\pi }}{4}>0.6,\\\\\\\\{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le 1} W(t) & =\\\\frac{2}{\\\\sqrt{2\\\\pi }}\\\\int _0^{+\\\\infty }xe^{-x^2/2}\\\\, dx = \\\\sqrt{\\\\frac{2}{\\\\pi }}<0.8,$ and (REF ) follows.\",\n \"After that, () is a re-statement of the Borell-TIS inequality (REF ).\",\n \"$\\\\Box $ We can now show that the number $C_X$ defined in (REF ) satisfies $1.4 < C_X < 14.$ For the lower bound, note that $C_X\\\\ge {\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{X(t)-X(0)}{\\\\sqrt{\\\\ln (2+t)}}\\\\right],$ whereas $\\\\sup _{t>0} \\\\frac{X(t)-X(0)}{\\\\sqrt{\\\\ln (2+t)}}\\\\ge \\\\limsup _{T\\\\rightarrow \\\\infty } \\\\frac{\\\\max \\\\limits _{0\\\\le t\\\\le T} \\\\big (X(t)-X(0)\\\\big )}{\\\\sqrt{\\\\ln (2+T)}},$ and it remains to apply (REF ).\",\n \"For the upper bound in (REF ), start by writing $C_X\\\\le {\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{|X(t)|}{\\\\sqrt{\\\\ln (2+t)}}\\\\right] + \\\\frac{{\\\\mathbb {E}}|X(0)|}{\\\\sqrt{\\\\ln 2}}; \\\\ \\\\ \\\\ \\\\frac{{\\\\mathbb {E}}|X(0)|}{\\\\sqrt{\\\\ln 2}}=\\\\sqrt{\\\\frac{2}{\\\\pi \\\\ln 2}}\\\\approx 0.96.$ Next, let $\\\\bar{X}$ be the process consisting of iid copies of $X(t),\\\\ t\\\\in [0,1),$ on each of the intervals $[k-1,k)$ , $k=1,2,\\\\ldots $ .\",\n \"Then ${\\\\mathbb {E}}X^2(t)={\\\\mathbb {E}}\\\\bar{X}^2(t)$ , ${\\\\mathbb {E}}X(t)X(s)\\\\ge {\\\\mathbb {E}}\\\\bar{X}(t)\\\\bar{X}(s)$ , and so ${\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{|X(t)|}{\\\\sqrt{\\\\ln (2+t)}}\\\\right] \\\\le 2{\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{X(t)}{\\\\sqrt{\\\\ln (2+t)}}\\\\right] \\\\le 2{\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{\\\\bar{X}(t)}{\\\\sqrt{\\\\ln (2+t)}}\\\\right],$ where the first inequality follows from [19], and the second, from (REF ).\",\n \"On the other hand, if $\\\\bar{\\\\eta }_k$ , $k\\\\ge 1$ , are iid copies of the random variable $\\\\bar{\\\\eta }$ from (REF ), then ${\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{\\\\bar{X}(t)}{\\\\sqrt{\\\\ln (2+t)}}\\\\right]\\\\le {\\\\mathbb {E}}\\\\left[\\\\sup _{k\\\\ge 1} \\\\frac{\\\\bar{\\\\eta }_k}{\\\\sqrt{\\\\ln (1+k)}}\\\\right].$ Using () with $x>5$ , $\\\\mathbb {P}\\\\Big (\\\\sup _{k\\\\ge 1} \\\\frac{\\\\bar{\\\\eta }_k}{\\\\sqrt{\\\\ln (1+k)}} > x\\\\Big ) &\\\\le \\\\sum _{k\\\\ge 1} \\\\mathbb {P}\\\\Big (\\\\bar{\\\\eta }_k>x\\\\sqrt{\\\\ln (1+k)}\\\\Big )\\\\\\\\&\\\\le \\\\sum _{k\\\\ge 1}\\\\frac{1}{(1+k)^{-(x-3.2)^2/2}}\\\\le \\\\frac{2}{(x-3.2)^2-2},$ and therefore ${\\\\mathbb {E}}\\\\left[\\\\sup _{k\\\\ge 1} \\\\frac{\\\\bar{\\\\eta }_k}{\\\\sqrt{\\\\ln (1+k)}}\\\\right]\\\\le 5+2\\\\int _5^{+\\\\infty } \\\\frac{dx}{(x-3.2)^2-2} <6.5.$ Then (REF ) follows from (REF ) – (REF ).\",\n \"Next, we will show that the number $c_X$ defined in (REF ) satisfies $0.2 < c_X < 0.8.$ Indeed, the upper bound follows immediately from (REF ).\",\n \"For the lower bound, start by noting that, for $N\\\\le \\\\kappa < N+1$ , ${\\\\mathbb {E}}\\\\left[ \\\\max \\\\limits _{0\\\\le t \\\\le \\\\kappa } X(t)\\\\right] \\\\ge {\\\\mathbb {E}}\\\\left[ \\\\max \\\\limits _{k=1,\\\\ldots , N} X(k)\\\\right]$ and ${\\\\mathbb {E}}|X(k)-X(m)|^2 = 2(1-e^{-2})> 1$ .\",\n \"Now take iid Gaussian $Y_k$ , $k=1,\\\\ldots , N$ with mean zero and variance $1/2$ .\",\n \"Then ${\\\\mathbb {E}}|Y(k)-Y(m)|^2=1$ and so ${\\\\mathbb {E}}|X(k)-X(m)|^2\\\\ge {\\\\mathbb {E}}|Y(k)-Y(m)|^2.$ By (REF ), ${\\\\mathbb {E}}\\\\left[ \\\\max \\\\limits _{k=1,\\\\ldots , N} X(k)\\\\right] \\\\ge {\\\\mathbb {E}}\\\\left[ \\\\max \\\\limits _{k=1,\\\\ldots , N} Y(k)\\\\right],$ and, by [19], ${\\\\mathbb {E}}\\\\left[ \\\\max \\\\limits _{k=1,\\\\ldots , N} Y(k)\\\\right] \\\\ge 0.4\\\\sqrt{\\\\ln N},$ leading to the lower bound in (REF ).\",\n \"To summarize, we can write (REF ) in a more explicit form $0.2\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}\\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0)\\\\le 14\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},\\\\ \\\\kappa \\\\ge 1.$ Theorem REF can be used to study convergence on a bounded interval.\",\n \"For $T>0$ , let ${\\\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\\\mu _{0}^T$ and $\\\\mu _{\\\\kappa }^T$ the measures on ${\\\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\\\kappa }$ .\",\n \"Theorem 2.4 There exist positive constants $C_{X,T}$ and $c_{_{X,T}}$ such that, for every $\\\\kappa \\\\ge 1$ , $c_{_{X,T}}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}\\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T)\\\\le C_{X,T}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ If $f\\\\in {\\\\mathcal {C}}_{(0)}$ , then, for $00$ , let $\\\\lfloor x \\\\rfloor $ denote the largest integer that is less than or equal to $x$ .\",\n \"Define the processes $W^{\\\\kappa }$ by $W^{\\\\kappa }(t)=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor } X_n+\\\\frac{\\\\kappa t-\\\\lfloor \\\\kappa t \\\\rfloor }{\\\\sqrt{\\\\kappa }} X_{\\\\lfloor \\\\kappa t \\\\rfloor +1},\\\\ t\\\\ge 0,\\\\ \\\\kappa \\\\ge 1.$ The second term on the right-hand side of (REF ) ensures that $W^{\\\\kappa }$ is a continuous function of $t$ .\",\n \"The case $a=0$ , that is, the Gaussian version of the original Donsker theorem, is of special interest; the corresponding process $W^{\\\\kappa }$ will be denoted by $S_{\\\\kappa }$ : $S_{\\\\kappa }(t)=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor } \\\\xi _n+\\\\frac{\\\\kappa t-\\\\lfloor \\\\kappa t \\\\rfloor }{\\\\sqrt{\\\\kappa }} \\\\xi _{\\\\lfloor \\\\kappa t \\\\rfloor +1},\\\\ t\\\\ge 0,\\\\ \\\\kappa \\\\ge 1.$ We have $\\\\mathbb {P}(W^{\\\\kappa }\\\\in {\\\\mathcal {C}}_{(0)})=1$ for every $\\\\kappa \\\\ge 1$ , and $a\\\\in (-1,1)$ , because the ergodic theorem implies $\\\\lim _{t\\\\rightarrow \\\\infty } \\\\frac{1}{\\\\lfloor \\\\kappa t \\\\rfloor }\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor } X_n = {\\\\mathbb {E}}X_0 = 0$ with probability one.\",\n \"Let $W=W(t),\\\\ t\\\\ge 0$ , be a standard Brownian motion, and denote by $\\\\mu _0$ and $\\\\mu _{\\\\kappa }$ the measures on ${\\\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\\\kappa }$ .\",\n \"The following is the discrete-time analog of Theorem REF .\",\n \"Theorem 3.1 There exist positive constants $C_a$ and $c_a$ such that, for every $\\\\kappa \\\\ge 1$ , $c_a\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2} \\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0)\\\\le C_a\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ The steps are the same as in the proof of Theorem REF .\",\n \"Substituting (REF ) in (REF ) and changing the order of summation, $W^{\\\\kappa }(t)=S_{\\\\kappa }(t)+\\\\frac{X^{\\\\kappa }(t)}{\\\\sqrt{\\\\kappa }},$ where $S_{\\\\kappa }$ is from (REF ) and $X^{\\\\kappa }(t) = \\\\left( (a-a^{\\\\lfloor \\\\kappa t \\\\rfloor })\\\\sum _{n=-\\\\infty }^{0}a^{-n}\\\\xi _n -\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor }a^{\\\\lfloor \\\\kappa t \\\\rfloor -n}\\\\xi _n\\\\right).$ As a result, it is enough to establish (REF ) when $a=0$ : $c_0\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}\\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0)\\\\le C_0\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Then, similar to the continuous time case, we see that ${\\\\mathbb {E}}\\\\Vert X^{\\\\kappa }\\\\Vert _{(0)} \\\\le \\\\bar{C}_a \\\\sqrt{\\\\ln (1+\\\\kappa )},$ with a suitable constant $\\\\bar{C}_a$ , and then (REF ) follows from (REF ) with $C_a=C_0+\\\\bar{C}_a$ and $c_a=c_0$ .\",\n \"To prove (REF ), we choose the random variables $\\\\xi _k$ in (REF ) as the increments of the Brownian motion $W$ : $\\\\frac{\\\\xi _n}{\\\\sqrt{\\\\kappa }} = W(n/\\\\kappa )- W((n-1)/\\\\kappa ).$ Then, for $(n-1)/\\\\kappa \\\\le t \\\\le n/\\\\kappa $ , the process $t\\\\mapsto S_{\\\\kappa }- W $ is a Brownian bridge, and, for every function $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ satisfying (REF ), $|{\\\\mathbb {E}}\\\\varphi (S_{\\\\kappa })-{\\\\mathbb {E}}\\\\varphi (W)|\\\\le {\\\\mathbb {E}}\\\\Vert B^{\\\\kappa }\\\\Vert _{(0)},$ where $B^{\\\\kappa }$ is a collection of independent Brownian bridges on $[(n-1)/\\\\kappa , n/\\\\kappa ]$ , $n=1,2,\\\\ldots $ .\",\n \"Direct computations show that, for $N=1,2,\\\\ldots $ , $& \\\\sqrt{\\\\kappa }\\\\;{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le N/\\\\kappa } |B^{\\\\kappa }(t)|\\\\le 8\\\\sqrt{{\\\\ln (N+1)}},\\\\\\\\&\\\\sqrt{\\\\kappa }\\\\;{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le N/\\\\kappa } B^{\\\\kappa }(t) \\\\ge 0.3\\\\sqrt{{\\\\ln (N+1)}};$ the numbers $0.3$ and 8 do not necessarily provide optimal bounds.\",\n \"Indeed, let $B=B(t),\\\\ t\\\\in [0,1],$ be the standard Brownian bridge, and let $\\\\eta =\\\\max _{0\\\\le t\\\\le 1} B(t).$ Then $\\\\sqrt{\\\\kappa } \\\\;{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le N/\\\\kappa } |B^{\\\\kappa }(t)| = {\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}|\\\\eta _k|,$ where $\\\\eta _k,\\\\ k=1,\\\\ldots , N$ , are iid copies of $\\\\eta $ .\",\n \"Also, ${\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}\\\\eta _k\\\\le {\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}|\\\\eta _k| \\\\le 2 {\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}\\\\eta _k.$ To derive (REF ), we repeat the arguments from the proof of Lemma 10.1 in [19] using (REF ) and conclude that ${\\\\mathbb {E}}\\\\max \\\\limits _{k=1,\\\\ldots ,N}\\\\eta _k\\\\le 4 \\\\sqrt{\\\\ln (N+1)}$ .\",\n \"Similarly, for (), we repeat the proof of Lemma 10.2 in [19].\",\n \"Next, denote by $\\\\bar{B}$ the process $B^{\\\\kappa }$ corresponding to $\\\\kappa =1$ , that is, the collection of independent standard Brownian bridges on $[n-1,n]$ , $n\\\\ge 1$ .\",\n \"Then we get the upper bound in (REF ), with $C_0={\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{|\\\\bar{B}(t)|}{\\\\sqrt{\\\\ln (2+t)}}\\\\right]< 14,$ by combining (REF ) and (REF ); the upper bound on $C_0$ is from (REF ), because, by (REF ), $C_0\\\\le C_X$ .\",\n \"The lower bound in (REF ), with $c_0=\\\\frac{1}{2}\\\\inf _{\\\\kappa \\\\ge 1} \\\\frac{{\\\\mathbb {E}}\\\\left[ \\\\max _{0\\\\le t \\\\le \\\\kappa } \\\\bar{B}(t)\\\\right]}{\\\\sqrt{\\\\ln (1+\\\\kappa )}}> 0.2,$ follows from () after the same arguments as in the proof of Theorem REF ; the lower bound on $c_0$ follows from ().\",\n \"$\\\\Box $ For $T>0$ , let ${\\\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\\\mu _{0}^T$ and $\\\\mu _{\\\\kappa }^T$ the measures on ${\\\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\\\kappa }$ .\",\n \"The discrete-time version of Theorem REF is obvious.\",\n \"When $a=0$ , and there is no continuous-time analog, we also have the following result (cf.\",\n \"[2]).\",\n \"Proposition 3.2 If $\\\\mu _{\\\\kappa }^T$ is the measure on ${\\\\mathcal {C}}(0,T)$ generated by the process $S_{\\\\kappa }$ from (REF ), then $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\sqrt{\\\\frac{\\\\kappa }{\\\\ln \\\\kappa }} \\\\,\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T) = {\\\\sqrt{2}}.$ Using the random variables $\\\\eta _k$ from the proof of Theorem REF , $\\\\limsup _{\\\\kappa \\\\rightarrow \\\\infty } \\\\sqrt{\\\\frac{ \\\\kappa }{\\\\ln \\\\kappa }}\\\\; \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T) \\\\le \\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{{\\\\mathbb {E}}\\\\left[\\\\max \\\\limits _{k=1,\\\\ldots ,N}|\\\\eta _k|\\\\right]}{\\\\sqrt{\\\\ln N}},$ and $\\\\liminf _{\\\\kappa \\\\rightarrow \\\\infty } \\\\sqrt{\\\\frac{ \\\\kappa }{\\\\ln \\\\kappa }} \\\\;\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T) \\\\ge \\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{{\\\\mathbb {E}}\\\\left[\\\\max \\\\limits _{k=1,\\\\ldots ,N}\\\\eta _k\\\\right]}{\\\\sqrt{\\\\ln N}}.$ By (REF ) and [24], $\\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{\\\\max \\\\limits _{k=1,\\\\ldots ,N}\\\\eta _k}{\\\\sqrt{\\\\ln N}}=\\\\lim _{N\\\\rightarrow \\\\infty } \\\\frac{\\\\max \\\\limits _{k=1,\\\\ldots ,N}|\\\\eta _k|}{\\\\sqrt{\\\\ln N}}={\\\\sqrt{2}}$ with probability 1; then uniform integrability [22] implies (REF ).\",\n \"$\\\\Box $ Applications Let $W$ be a standard Brownian motion and let $W^{\\\\kappa }$ be the process from (REF ) or (REF ).\",\n \"Consider a continuous mapping $\\\\Psi : {\\\\mathcal {C}}_{(0)} \\\\rightarrow {\\\\mathcal {C}}_{(0)}$ .\",\n \"Denote by $\\\\mu _{0,\\\\psi }$ and $\\\\mu _{\\\\kappa ,\\\\psi }$ the measures on ${\\\\mathcal {C}}_{(0)}$ generated by the processes $\\\\Psi (W)$ and $\\\\Psi (W^{\\\\kappa })$ .\",\n \"Proposition 4.1 We have weak convergence $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\mu _{\\\\kappa ,\\\\psi } = \\\\mu _{0,\\\\psi }$ .\",\n \"Moreover, if there exists a number $C_{\\\\psi }$ such that, for all $f,g\\\\in {\\\\mathcal {C}}_{(0)}$ , $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)}\\\\le C_{\\\\psi }\\\\Vert f-g\\\\Vert _{(0)},$ then $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa ,\\\\psi },\\\\mu _{0,\\\\psi })\\\\le \\\\bar{C}_XC_{\\\\psi }\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},$ with $\\\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\\\bar{C}_X=C_a$ from (REF ) in discrete time.\",\n \"Weak convergence follows by the continuous mapping theorem (e.g.\",\n \"[4]).\",\n \"To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ satisfies (REF ), then $|\\\\varphi (\\\\Psi (f))-\\\\varphi (\\\\Psi (g))|\\\\le C_{\\\\psi }\\\\Vert f-g\\\\Vert _{(0)}.$ $\\\\Box $ Example 1.\",\n \"Given ${\\\\alpha }>0$ , let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t) = -{\\\\alpha }\\\\int _0^t Y^{\\\\kappa }(s)\\\\, ds + W^{\\\\kappa }(t),\\\\ \\\\ Y(t)=-{\\\\alpha }\\\\int _0^t Y(s)\\\\, ds + W(t),\\\\ \\\\ t\\\\ge 0.$ Then $Y^{\\\\kappa }, Y\\\\in {\\\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\\\nu _{\\\\kappa }, \\\\nu $ on ${\\\\mathcal {C}}_{(0)}$ satisfy $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa },\\\\nu )\\\\le 2\\\\bar{C}_X\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},\\\\ \\\\kappa \\\\ge 1;$ the constant $\\\\bar{C}_X$ is from Proposition REF .\",\n \"Indeed, by direct computation, $Y^{\\\\kappa }(t)=\\\\Psi _{\\\\alpha }(W^{\\\\kappa })(t), \\\\ \\\\ Y(t)=\\\\Psi _{\\\\alpha }(W)(t),$ where $\\\\Psi _{\\\\alpha } : f(t)\\\\mapsto f(t)-{\\\\alpha }\\\\int _0^t e^{-{\\\\alpha }(t-s)} f(s) \\\\, ds,\\\\ \\\\ f \\\\in \\\\mathcal {C}_{(0)},$ is a linear operator.\",\n \"To see that $\\\\Psi _{\\\\alpha }$ maps ${\\\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\\\frac{|\\\\Psi _{\\\\alpha }(f)(t)|}{1+t}&\\\\le \\\\frac{|f(t)|}{1+t}+\\\\frac{{\\\\alpha }}{1+t}\\\\int _0^t e^{-{\\\\alpha }(t-s)}|f(s)|\\\\, ds\\\\\\\\& \\\\le \\\\frac{|f(t)|}{1+t}+\\\\frac{{\\\\alpha }}{1+t}\\\\int _0^T e^{-{\\\\alpha }(t-s)}|f(s)|\\\\, ds \\\\\\\\&+ \\\\frac{{\\\\alpha }e^{-{\\\\alpha }t}}{1+t}\\\\int _T^t (1+s)e^{{\\\\alpha }s}\\\\, \\\\frac{|f(s)|}{1+s}\\\\, ds.$ If $\\\\lim _{t\\\\rightarrow \\\\infty }|f(t)|/(1+t) = 0$ , then, for every $\\\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\\\varepsilon $ , $s>T$ .\",\n \"As a result, keeping in mind that $\\\\frac{{\\\\alpha }}{1+t}\\\\int _T^t (1+s) e^{{\\\\alpha }s}\\\\, ds \\\\le {\\\\alpha }\\\\int _0^t e^{{\\\\alpha }s}\\\\, ds \\\\le e^{{\\\\alpha }t},$ we compute $\\\\limsup _{t\\\\rightarrow \\\\infty } \\\\frac{|\\\\Psi _{\\\\alpha }(f)(t)|}{1+t} \\\\le \\\\varepsilon $ and conclude that $\\\\lim _{t\\\\rightarrow \\\\infty } |\\\\Psi _{\\\\alpha }(f)(t)|/(1+t)=0$ .\",\n \"Similarly, $\\\\Vert \\\\Psi _{\\\\alpha }(f)\\\\Vert _{(0)} \\\\le \\\\Vert f\\\\Vert _{(0)} \\\\left(1+\\\\alpha \\\\int _0^{+\\\\infty }e^{-\\\\alpha t}\\\\, dt \\\\right) =2\\\\Vert f\\\\Vert _{(0)},$ so that (REF ) holds with $C_{\\\\psi }=2$ .\",\n \"$\\\\Box $ The analog of Proposition REF on a bounded interval is as follows.\",\n \"Let $\\\\Psi ^T$ be a continuous mapping of ${\\\\mathcal {C}}(0,T)$ to itself.\",\n \"Denote by $\\\\mu _{0,\\\\psi }^T$ and $\\\\mu _{\\\\kappa ,\\\\psi }^T$ the measures on ${\\\\mathcal {C}}(0,T)$ generated by the processes $\\\\Psi ^T(W)$ and $\\\\Psi ^T(W^{\\\\kappa })$ .\",\n \"Proposition 4.2 We have weak convergence $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\mu _{\\\\kappa ,\\\\psi }^T = \\\\mu _{0,\\\\psi }^T$ .\",\n \"Moreover, if there exists a number $C_{\\\\psi }^T$ such that, for all $f,g\\\\in {\\\\mathcal {C}}(0,T)$ , $\\\\Vert \\\\Psi ^T(f)-\\\\Psi ^T(g)\\\\Vert _{{\\\\mathcal {C}}(0,T)}\\\\le C_{\\\\psi }^T\\\\Vert f-g\\\\Vert _{{\\\\mathcal {C}}(0,T)},$ then $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa ,\\\\psi }^T,\\\\mu _{0,\\\\psi }^T)\\\\le (1+T)\\\\bar{C}_{X}C_{\\\\psi }^T\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},$ with $\\\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\\\bar{C}_X=C_a$ from (REF ) in discrete time.\",\n \"Example 2.\",\n \"Let the function $b=b(x), \\\\ x\\\\in {\\\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\\\le K|x-y|,\\\\ x,y\\\\in {\\\\mathbb {R}},$ and let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t) = \\\\int _0^t b\\\\big (Y^{\\\\kappa }(s)\\\\big )\\\\, ds + W^{\\\\kappa }(t),\\\\ \\\\ Y(t)=\\\\int _0^t b\\\\big (Y(s)\\\\big )\\\\, ds + W(t),\\\\ \\\\ 0\\\\le t\\\\le T.$ If $\\\\nu _{\\\\kappa }^T, \\\\nu ^T$ are the corresponding measures on ${\\\\mathcal {C}}(0,T)$ , then $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa }^T,\\\\nu ^T)\\\\le (1+T)\\\\bar{C}_Xe^{KT}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Indeed, for $f\\\\in {\\\\mathcal {C}}(0,T)$ , define $\\\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\\\int _0^t b\\\\big (y(s)\\\\big )\\\\, ds + f(t),\\\\ 0\\\\le t\\\\le T.$ By direct computation (e.g.\",\n \"[10]), we have (REF ) with $C_{\\\\psi }^T = e^{KT}$ .\",\n \"$ $ $\\\\Box $ Example 3.\",\n \"Let us combine Examples 1 and 2.\",\n \"Take a positive number ${\\\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t)& = -{\\\\alpha }\\\\int _0^t Y^{\\\\kappa }(s)\\\\, ds+\\\\int _0^t b\\\\big (Y^{\\\\kappa }(s)\\\\big )\\\\, ds + W^{\\\\kappa }(t),\\\\\\\\Y(t)&= -{\\\\alpha }\\\\int _0^t Y(s)\\\\, ds+\\\\int _0^t b\\\\big (Y(s)\\\\big )\\\\, ds + W(t),\\\\ \\\\ t\\\\ge 0.$ If ${\\\\alpha }>K$ , then $Y^{\\\\kappa }, Y \\\\in {\\\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\\\nu _{\\\\kappa }, \\\\nu $ , $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa },\\\\nu )\\\\le \\\\frac{2{\\\\alpha }\\\\bar{C}_X}{{\\\\alpha }-K}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Indeed, for $f\\\\in {\\\\mathcal {C}}_{(0)}$ , define $\\\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\\\alpha }\\\\int _0^t y(s)\\\\, ds + \\\\int _0^t b\\\\big (y(s)\\\\big )\\\\, ds + f(t),\\\\ t\\\\ge 0.$ Using variation of parameters formula and (REF ), $\\\\Psi (f)(t)=\\\\int _0^t e^{-{\\\\alpha }(t-s)} b\\\\big (\\\\Psi (f)(s)\\\\big )\\\\, ds + \\\\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\\\Psi $ maps ${\\\\mathcal {C}}_{(0)}$ to itself.\",\n \"In particular, using (REF ) and (REF ), $\\\\Psi (f)(t)-\\\\Psi (g)(t) =\\\\int _0^t e^{-{\\\\alpha }(t-s)} \\\\Big ( b\\\\big (\\\\Psi (f)(s)\\\\big )-b\\\\big (\\\\Psi (g)(s)\\\\big )\\\\Big )\\\\, ds + \\\\Psi _{\\\\alpha }(f-g)(t)$ so that $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)} \\\\le K \\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)} \\\\int _0^{\\\\infty } e^{-as}\\\\, ds + 2\\\\Vert f-g\\\\Vert _{(0)}.$ As a result, if $\\\\alpha >K$ , then $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)}\\\\le \\\\frac{2{\\\\alpha }}{{\\\\alpha }-K}\\\\Vert f-g\\\\Vert _{(0)},$ and (REF ) follows from (REF ).\",\n \"$\\\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\\\kappa }$ .\",\n \"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.\",\n \"When the underlying processes are Gaussian, some of the approximation errors are not present.\",\n \"In continuous time, the first two steps are the equality (REF ).\",\n \"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.\",\n \"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.\",\n \"For convergence in the space of continuous functions, the $\\\\sqrt{\\\\ln \\\\kappa }$ correction to the classical rate $1/\\\\sqrt{\\\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.\",\n \"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\\\sqrt{\\\\kappa }$ is possible to achieve.\",\n \"For example, by considering $W$ and $S_{\\\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\\\mathbb {E}}\\\\int _0^1 |S_{\\\\kappa }(t)-W(t)|\\\\, dt=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\int _0^1 {\\\\mathbb {E}}|B(t)|\\\\, dt = \\\\sqrt{\\\\frac{2}{\\\\pi \\\\,\\\\kappa }} \\\\int _0^1\\\\sqrt{t(1-t)}\\\\, dt =\\\\sqrt{\\\\frac{\\\\pi }{32\\\\,\\\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\\\sqrt{\\\\kappa }$ ; see also [3].\",\n \"Given the variety of function spaces that can support $W$ and $W^{\\\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\\\sqrt{\\\\kappa }$ bound becomes an interesting challenge.\",\n \"For $\\\\mathrm {d}_{{w}}(W,W^{\\\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\\\sqrt{\\\\ln \\\\kappa /\\\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either.\"\n ],\n [\n \"Discrete Time\",\n \"Consider a stationary Gaussian sequence $X=\\\\lbrace X_n,\\\\ n\\\\ge 0\\\\rbrace ,$ with ${\\\\mathbb {E}}X_n=0$ and ${\\\\mathbb {E}}X_{k+n} X_k=(1-a)a^n/(1+a),\\\\ n,k\\\\ge 0$ , where $a\\\\in (-1,1)$ and $a=0$ corresponds to the sequence of iid standard Gaussian random variables.\",\n \"The variance of $X_n$ is chosen so that, for all $a\\\\in (-1,1)$ , the covariance function $R(n)={\\\\mathbb {E}}X_{k+n} X_n$ of $X$ satisfies $R(0)+2\\\\sum _{n=1}^{\\\\infty } R(n)=1.$ Using a collection $\\\\xi _k,\\\\ k=0,\\\\pm 1,\\\\pm 2,\\\\ldots $ of iid standard normal random variables, we get the discrete-time analogs of () and (): $X_n&=(1-a)\\\\sum _{k=-\\\\infty }^n a^{n-k}\\\\xi _k,\\\\\\\\X_{n+1}&=aX_n+\\\\xi _{n+1};$ in (), the initial condition $X_0$ is independent of $\\\\xi _k,\\\\ k\\\\ge 1,$ and is a normal random variable with mean 0 and variance $(1-a)/(1+a)$ .\",\n \"For $x>0$ , let $\\\\lfloor x \\\\rfloor $ denote the largest integer that is less than or equal to $x$ .\",\n \"Define the processes $W^{\\\\kappa }$ by $W^{\\\\kappa }(t)=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor } X_n+\\\\frac{\\\\kappa t-\\\\lfloor \\\\kappa t \\\\rfloor }{\\\\sqrt{\\\\kappa }} X_{\\\\lfloor \\\\kappa t \\\\rfloor +1},\\\\ t\\\\ge 0,\\\\ \\\\kappa \\\\ge 1.$ The second term on the right-hand side of (REF ) ensures that $W^{\\\\kappa }$ is a continuous function of $t$ .\",\n \"The case $a=0$ , that is, the Gaussian version of the original Donsker theorem, is of special interest; the corresponding process $W^{\\\\kappa }$ will be denoted by $S_{\\\\kappa }$ : $S_{\\\\kappa }(t)=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor } \\\\xi _n+\\\\frac{\\\\kappa t-\\\\lfloor \\\\kappa t \\\\rfloor }{\\\\sqrt{\\\\kappa }} \\\\xi _{\\\\lfloor \\\\kappa t \\\\rfloor +1},\\\\ t\\\\ge 0,\\\\ \\\\kappa \\\\ge 1.$ We have $\\\\mathbb {P}(W^{\\\\kappa }\\\\in {\\\\mathcal {C}}_{(0)})=1$ for every $\\\\kappa \\\\ge 1$ , and $a\\\\in (-1,1)$ , because the ergodic theorem implies $\\\\lim _{t\\\\rightarrow \\\\infty } \\\\frac{1}{\\\\lfloor \\\\kappa t \\\\rfloor }\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor } X_n = {\\\\mathbb {E}}X_0 = 0$ with probability one.\",\n \"Let $W=W(t),\\\\ t\\\\ge 0$ , be a standard Brownian motion, and denote by $\\\\mu _0$ and $\\\\mu _{\\\\kappa }$ the measures on ${\\\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\\\kappa }$ .\",\n \"The following is the discrete-time analog of Theorem REF .\",\n \"Theorem 3.1 There exist positive constants $C_a$ and $c_a$ such that, for every $\\\\kappa \\\\ge 1$ , $c_a\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2} \\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0)\\\\le C_a\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ The steps are the same as in the proof of Theorem REF .\",\n \"Substituting (REF ) in (REF ) and changing the order of summation, $W^{\\\\kappa }(t)=S_{\\\\kappa }(t)+\\\\frac{X^{\\\\kappa }(t)}{\\\\sqrt{\\\\kappa }},$ where $S_{\\\\kappa }$ is from (REF ) and $X^{\\\\kappa }(t) = \\\\left( (a-a^{\\\\lfloor \\\\kappa t \\\\rfloor })\\\\sum _{n=-\\\\infty }^{0}a^{-n}\\\\xi _n -\\\\sum _{n=1}^{\\\\lfloor \\\\kappa t \\\\rfloor }a^{\\\\lfloor \\\\kappa t \\\\rfloor -n}\\\\xi _n\\\\right).$ As a result, it is enough to establish (REF ) when $a=0$ : $c_0\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}\\\\le \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa },\\\\mu _0)\\\\le C_0\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Then, similar to the continuous time case, we see that ${\\\\mathbb {E}}\\\\Vert X^{\\\\kappa }\\\\Vert _{(0)} \\\\le \\\\bar{C}_a \\\\sqrt{\\\\ln (1+\\\\kappa )},$ with a suitable constant $\\\\bar{C}_a$ , and then (REF ) follows from (REF ) with $C_a=C_0+\\\\bar{C}_a$ and $c_a=c_0$ .\",\n \"To prove (REF ), we choose the random variables $\\\\xi _k$ in (REF ) as the increments of the Brownian motion $W$ : $\\\\frac{\\\\xi _n}{\\\\sqrt{\\\\kappa }} = W(n/\\\\kappa )- W((n-1)/\\\\kappa ).$ Then, for $(n-1)/\\\\kappa \\\\le t \\\\le n/\\\\kappa $ , the process $t\\\\mapsto S_{\\\\kappa }- W $ is a Brownian bridge, and, for every function $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ satisfying (REF ), $|{\\\\mathbb {E}}\\\\varphi (S_{\\\\kappa })-{\\\\mathbb {E}}\\\\varphi (W)|\\\\le {\\\\mathbb {E}}\\\\Vert B^{\\\\kappa }\\\\Vert _{(0)},$ where $B^{\\\\kappa }$ is a collection of independent Brownian bridges on $[(n-1)/\\\\kappa , n/\\\\kappa ]$ , $n=1,2,\\\\ldots $ .\",\n \"Direct computations show that, for $N=1,2,\\\\ldots $ , $& \\\\sqrt{\\\\kappa }\\\\;{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le N/\\\\kappa } |B^{\\\\kappa }(t)|\\\\le 8\\\\sqrt{{\\\\ln (N+1)}},\\\\\\\\&\\\\sqrt{\\\\kappa }\\\\;{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le N/\\\\kappa } B^{\\\\kappa }(t) \\\\ge 0.3\\\\sqrt{{\\\\ln (N+1)}};$ the numbers $0.3$ and 8 do not necessarily provide optimal bounds.\",\n \"Indeed, let $B=B(t),\\\\ t\\\\in [0,1],$ be the standard Brownian bridge, and let $\\\\eta =\\\\max _{0\\\\le t\\\\le 1} B(t).$ Then $\\\\sqrt{\\\\kappa } \\\\;{\\\\mathbb {E}}\\\\max _{0\\\\le t\\\\le N/\\\\kappa } |B^{\\\\kappa }(t)| = {\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}|\\\\eta _k|,$ where $\\\\eta _k,\\\\ k=1,\\\\ldots , N$ , are iid copies of $\\\\eta $ .\",\n \"Also, ${\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}\\\\eta _k\\\\le {\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}|\\\\eta _k| \\\\le 2 {\\\\mathbb {E}}\\\\max _{k=1,\\\\ldots ,N}\\\\eta _k.$ To derive (REF ), we repeat the arguments from the proof of Lemma 10.1 in [19] using (REF ) and conclude that ${\\\\mathbb {E}}\\\\max \\\\limits _{k=1,\\\\ldots ,N}\\\\eta _k\\\\le 4 \\\\sqrt{\\\\ln (N+1)}$ .\",\n \"Similarly, for (), we repeat the proof of Lemma 10.2 in [19].\",\n \"Next, denote by $\\\\bar{B}$ the process $B^{\\\\kappa }$ corresponding to $\\\\kappa =1$ , that is, the collection of independent standard Brownian bridges on $[n-1,n]$ , $n\\\\ge 1$ .\",\n \"Then we get the upper bound in (REF ), with $C_0={\\\\mathbb {E}}\\\\left[\\\\sup _{t>0} \\\\frac{|\\\\bar{B}(t)|}{\\\\sqrt{\\\\ln (2+t)}}\\\\right]< 14,$ by combining (REF ) and (REF ); the upper bound on $C_0$ is from (REF ), because, by (REF ), $C_0\\\\le C_X$ .\",\n \"The lower bound in (REF ), with $c_0=\\\\frac{1}{2}\\\\inf _{\\\\kappa \\\\ge 1} \\\\frac{{\\\\mathbb {E}}\\\\left[ \\\\max _{0\\\\le t \\\\le \\\\kappa } \\\\bar{B}(t)\\\\right]}{\\\\sqrt{\\\\ln (1+\\\\kappa )}}> 0.2,$ follows from () after the same arguments as in the proof of Theorem REF ; the lower bound on $c_0$ follows from ().\",\n \"$\\\\Box $ For $T>0$ , let ${\\\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\\\mu _{0}^T$ and $\\\\mu _{\\\\kappa }^T$ the measures on ${\\\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\\\kappa }$ .\",\n \"The discrete-time version of Theorem REF is obvious.\",\n \"When $a=0$ , and there is no continuous-time analog, we also have the following result (cf.\",\n \"[2]).\",\n \"Proposition 3.2 If $\\\\mu _{\\\\kappa }^T$ is the measure on ${\\\\mathcal {C}}(0,T)$ generated by the process $S_{\\\\kappa }$ from (REF ), then $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\sqrt{\\\\frac{\\\\kappa }{\\\\ln \\\\kappa }} \\\\,\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T) = {\\\\sqrt{2}}.$ Using the random variables $\\\\eta _k$ from the proof of Theorem REF , $\\\\limsup _{\\\\kappa \\\\rightarrow \\\\infty } \\\\sqrt{\\\\frac{ \\\\kappa }{\\\\ln \\\\kappa }}\\\\; \\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T) \\\\le \\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{{\\\\mathbb {E}}\\\\left[\\\\max \\\\limits _{k=1,\\\\ldots ,N}|\\\\eta _k|\\\\right]}{\\\\sqrt{\\\\ln N}},$ and $\\\\liminf _{\\\\kappa \\\\rightarrow \\\\infty } \\\\sqrt{\\\\frac{ \\\\kappa }{\\\\ln \\\\kappa }} \\\\;\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa }^T,\\\\mu _0^T) \\\\ge \\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{{\\\\mathbb {E}}\\\\left[\\\\max \\\\limits _{k=1,\\\\ldots ,N}\\\\eta _k\\\\right]}{\\\\sqrt{\\\\ln N}}.$ By (REF ) and [24], $\\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{\\\\max \\\\limits _{k=1,\\\\ldots ,N}\\\\eta _k}{\\\\sqrt{\\\\ln N}}=\\\\lim _{N\\\\rightarrow \\\\infty } \\\\frac{\\\\max \\\\limits _{k=1,\\\\ldots ,N}|\\\\eta _k|}{\\\\sqrt{\\\\ln N}}={\\\\sqrt{2}}$ with probability 1; then uniform integrability [22] implies (REF ).\",\n \"$\\\\Box $ Applications Let $W$ be a standard Brownian motion and let $W^{\\\\kappa }$ be the process from (REF ) or (REF ).\",\n \"Consider a continuous mapping $\\\\Psi : {\\\\mathcal {C}}_{(0)} \\\\rightarrow {\\\\mathcal {C}}_{(0)}$ .\",\n \"Denote by $\\\\mu _{0,\\\\psi }$ and $\\\\mu _{\\\\kappa ,\\\\psi }$ the measures on ${\\\\mathcal {C}}_{(0)}$ generated by the processes $\\\\Psi (W)$ and $\\\\Psi (W^{\\\\kappa })$ .\",\n \"Proposition 4.1 We have weak convergence $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\mu _{\\\\kappa ,\\\\psi } = \\\\mu _{0,\\\\psi }$ .\",\n \"Moreover, if there exists a number $C_{\\\\psi }$ such that, for all $f,g\\\\in {\\\\mathcal {C}}_{(0)}$ , $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)}\\\\le C_{\\\\psi }\\\\Vert f-g\\\\Vert _{(0)},$ then $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa ,\\\\psi },\\\\mu _{0,\\\\psi })\\\\le \\\\bar{C}_XC_{\\\\psi }\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},$ with $\\\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\\\bar{C}_X=C_a$ from (REF ) in discrete time.\",\n \"Weak convergence follows by the continuous mapping theorem (e.g.\",\n \"[4]).\",\n \"To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ satisfies (REF ), then $|\\\\varphi (\\\\Psi (f))-\\\\varphi (\\\\Psi (g))|\\\\le C_{\\\\psi }\\\\Vert f-g\\\\Vert _{(0)}.$ $\\\\Box $ Example 1.\",\n \"Given ${\\\\alpha }>0$ , let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t) = -{\\\\alpha }\\\\int _0^t Y^{\\\\kappa }(s)\\\\, ds + W^{\\\\kappa }(t),\\\\ \\\\ Y(t)=-{\\\\alpha }\\\\int _0^t Y(s)\\\\, ds + W(t),\\\\ \\\\ t\\\\ge 0.$ Then $Y^{\\\\kappa }, Y\\\\in {\\\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\\\nu _{\\\\kappa }, \\\\nu $ on ${\\\\mathcal {C}}_{(0)}$ satisfy $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa },\\\\nu )\\\\le 2\\\\bar{C}_X\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},\\\\ \\\\kappa \\\\ge 1;$ the constant $\\\\bar{C}_X$ is from Proposition REF .\",\n \"Indeed, by direct computation, $Y^{\\\\kappa }(t)=\\\\Psi _{\\\\alpha }(W^{\\\\kappa })(t), \\\\ \\\\ Y(t)=\\\\Psi _{\\\\alpha }(W)(t),$ where $\\\\Psi _{\\\\alpha } : f(t)\\\\mapsto f(t)-{\\\\alpha }\\\\int _0^t e^{-{\\\\alpha }(t-s)} f(s) \\\\, ds,\\\\ \\\\ f \\\\in \\\\mathcal {C}_{(0)},$ is a linear operator.\",\n \"To see that $\\\\Psi _{\\\\alpha }$ maps ${\\\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\\\frac{|\\\\Psi _{\\\\alpha }(f)(t)|}{1+t}&\\\\le \\\\frac{|f(t)|}{1+t}+\\\\frac{{\\\\alpha }}{1+t}\\\\int _0^t e^{-{\\\\alpha }(t-s)}|f(s)|\\\\, ds\\\\\\\\& \\\\le \\\\frac{|f(t)|}{1+t}+\\\\frac{{\\\\alpha }}{1+t}\\\\int _0^T e^{-{\\\\alpha }(t-s)}|f(s)|\\\\, ds \\\\\\\\&+ \\\\frac{{\\\\alpha }e^{-{\\\\alpha }t}}{1+t}\\\\int _T^t (1+s)e^{{\\\\alpha }s}\\\\, \\\\frac{|f(s)|}{1+s}\\\\, ds.$ If $\\\\lim _{t\\\\rightarrow \\\\infty }|f(t)|/(1+t) = 0$ , then, for every $\\\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\\\varepsilon $ , $s>T$ .\",\n \"As a result, keeping in mind that $\\\\frac{{\\\\alpha }}{1+t}\\\\int _T^t (1+s) e^{{\\\\alpha }s}\\\\, ds \\\\le {\\\\alpha }\\\\int _0^t e^{{\\\\alpha }s}\\\\, ds \\\\le e^{{\\\\alpha }t},$ we compute $\\\\limsup _{t\\\\rightarrow \\\\infty } \\\\frac{|\\\\Psi _{\\\\alpha }(f)(t)|}{1+t} \\\\le \\\\varepsilon $ and conclude that $\\\\lim _{t\\\\rightarrow \\\\infty } |\\\\Psi _{\\\\alpha }(f)(t)|/(1+t)=0$ .\",\n \"Similarly, $\\\\Vert \\\\Psi _{\\\\alpha }(f)\\\\Vert _{(0)} \\\\le \\\\Vert f\\\\Vert _{(0)} \\\\left(1+\\\\alpha \\\\int _0^{+\\\\infty }e^{-\\\\alpha t}\\\\, dt \\\\right) =2\\\\Vert f\\\\Vert _{(0)},$ so that (REF ) holds with $C_{\\\\psi }=2$ .\",\n \"$\\\\Box $ The analog of Proposition REF on a bounded interval is as follows.\",\n \"Let $\\\\Psi ^T$ be a continuous mapping of ${\\\\mathcal {C}}(0,T)$ to itself.\",\n \"Denote by $\\\\mu _{0,\\\\psi }^T$ and $\\\\mu _{\\\\kappa ,\\\\psi }^T$ the measures on ${\\\\mathcal {C}}(0,T)$ generated by the processes $\\\\Psi ^T(W)$ and $\\\\Psi ^T(W^{\\\\kappa })$ .\",\n \"Proposition 4.2 We have weak convergence $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\mu _{\\\\kappa ,\\\\psi }^T = \\\\mu _{0,\\\\psi }^T$ .\",\n \"Moreover, if there exists a number $C_{\\\\psi }^T$ such that, for all $f,g\\\\in {\\\\mathcal {C}}(0,T)$ , $\\\\Vert \\\\Psi ^T(f)-\\\\Psi ^T(g)\\\\Vert _{{\\\\mathcal {C}}(0,T)}\\\\le C_{\\\\psi }^T\\\\Vert f-g\\\\Vert _{{\\\\mathcal {C}}(0,T)},$ then $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa ,\\\\psi }^T,\\\\mu _{0,\\\\psi }^T)\\\\le (1+T)\\\\bar{C}_{X}C_{\\\\psi }^T\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},$ with $\\\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\\\bar{C}_X=C_a$ from (REF ) in discrete time.\",\n \"Example 2.\",\n \"Let the function $b=b(x), \\\\ x\\\\in {\\\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\\\le K|x-y|,\\\\ x,y\\\\in {\\\\mathbb {R}},$ and let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t) = \\\\int _0^t b\\\\big (Y^{\\\\kappa }(s)\\\\big )\\\\, ds + W^{\\\\kappa }(t),\\\\ \\\\ Y(t)=\\\\int _0^t b\\\\big (Y(s)\\\\big )\\\\, ds + W(t),\\\\ \\\\ 0\\\\le t\\\\le T.$ If $\\\\nu _{\\\\kappa }^T, \\\\nu ^T$ are the corresponding measures on ${\\\\mathcal {C}}(0,T)$ , then $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa }^T,\\\\nu ^T)\\\\le (1+T)\\\\bar{C}_Xe^{KT}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Indeed, for $f\\\\in {\\\\mathcal {C}}(0,T)$ , define $\\\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\\\int _0^t b\\\\big (y(s)\\\\big )\\\\, ds + f(t),\\\\ 0\\\\le t\\\\le T.$ By direct computation (e.g.\",\n \"[10]), we have (REF ) with $C_{\\\\psi }^T = e^{KT}$ .\",\n \"$ $ $\\\\Box $ Example 3.\",\n \"Let us combine Examples 1 and 2.\",\n \"Take a positive number ${\\\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t)& = -{\\\\alpha }\\\\int _0^t Y^{\\\\kappa }(s)\\\\, ds+\\\\int _0^t b\\\\big (Y^{\\\\kappa }(s)\\\\big )\\\\, ds + W^{\\\\kappa }(t),\\\\\\\\Y(t)&= -{\\\\alpha }\\\\int _0^t Y(s)\\\\, ds+\\\\int _0^t b\\\\big (Y(s)\\\\big )\\\\, ds + W(t),\\\\ \\\\ t\\\\ge 0.$ If ${\\\\alpha }>K$ , then $Y^{\\\\kappa }, Y \\\\in {\\\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\\\nu _{\\\\kappa }, \\\\nu $ , $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa },\\\\nu )\\\\le \\\\frac{2{\\\\alpha }\\\\bar{C}_X}{{\\\\alpha }-K}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Indeed, for $f\\\\in {\\\\mathcal {C}}_{(0)}$ , define $\\\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\\\alpha }\\\\int _0^t y(s)\\\\, ds + \\\\int _0^t b\\\\big (y(s)\\\\big )\\\\, ds + f(t),\\\\ t\\\\ge 0.$ Using variation of parameters formula and (REF ), $\\\\Psi (f)(t)=\\\\int _0^t e^{-{\\\\alpha }(t-s)} b\\\\big (\\\\Psi (f)(s)\\\\big )\\\\, ds + \\\\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\\\Psi $ maps ${\\\\mathcal {C}}_{(0)}$ to itself.\",\n \"In particular, using (REF ) and (REF ), $\\\\Psi (f)(t)-\\\\Psi (g)(t) =\\\\int _0^t e^{-{\\\\alpha }(t-s)} \\\\Big ( b\\\\big (\\\\Psi (f)(s)\\\\big )-b\\\\big (\\\\Psi (g)(s)\\\\big )\\\\Big )\\\\, ds + \\\\Psi _{\\\\alpha }(f-g)(t)$ so that $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)} \\\\le K \\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)} \\\\int _0^{\\\\infty } e^{-as}\\\\, ds + 2\\\\Vert f-g\\\\Vert _{(0)}.$ As a result, if $\\\\alpha >K$ , then $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)}\\\\le \\\\frac{2{\\\\alpha }}{{\\\\alpha }-K}\\\\Vert f-g\\\\Vert _{(0)},$ and (REF ) follows from (REF ).\",\n \"$\\\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\\\kappa }$ .\",\n \"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.\",\n \"When the underlying processes are Gaussian, some of the approximation errors are not present.\",\n \"In continuous time, the first two steps are the equality (REF ).\",\n \"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.\",\n \"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.\",\n \"For convergence in the space of continuous functions, the $\\\\sqrt{\\\\ln \\\\kappa }$ correction to the classical rate $1/\\\\sqrt{\\\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.\",\n \"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\\\sqrt{\\\\kappa }$ is possible to achieve.\",\n \"For example, by considering $W$ and $S_{\\\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\\\mathbb {E}}\\\\int _0^1 |S_{\\\\kappa }(t)-W(t)|\\\\, dt=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\int _0^1 {\\\\mathbb {E}}|B(t)|\\\\, dt = \\\\sqrt{\\\\frac{2}{\\\\pi \\\\,\\\\kappa }} \\\\int _0^1\\\\sqrt{t(1-t)}\\\\, dt =\\\\sqrt{\\\\frac{\\\\pi }{32\\\\,\\\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\\\sqrt{\\\\kappa }$ ; see also [3].\",\n \"Given the variety of function spaces that can support $W$ and $W^{\\\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\\\sqrt{\\\\kappa }$ bound becomes an interesting challenge.\",\n \"For $\\\\mathrm {d}_{{w}}(W,W^{\\\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\\\sqrt{\\\\ln \\\\kappa /\\\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either.\"\n ],\n [\n \"Applications\",\n \"Let $W$ be a standard Brownian motion and let $W^{\\\\kappa }$ be the process from (REF ) or (REF ).\",\n \"Consider a continuous mapping $\\\\Psi : {\\\\mathcal {C}}_{(0)} \\\\rightarrow {\\\\mathcal {C}}_{(0)}$ .\",\n \"Denote by $\\\\mu _{0,\\\\psi }$ and $\\\\mu _{\\\\kappa ,\\\\psi }$ the measures on ${\\\\mathcal {C}}_{(0)}$ generated by the processes $\\\\Psi (W)$ and $\\\\Psi (W^{\\\\kappa })$ .\",\n \"Proposition 4.1 We have weak convergence $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\mu _{\\\\kappa ,\\\\psi } = \\\\mu _{0,\\\\psi }$ .\",\n \"Moreover, if there exists a number $C_{\\\\psi }$ such that, for all $f,g\\\\in {\\\\mathcal {C}}_{(0)}$ , $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)}\\\\le C_{\\\\psi }\\\\Vert f-g\\\\Vert _{(0)},$ then $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa ,\\\\psi },\\\\mu _{0,\\\\psi })\\\\le \\\\bar{C}_XC_{\\\\psi }\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},$ with $\\\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\\\bar{C}_X=C_a$ from (REF ) in discrete time.\",\n \"Weak convergence follows by the continuous mapping theorem (e.g.\",\n \"[4]).\",\n \"To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\\\varphi : {\\\\mathcal {C}}_{(0)}\\\\rightarrow {\\\\mathbb {R}}$ satisfies (REF ), then $|\\\\varphi (\\\\Psi (f))-\\\\varphi (\\\\Psi (g))|\\\\le C_{\\\\psi }\\\\Vert f-g\\\\Vert _{(0)}.$ $\\\\Box $ Example 1.\",\n \"Given ${\\\\alpha }>0$ , let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t) = -{\\\\alpha }\\\\int _0^t Y^{\\\\kappa }(s)\\\\, ds + W^{\\\\kappa }(t),\\\\ \\\\ Y(t)=-{\\\\alpha }\\\\int _0^t Y(s)\\\\, ds + W(t),\\\\ \\\\ t\\\\ge 0.$ Then $Y^{\\\\kappa }, Y\\\\in {\\\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\\\nu _{\\\\kappa }, \\\\nu $ on ${\\\\mathcal {C}}_{(0)}$ satisfy $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa },\\\\nu )\\\\le 2\\\\bar{C}_X\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},\\\\ \\\\kappa \\\\ge 1;$ the constant $\\\\bar{C}_X$ is from Proposition REF .\",\n \"Indeed, by direct computation, $Y^{\\\\kappa }(t)=\\\\Psi _{\\\\alpha }(W^{\\\\kappa })(t), \\\\ \\\\ Y(t)=\\\\Psi _{\\\\alpha }(W)(t),$ where $\\\\Psi _{\\\\alpha } : f(t)\\\\mapsto f(t)-{\\\\alpha }\\\\int _0^t e^{-{\\\\alpha }(t-s)} f(s) \\\\, ds,\\\\ \\\\ f \\\\in \\\\mathcal {C}_{(0)},$ is a linear operator.\",\n \"To see that $\\\\Psi _{\\\\alpha }$ maps ${\\\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\\\frac{|\\\\Psi _{\\\\alpha }(f)(t)|}{1+t}&\\\\le \\\\frac{|f(t)|}{1+t}+\\\\frac{{\\\\alpha }}{1+t}\\\\int _0^t e^{-{\\\\alpha }(t-s)}|f(s)|\\\\, ds\\\\\\\\& \\\\le \\\\frac{|f(t)|}{1+t}+\\\\frac{{\\\\alpha }}{1+t}\\\\int _0^T e^{-{\\\\alpha }(t-s)}|f(s)|\\\\, ds \\\\\\\\&+ \\\\frac{{\\\\alpha }e^{-{\\\\alpha }t}}{1+t}\\\\int _T^t (1+s)e^{{\\\\alpha }s}\\\\, \\\\frac{|f(s)|}{1+s}\\\\, ds.$ If $\\\\lim _{t\\\\rightarrow \\\\infty }|f(t)|/(1+t) = 0$ , then, for every $\\\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\\\varepsilon $ , $s>T$ .\",\n \"As a result, keeping in mind that $\\\\frac{{\\\\alpha }}{1+t}\\\\int _T^t (1+s) e^{{\\\\alpha }s}\\\\, ds \\\\le {\\\\alpha }\\\\int _0^t e^{{\\\\alpha }s}\\\\, ds \\\\le e^{{\\\\alpha }t},$ we compute $\\\\limsup _{t\\\\rightarrow \\\\infty } \\\\frac{|\\\\Psi _{\\\\alpha }(f)(t)|}{1+t} \\\\le \\\\varepsilon $ and conclude that $\\\\lim _{t\\\\rightarrow \\\\infty } |\\\\Psi _{\\\\alpha }(f)(t)|/(1+t)=0$ .\",\n \"Similarly, $\\\\Vert \\\\Psi _{\\\\alpha }(f)\\\\Vert _{(0)} \\\\le \\\\Vert f\\\\Vert _{(0)} \\\\left(1+\\\\alpha \\\\int _0^{+\\\\infty }e^{-\\\\alpha t}\\\\, dt \\\\right) =2\\\\Vert f\\\\Vert _{(0)},$ so that (REF ) holds with $C_{\\\\psi }=2$ .\",\n \"$\\\\Box $ The analog of Proposition REF on a bounded interval is as follows.\",\n \"Let $\\\\Psi ^T$ be a continuous mapping of ${\\\\mathcal {C}}(0,T)$ to itself.\",\n \"Denote by $\\\\mu _{0,\\\\psi }^T$ and $\\\\mu _{\\\\kappa ,\\\\psi }^T$ the measures on ${\\\\mathcal {C}}(0,T)$ generated by the processes $\\\\Psi ^T(W)$ and $\\\\Psi ^T(W^{\\\\kappa })$ .\",\n \"Proposition 4.2 We have weak convergence $\\\\lim _{\\\\kappa \\\\rightarrow \\\\infty } \\\\mu _{\\\\kappa ,\\\\psi }^T = \\\\mu _{0,\\\\psi }^T$ .\",\n \"Moreover, if there exists a number $C_{\\\\psi }^T$ such that, for all $f,g\\\\in {\\\\mathcal {C}}(0,T)$ , $\\\\Vert \\\\Psi ^T(f)-\\\\Psi ^T(g)\\\\Vert _{{\\\\mathcal {C}}(0,T)}\\\\le C_{\\\\psi }^T\\\\Vert f-g\\\\Vert _{{\\\\mathcal {C}}(0,T)},$ then $\\\\mathrm {d}_{{w}}(\\\\mu _{\\\\kappa ,\\\\psi }^T,\\\\mu _{0,\\\\psi }^T)\\\\le (1+T)\\\\bar{C}_{X}C_{\\\\psi }^T\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2},$ with $\\\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\\\bar{C}_X=C_a$ from (REF ) in discrete time.\",\n \"Example 2.\",\n \"Let the function $b=b(x), \\\\ x\\\\in {\\\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\\\le K|x-y|,\\\\ x,y\\\\in {\\\\mathbb {R}},$ and let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t) = \\\\int _0^t b\\\\big (Y^{\\\\kappa }(s)\\\\big )\\\\, ds + W^{\\\\kappa }(t),\\\\ \\\\ Y(t)=\\\\int _0^t b\\\\big (Y(s)\\\\big )\\\\, ds + W(t),\\\\ \\\\ 0\\\\le t\\\\le T.$ If $\\\\nu _{\\\\kappa }^T, \\\\nu ^T$ are the corresponding measures on ${\\\\mathcal {C}}(0,T)$ , then $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa }^T,\\\\nu ^T)\\\\le (1+T)\\\\bar{C}_Xe^{KT}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Indeed, for $f\\\\in {\\\\mathcal {C}}(0,T)$ , define $\\\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\\\int _0^t b\\\\big (y(s)\\\\big )\\\\, ds + f(t),\\\\ 0\\\\le t\\\\le T.$ By direct computation (e.g.\",\n \"[10]), we have (REF ) with $C_{\\\\psi }^T = e^{KT}$ .\",\n \"$ $ $\\\\Box $ Example 3.\",\n \"Let us combine Examples 1 and 2.\",\n \"Take a positive number ${\\\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\\\kappa }, Y$ be the solutions of $Y^{\\\\kappa }(t)& = -{\\\\alpha }\\\\int _0^t Y^{\\\\kappa }(s)\\\\, ds+\\\\int _0^t b\\\\big (Y^{\\\\kappa }(s)\\\\big )\\\\, ds + W^{\\\\kappa }(t),\\\\\\\\Y(t)&= -{\\\\alpha }\\\\int _0^t Y(s)\\\\, ds+\\\\int _0^t b\\\\big (Y(s)\\\\big )\\\\, ds + W(t),\\\\ \\\\ t\\\\ge 0.$ If ${\\\\alpha }>K$ , then $Y^{\\\\kappa }, Y \\\\in {\\\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\\\nu _{\\\\kappa }, \\\\nu $ , $\\\\mathrm {d}_{{w}}(\\\\nu _{\\\\kappa },\\\\nu )\\\\le \\\\frac{2{\\\\alpha }\\\\bar{C}_X}{{\\\\alpha }-K}\\\\left(\\\\frac{\\\\ln (1+\\\\kappa )}{\\\\kappa }\\\\right)^{1/2}.$ Indeed, for $f\\\\in {\\\\mathcal {C}}_{(0)}$ , define $\\\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\\\alpha }\\\\int _0^t y(s)\\\\, ds + \\\\int _0^t b\\\\big (y(s)\\\\big )\\\\, ds + f(t),\\\\ t\\\\ge 0.$ Using variation of parameters formula and (REF ), $\\\\Psi (f)(t)=\\\\int _0^t e^{-{\\\\alpha }(t-s)} b\\\\big (\\\\Psi (f)(s)\\\\big )\\\\, ds + \\\\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\\\Psi $ maps ${\\\\mathcal {C}}_{(0)}$ to itself.\",\n \"In particular, using (REF ) and (REF ), $\\\\Psi (f)(t)-\\\\Psi (g)(t) =\\\\int _0^t e^{-{\\\\alpha }(t-s)} \\\\Big ( b\\\\big (\\\\Psi (f)(s)\\\\big )-b\\\\big (\\\\Psi (g)(s)\\\\big )\\\\Big )\\\\, ds + \\\\Psi _{\\\\alpha }(f-g)(t)$ so that $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)} \\\\le K \\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)} \\\\int _0^{\\\\infty } e^{-as}\\\\, ds + 2\\\\Vert f-g\\\\Vert _{(0)}.$ As a result, if $\\\\alpha >K$ , then $\\\\Vert \\\\Psi (f)-\\\\Psi (g)\\\\Vert _{(0)}\\\\le \\\\frac{2{\\\\alpha }}{{\\\\alpha }-K}\\\\Vert f-g\\\\Vert _{(0)},$ and (REF ) follows from (REF ).\",\n \"$\\\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\\\kappa }$ .\",\n \"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.\",\n \"When the underlying processes are Gaussian, some of the approximation errors are not present.\",\n \"In continuous time, the first two steps are the equality (REF ).\",\n \"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.\",\n \"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.\",\n \"For convergence in the space of continuous functions, the $\\\\sqrt{\\\\ln \\\\kappa }$ correction to the classical rate $1/\\\\sqrt{\\\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.\",\n \"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\\\sqrt{\\\\kappa }$ is possible to achieve.\",\n \"For example, by considering $W$ and $S_{\\\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\\\mathbb {E}}\\\\int _0^1 |S_{\\\\kappa }(t)-W(t)|\\\\, dt=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\int _0^1 {\\\\mathbb {E}}|B(t)|\\\\, dt = \\\\sqrt{\\\\frac{2}{\\\\pi \\\\,\\\\kappa }} \\\\int _0^1\\\\sqrt{t(1-t)}\\\\, dt =\\\\sqrt{\\\\frac{\\\\pi }{32\\\\,\\\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\\\sqrt{\\\\kappa }$ ; see also [3].\",\n \"Given the variety of function spaces that can support $W$ and $W^{\\\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\\\sqrt{\\\\kappa }$ bound becomes an interesting challenge.\",\n \"For $\\\\mathrm {d}_{{w}}(W,W^{\\\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\\\sqrt{\\\\ln \\\\kappa /\\\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either.\"\n ],\n [\n \"Concluding Remarks\",\n \"A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\\\kappa }$ .\",\n \"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.\",\n \"When the underlying processes are Gaussian, some of the approximation errors are not present.\",\n \"In continuous time, the first two steps are the equality (REF ).\",\n \"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.\",\n \"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.\",\n \"For convergence in the space of continuous functions, the $\\\\sqrt{\\\\ln \\\\kappa }$ correction to the classical rate $1/\\\\sqrt{\\\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.\",\n \"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\\\sqrt{\\\\kappa }$ is possible to achieve.\",\n \"For example, by considering $W$ and $S_{\\\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\\\mathbb {E}}\\\\int _0^1 |S_{\\\\kappa }(t)-W(t)|\\\\, dt=\\\\frac{1}{\\\\sqrt{\\\\kappa }}\\\\int _0^1 {\\\\mathbb {E}}|B(t)|\\\\, dt = \\\\sqrt{\\\\frac{2}{\\\\pi \\\\,\\\\kappa }} \\\\int _0^1\\\\sqrt{t(1-t)}\\\\, dt =\\\\sqrt{\\\\frac{\\\\pi }{32\\\\,\\\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\\\sqrt{\\\\kappa }$ ; see also [3].\",\n \"Given the variety of function spaces that can support $W$ and $W^{\\\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\\\sqrt{\\\\kappa }$ bound becomes an interesting challenge.\",\n \"For $\\\\mathrm {d}_{{w}}(W,W^{\\\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\\\sqrt{\\\\ln \\\\kappa /\\\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either.\"\n ]\n]"},"id":{"kind":"string","value":"2209.08249"}}},{"rowIdx":2390,"cells":{"text":{"kind":"list like","value":[["General History of X-Ray Polarimetry in Astrophysics"],["Abstract Soon after the discovery of the first extrasolar X-Ray sources it was suggested that polarimetry could play a major role as a diagnostic tool.","Attempts to measure polarization of X-Ray sources was performed by the team of Columbia University lead by Robert Novick.","The technique of Bragg diffraction at 45{\\deg} was successful to detect the polarization of the Crab with rockets and with OSO-8 satellite.","In the following evolution of X-Ray Astronomy, Polarimetry was too mismatched with the improved sensitivity of imaging and spectroscopy, based on the use of optics.","As a consequence no polarimeter was flown any more.","At the beginning of the century a new class of instruments based on the photoelectric effect were developed.","In the focus of an X-Ray telescope they can perform angular, energy and time resolved polarimetry and benefit of the large increase of sensitivity due to the optics.","The Imaging X-Ray Polarimetry Explorer, exploiting this technique, was launched at the end of 2021."],["The very early stage","Only 9 years after the discovery of X-Rays by Roentgen in 1904 Charles Glover Barkla, a student of Stokes, made experiments of scattering of these newly discovered particles and found that they follow the same rules of polarization of optical light and this demonstrated that these mysterious X-rays were electromagnetic radiation.","Physics and Polarimetry of X-Rays are born almost together.","When in 1962 Giacconi and Rossi found the first evidence for extrasolar X-ray sources theoreticians predicted that polarimetry would have in this wavelength domain a major role than in other domains.","The Soviet Physicist Vitali Ginzburg was very active in this promotion also toward the western community.","The easiest implementation of this concept would be an instrument based on the angular distribution of photons scattered from a target.","The difficulty is that the scattering prevails over the absorption at energies of few keV, where the majority of photons is, only with Hydrogen and Helium, that have a negligible stopping power for X-Ray photons.","The best dense scattering material is Lithium (that in any case must be encased with beryllium) for which the scattering prevails only above 10 keV.","In 1969 Herbert Schnopper suggested the use of Bragg crystals at incidence angles of $45^{0}$ as a good analyzer of linear polarization[1].","The technique would be very robust and simple.","The limit is the narrow band of the diffracted radiation that would result in a low effective area.","Moreover the detector would be of the same surface of the entrance window, and the poor resolution of the detectors would not allow to separate effectively by the pulse height the diffracted photons, so that a large background could be expected.","The team of Columbia University lead by Robert Novick carried on a systematic study to arrive to perform an astronomical experiment.","His co-workers Roger Angel and Martin Weisskopf found that the efficiency could be improved by using mosaic crystals of pyrolytic graphite (i.e.","a crystal with the spread of orientation of micro-domains artificially enhanced).","Moreover they found that a moderate bending of the crystal would allow a concentration of diffracted photons on a small detector, with a much lower background rate, while substantially preserving the modulated response to polarization[2],[3].","The Team of Columbia launched the first rocket searching for polarization of ScoX-1 on 1968[4].","The rocket included scattering targets of lithium encased in beryllium surrounded by proportional counters.","The polarization angle was derived from the angle between the scatterer and the detector and the spinning phase of the rocket.","The observation of ScoX1 (performed only 6 years after the discovery of the source!)","was unsuccessful in terms of polarimetry.","But the same rocket payload, was launched again to point the Crab Nebula.","The result was inconclusive but still compatible with the high polarization found in optical and radio bands.","When the technique of Bragg diffractors wih mosaic graphite was mature an improved version of the rocket was built.","Out of the atmosphere four panels mounting graphite crystals were deployed on the side at an angle of $45^{0}$ to the pointing/spinning axis of the rocket so that the photons parallel to the axis were diffracted toward proportional counters hosted in the rocket below the scattering stage of the payload that was similar to that of the first rockets.","Also this result was inconclusive.","But by overimposing the data of all flights Novick found at last a statistically significant evidence of polarization[5].","This was a paramount result in terms of physics of X-ray sources also confirming the diagnostic relevance of this subtopics, notwithstanding the experimental difficulties.","Another achievement for the future was that, in the few keV range, the Bragg approach, notwithstanding the very low efficiency, is better than Thomson in terms of both sensitivity and reliability While performing this first set of experiments the Novick Team also fixed the statistic frame that would be used for all the planned and the (few) performed experiments.","Both scattering and Bragg polarimeters would produce a histogram of phase of emerging photons.","The histogram can be fitted with a constant term plus a cos$^{2}$ term.","The ratio between the two terms carries the information about polarization degree while the phase of the second term carries the information n the polarization angle.","$M=\\dfrac{S_{max}-S_{min}}{S_{max}+S_{min}}$ The M is named modulation.","The modulation for a beam 100$ \\% $ polarized, conventionally named modulation factor ($\\mu $ ) is a feature of the detector and usually depends on the energy.","If counts follow Poisson Statistics, as usually do in X-ray detectors, the modulation, which is a positively defined quantity, follows a distribution of $ \\chi ^{2} $ for two degrees of freedom.","This is the basis to define the significance of a possible detection.","The Minimum Detectable Modulation can be computed starting from the statistics as the modulation that in only 1$\\%$ of occurrences can be arrived or passed by statistical fluctuation.","By dividing for the modulation factor we can derive the Minimum Detectable Polarization (MDP) that is the usual parameter to describe the sensitivity of an experiment of X-ray polarimetry when studying a source of a certain flux and spectrum.","$MDP=\\dfrac{4.29}{\\mu \\varepsilon S}\\times \\sqrt{\\dfrac{\\varepsilon S+B}{T}}$ The fact that a 99$ \\% $ confidence was chosen, instead than the most usual 3$ \\sigma $ (or 5$ \\sigma $ ), witnesses the awareness that polarimetry would be a tough challenge.","The Columbia team was also active in optical polarimetry of sources of potential interest for X-ray astronomy, so that it was familiar with the Stokes parameters formalism.","Actually the results of the first rockets were presented and discussed in this formal frame, although both Bragg and Thomson stages can only detect linear polarization.","The V parameter is unmeasurable so that the use of the Stokes is, in the frame of X-ray polarimetry only, of moderate usefulness.","To compare results in a sky map a display of the polarization degree and angle could be the most informative.","But in order to insert the X-ray result in a general frame of measurements in other wavelength or compare with theories the Stokes parameter representation would be the most suited.","In the topical paper where polarization is detected the discussion is concentrated on the technique to combine data of different experiments and flights and the related systematics, so that, in this case, the Stokes Parameters were ignored."],["Ariel-5 and OSO-8","The first two satellites with an X-ray polarimeter aboard, Ariel-5 and OSO-8, and the only ones for the following 45 years, were launched on 1974 and 1975 respectively, only 5 and 6 years after the first X-Ray survey mission UHURU.","The ARIEL-5 spectro-polarimeter designed and built by the Team of Leicester University lead by Richard Griffiths[6] was also based on Bragg Diffraction but was somehow different.","The direction of the radiation impinging on the instrument and that of the diffracted photons impinging on the detectors were defined by mechanical collimators as well.","But the diffracting crystals were of two kinds, one of mosaic graphite, like OSO-8, the second one of LiF.","Crystals were flat and their inclination, with respect to the incoming radiation could be adjusted in order to tune the mounting angle and the related diffraction energy.","The spinning satellite would be pointed not to the source but to a direction slightly off-set.","By this trick the photons from the source would impinge on the crystal with an angle slightly different at each phase of spin.","This would allow to perform high resolution spectroscopy of expected Si and S lines.","The LiF was used to make spectra of Fe lines.","At an angle of $45^{0}$ the diffractometer could be used as a polarimeter around an energy of 2.5 keV.","In OSO-8 experiment, lead by Novick and Weisskopf, both functions of spectroscopy and polarimetry were present but devoted to two dedicated instruments, both designed and built from the Team of Columbia University.","The spectrometer was based on a flat crystal of pyrolytic graphite with the unavoidably large detectors.","The polarimeter was based on graphite mosaic crystals, bent to concentrate photons on proportional counters.","The diffraction angles were confined in a range around $45^{0}$ from mechanical collimators.","The whole was in continuous rotation around the pointing axis through the satellite spinning.","The eventual yield of both spectrometers was very poor.","Only in one case a line was detected.","Upper limits on narrow lines were fixed of doubtful significance given that the observed sources were likely thick.","The real effect was to discourage the future instruments of high resolution spectroscopy.","The two polarimeter had different results.","Both OSO-8 and Ariel-5 used proportional counters with pulse shape and pulse height discrimination of background.","The areas of the two instruments were comparable but OSO-8 would be much more sensitive.","Thanks to the bent geometry the detectors of OSO-8 had a surface around 20 times smaller than the crystal, with an almost proportional reduction of the background.","In the Leicester experiment the main use for spectroscopy forced the flat geometry and as a consequence a much higher background.","Moreover the mosaic angular spread of the graphite was smaller than that of OSO-8, in order to preserve the spectral resolution and this means a lower efficiency for polarimetry.","Last but not least an important fraction of the surface was used for the LiF crystal, which had an efficiency much lower than that of graphite.","OSO-8 made a long pointing of the Crab.","Confirmed the results of the rockets measurement but with a very high significance (19$ \\sigma $ )[7].","The measured polarization was significantly above the so called Chandrasekhar limit (15.7$\\%$ vs 12$\\%$ ), namely the maximm polarization deriving from scattering on an asymmetric geometry.","This was the final evidence of a not thermal component of the X-ray emission, likely synchrotron.","A strong limit to ScoX1 polarization was also found[8].","But other results were of much lower impact.","These results were in general considered disappointing.","While the data on Crab rose a high interest, data on other sources were still consistent with theories but demonstrated that performing polarimetry was difficult, complex and technically cumbersome.","But the main trouble with polarimetry arrived from the great transformation that was occurring in X-Ray Astronomy in those years.","Soon after the first discovery of extrasolar sources, Giacconi outlined a long term planning that foresaw, after the first explorer mission (that would be UHURU), a first medium size mission based on an X-ray telescope with, in the focal plane a revolver capable to alternate different instruments in the focus.","This mission would become the Einstein Mission[9].","The High Resolution Imager produced impressive images of a limited sample of extended sources.","But in practice the Imaging Proportional Counter performed the large majority of the work for point-like sources.","The telescope collected the photons impinging on an area of 400 $cm^{2}$ and concentrated on a focal spot of the order of one $mm^{2}$ with the consequence that the background count was equivalent to that of a very weak source.","The increase in sensitivity for the detection of a source was impressive.","With an observation of few hours Einstein/IPC could detect a source at the limit of UHURU sensitivity.","Einstein disclosed the world of extragalactic sources that had been only touched with collimated missions.","The mismatching in sensitivity between imaging and polarimetry increased enormously.","Moreover the viable polarimeters implied a rotation of the instrument, easily achieved with spinning satellites.","On the contrary the new imaging/spectra instruments in the focus of the optics did not need any more rotation and a three axis pointing was much more useful for the measurement itself, for the use of power resources and for the pointing flexibility.","Therefore a polarimeter would be a rotating equipment harbored in the focal plane.","In other terms the addition of a polarimeter would represent a serious increase of complexity and a heavy share of observing time to make polarimetry of a small number of brighter sources belonging to classes that were no more the cutting edge of X-Ray astronomy.","Not surprisingly the originally foreseen polarimeters were excluded from both Einstein and Chandra.","Polarimetry was committed to hypothetic dedicated missions.","Some were proposed but none advanced in the way to approval."],["The Stellar X-Ray Polarimeter","The Stellar X-Ray Polarimeter (SXRP) was an important exception.","The Spectrum Roentgen Gamma (SRG) mission was planned by the Space Agency and Academy of Sciences of Soviet Union, under the scientific leadership of Rashid Sunyaev, one of the scientists who contributed to the highest level to predict the relevance of X-Ray Polarimetry.","SRG included many instruments in the band of soft and medium hard energy X-rays.","The largest instrument was the Soviet Danish Roentgen Telescope (SODART), a pair of large area, medium optical quality, optics, with a focal length of 10m and with an effective area of the order of 1200 cm$^{2}$ each.","Each telescope had a sliding platform in the focal plane capable to position different instruments.","SODART by itself and, a fortiori, the whole satellite were so complex and massive that could host a polarimeter of around 50 kg mass, included rotation, without a major perturbation.","SXRP was based on an optimal use of both Bragg and Thomson technique.","The difference in band allowed to stack the two stages.","Around 10 centimeters above the focal plane a flat graphite crystal at $45^{o}$ from the optical axis was diffracting 2.6 and 5.2 keV photons to a secondary focal plane.","Since the diffraction follow the same laws of optical reflection the whole was still preserving the imaging properties of the telescope in a kind of a Newtonian mounting[10].","The crystal was highly transparent to photons of higher energies that would impinge on a stick of lithium within a thin beryllium case.","The two analyzers, namely the diffractor and the scatterer, were hosted in the center of a well made with four detectors[11] .","The detectors were proportional counters filled with a mixture of Xe, A and $CO_{2}$ .","The window was 150$ \\mu $ m thick except for a circular sub-window in the positon of the secondary focus for the diffracted photons, only 50$ \\mu $ m thick to leave a reasonable transparency for 2.6 keV photons.","The detectors were position sensitive by exploiting the signal induced on a cathode plane subdivided according to a wedge and strip code.","The background was minimized by an anticoincidence plane and by pulse shape discrimination.","The whole instrument including the analyzers and the detectors was rotated around the optical axis.","The rotation would perform the measurement with the modulation of the diffracted image in the auxiliary detector and to compensate systematics in the scattering stage.","The Principal Investigator was Novic, replaced in a late stage in a late stage by Philip Kaaret.","An Italian team was part of the project and contributed the detectors.","This was likely the best way to use the two conventional techniques in combination with a grazing incidence telescope.","The telescope had a very large area and a band pass extended to 15 keV, namely with a reasonable overlap with the scattering cross section, so it was the best possible instrument with conventional (one-layer) optics.","The main difficulty was the large tolerances in the alignment of the instrument with the focus of the telescope.","This drove some design decisions with negative consequences.","A diameter of the scatterer larger than required from the convergence of the X-ray beam and from the tolerance on positioning increased the self-absorption in the scatterer with a loss of signal and, most relevant, an increase of the systematics in case of offset from the axis.","Also the distance of the detectors from the axis was larger than needed, to minimize the impact of possible offset of the lithium stick in the focus.","But in a scattering polarimeter the background is proportional to the surface of the detectors.","The predicted background rate would range from 200 to 400 mCrab[12].","This means that the benefit of having an instrument in the focus of a telescope would only apply for the observation of a limited subset of brighter sources.","The Bragg stage, thanks to the technique that preserves the imaging (although with a poor quality of the optics) would have a low background, virtually negligible.","But taking into account all the losses of signal the effective area at 2.6 keV was of the order of 10 $cm^{2}$ on a bandwidth of the order of 100 eV.","Polarimetry is a discipline in starvation of photons.","A reasonable MDP could be achieved only for sources of tens of mCrab.","In the baseline program for SRG the share of observing time with SXRP as a prime was of 5$ \\% $ .","So in practice a few brighter sources would be probed in the first years and the extragalactic sky was out of reach unless in the case of an extreme flare.","In any case SXRP would perform polarimetry of a certain number of bright galactic sources down to the level of a few $\\%$ , and this would be a serious step forward with respect to OSO-8.","The experiment was completed, integrated and fully calibrated.","It passed all the acceptance tests and was packed waiting to the delivery to be integrated in the SRG payload.","Unfortunately the making of SRG was slowed and eventually stopped by the general sinking of the Soviet System.","The fact that the end of the program was never declared had the effect that SXRP in a perpetual floating situation acted as a stopper for any other proposed X-ray polarimeter, as, for instance, for the one proposed for XMM.","One interesting heritage of SXRP was the calibration.","It was performed at Lawrence Radiation Laboratory in Berkley, by means of X-ray generators and crystal diffractors.","A particular attention was devoted to the calibration of the systematics foreseen for SXRP, such as the off-axis or a slight inclination.","A special attention was spent, with a large investment of time, to disentangle the intrinsic polarization of the calibrator from the measurement of the small effects of spurious modulation intrinsic to the instrument.","From the point of view of data analysis the technique of Fourier Analysis was widely used, with very promising results taking into account a starting point with a high level of systematics[13]."],["The quest for photoelectric polarimeter","The photoelectric effect had been studied in laboratory in the $^{\\prime }$ 20s essentially with two methods.","The first consisted in sending X-rays on a thin solid target and collect the emitted electrons with a detector on a goniometer geometry.","The method has a fundamental difficulty for astrophysical applications.","The modulation strongly depends on the penetration of the photon and this depends on the energy.","In a laboratory set up the measurement can be performed with photons of known energy.","In an astrophysical measurement the energy is unknown and the result is of difficult interpretation.","The other approach is to have a detector where the whole energy is lost in active regions.","In the laboratory this was performed with excellent imaging capability with cloud chambers.","The photographic read out was providing a very good image of the tracks, with all the details.","The quest for a photoelectric polarimeter for space is the search for a device with quality as similar as possible to a cloud chamber but self-triggered and suited to be hosted in a focal plane.","It is worth to mention the fact that for a short period the SXRP Team explored the possibility to introduce a new device based on the first of the two concepts described above.","Notwithstanding a relatively poor modulation factor of the order of 25$\\%$ , the full exploitation of the optics at low energies and the condition to be background-less down to very weak sources, promised an estimated sensitivity that would enable polarimetry of extragalactic sources.","Further measurements showed that the actual modulation was much lower[14] and this photoelectric polarimeter was no more adopted.","But this showed to the small community interested on this topic how relevant would be the implementation of a photoelectric imaging device for the focal plane.","SXRP was a big hope for X-ray polarimetry but, as the mission was continuously delayed and never stopped, it became a stopper for any new proposal.","Every proposal, such as that submitted for XMM, would face the statement that we should wait for the results of SXRP first.","In this phase of suspense, the idea that the next step would be a dedicated mission based on photoelectric effect started to spread.","In the very beginning of X-Ray Astronomy some laboratories tried to identify a method to perform polarimetry based on the photoelectric effect.","Given the limited imaging capability of typical wire chambers, these were based on the search of modulation of some macroscopic parameter deriving from the asymmetric angular distribution of the photoemission.","Riegler at GSFC[15] searched for a modulation with phase of coincident signals in nearby anodes for photons entering parallel to the wires.","Sanford at MSSL[16] searched for a difference of the rise-time of tracks parallel or perpendicular to the anode, in a proportional counter.","Both results were not conclusive and no further efforts were done after Novick results indicated the Bragg as the most effective technique.","With ASCA, Chandra and XMM-Newton the CCD took the place of Gas Proportional Counter as tow horse of X-Ray Astronomy.","Measurements showed a large majority of single hits but also a certain number of double hits, especially at higher energies.","The fluorescence yield for Si is only 3$\\%$ .","These hits were interpreted as a transfer of charge to the nearby pixel from photoelectrons created at a distance from the edge lower than the range.","Given the angular distribution of the photoelectrons the double hits along rows and those along columns have memory of the polarization of the X-rays.","This was verified by measurements and simulations by Osaka University[17], Max Planck[18] and Leicester[19] teams.","The method was obviously attractive because it does not require a dedicated instrument.","The measurement of polarization is the byproduct of the measurement of position and energy.","In truth this is only apparently true.","The range of the electrons is much shorter than the pixel size, so that the efficiency is very poor except at higher energies (>10 keV) where the efficiency of the optics is typically low and the chip is transparent to X-Rays.","Moreover the square pattern of the sensor is a problem itself.","A beam polarized at $45^{o}$ from the array and a beam unpolarized give exactly the same pattern and the system to work would need a rotation of the satellite or more detectors angularly misaligned.","But the most serious difficulty is that in these conditions the probability to have a double hit depends more on the distance of the absorption point from the edge then on polarization and with a point spread function that varies very sharply on the detector pattern the result would change completely for aspect change of seconds of arc.","For basic reasons the implementation of the photoelectric method was only possible with gas.","The team of Marshall Space Science Laboratory lead by Brian Ramsay made an important step forward with a gas detector with an absorption region with an electric field to drift the electrons of the track to a grid acting as a multiplication plane.","The gas mixture was based on Argon but included the TAE, a luminescent component.","During the multiplication optical photons were emitted so that the track became somehow luminous.","An optics would focus an image of the track on a CCD.","70 years after the images from cloud chambers these were the first images of photoelectron tracks.","They were acquired by optical methods, just like the cloud chamber experiments, with a CCD, that was taking the place of photographic films.","The method was very promising but the minimum energy was of 20 keV[20].","At the time the multi-layer optics were in a pioneering phase so that the path toward a realistic experiment was still to be defined.","But the technology of gas detectors was passing a season of great improvements.","The impetuous development of microelectronics also made feasible detection patterns with fine subdivision and all the simulations showed that a 2-dimension pixel imager would be the full exploitation of the photoelectric effect."],["The first gas pixel detectors","The team of IAPS (institutionally evolved from CNR to INAF), lead by the author and by Paolo Soffitta, another component of the SXRP collaboration, started a collaboration with the team of INFN-Pisa lead by Ronaldo Bellazzini, an outstanding personality in the field of gas detectors for particle physcs.","This INAF/INFN partnership could benefit of the convergence of two traditions and eventually succeeded to arrive to the first detector suited for the purpose.","A first testing on one dimension was performed with a microgap detector with a pitch of 200$ \\mu $ m filled with a Ne-DME mixture at 1 atmosphere pressure.","Data showed that the track length of electrons from photons of 5.4 polarized perpendicular to the strips were definitely longer than tracks from photons parallel[21].","To my knowledge this is the first unambiguous detection of an effect in the classical range 2-8 keV, viable for polarimetry.","The key was the use of a gas of low atomic number and the high spatial resolution, although in one dimension only.","Moreover the data were consistent with simulations software performed starting with a program for micro-analysis, given that at the epoch, general purpose software of the GEANT family were not yet adequate below 10 keV.","Encouraged from this result the INFN/INAF team made a first prototype that was the real turning point of photoelectric Polarimetry from a wishful thinking to a practical implementation.","The development of the GPD was articulated into various steps with some substantial commonality: A gas cell with a conductive window.","An electric field parallel to the optical axis to drift the electrons of the track to a multiplication stage.","A Gas Electron Multiplier (GEM) to perform a proportional multiplication of the electrons to amplify the track while preserving the shape.","In a first generation of devices the signal from the GEM gave the trigger for the data acquisition.","A sense plane of metal pads distributed on a honeycomb pattern to act as anodes to collect the charge multiplied in the gas above the pad itself.","A dedicated front end electronic chain independent for each pad.","An important driver of this design was the search for maximum axial symmetry, aimed to prevent any possible systematics and hopefully to avoid the need for rotation that till then was a mantra of X-ray polarimetry and one of the source of complications that made of polarimetry an odd companion in focal planes.","The first prototype based on trigger by GEM and collection plane built with multi-layer printed circuit technology.","The pad pitch was 150$ \\mu $ m. The GEM pitch 60$ \\mu $ m. Signals of each pad were routed horizontally to the input of an ASIC chips, relatively far from the gas cell.","The detector was filled with Ne and DME as a quencher.","The signal from the GEM was used to trigger the ASIC data acquisition.","Eventually the charge collected from each pixel was fetched to the output and A/D converted.","Photons of 5.4 and 5.9 keV generated ionization tracks in the gas.","From the analysis of the track images the direction of the emission was derived.","The histogram of angles was flat for unpolarized 5.9 keV photons and modulated for 5.4 keV photons, which had been polarized by scattering at 90$ ^{o} $ on a lithium target[22].","Although the statistics was not very high the papers presenting these results was the re-start of activities on X-ray polarimetry that can be classified into three groups.","1) An activity to improve and optimize the performances of prototypes and to build a real detector that could be the core of a space experiment 2) A rejuvenation of theoretical analysis to predict the potentiality of this new observable 3) The proposition of missions of polarimetry at every announcement of opportunity or in the context of large multi-instrument missions.","The results from the prototype demonstrated that the basic physics was correct but, for a realistic proposal, some problems had to be solved.","The major limits of the prototype were: 1) The multi-layer technique could allow for a limited number of pixels, of the order of one thousand.","2) The pads could not be smaller than 100$ \\mu $ m 3) The routing of the charge to chips far from the gas cell on an horizontal lay-out were the source of a noise much larger than that intrinsic to the anode read-out.","The total encumber of the detector and electronics, although already much better than that of the conventional experiments, was still relatively large compared with the small active sensing surface.","Notwithstanding these limitations at the Announcement of Opportunity from NASA for a Medium Size Explorer a Mission named INSPIRE was proposed including two telescopes one with a a Microcalorimeter.","The second step was the inclusion of the whole electronics in an ASIC chip.","The upper layer of the chip was the array of pads in hexagonal pattern.","The analog electronics was in the lower layers under the projection of the pad.","The signals were hold on the trigger from the GEM and routed to the output.","Two generations of these ASIC were made in sequence.","The first one had 2100 pixels, the second one 22000[23], in an hexagonal pattern.","The pixel size arrived to 80$ \\mu $ m.The most impressive evolution was to collapse the whole ingredients of the detecting system into a small volume around the active cell of gas.","This was basically different from other micro-pattern prototypes ad was named the Gas Pixel Detector (GPD).","The following evolution was a further decrease of the pixel size down to 50$ \\mu $ m and the increase of the number of pixels to 104000.","With the previous philosophy to trigger on the GEM and A/D convert the charge of each pixel dead time would have diverged.","The last step was the addition of self-triggering capability[24].","The ASIC was therefore playing different roles: the bottom of the detector, the sensing multi-pad plane, the front end electronics, the triggering electronics.","The chip was mounted on the bottom of a sealed ceramic case with a beryllium window and the whole weighted 80g.","In practice it was very close to what is needed for a space mission.","The chip could be used in principle with different gas mixtures, pressures and different thicknesses of the drift/absorption gap.","To operate in the 2-8 keV band Ne with Dymethylether (DME) quenching was used.","Following computation and tests a filling with DME and a 20$ \\% $ of He was found slightly more performing.","Eventually pure DME was used, although some problems could be expected with this chemically aggressive substance.","As a quencher and even more as the main detecting gas DME is outstanding from the point of view of diffusion.","Beside the choice of the gas mixture the pressure and the thickness of the absorbing cell was object of optimization.","A pressure around one atmosphere allowed for easier operation both in air and in vacuum and could be contained with a 50$ \\mu $ m thick Beryllium window.","The ingredients for the trade-off are: The thickness determines the efficiency.","The thickness determines the blurring of the track due to lateral diffusin in the drift.","The thickness determines a blurring of the image in the focal plane of the optics.","Point 1) is relevant because Polarimetry is a photons starving technique.","Point 2) impacts on the modulation factor, especially at lower energies.","Point 3) is relevant only for optics of good quality of the order of 10 arcseconds or better.","In fact the inclination angles are fixed by the band of the instrument and for medium quality telescopes (from 0.5 to 2 arcminutes) the defocusing effect is significantly minor in comparison with the telescope p.s.f..","The optimization of course depends on the band of optics and on the spectrum of a particular source.","The optimization for a typical optics of 4 m focal length and a spectrum E$ ^{-2} $ was a mixture of 80$\\%$ DME and 20$\\%$ He at one atmosphere.","Various sealed GPD with this filling and a window of 50 micron of Beryllium were built, studied, and tested to evaluate the performances included the resistance in a space environment.","This instrument was already mature for an actual mission and was proposed for various announcements to various Agencies (NASA, ESA, ASI, CAS)."],["The Time Projection Chamber ","At the NASA Announcement of Opportunity for a SMEX in 2003 a Team of Goddard Space Flight center lead by Jean Swank and based on the Laboratory for High Energy Astrophysics lead by Keith Jahoda and the Italian Teams that had developed the GPD lead by Ronaldo Bellazzini and Enrico Costa submitted a proposal for an Advanced X-Ray Polarimeter.","AXP was based on three conical optics like those for ASTRO-E2 three GPDs in the focus.","The proposed mission was not selected but NASA granted the LHEA with funds for the development of their own detectors for photoelectric polarimetry.","This was the Time Projecton Chamber (TPC)[25].","The process to optimize the performance of the GPD had shown the limitation in the axis-symmetric design.","The trade-off between efficiency and modulation factor even in mixtures with minimum diffusion was achieved at values around 10-15$\\%$ of efficiency at a peak of 2-3 keV.","On the other side in the GPD the ASIC must be orthogonal to the axis.","This forbids to recover the efficiency by stacking more GPDs.","The physics base of LHEA detector was the same of GPD and of any other modern photoelectric polarimeter, namely to image the ionization track produced by a photoelectron in a gas detector.","But in order to achieve a higher efficiency they decided to drop the axial symmetry.","In the TPC the electrons of the track are drifted on one side so that the drift path and the associated blurring is kept at an acceptable level.","The thickness of the absorption gap can be made very long in order to to increase the efficiency, potentially to values close to 1.","At the end of the drift a GEM, just like in the GPD, amplifies the track.","The multiplied electrons are collected by a set of vertical strips parallel to the optical axis.","The other dimension, namely that of the drift field, the track is measured by the Time Projection method.","Given that the the two coordinates of the images are derived with different methods and the drift method scales with the knowledge of the drift velocity an additional care is needed.","The track is imaged but, as usual in drift detectors, when a photon is absorbed no signal gives the start of the drift time.","The track is imaged, and the angular direction of the photoelectron can be measured, but the absorption point cannot be derived.","This impacts negatively in two ways: Image resolved polarimetry cannot be performed The reduction of background for the use of the optics is limited to one dimension only.",".","These limitations are intrinsic to the method.","Some other problems were associated to the specific implementation.","These include a large dimension that, combined with the need of rotation, made it more cumbersome and a relatively limited band width.","Conversely the TPC could be proposed from early times as a medium energy polarimeter.","In fact the lateral read-out was compatible with both a thick absorption gap or a thin absorption gap with a back window that would allow the higher energy photons to reach another polarimeter mounted in series (e.g.","a scatterig polarimeter).","To summarize the TPC is more efficient of the GPD but is not imaging.","The GPD should have a lower background.","Moreover the TPC needs rotation to compensate for systematics while the GPD, with hexagonal pads, by construction, at the first order could be free from systematic effects.","Actually in the implementation for IXPE sysiematics of the order of 1$\\%$ at lower energies were deteced and calibrated.","The constructive details and the performances of the GPD and the TPC are discussed in another part of this Handbook.","Here I give some hint on the following history."],["Toward a mission ","With the implementation of GPD and the alternative development of TPC X-Ray Polarimetry had reached the maturity to compete in the selections by the space agencies.","In theory, beside this baseline range of 2 - 8 keV, other ranges could be covered, such as the soft band or the hard X-Rays.","But in this classic band a large set of results could be expected for most classes of X-ray sources, on the basis of theoretical analysis and achievable sensitivity.","No comparable wealth of results was to be expected for other band, that would also require more challenging instrumental efforts.","This limited the path toward a mission to a comfrontation of GPD and TPC both filled with low Z gas.","This band was well matched with that of XEUS, the Large X-Ray mission studed for years by ESA.","A GPD polarimeter was added and was part of the baseline focal plane payload in all the various configurations of XEUS and IXO[26].","But the first implementation of this new technique could not be committed to such a remote horizon.","An Announcement Of Opportunity for Two Small Scientific Satellites was issued by the Italian Space Agency (ASI) on 2007.","A team lead by Enrico Costa, Ronaldo Bellazzini and Gianpiero Tagliaferri proposed POLARIX a mission with three telescopes, residual of another experiment foreseen for the SRG mission and three GPD[27].","The mission was selected for a phase A study and was ranked second in the following selection.","But eventually all the program of Small Scientific Missions was discontinued by ASI.","After the development of TPC the AXP collaboration splinted toward competing proposals.","At the further SMEX Announcement of Opportunity on 2007 the GSFC team, lead by Jean Swank proposed GEMS[28], based on the TPC, while the MSFC team lead by Martin Weisskopf with the Italian Team proposed the Imaging X-Ray Polarimetry Explorer[29] based on the GPD.","Both missions were based on three telescopes and a deployable boom.","NASA selected GEMS for an advanced study and eventually for flight.","But in 2012 GEMS (that meanwhile had been descooped to two telescopes) was discontinued by NASA for programmatic reasons.","For the second time a mission approved and eventually not launched acted as a stopper to other proposals.","After the suppression of GEMS instruments aimed to X-ray polarimetry were proposed again.","A descoped version of POLARIX was proposed at the first ESA Anouncement for a small mission[30].","The most advanced proposal was the X-ray Imaging Polarimetry Explorer proposed by an European team lead by Paolo Soffitta to the M4 AOO of ESA[30].","XIPE was selected for a phase A study together with two other missions.","XIPE was based on three telescopes with GPDs in the focus.","XIPE was similar to IXPE with a collecting area of the 3 mirrors around twice that of IXPE.","On the other side of the Ocean at the following AOO for a SMEX in 2014 many projects of X-Ray polarimetry were submitted and two of them were selected for an A phase study.","IXPE was a rejuvenated version of the homonym precursor with a major role of the Italian collaboration that would provide the whole instrument[31] in the focal plane of three telescopes built by the MSFC team.","Martin Weisskopf was the PI and Paolo Soffitta the Italian PI.","Praxys was based on the GEMS design, with two telescopes (following the GEMS design) and TPCs in the focus.","Keth Jahoda was the PI of this other proposal.","At the end of the phase A study in 2017 NASA selected IXPE for flight.","As a consequence XIPE was no more considered for the ESA M4 selection.","In parallel to the development of the hardware the analysis tools and the statistical context were better defined by Martin Weisskopf[32] and Stroheymer and Kalmann[33].","The use of Stokes Parameters for the analysis was better defined.","Interestingly it was proposed to apply the Stokes formalism also to single photons[34].","With management by NASA and ASI, IXPE passed all the program stages and was launched on December 9 2021[35]."],["Not only IXPE","IXPE, launched on 2021 is performing as hoped and planned.","During the years needed to have this mainstream experiment in orbit a certain level of activity was performed.","The activity of polarimetry in the hard X-Rays was performed with stratospheric balloons and was somehow decoupled from the activity at lower energies onboard satellites.","The two main experiments have been POGO and X-Calibur.","Both are scattering polarimeters but conceptually quite different.","POGO is a collimated detector.","The scattering element is plastic scintillator.","The background is reduced with an heavy passive shielding and with an active anti-coincidence including active collimator.","POGO evolved from a first design (POGOLite) to a larger version POGO+[36].","Scientific results include an observation of the Crab[37] and of CygX-1[38].","X-CALIBUR is based on an optics with 12m focal length and, in the focus, a scattering stick of Beryllium surrounded with an absorption well made of four strips of CZT detectors.","X-Calibur was launched from Antarctica.","It observed GX301-2 in a flare status and found an upper limit to polarization[39].","Following the results, the project evolved to an advanced version named XL-Calibur, with special attention is paid to improve the background shielding[40].","In general scattering polarimeters have larger background partially compensated with a higher modulation factor on a wider energy band.","This will be greatly improved when these instrumets will be hosted on satellites, allowing for a broader band and suitable observing time.","Also in the domain of Hard X-Rays and at the edge of the range of interest of this chapter a small satellite devoted to polarimetry is expected to be launched from ISRO in the next years.","POLIX is constituted of collimators, a Beryllium scatterer and proportional counters[41]l, heritage of the ASTROSAT design.","Three years before the launch of IXPE Polarlight, a cubesat with a GPD based on the same ASIC of IXPE and a collimator was launched on a sun synchronous orbit by Hua Feng of Tsing Hua University[42].","Polarlight by long integrations measured again after 40 years the polarization of the Crab[43], with a possible change of angle after a glitch, and of Sco X-1[44].","Interestlngly PolarLight measured a background rate at a level of one 20th of Crab.","This shows that a large number of galactic sources could be observed with collimated photoelectric detectors without the programmatic and financial loads of optics.","On the opposite approach the enhanced X-ray Timing and Polarimetry mission(eXTP), in an advanced state of approval, includes 4 telescopes devoted to polarimetry[45].","The general feature are similar to those of IXPE but the collecting area is 4 timeslarger and the observations will benefit of the simultaneous measurement with high throughput spectroscopy and timig instruments.","While the path toward new data was so slow and painful, some progress was done in the development of methods of analysis.","These include a better understanding of the parameters defining the sensitivity, more clear use of the Stokes Parameters, and the development of Bayesian algorithms.","POGO and PolarLight used for the first time the Bayesian analysis.","The methods of analysis are the subject of next chapters."]],"string":"[\n [\n \"General History of X-Ray Polarimetry in Astrophysics\"\n ],\n [\n \"Abstract Soon after the discovery of the first extrasolar X-Ray sources it was suggested that polarimetry could play a major role as a diagnostic tool.\",\n \"Attempts to measure polarization of X-Ray sources was performed by the team of Columbia University lead by Robert Novick.\",\n \"The technique of Bragg diffraction at 45{\\\\deg} was successful to detect the polarization of the Crab with rockets and with OSO-8 satellite.\",\n \"In the following evolution of X-Ray Astronomy, Polarimetry was too mismatched with the improved sensitivity of imaging and spectroscopy, based on the use of optics.\",\n \"As a consequence no polarimeter was flown any more.\",\n \"At the beginning of the century a new class of instruments based on the photoelectric effect were developed.\",\n \"In the focus of an X-Ray telescope they can perform angular, energy and time resolved polarimetry and benefit of the large increase of sensitivity due to the optics.\",\n \"The Imaging X-Ray Polarimetry Explorer, exploiting this technique, was launched at the end of 2021.\"\n ],\n [\n \"The very early stage\",\n \"Only 9 years after the discovery of X-Rays by Roentgen in 1904 Charles Glover Barkla, a student of Stokes, made experiments of scattering of these newly discovered particles and found that they follow the same rules of polarization of optical light and this demonstrated that these mysterious X-rays were electromagnetic radiation.\",\n \"Physics and Polarimetry of X-Rays are born almost together.\",\n \"When in 1962 Giacconi and Rossi found the first evidence for extrasolar X-ray sources theoreticians predicted that polarimetry would have in this wavelength domain a major role than in other domains.\",\n \"The Soviet Physicist Vitali Ginzburg was very active in this promotion also toward the western community.\",\n \"The easiest implementation of this concept would be an instrument based on the angular distribution of photons scattered from a target.\",\n \"The difficulty is that the scattering prevails over the absorption at energies of few keV, where the majority of photons is, only with Hydrogen and Helium, that have a negligible stopping power for X-Ray photons.\",\n \"The best dense scattering material is Lithium (that in any case must be encased with beryllium) for which the scattering prevails only above 10 keV.\",\n \"In 1969 Herbert Schnopper suggested the use of Bragg crystals at incidence angles of $45^{0}$ as a good analyzer of linear polarization[1].\",\n \"The technique would be very robust and simple.\",\n \"The limit is the narrow band of the diffracted radiation that would result in a low effective area.\",\n \"Moreover the detector would be of the same surface of the entrance window, and the poor resolution of the detectors would not allow to separate effectively by the pulse height the diffracted photons, so that a large background could be expected.\",\n \"The team of Columbia University lead by Robert Novick carried on a systematic study to arrive to perform an astronomical experiment.\",\n \"His co-workers Roger Angel and Martin Weisskopf found that the efficiency could be improved by using mosaic crystals of pyrolytic graphite (i.e.\",\n \"a crystal with the spread of orientation of micro-domains artificially enhanced).\",\n \"Moreover they found that a moderate bending of the crystal would allow a concentration of diffracted photons on a small detector, with a much lower background rate, while substantially preserving the modulated response to polarization[2],[3].\",\n \"The Team of Columbia launched the first rocket searching for polarization of ScoX-1 on 1968[4].\",\n \"The rocket included scattering targets of lithium encased in beryllium surrounded by proportional counters.\",\n \"The polarization angle was derived from the angle between the scatterer and the detector and the spinning phase of the rocket.\",\n \"The observation of ScoX1 (performed only 6 years after the discovery of the source!)\",\n \"was unsuccessful in terms of polarimetry.\",\n \"But the same rocket payload, was launched again to point the Crab Nebula.\",\n \"The result was inconclusive but still compatible with the high polarization found in optical and radio bands.\",\n \"When the technique of Bragg diffractors wih mosaic graphite was mature an improved version of the rocket was built.\",\n \"Out of the atmosphere four panels mounting graphite crystals were deployed on the side at an angle of $45^{0}$ to the pointing/spinning axis of the rocket so that the photons parallel to the axis were diffracted toward proportional counters hosted in the rocket below the scattering stage of the payload that was similar to that of the first rockets.\",\n \"Also this result was inconclusive.\",\n \"But by overimposing the data of all flights Novick found at last a statistically significant evidence of polarization[5].\",\n \"This was a paramount result in terms of physics of X-ray sources also confirming the diagnostic relevance of this subtopics, notwithstanding the experimental difficulties.\",\n \"Another achievement for the future was that, in the few keV range, the Bragg approach, notwithstanding the very low efficiency, is better than Thomson in terms of both sensitivity and reliability While performing this first set of experiments the Novick Team also fixed the statistic frame that would be used for all the planned and the (few) performed experiments.\",\n \"Both scattering and Bragg polarimeters would produce a histogram of phase of emerging photons.\",\n \"The histogram can be fitted with a constant term plus a cos$^{2}$ term.\",\n \"The ratio between the two terms carries the information about polarization degree while the phase of the second term carries the information n the polarization angle.\",\n \"$M=\\\\dfrac{S_{max}-S_{min}}{S_{max}+S_{min}}$ The M is named modulation.\",\n \"The modulation for a beam 100$ \\\\% $ polarized, conventionally named modulation factor ($\\\\mu $ ) is a feature of the detector and usually depends on the energy.\",\n \"If counts follow Poisson Statistics, as usually do in X-ray detectors, the modulation, which is a positively defined quantity, follows a distribution of $ \\\\chi ^{2} $ for two degrees of freedom.\",\n \"This is the basis to define the significance of a possible detection.\",\n \"The Minimum Detectable Modulation can be computed starting from the statistics as the modulation that in only 1$\\\\%$ of occurrences can be arrived or passed by statistical fluctuation.\",\n \"By dividing for the modulation factor we can derive the Minimum Detectable Polarization (MDP) that is the usual parameter to describe the sensitivity of an experiment of X-ray polarimetry when studying a source of a certain flux and spectrum.\",\n \"$MDP=\\\\dfrac{4.29}{\\\\mu \\\\varepsilon S}\\\\times \\\\sqrt{\\\\dfrac{\\\\varepsilon S+B}{T}}$ The fact that a 99$ \\\\% $ confidence was chosen, instead than the most usual 3$ \\\\sigma $ (or 5$ \\\\sigma $ ), witnesses the awareness that polarimetry would be a tough challenge.\",\n \"The Columbia team was also active in optical polarimetry of sources of potential interest for X-ray astronomy, so that it was familiar with the Stokes parameters formalism.\",\n \"Actually the results of the first rockets were presented and discussed in this formal frame, although both Bragg and Thomson stages can only detect linear polarization.\",\n \"The V parameter is unmeasurable so that the use of the Stokes is, in the frame of X-ray polarimetry only, of moderate usefulness.\",\n \"To compare results in a sky map a display of the polarization degree and angle could be the most informative.\",\n \"But in order to insert the X-ray result in a general frame of measurements in other wavelength or compare with theories the Stokes parameter representation would be the most suited.\",\n \"In the topical paper where polarization is detected the discussion is concentrated on the technique to combine data of different experiments and flights and the related systematics, so that, in this case, the Stokes Parameters were ignored.\"\n ],\n [\n \"Ariel-5 and OSO-8\",\n \"The first two satellites with an X-ray polarimeter aboard, Ariel-5 and OSO-8, and the only ones for the following 45 years, were launched on 1974 and 1975 respectively, only 5 and 6 years after the first X-Ray survey mission UHURU.\",\n \"The ARIEL-5 spectro-polarimeter designed and built by the Team of Leicester University lead by Richard Griffiths[6] was also based on Bragg Diffraction but was somehow different.\",\n \"The direction of the radiation impinging on the instrument and that of the diffracted photons impinging on the detectors were defined by mechanical collimators as well.\",\n \"But the diffracting crystals were of two kinds, one of mosaic graphite, like OSO-8, the second one of LiF.\",\n \"Crystals were flat and their inclination, with respect to the incoming radiation could be adjusted in order to tune the mounting angle and the related diffraction energy.\",\n \"The spinning satellite would be pointed not to the source but to a direction slightly off-set.\",\n \"By this trick the photons from the source would impinge on the crystal with an angle slightly different at each phase of spin.\",\n \"This would allow to perform high resolution spectroscopy of expected Si and S lines.\",\n \"The LiF was used to make spectra of Fe lines.\",\n \"At an angle of $45^{0}$ the diffractometer could be used as a polarimeter around an energy of 2.5 keV.\",\n \"In OSO-8 experiment, lead by Novick and Weisskopf, both functions of spectroscopy and polarimetry were present but devoted to two dedicated instruments, both designed and built from the Team of Columbia University.\",\n \"The spectrometer was based on a flat crystal of pyrolytic graphite with the unavoidably large detectors.\",\n \"The polarimeter was based on graphite mosaic crystals, bent to concentrate photons on proportional counters.\",\n \"The diffraction angles were confined in a range around $45^{0}$ from mechanical collimators.\",\n \"The whole was in continuous rotation around the pointing axis through the satellite spinning.\",\n \"The eventual yield of both spectrometers was very poor.\",\n \"Only in one case a line was detected.\",\n \"Upper limits on narrow lines were fixed of doubtful significance given that the observed sources were likely thick.\",\n \"The real effect was to discourage the future instruments of high resolution spectroscopy.\",\n \"The two polarimeter had different results.\",\n \"Both OSO-8 and Ariel-5 used proportional counters with pulse shape and pulse height discrimination of background.\",\n \"The areas of the two instruments were comparable but OSO-8 would be much more sensitive.\",\n \"Thanks to the bent geometry the detectors of OSO-8 had a surface around 20 times smaller than the crystal, with an almost proportional reduction of the background.\",\n \"In the Leicester experiment the main use for spectroscopy forced the flat geometry and as a consequence a much higher background.\",\n \"Moreover the mosaic angular spread of the graphite was smaller than that of OSO-8, in order to preserve the spectral resolution and this means a lower efficiency for polarimetry.\",\n \"Last but not least an important fraction of the surface was used for the LiF crystal, which had an efficiency much lower than that of graphite.\",\n \"OSO-8 made a long pointing of the Crab.\",\n \"Confirmed the results of the rockets measurement but with a very high significance (19$ \\\\sigma $ )[7].\",\n \"The measured polarization was significantly above the so called Chandrasekhar limit (15.7$\\\\%$ vs 12$\\\\%$ ), namely the maximm polarization deriving from scattering on an asymmetric geometry.\",\n \"This was the final evidence of a not thermal component of the X-ray emission, likely synchrotron.\",\n \"A strong limit to ScoX1 polarization was also found[8].\",\n \"But other results were of much lower impact.\",\n \"These results were in general considered disappointing.\",\n \"While the data on Crab rose a high interest, data on other sources were still consistent with theories but demonstrated that performing polarimetry was difficult, complex and technically cumbersome.\",\n \"But the main trouble with polarimetry arrived from the great transformation that was occurring in X-Ray Astronomy in those years.\",\n \"Soon after the first discovery of extrasolar sources, Giacconi outlined a long term planning that foresaw, after the first explorer mission (that would be UHURU), a first medium size mission based on an X-ray telescope with, in the focal plane a revolver capable to alternate different instruments in the focus.\",\n \"This mission would become the Einstein Mission[9].\",\n \"The High Resolution Imager produced impressive images of a limited sample of extended sources.\",\n \"But in practice the Imaging Proportional Counter performed the large majority of the work for point-like sources.\",\n \"The telescope collected the photons impinging on an area of 400 $cm^{2}$ and concentrated on a focal spot of the order of one $mm^{2}$ with the consequence that the background count was equivalent to that of a very weak source.\",\n \"The increase in sensitivity for the detection of a source was impressive.\",\n \"With an observation of few hours Einstein/IPC could detect a source at the limit of UHURU sensitivity.\",\n \"Einstein disclosed the world of extragalactic sources that had been only touched with collimated missions.\",\n \"The mismatching in sensitivity between imaging and polarimetry increased enormously.\",\n \"Moreover the viable polarimeters implied a rotation of the instrument, easily achieved with spinning satellites.\",\n \"On the contrary the new imaging/spectra instruments in the focus of the optics did not need any more rotation and a three axis pointing was much more useful for the measurement itself, for the use of power resources and for the pointing flexibility.\",\n \"Therefore a polarimeter would be a rotating equipment harbored in the focal plane.\",\n \"In other terms the addition of a polarimeter would represent a serious increase of complexity and a heavy share of observing time to make polarimetry of a small number of brighter sources belonging to classes that were no more the cutting edge of X-Ray astronomy.\",\n \"Not surprisingly the originally foreseen polarimeters were excluded from both Einstein and Chandra.\",\n \"Polarimetry was committed to hypothetic dedicated missions.\",\n \"Some were proposed but none advanced in the way to approval.\"\n ],\n [\n \"The Stellar X-Ray Polarimeter\",\n \"The Stellar X-Ray Polarimeter (SXRP) was an important exception.\",\n \"The Spectrum Roentgen Gamma (SRG) mission was planned by the Space Agency and Academy of Sciences of Soviet Union, under the scientific leadership of Rashid Sunyaev, one of the scientists who contributed to the highest level to predict the relevance of X-Ray Polarimetry.\",\n \"SRG included many instruments in the band of soft and medium hard energy X-rays.\",\n \"The largest instrument was the Soviet Danish Roentgen Telescope (SODART), a pair of large area, medium optical quality, optics, with a focal length of 10m and with an effective area of the order of 1200 cm$^{2}$ each.\",\n \"Each telescope had a sliding platform in the focal plane capable to position different instruments.\",\n \"SODART by itself and, a fortiori, the whole satellite were so complex and massive that could host a polarimeter of around 50 kg mass, included rotation, without a major perturbation.\",\n \"SXRP was based on an optimal use of both Bragg and Thomson technique.\",\n \"The difference in band allowed to stack the two stages.\",\n \"Around 10 centimeters above the focal plane a flat graphite crystal at $45^{o}$ from the optical axis was diffracting 2.6 and 5.2 keV photons to a secondary focal plane.\",\n \"Since the diffraction follow the same laws of optical reflection the whole was still preserving the imaging properties of the telescope in a kind of a Newtonian mounting[10].\",\n \"The crystal was highly transparent to photons of higher energies that would impinge on a stick of lithium within a thin beryllium case.\",\n \"The two analyzers, namely the diffractor and the scatterer, were hosted in the center of a well made with four detectors[11] .\",\n \"The detectors were proportional counters filled with a mixture of Xe, A and $CO_{2}$ .\",\n \"The window was 150$ \\\\mu $ m thick except for a circular sub-window in the positon of the secondary focus for the diffracted photons, only 50$ \\\\mu $ m thick to leave a reasonable transparency for 2.6 keV photons.\",\n \"The detectors were position sensitive by exploiting the signal induced on a cathode plane subdivided according to a wedge and strip code.\",\n \"The background was minimized by an anticoincidence plane and by pulse shape discrimination.\",\n \"The whole instrument including the analyzers and the detectors was rotated around the optical axis.\",\n \"The rotation would perform the measurement with the modulation of the diffracted image in the auxiliary detector and to compensate systematics in the scattering stage.\",\n \"The Principal Investigator was Novic, replaced in a late stage in a late stage by Philip Kaaret.\",\n \"An Italian team was part of the project and contributed the detectors.\",\n \"This was likely the best way to use the two conventional techniques in combination with a grazing incidence telescope.\",\n \"The telescope had a very large area and a band pass extended to 15 keV, namely with a reasonable overlap with the scattering cross section, so it was the best possible instrument with conventional (one-layer) optics.\",\n \"The main difficulty was the large tolerances in the alignment of the instrument with the focus of the telescope.\",\n \"This drove some design decisions with negative consequences.\",\n \"A diameter of the scatterer larger than required from the convergence of the X-ray beam and from the tolerance on positioning increased the self-absorption in the scatterer with a loss of signal and, most relevant, an increase of the systematics in case of offset from the axis.\",\n \"Also the distance of the detectors from the axis was larger than needed, to minimize the impact of possible offset of the lithium stick in the focus.\",\n \"But in a scattering polarimeter the background is proportional to the surface of the detectors.\",\n \"The predicted background rate would range from 200 to 400 mCrab[12].\",\n \"This means that the benefit of having an instrument in the focus of a telescope would only apply for the observation of a limited subset of brighter sources.\",\n \"The Bragg stage, thanks to the technique that preserves the imaging (although with a poor quality of the optics) would have a low background, virtually negligible.\",\n \"But taking into account all the losses of signal the effective area at 2.6 keV was of the order of 10 $cm^{2}$ on a bandwidth of the order of 100 eV.\",\n \"Polarimetry is a discipline in starvation of photons.\",\n \"A reasonable MDP could be achieved only for sources of tens of mCrab.\",\n \"In the baseline program for SRG the share of observing time with SXRP as a prime was of 5$ \\\\% $ .\",\n \"So in practice a few brighter sources would be probed in the first years and the extragalactic sky was out of reach unless in the case of an extreme flare.\",\n \"In any case SXRP would perform polarimetry of a certain number of bright galactic sources down to the level of a few $\\\\%$ , and this would be a serious step forward with respect to OSO-8.\",\n \"The experiment was completed, integrated and fully calibrated.\",\n \"It passed all the acceptance tests and was packed waiting to the delivery to be integrated in the SRG payload.\",\n \"Unfortunately the making of SRG was slowed and eventually stopped by the general sinking of the Soviet System.\",\n \"The fact that the end of the program was never declared had the effect that SXRP in a perpetual floating situation acted as a stopper for any other proposed X-ray polarimeter, as, for instance, for the one proposed for XMM.\",\n \"One interesting heritage of SXRP was the calibration.\",\n \"It was performed at Lawrence Radiation Laboratory in Berkley, by means of X-ray generators and crystal diffractors.\",\n \"A particular attention was devoted to the calibration of the systematics foreseen for SXRP, such as the off-axis or a slight inclination.\",\n \"A special attention was spent, with a large investment of time, to disentangle the intrinsic polarization of the calibrator from the measurement of the small effects of spurious modulation intrinsic to the instrument.\",\n \"From the point of view of data analysis the technique of Fourier Analysis was widely used, with very promising results taking into account a starting point with a high level of systematics[13].\"\n ],\n [\n \"The quest for photoelectric polarimeter\",\n \"The photoelectric effect had been studied in laboratory in the $^{\\\\prime }$ 20s essentially with two methods.\",\n \"The first consisted in sending X-rays on a thin solid target and collect the emitted electrons with a detector on a goniometer geometry.\",\n \"The method has a fundamental difficulty for astrophysical applications.\",\n \"The modulation strongly depends on the penetration of the photon and this depends on the energy.\",\n \"In a laboratory set up the measurement can be performed with photons of known energy.\",\n \"In an astrophysical measurement the energy is unknown and the result is of difficult interpretation.\",\n \"The other approach is to have a detector where the whole energy is lost in active regions.\",\n \"In the laboratory this was performed with excellent imaging capability with cloud chambers.\",\n \"The photographic read out was providing a very good image of the tracks, with all the details.\",\n \"The quest for a photoelectric polarimeter for space is the search for a device with quality as similar as possible to a cloud chamber but self-triggered and suited to be hosted in a focal plane.\",\n \"It is worth to mention the fact that for a short period the SXRP Team explored the possibility to introduce a new device based on the first of the two concepts described above.\",\n \"Notwithstanding a relatively poor modulation factor of the order of 25$\\\\%$ , the full exploitation of the optics at low energies and the condition to be background-less down to very weak sources, promised an estimated sensitivity that would enable polarimetry of extragalactic sources.\",\n \"Further measurements showed that the actual modulation was much lower[14] and this photoelectric polarimeter was no more adopted.\",\n \"But this showed to the small community interested on this topic how relevant would be the implementation of a photoelectric imaging device for the focal plane.\",\n \"SXRP was a big hope for X-ray polarimetry but, as the mission was continuously delayed and never stopped, it became a stopper for any new proposal.\",\n \"Every proposal, such as that submitted for XMM, would face the statement that we should wait for the results of SXRP first.\",\n \"In this phase of suspense, the idea that the next step would be a dedicated mission based on photoelectric effect started to spread.\",\n \"In the very beginning of X-Ray Astronomy some laboratories tried to identify a method to perform polarimetry based on the photoelectric effect.\",\n \"Given the limited imaging capability of typical wire chambers, these were based on the search of modulation of some macroscopic parameter deriving from the asymmetric angular distribution of the photoemission.\",\n \"Riegler at GSFC[15] searched for a modulation with phase of coincident signals in nearby anodes for photons entering parallel to the wires.\",\n \"Sanford at MSSL[16] searched for a difference of the rise-time of tracks parallel or perpendicular to the anode, in a proportional counter.\",\n \"Both results were not conclusive and no further efforts were done after Novick results indicated the Bragg as the most effective technique.\",\n \"With ASCA, Chandra and XMM-Newton the CCD took the place of Gas Proportional Counter as tow horse of X-Ray Astronomy.\",\n \"Measurements showed a large majority of single hits but also a certain number of double hits, especially at higher energies.\",\n \"The fluorescence yield for Si is only 3$\\\\%$ .\",\n \"These hits were interpreted as a transfer of charge to the nearby pixel from photoelectrons created at a distance from the edge lower than the range.\",\n \"Given the angular distribution of the photoelectrons the double hits along rows and those along columns have memory of the polarization of the X-rays.\",\n \"This was verified by measurements and simulations by Osaka University[17], Max Planck[18] and Leicester[19] teams.\",\n \"The method was obviously attractive because it does not require a dedicated instrument.\",\n \"The measurement of polarization is the byproduct of the measurement of position and energy.\",\n \"In truth this is only apparently true.\",\n \"The range of the electrons is much shorter than the pixel size, so that the efficiency is very poor except at higher energies (>10 keV) where the efficiency of the optics is typically low and the chip is transparent to X-Rays.\",\n \"Moreover the square pattern of the sensor is a problem itself.\",\n \"A beam polarized at $45^{o}$ from the array and a beam unpolarized give exactly the same pattern and the system to work would need a rotation of the satellite or more detectors angularly misaligned.\",\n \"But the most serious difficulty is that in these conditions the probability to have a double hit depends more on the distance of the absorption point from the edge then on polarization and with a point spread function that varies very sharply on the detector pattern the result would change completely for aspect change of seconds of arc.\",\n \"For basic reasons the implementation of the photoelectric method was only possible with gas.\",\n \"The team of Marshall Space Science Laboratory lead by Brian Ramsay made an important step forward with a gas detector with an absorption region with an electric field to drift the electrons of the track to a grid acting as a multiplication plane.\",\n \"The gas mixture was based on Argon but included the TAE, a luminescent component.\",\n \"During the multiplication optical photons were emitted so that the track became somehow luminous.\",\n \"An optics would focus an image of the track on a CCD.\",\n \"70 years after the images from cloud chambers these were the first images of photoelectron tracks.\",\n \"They were acquired by optical methods, just like the cloud chamber experiments, with a CCD, that was taking the place of photographic films.\",\n \"The method was very promising but the minimum energy was of 20 keV[20].\",\n \"At the time the multi-layer optics were in a pioneering phase so that the path toward a realistic experiment was still to be defined.\",\n \"But the technology of gas detectors was passing a season of great improvements.\",\n \"The impetuous development of microelectronics also made feasible detection patterns with fine subdivision and all the simulations showed that a 2-dimension pixel imager would be the full exploitation of the photoelectric effect.\"\n ],\n [\n \"The first gas pixel detectors\",\n \"The team of IAPS (institutionally evolved from CNR to INAF), lead by the author and by Paolo Soffitta, another component of the SXRP collaboration, started a collaboration with the team of INFN-Pisa lead by Ronaldo Bellazzini, an outstanding personality in the field of gas detectors for particle physcs.\",\n \"This INAF/INFN partnership could benefit of the convergence of two traditions and eventually succeeded to arrive to the first detector suited for the purpose.\",\n \"A first testing on one dimension was performed with a microgap detector with a pitch of 200$ \\\\mu $ m filled with a Ne-DME mixture at 1 atmosphere pressure.\",\n \"Data showed that the track length of electrons from photons of 5.4 polarized perpendicular to the strips were definitely longer than tracks from photons parallel[21].\",\n \"To my knowledge this is the first unambiguous detection of an effect in the classical range 2-8 keV, viable for polarimetry.\",\n \"The key was the use of a gas of low atomic number and the high spatial resolution, although in one dimension only.\",\n \"Moreover the data were consistent with simulations software performed starting with a program for micro-analysis, given that at the epoch, general purpose software of the GEANT family were not yet adequate below 10 keV.\",\n \"Encouraged from this result the INFN/INAF team made a first prototype that was the real turning point of photoelectric Polarimetry from a wishful thinking to a practical implementation.\",\n \"The development of the GPD was articulated into various steps with some substantial commonality: A gas cell with a conductive window.\",\n \"An electric field parallel to the optical axis to drift the electrons of the track to a multiplication stage.\",\n \"A Gas Electron Multiplier (GEM) to perform a proportional multiplication of the electrons to amplify the track while preserving the shape.\",\n \"In a first generation of devices the signal from the GEM gave the trigger for the data acquisition.\",\n \"A sense plane of metal pads distributed on a honeycomb pattern to act as anodes to collect the charge multiplied in the gas above the pad itself.\",\n \"A dedicated front end electronic chain independent for each pad.\",\n \"An important driver of this design was the search for maximum axial symmetry, aimed to prevent any possible systematics and hopefully to avoid the need for rotation that till then was a mantra of X-ray polarimetry and one of the source of complications that made of polarimetry an odd companion in focal planes.\",\n \"The first prototype based on trigger by GEM and collection plane built with multi-layer printed circuit technology.\",\n \"The pad pitch was 150$ \\\\mu $ m. The GEM pitch 60$ \\\\mu $ m. Signals of each pad were routed horizontally to the input of an ASIC chips, relatively far from the gas cell.\",\n \"The detector was filled with Ne and DME as a quencher.\",\n \"The signal from the GEM was used to trigger the ASIC data acquisition.\",\n \"Eventually the charge collected from each pixel was fetched to the output and A/D converted.\",\n \"Photons of 5.4 and 5.9 keV generated ionization tracks in the gas.\",\n \"From the analysis of the track images the direction of the emission was derived.\",\n \"The histogram of angles was flat for unpolarized 5.9 keV photons and modulated for 5.4 keV photons, which had been polarized by scattering at 90$ ^{o} $ on a lithium target[22].\",\n \"Although the statistics was not very high the papers presenting these results was the re-start of activities on X-ray polarimetry that can be classified into three groups.\",\n \"1) An activity to improve and optimize the performances of prototypes and to build a real detector that could be the core of a space experiment 2) A rejuvenation of theoretical analysis to predict the potentiality of this new observable 3) The proposition of missions of polarimetry at every announcement of opportunity or in the context of large multi-instrument missions.\",\n \"The results from the prototype demonstrated that the basic physics was correct but, for a realistic proposal, some problems had to be solved.\",\n \"The major limits of the prototype were: 1) The multi-layer technique could allow for a limited number of pixels, of the order of one thousand.\",\n \"2) The pads could not be smaller than 100$ \\\\mu $ m 3) The routing of the charge to chips far from the gas cell on an horizontal lay-out were the source of a noise much larger than that intrinsic to the anode read-out.\",\n \"The total encumber of the detector and electronics, although already much better than that of the conventional experiments, was still relatively large compared with the small active sensing surface.\",\n \"Notwithstanding these limitations at the Announcement of Opportunity from NASA for a Medium Size Explorer a Mission named INSPIRE was proposed including two telescopes one with a a Microcalorimeter.\",\n \"The second step was the inclusion of the whole electronics in an ASIC chip.\",\n \"The upper layer of the chip was the array of pads in hexagonal pattern.\",\n \"The analog electronics was in the lower layers under the projection of the pad.\",\n \"The signals were hold on the trigger from the GEM and routed to the output.\",\n \"Two generations of these ASIC were made in sequence.\",\n \"The first one had 2100 pixels, the second one 22000[23], in an hexagonal pattern.\",\n \"The pixel size arrived to 80$ \\\\mu $ m.The most impressive evolution was to collapse the whole ingredients of the detecting system into a small volume around the active cell of gas.\",\n \"This was basically different from other micro-pattern prototypes ad was named the Gas Pixel Detector (GPD).\",\n \"The following evolution was a further decrease of the pixel size down to 50$ \\\\mu $ m and the increase of the number of pixels to 104000.\",\n \"With the previous philosophy to trigger on the GEM and A/D convert the charge of each pixel dead time would have diverged.\",\n \"The last step was the addition of self-triggering capability[24].\",\n \"The ASIC was therefore playing different roles: the bottom of the detector, the sensing multi-pad plane, the front end electronics, the triggering electronics.\",\n \"The chip was mounted on the bottom of a sealed ceramic case with a beryllium window and the whole weighted 80g.\",\n \"In practice it was very close to what is needed for a space mission.\",\n \"The chip could be used in principle with different gas mixtures, pressures and different thicknesses of the drift/absorption gap.\",\n \"To operate in the 2-8 keV band Ne with Dymethylether (DME) quenching was used.\",\n \"Following computation and tests a filling with DME and a 20$ \\\\% $ of He was found slightly more performing.\",\n \"Eventually pure DME was used, although some problems could be expected with this chemically aggressive substance.\",\n \"As a quencher and even more as the main detecting gas DME is outstanding from the point of view of diffusion.\",\n \"Beside the choice of the gas mixture the pressure and the thickness of the absorbing cell was object of optimization.\",\n \"A pressure around one atmosphere allowed for easier operation both in air and in vacuum and could be contained with a 50$ \\\\mu $ m thick Beryllium window.\",\n \"The ingredients for the trade-off are: The thickness determines the efficiency.\",\n \"The thickness determines the blurring of the track due to lateral diffusin in the drift.\",\n \"The thickness determines a blurring of the image in the focal plane of the optics.\",\n \"Point 1) is relevant because Polarimetry is a photons starving technique.\",\n \"Point 2) impacts on the modulation factor, especially at lower energies.\",\n \"Point 3) is relevant only for optics of good quality of the order of 10 arcseconds or better.\",\n \"In fact the inclination angles are fixed by the band of the instrument and for medium quality telescopes (from 0.5 to 2 arcminutes) the defocusing effect is significantly minor in comparison with the telescope p.s.f..\",\n \"The optimization of course depends on the band of optics and on the spectrum of a particular source.\",\n \"The optimization for a typical optics of 4 m focal length and a spectrum E$ ^{-2} $ was a mixture of 80$\\\\%$ DME and 20$\\\\%$ He at one atmosphere.\",\n \"Various sealed GPD with this filling and a window of 50 micron of Beryllium were built, studied, and tested to evaluate the performances included the resistance in a space environment.\",\n \"This instrument was already mature for an actual mission and was proposed for various announcements to various Agencies (NASA, ESA, ASI, CAS).\"\n ],\n [\n \"The Time Projection Chamber \",\n \"At the NASA Announcement of Opportunity for a SMEX in 2003 a Team of Goddard Space Flight center lead by Jean Swank and based on the Laboratory for High Energy Astrophysics lead by Keith Jahoda and the Italian Teams that had developed the GPD lead by Ronaldo Bellazzini and Enrico Costa submitted a proposal for an Advanced X-Ray Polarimeter.\",\n \"AXP was based on three conical optics like those for ASTRO-E2 three GPDs in the focus.\",\n \"The proposed mission was not selected but NASA granted the LHEA with funds for the development of their own detectors for photoelectric polarimetry.\",\n \"This was the Time Projecton Chamber (TPC)[25].\",\n \"The process to optimize the performance of the GPD had shown the limitation in the axis-symmetric design.\",\n \"The trade-off between efficiency and modulation factor even in mixtures with minimum diffusion was achieved at values around 10-15$\\\\%$ of efficiency at a peak of 2-3 keV.\",\n \"On the other side in the GPD the ASIC must be orthogonal to the axis.\",\n \"This forbids to recover the efficiency by stacking more GPDs.\",\n \"The physics base of LHEA detector was the same of GPD and of any other modern photoelectric polarimeter, namely to image the ionization track produced by a photoelectron in a gas detector.\",\n \"But in order to achieve a higher efficiency they decided to drop the axial symmetry.\",\n \"In the TPC the electrons of the track are drifted on one side so that the drift path and the associated blurring is kept at an acceptable level.\",\n \"The thickness of the absorption gap can be made very long in order to to increase the efficiency, potentially to values close to 1.\",\n \"At the end of the drift a GEM, just like in the GPD, amplifies the track.\",\n \"The multiplied electrons are collected by a set of vertical strips parallel to the optical axis.\",\n \"The other dimension, namely that of the drift field, the track is measured by the Time Projection method.\",\n \"Given that the the two coordinates of the images are derived with different methods and the drift method scales with the knowledge of the drift velocity an additional care is needed.\",\n \"The track is imaged but, as usual in drift detectors, when a photon is absorbed no signal gives the start of the drift time.\",\n \"The track is imaged, and the angular direction of the photoelectron can be measured, but the absorption point cannot be derived.\",\n \"This impacts negatively in two ways: Image resolved polarimetry cannot be performed The reduction of background for the use of the optics is limited to one dimension only.\",\n \".\",\n \"These limitations are intrinsic to the method.\",\n \"Some other problems were associated to the specific implementation.\",\n \"These include a large dimension that, combined with the need of rotation, made it more cumbersome and a relatively limited band width.\",\n \"Conversely the TPC could be proposed from early times as a medium energy polarimeter.\",\n \"In fact the lateral read-out was compatible with both a thick absorption gap or a thin absorption gap with a back window that would allow the higher energy photons to reach another polarimeter mounted in series (e.g.\",\n \"a scatterig polarimeter).\",\n \"To summarize the TPC is more efficient of the GPD but is not imaging.\",\n \"The GPD should have a lower background.\",\n \"Moreover the TPC needs rotation to compensate for systematics while the GPD, with hexagonal pads, by construction, at the first order could be free from systematic effects.\",\n \"Actually in the implementation for IXPE sysiematics of the order of 1$\\\\%$ at lower energies were deteced and calibrated.\",\n \"The constructive details and the performances of the GPD and the TPC are discussed in another part of this Handbook.\",\n \"Here I give some hint on the following history.\"\n ],\n [\n \"Toward a mission \",\n \"With the implementation of GPD and the alternative development of TPC X-Ray Polarimetry had reached the maturity to compete in the selections by the space agencies.\",\n \"In theory, beside this baseline range of 2 - 8 keV, other ranges could be covered, such as the soft band or the hard X-Rays.\",\n \"But in this classic band a large set of results could be expected for most classes of X-ray sources, on the basis of theoretical analysis and achievable sensitivity.\",\n \"No comparable wealth of results was to be expected for other band, that would also require more challenging instrumental efforts.\",\n \"This limited the path toward a mission to a comfrontation of GPD and TPC both filled with low Z gas.\",\n \"This band was well matched with that of XEUS, the Large X-Ray mission studed for years by ESA.\",\n \"A GPD polarimeter was added and was part of the baseline focal plane payload in all the various configurations of XEUS and IXO[26].\",\n \"But the first implementation of this new technique could not be committed to such a remote horizon.\",\n \"An Announcement Of Opportunity for Two Small Scientific Satellites was issued by the Italian Space Agency (ASI) on 2007.\",\n \"A team lead by Enrico Costa, Ronaldo Bellazzini and Gianpiero Tagliaferri proposed POLARIX a mission with three telescopes, residual of another experiment foreseen for the SRG mission and three GPD[27].\",\n \"The mission was selected for a phase A study and was ranked second in the following selection.\",\n \"But eventually all the program of Small Scientific Missions was discontinued by ASI.\",\n \"After the development of TPC the AXP collaboration splinted toward competing proposals.\",\n \"At the further SMEX Announcement of Opportunity on 2007 the GSFC team, lead by Jean Swank proposed GEMS[28], based on the TPC, while the MSFC team lead by Martin Weisskopf with the Italian Team proposed the Imaging X-Ray Polarimetry Explorer[29] based on the GPD.\",\n \"Both missions were based on three telescopes and a deployable boom.\",\n \"NASA selected GEMS for an advanced study and eventually for flight.\",\n \"But in 2012 GEMS (that meanwhile had been descooped to two telescopes) was discontinued by NASA for programmatic reasons.\",\n \"For the second time a mission approved and eventually not launched acted as a stopper to other proposals.\",\n \"After the suppression of GEMS instruments aimed to X-ray polarimetry were proposed again.\",\n \"A descoped version of POLARIX was proposed at the first ESA Anouncement for a small mission[30].\",\n \"The most advanced proposal was the X-ray Imaging Polarimetry Explorer proposed by an European team lead by Paolo Soffitta to the M4 AOO of ESA[30].\",\n \"XIPE was selected for a phase A study together with two other missions.\",\n \"XIPE was based on three telescopes with GPDs in the focus.\",\n \"XIPE was similar to IXPE with a collecting area of the 3 mirrors around twice that of IXPE.\",\n \"On the other side of the Ocean at the following AOO for a SMEX in 2014 many projects of X-Ray polarimetry were submitted and two of them were selected for an A phase study.\",\n \"IXPE was a rejuvenated version of the homonym precursor with a major role of the Italian collaboration that would provide the whole instrument[31] in the focal plane of three telescopes built by the MSFC team.\",\n \"Martin Weisskopf was the PI and Paolo Soffitta the Italian PI.\",\n \"Praxys was based on the GEMS design, with two telescopes (following the GEMS design) and TPCs in the focus.\",\n \"Keth Jahoda was the PI of this other proposal.\",\n \"At the end of the phase A study in 2017 NASA selected IXPE for flight.\",\n \"As a consequence XIPE was no more considered for the ESA M4 selection.\",\n \"In parallel to the development of the hardware the analysis tools and the statistical context were better defined by Martin Weisskopf[32] and Stroheymer and Kalmann[33].\",\n \"The use of Stokes Parameters for the analysis was better defined.\",\n \"Interestingly it was proposed to apply the Stokes formalism also to single photons[34].\",\n \"With management by NASA and ASI, IXPE passed all the program stages and was launched on December 9 2021[35].\"\n ],\n [\n \"Not only IXPE\",\n \"IXPE, launched on 2021 is performing as hoped and planned.\",\n \"During the years needed to have this mainstream experiment in orbit a certain level of activity was performed.\",\n \"The activity of polarimetry in the hard X-Rays was performed with stratospheric balloons and was somehow decoupled from the activity at lower energies onboard satellites.\",\n \"The two main experiments have been POGO and X-Calibur.\",\n \"Both are scattering polarimeters but conceptually quite different.\",\n \"POGO is a collimated detector.\",\n \"The scattering element is plastic scintillator.\",\n \"The background is reduced with an heavy passive shielding and with an active anti-coincidence including active collimator.\",\n \"POGO evolved from a first design (POGOLite) to a larger version POGO+[36].\",\n \"Scientific results include an observation of the Crab[37] and of CygX-1[38].\",\n \"X-CALIBUR is based on an optics with 12m focal length and, in the focus, a scattering stick of Beryllium surrounded with an absorption well made of four strips of CZT detectors.\",\n \"X-Calibur was launched from Antarctica.\",\n \"It observed GX301-2 in a flare status and found an upper limit to polarization[39].\",\n \"Following the results, the project evolved to an advanced version named XL-Calibur, with special attention is paid to improve the background shielding[40].\",\n \"In general scattering polarimeters have larger background partially compensated with a higher modulation factor on a wider energy band.\",\n \"This will be greatly improved when these instrumets will be hosted on satellites, allowing for a broader band and suitable observing time.\",\n \"Also in the domain of Hard X-Rays and at the edge of the range of interest of this chapter a small satellite devoted to polarimetry is expected to be launched from ISRO in the next years.\",\n \"POLIX is constituted of collimators, a Beryllium scatterer and proportional counters[41]l, heritage of the ASTROSAT design.\",\n \"Three years before the launch of IXPE Polarlight, a cubesat with a GPD based on the same ASIC of IXPE and a collimator was launched on a sun synchronous orbit by Hua Feng of Tsing Hua University[42].\",\n \"Polarlight by long integrations measured again after 40 years the polarization of the Crab[43], with a possible change of angle after a glitch, and of Sco X-1[44].\",\n \"Interestlngly PolarLight measured a background rate at a level of one 20th of Crab.\",\n \"This shows that a large number of galactic sources could be observed with collimated photoelectric detectors without the programmatic and financial loads of optics.\",\n \"On the opposite approach the enhanced X-ray Timing and Polarimetry mission(eXTP), in an advanced state of approval, includes 4 telescopes devoted to polarimetry[45].\",\n \"The general feature are similar to those of IXPE but the collecting area is 4 timeslarger and the observations will benefit of the simultaneous measurement with high throughput spectroscopy and timig instruments.\",\n \"While the path toward new data was so slow and painful, some progress was done in the development of methods of analysis.\",\n \"These include a better understanding of the parameters defining the sensitivity, more clear use of the Stokes Parameters, and the development of Bayesian algorithms.\",\n \"POGO and PolarLight used for the first time the Bayesian analysis.\",\n \"The methods of analysis are the subject of next chapters.\"\n ]\n]"},"id":{"kind":"string","value":"2209.08181"}}},{"rowIdx":2391,"cells":{"text":{"kind":"list like","value":[["CompF2: Theoretical Calculations and Simulation Topical Group Report"],["Abstract This report summarizes the work of the Computational Frontier topical group on theoretical calculations and simulation for Snowmass 2021.","We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics).","The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements.","Calculations and simulations will need to keep up with these increased requirements.","The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs.","These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns."],["Executive Summary","We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics).","The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements.","Calculations and simulations will need to keep up with these increased requirements.","The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs.","These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns.","The adaptation of existing CPU-based software to specialized hardware can provide solutions to the experiment-driven challenges, but this requires additional expertise and substantial effort.","Artificial intelligence (AI) and machine learning (ML) can also provide solutions to some problems that are more immediately amenable to acceleration on coprocessors.","US HEP should, to the extent possible, guide continued development of computing hardware to be maximally useful for solving HEP computing challenges.","Important developments include: faster hardware, availability of general-purpose CPUs, universal programming interfaces and portability, automated memory hierarchy, and early access to new hardware.","Long-term support of careers for software developers and maintenance for common software tools is essential to the future of HEP calculations and simulations.","Efficient utilization of the aforementioned new compute accelerators will require substantial efforts to port existing software to these new paradigms and/or to adopt new AI/ML techniques.","Needs in this area include new organizations to ensure software sustainability, permanent software development staff at national labs, joint lab-university appointments for computationally-focused scientists, and increased training across the field.","Computing in HEP has long struggled with issues of diversity and inclusion.","As the intersection of two highly specialized topics, there is an especially high level of knowledge and skills required to contribute successfully.","To get to this level, individuals usually must seek expertise outside of traditional HEP education and training, which typically does not cover computing topics in detail.","This poses a discouraging burden, especially for those without connections to such expertise or without the resources to take on such additional work.","We hope that substantially expanded support for software and computing training and for software development as a viable career in the HEP ecosystem will alleviate this burden.","A larger, healthier HEP computing community can be more welcoming and accessible to participants from all backgrounds."],["Introduction","The Snowmass Theoretical Calculation and Simulation topical working group is cross-cutting and underpins multiple scientific domains.","The computing topics discussed overlap with almost all aspects of Snowmass science and almost every Frontier.","Increasingly, computation is central to most theoretical and experimental scientific endeavors.","The great challenges of science, and in particular physical science supported by the Department of Energy and the National Science Foundation, have a long history of pushing the boundaries of computational science and computing technology.","The current Snowmass study indicates this will continue on many fronts.","The Theoretical Calculation and Simulation group spans a number of sub-topical themes representing the American Physical Society (APS) Division of Particles and Fields (DPF) Snowmass scientific interests: Cosmic Calculations, Particle Accelerator Modeling, Detector Simulation, Physics (Event) Generators, Perturbative Calculations, and Lattice QCD (quantum chromodynamics).","This report first introduces the common challenges and the modern computing landscape in Section , discusses each subtopic in more detail in Sections –, and concludes with recommendations in Section ."],["Cross-cutting computational challenges","The computational landscape of scientific algorithms can broadly be divided into high-performance computing (HPC) and high-throughput computing (HTC) problems.","HPC requires the coordinated and cooperative processing of a single large computational problem across multiple nodes with data being exchanged as part of the calculation, often using high-performance interconnects.","In contrast, HTC processes a large volume of completely independent computational work units.","Depending on the problem type, both HPC and HTC may have a high volume of relatively simple bulk floating point arithmetic that can be accelerated using highly parallel special-purpose computing hardware.","Some calculation and simulation tasks, and particularly those that process large data arrays or are machine learning-based, may obtain substantial speed increases from these hardware accelerations.","In other cases, algorithms that involve substantial serial performance or many data dependent logical flow decisions will benefit most from hardware that is suited to efficient (and power efficient) serial execution.","Detector simulation, event generation, and continuum theoretical calculations are typically HTC tasks, while accelerator modeling, cosmic calculations, and lattice gauge theory are more often HPC tasks.","However, exceptions exist, and the use of different algorithms or programming paradigms can sometimes convert HTC to HPC.","Memory bandwidth has presented a fundamental rate limit to many calculations.","It is common to include computational accelerators that can provide substantially greater floating point throughput and memory bandwidth as coprocessor cards.","The most common of these are graphics processor units (GPUs), but Field Programmable Gate Arrays (FPGAs) and other technology options are also common.","These often provide distinctly addressable high performance memory to help deliver good performance.","Coprocessors are often highly parallel and have limited or fixed function operations, so they are only usable with certain classes of algorithms.","Machine learning (ML) algorithms are especially suited to be accelerated on these devices, as described in Section REF .","On the other hand, while event reconstruction is easy to parallelize by processing independent events independently on independent serial processor cores of any form, the standard GPU thread model serializes execution when different threads in a group make different data-dependent logical flow decisions (for example, making different event-specific patterns of subroutine calls).","From an electronics perspective, it is easier to provide multiple types of memory (small and fast, large and slow) organized in distinct circuit boards to support a high floating point density.","The memory technologies are currently split into on-chip caches, in-package DRAM or “high bandwidth memory”, off-package DRAM, and increasingly non-volatile memory.","These are listed in order of increasing capacity and decreasing memory bandwidth.","The appearance of 2.5D and 3D integration technologies has been transformational and this system-level innovation will likely continue to return greater memory bandwidth over time, giving continued practical gains in high performance computation.","Such hierarchical memory systems will likely remain in the future, as they are dictated by physics constraints.","The degree of hierarchy is increasing and may grow in the future, independent of the specific processor architecture.","In this sense, the trends identified are likely reasonable over a ten-year timeframe.","Scientific productivity will be enhanced if data placement is handled automatically and moved infrequently: for example, this should be automatically and efficiently handled by the Unix virtual memory system.","This would avoid a requirement that every single scientific programmer laboriously manage the placement and movement of data to specific locations, such as caches.","The managing of distinct pointers to distinct “host” and “device” memory spaces is both a recurrent source of programmer error and requires considerable care to ensure that the allocated GPU memory does not exceed the limited capacity, while automatic paging avoids this requirement.","Section REF discusses more considerations relevant to scientific programming with modern heterogeneous computing.","Interconnects, used for both message passing and filesystem access, are becoming an increasing bottleneck since accelerated computing nodes are becoming rapidly more powerful while interconnect performance gains have not always matched pace.","Technologies such as silicon photonics, allowing the integration of optical transceivers in standard silicon chips, are a promising basic technology that could help address interconnect bottlenecks.","Similarly, high speed links between GPUs within a computing node are a key enabling technology for many problems.","File systems are a key element of any scientific workflow.","Data access should be sufficiently fast not to present a bottleneck.","Modern computing systems use parallel file systems, such as Lustre, giving up to terabyte/s aggregate access rates to disk.","The use of non-volatile memory elements can both cache bursts of data and accelerate metadata access.","Data migration to cold or longer term storage can be carried out through either user input or hierarchical storage managers.","It is important that these keep pace with growing data volumes and increasing computation speed."],["Computing hardware acceleration","Fixed-function acceleration is possible for problems involving dense matrix multiplication, with tensor cores in GPUs and other hardware specifically designed to handle large matrices.","These operations are commonly performed using 64-bit and 32-bit floating point arithmetic, or even on 16-bit floating point arithmetic.","Mixed precision functional units are common.","Fixed-function dense matrix hardware such as tensor cores do not necessarily accelerate all algorithms.","ML is one common and growing source of matrix multiplication operations that can be accelerated in this way.","ML can broadly be decomposed into “training” and “inference” phases.","Training often involves scientist supervision for parameter tuning, with time to results being critical.","Distributed ML training parallelizes the training process.","The use of small “mini-batches” in stochastic gradient descent leaves the problem very sensitive to Ahmdahl's law slowdown.","In the massively distributed limit [1], the summation across an interconnect of a gradient vector calculated across many randomly selected training samples in a minibatch is a significant performance bottleneck [2].","Inference with a previously trained network is typically performed on many independent events of data and is in many cases a trivially parallel (HTC) problem.","However, when inference is used in either hard or soft real-time situations (as occurs in HEP experiments), the latency requirements impose challenging performance issues when traditional CPUs are used.","Accelerators from Graphcore and Cerebras are good examples of ML-optimized accelerators in addition to more standard GPUs by Nvidia, AMD, and Intel.","Reconfigurable computing hardware such as FPGAs or spatial accelerators are powerful options for certain algorithms.","They may be accessed via a dataflow model where a hardware circuit is configured in a non-von Neumann approach to stream data from memory, calculate, and store results in memory.","They may be particularly effective in eliminating software latency in real-time inference environments [3], [4].","Algorithms that are I/O-bound or executed in situations requiring reduced power consumption may also be more efficiently accelerated on FPGAs."],["Software challenges","The computing landscape at this time displays an exciting proliferation of competing computer architectures.","It opens vibrant and healthy opportunities for innovation and competition and for the introduction of new ideas.","However, it also risks introducing special-purpose or less-general features, compared to previous systems.","The memory spaces are often fragmented with different pointer types and data locations.","There are multiple instruction-set architectures in use, each with a different programming interface or model.","A single program frequently has different segments that target different types of instruction sets within a single computer node.","Indeed, some of the computing models available (such as FPGAs or spatial architectures) are not von Neumann computers.","Others have dedicated fixed-function acceleration of matrix-matrix multiplication for machine learning.","This diversity presents a significant challenge to the scientific programmer with either legacy code or the need to use more than one system.","There are a number of different programming models one must consider in the present computing landscape.","The following can all in principle be combined with MPI message passing.","Present Exascale and Pre-exascale systems support: traditional CPU core, SIMD floating point instructions, and OpenMP threading, CUDA-based GPU programming, HIP-based GPU programming, SYCL-based GPU and FPGA programming, and OpenMP 5.0 offload GPU programming.","Future systems may support standards-based C++ parallel STL offload programming; however, multi-system compiler support is currently absent.","The challenges of writing high-performance and portable code are threefold: syntactical differences, semantic differences, and data placement.","The proliferation of models represents a substantial cost and effort overhead for science exploitation, since multiple implementations of software must be developed and maintained.","At this time, due to the diversity of models, abstraction and support for multiple programming models is likely the safest option for portable efficiency on near-term computers.","For projects that wish to reuse interfaces to acceleration, several good portability options exist: Alpaka and Department of Energy ASCR projects Kokkos and RAJA.","Standards-based programming, such as SyCL, OpenMP, or C++ parallel STL, is to be encouraged.","It is important to recognize that not all algorithms are amenable to map to accelerated hardware.","This must be considered in the HEP computing landscape to avoid leaving important needs unaddressed.","There is much to be said for an element of the computing roadmap using general-purpose processor cores combined with advanced memory technologies such as high bandwidth or hierarchical memory.","In this way, the long tail of problems with computational challenges that would be either impossible or prohibitively expensive to port to accelerated hardware may continue to be addressed."],["Cosmic Calculations","In the next decade, a number of powerful observational facilities will start to operate.","These next-generation cosmological observations will provide great opportunities to study the early history of universe, the mysteries of dark matter and dark energy, the fundamental particle physics, and possible modifications of Einstein gravity.","In order to successfully achieve the observational goals, a state-of-the-art simulation and modeling program is needed [5].","Cosmological simulations provide an invaluable tool to optimize the observation design, to interpret observation data, and to help unearth the underlying physics.","Next-generation cosmological simulations will increasingly focus on physically rich high-fidelity simulations that can directly connect with observational outputs from multiple different surveys.","Several types of cosmological simulations have been employed in the cosmology study.","The first-principles N-body simulations that have no free parameters can accurately describe dark matter fluctuations from the largest observable scales down to scales deep into the nonlinear regime.","However, the N-body approach does not account for the physics of the baryonic sector.","Additional models such as the halo occupation distribution, sub-halo abundance matching, or other schemes are added on top of a simulation to reconstruct galaxies.","Hydrodynamical simulations provide a reasonably accurate description of the distribution of baryons, quantify the effects of baryons on various probes of large-scale structure, and provide useful results for the distribution and properties of galaxies and clusters.","However, the robustness of the hydrodynamical simulation results depends on the choices of subgrid models and the nature of the cosmic probe in the study.","In addition to the aforementioned simulations, other simulations such as beyond $\\Lambda $ CDM simulations that involve the solution of nonlinear Poisson equation in the modified gravity model and radiative transfer simulations that model reionization process have also been used in cosmological studies.","Different survey experiments include different observables in the study.","Signals from galaxy clustering and lensing are key observables for photometric galaxy surveys.","Spectroscopic instruments are used to measure redshifts, radial velocities, gas dynamics and chemical compositions of galaxies.","Signals from Baryonic Acoustic Oscillation (BAO) and redshift space distortions (RSD) measurements are used to extract useful cosmological information.","Signals from the cosmic microwave background lensing measure the integrated mass between the last scattering surface and us.","The Lyman-alpha forest signal in the spectra of distant quasars is used to constrain thermal properties of the intergalactic medium (IGM).","The thermal and kinematic Sunyaev Zeldovich (tSZ/kSZ) signals are used to infer the distribution of gas in the universe.","The emission from dusty star has been used to delens the cosmic microwave background (CMB).","X-ray observations are used to calibrate mass estimates.","High-redshift from interferometers can provide useful information about the thermal and ionization state of the intergalactic medium and can be used to learn about dark matter.","The diverse observational probes present significant challenges in cosmological simulations.","One challenge is the computational cost to include multiple probes with high precision.","These simulations need cover a large volume but with sufficient fine resolution for a large number of realizations.","In addition, computationally demanding hydrodynamical simulations are required to simulate some observables such as the thermal and kinetic Sunyaev-Zel'dovich effects, tSZ and kSZ, to infer the distribution of gas in the universe, or Lyman-$\\alpha $ , to constrain thermal properties of the intergalactic medium.","In some cases, the modeling of certain observables involves a very large dynamical range.","Semi-numerical simulations are used when the hydrodynamical simulations cannot cover large enough volume while reaching small enough halos.","Calibrations and benchmarks between different approaches are important to interpret new observation data.","Another challenge for cosmological simulations is to reproduce observations, for example, correlations between multi-wavelength observables, in consistent galaxy formation models.","It is necessary to develop simulation models that can be applied to gravity-only simulations and account for correlations between neutral and ionized gas, stars, and dust in galaxies and galaxy clusters.","Also, it is not trivial to simulate massive neutrinos since they make up a non-negligible fraction of the total energy but can be decoupled when relativistic and have a free streaming scale.","In the N-body simulations, thermal velocity distributions need to be accounted for in the structure formation calculation on smaller scales.","Covering both the large volume required by the weak lensing and maintaining the accuracy at small scales required by strong lensing presents another computational challenge in ray tracing.","In the modeling for future analyses of small scale measurements, baryonic feedback effects that can change the local matter density must be included in the cosmological simulations.","Besides advancing in terms of increased resolution, larger volumes and better treatments of known physics, additional new physics models need to be included as the reach of surveys expands.","These include models of modified gravity with nonlinear Poisson equation for dark energy studies and a variety of dark matter models (warm dark matter, interacting dark matter, self-interacting dark matter, ultralight dark matter, ultraheavy dark matter, multiple component dark matter [6]).","Dedicated cosmological simulations are needed for those new models.","Next-generation cosmological surveys demand better understanding, mitigation, and control of systematic uncertainties.","To attain these goals, it is important to create realistic “virtual universes” from cosmological simulations.","Such simulations can be used to solve the inverse problem to infer the physical parameters of different models from the sky survey observation data.","However, it is not computationally practical to model all relevant processes from a first-principles approach given the vast dynamic range in cosmology.","Fast surrogate models can be used as effective emulators to speed up the solution of this inverse problem.","The predictions for cosmological surveys from emulators have been widely used in cosmological studies.","It is desirable that these emulators can connect directly to the survey observables to extract cosmological parameters.","With the advance of multi-fidelity hydrodynamical simulations, it is also desirable to use emulators to optimize subgrid model parameters and to gain better understanding of the interplay between the subgrid model and cosmology parameters.","Cosmological simulations are computationally intensive and demand the use of state-of-the-art computer hardware.","With the arrival of exascale supercomputers, cosmological simulations will need to use computer accelerators efficiently and memory access patterns effectively, as well as performance portable programming models and scalable algorithms.","Progress has been made to address some of these challenges, for example under the Exascale Computing Project (ECP) led by the Office of Advanced Scientific Computing Research (ASCR) in collaboration with other DOE science program offices [7], [8].","Continuous development and support are extremely important to enable full use of these hardware systems.","The successful use of the exascale supercomputers will enable gravity-only simulations with unprecedented volume coverage and resolution and hydrodynamics simulations with exceptionally detailed baryonic physics modeling for the cosmological surveys.","Scalable analysis approaches are as important as the development of simulation codes since handling and processing of very large data sets generated by the simulation codes require large computing resources in their own right.","Besides the development of the simulation codes, the development of analysis tools faces the same challenges as the simulation codes with respect to the efficient usage of the available computer hardware.","Co-development of simulation codes and analysis tools is needed for a successful cosmological simulation program.","Cosmological simulations are critical for the success of next-generation cosmological observations.","The accuracy for these simulations needs to be carefully checked through verification by benchmark among codes and validation with direct comparison against observational data.","This requires a concerted effort between different code and analysis development teams and cosmic observation teams.","The detailed connection between the simulations and the survey observations has to be established for a successful validation program.","In summary, the cosmological simulation community recommended the following advances in order to meet the above challenges and needs [5].","Development of scientifically rich and broadly-scoped simulations, which capture the relevant physics and correlations between probes Accurate translation of simulation results into realistic image or spectral data to be directly compared with observations Improved emulators and/or data-driven methods serving as surrogates for expensive simulations, constructed from a finite set of full-physics simulations Detailed and transparent verification and validation programs for both simulations and analysis tools Another area of cosmic calculations is the study of numerical relativity for next-generation gravitational-wave probes of fundamental physics [9].","The next-generation gravitational-wave detectors will provide great opportunities to study the nuclear physics of dense matter from the signals of neutron-star or black hole mergers, the nonlinear dynamics of warped spacetime, and the physics beyond general relativity from the signals of binary black holes.","To attain these goals requires a new generation of numerical-relativity codes with dramatically improved accuracy that can make effective use of the exascale supercomputers.","The higher computational accuracy also results in a significant increase of needs for computational resources in comparison to the present typical weeks to months of runtime on tens to thousands of computer cores.","This is due to the higher simulation resolution and the longer simulation to cover the detectors' sensitive frequency spaces.","In addition, many simulations will be necessary to span the parameter space of potential signals from black holes and neutron stars.","A further topic in cosmic calculations is the modeling of neutrino transport in neutrino-driven core-collapse supernovae [10].","A kinetic description of these neutrinos is needed in this model.","The solution for the neutrino transport equation is computationally intensive, involving high-dimensional linear algebra.","These simulations have been ported to GPU-based supercomputers and will require more computing resources to improve the resolution of the calculations and to explore different physical scenarios."],["Particle Accelerator Modeling","Particle accelerators are regarded as one of the most important inventions of the $20^{\\mathrm {th}}$ century.","They provide an invaluable tool in high energy physics study.","However, particle accelerators are large and complicated scientific devices that can be thousands of meters long (e.g., the LHC's nearly 27 kilometer circumference) with hundreds of thousands of elements.","Their design, construction, and operation are both sophisticated and expensive, and depend critically on computer modeling In the particle accelerator modeling area, there are 26 related letters of intent submitted to this Snowmass process.","These letters of intent were summarized in a community-wide white paper that addresses the challenges and needs in modeling particle accelerators [11].","Next-generation high energy particle accelerators and colliders will be more complicated and expensive than the existing accelerators.","These accelerators may employ both conventional RF cavity and high gradient wakefields from either a plasma channel or a structure waveguide to accelerate charged particle beams to high energy and use strong superconducting magnets and sophisticated control systems to confine these beams.","Some potential next-generation particle accelerators include very high energy proton-proton colliders, multi-TeV electron-positron colliders, muon colliders, other very high energy machines such as gamma-gamma and high energy electron-ion colliders, and high intensity accelerators for neutrino physics study.","To meet the needs of these next-generation accelerators, it is important for particle accelerator modeling tools to include all types of accelerating structures (e.g., RF-based, plasma-based, structured-based wakefield, plasmonic) and different accelerator components (e.g.","superconducting magnets, structured plasmas), and to target the accelerator and beam physics thrust grand challenges on intensity, quality, control, safety, and prediction.","For RF-based accelerators, this requires modeling of collective effects with high fidelity over long time scales, improving computational speed to allow for statistical ensembles and design optimization including collective effects, and improved integration with realistic magnet and RF modeling.","For plasma-based accelerators, this requires supporting the development of modeling tools and methods that will enable start-to-end simulations that predict the full six-dimension (+ spin) evolution of beams in a plasma-based linear collider, from their creation to their final focusing at the interaction point, and include all the intermediate phases of acceleration, transport, manipulations, and collisions.","For structure-based accelerators, this requires the development of efficient and accurate algorithms capable of modeling the beam interaction with its wakefield over long interaction lengths in structures with arbitrary geometries and constitutive parameters.","For plasmonic accelerators, this requires the development of new approaches such as a quantum-kinetic method that incorporates the effects of localized petavolts per meter plasmonic fields on the underlying quantum mechanical processes such as the ionic lattice and the energy band structure and to incorporate the multi-physics nature of the unprecedented strongly electrostatic plasmonic modes.","For the structured plasma accelerator component, this requires the development and integration of fluid and kinetic codes to meet the modeling demands for a new class of structured plasma devices that couple macroscopic plasma properties with strict requirements on kinetic interactions.","The quest for higher accelerating gradients in next-generation particle accelerators puts increasing demands on better understanding the materials of the accelerating structure and relevant phenomena such as RF breakdown.","This needs the development of automated scale-bridging methods that can autonomously parameterize higher-level models from large numbers of lower-scale first-principles calculations to enable predictive materials studies over a broad range of materials and conditions.","Strong magnets will be highly demanded in the next-generation particle accelerators to reduce the machine size and cost for transverse focusing.","Some advanced modeling tools are currently utilized in the US Magnet Development Program.","Novel modeling tools with mixed finite element method are needed to improve design time, cost, and performance of future superconducting accelerator magnets.","Next-generation particle accelerators will occupy a large land space and consist of many different accelerator segments starting from charged particle sources and ending with final interaction colliders or targets.","Different segments of the accelerator are coupled with each other through the charged particle beam transporting through the entire accelerator system.","The final particle beam quality does not depends on the performance of a few elements, but on the performance of the entire accelerator system.","Without the modeling of the whole accelerator, no project can proceed to the building stage.","In order to evaluate the particle accelerator performance and to attain a global optimal design, end-to-end simulations are needed to establish a virtual particle accelerator.","The end-to-end simulation denotes a simulation starting with the source that generates charged particles and ending with either the final target or the interaction region.","These simulations should include all pertinent physical effects from both first-principles models for high precision accelerator design and fast machine learning-based surrogate models for online particle accelerator tuning.","Such a virtual particle accelerator can be used as an emulator of the real physical accelerator in the accelerator operation.","One challenge in the end-to-end simulation is to exchange particle distribution data from different components of the particle accelerator.","To solve this problem requires supporting efforts to establish standards that would ease the sharing of information and data across codes as well as standards for interfacing codes [12].","Another challenge associated with the high fidelity end-to-end accelerator simulation is that it is extremely time consuming (especially with the inclusion of a variety of collective effects), varying from days to weeks and months.","The cutting-edge and emerging computing techniques including advanced algorithms, AI/ML methods, and quantum computing may substantially reduce the computational time of accelerator simulation and enable a fast real-time optimization with end-to-end simulations.","Advanced algorithms play an important role in the accelerator modeling.","High-fidelity self-consistent simulations cannot be done within a reasonable return time without using these algorithms [13], [14].","It has also been shown that the advanced algorithms can lead to several orders of magnitude improvement on the computational speed [15].","Research and development in this area is needed to explore new algorithms that exhibit better properties, to refine the understanding of the properties and bottlenecks of existing algorithms, and to remove the bottlenecks on cutting-edge heterogeneous computing hardware (e.g., GPU, FPGA), and improve the speed and accuracy of accelerator modeling.","Differentiable simulations show great potential to reduce the number of simulations needed for optimization and design of particle accelerators.","Using AI/ML surrogate models of the physics simulation can save orders of magnitude computing time [16].","A single differentiable simulation can generate the sensitivity information of the final beam quality with respect to thousands of machine parameters.","Support is needed for the development of AI/ML modeling techniques and their integration into accelerator modeling and control systems, with an emphasis on fast executing (up to real-time) and differentiable models, continual and adaptive learning for time-varying systems, uncertainty quantification to assess confidence of model predictions, and physics-informed methods to enable broader model generalization to new application conditions.","Quantum computing is another area of emerging technology that has the potential to transform some areas particle accelerator simulations by providing exponential improvement of computational efficiency.","Support is needed for quantum computing algorithm and code development for accelerator modeling, feasibility study on quantum computing implementation in accelerator modeling.","Quantum computing education in the accelerator community is also needed.","Particle accelerator modeling is computationally intensive and has utilized supercomputers for many years.","It has used both high-performance computing to model collective effects and high-throughput computing to model single particle dynamics.","With the arrival of exascale supercomputers and new special purpose computers with hardware accelerators (e.g.","GPUs), it is a challenge to achieve the supercomputer peak performance for modeling collective effects in particle accelerators due to deep levels of independent memory caches, on top of global and neighboring communications that are used to exchange information across multiple computer nodes during the simulation.","As part of this, the workload needs to be uniformly distributed among many compute units in order to achieve a good parallel efficiency.","The current and next-generation supercomputers extensively use the computational accelerators such as GPUs and FPGAs to improve the computational speed of applications.","Applications related to single particle dynamics can be effectively simulated on GPUs.","It is a challenge for applications including multiple physics models and collective effects to use GPUs effectively.","Support is needed to adopt portable, state-of-the-art programming paradigms and frameworks that support both multi-level parallelization and effective dynamic load balancing over all levels of compute parallelism, so that simulations run efficiently on the latest computer architectures with scalable I/O, post-processing and in situ data analysis solutions to support accelerator machine design and operation.","Particle accelerator modeling codes were developed with the support of a variety of projects to simulate RF structures, magnets, beam dynamics inside the accelerator structure, laser and beam interaction with plasma, and beam beam and beam wave interactions.","These simulation tools are critical for the design and operation of next-generation high energy particle accelerators.","Sustainability, reliability, user support and training of these complicated codes are potential challenges in the accelerator community.","Further support is needed to maintain these codes with documentation, testing, and benchmark examples.","Training classes are needed at the U.S.","Particle Accelerator School or other accelerator schools or workshops to help improve the usage of these codes.","Another challenge for these accelerator simulation codes is validation with experimental measurements.","This is critical for establishing the confidence in the use of these simulation tools for next-generation particle accelerator design applications.","The direct comparison between code predictions and accelerator performance and output may be limited by a number of factors, such as not fully characterized machine settings, unknown initial phase space distribution, insufficient range or resolution of diagnostics, and incomplete or missing physics in simulations.","Support is needed to connect simulations with direct experimental diagnostics and to carry out detailed benchmark studies with experimental measurements.","The particle accelerator modeling codes include a variety of physics models for different accelerator components and involve different research groups.","In order to attain the end-to-end simulation for next-generation particle accelerators, different component models need to be integrated into a workflow and ported onto the latest exascale type computer hardware with GPUs.","New physical models and capabilities will be added in some of these codes to account for new accelerating and focusing methods.","It is desirable to organize the beam and accelerator modeling tools and community through the development of ecosystems of codes, modular libraries, and frameworks that are interoperable via open community data standards for both loosely integrated and tightly integrated workflows.","It is desirable to establish open access data repositories for reuse and community-wide AI/ML surrogate model training.","It is also desirable to have an organized particle accelerator modeling community (e.g.","in the form of centers and consortia) including both academic institutes and industry.","This will improve the efficiency of accelerator modeling for next-generation high energy accelerator applications.","In summary, to meet the above challenges and needs, the following high-level recommendations are provided from the particle accelerator modeling community [11]: Develop a comprehensive portfolio of particle accelerator and beam physics modeling tools in support of achieving Accelerator and Beam Physics Thrust Grand Challenges on intensity, quality, control, and prediction.","Develop software infrastructure to enable end-to-end virtual accelerator modeling and corresponding virtual twins of particle accelerators.","Develop advanced algorithms and methods including AI/ML modalities and quantum computing technologies.","Develop efficient and scalable software frameworks and associated tools to effectively leverage next-generation high-performance and high-throughput computing hardware.","Develop sustainable and reliable code maintenance practices, community benchmarking capabilities, and training opportunities to foster the cooperative application of accelerator software.","Organize the beam and accelerator modeling community in open efforts to (a) foster the development of ecosystems of interoperable codes and tools, libraries, and frameworks, based on API and data standards, (b) establish open access and data workflow repositories, and (c) have an organized governance structure."],["Detector Simulation","Accurate detector simulation is crucial for tasks ranging from detector design to data analysis.","Geant4 [17], [18], [19] is the primary tool used for detector simulation throughout HEP [20].","This C++-based software package was originally released in 1998 as a complete rewrite of the earlier FORTRAN-based Geant3 [21].","The Geant4 collaboration and the HEP community engage in continuous research and development to improve the computational performance, refine the models of physical interactions, and incorporate new technical features and models.","Geant4 is the center of an ecosystem that includes numerous software packages serving different needs: Geometry/material modeling and navigation: VecGeom, CADMesh, DD4hep, GDML; Physics models: Noble Element Simulation Technique (NEST) for excitation, ionization, etc.","; G4CMP for phonons and other condensed matter phenomena; Opticks for optical photon simulation; Other simulation packages (including radiation modeling): MARS, FLUKA, MCNP, SRIM; R&D projects: G4HepEM, AdePT, Celeritas; Fast simulation engines: DELPHES, TrackToy, bespoke experiment-specific solutions, emerging ML approaches.","Some of these packages are developed in very close coordination with Geant4, while others are more independent, using Geant4 as a reference or being used as cross-checks for Geant4 results.","In addition, Geant4 interfaces with event generators (see Section ) such as pythia 8 for decays of long-lived, metastable, or late-forming particles, and with ROOT and CLHEP for various data analysis and mathematical functions.","Custom simulations of electronics responses and other instrumental effects are developed and applied downstream from detector simulation [22].","Geant4 is one of the most successful examples of community-supported and -maintained software in HEP.","It is even used beyond HEP for medical and astronomical applications.","However, HEP needs are the primary drivers of R&D on the software, especially improvements to its computing performance.","In particular, the next generation of collider experiments, including the HL-LHC and the various future colliders proposed during the Snowmass process, will pose extreme computing challenges for detector simulation.","Compared to present-day experiments, data volumes will increase by at least an order of magnitude, requiring a corresponding increase in the number of simulated events.","Detectors will increase in complexity and precision, requiring more detailed geometries and more precise physics models that will use even more computation.","At the same time, other aspects of the event processing chain, such as reconstruction, will consume a growing proportion of available computing because of massive increases in simultaneous bunch crossings, or pileup, especially at hadron colliders.","Therefore, Geant4 will need to simulate an order of magnitude more events with higher precision while using a smaller proportion of HEP computing.","To ground this prediction in reality, Geant4 used 40% of all computing during the LHC Run 2 for CMS and ATLAS [23], and ${\\sim }70\\%$ for LHCb [24], [25].","For the former, it is projected that only 14–20% will be available for simulation at the HL-LHC [26], [27].","In addition to high energy colliders, experiments of various sizes that search for light or weakly interacting particles rely on Geant4 [28], [29], [30].","Such experiments include DUNE, FASER, MicroBOONE, Muon g-2, Mu2e, COHERENT, and direct dark matter detection experiments DAMIC, DarkSide, DARWIN, DEAP, LZ, NEWS-G, PandaX, PICO, SBC, SENSEI, SuperCDMS, and XENON.","While many of these experimental collaborations and apparatuses are smaller than those at the CERN LHC, they face similar computational challenges to process petabytes of data, including the corresponding simulation of background and signal processes.","The need to meet these challenges with substantially fewer dedicated personnel makes the availability of common software even more important, including both Geant4 and packages like NEST and Opticks that provide specialized physical interaction models.","Future neutrino, dark matter, and other similar experiments will obtain even larger data volumes and probe even more precise physics, increasing the computational challenges and the associated need for high-performing and well-supported software.","The increasing breadth and precision of next-generation HEP experiments will require corresponding improvements in the physics models used in Geant4 and related software [20].","Some areas of interest include liquid argon signal induction, scintillation materials, Cerenkov light propagation, condensed-matter effects, low-energy responses, and rare background processes and interactions.","In particular, hadronic interaction models are an area where agreement with data could be improved and access to more measurements, such as hadron-argon interactions, could lead to even further improvement.","In general, support for increased effort in these areas is needed to keep up with the demands of new experiments for more precise models, as well as to improve the computational efficiency of the model implementations.","The first major evolution in the Geant4 computing model was the introduction of multithreading to reduce memory usage by sharing read-only common data during otherwise embarrassingly parallel event processing [31].","This approach has been adopted by the LHC experiments and Mu2e as the experiment software frameworks have similarly evolved to incorporate multithreading in other event processing steps [32].","A recently concluded R&D project called GeantV introduced sub-event parallelism using vectorized instructions, primarily targeting many-core CPU architectures, in a new prototype transport engine.","The primary findings of this project were that the achievable speedup compared to Geant4 was limited to a factor $2\\pm 0.5$ and that a small percentage of the speedup actually arose from vectorization [33].","Nevertheless, the project produced several useful developments, such as VecGeom, and lessons, such as the importance of instruction cache locality.","VecGeom provides improved code and vectorization for geometry modeling, enabling a ${\\sim }15\\%$ speedup when used with Geant4.","As an illustration of the level of complexity of Geant4, it required 30 FTE-years, spread over a 5-year period, to re-implement its numerous components, even partially or as prototypes, in GeantV: particle transport, geometry modeling and navigation, magnetic field propagation, physical interaction models, and interfaces for experiment software frameworks.","The future of detector simulation computing is turning toward the use of GPUs, targeting in particular HPC facilities that are intended to provide a larger fraction of HEP computing for next-generation experiments.","Because detector simulation is naturally an HTC problem, nontrivial effort is required to adapt the necessary computations to use computer accelerators effectively.","Two ongoing projects, G4HepEM-AdePT and Celeritas, have produced viable prototypes of high-performing GPU-native simulation engines.","In particular, the Celeritas effort is led by US national laboratory personnel and was inspired by the earlier porting of the Shift MC radiation transport engine to GPUs [34].","Shift is a less general application than Geant4, focusing on neutral particles with few secondary particles produced, limited physical interactions, simple geometries, and no magnetic field propagation.","Despite the additional complications encountered in addressing the broad scope of HEP detector simulation, the prototypes have demonstrated promising results, achieving a factor of 40 speedup using 1 Nvidia V100 GPU compared to Geant4 using 7 Power9 CPUs on Summit [35].","This can be rephrased as a factor of 200 GPU:CPU equivalence comparing a V100 to a higher-end Xeon CPU, also taking into account power consumption and other relevant factors.","Variations in performance from different implementation choices were relatively small, a factor of 2 at most.","Scaling this up to the whole Summit supercomputer suggests that its 27 648 GPUs can correspond to 5.5 million CPU cores, an order of magnitude larger than the current worldwide LHC computing grid (WLCG) at 500 000 cores [36].","If this performance persists, it has the potential to resolve the major simulation computing challenges.","However, we must not underestimate the increased burden of developing and maintaining coprocessor-friendly code, including aspects such as portability to non-Nvidia GPUs and even future non-GPU architectures.","It should also be noted that the ongoing projects currently implement only some components of Geant4, including models of electromagnetic interactions, but not yet hadronic interactions, for example.","If a complete parallelization of all components is not ultimately possible, the observed speedup will be limited by the remaining serial components, according to Amdahl's law.","In terms of personpower, the Celeritas project has so far used 5 FTE-years spread over 2 years, and AdePT is similar.","Because this effort builds not just on Geant4, but also on the lessons learned from earlier projects including Shift and GeantV, as well as the products of those efforts such as VecGeom, the total personpower involved is much greater.","Another avenue to achieve faster simulation is the use of machine learning to replace or augment existing “classical” (rule-based) simulation engines [37].","Full replacement is most commonly pursued using generative models such as generative adversarial networks (GANs), variational autoencoders (VAEs), or normalizing flows (NFs).","Augmentation to improve lower-quality results, sometimes called refinement, can be applied to classical fast simulation engines or to generative models, using techniques similar to those above or regression-based approaches.","These approaches are largely still in development or just starting to be deployed [38], so their ability to provide an acceptable level of physics fidelity at scale must be assessed carefully.","Beyond these ML algorithms, there are proposals to explore the application of differentiable programming directly to simulation engines [37], [39].","ML algorithms whose training and inference are performed using industry-standard frameworks are more easily portable to coprocessors such as GPUs, including at HPCs.","The possibility also exists to use coprocessors more cost-effectively and efficiently by performing the ML inference as a service, avoiding the need for each CPU to be equipped with a GPU [3].","The above solutions, including GPU prototype transport engines and ML algorithms, are promising avenues for faster execution of detector simulation.","However, it is unlikely that these will completely supplant Geant4, which provides broad support for myriad use cases.","In addition, the results of any new approaches are most often compared to Geant4 to prove their validity.","Therefore, some percentage of general purpose von Neumann CPU-based computing will be needed in HEP for the decades to come.","The extent to which CPU-based computing can be shifted to less general architectures and devices such as GPUs depends strongly on the availability of experts to develop and maintain the necessary software.","There has recently been a severe decrease in funding for permanent positions in HEP to support software developers who work on both these common tools and experiment-specific matters.","This threatens the persistence of important knowledge, which must be passed directly from person to person, and the stability and future of these crucial software packages that serve as a foundation for the entire field.","There is a strong and vocal consensus throughout the field that this trend must be reversed, with funding for technical positions not just restored, but increased further: “...it is still necessary to support the existing code and make sure that sufficient funding and staffing is provided for maintenance and development of physics algorithms, as well as for adapting the code to any updated CPU hardware, operating systems and new compilers.” [31] “Unfortunately, the Bertini model [of hadronic interactions] has not been actively developed over the last few years due to the lack of personpower.” [6] “A team of highly-skilled physicists and engineers is required to provide the necessary support and developments for Geant4, and the packages extending it, to meet the needs and challenges within the scope of the US HEP experimental program.” [20] “In addition, Geant4's common software, such as physics models, no longer receives any US maintenance funding...","This is not sustainable for the long term viability of small experiments.","Long term, experiment-agnostic Geant4 support is critical for the success of small experiments...","It is therefore essential that there is funding for permanent software and computing experts.” [28] “However, U.S. funding for [the Geant4 toolkit] was discontinued in recent years...","Continued support for Geant4 is crucial to the design and construction of future experiments, and for the interpretation of their results...","The general challenge of maintaining software and computing talent is exacerbated in the direct detection community by the lack of long term, permanent positions within the experiments...","It is therefore essential to provide funding for permanent software and computing experts.” [29] “DM and neutrino collaborations need scientists trained in data acquisition, simulation, and analysis at both the user and developer levels to achieve their science goals.” [30]"],["Physics Generators","The generator software landscape is complex and varied, in order to provide solutions to different aspects: event (matrix element) generation, hadronization and parton shower modeling, underlying event tunes, matching/merging algorithms, multi-parton interactions, particle decays, cross section calculations, parton distribution functions, data formats, and analysis and reinterpretation.","There are many software packages that address some of these needs, which often include plugins to perform specific calculations and provide ways for users to plug in their own custom models and computations.","A rough and incomplete categorization includes: matrix element generators, which support leading-order (LO) as well as next-to-leading-order (NLO) and sometimes higher order calculations: MadGraph 5_amc@nlo, powheg, sherpa, alpgen, whizard, amegic, comix, helac; parton shower modelers, which also support LO event generation: pythia, herwig, sherpa; neutrino-nucleus interaction generators: genie, neut, NuWro, GiBUU, achilles; plugins for more specific tasks: EvtGen, tauola, photos, dire, vincia; higher-order differential cross section calculation: mcfm, matrix, NNLOJet; one-loop amplitude computation: MadLoop, OpenLoops, recola; utility software: lhapdf, HepMC; analysis and tuning: rivet, mcnntunes, professor, apprentice, nuisance.","This wide array of software packages includes a corresponding variety of programming languages and styles—primarily Fortran, C++, and Python—as well as development and maintenance approaches.","Unlike detector simulation, in which Geant4 serves as a locus for numerous related efforts as described in Section , most of the software packages listed here are developed largely independently by separate groups.","The usage of common data formats and exchange specifications, such as LHEF, UFO, and HepMC, provides some amount of generic interoperability.","The percentage of WLCG computing time used for event generation at the CMS and ATLAS experiments is estimated to be 5–12%, with similar or even more substantial requirements expected for ALICE and LHCb [40].","However, these summary values hide large variances in tasks and generator software.","For example, MadGraph 5_amc@nlo has been found to be substantially faster than sherpa for generation of W+jets events with 0–2 extra partons, but much of the difference depends on parameter choices that may impact physical accuracy in some regions of phase space.","In general, for a given task, generator CPU usage scales linearly with the number of events generated.","However, moving from LO to NLO and higher orders, or increasing the multiplicity with additional partons [41], for the same physics process typically causes a factorial increase in CPU usage, while different processes may still have very different baseline computational requirements.","The memory usage and multithreading abilities of different generator software are also highly variable and can lead to inefficiencies.","The range of programming languages and styles can make generators difficult to integrate directly in experiment software frameworks, another potential source of inefficiency.","The generator usage at neutrino frontier experiments has not been measured and profiled as systematically, and this subset of the field is generally smaller compared to the energy frontier.","Generator CPU usage is projected to increase to 8–20% in the HL-LHC era, when more events and higher precision will frequently be required [26], [27].","The increasing precision of upcoming neutrino experiments such as LBNF/DUNE and HyperK should be expected to require similar increases in experiment computing time devoted to event generation.","Further future experiments will have different generator needs depending on their initial states.","The different categories include high energy hadron colliders, lower (but still relatively high) energy lepton colliders, neutrino experiments, forward physics experiments (such as FASER and SND@LHC), transverse detectors (such as CODEX-b, AL3X, and MATHUSLA), the Electron-Ion Collider (EIC), and proposed muon colliders.","The details for the relevant tasks and generators in each category are vast and cannot be summarized here; Ref.","[42] presents a comprehensive treatment.","As a particular item of interest from the recent Snowmass effort, we highlight new techniques in development for muon collider physics: the use of the effective vector boson approximation for $2 \\rightarrow n$ processes [43] and vector boson fusion as the dominant tool to study the electroweak sector at multiple TeV [44].","These processes and approaches have been implemented in new versions of MadGraph 5_amc@nlo, sometimes requiring new features or more involved calculations to ensure stable and reliable results.","They illustrate the work that will be needed to model the physics at the next generations of HEP experiments.","The interdependence between lattice QCD, nuclear physics, and neutrino event generators is also important to highlight [45].","The computational burden of an event generator depends strongly on the precision required, as described above; the calculation techniques used; and any other requirements placed on the output [31], [40], [42].","For example, the automated computation of NLO and higher order diagrams, as used in mc@nlo, introduces events with negative weights, which can substantially reduce the statistical power of the final result.","There are also inefficiencies in phase space sampling; in slicing, biasing, or filtering to change kinematic features; and in merging to avoid double-counting between matrix element generators and parton shower modelers.","Further, experiments' use of generators may involve significant repetition of expensive calculations, especially to assess certain systematic uncertainties that cannot be represented as alternative event weights.","Recent efforts coordinated by the HEP Software Foundation (HSF) have identified and resolved several performance bottlenecks, some related to these issues [42].","There is ongoing work to continue to address these issues, especially negative weights, which can be reduced, but not yet eliminated, by certain new prescriptions that have not yet achieved wide deployment [40].","In general, the development of new theoretical and mathematical tools to perform relevant computations has an outsized impact on event generator computing requirements.","At the time of the previous Snowmass reports, generator software was in some ways ahead of other subfields.","Machine learning algorithms were employed to derive PDFs [46], and usage of MPI for efficient utilization of existing CPU-based HPC systems was incorporated in sherpa, alpgen, MadGraph 5_amc@nlo, and mcfm, among others.","However, more recent initial efforts to port MadGraph 5_amc@nlo to use GPUs did not lead to a production quality result.","Revitalizing this task has been identified as a priority, and recent progress is promising.","An opportunity comes from the possibility to parallelize the matrix element calculations, which take up more than 90% of the computing time for complex LHC physics processes.","Matrix element calculations are a perfect fit for CPU vectorization and hardware acceleration on GPUs.","The MadGraph4GPU project, which uses custom CUDA kernels, shows speed improvement factors of ${\\sim }1000$ on an Nvidia V100 GPU, as well as an improvement factor of 3 on CPUs by exploiting vectorization [47].","The MadFlow project, which uses a TensorFlow backend, shows a speed improvement of more than an order of magnitude on an Nvidia Titan V GPU and an improvement factor of up to 3 on CPUs [48].","A new approach called BlockGen for multi-gluon tree amplitudes shows similar improvements of roughly an order of magnitude on a V100 GPU and 2–6 on CPU, depending on the multiplicity [49].","Similar developments for other computationally-intensive generators such as sherpa also need to be prioritized and supported.","The use of ML for approximate matrix element calculations, including phase space integration and sampling as well as fully generative models similar to those discussed in Section , is also very promising, but still in relatively early R&D and not yet ready for deployment.","Additionally, there are ongoing investigations into ML for parton shower modeling, hadronization, event (un)weighting, and tuning, among other applications [42], [50].","Beyond technical developments, other possible improvements that require coordination between different groups have been identified.","A common interface, which could also be described as a modular framework, would standardize how generators are used and plug into each other and other software [45], [51], [42].","This would be useful not just for theoretical studies, but also for experiment use, especially small experiments with minimal personpower to implement, test, and maintain different interfaces.","It could also facilitate more collaboration among generator developers, for example to make faster progress on code modernization and related work.","The adoption of common data formats and standards, such as those provided by Les Houches, by more generators and communities could be encouraged as well.","While there are clearly a number of potential benefits, both coordination and personpower limitations would need to be overcome in order to realize them.","Sharing generator outputs, such as parton-level event records, is another idea to reduce both computation and disk usage [40].","At the LHC, ATLAS and CMS often duplicate the work to generate the same physics process at the same precision and to store the result.","Avoiding this duplication would require coordination to standardize data formats, host the output, and ensure that the choices of model and generator parameters satisfy different analysis needs.","The existing LHC working groups that provide resources for cross sections and other theoretical calculations could serve as a model to organize this kind of coordination.","Data preservation organizations could act as a central hub for storage, and an increased emphasis on data preservation and access would also improve the field's capabilities to reinterpret and combine results [42].","More broadly, it has been suggested that a diverse and cross-cutting collaboration, inspired to MCNet in Europe with a long-term outlook, could bring together the U.S. event generator community to share resources and ideas [42].","The breadth and diversity of generator software brings many advantages and opportunities to HEP.","However, it also brings challenges, especially when it comes to keeping up with the increasing demands for both physics precision and computing efficiency.","Historically, there has not been a single obvious locus to coordinate common activities, even if MCNet and, more recently, HSF have provided useful forums, especially in the areas of training and computational efficiency, respectively.","Towards the future, more emphasis should be placed on establishing and coordinating common activities.","and improvements in this area should continue to be emphasized.","Projects to reduce the computational burden of event generation, for example by adapting to use GPUs, need a substantial increase in effort in order to be successful.","Some of the difficulties that generators face in computing are intrinsically tied to the nature of the calculations being performed, so fundamental research may be required to develop better methods.","On the whole, event generators are usually supported by small teams that frequently have little to no dedicated funding and cannot adequately plan for succession.","As with detector simulation, it is vital to the entire field that funding is provided for permanent positions to support, improve, and expand these crucial components of HEP software.","This is echoed throughout the white papers written during the Snowmass process: “The physics Generators used in the field (eg.","Pythia, GENIE, Madgraph, Sherpa) also suffer from lack of stable funding in a similar way.” [28] “The common needs to all direct dark matter detection experiments include: ...","Continued support for event generators, including those developed as part of a national security program.” [29] “Participation in generator-related activities is poorly incentivized for both theorists and experimentalists, and opportunities to pursue neutrino generator development as one's primary research activity are rare... A need for greater coordination and prioritization of such activities is widely recognized in the neutrino scattering community.","Despite promising initial discussions that have taken place in a series of recent workshops [68, 529–531]; however, neither a clear leadership structure nor significant institutional support to carry out the related work have yet emerged...","The cooperation needed for success in neutrino generator development cannot occur to the extent that funding agencies impose rigid boundaries between [high energy and nuclear physics]... [Stitching together multiple models to achieve complete simulations] is ideally done in collaboration with theorists to minimize inconsistencies, but incentives for their direct involvement are currently poor.” [45] “[Theorists]... may not be motivated to work on software optimisations that are not “theoretical” enough to advance their careers.","Generator support tasks in the experiments may also not be enough to secure jobs or funding for experimentalists pursuing a career in research.” [31] Ref.","[51] specifically surveys the major collider event generators to report the expected level of support through the HL-LHC era: “Funding and career perspectives are a major common issue reported by many generator experts...","Generator support tasks in the experiments may also not be enough to secure jobs or funding for experimentalists pursuing a career in research.","It is still not clear for people who are doing work on generators or related tools whether it is really possible to have a dedicated career path from now throughout HL-LHC.","This is slowly being acknowledged from funding bodies, but the community has low expectations especially because there is no sign that hiring policies will be modified to provide reasonable funding for generator work.","The delay caused by the lack of funding for generator work has already started to cause a loss of know-how in some generator packages.” MadGraph 5_amc@nlo: “the collaboration are of the opinion, shared by other groups, that time to work on performance improvements, even by significant factors, is not well recognised by funding agencies...","There are no dedicated funds for HL-LHC.” powheg: “there are no funds specifically dedicated to the maintenance of the framework” sherpa: “Several of the Sherpa team are in permanent positions...","In the UK, Sherpa posts at IPPP are seen as part of IPPP's core mission and therefore are expected to be treated as core in future funding reviews.” herwig: “Funding is a serious issue... To fund the bread and butter of what the experiments need, one would need different specific funding.” pythia: “the continued development of Pythia8 throughout the HL-LHC era depends on funding agencies and laboratory priorities, though these are not expected to change.","The biggest challenge to the collaboration is finding permanent positions for junior people so that they can continue their valuable contributions.” EvtGen: “The core EvtGen team has 4 members who have a fraction of their time funded for EvtGen through an UK STFC grant...","There is some funding for a computer engineer to work on code redesign for multithreading.” The developers of the BeAGLE generator, which is used for inelastic scattering in nuclear collisions, state: “The future development of this code is uncertain primarily due to lack of manpower and reliable funding... As discussed above, funding/manpower issues are currently limiting our ability to implement many of these improvements.” [42]"],["Continuum Field Theory Calculations","Continuum field theory calculations include both precision high order perturbative calculations [52] and conformal bootstrap theoretical calculations [53]."],["Precision perturbative calculation","Precision measurements at the LHC and future colliders require theory predictions with uncertainties at the percent level for many observables.","Theory uncertainties due to the perturbative truncation are particularly relevant and must be reduced for a large variety of relevant processes.","At the high luminosity LHC, even rare processes like Higgs production require theoretical control of cross sections at the 1% level.","Reference [52] surveyed the theoretical community and received responses covering 53 scientific publications.","This Snowmass whitepaper gave clear motivation for precision as follows: “To fully exploit the physics potential of current and future collider experiments, in particular to unambiguously identify signals of new physics, it is crucial to improve the precision of theoretical predictions.” Since the 2013 Snowmass exercise, the state-of-the-art has moved dramatically and now next-to-next-to-next-to-leading order (N$^3$ LO) QCD calculations for $2\\rightarrow 1$ processes and NNLO QCD calculation for $2\\rightarrow 3$ processes and $2\\rightarrow 2$ processes with internal masses are becoming state-of-the-art and reasonably fast on contemporary computing resources, while automation and more complex final states at NNLO at N$^3$ LO are goals for the Snowmass period.","The computational challenge is to perform Feynman loop integrals within perturbative quantum field theory, giving access to the scattering matrix and its analytic structures, often involving non-standard special mathematical functions.","The aim is to, eventually, be able to numerically evaluate the expressions such that the error is reliable, the result has sufficient precision, and the computation consumes acceptable resources.","Suitable analytical or symbolic preparation of the integrals and amplitudes must be performed and the evaluation of these expressions during Monte Carlo integration over final state phase space, so that they may feed into event generators.","Analytical solutions of Feynman integrals in terms of special mathematical functions are preferred from an efficiency perspective, but often require process-specific knowledge or even new mathematical methods.","Numerical or semi-analytical methods can allow broader flexibility and better automation.","Numerical direct integration approaches are fully established and implemented in codes such as pySecDec and fiesta, but are typically slow in evaluation times.","Novel semi-analytical methods based on series expansions seem promising and first public packages such as DiffExp and AMFlow became available recently.","Such series expansion methods reduce evaluation times to the order of a minute per phase-space point for typical two-loop integrals.","QCD corrections to scattering amplitudes have been fully automated at NLO using libraries like Helac-NLO, MadGraph 5_amc@nlo, nlox, OpenLoops, and recola as well as many other public and private tools.","Two and higher loop calculations remain challenging but in the last five years significant progress has been made in many combinations of final state particle multiplicity and number of internal mass scales at two loops, and in more restrictive cases at three and four loops.","Further the complete five loop beta function has been calculated.","A core computational bottleneck for the calculation of both Feynman amplitudes and loop integrals is in the reduction of linear relations between Feynman integrals (integration-by-parts identities), with fire, reduze, LiteRed and kira being example codes.","From a computational perspective the use of modular (finite field) arithmetic in the calculation of integration by parts reduction is an interesting transformation of the computational requirement that may require radically different computing hardware solutions.","Reference [52] notes significant difficulties fitting within the constraints of common HPC centers: “These novel techniques have opened the door to perform computer algebraic computations for quantum field theory on HPC systems.","These novel methods typically use integer arithmetic rather than floating point arithmetic and, depending on the problem, they may require a significant amount of memory per core.","The development of (public) codes for challenging amplitude calculations in a HPC friendly way is an ongoing effort.","Currently, in some situations, one may need to resort to implementations that have large memory demands and require run-times that are hard to predict and possibly well beyond available batch limits on a given cluster.” Given the importance of precision theoretical calculation it is imperative that appropriate resources be available to enable them.","In 2013, a sample of then state-of-the-art calculations took between 50,000 and 1M core hours.","Since then, the per-core CPU performance has improved by about a factor of 5 and the per-node performance by about a factor of 30.","At the time, it was hoped to be able to use efficiently either many-core architectures such as Xeon Phi or GPUs.","The current state of the art has the additional complexity largely balanced by the growth in per-node performance, reaching up to about 10M core hours with current CPUs, as shown in Fig.","REF .","Parallelism is typically used within a node, and is indeed in part dictated by growth in core count relative to memory capacity.","Multinode parallelism is only exploited as throughput computing making use of trivial parallelism by the majority of codes in the area.","[50][h] Computational costs of current state-of-the-art multi-loop calculations in the community, from Ref. [52].","Figure: NO_CAPTION Although there is some role for GPUs in numerical integration, they only match a modest part of the scientific requirement [52]: “The use of GPUs is still unclear in our field, since many problems in our field rely on the numerical evaluation or handling of algebraic expressions which are large and/or require irregular memory access patterns.","So far, GPUs have found application to cutting-edge problems with the numerical integration of sector decomposed loop integrals.","A first step for future applications could include the efficient evaluation of one-loop amplitudes.","This would help the huge computational requirements for NLO high-multiplicity evaluations, but also for the real emission integrations for NNLO calculations.","Since the efficient use of GPUs is still unclear, future computing for our community will still need to focus on providing CPU resources without attached GPU resources.” Large memory and runtime limitations imposed by typical HPC environments mean that bespoke computing nodes and queues (wherever they are hosted) are required to enable these calculations.","Some of the job runtime constraints could be addressed with checkpointing virtual machines but since complex software is involved, some even commercial like Mathematica, a dedicated node environment may be the only practical solution.","Machine learning does not at this time play a large role, although some applications have been used in event generation.","The licensing of proprietary software like Mathematica or Maple also suggests dedicated computing nodes for this purpose may be cost-effective."],["Conformal bootstrap","The idea of the conformal bootstrap is to constrain and solve conformal field theories (CFTs) using physical consistency conditions like symmetry, unitarity, and causality.","By relying on nonperturbative structures, bootstrap methods can work even in strongly-coupled systems where traditional perturbative techniques fail.","Over the last few decades, the conformal bootstrap idea has crystallized into two concrete strategies: 1) exploiting exact solvability and 2) deriving bounds using sum rules with positivity constraints and using convex optimization to extract information.","When combined with numerical convex optimization, this leads to the numerical conformal bootstrap.","The conformal bootstrap is distinguished by being an intrinsically non-perturbative method that yields strict bounds on the theory.","Recent successes include precise determinations of critical exponents in physically-relevant theories such as the 3D Ising and O(N) models, constraints on theoretically important theories such as 4D N = 4 supersymmetric Yang-Mills theory and 6D $(2,0)$ SCFTs, and has inspired promising new ideas for bootstrapping S-matrices.","The numerical bootstrap is a rapidly-growing field and is anticipated to play a central role in CFTs over the next decade.","Targets include 3D scalar models, $O(N)$ vector models in 3D, and in the case of $N=1$ the 3D Ising CFT, while the 3D O(2) model describes the the superfluid transition in liquid helium as well as thin-film superconductors.","Here the bootstrap has helped address a controversy, where the results of experiment and the current best Monte Carlo simulations disagreed with each other with 8$\\sigma $ significance, giving results in remarkable agreement with those from Monte Carlo simulations.","A second target is 3D Fermionic Models, including 3D CFTs containing N Majorana fermions with Yukawa couplings to one or more scalar fields, often called Gross-Neveu-Yukawa model.","The simplest model with $N=1$ Majorana fermion coupled to a real scalar is believed to possess emergent supersymmetry and correspond to the minimal 3D $N=1$ supersymmetric extension of the Ising model, proposed to have a realization on the boundary of topological superconductors.","Preliminary studies applying the bootstrap to fermion 4-point functions showed the GNY models to be promising targets for the bootstrap, while further systems can be studied providing a supersymmetric systems including Wess Zumino models.","Progress in 3D and 4D gauge theories has been made including including the conformal window of 4D QCD with 12 flavors, establishing a hint of contact with lattice field theory near conformal window results and in $N=4$ supersymmetric Yang Mills theory.","From a computational perspective, the whitepaper identifies two key algorithmic challenges: i) to find faster optimization methods, or ways to scale up current tools, and ii) to find efficient methods for exploring high-dimensional spaces of CFT data.","High precision arithmetic is required, and often not in native hardware floating point, and used in bespoke semidefinite solver libraries such as SDPB.","SDPB can use distributed high precision linear algebra and parallelize over many HPC nodes and hundreds to thousands of cores.","The authors are optimistic that further progress will: “enable a new round of fundamental discoveries in theoretical physics, including the numerical classification of CFTs with a small number of relevant operators, definitive answers to longstanding questions about conformal windows of gauge theories in 3 and 4 dimensions, more-or-less complete numerical solutions of the known maximally-supersymmetric CFTs (and their holographic duals), and new robust starting points for Hamiltonian truncation studies of gapped phases.","Importantly, the numerical bootstrap can also be used as a discovery tool for finding previously unknown CFTs and as well as for identifying possible analytical solutions of known ones.” The computational requirements so far have involve the use of conventional cluster computers, such as XSEDE and up to 32 nodes and 768 cores.","Calculations of different theories range from 200,000 core hours for the 3D Ising model to a few million core hours testing 100 to 1000 points in the CFT parametric space for the 3D $O(3)$ model.","Large memory capacity is again significant requirement and is a barrier to efficiency, arising from the high precision (1000 bits) arithmetic and is required for accurately handling large cancellations in the calculation.","The authors identify significant software and computing hardware opportunities in future: “further scaling and improving this code, including implementing more sophisticated distributed linear algebra routines, improving memory management to allow scaling past ${\\sim }500$ cores, finding ways to reduce precision requirements, and leveraging new hardware like GPUs and FPGAs.","These engineering challenges are tightly coupled to physics: an increase in the scale of solvable semidefinite programs could allow one to explore larger systems of crossing equations and thereby access new CFTs.”"],["Continuum field theory summary","Both multi-loop perturbative calculations and numerical conformal bootstrap have a similar computer hardware requirement.","They both have a current software reliance on CPU cores, and a large memory per core requirement.","Perturbative calculations have a sizable barrier to the use of GPUs, while the conformal bootstrap approach, after significant software engineering, may be amenable to GPU or FPGA acceleration of high precision arithmetic."],["Lattice QCD","Lattice gauge theory is a systematically improvable theoretical tool for numerical evaluation of the Euclidean Feynman-path integral for Quantum Chromodynamics (QCD).","Worldwide lattice gauge theory efforts directly support numerous high-energy physics experiments by calculating properties of hadrons that are vital to interpretation of the experiments.","These efforts make significant use of supercomputers and are critically dependent upon continued computing advances.","The Flavour Lattice Averaging Group [54] performs a critical review of many important lattice QCD predictions every two to three years.","Snowmass Whitepapers have been contributed on the topics relevant to Lattice QCD [55], [45], [56], [57], [58], [59], [60], [61].","USQCD also published in 2019 comprehensive set of topical whitepapers and submitted these for consideration by Snowmass [62], [63], [64], [65], [66], [67], [68].","The search for new physics requires a joint experimental and theoretical effort.","Lattice QCD is an essential tool for obtaining precise model-free theoretical predictions of the hadronic processes underlying many key experimental searches.","The role of Lattice QCD in determining the hadronic contributions to the anomalous magnetic moment of the muon was discussed in a Snowmass white paper [59], by the international Muon $g-2$ Theory Initiative white paper [69] and in a white paper by the USQCD collaboration in 2019 [64].","These calculations will be critical to the interpretation of results from the Fermilab Muon $g-2$ Experiment [70], which is currently in excellent agreement with results from the earlier Brookhaven experiment [71].","Lattice QCD also plays a key role in ab initio prediction of nucleon structure and parton physics [57], [55].","Understanding the neutrino-nucleus interaction is critical to the analysis of the Deep Underground Neutrino Experiment.","The interaction between an intermediate W or Z boson with with a quark confined inside a neutron or proton within the nucleus is theoretically complex.","This was discussed by USQCD in a 2019 white paper [65] and holistically the interplay between lattice QCD, EFTs, nuclear physics, phenomenology, and neutrino event generators was discussed in a contributed Snowmass Whitepaper [45].","A comprehensive quark-flavor physics program has been discussed in a USQCD whitepaper [64].","Contributed Snowmass white papers include improving the understanding of anomalies in B physics [72], emerging from the the Large Hadron Collider and studied further at Belle II, as well as reconciling CP violation observed in kaon experiments with the standard model and exploring rare kaon decays [56].","These phenomena are typically highly suppressed in the standard model and therefore also offer promising avenues for the discovery of new physics.","As experimental measurements become more precise over the next decade, lattice QCD will play an increasing role in providing the needed matching theoretical precision.","Achieving the needed precision requires simulations with lattices with substantially increased resolution.","With finer lattice spacing comes an array of new challenges.","They include algorithmic and software-engineering challenges, challenges in computer technology and design, and challenges in maintaining the necessary human resources.","The simulations at finer lattice spacing and larger volumes required to realize these goals introduce new challenges [58]: “Meeting them requires new algorithmic research, novel computer hardware design beyond the exascale, improved software engineering, and attention to maintaining human resources.” As a computational problem, Lattice gauge theory is performed on structured Cartesian grids with a high degree of regularity and natural data parallelism.","The approach formulates the Feynman path integral for QCD as a statistical mechanical sampling of the related Euclidean space path integral.","The sampling is performed by Markov chain Monte Carlo (MCMC) sampling, using forms of the hybrid Monte Carlo (HMC) algorithm [73], [74], [75].","Present algorithms for both MCMC sampling and Dirac solvers display growing limitations as substantially greater ranges of energy scales are included in our problem, an algorithmic challenge called critical slowing down.","The development of numerical algorithms is a significant intellectual activity that spans physics, mathematics, and computer science.","The central repeated operation is the solution of the gauge covariant Dirac equation.","The solution is usually performed using iterative Krylov solvers, Newer multigrid [76], [77], [78] solvers have demonstrated order-of-magnitude gains for Wilson fermions by approximately and repeatedly handling degrees of freedom in the low lying eigenspace as a form of preconditioner.","The US HEP program is presently focused on the domain-wall and staggered approaches, but corresponding gains for staggered and domain-wall-fermion discretizations are a critical open research activity.","A second algorithmic direction is the critical slowing down of MCMC algorithms.","These are being studied under Exascale Computing and SciDAC projects and such support is critical to progress in the field.","Recent algorithmic directions with different ways of attacking the same problem include applications of machine learning to configuration sampling.","This possibility has been discussed in a dedicated Snowmass white paper [79] and will be covered in the report of topical group CompF3 Machine Learning[80].","Related Snowmass submissions on tensor networks and quantum simulation will be covered in the report of CompF6 Quantum Computing [81], [82].","The Lattice QCD workflow is divided into two phases.","First, a MCMC sampling phase generates an ensemble of the most likely gluon field configurations distributed according to the QCD action.","The ensemble generation is serially dependent and represents a strong scaling computational problem.","Ideally one would be able to use efficiently $O(10^4)$ computing nodes on $O(256^4)$ data points.","On the largest scales this becomes a halo-exchange communication problem with a very large interconnect bandwidth requirement since the local data bandwidths vastly exceed those of inter-node communication.","In the second phase hadronic observables are calculated on each sampled configuration where many thousands of quark propagators are calculated and assembled into hadronic correlation functions.","This both allows more scope for amortizing the setup cost of advanced algorithms like multigrid or deflation, and also has a high degree of trivial parallelism.","Scaling Monte Carlo sampling to many computational nodes requires strong scaling and high interconnect bandwidth.","Scaling the hadronic observable calculations is not as challenging as multiple configurations can be analyzed at the same time, each job running on a moderate number of nodes.","A number of specific simulations, Table REF , have been proposed with estimated costs in a Snowmass white paper [56], and the same methodology can be used to estimate the requirements of the ideal ensemble for flavor physics.","The final entry is associated with physics in the B-meson system indicated in Snowmass white paper [72].","A $256^3\\times 512$ lattice at a lattice spacing $a = 0.04$ fm ($a^{-1} \\sim 5$ GeV) would allow us to simulate up/down, strange, charm, and bottom quarks at their physical mass in a 10 fm box with $m_\\pi L = 7$ and requires more than a sustained Exaflop year.","Several such calculations would be sought by the whole community.","These are a modest subset of the many many proposed calculations across the Snowmass contributions, but set the scale of computing sources required in the area of Lattice QCD simulations in support of experiment at beyond-exascale.","Between flavor physics, nucleon structure, neutrino scattering, and beyond-the-standard model physics, one could project four or five times this requirement in aggregate across the entire field in the USA alone.","On this basis, Ref.","[58] estimated that: “These simulation goals therefore clearly demonstrate a need for computers at least 10x more capable than the coming Exaflop computers during the Snowmass period.","Since the performance is required to be delivered on a real-code performance basis... more than an order of magnitude improvement, perhaps, from both algorithms and computing are required.” Table: Proposed lattice volumes and cost estimatesin sustained Exaflop hours, scaled from currentsimulations on Cori (NERSC) and Summit (ORNL).Volumes and estimates are proposed in Snowmass white paper , while the finalentry uses the same methodology to estimate the costof the most expensive proposed B-physics capable simulation.The massive vector parallelism of lattice gauge theory is, in principle, amenable to GPU and possibly other acceleration.","This imposes a significant additional programmer overhead and the most commonly used packages receive sufficient investment to use the complete range of modern accelerated supercomputers and many of the largest projects use allocations on DOE supercomputer resources.","Commonly used Lattice QCD software has been supported by the DOE SciDAC and Exascale Computing Projects.","These include Grid [83], [84], [85], MILC [86], [87], CPS [88], Chroma [89] and Quda [90].","This has enabled major packages to support the most advanced GPU accelerated HPC computers using software interfaces such as HIP, SYCL, and CUDA APIs in addition to giving good performance on several CPU SIMD architectures.","Newer interfaces like OpenMP 5.0 offload and C++17 parallel STL are planned to be adopted as and when appropriate.","For smaller projects, especially where rapid development and programmer productivity are at a premium, it is better and more cost effective in terms of human effort to maintain access to a range of CPU resources.","USQCD institutional clusters at Brookhaven National Laboratory, Fermilab, and Jefferson Laboratory have been instrumental in supporting the significant number of smaller and experimental projects that would not achieve the return on investment to justify bespoke software development for multiple architectures.","Reference [58] notes the heavy reliance on high performance computing places a particularly large dependence on highly tuned bespoke software in this field: “One hopes that this might consolidate the proliferation of programming interfaces as the lack of standardization imposes a very significant software development burden on the science community with duplication of effort for multiple systems.","This software development underpins the entire community effort and requires support to make the Snowmass science goals feasible.” “The challenges exist on multiple fronts: intellectual in developing algorithms that evade critical slowing down, software engineering to develop well-performing and portable code on an evolving range of supercomputers and programming models, and technical to remain engaged with the DOE HPC community as systems are planned and developed.” “The community activity is dependent on the existence of bespoke software environments that enable efficient simulation on rapidly evolving supercomputers, which requires significant expertise to develop and sustain.” USQCD has had senior staff, postdoctoral researchers, and post-graduate students funded under the Exascale Computing Project (ECP) and the SciDAC program.","It has also funded the development of high performance software portable to Exascale hardware [83], [85].","It was noted as important that flexible high-performance software continues to be developed for a diverse range of architectures that tracks the DOE computing program: “the productivity of the community depends on large code bases... which do not have a secure model for development and support which places investments at risk.” Much of the bespoke software development is performed in US National Laboratories.","This is required to address HPC architectures as they emerge in the Computational Frontier, developing performance-portable high-level libraries to enable a write once and run anywhere approach that many domain scientists, both in laboratories and universities, can modify effectively.","This effort must track the evolution of computing.","Finally, it was noted that theoretical particle physics has been one of the last area of physics to recognize the importance of computation in forefront research [58].","Continued effort is urgently required to overcome this historical bias and create a vibrant pool of skilled young faculty, and around them their Ph.D. students and research groups.","Given the role of lattice gauge theory in confronting hadronic experiment, this field would particularly benefit from a program of joint Lab-University appointments."],["Conclusions and Recommendations","For scientific activities that consume a large amount of computing time and map well to emerging HPC architectures, there is a wonderful opportunity to apply enormous amounts of computation to solve incredibly complex problems.","However, common cross-cutting issues have been raised across many different scientific topics.","These issues are largely centered on the critical dependency between modern science and computation and software.","Computer architectures have necessarily become increasingly complex, diverse, and challenging to program.","In order to turn the potential of DOE computing into scientific deliverables, it is critical to develop skills in the community and to fund the required software and algorithmic development.","Support for widely used scientific software packages is required to enable the most effective return.","Further, accelerated architectures are not fully general purpose and present challenges to algorithms that are strongly serial without any intrinsic vectorization or parallelism potential.","Even for those algorithms for which it is possible in principle to use this hardware, the investment to develop software that runs efficiently across many different platforms and programming environments requires a real investment by the domain scientists.","This investment of effort may not always be a good value proposition.","For these two reasons, there is a continuing need for a mix of computing with both accelerated and general purpose architectures.","This mixture should be sized appropriately to accelerate large scale computation where possible but also to effectively support science whose software cannot exploit acceleration.","This may occur if it is not possible to port the underlying algorithms to compute accelerators or if a cost-benefit analysis identifies such a port as not being a cost-effective proposition.","The latter case likely comprises a “tail” of many smaller computing tasks that run diverse software with limited development effort available.","Our recommendations on computing hardware are as follows: Faster computer hardware: For lattice gauge theory applications, there is a need for HPC resources ten times or more faster than planned Exascale computing systems.","Support of right-sized CPU clusters: There should be a right-sized (as large as required, as small as possible) provisioning of general purpose computational cores with high performance memory for important algorithms that do not easily map to computational accelerators.","National Laboratory institutional (non-HPC) clusters contribute to this role.","NERSC is a welcome example of providing both CPU and GPU/accelerator resources to handle a wider variety of problems relevant to HEP.","These should accommodate a diverse range of requirements for software, large memory, and long running single node jobs.","Universal programming interface: The proliferation of programming interfaces is a barrier to portability and return on investment in software.","Simplification and consolidation of the interfaces will improve scientific output.","Software portability: The difficulty of programming highly accelerated hardware with performance portability is significant.","Appropriate support for software development and maintenance is essential to ensure success of much of the science discussed in this report.","Automation of memory hierarchy: Hierarchical memory is a significant burden on scientific programmer productivity.","Virtual memory paging systems can alleviate this and can be made efficient by using larger page sizes, if necessary.","This may need substantial investment by computer vendors with robust encouragement from the Department of Energy.","Early access to new computing hardware: A key element of managing the science program is the early engagement with DOE HPC laboratory sites during the development, and years prior to installation, of major new facilities.","The lead time for porting to new architectures lies in the region of multiple years, and early engagement with emerging architectures is required to ensure timely scientific exploitation.","The scientific software ecosystem should be nurtured to enable the proposed Snowmass science.","There are several steps that could be proactively taken to ensure the maximum scientific return is delivered.","Hardware accelerator-friendly, portable common tools: Specific projects to adapt common software packages to run efficiently on GPUs, including those at new HPCs, have shown initial promise with more than order of magnitude speed improvements.","These projects should be continued with sufficient effort to deliver complete products and to incorporate portability solutions to support different coprocessor devices and architectures.","Adoption of this new software will additionally require expertise devoted to integration and usage in experimental software frameworks, analysis tools, etc., which should be supported as discussed below.","The exploration of new machine learning-based methods, as well as technical improvements to existing (CPU-based) software, should also continue.","Best practices for common software: Whenever possible, interoperability and common data structures should be encouraged.","Existing standard libraries and underlying software technologies should be used when available and feasible.","Interoperability, portability, and common, shared software are fundamental pillars of international scientific collaboration and will become increasingly important in the future.","Lab-supported software development: Just as the large experiments require talented permanent staff at the Labs to engineer experiments, theoretical calculation and simulation in HEP experiment, in high energy theory, in cosmology, and in large particle accelerator building have large communities often dependent on sophisticated long-term software systems.","Career paths must be created to retain some of the most talented experts in software and algorithms.","A larger permanent laboratory staff of software development experts could help address HPC architectures as they increase in importance within the Computational Frontier.","Organization for long-term common software maintenance: Despite the numerous successes and broad adoption of common software tools across all topical areas in this report, the future of these projects is in jeopardy because of an extreme lack of funding.","One priority should be the development, maintenance, and support for common software tools in the areas of accelerator modeling, detector simulation, and physics generators.","Because this kind of long-term effort to sustain cross-cutting software does not fit easily into existing modes of funding, new processes and organizations should be established in cooperation with funding agencies.","Continuous collaboration with ASCR: A lot of progress has been made in high energy physics applications through collaborations with programs supported by ASCR under the SciDAC and ECP projects.","We believe that such collaborations should be encouraged and continuously supported for future computational modeling in high energy physics.","The community found the application development focus of the ECP project particularly effective.","Joint lab-university tenure track appointments: We believe the DOE should seek to foster the continued development of intellectual leaders in theoretical calculation and simulation.","The health of the field requires a similar cohort of individuals at the best universities, reflecting the intellectual vigor and potential of this area to contribute to DOE scientific goals.","The creation of such positions can be stimulated by DOE-funded joint, five-year, tenure-track appointments.","A good example might be the Jefferson Lab University Relations program of joint and bridged faculty appointments in nuclear physics.","Training: Accessible training in both novel architectures and AI/ML with a low barrier to access for graduate students, postdoctoral researchers, and domain scientists is critical to ensuring the skills base exists for productive science in theoretical calculation and simulation.","The use of cross-platform performance portable APIs should be encouraged to maximize return on investment in software.","Hackathons and code examples are particularly useful."],["Acknowledgments","P. Boyle has been funded by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under the Contract No.","DE-SC-0012704 (BNL).","K. Pedro is supported by the Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No.","DE-AC02-07CH11359 with the U.S. Department of Energy.","J. Qiang is supported by the U.S. Department of Energy under Contract No.","DE-AC02-05CH11231 (LBNL)."]],"string":"[\n [\n \"CompF2: Theoretical Calculations and Simulation Topical Group Report\"\n ],\n [\n \"Abstract This report summarizes the work of the Computational Frontier topical group on theoretical calculations and simulation for Snowmass 2021.\",\n \"We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics).\",\n \"The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements.\",\n \"Calculations and simulations will need to keep up with these increased requirements.\",\n \"The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs.\",\n \"These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns.\"\n ],\n [\n \"Executive Summary\",\n \"We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics).\",\n \"The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements.\",\n \"Calculations and simulations will need to keep up with these increased requirements.\",\n \"The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs.\",\n \"These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns.\",\n \"The adaptation of existing CPU-based software to specialized hardware can provide solutions to the experiment-driven challenges, but this requires additional expertise and substantial effort.\",\n \"Artificial intelligence (AI) and machine learning (ML) can also provide solutions to some problems that are more immediately amenable to acceleration on coprocessors.\",\n \"US HEP should, to the extent possible, guide continued development of computing hardware to be maximally useful for solving HEP computing challenges.\",\n \"Important developments include: faster hardware, availability of general-purpose CPUs, universal programming interfaces and portability, automated memory hierarchy, and early access to new hardware.\",\n \"Long-term support of careers for software developers and maintenance for common software tools is essential to the future of HEP calculations and simulations.\",\n \"Efficient utilization of the aforementioned new compute accelerators will require substantial efforts to port existing software to these new paradigms and/or to adopt new AI/ML techniques.\",\n \"Needs in this area include new organizations to ensure software sustainability, permanent software development staff at national labs, joint lab-university appointments for computationally-focused scientists, and increased training across the field.\",\n \"Computing in HEP has long struggled with issues of diversity and inclusion.\",\n \"As the intersection of two highly specialized topics, there is an especially high level of knowledge and skills required to contribute successfully.\",\n \"To get to this level, individuals usually must seek expertise outside of traditional HEP education and training, which typically does not cover computing topics in detail.\",\n \"This poses a discouraging burden, especially for those without connections to such expertise or without the resources to take on such additional work.\",\n \"We hope that substantially expanded support for software and computing training and for software development as a viable career in the HEP ecosystem will alleviate this burden.\",\n \"A larger, healthier HEP computing community can be more welcoming and accessible to participants from all backgrounds.\"\n ],\n [\n \"Introduction\",\n \"The Snowmass Theoretical Calculation and Simulation topical working group is cross-cutting and underpins multiple scientific domains.\",\n \"The computing topics discussed overlap with almost all aspects of Snowmass science and almost every Frontier.\",\n \"Increasingly, computation is central to most theoretical and experimental scientific endeavors.\",\n \"The great challenges of science, and in particular physical science supported by the Department of Energy and the National Science Foundation, have a long history of pushing the boundaries of computational science and computing technology.\",\n \"The current Snowmass study indicates this will continue on many fronts.\",\n \"The Theoretical Calculation and Simulation group spans a number of sub-topical themes representing the American Physical Society (APS) Division of Particles and Fields (DPF) Snowmass scientific interests: Cosmic Calculations, Particle Accelerator Modeling, Detector Simulation, Physics (Event) Generators, Perturbative Calculations, and Lattice QCD (quantum chromodynamics).\",\n \"This report first introduces the common challenges and the modern computing landscape in Section , discusses each subtopic in more detail in Sections –, and concludes with recommendations in Section .\"\n ],\n [\n \"Cross-cutting computational challenges\",\n \"The computational landscape of scientific algorithms can broadly be divided into high-performance computing (HPC) and high-throughput computing (HTC) problems.\",\n \"HPC requires the coordinated and cooperative processing of a single large computational problem across multiple nodes with data being exchanged as part of the calculation, often using high-performance interconnects.\",\n \"In contrast, HTC processes a large volume of completely independent computational work units.\",\n \"Depending on the problem type, both HPC and HTC may have a high volume of relatively simple bulk floating point arithmetic that can be accelerated using highly parallel special-purpose computing hardware.\",\n \"Some calculation and simulation tasks, and particularly those that process large data arrays or are machine learning-based, may obtain substantial speed increases from these hardware accelerations.\",\n \"In other cases, algorithms that involve substantial serial performance or many data dependent logical flow decisions will benefit most from hardware that is suited to efficient (and power efficient) serial execution.\",\n \"Detector simulation, event generation, and continuum theoretical calculations are typically HTC tasks, while accelerator modeling, cosmic calculations, and lattice gauge theory are more often HPC tasks.\",\n \"However, exceptions exist, and the use of different algorithms or programming paradigms can sometimes convert HTC to HPC.\",\n \"Memory bandwidth has presented a fundamental rate limit to many calculations.\",\n \"It is common to include computational accelerators that can provide substantially greater floating point throughput and memory bandwidth as coprocessor cards.\",\n \"The most common of these are graphics processor units (GPUs), but Field Programmable Gate Arrays (FPGAs) and other technology options are also common.\",\n \"These often provide distinctly addressable high performance memory to help deliver good performance.\",\n \"Coprocessors are often highly parallel and have limited or fixed function operations, so they are only usable with certain classes of algorithms.\",\n \"Machine learning (ML) algorithms are especially suited to be accelerated on these devices, as described in Section REF .\",\n \"On the other hand, while event reconstruction is easy to parallelize by processing independent events independently on independent serial processor cores of any form, the standard GPU thread model serializes execution when different threads in a group make different data-dependent logical flow decisions (for example, making different event-specific patterns of subroutine calls).\",\n \"From an electronics perspective, it is easier to provide multiple types of memory (small and fast, large and slow) organized in distinct circuit boards to support a high floating point density.\",\n \"The memory technologies are currently split into on-chip caches, in-package DRAM or “high bandwidth memory”, off-package DRAM, and increasingly non-volatile memory.\",\n \"These are listed in order of increasing capacity and decreasing memory bandwidth.\",\n \"The appearance of 2.5D and 3D integration technologies has been transformational and this system-level innovation will likely continue to return greater memory bandwidth over time, giving continued practical gains in high performance computation.\",\n \"Such hierarchical memory systems will likely remain in the future, as they are dictated by physics constraints.\",\n \"The degree of hierarchy is increasing and may grow in the future, independent of the specific processor architecture.\",\n \"In this sense, the trends identified are likely reasonable over a ten-year timeframe.\",\n \"Scientific productivity will be enhanced if data placement is handled automatically and moved infrequently: for example, this should be automatically and efficiently handled by the Unix virtual memory system.\",\n \"This would avoid a requirement that every single scientific programmer laboriously manage the placement and movement of data to specific locations, such as caches.\",\n \"The managing of distinct pointers to distinct “host” and “device” memory spaces is both a recurrent source of programmer error and requires considerable care to ensure that the allocated GPU memory does not exceed the limited capacity, while automatic paging avoids this requirement.\",\n \"Section REF discusses more considerations relevant to scientific programming with modern heterogeneous computing.\",\n \"Interconnects, used for both message passing and filesystem access, are becoming an increasing bottleneck since accelerated computing nodes are becoming rapidly more powerful while interconnect performance gains have not always matched pace.\",\n \"Technologies such as silicon photonics, allowing the integration of optical transceivers in standard silicon chips, are a promising basic technology that could help address interconnect bottlenecks.\",\n \"Similarly, high speed links between GPUs within a computing node are a key enabling technology for many problems.\",\n \"File systems are a key element of any scientific workflow.\",\n \"Data access should be sufficiently fast not to present a bottleneck.\",\n \"Modern computing systems use parallel file systems, such as Lustre, giving up to terabyte/s aggregate access rates to disk.\",\n \"The use of non-volatile memory elements can both cache bursts of data and accelerate metadata access.\",\n \"Data migration to cold or longer term storage can be carried out through either user input or hierarchical storage managers.\",\n \"It is important that these keep pace with growing data volumes and increasing computation speed.\"\n ],\n [\n \"Computing hardware acceleration\",\n \"Fixed-function acceleration is possible for problems involving dense matrix multiplication, with tensor cores in GPUs and other hardware specifically designed to handle large matrices.\",\n \"These operations are commonly performed using 64-bit and 32-bit floating point arithmetic, or even on 16-bit floating point arithmetic.\",\n \"Mixed precision functional units are common.\",\n \"Fixed-function dense matrix hardware such as tensor cores do not necessarily accelerate all algorithms.\",\n \"ML is one common and growing source of matrix multiplication operations that can be accelerated in this way.\",\n \"ML can broadly be decomposed into “training” and “inference” phases.\",\n \"Training often involves scientist supervision for parameter tuning, with time to results being critical.\",\n \"Distributed ML training parallelizes the training process.\",\n \"The use of small “mini-batches” in stochastic gradient descent leaves the problem very sensitive to Ahmdahl's law slowdown.\",\n \"In the massively distributed limit [1], the summation across an interconnect of a gradient vector calculated across many randomly selected training samples in a minibatch is a significant performance bottleneck [2].\",\n \"Inference with a previously trained network is typically performed on many independent events of data and is in many cases a trivially parallel (HTC) problem.\",\n \"However, when inference is used in either hard or soft real-time situations (as occurs in HEP experiments), the latency requirements impose challenging performance issues when traditional CPUs are used.\",\n \"Accelerators from Graphcore and Cerebras are good examples of ML-optimized accelerators in addition to more standard GPUs by Nvidia, AMD, and Intel.\",\n \"Reconfigurable computing hardware such as FPGAs or spatial accelerators are powerful options for certain algorithms.\",\n \"They may be accessed via a dataflow model where a hardware circuit is configured in a non-von Neumann approach to stream data from memory, calculate, and store results in memory.\",\n \"They may be particularly effective in eliminating software latency in real-time inference environments [3], [4].\",\n \"Algorithms that are I/O-bound or executed in situations requiring reduced power consumption may also be more efficiently accelerated on FPGAs.\"\n ],\n [\n \"Software challenges\",\n \"The computing landscape at this time displays an exciting proliferation of competing computer architectures.\",\n \"It opens vibrant and healthy opportunities for innovation and competition and for the introduction of new ideas.\",\n \"However, it also risks introducing special-purpose or less-general features, compared to previous systems.\",\n \"The memory spaces are often fragmented with different pointer types and data locations.\",\n \"There are multiple instruction-set architectures in use, each with a different programming interface or model.\",\n \"A single program frequently has different segments that target different types of instruction sets within a single computer node.\",\n \"Indeed, some of the computing models available (such as FPGAs or spatial architectures) are not von Neumann computers.\",\n \"Others have dedicated fixed-function acceleration of matrix-matrix multiplication for machine learning.\",\n \"This diversity presents a significant challenge to the scientific programmer with either legacy code or the need to use more than one system.\",\n \"There are a number of different programming models one must consider in the present computing landscape.\",\n \"The following can all in principle be combined with MPI message passing.\",\n \"Present Exascale and Pre-exascale systems support: traditional CPU core, SIMD floating point instructions, and OpenMP threading, CUDA-based GPU programming, HIP-based GPU programming, SYCL-based GPU and FPGA programming, and OpenMP 5.0 offload GPU programming.\",\n \"Future systems may support standards-based C++ parallel STL offload programming; however, multi-system compiler support is currently absent.\",\n \"The challenges of writing high-performance and portable code are threefold: syntactical differences, semantic differences, and data placement.\",\n \"The proliferation of models represents a substantial cost and effort overhead for science exploitation, since multiple implementations of software must be developed and maintained.\",\n \"At this time, due to the diversity of models, abstraction and support for multiple programming models is likely the safest option for portable efficiency on near-term computers.\",\n \"For projects that wish to reuse interfaces to acceleration, several good portability options exist: Alpaka and Department of Energy ASCR projects Kokkos and RAJA.\",\n \"Standards-based programming, such as SyCL, OpenMP, or C++ parallel STL, is to be encouraged.\",\n \"It is important to recognize that not all algorithms are amenable to map to accelerated hardware.\",\n \"This must be considered in the HEP computing landscape to avoid leaving important needs unaddressed.\",\n \"There is much to be said for an element of the computing roadmap using general-purpose processor cores combined with advanced memory technologies such as high bandwidth or hierarchical memory.\",\n \"In this way, the long tail of problems with computational challenges that would be either impossible or prohibitively expensive to port to accelerated hardware may continue to be addressed.\"\n ],\n [\n \"Cosmic Calculations\",\n \"In the next decade, a number of powerful observational facilities will start to operate.\",\n \"These next-generation cosmological observations will provide great opportunities to study the early history of universe, the mysteries of dark matter and dark energy, the fundamental particle physics, and possible modifications of Einstein gravity.\",\n \"In order to successfully achieve the observational goals, a state-of-the-art simulation and modeling program is needed [5].\",\n \"Cosmological simulations provide an invaluable tool to optimize the observation design, to interpret observation data, and to help unearth the underlying physics.\",\n \"Next-generation cosmological simulations will increasingly focus on physically rich high-fidelity simulations that can directly connect with observational outputs from multiple different surveys.\",\n \"Several types of cosmological simulations have been employed in the cosmology study.\",\n \"The first-principles N-body simulations that have no free parameters can accurately describe dark matter fluctuations from the largest observable scales down to scales deep into the nonlinear regime.\",\n \"However, the N-body approach does not account for the physics of the baryonic sector.\",\n \"Additional models such as the halo occupation distribution, sub-halo abundance matching, or other schemes are added on top of a simulation to reconstruct galaxies.\",\n \"Hydrodynamical simulations provide a reasonably accurate description of the distribution of baryons, quantify the effects of baryons on various probes of large-scale structure, and provide useful results for the distribution and properties of galaxies and clusters.\",\n \"However, the robustness of the hydrodynamical simulation results depends on the choices of subgrid models and the nature of the cosmic probe in the study.\",\n \"In addition to the aforementioned simulations, other simulations such as beyond $\\\\Lambda $ CDM simulations that involve the solution of nonlinear Poisson equation in the modified gravity model and radiative transfer simulations that model reionization process have also been used in cosmological studies.\",\n \"Different survey experiments include different observables in the study.\",\n \"Signals from galaxy clustering and lensing are key observables for photometric galaxy surveys.\",\n \"Spectroscopic instruments are used to measure redshifts, radial velocities, gas dynamics and chemical compositions of galaxies.\",\n \"Signals from Baryonic Acoustic Oscillation (BAO) and redshift space distortions (RSD) measurements are used to extract useful cosmological information.\",\n \"Signals from the cosmic microwave background lensing measure the integrated mass between the last scattering surface and us.\",\n \"The Lyman-alpha forest signal in the spectra of distant quasars is used to constrain thermal properties of the intergalactic medium (IGM).\",\n \"The thermal and kinematic Sunyaev Zeldovich (tSZ/kSZ) signals are used to infer the distribution of gas in the universe.\",\n \"The emission from dusty star has been used to delens the cosmic microwave background (CMB).\",\n \"X-ray observations are used to calibrate mass estimates.\",\n \"High-redshift from interferometers can provide useful information about the thermal and ionization state of the intergalactic medium and can be used to learn about dark matter.\",\n \"The diverse observational probes present significant challenges in cosmological simulations.\",\n \"One challenge is the computational cost to include multiple probes with high precision.\",\n \"These simulations need cover a large volume but with sufficient fine resolution for a large number of realizations.\",\n \"In addition, computationally demanding hydrodynamical simulations are required to simulate some observables such as the thermal and kinetic Sunyaev-Zel'dovich effects, tSZ and kSZ, to infer the distribution of gas in the universe, or Lyman-$\\\\alpha $ , to constrain thermal properties of the intergalactic medium.\",\n \"In some cases, the modeling of certain observables involves a very large dynamical range.\",\n \"Semi-numerical simulations are used when the hydrodynamical simulations cannot cover large enough volume while reaching small enough halos.\",\n \"Calibrations and benchmarks between different approaches are important to interpret new observation data.\",\n \"Another challenge for cosmological simulations is to reproduce observations, for example, correlations between multi-wavelength observables, in consistent galaxy formation models.\",\n \"It is necessary to develop simulation models that can be applied to gravity-only simulations and account for correlations between neutral and ionized gas, stars, and dust in galaxies and galaxy clusters.\",\n \"Also, it is not trivial to simulate massive neutrinos since they make up a non-negligible fraction of the total energy but can be decoupled when relativistic and have a free streaming scale.\",\n \"In the N-body simulations, thermal velocity distributions need to be accounted for in the structure formation calculation on smaller scales.\",\n \"Covering both the large volume required by the weak lensing and maintaining the accuracy at small scales required by strong lensing presents another computational challenge in ray tracing.\",\n \"In the modeling for future analyses of small scale measurements, baryonic feedback effects that can change the local matter density must be included in the cosmological simulations.\",\n \"Besides advancing in terms of increased resolution, larger volumes and better treatments of known physics, additional new physics models need to be included as the reach of surveys expands.\",\n \"These include models of modified gravity with nonlinear Poisson equation for dark energy studies and a variety of dark matter models (warm dark matter, interacting dark matter, self-interacting dark matter, ultralight dark matter, ultraheavy dark matter, multiple component dark matter [6]).\",\n \"Dedicated cosmological simulations are needed for those new models.\",\n \"Next-generation cosmological surveys demand better understanding, mitigation, and control of systematic uncertainties.\",\n \"To attain these goals, it is important to create realistic “virtual universes” from cosmological simulations.\",\n \"Such simulations can be used to solve the inverse problem to infer the physical parameters of different models from the sky survey observation data.\",\n \"However, it is not computationally practical to model all relevant processes from a first-principles approach given the vast dynamic range in cosmology.\",\n \"Fast surrogate models can be used as effective emulators to speed up the solution of this inverse problem.\",\n \"The predictions for cosmological surveys from emulators have been widely used in cosmological studies.\",\n \"It is desirable that these emulators can connect directly to the survey observables to extract cosmological parameters.\",\n \"With the advance of multi-fidelity hydrodynamical simulations, it is also desirable to use emulators to optimize subgrid model parameters and to gain better understanding of the interplay between the subgrid model and cosmology parameters.\",\n \"Cosmological simulations are computationally intensive and demand the use of state-of-the-art computer hardware.\",\n \"With the arrival of exascale supercomputers, cosmological simulations will need to use computer accelerators efficiently and memory access patterns effectively, as well as performance portable programming models and scalable algorithms.\",\n \"Progress has been made to address some of these challenges, for example under the Exascale Computing Project (ECP) led by the Office of Advanced Scientific Computing Research (ASCR) in collaboration with other DOE science program offices [7], [8].\",\n \"Continuous development and support are extremely important to enable full use of these hardware systems.\",\n \"The successful use of the exascale supercomputers will enable gravity-only simulations with unprecedented volume coverage and resolution and hydrodynamics simulations with exceptionally detailed baryonic physics modeling for the cosmological surveys.\",\n \"Scalable analysis approaches are as important as the development of simulation codes since handling and processing of very large data sets generated by the simulation codes require large computing resources in their own right.\",\n \"Besides the development of the simulation codes, the development of analysis tools faces the same challenges as the simulation codes with respect to the efficient usage of the available computer hardware.\",\n \"Co-development of simulation codes and analysis tools is needed for a successful cosmological simulation program.\",\n \"Cosmological simulations are critical for the success of next-generation cosmological observations.\",\n \"The accuracy for these simulations needs to be carefully checked through verification by benchmark among codes and validation with direct comparison against observational data.\",\n \"This requires a concerted effort between different code and analysis development teams and cosmic observation teams.\",\n \"The detailed connection between the simulations and the survey observations has to be established for a successful validation program.\",\n \"In summary, the cosmological simulation community recommended the following advances in order to meet the above challenges and needs [5].\",\n \"Development of scientifically rich and broadly-scoped simulations, which capture the relevant physics and correlations between probes Accurate translation of simulation results into realistic image or spectral data to be directly compared with observations Improved emulators and/or data-driven methods serving as surrogates for expensive simulations, constructed from a finite set of full-physics simulations Detailed and transparent verification and validation programs for both simulations and analysis tools Another area of cosmic calculations is the study of numerical relativity for next-generation gravitational-wave probes of fundamental physics [9].\",\n \"The next-generation gravitational-wave detectors will provide great opportunities to study the nuclear physics of dense matter from the signals of neutron-star or black hole mergers, the nonlinear dynamics of warped spacetime, and the physics beyond general relativity from the signals of binary black holes.\",\n \"To attain these goals requires a new generation of numerical-relativity codes with dramatically improved accuracy that can make effective use of the exascale supercomputers.\",\n \"The higher computational accuracy also results in a significant increase of needs for computational resources in comparison to the present typical weeks to months of runtime on tens to thousands of computer cores.\",\n \"This is due to the higher simulation resolution and the longer simulation to cover the detectors' sensitive frequency spaces.\",\n \"In addition, many simulations will be necessary to span the parameter space of potential signals from black holes and neutron stars.\",\n \"A further topic in cosmic calculations is the modeling of neutrino transport in neutrino-driven core-collapse supernovae [10].\",\n \"A kinetic description of these neutrinos is needed in this model.\",\n \"The solution for the neutrino transport equation is computationally intensive, involving high-dimensional linear algebra.\",\n \"These simulations have been ported to GPU-based supercomputers and will require more computing resources to improve the resolution of the calculations and to explore different physical scenarios.\"\n ],\n [\n \"Particle Accelerator Modeling\",\n \"Particle accelerators are regarded as one of the most important inventions of the $20^{\\\\mathrm {th}}$ century.\",\n \"They provide an invaluable tool in high energy physics study.\",\n \"However, particle accelerators are large and complicated scientific devices that can be thousands of meters long (e.g., the LHC's nearly 27 kilometer circumference) with hundreds of thousands of elements.\",\n \"Their design, construction, and operation are both sophisticated and expensive, and depend critically on computer modeling In the particle accelerator modeling area, there are 26 related letters of intent submitted to this Snowmass process.\",\n \"These letters of intent were summarized in a community-wide white paper that addresses the challenges and needs in modeling particle accelerators [11].\",\n \"Next-generation high energy particle accelerators and colliders will be more complicated and expensive than the existing accelerators.\",\n \"These accelerators may employ both conventional RF cavity and high gradient wakefields from either a plasma channel or a structure waveguide to accelerate charged particle beams to high energy and use strong superconducting magnets and sophisticated control systems to confine these beams.\",\n \"Some potential next-generation particle accelerators include very high energy proton-proton colliders, multi-TeV electron-positron colliders, muon colliders, other very high energy machines such as gamma-gamma and high energy electron-ion colliders, and high intensity accelerators for neutrino physics study.\",\n \"To meet the needs of these next-generation accelerators, it is important for particle accelerator modeling tools to include all types of accelerating structures (e.g., RF-based, plasma-based, structured-based wakefield, plasmonic) and different accelerator components (e.g.\",\n \"superconducting magnets, structured plasmas), and to target the accelerator and beam physics thrust grand challenges on intensity, quality, control, safety, and prediction.\",\n \"For RF-based accelerators, this requires modeling of collective effects with high fidelity over long time scales, improving computational speed to allow for statistical ensembles and design optimization including collective effects, and improved integration with realistic magnet and RF modeling.\",\n \"For plasma-based accelerators, this requires supporting the development of modeling tools and methods that will enable start-to-end simulations that predict the full six-dimension (+ spin) evolution of beams in a plasma-based linear collider, from their creation to their final focusing at the interaction point, and include all the intermediate phases of acceleration, transport, manipulations, and collisions.\",\n \"For structure-based accelerators, this requires the development of efficient and accurate algorithms capable of modeling the beam interaction with its wakefield over long interaction lengths in structures with arbitrary geometries and constitutive parameters.\",\n \"For plasmonic accelerators, this requires the development of new approaches such as a quantum-kinetic method that incorporates the effects of localized petavolts per meter plasmonic fields on the underlying quantum mechanical processes such as the ionic lattice and the energy band structure and to incorporate the multi-physics nature of the unprecedented strongly electrostatic plasmonic modes.\",\n \"For the structured plasma accelerator component, this requires the development and integration of fluid and kinetic codes to meet the modeling demands for a new class of structured plasma devices that couple macroscopic plasma properties with strict requirements on kinetic interactions.\",\n \"The quest for higher accelerating gradients in next-generation particle accelerators puts increasing demands on better understanding the materials of the accelerating structure and relevant phenomena such as RF breakdown.\",\n \"This needs the development of automated scale-bridging methods that can autonomously parameterize higher-level models from large numbers of lower-scale first-principles calculations to enable predictive materials studies over a broad range of materials and conditions.\",\n \"Strong magnets will be highly demanded in the next-generation particle accelerators to reduce the machine size and cost for transverse focusing.\",\n \"Some advanced modeling tools are currently utilized in the US Magnet Development Program.\",\n \"Novel modeling tools with mixed finite element method are needed to improve design time, cost, and performance of future superconducting accelerator magnets.\",\n \"Next-generation particle accelerators will occupy a large land space and consist of many different accelerator segments starting from charged particle sources and ending with final interaction colliders or targets.\",\n \"Different segments of the accelerator are coupled with each other through the charged particle beam transporting through the entire accelerator system.\",\n \"The final particle beam quality does not depends on the performance of a few elements, but on the performance of the entire accelerator system.\",\n \"Without the modeling of the whole accelerator, no project can proceed to the building stage.\",\n \"In order to evaluate the particle accelerator performance and to attain a global optimal design, end-to-end simulations are needed to establish a virtual particle accelerator.\",\n \"The end-to-end simulation denotes a simulation starting with the source that generates charged particles and ending with either the final target or the interaction region.\",\n \"These simulations should include all pertinent physical effects from both first-principles models for high precision accelerator design and fast machine learning-based surrogate models for online particle accelerator tuning.\",\n \"Such a virtual particle accelerator can be used as an emulator of the real physical accelerator in the accelerator operation.\",\n \"One challenge in the end-to-end simulation is to exchange particle distribution data from different components of the particle accelerator.\",\n \"To solve this problem requires supporting efforts to establish standards that would ease the sharing of information and data across codes as well as standards for interfacing codes [12].\",\n \"Another challenge associated with the high fidelity end-to-end accelerator simulation is that it is extremely time consuming (especially with the inclusion of a variety of collective effects), varying from days to weeks and months.\",\n \"The cutting-edge and emerging computing techniques including advanced algorithms, AI/ML methods, and quantum computing may substantially reduce the computational time of accelerator simulation and enable a fast real-time optimization with end-to-end simulations.\",\n \"Advanced algorithms play an important role in the accelerator modeling.\",\n \"High-fidelity self-consistent simulations cannot be done within a reasonable return time without using these algorithms [13], [14].\",\n \"It has also been shown that the advanced algorithms can lead to several orders of magnitude improvement on the computational speed [15].\",\n \"Research and development in this area is needed to explore new algorithms that exhibit better properties, to refine the understanding of the properties and bottlenecks of existing algorithms, and to remove the bottlenecks on cutting-edge heterogeneous computing hardware (e.g., GPU, FPGA), and improve the speed and accuracy of accelerator modeling.\",\n \"Differentiable simulations show great potential to reduce the number of simulations needed for optimization and design of particle accelerators.\",\n \"Using AI/ML surrogate models of the physics simulation can save orders of magnitude computing time [16].\",\n \"A single differentiable simulation can generate the sensitivity information of the final beam quality with respect to thousands of machine parameters.\",\n \"Support is needed for the development of AI/ML modeling techniques and their integration into accelerator modeling and control systems, with an emphasis on fast executing (up to real-time) and differentiable models, continual and adaptive learning for time-varying systems, uncertainty quantification to assess confidence of model predictions, and physics-informed methods to enable broader model generalization to new application conditions.\",\n \"Quantum computing is another area of emerging technology that has the potential to transform some areas particle accelerator simulations by providing exponential improvement of computational efficiency.\",\n \"Support is needed for quantum computing algorithm and code development for accelerator modeling, feasibility study on quantum computing implementation in accelerator modeling.\",\n \"Quantum computing education in the accelerator community is also needed.\",\n \"Particle accelerator modeling is computationally intensive and has utilized supercomputers for many years.\",\n \"It has used both high-performance computing to model collective effects and high-throughput computing to model single particle dynamics.\",\n \"With the arrival of exascale supercomputers and new special purpose computers with hardware accelerators (e.g.\",\n \"GPUs), it is a challenge to achieve the supercomputer peak performance for modeling collective effects in particle accelerators due to deep levels of independent memory caches, on top of global and neighboring communications that are used to exchange information across multiple computer nodes during the simulation.\",\n \"As part of this, the workload needs to be uniformly distributed among many compute units in order to achieve a good parallel efficiency.\",\n \"The current and next-generation supercomputers extensively use the computational accelerators such as GPUs and FPGAs to improve the computational speed of applications.\",\n \"Applications related to single particle dynamics can be effectively simulated on GPUs.\",\n \"It is a challenge for applications including multiple physics models and collective effects to use GPUs effectively.\",\n \"Support is needed to adopt portable, state-of-the-art programming paradigms and frameworks that support both multi-level parallelization and effective dynamic load balancing over all levels of compute parallelism, so that simulations run efficiently on the latest computer architectures with scalable I/O, post-processing and in situ data analysis solutions to support accelerator machine design and operation.\",\n \"Particle accelerator modeling codes were developed with the support of a variety of projects to simulate RF structures, magnets, beam dynamics inside the accelerator structure, laser and beam interaction with plasma, and beam beam and beam wave interactions.\",\n \"These simulation tools are critical for the design and operation of next-generation high energy particle accelerators.\",\n \"Sustainability, reliability, user support and training of these complicated codes are potential challenges in the accelerator community.\",\n \"Further support is needed to maintain these codes with documentation, testing, and benchmark examples.\",\n \"Training classes are needed at the U.S.\",\n \"Particle Accelerator School or other accelerator schools or workshops to help improve the usage of these codes.\",\n \"Another challenge for these accelerator simulation codes is validation with experimental measurements.\",\n \"This is critical for establishing the confidence in the use of these simulation tools for next-generation particle accelerator design applications.\",\n \"The direct comparison between code predictions and accelerator performance and output may be limited by a number of factors, such as not fully characterized machine settings, unknown initial phase space distribution, insufficient range or resolution of diagnostics, and incomplete or missing physics in simulations.\",\n \"Support is needed to connect simulations with direct experimental diagnostics and to carry out detailed benchmark studies with experimental measurements.\",\n \"The particle accelerator modeling codes include a variety of physics models for different accelerator components and involve different research groups.\",\n \"In order to attain the end-to-end simulation for next-generation particle accelerators, different component models need to be integrated into a workflow and ported onto the latest exascale type computer hardware with GPUs.\",\n \"New physical models and capabilities will be added in some of these codes to account for new accelerating and focusing methods.\",\n \"It is desirable to organize the beam and accelerator modeling tools and community through the development of ecosystems of codes, modular libraries, and frameworks that are interoperable via open community data standards for both loosely integrated and tightly integrated workflows.\",\n \"It is desirable to establish open access data repositories for reuse and community-wide AI/ML surrogate model training.\",\n \"It is also desirable to have an organized particle accelerator modeling community (e.g.\",\n \"in the form of centers and consortia) including both academic institutes and industry.\",\n \"This will improve the efficiency of accelerator modeling for next-generation high energy accelerator applications.\",\n \"In summary, to meet the above challenges and needs, the following high-level recommendations are provided from the particle accelerator modeling community [11]: Develop a comprehensive portfolio of particle accelerator and beam physics modeling tools in support of achieving Accelerator and Beam Physics Thrust Grand Challenges on intensity, quality, control, and prediction.\",\n \"Develop software infrastructure to enable end-to-end virtual accelerator modeling and corresponding virtual twins of particle accelerators.\",\n \"Develop advanced algorithms and methods including AI/ML modalities and quantum computing technologies.\",\n \"Develop efficient and scalable software frameworks and associated tools to effectively leverage next-generation high-performance and high-throughput computing hardware.\",\n \"Develop sustainable and reliable code maintenance practices, community benchmarking capabilities, and training opportunities to foster the cooperative application of accelerator software.\",\n \"Organize the beam and accelerator modeling community in open efforts to (a) foster the development of ecosystems of interoperable codes and tools, libraries, and frameworks, based on API and data standards, (b) establish open access and data workflow repositories, and (c) have an organized governance structure.\"\n ],\n [\n \"Detector Simulation\",\n \"Accurate detector simulation is crucial for tasks ranging from detector design to data analysis.\",\n \"Geant4 [17], [18], [19] is the primary tool used for detector simulation throughout HEP [20].\",\n \"This C++-based software package was originally released in 1998 as a complete rewrite of the earlier FORTRAN-based Geant3 [21].\",\n \"The Geant4 collaboration and the HEP community engage in continuous research and development to improve the computational performance, refine the models of physical interactions, and incorporate new technical features and models.\",\n \"Geant4 is the center of an ecosystem that includes numerous software packages serving different needs: Geometry/material modeling and navigation: VecGeom, CADMesh, DD4hep, GDML; Physics models: Noble Element Simulation Technique (NEST) for excitation, ionization, etc.\",\n \"; G4CMP for phonons and other condensed matter phenomena; Opticks for optical photon simulation; Other simulation packages (including radiation modeling): MARS, FLUKA, MCNP, SRIM; R&D projects: G4HepEM, AdePT, Celeritas; Fast simulation engines: DELPHES, TrackToy, bespoke experiment-specific solutions, emerging ML approaches.\",\n \"Some of these packages are developed in very close coordination with Geant4, while others are more independent, using Geant4 as a reference or being used as cross-checks for Geant4 results.\",\n \"In addition, Geant4 interfaces with event generators (see Section ) such as pythia 8 for decays of long-lived, metastable, or late-forming particles, and with ROOT and CLHEP for various data analysis and mathematical functions.\",\n \"Custom simulations of electronics responses and other instrumental effects are developed and applied downstream from detector simulation [22].\",\n \"Geant4 is one of the most successful examples of community-supported and -maintained software in HEP.\",\n \"It is even used beyond HEP for medical and astronomical applications.\",\n \"However, HEP needs are the primary drivers of R&D on the software, especially improvements to its computing performance.\",\n \"In particular, the next generation of collider experiments, including the HL-LHC and the various future colliders proposed during the Snowmass process, will pose extreme computing challenges for detector simulation.\",\n \"Compared to present-day experiments, data volumes will increase by at least an order of magnitude, requiring a corresponding increase in the number of simulated events.\",\n \"Detectors will increase in complexity and precision, requiring more detailed geometries and more precise physics models that will use even more computation.\",\n \"At the same time, other aspects of the event processing chain, such as reconstruction, will consume a growing proportion of available computing because of massive increases in simultaneous bunch crossings, or pileup, especially at hadron colliders.\",\n \"Therefore, Geant4 will need to simulate an order of magnitude more events with higher precision while using a smaller proportion of HEP computing.\",\n \"To ground this prediction in reality, Geant4 used 40% of all computing during the LHC Run 2 for CMS and ATLAS [23], and ${\\\\sim }70\\\\%$ for LHCb [24], [25].\",\n \"For the former, it is projected that only 14–20% will be available for simulation at the HL-LHC [26], [27].\",\n \"In addition to high energy colliders, experiments of various sizes that search for light or weakly interacting particles rely on Geant4 [28], [29], [30].\",\n \"Such experiments include DUNE, FASER, MicroBOONE, Muon g-2, Mu2e, COHERENT, and direct dark matter detection experiments DAMIC, DarkSide, DARWIN, DEAP, LZ, NEWS-G, PandaX, PICO, SBC, SENSEI, SuperCDMS, and XENON.\",\n \"While many of these experimental collaborations and apparatuses are smaller than those at the CERN LHC, they face similar computational challenges to process petabytes of data, including the corresponding simulation of background and signal processes.\",\n \"The need to meet these challenges with substantially fewer dedicated personnel makes the availability of common software even more important, including both Geant4 and packages like NEST and Opticks that provide specialized physical interaction models.\",\n \"Future neutrino, dark matter, and other similar experiments will obtain even larger data volumes and probe even more precise physics, increasing the computational challenges and the associated need for high-performing and well-supported software.\",\n \"The increasing breadth and precision of next-generation HEP experiments will require corresponding improvements in the physics models used in Geant4 and related software [20].\",\n \"Some areas of interest include liquid argon signal induction, scintillation materials, Cerenkov light propagation, condensed-matter effects, low-energy responses, and rare background processes and interactions.\",\n \"In particular, hadronic interaction models are an area where agreement with data could be improved and access to more measurements, such as hadron-argon interactions, could lead to even further improvement.\",\n \"In general, support for increased effort in these areas is needed to keep up with the demands of new experiments for more precise models, as well as to improve the computational efficiency of the model implementations.\",\n \"The first major evolution in the Geant4 computing model was the introduction of multithreading to reduce memory usage by sharing read-only common data during otherwise embarrassingly parallel event processing [31].\",\n \"This approach has been adopted by the LHC experiments and Mu2e as the experiment software frameworks have similarly evolved to incorporate multithreading in other event processing steps [32].\",\n \"A recently concluded R&D project called GeantV introduced sub-event parallelism using vectorized instructions, primarily targeting many-core CPU architectures, in a new prototype transport engine.\",\n \"The primary findings of this project were that the achievable speedup compared to Geant4 was limited to a factor $2\\\\pm 0.5$ and that a small percentage of the speedup actually arose from vectorization [33].\",\n \"Nevertheless, the project produced several useful developments, such as VecGeom, and lessons, such as the importance of instruction cache locality.\",\n \"VecGeom provides improved code and vectorization for geometry modeling, enabling a ${\\\\sim }15\\\\%$ speedup when used with Geant4.\",\n \"As an illustration of the level of complexity of Geant4, it required 30 FTE-years, spread over a 5-year period, to re-implement its numerous components, even partially or as prototypes, in GeantV: particle transport, geometry modeling and navigation, magnetic field propagation, physical interaction models, and interfaces for experiment software frameworks.\",\n \"The future of detector simulation computing is turning toward the use of GPUs, targeting in particular HPC facilities that are intended to provide a larger fraction of HEP computing for next-generation experiments.\",\n \"Because detector simulation is naturally an HTC problem, nontrivial effort is required to adapt the necessary computations to use computer accelerators effectively.\",\n \"Two ongoing projects, G4HepEM-AdePT and Celeritas, have produced viable prototypes of high-performing GPU-native simulation engines.\",\n \"In particular, the Celeritas effort is led by US national laboratory personnel and was inspired by the earlier porting of the Shift MC radiation transport engine to GPUs [34].\",\n \"Shift is a less general application than Geant4, focusing on neutral particles with few secondary particles produced, limited physical interactions, simple geometries, and no magnetic field propagation.\",\n \"Despite the additional complications encountered in addressing the broad scope of HEP detector simulation, the prototypes have demonstrated promising results, achieving a factor of 40 speedup using 1 Nvidia V100 GPU compared to Geant4 using 7 Power9 CPUs on Summit [35].\",\n \"This can be rephrased as a factor of 200 GPU:CPU equivalence comparing a V100 to a higher-end Xeon CPU, also taking into account power consumption and other relevant factors.\",\n \"Variations in performance from different implementation choices were relatively small, a factor of 2 at most.\",\n \"Scaling this up to the whole Summit supercomputer suggests that its 27 648 GPUs can correspond to 5.5 million CPU cores, an order of magnitude larger than the current worldwide LHC computing grid (WLCG) at 500 000 cores [36].\",\n \"If this performance persists, it has the potential to resolve the major simulation computing challenges.\",\n \"However, we must not underestimate the increased burden of developing and maintaining coprocessor-friendly code, including aspects such as portability to non-Nvidia GPUs and even future non-GPU architectures.\",\n \"It should also be noted that the ongoing projects currently implement only some components of Geant4, including models of electromagnetic interactions, but not yet hadronic interactions, for example.\",\n \"If a complete parallelization of all components is not ultimately possible, the observed speedup will be limited by the remaining serial components, according to Amdahl's law.\",\n \"In terms of personpower, the Celeritas project has so far used 5 FTE-years spread over 2 years, and AdePT is similar.\",\n \"Because this effort builds not just on Geant4, but also on the lessons learned from earlier projects including Shift and GeantV, as well as the products of those efforts such as VecGeom, the total personpower involved is much greater.\",\n \"Another avenue to achieve faster simulation is the use of machine learning to replace or augment existing “classical” (rule-based) simulation engines [37].\",\n \"Full replacement is most commonly pursued using generative models such as generative adversarial networks (GANs), variational autoencoders (VAEs), or normalizing flows (NFs).\",\n \"Augmentation to improve lower-quality results, sometimes called refinement, can be applied to classical fast simulation engines or to generative models, using techniques similar to those above or regression-based approaches.\",\n \"These approaches are largely still in development or just starting to be deployed [38], so their ability to provide an acceptable level of physics fidelity at scale must be assessed carefully.\",\n \"Beyond these ML algorithms, there are proposals to explore the application of differentiable programming directly to simulation engines [37], [39].\",\n \"ML algorithms whose training and inference are performed using industry-standard frameworks are more easily portable to coprocessors such as GPUs, including at HPCs.\",\n \"The possibility also exists to use coprocessors more cost-effectively and efficiently by performing the ML inference as a service, avoiding the need for each CPU to be equipped with a GPU [3].\",\n \"The above solutions, including GPU prototype transport engines and ML algorithms, are promising avenues for faster execution of detector simulation.\",\n \"However, it is unlikely that these will completely supplant Geant4, which provides broad support for myriad use cases.\",\n \"In addition, the results of any new approaches are most often compared to Geant4 to prove their validity.\",\n \"Therefore, some percentage of general purpose von Neumann CPU-based computing will be needed in HEP for the decades to come.\",\n \"The extent to which CPU-based computing can be shifted to less general architectures and devices such as GPUs depends strongly on the availability of experts to develop and maintain the necessary software.\",\n \"There has recently been a severe decrease in funding for permanent positions in HEP to support software developers who work on both these common tools and experiment-specific matters.\",\n \"This threatens the persistence of important knowledge, which must be passed directly from person to person, and the stability and future of these crucial software packages that serve as a foundation for the entire field.\",\n \"There is a strong and vocal consensus throughout the field that this trend must be reversed, with funding for technical positions not just restored, but increased further: “...it is still necessary to support the existing code and make sure that sufficient funding and staffing is provided for maintenance and development of physics algorithms, as well as for adapting the code to any updated CPU hardware, operating systems and new compilers.” [31] “Unfortunately, the Bertini model [of hadronic interactions] has not been actively developed over the last few years due to the lack of personpower.” [6] “A team of highly-skilled physicists and engineers is required to provide the necessary support and developments for Geant4, and the packages extending it, to meet the needs and challenges within the scope of the US HEP experimental program.” [20] “In addition, Geant4's common software, such as physics models, no longer receives any US maintenance funding...\",\n \"This is not sustainable for the long term viability of small experiments.\",\n \"Long term, experiment-agnostic Geant4 support is critical for the success of small experiments...\",\n \"It is therefore essential that there is funding for permanent software and computing experts.” [28] “However, U.S. funding for [the Geant4 toolkit] was discontinued in recent years...\",\n \"Continued support for Geant4 is crucial to the design and construction of future experiments, and for the interpretation of their results...\",\n \"The general challenge of maintaining software and computing talent is exacerbated in the direct detection community by the lack of long term, permanent positions within the experiments...\",\n \"It is therefore essential to provide funding for permanent software and computing experts.” [29] “DM and neutrino collaborations need scientists trained in data acquisition, simulation, and analysis at both the user and developer levels to achieve their science goals.” [30]\"\n ],\n [\n \"Physics Generators\",\n \"The generator software landscape is complex and varied, in order to provide solutions to different aspects: event (matrix element) generation, hadronization and parton shower modeling, underlying event tunes, matching/merging algorithms, multi-parton interactions, particle decays, cross section calculations, parton distribution functions, data formats, and analysis and reinterpretation.\",\n \"There are many software packages that address some of these needs, which often include plugins to perform specific calculations and provide ways for users to plug in their own custom models and computations.\",\n \"A rough and incomplete categorization includes: matrix element generators, which support leading-order (LO) as well as next-to-leading-order (NLO) and sometimes higher order calculations: MadGraph 5_amc@nlo, powheg, sherpa, alpgen, whizard, amegic, comix, helac; parton shower modelers, which also support LO event generation: pythia, herwig, sherpa; neutrino-nucleus interaction generators: genie, neut, NuWro, GiBUU, achilles; plugins for more specific tasks: EvtGen, tauola, photos, dire, vincia; higher-order differential cross section calculation: mcfm, matrix, NNLOJet; one-loop amplitude computation: MadLoop, OpenLoops, recola; utility software: lhapdf, HepMC; analysis and tuning: rivet, mcnntunes, professor, apprentice, nuisance.\",\n \"This wide array of software packages includes a corresponding variety of programming languages and styles—primarily Fortran, C++, and Python—as well as development and maintenance approaches.\",\n \"Unlike detector simulation, in which Geant4 serves as a locus for numerous related efforts as described in Section , most of the software packages listed here are developed largely independently by separate groups.\",\n \"The usage of common data formats and exchange specifications, such as LHEF, UFO, and HepMC, provides some amount of generic interoperability.\",\n \"The percentage of WLCG computing time used for event generation at the CMS and ATLAS experiments is estimated to be 5–12%, with similar or even more substantial requirements expected for ALICE and LHCb [40].\",\n \"However, these summary values hide large variances in tasks and generator software.\",\n \"For example, MadGraph 5_amc@nlo has been found to be substantially faster than sherpa for generation of W+jets events with 0–2 extra partons, but much of the difference depends on parameter choices that may impact physical accuracy in some regions of phase space.\",\n \"In general, for a given task, generator CPU usage scales linearly with the number of events generated.\",\n \"However, moving from LO to NLO and higher orders, or increasing the multiplicity with additional partons [41], for the same physics process typically causes a factorial increase in CPU usage, while different processes may still have very different baseline computational requirements.\",\n \"The memory usage and multithreading abilities of different generator software are also highly variable and can lead to inefficiencies.\",\n \"The range of programming languages and styles can make generators difficult to integrate directly in experiment software frameworks, another potential source of inefficiency.\",\n \"The generator usage at neutrino frontier experiments has not been measured and profiled as systematically, and this subset of the field is generally smaller compared to the energy frontier.\",\n \"Generator CPU usage is projected to increase to 8–20% in the HL-LHC era, when more events and higher precision will frequently be required [26], [27].\",\n \"The increasing precision of upcoming neutrino experiments such as LBNF/DUNE and HyperK should be expected to require similar increases in experiment computing time devoted to event generation.\",\n \"Further future experiments will have different generator needs depending on their initial states.\",\n \"The different categories include high energy hadron colliders, lower (but still relatively high) energy lepton colliders, neutrino experiments, forward physics experiments (such as FASER and SND@LHC), transverse detectors (such as CODEX-b, AL3X, and MATHUSLA), the Electron-Ion Collider (EIC), and proposed muon colliders.\",\n \"The details for the relevant tasks and generators in each category are vast and cannot be summarized here; Ref.\",\n \"[42] presents a comprehensive treatment.\",\n \"As a particular item of interest from the recent Snowmass effort, we highlight new techniques in development for muon collider physics: the use of the effective vector boson approximation for $2 \\\\rightarrow n$ processes [43] and vector boson fusion as the dominant tool to study the electroweak sector at multiple TeV [44].\",\n \"These processes and approaches have been implemented in new versions of MadGraph 5_amc@nlo, sometimes requiring new features or more involved calculations to ensure stable and reliable results.\",\n \"They illustrate the work that will be needed to model the physics at the next generations of HEP experiments.\",\n \"The interdependence between lattice QCD, nuclear physics, and neutrino event generators is also important to highlight [45].\",\n \"The computational burden of an event generator depends strongly on the precision required, as described above; the calculation techniques used; and any other requirements placed on the output [31], [40], [42].\",\n \"For example, the automated computation of NLO and higher order diagrams, as used in mc@nlo, introduces events with negative weights, which can substantially reduce the statistical power of the final result.\",\n \"There are also inefficiencies in phase space sampling; in slicing, biasing, or filtering to change kinematic features; and in merging to avoid double-counting between matrix element generators and parton shower modelers.\",\n \"Further, experiments' use of generators may involve significant repetition of expensive calculations, especially to assess certain systematic uncertainties that cannot be represented as alternative event weights.\",\n \"Recent efforts coordinated by the HEP Software Foundation (HSF) have identified and resolved several performance bottlenecks, some related to these issues [42].\",\n \"There is ongoing work to continue to address these issues, especially negative weights, which can be reduced, but not yet eliminated, by certain new prescriptions that have not yet achieved wide deployment [40].\",\n \"In general, the development of new theoretical and mathematical tools to perform relevant computations has an outsized impact on event generator computing requirements.\",\n \"At the time of the previous Snowmass reports, generator software was in some ways ahead of other subfields.\",\n \"Machine learning algorithms were employed to derive PDFs [46], and usage of MPI for efficient utilization of existing CPU-based HPC systems was incorporated in sherpa, alpgen, MadGraph 5_amc@nlo, and mcfm, among others.\",\n \"However, more recent initial efforts to port MadGraph 5_amc@nlo to use GPUs did not lead to a production quality result.\",\n \"Revitalizing this task has been identified as a priority, and recent progress is promising.\",\n \"An opportunity comes from the possibility to parallelize the matrix element calculations, which take up more than 90% of the computing time for complex LHC physics processes.\",\n \"Matrix element calculations are a perfect fit for CPU vectorization and hardware acceleration on GPUs.\",\n \"The MadGraph4GPU project, which uses custom CUDA kernels, shows speed improvement factors of ${\\\\sim }1000$ on an Nvidia V100 GPU, as well as an improvement factor of 3 on CPUs by exploiting vectorization [47].\",\n \"The MadFlow project, which uses a TensorFlow backend, shows a speed improvement of more than an order of magnitude on an Nvidia Titan V GPU and an improvement factor of up to 3 on CPUs [48].\",\n \"A new approach called BlockGen for multi-gluon tree amplitudes shows similar improvements of roughly an order of magnitude on a V100 GPU and 2–6 on CPU, depending on the multiplicity [49].\",\n \"Similar developments for other computationally-intensive generators such as sherpa also need to be prioritized and supported.\",\n \"The use of ML for approximate matrix element calculations, including phase space integration and sampling as well as fully generative models similar to those discussed in Section , is also very promising, but still in relatively early R&D and not yet ready for deployment.\",\n \"Additionally, there are ongoing investigations into ML for parton shower modeling, hadronization, event (un)weighting, and tuning, among other applications [42], [50].\",\n \"Beyond technical developments, other possible improvements that require coordination between different groups have been identified.\",\n \"A common interface, which could also be described as a modular framework, would standardize how generators are used and plug into each other and other software [45], [51], [42].\",\n \"This would be useful not just for theoretical studies, but also for experiment use, especially small experiments with minimal personpower to implement, test, and maintain different interfaces.\",\n \"It could also facilitate more collaboration among generator developers, for example to make faster progress on code modernization and related work.\",\n \"The adoption of common data formats and standards, such as those provided by Les Houches, by more generators and communities could be encouraged as well.\",\n \"While there are clearly a number of potential benefits, both coordination and personpower limitations would need to be overcome in order to realize them.\",\n \"Sharing generator outputs, such as parton-level event records, is another idea to reduce both computation and disk usage [40].\",\n \"At the LHC, ATLAS and CMS often duplicate the work to generate the same physics process at the same precision and to store the result.\",\n \"Avoiding this duplication would require coordination to standardize data formats, host the output, and ensure that the choices of model and generator parameters satisfy different analysis needs.\",\n \"The existing LHC working groups that provide resources for cross sections and other theoretical calculations could serve as a model to organize this kind of coordination.\",\n \"Data preservation organizations could act as a central hub for storage, and an increased emphasis on data preservation and access would also improve the field's capabilities to reinterpret and combine results [42].\",\n \"More broadly, it has been suggested that a diverse and cross-cutting collaboration, inspired to MCNet in Europe with a long-term outlook, could bring together the U.S. event generator community to share resources and ideas [42].\",\n \"The breadth and diversity of generator software brings many advantages and opportunities to HEP.\",\n \"However, it also brings challenges, especially when it comes to keeping up with the increasing demands for both physics precision and computing efficiency.\",\n \"Historically, there has not been a single obvious locus to coordinate common activities, even if MCNet and, more recently, HSF have provided useful forums, especially in the areas of training and computational efficiency, respectively.\",\n \"Towards the future, more emphasis should be placed on establishing and coordinating common activities.\",\n \"and improvements in this area should continue to be emphasized.\",\n \"Projects to reduce the computational burden of event generation, for example by adapting to use GPUs, need a substantial increase in effort in order to be successful.\",\n \"Some of the difficulties that generators face in computing are intrinsically tied to the nature of the calculations being performed, so fundamental research may be required to develop better methods.\",\n \"On the whole, event generators are usually supported by small teams that frequently have little to no dedicated funding and cannot adequately plan for succession.\",\n \"As with detector simulation, it is vital to the entire field that funding is provided for permanent positions to support, improve, and expand these crucial components of HEP software.\",\n \"This is echoed throughout the white papers written during the Snowmass process: “The physics Generators used in the field (eg.\",\n \"Pythia, GENIE, Madgraph, Sherpa) also suffer from lack of stable funding in a similar way.” [28] “The common needs to all direct dark matter detection experiments include: ...\",\n \"Continued support for event generators, including those developed as part of a national security program.” [29] “Participation in generator-related activities is poorly incentivized for both theorists and experimentalists, and opportunities to pursue neutrino generator development as one's primary research activity are rare... A need for greater coordination and prioritization of such activities is widely recognized in the neutrino scattering community.\",\n \"Despite promising initial discussions that have taken place in a series of recent workshops [68, 529–531]; however, neither a clear leadership structure nor significant institutional support to carry out the related work have yet emerged...\",\n \"The cooperation needed for success in neutrino generator development cannot occur to the extent that funding agencies impose rigid boundaries between [high energy and nuclear physics]... [Stitching together multiple models to achieve complete simulations] is ideally done in collaboration with theorists to minimize inconsistencies, but incentives for their direct involvement are currently poor.” [45] “[Theorists]... may not be motivated to work on software optimisations that are not “theoretical” enough to advance their careers.\",\n \"Generator support tasks in the experiments may also not be enough to secure jobs or funding for experimentalists pursuing a career in research.” [31] Ref.\",\n \"[51] specifically surveys the major collider event generators to report the expected level of support through the HL-LHC era: “Funding and career perspectives are a major common issue reported by many generator experts...\",\n \"Generator support tasks in the experiments may also not be enough to secure jobs or funding for experimentalists pursuing a career in research.\",\n \"It is still not clear for people who are doing work on generators or related tools whether it is really possible to have a dedicated career path from now throughout HL-LHC.\",\n \"This is slowly being acknowledged from funding bodies, but the community has low expectations especially because there is no sign that hiring policies will be modified to provide reasonable funding for generator work.\",\n \"The delay caused by the lack of funding for generator work has already started to cause a loss of know-how in some generator packages.” MadGraph 5_amc@nlo: “the collaboration are of the opinion, shared by other groups, that time to work on performance improvements, even by significant factors, is not well recognised by funding agencies...\",\n \"There are no dedicated funds for HL-LHC.” powheg: “there are no funds specifically dedicated to the maintenance of the framework” sherpa: “Several of the Sherpa team are in permanent positions...\",\n \"In the UK, Sherpa posts at IPPP are seen as part of IPPP's core mission and therefore are expected to be treated as core in future funding reviews.” herwig: “Funding is a serious issue... To fund the bread and butter of what the experiments need, one would need different specific funding.” pythia: “the continued development of Pythia8 throughout the HL-LHC era depends on funding agencies and laboratory priorities, though these are not expected to change.\",\n \"The biggest challenge to the collaboration is finding permanent positions for junior people so that they can continue their valuable contributions.” EvtGen: “The core EvtGen team has 4 members who have a fraction of their time funded for EvtGen through an UK STFC grant...\",\n \"There is some funding for a computer engineer to work on code redesign for multithreading.” The developers of the BeAGLE generator, which is used for inelastic scattering in nuclear collisions, state: “The future development of this code is uncertain primarily due to lack of manpower and reliable funding... As discussed above, funding/manpower issues are currently limiting our ability to implement many of these improvements.” [42]\"\n ],\n [\n \"Continuum Field Theory Calculations\",\n \"Continuum field theory calculations include both precision high order perturbative calculations [52] and conformal bootstrap theoretical calculations [53].\"\n ],\n [\n \"Precision perturbative calculation\",\n \"Precision measurements at the LHC and future colliders require theory predictions with uncertainties at the percent level for many observables.\",\n \"Theory uncertainties due to the perturbative truncation are particularly relevant and must be reduced for a large variety of relevant processes.\",\n \"At the high luminosity LHC, even rare processes like Higgs production require theoretical control of cross sections at the 1% level.\",\n \"Reference [52] surveyed the theoretical community and received responses covering 53 scientific publications.\",\n \"This Snowmass whitepaper gave clear motivation for precision as follows: “To fully exploit the physics potential of current and future collider experiments, in particular to unambiguously identify signals of new physics, it is crucial to improve the precision of theoretical predictions.” Since the 2013 Snowmass exercise, the state-of-the-art has moved dramatically and now next-to-next-to-next-to-leading order (N$^3$ LO) QCD calculations for $2\\\\rightarrow 1$ processes and NNLO QCD calculation for $2\\\\rightarrow 3$ processes and $2\\\\rightarrow 2$ processes with internal masses are becoming state-of-the-art and reasonably fast on contemporary computing resources, while automation and more complex final states at NNLO at N$^3$ LO are goals for the Snowmass period.\",\n \"The computational challenge is to perform Feynman loop integrals within perturbative quantum field theory, giving access to the scattering matrix and its analytic structures, often involving non-standard special mathematical functions.\",\n \"The aim is to, eventually, be able to numerically evaluate the expressions such that the error is reliable, the result has sufficient precision, and the computation consumes acceptable resources.\",\n \"Suitable analytical or symbolic preparation of the integrals and amplitudes must be performed and the evaluation of these expressions during Monte Carlo integration over final state phase space, so that they may feed into event generators.\",\n \"Analytical solutions of Feynman integrals in terms of special mathematical functions are preferred from an efficiency perspective, but often require process-specific knowledge or even new mathematical methods.\",\n \"Numerical or semi-analytical methods can allow broader flexibility and better automation.\",\n \"Numerical direct integration approaches are fully established and implemented in codes such as pySecDec and fiesta, but are typically slow in evaluation times.\",\n \"Novel semi-analytical methods based on series expansions seem promising and first public packages such as DiffExp and AMFlow became available recently.\",\n \"Such series expansion methods reduce evaluation times to the order of a minute per phase-space point for typical two-loop integrals.\",\n \"QCD corrections to scattering amplitudes have been fully automated at NLO using libraries like Helac-NLO, MadGraph 5_amc@nlo, nlox, OpenLoops, and recola as well as many other public and private tools.\",\n \"Two and higher loop calculations remain challenging but in the last five years significant progress has been made in many combinations of final state particle multiplicity and number of internal mass scales at two loops, and in more restrictive cases at three and four loops.\",\n \"Further the complete five loop beta function has been calculated.\",\n \"A core computational bottleneck for the calculation of both Feynman amplitudes and loop integrals is in the reduction of linear relations between Feynman integrals (integration-by-parts identities), with fire, reduze, LiteRed and kira being example codes.\",\n \"From a computational perspective the use of modular (finite field) arithmetic in the calculation of integration by parts reduction is an interesting transformation of the computational requirement that may require radically different computing hardware solutions.\",\n \"Reference [52] notes significant difficulties fitting within the constraints of common HPC centers: “These novel techniques have opened the door to perform computer algebraic computations for quantum field theory on HPC systems.\",\n \"These novel methods typically use integer arithmetic rather than floating point arithmetic and, depending on the problem, they may require a significant amount of memory per core.\",\n \"The development of (public) codes for challenging amplitude calculations in a HPC friendly way is an ongoing effort.\",\n \"Currently, in some situations, one may need to resort to implementations that have large memory demands and require run-times that are hard to predict and possibly well beyond available batch limits on a given cluster.” Given the importance of precision theoretical calculation it is imperative that appropriate resources be available to enable them.\",\n \"In 2013, a sample of then state-of-the-art calculations took between 50,000 and 1M core hours.\",\n \"Since then, the per-core CPU performance has improved by about a factor of 5 and the per-node performance by about a factor of 30.\",\n \"At the time, it was hoped to be able to use efficiently either many-core architectures such as Xeon Phi or GPUs.\",\n \"The current state of the art has the additional complexity largely balanced by the growth in per-node performance, reaching up to about 10M core hours with current CPUs, as shown in Fig.\",\n \"REF .\",\n \"Parallelism is typically used within a node, and is indeed in part dictated by growth in core count relative to memory capacity.\",\n \"Multinode parallelism is only exploited as throughput computing making use of trivial parallelism by the majority of codes in the area.\",\n \"[50][h] Computational costs of current state-of-the-art multi-loop calculations in the community, from Ref. [52].\",\n \"Figure: NO_CAPTION Although there is some role for GPUs in numerical integration, they only match a modest part of the scientific requirement [52]: “The use of GPUs is still unclear in our field, since many problems in our field rely on the numerical evaluation or handling of algebraic expressions which are large and/or require irregular memory access patterns.\",\n \"So far, GPUs have found application to cutting-edge problems with the numerical integration of sector decomposed loop integrals.\",\n \"A first step for future applications could include the efficient evaluation of one-loop amplitudes.\",\n \"This would help the huge computational requirements for NLO high-multiplicity evaluations, but also for the real emission integrations for NNLO calculations.\",\n \"Since the efficient use of GPUs is still unclear, future computing for our community will still need to focus on providing CPU resources without attached GPU resources.” Large memory and runtime limitations imposed by typical HPC environments mean that bespoke computing nodes and queues (wherever they are hosted) are required to enable these calculations.\",\n \"Some of the job runtime constraints could be addressed with checkpointing virtual machines but since complex software is involved, some even commercial like Mathematica, a dedicated node environment may be the only practical solution.\",\n \"Machine learning does not at this time play a large role, although some applications have been used in event generation.\",\n \"The licensing of proprietary software like Mathematica or Maple also suggests dedicated computing nodes for this purpose may be cost-effective.\"\n ],\n [\n \"Conformal bootstrap\",\n \"The idea of the conformal bootstrap is to constrain and solve conformal field theories (CFTs) using physical consistency conditions like symmetry, unitarity, and causality.\",\n \"By relying on nonperturbative structures, bootstrap methods can work even in strongly-coupled systems where traditional perturbative techniques fail.\",\n \"Over the last few decades, the conformal bootstrap idea has crystallized into two concrete strategies: 1) exploiting exact solvability and 2) deriving bounds using sum rules with positivity constraints and using convex optimization to extract information.\",\n \"When combined with numerical convex optimization, this leads to the numerical conformal bootstrap.\",\n \"The conformal bootstrap is distinguished by being an intrinsically non-perturbative method that yields strict bounds on the theory.\",\n \"Recent successes include precise determinations of critical exponents in physically-relevant theories such as the 3D Ising and O(N) models, constraints on theoretically important theories such as 4D N = 4 supersymmetric Yang-Mills theory and 6D $(2,0)$ SCFTs, and has inspired promising new ideas for bootstrapping S-matrices.\",\n \"The numerical bootstrap is a rapidly-growing field and is anticipated to play a central role in CFTs over the next decade.\",\n \"Targets include 3D scalar models, $O(N)$ vector models in 3D, and in the case of $N=1$ the 3D Ising CFT, while the 3D O(2) model describes the the superfluid transition in liquid helium as well as thin-film superconductors.\",\n \"Here the bootstrap has helped address a controversy, where the results of experiment and the current best Monte Carlo simulations disagreed with each other with 8$\\\\sigma $ significance, giving results in remarkable agreement with those from Monte Carlo simulations.\",\n \"A second target is 3D Fermionic Models, including 3D CFTs containing N Majorana fermions with Yukawa couplings to one or more scalar fields, often called Gross-Neveu-Yukawa model.\",\n \"The simplest model with $N=1$ Majorana fermion coupled to a real scalar is believed to possess emergent supersymmetry and correspond to the minimal 3D $N=1$ supersymmetric extension of the Ising model, proposed to have a realization on the boundary of topological superconductors.\",\n \"Preliminary studies applying the bootstrap to fermion 4-point functions showed the GNY models to be promising targets for the bootstrap, while further systems can be studied providing a supersymmetric systems including Wess Zumino models.\",\n \"Progress in 3D and 4D gauge theories has been made including including the conformal window of 4D QCD with 12 flavors, establishing a hint of contact with lattice field theory near conformal window results and in $N=4$ supersymmetric Yang Mills theory.\",\n \"From a computational perspective, the whitepaper identifies two key algorithmic challenges: i) to find faster optimization methods, or ways to scale up current tools, and ii) to find efficient methods for exploring high-dimensional spaces of CFT data.\",\n \"High precision arithmetic is required, and often not in native hardware floating point, and used in bespoke semidefinite solver libraries such as SDPB.\",\n \"SDPB can use distributed high precision linear algebra and parallelize over many HPC nodes and hundreds to thousands of cores.\",\n \"The authors are optimistic that further progress will: “enable a new round of fundamental discoveries in theoretical physics, including the numerical classification of CFTs with a small number of relevant operators, definitive answers to longstanding questions about conformal windows of gauge theories in 3 and 4 dimensions, more-or-less complete numerical solutions of the known maximally-supersymmetric CFTs (and their holographic duals), and new robust starting points for Hamiltonian truncation studies of gapped phases.\",\n \"Importantly, the numerical bootstrap can also be used as a discovery tool for finding previously unknown CFTs and as well as for identifying possible analytical solutions of known ones.” The computational requirements so far have involve the use of conventional cluster computers, such as XSEDE and up to 32 nodes and 768 cores.\",\n \"Calculations of different theories range from 200,000 core hours for the 3D Ising model to a few million core hours testing 100 to 1000 points in the CFT parametric space for the 3D $O(3)$ model.\",\n \"Large memory capacity is again significant requirement and is a barrier to efficiency, arising from the high precision (1000 bits) arithmetic and is required for accurately handling large cancellations in the calculation.\",\n \"The authors identify significant software and computing hardware opportunities in future: “further scaling and improving this code, including implementing more sophisticated distributed linear algebra routines, improving memory management to allow scaling past ${\\\\sim }500$ cores, finding ways to reduce precision requirements, and leveraging new hardware like GPUs and FPGAs.\",\n \"These engineering challenges are tightly coupled to physics: an increase in the scale of solvable semidefinite programs could allow one to explore larger systems of crossing equations and thereby access new CFTs.”\"\n ],\n [\n \"Continuum field theory summary\",\n \"Both multi-loop perturbative calculations and numerical conformal bootstrap have a similar computer hardware requirement.\",\n \"They both have a current software reliance on CPU cores, and a large memory per core requirement.\",\n \"Perturbative calculations have a sizable barrier to the use of GPUs, while the conformal bootstrap approach, after significant software engineering, may be amenable to GPU or FPGA acceleration of high precision arithmetic.\"\n ],\n [\n \"Lattice QCD\",\n \"Lattice gauge theory is a systematically improvable theoretical tool for numerical evaluation of the Euclidean Feynman-path integral for Quantum Chromodynamics (QCD).\",\n \"Worldwide lattice gauge theory efforts directly support numerous high-energy physics experiments by calculating properties of hadrons that are vital to interpretation of the experiments.\",\n \"These efforts make significant use of supercomputers and are critically dependent upon continued computing advances.\",\n \"The Flavour Lattice Averaging Group [54] performs a critical review of many important lattice QCD predictions every two to three years.\",\n \"Snowmass Whitepapers have been contributed on the topics relevant to Lattice QCD [55], [45], [56], [57], [58], [59], [60], [61].\",\n \"USQCD also published in 2019 comprehensive set of topical whitepapers and submitted these for consideration by Snowmass [62], [63], [64], [65], [66], [67], [68].\",\n \"The search for new physics requires a joint experimental and theoretical effort.\",\n \"Lattice QCD is an essential tool for obtaining precise model-free theoretical predictions of the hadronic processes underlying many key experimental searches.\",\n \"The role of Lattice QCD in determining the hadronic contributions to the anomalous magnetic moment of the muon was discussed in a Snowmass white paper [59], by the international Muon $g-2$ Theory Initiative white paper [69] and in a white paper by the USQCD collaboration in 2019 [64].\",\n \"These calculations will be critical to the interpretation of results from the Fermilab Muon $g-2$ Experiment [70], which is currently in excellent agreement with results from the earlier Brookhaven experiment [71].\",\n \"Lattice QCD also plays a key role in ab initio prediction of nucleon structure and parton physics [57], [55].\",\n \"Understanding the neutrino-nucleus interaction is critical to the analysis of the Deep Underground Neutrino Experiment.\",\n \"The interaction between an intermediate W or Z boson with with a quark confined inside a neutron or proton within the nucleus is theoretically complex.\",\n \"This was discussed by USQCD in a 2019 white paper [65] and holistically the interplay between lattice QCD, EFTs, nuclear physics, phenomenology, and neutrino event generators was discussed in a contributed Snowmass Whitepaper [45].\",\n \"A comprehensive quark-flavor physics program has been discussed in a USQCD whitepaper [64].\",\n \"Contributed Snowmass white papers include improving the understanding of anomalies in B physics [72], emerging from the the Large Hadron Collider and studied further at Belle II, as well as reconciling CP violation observed in kaon experiments with the standard model and exploring rare kaon decays [56].\",\n \"These phenomena are typically highly suppressed in the standard model and therefore also offer promising avenues for the discovery of new physics.\",\n \"As experimental measurements become more precise over the next decade, lattice QCD will play an increasing role in providing the needed matching theoretical precision.\",\n \"Achieving the needed precision requires simulations with lattices with substantially increased resolution.\",\n \"With finer lattice spacing comes an array of new challenges.\",\n \"They include algorithmic and software-engineering challenges, challenges in computer technology and design, and challenges in maintaining the necessary human resources.\",\n \"The simulations at finer lattice spacing and larger volumes required to realize these goals introduce new challenges [58]: “Meeting them requires new algorithmic research, novel computer hardware design beyond the exascale, improved software engineering, and attention to maintaining human resources.” As a computational problem, Lattice gauge theory is performed on structured Cartesian grids with a high degree of regularity and natural data parallelism.\",\n \"The approach formulates the Feynman path integral for QCD as a statistical mechanical sampling of the related Euclidean space path integral.\",\n \"The sampling is performed by Markov chain Monte Carlo (MCMC) sampling, using forms of the hybrid Monte Carlo (HMC) algorithm [73], [74], [75].\",\n \"Present algorithms for both MCMC sampling and Dirac solvers display growing limitations as substantially greater ranges of energy scales are included in our problem, an algorithmic challenge called critical slowing down.\",\n \"The development of numerical algorithms is a significant intellectual activity that spans physics, mathematics, and computer science.\",\n \"The central repeated operation is the solution of the gauge covariant Dirac equation.\",\n \"The solution is usually performed using iterative Krylov solvers, Newer multigrid [76], [77], [78] solvers have demonstrated order-of-magnitude gains for Wilson fermions by approximately and repeatedly handling degrees of freedom in the low lying eigenspace as a form of preconditioner.\",\n \"The US HEP program is presently focused on the domain-wall and staggered approaches, but corresponding gains for staggered and domain-wall-fermion discretizations are a critical open research activity.\",\n \"A second algorithmic direction is the critical slowing down of MCMC algorithms.\",\n \"These are being studied under Exascale Computing and SciDAC projects and such support is critical to progress in the field.\",\n \"Recent algorithmic directions with different ways of attacking the same problem include applications of machine learning to configuration sampling.\",\n \"This possibility has been discussed in a dedicated Snowmass white paper [79] and will be covered in the report of topical group CompF3 Machine Learning[80].\",\n \"Related Snowmass submissions on tensor networks and quantum simulation will be covered in the report of CompF6 Quantum Computing [81], [82].\",\n \"The Lattice QCD workflow is divided into two phases.\",\n \"First, a MCMC sampling phase generates an ensemble of the most likely gluon field configurations distributed according to the QCD action.\",\n \"The ensemble generation is serially dependent and represents a strong scaling computational problem.\",\n \"Ideally one would be able to use efficiently $O(10^4)$ computing nodes on $O(256^4)$ data points.\",\n \"On the largest scales this becomes a halo-exchange communication problem with a very large interconnect bandwidth requirement since the local data bandwidths vastly exceed those of inter-node communication.\",\n \"In the second phase hadronic observables are calculated on each sampled configuration where many thousands of quark propagators are calculated and assembled into hadronic correlation functions.\",\n \"This both allows more scope for amortizing the setup cost of advanced algorithms like multigrid or deflation, and also has a high degree of trivial parallelism.\",\n \"Scaling Monte Carlo sampling to many computational nodes requires strong scaling and high interconnect bandwidth.\",\n \"Scaling the hadronic observable calculations is not as challenging as multiple configurations can be analyzed at the same time, each job running on a moderate number of nodes.\",\n \"A number of specific simulations, Table REF , have been proposed with estimated costs in a Snowmass white paper [56], and the same methodology can be used to estimate the requirements of the ideal ensemble for flavor physics.\",\n \"The final entry is associated with physics in the B-meson system indicated in Snowmass white paper [72].\",\n \"A $256^3\\\\times 512$ lattice at a lattice spacing $a = 0.04$ fm ($a^{-1} \\\\sim 5$ GeV) would allow us to simulate up/down, strange, charm, and bottom quarks at their physical mass in a 10 fm box with $m_\\\\pi L = 7$ and requires more than a sustained Exaflop year.\",\n \"Several such calculations would be sought by the whole community.\",\n \"These are a modest subset of the many many proposed calculations across the Snowmass contributions, but set the scale of computing sources required in the area of Lattice QCD simulations in support of experiment at beyond-exascale.\",\n \"Between flavor physics, nucleon structure, neutrino scattering, and beyond-the-standard model physics, one could project four or five times this requirement in aggregate across the entire field in the USA alone.\",\n \"On this basis, Ref.\",\n \"[58] estimated that: “These simulation goals therefore clearly demonstrate a need for computers at least 10x more capable than the coming Exaflop computers during the Snowmass period.\",\n \"Since the performance is required to be delivered on a real-code performance basis... more than an order of magnitude improvement, perhaps, from both algorithms and computing are required.” Table: Proposed lattice volumes and cost estimatesin sustained Exaflop hours, scaled from currentsimulations on Cori (NERSC) and Summit (ORNL).Volumes and estimates are proposed in Snowmass white paper , while the finalentry uses the same methodology to estimate the costof the most expensive proposed B-physics capable simulation.The massive vector parallelism of lattice gauge theory is, in principle, amenable to GPU and possibly other acceleration.\",\n \"This imposes a significant additional programmer overhead and the most commonly used packages receive sufficient investment to use the complete range of modern accelerated supercomputers and many of the largest projects use allocations on DOE supercomputer resources.\",\n \"Commonly used Lattice QCD software has been supported by the DOE SciDAC and Exascale Computing Projects.\",\n \"These include Grid [83], [84], [85], MILC [86], [87], CPS [88], Chroma [89] and Quda [90].\",\n \"This has enabled major packages to support the most advanced GPU accelerated HPC computers using software interfaces such as HIP, SYCL, and CUDA APIs in addition to giving good performance on several CPU SIMD architectures.\",\n \"Newer interfaces like OpenMP 5.0 offload and C++17 parallel STL are planned to be adopted as and when appropriate.\",\n \"For smaller projects, especially where rapid development and programmer productivity are at a premium, it is better and more cost effective in terms of human effort to maintain access to a range of CPU resources.\",\n \"USQCD institutional clusters at Brookhaven National Laboratory, Fermilab, and Jefferson Laboratory have been instrumental in supporting the significant number of smaller and experimental projects that would not achieve the return on investment to justify bespoke software development for multiple architectures.\",\n \"Reference [58] notes the heavy reliance on high performance computing places a particularly large dependence on highly tuned bespoke software in this field: “One hopes that this might consolidate the proliferation of programming interfaces as the lack of standardization imposes a very significant software development burden on the science community with duplication of effort for multiple systems.\",\n \"This software development underpins the entire community effort and requires support to make the Snowmass science goals feasible.” “The challenges exist on multiple fronts: intellectual in developing algorithms that evade critical slowing down, software engineering to develop well-performing and portable code on an evolving range of supercomputers and programming models, and technical to remain engaged with the DOE HPC community as systems are planned and developed.” “The community activity is dependent on the existence of bespoke software environments that enable efficient simulation on rapidly evolving supercomputers, which requires significant expertise to develop and sustain.” USQCD has had senior staff, postdoctoral researchers, and post-graduate students funded under the Exascale Computing Project (ECP) and the SciDAC program.\",\n \"It has also funded the development of high performance software portable to Exascale hardware [83], [85].\",\n \"It was noted as important that flexible high-performance software continues to be developed for a diverse range of architectures that tracks the DOE computing program: “the productivity of the community depends on large code bases... which do not have a secure model for development and support which places investments at risk.” Much of the bespoke software development is performed in US National Laboratories.\",\n \"This is required to address HPC architectures as they emerge in the Computational Frontier, developing performance-portable high-level libraries to enable a write once and run anywhere approach that many domain scientists, both in laboratories and universities, can modify effectively.\",\n \"This effort must track the evolution of computing.\",\n \"Finally, it was noted that theoretical particle physics has been one of the last area of physics to recognize the importance of computation in forefront research [58].\",\n \"Continued effort is urgently required to overcome this historical bias and create a vibrant pool of skilled young faculty, and around them their Ph.D. students and research groups.\",\n \"Given the role of lattice gauge theory in confronting hadronic experiment, this field would particularly benefit from a program of joint Lab-University appointments.\"\n ],\n [\n \"Conclusions and Recommendations\",\n \"For scientific activities that consume a large amount of computing time and map well to emerging HPC architectures, there is a wonderful opportunity to apply enormous amounts of computation to solve incredibly complex problems.\",\n \"However, common cross-cutting issues have been raised across many different scientific topics.\",\n \"These issues are largely centered on the critical dependency between modern science and computation and software.\",\n \"Computer architectures have necessarily become increasingly complex, diverse, and challenging to program.\",\n \"In order to turn the potential of DOE computing into scientific deliverables, it is critical to develop skills in the community and to fund the required software and algorithmic development.\",\n \"Support for widely used scientific software packages is required to enable the most effective return.\",\n \"Further, accelerated architectures are not fully general purpose and present challenges to algorithms that are strongly serial without any intrinsic vectorization or parallelism potential.\",\n \"Even for those algorithms for which it is possible in principle to use this hardware, the investment to develop software that runs efficiently across many different platforms and programming environments requires a real investment by the domain scientists.\",\n \"This investment of effort may not always be a good value proposition.\",\n \"For these two reasons, there is a continuing need for a mix of computing with both accelerated and general purpose architectures.\",\n \"This mixture should be sized appropriately to accelerate large scale computation where possible but also to effectively support science whose software cannot exploit acceleration.\",\n \"This may occur if it is not possible to port the underlying algorithms to compute accelerators or if a cost-benefit analysis identifies such a port as not being a cost-effective proposition.\",\n \"The latter case likely comprises a “tail” of many smaller computing tasks that run diverse software with limited development effort available.\",\n \"Our recommendations on computing hardware are as follows: Faster computer hardware: For lattice gauge theory applications, there is a need for HPC resources ten times or more faster than planned Exascale computing systems.\",\n \"Support of right-sized CPU clusters: There should be a right-sized (as large as required, as small as possible) provisioning of general purpose computational cores with high performance memory for important algorithms that do not easily map to computational accelerators.\",\n \"National Laboratory institutional (non-HPC) clusters contribute to this role.\",\n \"NERSC is a welcome example of providing both CPU and GPU/accelerator resources to handle a wider variety of problems relevant to HEP.\",\n \"These should accommodate a diverse range of requirements for software, large memory, and long running single node jobs.\",\n \"Universal programming interface: The proliferation of programming interfaces is a barrier to portability and return on investment in software.\",\n \"Simplification and consolidation of the interfaces will improve scientific output.\",\n \"Software portability: The difficulty of programming highly accelerated hardware with performance portability is significant.\",\n \"Appropriate support for software development and maintenance is essential to ensure success of much of the science discussed in this report.\",\n \"Automation of memory hierarchy: Hierarchical memory is a significant burden on scientific programmer productivity.\",\n \"Virtual memory paging systems can alleviate this and can be made efficient by using larger page sizes, if necessary.\",\n \"This may need substantial investment by computer vendors with robust encouragement from the Department of Energy.\",\n \"Early access to new computing hardware: A key element of managing the science program is the early engagement with DOE HPC laboratory sites during the development, and years prior to installation, of major new facilities.\",\n \"The lead time for porting to new architectures lies in the region of multiple years, and early engagement with emerging architectures is required to ensure timely scientific exploitation.\",\n \"The scientific software ecosystem should be nurtured to enable the proposed Snowmass science.\",\n \"There are several steps that could be proactively taken to ensure the maximum scientific return is delivered.\",\n \"Hardware accelerator-friendly, portable common tools: Specific projects to adapt common software packages to run efficiently on GPUs, including those at new HPCs, have shown initial promise with more than order of magnitude speed improvements.\",\n \"These projects should be continued with sufficient effort to deliver complete products and to incorporate portability solutions to support different coprocessor devices and architectures.\",\n \"Adoption of this new software will additionally require expertise devoted to integration and usage in experimental software frameworks, analysis tools, etc., which should be supported as discussed below.\",\n \"The exploration of new machine learning-based methods, as well as technical improvements to existing (CPU-based) software, should also continue.\",\n \"Best practices for common software: Whenever possible, interoperability and common data structures should be encouraged.\",\n \"Existing standard libraries and underlying software technologies should be used when available and feasible.\",\n \"Interoperability, portability, and common, shared software are fundamental pillars of international scientific collaboration and will become increasingly important in the future.\",\n \"Lab-supported software development: Just as the large experiments require talented permanent staff at the Labs to engineer experiments, theoretical calculation and simulation in HEP experiment, in high energy theory, in cosmology, and in large particle accelerator building have large communities often dependent on sophisticated long-term software systems.\",\n \"Career paths must be created to retain some of the most talented experts in software and algorithms.\",\n \"A larger permanent laboratory staff of software development experts could help address HPC architectures as they increase in importance within the Computational Frontier.\",\n \"Organization for long-term common software maintenance: Despite the numerous successes and broad adoption of common software tools across all topical areas in this report, the future of these projects is in jeopardy because of an extreme lack of funding.\",\n \"One priority should be the development, maintenance, and support for common software tools in the areas of accelerator modeling, detector simulation, and physics generators.\",\n \"Because this kind of long-term effort to sustain cross-cutting software does not fit easily into existing modes of funding, new processes and organizations should be established in cooperation with funding agencies.\",\n \"Continuous collaboration with ASCR: A lot of progress has been made in high energy physics applications through collaborations with programs supported by ASCR under the SciDAC and ECP projects.\",\n \"We believe that such collaborations should be encouraged and continuously supported for future computational modeling in high energy physics.\",\n \"The community found the application development focus of the ECP project particularly effective.\",\n \"Joint lab-university tenure track appointments: We believe the DOE should seek to foster the continued development of intellectual leaders in theoretical calculation and simulation.\",\n \"The health of the field requires a similar cohort of individuals at the best universities, reflecting the intellectual vigor and potential of this area to contribute to DOE scientific goals.\",\n \"The creation of such positions can be stimulated by DOE-funded joint, five-year, tenure-track appointments.\",\n \"A good example might be the Jefferson Lab University Relations program of joint and bridged faculty appointments in nuclear physics.\",\n \"Training: Accessible training in both novel architectures and AI/ML with a low barrier to access for graduate students, postdoctoral researchers, and domain scientists is critical to ensuring the skills base exists for productive science in theoretical calculation and simulation.\",\n \"The use of cross-platform performance portable APIs should be encouraged to maximize return on investment in software.\",\n \"Hackathons and code examples are particularly useful.\"\n ],\n [\n \"Acknowledgments\",\n \"P. Boyle has been funded by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under the Contract No.\",\n \"DE-SC-0012704 (BNL).\",\n \"K. Pedro is supported by the Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No.\",\n \"DE-AC02-07CH11359 with the U.S. Department of Energy.\",\n \"J. Qiang is supported by the U.S. Department of Energy under Contract No.\",\n \"DE-AC02-05CH11231 (LBNL).\"\n ]\n]"},"id":{"kind":"string","value":"2209.08177"}}},{"rowIdx":2392,"cells":{"text":{"kind":"list like","value":[["A Robust and Constrained Multi-Agent Reinforcement Learning Framework\n for Electric Vehicle AMoD Systems"],["Abstract Electric vehicles (EVs) play critical roles in autonomous mobility-on-demand (AMoD) systems, but their unique charging patterns increase the model uncertainties in AMoD systems (e.g.","state transition probability).","Since there usually exists a mismatch between the training and test (true) environments, incorporating model uncertainty into system design is of critical importance in real-world applications.","However, model uncertainties have not been considered explicitly in EV AMoD system rebalancing by existing literature yet and remain an urgent and challenging task.","In this work, we design a robust and constrained multi-agent reinforcement learning (MARL) framework with transition kernel uncertainty for the EV rebalancing and charging problem.","We then propose a robust and constrained MARL algorithm (ROCOMA) that trains a robust EV rebalancing policy to balance the supply-demand ratio and the charging utilization rate across the whole city under state transition uncertainty.","Experiments show that the ROCOMA can learn an effective and robust rebalancing policy.","It outperforms non-robust MARL methods when there are model uncertainties.","It increases the system fairness by 19.6% and decreases the rebalancing costs by 75.8%."],["Introduction"," Autonomous mobility-on-demand (AMoD) system is one of the most promising energy-efficient transportation solutions as it provides people one-way rides from their origins to destinations , , .","Electric vehicles (EVs) are being adopted worldwide for environmental and economical benefits , and AMoD systems embrace this trend without exception.","However, the trips sporadically appear, and the origins and destinations are asymmetrically distributed and hard to predict in AMoD systems , .","Such spatial-temporal nature of urban mobility increases the management difficulty of a large-scale vehicle fleet and makes the system sensitive to disturbances , , .","Moreover, EVs' unique charging patterns (long charging time, high charging frequency, and unpredictable charging behaviors) increase the complexity and uncertainty of the EV AMoD system dynamics , , .","In real-world applications, we usually do not have perfectly accurate knowledge of the true system model, e.g., the state transition probability of the AMoD systems, and therefore, there usually exists a model mismatch between the simulator (training environment) and the real-world application (test environment).","Thus, existing EV AMoD vehicle coordination methods , , , , , , may have significant performance degradation in the test (true) environment.","Despite model-based methods considering prediction errors in mobility demand or vehicle supply , , , , uncertainty in system state transition remains largely unexplored in AMoD systems.","More importantly, practical deployment usually needs to satisfy constraints on, e.g.","safety and fairness, while maximizing the performance.","In this work, we propose a robust and constrained multi-agent reinforcement learning (MARL) framework for EV AMoD systems.","The goal is to achieve AMoD system fairness by finding robust rebalancing policies for idle and low-battery EVs with minimal rebalancing cost, under model uncertainty.","The advantages of our formulation are two-folds: (i) fairness constraints can still be satisfied even if there is model mismatch; and (ii) the rebalancing cost is still optimized when there is model mismatch.","Our Key Contributions are as follows: (1) To the best of our knowledge, this work is the first to formulate EV AMoD system vehicle rebalancing as a robust and constrained multi-agent reinforcement learning problem under model uncertainty.","Via a proper design of the state, action, reward, cost constraints, and uncertainty set, we set our goal as minimizing the rebalancing cost while balancing the city's charging utilization and service quality, under model uncertainty.","(2) We design a robust and constrained MARL algorithm (ROCOMA) to efficiently train robust policies.","The proposed algorithm adopts the centralized training and decentralized execution (CTDE) framework and develops the first robust natural policy gradient (NPG) to improve the efficiency of policy training.","(3) We run experiments based on real-world E-taxi system data.","We show that our proposed algorithm performs better in terms of reward and fairness, which are increased by 19.6%, and 75.8%, respectively, compared with a non-robust MARL-based method when model uncertainty is present."],["Related Work"," AMoD system vehicle rebalancing algorithms re-allocate idle vehicles, sometimes considering charging constraints .","Heuristics can lead to sub-optimal rebalancing solutions .","Other major categories of AMoD system rebalancing methods include optimization-based algorithms , Model Predictive Control (MPC) and Reinforcement Learning (RL) .","Optimization and MPC-based approaches usually formulate the AMoD system vehicle rebalancing problem as a convex optimization or mixed-integer programming problem, where the objective is to improve service quality , or maximize the number of served passengers with fewer vehicles , , .","These model-based approaches usually rely on precise knowledge of the probability transition model of the complex dynamics of AMoD systems and, consequently, are sensitive to model uncertainties.","Though robust and distributionally robust optimization-based methods have been designed to consider uncertainties caused by mobility demand, supply, or covariates predictions , , , the probability transition error or uncertainty in system dynamics has not been addressed yet.","RL-based approaches relax the dependency on system dynamic models.","Various RL-based methods, including DQN, A2C and their variants , , , , , , , have been proposed to solve the vehicle rebalancing problem.","However, RL suffers from the sim-to-real gap; that is, the gap between the simulator and the real world often leads to unsuccessful implementation if the learned policy is not robust to model uncertainties , .","None of the above RL-based rebalancing strategies consider this gap.","Given that designing an efficient and robust EV rebalancing method under model uncertainties is still an unsolved challenge, Robust RL aims to find a policy that maximizes the worst-case cumulative reward over an uncertainty set of MDPs , , , .","To achieve high system fairness while minimizing rebalancing cost under model uncertainty, we put the fairness constraints in our RL formulation, which is known as Constrained RL that aims to find a policy that maximizes an objective function while satisfying certain cost constraints , .","However, applying robust and constrained RL to AMoD rebalancing is still a challenge due to the high-dimensional state and action spaces commonly present in transportation systems.","And the problem of robust constrained RL itself is very challenging even in the simple tabular case.","Our proposed robust and constrained MARL formulation and algorithm explicitly consider model uncertainties and system fairness to learn robust rebalancing solutions for AMoD systems.","And we derive a natural policy gradient for robust and constrained MARL to improve the efficiency of policy training."],["Problem Statement","We consider the problem of managing a large-scale EV fleet to provide fair and robust AMoD service.","The goal is to (i) rebalance idle and energy-efficient EVs (we denote them as vacant EVs for notation convenience) among different regions to provide fair mobility service on the passenger's side; (ii) allocate low-battery EVs to charging stations for fair charging service on the EVs' side; (iii) minimize the managing cost of (i) and (ii); (iv) find rebalancing policies robust to model uncertainty, i.e.","uncertainty in transition kernel of the MDP.","We assume that a city is divided into $N$ regions according to a pre-defined partition method , , .","A day is divided into equal-length time intervals.","In each time interval $[t, t+1)$ , customers' ride requests and EVs' charging need are aggregated in each region.","After the location and status of each EV are observed, a local trip and charging assignment algorithm matches available EVs with passengers and low-battery EVs with charging stations, using existing methods in the literature , , .","Then the state information of each region is updated, including the numbers of vacant EVs and available charging spots in each region.","Each region then rebalances both vacant and low-battery EVs according to the trained MARL policy.","This work focuses on a robust EV rebalancing algorithm design under model uncertainties to maximize the worst-case expected reward of the system while satisfying fairness constraints.","For notational convenience, the parameters and variables defined in the following parts of this section omit the time index $t$ when there is no confusion."],["Preliminary: Multi-Agent Reinforcement Learning","We denote a Multi-Agent Reinforcement Learning (MARL) problem by a tuple $G = \\langle \\mathcal {N}, S, A, r,p,\\gamma \\rangle $ , in which $\\mathcal {N}$ is the set of $N$ agents.","Each agent is associated with an action $a^i \\in A^i$ and a state $s^i \\in S$ .","We use $A = A^1 \\times \\cdots \\times A^N$ to denote the joint action space, and $S = S^1 \\times \\cdots \\times S^N$ the joint state space.","At time $t$ , each agent chooses an action $a^i_t$ according to a policy $\\pi ^i: S^i \\rightarrow \\Delta (A^i)$ , where $\\Delta (A^i)$ represents the set of probability distributions over the action set $A^i$ .","We use $\\pi = \\prod _i^N \\pi ^i: S \\rightarrow \\Delta (A)$ to denote the joint policy.","After executing the joint action, the next state follows the state transition probability which depends on the current state and the joint action, i.e.","$p: S \\times A \\rightarrow \\Delta (S)$ .","And each agent receives a reward according to the reward function $r^i: S \\times A \\rightarrow \\mathbb {R}$ .","Each agent aims to learn a policy $\\pi ^i$ to maximize its expected total discounted reward, i.e.","$\\max _{\\pi ^i} v^{\\pi , i}_r(s)$ for all $s \\in S$ , where $v^{\\pi , i}_r(s) = \\mathbb {E}[\\sum _{t = 1}^{\\infty } \\gamma ^{t-1} r_t^i(s_t, a_t) | a_t \\sim \\pi (\\cdot | s_t), s_1 = s]$ which is also known as the state value function for agent $i$ .","$\\gamma \\in (0,1)$ is the discounted rate.","If these agents belong to a team, the objective of all agents is to collaboratively maximize the average expected total discounted reward over all agents, i.e.","$\\max _\\pi v^{\\pi }_{r}(s)$ for all $s \\in S$ , where $v^{\\pi }_{r}(s) = \\mathbb {E}_\\pi [\\sum _{t = 1}^{\\infty } \\gamma ^{t-1} \\sum _{i\\in \\mathcal {N}}r_t^i(s_t, a_t)/N | s_1 = s]$ ."],["Robust and Constrained Multi-Agent Reinforcement Learning Formulation for EV Rebalancing","We formulate the EV rebalancing problem as a robust and constrained MARL problem $G_{rc} = \\langle \\mathcal {N}, S, A, P, r, c, d, \\gamma \\rangle $ , and we define the agent, state, action, probability transition kernel uncertainty set, reward, and cost and fairness constraints as follows."],["We define for each region a region agent, who determines the rebalancing of vacant and low-battery EVs at every time step.","This distributed agent setting is more tractable for large-scale fleet management than a single agent setting because the action space can be prohibitively large if we use a single system-wide agent ."],["A state $s^{i}$ of a region agent $i$ consists two parts that indicate its spatiotemporal status from both the local view and global view of the city.","We define the state $s^i=\\lbrace s_{loc}^i, s_{glo}^i\\rbrace $ , where $s_{loc}^i = (V_i, L_i, D_i, E_i, C_i)$ is the state of region $i$ from the local view, denoting the number/amount of vacant EVs, low-battery EVs, mobility demand, empty charging spots, and total charging spots in region $i$ , respectively.","And $s_{glo}^i = (t, pos_i)$ , where $t$ is the time index (which time interval), $pos_i$ is region location information (longitudes, latitudes, region index).","The initial state distribution is $\\rho $ ."],["The rebalancing action for vacant EVs is denoted as $a^i_v = \\lbrace a^i_{v,j} \\rbrace _{j \\in \\text{Nebr}_i}$ , the charging action for low-battery EVs as $a^i_l = \\lbrace a^i_{l,j} \\rbrace _{j \\in \\text{Nebr}_i}$ , where $a^i_{v,j}, a^i_{l,j} \\in [0,1]$ is the percentage of currently vacant EVs and low-battery EVs to be assigned to region $j$ from region $i$ , respectively.","And $\\text{Nebr}_i$ is the set consisting of region $i$ and its adjacent regions as defined by the given partition.","Therefore $\\sum _{j \\in \\text{Nebr}_i} a^i_{v,j} = 1$ and $\\sum _{j \\in \\text{Nebr}_i} a^i_{l,j} = 1$ for all $i$ .","We denote $m^i_{v,j} = h( a^i_{v,j} v^i)$ the actual number of vacant EVs assigned from region $i$ to region $j$ , $m^i_{l,j} = h( a^i_{l,j} l^i)$ the actual number of low-battery EVs in region $i$ assigned to region $j$ .","The function $h(\\cdot )$ is used to ensure that the numbers remain as integers and the constraints $\\sum _j m^i_{v,j} = v^i, \\sum _j m^i_{l,j} = l^i$ hold for all $i$ ."],["We restrict the transition kernel $p$ to a $\\delta $ -contamination uncertainty set $P$ , , , in which the state transition could be arbitrarily perturbed by a small probability $\\delta $ .","Specifically, let $\\tilde{p} = \\lbrace \\tilde{p}^a_s \\mid s \\in S, a \\in A \\rbrace $ be the centroid transition kernel, from which training samples are generated.","The $\\delta $ -contamination uncertainty set centered at $\\tilde{p}$ is defined as $P := \\bigotimes _{s \\in S, a \\in A}P^a_s$ , where $P^a_s := \\lbrace (1-\\delta )\\tilde{p}^a_s + \\delta q \\mid q \\in \\Delta (S) \\rbrace , s \\in S, a \\in A$ ."],["Since one of our goals is to minimize the rebalancing cost, we define the shared reward as the negative value of the total rebalancing cost after EVs execute the decisions: $r(s,a) -[ c_v(s,a) + \\bar{\\alpha }c_l(s,a)]$ , where $\\bar{\\alpha }$ is a positive coefficient, and $c_v(s,a), c_l(s,a)$ are moving distances of all vacant and low-battery EVs under the joint state $s$ and action $a$ , respectively.","We then define the worst-case value function of a joint policy $\\pi $ as the worst-case expected total discounted reward under joint policy $\\pi $ over $P$ : $v^\\pi _r(s) = \\min _{p \\in P} \\mathbb {E}_\\pi \\left[ \\sum _{t=1}^\\infty \\gamma ^{t-1}{r}_t | s_1 = s \\right]$ .","The notation is the same as MARL without considering uncertainty.","By maximizing the shared worst-case value function, region agents are cooperating for the same goal."],["Another goal is to achieve the system-level benefit, i.e., balanced charging utilization and fair service.","We define the charging fairness $u_c$ and mobility fairness $u_m$ in Subsection REF .","If the values of these fairness metrics are higher than some thresholds by applying a rebalancing policy $\\pi $ , we say the policy $\\pi $ provides fair mobility and charging services among the city.","We then augment the MARL problem $G$ with an auxiliary cost function $c$ , and a limit $d$ .","The function $c: S \\times A \\rightarrow \\mathbb {R}$ maps transition tuples to cost, like the usual reward.","Similarly, we let $v^{\\pi }_c(s)$ denote the worst-case state value function of policy $\\pi $ with respect to cost function $c$ : $v^{\\pi }_c(s) = \\min _{p \\in P}\\mathbb {E}_\\pi [\\sum _{t=1}^{\\infty } \\gamma ^{t-1} c(s_{t}, a_{t}) | s_1 = s]$ .","The cost function $c$ is defined as the system fairness (a weighted sum of city's charging fairness $u_c$ and mobility fairness $u_m$ ), i.e., $c(s,a) u_c(s, a) + \\bar{\\beta }u_m(s, a)$ , where $\\bar{\\beta }$ is a positive coefficient.","Then the set of feasible joint policies for our robust and constrained MARL EV rebalancing problem is $\\Pi _{C} := \\lbrace \\pi : \\forall s\\in S, v^{\\pi }_c(s) \\ge d \\rbrace $ ."],["The goal of our robust and constrained MARL EV rebalancing problem is to find an optimal joint policy $\\pi ^*$ that maximizes the worst-case value function subject to constraints on the worst-case expected cost $\\pi ^* = \\operatornamewithlimits{argmax}_{\\pi \\in \\Pi _{C}} v^{\\pi }_r(\\rho )$ , where $v^{\\pi _\\theta }_{\\text{tp}}(\\rho )= \\mathbb {E}_{s\\sim \\rho }[v^{\\pi _\\theta }_{\\text{tp}}(s)]$ , $\\text{tp} \\in \\lbrace r,c\\rbrace $ .","We consider policies $\\pi (\\cdot | \\theta )$ parameterized by $\\theta $ and consider the following equivalent max-min problem based on the Lagrangian : $\\max _{\\theta } \\min _{\\lambda \\ge 0} J(\\theta , \\lambda ) := v^{\\pi _\\theta }_r(\\rho ) + \\lambda (v^{\\pi _\\theta }_c(\\rho ) - d),$"],["Fairness Definition","We consider both the mobility supply-demand ratio , , and the charging utilization rate , , , in each region as service quality metrics for the EV AMoD system.","However, with limited supply volume in a city, achieving high supply-demand ratios in all regions may not be possible.","Keeping the supply-demand ratio of each region at a similar level allows passengers in the city to receive fair service , .","Similarly, given a limited number of charging stations and spots, to improve the charging service quality and charging efficiency with limited infrastructure, balancing the charging utilization rate of all regions across the entire city is usually one objective in the scheduling of EV charging , , .","The fairness metrics of the charging utilization rate $u_c$ and supply-demand ratio $u_m$ are designed based on the difference between the local and global quantities: $u_c(s,a) = -\\sum ^N_{i = 1} \\left|\\frac{E_i}{C_i}- \\frac{\\sum ^N_{j = 1}E_j}{\\sum ^N_{j = 1}C_j} \\right|$ , $u_m(s,a)= -\\sum ^N_{i = 1} \\left|\\frac{D_i}{V_i^{ava}}- \\frac{\\sum ^N_{j = 1}D_j}{\\sum ^N_{j = 1}V_j^{ava}} \\right|$ , where $V^{ava}_i$ is the number of available EVs in region $i$ .","The fairness metrics $u_s(s,a)$ and $u_m(s,a)$ are calculated given the EVs rebalancing action $a$ , and the larger the better.","One advantage of the proposed robust and constrained MARL formulation is that the form of the reward/cost function does not need to satisfy the requirements as those of the robust optimization methods , , e.g., the objective/constraints do not need to be convex of the decision variable or concave of the uncertain parameters.","In this section, we derive robust natural policy gradient for robust and constrained MARL to efficiently train policies and alleviate overshooting/undershooting and high variance which results in slow convergence .","Then we present a robust and constrained MARL algorithm (ROCOMA) to solve the EV rebalancing problem under model uncertainties and fairness constraints."],["Robust Policy Gradient Descent Ascent","The problem (REF ) can be solved by Gradient Descent Ascent (GDA) , which currently is a widely used algorithm for solving minimax optimization problems.","At each iteration, GDA simultaneously performs gradient descent update on the variable $\\lambda $ and gradient ascent update on the variable $\\theta $ with step sizes $\\alpha _t$ and $\\beta _{t}$ : $\\theta _{t+1} = \\theta _t + \\alpha _t (\\nabla _\\pi v^{\\pi }_r(\\rho ) + \\lambda _t\\nabla _\\pi v^{\\pi }_c(\\rho )) $ , $\\lambda _{t+1} = \\operatorname{proj}_{\\lambda \\ge 0} \\left[ \\lambda _{t} - \\beta _{t} ( v^{\\pi }_c(\\rho ) - d) \\right]$ .","For notation convenience, we omit the subscripts $r$ and $c$ in the value functions when there is no confusion.","The robust policy gradient of the value function is given by $\\nabla _\\theta v^\\pi (s_1) = \\sum _{s,a} d^\\pi _{\\gamma ,\\delta ,s_1}(s) \\nabla _\\theta \\pi (a|s)\\phi ^\\pi (\\tau ) + b^\\pi \\propto \\mathbb {E}_{\\pi ,s_1}[\\phi ^\\pi (\\tau )\\nabla \\log \\pi (a|s)+b^\\pi ]$ , where $d^\\pi _{\\gamma , \\delta , s_1} \\propto \\sum _k\\gamma ^k(1-\\delta )^k p^\\pi (s_k=s|s_1)$ is the discounted visitation distribution of $s_{k}=s$ when initial state is $s_1$ and policy $\\pi $ is used; $\\tau $ denotes a trajectory $(s,a,r,c,s^\\prime )$ ; $\\phi ^\\pi (\\tau ) := r+ \\gamma \\delta \\min _sv^\\pi (s)+\\gamma (1-\\delta )v^\\pi (s^\\prime )-v^\\pi (s)$ is the TD residual; $b^\\pi :={\\gamma \\delta }/{(1-\\gamma +\\gamma \\delta )}\\partial _\\theta \\min _{s}v^\\pi (s)$ .","Similar to the vanilla policy gradient in the non-robust RL setting, robust policy gradient suffers from overshooting or undershooting and high variance, which results in slow convergence .","Hence we propose a robust natural policy gradient (RNPG) method as follows to update the policy along the steepest ascent direction in the policy space, ."],["Robust Natural Policy Gradient (RNPG)","Natural policy gradient (NPG) , , applies a preconditioning matrix to the gradient, and updates the policy along the steepest descent direction in the policy space, .","It has been proved that NPG moves toward choosing a greedy optimal action rather than just a better action in the literature .","Generally, for a function $L$ defined on a Riemannian manifold $\\Theta $ with a metric $M$ , the steepest descent direction of $L$ at $\\theta $ is given by $-M^{-1}(\\theta )\\nabla L(\\theta )$ , which is called the natural gradient of $L$ .","In the policy parameter space $\\left\\lbrace \\pi _\\theta \\right\\rbrace $ , the natural gradient of $L$ at $\\theta $ is given by $\\tilde{\\nabla } L(\\theta )=F(\\theta )^{-1}\\nabla L(\\theta )$ , where $F(\\theta ) := \\mathbb {E}_{s} \\left[ F_s(\\theta ) \\right]$ is the Fisher information matrix at $\\theta $ and $F_s(\\theta ) = \\mathbb {E}_{\\pi (a|s, \\theta )} \\left[ \\frac{\\partial \\log \\pi (a|s, \\theta )}{\\partial \\theta _i} \\frac{\\partial \\log \\pi (a|s, \\theta )}{\\partial \\theta _j} \\right]$ .","Although the natural gradient method has been studied in non-robust RL, it is not straightforward to efficiently find the NPG for a robust and constrained MARL problem.","We show the natural policy gradient for robust and constrained MARL in the following Theorem REF .","We first denote $\\psi ^\\pi (s,a) = \\nabla \\log \\pi (a|s,\\theta )$ and the Fisher information matrix is then given by $F(\\theta ) = \\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}(s)\\pi (a|s)\\psi ^\\pi (s,a)\\psi ^\\pi (s,a)^\\top $ .","Theorem 4.1 (Robust Natural Policy Gradient) Let $\\tilde{g}^*$ minimizes the objective $J(\\tilde{g},\\pi _\\theta )$ defined as follows: $\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s)[\\tilde{g}^\\top \\psi ^\\pi (s,a) - \\phi ^\\pi (\\tau ) - b^\\pi ]^2.$ Then $\\tilde{g}^*= F(\\theta )^{-1}\\nabla _\\theta v^\\pi (s_1)$ being the natural policy gradient of the objective function $v^\\pi (s_1)$ .","Since $\\tilde{g}^*$ minimizes (REF ), it satisfies the condition $\\partial J/\\partial \\tilde{g}_i = 0$ , which implies: $\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s) \\times \\psi ^\\pi (s,a)[ \\psi ^\\pi (s,a)^\\top \\tilde{g}^* - \\phi ^\\pi (\\tau ) -b^\\pi ] = 0$ or equivalently: $&\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s) \\psi ^\\pi (s,a)\\psi ^\\pi (s,a)^\\top \\tilde{g}^*\\\\= &\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s) \\psi ^\\pi (s,a)[ \\phi ^\\pi (\\tau ) +b^\\pi ]\\nonumber $ By the definition of Fisher information: $\\text{LHS} = F(\\theta )\\tilde{g}^*$ and $\\text{RHS} = \\nabla _\\theta v^\\pi (s_1)$ , which lead to: $F(\\theta ) \\tilde{g}^* = \\nabla _\\theta v^\\pi (s_1)$ .","Solving for $\\tilde{g}^*$ gives $\\tilde{g}^* = F(\\theta )^{-1}\\nabla _\\theta v^\\pi (s_1)$ which follows from the definition of the NPG."],["Robust and Constrained Multi-Agent Reinforcement Learning Algorithm (ROCOMA)","We then propose a robust and constrained MARL (ROCOMA) algorithm to solve the problem (REF ) and train a robust joint policy $\\pi $ using GDA and RNPG.","The proposed algorithm is shown in Algorithm REF .","Robust and Constrained Multi-Agent Reinforcement Learning Algorithm (ROCOMA) [1] Input $\\zeta ,\\alpha ,\\beta ,\\gamma ,\\delta $ .","Initialize $\\theta _0, \\lambda _0$ .","$t=0$ to $T$ Estimate $v_r^{\\theta _t}, v_c^{\\theta _t}$ using Algorithm 3 in $j=1$ to $M$ Sample $T_j \\sim \\textit {Geom}(1-\\gamma +\\gamma \\delta )$ , $s_1^j \\sim \\rho $ Sample trajectory from $s^j_1$ : $(s^j_1, a^j_1, \\cdots , s^j_{T_j})$ agent $i=1$ to $N$ $k=1$ to $W$ $\\tilde{g}^j_{t,k+1}(i)= \\arg \\min _{\\tilde{g}} ||\\tilde{g}^j_{t,k}(i)-\\tilde{g}-\\zeta \\nabla _{\\tilde{g}}\\mathcal {L}(\\tilde{g}_{t,k}(i), \\theta _t)||$ , $\\mathcal {L}$ is defined in (REF ) $\\tilde{g}^j_{t,k} = \\sum _{i=1}^N \\tilde{g}^j_{t,k}(i)/N$ $\\tilde{g}_t = \\sum _{j=1}^M\\sum _{k=1}^W \\tilde{g}^j_{t,k}/MW$ $\\theta _{t+1} = \\theta _t + \\alpha _t(\\tilde{g}_{r,t} - \\lambda _t \\tilde{g}_{c,t})$ $\\lambda _{t+1} = \\max \\lbrace \\lambda _t - \\beta _t( \\sum _{j}v_c^{\\theta _t}(s_1^j)/M-d),0\\rbrace $ Output $\\theta _T$ We first randomly initialize the actor neural network parameter $\\theta _0$ and the Lagrange multiplier parameter $\\lambda _0$ .","At each iteration $t$ , we first estimate the critic neural networks $v_r^{\\theta _t}, v_c^{\\theta _t}$ under policy $\\pi ^{\\theta _t}$ using Algorithm 3 in .","Line REF to line REF in Algorithm REF estimate the RNPG for $v_r^{\\theta _t}$ and $v_c^{\\theta _t}$ .","We then sample an initial state $s^j_1$ following the initial distribution $\\rho $ and a time horizon $T_j$ from the geometric distribution $\\textit {Geom}(1-\\gamma + \\gamma \\delta )$ at iteration $j=1, \\cdots , M$ .","These samples are used to estimate the RNPG.","As shown in Theorem REF , we can get the RNPG of $v^\\pi (s_1)$ by minimizing the objective defined in (REF ).","To minimize (REF ) and get the minimizer, we initialize $\\tilde{g}_{t,0}=0$ and use the following stochastic gradient descent (SGD) steps with $l_2$ -projection: $\\tilde{g}_{t,k+1} = \\arg \\min _{\\tilde{g}} ||\\tilde{g}_{t,k}-\\zeta \\nabla _{\\tilde{g}} \\mathcal {L}(\\tilde{g}_{t,k}, \\theta _t) - \\tilde{g}||$ , where $\\zeta $ is the learning rate and $\\mathcal {L}$ is defined in (REF ) as follows.","$\\mathcal {L}(\\tilde{g},\\theta ) = \\sum _{\\mathcal {D}(s^j_{T_j})}[\\tilde{g}^\\top \\psi ^{\\theta }(s,a) - \\phi ^\\theta (\\tau ) - b^\\theta ]^2/D,$ where $\\mathcal {D}(s^j_{T_j})$ is a set of trajectories $\\tau $ starting at $s^j_{T_j}$ using policy $\\pi ^{\\theta _t}$ , i.e.","$(s^j_{T_j},a,r,c,s^\\prime )$ , $D=|\\mathcal {D}(s^j_{T_j})|$ .","After $W$ steps of SDG iterations, the robust natural policy gradient for $v^{\\theta _t}(s^j_1)$ is estimated as $\\sum _{k=1}^W\\tilde{g}_{t,k}^j/W$ .","To reduce the computational complexity, we adopt the centralized training and decentralized execution (CTDE) framework in ROCOMA and assume all agents share the same policy $\\pi ^{\\theta ^i}(a^i|s^i)$ , where $\\theta ^1=\\cdots =\\theta ^N=\\theta $ .","Then we have $\\nabla \\pi (a|s) = \\sum _{i}^N \\psi ^{\\theta }_i(s,a)$ where $\\psi ^{\\theta }_i(s,a) := \\pi ^{-i}(a^{-i}|s^{-i})\\nabla \\pi ^i(a^i|s^i)$ , $\\pi ^{-i}(a^{-i}|s^{-i}):=\\prod _{j\\ne i}\\pi ^j(a^j|s^j)$ .","Therefore, in lines REF to REF , we address the high-dimensional action and state space issue in computing RNPG by using $\\psi ^{\\theta }_i(s,a)$ instead of $\\psi ^{\\theta }(s,a)$ in (REF ).","Finally, we update $\\theta _{t+1}$ and $\\lambda _{t+1}$ using GDA."],["Experiment Setup"," Three different data sets , including E-taxi GPS data, transaction data and charging station data are used to build an EV AMoD system simulator as the training and testing environment.","The simulated map is set as a grid city.","The policy networks and critic networks are two-layer fully-connected networks, both with 32 nodes.","We use Softplus as activations to ensure the output is positive.","The output of policy networks is used to be the concentration parameters of the Dirichlet distribution to satisfy the action constraints (sum to one).","We set the maximal training episode number $=20000$ , the maximal policy/critic estimation number $=2000$ , the NRPG SDG iteration number $=500$ , the discount rate $\\gamma = 0.99$ , the perturbed rate $\\delta =0.05$ , the coefficients $\\bar{\\alpha }=\\bar{\\beta }=1$ , the fairness constraint limit $d=-20$ for one simulation step, and use AdamOptimizer with a learning rate of $0.001$ for both policy/critic networks."],["Experiment Results"," Our goal of the experiments is to validate the following hypothesis: (1) The proposed ROCOMA can learn effective rebalancing policies; (2) Our proposed method is more robust than a non-robust MARL algorithm through our robust and constrained MARL design.","Other than Rebalancing cost: the total moving distance of vacant and low-battery EVs by using rebalancing policy (the lower the better); and System fairness: the weighted sum of mobility and charging fairness (the higher the better); we also monitor Number of expired orders: the total number of canceled orders due to waiting for more than 20 minutes (the lower the better) and Order response rate: the ratio between the number of served demands and the number of total passenger demand (the higher the better).","All metrics are calculated in every testing period which consists of 25 simulation steps.","We repeat testing for 10 times and show the average values.","Table: Conclusion"]],"string":"[\n [\n \"A Robust and Constrained Multi-Agent Reinforcement Learning Framework\\n for Electric Vehicle AMoD Systems\"\n ],\n [\n \"Abstract Electric vehicles (EVs) play critical roles in autonomous mobility-on-demand (AMoD) systems, but their unique charging patterns increase the model uncertainties in AMoD systems (e.g.\",\n \"state transition probability).\",\n \"Since there usually exists a mismatch between the training and test (true) environments, incorporating model uncertainty into system design is of critical importance in real-world applications.\",\n \"However, model uncertainties have not been considered explicitly in EV AMoD system rebalancing by existing literature yet and remain an urgent and challenging task.\",\n \"In this work, we design a robust and constrained multi-agent reinforcement learning (MARL) framework with transition kernel uncertainty for the EV rebalancing and charging problem.\",\n \"We then propose a robust and constrained MARL algorithm (ROCOMA) that trains a robust EV rebalancing policy to balance the supply-demand ratio and the charging utilization rate across the whole city under state transition uncertainty.\",\n \"Experiments show that the ROCOMA can learn an effective and robust rebalancing policy.\",\n \"It outperforms non-robust MARL methods when there are model uncertainties.\",\n \"It increases the system fairness by 19.6% and decreases the rebalancing costs by 75.8%.\"\n ],\n [\n \"Introduction\",\n \" Autonomous mobility-on-demand (AMoD) system is one of the most promising energy-efficient transportation solutions as it provides people one-way rides from their origins to destinations , , .\",\n \"Electric vehicles (EVs) are being adopted worldwide for environmental and economical benefits , and AMoD systems embrace this trend without exception.\",\n \"However, the trips sporadically appear, and the origins and destinations are asymmetrically distributed and hard to predict in AMoD systems , .\",\n \"Such spatial-temporal nature of urban mobility increases the management difficulty of a large-scale vehicle fleet and makes the system sensitive to disturbances , , .\",\n \"Moreover, EVs' unique charging patterns (long charging time, high charging frequency, and unpredictable charging behaviors) increase the complexity and uncertainty of the EV AMoD system dynamics , , .\",\n \"In real-world applications, we usually do not have perfectly accurate knowledge of the true system model, e.g., the state transition probability of the AMoD systems, and therefore, there usually exists a model mismatch between the simulator (training environment) and the real-world application (test environment).\",\n \"Thus, existing EV AMoD vehicle coordination methods , , , , , , may have significant performance degradation in the test (true) environment.\",\n \"Despite model-based methods considering prediction errors in mobility demand or vehicle supply , , , , uncertainty in system state transition remains largely unexplored in AMoD systems.\",\n \"More importantly, practical deployment usually needs to satisfy constraints on, e.g.\",\n \"safety and fairness, while maximizing the performance.\",\n \"In this work, we propose a robust and constrained multi-agent reinforcement learning (MARL) framework for EV AMoD systems.\",\n \"The goal is to achieve AMoD system fairness by finding robust rebalancing policies for idle and low-battery EVs with minimal rebalancing cost, under model uncertainty.\",\n \"The advantages of our formulation are two-folds: (i) fairness constraints can still be satisfied even if there is model mismatch; and (ii) the rebalancing cost is still optimized when there is model mismatch.\",\n \"Our Key Contributions are as follows: (1) To the best of our knowledge, this work is the first to formulate EV AMoD system vehicle rebalancing as a robust and constrained multi-agent reinforcement learning problem under model uncertainty.\",\n \"Via a proper design of the state, action, reward, cost constraints, and uncertainty set, we set our goal as minimizing the rebalancing cost while balancing the city's charging utilization and service quality, under model uncertainty.\",\n \"(2) We design a robust and constrained MARL algorithm (ROCOMA) to efficiently train robust policies.\",\n \"The proposed algorithm adopts the centralized training and decentralized execution (CTDE) framework and develops the first robust natural policy gradient (NPG) to improve the efficiency of policy training.\",\n \"(3) We run experiments based on real-world E-taxi system data.\",\n \"We show that our proposed algorithm performs better in terms of reward and fairness, which are increased by 19.6%, and 75.8%, respectively, compared with a non-robust MARL-based method when model uncertainty is present.\"\n ],\n [\n \"Related Work\",\n \" AMoD system vehicle rebalancing algorithms re-allocate idle vehicles, sometimes considering charging constraints .\",\n \"Heuristics can lead to sub-optimal rebalancing solutions .\",\n \"Other major categories of AMoD system rebalancing methods include optimization-based algorithms , Model Predictive Control (MPC) and Reinforcement Learning (RL) .\",\n \"Optimization and MPC-based approaches usually formulate the AMoD system vehicle rebalancing problem as a convex optimization or mixed-integer programming problem, where the objective is to improve service quality , or maximize the number of served passengers with fewer vehicles , , .\",\n \"These model-based approaches usually rely on precise knowledge of the probability transition model of the complex dynamics of AMoD systems and, consequently, are sensitive to model uncertainties.\",\n \"Though robust and distributionally robust optimization-based methods have been designed to consider uncertainties caused by mobility demand, supply, or covariates predictions , , , the probability transition error or uncertainty in system dynamics has not been addressed yet.\",\n \"RL-based approaches relax the dependency on system dynamic models.\",\n \"Various RL-based methods, including DQN, A2C and their variants , , , , , , , have been proposed to solve the vehicle rebalancing problem.\",\n \"However, RL suffers from the sim-to-real gap; that is, the gap between the simulator and the real world often leads to unsuccessful implementation if the learned policy is not robust to model uncertainties , .\",\n \"None of the above RL-based rebalancing strategies consider this gap.\",\n \"Given that designing an efficient and robust EV rebalancing method under model uncertainties is still an unsolved challenge, Robust RL aims to find a policy that maximizes the worst-case cumulative reward over an uncertainty set of MDPs , , , .\",\n \"To achieve high system fairness while minimizing rebalancing cost under model uncertainty, we put the fairness constraints in our RL formulation, which is known as Constrained RL that aims to find a policy that maximizes an objective function while satisfying certain cost constraints , .\",\n \"However, applying robust and constrained RL to AMoD rebalancing is still a challenge due to the high-dimensional state and action spaces commonly present in transportation systems.\",\n \"And the problem of robust constrained RL itself is very challenging even in the simple tabular case.\",\n \"Our proposed robust and constrained MARL formulation and algorithm explicitly consider model uncertainties and system fairness to learn robust rebalancing solutions for AMoD systems.\",\n \"And we derive a natural policy gradient for robust and constrained MARL to improve the efficiency of policy training.\"\n ],\n [\n \"Problem Statement\",\n \"We consider the problem of managing a large-scale EV fleet to provide fair and robust AMoD service.\",\n \"The goal is to (i) rebalance idle and energy-efficient EVs (we denote them as vacant EVs for notation convenience) among different regions to provide fair mobility service on the passenger's side; (ii) allocate low-battery EVs to charging stations for fair charging service on the EVs' side; (iii) minimize the managing cost of (i) and (ii); (iv) find rebalancing policies robust to model uncertainty, i.e.\",\n \"uncertainty in transition kernel of the MDP.\",\n \"We assume that a city is divided into $N$ regions according to a pre-defined partition method , , .\",\n \"A day is divided into equal-length time intervals.\",\n \"In each time interval $[t, t+1)$ , customers' ride requests and EVs' charging need are aggregated in each region.\",\n \"After the location and status of each EV are observed, a local trip and charging assignment algorithm matches available EVs with passengers and low-battery EVs with charging stations, using existing methods in the literature , , .\",\n \"Then the state information of each region is updated, including the numbers of vacant EVs and available charging spots in each region.\",\n \"Each region then rebalances both vacant and low-battery EVs according to the trained MARL policy.\",\n \"This work focuses on a robust EV rebalancing algorithm design under model uncertainties to maximize the worst-case expected reward of the system while satisfying fairness constraints.\",\n \"For notational convenience, the parameters and variables defined in the following parts of this section omit the time index $t$ when there is no confusion.\"\n ],\n [\n \"Preliminary: Multi-Agent Reinforcement Learning\",\n \"We denote a Multi-Agent Reinforcement Learning (MARL) problem by a tuple $G = \\\\langle \\\\mathcal {N}, S, A, r,p,\\\\gamma \\\\rangle $ , in which $\\\\mathcal {N}$ is the set of $N$ agents.\",\n \"Each agent is associated with an action $a^i \\\\in A^i$ and a state $s^i \\\\in S$ .\",\n \"We use $A = A^1 \\\\times \\\\cdots \\\\times A^N$ to denote the joint action space, and $S = S^1 \\\\times \\\\cdots \\\\times S^N$ the joint state space.\",\n \"At time $t$ , each agent chooses an action $a^i_t$ according to a policy $\\\\pi ^i: S^i \\\\rightarrow \\\\Delta (A^i)$ , where $\\\\Delta (A^i)$ represents the set of probability distributions over the action set $A^i$ .\",\n \"We use $\\\\pi = \\\\prod _i^N \\\\pi ^i: S \\\\rightarrow \\\\Delta (A)$ to denote the joint policy.\",\n \"After executing the joint action, the next state follows the state transition probability which depends on the current state and the joint action, i.e.\",\n \"$p: S \\\\times A \\\\rightarrow \\\\Delta (S)$ .\",\n \"And each agent receives a reward according to the reward function $r^i: S \\\\times A \\\\rightarrow \\\\mathbb {R}$ .\",\n \"Each agent aims to learn a policy $\\\\pi ^i$ to maximize its expected total discounted reward, i.e.\",\n \"$\\\\max _{\\\\pi ^i} v^{\\\\pi , i}_r(s)$ for all $s \\\\in S$ , where $v^{\\\\pi , i}_r(s) = \\\\mathbb {E}[\\\\sum _{t = 1}^{\\\\infty } \\\\gamma ^{t-1} r_t^i(s_t, a_t) | a_t \\\\sim \\\\pi (\\\\cdot | s_t), s_1 = s]$ which is also known as the state value function for agent $i$ .\",\n \"$\\\\gamma \\\\in (0,1)$ is the discounted rate.\",\n \"If these agents belong to a team, the objective of all agents is to collaboratively maximize the average expected total discounted reward over all agents, i.e.\",\n \"$\\\\max _\\\\pi v^{\\\\pi }_{r}(s)$ for all $s \\\\in S$ , where $v^{\\\\pi }_{r}(s) = \\\\mathbb {E}_\\\\pi [\\\\sum _{t = 1}^{\\\\infty } \\\\gamma ^{t-1} \\\\sum _{i\\\\in \\\\mathcal {N}}r_t^i(s_t, a_t)/N | s_1 = s]$ .\"\n ],\n [\n \"Robust and Constrained Multi-Agent Reinforcement Learning Formulation for EV Rebalancing\",\n \"We formulate the EV rebalancing problem as a robust and constrained MARL problem $G_{rc} = \\\\langle \\\\mathcal {N}, S, A, P, r, c, d, \\\\gamma \\\\rangle $ , and we define the agent, state, action, probability transition kernel uncertainty set, reward, and cost and fairness constraints as follows.\"\n ],\n [\n \"We define for each region a region agent, who determines the rebalancing of vacant and low-battery EVs at every time step.\",\n \"This distributed agent setting is more tractable for large-scale fleet management than a single agent setting because the action space can be prohibitively large if we use a single system-wide agent .\"\n ],\n [\n \"A state $s^{i}$ of a region agent $i$ consists two parts that indicate its spatiotemporal status from both the local view and global view of the city.\",\n \"We define the state $s^i=\\\\lbrace s_{loc}^i, s_{glo}^i\\\\rbrace $ , where $s_{loc}^i = (V_i, L_i, D_i, E_i, C_i)$ is the state of region $i$ from the local view, denoting the number/amount of vacant EVs, low-battery EVs, mobility demand, empty charging spots, and total charging spots in region $i$ , respectively.\",\n \"And $s_{glo}^i = (t, pos_i)$ , where $t$ is the time index (which time interval), $pos_i$ is region location information (longitudes, latitudes, region index).\",\n \"The initial state distribution is $\\\\rho $ .\"\n ],\n [\n \"The rebalancing action for vacant EVs is denoted as $a^i_v = \\\\lbrace a^i_{v,j} \\\\rbrace _{j \\\\in \\\\text{Nebr}_i}$ , the charging action for low-battery EVs as $a^i_l = \\\\lbrace a^i_{l,j} \\\\rbrace _{j \\\\in \\\\text{Nebr}_i}$ , where $a^i_{v,j}, a^i_{l,j} \\\\in [0,1]$ is the percentage of currently vacant EVs and low-battery EVs to be assigned to region $j$ from region $i$ , respectively.\",\n \"And $\\\\text{Nebr}_i$ is the set consisting of region $i$ and its adjacent regions as defined by the given partition.\",\n \"Therefore $\\\\sum _{j \\\\in \\\\text{Nebr}_i} a^i_{v,j} = 1$ and $\\\\sum _{j \\\\in \\\\text{Nebr}_i} a^i_{l,j} = 1$ for all $i$ .\",\n \"We denote $m^i_{v,j} = h( a^i_{v,j} v^i)$ the actual number of vacant EVs assigned from region $i$ to region $j$ , $m^i_{l,j} = h( a^i_{l,j} l^i)$ the actual number of low-battery EVs in region $i$ assigned to region $j$ .\",\n \"The function $h(\\\\cdot )$ is used to ensure that the numbers remain as integers and the constraints $\\\\sum _j m^i_{v,j} = v^i, \\\\sum _j m^i_{l,j} = l^i$ hold for all $i$ .\"\n ],\n [\n \"We restrict the transition kernel $p$ to a $\\\\delta $ -contamination uncertainty set $P$ , , , in which the state transition could be arbitrarily perturbed by a small probability $\\\\delta $ .\",\n \"Specifically, let $\\\\tilde{p} = \\\\lbrace \\\\tilde{p}^a_s \\\\mid s \\\\in S, a \\\\in A \\\\rbrace $ be the centroid transition kernel, from which training samples are generated.\",\n \"The $\\\\delta $ -contamination uncertainty set centered at $\\\\tilde{p}$ is defined as $P := \\\\bigotimes _{s \\\\in S, a \\\\in A}P^a_s$ , where $P^a_s := \\\\lbrace (1-\\\\delta )\\\\tilde{p}^a_s + \\\\delta q \\\\mid q \\\\in \\\\Delta (S) \\\\rbrace , s \\\\in S, a \\\\in A$ .\"\n ],\n [\n \"Since one of our goals is to minimize the rebalancing cost, we define the shared reward as the negative value of the total rebalancing cost after EVs execute the decisions: $r(s,a) -[ c_v(s,a) + \\\\bar{\\\\alpha }c_l(s,a)]$ , where $\\\\bar{\\\\alpha }$ is a positive coefficient, and $c_v(s,a), c_l(s,a)$ are moving distances of all vacant and low-battery EVs under the joint state $s$ and action $a$ , respectively.\",\n \"We then define the worst-case value function of a joint policy $\\\\pi $ as the worst-case expected total discounted reward under joint policy $\\\\pi $ over $P$ : $v^\\\\pi _r(s) = \\\\min _{p \\\\in P} \\\\mathbb {E}_\\\\pi \\\\left[ \\\\sum _{t=1}^\\\\infty \\\\gamma ^{t-1}{r}_t | s_1 = s \\\\right]$ .\",\n \"The notation is the same as MARL without considering uncertainty.\",\n \"By maximizing the shared worst-case value function, region agents are cooperating for the same goal.\"\n ],\n [\n \"Another goal is to achieve the system-level benefit, i.e., balanced charging utilization and fair service.\",\n \"We define the charging fairness $u_c$ and mobility fairness $u_m$ in Subsection REF .\",\n \"If the values of these fairness metrics are higher than some thresholds by applying a rebalancing policy $\\\\pi $ , we say the policy $\\\\pi $ provides fair mobility and charging services among the city.\",\n \"We then augment the MARL problem $G$ with an auxiliary cost function $c$ , and a limit $d$ .\",\n \"The function $c: S \\\\times A \\\\rightarrow \\\\mathbb {R}$ maps transition tuples to cost, like the usual reward.\",\n \"Similarly, we let $v^{\\\\pi }_c(s)$ denote the worst-case state value function of policy $\\\\pi $ with respect to cost function $c$ : $v^{\\\\pi }_c(s) = \\\\min _{p \\\\in P}\\\\mathbb {E}_\\\\pi [\\\\sum _{t=1}^{\\\\infty } \\\\gamma ^{t-1} c(s_{t}, a_{t}) | s_1 = s]$ .\",\n \"The cost function $c$ is defined as the system fairness (a weighted sum of city's charging fairness $u_c$ and mobility fairness $u_m$ ), i.e., $c(s,a) u_c(s, a) + \\\\bar{\\\\beta }u_m(s, a)$ , where $\\\\bar{\\\\beta }$ is a positive coefficient.\",\n \"Then the set of feasible joint policies for our robust and constrained MARL EV rebalancing problem is $\\\\Pi _{C} := \\\\lbrace \\\\pi : \\\\forall s\\\\in S, v^{\\\\pi }_c(s) \\\\ge d \\\\rbrace $ .\"\n ],\n [\n \"The goal of our robust and constrained MARL EV rebalancing problem is to find an optimal joint policy $\\\\pi ^*$ that maximizes the worst-case value function subject to constraints on the worst-case expected cost $\\\\pi ^* = \\\\operatornamewithlimits{argmax}_{\\\\pi \\\\in \\\\Pi _{C}} v^{\\\\pi }_r(\\\\rho )$ , where $v^{\\\\pi _\\\\theta }_{\\\\text{tp}}(\\\\rho )= \\\\mathbb {E}_{s\\\\sim \\\\rho }[v^{\\\\pi _\\\\theta }_{\\\\text{tp}}(s)]$ , $\\\\text{tp} \\\\in \\\\lbrace r,c\\\\rbrace $ .\",\n \"We consider policies $\\\\pi (\\\\cdot | \\\\theta )$ parameterized by $\\\\theta $ and consider the following equivalent max-min problem based on the Lagrangian : $\\\\max _{\\\\theta } \\\\min _{\\\\lambda \\\\ge 0} J(\\\\theta , \\\\lambda ) := v^{\\\\pi _\\\\theta }_r(\\\\rho ) + \\\\lambda (v^{\\\\pi _\\\\theta }_c(\\\\rho ) - d),$\"\n ],\n [\n \"Fairness Definition\",\n \"We consider both the mobility supply-demand ratio , , and the charging utilization rate , , , in each region as service quality metrics for the EV AMoD system.\",\n \"However, with limited supply volume in a city, achieving high supply-demand ratios in all regions may not be possible.\",\n \"Keeping the supply-demand ratio of each region at a similar level allows passengers in the city to receive fair service , .\",\n \"Similarly, given a limited number of charging stations and spots, to improve the charging service quality and charging efficiency with limited infrastructure, balancing the charging utilization rate of all regions across the entire city is usually one objective in the scheduling of EV charging , , .\",\n \"The fairness metrics of the charging utilization rate $u_c$ and supply-demand ratio $u_m$ are designed based on the difference between the local and global quantities: $u_c(s,a) = -\\\\sum ^N_{i = 1} \\\\left|\\\\frac{E_i}{C_i}- \\\\frac{\\\\sum ^N_{j = 1}E_j}{\\\\sum ^N_{j = 1}C_j} \\\\right|$ , $u_m(s,a)= -\\\\sum ^N_{i = 1} \\\\left|\\\\frac{D_i}{V_i^{ava}}- \\\\frac{\\\\sum ^N_{j = 1}D_j}{\\\\sum ^N_{j = 1}V_j^{ava}} \\\\right|$ , where $V^{ava}_i$ is the number of available EVs in region $i$ .\",\n \"The fairness metrics $u_s(s,a)$ and $u_m(s,a)$ are calculated given the EVs rebalancing action $a$ , and the larger the better.\",\n \"One advantage of the proposed robust and constrained MARL formulation is that the form of the reward/cost function does not need to satisfy the requirements as those of the robust optimization methods , , e.g., the objective/constraints do not need to be convex of the decision variable or concave of the uncertain parameters.\",\n \"In this section, we derive robust natural policy gradient for robust and constrained MARL to efficiently train policies and alleviate overshooting/undershooting and high variance which results in slow convergence .\",\n \"Then we present a robust and constrained MARL algorithm (ROCOMA) to solve the EV rebalancing problem under model uncertainties and fairness constraints.\"\n ],\n [\n \"Robust Policy Gradient Descent Ascent\",\n \"The problem (REF ) can be solved by Gradient Descent Ascent (GDA) , which currently is a widely used algorithm for solving minimax optimization problems.\",\n \"At each iteration, GDA simultaneously performs gradient descent update on the variable $\\\\lambda $ and gradient ascent update on the variable $\\\\theta $ with step sizes $\\\\alpha _t$ and $\\\\beta _{t}$ : $\\\\theta _{t+1} = \\\\theta _t + \\\\alpha _t (\\\\nabla _\\\\pi v^{\\\\pi }_r(\\\\rho ) + \\\\lambda _t\\\\nabla _\\\\pi v^{\\\\pi }_c(\\\\rho )) $ , $\\\\lambda _{t+1} = \\\\operatorname{proj}_{\\\\lambda \\\\ge 0} \\\\left[ \\\\lambda _{t} - \\\\beta _{t} ( v^{\\\\pi }_c(\\\\rho ) - d) \\\\right]$ .\",\n \"For notation convenience, we omit the subscripts $r$ and $c$ in the value functions when there is no confusion.\",\n \"The robust policy gradient of the value function is given by $\\\\nabla _\\\\theta v^\\\\pi (s_1) = \\\\sum _{s,a} d^\\\\pi _{\\\\gamma ,\\\\delta ,s_1}(s) \\\\nabla _\\\\theta \\\\pi (a|s)\\\\phi ^\\\\pi (\\\\tau ) + b^\\\\pi \\\\propto \\\\mathbb {E}_{\\\\pi ,s_1}[\\\\phi ^\\\\pi (\\\\tau )\\\\nabla \\\\log \\\\pi (a|s)+b^\\\\pi ]$ , where $d^\\\\pi _{\\\\gamma , \\\\delta , s_1} \\\\propto \\\\sum _k\\\\gamma ^k(1-\\\\delta )^k p^\\\\pi (s_k=s|s_1)$ is the discounted visitation distribution of $s_{k}=s$ when initial state is $s_1$ and policy $\\\\pi $ is used; $\\\\tau $ denotes a trajectory $(s,a,r,c,s^\\\\prime )$ ; $\\\\phi ^\\\\pi (\\\\tau ) := r+ \\\\gamma \\\\delta \\\\min _sv^\\\\pi (s)+\\\\gamma (1-\\\\delta )v^\\\\pi (s^\\\\prime )-v^\\\\pi (s)$ is the TD residual; $b^\\\\pi :={\\\\gamma \\\\delta }/{(1-\\\\gamma +\\\\gamma \\\\delta )}\\\\partial _\\\\theta \\\\min _{s}v^\\\\pi (s)$ .\",\n \"Similar to the vanilla policy gradient in the non-robust RL setting, robust policy gradient suffers from overshooting or undershooting and high variance, which results in slow convergence .\",\n \"Hence we propose a robust natural policy gradient (RNPG) method as follows to update the policy along the steepest ascent direction in the policy space, .\"\n ],\n [\n \"Robust Natural Policy Gradient (RNPG)\",\n \"Natural policy gradient (NPG) , , applies a preconditioning matrix to the gradient, and updates the policy along the steepest descent direction in the policy space, .\",\n \"It has been proved that NPG moves toward choosing a greedy optimal action rather than just a better action in the literature .\",\n \"Generally, for a function $L$ defined on a Riemannian manifold $\\\\Theta $ with a metric $M$ , the steepest descent direction of $L$ at $\\\\theta $ is given by $-M^{-1}(\\\\theta )\\\\nabla L(\\\\theta )$ , which is called the natural gradient of $L$ .\",\n \"In the policy parameter space $\\\\left\\\\lbrace \\\\pi _\\\\theta \\\\right\\\\rbrace $ , the natural gradient of $L$ at $\\\\theta $ is given by $\\\\tilde{\\\\nabla } L(\\\\theta )=F(\\\\theta )^{-1}\\\\nabla L(\\\\theta )$ , where $F(\\\\theta ) := \\\\mathbb {E}_{s} \\\\left[ F_s(\\\\theta ) \\\\right]$ is the Fisher information matrix at $\\\\theta $ and $F_s(\\\\theta ) = \\\\mathbb {E}_{\\\\pi (a|s, \\\\theta )} \\\\left[ \\\\frac{\\\\partial \\\\log \\\\pi (a|s, \\\\theta )}{\\\\partial \\\\theta _i} \\\\frac{\\\\partial \\\\log \\\\pi (a|s, \\\\theta )}{\\\\partial \\\\theta _j} \\\\right]$ .\",\n \"Although the natural gradient method has been studied in non-robust RL, it is not straightforward to efficiently find the NPG for a robust and constrained MARL problem.\",\n \"We show the natural policy gradient for robust and constrained MARL in the following Theorem REF .\",\n \"We first denote $\\\\psi ^\\\\pi (s,a) = \\\\nabla \\\\log \\\\pi (a|s,\\\\theta )$ and the Fisher information matrix is then given by $F(\\\\theta ) = \\\\sum _{s,a}d^\\\\pi _{\\\\gamma ,\\\\delta ,s_1}(s)\\\\pi (a|s)\\\\psi ^\\\\pi (s,a)\\\\psi ^\\\\pi (s,a)^\\\\top $ .\",\n \"Theorem 4.1 (Robust Natural Policy Gradient) Let $\\\\tilde{g}^*$ minimizes the objective $J(\\\\tilde{g},\\\\pi _\\\\theta )$ defined as follows: $\\\\sum _{s,a}d^\\\\pi _{\\\\gamma ,\\\\delta ,s_1}\\\\pi (a|s)[\\\\tilde{g}^\\\\top \\\\psi ^\\\\pi (s,a) - \\\\phi ^\\\\pi (\\\\tau ) - b^\\\\pi ]^2.$ Then $\\\\tilde{g}^*= F(\\\\theta )^{-1}\\\\nabla _\\\\theta v^\\\\pi (s_1)$ being the natural policy gradient of the objective function $v^\\\\pi (s_1)$ .\",\n \"Since $\\\\tilde{g}^*$ minimizes (REF ), it satisfies the condition $\\\\partial J/\\\\partial \\\\tilde{g}_i = 0$ , which implies: $\\\\sum _{s,a}d^\\\\pi _{\\\\gamma ,\\\\delta ,s_1}\\\\pi (a|s) \\\\times \\\\psi ^\\\\pi (s,a)[ \\\\psi ^\\\\pi (s,a)^\\\\top \\\\tilde{g}^* - \\\\phi ^\\\\pi (\\\\tau ) -b^\\\\pi ] = 0$ or equivalently: $&\\\\sum _{s,a}d^\\\\pi _{\\\\gamma ,\\\\delta ,s_1}\\\\pi (a|s) \\\\psi ^\\\\pi (s,a)\\\\psi ^\\\\pi (s,a)^\\\\top \\\\tilde{g}^*\\\\\\\\= &\\\\sum _{s,a}d^\\\\pi _{\\\\gamma ,\\\\delta ,s_1}\\\\pi (a|s) \\\\psi ^\\\\pi (s,a)[ \\\\phi ^\\\\pi (\\\\tau ) +b^\\\\pi ]\\\\nonumber $ By the definition of Fisher information: $\\\\text{LHS} = F(\\\\theta )\\\\tilde{g}^*$ and $\\\\text{RHS} = \\\\nabla _\\\\theta v^\\\\pi (s_1)$ , which lead to: $F(\\\\theta ) \\\\tilde{g}^* = \\\\nabla _\\\\theta v^\\\\pi (s_1)$ .\",\n \"Solving for $\\\\tilde{g}^*$ gives $\\\\tilde{g}^* = F(\\\\theta )^{-1}\\\\nabla _\\\\theta v^\\\\pi (s_1)$ which follows from the definition of the NPG.\"\n ],\n [\n \"Robust and Constrained Multi-Agent Reinforcement Learning Algorithm (ROCOMA)\",\n \"We then propose a robust and constrained MARL (ROCOMA) algorithm to solve the problem (REF ) and train a robust joint policy $\\\\pi $ using GDA and RNPG.\",\n \"The proposed algorithm is shown in Algorithm REF .\",\n \"Robust and Constrained Multi-Agent Reinforcement Learning Algorithm (ROCOMA) [1] Input $\\\\zeta ,\\\\alpha ,\\\\beta ,\\\\gamma ,\\\\delta $ .\",\n \"Initialize $\\\\theta _0, \\\\lambda _0$ .\",\n \"$t=0$ to $T$ Estimate $v_r^{\\\\theta _t}, v_c^{\\\\theta _t}$ using Algorithm 3 in $j=1$ to $M$ Sample $T_j \\\\sim \\\\textit {Geom}(1-\\\\gamma +\\\\gamma \\\\delta )$ , $s_1^j \\\\sim \\\\rho $ Sample trajectory from $s^j_1$ : $(s^j_1, a^j_1, \\\\cdots , s^j_{T_j})$ agent $i=1$ to $N$ $k=1$ to $W$ $\\\\tilde{g}^j_{t,k+1}(i)= \\\\arg \\\\min _{\\\\tilde{g}} ||\\\\tilde{g}^j_{t,k}(i)-\\\\tilde{g}-\\\\zeta \\\\nabla _{\\\\tilde{g}}\\\\mathcal {L}(\\\\tilde{g}_{t,k}(i), \\\\theta _t)||$ , $\\\\mathcal {L}$ is defined in (REF ) $\\\\tilde{g}^j_{t,k} = \\\\sum _{i=1}^N \\\\tilde{g}^j_{t,k}(i)/N$ $\\\\tilde{g}_t = \\\\sum _{j=1}^M\\\\sum _{k=1}^W \\\\tilde{g}^j_{t,k}/MW$ $\\\\theta _{t+1} = \\\\theta _t + \\\\alpha _t(\\\\tilde{g}_{r,t} - \\\\lambda _t \\\\tilde{g}_{c,t})$ $\\\\lambda _{t+1} = \\\\max \\\\lbrace \\\\lambda _t - \\\\beta _t( \\\\sum _{j}v_c^{\\\\theta _t}(s_1^j)/M-d),0\\\\rbrace $ Output $\\\\theta _T$ We first randomly initialize the actor neural network parameter $\\\\theta _0$ and the Lagrange multiplier parameter $\\\\lambda _0$ .\",\n \"At each iteration $t$ , we first estimate the critic neural networks $v_r^{\\\\theta _t}, v_c^{\\\\theta _t}$ under policy $\\\\pi ^{\\\\theta _t}$ using Algorithm 3 in .\",\n \"Line REF to line REF in Algorithm REF estimate the RNPG for $v_r^{\\\\theta _t}$ and $v_c^{\\\\theta _t}$ .\",\n \"We then sample an initial state $s^j_1$ following the initial distribution $\\\\rho $ and a time horizon $T_j$ from the geometric distribution $\\\\textit {Geom}(1-\\\\gamma + \\\\gamma \\\\delta )$ at iteration $j=1, \\\\cdots , M$ .\",\n \"These samples are used to estimate the RNPG.\",\n \"As shown in Theorem REF , we can get the RNPG of $v^\\\\pi (s_1)$ by minimizing the objective defined in (REF ).\",\n \"To minimize (REF ) and get the minimizer, we initialize $\\\\tilde{g}_{t,0}=0$ and use the following stochastic gradient descent (SGD) steps with $l_2$ -projection: $\\\\tilde{g}_{t,k+1} = \\\\arg \\\\min _{\\\\tilde{g}} ||\\\\tilde{g}_{t,k}-\\\\zeta \\\\nabla _{\\\\tilde{g}} \\\\mathcal {L}(\\\\tilde{g}_{t,k}, \\\\theta _t) - \\\\tilde{g}||$ , where $\\\\zeta $ is the learning rate and $\\\\mathcal {L}$ is defined in (REF ) as follows.\",\n \"$\\\\mathcal {L}(\\\\tilde{g},\\\\theta ) = \\\\sum _{\\\\mathcal {D}(s^j_{T_j})}[\\\\tilde{g}^\\\\top \\\\psi ^{\\\\theta }(s,a) - \\\\phi ^\\\\theta (\\\\tau ) - b^\\\\theta ]^2/D,$ where $\\\\mathcal {D}(s^j_{T_j})$ is a set of trajectories $\\\\tau $ starting at $s^j_{T_j}$ using policy $\\\\pi ^{\\\\theta _t}$ , i.e.\",\n \"$(s^j_{T_j},a,r,c,s^\\\\prime )$ , $D=|\\\\mathcal {D}(s^j_{T_j})|$ .\",\n \"After $W$ steps of SDG iterations, the robust natural policy gradient for $v^{\\\\theta _t}(s^j_1)$ is estimated as $\\\\sum _{k=1}^W\\\\tilde{g}_{t,k}^j/W$ .\",\n \"To reduce the computational complexity, we adopt the centralized training and decentralized execution (CTDE) framework in ROCOMA and assume all agents share the same policy $\\\\pi ^{\\\\theta ^i}(a^i|s^i)$ , where $\\\\theta ^1=\\\\cdots =\\\\theta ^N=\\\\theta $ .\",\n \"Then we have $\\\\nabla \\\\pi (a|s) = \\\\sum _{i}^N \\\\psi ^{\\\\theta }_i(s,a)$ where $\\\\psi ^{\\\\theta }_i(s,a) := \\\\pi ^{-i}(a^{-i}|s^{-i})\\\\nabla \\\\pi ^i(a^i|s^i)$ , $\\\\pi ^{-i}(a^{-i}|s^{-i}):=\\\\prod _{j\\\\ne i}\\\\pi ^j(a^j|s^j)$ .\",\n \"Therefore, in lines REF to REF , we address the high-dimensional action and state space issue in computing RNPG by using $\\\\psi ^{\\\\theta }_i(s,a)$ instead of $\\\\psi ^{\\\\theta }(s,a)$ in (REF ).\",\n \"Finally, we update $\\\\theta _{t+1}$ and $\\\\lambda _{t+1}$ using GDA.\"\n ],\n [\n \"Experiment Setup\",\n \" Three different data sets , including E-taxi GPS data, transaction data and charging station data are used to build an EV AMoD system simulator as the training and testing environment.\",\n \"The simulated map is set as a grid city.\",\n \"The policy networks and critic networks are two-layer fully-connected networks, both with 32 nodes.\",\n \"We use Softplus as activations to ensure the output is positive.\",\n \"The output of policy networks is used to be the concentration parameters of the Dirichlet distribution to satisfy the action constraints (sum to one).\",\n \"We set the maximal training episode number $=20000$ , the maximal policy/critic estimation number $=2000$ , the NRPG SDG iteration number $=500$ , the discount rate $\\\\gamma = 0.99$ , the perturbed rate $\\\\delta =0.05$ , the coefficients $\\\\bar{\\\\alpha }=\\\\bar{\\\\beta }=1$ , the fairness constraint limit $d=-20$ for one simulation step, and use AdamOptimizer with a learning rate of $0.001$ for both policy/critic networks.\"\n ],\n [\n \"Experiment Results\",\n \" Our goal of the experiments is to validate the following hypothesis: (1) The proposed ROCOMA can learn effective rebalancing policies; (2) Our proposed method is more robust than a non-robust MARL algorithm through our robust and constrained MARL design.\",\n \"Other than Rebalancing cost: the total moving distance of vacant and low-battery EVs by using rebalancing policy (the lower the better); and System fairness: the weighted sum of mobility and charging fairness (the higher the better); we also monitor Number of expired orders: the total number of canceled orders due to waiting for more than 20 minutes (the lower the better) and Order response rate: the ratio between the number of served demands and the number of total passenger demand (the higher the better).\",\n \"All metrics are calculated in every testing period which consists of 25 simulation steps.\",\n \"We repeat testing for 10 times and show the average values.\",\n \"Table: Conclusion\"\n ]\n]"},"id":{"kind":"string","value":"2209.08230"}}},{"rowIdx":2393,"cells":{"text":{"kind":"list like","value":[["Value Summation: A Novel Scoring Function for MPC-based Model-based\n Reinforcement Learning"],["Abstract This paper proposes a novel scoring function for the planning module of MPC-based model-based reinforcement learning methods to address the inherent bias of using the reward function to score trajectories.","The proposed method enhances the learning efficiency of existing MPC-based MBRL methods using the discounted sum of values.","The method utilizes optimal trajectories to guide policy learning and updates its state-action value function based on real-world and augmented on-board data.","The learning efficiency of the proposed method is evaluated in selected MuJoCo Gym environments as well as in learning locomotion skills for a simulated model of the Cassie robot.","The results demonstrate that the proposed method outperforms the current state-of-the-art algorithms in terms of learning efficiency and average reward return."],["Introduction","Reinforcement learning (RL) has recently received great attention from robotics and autonomous systems communities due to its power in handling system non-linearity, uncertainties and complexities.","The current RL methods can be classified as Model-free RL (MFRL) and Model-based RL (MBRL).","Although MFRL methods have demonstrated proficiency in complex tasks , , they are typically sample-inefficient and suffer from a prolonged training process.","In contrast, MBRL methods are usually more sample-efficient, since they learn the dynamic transition model and can use it for planning.","In order to learn the model of the environment, methods such as ensemble learning and latent models have been proposed, allowing for fast and robust learning of the environment dynamics.","Once the model has been obtained, it can be used together with a planner to achieve a wide variety of tasks and goals.","MBRL planner algorithms can be categorized as background planning and discrete-time planning .","The former uses the transition data obtained from the model to improve the policy or to learn the value function, while the latter mainly performs planning online for every state that the agent visits.","Background planning algorithms include dynamic programming , tabular Dyna , and prioritized sweeping .","Examples of discrete-time planning are Monte Carlo Tree Search (MCTS) and Model Predictive Control (MPC) .","MCTS algorithms showed superior performance in discrete spaces such as Atari games while MPC algorithms are mainly used for continuous applications such as robotics.","POLO is a prominent example of MPC-based MBRL, proposed by Lowrey et al., in which the concepts of value function approximation are combined with MPC to stabilize and accelerate the learning process of the value function .","The work on POLO was extended by Morgan et al.","through modelling MPC as an actor-critic framework and incorporating the approximation error when learning the dynamics model.","Another approach, given by Sikchi et al.","integrated the value function learned from model-free off-policy RL into MPC, and proposed actor regularized control to address the issue of actor divergence.","Hoeller et al.","defined the objective function of MPC in terms of an estimated value function and showed that MPC minimizes an upper bound of the cross-entropy sampling method to the state distribution of the optimal sampling policy.","Zong et al.","combined the concepts of iLQR control with MFRL, resulting in an algorithm with low computation overhead, high sample efficiency, and robustness against the errors of the model.","Hansen et al.","used a learned task-oriented dynamics model for MPC and a learned terminal value function, where both the model and the value function are learned jointly by temporal difference learning.","The concepts of Model Predictive Path Integral Control (MPPI) and MFRL were combined by Charlesworth et al.",", resulting in a high-performance algorithm in some challenging simulated manipulation tasks where current RL methods and MPC techniques perform poorly.","Typically, MPC methods are based on a random shooting process.","This process samples numerous trajectories composed of state-action sequences with a $H$ horizon and evaluates each trajectory with a scoring function .","The defined scoring functions in literature can be classified into two types.","The first one sums the immediate rewards along the trajectory to calculate the trajectory score $\\sum _{t=0}^{H} \\gamma ^{t}r(s_{t},a_{t})$ , , where $\\gamma \\in [0, 1)$ is the discount factor which balances the emphasis on imminent rewards over distant rewards.","The second one sums immediate rewards along the trajectory and adds the estimated value of the terminal state $\\sum _{t=0}^{H-1} \\gamma ^{t}r(s_{t},a_{t}) + \\gamma ^{H}\\hat{V}(s_{H})$ , , , , , , .","However, using immediate rewards within the trajectory may not be enough for action selection, since no information about the future is considered.","In addition, in environments with sparse reward functions, using immediate rewards within the planning horizon may result in the same score for all trajectories .","Moreover, in complex environments, the sampling time is small, and states are close to each other, making the immediate rewards not a good option for differentiating between states.","In order to address the aforementioned problems, we propose the use of the discounted summation of estimated state-action values as the scoring function for MPC-based MBRL algorithms, where the score of a trajectory is calculated by summing state-action values rather than rewards.","It should be noted that we use state-action values within the planning horizon instead of state values since state-action values pertain to the action taken in the trajectory, making the score accurate within the context of the trajectory.","Moreover, summing rewards to score trajectories emphasises actions with greater immediate reward, whereas summing state-action values emphasises actions with greater cumulative rewards, thereby increasing total return over the episode.","The results show that the proposed method improves learning efficiency and average return when compared to state-of-the-art baselines.","To the best of our knowledge, it is the first time a discounted sum of values has been used in RL.","The remainder of the paper is structured as follows: we discuss the related works in Section II while Section III describes preliminaries for key components of MBRL.","Section IV introduces the value summation function and its integration in MBRL.","The results are presented in Section V. Finally, our contributions and future works are summarized in Section VI.","In the context of RL, a problem can be formulated as a finite-horizon Markov Decision Process (MDP) $\\mathcal {M = }$ , where $\\mathcal {S}$ denotes the state space, $\\mathcal {A}$ denotes the action space, $\\mathcal {R : S \\times A} \\rightarrow \\mathbb {R}$ denotes the reward function, $\\mathcal {P : S \\times A \\times S} \\rightarrow \\mathbb {R}$ is the transition model, and $\\gamma \\in [0,1)$ is the discount factor.","A policy $\\pi \\in \\Pi : \\mathcal {S \\times A} \\rightarrow \\mathbb {R}$ describes a mapping from states to actions.","In RL problems, return is defined as the accumulative discounted reward, given by ${G} = \\sum _{t=0}^{\\infty } {\\gamma ^t r(s_t,\\pi (s_t))}$ .","An MDP aims to find the optimal policy $\\pi ^*$ that maximizes the expected return.","$V(s)$ denotes the state value, which is defined as the expected return by following the policy $\\pi $ from the state $s$ : $V(s) = \\mathbb {E}_{\\pi } \\left[ \\sum _{t=0}^{\\infty } {\\gamma ^t r(s_t,\\pi (s_t))} \\right], s_0=s$ Similarly, $Q(s,a)$ denotes the state-action value, which is defined as the expected return by taking the action $a$ at the state $s$ and then following the policy $\\pi $ : ${Q(s,a)} = \\mathbb {E}_{\\pi } \\left[ \\sum _{t=0}^{\\infty } {\\gamma ^t r(s_t,\\pi (s_t))} \\right], s_0=s, a_0=a$ In continuous MDPs, the policy and state-action value function are parameterized as $\\pi _{\\theta }$ and $Q_{\\phi }$ , where $\\theta $ and $\\phi $ denote the vector of parameters.","The parameters of the state-action value function $Q_{\\phi }$ can be estimated using the following loss function: $J_{\\phi } = \\mathbb {E}_{s,a,r,s^{\\prime } \\sim \\mathcal {D}} \\left[{(r + {\\gamma } {\\max _{a^{\\prime }} {Q_{\\phi }(s^{\\prime },a^{\\prime })}} - Q_{\\phi }(s,a))^2} \\right]$ where $\\mathcal {D}$ is a dataset containing a series of transitions $(s,a,r,s^{\\prime })$ collected by the policy $\\pi _{\\theta }$ .","After the state-action value function is approximated, the parameters of the policy are updated using the following function: $\\nabla _\\theta {J_{\\theta }} = \\mathbb {E}_{\\pi _\\theta } \\left[ {\\nabla _\\theta \\log {\\pi _\\theta (s,a)} Q_\\phi (s, a)} \\right]$"],["MPC-based Model-based Reinforcement Learning","MBRL approaches benefit from the model of the environment to plan the best action sequence at each state.","In the context of MBRL, the transition model can be known or learned.","When the transition model is available, it is used for planning the action sequence at each time step.","In this regard, MPC is mostly used to plan the best action sequence.","The MPC policy $\\pi _{MPC}$ computes the best action sequence at each state using the transition model in three stages: trajectory sampling, trajectory evaluation, and action selection.","In general, an MPC policy is computed as follows: $\\begin{aligned}\\pi _{MPC}(s) = \\arg \\max _{a_{0:H-1}} {\\mathbb {E} \\left[ \\sum _{t=0}^{H-1} {\\gamma ^t r(s_t,a_t) + \\gamma ^H r_f(s_H)} \\right]},\\\\a_t = \\pi _t(s_t), s_0=s\\end{aligned}$ where states evolve according to the on-board model $s_{t+1}=\\hat{f}(s_t,a_t)$ and $\\pi _t$ is a distribution for sampling action at each state.","After the optimal action sequence is computed, the first action is executed in the environment and the procedure is repeated at the next time step.","It is worth mentioning that, in some cases where MBRL is combined with MFRL, the distribution $\\pi _t$ can be the parameterized policy $\\pi _\\theta $ , called the behavioral policy.","Figure: The discounted value summation scoring function assigns greater importance to future rewards than the conventional definition of reward-value functions.","The hatched area shows that future rewards are emphasized by the proposed method (γ=0.9,H=100\\gamma =0.9, H=100)."],["Proposed MBRL Method","The proposed Value Summation Model-Based Reinforcement Learning (VS-MBRL) method is presented in this section.","The concept of discounted value summation and its boundedness over the infinite horizon is explained.","Then, the integration of the proposed scoring function in trajectory evaluation and the corresponding VS-MBRL is discussed."],["Scoring Function","The use of reward-based scoring functions to evaluate trajectories and selecting the best action out of different trajectories can be misleading.","The problem with the application of reward-based scoring functions is that the discounted summation of finite-horizon immediate rewards can be significantly biased against future decisions because the impact of future rewards is not considered in the scoring function.","Even the selection of greater horizons $H$ cannot alleviate this problem, because they may impose cumulative errors and reduce planning efficiency.","Another problem regarding the application of reward-based planning is that in environments with sparse reward feedback, where the total reward is received upon the completion of an episode, the summation of rewards within a planning horizon can be the same across all trajectories .","Last but not least, a problem occurs in environments with short sampling times, where the states' difference across each increment of time is too short to have a considerable impact on reward function $r(s_{t},a_{t})$ .","For these reasons, incorporating the estimated state-value function in the scoring function can be more insightful.","To address these issues, we propose the discounted sum of estimated state-action values to score trajectories as: $S = \\sum _{t=0}^{H}\\gamma ^{t}{Q^{\\pi }}(s_{t},a_{t})$ where the estimated state-action values over a trajectory with the planning horizon $H$ are summed up to determine the trajectory score $S$ .","Using the Bellman equation, we can roughly estimate the state-action value of each time-step as follows: $Q^{\\pi }(s_{t},a_{t}) = r(s_{t},a_{t}) + \\gamma Q^{\\pi }(s_{t+1},a_{t+1})$ Using (REF ) iteratively for all time steps within the planning horizon, we can write the scoring function in (REF ) in terms of the immediate rewards as below: $\\begin{aligned}S = & \\sum _{t=0}^{H}\\gamma ^{t}(t+1)r(s_{t}, a_{t}) + \\sum _{t=H+1}^{T}(H+1)\\gamma ^{t}r(s_{t}, a_{t})\\end{aligned}$ See Appendix A for a detailed derivation of (8).","The obtained scoring function (REF ) means the rewards within the planning horizon are emphasized by $\\gamma ^{t}(t+1)$ factor, and the rewards outside of the planning horizon are discounted by $(H+1)\\gamma ^{t}$ term.","Fig.","REF indicates that the value summation method imposes a greater weight on rewards than the reward-value scoring function proposed in .","This highlights the potential of the proposed scoring function in better evaluating future rewards and in making future-oriented decisions.","It is worth mentioning that over a specific range of horizon $t \\in \\left[2,34 \\right]$ (See Fig.","REF ), rewards are weighted by factors greater than 1.","We demonstrate that the proposed scoring function (REF ) would not violate the boundedness of summation, even in an infinite-horizon ($H=\\infty $ ) planning case (See Lemma 1).","Value Summation MBRL [1] Given $\\hat{f}(s_t,a_t)$ : on-board model, $D$ : replay buffer, $\\pi _{\\theta }(s_{t})$ : actor-network, $Q_{\\phi }(s_{t},a_{t})$ : critic network, $\\gamma $ : discounted factor; ${H}$ : Horizon length; ${N}$ : Number of trajectories; Task Not Completed $state \\leftarrow ReadSensors()$ $n \\leftarrow 0$ to $N$ ${s_{0}} = state$ $S(n) = 0$ $h \\leftarrow 0$ to $H$ $a_{h} = \\pi _{\\theta }(s_{h})$ $s_{h+1} = \\hat{f}(s_{h}, a_{h})$ $q_{h} = Q_{\\phi }(s_{h}, a_{h})$ $S(n) += \\gamma ^{h}q_{h}$ Store $(s_{h}, a_{h})$ in ${D}$ $Train(Q_{\\phi })$ $a_{t} = \\underset{a_{t:t+H}}{\\mathrm {argmax}}\\ S$ $Execute(a_{t})$ $Train(\\pi _{\\theta })$ Figure: The schematic of the Value Summation MBRL algorithm.","The al policy generates several trajectories, and the scoring mechanism selects the optimal trajectory with respect to the value summation function.Lemma 1 Consider an infinite horizon action sequence ${A}= \\lbrace a_{0},a_{1},a_{2}, ...\\rbrace $ and its resultant reward sequence ${R} = \\lbrace r_{0}, r_{1}, r_{2}, ... \\rbrace $ .","With these assumptions, (REF ) is expressed as below: $S = \\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r(s_{t},a_{t})$ Considering the maximum reward from the infinite reward sequence $R$ as $r_{max} = max \\lbrace r(s_{0},a_{0}), r(s_{1},a_{1}), r(s_{2},a_{2}), ... \\rbrace $ , the scoring function is upper bounded by: $\\begin{aligned}\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r(s_{t},a_{t}) & \\le \\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r_{max}\\\\& = r_{max}\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)\\end{aligned}$ where $\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)$ has an analytical solution as follows (See Appendix B): $\\sum _{t=0}^{\\infty }{\\gamma ^{t}(t+1)} = \\frac{1}{(1-\\gamma )^{2}}$ Substituting (REF ) into (REF ), we have the following relation for the scoring function: $\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r(s_{t},a_{t}) \\le \\frac{1}{(1-\\gamma )^{2}}r_{max}$ Therefore the infinite horizon scoring function has an upper bound.","Figure: The performance of value summation model-based reinforcement learning algorithm in comparison to well-known scoring functions.","(i) Sum-Value: ∑ t ' =t t+H γ t ' Q π (s t ,a t )\\sum _{t^{\\prime }=t}^{t+H} \\gamma ^{t^{\\prime }}Q^{\\pi }(s_{t},a_{t}) (ii) Sum-Reward: ∑ t ' =t t+H γ t ' r π (s t ,a t )\\sum _{t^{\\prime }=t}^{t+H} \\gamma ^{t^{\\prime }}r^{\\pi }(s_{t},a_{t}) and (iii) Sum-Reward-Value ∑ t ' =t t+H-1 γ t ' r π (s t ,a t )+γ H Q π (s H ,a H )\\sum _{t^{\\prime }=t}^{t+H-1} \\gamma ^{t^{\\prime }}r^{\\pi }(s_{t},a_{t}) + \\gamma ^{H}Q^{\\pi }(s_{H},a_{H})"],["Value Summation MBRL","This subsection presents the VS-MBRL method, which integrates the value summation scoring function into planning (Fig.","REF ).","This algorithm consists of two sections: gathering trajectories and scoring them.","After receiving the current state $s_{t}$ and the instant reward $r_{t}$ from the environment, the on-board model $\\hat{f}(s_{t}, a_{t})$ is used to sample several trajectories with the planning horizon $H$ from the policy $\\pi _{\\theta }$ .","The collected trajectories, then, are scored by the value summation function, and the first action from the highest-scored trajectory is executed in the environment.","VS-MBRL employs Soft Actor-Critic (SAC) as the underlying actor-critic algorithm to take advantage of the exploration induced by soft policy updates based on the maximum entropy principle.","This property makes SAC an excellent candidate for MBRL applications since actions can be sampled directly from the policy $\\pi _{\\theta }$ , and no additional noise is required.","However, VS-MBRL can be applied to any developed off-policy actor-critic algorithm.","A soft policy evaluation step is conducted in SAC using the soft Bellman backup operator , followed by a soft policy improvement step in which the expected KL divergence is minimized.","A description of the VS-MBRL algorithm can be found in Algorithm REF .","Using the on-board model $\\hat{f}(s_{t}, a_{t})$ with planning horizon $H$ , the agent shoots a number of trajectories at each time step (line code: $6-17$ ).","Eq.","(REF ) is applied to evaluate each pair $(s_{t:t+H}, a_{t:t+H})$ (line code: $12-13$ ) such that the corresponding trajectory can be evaluated.","The augmented data is collected in the replay buffer ${D}$ , which is then applied to train the state-action value function (line code: 16).","The first action from the best-scored trajectory is then executed in the environment (line code: 19).","To reduce computation costs, the augmented data is only employed for updating state-action value functions, while the actor-network is updated by $1/{N}^{th}$ that of the critic network, where $N$ is the number of trajectories (line code: 20)."],["Simulation","In this section, we compare the proposed value-summation scoring function with the most recent proposed scoring functions in the literature.","The scoring functions used for comparison purposes are: 1) Sum-Reward: Discounted sum of rewards $\\sum _{t=0}^{H} \\gamma ^{t}r(s_{t},a_{t})$ , proposed by .","2) Sum-Reward-Value: Discounted sum of rewards added with the estimated value of the terminal state, given by $\\sum _{t=0}^{H-1} \\gamma ^{t}r(s_{t},a_{t}) + \\gamma ^{H}{Q}^{\\pi }(s_{H}, a_{H})$ , proposed by .","3) Sum-Value: Our proposed discounted sum of state-action values $\\sum _{t=0}^{H} \\gamma ^{t}Q^{\\pi }(s_{t},a_{t})$ .","First, the performance of VS-MBRL is evaluated and compared in standard Gym environments.","Then, VS-MBRL is integrated into the process of learning locomotion skills for the Cassie robot and its performance is compared with other baselines.","The results indicate that the proposed scoring function outperforms other scoring functions in terms of learning efficiency and average return."],["Standard Gym Environments","We evaluate VS-MBRL in the simulated control tasks of MuJoCo included in the OpenAI-gym library : Ant-v3, Hopper-v3, HalfCheetah-v3, Walker2d-v3, Humanoid-v3, and InvertedDoublePendulum-v2.","We compare the average return of VS-MBRL over three trials against other scoring functions.","As shown in Fig.","REF , the average return of each scoring function is plotted against the time step for each environment.","In all environments, except Half-Cheetah-v3 and InvertedDoublePendulum-v2, VS-MBRL outperforms other scoring functions, proving the benefit of using value summation over other methods (See Table REF ).","It is speculated that in Half-Cheetah-v3, the estimation of the state-action value function imposes a significant error in planning, especially at the first 200,000 time steps, when the state-action value neural network needs more data.","Also, rewards are more useful in simple problems, such as InvertedDoublePendulum-v2, since the state-action value network requires a significant amount of training data.","However, these events prove negligible in complex problems, such as Humanoid-v3, Walker2d-v3 and Hopper-v3, where value summation results in fast and stable learning.","According to Ant-v3 and HalfCheetah-v3, relying only on reward summation can impede learning for a long time, resulting in poor performance.","Table: Performance of Scoring Functions in Different Gym Environments"],["Cassie's Learning Locomotion Skills","Additionally, we show that VS-MBRL is more effective than other scoring functions for the learning of Cassie's locomotion skills.","Initially designed and built by Agility Robotics, the Cassie robot is approximately one meter tall and weighs 33 kg.","Most of Cassie's mass is centred on the pelvis and each leg has two leaf springs to make it more flexible.","Traditional controller design becomes more challenging as a result of underactuation.","Xie et al.","propose an iterative RL algorithm to increase the stability of locomotion skills through a policy distillation mechanism.","In this way, the trained policy is used in the next iteration as an expert $\\pi _{e}$ .","The expert's action is then added to that of the behavioral policy $\\pi _{\\theta }$ and fed into the PD controller.","Each joint is controlled by a PD control loop, executed at the total rate of 2 kHz, with the targets updated every 30 ms from the policy.","We combine VS-MBRL with the controller proposed in , in which VS-MBRL is incorporated into the feedback control loop (Fig.","REF ).","The controller, therefore, contains an internal loop in which the behavioral policy generates trajectories and scores them.","Following this, the optimized action $a_{t}$ is summed up with the reference motion $\\delta a$ and fed into the PD controller.","It was decided to set the horizon length $H$ and the number of trajectories $N$ to 3 to reduce computation time.","Figure REF compares the performance of VS-MBRL with traditional scoring functions.","In the first 400 episodes, there is a negligible difference in performance.","Throughout the next 400 episodes, the average return exponentially increases until it reaches a maximum of 250.","In comparison to other gym environments, there is a substantial performance gap between scoring functions.","This is mainly because, in the simulated model of the Cassie robot for learning locomotion skills, the sampling time is $0.5$ millisecond, which is smaller than that in gym environments (Humanoid's sampling time is 3 millisecond), thus reducing the variation of states and rewards along each trajectory, resulting in difficult reward-based planning.","Figure: The application of value summation model-based reinforcement learning algorithm in feedback controller.","Reference motions from the expert policy are fed into the algorithm and added to the output.Figure: Value Summation MBRL algorithm (Sum-Value) outperforms other scoring functions in terms of sample efficiency and average return"],["Conclusion and Discussion","This paper presents a model-based reinforcement learning algorithm incorporating discounted value summation into planning.","In addition, Soft Actor-Critic was selected as the policy optimization algorithm, in which actions were directly sampled from a behavioral policy.","We compared the proposed scoring function with two prevalent planning methods: discounted sum of rewards and discounted sum of rewards added with the estimated value of the terminal state.","The performance over some standard baselines indicated the superiority of the proposed method.","Moreover, VS-MBRL is applied to learning locomotion skills for a simulated model of the Cassie robot.","VS-MBRL proves to be a preferable alternative to current scoring functions, offering considerably greater returns for Cassie's locomotion skills as well as MuJoCo Gym environments."],["APPENDIX","A.","Proof of (REF ).","Consider the case that we have an MDP with a finite horizon $T$ and a finite planning horizon $H$ .","For simplicity, we denote $Q^\\pi (s_t,a_t)$ and $r(s_t,a_t)$ as $Q_t$ and $r_t$ , respectively.","Using (REF ) iteratively, we can expand the state-action values of time steps within the planning horizon as follows: $\\begin{aligned}Q_H & = r_H + \\gamma Q_{H+1}\\\\Q_{H-1} & = r_{H-1} + \\gamma r_{H} + \\gamma ^2 Q_{H+1}\\\\& \\vdots \\\\Q_1 & = r_1 + \\gamma r_2 + \\gamma ^2 r_3 + ... + \\gamma ^{H-1} r_{H} + \\gamma ^H Q_{H+1}\\\\Q_0 & = r_0 + \\gamma r_1 + \\gamma ^2 r_2 + ... + \\gamma ^H r_H + \\gamma ^{H+1} Q_{H+1}\\end{aligned}$ The scoring function in (REF ) can be expanded as follows: $S = Q_0 + \\gamma Q_1 + ... + \\gamma ^{H-1} Q_{H-1} + \\gamma ^H Q_H$ Substituting the state-action values from (REF ) in (REF ), we have: $\\begin{aligned}S & = r_0 + \\gamma r_1 + \\gamma ^2 r_2 + ... + \\gamma ^H r_H + \\gamma ^{H+1} Q_{H+1} \\\\& + \\gamma r_1 + \\gamma ^2 r_2 + \\gamma ^3 r_3 + ... + \\gamma ^{H} r_{H} + \\gamma ^{H+1} Q_{H+1} \\\\& \\vdots \\\\& + \\gamma ^{H-1} r_{H-1} + \\gamma ^H r_{H} + \\gamma ^{H+1} Q_{H+1}\\\\& + \\gamma ^H r_H + \\gamma ^{H+1} Q_{H+1}\\\\& = r_0 + 2 \\gamma r_1 + ... + (H+1) \\gamma ^H r_H + (H+1) \\gamma ^{H+1} Q_{H+1}\\\\& = \\sum _{t=0}^{H} \\gamma ^{t}(t+1)r_t + (H+1) \\gamma ^{H+1} Q_{H+1}\\end{aligned}$ Moreover, using the definition of the state-action value, we can write the $Q_{H+1}$ as follows: $Q_{H+1} = \\sum _{t=H+1}^{T} \\gamma ^{t-H-1}r_t$ Finally, after substituting (REF ) in (REF ), we obtain the scoring function in terms of the immediate rewards as follows: $S = \\sum _{t=0}^{H} \\gamma ^{t}(t+1)r_t + \\sum _{t=H+1}^{T} (H+1) \\gamma ^{t}r_t$ B.","Proof of the Time Series.","From mathematics, we have the following time series with its solution: $\\sum _{t=0}^{\\infty }{\\gamma ^{t}} = \\frac{1}{1-\\gamma }$ The derivative of the time series in (REF ) with respect to $\\gamma $ results in another time series as follows: $\\begin{aligned}\\frac{d}{d\\gamma } \\sum _{t=0}^{\\infty }{\\gamma ^{t}} & = \\frac{d}{dt} (1 + \\gamma + \\gamma ^2 + \\gamma ^3 + ...) \\\\& = 1 + 2 \\gamma + 3 \\gamma ^2 + ... \\\\& = \\sum _{t=0}^{\\infty }{\\gamma ^{t}(t+1)}\\end{aligned}$ Thus, if we take the derivative of both sides of (REF ) with respect to $\\gamma $ , the following relation is obtained: $\\sum _{t=0}^{\\infty }{\\gamma ^{t}(t+1)} = \\frac{1}{(1-\\gamma )^{2}}$ [heading=bibintoc]"]],"string":"[\n [\n \"Value Summation: A Novel Scoring Function for MPC-based Model-based\\n Reinforcement Learning\"\n ],\n [\n \"Abstract This paper proposes a novel scoring function for the planning module of MPC-based model-based reinforcement learning methods to address the inherent bias of using the reward function to score trajectories.\",\n \"The proposed method enhances the learning efficiency of existing MPC-based MBRL methods using the discounted sum of values.\",\n \"The method utilizes optimal trajectories to guide policy learning and updates its state-action value function based on real-world and augmented on-board data.\",\n \"The learning efficiency of the proposed method is evaluated in selected MuJoCo Gym environments as well as in learning locomotion skills for a simulated model of the Cassie robot.\",\n \"The results demonstrate that the proposed method outperforms the current state-of-the-art algorithms in terms of learning efficiency and average reward return.\"\n ],\n [\n \"Introduction\",\n \"Reinforcement learning (RL) has recently received great attention from robotics and autonomous systems communities due to its power in handling system non-linearity, uncertainties and complexities.\",\n \"The current RL methods can be classified as Model-free RL (MFRL) and Model-based RL (MBRL).\",\n \"Although MFRL methods have demonstrated proficiency in complex tasks , , they are typically sample-inefficient and suffer from a prolonged training process.\",\n \"In contrast, MBRL methods are usually more sample-efficient, since they learn the dynamic transition model and can use it for planning.\",\n \"In order to learn the model of the environment, methods such as ensemble learning and latent models have been proposed, allowing for fast and robust learning of the environment dynamics.\",\n \"Once the model has been obtained, it can be used together with a planner to achieve a wide variety of tasks and goals.\",\n \"MBRL planner algorithms can be categorized as background planning and discrete-time planning .\",\n \"The former uses the transition data obtained from the model to improve the policy or to learn the value function, while the latter mainly performs planning online for every state that the agent visits.\",\n \"Background planning algorithms include dynamic programming , tabular Dyna , and prioritized sweeping .\",\n \"Examples of discrete-time planning are Monte Carlo Tree Search (MCTS) and Model Predictive Control (MPC) .\",\n \"MCTS algorithms showed superior performance in discrete spaces such as Atari games while MPC algorithms are mainly used for continuous applications such as robotics.\",\n \"POLO is a prominent example of MPC-based MBRL, proposed by Lowrey et al., in which the concepts of value function approximation are combined with MPC to stabilize and accelerate the learning process of the value function .\",\n \"The work on POLO was extended by Morgan et al.\",\n \"through modelling MPC as an actor-critic framework and incorporating the approximation error when learning the dynamics model.\",\n \"Another approach, given by Sikchi et al.\",\n \"integrated the value function learned from model-free off-policy RL into MPC, and proposed actor regularized control to address the issue of actor divergence.\",\n \"Hoeller et al.\",\n \"defined the objective function of MPC in terms of an estimated value function and showed that MPC minimizes an upper bound of the cross-entropy sampling method to the state distribution of the optimal sampling policy.\",\n \"Zong et al.\",\n \"combined the concepts of iLQR control with MFRL, resulting in an algorithm with low computation overhead, high sample efficiency, and robustness against the errors of the model.\",\n \"Hansen et al.\",\n \"used a learned task-oriented dynamics model for MPC and a learned terminal value function, where both the model and the value function are learned jointly by temporal difference learning.\",\n \"The concepts of Model Predictive Path Integral Control (MPPI) and MFRL were combined by Charlesworth et al.\",\n \", resulting in a high-performance algorithm in some challenging simulated manipulation tasks where current RL methods and MPC techniques perform poorly.\",\n \"Typically, MPC methods are based on a random shooting process.\",\n \"This process samples numerous trajectories composed of state-action sequences with a $H$ horizon and evaluates each trajectory with a scoring function .\",\n \"The defined scoring functions in literature can be classified into two types.\",\n \"The first one sums the immediate rewards along the trajectory to calculate the trajectory score $\\\\sum _{t=0}^{H} \\\\gamma ^{t}r(s_{t},a_{t})$ , , where $\\\\gamma \\\\in [0, 1)$ is the discount factor which balances the emphasis on imminent rewards over distant rewards.\",\n \"The second one sums immediate rewards along the trajectory and adds the estimated value of the terminal state $\\\\sum _{t=0}^{H-1} \\\\gamma ^{t}r(s_{t},a_{t}) + \\\\gamma ^{H}\\\\hat{V}(s_{H})$ , , , , , , .\",\n \"However, using immediate rewards within the trajectory may not be enough for action selection, since no information about the future is considered.\",\n \"In addition, in environments with sparse reward functions, using immediate rewards within the planning horizon may result in the same score for all trajectories .\",\n \"Moreover, in complex environments, the sampling time is small, and states are close to each other, making the immediate rewards not a good option for differentiating between states.\",\n \"In order to address the aforementioned problems, we propose the use of the discounted summation of estimated state-action values as the scoring function for MPC-based MBRL algorithms, where the score of a trajectory is calculated by summing state-action values rather than rewards.\",\n \"It should be noted that we use state-action values within the planning horizon instead of state values since state-action values pertain to the action taken in the trajectory, making the score accurate within the context of the trajectory.\",\n \"Moreover, summing rewards to score trajectories emphasises actions with greater immediate reward, whereas summing state-action values emphasises actions with greater cumulative rewards, thereby increasing total return over the episode.\",\n \"The results show that the proposed method improves learning efficiency and average return when compared to state-of-the-art baselines.\",\n \"To the best of our knowledge, it is the first time a discounted sum of values has been used in RL.\",\n \"The remainder of the paper is structured as follows: we discuss the related works in Section II while Section III describes preliminaries for key components of MBRL.\",\n \"Section IV introduces the value summation function and its integration in MBRL.\",\n \"The results are presented in Section V. Finally, our contributions and future works are summarized in Section VI.\",\n \"In the context of RL, a problem can be formulated as a finite-horizon Markov Decision Process (MDP) $\\\\mathcal {M = }$ , where $\\\\mathcal {S}$ denotes the state space, $\\\\mathcal {A}$ denotes the action space, $\\\\mathcal {R : S \\\\times A} \\\\rightarrow \\\\mathbb {R}$ denotes the reward function, $\\\\mathcal {P : S \\\\times A \\\\times S} \\\\rightarrow \\\\mathbb {R}$ is the transition model, and $\\\\gamma \\\\in [0,1)$ is the discount factor.\",\n \"A policy $\\\\pi \\\\in \\\\Pi : \\\\mathcal {S \\\\times A} \\\\rightarrow \\\\mathbb {R}$ describes a mapping from states to actions.\",\n \"In RL problems, return is defined as the accumulative discounted reward, given by ${G} = \\\\sum _{t=0}^{\\\\infty } {\\\\gamma ^t r(s_t,\\\\pi (s_t))}$ .\",\n \"An MDP aims to find the optimal policy $\\\\pi ^*$ that maximizes the expected return.\",\n \"$V(s)$ denotes the state value, which is defined as the expected return by following the policy $\\\\pi $ from the state $s$ : $V(s) = \\\\mathbb {E}_{\\\\pi } \\\\left[ \\\\sum _{t=0}^{\\\\infty } {\\\\gamma ^t r(s_t,\\\\pi (s_t))} \\\\right], s_0=s$ Similarly, $Q(s,a)$ denotes the state-action value, which is defined as the expected return by taking the action $a$ at the state $s$ and then following the policy $\\\\pi $ : ${Q(s,a)} = \\\\mathbb {E}_{\\\\pi } \\\\left[ \\\\sum _{t=0}^{\\\\infty } {\\\\gamma ^t r(s_t,\\\\pi (s_t))} \\\\right], s_0=s, a_0=a$ In continuous MDPs, the policy and state-action value function are parameterized as $\\\\pi _{\\\\theta }$ and $Q_{\\\\phi }$ , where $\\\\theta $ and $\\\\phi $ denote the vector of parameters.\",\n \"The parameters of the state-action value function $Q_{\\\\phi }$ can be estimated using the following loss function: $J_{\\\\phi } = \\\\mathbb {E}_{s,a,r,s^{\\\\prime } \\\\sim \\\\mathcal {D}} \\\\left[{(r + {\\\\gamma } {\\\\max _{a^{\\\\prime }} {Q_{\\\\phi }(s^{\\\\prime },a^{\\\\prime })}} - Q_{\\\\phi }(s,a))^2} \\\\right]$ where $\\\\mathcal {D}$ is a dataset containing a series of transitions $(s,a,r,s^{\\\\prime })$ collected by the policy $\\\\pi _{\\\\theta }$ .\",\n \"After the state-action value function is approximated, the parameters of the policy are updated using the following function: $\\\\nabla _\\\\theta {J_{\\\\theta }} = \\\\mathbb {E}_{\\\\pi _\\\\theta } \\\\left[ {\\\\nabla _\\\\theta \\\\log {\\\\pi _\\\\theta (s,a)} Q_\\\\phi (s, a)} \\\\right]$\"\n ],\n [\n \"MPC-based Model-based Reinforcement Learning\",\n \"MBRL approaches benefit from the model of the environment to plan the best action sequence at each state.\",\n \"In the context of MBRL, the transition model can be known or learned.\",\n \"When the transition model is available, it is used for planning the action sequence at each time step.\",\n \"In this regard, MPC is mostly used to plan the best action sequence.\",\n \"The MPC policy $\\\\pi _{MPC}$ computes the best action sequence at each state using the transition model in three stages: trajectory sampling, trajectory evaluation, and action selection.\",\n \"In general, an MPC policy is computed as follows: $\\\\begin{aligned}\\\\pi _{MPC}(s) = \\\\arg \\\\max _{a_{0:H-1}} {\\\\mathbb {E} \\\\left[ \\\\sum _{t=0}^{H-1} {\\\\gamma ^t r(s_t,a_t) + \\\\gamma ^H r_f(s_H)} \\\\right]},\\\\\\\\a_t = \\\\pi _t(s_t), s_0=s\\\\end{aligned}$ where states evolve according to the on-board model $s_{t+1}=\\\\hat{f}(s_t,a_t)$ and $\\\\pi _t$ is a distribution for sampling action at each state.\",\n \"After the optimal action sequence is computed, the first action is executed in the environment and the procedure is repeated at the next time step.\",\n \"It is worth mentioning that, in some cases where MBRL is combined with MFRL, the distribution $\\\\pi _t$ can be the parameterized policy $\\\\pi _\\\\theta $ , called the behavioral policy.\",\n \"Figure: The discounted value summation scoring function assigns greater importance to future rewards than the conventional definition of reward-value functions.\",\n \"The hatched area shows that future rewards are emphasized by the proposed method (γ=0.9,H=100\\\\gamma =0.9, H=100).\"\n ],\n [\n \"Proposed MBRL Method\",\n \"The proposed Value Summation Model-Based Reinforcement Learning (VS-MBRL) method is presented in this section.\",\n \"The concept of discounted value summation and its boundedness over the infinite horizon is explained.\",\n \"Then, the integration of the proposed scoring function in trajectory evaluation and the corresponding VS-MBRL is discussed.\"\n ],\n [\n \"Scoring Function\",\n \"The use of reward-based scoring functions to evaluate trajectories and selecting the best action out of different trajectories can be misleading.\",\n \"The problem with the application of reward-based scoring functions is that the discounted summation of finite-horizon immediate rewards can be significantly biased against future decisions because the impact of future rewards is not considered in the scoring function.\",\n \"Even the selection of greater horizons $H$ cannot alleviate this problem, because they may impose cumulative errors and reduce planning efficiency.\",\n \"Another problem regarding the application of reward-based planning is that in environments with sparse reward feedback, where the total reward is received upon the completion of an episode, the summation of rewards within a planning horizon can be the same across all trajectories .\",\n \"Last but not least, a problem occurs in environments with short sampling times, where the states' difference across each increment of time is too short to have a considerable impact on reward function $r(s_{t},a_{t})$ .\",\n \"For these reasons, incorporating the estimated state-value function in the scoring function can be more insightful.\",\n \"To address these issues, we propose the discounted sum of estimated state-action values to score trajectories as: $S = \\\\sum _{t=0}^{H}\\\\gamma ^{t}{Q^{\\\\pi }}(s_{t},a_{t})$ where the estimated state-action values over a trajectory with the planning horizon $H$ are summed up to determine the trajectory score $S$ .\",\n \"Using the Bellman equation, we can roughly estimate the state-action value of each time-step as follows: $Q^{\\\\pi }(s_{t},a_{t}) = r(s_{t},a_{t}) + \\\\gamma Q^{\\\\pi }(s_{t+1},a_{t+1})$ Using (REF ) iteratively for all time steps within the planning horizon, we can write the scoring function in (REF ) in terms of the immediate rewards as below: $\\\\begin{aligned}S = & \\\\sum _{t=0}^{H}\\\\gamma ^{t}(t+1)r(s_{t}, a_{t}) + \\\\sum _{t=H+1}^{T}(H+1)\\\\gamma ^{t}r(s_{t}, a_{t})\\\\end{aligned}$ See Appendix A for a detailed derivation of (8).\",\n \"The obtained scoring function (REF ) means the rewards within the planning horizon are emphasized by $\\\\gamma ^{t}(t+1)$ factor, and the rewards outside of the planning horizon are discounted by $(H+1)\\\\gamma ^{t}$ term.\",\n \"Fig.\",\n \"REF indicates that the value summation method imposes a greater weight on rewards than the reward-value scoring function proposed in .\",\n \"This highlights the potential of the proposed scoring function in better evaluating future rewards and in making future-oriented decisions.\",\n \"It is worth mentioning that over a specific range of horizon $t \\\\in \\\\left[2,34 \\\\right]$ (See Fig.\",\n \"REF ), rewards are weighted by factors greater than 1.\",\n \"We demonstrate that the proposed scoring function (REF ) would not violate the boundedness of summation, even in an infinite-horizon ($H=\\\\infty $ ) planning case (See Lemma 1).\",\n \"Value Summation MBRL [1] Given $\\\\hat{f}(s_t,a_t)$ : on-board model, $D$ : replay buffer, $\\\\pi _{\\\\theta }(s_{t})$ : actor-network, $Q_{\\\\phi }(s_{t},a_{t})$ : critic network, $\\\\gamma $ : discounted factor; ${H}$ : Horizon length; ${N}$ : Number of trajectories; Task Not Completed $state \\\\leftarrow ReadSensors()$ $n \\\\leftarrow 0$ to $N$ ${s_{0}} = state$ $S(n) = 0$ $h \\\\leftarrow 0$ to $H$ $a_{h} = \\\\pi _{\\\\theta }(s_{h})$ $s_{h+1} = \\\\hat{f}(s_{h}, a_{h})$ $q_{h} = Q_{\\\\phi }(s_{h}, a_{h})$ $S(n) += \\\\gamma ^{h}q_{h}$ Store $(s_{h}, a_{h})$ in ${D}$ $Train(Q_{\\\\phi })$ $a_{t} = \\\\underset{a_{t:t+H}}{\\\\mathrm {argmax}}\\\\ S$ $Execute(a_{t})$ $Train(\\\\pi _{\\\\theta })$ Figure: The schematic of the Value Summation MBRL algorithm.\",\n \"The al policy generates several trajectories, and the scoring mechanism selects the optimal trajectory with respect to the value summation function.Lemma 1 Consider an infinite horizon action sequence ${A}= \\\\lbrace a_{0},a_{1},a_{2}, ...\\\\rbrace $ and its resultant reward sequence ${R} = \\\\lbrace r_{0}, r_{1}, r_{2}, ... \\\\rbrace $ .\",\n \"With these assumptions, (REF ) is expressed as below: $S = \\\\sum _{t=0}^{\\\\infty }\\\\gamma ^{t}(t+1)r(s_{t},a_{t})$ Considering the maximum reward from the infinite reward sequence $R$ as $r_{max} = max \\\\lbrace r(s_{0},a_{0}), r(s_{1},a_{1}), r(s_{2},a_{2}), ... \\\\rbrace $ , the scoring function is upper bounded by: $\\\\begin{aligned}\\\\sum _{t=0}^{\\\\infty }\\\\gamma ^{t}(t+1)r(s_{t},a_{t}) & \\\\le \\\\sum _{t=0}^{\\\\infty }\\\\gamma ^{t}(t+1)r_{max}\\\\\\\\& = r_{max}\\\\sum _{t=0}^{\\\\infty }\\\\gamma ^{t}(t+1)\\\\end{aligned}$ where $\\\\sum _{t=0}^{\\\\infty }\\\\gamma ^{t}(t+1)$ has an analytical solution as follows (See Appendix B): $\\\\sum _{t=0}^{\\\\infty }{\\\\gamma ^{t}(t+1)} = \\\\frac{1}{(1-\\\\gamma )^{2}}$ Substituting (REF ) into (REF ), we have the following relation for the scoring function: $\\\\sum _{t=0}^{\\\\infty }\\\\gamma ^{t}(t+1)r(s_{t},a_{t}) \\\\le \\\\frac{1}{(1-\\\\gamma )^{2}}r_{max}$ Therefore the infinite horizon scoring function has an upper bound.\",\n \"Figure: The performance of value summation model-based reinforcement learning algorithm in comparison to well-known scoring functions.\",\n \"(i) Sum-Value: ∑ t ' =t t+H γ t ' Q π (s t ,a t )\\\\sum _{t^{\\\\prime }=t}^{t+H} \\\\gamma ^{t^{\\\\prime }}Q^{\\\\pi }(s_{t},a_{t}) (ii) Sum-Reward: ∑ t ' =t t+H γ t ' r π (s t ,a t )\\\\sum _{t^{\\\\prime }=t}^{t+H} \\\\gamma ^{t^{\\\\prime }}r^{\\\\pi }(s_{t},a_{t}) and (iii) Sum-Reward-Value ∑ t ' =t t+H-1 γ t ' r π (s t ,a t )+γ H Q π (s H ,a H )\\\\sum _{t^{\\\\prime }=t}^{t+H-1} \\\\gamma ^{t^{\\\\prime }}r^{\\\\pi }(s_{t},a_{t}) + \\\\gamma ^{H}Q^{\\\\pi }(s_{H},a_{H})\"\n ],\n [\n \"Value Summation MBRL\",\n \"This subsection presents the VS-MBRL method, which integrates the value summation scoring function into planning (Fig.\",\n \"REF ).\",\n \"This algorithm consists of two sections: gathering trajectories and scoring them.\",\n \"After receiving the current state $s_{t}$ and the instant reward $r_{t}$ from the environment, the on-board model $\\\\hat{f}(s_{t}, a_{t})$ is used to sample several trajectories with the planning horizon $H$ from the policy $\\\\pi _{\\\\theta }$ .\",\n \"The collected trajectories, then, are scored by the value summation function, and the first action from the highest-scored trajectory is executed in the environment.\",\n \"VS-MBRL employs Soft Actor-Critic (SAC) as the underlying actor-critic algorithm to take advantage of the exploration induced by soft policy updates based on the maximum entropy principle.\",\n \"This property makes SAC an excellent candidate for MBRL applications since actions can be sampled directly from the policy $\\\\pi _{\\\\theta }$ , and no additional noise is required.\",\n \"However, VS-MBRL can be applied to any developed off-policy actor-critic algorithm.\",\n \"A soft policy evaluation step is conducted in SAC using the soft Bellman backup operator , followed by a soft policy improvement step in which the expected KL divergence is minimized.\",\n \"A description of the VS-MBRL algorithm can be found in Algorithm REF .\",\n \"Using the on-board model $\\\\hat{f}(s_{t}, a_{t})$ with planning horizon $H$ , the agent shoots a number of trajectories at each time step (line code: $6-17$ ).\",\n \"Eq.\",\n \"(REF ) is applied to evaluate each pair $(s_{t:t+H}, a_{t:t+H})$ (line code: $12-13$ ) such that the corresponding trajectory can be evaluated.\",\n \"The augmented data is collected in the replay buffer ${D}$ , which is then applied to train the state-action value function (line code: 16).\",\n \"The first action from the best-scored trajectory is then executed in the environment (line code: 19).\",\n \"To reduce computation costs, the augmented data is only employed for updating state-action value functions, while the actor-network is updated by $1/{N}^{th}$ that of the critic network, where $N$ is the number of trajectories (line code: 20).\"\n ],\n [\n \"Simulation\",\n \"In this section, we compare the proposed value-summation scoring function with the most recent proposed scoring functions in the literature.\",\n \"The scoring functions used for comparison purposes are: 1) Sum-Reward: Discounted sum of rewards $\\\\sum _{t=0}^{H} \\\\gamma ^{t}r(s_{t},a_{t})$ , proposed by .\",\n \"2) Sum-Reward-Value: Discounted sum of rewards added with the estimated value of the terminal state, given by $\\\\sum _{t=0}^{H-1} \\\\gamma ^{t}r(s_{t},a_{t}) + \\\\gamma ^{H}{Q}^{\\\\pi }(s_{H}, a_{H})$ , proposed by .\",\n \"3) Sum-Value: Our proposed discounted sum of state-action values $\\\\sum _{t=0}^{H} \\\\gamma ^{t}Q^{\\\\pi }(s_{t},a_{t})$ .\",\n \"First, the performance of VS-MBRL is evaluated and compared in standard Gym environments.\",\n \"Then, VS-MBRL is integrated into the process of learning locomotion skills for the Cassie robot and its performance is compared with other baselines.\",\n \"The results indicate that the proposed scoring function outperforms other scoring functions in terms of learning efficiency and average return.\"\n ],\n [\n \"Standard Gym Environments\",\n \"We evaluate VS-MBRL in the simulated control tasks of MuJoCo included in the OpenAI-gym library : Ant-v3, Hopper-v3, HalfCheetah-v3, Walker2d-v3, Humanoid-v3, and InvertedDoublePendulum-v2.\",\n \"We compare the average return of VS-MBRL over three trials against other scoring functions.\",\n \"As shown in Fig.\",\n \"REF , the average return of each scoring function is plotted against the time step for each environment.\",\n \"In all environments, except Half-Cheetah-v3 and InvertedDoublePendulum-v2, VS-MBRL outperforms other scoring functions, proving the benefit of using value summation over other methods (See Table REF ).\",\n \"It is speculated that in Half-Cheetah-v3, the estimation of the state-action value function imposes a significant error in planning, especially at the first 200,000 time steps, when the state-action value neural network needs more data.\",\n \"Also, rewards are more useful in simple problems, such as InvertedDoublePendulum-v2, since the state-action value network requires a significant amount of training data.\",\n \"However, these events prove negligible in complex problems, such as Humanoid-v3, Walker2d-v3 and Hopper-v3, where value summation results in fast and stable learning.\",\n \"According to Ant-v3 and HalfCheetah-v3, relying only on reward summation can impede learning for a long time, resulting in poor performance.\",\n \"Table: Performance of Scoring Functions in Different Gym Environments\"\n ],\n [\n \"Cassie's Learning Locomotion Skills\",\n \"Additionally, we show that VS-MBRL is more effective than other scoring functions for the learning of Cassie's locomotion skills.\",\n \"Initially designed and built by Agility Robotics, the Cassie robot is approximately one meter tall and weighs 33 kg.\",\n \"Most of Cassie's mass is centred on the pelvis and each leg has two leaf springs to make it more flexible.\",\n \"Traditional controller design becomes more challenging as a result of underactuation.\",\n \"Xie et al.\",\n \"propose an iterative RL algorithm to increase the stability of locomotion skills through a policy distillation mechanism.\",\n \"In this way, the trained policy is used in the next iteration as an expert $\\\\pi _{e}$ .\",\n \"The expert's action is then added to that of the behavioral policy $\\\\pi _{\\\\theta }$ and fed into the PD controller.\",\n \"Each joint is controlled by a PD control loop, executed at the total rate of 2 kHz, with the targets updated every 30 ms from the policy.\",\n \"We combine VS-MBRL with the controller proposed in , in which VS-MBRL is incorporated into the feedback control loop (Fig.\",\n \"REF ).\",\n \"The controller, therefore, contains an internal loop in which the behavioral policy generates trajectories and scores them.\",\n \"Following this, the optimized action $a_{t}$ is summed up with the reference motion $\\\\delta a$ and fed into the PD controller.\",\n \"It was decided to set the horizon length $H$ and the number of trajectories $N$ to 3 to reduce computation time.\",\n \"Figure REF compares the performance of VS-MBRL with traditional scoring functions.\",\n \"In the first 400 episodes, there is a negligible difference in performance.\",\n \"Throughout the next 400 episodes, the average return exponentially increases until it reaches a maximum of 250.\",\n \"In comparison to other gym environments, there is a substantial performance gap between scoring functions.\",\n \"This is mainly because, in the simulated model of the Cassie robot for learning locomotion skills, the sampling time is $0.5$ millisecond, which is smaller than that in gym environments (Humanoid's sampling time is 3 millisecond), thus reducing the variation of states and rewards along each trajectory, resulting in difficult reward-based planning.\",\n \"Figure: The application of value summation model-based reinforcement learning algorithm in feedback controller.\",\n \"Reference motions from the expert policy are fed into the algorithm and added to the output.Figure: Value Summation MBRL algorithm (Sum-Value) outperforms other scoring functions in terms of sample efficiency and average return\"\n ],\n [\n \"Conclusion and Discussion\",\n \"This paper presents a model-based reinforcement learning algorithm incorporating discounted value summation into planning.\",\n \"In addition, Soft Actor-Critic was selected as the policy optimization algorithm, in which actions were directly sampled from a behavioral policy.\",\n \"We compared the proposed scoring function with two prevalent planning methods: discounted sum of rewards and discounted sum of rewards added with the estimated value of the terminal state.\",\n \"The performance over some standard baselines indicated the superiority of the proposed method.\",\n \"Moreover, VS-MBRL is applied to learning locomotion skills for a simulated model of the Cassie robot.\",\n \"VS-MBRL proves to be a preferable alternative to current scoring functions, offering considerably greater returns for Cassie's locomotion skills as well as MuJoCo Gym environments.\"\n ],\n [\n \"APPENDIX\",\n \"A.\",\n \"Proof of (REF ).\",\n \"Consider the case that we have an MDP with a finite horizon $T$ and a finite planning horizon $H$ .\",\n \"For simplicity, we denote $Q^\\\\pi (s_t,a_t)$ and $r(s_t,a_t)$ as $Q_t$ and $r_t$ , respectively.\",\n \"Using (REF ) iteratively, we can expand the state-action values of time steps within the planning horizon as follows: $\\\\begin{aligned}Q_H & = r_H + \\\\gamma Q_{H+1}\\\\\\\\Q_{H-1} & = r_{H-1} + \\\\gamma r_{H} + \\\\gamma ^2 Q_{H+1}\\\\\\\\& \\\\vdots \\\\\\\\Q_1 & = r_1 + \\\\gamma r_2 + \\\\gamma ^2 r_3 + ... + \\\\gamma ^{H-1} r_{H} + \\\\gamma ^H Q_{H+1}\\\\\\\\Q_0 & = r_0 + \\\\gamma r_1 + \\\\gamma ^2 r_2 + ... + \\\\gamma ^H r_H + \\\\gamma ^{H+1} Q_{H+1}\\\\end{aligned}$ The scoring function in (REF ) can be expanded as follows: $S = Q_0 + \\\\gamma Q_1 + ... + \\\\gamma ^{H-1} Q_{H-1} + \\\\gamma ^H Q_H$ Substituting the state-action values from (REF ) in (REF ), we have: $\\\\begin{aligned}S & = r_0 + \\\\gamma r_1 + \\\\gamma ^2 r_2 + ... + \\\\gamma ^H r_H + \\\\gamma ^{H+1} Q_{H+1} \\\\\\\\& + \\\\gamma r_1 + \\\\gamma ^2 r_2 + \\\\gamma ^3 r_3 + ... + \\\\gamma ^{H} r_{H} + \\\\gamma ^{H+1} Q_{H+1} \\\\\\\\& \\\\vdots \\\\\\\\& + \\\\gamma ^{H-1} r_{H-1} + \\\\gamma ^H r_{H} + \\\\gamma ^{H+1} Q_{H+1}\\\\\\\\& + \\\\gamma ^H r_H + \\\\gamma ^{H+1} Q_{H+1}\\\\\\\\& = r_0 + 2 \\\\gamma r_1 + ... + (H+1) \\\\gamma ^H r_H + (H+1) \\\\gamma ^{H+1} Q_{H+1}\\\\\\\\& = \\\\sum _{t=0}^{H} \\\\gamma ^{t}(t+1)r_t + (H+1) \\\\gamma ^{H+1} Q_{H+1}\\\\end{aligned}$ Moreover, using the definition of the state-action value, we can write the $Q_{H+1}$ as follows: $Q_{H+1} = \\\\sum _{t=H+1}^{T} \\\\gamma ^{t-H-1}r_t$ Finally, after substituting (REF ) in (REF ), we obtain the scoring function in terms of the immediate rewards as follows: $S = \\\\sum _{t=0}^{H} \\\\gamma ^{t}(t+1)r_t + \\\\sum _{t=H+1}^{T} (H+1) \\\\gamma ^{t}r_t$ B.\",\n \"Proof of the Time Series.\",\n \"From mathematics, we have the following time series with its solution: $\\\\sum _{t=0}^{\\\\infty }{\\\\gamma ^{t}} = \\\\frac{1}{1-\\\\gamma }$ The derivative of the time series in (REF ) with respect to $\\\\gamma $ results in another time series as follows: $\\\\begin{aligned}\\\\frac{d}{d\\\\gamma } \\\\sum _{t=0}^{\\\\infty }{\\\\gamma ^{t}} & = \\\\frac{d}{dt} (1 + \\\\gamma + \\\\gamma ^2 + \\\\gamma ^3 + ...) \\\\\\\\& = 1 + 2 \\\\gamma + 3 \\\\gamma ^2 + ... \\\\\\\\& = \\\\sum _{t=0}^{\\\\infty }{\\\\gamma ^{t}(t+1)}\\\\end{aligned}$ Thus, if we take the derivative of both sides of (REF ) with respect to $\\\\gamma $ , the following relation is obtained: $\\\\sum _{t=0}^{\\\\infty }{\\\\gamma ^{t}(t+1)} = \\\\frac{1}{(1-\\\\gamma )^{2}}$ [heading=bibintoc]\"\n ]\n]"},"id":{"kind":"string","value":"2209.08169"}}},{"rowIdx":2394,"cells":{"text":{"kind":"list like","value":[["Predicting the Mpemba Effect Using Machine Learning"],["Abstract The Mpemba Effect -- when a system that is further from equilibrium relaxes faster than a system that is closer -- can be studied with Markovian dynamics in a non-equilibrium thermodynamics framework.","The Markovian Mpemba Effect can be observed in a variety of systems including the Ising model.","We demonstrate that the Markovian Mpemba Effect can be predicted in the Ising model with several machine learning methods: the decision tree algorithm, neural networks, linear regression, and non-linear regression with the LASSO method.","The effectiveness of these methods are compared.","Additionally, we find that machine learning methods can be used to accurately extrapolate to data outside the range which they were trained.","Neural Networks can even predict the existence of the Mpemba Effect when they are trained only on data in which the Mpemba Effect does not occur.","This indicates that information about the effect is contained even in systems where it is not present.","All of these results demonstrate that the Mpemba Effect can be predicted in complex, computationally expensive systems, without performing full calculations."],["Mpemba Effect","The Mpemba Effect involves two systems that are not in equilibrium.","It occurs when the system that is initially farther from equilibrium becomes closer to equilibrium.","The best known example is the claim that hot water can freeze faster than cool water in the same environment.","This was named after Erasto Mpemba, who most famously brought it to the attention of the scientific community [1] although it has been debated for thousands of years [2], [3], [4].","Several recent theoretical explanations of the effect have been demonstrated.","One shows that the Mpemba Effect can be predicted from statistical mechanics [5], [6], [7], [8], [9].","Another explanation, Refs.","[10], [11] makes use of the framework developed in Ref.","[10], which considers the Markovian dynamics as cooling process in the framework of non-equilibrium thermodynamics.","We refer to this process as the Markovian Mpemba Effect (MME).","The MME is further studied in Refs.","[12], [13] Ref.","[10] applies the MME to an antiferromagnetic nearest-neighbor interacting Ising spin chain, generalizing the Mpemba Effect to other systems.","The one-dimensional Ising spin chain will be the main model used in this paper, enabling us to use machine learning methods to predict the effect, as described in Sec. .","The Mpemba Effect traditionally describes systems relaxing towards an equilibrium temperature that is lower than their initial temperature.","However, a reverse effect occurs when two systems relax towards equilibrium at a higher temperature.","Sometimes the former is referred to as the Mpemba Effect and the latter is referred to as the Inverse Mpemba Effect, but both cases meet the definition of the Mpemba Effect used in this paper, namely that the system that is initially farther from equilibrium becomes closer after a finite time.","Unless otherwise specified we refer to the Inverse Markovian Mpemba Effect simply as the Mpemba Effect.","This paper is organized as follows.","Sec.","REF discusses recent works with results that can be related to our results.","Sec.",", describe the formalism (laid out by Ref.","[10], [11]) used to determine the Mpemba Effect in the Ising model.","Sec.",", summarizes the various machine learning methods employed.","The main results of this work are presented and discussed in Sec. .","We conclude in Sec.","."],["Applications and Impact","The Mpemba Effect has been observed in a number of systems.","In addition to water, [14], [15], it has been predicted in magnetoresistance alloys [16], numerically predicted in colloids [17], predicted analytically and numerically in gasses [18], [19], predicted in particles bounded by anharmonic potentials [20], observed in computational simulations and experiments in gas hydrates [21], and in computational simulations of carbon nanotube resonators [22].","It has been demonstrated to have application for faster heating with precooling [23], that the effect happens for temperatures dependent on the maximum work that can be extracted from the system [24], and that the Mpemba Effect can lead to quantum heat engines with greater power output and stability [25].","Furthermore, the Mpemba Effect is closely linked to the Kovacs Memory Effect, by which systems out of equilibrium cannot merely be described by macroscopic thermodynamic variables.","Knowledge of the Mpemba Effect can be applied when studying memory effects in Refs.","[26], [27], [28], [29], [30].","These works demonstrate the potential applications of the effect and motivate better statistical understanding.","To our knowledge, no work has used machine methods to predict the occurence of the ME in any system.","The work presented in this paper uses learning methods to predict the Mpemba Effect in the Ising model.","The comparison of methods done in this analysis could be applied to the prediction of the ME in other systems.","Machine learning has previously been applied to the Ising model in a number of ways other than the Mpemba Effect: using autoencoding neural networks, [31], using neural networks [32] to predict probability distribution, and comparison of various classification methods such as random forests, decision trees, k nearest neighbors and artificial neural networks [33].","These works find several interesting results that can be compared to the work in this paper.","Firstly, Ref.","[32] found that only the number of nodes in the first layer of the deep neural network could improve the accuracy of the prediction.","Secondly, Ref.","[33] found that the Decision Tree method was the most accurate predictor.","Even though previous results predict different things, comparison with our work could show general trends in machine learning applied to the Ising model.","This work uses the formalism developed in Refs.","[10], [11] for the Mpemba Effect in the Ising model.","We summarize the key equations in the following.","The one-dimensional Ising model consists of a chain of $N$ spins, $s_i$ with values of 1 or -1.","The energy for a given set of spins is defined to be $E=-J\\sum _{k=0}^N s_k s_{k+1}-h\\sum _{k=1}^N s_k,$ where $J$ and $h$ are parameters respectively giving the field of interaction between neighboring spins and the strength of the external field.","A probabilistic distribution $\\vec{p}(t)$ that represents the probability of finding the system in one of the $2^N$ micro-states is defined for the case of a finite number of states, by the equation: $\\frac{d \\vec{p}(t)}{dt}= R(T_b)\\vec{p}(t),$ for $i=1,2,\\cdots , n$ .","Matrix multiplication is assumed.","Here $R_{ij}(T_b)$ is the transition rate matrix from a state $j$ to state $i$ defined as: $R_{ij} = {\\left\\lbrace \\begin{array}{ll}\\Gamma e^{-\\frac{1}{2}\\beta _b (E_{i} - E_j)},& \\text{if i and j differ by one spin flip} \\\\0,& \\text{if i and j differ by} >1 \\text{ spin flip} \\\\-\\sum _{k\\ne j}R_{{kj}}, & i=j\\\\\\end{array}\\right.","}$ At a given temperature, a system will have an equilibrium state, $\\vec{\\pi }(T)=\\frac{e^{-E_i/k_BT}}{\\sum _{i}e^{-E_i/k_BT}}.$ Because the eigenvalues form a complete basis, any state can be expressed as, $\\vec{p}=\\vec{\\pi }(T)+\\sum _{i>1} a_i \\vec{v}_i$ where $v_i$ are the eigenvectors of the rate transition matrix and $a_i$ are coefficients.","$\\vec{\\pi }$ is the eigenvector corresponding to an eigenvalue of 0.","The general solution to Eq.","(REF ) is given by $\\vec{ p}(T;t) = e^{Rt}\\vec{\\pi }(T)= \\vec{\\pi }(T_b) + \\sum _{i > 1} a_i(T) e^{\\lambda _{i}t} \\mathbf {v}_i,$ and $\\lambda _i$ are the eigenvalues of the transition rate matrix.","The distance from equilibrium function, $D[\\vec{p}(t); T_b]$ , is given by $D[\\vec{p}(t); T_b]=\\sum _i \\left( \\frac{E_i (p_i-\\pi _i^b)}{T_b}+p_i \\ln {p_i}-\\pi ^b_i \\ln {\\pi _i^b} \\right).$ The Mpemba Effect occurs for any two probability distributions $\\vec{p}_H$ and $\\vec{p}_C$ if $D[\\vec{p}_H(0); T_b]>D[\\vec{p}_C(0); T_b]$ but $D[\\vec{p}_H(t^{\\prime }) T_b]1 \\\\text{ spin flip} \\\\\\\\-\\\\sum _{k\\\\ne j}R_{{kj}}, & i=j\\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ At a given temperature, a system will have an equilibrium state, $\\\\vec{\\\\pi }(T)=\\\\frac{e^{-E_i/k_BT}}{\\\\sum _{i}e^{-E_i/k_BT}}.$ Because the eigenvalues form a complete basis, any state can be expressed as, $\\\\vec{p}=\\\\vec{\\\\pi }(T)+\\\\sum _{i>1} a_i \\\\vec{v}_i$ where $v_i$ are the eigenvectors of the rate transition matrix and $a_i$ are coefficients.\",\n \"$\\\\vec{\\\\pi }$ is the eigenvector corresponding to an eigenvalue of 0.\",\n \"The general solution to Eq.\",\n \"(REF ) is given by $\\\\vec{ p}(T;t) = e^{Rt}\\\\vec{\\\\pi }(T)= \\\\vec{\\\\pi }(T_b) + \\\\sum _{i > 1} a_i(T) e^{\\\\lambda _{i}t} \\\\mathbf {v}_i,$ and $\\\\lambda _i$ are the eigenvalues of the transition rate matrix.\",\n \"The distance from equilibrium function, $D[\\\\vec{p}(t); T_b]$ , is given by $D[\\\\vec{p}(t); T_b]=\\\\sum _i \\\\left( \\\\frac{E_i (p_i-\\\\pi _i^b)}{T_b}+p_i \\\\ln {p_i}-\\\\pi ^b_i \\\\ln {\\\\pi _i^b} \\\\right).$ The Mpemba Effect occurs for any two probability distributions $\\\\vec{p}_H$ and $\\\\vec{p}_C$ if $D[\\\\vec{p}_H(0); T_b]>D[\\\\vec{p}_C(0); T_b]$ but $D[\\\\vec{p}_H(t^{\\\\prime }) T_b]0$ corresponds to compressing the top layer while streching the bottom layer along the direction determined by $\\varphi $ , as illustrated in Fig.","REF (a).","A relative deformation $\\mathcal {E}$ between the graphene bilayers generates a moiré superlattice, with moiré reciprocal lattice vectors $\\mathbf {g}_{i=1,2}=\\mathcal {E}^T\\mathbf {G}_{i=1,2}$ , where $\\mathbf {G}_{i}$ are reciprocal lattice vectors of the undeformed monolayer graphene.","The moiré lattice vectors $\\mathbf {L}_{i=1,2}$ are uniquely defined through the relation $\\mathbf {L}_i\\cdot \\mathbf {g}_j=2\\pi \\delta _{ij}$ .","It is important to note that only relative deformations generate the moiré superlattice.","Homogenous deformations do not play an important role in the narrow band physics, and we neglect it in this work We checked numerically that adding a small homogeneous strain in addition to a heterostrain of similar strength yields almost identical band and transport properties to the case of adding a heterostrain alone..","Under rotation $R_\\varphi $ , the strain tensor transforms as a headless vector that remains invariant under $\\varphi \\rightarrow \\varphi +180^\\circ $ .","Combined with the $C_{3z}$ symmetry of the undeformed graphene lattice, the strained electronic dispersion within a given graphene valley simply rotates $60^\\circ $ under $\\varphi \\rightarrow \\varphi + 60^\\circ $ .","We hereby will only report results for $\\varphi \\in [0^\\circ ,60^\\circ )$ .","For concreteness we define the microscopic unit cell vectors $\\mathbf {a}_{i=1,2}$ of undeformed graphene lattice as $ \\mathbf {a}_1 = a(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}),\\ \\mathbf {a}_2 = a(1,0)$ , where $a\\approx 2.46Å$ is the lattice constant.","The positions of the sublattice A,B within a unit cell are chosen as $\\vec{\\tau }_A = (0,0)$ and $\\vec{\\tau }_B = \\frac{a}{\\sqrt{3}}(0,1)$ .","The reciprocal lattice vectors are $\\mathbf {G}_1 = \\frac{4\\pi }{\\sqrt{3}a}(0,-1)$ and $\\mathbf {G}_2 = \\frac{4\\pi }{\\sqrt{3}a}(\\frac{\\sqrt{3}}{2},\\frac{1}{2})$ .","Different conventions lead to different definitions of the Dirac Hamiltonian (see for instance Ref.","[38]), but the physics is consistent.","Fig.","REF (a-b) illustrates the geometric effects of heterostrain for twist angle $\\theta =1.38^\\circ $ .","For $\\epsilon =0.2\\%$ , typical in these systems [3], [9], [11], there is a large change in the bond length of the neighboring AA-stacked regions ($\\mathbf {L}_{i=1,2,3}$ ) of the moiré triangular superlattice, which used to form an equilateral triangle at $\\epsilon =0$ .","The effect of heterostrain on the moiré unit cell vectors can be estimated to be as large as $\\epsilon /\\theta \\approx 8\\%$ .","However, the effect on the moiré unit cell area is much smaller at $\\nu ^2\\epsilon ^2/\\theta ^2$ (see Supplementary Material (SM) Sec.","I).","Such dramatic amplification of the microscopic strain makes moiré materials ideal for strain engineering not achievable in conventional materials due to structural instability.","We proceed to discuss the energetic effects in the context of the continuum BM model [18].","We work in the limit where both $\\mathcal {E}_l$ and the wavevector $\\mathbf {k}$ in the moiré Brillouin zone are small, and consider only the leading order terms in both.","This would mean, for instance, that terms such as $\\mathcal {E}\\mathbf {k}$ are omitted as higher order terms.","This treatment is generally justified away from the magic angle, because higher order terms can play an important role only close to the magic angle where the narrow-band bandwidth is suppressed to a similar energy scale [43], [44].","Furthermore, we checked that at $\\theta \\approx 1.38^\\circ $ the effects of such higher order terms are indeed negligibly small.","To leading order, the strained BM Hamiltonian for a given valley is given by: $H_{\\eta } = (\\sum _{l=t,b} H_{\\eta ,l}^{intra}) + H_{\\eta }^{inter} ,$ where $\\eta =\\pm 1$ labels $\\mathbf {K}\\ (\\mathbf {K}^{\\prime })$ valleys of monolayer graphene.","The interlayer Hamiltonian is given by: $H_{\\eta ,l}^{inter} \\approx \\int \\mathrm {d}^2\\mathbf {r}\\psi ^\\dagger _{\\eta ,t} \\left(\\sum _{j=1,2,3} T_{\\eta ,j} e^{-i\\eta \\mathbf {q}_{j}\\cdot \\mathbf {r}}\\right)\\psi _{\\eta ,b}(\\mathbf {r}) + h.c.,$ where $\\psi _{\\eta ,l}(\\mathbf {r})\\equiv (\\psi _{\\eta ,l,A}(\\mathbf {r}),\\psi _{\\eta ,l,B}(\\mathbf {r}))^T$ is a spinor in the sublattice basis for a given valley and layer.","We have suppressed the spin index for simplicity.","$\\mathbf {q}_{j=1,2,3}$ are the three nearest neighbor bonds of the reciprocal honeycomb lattice, and $T_{\\eta ,j} = w_0\\sigma _0+w_1\\left(\\cos \\frac{2\\pi (j-1)}{3} \\sigma _x+ \\eta \\sin \\frac{2\\pi (j-1)}{3}\\sigma _y\\right).$ $(\\sigma _0,\\sigma _x,\\sigma _y)$ are Pauli matrices acting on sublattice degrees of freedom.","The intra-layer Hamiltonian is given by: $\\begin{split}& H_{\\eta ,l}^{intra} = \\alpha \\sum _{\\mathbf {k}} \\psi ^\\dagger _{\\eta ,l}(\\mathbf {r}) ( [\\mathcal {E}_l] \\sigma _0)\\psi _{\\eta ,l}(\\mathbf {r})\\\\&- \\frac{\\hbar v_F}{a}\\sum _{\\mathbf {k}} \\psi ^\\dagger _{\\eta ,l}(\\mathbf {r}) \\left[ (\\mathbf {k}- \\mathbf {A}_{\\eta ,l})\\cdot (\\eta \\sigma _x,\\sigma _y) \\right]\\psi _{\\eta ,l}(\\mathbf {r}).\\end{split}$ Here the first term is the deformation potential that couples to the electron density.","Its value is not precisely known in the literature, with numbers ranging from $-4.1\\ \\mathrm {eV}$ to $30\\ \\mathrm {eV}$ depending on the methodology [45], [46], [47], [48].","We use $\\alpha =-4.1\\ \\mathrm {eV}$ in this work based on first principles calculations [48], although the deformation potential does not have an important effect on the band dispersions for heterostrain $\\epsilon \\approx 0.2\\%$ , and only leads to minor quantitative differences.","$\\mathbf {A}_{\\eta ,l}$ is the pseudovector potential that comes from changes in the inter-sublattice hopping due to deformations, and changes sign between graphene valleys.","It is given as [40], [41]: $\\mathbf {A}_{\\eta ,l} = \\frac{\\sqrt{3}\\beta }{2a}\\eta (\\epsilon _{l,xx}-\\epsilon _{l,yy},-2\\epsilon _{l,xy})$ , where we choose $\\beta \\approx 3.14$ from Refs.","[3], [38].","We shall further fix $\\hbar v_F/a=2.68\\mathrm {eV}$ , $w_0=88\\mathrm {meV}$ , and $w_1=110\\mathrm {meV}$ in our calculations, and also set $\\hbar =1$ in the remainder of the paper.","To leading order approximation, the strained BM Hamiltonian in a given valley (Eq.","(REF )) has particle-hole symmetry under $P \\psi _{l}(\\mathbf {r}) = \\sum _{l^{\\prime }}i(\\mu _y)_{ll^{\\prime }}\\psi _{l^{\\prime }}(-\\mathbf {r})$ [49], where $\\mu _y$ is a Pauli matrix acting on the layer degrees of freedom.","This means that for every single electron state at energy $E$ and wavevector $\\mathbf {k}$ , there is a state at energy $-E$ and wavevector $-\\mathbf {k}$ .","This particle-hole symmetry has been investigated extensively for the unstrained BM model, e.g., Refs.","[25], [50], and here it is generalized to the strained case.","Since in experiments particle-hole asymmetry is evident for the off-magic-angle device [39], they would come from either higher order gradient terms beyond what's captured in the BM model in Eq.","(REF ), or due to interaction effects [51], [52], [53], [54], or their combination.","We proceed to discuss the heterostrain effects on the band structure with $\\epsilon =0.2\\%$ and varying direction specified by $\\varphi \\in [0^\\circ ,60^\\circ )$ , depicted in Fig.","REF (d-f).","For simplicity we only show contour maps of the upper band from valley $\\mathbf {K}$ in the moiré Brillouin zone specified by $\\mathbf {k}=k_1\\mathbf {g}_1+k_2\\mathbf {g}_2$ , where $k_{1,2}\\in [0,1)$ .","Heterostrain preserves $C_2T$ and valley $U(1)$ [22] and therefore the lower and upper bands remain connected via two Dirac points.","The upper band features six special points — two Dirac points (black stars), three van Hove points (colored dots), and one band maximum (black cross).","The six special points of a given band are related to “critical points\" in the context of the Morse theory, which states that $\\sum _{i}(-1)^{\\gamma _i}=\\chi ,$ where $\\gamma _i$ is the index of the $i$ -th critical point, and $\\chi $ is the Euler characteristic of a manifold [55]; $\\chi $ vanishes for the Brillouin zone which is a torus.","Although a Dirac point is strictly a point of non-analyticity and is not directly covered by Morse theory, if we imagine adding a tiny gap term it will become a legitimate band extremum and Morse theory applies.","Whereas the two band minima (Dirac points) and the band maximum have even $\\gamma $ and so each contributes $+1$ to the sum, every conventional van Hove point (i.e.","not a higher order) has an odd $\\gamma $ and contributes $-1$ .","Their sum thus vanishes.","Therefore, the van Hove points can only be annihilated/created by colliding with local minima/maxima.","For a relatively small heterostrain as shown in Fig.","REF , the number of special points per band is the same as at $\\epsilon =0$ .","However for larger heterostrain (e.g., $\\epsilon =0.5\\%$ , see SM Fig.","1), more striking behavior of the special points can occur, such as a change in their total number via afore mentioned collisions and the appearance of tilted type II Dirac cones [56], [57].","A key finding of the present work is that the respective energy degeneracies of the two Dirac points and the three van Hove points are lifted by uniaxial heterostrain, and depend sensitively on $\\varphi $ .","In the absence of strain [Fig.","REF (c)], the three van Hove points are at equal energy, and separate closed contours of constant energy centered around the Dirac points from closed contours centered around band maximum.","As illustrated in Fig.","REF (d-f), uniaxial heterostrain splits the energy degeneracy of the two Dirac points, leading to a semimetallic state with small Fermi pockets near CNP [38].","The three van Hove points also split in energy.","The two outermost van Hove points (i.e., closer to the band maximum) bound a filling range of open FSs near $\\nu =2$ , while the innermost van Hove point moves closer to one of the Dirac points.","If we continue increasing $\\epsilon $ , a collision of the critical points occurs, the innermost van Hove disappears, the two Dirac points become type-II tilted, and a new ordinary minimum is created.","Note that a small mass added to type-II tilted Dirac points won't introduce band extrema and as a consequence type-II tilted Dirac points are not critical points of Morse theory, therefore after the collision Eq.","(REF ) still holds.","Interestingly, the elongation of the FSs shows a strong filling dependence.","Close to the CNP, the bigger Fermi pocket that encloses a Dirac point is stretched along a perpendicular direction to that of the open FSs, see Figs.","REF (d-f).","As explained later, this leads to a dramatic rotation of the principal transport axis when the filling is tuned from the CNP to the open FS range.","The dependence of the energy and filling of the band structure special points on $\\varphi $ at a fixed $\\epsilon $ is shown in Fig.","REF (g-h).","Of notable interest is the sensitivity of the filling range with open FSs to $\\varphi $ .","This filling range must in fact vanish at some $\\varphi $ between $0^\\circ $ and $60^\\circ $ , when the energies of the two outermost van Hove points cross.","As seen in Fig.","REF (d-f), this also alters the elongation of the open FSs."],["Boltzmann equation and Magnetoresistivity in TBG","Having understood the heterostrain effects on the bandstructure, we proceed to discuss the implications for magnetotransport.","We begin by considering the general structure of the two-dimensional resistivity tensor ${\\rho }$ subject to heterostrain.","The resistivity tensor is defined via: $\\begin{pmatrix}E_x \\\\ E_y\\end{pmatrix} =\\begin{pmatrix}\\rho _{xx} & \\rho _{xy}\\\\\\rho _{yx} & \\rho _{yy}\\end{pmatrix}\\begin{pmatrix}j_x \\\\ j_y\\end{pmatrix},$ where $\\mathbf {E}=(E_x,E_y)^T$ and $\\mathbf {j}=(j_x,j_y)^T$ are electric field and current vectors respectively.","Under rotation by $\\delta \\theta $ , the resistivity tensor transform as: ${\\rho }^{\\prime } = R_{\\delta \\theta }^T {\\rho } R_{\\delta \\theta },\\ R_{\\delta \\theta } = \\begin{pmatrix}\\cos \\delta \\theta & -\\sin \\delta \\theta \\\\\\sin \\delta \\theta & \\cos \\delta \\theta \\end{pmatrix}.$ If the underlying system has a point group symmetry that is higher than $C_{2z}$ (e.g., $C_{3z},C_{6z}$ ), then ${\\rho }=\\rho _0\\mathbb {I} - i \\rho _H \\tau _y$ is the most general form of ${\\rho }$ invariant under such rotations.","Here $\\tau _y$ is the Pauli matrix acting in the two-dimensional coordinate basis, $\\rho _0(-B)=\\rho _0(B)$ is the longitudinal resistivity, and $\\rho _H(-B)=-\\rho _H(B)$ is the Hall resistivity.","The even/odd parity under time reversal is guaranteed by the Onsager reciprocal relations.","Since heterostrain breaks the point group symmetry down to $C_{2z}$ , we generally expect $\\rho _{xx}\\ne \\rho _{yy},\\ \\rho _{xy}\\ne -\\rho _{yx}$ .","Nevertheless, it is always possible to define transport principal axes after a suitable rotation $\\delta \\theta $ of the coordinate system, such that: $ {\\rho }_{\\text{principal}}= \\frac{1}{2}(\\rho _1+\\rho _2)\\mathbb {I}+\\frac{1}{2}(\\rho _1-\\rho _2)\\tau _z + \\rho _H i\\tau _y.$ Here $\\rho _{1,2}$ are longitudinal resistivities along the principal transport directions $\\hat{e}_{1,2}$ respectively.","The rotation angle $\\delta \\theta $ is determined up to $180^\\circ $ by requiring $\\rho _1< \\rho _2$ .","Below we first derive the MR tensor using Boltzmann approach for a general non-interacting electronic system within the relaxation time approximation.","Since there is currently insufficient understanding of the scattering mechanisms determining electrical transport in TBG, here we follow Ref.","[58] and use relaxation time approximation.","We will then present the results for heterostrained TBG, showing that in the open FS region, the low resistivity principal axis ($\\hat{e}_{1}$ ) is nearly perfectly aligned with the shortest moiré bond direction.","However there is a dramatic rotation of the principal axis as the filling moves towards the CNP.","We further show that the open FSs lead to a $B^2$ non-saturating MR along $\\hat{e}_{2}$ , and a saturating resistivity along $\\hat{e}_{1}$ .","For random orientation ($\\theta _0$ ) of the principal axis to the electrical current axis in the Hall bar geometry, e.g., as in Ref.","[39], the longitudinal resistivity is given by: $\\rho _{xx} =\\rho _1\\cos ^2 \\theta _0+\\rho _2\\sin ^2\\theta _0$ .","It is dominated by the $\\rho _2\\sim B^2$ component, and as a result, the experimental measurements should observe the non-saturating MR component if there is a misalignment with respect to the principal transport axis."],["Boltzmann equation and method of characteristics","We begin with a brief description of the method of characteristics used to solve the Boltzmann equation perturbatively in electric field ${\\bf E}$ but without a restriction on the strength of the perpendicular magnetic field ${\\bf B}=B\\hat{z}$ , as long as the semiclassical regime holds [59].","Due to $C_{2z}T$ symmetry of TBG at ${\\bf B}=0$ , there is no Berry curvature contribution to the semiclassical equations of motion.","Then, within the relaxation time approximation, the Boltzmann equation for a given energy band becomes $\\frac{\\partial n_\\mathbf {k}}{\\partial t} + (q\\mathbf {E}+q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B}) \\cdot \\frac{\\partial n_\\mathbf {k}}{\\partial \\mathbf {k}} = - \\frac{n_{\\mathbf {k}}-n_{0,\\mathbf {k}}}{\\tau },$ where $q\\mathbf {E}+q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B}$ is the total force on the Bloch electrons, with $\\mathbf {v}_\\mathbf {k}\\equiv \\nabla _{\\mathbf {k}} \\varepsilon _{\\mathbf {k}}$ and charge $q$ ; $n_{0,\\mathbf {k}}$ is the equilibrium Fermi-Dirac distribution and $n_{\\mathbf {k}}$ is the desired non-equilibrium distribution function.","We consider a stationary solution to the Boltzmann equation by parameterizing the distribution function as: $n_{\\mathbf {k}} = n_{0,\\mathbf {k}} + n_{1,\\mathbf {k}}.$ As a result, the Boltzmann equation for the deviation of the distribution function from equilibrium is: $(q\\mathbf {E}\\cdot \\mathbf {v}_{\\mathbf {k}}) \\frac{\\partial n_{0,\\mathbf {k}}}{\\partial \\varepsilon _\\mathbf {k}}+(q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B})\\cdot \\frac{\\partial n_{1,\\mathbf {k}}}{\\partial \\mathbf {k}} = - \\frac{n_{1,\\mathbf {k}}}{\\tau }.$ Note that the magnetic field only couples to $n_1$ since $(q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B})\\cdot \\nabla _\\mathbf {k}n_{0,\\mathbf {k}}=(q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B})\\cdot \\mathbf {v}_{\\mathbf {k}} \\partial _{\\varepsilon _{\\mathbf {k}}} n_{0,\\mathbf {k}}=0$ .","In order to solve the above partial differential equation (PDE), we seek a family of curves covering the $\\mathbf {k}$ -space which we parameterize as $\\mathbf {k}(s)$ with $s\\in [0,s_0)$ , such that along these curves the PDE becomes an ordinary differential equation (ODE).","If a curve $\\mathbf {k}(s)$ satisfies $ \\frac{d\\mathbf {k}(s)}{d s} = q\\mathbf {v}(s)\\times \\mathbf {B},$ then $n_{1,\\mathbf {k}(s)}\\equiv n_{1}(s)$ satisfies $(q\\mathbf {E}\\cdot \\mathbf {v}_{\\mathbf {k}}) \\frac{\\partial n_{0,\\mathbf {k}}}{\\partial \\varepsilon _\\mathbf {k}}|_{\\mathbf {k}=\\mathbf {k}(s)}+ \\frac{d n_{1}(s)}{d s} = - \\frac{n_{1}(s)}{\\tau }.$ Because $\\frac{d \\varepsilon (s)}{d s} = \\mathbf {v}(s) \\cdot \\frac{d \\mathbf {k}(s)}{d s} = 0,$ the curve $\\mathbf {k}(s)$ must coincide with the contour of constant energy.","Thus, the Boltzmann equation becomes: $[q\\mathbf {E}\\cdot \\mathbf {v}(s)] \\frac{\\partial n_{0}(s)}{\\partial \\varepsilon (s)}+ \\frac{d n_{1}(s)}{d s} = - \\frac{n_{1}(s)}{\\tau }.$ The ODE is readily solved with: $n_{1}(s) = \\chi _0 e^{-s/\\tau } - e^{-s/\\tau }\\int _0^{s}\\mathrm {d}s^{\\prime } e^{s^{\\prime }/\\tau } [q\\mathbf {E}\\cdot \\mathbf {v}(s^{\\prime })] \\frac{\\partial n_{0}(s^{\\prime })}{\\partial \\varepsilon (s^{\\prime })}.$ where $\\chi _0$ is a constant determined by the following argument.","Since $\\mathbf {k}(s)$ describes a constant energy contour in a two-dimensional Brillouin zone, it is either a closed contour, or several open contours that terminate on boundaries of the Brillouin zone such that they form a closed loop on a torus.","In either case, $\\mathbf {k}(s)$ is periodic under $s\\rightarrow s+s_0$ modulo a moiré reciprocal lattice vector, where $s_0$ is the periodicity.","The periodicity condition $n_{1}(s_0)=n_{1}(0)$ leads to $\\chi _0 = \\frac{1}{1-e^{s_0/\\tau }}\\int _0^{s_0}\\mathrm {d}s^{\\prime } e^{s^{\\prime }/\\tau }(q\\mathbf {E}\\cdot \\mathbf {v}(s^{\\prime })) \\frac{\\partial n_{0}(s^{\\prime })}{\\partial \\varepsilon (s^{\\prime })},$ which determines the desired $n_1(s)$ .","In the low temperature limit, the steady state current from a given energy band is calculated as: $ \\begin{split}j^\\mu & = q\\int \\frac{\\mathrm {d}^2\\mathbf {k}}{(2\\pi )^2} v^{\\mu }_{\\mathbf {k}} n_{1,\\mathbf {k}} \\\\& =\\frac{q^2B}{(2\\pi )^2} \\int \\mathrm {d} {\\varepsilon }\\int _0^{s_0} \\mathrm {d} s v^{\\mu }(s) n_{1}(s)\\\\& = \\frac{q^3B}{(2\\pi )} \\frac{\\tau }{\\omega _c} \\sum _{n=-\\infty }^{\\infty } \\frac{v^{\\mu }_nv^{\\nu }_{-n}}{1+in \\omega _c \\tau }E^{\\nu },\\end{split}$ where $(\\mu ,\\nu )=x,y$ , and we have defined the cyclotron frequency as: $\\omega _c \\equiv 2\\pi /s_0.$ We have also made use of the periodicity of velocity under $s\\rightarrow s+s_0$ to write it in terms of Fourier series, $\\mathbf {v}(s)=\\sum _{n=-\\infty }^{\\infty }\\mathbf {v}_ne^{-in \\omega _c s }$ .","To show that the second line of Eq.","(REF ) holds, note that at every $\\mathbf {k}$ we can define a local coordinate system $(\\hat{e}_{v},\\hat{e}_{s})$ such that $\\mathbf {v}\\equiv v\\hat{e}_{\\mathbf {v}}$ where $v\\ge 0$ , and $\\hat{e}_{s}=\\hat{e}_{\\mathbf {v}}\\times \\hat{z}$ .","The infinitesimal wavevector can be equivalently written as: $\\mathrm {d}\\mathbf {k}=\\mathrm {d}k_x \\hat{e}_x + \\mathrm {d}k_y \\hat{e}_y= \\mathrm {d}k_s \\hat{e}_s + \\mathrm {d}k_v \\hat{e}_v.$ Eq.","(REF ) can then be written as ${\\mathrm {d}\\mathbf {k}}/{\\mathrm {d} s} = qvB \\hat{e}_s$ , or equivalently $\\mathrm {d} k_s = qvB\\mathrm {d}s$ .","As a result, $\\int \\mathrm {d} k_x \\mathrm {d}k_y = \\int \\mathrm {d} k_s \\mathrm {d}k_v = qB \\int \\mathrm {d}\\varepsilon \\mathrm {d} s.$ The conductivity tensor is therefore given by the following expression: $ \\sigma ^{\\mu \\nu } = \\frac{q^3B}{2\\pi } \\frac{\\tau }{\\omega _c} \\sum _{n=-\\infty }^{\\infty } \\frac{v^{(\\mu )}_nv^{(\\nu )}_{-n}}{1+in \\omega _c\\tau }.$ Eq.","(REF ) gives the magnetoconductivity for a given FS contour.","In the case of multiple FS contours and multiple bands –as due to spin and valley degeneracy in TBG– conductivities from different FS contours and bands add.","Finally, the MR tensor is obtained by inverting the conductivity tensor, i.e., $\\rho = \\left(\\sum _{n,i}\\sigma _{n,i}\\right)^{-1}$ , where $n,i$ are band and contour labels respectively for a given energy level.","To better understand Eq.","(REF ) consider an example of a parabolic dipsersion with $\\varepsilon _{\\mathbf {k}}=\\frac{1}{2m_0}(k_x^2+k_y^2)$ , where $m_0$ is the bare electron mass.","At a fixed energy $\\mu $ the contour is a circle parameterized as: $(k_x,k_y)=\\sqrt{2m_0\\mu }(\\cos \\theta ,\\sin \\theta ),\\ \\theta \\in [0,2\\pi )$ .","Using method of characteristics, we get: $\\frac{\\mathrm {d}\\theta }{\\mathrm {d} s} = -\\frac{qB}{m_0}$ , or $\\theta = \\theta _0 - \\omega _0 s$ , where $\\omega _0\\equiv \\frac{qB}{m_0}$ is the cyclotron frequency of bare electrons.","This leads to the periodicity in $s$ to be $s_0=2\\pi /\\omega _0$ , where we have chosen the clockwise trajectory such that $s_0>0$ .","The Fourier series of the velocity along the constant energy contour is given by: $v_{x}(s)=\\sqrt{\\frac{\\mu }{2m}}\\left(e^{-i \\omega _0 s}+e^{i\\omega _0 s}\\right)$ , and $v_{y}(s)=\\sqrt{\\frac{\\mu }{2m}}\\frac{1}{i}\\left(e^{-i \\omega _0 s}-e^{i\\omega _0 s}\\right)$ .","Substituting into Eq.","(REF ), we obtain the conductivity tensor: $\\sigma = q^2\\tau \\frac{\\mu }{2\\pi } \\frac{1}{1+\\omega _0^2\\tau ^2}\\begin{pmatrix} 1 & -\\omega _0\\tau \\\\ \\omega _0\\tau & 1 \\end{pmatrix}.$ Note that the total number density of filled electrons is given by $n = \\int \\frac{\\mathrm {d}^2\\mathbf {k}}{(2\\pi )^2} \\Theta (\\mu -\\varepsilon _{\\mathbf {k}}) = \\frac{m_0\\mu }{2\\pi }$ .","We therefore reproduce the well known magnetoconductivity tensor: $\\sigma = \\frac{nq^2\\tau }{m_0} \\frac{1}{1+\\omega _0^2\\tau ^2}\\begin{pmatrix} 1 & -\\omega _0\\tau \\\\ \\omega _0\\tau & 1 \\end{pmatrix}.$ In this simple example of a closed FS, the longitudinal resistivity is given by $\\frac{m_0}{nq^2\\tau }$ , independent of the magnetic field.","The average of the velocity field, $\\mathbf {v}_{n=0}\\equiv \\frac{1}{s_0}\\int _0^{s_0}\\mathrm {d}s \\mathbf {v}(s)$ , vanishes.","However, for an open FS generally $\\mathbf {v}_{n=0} \\ne \\mathbf {0}$ , i.e.. electrons have a finite drift velocity when traversing the contour due to a magnetic field (see SM Fig. 2).","The impact of such a finite drift velocity on the magnetotransport can be qualitatively understood using the following example: in the expression for the conductivity tensor (Eq.","(REF )), we consider $v^{x}_{n=0}\\ne 0$ but $v^{y}_{n=0}= 0$ .","This corresponds to an open FS with a drift velocity along the $x$ direction.","In the high field limit ( $\\omega _c\\tau \\propto B \\gg 1)$ , we truncate the Fourier series at the leading order, and as a result, $\\sigma _{\\text{open FS}} \\approx \\frac{q^3B}{2\\pi }\\frac{\\tau }{\\omega _c}\\begin{pmatrix}{(v^{x}_0)^2} & -\\frac{2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau }\\\\ \\frac{2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau } & \\frac{|v^{y}_1|^2}{\\omega _c^2\\tau ^2}\\end{pmatrix},$ where we made use of the equality: $\\mathbf {v}_{-n}=\\mathbf {v}_{n}^*$ .","Inverting the matrix, we obtain the MR tensor: $\\begin{split}{\\rho }_{\\text{open FS}} &\\approx \\frac{(2\\pi )\\omega _c}{q^3B \\tau } \\frac{1}{4\\text{Im}(v_{-1}^{x}v_{1}^{y})^2+(v_0^{x})^2|v_{1}^{y}|^2} \\\\& \\times \\begin{pmatrix}|v^{y}_1|^2 & {2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau }\\\\ -{2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau } & (v^{(x)}_0)^2\\left(\\omega _c\\tau \\right)^2\\end{pmatrix}.\\end{split}$ It is clear that $\\rho _{yy}\\propto B^2$ whereas $\\rho _{xx}\\sim \\mathcal {O}(1)$ .","We therefore arrive at the important conclusion that for an open FS, the longitudinal MR has non-saturating $B^2$ behavior along the axis with a zero drift velocity ($\\hat{y}$ in the above example), and saturating behavior along the other axis."],["Magnetotransport in TBG under heterostrain","We proceed to apply the above results to analyze the magnetotransport in TBG.","The theory satisfactorily explains the weak-field magnetotransport measurements presented in Ref. [39].","We then present two predictions of the theory that we did not anticipate prior to starting this work: the dependence of the principle axis of transport on filling, and the behavior of magnetoresistance and quantum oscillations at densities between the CNP and the onset of quadratic MR.","The former is of academic interest, however it cannot be confirmed with our present data sets because of limitations of the Hall bar geometry.","The latter can be considered smoking gun evidence for the the presence of the lowest-energy van Hove point and the energetic splitting of the Dirac cones.","We do not expect our strained BM model in Eq.","(REF ) to yield precise agreement with experiment, so we do not perform fine-tuning of its input parameters.","Specifically, the model has particle-hole symmetry, which is absent from experimental measurements.","More sophisticated non-interacting model calculations [44], [43] as well as interaction renormalizations [54] are likely necessary to properly account for such details.","Although the general phenomena of open FSs and quadratic MR holds for a broad range of heterostrain parameters, we present calculations for $\\theta =1.38^\\circ $ , $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ , parameters chosen to yield reasonable quantitative agreement between the theoretical and experimental results both on the filling range of open FSs, as well as on the frequencies of magnetoresistance oscillations to be presented later.","In Fig.","REF , we show the computed MR along the principal transport axes (a) and the Hall number (b).","For comparison, we plot the experimentally measured longitudinal and transverse resistivities (c) and Hall number (d) for the TBG device studied in Ref. [39].","In the filling ranges with open FSs, the calculated $\\rho _2(B)$ exhibits quadratic non-saturating MR, whereas $\\rho _1(B)$ saturates.","The filling range for which quadratic MR occurs is bounded by the two outermost van Hove points of the zero-field strained band structure.","In experiment, we observe quadratic MR in longitudinal resistivity within a similar range of fillings.","More strikingly, we observe quadratic MR in the transverse resistivity as well.","In some cases, the symmetric part of the transverse resistivity becomes larger than that of the longitudinal resistivity with field.","As discussed earlier, this degree of mixing can be attributed to the misalignment between the strain-induced principal axis of transport and the direction of current flow in the Hall bar geometry.","At the first van Hove point ($\\nu \\approx \\pm 0.6$ ), the non-analyticity in the density of states leads to a cusp in the first derivative of the zero-field resistivity with respect to filling (see SM Fig. 6).","As shown in Fig.","REF (a), at $B\\ne 0$ the longitudinal resistance develops a cusp as a function of filling at the first van Hove point.","The cusp becomes more pronounced with increasing $B$ .","Experimentally as shown in Fig.","REF (c), there is a cusp-like feature developing at $|\\nu |\\sim 0.5-0.8$ depending on the contact pair within the device used, consistent with theoretical predictions.","In many contact pairs, this feature presents as a shoulder at $B=0$ , only developing into a cusp at $B \\sim 0.1$ T (see SM Fig. 7).","As depicted in Fig.","REF (b), the calculated filling dependence of the Hall number shows two singular sign changes inside the open FS regions near $\\nu \\approx \\pm 2$ .","The sign changing singularity in the open FS region is $B$ -independent, and is not directly associated with any van Hove point (see SM Fig.","5 for a plot of $\\rho _H(B)$ , which crosses zero at the same filling fraction inside the open FS filling range for varying field strength).","Moreover, the filling dependence of the Hall number $n_H$ tracks the filling fraction in a broad filling range near the CNP, with the filling range being extended upon increasing $B$ .","In Fig.","REF (d), we observe the same general shape of the Hall number.","Within the open FS filling range, however, the measured Hall number qualitatively deviates from the theoretical curves.","We attribute this to a small constant offset in the magnetic field of order 10-20 mT, likely resulting from trapped flux in the superconducting magnet.","Here a large quadratic symmetric component of the transverse resistivity is concurrent with a vanishing antisymmetric component.","An offset of only a few mT will lead to a small part of the symmetric component mixing into the antisymmetric component, leading to these deviations from theory (See SM Fig. 8).","Figure: (a) Rotation of the transport principal axis e ^ 1 \\hat{e}_1 with respect to the global coordinate system for strained BM with ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .","The three horizontal dashed lines are the bond directions.","In the open FS region, the saturating MR axis is locked to the shortest bond (𝐋 1 \\mathbf {L}_1) direction.","However, it rapidly rotates in the closed FS region upon approaching the CNP.","(b) Principal transport axes e ^ 1 \\hat{e}_1 (red) and e ^ 2 \\hat{e}_2 (blue) for a few filling fractions.","Near the CNP, e ^ 1 \\hat{e}_1 is perpendicular to the shortest moiré bond direction.","In the open FS filling range (e.g.","ν≈2.13\\nu \\approx 2.13) it is rotated to be parallel to the shortest bond direction.Our calculation finds a dramatic rotation of the principal axis with filling, as illustrated in Fig.","REF .","In the filling range with open FSs, the principal axis with saturating MR ($\\hat{e}_1$ ) is aligned with direction of the shortest moiré triangular bond, suggesting that the electrons are hopping more efficiently along the shortest bond, which leads to a larger conductivity and therefore a smaller resistivity.","Interestingly, when filling is changed from the second van Hove point ($\\nu \\approx \\pm 1.3$ ) to the vicinity of the CNP, $\\hat{e}_1$ rotates dramatically to the perpendicular direction compared to the filling range with open FSs.","The rotation of the principal axis is likely due to the opposite elongation of the larger Fermi pocket encircling a Dirac point compared to the open FS contours, see for example Figs.","REF (d-f).","The rotation of the transport axis with filling purely due to strain-induced bandstructure effects demonstrates that filling dependence of the principal axes orientation need not be associated with interaction induced nematicity [14].","Such a filling-dependent rotation of the principal transport axis was not possible to observe in Ref.","[39] using the Hall bar geometry, where only $\\rho _{xx}$ and $\\rho _{yx}$ are measured but not $\\rho _{yy}$ .","Additional transport measurements are needed, where the filling-dependence of the entire resistivity tensor can be mapped out.","Figure: (a) Line cuts of MR near the CNP taken at 26 mK in contact pair 4 - 5 at the indicated field strengths, in Tesla.","Vertical dashed lines indicate our estimated location of the lowest-energy van Hove points, based on the cusps in resistivity at low field.","Within the region bounded by these points, the quantum oscillations show up before 0.4 T, and their relative strengths do not follow a simple pattern.","Outside of this region, the quantum oscillations onset at higher field, and every multiple of 4 quantum Hall filling fraction is observed relatively equally.","(b) Fourier transform of the quantum oscillation data with respect to 1/B1/B.","It reveals a transition from two pockets to one pocket at the lowest-energy van Hove points.","(c) Schematic description of the frequencies observed in panel (b).","Red dashed lines are frequencies from the experimental data.","Solid black lines are predictions from the theory for ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .","The two frequencies f 1 f_1 and f 2 f_2 sum to the one-pocket frequency f 3 f_3 that extends beyond the first van Hove point.","They additionally account for the nontrivial relative strengths of the quantum oscillations within the bounds of the first van Hove points.","As with other details of this work, the theory predicts electron-hole symmetry, while some asymmetry is observed in experiment.Since this theory predicts a third van Hove point between the CNP and the filling range with open FSs, a direct measurement of this van Hove point is desired.","In Fig.","REF we reanalyze quantum oscillation measurements of the TBG device discussed in Ref. [39].","The effective cyclotron mass $m^*$ is light in the filling range with two small closed Fermi pockets, and dramatically heavier in the filling range with only one closed pocket (Figs.","REF (d-f) and SM Fig. 5).","The large difference in masses on either side of the innermost van Hove singularity can be used to explain the substantially earlier onset of quantum oscillations with increasing field close to the CNP than away from it, as shown in Fig.","REF (a).","Fig.","REF (b) is a Fourier transform of the quantum oscillation data with respect to $1/B$ .","In the filling range of $-0.7\\le \\nu \\le 0.8$ three distinct frequencies $f_{i=1,2,3}$ are clearly observed in the data, with $f_1$ and $f_2$ corresponding to two small Fermi pockets, and $f_3=f_1+f_2$ to the breakdown orbit when the inverse magnetic length is comparable to the momentum space distance between the two small Fermi pockets [60].","Outside of the filling range only $f_3$ is observed, showing that there are Lifshitz transitions, one on either side of the CNP, that we ascribe to crossing the lowest-energy van Hove points.","Furthermore, these filling fractions also correspond to the cusp-like features in the longitudinal MR data shown in Fig.","REF (c), consistent with theoretical predictions for its behavior at van Hove singularities.","Therefore, the quantum oscillation data unambiguously demonstrates the existence of a third van Hove singularity at filling fractions between the CNP and the filling range of $B^2$ MR.","It is interesting to note again, that the Hall number does not show a sign-changing singularity at this van Hove point, as illustrated in Fig.","REF (b) and (d).","The frequencies $f_{1,2}$ are a strong constraint on the amount of heterostrain in the TBG sample.","Specifically, as illustrated in Fig.","REF (c), the frequency $f_2$ is roughly two times $f_1$ , showing that the two small Fermi pockets have an area ratio $\\sim 2:1$ .","Theoretically as illustrated by the solid black lines in Fig.","REF (c), for a heterostrain strength $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ , the areas $A_{i=1,2}$ of the two small pockets, when converted to frequency via $f_i^{-1}\\equiv (\\Delta \\frac{1}{B})_i=\\frac{2\\pi e}{\\hbar A_i}$ , are in good agreement with experiment.","We observe behavior qualitatively similar in all respects to that in Fig.","REF (a) in all 3 longitudinal contact pairs for which we have dilution-fridge measurements (see SM Fig. 11).","In addition to the quantum oscillation measurements above, we propose an additional experimental procedure for identifying the van Hove points.","As usual, at van Hove singularities there are non-analyticities in the electronic density of states.","Such non-analyticities will lead to cusps in the first derivative of the zero-field resistivity with respect to filling (see SM Fig. 6).","This can be probed via transport measurements, for example, by adding a small ac modulation of the filling or by numerical differentiation of the dc data."],["Summary and Outlook","In summary, we have shown that due to the large size of the moiré unit cell at small twist angles, even a small amount of uniaxial heterostrain on the microscopic scale can lead to dramatic changes in the narrow bands of twisted bilayer graphene.","A key feature of the strained bandstructure is the splitting of the respective energetic degeneracies of the two Dirac points and the three van Hove points.","The splitting of the two Dirac points leads to a semimetallic state with two small Fermi pockets at the CNP.","On the other hand, the two outermost van Hove points bound a broad filling range near $\\nu =\\pm 2$ where the constant energy contours become open.","Interestingly, the elongation of the larger Fermi pocket near the CNP is perpendicular to that of the open FSs, the latter being perpendicular to the direction of the shortest moiré triangular bond.","We have analyzed the resulting magnetotransport in strained TBG in the framework of the Boltzmann equation using the method of characteristics, treating the magnetic field non-perturbatively.","We showed that a non-saturating quadratic longitudinal magnetoresistance in a broad filling range near $\\nu =\\pm 2$ naturally arises due to the heterostrain-induced open Fermi surfaces, therefore explaining the experimental results in the off-magic-angle devices [39].","We have also shown that the sign-changing singularities in the Hall number occur in the open FS filling range and are not directly associated with any van Hove singularity as commonly assumed, e.g., in Ref. [61].","Furthermore, our results reveal a dramatic rotation of the transport principal axis as the filling is tuned from the charge neutrality point to the filling range of open Fermi surfaces.","This is entirely attributed to the strained non-interacting bandstructure effects, and does not require interaction-induced electronic nematicity for explanation.","Given the importance of energy-shifted van Hove points in the transport properties of TBG devices, we have analyzed previous quantum oscillation data, which has revealed a Lifshitz transition from two pockets to one pocket at a filling fraction where the innermost van Hove singularity is predicted to occur based on theoretical calculations, therefore offering strong evidence of heterostrain effects on these devices.","We have further proposed several additional signatures to look for in future experiments.","These include cusps in the derivative of zero field resistivity with respect to filling, a significant difference in cyclotron mass on either side of the innermost van Hove singularity, and a principal transport axis with saturating magnetoresistance in the open Fermi surface filling range.","Finally, given the amplifying effect of a small strain at the underlying carbon lattice scale on the moiré lattice scale, the latter of which controls the electronic behavior within the narrow bands, it is tantalizing to consider strain engineering of such devices to achieve effects which would be impossible in regular solids due to structural instabilities."],["Acknowledgement","Funding: X.W.","acknowledges financial support from National MagLab through Dirac fellowship, which is funded by the National Science Foundation (Grant No.","DMR-1644779) and the state of Florida.","O.V.","was supported by NSF Grant No.","DMR-1916958 and is partially funded by the Gordon and Betty Moore Foundation's EPiQS Initiative Grant GBMF11070, National High Magnetic Field Laboratory through NSF Grant No.","DMR-1157490 and the State of Florida.","Device measurements and analysis were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-76SF00515.","Measurement infrastructure was funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF3429 and grant GBMF9460.","D.G.-G. gratefully acknowledges support from the Ross M. Brown Family Foundation.","Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.","K.W.","and T.T.","acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).","Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822.","Author contributions: O.V.","conceived the theoretical explanation of the experiments.","X.W.","and O.V.","performed calculations.","J.F., L. R., and C.H.","fabricated devices.","J.F.","and X.W.","analysed the data.","K.W.","and T.T.","generously supplied the hBN crystals.","A.L.S., M.A.K., O.V., and D.G.-G. supervised the experiments and analysis.","The manuscript was prepared by X.W.","and J.F.","with input from all authors.","Supplementary Materials for “Unusual magnetoresistance in twisted bilayer graphene from strain induced open Fermi surfaces\" We present additional theoretical results and experimental measurements in support of the main text."],["Heterostrain effects on the geometry of moiré superlattice","Our off-magic-angle twisted bilayer graphene (TBG) devices in Ref.","[39] are prepared using the “tear-and-stack\" procedure, and as a result, strain is inevitably introduced.","Here we first show that while the moiré unit cell vectors are strongly deformed by even an infinitesimal amount of uniaxial heterostrain in the device, the unit cell area is much less affected.","As a result, for the off-magic-angle device studied in Ref.","[39], we can have a good estimate of the twist angle ($\\theta $ ) based on the moiré unit cell area alone.","In the limit of small deformations, both the uniaxial heterostrain and a small twist angle are captured via a coordinate transformation: $\\mathbf {r}^{\\prime }_l = \\mathbf {r}+ \\mathbf {u}_l(\\mathbf {r})$ , where $l=t,b$ labels the top (bottom) graphene layers, and $\\mathbf {u}_l(\\mathbf {r}) \\approx \\mathcal {E}_l \\mathbf {r}$ is the local deformation field.","The symmetric and antisymmetric part of the $2\\times 2$ tensor $\\mathcal {E}_l$ describes strain and rotation respectively.","For twist angle $(\\theta )$ and a uniaxial heterostrain of strength $(\\epsilon )$ and direction $(\\varphi )$ , we parameterize $\\mathcal {E}_{t}=-\\mathcal {E}_{b}\\equiv \\mathcal {E}/2$ , where $\\mathcal {E}\\equiv \\mathcal {T}(\\theta )+ \\mathcal {S}(\\epsilon ,\\varphi )$ , and given by: $\\mathcal {T}(\\theta ) = \\begin{pmatrix} 0 & -\\theta \\\\ \\theta & 0\\end{pmatrix},\\ \\mathcal {S}(\\epsilon ,\\varphi ) = R_{\\varphi }^T \\begin{pmatrix} -\\epsilon & 0\\\\ 0 & \\nu \\epsilon \\end{pmatrix}R_{\\varphi }.$ Here $R_\\varphi $ is the two-dimensional rotation matrix, and $\\nu \\approx 0.16$ is the Poisson ratio [3].","Physically, $\\epsilon >0$ corresponds to compressing the top layer while streching the bottom layer along the $x$ -axis.","A relative deformation $\\mathcal {E}$ between the graphene bilayers generate a moiré superlattice, with moiré reciprocal lattice vectors given by: $\\mathbf {g}_{i=1,2}=\\mathcal {E}^T\\mathbf {G}_{i=1,2},$ where $\\mathbf {G}_{i}$ are reciprocal lattice vectors of the undeformed monolayer graphene.","Eq.","(REF ) can be used to uniquely determine the three parameters $(\\theta ,\\epsilon ,\\varphi )$ .","Additionally it also determines a global angle $\\alpha $ that measures the rotation between the lab and theoretical coordinate systems.","Uniaxial heterostrain has a dramatic effect on the distortion of the moiré unit cell vectors, as $|\\delta \\mathbf {g}|/|\\mathbf {g}|\\sim \\mathcal {O}(\\epsilon /\\theta )$ .","However, its effect on the moiré unit cell area is much smaller.","To show this, note that the area of the moiré Brillouin zone is calculated as: $A_{mBZ} = \\left| (\\mathbf {g}_1 \\times \\mathbf {g}_2 )\\cdot \\hat{z}\\right| = \\left| \\mathbf {g}_1^T (i\\sigma _y) \\mathbf {g}_2 \\right|,$ where on the second equality we have used a vector notation $\\mathbf {g}_i\\equiv (g_{i,x},g_{i,y})^T$ .","Following Eq.","(REF ), we obtain that the area of the moiré Brillouin zone is independent on $\\varphi $ , and calculated as: $A_{mBZ} = (\\theta ^2-\\nu ^2\\epsilon ^2)A_{BZ},$ where $A_{BZ}\\equiv \\left|(\\mathbf {G}_1\\times \\mathbf {G}_2)\\cdot \\hat{z}\\right|$ is the Brillouin zone area of the undeformed monolayer graphene.","The area of the strained moiré unit cell can be calculated in a similar manner, and we get: $A_{m.u.c.}=A_{u.c.","}/(\\theta ^2-\\nu ^2\\epsilon ^2)$ , where $A_{u.c.","}$ is the unit cell area of undeformed monolayer graphene.","Observe that the heterostrain only affects the area of the moiré unit cell by $\\mathcal {O}(\\nu ^2\\epsilon ^2/\\theta ^2)$ which is much smaller than the linear distortion of moiré unit cell vectors.","With only a knowledge of the moiré unit cell areas in Ref.","[39] (see Table REF ), we estimate the twist angle to be $\\theta \\sim 1.35^\\circ - 1.39^\\circ $ for various contact pairs studied using the Hall bar geometry."],["Constraining heterostrain from transport measurements","For the TBG device studied in Ref.","[39], the deformed moiré lattice vectors were not measured.","Nevertheless, here we show that magnetotransport measurements, along with theoretical calculations based on the strained Bistrizer-MacDonald (BM) Hamiltonian, offer strong constraints on the heterostrain in the device.","We caution, however, that since the strained BM model is an approximate description of the narrow bands of TBG, a precise determination of heterostrain from model calculations is not feasible.","First of all, as predicted by theoretical calculations, the van Hove singularities of the band structure lead to non-analytic behavior for the longitudinal magnetoresistance as a function of electron filling.","The filling fractions for the six van Hove singularities in the narrow band are listed in Table REF for various contact pairs.","Secondly, magnetic oscillations show a Lifshitz transition at the inntermost van Hove singularities ($\\nu _3,\\nu _4$ ), from two small Fermi pockets closer to the charge neutrality point to one Fermi pocket away from it.","Furthermore, the areas of the two small Fermi pockets, as revealed by the frequencies of magnetic oscillations, show a $2:1$ or smaller ratio.","Both the filling fractions for van Hove singularities and the pocket area size offer strong constraints for the heterostrain.","Qualitatively, on the one hand, a broader filling range of open Fermi surfaces can be achieved by increasing the strength of uniaxial heterostrain.","On the other hand, to obtain Fermi pocket area sizes near $2:1$ ratio or smaller, a smaller heterostrain is necessary as it leads to a weaker splitting of the two Dirac cones.","For theoretical calculations presented in the main text, we find $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ to give reasonably good agreements with both experimental observations described above.","A larger heterostrain strength ($\\epsilon =0.3\\%$ ) will lead to a much larger pocket area ratio ( $4:1$ for $\\epsilon =0.3\\%$ and $\\varphi =0^\\circ $ ), inconsistent with magnetic oscillation measurements.","On the other hand, a smaller heterostrain strength $\\epsilon =0.1\\%$ decreases the filling range of open Fermi surfaces dramatically, inconsistent with the longitudinal magnetoresistance measurements."],["Detailed band structure analysis for varying uniaxial heterostrain","In the main text we discussed the band structure of the strained TBG for $\\epsilon =0.2\\%$ .","The main effect of uniaxial heterostrain is to break the respective energetic degeneracies of the two Dirac points and three van Hove points of a given band, therefore giving rise to a semimetallic state at charge neutrality point, and open Fermi surface regions bounded by the two outermost van Hove points.","However for a larger heterostrain, the innermost van Hove point moves closer to one of the Dirac point.","As a result, both Dirac cones become type II titled, and the innermost van Hove points of both the upper and lower bands are annihilated.","In turn two new band extrema are formed.","This is illustrated in Fig.","REF .","We also explore the possibilities of heterostrain-induced higher order van Hove singularities which is possible for the magic-angle TBG as discussed in Ref. [38].","We checked that for $\\theta =1.38^\\circ $ , and up to uniaxial heterostrain strength of $\\epsilon =0.7\\%$ , no higher order van Hove singularities are found.","This shows that the band flattening effect at the magic angle may be important for strain engineering of higher order van Hove points.","Figure: Velocity field of typical closed and open Fermi surfaces.","Whereas for a closed Fermi surface the averaged velocity vanishes, for open Fermi surfaces this is generally violated.In Fig.","REF we plot the velocity fields $v_x(s)$ and $v_y(s)$ on typical open and closed Fermi surfaces for the strained TBG, parameterized by $s\\in [0,s_0)$ as defined in the main text.","For the closed Fermi surface contours, the averaged velocity, $\\mathbf {v}_{n=0}\\equiv \\frac{1}{s_0}\\int _0^{s_0}\\mathrm {d}s \\mathbf {v}(s)$ , is zero.","On the other hand, for a typical open Fermi surface contour, it is finite, and as a result the electron traversing the open Fermi surface contour in the presence of a magnetic field has a finite drift velocity.","As discussed in the main text, this is the reason for the non-saturating $B^2$ magnetoresistivity (MR) observed in strained TBG devices."],["More details on magnetotransport in TBG","Here we show that while $B^2$ longitudinal MR generally occurs for strained TBG due to open Fermi surfaces, it does not occur for unstrained devices.","In Fig.","REF , the longitudinal MRs $\\rho _{xx}$ and $\\rho _{yy}$ as well as the Hall number $n_H$ are plotted for an unstrained BM model calculation.","First of all, $\\rho _{xx}=\\rho _{yy}$ since the unstrained TBG has $C_{3z}$ rotational symmetry.","Secondly, cusp-like features develop at the triply-degenerate van Hove point at filling fractions $\\nu \\approx \\pm 1.4$ , and are attributed to the non-analyticities in the density of states behavior at the van Hove singularities.","Finally, unstrained TBG has saturating MR across all filling range, as illustrated in the inset to Fig.","REF (a).","Figure: (a) Longitudinal MR ρ xx \\rho _{xx} (dashed) and ρ yy \\rho _{yy} (solid) for strained BM with ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .","Different colors represent varying magnetic field strength.","Vertical dashed lines are positions of the van Hove points.","Shaded areas are open Fermi surface regions.","(b) In the closed Fermi surface region, MR saturates at large magnetic fields.","(c) In the open Fermi surface region, MR exhibit non-saturating B 2 B^2 dependence along both directions.In Fig.","REF we show that for the globally defined coordinate system which is misaligned from the principal transport axis, the $B^2$ behavior generally dominates the MR, and therefore will show up in both $\\rho _{xx}$ and $\\rho _{yy}$ measurements.","This remains true for a generic misalignment between the transport axis from experiment and the principal transport axis.","Figure: Hall resistivity ρ H \\rho _H at varying magnetic fields.","Note that it crosses zero within the filling range of open Fermi surfaces on both sides of the charge neutrality point.","These mark the sign-changing singularities in the Hall number depicted in Fig.","2(b) of the main text.In Fig.","REF we show the Hall resistivity $\\rho _H(B)$ for varying magnetic field strength.","Since $\\rho _H = B/n_Hq$ , wherever $\\rho _H(B)$ crosses zero and changes sign, the Hall number displays a sign-changing signularity.","Fig.","REF clearly shows that $\\rho _H(B)$ crosses zero in the open Fermi surface regions on both sides of the charge neutrality point, and independent on the strength of the $B$ -field.","Figure: Filling dependence of the averaged inverse cyclotron mass 1/m * 1/m^*, extracted from the averaged cyclotron frequency ω c ¯=eB m * \\bar{\\omega _c}=\\frac{eB}{m^*}.","Here ω ¯ c =1 N∑ n,i ω c,n,i \\bar{\\omega }_c=\\frac{1}{N}\\sum _{n,i}\\omega _{c,n,i}, where nn and ii label the FS (ii) coming from a given band (nn), in units of the bare electron mass.","1/m * 1/m^* is larger near the charge neutrality and band edges, explaining the earlier onset of quantum oscillations in these filling regions.In Fig.","REF we illustrate the filling-dependent inverse cyclotron mass $1/m^*$ for strained BM model with $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ .","This is to highlight the dichotomy of light-heavy masses on either side of the innermost van Hove singularities closest to the charge neutrality point.","This is consistent with the experimental observation of a much earlier onset field of quantum oscillations in filling range below the innermost van Hove point than above.","Figure: (a) Longitudinal resistivitities ρ xx \\rho _{xx} (blue) and ρ yy \\rho _{yy} (orange) at B=0B=0 for strained BM model, with ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .","(b) The derivative of log resistivity with respect to filling.","Gray dotted vertical lines mark positions of the van Hove points, and the yellow shaded area marks the open Fermi surface region.In Fig.","REF we also show the filling dependence of the $B=0$ longitudinal resistivities and their derivatives with respect to filling.","A key highlight is that the non-analyticities in the density of states at the van Hove points lead to kink-like features in the derivatives, but nearly invisible in the resistitivies themselves."],["Error analysis of antisymmetrization","In Fig.","REF we show that the bump-like features in the experimental Hall number plots in filling range of open Fermi surfaces (Fig.","2(d) of main text and Fig.","REF in the SM) may be attributed to improper antisymmetrization with respect to the $B$ -field, namely, $\\tilde{\\rho }_H(B) = \\frac{\\rho _{yx}(B+\\delta B) - \\rho _{yx}(-B+\\delta B)}{2},$ where $\\delta B$ is a systematic error.","The error may be attributed to a small trapped flux of 10 mT in the superconducting magnet, or perhaps an offset in the magnet power supply.","Due to the misalignment of transport principal axis with the Hall bar geometry, longitudinal MR also contributes to $\\rho _{yx}(B)$ .","In the filling range with open Fermi surfaces, the longitudinal resistance exhibits non-saturating quadratic MR, and will mix into the Hall component which is odd in B.","As a result, one expects the improper antisymmetrization error to be largest in this filling range.","We investigate this possibility by first fitting a polynomial to the low-field transverse resistivity.","This allows us to interpolate the data and add small constant offsets prior to antisymmetrization.","Accounting for an offset of roughly 20 mT largely removes the bumps from the data.","This offset is larger than what we would expect from trapped flux in a superconducting magnet, however we do not expect the procedure to be accurate to such a fine degree, simply because we do not have fine enough resolution in field to get an accurate polynomial fit."],["More experimental measurements based on various contact pairs of the Hall bar geometry","The device has nine voltage probes on each side.","We observe quadratic magnetoresistance regions in roughly half of the device, between the fourth and eighth contacts.","We present longitudinal resistivities of these pairs in Fig.","REF .","In each of these pairs, we observe behavior qualitatively consistent with that presented in the main text.","Our Hall measurements (Fig.","REF ) are similarly consistent.","In Fig.","REF , we show quantum oscillations and their Fourier transforms for all three contact pairs for which we have dilution refridgerator data.","In all cases, we observe behavior consistent with what we present in the main text: 1) quantum oscillation onset at lower field close to CNP, 2) an irregular pattern of resistivity minima close to CNP, and 3) extra features in the FFT of the quantum oscillations that end at vH1.","The density of the first van Hove point is closer to the CNP in the other two contact pairs, and the extra features in the FFT are not as clear."]],"string":"[\n [\n \"Unusual magnetotransport in twisted bilayer graphene from strain-induced\\n open Fermi surfaces\"\n ],\n [\n \"Abstract Anisotropic hopping in a toy Hofstadter model was recently invoked to explain a rich and surprising Landau spectrum measured in twisted bilayer graphene away from the magic angle.\",\n \"Suspecting that such anisotropy could arise from unintended uniaxial strain, we extend the Bistritzer-MacDonald model to include uniaxial heterostrain.\",\n \"We find that such strain strongly influences band structure, shifting the three otherwise-degenerate van Hove points to different energies.\",\n \"Coupled to a Boltzmann magnetotransport calculation, this reproduces previously-unexplained non-saturating $B^2$ magnetoresistance over broad ranges of density near filling $\\\\nu=\\\\pm 2$, and predicts subtler features that had not been noticed in the experimental data.\",\n \"In contrast to these distinctive signatures in longitudinal resistivity, the Hall coefficient is barely influenced by strain, to the extent that it still shows a single sign change on each side of the charge neutrality point -- surprisingly, this sign change no longer occurs at a van Hove point.\",\n \"The theory also predicts a marked rotation of the electrical transport principal axes as a function of filling even for fixed strain and for rigid bands.\",\n \"More careful examination of interaction-induced nematic order versus strain effects in twisted bilayer graphene could thus be in order.\"\n ],\n [\n \"Introduction\",\n \"The discovery of superconductivity and correlated insulating states in magic-angle twisted bilayer graphene (TBG) [1], [2] placed the material at the forefront of condensed matter physics research [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].\",\n \"The moiré superlattice potential of TBG, resulting from a small relative twist angle $\\\\theta $ between the graphene layers, can induce nearly flat, topologically non-trivial, isolated bands, consisting of electronic states near the Dirac points of each monolayer of graphene [18].\",\n \"As a result, TBG is an exceptional platform for studying the interplay of electron correlations and band topology [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37].\",\n \"Strain – especially heterostrain consisting of differing lattice distortions in the two layers – is believed to play an important role in the phase diagram of TBG [38], [37], [36].\",\n \"Scanning probe measurements typically find uniaxial heterostrain in the range of $0.1 - 0.7\\\\%$ in samples fabricated with the tear-and-stack method [3], [9], [11].\",\n \"For heterostrain, as opposed to homostrain, the linear distortion of the moiré unit cell is amplified by a factor of $\\\\sim 1/\\\\theta $ relative to the linear distortion of the microscopic atomic lattice.\",\n \"Because we infer twist angle from moiré unit cell area in transport, this effect leads to underestimates of the uncertainty in twist angles presented in transport literature, as noted in Ref.\",\n \"[3].\",\n \"For example, $0.2\\\\%$ uniaxial heterostrain causes a $\\\\sim 8\\\\%$ change in the linear size of the moiré unit cell for a twist angle of $1.38^\\\\circ $ .\",\n \"However, the effect on the moiré unit cell area is much reduced.\",\n \"In a recent report by some of the authors [Finney et al, Ref.\",\n \"[39]], a TBG sample with a moiré unit cell area of 90 nm$^2$ (corresponding to $\\\\theta =1.38^\\\\circ $ , well above the magic angle) displayed several unusual phenomena in magnetotransport.\",\n \"The sample did not exhibit the strong interaction driven effects typically observed in near-magic-angle devices.\",\n \"Rather, over a broad filling range near half filling, the longitudinal magnetoresistivity (MR) exhibited a $B^2$ increase up to $\\\\approx 5$ T, after which quantum oscillations set in.\",\n \"Such $\\\\sim 100$ -fold increase in MR was not explained, although the authors conjectured that strain may have played a role based on comparison of a toy Hofstadter model with anisotropy, over a broader range of magnetic field.\",\n \"In this work, we present a systematic theoretical study of the impact of uniaxial heterostrain on the narrow-band dispersion of TBG above the magic angle, analyze its consequences for weak field magnetotransport, and compare it with experimental data from Ref. [39].\",\n \"We base our theory on the Bistritzer-MacDonald (BM) continuum model [18], incorporating heterostrain in the form of a deformation potential, a pseudo-magnetic field [40], [41], and a distortion of the moiré pattern in the interlayer tunneling.\",\n \"Our key theoretical result is that heterostrain lifts the energetic degeneracy of the two Dirac points as well as that of the three van Hove points of a given band.\",\n \"The splitting of the two Dirac points leads to a semimetallic state near the charge neurality point (CNP) with small Fermi pockets.\",\n \"More interestingly, the splitting of the van Hove points leads to open Fermi surfaces (FS) in the filling range bounded by two of the van Hove points.\",\n \"In the weak field semiclassical regime governed by the Boltzmann equation, the open FSs generally lead to a non-saturating $B^2$ MR, explaining the low-field experimental findings of Ref. [39].\",\n \"This theory makes a number of falsifiable predictions.\",\n \"Of note, it predicts a large degree of mixing between longitudinal and transverse MRs within the open FS regime, due to an uncontrolled misalignment of the strain-induced principle axis of transport and the direction of current flow in the Hall bar.\",\n \"It predicts a subtle cusp in resistivity corresponding to the crossing of the lowest-energy van Hove point.\",\n \"Finally, it predicts a Lifshitz transition from two FS pockets to one upon crossing this lower van Hove point.\",\n \"We reanalyze experimental data from [39], and find that these predictions are verified.\",\n \"The theory does not capture the electron-hole asymmetry in the experimental data.\",\n \"The theory also has a few unexpected features.\",\n \"Firstly, the sign change singularity in the Hall number, one on each side of CNP, does not coincide with any of the van Hove points and instead occurs inside the filling range with open FSs.\",\n \"Secondly, the transport principal axis continuously rotates by up to $90^\\\\circ $ as density is tuned from the CNP to the open FS regime.\",\n \"Such rotation of the transport axes is generally associated with interaction-induced nematic order [14], but here we find that it can arise purely due to strain-induced band structure effects.\",\n \"This work clearly demonstrates that the effects of even miniscule amounts of heterostrain in TBG cannot be neglected.\",\n \"Dramatic and unexpected phenomena occur in strained TBG even in the single-particle regime, without the strong correlation effects that arise near the magic angle.\",\n \"Given the amplifying effect of a small heterostrain on the moiré length scale, it is tantalizing to consider strain engineering of such devices to achieve effects that would be impossible in regular solids due to structural instabilities.\"\n ],\n [\n \"Geometric and energetic effects of uniaxial heterostrain on TBG\",\n \"In the limit of small deformations, both the uniaxial heterostrain and a small twist angle are captured via a coordinate transformation: $\\\\mathbf {r}^{\\\\prime }_l = \\\\mathbf {r}+ \\\\mathbf {u}_l(\\\\mathbf {r})$ , where $l=t,b$ labels the top (bottom) graphene layers, and $\\\\mathbf {u}_l(\\\\mathbf {r}) \\\\approx \\\\mathcal {E}_l \\\\mathbf {r}$ is the local deformation field.\",\n \"The symmetric and antisymmetric part of the $2\\\\times 2$ tensor $\\\\mathcal {E}_l$ describes strain and rotation respectively.\",\n \"For twist angle $(\\\\theta )$ and a uniaxial heterostrain of strength $(\\\\epsilon )$ and direction $(\\\\varphi )$ , we parameterize $\\\\mathcal {E}_{t}=-\\\\mathcal {E}_{b}\\\\equiv \\\\mathcal {E}/2$ , where $\\\\mathcal {E}\\\\equiv \\\\mathcal {T}(\\\\theta )+ \\\\mathcal {S}(\\\\epsilon ,\\\\varphi )$ , and given by: $\\\\mathcal {T}(\\\\theta ) = \\\\begin{pmatrix} 0 & -\\\\theta \\\\\\\\ \\\\theta & 0\\\\end{pmatrix},\\\\ \\\\mathcal {S}(\\\\epsilon ,\\\\varphi ) = R_{\\\\varphi }^T \\\\begin{pmatrix} -\\\\epsilon & 0\\\\\\\\ 0 & \\\\nu \\\\epsilon \\\\end{pmatrix}R_{\\\\varphi }.$ Here $R_\\\\varphi $ is the two-dimensional rotation matrix, and $\\\\nu \\\\approx 0.16$ is the Poisson ratio [3].\",\n \"Physically, $\\\\epsilon >0$ corresponds to compressing the top layer while streching the bottom layer along the direction determined by $\\\\varphi $ , as illustrated in Fig.\",\n \"REF (a).\",\n \"A relative deformation $\\\\mathcal {E}$ between the graphene bilayers generates a moiré superlattice, with moiré reciprocal lattice vectors $\\\\mathbf {g}_{i=1,2}=\\\\mathcal {E}^T\\\\mathbf {G}_{i=1,2}$ , where $\\\\mathbf {G}_{i}$ are reciprocal lattice vectors of the undeformed monolayer graphene.\",\n \"The moiré lattice vectors $\\\\mathbf {L}_{i=1,2}$ are uniquely defined through the relation $\\\\mathbf {L}_i\\\\cdot \\\\mathbf {g}_j=2\\\\pi \\\\delta _{ij}$ .\",\n \"It is important to note that only relative deformations generate the moiré superlattice.\",\n \"Homogenous deformations do not play an important role in the narrow band physics, and we neglect it in this work We checked numerically that adding a small homogeneous strain in addition to a heterostrain of similar strength yields almost identical band and transport properties to the case of adding a heterostrain alone..\",\n \"Under rotation $R_\\\\varphi $ , the strain tensor transforms as a headless vector that remains invariant under $\\\\varphi \\\\rightarrow \\\\varphi +180^\\\\circ $ .\",\n \"Combined with the $C_{3z}$ symmetry of the undeformed graphene lattice, the strained electronic dispersion within a given graphene valley simply rotates $60^\\\\circ $ under $\\\\varphi \\\\rightarrow \\\\varphi + 60^\\\\circ $ .\",\n \"We hereby will only report results for $\\\\varphi \\\\in [0^\\\\circ ,60^\\\\circ )$ .\",\n \"For concreteness we define the microscopic unit cell vectors $\\\\mathbf {a}_{i=1,2}$ of undeformed graphene lattice as $ \\\\mathbf {a}_1 = a(\\\\frac{1}{2},-\\\\frac{\\\\sqrt{3}}{2}),\\\\ \\\\mathbf {a}_2 = a(1,0)$ , where $a\\\\approx 2.46Å$ is the lattice constant.\",\n \"The positions of the sublattice A,B within a unit cell are chosen as $\\\\vec{\\\\tau }_A = (0,0)$ and $\\\\vec{\\\\tau }_B = \\\\frac{a}{\\\\sqrt{3}}(0,1)$ .\",\n \"The reciprocal lattice vectors are $\\\\mathbf {G}_1 = \\\\frac{4\\\\pi }{\\\\sqrt{3}a}(0,-1)$ and $\\\\mathbf {G}_2 = \\\\frac{4\\\\pi }{\\\\sqrt{3}a}(\\\\frac{\\\\sqrt{3}}{2},\\\\frac{1}{2})$ .\",\n \"Different conventions lead to different definitions of the Dirac Hamiltonian (see for instance Ref.\",\n \"[38]), but the physics is consistent.\",\n \"Fig.\",\n \"REF (a-b) illustrates the geometric effects of heterostrain for twist angle $\\\\theta =1.38^\\\\circ $ .\",\n \"For $\\\\epsilon =0.2\\\\%$ , typical in these systems [3], [9], [11], there is a large change in the bond length of the neighboring AA-stacked regions ($\\\\mathbf {L}_{i=1,2,3}$ ) of the moiré triangular superlattice, which used to form an equilateral triangle at $\\\\epsilon =0$ .\",\n \"The effect of heterostrain on the moiré unit cell vectors can be estimated to be as large as $\\\\epsilon /\\\\theta \\\\approx 8\\\\%$ .\",\n \"However, the effect on the moiré unit cell area is much smaller at $\\\\nu ^2\\\\epsilon ^2/\\\\theta ^2$ (see Supplementary Material (SM) Sec.\",\n \"I).\",\n \"Such dramatic amplification of the microscopic strain makes moiré materials ideal for strain engineering not achievable in conventional materials due to structural instability.\",\n \"We proceed to discuss the energetic effects in the context of the continuum BM model [18].\",\n \"We work in the limit where both $\\\\mathcal {E}_l$ and the wavevector $\\\\mathbf {k}$ in the moiré Brillouin zone are small, and consider only the leading order terms in both.\",\n \"This would mean, for instance, that terms such as $\\\\mathcal {E}\\\\mathbf {k}$ are omitted as higher order terms.\",\n \"This treatment is generally justified away from the magic angle, because higher order terms can play an important role only close to the magic angle where the narrow-band bandwidth is suppressed to a similar energy scale [43], [44].\",\n \"Furthermore, we checked that at $\\\\theta \\\\approx 1.38^\\\\circ $ the effects of such higher order terms are indeed negligibly small.\",\n \"To leading order, the strained BM Hamiltonian for a given valley is given by: $H_{\\\\eta } = (\\\\sum _{l=t,b} H_{\\\\eta ,l}^{intra}) + H_{\\\\eta }^{inter} ,$ where $\\\\eta =\\\\pm 1$ labels $\\\\mathbf {K}\\\\ (\\\\mathbf {K}^{\\\\prime })$ valleys of monolayer graphene.\",\n \"The interlayer Hamiltonian is given by: $H_{\\\\eta ,l}^{inter} \\\\approx \\\\int \\\\mathrm {d}^2\\\\mathbf {r}\\\\psi ^\\\\dagger _{\\\\eta ,t} \\\\left(\\\\sum _{j=1,2,3} T_{\\\\eta ,j} e^{-i\\\\eta \\\\mathbf {q}_{j}\\\\cdot \\\\mathbf {r}}\\\\right)\\\\psi _{\\\\eta ,b}(\\\\mathbf {r}) + h.c.,$ where $\\\\psi _{\\\\eta ,l}(\\\\mathbf {r})\\\\equiv (\\\\psi _{\\\\eta ,l,A}(\\\\mathbf {r}),\\\\psi _{\\\\eta ,l,B}(\\\\mathbf {r}))^T$ is a spinor in the sublattice basis for a given valley and layer.\",\n \"We have suppressed the spin index for simplicity.\",\n \"$\\\\mathbf {q}_{j=1,2,3}$ are the three nearest neighbor bonds of the reciprocal honeycomb lattice, and $T_{\\\\eta ,j} = w_0\\\\sigma _0+w_1\\\\left(\\\\cos \\\\frac{2\\\\pi (j-1)}{3} \\\\sigma _x+ \\\\eta \\\\sin \\\\frac{2\\\\pi (j-1)}{3}\\\\sigma _y\\\\right).$ $(\\\\sigma _0,\\\\sigma _x,\\\\sigma _y)$ are Pauli matrices acting on sublattice degrees of freedom.\",\n \"The intra-layer Hamiltonian is given by: $\\\\begin{split}& H_{\\\\eta ,l}^{intra} = \\\\alpha \\\\sum _{\\\\mathbf {k}} \\\\psi ^\\\\dagger _{\\\\eta ,l}(\\\\mathbf {r}) ( [\\\\mathcal {E}_l] \\\\sigma _0)\\\\psi _{\\\\eta ,l}(\\\\mathbf {r})\\\\\\\\&- \\\\frac{\\\\hbar v_F}{a}\\\\sum _{\\\\mathbf {k}} \\\\psi ^\\\\dagger _{\\\\eta ,l}(\\\\mathbf {r}) \\\\left[ (\\\\mathbf {k}- \\\\mathbf {A}_{\\\\eta ,l})\\\\cdot (\\\\eta \\\\sigma _x,\\\\sigma _y) \\\\right]\\\\psi _{\\\\eta ,l}(\\\\mathbf {r}).\\\\end{split}$ Here the first term is the deformation potential that couples to the electron density.\",\n \"Its value is not precisely known in the literature, with numbers ranging from $-4.1\\\\ \\\\mathrm {eV}$ to $30\\\\ \\\\mathrm {eV}$ depending on the methodology [45], [46], [47], [48].\",\n \"We use $\\\\alpha =-4.1\\\\ \\\\mathrm {eV}$ in this work based on first principles calculations [48], although the deformation potential does not have an important effect on the band dispersions for heterostrain $\\\\epsilon \\\\approx 0.2\\\\%$ , and only leads to minor quantitative differences.\",\n \"$\\\\mathbf {A}_{\\\\eta ,l}$ is the pseudovector potential that comes from changes in the inter-sublattice hopping due to deformations, and changes sign between graphene valleys.\",\n \"It is given as [40], [41]: $\\\\mathbf {A}_{\\\\eta ,l} = \\\\frac{\\\\sqrt{3}\\\\beta }{2a}\\\\eta (\\\\epsilon _{l,xx}-\\\\epsilon _{l,yy},-2\\\\epsilon _{l,xy})$ , where we choose $\\\\beta \\\\approx 3.14$ from Refs.\",\n \"[3], [38].\",\n \"We shall further fix $\\\\hbar v_F/a=2.68\\\\mathrm {eV}$ , $w_0=88\\\\mathrm {meV}$ , and $w_1=110\\\\mathrm {meV}$ in our calculations, and also set $\\\\hbar =1$ in the remainder of the paper.\",\n \"To leading order approximation, the strained BM Hamiltonian in a given valley (Eq.\",\n \"(REF )) has particle-hole symmetry under $P \\\\psi _{l}(\\\\mathbf {r}) = \\\\sum _{l^{\\\\prime }}i(\\\\mu _y)_{ll^{\\\\prime }}\\\\psi _{l^{\\\\prime }}(-\\\\mathbf {r})$ [49], where $\\\\mu _y$ is a Pauli matrix acting on the layer degrees of freedom.\",\n \"This means that for every single electron state at energy $E$ and wavevector $\\\\mathbf {k}$ , there is a state at energy $-E$ and wavevector $-\\\\mathbf {k}$ .\",\n \"This particle-hole symmetry has been investigated extensively for the unstrained BM model, e.g., Refs.\",\n \"[25], [50], and here it is generalized to the strained case.\",\n \"Since in experiments particle-hole asymmetry is evident for the off-magic-angle device [39], they would come from either higher order gradient terms beyond what's captured in the BM model in Eq.\",\n \"(REF ), or due to interaction effects [51], [52], [53], [54], or their combination.\",\n \"We proceed to discuss the heterostrain effects on the band structure with $\\\\epsilon =0.2\\\\%$ and varying direction specified by $\\\\varphi \\\\in [0^\\\\circ ,60^\\\\circ )$ , depicted in Fig.\",\n \"REF (d-f).\",\n \"For simplicity we only show contour maps of the upper band from valley $\\\\mathbf {K}$ in the moiré Brillouin zone specified by $\\\\mathbf {k}=k_1\\\\mathbf {g}_1+k_2\\\\mathbf {g}_2$ , where $k_{1,2}\\\\in [0,1)$ .\",\n \"Heterostrain preserves $C_2T$ and valley $U(1)$ [22] and therefore the lower and upper bands remain connected via two Dirac points.\",\n \"The upper band features six special points — two Dirac points (black stars), three van Hove points (colored dots), and one band maximum (black cross).\",\n \"The six special points of a given band are related to “critical points\\\" in the context of the Morse theory, which states that $\\\\sum _{i}(-1)^{\\\\gamma _i}=\\\\chi ,$ where $\\\\gamma _i$ is the index of the $i$ -th critical point, and $\\\\chi $ is the Euler characteristic of a manifold [55]; $\\\\chi $ vanishes for the Brillouin zone which is a torus.\",\n \"Although a Dirac point is strictly a point of non-analyticity and is not directly covered by Morse theory, if we imagine adding a tiny gap term it will become a legitimate band extremum and Morse theory applies.\",\n \"Whereas the two band minima (Dirac points) and the band maximum have even $\\\\gamma $ and so each contributes $+1$ to the sum, every conventional van Hove point (i.e.\",\n \"not a higher order) has an odd $\\\\gamma $ and contributes $-1$ .\",\n \"Their sum thus vanishes.\",\n \"Therefore, the van Hove points can only be annihilated/created by colliding with local minima/maxima.\",\n \"For a relatively small heterostrain as shown in Fig.\",\n \"REF , the number of special points per band is the same as at $\\\\epsilon =0$ .\",\n \"However for larger heterostrain (e.g., $\\\\epsilon =0.5\\\\%$ , see SM Fig.\",\n \"1), more striking behavior of the special points can occur, such as a change in their total number via afore mentioned collisions and the appearance of tilted type II Dirac cones [56], [57].\",\n \"A key finding of the present work is that the respective energy degeneracies of the two Dirac points and the three van Hove points are lifted by uniaxial heterostrain, and depend sensitively on $\\\\varphi $ .\",\n \"In the absence of strain [Fig.\",\n \"REF (c)], the three van Hove points are at equal energy, and separate closed contours of constant energy centered around the Dirac points from closed contours centered around band maximum.\",\n \"As illustrated in Fig.\",\n \"REF (d-f), uniaxial heterostrain splits the energy degeneracy of the two Dirac points, leading to a semimetallic state with small Fermi pockets near CNP [38].\",\n \"The three van Hove points also split in energy.\",\n \"The two outermost van Hove points (i.e., closer to the band maximum) bound a filling range of open FSs near $\\\\nu =2$ , while the innermost van Hove point moves closer to one of the Dirac points.\",\n \"If we continue increasing $\\\\epsilon $ , a collision of the critical points occurs, the innermost van Hove disappears, the two Dirac points become type-II tilted, and a new ordinary minimum is created.\",\n \"Note that a small mass added to type-II tilted Dirac points won't introduce band extrema and as a consequence type-II tilted Dirac points are not critical points of Morse theory, therefore after the collision Eq.\",\n \"(REF ) still holds.\",\n \"Interestingly, the elongation of the FSs shows a strong filling dependence.\",\n \"Close to the CNP, the bigger Fermi pocket that encloses a Dirac point is stretched along a perpendicular direction to that of the open FSs, see Figs.\",\n \"REF (d-f).\",\n \"As explained later, this leads to a dramatic rotation of the principal transport axis when the filling is tuned from the CNP to the open FS range.\",\n \"The dependence of the energy and filling of the band structure special points on $\\\\varphi $ at a fixed $\\\\epsilon $ is shown in Fig.\",\n \"REF (g-h).\",\n \"Of notable interest is the sensitivity of the filling range with open FSs to $\\\\varphi $ .\",\n \"This filling range must in fact vanish at some $\\\\varphi $ between $0^\\\\circ $ and $60^\\\\circ $ , when the energies of the two outermost van Hove points cross.\",\n \"As seen in Fig.\",\n \"REF (d-f), this also alters the elongation of the open FSs.\"\n ],\n [\n \"Boltzmann equation and Magnetoresistivity in TBG\",\n \"Having understood the heterostrain effects on the bandstructure, we proceed to discuss the implications for magnetotransport.\",\n \"We begin by considering the general structure of the two-dimensional resistivity tensor ${\\\\rho }$ subject to heterostrain.\",\n \"The resistivity tensor is defined via: $\\\\begin{pmatrix}E_x \\\\\\\\ E_y\\\\end{pmatrix} =\\\\begin{pmatrix}\\\\rho _{xx} & \\\\rho _{xy}\\\\\\\\\\\\rho _{yx} & \\\\rho _{yy}\\\\end{pmatrix}\\\\begin{pmatrix}j_x \\\\\\\\ j_y\\\\end{pmatrix},$ where $\\\\mathbf {E}=(E_x,E_y)^T$ and $\\\\mathbf {j}=(j_x,j_y)^T$ are electric field and current vectors respectively.\",\n \"Under rotation by $\\\\delta \\\\theta $ , the resistivity tensor transform as: ${\\\\rho }^{\\\\prime } = R_{\\\\delta \\\\theta }^T {\\\\rho } R_{\\\\delta \\\\theta },\\\\ R_{\\\\delta \\\\theta } = \\\\begin{pmatrix}\\\\cos \\\\delta \\\\theta & -\\\\sin \\\\delta \\\\theta \\\\\\\\\\\\sin \\\\delta \\\\theta & \\\\cos \\\\delta \\\\theta \\\\end{pmatrix}.$ If the underlying system has a point group symmetry that is higher than $C_{2z}$ (e.g., $C_{3z},C_{6z}$ ), then ${\\\\rho }=\\\\rho _0\\\\mathbb {I} - i \\\\rho _H \\\\tau _y$ is the most general form of ${\\\\rho }$ invariant under such rotations.\",\n \"Here $\\\\tau _y$ is the Pauli matrix acting in the two-dimensional coordinate basis, $\\\\rho _0(-B)=\\\\rho _0(B)$ is the longitudinal resistivity, and $\\\\rho _H(-B)=-\\\\rho _H(B)$ is the Hall resistivity.\",\n \"The even/odd parity under time reversal is guaranteed by the Onsager reciprocal relations.\",\n \"Since heterostrain breaks the point group symmetry down to $C_{2z}$ , we generally expect $\\\\rho _{xx}\\\\ne \\\\rho _{yy},\\\\ \\\\rho _{xy}\\\\ne -\\\\rho _{yx}$ .\",\n \"Nevertheless, it is always possible to define transport principal axes after a suitable rotation $\\\\delta \\\\theta $ of the coordinate system, such that: $ {\\\\rho }_{\\\\text{principal}}= \\\\frac{1}{2}(\\\\rho _1+\\\\rho _2)\\\\mathbb {I}+\\\\frac{1}{2}(\\\\rho _1-\\\\rho _2)\\\\tau _z + \\\\rho _H i\\\\tau _y.$ Here $\\\\rho _{1,2}$ are longitudinal resistivities along the principal transport directions $\\\\hat{e}_{1,2}$ respectively.\",\n \"The rotation angle $\\\\delta \\\\theta $ is determined up to $180^\\\\circ $ by requiring $\\\\rho _1< \\\\rho _2$ .\",\n \"Below we first derive the MR tensor using Boltzmann approach for a general non-interacting electronic system within the relaxation time approximation.\",\n \"Since there is currently insufficient understanding of the scattering mechanisms determining electrical transport in TBG, here we follow Ref.\",\n \"[58] and use relaxation time approximation.\",\n \"We will then present the results for heterostrained TBG, showing that in the open FS region, the low resistivity principal axis ($\\\\hat{e}_{1}$ ) is nearly perfectly aligned with the shortest moiré bond direction.\",\n \"However there is a dramatic rotation of the principal axis as the filling moves towards the CNP.\",\n \"We further show that the open FSs lead to a $B^2$ non-saturating MR along $\\\\hat{e}_{2}$ , and a saturating resistivity along $\\\\hat{e}_{1}$ .\",\n \"For random orientation ($\\\\theta _0$ ) of the principal axis to the electrical current axis in the Hall bar geometry, e.g., as in Ref.\",\n \"[39], the longitudinal resistivity is given by: $\\\\rho _{xx} =\\\\rho _1\\\\cos ^2 \\\\theta _0+\\\\rho _2\\\\sin ^2\\\\theta _0$ .\",\n \"It is dominated by the $\\\\rho _2\\\\sim B^2$ component, and as a result, the experimental measurements should observe the non-saturating MR component if there is a misalignment with respect to the principal transport axis.\"\n ],\n [\n \"Boltzmann equation and method of characteristics\",\n \"We begin with a brief description of the method of characteristics used to solve the Boltzmann equation perturbatively in electric field ${\\\\bf E}$ but without a restriction on the strength of the perpendicular magnetic field ${\\\\bf B}=B\\\\hat{z}$ , as long as the semiclassical regime holds [59].\",\n \"Due to $C_{2z}T$ symmetry of TBG at ${\\\\bf B}=0$ , there is no Berry curvature contribution to the semiclassical equations of motion.\",\n \"Then, within the relaxation time approximation, the Boltzmann equation for a given energy band becomes $\\\\frac{\\\\partial n_\\\\mathbf {k}}{\\\\partial t} + (q\\\\mathbf {E}+q\\\\mathbf {v}_\\\\mathbf {k}\\\\times \\\\mathbf {B}) \\\\cdot \\\\frac{\\\\partial n_\\\\mathbf {k}}{\\\\partial \\\\mathbf {k}} = - \\\\frac{n_{\\\\mathbf {k}}-n_{0,\\\\mathbf {k}}}{\\\\tau },$ where $q\\\\mathbf {E}+q\\\\mathbf {v}_\\\\mathbf {k}\\\\times \\\\mathbf {B}$ is the total force on the Bloch electrons, with $\\\\mathbf {v}_\\\\mathbf {k}\\\\equiv \\\\nabla _{\\\\mathbf {k}} \\\\varepsilon _{\\\\mathbf {k}}$ and charge $q$ ; $n_{0,\\\\mathbf {k}}$ is the equilibrium Fermi-Dirac distribution and $n_{\\\\mathbf {k}}$ is the desired non-equilibrium distribution function.\",\n \"We consider a stationary solution to the Boltzmann equation by parameterizing the distribution function as: $n_{\\\\mathbf {k}} = n_{0,\\\\mathbf {k}} + n_{1,\\\\mathbf {k}}.$ As a result, the Boltzmann equation for the deviation of the distribution function from equilibrium is: $(q\\\\mathbf {E}\\\\cdot \\\\mathbf {v}_{\\\\mathbf {k}}) \\\\frac{\\\\partial n_{0,\\\\mathbf {k}}}{\\\\partial \\\\varepsilon _\\\\mathbf {k}}+(q\\\\mathbf {v}_\\\\mathbf {k}\\\\times \\\\mathbf {B})\\\\cdot \\\\frac{\\\\partial n_{1,\\\\mathbf {k}}}{\\\\partial \\\\mathbf {k}} = - \\\\frac{n_{1,\\\\mathbf {k}}}{\\\\tau }.$ Note that the magnetic field only couples to $n_1$ since $(q\\\\mathbf {v}_\\\\mathbf {k}\\\\times \\\\mathbf {B})\\\\cdot \\\\nabla _\\\\mathbf {k}n_{0,\\\\mathbf {k}}=(q\\\\mathbf {v}_\\\\mathbf {k}\\\\times \\\\mathbf {B})\\\\cdot \\\\mathbf {v}_{\\\\mathbf {k}} \\\\partial _{\\\\varepsilon _{\\\\mathbf {k}}} n_{0,\\\\mathbf {k}}=0$ .\",\n \"In order to solve the above partial differential equation (PDE), we seek a family of curves covering the $\\\\mathbf {k}$ -space which we parameterize as $\\\\mathbf {k}(s)$ with $s\\\\in [0,s_0)$ , such that along these curves the PDE becomes an ordinary differential equation (ODE).\",\n \"If a curve $\\\\mathbf {k}(s)$ satisfies $ \\\\frac{d\\\\mathbf {k}(s)}{d s} = q\\\\mathbf {v}(s)\\\\times \\\\mathbf {B},$ then $n_{1,\\\\mathbf {k}(s)}\\\\equiv n_{1}(s)$ satisfies $(q\\\\mathbf {E}\\\\cdot \\\\mathbf {v}_{\\\\mathbf {k}}) \\\\frac{\\\\partial n_{0,\\\\mathbf {k}}}{\\\\partial \\\\varepsilon _\\\\mathbf {k}}|_{\\\\mathbf {k}=\\\\mathbf {k}(s)}+ \\\\frac{d n_{1}(s)}{d s} = - \\\\frac{n_{1}(s)}{\\\\tau }.$ Because $\\\\frac{d \\\\varepsilon (s)}{d s} = \\\\mathbf {v}(s) \\\\cdot \\\\frac{d \\\\mathbf {k}(s)}{d s} = 0,$ the curve $\\\\mathbf {k}(s)$ must coincide with the contour of constant energy.\",\n \"Thus, the Boltzmann equation becomes: $[q\\\\mathbf {E}\\\\cdot \\\\mathbf {v}(s)] \\\\frac{\\\\partial n_{0}(s)}{\\\\partial \\\\varepsilon (s)}+ \\\\frac{d n_{1}(s)}{d s} = - \\\\frac{n_{1}(s)}{\\\\tau }.$ The ODE is readily solved with: $n_{1}(s) = \\\\chi _0 e^{-s/\\\\tau } - e^{-s/\\\\tau }\\\\int _0^{s}\\\\mathrm {d}s^{\\\\prime } e^{s^{\\\\prime }/\\\\tau } [q\\\\mathbf {E}\\\\cdot \\\\mathbf {v}(s^{\\\\prime })] \\\\frac{\\\\partial n_{0}(s^{\\\\prime })}{\\\\partial \\\\varepsilon (s^{\\\\prime })}.$ where $\\\\chi _0$ is a constant determined by the following argument.\",\n \"Since $\\\\mathbf {k}(s)$ describes a constant energy contour in a two-dimensional Brillouin zone, it is either a closed contour, or several open contours that terminate on boundaries of the Brillouin zone such that they form a closed loop on a torus.\",\n \"In either case, $\\\\mathbf {k}(s)$ is periodic under $s\\\\rightarrow s+s_0$ modulo a moiré reciprocal lattice vector, where $s_0$ is the periodicity.\",\n \"The periodicity condition $n_{1}(s_0)=n_{1}(0)$ leads to $\\\\chi _0 = \\\\frac{1}{1-e^{s_0/\\\\tau }}\\\\int _0^{s_0}\\\\mathrm {d}s^{\\\\prime } e^{s^{\\\\prime }/\\\\tau }(q\\\\mathbf {E}\\\\cdot \\\\mathbf {v}(s^{\\\\prime })) \\\\frac{\\\\partial n_{0}(s^{\\\\prime })}{\\\\partial \\\\varepsilon (s^{\\\\prime })},$ which determines the desired $n_1(s)$ .\",\n \"In the low temperature limit, the steady state current from a given energy band is calculated as: $ \\\\begin{split}j^\\\\mu & = q\\\\int \\\\frac{\\\\mathrm {d}^2\\\\mathbf {k}}{(2\\\\pi )^2} v^{\\\\mu }_{\\\\mathbf {k}} n_{1,\\\\mathbf {k}} \\\\\\\\& =\\\\frac{q^2B}{(2\\\\pi )^2} \\\\int \\\\mathrm {d} {\\\\varepsilon }\\\\int _0^{s_0} \\\\mathrm {d} s v^{\\\\mu }(s) n_{1}(s)\\\\\\\\& = \\\\frac{q^3B}{(2\\\\pi )} \\\\frac{\\\\tau }{\\\\omega _c} \\\\sum _{n=-\\\\infty }^{\\\\infty } \\\\frac{v^{\\\\mu }_nv^{\\\\nu }_{-n}}{1+in \\\\omega _c \\\\tau }E^{\\\\nu },\\\\end{split}$ where $(\\\\mu ,\\\\nu )=x,y$ , and we have defined the cyclotron frequency as: $\\\\omega _c \\\\equiv 2\\\\pi /s_0.$ We have also made use of the periodicity of velocity under $s\\\\rightarrow s+s_0$ to write it in terms of Fourier series, $\\\\mathbf {v}(s)=\\\\sum _{n=-\\\\infty }^{\\\\infty }\\\\mathbf {v}_ne^{-in \\\\omega _c s }$ .\",\n \"To show that the second line of Eq.\",\n \"(REF ) holds, note that at every $\\\\mathbf {k}$ we can define a local coordinate system $(\\\\hat{e}_{v},\\\\hat{e}_{s})$ such that $\\\\mathbf {v}\\\\equiv v\\\\hat{e}_{\\\\mathbf {v}}$ where $v\\\\ge 0$ , and $\\\\hat{e}_{s}=\\\\hat{e}_{\\\\mathbf {v}}\\\\times \\\\hat{z}$ .\",\n \"The infinitesimal wavevector can be equivalently written as: $\\\\mathrm {d}\\\\mathbf {k}=\\\\mathrm {d}k_x \\\\hat{e}_x + \\\\mathrm {d}k_y \\\\hat{e}_y= \\\\mathrm {d}k_s \\\\hat{e}_s + \\\\mathrm {d}k_v \\\\hat{e}_v.$ Eq.\",\n \"(REF ) can then be written as ${\\\\mathrm {d}\\\\mathbf {k}}/{\\\\mathrm {d} s} = qvB \\\\hat{e}_s$ , or equivalently $\\\\mathrm {d} k_s = qvB\\\\mathrm {d}s$ .\",\n \"As a result, $\\\\int \\\\mathrm {d} k_x \\\\mathrm {d}k_y = \\\\int \\\\mathrm {d} k_s \\\\mathrm {d}k_v = qB \\\\int \\\\mathrm {d}\\\\varepsilon \\\\mathrm {d} s.$ The conductivity tensor is therefore given by the following expression: $ \\\\sigma ^{\\\\mu \\\\nu } = \\\\frac{q^3B}{2\\\\pi } \\\\frac{\\\\tau }{\\\\omega _c} \\\\sum _{n=-\\\\infty }^{\\\\infty } \\\\frac{v^{(\\\\mu )}_nv^{(\\\\nu )}_{-n}}{1+in \\\\omega _c\\\\tau }.$ Eq.\",\n \"(REF ) gives the magnetoconductivity for a given FS contour.\",\n \"In the case of multiple FS contours and multiple bands –as due to spin and valley degeneracy in TBG– conductivities from different FS contours and bands add.\",\n \"Finally, the MR tensor is obtained by inverting the conductivity tensor, i.e., $\\\\rho = \\\\left(\\\\sum _{n,i}\\\\sigma _{n,i}\\\\right)^{-1}$ , where $n,i$ are band and contour labels respectively for a given energy level.\",\n \"To better understand Eq.\",\n \"(REF ) consider an example of a parabolic dipsersion with $\\\\varepsilon _{\\\\mathbf {k}}=\\\\frac{1}{2m_0}(k_x^2+k_y^2)$ , where $m_0$ is the bare electron mass.\",\n \"At a fixed energy $\\\\mu $ the contour is a circle parameterized as: $(k_x,k_y)=\\\\sqrt{2m_0\\\\mu }(\\\\cos \\\\theta ,\\\\sin \\\\theta ),\\\\ \\\\theta \\\\in [0,2\\\\pi )$ .\",\n \"Using method of characteristics, we get: $\\\\frac{\\\\mathrm {d}\\\\theta }{\\\\mathrm {d} s} = -\\\\frac{qB}{m_0}$ , or $\\\\theta = \\\\theta _0 - \\\\omega _0 s$ , where $\\\\omega _0\\\\equiv \\\\frac{qB}{m_0}$ is the cyclotron frequency of bare electrons.\",\n \"This leads to the periodicity in $s$ to be $s_0=2\\\\pi /\\\\omega _0$ , where we have chosen the clockwise trajectory such that $s_0>0$ .\",\n \"The Fourier series of the velocity along the constant energy contour is given by: $v_{x}(s)=\\\\sqrt{\\\\frac{\\\\mu }{2m}}\\\\left(e^{-i \\\\omega _0 s}+e^{i\\\\omega _0 s}\\\\right)$ , and $v_{y}(s)=\\\\sqrt{\\\\frac{\\\\mu }{2m}}\\\\frac{1}{i}\\\\left(e^{-i \\\\omega _0 s}-e^{i\\\\omega _0 s}\\\\right)$ .\",\n \"Substituting into Eq.\",\n \"(REF ), we obtain the conductivity tensor: $\\\\sigma = q^2\\\\tau \\\\frac{\\\\mu }{2\\\\pi } \\\\frac{1}{1+\\\\omega _0^2\\\\tau ^2}\\\\begin{pmatrix} 1 & -\\\\omega _0\\\\tau \\\\\\\\ \\\\omega _0\\\\tau & 1 \\\\end{pmatrix}.$ Note that the total number density of filled electrons is given by $n = \\\\int \\\\frac{\\\\mathrm {d}^2\\\\mathbf {k}}{(2\\\\pi )^2} \\\\Theta (\\\\mu -\\\\varepsilon _{\\\\mathbf {k}}) = \\\\frac{m_0\\\\mu }{2\\\\pi }$ .\",\n \"We therefore reproduce the well known magnetoconductivity tensor: $\\\\sigma = \\\\frac{nq^2\\\\tau }{m_0} \\\\frac{1}{1+\\\\omega _0^2\\\\tau ^2}\\\\begin{pmatrix} 1 & -\\\\omega _0\\\\tau \\\\\\\\ \\\\omega _0\\\\tau & 1 \\\\end{pmatrix}.$ In this simple example of a closed FS, the longitudinal resistivity is given by $\\\\frac{m_0}{nq^2\\\\tau }$ , independent of the magnetic field.\",\n \"The average of the velocity field, $\\\\mathbf {v}_{n=0}\\\\equiv \\\\frac{1}{s_0}\\\\int _0^{s_0}\\\\mathrm {d}s \\\\mathbf {v}(s)$ , vanishes.\",\n \"However, for an open FS generally $\\\\mathbf {v}_{n=0} \\\\ne \\\\mathbf {0}$ , i.e.. electrons have a finite drift velocity when traversing the contour due to a magnetic field (see SM Fig. 2).\",\n \"The impact of such a finite drift velocity on the magnetotransport can be qualitatively understood using the following example: in the expression for the conductivity tensor (Eq.\",\n \"(REF )), we consider $v^{x}_{n=0}\\\\ne 0$ but $v^{y}_{n=0}= 0$ .\",\n \"This corresponds to an open FS with a drift velocity along the $x$ direction.\",\n \"In the high field limit ( $\\\\omega _c\\\\tau \\\\propto B \\\\gg 1)$ , we truncate the Fourier series at the leading order, and as a result, $\\\\sigma _{\\\\text{open FS}} \\\\approx \\\\frac{q^3B}{2\\\\pi }\\\\frac{\\\\tau }{\\\\omega _c}\\\\begin{pmatrix}{(v^{x}_0)^2} & -\\\\frac{2\\\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\\\omega _c\\\\tau }\\\\\\\\ \\\\frac{2\\\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\\\omega _c\\\\tau } & \\\\frac{|v^{y}_1|^2}{\\\\omega _c^2\\\\tau ^2}\\\\end{pmatrix},$ where we made use of the equality: $\\\\mathbf {v}_{-n}=\\\\mathbf {v}_{n}^*$ .\",\n \"Inverting the matrix, we obtain the MR tensor: $\\\\begin{split}{\\\\rho }_{\\\\text{open FS}} &\\\\approx \\\\frac{(2\\\\pi )\\\\omega _c}{q^3B \\\\tau } \\\\frac{1}{4\\\\text{Im}(v_{-1}^{x}v_{1}^{y})^2+(v_0^{x})^2|v_{1}^{y}|^2} \\\\\\\\& \\\\times \\\\begin{pmatrix}|v^{y}_1|^2 & {2\\\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\\\omega _c\\\\tau }\\\\\\\\ -{2\\\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\\\omega _c\\\\tau } & (v^{(x)}_0)^2\\\\left(\\\\omega _c\\\\tau \\\\right)^2\\\\end{pmatrix}.\\\\end{split}$ It is clear that $\\\\rho _{yy}\\\\propto B^2$ whereas $\\\\rho _{xx}\\\\sim \\\\mathcal {O}(1)$ .\",\n \"We therefore arrive at the important conclusion that for an open FS, the longitudinal MR has non-saturating $B^2$ behavior along the axis with a zero drift velocity ($\\\\hat{y}$ in the above example), and saturating behavior along the other axis.\"\n ],\n [\n \"Magnetotransport in TBG under heterostrain\",\n \"We proceed to apply the above results to analyze the magnetotransport in TBG.\",\n \"The theory satisfactorily explains the weak-field magnetotransport measurements presented in Ref. [39].\",\n \"We then present two predictions of the theory that we did not anticipate prior to starting this work: the dependence of the principle axis of transport on filling, and the behavior of magnetoresistance and quantum oscillations at densities between the CNP and the onset of quadratic MR.\",\n \"The former is of academic interest, however it cannot be confirmed with our present data sets because of limitations of the Hall bar geometry.\",\n \"The latter can be considered smoking gun evidence for the the presence of the lowest-energy van Hove point and the energetic splitting of the Dirac cones.\",\n \"We do not expect our strained BM model in Eq.\",\n \"(REF ) to yield precise agreement with experiment, so we do not perform fine-tuning of its input parameters.\",\n \"Specifically, the model has particle-hole symmetry, which is absent from experimental measurements.\",\n \"More sophisticated non-interacting model calculations [44], [43] as well as interaction renormalizations [54] are likely necessary to properly account for such details.\",\n \"Although the general phenomena of open FSs and quadratic MR holds for a broad range of heterostrain parameters, we present calculations for $\\\\theta =1.38^\\\\circ $ , $\\\\epsilon =0.2\\\\%$ and $\\\\varphi =0^\\\\circ $ , parameters chosen to yield reasonable quantitative agreement between the theoretical and experimental results both on the filling range of open FSs, as well as on the frequencies of magnetoresistance oscillations to be presented later.\",\n \"In Fig.\",\n \"REF , we show the computed MR along the principal transport axes (a) and the Hall number (b).\",\n \"For comparison, we plot the experimentally measured longitudinal and transverse resistivities (c) and Hall number (d) for the TBG device studied in Ref. [39].\",\n \"In the filling ranges with open FSs, the calculated $\\\\rho _2(B)$ exhibits quadratic non-saturating MR, whereas $\\\\rho _1(B)$ saturates.\",\n \"The filling range for which quadratic MR occurs is bounded by the two outermost van Hove points of the zero-field strained band structure.\",\n \"In experiment, we observe quadratic MR in longitudinal resistivity within a similar range of fillings.\",\n \"More strikingly, we observe quadratic MR in the transverse resistivity as well.\",\n \"In some cases, the symmetric part of the transverse resistivity becomes larger than that of the longitudinal resistivity with field.\",\n \"As discussed earlier, this degree of mixing can be attributed to the misalignment between the strain-induced principal axis of transport and the direction of current flow in the Hall bar geometry.\",\n \"At the first van Hove point ($\\\\nu \\\\approx \\\\pm 0.6$ ), the non-analyticity in the density of states leads to a cusp in the first derivative of the zero-field resistivity with respect to filling (see SM Fig. 6).\",\n \"As shown in Fig.\",\n \"REF (a), at $B\\\\ne 0$ the longitudinal resistance develops a cusp as a function of filling at the first van Hove point.\",\n \"The cusp becomes more pronounced with increasing $B$ .\",\n \"Experimentally as shown in Fig.\",\n \"REF (c), there is a cusp-like feature developing at $|\\\\nu |\\\\sim 0.5-0.8$ depending on the contact pair within the device used, consistent with theoretical predictions.\",\n \"In many contact pairs, this feature presents as a shoulder at $B=0$ , only developing into a cusp at $B \\\\sim 0.1$ T (see SM Fig. 7).\",\n \"As depicted in Fig.\",\n \"REF (b), the calculated filling dependence of the Hall number shows two singular sign changes inside the open FS regions near $\\\\nu \\\\approx \\\\pm 2$ .\",\n \"The sign changing singularity in the open FS region is $B$ -independent, and is not directly associated with any van Hove point (see SM Fig.\",\n \"5 for a plot of $\\\\rho _H(B)$ , which crosses zero at the same filling fraction inside the open FS filling range for varying field strength).\",\n \"Moreover, the filling dependence of the Hall number $n_H$ tracks the filling fraction in a broad filling range near the CNP, with the filling range being extended upon increasing $B$ .\",\n \"In Fig.\",\n \"REF (d), we observe the same general shape of the Hall number.\",\n \"Within the open FS filling range, however, the measured Hall number qualitatively deviates from the theoretical curves.\",\n \"We attribute this to a small constant offset in the magnetic field of order 10-20 mT, likely resulting from trapped flux in the superconducting magnet.\",\n \"Here a large quadratic symmetric component of the transverse resistivity is concurrent with a vanishing antisymmetric component.\",\n \"An offset of only a few mT will lead to a small part of the symmetric component mixing into the antisymmetric component, leading to these deviations from theory (See SM Fig. 8).\",\n \"Figure: (a) Rotation of the transport principal axis e ^ 1 \\\\hat{e}_1 with respect to the global coordinate system for strained BM with ϵ=0.2%\\\\epsilon =0.2\\\\% and ϕ=0 ∘ \\\\varphi =0^\\\\circ .\",\n \"The three horizontal dashed lines are the bond directions.\",\n \"In the open FS region, the saturating MR axis is locked to the shortest bond (𝐋 1 \\\\mathbf {L}_1) direction.\",\n \"However, it rapidly rotates in the closed FS region upon approaching the CNP.\",\n \"(b) Principal transport axes e ^ 1 \\\\hat{e}_1 (red) and e ^ 2 \\\\hat{e}_2 (blue) for a few filling fractions.\",\n \"Near the CNP, e ^ 1 \\\\hat{e}_1 is perpendicular to the shortest moiré bond direction.\",\n \"In the open FS filling range (e.g.\",\n \"ν≈2.13\\\\nu \\\\approx 2.13) it is rotated to be parallel to the shortest bond direction.Our calculation finds a dramatic rotation of the principal axis with filling, as illustrated in Fig.\",\n \"REF .\",\n \"In the filling range with open FSs, the principal axis with saturating MR ($\\\\hat{e}_1$ ) is aligned with direction of the shortest moiré triangular bond, suggesting that the electrons are hopping more efficiently along the shortest bond, which leads to a larger conductivity and therefore a smaller resistivity.\",\n \"Interestingly, when filling is changed from the second van Hove point ($\\\\nu \\\\approx \\\\pm 1.3$ ) to the vicinity of the CNP, $\\\\hat{e}_1$ rotates dramatically to the perpendicular direction compared to the filling range with open FSs.\",\n \"The rotation of the principal axis is likely due to the opposite elongation of the larger Fermi pocket encircling a Dirac point compared to the open FS contours, see for example Figs.\",\n \"REF (d-f).\",\n \"The rotation of the transport axis with filling purely due to strain-induced bandstructure effects demonstrates that filling dependence of the principal axes orientation need not be associated with interaction induced nematicity [14].\",\n \"Such a filling-dependent rotation of the principal transport axis was not possible to observe in Ref.\",\n \"[39] using the Hall bar geometry, where only $\\\\rho _{xx}$ and $\\\\rho _{yx}$ are measured but not $\\\\rho _{yy}$ .\",\n \"Additional transport measurements are needed, where the filling-dependence of the entire resistivity tensor can be mapped out.\",\n \"Figure: (a) Line cuts of MR near the CNP taken at 26 mK in contact pair 4 - 5 at the indicated field strengths, in Tesla.\",\n \"Vertical dashed lines indicate our estimated location of the lowest-energy van Hove points, based on the cusps in resistivity at low field.\",\n \"Within the region bounded by these points, the quantum oscillations show up before 0.4 T, and their relative strengths do not follow a simple pattern.\",\n \"Outside of this region, the quantum oscillations onset at higher field, and every multiple of 4 quantum Hall filling fraction is observed relatively equally.\",\n \"(b) Fourier transform of the quantum oscillation data with respect to 1/B1/B.\",\n \"It reveals a transition from two pockets to one pocket at the lowest-energy van Hove points.\",\n \"(c) Schematic description of the frequencies observed in panel (b).\",\n \"Red dashed lines are frequencies from the experimental data.\",\n \"Solid black lines are predictions from the theory for ϵ=0.2%\\\\epsilon =0.2\\\\% and ϕ=0 ∘ \\\\varphi =0^\\\\circ .\",\n \"The two frequencies f 1 f_1 and f 2 f_2 sum to the one-pocket frequency f 3 f_3 that extends beyond the first van Hove point.\",\n \"They additionally account for the nontrivial relative strengths of the quantum oscillations within the bounds of the first van Hove points.\",\n \"As with other details of this work, the theory predicts electron-hole symmetry, while some asymmetry is observed in experiment.Since this theory predicts a third van Hove point between the CNP and the filling range with open FSs, a direct measurement of this van Hove point is desired.\",\n \"In Fig.\",\n \"REF we reanalyze quantum oscillation measurements of the TBG device discussed in Ref. [39].\",\n \"The effective cyclotron mass $m^*$ is light in the filling range with two small closed Fermi pockets, and dramatically heavier in the filling range with only one closed pocket (Figs.\",\n \"REF (d-f) and SM Fig. 5).\",\n \"The large difference in masses on either side of the innermost van Hove singularity can be used to explain the substantially earlier onset of quantum oscillations with increasing field close to the CNP than away from it, as shown in Fig.\",\n \"REF (a).\",\n \"Fig.\",\n \"REF (b) is a Fourier transform of the quantum oscillation data with respect to $1/B$ .\",\n \"In the filling range of $-0.7\\\\le \\\\nu \\\\le 0.8$ three distinct frequencies $f_{i=1,2,3}$ are clearly observed in the data, with $f_1$ and $f_2$ corresponding to two small Fermi pockets, and $f_3=f_1+f_2$ to the breakdown orbit when the inverse magnetic length is comparable to the momentum space distance between the two small Fermi pockets [60].\",\n \"Outside of the filling range only $f_3$ is observed, showing that there are Lifshitz transitions, one on either side of the CNP, that we ascribe to crossing the lowest-energy van Hove points.\",\n \"Furthermore, these filling fractions also correspond to the cusp-like features in the longitudinal MR data shown in Fig.\",\n \"REF (c), consistent with theoretical predictions for its behavior at van Hove singularities.\",\n \"Therefore, the quantum oscillation data unambiguously demonstrates the existence of a third van Hove singularity at filling fractions between the CNP and the filling range of $B^2$ MR.\",\n \"It is interesting to note again, that the Hall number does not show a sign-changing singularity at this van Hove point, as illustrated in Fig.\",\n \"REF (b) and (d).\",\n \"The frequencies $f_{1,2}$ are a strong constraint on the amount of heterostrain in the TBG sample.\",\n \"Specifically, as illustrated in Fig.\",\n \"REF (c), the frequency $f_2$ is roughly two times $f_1$ , showing that the two small Fermi pockets have an area ratio $\\\\sim 2:1$ .\",\n \"Theoretically as illustrated by the solid black lines in Fig.\",\n \"REF (c), for a heterostrain strength $\\\\epsilon =0.2\\\\%$ and $\\\\varphi =0^\\\\circ $ , the areas $A_{i=1,2}$ of the two small pockets, when converted to frequency via $f_i^{-1}\\\\equiv (\\\\Delta \\\\frac{1}{B})_i=\\\\frac{2\\\\pi e}{\\\\hbar A_i}$ , are in good agreement with experiment.\",\n \"We observe behavior qualitatively similar in all respects to that in Fig.\",\n \"REF (a) in all 3 longitudinal contact pairs for which we have dilution-fridge measurements (see SM Fig. 11).\",\n \"In addition to the quantum oscillation measurements above, we propose an additional experimental procedure for identifying the van Hove points.\",\n \"As usual, at van Hove singularities there are non-analyticities in the electronic density of states.\",\n \"Such non-analyticities will lead to cusps in the first derivative of the zero-field resistivity with respect to filling (see SM Fig. 6).\",\n \"This can be probed via transport measurements, for example, by adding a small ac modulation of the filling or by numerical differentiation of the dc data.\"\n ],\n [\n \"Summary and Outlook\",\n \"In summary, we have shown that due to the large size of the moiré unit cell at small twist angles, even a small amount of uniaxial heterostrain on the microscopic scale can lead to dramatic changes in the narrow bands of twisted bilayer graphene.\",\n \"A key feature of the strained bandstructure is the splitting of the respective energetic degeneracies of the two Dirac points and the three van Hove points.\",\n \"The splitting of the two Dirac points leads to a semimetallic state with two small Fermi pockets at the CNP.\",\n \"On the other hand, the two outermost van Hove points bound a broad filling range near $\\\\nu =\\\\pm 2$ where the constant energy contours become open.\",\n \"Interestingly, the elongation of the larger Fermi pocket near the CNP is perpendicular to that of the open FSs, the latter being perpendicular to the direction of the shortest moiré triangular bond.\",\n \"We have analyzed the resulting magnetotransport in strained TBG in the framework of the Boltzmann equation using the method of characteristics, treating the magnetic field non-perturbatively.\",\n \"We showed that a non-saturating quadratic longitudinal magnetoresistance in a broad filling range near $\\\\nu =\\\\pm 2$ naturally arises due to the heterostrain-induced open Fermi surfaces, therefore explaining the experimental results in the off-magic-angle devices [39].\",\n \"We have also shown that the sign-changing singularities in the Hall number occur in the open FS filling range and are not directly associated with any van Hove singularity as commonly assumed, e.g., in Ref. [61].\",\n \"Furthermore, our results reveal a dramatic rotation of the transport principal axis as the filling is tuned from the charge neutrality point to the filling range of open Fermi surfaces.\",\n \"This is entirely attributed to the strained non-interacting bandstructure effects, and does not require interaction-induced electronic nematicity for explanation.\",\n \"Given the importance of energy-shifted van Hove points in the transport properties of TBG devices, we have analyzed previous quantum oscillation data, which has revealed a Lifshitz transition from two pockets to one pocket at a filling fraction where the innermost van Hove singularity is predicted to occur based on theoretical calculations, therefore offering strong evidence of heterostrain effects on these devices.\",\n \"We have further proposed several additional signatures to look for in future experiments.\",\n \"These include cusps in the derivative of zero field resistivity with respect to filling, a significant difference in cyclotron mass on either side of the innermost van Hove singularity, and a principal transport axis with saturating magnetoresistance in the open Fermi surface filling range.\",\n \"Finally, given the amplifying effect of a small strain at the underlying carbon lattice scale on the moiré lattice scale, the latter of which controls the electronic behavior within the narrow bands, it is tantalizing to consider strain engineering of such devices to achieve effects which would be impossible in regular solids due to structural instabilities.\"\n ],\n [\n \"Acknowledgement\",\n \"Funding: X.W.\",\n \"acknowledges financial support from National MagLab through Dirac fellowship, which is funded by the National Science Foundation (Grant No.\",\n \"DMR-1644779) and the state of Florida.\",\n \"O.V.\",\n \"was supported by NSF Grant No.\",\n \"DMR-1916958 and is partially funded by the Gordon and Betty Moore Foundation's EPiQS Initiative Grant GBMF11070, National High Magnetic Field Laboratory through NSF Grant No.\",\n \"DMR-1157490 and the State of Florida.\",\n \"Device measurements and analysis were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-76SF00515.\",\n \"Measurement infrastructure was funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF3429 and grant GBMF9460.\",\n \"D.G.-G. gratefully acknowledges support from the Ross M. Brown Family Foundation.\",\n \"Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.\",\n \"K.W.\",\n \"and T.T.\",\n \"acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).\",\n \"Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822.\",\n \"Author contributions: O.V.\",\n \"conceived the theoretical explanation of the experiments.\",\n \"X.W.\",\n \"and O.V.\",\n \"performed calculations.\",\n \"J.F., L. R., and C.H.\",\n \"fabricated devices.\",\n \"J.F.\",\n \"and X.W.\",\n \"analysed the data.\",\n \"K.W.\",\n \"and T.T.\",\n \"generously supplied the hBN crystals.\",\n \"A.L.S., M.A.K., O.V., and D.G.-G. supervised the experiments and analysis.\",\n \"The manuscript was prepared by X.W.\",\n \"and J.F.\",\n \"with input from all authors.\",\n \"Supplementary Materials for “Unusual magnetoresistance in twisted bilayer graphene from strain induced open Fermi surfaces\\\" We present additional theoretical results and experimental measurements in support of the main text.\"\n ],\n [\n \"Heterostrain effects on the geometry of moiré superlattice\",\n \"Our off-magic-angle twisted bilayer graphene (TBG) devices in Ref.\",\n \"[39] are prepared using the “tear-and-stack\\\" procedure, and as a result, strain is inevitably introduced.\",\n \"Here we first show that while the moiré unit cell vectors are strongly deformed by even an infinitesimal amount of uniaxial heterostrain in the device, the unit cell area is much less affected.\",\n \"As a result, for the off-magic-angle device studied in Ref.\",\n \"[39], we can have a good estimate of the twist angle ($\\\\theta $ ) based on the moiré unit cell area alone.\",\n \"In the limit of small deformations, both the uniaxial heterostrain and a small twist angle are captured via a coordinate transformation: $\\\\mathbf {r}^{\\\\prime }_l = \\\\mathbf {r}+ \\\\mathbf {u}_l(\\\\mathbf {r})$ , where $l=t,b$ labels the top (bottom) graphene layers, and $\\\\mathbf {u}_l(\\\\mathbf {r}) \\\\approx \\\\mathcal {E}_l \\\\mathbf {r}$ is the local deformation field.\",\n \"The symmetric and antisymmetric part of the $2\\\\times 2$ tensor $\\\\mathcal {E}_l$ describes strain and rotation respectively.\",\n \"For twist angle $(\\\\theta )$ and a uniaxial heterostrain of strength $(\\\\epsilon )$ and direction $(\\\\varphi )$ , we parameterize $\\\\mathcal {E}_{t}=-\\\\mathcal {E}_{b}\\\\equiv \\\\mathcal {E}/2$ , where $\\\\mathcal {E}\\\\equiv \\\\mathcal {T}(\\\\theta )+ \\\\mathcal {S}(\\\\epsilon ,\\\\varphi )$ , and given by: $\\\\mathcal {T}(\\\\theta ) = \\\\begin{pmatrix} 0 & -\\\\theta \\\\\\\\ \\\\theta & 0\\\\end{pmatrix},\\\\ \\\\mathcal {S}(\\\\epsilon ,\\\\varphi ) = R_{\\\\varphi }^T \\\\begin{pmatrix} -\\\\epsilon & 0\\\\\\\\ 0 & \\\\nu \\\\epsilon \\\\end{pmatrix}R_{\\\\varphi }.$ Here $R_\\\\varphi $ is the two-dimensional rotation matrix, and $\\\\nu \\\\approx 0.16$ is the Poisson ratio [3].\",\n \"Physically, $\\\\epsilon >0$ corresponds to compressing the top layer while streching the bottom layer along the $x$ -axis.\",\n \"A relative deformation $\\\\mathcal {E}$ between the graphene bilayers generate a moiré superlattice, with moiré reciprocal lattice vectors given by: $\\\\mathbf {g}_{i=1,2}=\\\\mathcal {E}^T\\\\mathbf {G}_{i=1,2},$ where $\\\\mathbf {G}_{i}$ are reciprocal lattice vectors of the undeformed monolayer graphene.\",\n \"Eq.\",\n \"(REF ) can be used to uniquely determine the three parameters $(\\\\theta ,\\\\epsilon ,\\\\varphi )$ .\",\n \"Additionally it also determines a global angle $\\\\alpha $ that measures the rotation between the lab and theoretical coordinate systems.\",\n \"Uniaxial heterostrain has a dramatic effect on the distortion of the moiré unit cell vectors, as $|\\\\delta \\\\mathbf {g}|/|\\\\mathbf {g}|\\\\sim \\\\mathcal {O}(\\\\epsilon /\\\\theta )$ .\",\n \"However, its effect on the moiré unit cell area is much smaller.\",\n \"To show this, note that the area of the moiré Brillouin zone is calculated as: $A_{mBZ} = \\\\left| (\\\\mathbf {g}_1 \\\\times \\\\mathbf {g}_2 )\\\\cdot \\\\hat{z}\\\\right| = \\\\left| \\\\mathbf {g}_1^T (i\\\\sigma _y) \\\\mathbf {g}_2 \\\\right|,$ where on the second equality we have used a vector notation $\\\\mathbf {g}_i\\\\equiv (g_{i,x},g_{i,y})^T$ .\",\n \"Following Eq.\",\n \"(REF ), we obtain that the area of the moiré Brillouin zone is independent on $\\\\varphi $ , and calculated as: $A_{mBZ} = (\\\\theta ^2-\\\\nu ^2\\\\epsilon ^2)A_{BZ},$ where $A_{BZ}\\\\equiv \\\\left|(\\\\mathbf {G}_1\\\\times \\\\mathbf {G}_2)\\\\cdot \\\\hat{z}\\\\right|$ is the Brillouin zone area of the undeformed monolayer graphene.\",\n \"The area of the strained moiré unit cell can be calculated in a similar manner, and we get: $A_{m.u.c.}=A_{u.c.\",\n \"}/(\\\\theta ^2-\\\\nu ^2\\\\epsilon ^2)$ , where $A_{u.c.\",\n \"}$ is the unit cell area of undeformed monolayer graphene.\",\n \"Observe that the heterostrain only affects the area of the moiré unit cell by $\\\\mathcal {O}(\\\\nu ^2\\\\epsilon ^2/\\\\theta ^2)$ which is much smaller than the linear distortion of moiré unit cell vectors.\",\n \"With only a knowledge of the moiré unit cell areas in Ref.\",\n \"[39] (see Table REF ), we estimate the twist angle to be $\\\\theta \\\\sim 1.35^\\\\circ - 1.39^\\\\circ $ for various contact pairs studied using the Hall bar geometry.\"\n ],\n [\n \"Constraining heterostrain from transport measurements\",\n \"For the TBG device studied in Ref.\",\n \"[39], the deformed moiré lattice vectors were not measured.\",\n \"Nevertheless, here we show that magnetotransport measurements, along with theoretical calculations based on the strained Bistrizer-MacDonald (BM) Hamiltonian, offer strong constraints on the heterostrain in the device.\",\n \"We caution, however, that since the strained BM model is an approximate description of the narrow bands of TBG, a precise determination of heterostrain from model calculations is not feasible.\",\n \"First of all, as predicted by theoretical calculations, the van Hove singularities of the band structure lead to non-analytic behavior for the longitudinal magnetoresistance as a function of electron filling.\",\n \"The filling fractions for the six van Hove singularities in the narrow band are listed in Table REF for various contact pairs.\",\n \"Secondly, magnetic oscillations show a Lifshitz transition at the inntermost van Hove singularities ($\\\\nu _3,\\\\nu _4$ ), from two small Fermi pockets closer to the charge neutrality point to one Fermi pocket away from it.\",\n \"Furthermore, the areas of the two small Fermi pockets, as revealed by the frequencies of magnetic oscillations, show a $2:1$ or smaller ratio.\",\n \"Both the filling fractions for van Hove singularities and the pocket area size offer strong constraints for the heterostrain.\",\n \"Qualitatively, on the one hand, a broader filling range of open Fermi surfaces can be achieved by increasing the strength of uniaxial heterostrain.\",\n \"On the other hand, to obtain Fermi pocket area sizes near $2:1$ ratio or smaller, a smaller heterostrain is necessary as it leads to a weaker splitting of the two Dirac cones.\",\n \"For theoretical calculations presented in the main text, we find $\\\\epsilon =0.2\\\\%$ and $\\\\varphi =0^\\\\circ $ to give reasonably good agreements with both experimental observations described above.\",\n \"A larger heterostrain strength ($\\\\epsilon =0.3\\\\%$ ) will lead to a much larger pocket area ratio ( $4:1$ for $\\\\epsilon =0.3\\\\%$ and $\\\\varphi =0^\\\\circ $ ), inconsistent with magnetic oscillation measurements.\",\n \"On the other hand, a smaller heterostrain strength $\\\\epsilon =0.1\\\\%$ decreases the filling range of open Fermi surfaces dramatically, inconsistent with the longitudinal magnetoresistance measurements.\"\n ],\n [\n \"Detailed band structure analysis for varying uniaxial heterostrain\",\n \"In the main text we discussed the band structure of the strained TBG for $\\\\epsilon =0.2\\\\%$ .\",\n \"The main effect of uniaxial heterostrain is to break the respective energetic degeneracies of the two Dirac points and three van Hove points of a given band, therefore giving rise to a semimetallic state at charge neutrality point, and open Fermi surface regions bounded by the two outermost van Hove points.\",\n \"However for a larger heterostrain, the innermost van Hove point moves closer to one of the Dirac point.\",\n \"As a result, both Dirac cones become type II titled, and the innermost van Hove points of both the upper and lower bands are annihilated.\",\n \"In turn two new band extrema are formed.\",\n \"This is illustrated in Fig.\",\n \"REF .\",\n \"We also explore the possibilities of heterostrain-induced higher order van Hove singularities which is possible for the magic-angle TBG as discussed in Ref. [38].\",\n \"We checked that for $\\\\theta =1.38^\\\\circ $ , and up to uniaxial heterostrain strength of $\\\\epsilon =0.7\\\\%$ , no higher order van Hove singularities are found.\",\n \"This shows that the band flattening effect at the magic angle may be important for strain engineering of higher order van Hove points.\",\n \"Figure: Velocity field of typical closed and open Fermi surfaces.\",\n \"Whereas for a closed Fermi surface the averaged velocity vanishes, for open Fermi surfaces this is generally violated.In Fig.\",\n \"REF we plot the velocity fields $v_x(s)$ and $v_y(s)$ on typical open and closed Fermi surfaces for the strained TBG, parameterized by $s\\\\in [0,s_0)$ as defined in the main text.\",\n \"For the closed Fermi surface contours, the averaged velocity, $\\\\mathbf {v}_{n=0}\\\\equiv \\\\frac{1}{s_0}\\\\int _0^{s_0}\\\\mathrm {d}s \\\\mathbf {v}(s)$ , is zero.\",\n \"On the other hand, for a typical open Fermi surface contour, it is finite, and as a result the electron traversing the open Fermi surface contour in the presence of a magnetic field has a finite drift velocity.\",\n \"As discussed in the main text, this is the reason for the non-saturating $B^2$ magnetoresistivity (MR) observed in strained TBG devices.\"\n ],\n [\n \"More details on magnetotransport in TBG\",\n \"Here we show that while $B^2$ longitudinal MR generally occurs for strained TBG due to open Fermi surfaces, it does not occur for unstrained devices.\",\n \"In Fig.\",\n \"REF , the longitudinal MRs $\\\\rho _{xx}$ and $\\\\rho _{yy}$ as well as the Hall number $n_H$ are plotted for an unstrained BM model calculation.\",\n \"First of all, $\\\\rho _{xx}=\\\\rho _{yy}$ since the unstrained TBG has $C_{3z}$ rotational symmetry.\",\n \"Secondly, cusp-like features develop at the triply-degenerate van Hove point at filling fractions $\\\\nu \\\\approx \\\\pm 1.4$ , and are attributed to the non-analyticities in the density of states behavior at the van Hove singularities.\",\n \"Finally, unstrained TBG has saturating MR across all filling range, as illustrated in the inset to Fig.\",\n \"REF (a).\",\n \"Figure: (a) Longitudinal MR ρ xx \\\\rho _{xx} (dashed) and ρ yy \\\\rho _{yy} (solid) for strained BM with ϵ=0.2%\\\\epsilon =0.2\\\\% and ϕ=0 ∘ \\\\varphi =0^\\\\circ .\",\n \"Different colors represent varying magnetic field strength.\",\n \"Vertical dashed lines are positions of the van Hove points.\",\n \"Shaded areas are open Fermi surface regions.\",\n \"(b) In the closed Fermi surface region, MR saturates at large magnetic fields.\",\n \"(c) In the open Fermi surface region, MR exhibit non-saturating B 2 B^2 dependence along both directions.In Fig.\",\n \"REF we show that for the globally defined coordinate system which is misaligned from the principal transport axis, the $B^2$ behavior generally dominates the MR, and therefore will show up in both $\\\\rho _{xx}$ and $\\\\rho _{yy}$ measurements.\",\n \"This remains true for a generic misalignment between the transport axis from experiment and the principal transport axis.\",\n \"Figure: Hall resistivity ρ H \\\\rho _H at varying magnetic fields.\",\n \"Note that it crosses zero within the filling range of open Fermi surfaces on both sides of the charge neutrality point.\",\n \"These mark the sign-changing singularities in the Hall number depicted in Fig.\",\n \"2(b) of the main text.In Fig.\",\n \"REF we show the Hall resistivity $\\\\rho _H(B)$ for varying magnetic field strength.\",\n \"Since $\\\\rho _H = B/n_Hq$ , wherever $\\\\rho _H(B)$ crosses zero and changes sign, the Hall number displays a sign-changing signularity.\",\n \"Fig.\",\n \"REF clearly shows that $\\\\rho _H(B)$ crosses zero in the open Fermi surface regions on both sides of the charge neutrality point, and independent on the strength of the $B$ -field.\",\n \"Figure: Filling dependence of the averaged inverse cyclotron mass 1/m * 1/m^*, extracted from the averaged cyclotron frequency ω c ¯=eB m * \\\\bar{\\\\omega _c}=\\\\frac{eB}{m^*}.\",\n \"Here ω ¯ c =1 N∑ n,i ω c,n,i \\\\bar{\\\\omega }_c=\\\\frac{1}{N}\\\\sum _{n,i}\\\\omega _{c,n,i}, where nn and ii label the FS (ii) coming from a given band (nn), in units of the bare electron mass.\",\n \"1/m * 1/m^* is larger near the charge neutrality and band edges, explaining the earlier onset of quantum oscillations in these filling regions.In Fig.\",\n \"REF we illustrate the filling-dependent inverse cyclotron mass $1/m^*$ for strained BM model with $\\\\epsilon =0.2\\\\%$ and $\\\\varphi =0^\\\\circ $ .\",\n \"This is to highlight the dichotomy of light-heavy masses on either side of the innermost van Hove singularities closest to the charge neutrality point.\",\n \"This is consistent with the experimental observation of a much earlier onset field of quantum oscillations in filling range below the innermost van Hove point than above.\",\n \"Figure: (a) Longitudinal resistivitities ρ xx \\\\rho _{xx} (blue) and ρ yy \\\\rho _{yy} (orange) at B=0B=0 for strained BM model, with ϵ=0.2%\\\\epsilon =0.2\\\\% and ϕ=0 ∘ \\\\varphi =0^\\\\circ .\",\n \"(b) The derivative of log resistivity with respect to filling.\",\n \"Gray dotted vertical lines mark positions of the van Hove points, and the yellow shaded area marks the open Fermi surface region.In Fig.\",\n \"REF we also show the filling dependence of the $B=0$ longitudinal resistivities and their derivatives with respect to filling.\",\n \"A key highlight is that the non-analyticities in the density of states at the van Hove points lead to kink-like features in the derivatives, but nearly invisible in the resistitivies themselves.\"\n ],\n [\n \"Error analysis of antisymmetrization\",\n \"In Fig.\",\n \"REF we show that the bump-like features in the experimental Hall number plots in filling range of open Fermi surfaces (Fig.\",\n \"2(d) of main text and Fig.\",\n \"REF in the SM) may be attributed to improper antisymmetrization with respect to the $B$ -field, namely, $\\\\tilde{\\\\rho }_H(B) = \\\\frac{\\\\rho _{yx}(B+\\\\delta B) - \\\\rho _{yx}(-B+\\\\delta B)}{2},$ where $\\\\delta B$ is a systematic error.\",\n \"The error may be attributed to a small trapped flux of 10 mT in the superconducting magnet, or perhaps an offset in the magnet power supply.\",\n \"Due to the misalignment of transport principal axis with the Hall bar geometry, longitudinal MR also contributes to $\\\\rho _{yx}(B)$ .\",\n \"In the filling range with open Fermi surfaces, the longitudinal resistance exhibits non-saturating quadratic MR, and will mix into the Hall component which is odd in B.\",\n \"As a result, one expects the improper antisymmetrization error to be largest in this filling range.\",\n \"We investigate this possibility by first fitting a polynomial to the low-field transverse resistivity.\",\n \"This allows us to interpolate the data and add small constant offsets prior to antisymmetrization.\",\n \"Accounting for an offset of roughly 20 mT largely removes the bumps from the data.\",\n \"This offset is larger than what we would expect from trapped flux in a superconducting magnet, however we do not expect the procedure to be accurate to such a fine degree, simply because we do not have fine enough resolution in field to get an accurate polynomial fit.\"\n ],\n [\n \"More experimental measurements based on various contact pairs of the Hall bar geometry\",\n \"The device has nine voltage probes on each side.\",\n \"We observe quadratic magnetoresistance regions in roughly half of the device, between the fourth and eighth contacts.\",\n \"We present longitudinal resistivities of these pairs in Fig.\",\n \"REF .\",\n \"In each of these pairs, we observe behavior qualitatively consistent with that presented in the main text.\",\n \"Our Hall measurements (Fig.\",\n \"REF ) are similarly consistent.\",\n \"In Fig.\",\n \"REF , we show quantum oscillations and their Fourier transforms for all three contact pairs for which we have dilution refridgerator data.\",\n \"In all cases, we observe behavior consistent with what we present in the main text: 1) quantum oscillation onset at lower field close to CNP, 2) an irregular pattern of resistivity minima close to CNP, and 3) extra features in the FFT of the quantum oscillations that end at vH1.\",\n \"The density of the first van Hove point is closer to the CNP in the other two contact pairs, and the extra features in the FFT are not as clear.\"\n ]\n]"},"id":{"kind":"string","value":"2209.08204"}}},{"rowIdx":2396,"cells":{"text":{"kind":"list like","value":[["The trace reconstruction problem for spider graphs"],["Abstract We study the trace reconstruction problem for spider graphs.","Let $n$ be the number of nodes of a spider and $d$ be the length of each leg, and suppose that we are given independent traces of the spider from a deletion channel in which each non-root node is deleted with probability $q$.","This is a natural generalization of the string trace reconstruction problem in theoretical computer science, which corresponds to the special case where the spider has one leg.","In the regime where $d\\ge \\log_{1/q}(n)$, the problem can be reduced to the vanilla string trace reconstruction problem.","We thus study the more interesting regime $d\\le \\log_{1/q}(n)$, in which entire legs of the spider are deleted with non-negligible probability.","We describe an algorithm that reconstructs spiders with high probability using $\\exp\\left(\\mathcal{O}\\left(\\frac{(nq^d)^{1/3}}{d^{1/3}}(\\log n)^{2/3}\\right)\\right)$ traces.","Our algorithm works for all deletion probabilities $q\\in(0,1)$."],["Introduction","The string trace reconstruction problem, first introduced in 1997 by Levenshtein [17], is concerned with reconstructing an unknown seed string using only noisy samples of the data.","The unknown seed string is passed into some noisy channel multiple times, and the resulting error-prone copies are referred to as traces.","The goal is to use multiple traces to reconstruct the original seed string with high probability.","Levenshtein solved the trace reconstruction problem for a substitution channel, where each symbol of the seed string is mutated independently with constant probability.","In 2004, Batu, Kannan, Khanna, and McGregor [2] analyzed the problem for a deletion channel, where symbols of the seed string are each deleted independently with constant probability.","The string trace reconstruction problem has applications to computational biology, specifically in the new rapidly-evolving fields of DNA data storage and personalized immunogenics.","For example, one might want to reconstruct the correct sequence of nucleotides of a DNA sequence from several traces, each of which has many deletion mutations.","It is critical to minimize the number of traces required to reconstruct the seed string with high probability.","For example, in the application of DNA data storage, reducing the number of traces results in lower sequencing cost and time [3].","However, despite a wealth of recent work and attention on the deletion channel string trace reconstruction problem, for example [5], [10], [11], [12], [13], [14], [18], [20], [21], the current best upper and lower bounds for the number of traces necessary to reconstruct the seed string with high probability remain at $\\exp (\\mathcal {O}(n^{1/5}))$ [4] and $\\tilde{\\Omega }(n^{3/2})$ [5], [12], respectively, where $n$ is the length of the seed string.","We remark that a lower bound of $\\exp (\\mathcal {O}(n^{1/3}))$ traces was shown for mean-based algorithms, which are algorithms that only use the empirical means of individual bits in the traces for reconstruction [10], [20].","The exponential gap between upper and lower bounds for string trace reconstruction motivates studying variants of the problem for which one may be able to close the gap.","Many variants have been recently proposed and studied, for example [1], [6], [7], [9], [16], [19].","We focus on a variant known as the tree trace reconstruction problem introduced by Davies, Rácz, and Rashtchian [8].","This is a generalization of the vanilla string trace reconstruction problem where the goal is to learn a node-labeled tree, rather than a single string, using traces from a suitably-defined deletion channel.","The tree trace reconstruction problem may be directly applicable as well, as research on DNA nanotechnology has demonstrated that DNA molecule structures can be assembled into trees.","Recent research has also shown how to distinguish different molecular topologies, such as spiders with three arms from line DNA, using nanopores [15].","Davies et al.","[8] studied the tree trace reconstruction problem for two special classes of trees: complete $k$ -ary trees and spiders.","This paper extends their work on spiders.","An $(n,d)$ -spider consists of a single unlabeled root node with paths of $d$ labeled nodes attached to it.","In total, there are $n$ labeled nodes.","Consider a deletion channel, formally defined in deletion channel, in which every node is independently deleted with probability $q$ .","When $d\\ge \\log _{1/q}(n)$ , solving the spider trace reconstruction problem directly reduces to the string trace reconstruction problem [8].","This is because in this regime, the legs of the spider are long enough for all of the legs to survive the deletion channel with high probability, so each leg can be considered independently as its own string trace reconstruction problem.","Therefore, we assume that $d\\le \\log _{1/q}(n)$ .","In this more interesting regime, entire legs are deleted with non-negligible probability.","Hence, if one looks at a single trace, it is unclear which of the legs in the seed spider the legs in the trace come from.","Davies et al.","[8] proved that for deletion probabilities $q<0.7$ , there is some constant $C>0$ that depends only on $q$ such that $\\exp (C\\cdot d(nq^d)^{1/3})$ traces suffice to reconstruct an $(n,d)$ -spider with probability $1-\\mathcal {O}(1/n)$ (we refer to this as with high probability).","In this paper, we match this upper bound, up to polylogarithmic factors, but for the full range of deletion probabilities $q\\in (0,1)$ .","Furthermore, while Davies et al.","[8] used a single variable generating function alongside harmonic analysis, we consider a bivariate generating function, which results in considerably simpler analysis.","We use a best-match algorithm coupled with some results about bivariate Littlewood polynomials.","We remark that Littlewood polynomials have also been used to analyze a different variant of trace reconstruction known as the matrix reconstruction problem [16].","Our main result is the following theorem: Theorem 1.1 Assume that $d\\le \\log _{1/q}(n)$ .","For any fixed deletion probability $q<1$ , there exists some constant $C>0$ that depends only on $q$ such that $\\exp \\left(C\\cdot \\frac{(nq^d)^{1/3}}{d^{1/3}}(\\log n)^{2/3}\\right)$ traces suffice to reconstruct an $(n,d)$ -spider with high probability.","Note that the upper bound in main result matches the upper bound $\\exp (C\\cdot d(nq^d)^{1/3})$ in [8] up to polylogarithmic factors and works for all deletion probabilities $q\\in (0, 1)$ , not just $q<0.7$ .","Furthermore, main result strictly improves upon the upper bound $\\exp (C\\cdot d(nq^d)^{1/3})$ for all $q\\in (0,1)$ when $d = \\omega (\\sqrt{\\log n})$ ."],["Acknowledgements","The authors would like to thank Shyam Narayanan for suggesting the problem."],["Rooted spiders","In this section, we define the objects to be reconstructed: rooted binary-labeled spiders $X$ , as well as an indexing system for their nodes.","Definition 2.1 Let $n$ and $d$ be positive integers, and for convenience assume that $d\\mid n$ .","An $(n,d)$ -spider $X$ consists of a single unlabeled root node with $\\frac{n}{d}$ paths of $d$ nodes with binary labels from $\\lbrace 0,1\\rbrace $ emanating from it, so there are $n$ labeled nodes in total.","We refer to these paths as the legs of the spider.","Figure: A binary-labeled (12,4)(12,4) spider, with indexing system drawn in blue to the bottom left of each vertex.An example of a binary-labeled $(12,4)$ -spider is shown in fig:spider.","We index each node using two coordinates, where the first coordinate denotes the leg of the spider the node is on, and the second its depth down that leg.","This is in contrast to the depth-first-search labelling in [8].","In general, we may denote by $a_{i,j}\\in \\lbrace 0,1\\rbrace $ the label of the $(i,j)$ -node and the set of all labels of $X$ as $a=\\lbrace a_{i,j}\\rbrace _{0\\le i<\\frac{n}{d}, 0\\le j0$ that depends only on $q$ such that $\\\\exp (C\\\\cdot d(nq^d)^{1/3})$ traces suffice to reconstruct an $(n,d)$ -spider with probability $1-\\\\mathcal {O}(1/n)$ (we refer to this as with high probability).\",\n \"In this paper, we match this upper bound, up to polylogarithmic factors, but for the full range of deletion probabilities $q\\\\in (0,1)$ .\",\n \"Furthermore, while Davies et al.\",\n \"[8] used a single variable generating function alongside harmonic analysis, we consider a bivariate generating function, which results in considerably simpler analysis.\",\n \"We use a best-match algorithm coupled with some results about bivariate Littlewood polynomials.\",\n \"We remark that Littlewood polynomials have also been used to analyze a different variant of trace reconstruction known as the matrix reconstruction problem [16].\",\n \"Our main result is the following theorem: Theorem 1.1 Assume that $d\\\\le \\\\log _{1/q}(n)$ .\",\n \"For any fixed deletion probability $q<1$ , there exists some constant $C>0$ that depends only on $q$ such that $\\\\exp \\\\left(C\\\\cdot \\\\frac{(nq^d)^{1/3}}{d^{1/3}}(\\\\log n)^{2/3}\\\\right)$ traces suffice to reconstruct an $(n,d)$ -spider with high probability.\",\n \"Note that the upper bound in main result matches the upper bound $\\\\exp (C\\\\cdot d(nq^d)^{1/3})$ in [8] up to polylogarithmic factors and works for all deletion probabilities $q\\\\in (0, 1)$ , not just $q<0.7$ .\",\n \"Furthermore, main result strictly improves upon the upper bound $\\\\exp (C\\\\cdot d(nq^d)^{1/3})$ for all $q\\\\in (0,1)$ when $d = \\\\omega (\\\\sqrt{\\\\log n})$ .\"\n ],\n [\n \"Acknowledgements\",\n \"The authors would like to thank Shyam Narayanan for suggesting the problem.\"\n ],\n [\n \"Rooted spiders\",\n \"In this section, we define the objects to be reconstructed: rooted binary-labeled spiders $X$ , as well as an indexing system for their nodes.\",\n \"Definition 2.1 Let $n$ and $d$ be positive integers, and for convenience assume that $d\\\\mid n$ .\",\n \"An $(n,d)$ -spider $X$ consists of a single unlabeled root node with $\\\\frac{n}{d}$ paths of $d$ nodes with binary labels from $\\\\lbrace 0,1\\\\rbrace $ emanating from it, so there are $n$ labeled nodes in total.\",\n \"We refer to these paths as the legs of the spider.\",\n \"Figure: A binary-labeled (12,4)(12,4) spider, with indexing system drawn in blue to the bottom left of each vertex.An example of a binary-labeled $(12,4)$ -spider is shown in fig:spider.\",\n \"We index each node using two coordinates, where the first coordinate denotes the leg of the spider the node is on, and the second its depth down that leg.\",\n \"This is in contrast to the depth-first-search labelling in [8].\",\n \"In general, we may denote by $a_{i,j}\\\\in \\\\lbrace 0,1\\\\rbrace $ the label of the $(i,j)$ -node and the set of all labels of $X$ as $a=\\\\lbrace a_{i,j}\\\\rbrace _{0\\\\le i<\\\\frac{n}{d}, 0\\\\le j -1$, on the conic surface $\\{(x,t): \\|x\\| = t, \\, x \\in \\mathbb{R}^d, \\, t \\le 1\\}$ are studied recently, and they are shown to be eigenfunctions of a second order differential operator $\\mathcal{D}_\\gamma$ when $\\beta =-1$.","We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th partial derivatives in $t$ variable with respect to $w_{\\beta+s,0}$ over the conic surface plus a sum of integrals over the rim of the cone.","Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series.","The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\\beta, -s}$ for a positive integer $s$.","It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\\mathcal{D}_{\\gamma}$ when $\\gamma = -1$."],["Introduction","Orthogonal polynomials on the conic surface of the revolution were studied recently, which are shown to possess properties parallel to those of spherical harmonics on the unit sphere.","Let ${\\mathbb {V}}_0^{d+1}$ be the conic surface ${\\mathbb {V}}_0^{d+1} = \\lbrace (x,t) \\in {\\mathbb {R}}^{d+1}: \\, \\Vert x\\Vert = t, \\, x \\in {\\mathbb {R}}^d, \\, 0 \\le t \\le 1\\rbrace $ in ${\\mathbb {R}}^{d+1}$ .","For the weight function ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t) = t^{\\beta }(1-t)^{\\gamma }$ , ${\\beta }> -d$ and ${\\gamma }> -1$ , the orthogonal polynomials with respect to the inner product ${\\langle }f,g{\\rangle }_{{\\beta },{\\gamma }} = b_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\mathsf {w}}_{{\\beta },{\\gamma }}(t) \\mathrm {d}{\\mathsf {m}}(x,t)$ are called the Jacobi polynomials on the cone.","These polynomials are studied in [12], [21], [22], [23].","It was shown in [21] that these polynomials share many properties of spherical harmonics, including explicit orthogonal basis and an addition formula, which provides essential tools for an extensive study in approximation theory and computational analysis over the cone in [23].","Another remarkable property of the Jacobi polynomials on the cone is that they are eigenfunctions of a second-order linear differential operator ${\\mathcal {D}}_{\\gamma }$ when ${\\beta }= -1$ , which is an analog of the Laplace-Beltrami operator on the unit sphere.","The purpose of the present paper is to study Sobolev orthogonal polynomials on the conic surface, which are orthogonal with respect to an inner product that contains derivatives.","The first case is ${\\langle }f,g{\\rangle }_{{\\beta }, -1} = \\frac{1}{{\\omega }_d}\\int _{{\\mathbb {V}}_0^{d+1}} \\frac{\\partial }{\\partial t} f (x,t)\\frac{\\partial }{\\partial t} g(x,t) t^{{\\beta }+1} \\mathrm {d}{\\mathsf {m}}(x,t)+ \\frac{{\\lambda }}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}}f(\\xi ,1) g(\\xi ,1) \\mathrm {d}\\sigma (\\xi ),$ and we also consider ${\\langle }f,g{\\rangle }_{{\\beta }, -s}$ that involves derivatives up to $s$ order for a positive integer $s$ .","There is a reason that we consider only derivatives in the $t$ variable but not the $x$ variable; see the discussion in Section 4.","Like in the case of ${\\gamma }> -1$ , our main result provides explicit construction of orthogonal bases and a closed-form formula for the orthogonal projection operator.","The study requires an extension of the Jacobi polynomials with a parameter being a negative integer, which needs to satisfy the Sobolev orthogonality of one variable that is inherited from ${\\langle }\\cdot ,\\cdot {\\rangle }_{{\\beta },-s}$ when we restrict the inner product to polynomials depending only on the $t$ variable.","Such Sobolev orthogonal polynomials of one variable have been studied by several authors; see, for example, [1], [2], [7], [15], [16], [20] and [11].","We shall follow the approach in [20] since it is more convenient for studying orthogonal projection operators and provides a link, in particular, between the Sobolev orthogonal structure and the ordinary orthogonal structure, which is useful for studying the convergence of the Fourier orthogonal series in the Sobolev orthogonal polynomials.","In the framework of polynomial approximation theory on the ball and standard or Sobolev orthogonal polynomials, we can refer to [3], [4], [5], [8], [9], [10], [14], [18], among others.","In more than one way, our study extends the Jacobi polynomials for ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ on the cone from ${\\gamma }> -1$ to ${\\gamma }= -s$ with $s \\in {\\mathbb {N}}$ .","We will show, in particular, that the spectral operator ${\\mathcal {D}}_{-s}$ has the Sobolev orthogonal polynomials as eigenfunctions if $s = 1$ .","While the latter fails for $s > 1$ , we do have a clear understanding of what the eigenspaces of ${\\mathcal {D}}_{-s}$ are.","For orthogonal polynomials in several variables, our study is also closely related to the Sobolev orthogonal polynomials on the unit ball, which have been extensively studied (see [6] and its references therein).","In particular, the description of the eigenspaces of ${\\mathcal {D}}_{\\gamma }$ is similar to the study on the unit ball in [13].","The paper is organized as follows.","The next section is preliminary, in which we recall two essential ingredients needed for our study, the Jacobi polynomials with negative parameters and spherical harmonics.","In Section 3 we review results on ordinary orthogonal polynomials on the conic surface and discuss further properties of the orthogonal projection operators.","The Sobolev orthogonal polynomials are defined and studied in Section 4.","Finally, the eigenspaces of the operator ${\\mathcal {D}}_{\\gamma }$ are discussed in Section 5."],["Preliminary","The study for orthogonal polynomials on the conic surface follows that of spherical harmonics on the unit sphere.","The latter will also be essential for constructing an orthogonal basis on the cone.","Another ingredient is the Jacobi polynomial, which we often need an extension to negative parameters in the study of the Sobolev orthogonal polynomials."],["Jacobi polynomials with negative paramters","The Jacobi polynomials $P_n^{({\\alpha },{\\beta })}$ are given explicitly by the hypergeometric function $P_n^{({\\alpha },{\\beta })}(t) = \\frac{({\\alpha }+1)_n}{n!}","{}_2F_1 \\left( \\begin{matrix}-n, n+{\\alpha }+{\\beta }+1 \\\\ {\\alpha }+1\\end{matrix}; \\frac{1-x}{2}\\right)$ for ${\\alpha },{\\beta }> -1$ and $n = 0, 1, 2, \\ldots $ .","They are orthogonal with respect to the weight function $w_{{\\alpha },{\\beta }}(t) = (1-t)^{\\alpha }(1+t)^{\\beta }$ on $[-1,1]$ with ${\\alpha }, {\\beta }> -1$ , and they satisfy $\\frac{c_{{\\alpha },{\\beta }}}{2^{{\\alpha }+{\\beta }+1}}\\int _{-1}^1 P_n^{({\\alpha },{\\beta })}(t) P_n^{({\\alpha },{\\beta })}(t) w_{{\\alpha },{\\beta }}(t) \\mathrm {d}t = h_n^{({\\alpha },{\\beta })} \\delta _{n,m},$ where $c_{{\\alpha },{\\beta }}$ is the constant so that $h_0^{({\\alpha },{\\beta })} =1$ , $c_{{\\alpha },{\\beta }} = \\frac{\\Gamma ({\\alpha }+{\\beta }+2)}{\\Gamma ({\\alpha }+1)\\Gamma ({\\beta }+1)} \\quad \\hbox{and}\\quad h_n^{({\\alpha },{\\beta })} = \\frac{({\\alpha }+1)_n({\\beta }+1)_n ({\\alpha }+{\\beta }+n+1)}{n!","({\\alpha }+{\\beta }+2)_n ({\\alpha }+{\\beta }+2n+1)}.$ For studying the Sobolev orthogonal polynomials, we often need the parameters ${\\alpha }$ or ${\\beta }$ to be negative integers.","Such polynomials are discussed already in [17] but they are no longer orthogonal with respect to $w_{{\\alpha },{\\beta }}$ , since the weight function $(1-t)^a(1+t)^{b}$ is no longer integrable if $a$ or $b \\le -1$ .","Moreover, $P_n^{({\\alpha },{\\beta })}$ has a degree reduction if $n+{\\alpha }+{\\beta }$ is a negative integer between 1 to $n$ , which causes problems for studying the Sobolev orthogonal polynomials, especially in several variables, since such polynomials are needed for all $n \\in {\\mathbb {N}}_0$ .","What we need in this paper are the polynomials $P_n^{({\\alpha },-s)}$ with ${\\alpha }> -1$ and $s \\in {\\mathbb {N}}$ .","These polynomials are well defined if $n \\ge s$ and satisfy [17] $P_n^{({\\alpha },-s)} (t) = \\frac{(-{\\alpha }-n)_s}{2^s (-n)_s} (1+t)^s P_{n-s}^{({\\alpha },s)}(t), \\quad n \\ge s,$ which follows from [17] by using $P_n^{({\\alpha },{\\beta })}(-t) = (-1)^n P_n^{({\\beta },{\\alpha })}(t)$ .","The definition of $P_n^{({\\alpha },-s)}$ for $n < s$ could be problematic because of the degree reduction.","Such polynomials have been studied in the setting of the Sobolev orthogonal polynomials; for example, the Sobolev orthogonality defined via the inner product $ [f,g]_{{\\alpha },{\\beta }}^{-s} : = \\int _{-1}^1 f^{(s)}(t) g^{(s)}(t) w_{{\\alpha },{\\beta }}(t) \\mathrm {d}t + \\sum _{k=0}^{s-1} \\mu _k f^{(k)}(1)g^{(k)}(1),$ where $\\mu _k$ are fixed positive constants and ${\\alpha },{\\beta }> -1$ .","The study of such polynomials and their orthogonality has appeared in several papers; see, for example, [1], [7], [2], [15], [16], [19], [20] and [11].","There are several ways to define a complete set of orthogonal polynomials for the inner product (REF ).","We shall follow the approach given in [20], see also [15], [16], which is more suitable for studying the Fourier orthogonal series.","We now recall the necessary result from [20].","For convenience, we first define a renormalization of the Jacobi polynomials, $ \\widehat{P}^{(\\alpha ,\\beta )}_{n}(t) = A_n^{({\\alpha },{\\beta })} {P}^{(\\alpha ,\\beta )}_{n}(t) \\quad \\hbox{with}\\quad A_n^{({\\alpha },{\\beta })} = \\frac{2^n}{(n+{\\alpha }+\\beta +1)_{n}}$ for ${\\alpha },{\\beta }> -1$ .","This normalization has the advantage that it satisfies, by [17], $ \\frac{d}{dt} \\widehat{P}_n^{({\\alpha },{\\beta })}(t) = \\widehat{P}_{n-1}^{({\\alpha }+1,{\\beta }+1)}(t).$ Now, for ${\\alpha },{\\beta }> -1$ , $s \\in {\\mathbb {N}}$ and $n = 0,1,2,\\ldots ,$ we define a new sequence of polynomials $ \\begin{split}J_n^{({\\alpha }-s,{\\beta }-s)}(t) := {\\left\\lbrace \\begin{array}{ll} \\dfrac{(t+1)^n}{n!","}, & 0 \\le n \\le s-1, \\vspace{3.61371pt} \\\\\\displaystyle {\\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}","\\widehat{P}_{n-s}^{({\\alpha },{\\beta })}(u) \\mathrm {d}u}, & n \\ge s.\\end{array}\\right.","}\\end{split}$ It is easy to see that $J_n^{({\\alpha }-s,{\\beta }-s)}$ is a polynomial of degree $n$ and it satisfies $\\partial ^s J_n^{({\\alpha }-s,{\\beta }-s)}(t) &\\ = \\widehat{P}_{n-s}^{({\\alpha },{\\beta })}(t), \\qquad n \\ge s; \\\\\\partial ^k J_n^{({\\alpha }-s,{\\beta }-s)}(-1) &\\ = {\\left\\lbrace \\begin{array}{ll} \\delta _{k,n}, & n \\le s-1, \\\\0, & n \\ge s,\\end{array}\\right.}","\\qquad 0\\le k \\le s-1, $ where $\\partial ^k$ denotes the $k$ -th derivative.","These are our polynomials that extend the definition of the Jacobi polynomials to allow negative parameters, which are also orthogonal polynomials with respect to the Sobolev inner product (REF ).","More precisely, we have the following [20]: Theorem 2.1 For ${\\alpha },{\\beta }> -1$ and $s \\in {\\mathbb {N}}$ .","The polynomial $J_n^{({\\alpha }-s,{\\beta }-s)}$ is orthogonal with respect to the inner product ${\\langle }\\cdot , \\cdot {\\rangle }_{{\\alpha },{\\beta }}^{-s}$ and its norm square is given by $\\left[J_n^{({\\alpha }-s,{\\beta }-s)},J_n^{({\\alpha }-s,{\\beta }-s)}\\right]_{{\\alpha },{\\beta }}^{-s}= {\\left\\lbrace \\begin{array}{ll}\\mu _n & 0 \\le n \\le s-1 \\\\ \\widehat{h}_{n-s}^{({\\alpha },{\\beta })} & n \\ge s \\end{array}\\right.","},$ where $\\mu _n$ comes from (REF ), and $\\widehat{h}_n^{({\\alpha },{\\beta })}$ is the norm square of $\\widehat{P}_n^{({\\alpha },{\\beta })}$ , which is given in terms of $h_n^{({\\alpha },{\\beta })}$ by $\\widehat{h}_n^{({\\alpha },{\\beta })} = \\frac{2^{{\\alpha }+{\\beta }+1}}{c_{{\\alpha },{\\beta }}} \\left[A_n^{({\\alpha },{\\beta })}\\right]^2 h_n^{({\\alpha },{\\beta })}.$ Our next proposition shows that, if ${\\alpha }> -1$ and $s\\in {\\mathbb {N}}$ , then the definition $J_n^{({\\alpha },-s)}$ in (REF ) agrees with that of (REF ) when $n\\ge s$ .","Proposition 2.2 For ${\\alpha }> -1$ and $s \\in {\\mathbb {N}}$ , $J_n^{({\\alpha },-s)} (t) = \\frac{(n-s)!}{n!}","(1+t)^s \\widehat{P}_{n-s}^{({\\alpha },s)}(t), \\qquad n \\ge s.$ In particular, for $n \\ge s$ , $J_n^{({\\alpha },-s)} (t) = \\frac{(-1)^s 2^s}{(-{\\alpha }-n)_s}A_{n-s}^{({\\alpha },s)} P_n^{({\\alpha },-s)}(t).$ We use the hypergeometric expression of the Jacobi polynomials, $P_n^{({\\alpha },{\\beta })}(t) = \\frac{({\\alpha }+1)_n}{n!}","{}_2F_1\\left(\\begin{matrix} -n \\,\\,\\, n+{\\alpha }+{\\beta }+1\\\\ {\\alpha }+1 \\end{matrix}; \\frac{1-t}{2} \\right)$ and the fact that $P_n^{({\\alpha },{\\beta })}(t) = (-1)^n P_n^{({\\beta },{\\alpha })}(-t)$ .","It follows that $& \\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}","P_n^{({\\alpha }+s,0)}(u) \\mathrm {d}u= (-1)^n \\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}","P_n^{(0, {\\alpha }+s)}(-u) \\mathrm {d}u \\\\& \\qquad \\quad = (-1)^n \\sum _{k=0}^n \\frac{(-n)_k (n+{\\alpha }+s+1)_k}{k!","k!","}\\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}","\\left(\\frac{1+u}{2} \\right)^k \\mathrm {d}u \\\\& \\qquad \\quad = \\frac{(-1)^n}{s!}","(1+t)^s \\sum _{k=0}^n \\frac{(-n)_k (n+{\\alpha }+s+1)_k}{k!","(s+1)_k} \\left(\\frac{1+t}{2} \\right)^k \\\\& \\qquad \\quad = \\frac{(-1)^n}{s!}","(1+t)^s \\frac{n!","}{(s+1)_n} P_n^{(s,{\\alpha })}(-t)= \\frac{n!}{(n+s)!}","(1+t)^sP_n^{({\\alpha },s)}(t).$ Replacing $n$ by $n-s$ and using $A_n^{({\\alpha },s)} = A_n^{({\\alpha }+s,0)}$ , the identity (REF ) then follows from (REF ).","The second identity follows from the identity (REF ).","It should be pointed out that the integral expression of $J_n^{({\\alpha },-s)}$ for $n \\ge s$ in (REF ) is more convenient for studying the Fourier orthogonal series, as shown in [20] and as we shall see in Section 4 below."],["Spherical harmonics","A homogeneous polynomial $Y$ of $d$ variables is called a solid harmonic if $\\Delta Y =0$ , where $\\Delta $ is the Laplace operator on ${\\mathbb {R}}^d$ .","We denote by ${\\mathcal {H}}_m^{d,0}$ the space of homogeneous solid harmonics of degree $m$ in $d$ variables.","Thus, if $Y \\in {\\mathcal {H}}_m^{d,0}$ , then $Y(r \\xi ) = r^m Y(\\xi )$ for $\\xi \\in {\\mathbb {S}^{d-1}}$ .","Spherical harmonics are restrictions of solid harmonics on the unit ball.","We denote the space of spherical harmonics of degree $m$ by ${\\mathcal {H}}_m^d$ .","Thus, ${\\mathcal {H}}_m^d = {\\mathcal {H}}_m^{d,0} \\vert _{\\mathbb {S}^{d-1}}$ .","It is a common practice to identify ${\\mathcal {H}}_m^{d,0}$ and ${\\mathcal {H}}_m^d$ , we distinguish them to emphasize the dependence on variables for the orthogonal polynomials on the conic surface.","It is well known that $\\dim {\\mathcal {H}}_m^d = \\binom{m+d-2}{n}+\\binom{m+d-3}{n-1},\\quad m=1,2,3,\\ldots ,$ and spherical harmonics of different degrees are orthogonal with respect to the surface measure on the unit sphere.","Throughout the paper, we denote by $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d\\rbrace $ an orthonormal basis of ${\\mathcal {H}}_m^d$ , so that $\\frac{1}{{\\omega }_d} \\int _{\\mathbb {S}^{d-1}}Y_\\ell ^n(\\xi ) Y_{\\ell ^{\\prime }}^m(\\xi ) \\mathrm {d}\\sigma (\\xi ) = \\delta _{\\ell ,\\ell ^{\\prime }} \\delta _{n,m},$ where $\\mathrm {d}\\sigma $ is the surface measure of ${\\mathbb {S}^{d-1}}$ and ${\\omega }_d$ is the surface area of ${\\mathbb {S}^{d-1}}$ .","Let $\\operatorname{proj}_n^{\\mathbb {S}^{d-1}}: L^2({\\mathbb {S}^{d-1}}) \\rightarrow {\\mathcal {H}}_n^d$ be the orthogonal projection operator from $L^2({\\mathbb {S}^{d-1}})$ onto ${\\mathcal {H}}_n^d$ .","If $\\lbrace Y_\\ell ^n: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n^{d-1} \\rbrace $ is an orthonormal basis of ${\\mathcal {H}}_n^d$ , then $\\operatorname{proj}_n^{\\mathbb {S}^{d-1}}f (x) = \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_n^d} \\hat{f}_{\\ell }^n Y_\\ell ^n, \\quad \\hat{f}_\\ell ^n = \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} f(\\xi ) Y_\\ell ^n(\\xi ) \\mathrm {d}\\sigma (\\xi ).$ For $f \\in L^2({\\mathbb {S}^{d-1}})$ , its Fourier expansion in spherical harmonics is defined by $f = \\sum _{n=0}^\\infty \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_n^d} \\hat{f}_{\\ell }^n Y_\\ell ^n =\\sum _{n=0}^\\infty \\operatorname{proj}_n^{\\mathbb {S}^{d-1}}f.$ Let $\\Delta _0$ be the Laplace-Beltrami operator, which is the restriction of the Laplacian $\\Delta $ on the unit sphere.","Under the spherical polar coordinates $x = r \\xi $ , $r > 0$ , $\\xi \\in {\\mathbb {S}^{d-1}}$ , $\\Delta = \\frac{\\mathrm {d}^2}{\\mathrm {d}r^2} + \\frac{d-1}{r} \\frac{\\mathrm {d}}{\\mathrm {d}r} + \\frac{1}{r^2} \\Delta _0.$ The spherical harmonics are the eigenfunctions of $\\Delta _0$ .","More precisely, $\\Delta _0 Y = - n (n+d-2)Y, \\qquad Y \\in {\\mathcal {H}}_n^d, \\quad n=0,1,2,\\ldots .$ The $\\Delta _0$ is a second-order differential operator on the unit sphere.","One can also consider spherical gradient $\\nabla _0$ , the first-order differential operators, on the sphere, which is defined by $\\nabla = \\frac{1}{r} \\nabla _0 + \\xi \\frac{\\mathrm {d}}{\\mathrm {d}r}, \\quad x = r \\xi , \\quad \\xi \\in {\\mathbb {S}^{d-1}}.$ The integration by parts formula holds and gives [5], $\\int _{{\\mathbb {S}^{d-1}}} \\nabla _0 f(\\xi ) \\cdot \\nabla _0 g(\\xi ) \\mathrm {d}\\sigma (\\xi ) = - \\int _{{\\mathbb {S}^{d-1}}} \\Delta _0 f(\\xi ) g(\\xi ) \\mathrm {d}\\sigma (\\xi ).$ Together with (REF ), this identity implies that if $\\lbrace Y_\\ell ^n : 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n^d\\rbrace $ is an orthogonal basis of ${\\mathcal {H}}_n^d$ , then $\\int _{{\\mathbb {S}^{d-1}}} \\nabla _0 Y_\\ell ^n(\\xi ) \\cdot \\nabla _0 Y_{\\ell ^{\\prime }}^{n^{\\prime }}(\\xi ) \\mathrm {d}\\sigma (\\xi )={\\lambda }_n \\int _{{\\mathbb {S}^{d-1}}} Y_\\ell ^n(\\xi ) \\cdot Y_{\\ell ^{\\prime }}^{n^{\\prime }}(\\xi ) \\mathrm {d}\\sigma (\\xi ) = {\\lambda }_n \\delta _{\\ell ,\\ell ^{\\prime }} \\delta _{n,n^{\\prime }},$ where ${\\lambda }_n = n(n+d-2)$ , which implies that $\\lbrace Y_\\ell ^n : 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n^d\\rbrace $ is also a family of orthogonal polynomials for the Sobolev inner product defined by, for example, ${\\langle }f,g {\\rangle }_{\\nabla } = \\sum _{k=1}^s \\mu _k\\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\nabla _0^k f(\\xi ) \\cdot \\nabla _0^k g(\\xi ) \\mathrm {d}\\sigma (\\xi ) + \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} f(\\xi ) g(\\xi ) \\mathrm {d}\\sigma (\\xi ),$ where $\\mu _k \\ge 0$ and $s$ is a fixed positive integer, and $\\nabla _0^{2m} := \\Delta _0^m \\quad \\hbox{and} \\quad \\nabla _0^{2m+1} = \\Delta _0^m \\nabla _0.$ In other words, the Sobolev orthogonal polynomials for ${\\langle }\\cdot ,\\cdot {\\rangle }_\\nabla $ on the unit sphere are trivially spherical harmonics themselves."],["Orthogonal polynomials on the conic surface","Orthogonal polynomials on the conic surface ${\\mathbb {V}}_0^{d+1}$ are studied in [21] for the inner product defined by ${\\langle }f, g {\\rangle }_{{\\beta },{\\gamma }} = b_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\mathsf {w}}_{{\\beta },{\\gamma }}(t) \\mathrm {d}{\\mathsf {m}}(x,t),$ where $\\mathrm {d}{\\mathsf {m}}$ is the Lebesgue measure on the conic surface and the weight function ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ is the Jacobi weight function on $[0,1]$ , ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t) = t^{\\beta }(1-t)^{\\gamma }, \\quad {\\beta }> -d, \\quad {\\gamma }> -1$ and $b_{{\\beta },{\\gamma }}$ is the normalization constant so that ${\\langle }1, 1 {\\rangle }_{{\\beta },g} =1$ , which is determined by $\\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) \\mathrm {d}{\\mathsf {m}}(x,t) = \\int _{0}^1 t^{d-1} \\int _{{\\mathbb {S}^{d-1}}} f(t \\xi ,t)\\mathrm {d}\\sigma (\\xi ) \\mathrm {d}t,$ where $\\mathrm {d}\\sigma $ denotes the Lebesgue measure on the unit sphere ${\\mathbb {S}^{d-1}}$ .","Then, $b_{\\beta ,\\gamma }= \\frac{1}{\\omega _d} c_{{\\beta }+d-1,{\\gamma }} \\quad \\hbox{with}\\quad c_{{\\beta },{\\gamma }} = \\frac{\\Gamma (\\beta +\\gamma +2)}{\\Gamma (\\beta +1)\\Gamma (\\gamma +1)} \\quad \\hbox{and}\\quad \\omega _{d}=\\frac{2\\pi ^{d/2}}{\\Gamma (d/2)},$ where $\\omega _{d}$ is the surface area of ${\\mathbb {S}^{d-1}}$ .","The inner product is well defined for the space ${\\mathbb {R}}[x,t] / {\\langle }t^2 - \\Vert x\\Vert ^2{\\rangle }$ of polynomials modulus the ideal generated by $t^2 -\\Vert x\\Vert ^2$ .","For $n \\in {\\mathbb {N}}_0$ , let ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ denote the space of orthogonal polynomials of degree $n$ .","Since ${\\mathbb {V}}_0^{d+1}$ is a quadratic surface, so the dimension of the space is the same as that of ${\\mathcal {H}}_n^{d+1}$ .","Thus, $\\dim {\\mathcal {V}}_0({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }}) =1$ and $\\dim {\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }}) = \\binom{n+d-1}{n}+\\binom{n+d-2}{n-1},\\quad n=1,2,3,\\ldots .$ An orthogonal basis for ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ is given in [21] in terms of the Jacobi polynomials and spherical harmonics.","Let $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d\\rbrace $ be an orthonormal basis of ${\\mathcal {H}}_m^d$ .","Define the polynomials, called the Jacobi polynomials on the conic surface, by $ {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })} (x,t) := P_{n-m}^{(2m + {\\beta }+ d-1,{\\gamma })} (1-2t) Y_\\ell ^m (x),$ where, for $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , $Y_\\ell ^m(x)$ is a solid harmonic in ${\\mathcal {H}}_m^{d,0}$ and $Y_\\ell ^m (t\\xi ) = t^m Y_\\ell ^m(\\xi )$ .","Then $\\lbrace {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })}: 0 \\le m \\le n, \\,\\, 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m({\\mathbb {S}^{d-1}})\\rbrace $ is an orthogonal basis of ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ , which satisfies $b_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })} (x,t) {\\mathsf {S}}_{m^{\\prime }, \\ell ^{\\prime }}^{n^{\\prime },({\\beta },{\\gamma })} (x,t){\\mathsf {w}}_{{\\beta },{\\gamma }}(t) \\mathrm {d}{\\mathsf {m}}(x,t) = h_{n,m}^{{\\beta },{\\gamma }}\\delta _{n,n^{\\prime }}\\delta _{m,m^{\\prime }}\\delta _{\\ell ,\\ell ^{\\prime }},$ where $h_{n,m}^{{\\beta },{\\gamma }}$ is the square of the $L^2({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }})$ norm of ${\\mathsf {S}}_{m, \\ell }^n$ and $h_{n,m}^{{\\beta },{\\gamma }} = \\frac{({\\beta }+d)_{2m}}{({\\beta }+{\\gamma }+d+1)_{2m}} h_{n-m}^{(2m+{\\beta }+d-1,{\\gamma })}$ with $h_{n-m}^{(2m+{\\beta }+d-1,{\\gamma })}$ defined in (REF ).","Of particular interest is the case ${\\beta }= -1$ , for which the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ is an eigenspace of a second order linear differential operator.","Parametrizing the space ${\\mathbb {V}}_0^{d+1}$ by $(x,t) = (t\\xi , t)$ and let $\\Delta _0^{(\\xi )}$ be the Laplace-Beltrami operator on the unit sphere in the $\\xi $ variable, which is the restriction of the Laplace operator $\\Delta $ on ${\\mathbb {S}^{d-1}}$ .","For ${\\gamma }> -1$ , define $ {\\mathcal {D}}_{\\gamma }= t(1-t)\\frac{\\mathrm {d}^2}{\\mathrm {d}t^2} + \\big ( d-1 - (d+{\\gamma })t \\big ) \\frac{\\mathrm {d}}{\\mathrm {d}t}+ t^{-1} \\Delta _0^{(\\xi )}.$ Theorem 3.1 Let $d\\ge 2$ and ${\\gamma }> -1$ .","The orthogonal polynomials in ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ are eigenfunctions of ${\\mathcal {D}}_{{\\gamma }}$ ; more precisely, ${\\mathcal {D}}_{{\\gamma }} u = -n (n+{\\gamma }+d-1) u, \\qquad \\forall u \\in {\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }}).$ The differential operator ${\\mathcal {D}}_{{\\gamma }}$ plays an important role in the study of the Fourier orthogonal series and the best approximation by polynomials on the conic surface [23].","For $f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , the Fourier orthogonal series of $f$ is defined by $f = \\sum _{n=0}^\\infty \\operatorname{proj}_n^{{\\beta },{\\gamma }} f,$ where $\\operatorname{proj}_n^{{\\beta },{\\gamma }}: L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }}) \\rightarrow {\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ is the orthogonal projection operator, which satisfies, using the orthogonal basis given above, $\\operatorname{proj}_n^{{\\beta },{\\gamma }} f = \\sum _{m=0}^n \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_m({\\mathbb {S}^{d-1}})}\\hat{f}_{m,\\ell }^{n, ({\\beta },{\\gamma })} {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },{\\gamma })},\\quad \\hbox{where}\\quad \\hat{f}_{m,\\ell }^{n, ({\\beta },{\\gamma })} := \\frac{{\\langle }f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })}{\\rangle }_{{\\beta },{\\gamma }}}{h_{m,n}^{{\\beta },{\\gamma }}}.$ In the rest of this section, we consider a property on the derivative of the reproducing kernel.","Parametrizing the function $f: {\\mathbb {V}}_0^{d+1} \\rightarrow {\\mathbb {R}}$ by $(x,t) = (t\\xi , t)$ with $\\xi \\in {\\mathbb {S}^{d-1}}$ , it follows that $ \\frac{\\mathrm {d}}{\\mathrm {d}t} f(x,t) = \\frac{\\mathrm {d}}{\\mathrm {d}t} f(t\\xi ,t)= \\xi \\cdot \\nabla _x f(t\\xi ,t) + \\frac{\\partial }{\\partial t} f(t\\xi ,t),$ where $\\frac{\\partial }{\\partial t} = \\partial _{d+1}$ denotes the partial derivative with respect to the $d+1$ variable of $f$ , which should not be confused with the $\\frac{\\mathrm {d}}{\\mathrm {d}t}$ on the left-hand side.","Throughout the rest of this paper, we shall adopt the notation $\\partial _t = \\partial _{d+1} = \\frac{\\partial }{\\partial t}$ for functions $f(x,t)$ .","We consider the action of $\\partial _t$ on the projection operator.","Theorem 3.2 For ${\\beta }> -d$ and ${\\gamma }> -1$ , let $f$ be a differentiable function such that $f\\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ and $\\partial _t f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+1,{\\gamma }+1})$ .","Then, for $n =0,1,2,\\ldots $ , $\\frac{\\partial }{\\partial t} \\operatorname{proj}_n^{{\\beta },{\\gamma }} f(x,t) = \\operatorname{proj}_{n-1}^{{\\beta }+1,{\\gamma }+1} \\partial _t f (x,t).$ For $m = n$ , the polynomial ${\\mathsf {S}}_{n,\\ell }^{n,({\\beta },{\\gamma })}(x,t) = Y_\\ell ^n(x)$ , so that $\\frac{\\partial }{\\partial t} {\\mathsf {S}}_{n,\\ell }^{n,({\\beta },{\\gamma })}(x,t) =0$ .","For $0 \\le m \\le n-1$ , using the well-known formula for the derivative of the Jacobi polynomials, we obtain $\\frac{\\partial }{\\partial t} {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },{\\gamma })}(x,t) \\, & = \\frac{\\partial }{\\partial t} P_{n-m}^{(2m + {\\beta }+ d-1,{\\gamma })} (1-2t)Y_\\ell ^m (x) \\\\& =\\tau _{n,m} P_{n-m-1}^{(2m + {\\beta }+ d,{\\gamma }+1)} (1-2t) Y_\\ell ^m (x)= \\tau _{n,m} {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}(x,t), $ where $\\tau _{n,m} = - (n+m + {\\beta }+ {\\gamma }+ d)$ .","As a consequence of these identities, we see that $\\partial _t \\operatorname{proj}_n^{{\\beta },{\\gamma }} f \\in {\\mathcal {V}}_{n-1}^d({\\mathsf {w}}_{{\\beta }+1,{\\gamma }+1})$ .","Consequently, it follows that $& \\left\\langle \\partial _t f, {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}\\right\\rangle _{{\\beta }+1,{\\gamma }+1} =\\left\\langle \\partial _t \\operatorname{proj}_n^{{\\beta },{\\gamma }} f, {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)} \\right\\rangle _{{\\beta }+1,{\\gamma }+1} \\\\& \\qquad \\qquad = \\sum _{k=0}^{n-1} \\sum _{\\nu } \\hat{f}_{k,\\nu }^{n,({\\beta },{\\gamma })} \\tau _{n,k} \\left\\langle {\\mathsf {S}}_{k,\\nu }^{n-1,({\\beta }+1,{\\gamma }+1)},{\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)} \\right\\rangle _{{\\beta }+1,{\\gamma }+1} \\\\& \\qquad \\qquad = \\tau _{n,m} \\hat{f}_{m,\\ell }^{n,({\\beta },{\\gamma })} h_{n-1,m}^{{\\beta }+1,{\\gamma }+1},$ which implies immediately that $\\widehat{\\partial _t f}_{m,\\ell }^{n-1, ({\\beta }+1,{\\gamma }+1)} = \\tau _{n,m} \\hat{f}_{m,\\ell }^{n,({\\beta },{\\gamma })}, \\quad 0 \\le m \\le n-1.$ Consequently, we obtain $\\frac{\\partial }{\\partial t} \\operatorname{proj}_n^{{\\beta },{\\gamma }} f (x,t)\\, & = \\sum _{m=0}^{n-1} \\sum _\\ell \\hat{f}_{m,\\ell }^{n,({\\beta },{\\gamma })}\\tau _{n,m} {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}(x,t) \\\\& = \\sum _{m=0}^{n-1} \\sum _\\ell \\widehat{\\partial _t f}_{m,\\ell }^{n-1, ({\\beta }+1,{\\gamma }+1)} {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}(x,t) \\\\& = \\operatorname{proj}_{n-1}^{{\\beta }+1,{\\gamma }+1} \\partial _t f (x,t),$ where in the first step we used again $\\partial _t {\\mathsf {S}}_{n,\\ell }^n(x,t) = 0$ .","For $1 \\le p \\le \\infty $ , let $\\Vert f\\Vert _{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}}$ denote the norm of the space $L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , and we adopt the convention that the space is $C({\\mathbb {V}}_0^{d+1})$ with the norm taken as the uniform norm when $p = \\infty $ .","Let $\\Pi _n({\\mathbb {V}}_0^{d+1})$ denote the space of polynomials of degree $n$ restricted on the ${\\mathbb {V}}_0^{d+1}$ .","For $f \\in L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , the quantity $E_n(f)_{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}} := \\inf _{P \\in \\Pi _n({\\mathbb {V}}_0^d)} \\Vert f - P \\Vert _{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}}$ is the error of the best approximation by polynomials of degree at most $n$ in the norm of $L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ .","We call $\\eta \\in C^\\infty $ an admissible cut-off function if it is supported on $[0,2]$ and satisfies $\\eta (t) = 1$ if $0 \\le t \\le 1$ .","Let $\\eta $ be such a function; we define $ Q_{n,\\eta }^{({\\beta },{\\gamma })} f = \\sum _{k=0}^{2n} \\eta \\left(\\frac{k}{n} \\right) \\operatorname{proj}_k^{{\\beta },{\\gamma }} f.$ Then it is known [23] that $Q_{n,\\eta }^{({\\beta },{\\gamma })}$ is a bounded operator in $L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ and it is a polynomial of near best approximation in the sense that $ \\left\\Vert f - Q_{n,\\eta }^{({\\beta },{\\gamma })} f\\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}} \\le c E_n (f)_{p,{\\mathsf {w}}_{{\\beta },{\\gamma }}},$ where $c$ is a constant that depends only on $\\eta $ , $p$ , ${\\beta }$ and ${\\gamma }$ .","In particular, we obtain the following as a corollary of Theorem REF .","Corollary 3.3 Let ${\\beta }> -d$ and ${\\gamma }> -1$ .","Let $r$ be a postive integer and let $f \\in C^r({\\mathbb {V}}_0^{d+1})$ such that $\\partial _t^k f \\in L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+k,{\\gamma }+k})$ for $0 \\le k \\le r$ .","Then, for $1 \\le p \\le \\infty $ , $\\left\\Vert \\partial _t^k f - \\partial _t^k Q_{n,\\eta }^{({\\beta },{\\gamma })}f\\right\\Vert _{p, {\\mathsf {w}}_{{\\beta }+k,{\\gamma }+k} } \\le c_kE_{n-k} (\\partial _t^k f)_{p, {\\mathsf {w}}_{{\\beta }+k,{\\gamma }+k} }, \\qquad 0 \\le k \\le r.$ This follows immediately from $\\partial _t^k Q_{n,\\eta }^{({\\beta },{\\gamma })} f & = \\sum _{j=0}^{2n} \\eta \\left(\\frac{j}{n} \\right) \\partial _t^j\\operatorname{proj}_j^{{\\beta },{\\gamma }} f= \\sum _{j=k}^{2n} \\eta \\left(\\frac{j}{n} \\right) \\operatorname{proj}_{j-k}^{{\\beta }+k,{\\gamma }+k} f \\\\& = \\sum _{j=0}^{2n-k} \\eta \\left(\\frac{j+k}{n} \\right) \\operatorname{proj}_{j}^{{\\beta }+k,{\\gamma }+k} \\partial _t^k f= {\\mathsf {Q}}_{n,\\eta }^{({\\beta }+k,{\\gamma }+k)} \\partial _t^k f,$ where we define $ {\\mathsf {Q}}_{ n,\\eta }^{({\\beta },{\\gamma })} g = \\sum _{j=0}^{2n-k} \\eta \\left(\\frac{j+k}{n} \\right) \\operatorname{proj}_{j}^{{\\beta },{\\gamma }} g, \\quad 0 \\le k < 2n.$ For fixed $k \\le r$ independent of $n$ , the function $\\eta \\left(\\frac{j+k}{n} \\right)$ plays essentially the same role as $\\eta \\left(\\frac{j}{n} \\right)$ , so that (REF ) holds with ${\\mathsf {Q}}_{n,\\eta }^{({\\beta },{\\gamma })}\\partial _t^k f$ in place of $Q_{n,\\eta }^{({\\beta },{\\gamma })}f$ , form which the proof follows readily."],["Sobolev orthogonal polynomials on the conic surface","As seen in the differential operator (REF ), the derivatives on the conic surface ${\\mathbb {V}}_0^{d+1}$ are partial derivatives with respect to $\\xi $ and $t$ with $x = t \\xi $ for $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ .","For the Sobolev inner product, the derivatives in $\\xi \\in {\\mathbb {S}^{d-1}}$ will act on the spherical harmonics and lead to the same orthogonal basis as discussed at the end of Subsection 2.2.","Hence, we consider only the derivatives with respect to $\\partial _t$ ; see Remark REF below.","For $s \\in {\\mathbb {N}}$ , let us define $W_p^s({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})= \\left\\lbrace f\\in C({\\mathbb {V}}_0^{d+1}): \\partial _t^s f\\in L^p({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta }+s,0})\\right\\rbrace ,$ where $1 \\le p \\le \\infty $ and the space is taken as the $C^s({\\mathbb {V}}_0^{d+1})$ if $p = \\infty $ .","Let $s$ be a positive integer and ${\\beta }> -s-d$ .","Let ${\\lambda }_1,\\ldots ,{\\lambda }_{r-1}$ be fixed positive numbers.","We consider the Sobolev inner product defined by $ {\\langle }f,g{\\rangle }_{{\\beta }, -s} =\\,& \\frac{1}{{\\omega }_d}\\int _{{\\mathbb {V}}_0^{d+1}} \\frac{\\partial ^s}{\\partial t^s} f (x,t)\\frac{\\partial ^s}{\\partial t^s} g(x,t) t^{{\\beta }+s} \\mathrm {d}{\\mathsf {m}}(x,t)\\\\\\ & + \\sum _{k=0}^{s-1} \\frac{{\\lambda }_k}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}}\\frac{\\partial ^k f}{\\partial t^k} (\\xi ,1) \\frac{\\partial ^k g}{\\partial t^k}(\\xi ,1) \\mathrm {d}\\sigma (\\xi ), $ which is evidently an inner product on the space $W_2^s({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ .","We denote by ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ the space of orthogonal polynomials of degree $n$ with respect to this inner product.","Our first task is to find an orthogonal basis for this space.","Recall the modified Jacobi polynomial $J_n^{({\\alpha },-s)}$ defined in (REF ).","Let $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d\\rbrace $ be an orthonormal basis of ${\\mathcal {H}}_m^d$ .","For $1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d$ , $0\\le m \\le n$ , we define $ {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} (x,t) := J_{n-m}^{(2m+{\\beta }+d-1,-s)}(1-2 t) Y_\\ell ^m(x).$ Theorem 4.1 Let $s \\in {\\mathbb {N}}$ and ${\\beta }> -d-s$ .","The polynomials ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}$ , $1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d$ and $0\\le m \\le n$ consist of a basis of ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ .","Moreover, for all $\\ell $ , ${\\mathsf {h}}_{m,n}^{({\\beta },-s)} =\\langle {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}, {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} \\rangle _{{\\beta },-s}$ satisfies ${\\mathsf {h}}_{m,n}^{({\\beta },-s)} = {\\left\\lbrace \\begin{array}{ll}2^{2n-2m} {\\lambda }_{n-m} & 0 \\le n -m \\le s-1, \\\\2^{s - {\\beta }-2m-d}\\, \\widehat{h}_{n-m-s}^{(s+2m+{\\beta }+d-1,0)} & n-m \\ge s.","\\end{array}\\right.","}$ Let $q_{n-m}(t) = J_{n-m}^{(2m+{\\beta }+d-1, -s)}(t)$ .","Changing variable $x = t \\xi $ and using the homogeneity of $Y_\\ell ^m$ , we obtain $\\left\\langle {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}, {\\mathsf {S}}_{m^{\\prime },\\ell ^{\\prime }}^{n^{\\prime },({\\beta },-s)} \\right\\rangle _{{\\beta },-s}= \\, & 2^{ 2s} \\int _0^1 t^{d-1} q_{n-m}^{(s)} (1-2t) q_{n^{\\prime }-m^{\\prime }}^{(s)}(1-2t) t^{{\\beta }+s+m+m^{\\prime }} \\mathrm {d}t \\\\&\\qquad \\times \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}} Y_\\ell ^m (\\xi )Y_{\\ell ^{\\prime }}^{m^{\\prime }} (\\xi ) \\mathrm {d}\\sigma (\\xi ) \\\\& + \\sum _{k=0}^{s-1} {\\lambda }_k 2^{2k} q_{n-m}^{(k)}(-1) q_{n^{\\prime }-m^{\\prime }}^{(k)}(-1) \\\\&\\qquad \\times \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}} Y_\\ell ^m (\\xi )Y_{\\ell ^{\\prime }}^{m^{\\prime }} (\\xi ) \\mathrm {d}\\sigma (\\xi ).$ Using the orthogonality of $Y_\\ell ^m$ and changing variable $t \\mapsto (1-t)/2$ in the first integral, we see that the expression containing $q_{n-m}$ can be written in terms of the Sobolev inner product $[\\cdot ,\\cdot ]_{{\\alpha },{\\beta }}^{-s}$ defined in (REF ).","More precisely, we obtain $\\left\\langle {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}, {\\mathsf {S}}_{m^{\\prime },\\ell ^{\\prime }}^{n^{\\prime },({\\beta },-s)} \\right\\rangle _{{\\beta },-s} =2^{s - {\\alpha }-1} [q_{n-m}, q_{n^{\\prime }-m}]_{{\\alpha },0}^{-s} \\delta _{m,m^{\\prime }} \\delta _{\\ell ,\\ell ^{\\prime }},$ where ${\\alpha }= s+{\\beta }+2m+d-1$ and the inner product $[\\cdot , \\cdot ]_{{\\alpha },0}^{-s}$ is defined as in (REF ) but with $\\mu _k = 2^{2k} {\\lambda }_k/2^{s-{\\alpha }-1}$ .","As shown in Theorem REF , the polynomials $J_n^{({\\alpha }-s,-s)}$ are orthogonal with respect to this inner product, regardless of the values of $\\mu _k$ .","This proves the orthogonality of ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ .","Moreover, the norm ${\\mathsf {h}}_{m,n}^{({\\beta },-s)}$ follows from the norm given in Theorem REF .","Remark 4.1 We could also include an additional term in the right-hand side of ${\\langle }\\cdot ,\\cdot {\\rangle }_{{\\beta },-s}$ defined by $\\int _{{\\mathbb {V}}_0^{d+1}} \\nabla _0^s f(x,t) \\cdot \\nabla _0^s g(x,t) \\mathrm {d}{\\mathsf {m}}(x,t),$ where $\\nabla _0$ acts on $\\xi $ .","By the discussion for the inner product (REF ) on the unit sphere, the inner product with this additional term has the same orthogonal basis.","Recall that the constant $A_n^{({\\alpha },{\\beta })}$ is defined in (REF ).","Corollary 4.2 For $0 \\le m \\le n-s$ , $ \\frac{\\partial ^s}{\\partial t^s} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}(x,t) = (-2)^s A_{n-s-m}^{(s+2m+{\\beta }+d-1,0)} S_{m,\\ell }^{n-s, ({\\beta }+s,0)}(x,t),$ and it is equal to zero if $m > n-s$ .","Moreover, for $1 \\le k \\le s-1$ and $\\xi \\in {\\mathbb {S}^{d-1}}$ , $ \\frac{\\partial ^k}{\\partial t^k} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}(\\xi ,1) = {\\left\\lbrace \\begin{array}{ll}(-2)^k Y_\\ell ^m(\\xi ) \\delta _{k,n-m}, & m > n-s \\\\0, & m \\le n-s.\\end{array}\\right.","}$ If $m \\le n-s$ , by our convention of the derivative $\\partial _t$ over the conic surface and the identity (REF ), $\\frac{\\partial ^s}{\\partial t^s} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}(x,t) \\,& =\\frac{\\partial ^s}{\\partial t^s}\\left[J_{n-m}^{(2m+{\\beta }+d-1,-s)}(1-2t)\\right]Y_\\ell ^m(x) \\\\& = (-2)^{s} \\widehat{P}_{n-m-s}^{(s+2m+{\\beta }+d-1,0)}(1-2t)Y_\\ell ^m(x),$ from which (REF ) follows from (REF ) and the definition of ${\\mathsf {S}}_{\\ell ,m}^{n,({\\beta }+s,0)}$ .","Moreover, since $n-m \\ge s$ , for $1 \\le k \\le s-1$ it follows immediately by () that (REF ) holds.","If $m > n-s$ , then the derivative in the identity (REF ) is evidently zero since the Jacobi polynomial in ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}$ is of degree $n-m -d-s$ , let $f$ be a differentiable function such that $\\partial _t^s f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ .","Then, for $n =0,1,2,\\ldots $ and $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , $\\operatorname{proj}_n^{{\\beta },-s} f(x,t) = \\,& \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!}","\\operatorname{proj}_{n-m}^{\\mathbb {S}^{d-1}}\\left[\\partial _t^m f(\\cdot ,1)\\right](\\xi ) \\\\& + (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}","\\operatorname{proj}_{n-s}^{{\\beta }+s,0} \\left(\\partial _t^s f\\right) (x, v) \\mathrm {d}v. $ Let ${\\alpha }= s+{\\beta }+d-1$ .","For $0 \\le m \\le n-s$ , it follows from (REF ) that $\\left\\langle f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}\\right\\rangle _{{\\beta },-s} \\,& =(-2)^s A_{n-m}^{(2m+{\\alpha },0)}\\frac{1}{{\\omega }_d} \\int _{{\\mathbb {V}}_0^{d+1}} \\frac{\\partial ^s}{\\partial t^s} f(x,t) {\\mathsf {S}}_{m,\\ell }^{n-s,({\\beta }+s,0)}(x,t) t^{{\\beta }+s} \\mathrm {d}{\\mathsf {m}}(x,t)\\\\& = (-2)^s A_{n-m}^{(2m+{\\alpha },0)} \\frac{1}{c_{{\\alpha },0}}\\left\\langle \\partial _t^s f, {\\mathsf {S}}_{m, \\ell }^{n-s,({\\beta }+s,0)}\\right\\rangle _{{\\beta }+s,0}$ and the norm of ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ satisfies ${\\mathsf {h}}_{m,n}^{({\\beta },-s)} = 2^{s - {\\beta }-2m-d} \\widehat{h}_{n-m-s}^{({\\alpha }+2m,0)} \\, & =2^{2s} \\frac{1}{c_{2m+{\\alpha },0}} \\left[A_{n-m-s}^{(2m+{\\alpha },0)}\\right]^2 h_{n-s-m}^{(2m+{\\alpha },0)} \\\\& = \\frac{1}{c_{{\\alpha },0}} 2^{2s} \\left[A_{n-m-s}^{(2m+{\\alpha },0)}\\right]^2 \\frac{({\\alpha }+1)_{2m}}{({\\alpha }+2)_{2m}}h_{n-s-m}^{(2m+{\\alpha },0)}\\\\& = \\frac{1}{c_{{\\alpha },0}} 2^{2s} \\left[A_{n-m-s}^{(2m+{\\alpha },0)}\\right]^2 h_{n-s, m}^{{\\beta }+s,0}.$ Consequently, it follows that $\\hat{f}_{m,\\ell }^{n, ({\\beta },-s)} = \\frac{c_{2m+{\\alpha },0}}{c_{{\\beta }+s,0}} \\frac{\\left\\langle \\partial _t^s f, {\\mathsf {S}}_{m, \\ell }^{n-s,({\\beta }+s,0)}\\right\\rangle _{{\\beta }+s,0}}{ (-2)^{s} A_{n-m-s}^{(2m+{\\alpha },0)} h_{n-s, m}^{{\\beta }+s,0}}= \\frac{ \\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)}}{(-2)^{s} A_{n-m-s}^{(2m+{\\alpha },0)}}.$ Now, for $n-s < m \\le n$ , it follows by Corollary REF that $\\left\\langle f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}\\right\\rangle _{{\\beta },-s}\\,& = \\sum _{k=0}^{s-1} \\frac{{\\lambda }_k}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\partial _t^k f(\\xi ,1) \\partial _t^k {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}(\\xi ,1)\\mathrm {d}\\sigma (\\xi ) \\\\& = (-2)^{n-m} \\frac{{\\lambda }_{n-m}}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\partial _t^{n-m} f(\\xi ,1) Y_\\ell ^{m} (\\xi )\\mathrm {d}\\sigma (\\xi ),$ so that, using the norm of ${\\mathsf {h}}_{n,m}^{({\\beta },-s)}$ in Theorem REF , $\\hat{f}_{m,\\ell }^{n, ({\\beta },-s)} = \\frac{1}{(-2)^{n-m}} \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\partial _t^{n-m} f(\\xi ,1) Y_\\ell ^{m} (\\xi )\\mathrm {d}\\sigma (\\xi ),\\quad 0\\le n-m \\le s-1.$ Consequently, by (REF ) and using ${\\alpha }= s+{\\beta }+d-1$ again, it follows that $& \\operatorname{proj}_n^{{\\beta },-s} f(x,t) \\, = \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!","}\\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_{n-m}^d} \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}}\\partial _t^{m} f(\\eta ,1) Y_\\ell ^{n-m} (\\eta )\\mathrm {d}\\sigma (\\eta )Y_\\ell ^{n-m} (\\xi ) \\\\& \\qquad \\quad + \\sum _{m = 0}^{n-s} \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_{m}^d}\\frac{ \\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)}}{(-2)^{s} A_{n-m-s}^{(2m+{\\alpha },0)}}\\int _{-1}^{1-2t} \\frac{(1-2t -u)^{s-1}}{(s-1)!}","\\widehat{P}_{n-s}^{({\\alpha }+2m,0)}(u) \\mathrm {d}u Y_\\ell ^m(x).$ The inner sum of the first term in the right-hand side is exactly $\\operatorname{proj}_{n-m}^{\\mathbb {S}^{d-1}}[\\partial _t^m f(\\cdot ,1)](\\xi )$ for the function $\\xi \\mapsto \\partial _t^m f(\\xi ,1)$ on the unit sphere.","Changing variable $u = 1-2v$ and using (REF ) in the integral, the second term on the right-hand side becomes, $\\sum _{m = 0}^{n-s} \\sum _{\\ell } & \\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)} (-1)^s\\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}","P_{n-s}^{({\\alpha }+2m,0)}(1-2v) \\mathrm {d}v Y_\\ell ^m(x) \\\\& = (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}","\\sum _{m = 0}^{n-s} \\sum _{\\ell }\\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)} {\\mathsf {S}}_{m,\\ell }^{n-s, ({\\beta }+s,0)} (x,v) \\mathrm {d}v\\\\& = (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}","\\operatorname{proj}_{n-s}^{{\\beta }+s,0}\\partial _t^s f (x, v) \\mathrm {d}v.$ Putting the two terms together completes the proof.","Corollary 4.5 Let $s$ be a positive integer.","For ${\\beta }> -d-s$ , let $f$ be a differentiable function such that $\\partial _t^s f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ .","Then, for $n \\ge s$ , $\\frac{\\partial ^s}{\\partial t^s} \\operatorname{proj}_n^{{\\beta },-s} f(x,t) = \\operatorname{proj}_{n-s}^{{\\beta }+s,0}\\left(\\partial _t^s f\\right) (x,t).$ Taking the derivative $\\partial _t^s$ of (REF ), we see that the first term in the right-hand side is zero, while the second term gives the stated identity.","Another consequence of Theorem REF is an expression for the error of approximation.","Let $\\eta \\in C^\\infty ({\\mathbb {R}}_+)$ be an admissible cut-off function.","We denote by ${\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}$ the near best approximation operator defined in (REF ) but with ${\\gamma }= -s$ .","Furthermore, let ${\\mathsf {Q}}_{n-m}^{\\mathbb {S}^{d-1}}f (\\xi )= \\sum _{k=0}^{2(n-m)} \\eta \\left(\\frac{k}{n}\\right) \\operatorname{proj}_k^{\\mathbb {S}^{d-1}}f(\\xi ), \\quad 0 \\le m \\le n,$ be a near best approximation operator of degree $n-m$ on the unit sphere.","For $f\\in L^p({\\mathbb {S}^{d-1}})$ , let $E_n^{\\mathbb {S}^{d-1}}(f)_p = \\inf _{Y \\in \\Pi _n({\\mathbb {S}^{d-1}})} \\Vert f - Y\\Vert _{L^p({\\mathbb {S}^{d-1}})}, \\qquad 1 \\le p \\le \\infty $ be the error of best approximation by polynomials on the unit sphere, where the norm is the uniform norm for $p = \\infty $ and $\\Pi _n({\\mathbb {S}^{d-1}})$ denote the space of polynomial of degree at most $n$ restricted on the unit sphere ${\\mathbb {S}^{d-1}}$ .","Then (cf.","[5]) $\\left\\Vert f - {\\mathsf {Q}}_{n}^{\\mathbb {S}^{d-1}}f \\right\\Vert _p \\le c E_n^{{\\mathbb {S}^{d-1}}}(f)_p, \\qquad 1 \\le p \\le \\infty .$ Theorem 4.6 For $f \\in C^s({\\mathbb {V}}_0^{d+1})$ , $f(x,t) - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f(x,t) \\,& = \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!}","\\left[ \\partial _t^m [f(\\cdot ,1)] (\\xi )- {\\mathsf {Q}}_{n-m,\\eta }^{{\\mathbb {S}^{d-1}}} \\left[\\partial _t^m f(\\cdot ,1)\\right](\\xi ) \\right] \\\\& + (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}","\\left[ \\partial _t^s f (x, v)-{\\mathsf {Q}}_{n-s,\\eta }^{{\\beta }+s,0}\\left(\\partial _t^s f\\right) (x, v) \\right] \\mathrm {d}v.$ Multiplying (REF ) by $\\eta \\left( \\frac{k}{n} \\right)$ and summing up over $k$ gives an expression of ${\\mathsf {Q}}_{n,\\eta }^{({\\beta },-s)} f$ .","Furthermore, using the Tylor expansion of $t\\rightarrow f(x,t)$ with respect to the $t$ variable at $t =1$ , together with its remainder formula, we can write $f(x,t) = \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!}","\\partial _t^m [f(\\cdot ,t)] (\\xi ,1) +(-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}","\\partial _t^s f (x, v) \\mathrm {d}v.$ Together the difference of the two identities gives the error formula for $f -{\\mathsf {Q}}_{n,\\eta }^{({\\beta },-s)} f$ and completes the proof.","Taking the $k$ -th order derivative with respect to the $t$ variable for $ 0 \\le t \\le s-1$ , we obtain $\\partial _t^k \\left(f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\right)&(x,t) = \\sum _{m= k}^{s-1} \\frac{(t-1)^{m-k}}{(m-k)!","}\\left[ \\partial _t^m [f(\\cdot ,1)] (\\xi )- {\\mathsf {Q}}_{n-m,\\eta }^{{\\mathbb {S}^{d-1}}} \\left[\\partial _t^m f(\\cdot ,1)\\right](\\xi ) \\right] \\\\& + (-1)^{s-k} \\int _{t}^{1} \\frac{(v-t)^{s-k-1}}{(s-k-1)!}","\\left[ \\partial _t^s f (x, v)-{\\mathsf {Q}}_{n-s,\\eta }^{({\\beta }+s,0)}\\left(\\partial _t^s f\\right) (x, v) \\right] \\mathrm {d}v,$ moreover, taking the $s$ -th derivative in the $t$ variable, we obtain $ \\partial _t^s \\left(f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\right)(x,t) = {\\mathsf {Q}}_{n-s,\\eta }^{({\\beta }+s,0)}\\left(\\partial _t^s f\\right) (x, t).$ In the above error formulas for $1\\le k \\le s-1$ , the integral is over $v \\in [t,1]$ while the variable is $(x,v) = (t \\xi , v)$ .","As it is, we cannot deduce the error estimate for $\\partial _t^k \\big (f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\big )$ immediately from that of $f - {\\mathsf {Q}}_{n-s,\\eta }^{({\\beta }+s,0)}\\big (\\partial _t^s f\\big )$ over ${\\mathbb {V}}_0^{d+1}$ if $1 \\le k \\le s-1$ , although we can if $k =s$ by (REF ).","We state the latter case as a corollary.","Corollary 4.7 Let $f \\in W_p^s({\\mathsf {w}}_{{\\beta }+s,0})$ .","Then, for $1 \\le p \\le \\infty $ , $\\left\\Vert \\partial _t^s f - \\partial _t^s {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta }+s,0}}\\le c E_n\\left( \\partial _t^s f \\right)_{p, {\\mathsf {w}}_{{\\beta }+s,0}}.$ We can derive another interesting property of the projection operator for the SOPs, which relies on the property that the Jacobi polynomial $P_n^{({\\alpha },-s)}$ is related to $P_{n-s}^{({\\alpha },s)}$ , as seen in [17].","The following lemma shows that our extension $J_n^{{\\alpha },-s}$ satisfies the same property.","Theorem 4.8 If $f(x,t) = (1-t)^s g(x,t)$ , then $\\operatorname{proj}_n^{{\\beta },-s} f = (1-t)^s \\operatorname{proj}_{n-s}^{{\\beta },s} g.$ By the definition of the ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ , it follows from (REF ) that ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} (x,t)= \\sigma _{m,n} (1-t)^s {\\mathsf {S}}_{m,\\ell }^{n-s, ({\\beta },s)}(x,t), \\quad \\sigma _{m,n} := \\frac{(n-s-m)!}{(n-m)!}","A_{n-m}^{(2m+{\\beta }+d-1,0)}.$ Expanding $g$ in terms of the Fourier orthogonal series in $L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },1})$ , we obtain $f(x,t) = (1-t)^s g(x,t) \\,& = (1-t)^s \\sum _{n=0}^\\infty \\sum _{m=0}^n \\sum _\\ell \\hat{g}_{m,\\ell }^{n, ({\\beta },s)}{\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },s)}(x,t) \\\\& = \\sum _{n=0}^\\infty \\sum _{m=0}^n \\sum _\\ell \\hat{g}_{m,\\ell }^{n, ({\\beta },s)}\\sigma _{n+s,m}^{-1} {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)}(x,t).$ Applying the orthogonality in the Sobolev inner product, it follows immediately that $\\left\\langle f, {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)} \\right\\rangle _{{\\beta },-s} =\\sigma _{n+1,m}^{-1} \\hat{g}_{m,\\ell }^{n,({\\beta },s)} \\left\\langle {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)}, {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)} \\right\\rangle _{{\\beta },-s}, \\quad 0 \\le m \\le n;$ in particular, with $n$ replaced by $n-s$ , we then obtain $ \\hat{f}_{m,\\ell }^{n,({\\beta },-s)} = \\sigma _{n,m}^{-s} \\hat{g}_{m,\\ell }^{n-s,({\\beta },s)}, \\quad 0 \\le m \\le n-s.$ For $n-s < m \\le n$ , the polynomial $J_{n-m}^{(2m+{\\beta }+d-1,-s)}(t)$ is a polynomial of degree at most $s-1$ , so that $\\partial _t^s {\\mathsf {S}}_{n,\\ell }^{n, ({\\beta },-1)}(x,t) = 0$ ; moreover, $\\partial _t^k f(\\xi ,1) =0$ for $0 \\le k \\le s-1$ , since $f$ contains a factor $(1-t)^s$ .","It follows that $\\hat{f}_{n,\\ell }^{n,({\\beta },-s)} = 0$ for $n -s < m \\le n$ by the definition of the Sobolev inner product.","Consequently, by (REF ) and (REF ) we obtain $(1-t) \\operatorname{proj}_{n-s}^{{\\beta },s} g\\, & = (1-t) \\sum _{m=0}^{n-s} \\sum _\\ell \\hat{g}_{m,\\ell }^{n-1,({\\beta },s)} {\\mathsf {S}}_{m,\\ell }^{n-s, ({\\beta },s)} \\\\& = \\sum _{m=0}^{n} \\sum _\\ell \\hat{f}_{m,\\ell }^{n,({\\beta },-s)} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} = \\operatorname{proj}_{n}^{{\\beta },-s}f.$ The proof is completed.","Corollary 4.9 Let $f \\in L^p({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },0})$ and $f(t) = (1-t) g(t)$ .","Then, for $1 \\le p \\le \\infty $ , $\\left\\Vert f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)} f \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },0}}\\le c E_n\\left(g \\right)_{p, {\\mathsf {w}}_{{\\beta },s}} \\le c E_n\\left(g \\right)_{p, {\\mathsf {w}}_{{\\beta },0}}$ Since $f = (1-t)^s g$ , we see that ${\\mathsf {Q}}_{n,\\eta }^{(\\beta , -1)} f = \\sum _{k=0}^{2n} \\eta \\left( \\frac{k}{n}\\right) (1-t)^s \\operatorname{proj}_{k-1}^{{\\beta },s} g= (1-t)^s \\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{\\beta ,s} g,$ where $\\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{(\\beta ,s)}$ is defined as in the proof of Corollary REF .","For $p \\ge 1$ , we then obtain $\\left\\Vert f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)} f \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },0}}& = \\left\\Vert (1-t)^s \\left[ g - \\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{(\\beta , s)} g \\right] \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },0}}\\\\& \\le \\left\\Vert g - \\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{(\\beta , s)} g \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },s}} \\le c E_n(g)_{p, {\\mathsf {w}}_{{\\beta },s}},$ By the definition of $E_n(g)_{p, {\\mathsf {w}}}$ , it follows immediately $E_n(g)_{p, {\\mathsf {w}}_{{\\beta },s}} \\le E_n(g)_{p, {\\mathsf {w}}_{{\\beta },0}}$ .","The proof is completed."],["Partial differential equation for the Sobolev orthogonal polynomials","This section considers partial differential equations satisfied by the Sobolev orthogonal polynomials.","As we mentioned in Theorem REF proved in [21], the ordinary orthogonal polynomials in ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ , ${\\gamma }> -1$ , are eigenfunctions of the differential operator ${\\mathcal {D}}_{{\\gamma }}$ , defined in (REF ).","For the Sobolev orthogonal polynomials, we need ${\\gamma }$ to be negative integers.","We first consider the action of the differential operator when ${\\gamma }$ is a real number."],["Eigenfunctions of ${\\mathcal {D}}_{{\\gamma }}$ for {{formula:3f962e14-6e6a-454f-99e0-f53f9b58f637}}","Recall that the operator ${\\mathcal {D}}_{{\\gamma }}$ is given by ${\\mathcal {D}}_{{\\gamma }} = t(1-t)\\frac{\\mathrm {d}^2}{\\mathrm {d}t^2} + \\big ( d-1 - (d+{\\gamma })t \\big ) \\frac{\\mathrm {d}}{\\mathrm {d}t}+ t^{-1} \\Delta _0^{(\\xi )}.$ We start with a lemma for the action of this operator.","Lemma 5.1 Let $p(t)$ be a polynomial in the $t$ variable and $Y(\\xi )$ be a function defined on the unit sphere ${\\mathbb {S}^{d-1}}$ .","For $k \\ge 1$ , ${\\mathcal {D}}_{\\gamma } \\left[(1-t)^k\\,p(t) Y(\\xi ) \\right] \\, & = (1-t)^k {\\mathcal {D}}_{\\gamma +2k}\\left[p(t) Y (\\xi )\\right] \\\\& - k\\,(1-t)^{k-1} [d-1-(d+{\\gamma }+k-1)t] p(t)Y(\\xi ).","$ For $k=1$ , a quick computation from the definition of ${\\mathcal {D}}_{{\\gamma }}$ shows ${\\mathcal {D}}_{{\\gamma }}[(1-t) p(t) Y(\\xi )]=\\, & \\big ((1-t) \\left[t(1-t)p^{\\prime \\prime }(t) + [d-1-(d+\\gamma +2)t]p^{\\prime }(t)\\right] \\\\& - [d-1-(d+\\gamma )t]p(t)\\big ) Y(\\xi )+ t^{-1}\\,(1-t) p(t) \\, \\Delta _0^{(\\xi )}Y(\\xi ) \\\\= \\, & (1-t){\\mathcal {D}}_{\\gamma +2}[p(t)Y(\\xi )] - [d-1-(d+\\gamma )t]p(t)Y(\\xi ),$ which verifies (REF ) for $k=1$ .","Assume that (REF ) holds for $k\\ge 1$ .","Then $& {\\mathcal {D}}_{\\gamma }[(1-t)^{k+1} p(t) Y(\\xi )] = {\\mathcal {D}}_{{\\gamma }} [(1-t)^k (1-t) p(t) Y(\\xi )] \\\\& \\quad = (1-t)^k {\\mathcal {D}}_{{\\gamma }+2k}[(1-t)p(t) Y(\\xi )] \\\\&\\qquad - k\\,(1-t)^{k-1} [d-1-(d+\\gamma +k-1)t][(1-t)p(t) Y(\\xi )]\\\\& \\quad = (1-t)^k \\left\\lbrace (1-t){\\mathcal {D}}_{{\\gamma }+2k+2}[p(t) Y(\\xi )] - [d-1-(d+\\gamma +2k)t]p(t) Y(\\xi )\\right\\rbrace \\\\&\\qquad - k\\,(1-t)^{k} [d-1-(d+\\gamma +k-1)t]p(t) Y(\\xi )\\\\& \\quad = (1-t)^{k+1} {\\mathcal {D}}_{{\\gamma }+2k+2}[p(t) Y(\\xi )] - (k+1)(1-t)^k [d-1-(d+\\gamma +k)t]p(t) Y(\\xi ),$ so that (REF ) holds for $k+1$ and the proof is completed by induction.","Theorem 5.2 For $0\\le j \\le n$ , let $Z_{j,n}(x,t) = p(t) Y(x)$ with $p$ being a polynomial of degree $j$ in one variable and $Y$ be a solid harmonic polynomial in ${\\mathcal {H}}_{n-j}^{d,0}$ .","For $\\gamma \\in {\\mathbb {R}}$ , the only polynomial $p$ for which $Z_{j,n}$ satisfies ${\\mathcal {D}}_{{\\gamma }} Z (x,t) = \\lambda _n^{(\\gamma )} Z(x,t), \\qquad {\\lambda }_n^{({\\gamma })} = -n(n+{\\gamma }+d-1),$ is the Jacobi polynomial $p(t) = c_j\\,P_j^{(2n-2j+d-2, \\gamma )}(1-2t)$ , where $c_j$ is an appropriate constant, if $2n+\\gamma +d-r-1\\ne 0$ for $0\\le r\\le j$ .","Without losing generality, we assume $p$ is a monic polynomial, $p(t) = \\sum _{i=0}^j \\,(1-t)^i\\,a_{j,i}, \\quad a_{j,j} = 1.$ Our objective is to determine the coefficients $a_{j,i}$ such that $Z_j$ satisfies (REF ).","First, we observe that (REF ) holds if $p(t) =1$ .","In this case $Y \\in {\\mathcal {H}}_n^{d,0}$ and $Y(x) = t^n Y(\\xi )$ .","Hence, taking the derivative over $t$ and using the relation (REF ), a quick computation shows that $ {\\mathcal {D}}_{{\\gamma }} Y(x)= &\\, {\\mathcal {D}}_{{\\gamma }}[t^n Y(\\xi )] \\\\= &\\, n (n+d-2)Y(\\xi ) - m (m+{\\gamma }+d-1) t^m Y(\\xi ) + t^{m-1} \\Delta _0 Y(\\xi ) \\\\= &\\, -m(m+d+{\\gamma }-1) t^m Y(\\xi ) = -m(m+d+{\\gamma }-1) Y(x).","$ For $j \\ge 0$ , we use the identity (REF ) to obtain $& {\\mathcal {D}}_{{\\gamma }} Z_j(x,t) = \\sum _{i=0}^j a_{j,i} {\\mathcal {D}}_{{\\gamma }}\\left[(1-t)^i t^{n-j} Y(\\xi )\\right] \\\\& = \\sum _{i=0}^j a_{j,i} \\left[(1-t)^i {\\mathcal {D}}_{{\\gamma }+ 2i}[Y(x)] - i(1-t)^{i-1}[d-1-(d+\\gamma +i-1)t] Y(x)\\right]$ and then apply (REF ) for $Y\\in {\\mathcal {H}}_{n-j}^{d,0}$ and simplify to obtain ${\\mathcal {D}}_{{\\gamma }} Z_j(x,t) =\\, & \\sum _{i=0}^{j} \\,(1-t)^i \\,a_{j,i} [\\lambda _{n-j}^{(\\gamma +2i)} - i\\,(d+\\gamma +i-1)]\\,Y(x)\\\\&+\\sum _{i=0}^{j-1} (i+1)(1-t)^{i}\\,(\\gamma +i+1) \\,a_{j,i+1}\\,Y(x)\\\\= \\, & (1-t)^j \\,\\lambda _n^{(\\gamma )}\\,Y(x) +\\sum _{i=0}^{j-1} \\,(1-t)^i \\\\& \\times \\lbrace a_{j,i} \\,[\\lambda _{n-j}^{(\\gamma +2i)} - i(d+\\gamma +i-1)] + a_{j,i+1}(i+1)(\\gamma +i+1)\\rbrace Y(x),$ since $a_{j,j} = 1$ .","Thus, in order to have $Z_{j,n} (x,t)$ satisfying (REF ), we must have $a_{j,i} \\,[\\lambda _{n-j}^{(\\gamma +2i)} - i(d+\\gamma +i-1)] + a_{j,i+1}(i+1)(\\gamma +i+1) = a_{j,i} \\lambda _n^{(\\gamma )}$ for $0 \\le i \\le j-1$ , which leads to the recurrence relation $a_{j,i} = - \\frac{(i+1)(\\gamma +i+1)}{(j-i)(2n+\\gamma +d-1-j+i)}a_{j,i+1} , \\quad i=0, 1, \\ldots , j-1.$ Solving this recurrence relation and using $a_{j,j} = 1$ , we obtain $a_{j,i} = (-1)^{j-i}\\frac{(i+1)_{j-i}(\\gamma +i+1)_{j-i}}{(j-i)!","(2n+\\gamma +d- 1-j+i)_{j-i}}, \\quad 0\\le i \\le j.$ Using the identity $(a+i)_{j-i} = (a)_j/(a)_i$ and $(j-i)!","= (-1)^i j!","/ (-j)_i$ , we can rewrite it as $a_{j,i} = (-1)^j c_j \\frac{(-j)_i (2n+{\\gamma }+d-j-1)_i}{i!","({\\gamma }+1)_i}, \\quad c_j= \\frac{({\\gamma }+1)_j}{(2n+{\\gamma }+d-j-1)_j},$ which is well-defined if $(2n+{\\gamma }+d-j-1)_j \\ne 0$ .","Consequently, the polynomial $p$ must be given by $p(x) \\,& = (-1)^j c_j\\phantom{i}_2F_1\\left( \\begin{matrix} -j, 2n+{\\gamma }+d-j-1 \\\\ {\\gamma }+1 \\end{matrix}; 1-t \\right) \\\\& = (-1)^jc_j P_j^{({\\gamma }, 2n-2j+d-2)}(2t-1) =c_j P_j^{(2n-2j+d-2, {\\gamma })}(1-2t),$ which completes the proof.","For ${\\gamma }> -1$ , the theorem recovers Theorem REF established in [21].","We discuss the case when ${\\gamma }= -s$ and $s \\in {\\mathbb {N}}$ in the next subsection."],["Eigenfunctions for ${\\mathcal {D}}_{-s}$","As a corollary of Theorem REF , we obtain the following result for ${\\mathcal {D}}_{-s}$ .","Proposition 5.3 Let $s \\in {\\mathbb {N}}$ and $n \\in {\\mathbb {N}}_0$ .","Then the polynomials ${\\mathsf {S}}_{m,\\ell }^{n, (-1,-s)}$ are eigenfunctions of ${\\mathcal {D}}_{-s}$ if and only if $m \\le n-s$ and $m =n$ .","By (REF ), the polynomials ${\\mathsf {S}}_{m,-\\ell }^{n,(-1,-s)}$ can be written as ${\\mathsf {S}}_{m,\\ell }^{n,(-1,-s)}(x,y) = c_{n-m} P_{n-m}^{(2m+d-2,-s)}(1-2t) Y_\\ell ^m(x), \\quad n-m \\ge s,$ so that Theorem REF applies for $n - m \\ge s$ , thus ${\\mathsf {S}}_{m,\\ell }^{n, (-1,-s)}$ satisfies (REF ).","If $n = m$ , then ${\\mathsf {S}}_{n, \\ell }^{n, (-1,-s)}(x,t) = Y_\\ell ^n(x)$ .","As a result, (REF ) holds with $p(z) =1$ .","For $1 \\le n-m \\le s-1$ , however, the polynomial $J_{n-m}^{(2m+d-2,-s)}(t) = (1-t)^{n-m}/(n-m)!$ is not the Jacobi polynomial $P_{n-m}^{(2m+d-2,-s)}(t)$ , hence ${\\mathsf {S}}_{n, \\ell }^{n, (-1,-s)}$ is not a solution of (REF ).","In the case $s =1$ , the set $\\lbrace m: n -m \\ge s\\rbrace \\cup \\lbrace n\\rbrace = \\lbrace m: 1 \\le m \\le n\\rbrace $ .","Hence, we obtain the following corollary.","Theorem 5.4 For ${\\gamma }= -1$ , all elements in ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1})$ are eigenfunctions of ${\\mathcal {D}}_{-1}$ ; that is, ${\\mathcal {D}}_{-1} Z = - n (n+ d-2) Z, \\qquad \\forall Z \\in {\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1}).$ Moreover, the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1})$ satisfies a decomposition ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1}) = {\\mathcal {H}}_n^{d,0} \\cup (1-t) {\\mathcal {V}}_{n-1}^d({\\mathsf {w}}_{-1,1}), \\qquad n =0,1,\\ldots .$ We only need to prove the decomposition.","If $m \\le n-1$ , it follows from from (REF ) that ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-1)} = c (1-t) {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta },1)} \\in (1-t) {\\mathcal {V}}_{n-1}^d({\\mathsf {w}}_{-1,1})$ , whereas if $m = n$ , then ${\\mathsf {S}}_{n,\\ell }^{n,({\\beta },-s)} = Y_\\ell ^n \\in {\\mathcal {H}}_n^{d,0}$ .","The theorem shows, in particular, that ${\\mathcal {D}}_{{\\gamma }}$ has the complete eigenspaces if ${\\gamma }\\ge -1$ .","For $s = 2,3,\\ldots $ , however, not every element of the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-s})$ is an eigenfunction of ${\\mathcal {D}}_{ -s}$ by Theorem REF .","We can, however, identify the eigenspaces of the operator as follows.","Theorem 5.5 For $s = 2,3,\\ldots $ , define ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s}):= {\\mathcal {H}}_{n}^{d,0} \\bigcup _{j=1}^{s-1}P_j^{(2n-2j+d-2,-s)}(1-2t){\\mathcal {H}}_{n-j}^{d,0}\\cup (1-t)^s {\\mathcal {V}}_{n-s}^d ({\\mathsf {w}}_{-1,s}).$ Then ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ is the eigenspace of ${\\mathcal {D}}_{-s}$ ; more precisely, ${\\mathcal {D}}_{-s} Z=\\lambda _n^{(-s)}Z$ for all $Z \\in {\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ .","Moreover, the space satisfies $\\dim {\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s}) = \\binom{n+d-1}{d}$ if $-2n-d+j+s+2$ is not a positive integer between 1 and $j$ for $1 \\le j \\le s-1$ and $n \\ge j$ .","As in the case of $s = 1$ , using (REF ), the statement on the eigenspace follows readily from Theorem REF .","As it is shown for the dimension of ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ , the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ has the full dimension $\\binom{n+d-1}{d}$ if the Jacobi polynomials $P_j^{(2n-2j+d-2,-s)}$ does not have the degree reduction, which holds if $2n-j+d-2-s +k =0$ for a certain integer $k$ , $1 \\le k \\le j$ , by [17].","The theorem shows that the operator ${\\mathcal {D}}_{-s}$ has a complete basis of polynomials as eigenfunctions only if the restriction on $-2n-d+j+s+2$ as stated holds.","Example 5.6 If $s = 2$ , then the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-2})$ is given by ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-2}) = {\\mathcal {H}}_n^d \\cup \\left(1+ (2n+d-4)(1-t) \\right) {\\mathcal {H}}_{n-1}^d\\cup (1-t)^s {\\mathcal {V}}_{n-2}^d ({\\mathsf {w}}_{-1,2})$ and it has the full dimension if $-2n-d+5 \\ne 1$ or $2n-d+4 \\ne 0$ for $n \\ge 1$ .","Thus, ${\\mathcal {D}}_{-2}$ has a complete basis of polynomials as eigenfunctions if $d$ is odd.","In the case that $d$ is even, $ {\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-2})$ is not well-defined for $n = d/2-2$ .","We note that the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ does not coincide with ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-s})$ for $s =2,3,\\ldots $ .","In the case of $s = -2$ and $d$ is odd, one may ask if there is a Sobolev inner product for which the polynomials in the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ are orthogonal.","The authors have no relevant financial or non-financial interests to disclose."]],"string":"[\n [\n \"Sobolev orthogonal polynomials on the conic surface\"\n ],\n [\n \"Abstract Orthogonal polynomials with respect to the weight function $w_{\\\\beta,\\\\gamma}(t) = t^\\\\beta (1-t)^\\\\gamma$, $\\\\gamma > -1$, on the conic surface $\\\\{(x,t): \\\\|x\\\\| = t, \\\\, x \\\\in \\\\mathbb{R}^d, \\\\, t \\\\le 1\\\\}$ are studied recently, and they are shown to be eigenfunctions of a second order differential operator $\\\\mathcal{D}_\\\\gamma$ when $\\\\beta =-1$.\",\n \"We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th partial derivatives in $t$ variable with respect to $w_{\\\\beta+s,0}$ over the conic surface plus a sum of integrals over the rim of the cone.\",\n \"Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series.\",\n \"The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\\\\beta, -s}$ for a positive integer $s$.\",\n \"It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\\\\mathcal{D}_{\\\\gamma}$ when $\\\\gamma = -1$.\"\n ],\n [\n \"Introduction\",\n \"Orthogonal polynomials on the conic surface of the revolution were studied recently, which are shown to possess properties parallel to those of spherical harmonics on the unit sphere.\",\n \"Let ${\\\\mathbb {V}}_0^{d+1}$ be the conic surface ${\\\\mathbb {V}}_0^{d+1} = \\\\lbrace (x,t) \\\\in {\\\\mathbb {R}}^{d+1}: \\\\, \\\\Vert x\\\\Vert = t, \\\\, x \\\\in {\\\\mathbb {R}}^d, \\\\, 0 \\\\le t \\\\le 1\\\\rbrace $ in ${\\\\mathbb {R}}^{d+1}$ .\",\n \"For the weight function ${\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}(t) = t^{\\\\beta }(1-t)^{\\\\gamma }$ , ${\\\\beta }> -d$ and ${\\\\gamma }> -1$ , the orthogonal polynomials with respect to the inner product ${\\\\langle }f,g{\\\\rangle }_{{\\\\beta },{\\\\gamma }} = b_{{\\\\beta },{\\\\gamma }} \\\\int _{{\\\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}(t) \\\\mathrm {d}{\\\\mathsf {m}}(x,t)$ are called the Jacobi polynomials on the cone.\",\n \"These polynomials are studied in [12], [21], [22], [23].\",\n \"It was shown in [21] that these polynomials share many properties of spherical harmonics, including explicit orthogonal basis and an addition formula, which provides essential tools for an extensive study in approximation theory and computational analysis over the cone in [23].\",\n \"Another remarkable property of the Jacobi polynomials on the cone is that they are eigenfunctions of a second-order linear differential operator ${\\\\mathcal {D}}_{\\\\gamma }$ when ${\\\\beta }= -1$ , which is an analog of the Laplace-Beltrami operator on the unit sphere.\",\n \"The purpose of the present paper is to study Sobolev orthogonal polynomials on the conic surface, which are orthogonal with respect to an inner product that contains derivatives.\",\n \"The first case is ${\\\\langle }f,g{\\\\rangle }_{{\\\\beta }, -1} = \\\\frac{1}{{\\\\omega }_d}\\\\int _{{\\\\mathbb {V}}_0^{d+1}} \\\\frac{\\\\partial }{\\\\partial t} f (x,t)\\\\frac{\\\\partial }{\\\\partial t} g(x,t) t^{{\\\\beta }+1} \\\\mathrm {d}{\\\\mathsf {m}}(x,t)+ \\\\frac{{\\\\lambda }}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}}^{d-1}}f(\\\\xi ,1) g(\\\\xi ,1) \\\\mathrm {d}\\\\sigma (\\\\xi ),$ and we also consider ${\\\\langle }f,g{\\\\rangle }_{{\\\\beta }, -s}$ that involves derivatives up to $s$ order for a positive integer $s$ .\",\n \"There is a reason that we consider only derivatives in the $t$ variable but not the $x$ variable; see the discussion in Section 4.\",\n \"Like in the case of ${\\\\gamma }> -1$ , our main result provides explicit construction of orthogonal bases and a closed-form formula for the orthogonal projection operator.\",\n \"The study requires an extension of the Jacobi polynomials with a parameter being a negative integer, which needs to satisfy the Sobolev orthogonality of one variable that is inherited from ${\\\\langle }\\\\cdot ,\\\\cdot {\\\\rangle }_{{\\\\beta },-s}$ when we restrict the inner product to polynomials depending only on the $t$ variable.\",\n \"Such Sobolev orthogonal polynomials of one variable have been studied by several authors; see, for example, [1], [2], [7], [15], [16], [20] and [11].\",\n \"We shall follow the approach in [20] since it is more convenient for studying orthogonal projection operators and provides a link, in particular, between the Sobolev orthogonal structure and the ordinary orthogonal structure, which is useful for studying the convergence of the Fourier orthogonal series in the Sobolev orthogonal polynomials.\",\n \"In the framework of polynomial approximation theory on the ball and standard or Sobolev orthogonal polynomials, we can refer to [3], [4], [5], [8], [9], [10], [14], [18], among others.\",\n \"In more than one way, our study extends the Jacobi polynomials for ${\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}$ on the cone from ${\\\\gamma }> -1$ to ${\\\\gamma }= -s$ with $s \\\\in {\\\\mathbb {N}}$ .\",\n \"We will show, in particular, that the spectral operator ${\\\\mathcal {D}}_{-s}$ has the Sobolev orthogonal polynomials as eigenfunctions if $s = 1$ .\",\n \"While the latter fails for $s > 1$ , we do have a clear understanding of what the eigenspaces of ${\\\\mathcal {D}}_{-s}$ are.\",\n \"For orthogonal polynomials in several variables, our study is also closely related to the Sobolev orthogonal polynomials on the unit ball, which have been extensively studied (see [6] and its references therein).\",\n \"In particular, the description of the eigenspaces of ${\\\\mathcal {D}}_{\\\\gamma }$ is similar to the study on the unit ball in [13].\",\n \"The paper is organized as follows.\",\n \"The next section is preliminary, in which we recall two essential ingredients needed for our study, the Jacobi polynomials with negative parameters and spherical harmonics.\",\n \"In Section 3 we review results on ordinary orthogonal polynomials on the conic surface and discuss further properties of the orthogonal projection operators.\",\n \"The Sobolev orthogonal polynomials are defined and studied in Section 4.\",\n \"Finally, the eigenspaces of the operator ${\\\\mathcal {D}}_{\\\\gamma }$ are discussed in Section 5.\"\n ],\n [\n \"Preliminary\",\n \"The study for orthogonal polynomials on the conic surface follows that of spherical harmonics on the unit sphere.\",\n \"The latter will also be essential for constructing an orthogonal basis on the cone.\",\n \"Another ingredient is the Jacobi polynomial, which we often need an extension to negative parameters in the study of the Sobolev orthogonal polynomials.\"\n ],\n [\n \"Jacobi polynomials with negative paramters\",\n \"The Jacobi polynomials $P_n^{({\\\\alpha },{\\\\beta })}$ are given explicitly by the hypergeometric function $P_n^{({\\\\alpha },{\\\\beta })}(t) = \\\\frac{({\\\\alpha }+1)_n}{n!}\",\n \"{}_2F_1 \\\\left( \\\\begin{matrix}-n, n+{\\\\alpha }+{\\\\beta }+1 \\\\\\\\ {\\\\alpha }+1\\\\end{matrix}; \\\\frac{1-x}{2}\\\\right)$ for ${\\\\alpha },{\\\\beta }> -1$ and $n = 0, 1, 2, \\\\ldots $ .\",\n \"They are orthogonal with respect to the weight function $w_{{\\\\alpha },{\\\\beta }}(t) = (1-t)^{\\\\alpha }(1+t)^{\\\\beta }$ on $[-1,1]$ with ${\\\\alpha }, {\\\\beta }> -1$ , and they satisfy $\\\\frac{c_{{\\\\alpha },{\\\\beta }}}{2^{{\\\\alpha }+{\\\\beta }+1}}\\\\int _{-1}^1 P_n^{({\\\\alpha },{\\\\beta })}(t) P_n^{({\\\\alpha },{\\\\beta })}(t) w_{{\\\\alpha },{\\\\beta }}(t) \\\\mathrm {d}t = h_n^{({\\\\alpha },{\\\\beta })} \\\\delta _{n,m},$ where $c_{{\\\\alpha },{\\\\beta }}$ is the constant so that $h_0^{({\\\\alpha },{\\\\beta })} =1$ , $c_{{\\\\alpha },{\\\\beta }} = \\\\frac{\\\\Gamma ({\\\\alpha }+{\\\\beta }+2)}{\\\\Gamma ({\\\\alpha }+1)\\\\Gamma ({\\\\beta }+1)} \\\\quad \\\\hbox{and}\\\\quad h_n^{({\\\\alpha },{\\\\beta })} = \\\\frac{({\\\\alpha }+1)_n({\\\\beta }+1)_n ({\\\\alpha }+{\\\\beta }+n+1)}{n!\",\n \"({\\\\alpha }+{\\\\beta }+2)_n ({\\\\alpha }+{\\\\beta }+2n+1)}.$ For studying the Sobolev orthogonal polynomials, we often need the parameters ${\\\\alpha }$ or ${\\\\beta }$ to be negative integers.\",\n \"Such polynomials are discussed already in [17] but they are no longer orthogonal with respect to $w_{{\\\\alpha },{\\\\beta }}$ , since the weight function $(1-t)^a(1+t)^{b}$ is no longer integrable if $a$ or $b \\\\le -1$ .\",\n \"Moreover, $P_n^{({\\\\alpha },{\\\\beta })}$ has a degree reduction if $n+{\\\\alpha }+{\\\\beta }$ is a negative integer between 1 to $n$ , which causes problems for studying the Sobolev orthogonal polynomials, especially in several variables, since such polynomials are needed for all $n \\\\in {\\\\mathbb {N}}_0$ .\",\n \"What we need in this paper are the polynomials $P_n^{({\\\\alpha },-s)}$ with ${\\\\alpha }> -1$ and $s \\\\in {\\\\mathbb {N}}$ .\",\n \"These polynomials are well defined if $n \\\\ge s$ and satisfy [17] $P_n^{({\\\\alpha },-s)} (t) = \\\\frac{(-{\\\\alpha }-n)_s}{2^s (-n)_s} (1+t)^s P_{n-s}^{({\\\\alpha },s)}(t), \\\\quad n \\\\ge s,$ which follows from [17] by using $P_n^{({\\\\alpha },{\\\\beta })}(-t) = (-1)^n P_n^{({\\\\beta },{\\\\alpha })}(t)$ .\",\n \"The definition of $P_n^{({\\\\alpha },-s)}$ for $n < s$ could be problematic because of the degree reduction.\",\n \"Such polynomials have been studied in the setting of the Sobolev orthogonal polynomials; for example, the Sobolev orthogonality defined via the inner product $ [f,g]_{{\\\\alpha },{\\\\beta }}^{-s} : = \\\\int _{-1}^1 f^{(s)}(t) g^{(s)}(t) w_{{\\\\alpha },{\\\\beta }}(t) \\\\mathrm {d}t + \\\\sum _{k=0}^{s-1} \\\\mu _k f^{(k)}(1)g^{(k)}(1),$ where $\\\\mu _k$ are fixed positive constants and ${\\\\alpha },{\\\\beta }> -1$ .\",\n \"The study of such polynomials and their orthogonality has appeared in several papers; see, for example, [1], [7], [2], [15], [16], [19], [20] and [11].\",\n \"There are several ways to define a complete set of orthogonal polynomials for the inner product (REF ).\",\n \"We shall follow the approach given in [20], see also [15], [16], which is more suitable for studying the Fourier orthogonal series.\",\n \"We now recall the necessary result from [20].\",\n \"For convenience, we first define a renormalization of the Jacobi polynomials, $ \\\\widehat{P}^{(\\\\alpha ,\\\\beta )}_{n}(t) = A_n^{({\\\\alpha },{\\\\beta })} {P}^{(\\\\alpha ,\\\\beta )}_{n}(t) \\\\quad \\\\hbox{with}\\\\quad A_n^{({\\\\alpha },{\\\\beta })} = \\\\frac{2^n}{(n+{\\\\alpha }+\\\\beta +1)_{n}}$ for ${\\\\alpha },{\\\\beta }> -1$ .\",\n \"This normalization has the advantage that it satisfies, by [17], $ \\\\frac{d}{dt} \\\\widehat{P}_n^{({\\\\alpha },{\\\\beta })}(t) = \\\\widehat{P}_{n-1}^{({\\\\alpha }+1,{\\\\beta }+1)}(t).$ Now, for ${\\\\alpha },{\\\\beta }> -1$ , $s \\\\in {\\\\mathbb {N}}$ and $n = 0,1,2,\\\\ldots ,$ we define a new sequence of polynomials $ \\\\begin{split}J_n^{({\\\\alpha }-s,{\\\\beta }-s)}(t) := {\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\dfrac{(t+1)^n}{n!\",\n \"}, & 0 \\\\le n \\\\le s-1, \\\\vspace{3.61371pt} \\\\\\\\\\\\displaystyle {\\\\int _{-1}^t \\\\frac{(t-u)^{s-1}}{(s-1)!}\",\n \"\\\\widehat{P}_{n-s}^{({\\\\alpha },{\\\\beta })}(u) \\\\mathrm {d}u}, & n \\\\ge s.\\\\end{array}\\\\right.\",\n \"}\\\\end{split}$ It is easy to see that $J_n^{({\\\\alpha }-s,{\\\\beta }-s)}$ is a polynomial of degree $n$ and it satisfies $\\\\partial ^s J_n^{({\\\\alpha }-s,{\\\\beta }-s)}(t) &\\\\ = \\\\widehat{P}_{n-s}^{({\\\\alpha },{\\\\beta })}(t), \\\\qquad n \\\\ge s; \\\\\\\\\\\\partial ^k J_n^{({\\\\alpha }-s,{\\\\beta }-s)}(-1) &\\\\ = {\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\delta _{k,n}, & n \\\\le s-1, \\\\\\\\0, & n \\\\ge s,\\\\end{array}\\\\right.}\",\n \"\\\\qquad 0\\\\le k \\\\le s-1, $ where $\\\\partial ^k$ denotes the $k$ -th derivative.\",\n \"These are our polynomials that extend the definition of the Jacobi polynomials to allow negative parameters, which are also orthogonal polynomials with respect to the Sobolev inner product (REF ).\",\n \"More precisely, we have the following [20]: Theorem 2.1 For ${\\\\alpha },{\\\\beta }> -1$ and $s \\\\in {\\\\mathbb {N}}$ .\",\n \"The polynomial $J_n^{({\\\\alpha }-s,{\\\\beta }-s)}$ is orthogonal with respect to the inner product ${\\\\langle }\\\\cdot , \\\\cdot {\\\\rangle }_{{\\\\alpha },{\\\\beta }}^{-s}$ and its norm square is given by $\\\\left[J_n^{({\\\\alpha }-s,{\\\\beta }-s)},J_n^{({\\\\alpha }-s,{\\\\beta }-s)}\\\\right]_{{\\\\alpha },{\\\\beta }}^{-s}= {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mu _n & 0 \\\\le n \\\\le s-1 \\\\\\\\ \\\\widehat{h}_{n-s}^{({\\\\alpha },{\\\\beta })} & n \\\\ge s \\\\end{array}\\\\right.\",\n \"},$ where $\\\\mu _n$ comes from (REF ), and $\\\\widehat{h}_n^{({\\\\alpha },{\\\\beta })}$ is the norm square of $\\\\widehat{P}_n^{({\\\\alpha },{\\\\beta })}$ , which is given in terms of $h_n^{({\\\\alpha },{\\\\beta })}$ by $\\\\widehat{h}_n^{({\\\\alpha },{\\\\beta })} = \\\\frac{2^{{\\\\alpha }+{\\\\beta }+1}}{c_{{\\\\alpha },{\\\\beta }}} \\\\left[A_n^{({\\\\alpha },{\\\\beta })}\\\\right]^2 h_n^{({\\\\alpha },{\\\\beta })}.$ Our next proposition shows that, if ${\\\\alpha }> -1$ and $s\\\\in {\\\\mathbb {N}}$ , then the definition $J_n^{({\\\\alpha },-s)}$ in (REF ) agrees with that of (REF ) when $n\\\\ge s$ .\",\n \"Proposition 2.2 For ${\\\\alpha }> -1$ and $s \\\\in {\\\\mathbb {N}}$ , $J_n^{({\\\\alpha },-s)} (t) = \\\\frac{(n-s)!}{n!}\",\n \"(1+t)^s \\\\widehat{P}_{n-s}^{({\\\\alpha },s)}(t), \\\\qquad n \\\\ge s.$ In particular, for $n \\\\ge s$ , $J_n^{({\\\\alpha },-s)} (t) = \\\\frac{(-1)^s 2^s}{(-{\\\\alpha }-n)_s}A_{n-s}^{({\\\\alpha },s)} P_n^{({\\\\alpha },-s)}(t).$ We use the hypergeometric expression of the Jacobi polynomials, $P_n^{({\\\\alpha },{\\\\beta })}(t) = \\\\frac{({\\\\alpha }+1)_n}{n!}\",\n \"{}_2F_1\\\\left(\\\\begin{matrix} -n \\\\,\\\\,\\\\, n+{\\\\alpha }+{\\\\beta }+1\\\\\\\\ {\\\\alpha }+1 \\\\end{matrix}; \\\\frac{1-t}{2} \\\\right)$ and the fact that $P_n^{({\\\\alpha },{\\\\beta })}(t) = (-1)^n P_n^{({\\\\beta },{\\\\alpha })}(-t)$ .\",\n \"It follows that $& \\\\int _{-1}^t \\\\frac{(t-u)^{s-1}}{(s-1)!}\",\n \"P_n^{({\\\\alpha }+s,0)}(u) \\\\mathrm {d}u= (-1)^n \\\\int _{-1}^t \\\\frac{(t-u)^{s-1}}{(s-1)!}\",\n \"P_n^{(0, {\\\\alpha }+s)}(-u) \\\\mathrm {d}u \\\\\\\\& \\\\qquad \\\\quad = (-1)^n \\\\sum _{k=0}^n \\\\frac{(-n)_k (n+{\\\\alpha }+s+1)_k}{k!\",\n \"k!\",\n \"}\\\\int _{-1}^t \\\\frac{(t-u)^{s-1}}{(s-1)!}\",\n \"\\\\left(\\\\frac{1+u}{2} \\\\right)^k \\\\mathrm {d}u \\\\\\\\& \\\\qquad \\\\quad = \\\\frac{(-1)^n}{s!}\",\n \"(1+t)^s \\\\sum _{k=0}^n \\\\frac{(-n)_k (n+{\\\\alpha }+s+1)_k}{k!\",\n \"(s+1)_k} \\\\left(\\\\frac{1+t}{2} \\\\right)^k \\\\\\\\& \\\\qquad \\\\quad = \\\\frac{(-1)^n}{s!}\",\n \"(1+t)^s \\\\frac{n!\",\n \"}{(s+1)_n} P_n^{(s,{\\\\alpha })}(-t)= \\\\frac{n!}{(n+s)!}\",\n \"(1+t)^sP_n^{({\\\\alpha },s)}(t).$ Replacing $n$ by $n-s$ and using $A_n^{({\\\\alpha },s)} = A_n^{({\\\\alpha }+s,0)}$ , the identity (REF ) then follows from (REF ).\",\n \"The second identity follows from the identity (REF ).\",\n \"It should be pointed out that the integral expression of $J_n^{({\\\\alpha },-s)}$ for $n \\\\ge s$ in (REF ) is more convenient for studying the Fourier orthogonal series, as shown in [20] and as we shall see in Section 4 below.\"\n ],\n [\n \"Spherical harmonics\",\n \"A homogeneous polynomial $Y$ of $d$ variables is called a solid harmonic if $\\\\Delta Y =0$ , where $\\\\Delta $ is the Laplace operator on ${\\\\mathbb {R}}^d$ .\",\n \"We denote by ${\\\\mathcal {H}}_m^{d,0}$ the space of homogeneous solid harmonics of degree $m$ in $d$ variables.\",\n \"Thus, if $Y \\\\in {\\\\mathcal {H}}_m^{d,0}$ , then $Y(r \\\\xi ) = r^m Y(\\\\xi )$ for $\\\\xi \\\\in {\\\\mathbb {S}^{d-1}}$ .\",\n \"Spherical harmonics are restrictions of solid harmonics on the unit ball.\",\n \"We denote the space of spherical harmonics of degree $m$ by ${\\\\mathcal {H}}_m^d$ .\",\n \"Thus, ${\\\\mathcal {H}}_m^d = {\\\\mathcal {H}}_m^{d,0} \\\\vert _{\\\\mathbb {S}^{d-1}}$ .\",\n \"It is a common practice to identify ${\\\\mathcal {H}}_m^{d,0}$ and ${\\\\mathcal {H}}_m^d$ , we distinguish them to emphasize the dependence on variables for the orthogonal polynomials on the conic surface.\",\n \"It is well known that $\\\\dim {\\\\mathcal {H}}_m^d = \\\\binom{m+d-2}{n}+\\\\binom{m+d-3}{n-1},\\\\quad m=1,2,3,\\\\ldots ,$ and spherical harmonics of different degrees are orthogonal with respect to the surface measure on the unit sphere.\",\n \"Throughout the paper, we denote by $\\\\lbrace Y_\\\\ell ^m: 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_m^d\\\\rbrace $ an orthonormal basis of ${\\\\mathcal {H}}_m^d$ , so that $\\\\frac{1}{{\\\\omega }_d} \\\\int _{\\\\mathbb {S}^{d-1}}Y_\\\\ell ^n(\\\\xi ) Y_{\\\\ell ^{\\\\prime }}^m(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ) = \\\\delta _{\\\\ell ,\\\\ell ^{\\\\prime }} \\\\delta _{n,m},$ where $\\\\mathrm {d}\\\\sigma $ is the surface measure of ${\\\\mathbb {S}^{d-1}}$ and ${\\\\omega }_d$ is the surface area of ${\\\\mathbb {S}^{d-1}}$ .\",\n \"Let $\\\\operatorname{proj}_n^{\\\\mathbb {S}^{d-1}}: L^2({\\\\mathbb {S}^{d-1}}) \\\\rightarrow {\\\\mathcal {H}}_n^d$ be the orthogonal projection operator from $L^2({\\\\mathbb {S}^{d-1}})$ onto ${\\\\mathcal {H}}_n^d$ .\",\n \"If $\\\\lbrace Y_\\\\ell ^n: 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_n^{d-1} \\\\rbrace $ is an orthonormal basis of ${\\\\mathcal {H}}_n^d$ , then $\\\\operatorname{proj}_n^{\\\\mathbb {S}^{d-1}}f (x) = \\\\sum _{\\\\ell =1}^{\\\\dim {\\\\mathcal {H}}_n^d} \\\\hat{f}_{\\\\ell }^n Y_\\\\ell ^n, \\\\quad \\\\hat{f}_\\\\ell ^n = \\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}} f(\\\\xi ) Y_\\\\ell ^n(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ).$ For $f \\\\in L^2({\\\\mathbb {S}^{d-1}})$ , its Fourier expansion in spherical harmonics is defined by $f = \\\\sum _{n=0}^\\\\infty \\\\sum _{\\\\ell =1}^{\\\\dim {\\\\mathcal {H}}_n^d} \\\\hat{f}_{\\\\ell }^n Y_\\\\ell ^n =\\\\sum _{n=0}^\\\\infty \\\\operatorname{proj}_n^{\\\\mathbb {S}^{d-1}}f.$ Let $\\\\Delta _0$ be the Laplace-Beltrami operator, which is the restriction of the Laplacian $\\\\Delta $ on the unit sphere.\",\n \"Under the spherical polar coordinates $x = r \\\\xi $ , $r > 0$ , $\\\\xi \\\\in {\\\\mathbb {S}^{d-1}}$ , $\\\\Delta = \\\\frac{\\\\mathrm {d}^2}{\\\\mathrm {d}r^2} + \\\\frac{d-1}{r} \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}r} + \\\\frac{1}{r^2} \\\\Delta _0.$ The spherical harmonics are the eigenfunctions of $\\\\Delta _0$ .\",\n \"More precisely, $\\\\Delta _0 Y = - n (n+d-2)Y, \\\\qquad Y \\\\in {\\\\mathcal {H}}_n^d, \\\\quad n=0,1,2,\\\\ldots .$ The $\\\\Delta _0$ is a second-order differential operator on the unit sphere.\",\n \"One can also consider spherical gradient $\\\\nabla _0$ , the first-order differential operators, on the sphere, which is defined by $\\\\nabla = \\\\frac{1}{r} \\\\nabla _0 + \\\\xi \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}r}, \\\\quad x = r \\\\xi , \\\\quad \\\\xi \\\\in {\\\\mathbb {S}^{d-1}}.$ The integration by parts formula holds and gives [5], $\\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\nabla _0 f(\\\\xi ) \\\\cdot \\\\nabla _0 g(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ) = - \\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\Delta _0 f(\\\\xi ) g(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ).$ Together with (REF ), this identity implies that if $\\\\lbrace Y_\\\\ell ^n : 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_n^d\\\\rbrace $ is an orthogonal basis of ${\\\\mathcal {H}}_n^d$ , then $\\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\nabla _0 Y_\\\\ell ^n(\\\\xi ) \\\\cdot \\\\nabla _0 Y_{\\\\ell ^{\\\\prime }}^{n^{\\\\prime }}(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi )={\\\\lambda }_n \\\\int _{{\\\\mathbb {S}^{d-1}}} Y_\\\\ell ^n(\\\\xi ) \\\\cdot Y_{\\\\ell ^{\\\\prime }}^{n^{\\\\prime }}(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ) = {\\\\lambda }_n \\\\delta _{\\\\ell ,\\\\ell ^{\\\\prime }} \\\\delta _{n,n^{\\\\prime }},$ where ${\\\\lambda }_n = n(n+d-2)$ , which implies that $\\\\lbrace Y_\\\\ell ^n : 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_n^d\\\\rbrace $ is also a family of orthogonal polynomials for the Sobolev inner product defined by, for example, ${\\\\langle }f,g {\\\\rangle }_{\\\\nabla } = \\\\sum _{k=1}^s \\\\mu _k\\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\nabla _0^k f(\\\\xi ) \\\\cdot \\\\nabla _0^k g(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ) + \\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}} f(\\\\xi ) g(\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ),$ where $\\\\mu _k \\\\ge 0$ and $s$ is a fixed positive integer, and $\\\\nabla _0^{2m} := \\\\Delta _0^m \\\\quad \\\\hbox{and} \\\\quad \\\\nabla _0^{2m+1} = \\\\Delta _0^m \\\\nabla _0.$ In other words, the Sobolev orthogonal polynomials for ${\\\\langle }\\\\cdot ,\\\\cdot {\\\\rangle }_\\\\nabla $ on the unit sphere are trivially spherical harmonics themselves.\"\n ],\n [\n \"Orthogonal polynomials on the conic surface\",\n \"Orthogonal polynomials on the conic surface ${\\\\mathbb {V}}_0^{d+1}$ are studied in [21] for the inner product defined by ${\\\\langle }f, g {\\\\rangle }_{{\\\\beta },{\\\\gamma }} = b_{{\\\\beta },{\\\\gamma }} \\\\int _{{\\\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}(t) \\\\mathrm {d}{\\\\mathsf {m}}(x,t),$ where $\\\\mathrm {d}{\\\\mathsf {m}}$ is the Lebesgue measure on the conic surface and the weight function ${\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}$ is the Jacobi weight function on $[0,1]$ , ${\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}(t) = t^{\\\\beta }(1-t)^{\\\\gamma }, \\\\quad {\\\\beta }> -d, \\\\quad {\\\\gamma }> -1$ and $b_{{\\\\beta },{\\\\gamma }}$ is the normalization constant so that ${\\\\langle }1, 1 {\\\\rangle }_{{\\\\beta },g} =1$ , which is determined by $\\\\int _{{\\\\mathbb {V}}_0^{d+1}} f(x,t) \\\\mathrm {d}{\\\\mathsf {m}}(x,t) = \\\\int _{0}^1 t^{d-1} \\\\int _{{\\\\mathbb {S}^{d-1}}} f(t \\\\xi ,t)\\\\mathrm {d}\\\\sigma (\\\\xi ) \\\\mathrm {d}t,$ where $\\\\mathrm {d}\\\\sigma $ denotes the Lebesgue measure on the unit sphere ${\\\\mathbb {S}^{d-1}}$ .\",\n \"Then, $b_{\\\\beta ,\\\\gamma }= \\\\frac{1}{\\\\omega _d} c_{{\\\\beta }+d-1,{\\\\gamma }} \\\\quad \\\\hbox{with}\\\\quad c_{{\\\\beta },{\\\\gamma }} = \\\\frac{\\\\Gamma (\\\\beta +\\\\gamma +2)}{\\\\Gamma (\\\\beta +1)\\\\Gamma (\\\\gamma +1)} \\\\quad \\\\hbox{and}\\\\quad \\\\omega _{d}=\\\\frac{2\\\\pi ^{d/2}}{\\\\Gamma (d/2)},$ where $\\\\omega _{d}$ is the surface area of ${\\\\mathbb {S}^{d-1}}$ .\",\n \"The inner product is well defined for the space ${\\\\mathbb {R}}[x,t] / {\\\\langle }t^2 - \\\\Vert x\\\\Vert ^2{\\\\rangle }$ of polynomials modulus the ideal generated by $t^2 -\\\\Vert x\\\\Vert ^2$ .\",\n \"For $n \\\\in {\\\\mathbb {N}}_0$ , let ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ denote the space of orthogonal polynomials of degree $n$ .\",\n \"Since ${\\\\mathbb {V}}_0^{d+1}$ is a quadratic surface, so the dimension of the space is the same as that of ${\\\\mathcal {H}}_n^{d+1}$ .\",\n \"Thus, $\\\\dim {\\\\mathcal {V}}_0({\\\\mathbb {V}}_0^{d+1},{\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}) =1$ and $\\\\dim {\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}) = \\\\binom{n+d-1}{n}+\\\\binom{n+d-2}{n-1},\\\\quad n=1,2,3,\\\\ldots .$ An orthogonal basis for ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ is given in [21] in terms of the Jacobi polynomials and spherical harmonics.\",\n \"Let $\\\\lbrace Y_\\\\ell ^m: 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_m^d\\\\rbrace $ be an orthonormal basis of ${\\\\mathcal {H}}_m^d$ .\",\n \"Define the polynomials, called the Jacobi polynomials on the conic surface, by $ {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },{\\\\gamma })} (x,t) := P_{n-m}^{(2m + {\\\\beta }+ d-1,{\\\\gamma })} (1-2t) Y_\\\\ell ^m (x),$ where, for $(x,t) \\\\in {\\\\mathbb {V}}_0^{d+1}$ , $Y_\\\\ell ^m(x)$ is a solid harmonic in ${\\\\mathcal {H}}_m^{d,0}$ and $Y_\\\\ell ^m (t\\\\xi ) = t^m Y_\\\\ell ^m(\\\\xi )$ .\",\n \"Then $\\\\lbrace {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },{\\\\gamma })}: 0 \\\\le m \\\\le n, \\\\,\\\\, 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_m({\\\\mathbb {S}^{d-1}})\\\\rbrace $ is an orthogonal basis of ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ , which satisfies $b_{{\\\\beta },{\\\\gamma }} \\\\int _{{\\\\mathbb {V}}_0^{d+1}} {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },{\\\\gamma })} (x,t) {\\\\mathsf {S}}_{m^{\\\\prime }, \\\\ell ^{\\\\prime }}^{n^{\\\\prime },({\\\\beta },{\\\\gamma })} (x,t){\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}(t) \\\\mathrm {d}{\\\\mathsf {m}}(x,t) = h_{n,m}^{{\\\\beta },{\\\\gamma }}\\\\delta _{n,n^{\\\\prime }}\\\\delta _{m,m^{\\\\prime }}\\\\delta _{\\\\ell ,\\\\ell ^{\\\\prime }},$ where $h_{n,m}^{{\\\\beta },{\\\\gamma }}$ is the square of the $L^2({\\\\mathbb {V}}_0^{d+1},{\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ norm of ${\\\\mathsf {S}}_{m, \\\\ell }^n$ and $h_{n,m}^{{\\\\beta },{\\\\gamma }} = \\\\frac{({\\\\beta }+d)_{2m}}{({\\\\beta }+{\\\\gamma }+d+1)_{2m}} h_{n-m}^{(2m+{\\\\beta }+d-1,{\\\\gamma })}$ with $h_{n-m}^{(2m+{\\\\beta }+d-1,{\\\\gamma })}$ defined in (REF ).\",\n \"Of particular interest is the case ${\\\\beta }= -1$ , for which the space ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ is an eigenspace of a second order linear differential operator.\",\n \"Parametrizing the space ${\\\\mathbb {V}}_0^{d+1}$ by $(x,t) = (t\\\\xi , t)$ and let $\\\\Delta _0^{(\\\\xi )}$ be the Laplace-Beltrami operator on the unit sphere in the $\\\\xi $ variable, which is the restriction of the Laplace operator $\\\\Delta $ on ${\\\\mathbb {S}^{d-1}}$ .\",\n \"For ${\\\\gamma }> -1$ , define $ {\\\\mathcal {D}}_{\\\\gamma }= t(1-t)\\\\frac{\\\\mathrm {d}^2}{\\\\mathrm {d}t^2} + \\\\big ( d-1 - (d+{\\\\gamma })t \\\\big ) \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}+ t^{-1} \\\\Delta _0^{(\\\\xi )}.$ Theorem 3.1 Let $d\\\\ge 2$ and ${\\\\gamma }> -1$ .\",\n \"The orthogonal polynomials in ${\\\\mathcal {V}}_n({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{-1,{\\\\gamma }})$ are eigenfunctions of ${\\\\mathcal {D}}_{{\\\\gamma }}$ ; more precisely, ${\\\\mathcal {D}}_{{\\\\gamma }} u = -n (n+{\\\\gamma }+d-1) u, \\\\qquad \\\\forall u \\\\in {\\\\mathcal {V}}_n({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{-1,{\\\\gamma }}).$ The differential operator ${\\\\mathcal {D}}_{{\\\\gamma }}$ plays an important role in the study of the Fourier orthogonal series and the best approximation by polynomials on the conic surface [23].\",\n \"For $f \\\\in L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ , the Fourier orthogonal series of $f$ is defined by $f = \\\\sum _{n=0}^\\\\infty \\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }} f,$ where $\\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }}: L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}) \\\\rightarrow {\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ is the orthogonal projection operator, which satisfies, using the orthogonal basis given above, $\\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }} f = \\\\sum _{m=0}^n \\\\sum _{\\\\ell =1}^{\\\\dim {\\\\mathcal {H}}_m({\\\\mathbb {S}^{d-1}})}\\\\hat{f}_{m,\\\\ell }^{n, ({\\\\beta },{\\\\gamma })} {\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },{\\\\gamma })},\\\\quad \\\\hbox{where}\\\\quad \\\\hat{f}_{m,\\\\ell }^{n, ({\\\\beta },{\\\\gamma })} := \\\\frac{{\\\\langle }f, {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },{\\\\gamma })}{\\\\rangle }_{{\\\\beta },{\\\\gamma }}}{h_{m,n}^{{\\\\beta },{\\\\gamma }}}.$ In the rest of this section, we consider a property on the derivative of the reproducing kernel.\",\n \"Parametrizing the function $f: {\\\\mathbb {V}}_0^{d+1} \\\\rightarrow {\\\\mathbb {R}}$ by $(x,t) = (t\\\\xi , t)$ with $\\\\xi \\\\in {\\\\mathbb {S}^{d-1}}$ , it follows that $ \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t} f(x,t) = \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t} f(t\\\\xi ,t)= \\\\xi \\\\cdot \\\\nabla _x f(t\\\\xi ,t) + \\\\frac{\\\\partial }{\\\\partial t} f(t\\\\xi ,t),$ where $\\\\frac{\\\\partial }{\\\\partial t} = \\\\partial _{d+1}$ denotes the partial derivative with respect to the $d+1$ variable of $f$ , which should not be confused with the $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}$ on the left-hand side.\",\n \"Throughout the rest of this paper, we shall adopt the notation $\\\\partial _t = \\\\partial _{d+1} = \\\\frac{\\\\partial }{\\\\partial t}$ for functions $f(x,t)$ .\",\n \"We consider the action of $\\\\partial _t$ on the projection operator.\",\n \"Theorem 3.2 For ${\\\\beta }> -d$ and ${\\\\gamma }> -1$ , let $f$ be a differentiable function such that $f\\\\in L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ and $\\\\partial _t f \\\\in L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta }+1,{\\\\gamma }+1})$ .\",\n \"Then, for $n =0,1,2,\\\\ldots $ , $\\\\frac{\\\\partial }{\\\\partial t} \\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }} f(x,t) = \\\\operatorname{proj}_{n-1}^{{\\\\beta }+1,{\\\\gamma }+1} \\\\partial _t f (x,t).$ For $m = n$ , the polynomial ${\\\\mathsf {S}}_{n,\\\\ell }^{n,({\\\\beta },{\\\\gamma })}(x,t) = Y_\\\\ell ^n(x)$ , so that $\\\\frac{\\\\partial }{\\\\partial t} {\\\\mathsf {S}}_{n,\\\\ell }^{n,({\\\\beta },{\\\\gamma })}(x,t) =0$ .\",\n \"For $0 \\\\le m \\\\le n-1$ , using the well-known formula for the derivative of the Jacobi polynomials, we obtain $\\\\frac{\\\\partial }{\\\\partial t} {\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },{\\\\gamma })}(x,t) \\\\, & = \\\\frac{\\\\partial }{\\\\partial t} P_{n-m}^{(2m + {\\\\beta }+ d-1,{\\\\gamma })} (1-2t)Y_\\\\ell ^m (x) \\\\\\\\& =\\\\tau _{n,m} P_{n-m-1}^{(2m + {\\\\beta }+ d,{\\\\gamma }+1)} (1-2t) Y_\\\\ell ^m (x)= \\\\tau _{n,m} {\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)}(x,t), $ where $\\\\tau _{n,m} = - (n+m + {\\\\beta }+ {\\\\gamma }+ d)$ .\",\n \"As a consequence of these identities, we see that $\\\\partial _t \\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }} f \\\\in {\\\\mathcal {V}}_{n-1}^d({\\\\mathsf {w}}_{{\\\\beta }+1,{\\\\gamma }+1})$ .\",\n \"Consequently, it follows that $& \\\\left\\\\langle \\\\partial _t f, {\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)}\\\\right\\\\rangle _{{\\\\beta }+1,{\\\\gamma }+1} =\\\\left\\\\langle \\\\partial _t \\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }} f, {\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)} \\\\right\\\\rangle _{{\\\\beta }+1,{\\\\gamma }+1} \\\\\\\\& \\\\qquad \\\\qquad = \\\\sum _{k=0}^{n-1} \\\\sum _{\\\\nu } \\\\hat{f}_{k,\\\\nu }^{n,({\\\\beta },{\\\\gamma })} \\\\tau _{n,k} \\\\left\\\\langle {\\\\mathsf {S}}_{k,\\\\nu }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)},{\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)} \\\\right\\\\rangle _{{\\\\beta }+1,{\\\\gamma }+1} \\\\\\\\& \\\\qquad \\\\qquad = \\\\tau _{n,m} \\\\hat{f}_{m,\\\\ell }^{n,({\\\\beta },{\\\\gamma })} h_{n-1,m}^{{\\\\beta }+1,{\\\\gamma }+1},$ which implies immediately that $\\\\widehat{\\\\partial _t f}_{m,\\\\ell }^{n-1, ({\\\\beta }+1,{\\\\gamma }+1)} = \\\\tau _{n,m} \\\\hat{f}_{m,\\\\ell }^{n,({\\\\beta },{\\\\gamma })}, \\\\quad 0 \\\\le m \\\\le n-1.$ Consequently, we obtain $\\\\frac{\\\\partial }{\\\\partial t} \\\\operatorname{proj}_n^{{\\\\beta },{\\\\gamma }} f (x,t)\\\\, & = \\\\sum _{m=0}^{n-1} \\\\sum _\\\\ell \\\\hat{f}_{m,\\\\ell }^{n,({\\\\beta },{\\\\gamma })}\\\\tau _{n,m} {\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)}(x,t) \\\\\\\\& = \\\\sum _{m=0}^{n-1} \\\\sum _\\\\ell \\\\widehat{\\\\partial _t f}_{m,\\\\ell }^{n-1, ({\\\\beta }+1,{\\\\gamma }+1)} {\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta }+1,{\\\\gamma }+1)}(x,t) \\\\\\\\& = \\\\operatorname{proj}_{n-1}^{{\\\\beta }+1,{\\\\gamma }+1} \\\\partial _t f (x,t),$ where in the first step we used again $\\\\partial _t {\\\\mathsf {S}}_{n,\\\\ell }^n(x,t) = 0$ .\",\n \"For $1 \\\\le p \\\\le \\\\infty $ , let $\\\\Vert f\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}}$ denote the norm of the space $L^p({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ , and we adopt the convention that the space is $C({\\\\mathbb {V}}_0^{d+1})$ with the norm taken as the uniform norm when $p = \\\\infty $ .\",\n \"Let $\\\\Pi _n({\\\\mathbb {V}}_0^{d+1})$ denote the space of polynomials of degree $n$ restricted on the ${\\\\mathbb {V}}_0^{d+1}$ .\",\n \"For $f \\\\in L^p({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ , the quantity $E_n(f)_{p, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}} := \\\\inf _{P \\\\in \\\\Pi _n({\\\\mathbb {V}}_0^d)} \\\\Vert f - P \\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}}$ is the error of the best approximation by polynomials of degree at most $n$ in the norm of $L^p({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ .\",\n \"We call $\\\\eta \\\\in C^\\\\infty $ an admissible cut-off function if it is supported on $[0,2]$ and satisfies $\\\\eta (t) = 1$ if $0 \\\\le t \\\\le 1$ .\",\n \"Let $\\\\eta $ be such a function; we define $ Q_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })} f = \\\\sum _{k=0}^{2n} \\\\eta \\\\left(\\\\frac{k}{n} \\\\right) \\\\operatorname{proj}_k^{{\\\\beta },{\\\\gamma }} f.$ Then it is known [23] that $Q_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })}$ is a bounded operator in $L^p({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }})$ and it is a polynomial of near best approximation in the sense that $ \\\\left\\\\Vert f - Q_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })} f\\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}} \\\\le c E_n (f)_{p,{\\\\mathsf {w}}_{{\\\\beta },{\\\\gamma }}},$ where $c$ is a constant that depends only on $\\\\eta $ , $p$ , ${\\\\beta }$ and ${\\\\gamma }$ .\",\n \"In particular, we obtain the following as a corollary of Theorem REF .\",\n \"Corollary 3.3 Let ${\\\\beta }> -d$ and ${\\\\gamma }> -1$ .\",\n \"Let $r$ be a postive integer and let $f \\\\in C^r({\\\\mathbb {V}}_0^{d+1})$ such that $\\\\partial _t^k f \\\\in L^p({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta }+k,{\\\\gamma }+k})$ for $0 \\\\le k \\\\le r$ .\",\n \"Then, for $1 \\\\le p \\\\le \\\\infty $ , $\\\\left\\\\Vert \\\\partial _t^k f - \\\\partial _t^k Q_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })}f\\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta }+k,{\\\\gamma }+k} } \\\\le c_kE_{n-k} (\\\\partial _t^k f)_{p, {\\\\mathsf {w}}_{{\\\\beta }+k,{\\\\gamma }+k} }, \\\\qquad 0 \\\\le k \\\\le r.$ This follows immediately from $\\\\partial _t^k Q_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })} f & = \\\\sum _{j=0}^{2n} \\\\eta \\\\left(\\\\frac{j}{n} \\\\right) \\\\partial _t^j\\\\operatorname{proj}_j^{{\\\\beta },{\\\\gamma }} f= \\\\sum _{j=k}^{2n} \\\\eta \\\\left(\\\\frac{j}{n} \\\\right) \\\\operatorname{proj}_{j-k}^{{\\\\beta }+k,{\\\\gamma }+k} f \\\\\\\\& = \\\\sum _{j=0}^{2n-k} \\\\eta \\\\left(\\\\frac{j+k}{n} \\\\right) \\\\operatorname{proj}_{j}^{{\\\\beta }+k,{\\\\gamma }+k} \\\\partial _t^k f= {\\\\mathsf {Q}}_{n,\\\\eta }^{({\\\\beta }+k,{\\\\gamma }+k)} \\\\partial _t^k f,$ where we define $ {\\\\mathsf {Q}}_{ n,\\\\eta }^{({\\\\beta },{\\\\gamma })} g = \\\\sum _{j=0}^{2n-k} \\\\eta \\\\left(\\\\frac{j+k}{n} \\\\right) \\\\operatorname{proj}_{j}^{{\\\\beta },{\\\\gamma }} g, \\\\quad 0 \\\\le k < 2n.$ For fixed $k \\\\le r$ independent of $n$ , the function $\\\\eta \\\\left(\\\\frac{j+k}{n} \\\\right)$ plays essentially the same role as $\\\\eta \\\\left(\\\\frac{j}{n} \\\\right)$ , so that (REF ) holds with ${\\\\mathsf {Q}}_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })}\\\\partial _t^k f$ in place of $Q_{n,\\\\eta }^{({\\\\beta },{\\\\gamma })}f$ , form which the proof follows readily.\"\n ],\n [\n \"Sobolev orthogonal polynomials on the conic surface\",\n \"As seen in the differential operator (REF ), the derivatives on the conic surface ${\\\\mathbb {V}}_0^{d+1}$ are partial derivatives with respect to $\\\\xi $ and $t$ with $x = t \\\\xi $ for $(x,t) \\\\in {\\\\mathbb {V}}_0^{d+1}$ .\",\n \"For the Sobolev inner product, the derivatives in $\\\\xi \\\\in {\\\\mathbb {S}^{d-1}}$ will act on the spherical harmonics and lead to the same orthogonal basis as discussed at the end of Subsection 2.2.\",\n \"Hence, we consider only the derivatives with respect to $\\\\partial _t$ ; see Remark REF below.\",\n \"For $s \\\\in {\\\\mathbb {N}}$ , let us define $W_p^s({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta }+s,0})= \\\\left\\\\lbrace f\\\\in C({\\\\mathbb {V}}_0^{d+1}): \\\\partial _t^s f\\\\in L^p({\\\\mathbb {V}}_0^{d+1},{\\\\mathsf {w}}_{{\\\\beta }+s,0})\\\\right\\\\rbrace ,$ where $1 \\\\le p \\\\le \\\\infty $ and the space is taken as the $C^s({\\\\mathbb {V}}_0^{d+1})$ if $p = \\\\infty $ .\",\n \"Let $s$ be a positive integer and ${\\\\beta }> -s-d$ .\",\n \"Let ${\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_{r-1}$ be fixed positive numbers.\",\n \"We consider the Sobolev inner product defined by $ {\\\\langle }f,g{\\\\rangle }_{{\\\\beta }, -s} =\\\\,& \\\\frac{1}{{\\\\omega }_d}\\\\int _{{\\\\mathbb {V}}_0^{d+1}} \\\\frac{\\\\partial ^s}{\\\\partial t^s} f (x,t)\\\\frac{\\\\partial ^s}{\\\\partial t^s} g(x,t) t^{{\\\\beta }+s} \\\\mathrm {d}{\\\\mathsf {m}}(x,t)\\\\\\\\\\\\ & + \\\\sum _{k=0}^{s-1} \\\\frac{{\\\\lambda }_k}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}}^{d-1}}\\\\frac{\\\\partial ^k f}{\\\\partial t^k} (\\\\xi ,1) \\\\frac{\\\\partial ^k g}{\\\\partial t^k}(\\\\xi ,1) \\\\mathrm {d}\\\\sigma (\\\\xi ), $ which is evidently an inner product on the space $W_2^s({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta }+s,0})$ .\",\n \"We denote by ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },-s})$ the space of orthogonal polynomials of degree $n$ with respect to this inner product.\",\n \"Our first task is to find an orthogonal basis for this space.\",\n \"Recall the modified Jacobi polynomial $J_n^{({\\\\alpha },-s)}$ defined in (REF ).\",\n \"Let $\\\\lbrace Y_\\\\ell ^m: 1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_m^d\\\\rbrace $ be an orthonormal basis of ${\\\\mathcal {H}}_m^d$ .\",\n \"For $1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_m^d$ , $0\\\\le m \\\\le n$ , we define $ {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)} (x,t) := J_{n-m}^{(2m+{\\\\beta }+d-1,-s)}(1-2 t) Y_\\\\ell ^m(x).$ Theorem 4.1 Let $s \\\\in {\\\\mathbb {N}}$ and ${\\\\beta }> -d-s$ .\",\n \"The polynomials ${\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)}$ , $1 \\\\le \\\\ell \\\\le \\\\dim {\\\\mathcal {H}}_m^d$ and $0\\\\le m \\\\le n$ consist of a basis of ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },-s})$ .\",\n \"Moreover, for all $\\\\ell $ , ${\\\\mathsf {h}}_{m,n}^{({\\\\beta },-s)} =\\\\langle {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)}, {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)} \\\\rangle _{{\\\\beta },-s}$ satisfies ${\\\\mathsf {h}}_{m,n}^{({\\\\beta },-s)} = {\\\\left\\\\lbrace \\\\begin{array}{ll}2^{2n-2m} {\\\\lambda }_{n-m} & 0 \\\\le n -m \\\\le s-1, \\\\\\\\2^{s - {\\\\beta }-2m-d}\\\\, \\\\widehat{h}_{n-m-s}^{(s+2m+{\\\\beta }+d-1,0)} & n-m \\\\ge s.\",\n \"\\\\end{array}\\\\right.\",\n \"}$ Let $q_{n-m}(t) = J_{n-m}^{(2m+{\\\\beta }+d-1, -s)}(t)$ .\",\n \"Changing variable $x = t \\\\xi $ and using the homogeneity of $Y_\\\\ell ^m$ , we obtain $\\\\left\\\\langle {\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },-s)}, {\\\\mathsf {S}}_{m^{\\\\prime },\\\\ell ^{\\\\prime }}^{n^{\\\\prime },({\\\\beta },-s)} \\\\right\\\\rangle _{{\\\\beta },-s}= \\\\, & 2^{ 2s} \\\\int _0^1 t^{d-1} q_{n-m}^{(s)} (1-2t) q_{n^{\\\\prime }-m^{\\\\prime }}^{(s)}(1-2t) t^{{\\\\beta }+s+m+m^{\\\\prime }} \\\\mathrm {d}t \\\\\\\\&\\\\qquad \\\\times \\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}}^{d-1}} Y_\\\\ell ^m (\\\\xi )Y_{\\\\ell ^{\\\\prime }}^{m^{\\\\prime }} (\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ) \\\\\\\\& + \\\\sum _{k=0}^{s-1} {\\\\lambda }_k 2^{2k} q_{n-m}^{(k)}(-1) q_{n^{\\\\prime }-m^{\\\\prime }}^{(k)}(-1) \\\\\\\\&\\\\qquad \\\\times \\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}}^{d-1}} Y_\\\\ell ^m (\\\\xi )Y_{\\\\ell ^{\\\\prime }}^{m^{\\\\prime }} (\\\\xi ) \\\\mathrm {d}\\\\sigma (\\\\xi ).$ Using the orthogonality of $Y_\\\\ell ^m$ and changing variable $t \\\\mapsto (1-t)/2$ in the first integral, we see that the expression containing $q_{n-m}$ can be written in terms of the Sobolev inner product $[\\\\cdot ,\\\\cdot ]_{{\\\\alpha },{\\\\beta }}^{-s}$ defined in (REF ).\",\n \"More precisely, we obtain $\\\\left\\\\langle {\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },-s)}, {\\\\mathsf {S}}_{m^{\\\\prime },\\\\ell ^{\\\\prime }}^{n^{\\\\prime },({\\\\beta },-s)} \\\\right\\\\rangle _{{\\\\beta },-s} =2^{s - {\\\\alpha }-1} [q_{n-m}, q_{n^{\\\\prime }-m}]_{{\\\\alpha },0}^{-s} \\\\delta _{m,m^{\\\\prime }} \\\\delta _{\\\\ell ,\\\\ell ^{\\\\prime }},$ where ${\\\\alpha }= s+{\\\\beta }+2m+d-1$ and the inner product $[\\\\cdot , \\\\cdot ]_{{\\\\alpha },0}^{-s}$ is defined as in (REF ) but with $\\\\mu _k = 2^{2k} {\\\\lambda }_k/2^{s-{\\\\alpha }-1}$ .\",\n \"As shown in Theorem REF , the polynomials $J_n^{({\\\\alpha }-s,-s)}$ are orthogonal with respect to this inner product, regardless of the values of $\\\\mu _k$ .\",\n \"This proves the orthogonality of ${\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },-s)}$ .\",\n \"Moreover, the norm ${\\\\mathsf {h}}_{m,n}^{({\\\\beta },-s)}$ follows from the norm given in Theorem REF .\",\n \"Remark 4.1 We could also include an additional term in the right-hand side of ${\\\\langle }\\\\cdot ,\\\\cdot {\\\\rangle }_{{\\\\beta },-s}$ defined by $\\\\int _{{\\\\mathbb {V}}_0^{d+1}} \\\\nabla _0^s f(x,t) \\\\cdot \\\\nabla _0^s g(x,t) \\\\mathrm {d}{\\\\mathsf {m}}(x,t),$ where $\\\\nabla _0$ acts on $\\\\xi $ .\",\n \"By the discussion for the inner product (REF ) on the unit sphere, the inner product with this additional term has the same orthogonal basis.\",\n \"Recall that the constant $A_n^{({\\\\alpha },{\\\\beta })}$ is defined in (REF ).\",\n \"Corollary 4.2 For $0 \\\\le m \\\\le n-s$ , $ \\\\frac{\\\\partial ^s}{\\\\partial t^s} {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)}(x,t) = (-2)^s A_{n-s-m}^{(s+2m+{\\\\beta }+d-1,0)} S_{m,\\\\ell }^{n-s, ({\\\\beta }+s,0)}(x,t),$ and it is equal to zero if $m > n-s$ .\",\n \"Moreover, for $1 \\\\le k \\\\le s-1$ and $\\\\xi \\\\in {\\\\mathbb {S}^{d-1}}$ , $ \\\\frac{\\\\partial ^k}{\\\\partial t^k} {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)}(\\\\xi ,1) = {\\\\left\\\\lbrace \\\\begin{array}{ll}(-2)^k Y_\\\\ell ^m(\\\\xi ) \\\\delta _{k,n-m}, & m > n-s \\\\\\\\0, & m \\\\le n-s.\\\\end{array}\\\\right.\",\n \"}$ If $m \\\\le n-s$ , by our convention of the derivative $\\\\partial _t$ over the conic surface and the identity (REF ), $\\\\frac{\\\\partial ^s}{\\\\partial t^s} {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)}(x,t) \\\\,& =\\\\frac{\\\\partial ^s}{\\\\partial t^s}\\\\left[J_{n-m}^{(2m+{\\\\beta }+d-1,-s)}(1-2t)\\\\right]Y_\\\\ell ^m(x) \\\\\\\\& = (-2)^{s} \\\\widehat{P}_{n-m-s}^{(s+2m+{\\\\beta }+d-1,0)}(1-2t)Y_\\\\ell ^m(x),$ from which (REF ) follows from (REF ) and the definition of ${\\\\mathsf {S}}_{\\\\ell ,m}^{n,({\\\\beta }+s,0)}$ .\",\n \"Moreover, since $n-m \\\\ge s$ , for $1 \\\\le k \\\\le s-1$ it follows immediately by () that (REF ) holds.\",\n \"If $m > n-s$ , then the derivative in the identity (REF ) is evidently zero since the Jacobi polynomial in ${\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)}$ is of degree $n-m -d-s$ , let $f$ be a differentiable function such that $\\\\partial _t^s f \\\\in L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta }+s,0})$ .\",\n \"Then, for $n =0,1,2,\\\\ldots $ and $(x,t) \\\\in {\\\\mathbb {V}}_0^{d+1}$ , $\\\\operatorname{proj}_n^{{\\\\beta },-s} f(x,t) = \\\\,& \\\\sum _{m=0}^{s-1} \\\\frac{(t-1)^m}{m!}\",\n \"\\\\operatorname{proj}_{n-m}^{\\\\mathbb {S}^{d-1}}\\\\left[\\\\partial _t^m f(\\\\cdot ,1)\\\\right](\\\\xi ) \\\\\\\\& + (-1)^s \\\\int _{t}^{1} \\\\frac{(v-t)^{s-1}}{(s-1)!}\",\n \"\\\\operatorname{proj}_{n-s}^{{\\\\beta }+s,0} \\\\left(\\\\partial _t^s f\\\\right) (x, v) \\\\mathrm {d}v. $ Let ${\\\\alpha }= s+{\\\\beta }+d-1$ .\",\n \"For $0 \\\\le m \\\\le n-s$ , it follows from (REF ) that $\\\\left\\\\langle f, {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },-s)}\\\\right\\\\rangle _{{\\\\beta },-s} \\\\,& =(-2)^s A_{n-m}^{(2m+{\\\\alpha },0)}\\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {V}}_0^{d+1}} \\\\frac{\\\\partial ^s}{\\\\partial t^s} f(x,t) {\\\\mathsf {S}}_{m,\\\\ell }^{n-s,({\\\\beta }+s,0)}(x,t) t^{{\\\\beta }+s} \\\\mathrm {d}{\\\\mathsf {m}}(x,t)\\\\\\\\& = (-2)^s A_{n-m}^{(2m+{\\\\alpha },0)} \\\\frac{1}{c_{{\\\\alpha },0}}\\\\left\\\\langle \\\\partial _t^s f, {\\\\mathsf {S}}_{m, \\\\ell }^{n-s,({\\\\beta }+s,0)}\\\\right\\\\rangle _{{\\\\beta }+s,0}$ and the norm of ${\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },-s)}$ satisfies ${\\\\mathsf {h}}_{m,n}^{({\\\\beta },-s)} = 2^{s - {\\\\beta }-2m-d} \\\\widehat{h}_{n-m-s}^{({\\\\alpha }+2m,0)} \\\\, & =2^{2s} \\\\frac{1}{c_{2m+{\\\\alpha },0}} \\\\left[A_{n-m-s}^{(2m+{\\\\alpha },0)}\\\\right]^2 h_{n-s-m}^{(2m+{\\\\alpha },0)} \\\\\\\\& = \\\\frac{1}{c_{{\\\\alpha },0}} 2^{2s} \\\\left[A_{n-m-s}^{(2m+{\\\\alpha },0)}\\\\right]^2 \\\\frac{({\\\\alpha }+1)_{2m}}{({\\\\alpha }+2)_{2m}}h_{n-s-m}^{(2m+{\\\\alpha },0)}\\\\\\\\& = \\\\frac{1}{c_{{\\\\alpha },0}} 2^{2s} \\\\left[A_{n-m-s}^{(2m+{\\\\alpha },0)}\\\\right]^2 h_{n-s, m}^{{\\\\beta }+s,0}.$ Consequently, it follows that $\\\\hat{f}_{m,\\\\ell }^{n, ({\\\\beta },-s)} = \\\\frac{c_{2m+{\\\\alpha },0}}{c_{{\\\\beta }+s,0}} \\\\frac{\\\\left\\\\langle \\\\partial _t^s f, {\\\\mathsf {S}}_{m, \\\\ell }^{n-s,({\\\\beta }+s,0)}\\\\right\\\\rangle _{{\\\\beta }+s,0}}{ (-2)^{s} A_{n-m-s}^{(2m+{\\\\alpha },0)} h_{n-s, m}^{{\\\\beta }+s,0}}= \\\\frac{ \\\\widehat{\\\\partial _t^s f}_{m,\\\\ell }^{n-s, ({\\\\beta }+s,0)}}{(-2)^{s} A_{n-m-s}^{(2m+{\\\\alpha },0)}}.$ Now, for $n-s < m \\\\le n$ , it follows by Corollary REF that $\\\\left\\\\langle f, {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },-s)}\\\\right\\\\rangle _{{\\\\beta },-s}\\\\,& = \\\\sum _{k=0}^{s-1} \\\\frac{{\\\\lambda }_k}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\partial _t^k f(\\\\xi ,1) \\\\partial _t^k {\\\\mathsf {S}}_{m, \\\\ell }^{n,({\\\\beta },-s)}(\\\\xi ,1)\\\\mathrm {d}\\\\sigma (\\\\xi ) \\\\\\\\& = (-2)^{n-m} \\\\frac{{\\\\lambda }_{n-m}}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\partial _t^{n-m} f(\\\\xi ,1) Y_\\\\ell ^{m} (\\\\xi )\\\\mathrm {d}\\\\sigma (\\\\xi ),$ so that, using the norm of ${\\\\mathsf {h}}_{n,m}^{({\\\\beta },-s)}$ in Theorem REF , $\\\\hat{f}_{m,\\\\ell }^{n, ({\\\\beta },-s)} = \\\\frac{1}{(-2)^{n-m}} \\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}} \\\\partial _t^{n-m} f(\\\\xi ,1) Y_\\\\ell ^{m} (\\\\xi )\\\\mathrm {d}\\\\sigma (\\\\xi ),\\\\quad 0\\\\le n-m \\\\le s-1.$ Consequently, by (REF ) and using ${\\\\alpha }= s+{\\\\beta }+d-1$ again, it follows that $& \\\\operatorname{proj}_n^{{\\\\beta },-s} f(x,t) \\\\, = \\\\sum _{m=0}^{s-1} \\\\frac{(t-1)^m}{m!\",\n \"}\\\\sum _{\\\\ell =1}^{\\\\dim {\\\\mathcal {H}}_{n-m}^d} \\\\frac{1}{{\\\\omega }_d} \\\\int _{{\\\\mathbb {S}^{d-1}}}\\\\partial _t^{m} f(\\\\eta ,1) Y_\\\\ell ^{n-m} (\\\\eta )\\\\mathrm {d}\\\\sigma (\\\\eta )Y_\\\\ell ^{n-m} (\\\\xi ) \\\\\\\\& \\\\qquad \\\\quad + \\\\sum _{m = 0}^{n-s} \\\\sum _{\\\\ell =1}^{\\\\dim {\\\\mathcal {H}}_{m}^d}\\\\frac{ \\\\widehat{\\\\partial _t^s f}_{m,\\\\ell }^{n-s, ({\\\\beta }+s,0)}}{(-2)^{s} A_{n-m-s}^{(2m+{\\\\alpha },0)}}\\\\int _{-1}^{1-2t} \\\\frac{(1-2t -u)^{s-1}}{(s-1)!}\",\n \"\\\\widehat{P}_{n-s}^{({\\\\alpha }+2m,0)}(u) \\\\mathrm {d}u Y_\\\\ell ^m(x).$ The inner sum of the first term in the right-hand side is exactly $\\\\operatorname{proj}_{n-m}^{\\\\mathbb {S}^{d-1}}[\\\\partial _t^m f(\\\\cdot ,1)](\\\\xi )$ for the function $\\\\xi \\\\mapsto \\\\partial _t^m f(\\\\xi ,1)$ on the unit sphere.\",\n \"Changing variable $u = 1-2v$ and using (REF ) in the integral, the second term on the right-hand side becomes, $\\\\sum _{m = 0}^{n-s} \\\\sum _{\\\\ell } & \\\\widehat{\\\\partial _t^s f}_{m,\\\\ell }^{n-s, ({\\\\beta }+s,0)} (-1)^s\\\\int _{t}^{1} \\\\frac{(v-t)^{s-1}}{(s-1)!}\",\n \"P_{n-s}^{({\\\\alpha }+2m,0)}(1-2v) \\\\mathrm {d}v Y_\\\\ell ^m(x) \\\\\\\\& = (-1)^s \\\\int _{t}^{1} \\\\frac{(v-t)^{s-1}}{(s-1)!}\",\n \"\\\\sum _{m = 0}^{n-s} \\\\sum _{\\\\ell }\\\\widehat{\\\\partial _t^s f}_{m,\\\\ell }^{n-s, ({\\\\beta }+s,0)} {\\\\mathsf {S}}_{m,\\\\ell }^{n-s, ({\\\\beta }+s,0)} (x,v) \\\\mathrm {d}v\\\\\\\\& = (-1)^s \\\\int _{t}^{1} \\\\frac{(v-t)^{s-1}}{(s-1)!}\",\n \"\\\\operatorname{proj}_{n-s}^{{\\\\beta }+s,0}\\\\partial _t^s f (x, v) \\\\mathrm {d}v.$ Putting the two terms together completes the proof.\",\n \"Corollary 4.5 Let $s$ be a positive integer.\",\n \"For ${\\\\beta }> -d-s$ , let $f$ be a differentiable function such that $\\\\partial _t^s f \\\\in L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta }+s,0})$ .\",\n \"Then, for $n \\\\ge s$ , $\\\\frac{\\\\partial ^s}{\\\\partial t^s} \\\\operatorname{proj}_n^{{\\\\beta },-s} f(x,t) = \\\\operatorname{proj}_{n-s}^{{\\\\beta }+s,0}\\\\left(\\\\partial _t^s f\\\\right) (x,t).$ Taking the derivative $\\\\partial _t^s$ of (REF ), we see that the first term in the right-hand side is zero, while the second term gives the stated identity.\",\n \"Another consequence of Theorem REF is an expression for the error of approximation.\",\n \"Let $\\\\eta \\\\in C^\\\\infty ({\\\\mathbb {R}}_+)$ be an admissible cut-off function.\",\n \"We denote by ${\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)}$ the near best approximation operator defined in (REF ) but with ${\\\\gamma }= -s$ .\",\n \"Furthermore, let ${\\\\mathsf {Q}}_{n-m}^{\\\\mathbb {S}^{d-1}}f (\\\\xi )= \\\\sum _{k=0}^{2(n-m)} \\\\eta \\\\left(\\\\frac{k}{n}\\\\right) \\\\operatorname{proj}_k^{\\\\mathbb {S}^{d-1}}f(\\\\xi ), \\\\quad 0 \\\\le m \\\\le n,$ be a near best approximation operator of degree $n-m$ on the unit sphere.\",\n \"For $f\\\\in L^p({\\\\mathbb {S}^{d-1}})$ , let $E_n^{\\\\mathbb {S}^{d-1}}(f)_p = \\\\inf _{Y \\\\in \\\\Pi _n({\\\\mathbb {S}^{d-1}})} \\\\Vert f - Y\\\\Vert _{L^p({\\\\mathbb {S}^{d-1}})}, \\\\qquad 1 \\\\le p \\\\le \\\\infty $ be the error of best approximation by polynomials on the unit sphere, where the norm is the uniform norm for $p = \\\\infty $ and $\\\\Pi _n({\\\\mathbb {S}^{d-1}})$ denote the space of polynomial of degree at most $n$ restricted on the unit sphere ${\\\\mathbb {S}^{d-1}}$ .\",\n \"Then (cf.\",\n \"[5]) $\\\\left\\\\Vert f - {\\\\mathsf {Q}}_{n}^{\\\\mathbb {S}^{d-1}}f \\\\right\\\\Vert _p \\\\le c E_n^{{\\\\mathbb {S}^{d-1}}}(f)_p, \\\\qquad 1 \\\\le p \\\\le \\\\infty .$ Theorem 4.6 For $f \\\\in C^s({\\\\mathbb {V}}_0^{d+1})$ , $f(x,t) - {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)}f(x,t) \\\\,& = \\\\sum _{m=0}^{s-1} \\\\frac{(t-1)^m}{m!}\",\n \"\\\\left[ \\\\partial _t^m [f(\\\\cdot ,1)] (\\\\xi )- {\\\\mathsf {Q}}_{n-m,\\\\eta }^{{\\\\mathbb {S}^{d-1}}} \\\\left[\\\\partial _t^m f(\\\\cdot ,1)\\\\right](\\\\xi ) \\\\right] \\\\\\\\& + (-1)^s \\\\int _{t}^{1} \\\\frac{(v-t)^{s-1}}{(s-1)!}\",\n \"\\\\left[ \\\\partial _t^s f (x, v)-{\\\\mathsf {Q}}_{n-s,\\\\eta }^{{\\\\beta }+s,0}\\\\left(\\\\partial _t^s f\\\\right) (x, v) \\\\right] \\\\mathrm {d}v.$ Multiplying (REF ) by $\\\\eta \\\\left( \\\\frac{k}{n} \\\\right)$ and summing up over $k$ gives an expression of ${\\\\mathsf {Q}}_{n,\\\\eta }^{({\\\\beta },-s)} f$ .\",\n \"Furthermore, using the Tylor expansion of $t\\\\rightarrow f(x,t)$ with respect to the $t$ variable at $t =1$ , together with its remainder formula, we can write $f(x,t) = \\\\sum _{m=0}^{s-1} \\\\frac{(t-1)^m}{m!}\",\n \"\\\\partial _t^m [f(\\\\cdot ,t)] (\\\\xi ,1) +(-1)^s \\\\int _{t}^{1} \\\\frac{(v-t)^{s-1}}{(s-1)!}\",\n \"\\\\partial _t^s f (x, v) \\\\mathrm {d}v.$ Together the difference of the two identities gives the error formula for $f -{\\\\mathsf {Q}}_{n,\\\\eta }^{({\\\\beta },-s)} f$ and completes the proof.\",\n \"Taking the $k$ -th order derivative with respect to the $t$ variable for $ 0 \\\\le t \\\\le s-1$ , we obtain $\\\\partial _t^k \\\\left(f - {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)}f \\\\right)&(x,t) = \\\\sum _{m= k}^{s-1} \\\\frac{(t-1)^{m-k}}{(m-k)!\",\n \"}\\\\left[ \\\\partial _t^m [f(\\\\cdot ,1)] (\\\\xi )- {\\\\mathsf {Q}}_{n-m,\\\\eta }^{{\\\\mathbb {S}^{d-1}}} \\\\left[\\\\partial _t^m f(\\\\cdot ,1)\\\\right](\\\\xi ) \\\\right] \\\\\\\\& + (-1)^{s-k} \\\\int _{t}^{1} \\\\frac{(v-t)^{s-k-1}}{(s-k-1)!}\",\n \"\\\\left[ \\\\partial _t^s f (x, v)-{\\\\mathsf {Q}}_{n-s,\\\\eta }^{({\\\\beta }+s,0)}\\\\left(\\\\partial _t^s f\\\\right) (x, v) \\\\right] \\\\mathrm {d}v,$ moreover, taking the $s$ -th derivative in the $t$ variable, we obtain $ \\\\partial _t^s \\\\left(f - {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)}f \\\\right)(x,t) = {\\\\mathsf {Q}}_{n-s,\\\\eta }^{({\\\\beta }+s,0)}\\\\left(\\\\partial _t^s f\\\\right) (x, t).$ In the above error formulas for $1\\\\le k \\\\le s-1$ , the integral is over $v \\\\in [t,1]$ while the variable is $(x,v) = (t \\\\xi , v)$ .\",\n \"As it is, we cannot deduce the error estimate for $\\\\partial _t^k \\\\big (f - {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)}f \\\\big )$ immediately from that of $f - {\\\\mathsf {Q}}_{n-s,\\\\eta }^{({\\\\beta }+s,0)}\\\\big (\\\\partial _t^s f\\\\big )$ over ${\\\\mathbb {V}}_0^{d+1}$ if $1 \\\\le k \\\\le s-1$ , although we can if $k =s$ by (REF ).\",\n \"We state the latter case as a corollary.\",\n \"Corollary 4.7 Let $f \\\\in W_p^s({\\\\mathsf {w}}_{{\\\\beta }+s,0})$ .\",\n \"Then, for $1 \\\\le p \\\\le \\\\infty $ , $\\\\left\\\\Vert \\\\partial _t^s f - \\\\partial _t^s {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)}f \\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta }+s,0}}\\\\le c E_n\\\\left( \\\\partial _t^s f \\\\right)_{p, {\\\\mathsf {w}}_{{\\\\beta }+s,0}}.$ We can derive another interesting property of the projection operator for the SOPs, which relies on the property that the Jacobi polynomial $P_n^{({\\\\alpha },-s)}$ is related to $P_{n-s}^{({\\\\alpha },s)}$ , as seen in [17].\",\n \"The following lemma shows that our extension $J_n^{{\\\\alpha },-s}$ satisfies the same property.\",\n \"Theorem 4.8 If $f(x,t) = (1-t)^s g(x,t)$ , then $\\\\operatorname{proj}_n^{{\\\\beta },-s} f = (1-t)^s \\\\operatorname{proj}_{n-s}^{{\\\\beta },s} g.$ By the definition of the ${\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },-s)}$ , it follows from (REF ) that ${\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)} (x,t)= \\\\sigma _{m,n} (1-t)^s {\\\\mathsf {S}}_{m,\\\\ell }^{n-s, ({\\\\beta },s)}(x,t), \\\\quad \\\\sigma _{m,n} := \\\\frac{(n-s-m)!}{(n-m)!}\",\n \"A_{n-m}^{(2m+{\\\\beta }+d-1,0)}.$ Expanding $g$ in terms of the Fourier orthogonal series in $L^2({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{{\\\\beta },1})$ , we obtain $f(x,t) = (1-t)^s g(x,t) \\\\,& = (1-t)^s \\\\sum _{n=0}^\\\\infty \\\\sum _{m=0}^n \\\\sum _\\\\ell \\\\hat{g}_{m,\\\\ell }^{n, ({\\\\beta },s)}{\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },s)}(x,t) \\\\\\\\& = \\\\sum _{n=0}^\\\\infty \\\\sum _{m=0}^n \\\\sum _\\\\ell \\\\hat{g}_{m,\\\\ell }^{n, ({\\\\beta },s)}\\\\sigma _{n+s,m}^{-1} {\\\\mathsf {S}}_{m,\\\\ell }^{n+s, ({\\\\beta },-s)}(x,t).$ Applying the orthogonality in the Sobolev inner product, it follows immediately that $\\\\left\\\\langle f, {\\\\mathsf {S}}_{m,\\\\ell }^{n+s, ({\\\\beta },-s)} \\\\right\\\\rangle _{{\\\\beta },-s} =\\\\sigma _{n+1,m}^{-1} \\\\hat{g}_{m,\\\\ell }^{n,({\\\\beta },s)} \\\\left\\\\langle {\\\\mathsf {S}}_{m,\\\\ell }^{n+s, ({\\\\beta },-s)}, {\\\\mathsf {S}}_{m,\\\\ell }^{n+s, ({\\\\beta },-s)} \\\\right\\\\rangle _{{\\\\beta },-s}, \\\\quad 0 \\\\le m \\\\le n;$ in particular, with $n$ replaced by $n-s$ , we then obtain $ \\\\hat{f}_{m,\\\\ell }^{n,({\\\\beta },-s)} = \\\\sigma _{n,m}^{-s} \\\\hat{g}_{m,\\\\ell }^{n-s,({\\\\beta },s)}, \\\\quad 0 \\\\le m \\\\le n-s.$ For $n-s < m \\\\le n$ , the polynomial $J_{n-m}^{(2m+{\\\\beta }+d-1,-s)}(t)$ is a polynomial of degree at most $s-1$ , so that $\\\\partial _t^s {\\\\mathsf {S}}_{n,\\\\ell }^{n, ({\\\\beta },-1)}(x,t) = 0$ ; moreover, $\\\\partial _t^k f(\\\\xi ,1) =0$ for $0 \\\\le k \\\\le s-1$ , since $f$ contains a factor $(1-t)^s$ .\",\n \"It follows that $\\\\hat{f}_{n,\\\\ell }^{n,({\\\\beta },-s)} = 0$ for $n -s < m \\\\le n$ by the definition of the Sobolev inner product.\",\n \"Consequently, by (REF ) and (REF ) we obtain $(1-t) \\\\operatorname{proj}_{n-s}^{{\\\\beta },s} g\\\\, & = (1-t) \\\\sum _{m=0}^{n-s} \\\\sum _\\\\ell \\\\hat{g}_{m,\\\\ell }^{n-1,({\\\\beta },s)} {\\\\mathsf {S}}_{m,\\\\ell }^{n-s, ({\\\\beta },s)} \\\\\\\\& = \\\\sum _{m=0}^{n} \\\\sum _\\\\ell \\\\hat{f}_{m,\\\\ell }^{n,({\\\\beta },-s)} {\\\\mathsf {S}}_{m,\\\\ell }^{n, ({\\\\beta },-s)} = \\\\operatorname{proj}_{n}^{{\\\\beta },-s}f.$ The proof is completed.\",\n \"Corollary 4.9 Let $f \\\\in L^p({\\\\mathbb {V}}_0^{d+1},{\\\\mathsf {w}}_{{\\\\beta },0})$ and $f(t) = (1-t) g(t)$ .\",\n \"Then, for $1 \\\\le p \\\\le \\\\infty $ , $\\\\left\\\\Vert f - {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)} f \\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },0}}\\\\le c E_n\\\\left(g \\\\right)_{p, {\\\\mathsf {w}}_{{\\\\beta },s}} \\\\le c E_n\\\\left(g \\\\right)_{p, {\\\\mathsf {w}}_{{\\\\beta },0}}$ Since $f = (1-t)^s g$ , we see that ${\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -1)} f = \\\\sum _{k=0}^{2n} \\\\eta \\\\left( \\\\frac{k}{n}\\\\right) (1-t)^s \\\\operatorname{proj}_{k-1}^{{\\\\beta },s} g= (1-t)^s \\\\widetilde{{\\\\mathsf {Q}}}_{n,\\\\eta }^{\\\\beta ,s} g,$ where $\\\\widetilde{{\\\\mathsf {Q}}}_{n,\\\\eta }^{(\\\\beta ,s)}$ is defined as in the proof of Corollary REF .\",\n \"For $p \\\\ge 1$ , we then obtain $\\\\left\\\\Vert f - {\\\\mathsf {Q}}_{n,\\\\eta }^{(\\\\beta , -s)} f \\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },0}}& = \\\\left\\\\Vert (1-t)^s \\\\left[ g - \\\\widetilde{{\\\\mathsf {Q}}}_{n,\\\\eta }^{(\\\\beta , s)} g \\\\right] \\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },0}}\\\\\\\\& \\\\le \\\\left\\\\Vert g - \\\\widetilde{{\\\\mathsf {Q}}}_{n,\\\\eta }^{(\\\\beta , s)} g \\\\right\\\\Vert _{p, {\\\\mathsf {w}}_{{\\\\beta },s}} \\\\le c E_n(g)_{p, {\\\\mathsf {w}}_{{\\\\beta },s}},$ By the definition of $E_n(g)_{p, {\\\\mathsf {w}}}$ , it follows immediately $E_n(g)_{p, {\\\\mathsf {w}}_{{\\\\beta },s}} \\\\le E_n(g)_{p, {\\\\mathsf {w}}_{{\\\\beta },0}}$ .\",\n \"The proof is completed.\"\n ],\n [\n \"Partial differential equation for the Sobolev orthogonal polynomials\",\n \"This section considers partial differential equations satisfied by the Sobolev orthogonal polynomials.\",\n \"As we mentioned in Theorem REF proved in [21], the ordinary orthogonal polynomials in ${\\\\mathcal {V}}_n({\\\\mathbb {V}}_0^{d+1}, {\\\\mathsf {w}}_{-1,{\\\\gamma }})$ , ${\\\\gamma }> -1$ , are eigenfunctions of the differential operator ${\\\\mathcal {D}}_{{\\\\gamma }}$ , defined in (REF ).\",\n \"For the Sobolev orthogonal polynomials, we need ${\\\\gamma }$ to be negative integers.\",\n \"We first consider the action of the differential operator when ${\\\\gamma }$ is a real number.\"\n ],\n [\n \"Eigenfunctions of ${\\\\mathcal {D}}_{{\\\\gamma }}$ for {{formula:3f962e14-6e6a-454f-99e0-f53f9b58f637}}\",\n \"Recall that the operator ${\\\\mathcal {D}}_{{\\\\gamma }}$ is given by ${\\\\mathcal {D}}_{{\\\\gamma }} = t(1-t)\\\\frac{\\\\mathrm {d}^2}{\\\\mathrm {d}t^2} + \\\\big ( d-1 - (d+{\\\\gamma })t \\\\big ) \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}t}+ t^{-1} \\\\Delta _0^{(\\\\xi )}.$ We start with a lemma for the action of this operator.\",\n \"Lemma 5.1 Let $p(t)$ be a polynomial in the $t$ variable and $Y(\\\\xi )$ be a function defined on the unit sphere ${\\\\mathbb {S}^{d-1}}$ .\",\n \"For $k \\\\ge 1$ , ${\\\\mathcal {D}}_{\\\\gamma } \\\\left[(1-t)^k\\\\,p(t) Y(\\\\xi ) \\\\right] \\\\, & = (1-t)^k {\\\\mathcal {D}}_{\\\\gamma +2k}\\\\left[p(t) Y (\\\\xi )\\\\right] \\\\\\\\& - k\\\\,(1-t)^{k-1} [d-1-(d+{\\\\gamma }+k-1)t] p(t)Y(\\\\xi ).\",\n \"$ For $k=1$ , a quick computation from the definition of ${\\\\mathcal {D}}_{{\\\\gamma }}$ shows ${\\\\mathcal {D}}_{{\\\\gamma }}[(1-t) p(t) Y(\\\\xi )]=\\\\, & \\\\big ((1-t) \\\\left[t(1-t)p^{\\\\prime \\\\prime }(t) + [d-1-(d+\\\\gamma +2)t]p^{\\\\prime }(t)\\\\right] \\\\\\\\& - [d-1-(d+\\\\gamma )t]p(t)\\\\big ) Y(\\\\xi )+ t^{-1}\\\\,(1-t) p(t) \\\\, \\\\Delta _0^{(\\\\xi )}Y(\\\\xi ) \\\\\\\\= \\\\, & (1-t){\\\\mathcal {D}}_{\\\\gamma +2}[p(t)Y(\\\\xi )] - [d-1-(d+\\\\gamma )t]p(t)Y(\\\\xi ),$ which verifies (REF ) for $k=1$ .\",\n \"Assume that (REF ) holds for $k\\\\ge 1$ .\",\n \"Then $& {\\\\mathcal {D}}_{\\\\gamma }[(1-t)^{k+1} p(t) Y(\\\\xi )] = {\\\\mathcal {D}}_{{\\\\gamma }} [(1-t)^k (1-t) p(t) Y(\\\\xi )] \\\\\\\\& \\\\quad = (1-t)^k {\\\\mathcal {D}}_{{\\\\gamma }+2k}[(1-t)p(t) Y(\\\\xi )] \\\\\\\\&\\\\qquad - k\\\\,(1-t)^{k-1} [d-1-(d+\\\\gamma +k-1)t][(1-t)p(t) Y(\\\\xi )]\\\\\\\\& \\\\quad = (1-t)^k \\\\left\\\\lbrace (1-t){\\\\mathcal {D}}_{{\\\\gamma }+2k+2}[p(t) Y(\\\\xi )] - [d-1-(d+\\\\gamma +2k)t]p(t) Y(\\\\xi )\\\\right\\\\rbrace \\\\\\\\&\\\\qquad - k\\\\,(1-t)^{k} [d-1-(d+\\\\gamma +k-1)t]p(t) Y(\\\\xi )\\\\\\\\& \\\\quad = (1-t)^{k+1} {\\\\mathcal {D}}_{{\\\\gamma }+2k+2}[p(t) Y(\\\\xi )] - (k+1)(1-t)^k [d-1-(d+\\\\gamma +k)t]p(t) Y(\\\\xi ),$ so that (REF ) holds for $k+1$ and the proof is completed by induction.\",\n \"Theorem 5.2 For $0\\\\le j \\\\le n$ , let $Z_{j,n}(x,t) = p(t) Y(x)$ with $p$ being a polynomial of degree $j$ in one variable and $Y$ be a solid harmonic polynomial in ${\\\\mathcal {H}}_{n-j}^{d,0}$ .\",\n \"For $\\\\gamma \\\\in {\\\\mathbb {R}}$ , the only polynomial $p$ for which $Z_{j,n}$ satisfies ${\\\\mathcal {D}}_{{\\\\gamma }} Z (x,t) = \\\\lambda _n^{(\\\\gamma )} Z(x,t), \\\\qquad {\\\\lambda }_n^{({\\\\gamma })} = -n(n+{\\\\gamma }+d-1),$ is the Jacobi polynomial $p(t) = c_j\\\\,P_j^{(2n-2j+d-2, \\\\gamma )}(1-2t)$ , where $c_j$ is an appropriate constant, if $2n+\\\\gamma +d-r-1\\\\ne 0$ for $0\\\\le r\\\\le j$ .\",\n \"Without losing generality, we assume $p$ is a monic polynomial, $p(t) = \\\\sum _{i=0}^j \\\\,(1-t)^i\\\\,a_{j,i}, \\\\quad a_{j,j} = 1.$ Our objective is to determine the coefficients $a_{j,i}$ such that $Z_j$ satisfies (REF ).\",\n \"First, we observe that (REF ) holds if $p(t) =1$ .\",\n \"In this case $Y \\\\in {\\\\mathcal {H}}_n^{d,0}$ and $Y(x) = t^n Y(\\\\xi )$ .\",\n \"Hence, taking the derivative over $t$ and using the relation (REF ), a quick computation shows that $ {\\\\mathcal {D}}_{{\\\\gamma }} Y(x)= &\\\\, {\\\\mathcal {D}}_{{\\\\gamma }}[t^n Y(\\\\xi )] \\\\\\\\= &\\\\, n (n+d-2)Y(\\\\xi ) - m (m+{\\\\gamma }+d-1) t^m Y(\\\\xi ) + t^{m-1} \\\\Delta _0 Y(\\\\xi ) \\\\\\\\= &\\\\, -m(m+d+{\\\\gamma }-1) t^m Y(\\\\xi ) = -m(m+d+{\\\\gamma }-1) Y(x).\",\n \"$ For $j \\\\ge 0$ , we use the identity (REF ) to obtain $& {\\\\mathcal {D}}_{{\\\\gamma }} Z_j(x,t) = \\\\sum _{i=0}^j a_{j,i} {\\\\mathcal {D}}_{{\\\\gamma }}\\\\left[(1-t)^i t^{n-j} Y(\\\\xi )\\\\right] \\\\\\\\& = \\\\sum _{i=0}^j a_{j,i} \\\\left[(1-t)^i {\\\\mathcal {D}}_{{\\\\gamma }+ 2i}[Y(x)] - i(1-t)^{i-1}[d-1-(d+\\\\gamma +i-1)t] Y(x)\\\\right]$ and then apply (REF ) for $Y\\\\in {\\\\mathcal {H}}_{n-j}^{d,0}$ and simplify to obtain ${\\\\mathcal {D}}_{{\\\\gamma }} Z_j(x,t) =\\\\, & \\\\sum _{i=0}^{j} \\\\,(1-t)^i \\\\,a_{j,i} [\\\\lambda _{n-j}^{(\\\\gamma +2i)} - i\\\\,(d+\\\\gamma +i-1)]\\\\,Y(x)\\\\\\\\&+\\\\sum _{i=0}^{j-1} (i+1)(1-t)^{i}\\\\,(\\\\gamma +i+1) \\\\,a_{j,i+1}\\\\,Y(x)\\\\\\\\= \\\\, & (1-t)^j \\\\,\\\\lambda _n^{(\\\\gamma )}\\\\,Y(x) +\\\\sum _{i=0}^{j-1} \\\\,(1-t)^i \\\\\\\\& \\\\times \\\\lbrace a_{j,i} \\\\,[\\\\lambda _{n-j}^{(\\\\gamma +2i)} - i(d+\\\\gamma +i-1)] + a_{j,i+1}(i+1)(\\\\gamma +i+1)\\\\rbrace Y(x),$ since $a_{j,j} = 1$ .\",\n \"Thus, in order to have $Z_{j,n} (x,t)$ satisfying (REF ), we must have $a_{j,i} \\\\,[\\\\lambda _{n-j}^{(\\\\gamma +2i)} - i(d+\\\\gamma +i-1)] + a_{j,i+1}(i+1)(\\\\gamma +i+1) = a_{j,i} \\\\lambda _n^{(\\\\gamma )}$ for $0 \\\\le i \\\\le j-1$ , which leads to the recurrence relation $a_{j,i} = - \\\\frac{(i+1)(\\\\gamma +i+1)}{(j-i)(2n+\\\\gamma +d-1-j+i)}a_{j,i+1} , \\\\quad i=0, 1, \\\\ldots , j-1.$ Solving this recurrence relation and using $a_{j,j} = 1$ , we obtain $a_{j,i} = (-1)^{j-i}\\\\frac{(i+1)_{j-i}(\\\\gamma +i+1)_{j-i}}{(j-i)!\",\n \"(2n+\\\\gamma +d- 1-j+i)_{j-i}}, \\\\quad 0\\\\le i \\\\le j.$ Using the identity $(a+i)_{j-i} = (a)_j/(a)_i$ and $(j-i)!\",\n \"= (-1)^i j!\",\n \"/ (-j)_i$ , we can rewrite it as $a_{j,i} = (-1)^j c_j \\\\frac{(-j)_i (2n+{\\\\gamma }+d-j-1)_i}{i!\",\n \"({\\\\gamma }+1)_i}, \\\\quad c_j= \\\\frac{({\\\\gamma }+1)_j}{(2n+{\\\\gamma }+d-j-1)_j},$ which is well-defined if $(2n+{\\\\gamma }+d-j-1)_j \\\\ne 0$ .\",\n \"Consequently, the polynomial $p$ must be given by $p(x) \\\\,& = (-1)^j c_j\\\\phantom{i}_2F_1\\\\left( \\\\begin{matrix} -j, 2n+{\\\\gamma }+d-j-1 \\\\\\\\ {\\\\gamma }+1 \\\\end{matrix}; 1-t \\\\right) \\\\\\\\& = (-1)^jc_j P_j^{({\\\\gamma }, 2n-2j+d-2)}(2t-1) =c_j P_j^{(2n-2j+d-2, {\\\\gamma })}(1-2t),$ which completes the proof.\",\n \"For ${\\\\gamma }> -1$ , the theorem recovers Theorem REF established in [21].\",\n \"We discuss the case when ${\\\\gamma }= -s$ and $s \\\\in {\\\\mathbb {N}}$ in the next subsection.\"\n ],\n [\n \"Eigenfunctions for ${\\\\mathcal {D}}_{-s}$\",\n \"As a corollary of Theorem REF , we obtain the following result for ${\\\\mathcal {D}}_{-s}$ .\",\n \"Proposition 5.3 Let $s \\\\in {\\\\mathbb {N}}$ and $n \\\\in {\\\\mathbb {N}}_0$ .\",\n \"Then the polynomials ${\\\\mathsf {S}}_{m,\\\\ell }^{n, (-1,-s)}$ are eigenfunctions of ${\\\\mathcal {D}}_{-s}$ if and only if $m \\\\le n-s$ and $m =n$ .\",\n \"By (REF ), the polynomials ${\\\\mathsf {S}}_{m,-\\\\ell }^{n,(-1,-s)}$ can be written as ${\\\\mathsf {S}}_{m,\\\\ell }^{n,(-1,-s)}(x,y) = c_{n-m} P_{n-m}^{(2m+d-2,-s)}(1-2t) Y_\\\\ell ^m(x), \\\\quad n-m \\\\ge s,$ so that Theorem REF applies for $n - m \\\\ge s$ , thus ${\\\\mathsf {S}}_{m,\\\\ell }^{n, (-1,-s)}$ satisfies (REF ).\",\n \"If $n = m$ , then ${\\\\mathsf {S}}_{n, \\\\ell }^{n, (-1,-s)}(x,t) = Y_\\\\ell ^n(x)$ .\",\n \"As a result, (REF ) holds with $p(z) =1$ .\",\n \"For $1 \\\\le n-m \\\\le s-1$ , however, the polynomial $J_{n-m}^{(2m+d-2,-s)}(t) = (1-t)^{n-m}/(n-m)!$ is not the Jacobi polynomial $P_{n-m}^{(2m+d-2,-s)}(t)$ , hence ${\\\\mathsf {S}}_{n, \\\\ell }^{n, (-1,-s)}$ is not a solution of (REF ).\",\n \"In the case $s =1$ , the set $\\\\lbrace m: n -m \\\\ge s\\\\rbrace \\\\cup \\\\lbrace n\\\\rbrace = \\\\lbrace m: 1 \\\\le m \\\\le n\\\\rbrace $ .\",\n \"Hence, we obtain the following corollary.\",\n \"Theorem 5.4 For ${\\\\gamma }= -1$ , all elements in ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{-1,-1})$ are eigenfunctions of ${\\\\mathcal {D}}_{-1}$ ; that is, ${\\\\mathcal {D}}_{-1} Z = - n (n+ d-2) Z, \\\\qquad \\\\forall Z \\\\in {\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{-1,-1}).$ Moreover, the space ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{-1,-1})$ satisfies a decomposition ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{-1,-1}) = {\\\\mathcal {H}}_n^{d,0} \\\\cup (1-t) {\\\\mathcal {V}}_{n-1}^d({\\\\mathsf {w}}_{-1,1}), \\\\qquad n =0,1,\\\\ldots .$ We only need to prove the decomposition.\",\n \"If $m \\\\le n-1$ , it follows from from (REF ) that ${\\\\mathsf {S}}_{m,\\\\ell }^{n,({\\\\beta },-1)} = c (1-t) {\\\\mathsf {S}}_{m,\\\\ell }^{n-1,({\\\\beta },1)} \\\\in (1-t) {\\\\mathcal {V}}_{n-1}^d({\\\\mathsf {w}}_{-1,1})$ , whereas if $m = n$ , then ${\\\\mathsf {S}}_{n,\\\\ell }^{n,({\\\\beta },-s)} = Y_\\\\ell ^n \\\\in {\\\\mathcal {H}}_n^{d,0}$ .\",\n \"The theorem shows, in particular, that ${\\\\mathcal {D}}_{{\\\\gamma }}$ has the complete eigenspaces if ${\\\\gamma }\\\\ge -1$ .\",\n \"For $s = 2,3,\\\\ldots $ , however, not every element of the space ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{-1,-s})$ is an eigenfunction of ${\\\\mathcal {D}}_{ -s}$ by Theorem REF .\",\n \"We can, however, identify the eigenspaces of the operator as follows.\",\n \"Theorem 5.5 For $s = 2,3,\\\\ldots $ , define ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s}):= {\\\\mathcal {H}}_{n}^{d,0} \\\\bigcup _{j=1}^{s-1}P_j^{(2n-2j+d-2,-s)}(1-2t){\\\\mathcal {H}}_{n-j}^{d,0}\\\\cup (1-t)^s {\\\\mathcal {V}}_{n-s}^d ({\\\\mathsf {w}}_{-1,s}).$ Then ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s})$ is the eigenspace of ${\\\\mathcal {D}}_{-s}$ ; more precisely, ${\\\\mathcal {D}}_{-s} Z=\\\\lambda _n^{(-s)}Z$ for all $Z \\\\in {\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s})$ .\",\n \"Moreover, the space satisfies $\\\\dim {\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s}) = \\\\binom{n+d-1}{d}$ if $-2n-d+j+s+2$ is not a positive integer between 1 and $j$ for $1 \\\\le j \\\\le s-1$ and $n \\\\ge j$ .\",\n \"As in the case of $s = 1$ , using (REF ), the statement on the eigenspace follows readily from Theorem REF .\",\n \"As it is shown for the dimension of ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{{\\\\beta },-s})$ , the space ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s})$ has the full dimension $\\\\binom{n+d-1}{d}$ if the Jacobi polynomials $P_j^{(2n-2j+d-2,-s)}$ does not have the degree reduction, which holds if $2n-j+d-2-s +k =0$ for a certain integer $k$ , $1 \\\\le k \\\\le j$ , by [17].\",\n \"The theorem shows that the operator ${\\\\mathcal {D}}_{-s}$ has a complete basis of polynomials as eigenfunctions only if the restriction on $-2n-d+j+s+2$ as stated holds.\",\n \"Example 5.6 If $s = 2$ , then the space ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-2})$ is given by ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-2}) = {\\\\mathcal {H}}_n^d \\\\cup \\\\left(1+ (2n+d-4)(1-t) \\\\right) {\\\\mathcal {H}}_{n-1}^d\\\\cup (1-t)^s {\\\\mathcal {V}}_{n-2}^d ({\\\\mathsf {w}}_{-1,2})$ and it has the full dimension if $-2n-d+5 \\\\ne 1$ or $2n-d+4 \\\\ne 0$ for $n \\\\ge 1$ .\",\n \"Thus, ${\\\\mathcal {D}}_{-2}$ has a complete basis of polynomials as eigenfunctions if $d$ is odd.\",\n \"In the case that $d$ is even, $ {\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-2})$ is not well-defined for $n = d/2-2$ .\",\n \"We note that the space ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s})$ does not coincide with ${\\\\mathcal {V}}_n^d({\\\\mathsf {w}}_{-1,-s})$ for $s =2,3,\\\\ldots $ .\",\n \"In the case of $s = -2$ and $d$ is odd, one may ask if there is a Sobolev inner product for which the polynomials in the space ${\\\\mathcal {U}}_n^d({\\\\mathsf {w}}_{-1,-s})$ are orthogonal.\",\n \"The authors have no relevant financial or non-financial interests to disclose.\"\n ]\n]"},"id":{"kind":"string","value":"2209.08186"}}},{"rowIdx":2398,"cells":{"text":{"kind":"list like","value":[["Computing analytic Bayes factors from summary statistics in\n repeated-measures designs"],["Abstract Bayes factors are an increasingly popular tool for indexing evidence from experiments.","For two competing population models, the Bayes factor reflects the relative likelihood of observing some data under one model compared to the other.","In general, computing a Bayes factor is difficult, because computing the marginal likelihood of each model requires integrating the product of the likelihood and a prior distribution on the population parameter(s).","In this paper, we develop a new analytic formula for computing Bayes factors directly from minimal summary statistics in repeated-measures designs.","This work is an improvement on previous methods for computing Bayes factors from summary statistics (e.g., the BIC method), which produce Bayes factors that violate the Sellke upper bound of evidence for smaller sample sizes.","The new approach taken in this paper extends requires knowing only the $F$-statistic and degrees of freedom, both of which are commonly reported in most empirical work.","In addition to providing computational examples, we report a simulation study that benchmarks the new formula against other methods for computing Bayes factors in repeated-measures designs.","Our new method provides an easy way for researchers to compute Bayes factors directly from a minimal set of summary statistics, allowing users to index the evidential value of their own data, as well as data reported in published studies."],["Introduction","In this paper, we develop an analytic formula to compute Bayes factors for repeated-measures designs using only minimal summary statistics from the analysis of variance.","Previous attempts to quantify evidence from summary statistics in repeated-measures designs have all relied upon the BIC approximation [21], [4], [6].","In contrast, our new method avoids the need for approximation and produces an exact (analytic) Bayes factor directly from the observed $F$ -statistic and associated degrees of freedom.","This paper extends the development of the between-subjects Pearson Bayes factor [7] to also consider repeated-measures designs, thus widening the scope and its use for indexing evidential value from summary statistics.","Further, our formula is gives researchers a way to compute Bayes factors in repeated-measures designs that does not involve integration [17] or approximation [15], [6].","As such, users can easily compute Bayes factors for their own repeated-measures data, and also for any results that are reported in the scientific literature – even null effects.","Our method thus further affords researchers the ability to gauge the evidential value of a collection of published results in a straightforward way without the need for raw data."],["Background","We begin with some background on inference in experimental designs with repeated measurements.","Let us consider an experiment where $k$ repeated measurements are taken from each of $n$ experimental subjects.","We then have a total of $N=nk$ observations.","We then place a linear mixed model structure on these observations: $\\mathbf {y} = y_{ij} = \\mu + \\alpha _j + \\pi _i + \\varepsilon _{ij}; \\hspace{8.53581pt} i=1,\\dots ,n;\\hspace{2.84526pt}j=1\\cdots ,k,$ where $\\mu $ represents the grand mean, $\\alpha _j$ represents the treatment effect associated with group $j$ , $\\pi _i$ represents the effect of subject $i$ , and $\\varepsilon _{ij} \\sim \\mathcal {N}(0,\\sigma _{\\varepsilon }^2)$ .","Note that the repeated-measures design induces a correlated structure on the data, so the $N=nk$ observations are not independent.","As a result, we have $n(k-1)$ independent observations [14].","Ultimately, we want to know whether there is a treatment effect (i.e., if there are differences among the treatment groups induced by the repeated measurements).","To proceed with hypothesis testing, we define two competing models: $\\mathcal {H}_0:\\alpha _j=0\\text{ for }j=1,\\dots ,k\\\\\\mathcal {H}_1:\\alpha _j\\ne 0\\text{ for some }j$ A classical approach to model selection is the analysis of variance (ANOVA) procedure [9].","The analysis of variance procedure works by initially partitioning the total variance in the data $\\mathbf {y}$ into two sources: the variance between the treatment groups, and the residual variance that is left over after accounting for this treatment variability.","In repeated-measures designs, we further partition the residual variance by separately breaking out the variance between subjects.","Technically, this is done by computing $SS$ (sum of squares) terms, where $SSA = n\\sum _{j=1}^k(\\overline{y}_{\\cdot j}-\\overline{y}_{\\cdot \\cdot })^2, \\hspace{28.45274pt}SSB = k\\sum _{i=1}^n(\\overline{y}_{i\\cdot }-\\overline{y}_{\\cdot \\cdot })^2$ represent the sums of squares corresponding to the treatment effect and the subject effect, respectively; $SST = \\sum _{i=1}^n \\sum _{j=1}^k (y_{ij} - \\overline{y}_{\\cdot \\cdot })^2$ represents the total sum of squares, and $SSR = SST - SSA - SSB$ represents the residual sum of squares left over after accounting for both treatment and subject effects.","From here, we compute the $F$ -statistic for the treatment effect in our design as $F=\\frac{SSA}{SSR}\\cdot \\frac{df_{\\text{residual}}}{df_{\\text{treatment}}} = \\frac{SSA}{SSR}\\cdot \\frac{(n-1)(k-1)}{k-1} = \\frac{SSA}{SSR}\\cdot (n-1).$ Conceptually, the $F$ statistic represents the ratio of between-groups variance to the residual variance left over after removing the subject variability.","For inference, we compute the likelihood of the observed data $\\mathbf {y}$ under the null hypothesis $\\mathcal {H}_0$ .","Specifically, we find the probability of obtaining the observed $F$ statistic (or greater) under $\\mathcal {H}_0$ .","If this probability, called the $p$ -value, is small, this indicates that the data $\\mathbf {y}$ are rare under $\\mathcal {H}_0$ , so we can reject $\\mathcal {H}_0$ in favor of the alternative hypothesis $\\mathcal {H}_1$ .","Unfortunately, this classical approach to inference is fraught with some issues that undermine its use [21].","First, the $p$ -value is not equivalent to the posterior probability $p(\\mathcal {H}_0\\mid \\mathbf {y})$ .","Regardless, many researchers believe (incorrectly) that a $p$ -value represents the probability that $\\mathcal {H}_0$ is true [11], and thus take a small $p$ -value to represent evidence for $\\mathcal {H}_1$ .","However, [1] demonstrated that $p$ -values classically overestimate this evidence.","For example, with a $t$ -test performed on a sample size of 100, a $p$ -value of 0.05 transforms to $p(\\mathcal {H}_0\\mid \\mathbf {y})=0.52$ – rather than reflecting evidence for $\\mathcal {H}_1$ , this small $p$ -value reflects only a slight preference for $\\mathcal {H}_0$ .","Second, the evidence provided for $\\mathcal {H}_1$ from this procedure is only indirect – whereas the $p$ -value does measures the predictive adequacy of $\\mathcal {H}_0$ , it makes no such measurement of predictive adequacy for $\\mathcal {H}_1$ .","Thus, in this paper we consider a Bayesian approach to model selection.","Instead of computing a $p$ -value, we instead compute a Bayes factor [12].","The Bayes factor, denoted $\\text{BF}_{01}$ is defined as the ratio of marginal likelihoods for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , respectively.","That is, $\\text{BF}_{01} = \\frac{p(\\mathbf {y}\\mid \\mathcal {H}_0)}{p(\\mathbf {y}\\mid \\mathcal {H}_1)}.$ As such, $\\text{BF}_{01}$ indexes the relative likelihood of observing data $\\mathbf {y}$ under $\\mathcal {H}_0$ compared to $\\mathcal {H}_1$ , so $\\text{BF}_{01}>1$ is taken as evidence for $\\mathcal {H}_0$ over $\\mathcal {H}_1$ .","Similarly, $\\text{BF}_{01}<1$ is taken as evidence for $\\mathcal {H}_1$ .","Note that $\\text{BF}_{01} = 1/\\text{BF}_{10}$ , so if $\\text{BF}_{01}<1$ , we can take the reciprocal and equivalently write $BF_{10} > 1$ .","For this reason, it is canonical to report a Bayes factor as a number greater than 1.","For example, instead of reporting $\\text{BF}_{01} = 0.25$ , we will take the reciprocal and report $\\text{BF}_{10} = 1/0.25 = 4$ .","Both representations imply that the observed data are 4 times more likely under $\\mathcal {H}_1$ than under $\\mathcal {H}_0$ .","An equivalent definition of the Bayes factor $\\text{BF}_{01}$ is the extent to which the prior odds for $\\mathcal {H}_0$ over $\\mathcal {H}_1$ are updated after observing data.","Thus, the ratio of posterior probabilities for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ can be found by multiplying the ratio of prior probabilities by $\\text{BF}_{01}$ : $\\frac{p(\\mathcal {H}_0\\mid \\mathbf {y})}{p(\\mathcal {H}_1\\mid \\mathbf {y})} = \\text{BF}_{01}\\cdot \\frac{p(\\mathcal {H}_0)}{p(\\mathcal {H}_1)}.$ One immediate consequence of Equation REF is that the Bayes factor can be directly transformed into posterior probability of $\\mathcal {H}_0$ (or if desired, $\\mathcal {H}_1$ ).","To see this, we simply solve Equation REF for the posterior probability $p(\\mathcal {H}_0\\mid \\mathbf {y})$ and then apply Bayes' theorem.","This gives $p(\\mathcal {H}_0\\mid \\mathbf {y}) &= \\text{BF}_{01} \\cdot \\frac{p(\\mathcal {H}_0)}{p(\\mathcal {H}_1)} \\cdot p(\\mathcal {H}_1\\mid \\mathbf {y})\\\\&= \\frac{\\text{BF}_{01} \\cdot p(\\mathcal {H}_0) \\cdot p(\\mathbf {y}\\mid \\mathcal {H}_1)\\cdot p(\\mathcal {H}_1)}{p(\\mathcal {H}_1)\\cdot p(\\mathbf {y})}\\\\&= \\frac{\\text{BF}_{01}\\cdot p(\\mathcal {H}_0)\\cdot p(\\mathbf {y}\\mid \\mathcal {H}_1)}{p(\\mathbf {y}\\mid \\mathcal {H}_0)\\cdot p(\\mathcal {H}_0) + p(\\mathbf {y}\\mid \\mathcal {H}_1)\\cdot p(\\mathcal {H}_1)}.$ Dividing numerator and denominator by the marginal likelihood $p(\\mathbf {y}\\mid \\mathcal {H}_1)$ yields $p(\\mathcal {H}_0\\mid \\mathbf {y}) = \\frac{\\text{BF}_{01}\\cdot p(\\mathcal {H}_0)}{\\text{BF}_{01}\\cdot p(\\mathcal {H}_0) + p(\\mathcal {H}_1)}.$ By a similar argument, we can show $p(\\mathcal {H}_1\\mid \\mathbf {y}) = \\frac{\\text{BF}_{10}\\cdot p(\\mathcal {H}_1)}{\\text{BF}_{10}\\cdot p(\\mathcal {H}_1) + p(\\mathcal {H}_0)}.$ Commonly, we take a default assumption that both models are equally likely a priori, permitting us to set $p(\\mathcal {H}_0)=p(\\mathcal {H}_1) = 0.5$ .","In this case, we get the following simplified formulas: $p(\\mathcal {H}_0\\mid \\mathbf {y}) = \\frac{\\text{BF}_{01}}{\\text{BF}_{01}+1}, \\hspace{28.45274pt} p(\\mathcal {H}_1\\mid \\mathbf {y}) = \\frac{\\text{BF}_{10}}{\\text{BF}_{10}+1}.$ Despite the conceptual simplicity of the Bayes factor, actually computing $\\text{BF}_{01}$ is usually quite difficult.","From the conceptual definition, we see that the Bayes factor $\\operatorname{BF}_{01}$ is the ratio of likelihoods for $\\mathbf {y}$ under 0 and 1, respectively.","These likelihoods are marginal likelihoods, where each likelihood function $f_i(\\mathbf {y} \\mid \\theta _i)=f(\\mathbf {y} \\mid \\theta _i, i)$ ($i=0,1$ ) is integrated over the prior distribution $\\pi _i(\\theta _i)=\\pi (\\theta _i,i)$ for the vector of model parameters $\\theta _i$ associated with model $i$ .","That is, $\\operatorname{BF}_{01} = \\frac{\\int f_0(\\theta _0)\\pi _0(\\theta _0)d\\theta _0}{\\int f_1(\\theta _1)\\pi _1(\\theta _1)d\\theta _1} \\;.$ The marginal likelihoods that form the numerator and denominator of the Bayes factor are in general very difficult to compute.","Even if a researcher has sufficient mathematical background to consider this computation, most techniques of integration are difficult to apply.","As sample size increases, the likelihood becomes highly peaked at its maximum.","But with no prior knowledge of where the maximum occurs, numerical integration methods often fail to converge around this highly peaked area.","Also, the parameter vector $\\theta _i$ is often of high dimension.","Combined with the peaked integrand, getting numerical methodsto converge is like “finding a needle in a haystack” [20].","Thus, our primary goal in this work is to make simplifying assumptions which lead to analytic solutions (i.e., without integral representation) for these marginal likelihoods."],["The BIC approximation","One classic approach to avoid computing marginal likelihoods is to use a calculus trick to approximate them instead.","This method is called the BIC approximation [16], [12], [21], [14], and it works by constructing a second-order Taylor approximation of $\\ln m_i(\\mathbf {y}) = \\ln P(\\mathbf {y} \\mid \\theta _i, i)$ centered at the posterior mode for $\\theta _i$ .","Since all terms in the Taylor approximation containing the second-derivative or higher are removed, the result only approximates the log marginal likelihood.","However, [16] showed that placing a noninformative unit information prior on $\\theta _i$ , the order of the error term in $\\mathcal {O}(1/\\sqrt{N})$ .","This results in the approximation $\\ln m_i(\\mathbf {y}) \\approx \\ln P(\\mathbf {y} \\mid \\hat{\\theta }_i,i) - \\frac{k_i}{2}\\ln N$ where $\\hat{\\theta }_i$ is the maximum likelihood estimate for $\\theta _i$ under $i$ , $k_i$ is the number of parameters in $i$ , and $N$ is the total number of independent observations in $\\mathbf {y}$ .","Equation REF relates quite well to the Bayesian information criterion (BIC) of [18]: $\\operatorname{BIC}(i)=-2\\ln L_i + k_i\\ln N,$ where $L$ is the maximum likelihood estimate for model $i$ .","Combining these two equations yields $\\ln m_i(\\mathbf {y}) \\approx -\\frac{1}{2}\\operatorname{BIC}(i) \\; ,$ or equivalently $m_i(\\mathbf {y}) \\approx \\exp \\Bigl (-\\frac{1}{2}\\operatorname{BIC}(i)\\Bigr ).$ It is then simple to derive the following approximation: $\\operatorname{BF}_{01} = \\frac{m_0(\\mathbf {y})}{m_1(\\mathbf {y})} & \\approx \\frac{\\exp \\Bigl (-\\frac{1}{2}\\operatorname{BIC}(0)\\Bigr )}{\\exp \\Bigl (-\\frac{1}{2}\\operatorname{BIC}(1)\\Bigr )}\\\\ \\nonumber &= \\exp \\Biggl (\\frac{\\operatorname{BIC}(1) - \\operatorname{BIC}(0)}{2}\\Biggr ).$ To use Equation REF for computing Bayes factors, we need only know the BIC values for models 0 and 1.","In the context of analysis of variance presented earlier, the BIC can be calculated [16] as $BIC &= N\\ln (1-R^2)+k\\ln N\\\\& = N\\ln \\Biggl (\\frac{SSR}{SST}\\Biggr ) + k\\ln N\\\\ ; .$ One downside to the BIC method is that it requires users to have the “raw” data available in order to compute $SSR$ and $SST$ .","One way to improve Equation REF is recast the Bayes factor computation to a form that requires only summary statistics.","[4] initially did this for between-subjects designs, and then extended that result to the context of repeated-measures designs [6] by deriving the formula: $\\text{BF}_{01} \\approx \\sqrt{(nk-n)^{k-1}\\cdot \\Biggl (1+\\frac{F}{n-1}\\Biggr )^{n-nk}} \\; .$ The following example nicely illustrates the use of (and a problem with) Equation REF .","In a repeated-measures ($k=2$ ) study with $n=18$ participants, [8] observed that mean solution times to subtraction problems were 43 milliseconds faster when the subtraction sign was briefly presented before the actual problem itself, $F(1,17) = 27.17$ , $p<0.001$ .","Using Equation REF , we can compute the BIC Bayes factor for these observed data as: $\\text{BF}_{01} & \\approx \\sqrt{(nk-n)^{k-1}\\cdot \\Biggl (1+\\frac{F}{n-1}\\Biggr )^{n-nk}}\\\\& = \\sqrt{(18\\cdot 2-18)^{2-1}\\cdot \\Biggl (1+\\frac{27.17}{18-1}\\Biggr )^{18-18\\cdot 2}}\\\\& = \\sqrt{18^1 \\cdot \\Biggl (1 + \\frac{27.17}{17}\\Biggr )^{-18}}\\\\& = \\sqrt{18\\cdot (2.5982)^{-18}}\\\\& = 0.0007863 \\;.$ By taking the reciprocal and casting this Bayes factor as support for $\\mathcal {H}_1$ , we have $\\text{BF}_{10} \\approx 1/0.0007863 = 1271.79$ .","Thus, the BIC Bayes factor tells us that the observed data are approximately 1272 times more likely under $\\mathcal {H}_1$ than under $\\mathcal {H}_0$ .","It is important to remember that this Bayes factor is an approximation.","This fact is made especially salient by considering the following.","[19] showed that under a reasonable class of prior distributions for $p$ -values, an upper bound for the Bayes factor can be computed directly from the $p$ -value as $\\operatorname{BF}_{10} \\le -\\frac{1}{e\\cdot p\\ln (p)}.$ We will now compute this Sellke bound for our previous example.","First, we note that for $F(1,17)=27.17$ , the associated $p$ -value is $p=0.0000704$ .","This gives us the upper bound $\\operatorname{BF}_{10} & \\le -\\frac{1}{e\\cdot 0.0000704\\cdot \\ln (0.0000704)}\\\\&=546.53.$ From this, the limitation of the BIC approximation is clear.","Our computed Bayes factor of 1272 greatly exceeds the Sellke bound of 546.53.","In fact, Figure REF shows that this problem persists over a large range of $p$ -values.","In the figure, the solid line represents the Sellke bound $B(p)$ for $p$ -values ranging between 0 and 0.02.","The dashed line represents the associated repeated-measures BIC Bayes factor of [6], given design parameters equivalent to [8] (i.e., $k=2$ repeated-measures conditions and $n=18$ subjects).","Figure: Plot showing that the repeated-measures BIC Bayes factor (dashed line) of is greater than the Sellke bound (solid line) for pp-values ranging between p≈0p\\approx 0 and p=0.02p=0.02."],["Analytic Bayes factors for repeated-measures designs","Against the background of the previous section, we think the need for an analytic Bayes factor for repeated-measures designs is well motivated.","Fortunately, recent work by [22] takes an important first step toward this goal.","Wang and Sun, motivated by the the work of [10], started with a random effects linear model on the observed data $\\mathbf {y}$ : $\\mathbf {y} = y_{ij}=\\mu + \\alpha _j + \\pi _i + \\varepsilon _{ij},$ where $\\alpha _j \\sim \\mathcal {N}(0,\\sigma ^2_a)$ , $\\pi _i \\sim \\mathcal {N}(0,\\sigma ^2_p)$ and $\\varepsilon _{ij} \\sim \\mathcal {N}(0,\\sigma ^2)$ .","Cast in this slightly different context of random effects versus the classical fixed effects model, the analysis of variance procedure amounts to testing whether the random effects term $\\alpha _j$ is identically 0.","To capture this constraint, the competing models are defined in a slightly different manner: $0:\\sigma ^2_a=0 \\text{ versus }1:\\sigma _a^2\\ne 0.$ With this setup, [10] placed noninformative priors on $\\mu $ and $\\sigma $ under both 0 and 1 and considered a proper prior on the ratio of variance components $\\tau = \\sigma ^2_a/\\sigma ^2$ under 1.","With this prior specification, Garcia-Donato and Sun showed $BF_{10} = \\int _0^{\\infty }(1+\\tau n)^{\\frac{1-k}{2}}\\Biggl (1-\\frac{\\tau n}{1+\\tau n}\\cdot \\frac{SSA}{SST}\\Biggr )^{\\frac{1-N}{2}}\\cdot \\pi (\\tau )d\\tau $ where $\\pi (\\tau )$ is left up to the analyst to choose.","[22] used a Pearson Type VI distribution, given by: $\\pi ^{PT}(\\tau ) = \\frac{\\kappa (\\kappa \\tau )^{\\beta }(1+\\kappa \\tau )^{-\\alpha -\\beta -2}}{\\mathcal {B}(\\alpha +1, \\beta +1)}I_{(0,\\infty )}(\\tau )$ where $\\alpha >-1$ and $\\beta >-1$ are shape parameters and $\\kappa >0$ is a scale parameter, and $\\mathcal {B}(x,y) = \\int _0^1t^{x-1}(1-t)^{y-1}dt$ is the standard Beta function.","Wang and Sun further reduced the problem of specifying the prior $\\pi ^{PT}$ to choosing one single parameter $\\alpha \\in [-\\frac{1}{2}, 0]$ .","They did this by taking $\\kappa =n$ and $\\beta = \\frac{N-k}{2}-\\alpha -2$ .","A plot of this prior can be seen in Figure REF ; here, we take the values $n=18$ , $k=2$ , and $N=36$ from our example above, thus setting $\\kappa =18$ and $\\beta =\\frac{36-2}{2}-\\alpha -2$ , where $\\alpha $ ranges among the values $-\\frac{1}{2}, -\\frac{1}{4}, -\\frac{1}{10}, 0$ .","As we can see, as $\\alpha $ decreases from to 0 to $-\\frac{1}{2}$ , $\\tau $ becomes more dispersed and less peaked around the mode.","This places more prior mass on larger treatment effects than we would see for values of $\\alpha $ closer to 0.","Figure: A Pearson Type VI prior for τ\\tau , plotted as a function of shape parameter α\\alpha .Given this choice of prior and simplified parameterization, [22] proved that the Bayes factor derived by Garcia-Donato and Sun (Equation REF ) simplifies to an analytic expression without the need for integral representation: $\\operatorname{BF}_{10} = \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{N-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{N-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{N-k-2}{2}} \\; .$ To apply the Wang and Sun method for computing $\\text{BF}_{10}$ (Equation REF ) in a repeated-measures context, we apply a method of [14] and replace $N$ by $n(k-1)$ [3], [2], [6].","This readily gives: $ \\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{n(k-1)-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{n(k-1)-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{n(k-1)-k-2}{2}} \\nonumber \\\\&= \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{nk-n-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{nk-n-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{nk-n-k-2}{2}} \\; .$ We now consider some identities that will greatly simplify Equation REF .","First, let $x$ denote the between-groups degrees of freedom, $df_{\\text{treatment}}$ , and let $y$ denote the residual degrees of freedom, $df_{\\text{residual}}$ .","As seen earlier, this gives $x=k-1$ and $y=(n-1)(k-1) = nk - n - k +1$ .","From here, we can derive the following three identities: $ k=x+1 \\; ; $ $nk-n-k &= (nk-n-k+1)-1\\\\&= y-1 \\; ;$ $nk-n-1 &= (nk-n-k+1)+k-2\\\\&= y+(x+1)-2\\\\&= x + y - 1 \\; .$ We now substitute these identities into Equation REF , giving: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{nk-n-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{nk-n-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{nk-n-k-2}{2}} \\nonumber \\\\&= \\frac{\\Gamma \\Bigl (\\frac{x+1}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{(y-1)-2}{2}}\\; .$ By definition, we have $F =\\frac{SSA}{SSR}\\cdot \\frac{df_{\\text{residual}}}{df_{\\text{treatment}}} = \\frac{SSA}{SSR}\\cdot \\frac{y}{x} \\; .$ Also, in a repeated-measures design, subject variability is removed from the total sum of squares term $SST$ .","Thus, we can replace $SST$ with $SSA+SSR$ , giving $\\frac{SST}{SSR} &= \\frac{SSA+SSR}{SSR}\\\\&= \\frac{SSA}{SSR}+1\\\\&= \\frac{xF}{y} + 1\\\\&=\\frac{y+xF}{y} \\; .$ Taking the reciprocal of this identity and subsituting back into Equation REF proves the following proposition: Given a repeated-measures analysis of variance summary reported in the form $F(x,y)$ , where $x$ equals the between-treatments degrees of freedom and $y$ equals the residual degrees of freedom, the Bayes factor can be expressed in analytic form as $\\text{BF}_{10} = \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}} \\; ,$ where $\\alpha \\in \\Bigl [-\\frac{1}{2},0\\Bigr ]$ ."],["Example computations","In this section, we provide two examples of computation of the repeated-measures analytic Bayes factor given in Proposition .","Both examples come from [8], which was briefly mentioned in the previous section on BIC Bayes factors.","In their study, [8] measured adults' response times on a computerized single-digit mental arithmetic task.","On some problems, the arithmetic operator (e.g., the addition sign or the multiplication sign) appeared on a computer screen 150 milliseconds before the operands, whereas on other problems, the operator and operands appeared simultaneously.","Two critical results appeared and are worth further consideration in our example computations.","First, [8] observed that addition problems for which the operator appeared 150 milliseconds before the operands were solved significantly faster than those for which the operator and operands appeared simultaneously, $F(1,17) = 52.36$ , $p<0.001$ .","Second, they observed that this pattern did not occur on multiplication problems, $F(1,17) = 1.75$ , $p=0.20$ .","On the basis of this claimed null effect, Fayol and Thevenot reasoned that mental processes for addition must be fundamentally different from those involved in multiplication.","We can use our repeated-measures analytic Bayes factor formula to index the evidence for $\\mathcal {H}_1$ and $\\mathcal {H}_0$ that come from the addition and multiplication data, respectively.","First, let's compute the Bayes factor for the addition result.","In addition to the observed $F$ -statistic (52.36) and the relevant degrees of freedom ($x=1$ , $y=17$ ), we must also specify $\\alpha $ , the width of the prior distribution on the variance ratio $\\tau $ .","We will employ a “bracketing” approach and compute the Bayes factor at both ends of its consistency range; that is, we will use both $\\alpha =-1/2$ and $\\alpha =0$ .","Beginning with $\\alpha =-1/2$ , we substitute the summary statistics into the formula, obtaining the following: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{1}{2}-\\frac{1}{2}+1\\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (-\\frac{1}{2}+1\\Bigr )}\\Biggl (\\frac{17}{17+1\\cdot 52.36}\\Biggr )^{-\\frac{1}{2}-\\Bigl (\\frac{17-3}{2}\\Bigr )}\\\\&=\\frac{\\Gamma \\bigl (1\\bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{1}{2}\\Bigr )}\\Biggl (\\frac{17}{69.36}\\Biggr )^{-\\frac{15}{2}}\\\\&=\\frac{1 \\cdot 5040}{14034.41 \\cdot 1.772454} \\bigl ( 0.245098 \\bigr )^{-7.5}\\\\&=7702.17 \\; .$ Using Equation REF , we can convert this Bayes factor to a posterior probability.","Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_1$ is: $p(\\mathcal {H}_1\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{10}}{\\text{BF}_{10}+1}\\\\&= \\frac{7702.17}{7702.17+1}\\\\&= 0.99987 \\;.$ Now, we repeat this calculation, but with $\\alpha =0$ .","This gives the following: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{1}{2}+0+1\\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\bigl (0+1\\bigr )}\\Biggl (\\frac{17}{17+1\\cdot 52.36}\\Biggr )^{0-\\Bigl (\\frac{17-3}{2}\\Bigr )}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{3}{2}\\Bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\bigl (1 \\bigr )}\\Biggl (\\frac{17}{69.36}\\Biggr )^{-\\frac{14}{2}}\\\\&=\\frac{0.8862269 \\cdot 5040}{14034.41 \\cdot 1} \\bigl ( 0.245098 \\bigr )^{-7}\\\\&=5989.80 \\; .$ Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_1$ can be computed (using Equation REF ) as: $p(\\mathcal {H}_1\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{10}}{\\text{BF}_{10}+1}\\\\&= \\frac{5989.80}{5989.80+1}\\\\&= 0.99983 \\;.$ Thus, the data observed by [8] are between 5989.80 and 7702.17 times more likely under $\\mathcal {H}_1$ than under $\\mathcal {H}_0$ , with posterior probability for $\\mathcal {H}_1$ between 0.99983 and 0.99987.","In all, these data give substantial evidence for an operator priming effect on addition.","What can be said about the evidence for $\\mathcal {H}_0$ from their data for multiplication?","Recall that they did not find a significant operator priming effect for multiplication, $F(1,17)=1.75$ , $p=0.20$ .","We can now perform a similar computation to gauge this evidence exactly.","Proceeding as before with $\\alpha =-1/2$ , we obtain: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&= \\frac{\\Gamma \\Bigl (\\frac{1}{2}-\\frac{1}{2} + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (-\\frac{1}{2}+1\\Bigr )} \\cdot \\Biggl (\\frac{17}{17+1\\cdot 1.75}\\Biggr )^{-\\frac{1}{2}-\\frac{17-3}{2}}\\\\&=\\frac{\\Gamma \\bigl (1\\bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{1}{2}\\Bigr )}\\Biggl (\\frac{17}{18.75}\\Biggr )^{-\\frac{15}{2}}\\\\&=\\frac{1 \\cdot 5040}{14034.41 \\cdot 1.772454} \\bigl ( 0.90666667 \\bigr )^{-7.5}\\\\&=0.4225 \\; .$ Since the obtained Bayes factor is less than 1, we take the reciprocal to cast it as evidence for $\\mathcal {H}_0$ : $\\text{BF}_{01} &= \\frac{1}{\\text{BF}_{10}}\\\\&= \\frac{1}{0.4225}\\\\&=2.37 \\;.$ Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_0$ can be computed via Equation REF : $p(\\mathcal {H}_0\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{01}}{\\text{BF}_{01}+1}\\\\&= \\frac{2.37}{2.37+1}\\\\&= 0.70326 \\;.$ Similarly, we can repeat the computation with $\\alpha =0$ : $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{1}{2}+0+1\\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\bigl (0+1\\bigr )}\\Biggl (\\frac{17}{17+1\\cdot 1.75}\\Biggr )^{0-\\Bigl (\\frac{17-3}{2}\\Bigr )}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{3}{2}\\Bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\bigl (1 \\bigr )}\\Biggl (\\frac{17}{18.75}\\Biggr )^{-\\frac{14}{2}}\\\\&=\\frac{0.8862269 \\cdot 5040}{14034.41 \\cdot 1} \\bigl ( 0.90666667 \\bigr )^{-7}\\\\&=0.6319 \\; .$ Taking the reciprocal gives: $\\text{BF}_{01} &= \\frac{1}{\\text{BF}_{10}}\\\\&= \\frac{1}{0.6319}\\\\&=1.58 \\;.$ Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_0$ can be computed using Equation REF to be: $p(\\mathcal {H}_0\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{01}}{\\text{BF}_{01}+1}\\\\&= \\frac{1.58}{1.58+1}\\\\&= 0.61240 \\;.$ Thus, data observed by [8] are between 1.58 and 2.37 times more likely under $\\mathcal {H}_0$ than under $\\mathcal {H}_1$ , with posterior probability for $\\mathcal {H}_0$ between 0.61240 and 0.70326.","Despite their claim for a null priming effect on multiplication, the evidence for this claim appears to be anecdotal at best."],["Simulation","In this section, we report the results of a simulation study that was designed to benchmark the performance of the analytic Bayes factor in Proposition against two other repeated-measures Bayes factors: the BIC approximation of [6] and the JZS Bayes factor of [17].","In this simulation, we used randomly generated datasets that represented several different types of repeated-measures structure.","Specifically, our datasets were generated from the linear mixed model $\\mathbf {y} = y_{ij} = \\mu + \\alpha _j + \\pi _i + \\varepsilon _{ij};\\hspace{14.22636pt}i=1,\\dots ,n;\\hspace{8.53581pt}j=1,\\dots ,k,$ where $\\mu $ represents a grand mean, $a_j \\sim \\mathcal {N}(0,\\sigma _{a})$ represent each of the $k$ randomly drawn treatment effects, $\\pi _i \\sim \\mathcal {N}(0,\\sigma ^2_p)$ represent each of the $n$ randomly drawn participant effects, and $\\varepsilon _{ij} \\sim \\mathcal {N}(0,\\sigma _{\\varepsilon }^2)$ represent the normally-distributed error terms.","For convenience and brevity of exposition we set $k=3$ , though we saw similar results with other values of $k$ .","Also, without loss of generality we set $\\mu =0$ and $\\sigma _{\\varepsilon }=1$ .","We then systematically varied the following components of the model: The number of experimental subjects $n$ was set to either $n=10$ , $n=30$ , or $n=80$ ; The intraclass correlation $\\rho $ among the subjects' repeated measurements was set to be either low ($\\rho =0.2$ ) or high ($\\rho =0.8$ ); The size of the treatment effect was manipulated by setting $\\tau = \\sigma _a^2/\\sigma ^2$ to be either $\\tau =0$ , $\\tau =0.5$ , or $\\tau =1$ .","For data generated under the condition $\\tau =0$ , the correct model is the null model $0:\\tau =0$ , whereas for data generated under $\\tau =0.5$ and $\\tau =1.0$ the correct model is the alternative model $1:\\tau >0$ .","For each combination of number of subjects ($n=10,50,80$ ), treatment effect size ($\\tau =0,0.5,1.0$ ), repeated-measures correlation ($\\rho =0.2,0.8$ ), we generated 1000 simulated datasets.","For each of the datasets, we performed a repeated-measures analysis of variance, extracting the $F$ statistic and relevant degrees of freedom ($x$ =between-treatments degrees of freedom and $y$ =residual degrees of freedom).","Then we used these values to compute two analytic Bayes factors from Proposition : one using $\\alpha =-\\frac{1}{2}$ and another using $\\alpha =0$ .","We also computed the BIC Bayes factor from [6].","Finally, we computed the JZS Bayes factor from [17]; note that this Bayes factor can only be computed from the raw data, as the summary statistics alone are not sufficient for its estimation.","However, as we wish to show that our analytic Bayes factor outperforms the BIC Bayes factor previously obtained in [6], the JZS Bayes factor is an important benchmark to assess against.","All obtained Bayes factors were converted to posterior probabilities via Equation REF , assuming 1-1 prior model odds.","To compare the performance of the various computation methods in the simulation, we considered three analyses: we visualized the distribution of posterior probabilities $p(\\mathcal {H}_1\\mid \\mathbf {y})$ ; we calculated the proportion of simulated trials for which the correct model was chosen (i.e., model choice accuracy); we calculated of the proportion of simulated trials for which both methods chose the same model (i.e., model choice consistency).","First, let's consider the distribution of posterior probabilities $p(\\mathcal {H}_1\\mid \\mathbf {y})$ (Figure REF ).","Here, we constructed boxplots of the posterior probabilities for each of the four Bayes factor methods, split within plots by the number of subjects $n$ , and split across plots to represent all possible combinations of effect size $\\tau $ and repeated-measures correlation $rho$ .","We can see that the variability of the posterior probability estimates decreases substantially as the number of subjects $n$ increases.","For all simulated datasets in which 0 was the correct model (i.e., $\\tau =0$ , depicted in the first row of Figure REF ), all methods produced posterior probabilities for 1 that were reasonably small.","It is striking that in this case, the posterior probabilities derived from our new analytic Bayes factors as well as the BIC approximation of [6] are less than the posterior probabilities derived from the JZS Bayes factor.","Note also that setting the prior width to $\\alpha =-1/2$ produces the smallest posterior probabilities.","We also note that the separation in performance between the four methods decreases with increasing numbers of subjects $n$ .","For datasets in which 1 was the correct model (i.e., $\\tau =0.5$ and $\\tau =1.0$ ; rows 2 and 3 of Figure REF ), a different pattern of results emerged.","In these cases, the JZS Bayes factor produced posterior probabilities closer to 1 than did the analytic or BIC Bayes factors.","Whereas setting $\\alpha =-1/2$ was preferred in the $\\tau =0$ case, these data reveal that $\\alpha =0$ was the preferred setting when $\\tau > 0$ .","Finally, we note that repeated-measures correlation $\\rho $ had little effect on the pattern of posterior probabilities that was observed.","Figure: Boxplots depicting the distribution of the posterior probabilities p(ℋ 0 ∣𝐲)p(\\mathcal {H}_0\\mid \\mathbf {y}) for 1000 simulated datasets, split by number of subjects nn (within plots) and effect size τ\\tau and repeated-measures correlation ρ\\rho (across plots).","White and light-gray boxes represent the analytic Bayes factor with α=-1 2\\alpha =-\\frac{1}{2} and 0, respectively.","Medium gray boxes represent the BIC Bayes factor of .","Black boxes represent the JZS Bayes factor of .For the next analysis, we note that even though the distributions of posterior probabilities appear to follow the same pattern across methods, it is unclear whether the analytic Bayes factor proposed in this paper provides the user with correct model choice.","Since the data are simulated from target parameters, it is possible for us to gauge this performance exactly.","For simulated datasets where $\\tau =0$ , the correct model is $\\mathcal {H}_0$ , whereas when $\\tau =0.5$ or $\\delta =1.0$ , the correct model is $\\mathcal {H}_1$ .","Thus, to compare performance of these Bayes factor methods, we calculated model choice accuracy, defined simply as the proportion of the 1000 simulated datasets for which the correct model (0 or 1 was chosen.","Model choice was defined by considering $\\mathcal {H}_0$ to be chosen whenever $\\text{BF}_{01}>1$ and $\\mathcal {H}_1$ to be chosen whenever $\\text{BF}_{01}<1$ .","The results are displayed in Table REF .","Table: Model choice accuracy calculated as the proportion of simulated datasets for which the correct model was chosen.","Accuracies are presented as a function of Bayes factor method (analytic with α=-1 2\\alpha =-\\frac{1}{2}, analytic with α=0\\alpha =0, BIC, and JZS), numbers of subjects (n=10,30,80n=10,30,80), effect size (τ=0,0.5,1.0\\tau =0,0.5,1.0), and repeated-measures correlation (ρ=0.2,0.8\\rho =0.2,0.8).All methods were reasonably accurate at choosing the correct model.","We observed the highest accuracy for the case where $\\tau =0$ ; here, the analytic methods and the BIC method outperformed the JZS method (mirroring what we saw with the distributions of posterior probabilities in Figure REF ).","As before, setting $\\alpha =-1/2$ for our analytic Bayes factor was the most accurate in this case.","Accuracy when choosing 1 declined for all methods when $\\tau >0$ .","Again, we see that when 1 is the correct model, the JZS Bayes factor is more accurate with respect to model selection.","However, we note that setting $\\alpha =0$ increased accuracy relative to $\\alpha =-1/2$ .","Finally, we note that accuracy increased with increasing number of subjects $n$ , and there was very little variation between our two repeated-measures correlation settings ($\\rho =0.2$ and $\\rho =0.8$ ).","Finally, we calculated model choice consistency, defined as the proportion of simulated datasets for which each of the methods based on summary statistics alone (analytic and BIC) chose the same model as the JZS Bayes factor.","This analysis confirms what we observed in the previous two analyses: (1) consistency across methods increases with increasing numbers of subjects $n$ ; and (2) setting $\\alpha =0$ gives Bayes factors that are more consistent with the JZS Bayes factor.","Table: Model choice consistency calculated as the proportion of simulated datasets for which each method chose the same model as the JZS Bayes factor.","Proportions are presented as a function of Bayes factor method (analytic with α=-1 2\\alpha =-\\frac{1}{2}, analytic with α=0\\alpha =0, and BIC), numbers of subjects (n=10,30,80n=10,30,80), effect size (τ=0,0.5,1.0\\tau =0,0.5,1.0), and repeated-measures correlation (ρ=0.2,0.8\\rho =0.2,0.8)."],["Conclusion","In this paper, we developed an analytic Bayes factor for repeated-measures analysis of variance designs.","This Bayes factor gives researchers the ability to obtain Bayes factors directly from a minimal set of summary statistics (namely, the $F$ score and the degrees of freedom).","This formula improves upon the repeated-measures BIC Bayes factor formula of [6] in two ways.","First, this new analytic formula provides the user with an exact Bayes factor instead of an approximation.","Second, our method gives the user the ability to tune the prior used in the computation of the Bayes factor by specifying a hyperparameter $\\alpha $ , which controls the width of the prior distribution of effect sizes $\\tau =\\sigma ^2_a/\\sigma ^2$ .","Our simulation study shows that our analytic Bayes factor performs well compared to the JZS Bayes factor of [17] (especially when the prior parameter $\\alpha $ is set to 0).","Remarkably, our analytic Bayes factor outperforms the JZS Bayes factor when data are generated under 0.","We note that the analytic Bayes factor did not perform quite as well as the JZS Bayes on data generated under 1, but we think this limitation is far outweighed by ease of use of our method.","First, our analytic Bayes factor has the unique ability to be computed directly from summary statistics, with no need for raw data or the need to compute a multi-dimensional integral.","For this reason, we propose that our analytic Bayes factor will be an invaluable tool for anyone who wishes to assess the evidential value of their own data, as well as data from published studies where raw data is not readily available."]],"string":"[\n [\n \"Computing analytic Bayes factors from summary statistics in\\n repeated-measures designs\"\n ],\n [\n \"Abstract Bayes factors are an increasingly popular tool for indexing evidence from experiments.\",\n \"For two competing population models, the Bayes factor reflects the relative likelihood of observing some data under one model compared to the other.\",\n \"In general, computing a Bayes factor is difficult, because computing the marginal likelihood of each model requires integrating the product of the likelihood and a prior distribution on the population parameter(s).\",\n \"In this paper, we develop a new analytic formula for computing Bayes factors directly from minimal summary statistics in repeated-measures designs.\",\n \"This work is an improvement on previous methods for computing Bayes factors from summary statistics (e.g., the BIC method), which produce Bayes factors that violate the Sellke upper bound of evidence for smaller sample sizes.\",\n \"The new approach taken in this paper extends requires knowing only the $F$-statistic and degrees of freedom, both of which are commonly reported in most empirical work.\",\n \"In addition to providing computational examples, we report a simulation study that benchmarks the new formula against other methods for computing Bayes factors in repeated-measures designs.\",\n \"Our new method provides an easy way for researchers to compute Bayes factors directly from a minimal set of summary statistics, allowing users to index the evidential value of their own data, as well as data reported in published studies.\"\n ],\n [\n \"Introduction\",\n \"In this paper, we develop an analytic formula to compute Bayes factors for repeated-measures designs using only minimal summary statistics from the analysis of variance.\",\n \"Previous attempts to quantify evidence from summary statistics in repeated-measures designs have all relied upon the BIC approximation [21], [4], [6].\",\n \"In contrast, our new method avoids the need for approximation and produces an exact (analytic) Bayes factor directly from the observed $F$ -statistic and associated degrees of freedom.\",\n \"This paper extends the development of the between-subjects Pearson Bayes factor [7] to also consider repeated-measures designs, thus widening the scope and its use for indexing evidential value from summary statistics.\",\n \"Further, our formula is gives researchers a way to compute Bayes factors in repeated-measures designs that does not involve integration [17] or approximation [15], [6].\",\n \"As such, users can easily compute Bayes factors for their own repeated-measures data, and also for any results that are reported in the scientific literature – even null effects.\",\n \"Our method thus further affords researchers the ability to gauge the evidential value of a collection of published results in a straightforward way without the need for raw data.\"\n ],\n [\n \"Background\",\n \"We begin with some background on inference in experimental designs with repeated measurements.\",\n \"Let us consider an experiment where $k$ repeated measurements are taken from each of $n$ experimental subjects.\",\n \"We then have a total of $N=nk$ observations.\",\n \"We then place a linear mixed model structure on these observations: $\\\\mathbf {y} = y_{ij} = \\\\mu + \\\\alpha _j + \\\\pi _i + \\\\varepsilon _{ij}; \\\\hspace{8.53581pt} i=1,\\\\dots ,n;\\\\hspace{2.84526pt}j=1\\\\cdots ,k,$ where $\\\\mu $ represents the grand mean, $\\\\alpha _j$ represents the treatment effect associated with group $j$ , $\\\\pi _i$ represents the effect of subject $i$ , and $\\\\varepsilon _{ij} \\\\sim \\\\mathcal {N}(0,\\\\sigma _{\\\\varepsilon }^2)$ .\",\n \"Note that the repeated-measures design induces a correlated structure on the data, so the $N=nk$ observations are not independent.\",\n \"As a result, we have $n(k-1)$ independent observations [14].\",\n \"Ultimately, we want to know whether there is a treatment effect (i.e., if there are differences among the treatment groups induced by the repeated measurements).\",\n \"To proceed with hypothesis testing, we define two competing models: $\\\\mathcal {H}_0:\\\\alpha _j=0\\\\text{ for }j=1,\\\\dots ,k\\\\\\\\\\\\mathcal {H}_1:\\\\alpha _j\\\\ne 0\\\\text{ for some }j$ A classical approach to model selection is the analysis of variance (ANOVA) procedure [9].\",\n \"The analysis of variance procedure works by initially partitioning the total variance in the data $\\\\mathbf {y}$ into two sources: the variance between the treatment groups, and the residual variance that is left over after accounting for this treatment variability.\",\n \"In repeated-measures designs, we further partition the residual variance by separately breaking out the variance between subjects.\",\n \"Technically, this is done by computing $SS$ (sum of squares) terms, where $SSA = n\\\\sum _{j=1}^k(\\\\overline{y}_{\\\\cdot j}-\\\\overline{y}_{\\\\cdot \\\\cdot })^2, \\\\hspace{28.45274pt}SSB = k\\\\sum _{i=1}^n(\\\\overline{y}_{i\\\\cdot }-\\\\overline{y}_{\\\\cdot \\\\cdot })^2$ represent the sums of squares corresponding to the treatment effect and the subject effect, respectively; $SST = \\\\sum _{i=1}^n \\\\sum _{j=1}^k (y_{ij} - \\\\overline{y}_{\\\\cdot \\\\cdot })^2$ represents the total sum of squares, and $SSR = SST - SSA - SSB$ represents the residual sum of squares left over after accounting for both treatment and subject effects.\",\n \"From here, we compute the $F$ -statistic for the treatment effect in our design as $F=\\\\frac{SSA}{SSR}\\\\cdot \\\\frac{df_{\\\\text{residual}}}{df_{\\\\text{treatment}}} = \\\\frac{SSA}{SSR}\\\\cdot \\\\frac{(n-1)(k-1)}{k-1} = \\\\frac{SSA}{SSR}\\\\cdot (n-1).$ Conceptually, the $F$ statistic represents the ratio of between-groups variance to the residual variance left over after removing the subject variability.\",\n \"For inference, we compute the likelihood of the observed data $\\\\mathbf {y}$ under the null hypothesis $\\\\mathcal {H}_0$ .\",\n \"Specifically, we find the probability of obtaining the observed $F$ statistic (or greater) under $\\\\mathcal {H}_0$ .\",\n \"If this probability, called the $p$ -value, is small, this indicates that the data $\\\\mathbf {y}$ are rare under $\\\\mathcal {H}_0$ , so we can reject $\\\\mathcal {H}_0$ in favor of the alternative hypothesis $\\\\mathcal {H}_1$ .\",\n \"Unfortunately, this classical approach to inference is fraught with some issues that undermine its use [21].\",\n \"First, the $p$ -value is not equivalent to the posterior probability $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y})$ .\",\n \"Regardless, many researchers believe (incorrectly) that a $p$ -value represents the probability that $\\\\mathcal {H}_0$ is true [11], and thus take a small $p$ -value to represent evidence for $\\\\mathcal {H}_1$ .\",\n \"However, [1] demonstrated that $p$ -values classically overestimate this evidence.\",\n \"For example, with a $t$ -test performed on a sample size of 100, a $p$ -value of 0.05 transforms to $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y})=0.52$ – rather than reflecting evidence for $\\\\mathcal {H}_1$ , this small $p$ -value reflects only a slight preference for $\\\\mathcal {H}_0$ .\",\n \"Second, the evidence provided for $\\\\mathcal {H}_1$ from this procedure is only indirect – whereas the $p$ -value does measures the predictive adequacy of $\\\\mathcal {H}_0$ , it makes no such measurement of predictive adequacy for $\\\\mathcal {H}_1$ .\",\n \"Thus, in this paper we consider a Bayesian approach to model selection.\",\n \"Instead of computing a $p$ -value, we instead compute a Bayes factor [12].\",\n \"The Bayes factor, denoted $\\\\text{BF}_{01}$ is defined as the ratio of marginal likelihoods for $\\\\mathcal {H}_0$ and $\\\\mathcal {H}_1$ , respectively.\",\n \"That is, $\\\\text{BF}_{01} = \\\\frac{p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_0)}{p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_1)}.$ As such, $\\\\text{BF}_{01}$ indexes the relative likelihood of observing data $\\\\mathbf {y}$ under $\\\\mathcal {H}_0$ compared to $\\\\mathcal {H}_1$ , so $\\\\text{BF}_{01}>1$ is taken as evidence for $\\\\mathcal {H}_0$ over $\\\\mathcal {H}_1$ .\",\n \"Similarly, $\\\\text{BF}_{01}<1$ is taken as evidence for $\\\\mathcal {H}_1$ .\",\n \"Note that $\\\\text{BF}_{01} = 1/\\\\text{BF}_{10}$ , so if $\\\\text{BF}_{01}<1$ , we can take the reciprocal and equivalently write $BF_{10} > 1$ .\",\n \"For this reason, it is canonical to report a Bayes factor as a number greater than 1.\",\n \"For example, instead of reporting $\\\\text{BF}_{01} = 0.25$ , we will take the reciprocal and report $\\\\text{BF}_{10} = 1/0.25 = 4$ .\",\n \"Both representations imply that the observed data are 4 times more likely under $\\\\mathcal {H}_1$ than under $\\\\mathcal {H}_0$ .\",\n \"An equivalent definition of the Bayes factor $\\\\text{BF}_{01}$ is the extent to which the prior odds for $\\\\mathcal {H}_0$ over $\\\\mathcal {H}_1$ are updated after observing data.\",\n \"Thus, the ratio of posterior probabilities for $\\\\mathcal {H}_0$ and $\\\\mathcal {H}_1$ can be found by multiplying the ratio of prior probabilities by $\\\\text{BF}_{01}$ : $\\\\frac{p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y})}{p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y})} = \\\\text{BF}_{01}\\\\cdot \\\\frac{p(\\\\mathcal {H}_0)}{p(\\\\mathcal {H}_1)}.$ One immediate consequence of Equation REF is that the Bayes factor can be directly transformed into posterior probability of $\\\\mathcal {H}_0$ (or if desired, $\\\\mathcal {H}_1$ ).\",\n \"To see this, we simply solve Equation REF for the posterior probability $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y})$ and then apply Bayes' theorem.\",\n \"This gives $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y}) &= \\\\text{BF}_{01} \\\\cdot \\\\frac{p(\\\\mathcal {H}_0)}{p(\\\\mathcal {H}_1)} \\\\cdot p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y})\\\\\\\\&= \\\\frac{\\\\text{BF}_{01} \\\\cdot p(\\\\mathcal {H}_0) \\\\cdot p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_1)\\\\cdot p(\\\\mathcal {H}_1)}{p(\\\\mathcal {H}_1)\\\\cdot p(\\\\mathbf {y})}\\\\\\\\&= \\\\frac{\\\\text{BF}_{01}\\\\cdot p(\\\\mathcal {H}_0)\\\\cdot p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_1)}{p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_0)\\\\cdot p(\\\\mathcal {H}_0) + p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_1)\\\\cdot p(\\\\mathcal {H}_1)}.$ Dividing numerator and denominator by the marginal likelihood $p(\\\\mathbf {y}\\\\mid \\\\mathcal {H}_1)$ yields $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y}) = \\\\frac{\\\\text{BF}_{01}\\\\cdot p(\\\\mathcal {H}_0)}{\\\\text{BF}_{01}\\\\cdot p(\\\\mathcal {H}_0) + p(\\\\mathcal {H}_1)}.$ By a similar argument, we can show $p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y}) = \\\\frac{\\\\text{BF}_{10}\\\\cdot p(\\\\mathcal {H}_1)}{\\\\text{BF}_{10}\\\\cdot p(\\\\mathcal {H}_1) + p(\\\\mathcal {H}_0)}.$ Commonly, we take a default assumption that both models are equally likely a priori, permitting us to set $p(\\\\mathcal {H}_0)=p(\\\\mathcal {H}_1) = 0.5$ .\",\n \"In this case, we get the following simplified formulas: $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y}) = \\\\frac{\\\\text{BF}_{01}}{\\\\text{BF}_{01}+1}, \\\\hspace{28.45274pt} p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y}) = \\\\frac{\\\\text{BF}_{10}}{\\\\text{BF}_{10}+1}.$ Despite the conceptual simplicity of the Bayes factor, actually computing $\\\\text{BF}_{01}$ is usually quite difficult.\",\n \"From the conceptual definition, we see that the Bayes factor $\\\\operatorname{BF}_{01}$ is the ratio of likelihoods for $\\\\mathbf {y}$ under 0 and 1, respectively.\",\n \"These likelihoods are marginal likelihoods, where each likelihood function $f_i(\\\\mathbf {y} \\\\mid \\\\theta _i)=f(\\\\mathbf {y} \\\\mid \\\\theta _i, i)$ ($i=0,1$ ) is integrated over the prior distribution $\\\\pi _i(\\\\theta _i)=\\\\pi (\\\\theta _i,i)$ for the vector of model parameters $\\\\theta _i$ associated with model $i$ .\",\n \"That is, $\\\\operatorname{BF}_{01} = \\\\frac{\\\\int f_0(\\\\theta _0)\\\\pi _0(\\\\theta _0)d\\\\theta _0}{\\\\int f_1(\\\\theta _1)\\\\pi _1(\\\\theta _1)d\\\\theta _1} \\\\;.$ The marginal likelihoods that form the numerator and denominator of the Bayes factor are in general very difficult to compute.\",\n \"Even if a researcher has sufficient mathematical background to consider this computation, most techniques of integration are difficult to apply.\",\n \"As sample size increases, the likelihood becomes highly peaked at its maximum.\",\n \"But with no prior knowledge of where the maximum occurs, numerical integration methods often fail to converge around this highly peaked area.\",\n \"Also, the parameter vector $\\\\theta _i$ is often of high dimension.\",\n \"Combined with the peaked integrand, getting numerical methodsto converge is like “finding a needle in a haystack” [20].\",\n \"Thus, our primary goal in this work is to make simplifying assumptions which lead to analytic solutions (i.e., without integral representation) for these marginal likelihoods.\"\n ],\n [\n \"The BIC approximation\",\n \"One classic approach to avoid computing marginal likelihoods is to use a calculus trick to approximate them instead.\",\n \"This method is called the BIC approximation [16], [12], [21], [14], and it works by constructing a second-order Taylor approximation of $\\\\ln m_i(\\\\mathbf {y}) = \\\\ln P(\\\\mathbf {y} \\\\mid \\\\theta _i, i)$ centered at the posterior mode for $\\\\theta _i$ .\",\n \"Since all terms in the Taylor approximation containing the second-derivative or higher are removed, the result only approximates the log marginal likelihood.\",\n \"However, [16] showed that placing a noninformative unit information prior on $\\\\theta _i$ , the order of the error term in $\\\\mathcal {O}(1/\\\\sqrt{N})$ .\",\n \"This results in the approximation $\\\\ln m_i(\\\\mathbf {y}) \\\\approx \\\\ln P(\\\\mathbf {y} \\\\mid \\\\hat{\\\\theta }_i,i) - \\\\frac{k_i}{2}\\\\ln N$ where $\\\\hat{\\\\theta }_i$ is the maximum likelihood estimate for $\\\\theta _i$ under $i$ , $k_i$ is the number of parameters in $i$ , and $N$ is the total number of independent observations in $\\\\mathbf {y}$ .\",\n \"Equation REF relates quite well to the Bayesian information criterion (BIC) of [18]: $\\\\operatorname{BIC}(i)=-2\\\\ln L_i + k_i\\\\ln N,$ where $L$ is the maximum likelihood estimate for model $i$ .\",\n \"Combining these two equations yields $\\\\ln m_i(\\\\mathbf {y}) \\\\approx -\\\\frac{1}{2}\\\\operatorname{BIC}(i) \\\\; ,$ or equivalently $m_i(\\\\mathbf {y}) \\\\approx \\\\exp \\\\Bigl (-\\\\frac{1}{2}\\\\operatorname{BIC}(i)\\\\Bigr ).$ It is then simple to derive the following approximation: $\\\\operatorname{BF}_{01} = \\\\frac{m_0(\\\\mathbf {y})}{m_1(\\\\mathbf {y})} & \\\\approx \\\\frac{\\\\exp \\\\Bigl (-\\\\frac{1}{2}\\\\operatorname{BIC}(0)\\\\Bigr )}{\\\\exp \\\\Bigl (-\\\\frac{1}{2}\\\\operatorname{BIC}(1)\\\\Bigr )}\\\\\\\\ \\\\nonumber &= \\\\exp \\\\Biggl (\\\\frac{\\\\operatorname{BIC}(1) - \\\\operatorname{BIC}(0)}{2}\\\\Biggr ).$ To use Equation REF for computing Bayes factors, we need only know the BIC values for models 0 and 1.\",\n \"In the context of analysis of variance presented earlier, the BIC can be calculated [16] as $BIC &= N\\\\ln (1-R^2)+k\\\\ln N\\\\\\\\& = N\\\\ln \\\\Biggl (\\\\frac{SSR}{SST}\\\\Biggr ) + k\\\\ln N\\\\\\\\ ; .$ One downside to the BIC method is that it requires users to have the “raw” data available in order to compute $SSR$ and $SST$ .\",\n \"One way to improve Equation REF is recast the Bayes factor computation to a form that requires only summary statistics.\",\n \"[4] initially did this for between-subjects designs, and then extended that result to the context of repeated-measures designs [6] by deriving the formula: $\\\\text{BF}_{01} \\\\approx \\\\sqrt{(nk-n)^{k-1}\\\\cdot \\\\Biggl (1+\\\\frac{F}{n-1}\\\\Biggr )^{n-nk}} \\\\; .$ The following example nicely illustrates the use of (and a problem with) Equation REF .\",\n \"In a repeated-measures ($k=2$ ) study with $n=18$ participants, [8] observed that mean solution times to subtraction problems were 43 milliseconds faster when the subtraction sign was briefly presented before the actual problem itself, $F(1,17) = 27.17$ , $p<0.001$ .\",\n \"Using Equation REF , we can compute the BIC Bayes factor for these observed data as: $\\\\text{BF}_{01} & \\\\approx \\\\sqrt{(nk-n)^{k-1}\\\\cdot \\\\Biggl (1+\\\\frac{F}{n-1}\\\\Biggr )^{n-nk}}\\\\\\\\& = \\\\sqrt{(18\\\\cdot 2-18)^{2-1}\\\\cdot \\\\Biggl (1+\\\\frac{27.17}{18-1}\\\\Biggr )^{18-18\\\\cdot 2}}\\\\\\\\& = \\\\sqrt{18^1 \\\\cdot \\\\Biggl (1 + \\\\frac{27.17}{17}\\\\Biggr )^{-18}}\\\\\\\\& = \\\\sqrt{18\\\\cdot (2.5982)^{-18}}\\\\\\\\& = 0.0007863 \\\\;.$ By taking the reciprocal and casting this Bayes factor as support for $\\\\mathcal {H}_1$ , we have $\\\\text{BF}_{10} \\\\approx 1/0.0007863 = 1271.79$ .\",\n \"Thus, the BIC Bayes factor tells us that the observed data are approximately 1272 times more likely under $\\\\mathcal {H}_1$ than under $\\\\mathcal {H}_0$ .\",\n \"It is important to remember that this Bayes factor is an approximation.\",\n \"This fact is made especially salient by considering the following.\",\n \"[19] showed that under a reasonable class of prior distributions for $p$ -values, an upper bound for the Bayes factor can be computed directly from the $p$ -value as $\\\\operatorname{BF}_{10} \\\\le -\\\\frac{1}{e\\\\cdot p\\\\ln (p)}.$ We will now compute this Sellke bound for our previous example.\",\n \"First, we note that for $F(1,17)=27.17$ , the associated $p$ -value is $p=0.0000704$ .\",\n \"This gives us the upper bound $\\\\operatorname{BF}_{10} & \\\\le -\\\\frac{1}{e\\\\cdot 0.0000704\\\\cdot \\\\ln (0.0000704)}\\\\\\\\&=546.53.$ From this, the limitation of the BIC approximation is clear.\",\n \"Our computed Bayes factor of 1272 greatly exceeds the Sellke bound of 546.53.\",\n \"In fact, Figure REF shows that this problem persists over a large range of $p$ -values.\",\n \"In the figure, the solid line represents the Sellke bound $B(p)$ for $p$ -values ranging between 0 and 0.02.\",\n \"The dashed line represents the associated repeated-measures BIC Bayes factor of [6], given design parameters equivalent to [8] (i.e., $k=2$ repeated-measures conditions and $n=18$ subjects).\",\n \"Figure: Plot showing that the repeated-measures BIC Bayes factor (dashed line) of is greater than the Sellke bound (solid line) for pp-values ranging between p≈0p\\\\approx 0 and p=0.02p=0.02.\"\n ],\n [\n \"Analytic Bayes factors for repeated-measures designs\",\n \"Against the background of the previous section, we think the need for an analytic Bayes factor for repeated-measures designs is well motivated.\",\n \"Fortunately, recent work by [22] takes an important first step toward this goal.\",\n \"Wang and Sun, motivated by the the work of [10], started with a random effects linear model on the observed data $\\\\mathbf {y}$ : $\\\\mathbf {y} = y_{ij}=\\\\mu + \\\\alpha _j + \\\\pi _i + \\\\varepsilon _{ij},$ where $\\\\alpha _j \\\\sim \\\\mathcal {N}(0,\\\\sigma ^2_a)$ , $\\\\pi _i \\\\sim \\\\mathcal {N}(0,\\\\sigma ^2_p)$ and $\\\\varepsilon _{ij} \\\\sim \\\\mathcal {N}(0,\\\\sigma ^2)$ .\",\n \"Cast in this slightly different context of random effects versus the classical fixed effects model, the analysis of variance procedure amounts to testing whether the random effects term $\\\\alpha _j$ is identically 0.\",\n \"To capture this constraint, the competing models are defined in a slightly different manner: $0:\\\\sigma ^2_a=0 \\\\text{ versus }1:\\\\sigma _a^2\\\\ne 0.$ With this setup, [10] placed noninformative priors on $\\\\mu $ and $\\\\sigma $ under both 0 and 1 and considered a proper prior on the ratio of variance components $\\\\tau = \\\\sigma ^2_a/\\\\sigma ^2$ under 1.\",\n \"With this prior specification, Garcia-Donato and Sun showed $BF_{10} = \\\\int _0^{\\\\infty }(1+\\\\tau n)^{\\\\frac{1-k}{2}}\\\\Biggl (1-\\\\frac{\\\\tau n}{1+\\\\tau n}\\\\cdot \\\\frac{SSA}{SST}\\\\Biggr )^{\\\\frac{1-N}{2}}\\\\cdot \\\\pi (\\\\tau )d\\\\tau $ where $\\\\pi (\\\\tau )$ is left up to the analyst to choose.\",\n \"[22] used a Pearson Type VI distribution, given by: $\\\\pi ^{PT}(\\\\tau ) = \\\\frac{\\\\kappa (\\\\kappa \\\\tau )^{\\\\beta }(1+\\\\kappa \\\\tau )^{-\\\\alpha -\\\\beta -2}}{\\\\mathcal {B}(\\\\alpha +1, \\\\beta +1)}I_{(0,\\\\infty )}(\\\\tau )$ where $\\\\alpha >-1$ and $\\\\beta >-1$ are shape parameters and $\\\\kappa >0$ is a scale parameter, and $\\\\mathcal {B}(x,y) = \\\\int _0^1t^{x-1}(1-t)^{y-1}dt$ is the standard Beta function.\",\n \"Wang and Sun further reduced the problem of specifying the prior $\\\\pi ^{PT}$ to choosing one single parameter $\\\\alpha \\\\in [-\\\\frac{1}{2}, 0]$ .\",\n \"They did this by taking $\\\\kappa =n$ and $\\\\beta = \\\\frac{N-k}{2}-\\\\alpha -2$ .\",\n \"A plot of this prior can be seen in Figure REF ; here, we take the values $n=18$ , $k=2$ , and $N=36$ from our example above, thus setting $\\\\kappa =18$ and $\\\\beta =\\\\frac{36-2}{2}-\\\\alpha -2$ , where $\\\\alpha $ ranges among the values $-\\\\frac{1}{2}, -\\\\frac{1}{4}, -\\\\frac{1}{10}, 0$ .\",\n \"As we can see, as $\\\\alpha $ decreases from to 0 to $-\\\\frac{1}{2}$ , $\\\\tau $ becomes more dispersed and less peaked around the mode.\",\n \"This places more prior mass on larger treatment effects than we would see for values of $\\\\alpha $ closer to 0.\",\n \"Figure: A Pearson Type VI prior for τ\\\\tau , plotted as a function of shape parameter α\\\\alpha .Given this choice of prior and simplified parameterization, [22] proved that the Bayes factor derived by Garcia-Donato and Sun (Equation REF ) simplifies to an analytic expression without the need for integral representation: $\\\\operatorname{BF}_{10} = \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{k}{2}+\\\\alpha + \\\\frac{1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{N-k}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{N-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{SSR}{SST}\\\\Biggr )^{\\\\alpha -\\\\frac{N-k-2}{2}} \\\\; .$ To apply the Wang and Sun method for computing $\\\\text{BF}_{10}$ (Equation REF ) in a repeated-measures context, we apply a method of [14] and replace $N$ by $n(k-1)$ [3], [2], [6].\",\n \"This readily gives: $ \\\\text{BF}_{10} &= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{k}{2}+\\\\alpha + \\\\frac{1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{n(k-1)-k}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{n(k-1)-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{SSR}{SST}\\\\Biggr )^{\\\\alpha -\\\\frac{n(k-1)-k-2}{2}} \\\\nonumber \\\\\\\\&= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{k}{2}+\\\\alpha + \\\\frac{1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{nk-n-k}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{nk-n-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{SSR}{SST}\\\\Biggr )^{\\\\alpha -\\\\frac{nk-n-k-2}{2}} \\\\; .$ We now consider some identities that will greatly simplify Equation REF .\",\n \"First, let $x$ denote the between-groups degrees of freedom, $df_{\\\\text{treatment}}$ , and let $y$ denote the residual degrees of freedom, $df_{\\\\text{residual}}$ .\",\n \"As seen earlier, this gives $x=k-1$ and $y=(n-1)(k-1) = nk - n - k +1$ .\",\n \"From here, we can derive the following three identities: $ k=x+1 \\\\; ; $ $nk-n-k &= (nk-n-k+1)-1\\\\\\\\&= y-1 \\\\; ;$ $nk-n-1 &= (nk-n-k+1)+k-2\\\\\\\\&= y+(x+1)-2\\\\\\\\&= x + y - 1 \\\\; .$ We now substitute these identities into Equation REF , giving: $\\\\text{BF}_{10} &= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{k}{2}+\\\\alpha + \\\\frac{1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{nk-n-k}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{nk-n-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{SSR}{SST}\\\\Biggr )^{\\\\alpha -\\\\frac{nk-n-k-2}{2}} \\\\nonumber \\\\\\\\&= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{x+1}{2}+\\\\alpha + \\\\frac{1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{y-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{x+y-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{SSR}{SST}\\\\Biggr )^{\\\\alpha -\\\\frac{(y-1)-2}{2}}\\\\; .$ By definition, we have $F =\\\\frac{SSA}{SSR}\\\\cdot \\\\frac{df_{\\\\text{residual}}}{df_{\\\\text{treatment}}} = \\\\frac{SSA}{SSR}\\\\cdot \\\\frac{y}{x} \\\\; .$ Also, in a repeated-measures design, subject variability is removed from the total sum of squares term $SST$ .\",\n \"Thus, we can replace $SST$ with $SSA+SSR$ , giving $\\\\frac{SST}{SSR} &= \\\\frac{SSA+SSR}{SSR}\\\\\\\\&= \\\\frac{SSA}{SSR}+1\\\\\\\\&= \\\\frac{xF}{y} + 1\\\\\\\\&=\\\\frac{y+xF}{y} \\\\; .$ Taking the reciprocal of this identity and subsituting back into Equation REF proves the following proposition: Given a repeated-measures analysis of variance summary reported in the form $F(x,y)$ , where $x$ equals the between-treatments degrees of freedom and $y$ equals the residual degrees of freedom, the Bayes factor can be expressed in analytic form as $\\\\text{BF}_{10} = \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{x}{2}+\\\\alpha + 1\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{y-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{x+y-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{y}{y+xF}\\\\Biggr )^{\\\\alpha -\\\\frac{y-3}{2}} \\\\; ,$ where $\\\\alpha \\\\in \\\\Bigl [-\\\\frac{1}{2},0\\\\Bigr ]$ .\"\n ],\n [\n \"Example computations\",\n \"In this section, we provide two examples of computation of the repeated-measures analytic Bayes factor given in Proposition .\",\n \"Both examples come from [8], which was briefly mentioned in the previous section on BIC Bayes factors.\",\n \"In their study, [8] measured adults' response times on a computerized single-digit mental arithmetic task.\",\n \"On some problems, the arithmetic operator (e.g., the addition sign or the multiplication sign) appeared on a computer screen 150 milliseconds before the operands, whereas on other problems, the operator and operands appeared simultaneously.\",\n \"Two critical results appeared and are worth further consideration in our example computations.\",\n \"First, [8] observed that addition problems for which the operator appeared 150 milliseconds before the operands were solved significantly faster than those for which the operator and operands appeared simultaneously, $F(1,17) = 52.36$ , $p<0.001$ .\",\n \"Second, they observed that this pattern did not occur on multiplication problems, $F(1,17) = 1.75$ , $p=0.20$ .\",\n \"On the basis of this claimed null effect, Fayol and Thevenot reasoned that mental processes for addition must be fundamentally different from those involved in multiplication.\",\n \"We can use our repeated-measures analytic Bayes factor formula to index the evidence for $\\\\mathcal {H}_1$ and $\\\\mathcal {H}_0$ that come from the addition and multiplication data, respectively.\",\n \"First, let's compute the Bayes factor for the addition result.\",\n \"In addition to the observed $F$ -statistic (52.36) and the relevant degrees of freedom ($x=1$ , $y=17$ ), we must also specify $\\\\alpha $ , the width of the prior distribution on the variance ratio $\\\\tau $ .\",\n \"We will employ a “bracketing” approach and compute the Bayes factor at both ends of its consistency range; that is, we will use both $\\\\alpha =-1/2$ and $\\\\alpha =0$ .\",\n \"Beginning with $\\\\alpha =-1/2$ , we substitute the summary statistics into the formula, obtaining the following: $\\\\text{BF}_{10} &= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{x}{2}+\\\\alpha + 1\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{y-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{x+y-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{y}{y+xF}\\\\Biggr )^{\\\\alpha -\\\\frac{y-3}{2}}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{1}{2}-\\\\frac{1}{2}+1\\\\Bigr ) \\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{17-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{1+17-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (-\\\\frac{1}{2}+1\\\\Bigr )}\\\\Biggl (\\\\frac{17}{17+1\\\\cdot 52.36}\\\\Biggr )^{-\\\\frac{1}{2}-\\\\Bigl (\\\\frac{17-3}{2}\\\\Bigr )}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\bigl (1\\\\bigr ) \\\\cdot \\\\Gamma \\\\bigl (8\\\\bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{17}{2} \\\\Bigr ) \\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{1}{2}\\\\Bigr )}\\\\Biggl (\\\\frac{17}{69.36}\\\\Biggr )^{-\\\\frac{15}{2}}\\\\\\\\&=\\\\frac{1 \\\\cdot 5040}{14034.41 \\\\cdot 1.772454} \\\\bigl ( 0.245098 \\\\bigr )^{-7.5}\\\\\\\\&=7702.17 \\\\; .$ Using Equation REF , we can convert this Bayes factor to a posterior probability.\",\n \"Assuming equal prior odds for $\\\\mathcal {H}_0$ and $\\\\mathcal {H}_1$ , the posterior probability for $\\\\mathcal {H}_1$ is: $p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y}) &= \\\\frac{\\\\text{BF}_{10}}{\\\\text{BF}_{10}+1}\\\\\\\\&= \\\\frac{7702.17}{7702.17+1}\\\\\\\\&= 0.99987 \\\\;.$ Now, we repeat this calculation, but with $\\\\alpha =0$ .\",\n \"This gives the following: $\\\\text{BF}_{10} &= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{x}{2}+\\\\alpha + 1\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{y-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{x+y-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{y}{y+xF}\\\\Biggr )^{\\\\alpha -\\\\frac{y-3}{2}}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{1}{2}+0+1\\\\Bigr ) \\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{17-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{1+17-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\bigl (0+1\\\\bigr )}\\\\Biggl (\\\\frac{17}{17+1\\\\cdot 52.36}\\\\Biggr )^{0-\\\\Bigl (\\\\frac{17-3}{2}\\\\Bigr )}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{3}{2}\\\\Bigr ) \\\\cdot \\\\Gamma \\\\bigl (8\\\\bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{17}{2} \\\\Bigr ) \\\\cdot \\\\Gamma \\\\bigl (1 \\\\bigr )}\\\\Biggl (\\\\frac{17}{69.36}\\\\Biggr )^{-\\\\frac{14}{2}}\\\\\\\\&=\\\\frac{0.8862269 \\\\cdot 5040}{14034.41 \\\\cdot 1} \\\\bigl ( 0.245098 \\\\bigr )^{-7}\\\\\\\\&=5989.80 \\\\; .$ Assuming equal prior odds for $\\\\mathcal {H}_0$ and $\\\\mathcal {H}_1$ , the posterior probability for $\\\\mathcal {H}_1$ can be computed (using Equation REF ) as: $p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y}) &= \\\\frac{\\\\text{BF}_{10}}{\\\\text{BF}_{10}+1}\\\\\\\\&= \\\\frac{5989.80}{5989.80+1}\\\\\\\\&= 0.99983 \\\\;.$ Thus, the data observed by [8] are between 5989.80 and 7702.17 times more likely under $\\\\mathcal {H}_1$ than under $\\\\mathcal {H}_0$ , with posterior probability for $\\\\mathcal {H}_1$ between 0.99983 and 0.99987.\",\n \"In all, these data give substantial evidence for an operator priming effect on addition.\",\n \"What can be said about the evidence for $\\\\mathcal {H}_0$ from their data for multiplication?\",\n \"Recall that they did not find a significant operator priming effect for multiplication, $F(1,17)=1.75$ , $p=0.20$ .\",\n \"We can now perform a similar computation to gauge this evidence exactly.\",\n \"Proceeding as before with $\\\\alpha =-1/2$ , we obtain: $\\\\text{BF}_{10} &= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{x}{2}+\\\\alpha + 1\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{y-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{x+y-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{y}{y+xF}\\\\Biggr )^{\\\\alpha -\\\\frac{y-3}{2}}\\\\\\\\&= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{1}{2}-\\\\frac{1}{2} + 1\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{17-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{1+17-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (-\\\\frac{1}{2}+1\\\\Bigr )} \\\\cdot \\\\Biggl (\\\\frac{17}{17+1\\\\cdot 1.75}\\\\Biggr )^{-\\\\frac{1}{2}-\\\\frac{17-3}{2}}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\bigl (1\\\\bigr ) \\\\cdot \\\\Gamma \\\\bigl (8\\\\bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{17}{2} \\\\Bigr ) \\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{1}{2}\\\\Bigr )}\\\\Biggl (\\\\frac{17}{18.75}\\\\Biggr )^{-\\\\frac{15}{2}}\\\\\\\\&=\\\\frac{1 \\\\cdot 5040}{14034.41 \\\\cdot 1.772454} \\\\bigl ( 0.90666667 \\\\bigr )^{-7.5}\\\\\\\\&=0.4225 \\\\; .$ Since the obtained Bayes factor is less than 1, we take the reciprocal to cast it as evidence for $\\\\mathcal {H}_0$ : $\\\\text{BF}_{01} &= \\\\frac{1}{\\\\text{BF}_{10}}\\\\\\\\&= \\\\frac{1}{0.4225}\\\\\\\\&=2.37 \\\\;.$ Assuming equal prior odds for $\\\\mathcal {H}_0$ and $\\\\mathcal {H}_1$ , the posterior probability for $\\\\mathcal {H}_0$ can be computed via Equation REF : $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y}) &= \\\\frac{\\\\text{BF}_{01}}{\\\\text{BF}_{01}+1}\\\\\\\\&= \\\\frac{2.37}{2.37+1}\\\\\\\\&= 0.70326 \\\\;.$ Similarly, we can repeat the computation with $\\\\alpha =0$ : $\\\\text{BF}_{10} &= \\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{x}{2}+\\\\alpha + 1\\\\Bigr )\\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{y-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{x+y-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma (\\\\alpha +1)} \\\\cdot \\\\Biggl (\\\\frac{y}{y+xF}\\\\Biggr )^{\\\\alpha -\\\\frac{y-3}{2}}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{1}{2}+0+1\\\\Bigr ) \\\\cdot \\\\Gamma \\\\Bigl (\\\\frac{17-1}{2}\\\\Bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{1+17-1}{2}\\\\Bigr )\\\\cdot \\\\Gamma \\\\bigl (0+1\\\\bigr )}\\\\Biggl (\\\\frac{17}{17+1\\\\cdot 1.75}\\\\Biggr )^{0-\\\\Bigl (\\\\frac{17-3}{2}\\\\Bigr )}\\\\\\\\&=\\\\frac{\\\\Gamma \\\\Bigl (\\\\frac{3}{2}\\\\Bigr ) \\\\cdot \\\\Gamma \\\\bigl (8\\\\bigr )}{\\\\Gamma \\\\Bigl (\\\\frac{17}{2} \\\\Bigr ) \\\\cdot \\\\Gamma \\\\bigl (1 \\\\bigr )}\\\\Biggl (\\\\frac{17}{18.75}\\\\Biggr )^{-\\\\frac{14}{2}}\\\\\\\\&=\\\\frac{0.8862269 \\\\cdot 5040}{14034.41 \\\\cdot 1} \\\\bigl ( 0.90666667 \\\\bigr )^{-7}\\\\\\\\&=0.6319 \\\\; .$ Taking the reciprocal gives: $\\\\text{BF}_{01} &= \\\\frac{1}{\\\\text{BF}_{10}}\\\\\\\\&= \\\\frac{1}{0.6319}\\\\\\\\&=1.58 \\\\;.$ Assuming equal prior odds for $\\\\mathcal {H}_0$ and $\\\\mathcal {H}_1$ , the posterior probability for $\\\\mathcal {H}_0$ can be computed using Equation REF to be: $p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y}) &= \\\\frac{\\\\text{BF}_{01}}{\\\\text{BF}_{01}+1}\\\\\\\\&= \\\\frac{1.58}{1.58+1}\\\\\\\\&= 0.61240 \\\\;.$ Thus, data observed by [8] are between 1.58 and 2.37 times more likely under $\\\\mathcal {H}_0$ than under $\\\\mathcal {H}_1$ , with posterior probability for $\\\\mathcal {H}_0$ between 0.61240 and 0.70326.\",\n \"Despite their claim for a null priming effect on multiplication, the evidence for this claim appears to be anecdotal at best.\"\n ],\n [\n \"Simulation\",\n \"In this section, we report the results of a simulation study that was designed to benchmark the performance of the analytic Bayes factor in Proposition against two other repeated-measures Bayes factors: the BIC approximation of [6] and the JZS Bayes factor of [17].\",\n \"In this simulation, we used randomly generated datasets that represented several different types of repeated-measures structure.\",\n \"Specifically, our datasets were generated from the linear mixed model $\\\\mathbf {y} = y_{ij} = \\\\mu + \\\\alpha _j + \\\\pi _i + \\\\varepsilon _{ij};\\\\hspace{14.22636pt}i=1,\\\\dots ,n;\\\\hspace{8.53581pt}j=1,\\\\dots ,k,$ where $\\\\mu $ represents a grand mean, $a_j \\\\sim \\\\mathcal {N}(0,\\\\sigma _{a})$ represent each of the $k$ randomly drawn treatment effects, $\\\\pi _i \\\\sim \\\\mathcal {N}(0,\\\\sigma ^2_p)$ represent each of the $n$ randomly drawn participant effects, and $\\\\varepsilon _{ij} \\\\sim \\\\mathcal {N}(0,\\\\sigma _{\\\\varepsilon }^2)$ represent the normally-distributed error terms.\",\n \"For convenience and brevity of exposition we set $k=3$ , though we saw similar results with other values of $k$ .\",\n \"Also, without loss of generality we set $\\\\mu =0$ and $\\\\sigma _{\\\\varepsilon }=1$ .\",\n \"We then systematically varied the following components of the model: The number of experimental subjects $n$ was set to either $n=10$ , $n=30$ , or $n=80$ ; The intraclass correlation $\\\\rho $ among the subjects' repeated measurements was set to be either low ($\\\\rho =0.2$ ) or high ($\\\\rho =0.8$ ); The size of the treatment effect was manipulated by setting $\\\\tau = \\\\sigma _a^2/\\\\sigma ^2$ to be either $\\\\tau =0$ , $\\\\tau =0.5$ , or $\\\\tau =1$ .\",\n \"For data generated under the condition $\\\\tau =0$ , the correct model is the null model $0:\\\\tau =0$ , whereas for data generated under $\\\\tau =0.5$ and $\\\\tau =1.0$ the correct model is the alternative model $1:\\\\tau >0$ .\",\n \"For each combination of number of subjects ($n=10,50,80$ ), treatment effect size ($\\\\tau =0,0.5,1.0$ ), repeated-measures correlation ($\\\\rho =0.2,0.8$ ), we generated 1000 simulated datasets.\",\n \"For each of the datasets, we performed a repeated-measures analysis of variance, extracting the $F$ statistic and relevant degrees of freedom ($x$ =between-treatments degrees of freedom and $y$ =residual degrees of freedom).\",\n \"Then we used these values to compute two analytic Bayes factors from Proposition : one using $\\\\alpha =-\\\\frac{1}{2}$ and another using $\\\\alpha =0$ .\",\n \"We also computed the BIC Bayes factor from [6].\",\n \"Finally, we computed the JZS Bayes factor from [17]; note that this Bayes factor can only be computed from the raw data, as the summary statistics alone are not sufficient for its estimation.\",\n \"However, as we wish to show that our analytic Bayes factor outperforms the BIC Bayes factor previously obtained in [6], the JZS Bayes factor is an important benchmark to assess against.\",\n \"All obtained Bayes factors were converted to posterior probabilities via Equation REF , assuming 1-1 prior model odds.\",\n \"To compare the performance of the various computation methods in the simulation, we considered three analyses: we visualized the distribution of posterior probabilities $p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y})$ ; we calculated the proportion of simulated trials for which the correct model was chosen (i.e., model choice accuracy); we calculated of the proportion of simulated trials for which both methods chose the same model (i.e., model choice consistency).\",\n \"First, let's consider the distribution of posterior probabilities $p(\\\\mathcal {H}_1\\\\mid \\\\mathbf {y})$ (Figure REF ).\",\n \"Here, we constructed boxplots of the posterior probabilities for each of the four Bayes factor methods, split within plots by the number of subjects $n$ , and split across plots to represent all possible combinations of effect size $\\\\tau $ and repeated-measures correlation $rho$ .\",\n \"We can see that the variability of the posterior probability estimates decreases substantially as the number of subjects $n$ increases.\",\n \"For all simulated datasets in which 0 was the correct model (i.e., $\\\\tau =0$ , depicted in the first row of Figure REF ), all methods produced posterior probabilities for 1 that were reasonably small.\",\n \"It is striking that in this case, the posterior probabilities derived from our new analytic Bayes factors as well as the BIC approximation of [6] are less than the posterior probabilities derived from the JZS Bayes factor.\",\n \"Note also that setting the prior width to $\\\\alpha =-1/2$ produces the smallest posterior probabilities.\",\n \"We also note that the separation in performance between the four methods decreases with increasing numbers of subjects $n$ .\",\n \"For datasets in which 1 was the correct model (i.e., $\\\\tau =0.5$ and $\\\\tau =1.0$ ; rows 2 and 3 of Figure REF ), a different pattern of results emerged.\",\n \"In these cases, the JZS Bayes factor produced posterior probabilities closer to 1 than did the analytic or BIC Bayes factors.\",\n \"Whereas setting $\\\\alpha =-1/2$ was preferred in the $\\\\tau =0$ case, these data reveal that $\\\\alpha =0$ was the preferred setting when $\\\\tau > 0$ .\",\n \"Finally, we note that repeated-measures correlation $\\\\rho $ had little effect on the pattern of posterior probabilities that was observed.\",\n \"Figure: Boxplots depicting the distribution of the posterior probabilities p(ℋ 0 ∣𝐲)p(\\\\mathcal {H}_0\\\\mid \\\\mathbf {y}) for 1000 simulated datasets, split by number of subjects nn (within plots) and effect size τ\\\\tau and repeated-measures correlation ρ\\\\rho (across plots).\",\n \"White and light-gray boxes represent the analytic Bayes factor with α=-1 2\\\\alpha =-\\\\frac{1}{2} and 0, respectively.\",\n \"Medium gray boxes represent the BIC Bayes factor of .\",\n \"Black boxes represent the JZS Bayes factor of .For the next analysis, we note that even though the distributions of posterior probabilities appear to follow the same pattern across methods, it is unclear whether the analytic Bayes factor proposed in this paper provides the user with correct model choice.\",\n \"Since the data are simulated from target parameters, it is possible for us to gauge this performance exactly.\",\n \"For simulated datasets where $\\\\tau =0$ , the correct model is $\\\\mathcal {H}_0$ , whereas when $\\\\tau =0.5$ or $\\\\delta =1.0$ , the correct model is $\\\\mathcal {H}_1$ .\",\n \"Thus, to compare performance of these Bayes factor methods, we calculated model choice accuracy, defined simply as the proportion of the 1000 simulated datasets for which the correct model (0 or 1 was chosen.\",\n \"Model choice was defined by considering $\\\\mathcal {H}_0$ to be chosen whenever $\\\\text{BF}_{01}>1$ and $\\\\mathcal {H}_1$ to be chosen whenever $\\\\text{BF}_{01}<1$ .\",\n \"The results are displayed in Table REF .\",\n \"Table: Model choice accuracy calculated as the proportion of simulated datasets for which the correct model was chosen.\",\n \"Accuracies are presented as a function of Bayes factor method (analytic with α=-1 2\\\\alpha =-\\\\frac{1}{2}, analytic with α=0\\\\alpha =0, BIC, and JZS), numbers of subjects (n=10,30,80n=10,30,80), effect size (τ=0,0.5,1.0\\\\tau =0,0.5,1.0), and repeated-measures correlation (ρ=0.2,0.8\\\\rho =0.2,0.8).All methods were reasonably accurate at choosing the correct model.\",\n \"We observed the highest accuracy for the case where $\\\\tau =0$ ; here, the analytic methods and the BIC method outperformed the JZS method (mirroring what we saw with the distributions of posterior probabilities in Figure REF ).\",\n \"As before, setting $\\\\alpha =-1/2$ for our analytic Bayes factor was the most accurate in this case.\",\n \"Accuracy when choosing 1 declined for all methods when $\\\\tau >0$ .\",\n \"Again, we see that when 1 is the correct model, the JZS Bayes factor is more accurate with respect to model selection.\",\n \"However, we note that setting $\\\\alpha =0$ increased accuracy relative to $\\\\alpha =-1/2$ .\",\n \"Finally, we note that accuracy increased with increasing number of subjects $n$ , and there was very little variation between our two repeated-measures correlation settings ($\\\\rho =0.2$ and $\\\\rho =0.8$ ).\",\n \"Finally, we calculated model choice consistency, defined as the proportion of simulated datasets for which each of the methods based on summary statistics alone (analytic and BIC) chose the same model as the JZS Bayes factor.\",\n \"This analysis confirms what we observed in the previous two analyses: (1) consistency across methods increases with increasing numbers of subjects $n$ ; and (2) setting $\\\\alpha =0$ gives Bayes factors that are more consistent with the JZS Bayes factor.\",\n \"Table: Model choice consistency calculated as the proportion of simulated datasets for which each method chose the same model as the JZS Bayes factor.\",\n \"Proportions are presented as a function of Bayes factor method (analytic with α=-1 2\\\\alpha =-\\\\frac{1}{2}, analytic with α=0\\\\alpha =0, and BIC), numbers of subjects (n=10,30,80n=10,30,80), effect size (τ=0,0.5,1.0\\\\tau =0,0.5,1.0), and repeated-measures correlation (ρ=0.2,0.8\\\\rho =0.2,0.8).\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we developed an analytic Bayes factor for repeated-measures analysis of variance designs.\",\n \"This Bayes factor gives researchers the ability to obtain Bayes factors directly from a minimal set of summary statistics (namely, the $F$ score and the degrees of freedom).\",\n \"This formula improves upon the repeated-measures BIC Bayes factor formula of [6] in two ways.\",\n \"First, this new analytic formula provides the user with an exact Bayes factor instead of an approximation.\",\n \"Second, our method gives the user the ability to tune the prior used in the computation of the Bayes factor by specifying a hyperparameter $\\\\alpha $ , which controls the width of the prior distribution of effect sizes $\\\\tau =\\\\sigma ^2_a/\\\\sigma ^2$ .\",\n \"Our simulation study shows that our analytic Bayes factor performs well compared to the JZS Bayes factor of [17] (especially when the prior parameter $\\\\alpha $ is set to 0).\",\n \"Remarkably, our analytic Bayes factor outperforms the JZS Bayes factor when data are generated under 0.\",\n \"We note that the analytic Bayes factor did not perform quite as well as the JZS Bayes on data generated under 1, but we think this limitation is far outweighed by ease of use of our method.\",\n \"First, our analytic Bayes factor has the unique ability to be computed directly from summary statistics, with no need for raw data or the need to compute a multi-dimensional integral.\",\n \"For this reason, we propose that our analytic Bayes factor will be an invaluable tool for anyone who wishes to assess the evidential value of their own data, as well as data from published studies where raw data is not readily available.\"\n ]\n]"},"id":{"kind":"string","value":"2209.08159"}}},{"rowIdx":2399,"cells":{"text":{"kind":"list like","value":[["Modeling Quantum Enhanced Sensing on a Quantum Computer"],["Abstract Quantum computers allow for direct simulation of the quantum interference and entanglement used in modern interferometry experiments with applications ranging from biological sensing to gravitational wave detection.","Inspired by recent developments in quantum sensing at the Laser Interferometer Gravitational-wave Observatory (LIGO), here we present two quantum circuit models that demonstrate the role of quantum mechanics and entanglement in modern precision sensors.","We implemented these quantum circuits on IBM quantum processors, using a single qubit to represent independent photons traveling through the LIGO interferometer and two entangled qubits to illustrate the improved sensitivity that LIGO has achieved by using non-classical states of light.","The one-qubit interferometer illustrates how projection noise in the measurement of independent photons corresponds to phase sensitivity at the standard quantum limit.","In the presence of technical noise on a real quantum computer, this interferometer achieves the sensitivity of 11\\% above the standard quantum limit.","The two-qubit interferometer demonstrates how entanglement circumvents the limits imposed by the quantum shot noise, achieving the phase sensitivity 17\\% below the standard quantum limit.","These experiments illustrate the role that quantum mechanics plays in setting new records for precision measurements on platforms like LIGO.","The experiments are broadly accessible, remotely executable activities that are well suited for introducing undergraduate students to quantum computation, error propagation, and quantum sensing on real quantum hardware."],["Introduction","Quantum systems at the forefront of modern research increasingly utilize entanglement for goals ranging from precision sensing to quantum simulation and computation.","Prominently, quantum computers employ entanglement to push beyond the capabilities of classical computers, providing a new set of tools that promise to solve complex problems across multiple fields.","On the way to more advanced research goals, first generation quantum computers also serve as educational tools.","The IBM Quantum Experience, for example, is an online quantum computing platform that provides the opportunity for students to interact with real quantum hardware [1], [2], [3], [4].","Here we use the IBM Quantum Experience to teach students about quantum metrology, one of the first areas of quantum science where entanglement has found practical applications.","Applications for entanglement in quantum metrology include spin squeezed clocks[5], enhanced biological imaging[6], remote sensing with EPR states [7], and the detection of gravitational waves [8], [9], [10].","Our focus for this paper will be on the particularly charismatic example of LIGO [11], which made the first detection of gravitational waves in 2015, realizing a century-old prediction by Einstein [8].","At the heart of LIGO is the most sensitive interferometer ever built, whose sensitivity has recently been enhanced by quantum squeezing of light [9].","LIGO clearly illustrates the quantum nature of interferometers and the role that entanglement can play in modern metrology.","Here, we describe how to use the IBM Quantum Experience to understand the quantum physics underpinning discoveries at LIGO.","In Sec.",", we construct a single-qubit quantum circuit representation of the LIGO interferometer by analyzing the behavior of a photon traveling through the interferometer.","In Sec.",", we present two representations of this quantum circuit - matrices and Bloch spheres - which provide precise mathematical intuition for the behavior of the quantum circuit.","We test predictions made by these representations against the results of experiments performed on the IBM Quantum Experience in Sec. .","These results allow us to model the nature of quantum noise in the interferometer.","We present a calibrated measurement of the sensitivity of the interferometer in Sec. .","The tools developed through analysis of the single-qubit interferometer circuit allow us to efficiently assess the enhancement that can be provided by quantum entanglement.","In Sec.",", we present an interferometer circuit with entanglement between two qubits: the simplest possible example of quantum-enhanced metrology.","We model the expected behavior of this circuit using matrix mechanics and visualize the expected dynamics using spin Wigner functions [12] before implementing the circuit on a quantum computer.","We show in Sec.","that entanglement reduces the noise of this interferometer below the standard quantum limit, even in the presence of experimental imperfections.","We present our conclusions in Sec.","."],["LIGO as a quantum circuit","LIGO utilizes laser interferometry to measure the microscopic spacetime distortions caused by gravitational waves.","As shown in Fig.","REF , these distortions change the relative length of LIGO's two arms and thus change the interference signal on the detector.","The resulting signals are so small that measurements in the frequency bands relevant to the final moments of black hole and neutron star mergers are limited by the inherent quantum shot noise of the laser used in the interferometer [13].","Thus there is value in incorporating quantum mechanics into the description of the interferometer, going beyond the classical framework [11] of understanding interferometers with electromagnetic waves.","With only one qubit, a quantum circuit model can capture all the salient features of a classical model and demonstrate the effects of quantum projection noise.","Repeatedly executing this circuit simulates measurements of the independent photons that make up a coherent laser beam.","The quantum circuit model also provides a scalable framework that can incorporate entanglement, which is necessary for quantum enhanced measurement [14].","To construct a quantum circuit model of LIGO, we pare the complicated optical system down to a Michelson interferometer, consisting of only a laser, a beam splitter and two mirrors.","This geometry, depicted in Fig.","REF , captures the essential physics of the system and is often used to illustrate the operation of LIGO [11], [15].","We map the two directions that a photon travels in the interferometer to the two states of a single qubit in the quantum circuit.","A photon traveling parallel to the laser corresponds to the qubit state $|0\\rangle $ , and a photon traveling in the perpendicular direction corresponds to the state $|1\\rangle $ .","We deduce the gates that make up the quantum circuit by analyzing the behavior of each of the optical elements that a photon encounters while traveling through the interferometer.","This procedure is similar to that used by Nielsen and Chuang to construct a Mach–Zehnder interferometer in the context of dual-rail quantum computation (Section 7.4.2) [16].","A photon in the interferometer is initially emitted by a monochromatic laser source and travels towards the beam splitter.","The initial direction of the photon is represented by the quantum state $|0\\rangle $ .","At the beam splitter, the photon has 50% chance of reflection.","This creates an equal superposition of the photon continuing to travel in the same direction (quantum state $|0\\rangle $ ) and being reflected into the perpendicular direction (quantum state $|1\\rangle $ ), which precisely maps onto the action of a Hadamard gate on a single qubit.","After the beam splitter, each state accumulates a phase that is proportional to the distance traveled in the respective arm.","A difference in length of the two arms, which can be caused by a gravitational wave, introduces a phase difference $\\phi $ between the two states.","This is analogous to the action of an $R_Z(\\phi )$ gate.","After retro-reflection at the end of each arm, the photon returns for a second pass through the beam splitter, which is again represented by a Hadamard gate.","After the beam splitter, photons in the state corresponding to $|1\\rangle $ are measured on the detector, constituting a measurement in the qubit's computational ($z$ ) basis.","Putting together the steps of the photon's journey, the full quantum circuit that represents the LIGO interferometer is $@C=1.0em @R=0.2em @!R {{ {q}_{0} : } & {\\left|0\\right\\rangle } & {\\mathrm {H}} & {\\mathrm {R}_\\mathrm {Z}\\,\\mathrm {(}\\mathrm {\\phi }\\mathrm {)}} & {\\mathrm {H}} & & & \\\\{c:} & {/_{_{1}}} & & & & {_{_{0}}} [-1] & & \\\\}$ This circuit allows us to directly simulate the behavior of a Michelson interferometer on a quantum computer."],["Expected Behavior of the Single-Qubit Circuit","In order to formulate a precise quantum picture of a Michelson interferometer like LIGO, we require a mathematical framework that can predict the behavior of the quantum circuit presented in the prior section.","Quantum matrix mechanics provides this framework, but is mathematically removed from a physical or graphical interpretation.","We thus present the Bloch sphere representation alongside the matrix representation to help visualize the action of the quantum gates on the qubit.","For a single qubit, both complementary representations independently provide a precise accounting of the quantum dynamics.","The matrix notation for quantum states represents a qubit as a $2\\times 1$ vector whose entries are complex numbers.","Quantum gates are represented by $2\\times 2$ matrices, which act on the state vector through matrix multiplication.","Following the convention of Nielsen and Chuang [16], the two basis states $|0\\rangle $ and $|1\\rangle $ are given by $|0\\rangle = \\begin{bmatrix}1 \\\\0\\end{bmatrix} \\text{and }|1\\rangle = \\begin{bmatrix}0 \\\\1\\end{bmatrix}.$ An arbitrary superposition of $|0\\rangle $ and $|1\\rangle $ is given by $|\\psi \\rangle = c_0|0\\rangle + c_1|1\\rangle $ and can be written as the vector $|\\psi \\rangle = \\begin{bmatrix}c_0 \\\\c_1\\end{bmatrix}.$ The squared magnitudes of the coefficients $|c_0|^2$ and $|c_1|^2$ give the probabilities of measuring the qubit in the $|0\\rangle $ and $|1\\rangle $ states, respectively.","In the context of the interferometer, only photons in the $|1\\rangle $ state (traveling perpendicularly to the laser output beam) reach the detector whereas photons in the $|0\\rangle $ state (traveling along the laser output beam) return to the laser without being detected.","The probabilities $p_1 = |c_1|^2$ and $p_0 = |c_0|^2$ hence respectively correspond to the probabilities of detecting and not detecting each photon.","Information about the quantum superposition is encapsulated in the relative phase $\\phi = \\phi _1 - \\phi _0$ between the two complex coefficients, where $\\phi _0$ and $\\phi _1$ are defined by the expression of the complex coefficients in polar form: $c_0 = |c_0|e^{i\\phi _0}$ and $c_1 = |c_1|e^{i\\phi _1}$ .","The quantum state $|\\psi \\rangle $ of a qubit can also be represented on the Bloch sphere as seen in Fig.","REF .","The Bloch sphere is a unit sphere where a vector from the origin to the surface of the sphere represents a single-qubit state.","This vector is known as the Bloch vector.","The state $|0\\rangle $ is represented by a spin pointing up ($+z$ ) and $|1\\rangle $ is represented by a spin pointing down ($-z$ ).","For superpositions of $|0\\rangle $ and $|1\\rangle $ , the $z$ coordinate is given by the relative probability of measuring each of the two states, $p_0 - p_1$ .","The angle in the $xy$ -plane is given by the quantum phase $\\phi $ , with $\\phi =0$ corresponding to the $x$ -axis.","Quantum gates generate rotations of the state vector on the Bloch sphere.","Figure: Visualization on the Bloch sphere of the quantum state of the single-qubit circuit after each gate is applied.","The arrow shown on each sphere represents the quantum state |ψ〉=c 0 |0〉+c 1 |1〉.|\\psi \\rangle = c_0|0\\rangle + c_1|1\\rangle .","The zz coordinate is the relative probability p 0 -p 1 p_0 - p_1 of measuring |0〉|0\\rangle or |1〉|1\\rangle while the azimuthal angle is the relative phase φ\\phi between the complex coefficients c 0 c_0 and c 1 c_1.","The initial state |0〉|0\\rangle is a vector pointing up.","The Hadamard gate rotates the state by π\\pi about the zz-axis, then by π/2\\pi /2 about the yy-axis.","The R Z (φ)R_Z(\\phi ) gate rotates the state about the zz-axis, with φ=π/3\\phi =\\pi /3 chosen for illustration.","The final Hadamard gate maps xx-axis back onto the z-axis, which allows φ\\phi to be measured.During the sequence, the qubit is initialized in the state $|0\\rangle $ , represented by a vector pointing up on the Bloch sphere in Fig.","REF a.","The first gate applied to this initial state is a Hadamard gate $H$ .","The matrix form of $H$ is $H = \\frac{1}{\\sqrt{2}}\\begin{bmatrix}1 & 1 \\\\1 & -1\\end{bmatrix}.$ On the Bloch sphere, the Hadamard gate first rotates the state by $\\pi $ about $z$ and then by $\\pi /2$ about $y$ .","This transforms the initial state $|0\\rangle $ into an equal superposition of $|0\\rangle $ and $|1\\rangle $ with a relative phase of 0, $|\\psi \\rangle = \\frac{1}{\\sqrt{2}}\\begin{bmatrix}1 \\\\1\\end{bmatrix}.$ This state vector lies along the $x$ -axis of the Bloch sphere.","Following the Hadamard gate, the $R_Z(\\phi )$ gate rotates the state about the $z$ -axis by an angle $\\phi $ .","As an example, a rotation by $\\phi = \\pi /3$ is depicted in Fig.","REF .","In the matrix formulation, the $R_Z(\\phi )$ gate introduces a relative phase of $\\phi $ between the two complex coefficients $c_0$ and $c_1$ via the action of the matrix $R_Z(\\phi ) = \\begin{bmatrix}e^{-i\\phi /2} & 0 \\\\0 & e^{i\\phi /2}\\end{bmatrix}.$ For an interferometer like LIGO, this relative phase $\\phi $ is induced by a path length difference of $\\Delta L = \\phi \\lambda /2\\pi $ where $\\lambda $ is the wavelength of light used in the interferometer.","Because the quantum state changes based on the value of $\\phi $ , the detector is sensitive to the effects of gravitational waves.","In order to map the phase accumulated due to the $R_Z(\\phi )$ back into the measurement basis ($z$ ), we apply a final Hadamard gate.","This gate maps the $x$ -axis back to the $z$ -axis so that phase accumulations in the $xy$ -plane manifest as changes in expectation value for measurements in the standard basis of $|0\\rangle $ and $|1\\rangle $ .","Mathematically, we compute the resulting state of the quantum circuit by multiplying together all of the gates so that the final state is $|\\psi \\rangle = HR_Z(\\phi )H|0\\rangle $ , which evaluates to $|\\psi \\rangle =\\frac{1}{2}\\begin{bmatrix}e^{-i\\phi /2} + e^{i\\phi /2}\\\\e^{-i\\phi /2} - e^{i\\phi /2}\\end{bmatrix}=\\begin{bmatrix}\\cos {(\\phi /2)}\\\\-i\\sin {(\\phi /2)}\\end{bmatrix}.$ Here, the probabilities of measuring the final state of the circuit as $|0\\rangle $ and $|1\\rangle $ are given by $p_0 = \\cos ^2(\\phi /2)$ and $p_1 = \\sin ^2(\\phi /2)$ , respectively.","We summarize the result of this measurement by defining the polarization operator $\\hat{P}$ such that the polarization is equal to -1 if we measure $|0\\rangle $ and equal to 1 if we measure $|1\\rangle $ .","Up to the overall sign, this is the $z$ coordinate of the Bloch vector.","In matrix form, the polarization operator is $\\hat{P} = \\begin{bmatrix}-1 & 0 \\\\0 & 1\\end{bmatrix}.$ For the final state $|\\psi \\rangle $ from Eq.","(REF ), the expectation value of the polarization operator is $\\langle \\psi | \\hat{P} |\\psi \\rangle = p_1 - p_0 = \\frac{1}{2}\\sin ^2(\\phi /2) - \\frac{1}{2}\\cos ^2(\\phi /2) = -\\cos {\\phi }$ This sinusoidal dependence on $\\phi $ is consistent with the behavior of the $z$ -coordinate of the final state visualized in Fig.","REF , where the final state's Bloch vector makes an angle $\\phi $ with respect to the $z$ axis.","In the context of LIGO, this dependence on $\\phi $ facilitates sensitivity to signals from gravitational waves.","The mapping between the LIGO interferometer and the quantum circuit allows us to simulate the detection of gravitational waves on the quantum computer and better understand the impacts of both fundamental and technical noise sources."],["Implementing on a quantum computer","We implemented the quantum circuit model for LIGO on the IBM Quantum Experience.","Quantum circuits were initially constructed manually via the circuit composer interface, which allows quick prototyping without the need for any programming experience.","To scale up the number experimental trials and systematically scan through experimental parameters, we used the Qiskit interface, which allows users to program quantum circuits in Python [17].","The code for interfacing with the quantum computers was adapted from existing examples utilizing IBM Quantum Experience [1], [3], [17].","The code used for all experiments in this paper is available online [18].","Experiments were performed on both the quantum computer ibmq_manila and the state vector simulator ibmq_qasm_simulator.","The quantum computer ibmq_manila was chosen because, with a quantum volume of 32, it had the largest quantum volume among freely accessible quantum computers [19].","The quantum volume is IBM's preferred metric for its quantum computers and is a proxy for overall error rates; it measures the size of the largest quantum circuit that can be executed with acceptable fidelity [20].","The simulator ibmq_qasm_simulator explicitly computes the final qubit state and then samples outcomes from the corresponding probability distribution.","This provides a reference point for the effects of finite quantum volume and allows us to distinguish the effects of statistical noise from technical noise sources on the quantum computer.","Figure REF shows plots of polarization $P$ as a function of $\\phi $ , measured by running the single-qubit circuit from Eq.","(REF ) on both the quantum computer ibmq_manila and the simulator ibmq_qasm_simulator.","Each data point is obtained by repeatedly executing the quantum circuit and tallying the measurement outcomes 0 and 1 from each shot.","A single trial consists of 1024 shots from which we calculate the proportions of shots $p_0$ and $p_1$ with each measurement outcome.","Each data point shown is the average polarization $P = p_1 - p_0$ from 5 trials conducted at a fixed value of $\\phi $ .","The variability among the 5 trials is smaller than the size of the data points.","Figure: Measured polarization P=p 1 -p 0 P = p_1 - p_0 as a function of φ\\phi , computed from the data taken on the quantum computer (purple circle) and the simulator (green square) at 13 angles between 0 and 2π2\\pi , along with the best-fitted curves.","Error bars are smaller than the data points.","The curves are fitted to the data with amplitude and offset as free parameters.To analyze the relationship between $P$ and $\\phi $ , we fit a cosine curve to the data, with amplitude and vertical offset as free parameters.","The fit to the simulator data has an amplitude of 0.999(1) and an offset of -0.0001(7), which are consistent with the expected values of 1 and 0 to within the standard errors reported by the fitting routine (shown in parentheses).","This affirms the validity of the fitting routine.","The fit to the experimental data from the quantum computer has an amplitude of 0.932(5) and an offset of -0.042(4).","The amplitude is reduced from the expected value, mainly due to deviations from theory at positive values of polarization.","The majority of these deviations can be explained by readout errors.","The readout errors on ibmq_manila vary between qubits and over time, but the overall error rate is typically reported between 2 and 3 percent [19], which corresponds to 4-6% reductions in the amplitude of the polarization.","The offset can be explained by unequal rates of the two types of readout error, where the dominant source of error is relaxation from the excited state $|1\\rangle $ to the ground state $|0\\rangle $ during measurement [19].","IBM reports an error rates of $2\\times 10^{-4}$ for the main component of Hadamard gate, the $\\sqrt{X}$ gate.","This error rate is approximately one hundred times smaller than the readout error rates and thus a negligible contribution to the overall error.","Additional errors can come from the $R_z$ gate, whose error rate is not reported by IBM.","These results verify that measurements of the single-qubit interferometer circuit depend on the interferometer phase $\\phi $ as expected after incorporating corrections due to dissipation in the real quantum system.","Dissipation primarily manifests as readout errors that reduce the amplitude of the polarization curve.","The corrections to the curve are important to note as we next seek to invert the relationship and extract the phase of the interferometer from measurement outcomes."],["Shot noise and the Measurement of Gravitational Waves","The sensitivity of an interferometer is defined by the smallest phase shift that can be reliably detected in the measured output of the interferometer.","In general, this phase shift is a change from an initial interferometer phase $\\phi _0$ to a new interferometer phase $\\phi = \\phi _0 + \\delta \\phi $ .","At LIGO, $\\delta \\phi $ is induced by gravitational waves and $\\phi _0$ corresponds to the initial phase in the absence of gravitational waves.","On the quantum computer we can choose $\\delta \\phi $ and $\\phi _0$ arbitrarily to illustrate how shot noise limits the precision with which an interferometer can measure phase shifts.","Since $\\delta \\phi $ is small, we can usefully approximate the cosine relationship between polarization and $\\phi $ as locally linear, as depicted in the center panel of Fig.","REF .","Using this linear relationship, polarization measurements can be used to produce an estimate $\\tilde{\\phi }$ of the phase of the interferometer $\\phi = \\phi _0 + \\delta \\phi $ .","Error propagates from measurements of the polarization to the inferred values of $\\tilde{\\phi }$ with a scale factor given by the slope $m$ of the line, where $m=\\sin (\\phi _0)$ for an ideal interferometer.","We can express the standard error of the inferred phase $\\tilde{\\phi }$ in terms of the standard deviation of polarization measurement as $\\sigma _{\\tilde{\\phi }} = \\frac{1}{m}\\sigma _{P}.$ Using this expression for the standard deviation of the inferred phase, we can determine theoretically how the precision of the interferometer should scale with number of qubits $N$ that are included in the measurement of average polarization $P$ from a single trial.","Here $P = \\frac{1}{N}\\sum _{i=1}^N P_i$ , where $P_i$ represents the measurement of an individual qubit.","We can compute $\\sigma _{P} = \\sqrt{\\operatorname{Var}(P)}$ by determining the variance of a polarization measurement.","Since each photon is independent and $1/N$ is a constant factor, the variance can be expressed as a sum, $\\operatorname{Var}(P) = \\frac{1}{N^2}\\sum _{i=1}^N \\operatorname{Var}(P_i).$ Using expectation values of the single-qubit polarization operator, we deduce that the variance of a single measurement of an ideal interferometer is $\\operatorname{Var}(P_i) = \\langle P_i^2 \\rangle - \\langle P_i \\rangle ^2 = 1 - \\cos ^2\\phi = \\sin ^2 \\phi .$ Here $\\langle P_i^2 \\rangle = 1$ because both polarization measurement outcomes square to 1, and $\\langle P_i \\rangle = -\\cos {\\phi }$ as derived in Eq.","(REF ).","Thus, $\\operatorname{Var}(P) = \\sin ^2 \\phi /N$ and $\\sigma _P = \\sin \\phi /\\sqrt{N}$ .","Substituting back into Eq.","(REF ), the phase sensitivity is $\\sigma _\\phi = \\frac{1}{\\sqrt{N}},$ which is equal to the standard quantum limit and is independent of $\\phi $ .","Figure: Polarization measurements at φ=π/2+0.2\\phi = \\pi /2 + 0.2 (right histogram) and the corresponding inferred phases φ ˜\\tilde{\\phi } (top histogram) obtained by inverting the linear approximation (purple solid line) of the best-fitted PP vs φ\\phi cosine curve (black solid curve) at φ 0 =π/2\\phi _0 = \\pi /2.","The vertical and horizontal dotted lines represent the original phase value φ 0 =π/2\\phi _0 = \\pi /2 and the corresponding polarization P(φ=π/2)P(\\phi = \\pi /2), respectively.","The black curve is the ibmq_manila cosine fit shown earlier in Fig. .","The data were taken from 75 sequential polarization measurements, each consisting of N=100N=100 shots.Figure REF shows how shot noise manifests in an interferometer implemented on a quantum computer.","The central panel of the figure depicts the experimentally determined relationship between interferometer phase and polarization and a linear approximation of this curve around $\\phi _0 = \\pi /2$ .","This initial phase and the corresponding polarization are depicted as dashed lines in the figure.","We illustrate the sensitivity of the interferometer to a phase shift $\\delta \\phi = 0.2$ by repeatedly measuring the polarization at the output of a quantum circuit where the interferometer phase was set to $\\phi = \\pi /2 + 0.2$ .","From each measurement of the polarization, we obtain an estimate $\\tilde{\\phi }$ using the previously determined relationship between the two variables.","Repeatedly measuring polarization illustrates the uncertainty in each measurement due to shot noise and other technical noise sources.","Each polarization measurement consists of $N=100$ shots.","The histogram shown on the right margin is constructed from 75 independent measurements of probability to illustrate variability.","The measured polarization has a standard deviation of 0.109(9), which is within error of theoretical value $\\sigma _P = \\sin {\\phi }/\\sqrt{N}$ at $\\phi = \\pi /2 + 0.2$ .","The distribution of polarization maps to a distribution of the inferred values of $\\tilde{\\phi }$ which has a mean of $\\pi /2 + 0.20(1)$ and standard deviation of 0.117(10).","The mean is consistent with the initially chosen $\\delta \\phi $ , showing that $\\tilde{\\phi }$ is an unbiased estimator for the phase shifts.","The width of the distribution of $\\tilde{\\phi }$ is the sensitivity of the interferometer to phase shifts.","The estimated phase sensitivity of 0.117(10) is consistent with shot noise for $N=100$ shots combined with the factor of $1/m = 1.073$ due to the reduced amplitude of the polarization curve as compared to Eq.","(REF ).","The model we derived in Eq.","(REF ) for the shot noise of independent photons predicts that the phase sensitivity scales with the number of shots $N$ as $\\sigma _{\\tilde{\\phi }} \\propto 1/\\sqrt{N}$ .","To verify this scaling, we repeat the procedure performed in Fig.","REF for values of $N$ between 1 and 1024.","We plot the measured sensitivities $\\sigma _{\\tilde{\\phi }}$ in Fig.","REF .","Shot noise corresponds to a slope of $-1/2$ on the log-log scale depicted in this figure.","For all $N$ , the measured sensitivities are slightly above shot noise, but follow the expected scaling, with a slope of -0.506(5) on the log-log plot.","On average, the plotted sensitivities are 11(2)% above shot noise, of which 7.3% is due to the reduced amplitude of polarization curve from Fig.","REF .","The remaining deviation is not statistically significant, but may be due to weak correlations between successive shots, which violate the assumption of independence required for Eq.","(REF ).","Overall, the results are consistent with the expected scaling for a predominantly shot noise limited device in the presence of some technical noise.","The sensitivity of this interferometer is thus primarily limited by the number of qubits queried.","This corresponds to noise scaling with the photons or optical power at LIGO.","Figure: Standard deviation of inferred φ ˜\\tilde{\\phi } from polarization data taken on ibmq_manila as a function of number of qubits NN.","The least-squares best fit line for the log-log plot is shown in solid purple.","All data points lie slightly above the standard quantum limit (dashed line) 1/N1/\\sqrt{N}."],["Modeling Quantum Enhanced Measurement","LIGO now utilizes entanglement to improve sensitivity without further increasing optical power.","Specifically, LIGO now uses light in a squeezed quantum state where photons are entangled to each other [9].","The correlations between the photons in this state improve sensitivity to phase at the expense of amplitude uncertainty.","Due to these correlations, the photons are not independent and can circumvent the standard quantum limit.","This manipulation of quantum uncertainty is widely used in quantum metrology.","Many examples of metrologically useful states can modeled using the language of quantum circuits [14].","The simplest form of entanglement involves two qubits.","We can create a two-qubit entangled state on a quantum computer by using a combination of $H$ , $X$ , and CNOT gates.","Specifically, the controlled not gate (CNOT) flips a target qubit (labeled with $\\bigoplus $ ) based on the state of a control qubit (labeled with $\\bullet $ ).","This allows the two qubits to interact, producing an entangled state that can be used as the input to an interferometer circuit in order to achieve improved sensitivity [21].","The full circuit is $@C=0.75em @R=0.2em @!R { \\\\{ {q}_{0} : } & { {q}_{0} : } & {\\mathrm {\\left|0\\right\\rangle }} & {\\mathrm {H}} & {1} & {\\mathrm {H}} & {\\mathrm {R_Z}\\,(\\mathrm {\\phi })} & {\\mathrm {H}} & & & & \\\\{ {q}_{1} : } & { {q}_{1} : } & {\\mathrm {\\left|0\\right\\rangle }} & {\\mathrm {X}} & & {\\mathrm {H}} & {\\mathrm {R_Z}\\,(\\mathrm {\\phi })} & {\\mathrm {H}} & & & & \\\\{c:} & {c:} & {/_{_{2}}} & & & & & & {_{_{0}}} [-2] & {_{_{1}}} [-1] & & \\\\}$ To understand this circuit's dynamics, we first construct a mathematical representation of two-qubit states.","An arbitrary two-qubit state can be expressed as a superposition of four two-qubit basis states $|\\psi \\rangle = c_0 |00\\rangle + c_1|01\\rangle + c_2|10\\rangle + c_3 |11\\rangle $ .","The corresponding matrix notation is the $4 \\times 1$ vector $[c_0, c_1, c_2, c_3]^T$ .","Here each basis state is the tensor product of single-qubit basis states for qubits $q_0$ and $q_1$ .","For example, $|01\\rangle = |0\\rangle _0 \\otimes |1\\rangle _1$ .","In general, states that can be represented as a tensor product $|\\psi \\rangle = |\\varphi _0\\rangle \\otimes |\\varphi _1\\rangle $ are not entangled.","If a two-qubit state cannot be factorized as a tensor product of single-qubit states, the two qubits are entangled [16].","The first three gates in the two-qubit circuit (Eq.","(REF )) transform the initial state $|00\\rangle $ into an entangled state $\\frac{1}{\\sqrt{2}} (|01\\rangle + |10\\rangle )$ , known as a Fock state, to be used for quantum enhanced measurements.","The first Hadamard gate and the $X$ gate act independently on the two qubits, preparing the product state $\\frac{1}{\\sqrt{2}}(|0\\rangle _0 + |1\\rangle _0) \\otimes |1\\rangle _1 = \\frac{1}{\\sqrt{2}} (|01\\rangle + |11\\rangle )$ .","Next, the CNOT gate produces entanglement by flipping the target bit $q_1$ when the control bit $q_0$ is in the state $|1\\rangle $ .","This operation can be represented by the $4 \\times 4$ matrix $\\text{CNOT} = \\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 0 \\\\\\end{bmatrix}.$ Applying this matrix to the state $\\frac{1}{\\sqrt{2}}[0, 1, 0, 1]^T$ produces the Fock state $\\frac{1}{\\sqrt{2}}[0, 1, 1, 0]^T = \\frac{1}{\\sqrt{2}} (|01\\rangle + |10\\rangle )$ .","The value of using an entangled state like the Fock state for quantum metrology can be understood visually through the Wigner representation on a spin sphere [12].","The Wigner function is a quasi-probability distribution that maps the uncertainty of the quantum state onto the spin sphere.","Similar to the Bloch sphere representation, single-qubit gates applied to all qubits rotate the Wigner function without changing the shape of the distribution.","This allows for a clean representation of the behavior of each state as it evolves through the interferometer circuit.","Figure REF shows the evolution of the two-qubit state after each gate using the Wigner distribution.","The single-qubit Bloch spheres from Fig.","REF are reproduced in the first row for comparison.","The second and third row of Fig.","REF compare the evolution of the Wigner distribution when the two qubits start in an untangled state $|00\\rangle $ and when they start in the Fock state $\\frac{1}{\\sqrt{2}}(|01\\rangle + |10\\rangle )$ , respectively.","The Wigner function of the unentangled pair is a localized distribution centered about the Bloch vector, with equal uncertainty in both x and y directions.","The width of this local distribution (dark blue region) on the unit sphere is $1/\\sqrt{2}$ radians [12], corresponding to the shot noise of two unentangled qubits.","For an entangled state like the Fock state, the Wigner distribution is not necessarily symmetric and can have less uncertainty in one direction.","The Fock state Wigner function appears as a horizontal ring centered about z=0 in the bottom left sphere in Fig.","REF .","The thickness of the ring, denoting the uncertainty in the z-direction, is smaller than the width of the unentangled Wigner distribution, at the expense of increased uncertainty in the azimuthal direction.","Entanglement also leads to negative Wigner function values at the poles.","When properly rotated during the interferometer sequence, this entangled state can exhibit increased sensitivity to the interferometer phase $\\phi $ due to the reduced width and increased spatial structure of Wigner distribution.","Figure: Visualization of the quantum state over time in the two-qubit circuit using the spin Wigner distribution W(θ,φ)W(\\theta ,\\phi ).","Successive columns depict evolution of the quantum state of the system after each gate in the interferometry sequence is applied.","The first row reproduces the single-qubit Bloch spheres from Fig.","for comparison.","The second row depicts the Wigner distribution for an unentangled initial state |ψ 0 〉=|00〉|\\psi _0\\rangle = |00\\rangle , which is localized about the single-qubit Bloch vector.","The third row depicts the Wigner distribution for an entangled pair of qubits initialized in the Fock state |ψ 0 〉=1 2(|01〉+|10〉)|\\psi _0\\rangle = \\frac{1}{\\sqrt{2}}(|01\\rangle + |10\\rangle ), appearing as a horizontal ring.","The shape of the Wigner quasi-probability distribution illustrates measurement uncertainty for each quantum state and is rotated under the application of gates in the interferometer sequence in same way as the Bloch vector.To precisely model the behavior of the Fock state in the interferometer requires matrix representations of the three gates used in the interferometer sequence.","Since these gates act independently on the two qubits, we can represent each step of the interferometer as the tensor product of two single-qubit gates.","The interferometer's beam splitter is represented by a pair of Hadamard gates, $H \\otimes H = \\frac{1}{2}\\begin{bmatrix}1 & 1 & 1 & 1 \\\\1 & -1 & 1 & -1 \\\\1 & 1 & -1 & -1 \\\\1 & -1 & -1 & 1 \\\\\\end{bmatrix}.$ The application of this Hadamard gate transforms the Fock state $\\frac{1}{\\sqrt{2}}(|01\\rangle + |10\\rangle )$ into a cat state $\\frac{1}{\\sqrt{2}}(|00\\rangle - |11\\rangle )$ .","Graphically this corresponds to rotating the Wigner distribution by $\\pi /2$ such that the ring is oriented vertically and is sensitive to azimuthal rotations due the interferometer phase.","The phase shift portion on the interferometer is given by the product of individual qubit rotations, $R_Z(\\phi ) \\otimes R_Z(\\phi ) = \\begin{bmatrix}e^{-i\\phi } & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & e^{i\\phi } \\\\\\end{bmatrix}.$ The application of this gate to the cat state results in the state $|\\psi \\rangle =e^{-i\\phi }|00\\rangle - e^{i\\phi } |11\\rangle $ , introducing a phase shift of $2\\phi $ between the basis states.","The state oscillates as a function of $\\phi $ between the two cat states $\\frac{1}{\\sqrt{2}}(|00\\rangle - |11\\rangle )$ and $\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle )$ with twice the frequency at which states in the single-qubit interferometer oscillate around the Bloch sphere.","The final pair of Hadamard gates represents the second pass through a beam splitter and maps the phase shift from the $R_Z$ gates onto the computation basis.","The Hadamard gates map one cat state $\\frac{1}{\\sqrt{2}}(|00\\rangle - |11\\rangle )$ back to the initial Fock state, while leaving the other cat state $\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle )$ unchanged.","Thus the final state is an oscillation between the Fock state and the cat state, $|\\psi \\rangle = \\frac{1}{2\\sqrt{2}}\\begin{bmatrix}e^{-i\\phi } - e^{i\\phi } \\\\e^{-i\\phi } + e^{i\\phi } \\\\e^{-i\\phi } + e^{i\\phi } \\\\e^{-i\\phi } - e^{i\\phi }\\end{bmatrix}= \\frac{\\cos {\\phi }}{\\sqrt{2}} \\begin{bmatrix}0 \\\\1 \\\\1 \\\\0\\end{bmatrix} - \\frac{i\\sin {\\phi }}{\\sqrt{2}}\\begin{bmatrix}1 \\\\0 \\\\0 \\\\1\\end{bmatrix}.$ The Fock state is detected when the measurements of the two qubits are different, which has a total probability of $p_{01} + p_{10} = \\cos ^2{\\phi }$ .","The cat state corresponds to both qubits being in the same state, which has a total probability of $p_{00} + p_{11} = \\sin ^2{\\phi }$ .","To encapsulate the oscillation between the cat state and Fock state in a single number, we define a parity operator such that parity is 1 when both qubits are in the same state (e.g., cat state), and parity is -1 when the qubits are in different states (e.g., Fock state).","In matrix form, this parity operator is $\\hat{\\Pi } = \\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 \\\\0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}.$ The expectation value of the operator is $\\begin{aligned}\\langle \\Pi \\rangle &= \\langle \\psi |\\hat{\\Pi }| \\psi \\rangle \\\\&= (p_{00} + p_{11}) - (p_{01} + p_{10}) \\\\&= -\\cos {(2\\phi )},\\end{aligned}$ in close analogy to the polarization measurement for a single-qubit interferometer.","The key difference between the expectation value of the two-qubit parity measurement (Eq.","(REF )) and the expectation value of the average polarization of independent qubits (Eq.","(REF )) is the frequency of oscillation when $\\phi $ is varied continuously.","As illustrated by Wigner distributions in Fig.","REF , an untangled state returns to its original orientation after a rotation of $\\phi = 2\\pi $ whereas the ring-shaped Wigner distribution for an entangled pair of qubits is indistinguishable from its initial state after a rotation by $\\phi =\\pi $ .","The increased frequency of oscillation leads to an increased slope $m=2\\sin (2\\phi )$ of the parity vs $\\phi $ curve as compared to the slope $m=\\sin (\\phi )$ of the polarization curve for the single-qubit interferometer.","This improves phase sensitivity.","In analogy to Eq.","(REF ), the phase sensitivity achievable by parity measurement is $\\sigma _{\\tilde{\\phi }} = \\frac{\\sigma _\\Pi }{2\\sin {(2\\phi )}},$ where $\\sigma _{\\Pi }$ is the standard deviation of the average parity.","The standard deviation of the average parity in measurement consisting of $n$ shots of the two-qubit interferometer can be computed using the matrix form of the single-shot parity operator $\\hat{\\Pi }_i$ .","Assuming all shots are independent, $\\sigma _{\\Pi }^2 = \\frac{\\operatorname{Var}({\\Pi _i})}{n} = \\frac{\\langle \\hat{\\Pi }_i^2 \\rangle - \\langle \\hat{\\Pi }_i \\rangle ^2}{n} = \\frac{\\sin ^2{(2\\phi )}}{n}.$ For a fair comparison to the single-qubit interferometer, we note that $n$ shots involve measurements of $N=2n$ qubits so the properly normalized phase sensitivity is $\\sigma _{\\tilde{\\phi }} = \\frac{1}{\\sqrt{2N}}.$ This is an overall improvement by a factor of $\\sqrt{2}$ as compared to the shot noise in an unentangled system."],["Quantifying Benefits of Entanglement","To demonstrate the improvement in sensitivity introduced by entanglement, we implemented the two-qubit interferometer circuit on the same quantum computer, ibmq_manila, that was used earlier to implement the single-qubit interferometer.","We additionally implemented the circuit on the state vector simulator, ibmq_qasm_simulator, to provide a comparison without the effects of technical noise.","Figure REF shows the results of parity measurements for each of the two implementations.","Each measurement of parity $\\Pi = p_{00} + p_{11} - p_{01} - p_{10}$ is computed from $n = 1024$ shots of the 2-qubit interferometer circuit.","The results show the expected sinusoidal dependence on $\\phi $ with a frequency twice that of the single-qubit interferometer.","Figure: Measured parity, Π=p 00 +p 11 -p 01 -p 10 \\Pi = p_{00} + p_{11} - p_{01} - p_{10}, as a function of φ\\phi .","Purple circles show data from the two-qubit interferometer taken on the quantum computer ibmq_manila at 13 angles between 0 and 2π2\\pi .","Green squares show data from the simulator ibmq_qasm_simulator at the same angles.","Error bars depict the standard deviation of 5 trials.","The curves are fit to the data with amplitude and offset as free parameters.We fit the Manila and simulator data to the expected functional form $\\Pi = -A\\cos {(2\\phi )}+B$ with amplitude and vertical offset as free parameters.","The simulator fit has an offset of -0.001(2) and amplitude of 0.997(3), both of which agree well with the expected values of 0 and 1.","The Manila fit's offset of -0.007(6) is also consistent with 0, while the amplitude of 0.852(7) corresponds to a 15% reduction from the ideal value.","This deviation in amplitude is larger than the corresponding value for the single-qubit interferometer (6.8%) because readout and gate errors in either qubit can flip overall parity, doubling the contribution from these errors.","Additionally, the CNOT gate has an error of up to 1% [19], (varying between daily calibrations), which further contributes to the reduction in amplitude.","Curves of parity $\\Pi $ vs phase $\\phi $ can be used to infer phase $\\tilde{\\phi }$ from measurements of the two-qubit, in analogy to the procedure illustrated for single-qubit polarization measurements in Fig.","REF and Fig.","REF .","To analyze the phase sensitivity quantitatively, like we did for the one-qubit interferometer, we collected parity data on both the quantum computer, ibmq_manila, and the simulator.","Each measurement of parity came from a trial consisting of 1 to 512 sequential shots (corresponding to $N=$ 2 to 1024 qubits), conducted with an interferometer phase $\\phi = \\pi /4$ .","From each parity measurement, we inferred a value for the interferometer phase $\\tilde{\\phi }$ using the slope of the cosine fits from Fig.","REF .","We computed the standard deviation of these phase measurements $\\sigma _{\\tilde{\\phi }}$ using data from 600 trials.","The resulting relationship between phase uncertainty and number of qubits contributing to each measurement is plotted in Fig.","REF .","The measured phase sensitivities for both ibmq_manila and simulator data are consistently below the standard quantum limit $1/\\sqrt{N}$ for unentangled qubits.","Fitting the simulator data verifies that the phase sensitivity is $1/\\sqrt{2N}$ , as expected for $N/2$ pairs of entangled qubits.","The fit for measurements made on the quantum computer is $(0.83 \\pm 0.02) N^{-0.495(6)}$ , which lies between the standard quantum limit and the limit for entangled pairs of qubits.","Overall, the two-qubit interferometer shows an improvement of 17(2)% as compared to the standard quantum limit and 25(3)% when compared to the single-qubit interferometer implemented on the same hardware.","This illustrates the benefits that entanglement can provide for quantum measurement, even in the smallest possible system sizes.","Figure: Standard deviation of inferred φ ˜\\tilde{\\phi } from polarization data taken on the quantum computer Manila and the state vector simulator as a function of number of qubits NN along with their best-fit lines.","All data points lie below the standard quantum limit (dashed line) 1/N1/\\sqrt{N}."],["Conclusions","We have demonstrated how entanglement between two qubits enables measurement with sensitivity below the standard quantum limit.","Students working on these experiments gain better comprehension of quantum mechanics, quantum metrology, and entanglement.","Running these experiments on a real quantum computer, students practice quantum programming, analyze experimental data, and assess the effects of both statistical uncertainty and technical noise sources.","This prepares students with the tools needed to engage in modern research in quantum science and technology.","The project presented here requires minimal prerequisites.","The current work began as an independent student project through a STEM outreach program that pairs high school student researchers with graduate student mentors [22].","This illustrates how the material can be accessible to motivated students for whom this is their first introduction to quantum mechanics.","We believe this project would fit well in the undergraduate curriculum, either as a demonstration for an introductory quantum information course or as a project in a modern physics laboratory class.","These experiments can be accomplished with minimal resources as the IBM Quantum Experience allows for free remote access.","Additionally, students working independently for a class project could extend the techniques presented here to circuits with more than two qubits [4] in order to demonstrate different forms of entanglement and implement a variety of quantum squeezing techniques.","We thank Stanford FAST for facilitating our collaboration and providing support for this research.","TN acknowledges support from NSF Q-SEnSE and ESC acknowledges the support of the NSF Graduate Research Fellowship.","We additionally thank Victoria Borish, Monika Schleier-Smith and Patrick Allamandola for useful discussions."],["Author contributions","All authors contributed equally to this project.","CT conducted experiments with guidance from TN and ESC.","All authors participated in the analysis of experimental data and the preparation of this manuscript."]],"string":"[\n [\n \"Modeling Quantum Enhanced Sensing on a Quantum Computer\"\n ],\n [\n \"Abstract Quantum computers allow for direct simulation of the quantum interference and entanglement used in modern interferometry experiments with applications ranging from biological sensing to gravitational wave detection.\",\n \"Inspired by recent developments in quantum sensing at the Laser Interferometer Gravitational-wave Observatory (LIGO), here we present two quantum circuit models that demonstrate the role of quantum mechanics and entanglement in modern precision sensors.\",\n \"We implemented these quantum circuits on IBM quantum processors, using a single qubit to represent independent photons traveling through the LIGO interferometer and two entangled qubits to illustrate the improved sensitivity that LIGO has achieved by using non-classical states of light.\",\n \"The one-qubit interferometer illustrates how projection noise in the measurement of independent photons corresponds to phase sensitivity at the standard quantum limit.\",\n \"In the presence of technical noise on a real quantum computer, this interferometer achieves the sensitivity of 11\\\\% above the standard quantum limit.\",\n \"The two-qubit interferometer demonstrates how entanglement circumvents the limits imposed by the quantum shot noise, achieving the phase sensitivity 17\\\\% below the standard quantum limit.\",\n \"These experiments illustrate the role that quantum mechanics plays in setting new records for precision measurements on platforms like LIGO.\",\n \"The experiments are broadly accessible, remotely executable activities that are well suited for introducing undergraduate students to quantum computation, error propagation, and quantum sensing on real quantum hardware.\"\n ],\n [\n \"Introduction\",\n \"Quantum systems at the forefront of modern research increasingly utilize entanglement for goals ranging from precision sensing to quantum simulation and computation.\",\n \"Prominently, quantum computers employ entanglement to push beyond the capabilities of classical computers, providing a new set of tools that promise to solve complex problems across multiple fields.\",\n \"On the way to more advanced research goals, first generation quantum computers also serve as educational tools.\",\n \"The IBM Quantum Experience, for example, is an online quantum computing platform that provides the opportunity for students to interact with real quantum hardware [1], [2], [3], [4].\",\n \"Here we use the IBM Quantum Experience to teach students about quantum metrology, one of the first areas of quantum science where entanglement has found practical applications.\",\n \"Applications for entanglement in quantum metrology include spin squeezed clocks[5], enhanced biological imaging[6], remote sensing with EPR states [7], and the detection of gravitational waves [8], [9], [10].\",\n \"Our focus for this paper will be on the particularly charismatic example of LIGO [11], which made the first detection of gravitational waves in 2015, realizing a century-old prediction by Einstein [8].\",\n \"At the heart of LIGO is the most sensitive interferometer ever built, whose sensitivity has recently been enhanced by quantum squeezing of light [9].\",\n \"LIGO clearly illustrates the quantum nature of interferometers and the role that entanglement can play in modern metrology.\",\n \"Here, we describe how to use the IBM Quantum Experience to understand the quantum physics underpinning discoveries at LIGO.\",\n \"In Sec.\",\n \", we construct a single-qubit quantum circuit representation of the LIGO interferometer by analyzing the behavior of a photon traveling through the interferometer.\",\n \"In Sec.\",\n \", we present two representations of this quantum circuit - matrices and Bloch spheres - which provide precise mathematical intuition for the behavior of the quantum circuit.\",\n \"We test predictions made by these representations against the results of experiments performed on the IBM Quantum Experience in Sec. .\",\n \"These results allow us to model the nature of quantum noise in the interferometer.\",\n \"We present a calibrated measurement of the sensitivity of the interferometer in Sec. .\",\n \"The tools developed through analysis of the single-qubit interferometer circuit allow us to efficiently assess the enhancement that can be provided by quantum entanglement.\",\n \"In Sec.\",\n \", we present an interferometer circuit with entanglement between two qubits: the simplest possible example of quantum-enhanced metrology.\",\n \"We model the expected behavior of this circuit using matrix mechanics and visualize the expected dynamics using spin Wigner functions [12] before implementing the circuit on a quantum computer.\",\n \"We show in Sec.\",\n \"that entanglement reduces the noise of this interferometer below the standard quantum limit, even in the presence of experimental imperfections.\",\n \"We present our conclusions in Sec.\",\n \".\"\n ],\n [\n \"LIGO as a quantum circuit\",\n \"LIGO utilizes laser interferometry to measure the microscopic spacetime distortions caused by gravitational waves.\",\n \"As shown in Fig.\",\n \"REF , these distortions change the relative length of LIGO's two arms and thus change the interference signal on the detector.\",\n \"The resulting signals are so small that measurements in the frequency bands relevant to the final moments of black hole and neutron star mergers are limited by the inherent quantum shot noise of the laser used in the interferometer [13].\",\n \"Thus there is value in incorporating quantum mechanics into the description of the interferometer, going beyond the classical framework [11] of understanding interferometers with electromagnetic waves.\",\n \"With only one qubit, a quantum circuit model can capture all the salient features of a classical model and demonstrate the effects of quantum projection noise.\",\n \"Repeatedly executing this circuit simulates measurements of the independent photons that make up a coherent laser beam.\",\n \"The quantum circuit model also provides a scalable framework that can incorporate entanglement, which is necessary for quantum enhanced measurement [14].\",\n \"To construct a quantum circuit model of LIGO, we pare the complicated optical system down to a Michelson interferometer, consisting of only a laser, a beam splitter and two mirrors.\",\n \"This geometry, depicted in Fig.\",\n \"REF , captures the essential physics of the system and is often used to illustrate the operation of LIGO [11], [15].\",\n \"We map the two directions that a photon travels in the interferometer to the two states of a single qubit in the quantum circuit.\",\n \"A photon traveling parallel to the laser corresponds to the qubit state $|0\\\\rangle $ , and a photon traveling in the perpendicular direction corresponds to the state $|1\\\\rangle $ .\",\n \"We deduce the gates that make up the quantum circuit by analyzing the behavior of each of the optical elements that a photon encounters while traveling through the interferometer.\",\n \"This procedure is similar to that used by Nielsen and Chuang to construct a Mach–Zehnder interferometer in the context of dual-rail quantum computation (Section 7.4.2) [16].\",\n \"A photon in the interferometer is initially emitted by a monochromatic laser source and travels towards the beam splitter.\",\n \"The initial direction of the photon is represented by the quantum state $|0\\\\rangle $ .\",\n \"At the beam splitter, the photon has 50% chance of reflection.\",\n \"This creates an equal superposition of the photon continuing to travel in the same direction (quantum state $|0\\\\rangle $ ) and being reflected into the perpendicular direction (quantum state $|1\\\\rangle $ ), which precisely maps onto the action of a Hadamard gate on a single qubit.\",\n \"After the beam splitter, each state accumulates a phase that is proportional to the distance traveled in the respective arm.\",\n \"A difference in length of the two arms, which can be caused by a gravitational wave, introduces a phase difference $\\\\phi $ between the two states.\",\n \"This is analogous to the action of an $R_Z(\\\\phi )$ gate.\",\n \"After retro-reflection at the end of each arm, the photon returns for a second pass through the beam splitter, which is again represented by a Hadamard gate.\",\n \"After the beam splitter, photons in the state corresponding to $|1\\\\rangle $ are measured on the detector, constituting a measurement in the qubit's computational ($z$ ) basis.\",\n \"Putting together the steps of the photon's journey, the full quantum circuit that represents the LIGO interferometer is $@C=1.0em @R=0.2em @!R {{ {q}_{0} : } & {\\\\left|0\\\\right\\\\rangle } & {\\\\mathrm {H}} & {\\\\mathrm {R}_\\\\mathrm {Z}\\\\,\\\\mathrm {(}\\\\mathrm {\\\\phi }\\\\mathrm {)}} & {\\\\mathrm {H}} & & & \\\\\\\\{c:} & {/_{_{1}}} & & & & {_{_{0}}} [-1] & & \\\\\\\\}$ This circuit allows us to directly simulate the behavior of a Michelson interferometer on a quantum computer.\"\n ],\n [\n \"Expected Behavior of the Single-Qubit Circuit\",\n \"In order to formulate a precise quantum picture of a Michelson interferometer like LIGO, we require a mathematical framework that can predict the behavior of the quantum circuit presented in the prior section.\",\n \"Quantum matrix mechanics provides this framework, but is mathematically removed from a physical or graphical interpretation.\",\n \"We thus present the Bloch sphere representation alongside the matrix representation to help visualize the action of the quantum gates on the qubit.\",\n \"For a single qubit, both complementary representations independently provide a precise accounting of the quantum dynamics.\",\n \"The matrix notation for quantum states represents a qubit as a $2\\\\times 1$ vector whose entries are complex numbers.\",\n \"Quantum gates are represented by $2\\\\times 2$ matrices, which act on the state vector through matrix multiplication.\",\n \"Following the convention of Nielsen and Chuang [16], the two basis states $|0\\\\rangle $ and $|1\\\\rangle $ are given by $|0\\\\rangle = \\\\begin{bmatrix}1 \\\\\\\\0\\\\end{bmatrix} \\\\text{and }|1\\\\rangle = \\\\begin{bmatrix}0 \\\\\\\\1\\\\end{bmatrix}.$ An arbitrary superposition of $|0\\\\rangle $ and $|1\\\\rangle $ is given by $|\\\\psi \\\\rangle = c_0|0\\\\rangle + c_1|1\\\\rangle $ and can be written as the vector $|\\\\psi \\\\rangle = \\\\begin{bmatrix}c_0 \\\\\\\\c_1\\\\end{bmatrix}.$ The squared magnitudes of the coefficients $|c_0|^2$ and $|c_1|^2$ give the probabilities of measuring the qubit in the $|0\\\\rangle $ and $|1\\\\rangle $ states, respectively.\",\n \"In the context of the interferometer, only photons in the $|1\\\\rangle $ state (traveling perpendicularly to the laser output beam) reach the detector whereas photons in the $|0\\\\rangle $ state (traveling along the laser output beam) return to the laser without being detected.\",\n \"The probabilities $p_1 = |c_1|^2$ and $p_0 = |c_0|^2$ hence respectively correspond to the probabilities of detecting and not detecting each photon.\",\n \"Information about the quantum superposition is encapsulated in the relative phase $\\\\phi = \\\\phi _1 - \\\\phi _0$ between the two complex coefficients, where $\\\\phi _0$ and $\\\\phi _1$ are defined by the expression of the complex coefficients in polar form: $c_0 = |c_0|e^{i\\\\phi _0}$ and $c_1 = |c_1|e^{i\\\\phi _1}$ .\",\n \"The quantum state $|\\\\psi \\\\rangle $ of a qubit can also be represented on the Bloch sphere as seen in Fig.\",\n \"REF .\",\n \"The Bloch sphere is a unit sphere where a vector from the origin to the surface of the sphere represents a single-qubit state.\",\n \"This vector is known as the Bloch vector.\",\n \"The state $|0\\\\rangle $ is represented by a spin pointing up ($+z$ ) and $|1\\\\rangle $ is represented by a spin pointing down ($-z$ ).\",\n \"For superpositions of $|0\\\\rangle $ and $|1\\\\rangle $ , the $z$ coordinate is given by the relative probability of measuring each of the two states, $p_0 - p_1$ .\",\n \"The angle in the $xy$ -plane is given by the quantum phase $\\\\phi $ , with $\\\\phi =0$ corresponding to the $x$ -axis.\",\n \"Quantum gates generate rotations of the state vector on the Bloch sphere.\",\n \"Figure: Visualization on the Bloch sphere of the quantum state of the single-qubit circuit after each gate is applied.\",\n \"The arrow shown on each sphere represents the quantum state |ψ〉=c 0 |0〉+c 1 |1〉.|\\\\psi \\\\rangle = c_0|0\\\\rangle + c_1|1\\\\rangle .\",\n \"The zz coordinate is the relative probability p 0 -p 1 p_0 - p_1 of measuring |0〉|0\\\\rangle or |1〉|1\\\\rangle while the azimuthal angle is the relative phase φ\\\\phi between the complex coefficients c 0 c_0 and c 1 c_1.\",\n \"The initial state |0〉|0\\\\rangle is a vector pointing up.\",\n \"The Hadamard gate rotates the state by π\\\\pi about the zz-axis, then by π/2\\\\pi /2 about the yy-axis.\",\n \"The R Z (φ)R_Z(\\\\phi ) gate rotates the state about the zz-axis, with φ=π/3\\\\phi =\\\\pi /3 chosen for illustration.\",\n \"The final Hadamard gate maps xx-axis back onto the z-axis, which allows φ\\\\phi to be measured.During the sequence, the qubit is initialized in the state $|0\\\\rangle $ , represented by a vector pointing up on the Bloch sphere in Fig.\",\n \"REF a.\",\n \"The first gate applied to this initial state is a Hadamard gate $H$ .\",\n \"The matrix form of $H$ is $H = \\\\frac{1}{\\\\sqrt{2}}\\\\begin{bmatrix}1 & 1 \\\\\\\\1 & -1\\\\end{bmatrix}.$ On the Bloch sphere, the Hadamard gate first rotates the state by $\\\\pi $ about $z$ and then by $\\\\pi /2$ about $y$ .\",\n \"This transforms the initial state $|0\\\\rangle $ into an equal superposition of $|0\\\\rangle $ and $|1\\\\rangle $ with a relative phase of 0, $|\\\\psi \\\\rangle = \\\\frac{1}{\\\\sqrt{2}}\\\\begin{bmatrix}1 \\\\\\\\1\\\\end{bmatrix}.$ This state vector lies along the $x$ -axis of the Bloch sphere.\",\n \"Following the Hadamard gate, the $R_Z(\\\\phi )$ gate rotates the state about the $z$ -axis by an angle $\\\\phi $ .\",\n \"As an example, a rotation by $\\\\phi = \\\\pi /3$ is depicted in Fig.\",\n \"REF .\",\n \"In the matrix formulation, the $R_Z(\\\\phi )$ gate introduces a relative phase of $\\\\phi $ between the two complex coefficients $c_0$ and $c_1$ via the action of the matrix $R_Z(\\\\phi ) = \\\\begin{bmatrix}e^{-i\\\\phi /2} & 0 \\\\\\\\0 & e^{i\\\\phi /2}\\\\end{bmatrix}.$ For an interferometer like LIGO, this relative phase $\\\\phi $ is induced by a path length difference of $\\\\Delta L = \\\\phi \\\\lambda /2\\\\pi $ where $\\\\lambda $ is the wavelength of light used in the interferometer.\",\n \"Because the quantum state changes based on the value of $\\\\phi $ , the detector is sensitive to the effects of gravitational waves.\",\n \"In order to map the phase accumulated due to the $R_Z(\\\\phi )$ back into the measurement basis ($z$ ), we apply a final Hadamard gate.\",\n \"This gate maps the $x$ -axis back to the $z$ -axis so that phase accumulations in the $xy$ -plane manifest as changes in expectation value for measurements in the standard basis of $|0\\\\rangle $ and $|1\\\\rangle $ .\",\n \"Mathematically, we compute the resulting state of the quantum circuit by multiplying together all of the gates so that the final state is $|\\\\psi \\\\rangle = HR_Z(\\\\phi )H|0\\\\rangle $ , which evaluates to $|\\\\psi \\\\rangle =\\\\frac{1}{2}\\\\begin{bmatrix}e^{-i\\\\phi /2} + e^{i\\\\phi /2}\\\\\\\\e^{-i\\\\phi /2} - e^{i\\\\phi /2}\\\\end{bmatrix}=\\\\begin{bmatrix}\\\\cos {(\\\\phi /2)}\\\\\\\\-i\\\\sin {(\\\\phi /2)}\\\\end{bmatrix}.$ Here, the probabilities of measuring the final state of the circuit as $|0\\\\rangle $ and $|1\\\\rangle $ are given by $p_0 = \\\\cos ^2(\\\\phi /2)$ and $p_1 = \\\\sin ^2(\\\\phi /2)$ , respectively.\",\n \"We summarize the result of this measurement by defining the polarization operator $\\\\hat{P}$ such that the polarization is equal to -1 if we measure $|0\\\\rangle $ and equal to 1 if we measure $|1\\\\rangle $ .\",\n \"Up to the overall sign, this is the $z$ coordinate of the Bloch vector.\",\n \"In matrix form, the polarization operator is $\\\\hat{P} = \\\\begin{bmatrix}-1 & 0 \\\\\\\\0 & 1\\\\end{bmatrix}.$ For the final state $|\\\\psi \\\\rangle $ from Eq.\",\n \"(REF ), the expectation value of the polarization operator is $\\\\langle \\\\psi | \\\\hat{P} |\\\\psi \\\\rangle = p_1 - p_0 = \\\\frac{1}{2}\\\\sin ^2(\\\\phi /2) - \\\\frac{1}{2}\\\\cos ^2(\\\\phi /2) = -\\\\cos {\\\\phi }$ This sinusoidal dependence on $\\\\phi $ is consistent with the behavior of the $z$ -coordinate of the final state visualized in Fig.\",\n \"REF , where the final state's Bloch vector makes an angle $\\\\phi $ with respect to the $z$ axis.\",\n \"In the context of LIGO, this dependence on $\\\\phi $ facilitates sensitivity to signals from gravitational waves.\",\n \"The mapping between the LIGO interferometer and the quantum circuit allows us to simulate the detection of gravitational waves on the quantum computer and better understand the impacts of both fundamental and technical noise sources.\"\n ],\n [\n \"Implementing on a quantum computer\",\n \"We implemented the quantum circuit model for LIGO on the IBM Quantum Experience.\",\n \"Quantum circuits were initially constructed manually via the circuit composer interface, which allows quick prototyping without the need for any programming experience.\",\n \"To scale up the number experimental trials and systematically scan through experimental parameters, we used the Qiskit interface, which allows users to program quantum circuits in Python [17].\",\n \"The code for interfacing with the quantum computers was adapted from existing examples utilizing IBM Quantum Experience [1], [3], [17].\",\n \"The code used for all experiments in this paper is available online [18].\",\n \"Experiments were performed on both the quantum computer ibmq_manila and the state vector simulator ibmq_qasm_simulator.\",\n \"The quantum computer ibmq_manila was chosen because, with a quantum volume of 32, it had the largest quantum volume among freely accessible quantum computers [19].\",\n \"The quantum volume is IBM's preferred metric for its quantum computers and is a proxy for overall error rates; it measures the size of the largest quantum circuit that can be executed with acceptable fidelity [20].\",\n \"The simulator ibmq_qasm_simulator explicitly computes the final qubit state and then samples outcomes from the corresponding probability distribution.\",\n \"This provides a reference point for the effects of finite quantum volume and allows us to distinguish the effects of statistical noise from technical noise sources on the quantum computer.\",\n \"Figure REF shows plots of polarization $P$ as a function of $\\\\phi $ , measured by running the single-qubit circuit from Eq.\",\n \"(REF ) on both the quantum computer ibmq_manila and the simulator ibmq_qasm_simulator.\",\n \"Each data point is obtained by repeatedly executing the quantum circuit and tallying the measurement outcomes 0 and 1 from each shot.\",\n \"A single trial consists of 1024 shots from which we calculate the proportions of shots $p_0$ and $p_1$ with each measurement outcome.\",\n \"Each data point shown is the average polarization $P = p_1 - p_0$ from 5 trials conducted at a fixed value of $\\\\phi $ .\",\n \"The variability among the 5 trials is smaller than the size of the data points.\",\n \"Figure: Measured polarization P=p 1 -p 0 P = p_1 - p_0 as a function of φ\\\\phi , computed from the data taken on the quantum computer (purple circle) and the simulator (green square) at 13 angles between 0 and 2π2\\\\pi , along with the best-fitted curves.\",\n \"Error bars are smaller than the data points.\",\n \"The curves are fitted to the data with amplitude and offset as free parameters.To analyze the relationship between $P$ and $\\\\phi $ , we fit a cosine curve to the data, with amplitude and vertical offset as free parameters.\",\n \"The fit to the simulator data has an amplitude of 0.999(1) and an offset of -0.0001(7), which are consistent with the expected values of 1 and 0 to within the standard errors reported by the fitting routine (shown in parentheses).\",\n \"This affirms the validity of the fitting routine.\",\n \"The fit to the experimental data from the quantum computer has an amplitude of 0.932(5) and an offset of -0.042(4).\",\n \"The amplitude is reduced from the expected value, mainly due to deviations from theory at positive values of polarization.\",\n \"The majority of these deviations can be explained by readout errors.\",\n \"The readout errors on ibmq_manila vary between qubits and over time, but the overall error rate is typically reported between 2 and 3 percent [19], which corresponds to 4-6% reductions in the amplitude of the polarization.\",\n \"The offset can be explained by unequal rates of the two types of readout error, where the dominant source of error is relaxation from the excited state $|1\\\\rangle $ to the ground state $|0\\\\rangle $ during measurement [19].\",\n \"IBM reports an error rates of $2\\\\times 10^{-4}$ for the main component of Hadamard gate, the $\\\\sqrt{X}$ gate.\",\n \"This error rate is approximately one hundred times smaller than the readout error rates and thus a negligible contribution to the overall error.\",\n \"Additional errors can come from the $R_z$ gate, whose error rate is not reported by IBM.\",\n \"These results verify that measurements of the single-qubit interferometer circuit depend on the interferometer phase $\\\\phi $ as expected after incorporating corrections due to dissipation in the real quantum system.\",\n \"Dissipation primarily manifests as readout errors that reduce the amplitude of the polarization curve.\",\n \"The corrections to the curve are important to note as we next seek to invert the relationship and extract the phase of the interferometer from measurement outcomes.\"\n ],\n [\n \"Shot noise and the Measurement of Gravitational Waves\",\n \"The sensitivity of an interferometer is defined by the smallest phase shift that can be reliably detected in the measured output of the interferometer.\",\n \"In general, this phase shift is a change from an initial interferometer phase $\\\\phi _0$ to a new interferometer phase $\\\\phi = \\\\phi _0 + \\\\delta \\\\phi $ .\",\n \"At LIGO, $\\\\delta \\\\phi $ is induced by gravitational waves and $\\\\phi _0$ corresponds to the initial phase in the absence of gravitational waves.\",\n \"On the quantum computer we can choose $\\\\delta \\\\phi $ and $\\\\phi _0$ arbitrarily to illustrate how shot noise limits the precision with which an interferometer can measure phase shifts.\",\n \"Since $\\\\delta \\\\phi $ is small, we can usefully approximate the cosine relationship between polarization and $\\\\phi $ as locally linear, as depicted in the center panel of Fig.\",\n \"REF .\",\n \"Using this linear relationship, polarization measurements can be used to produce an estimate $\\\\tilde{\\\\phi }$ of the phase of the interferometer $\\\\phi = \\\\phi _0 + \\\\delta \\\\phi $ .\",\n \"Error propagates from measurements of the polarization to the inferred values of $\\\\tilde{\\\\phi }$ with a scale factor given by the slope $m$ of the line, where $m=\\\\sin (\\\\phi _0)$ for an ideal interferometer.\",\n \"We can express the standard error of the inferred phase $\\\\tilde{\\\\phi }$ in terms of the standard deviation of polarization measurement as $\\\\sigma _{\\\\tilde{\\\\phi }} = \\\\frac{1}{m}\\\\sigma _{P}.$ Using this expression for the standard deviation of the inferred phase, we can determine theoretically how the precision of the interferometer should scale with number of qubits $N$ that are included in the measurement of average polarization $P$ from a single trial.\",\n \"Here $P = \\\\frac{1}{N}\\\\sum _{i=1}^N P_i$ , where $P_i$ represents the measurement of an individual qubit.\",\n \"We can compute $\\\\sigma _{P} = \\\\sqrt{\\\\operatorname{Var}(P)}$ by determining the variance of a polarization measurement.\",\n \"Since each photon is independent and $1/N$ is a constant factor, the variance can be expressed as a sum, $\\\\operatorname{Var}(P) = \\\\frac{1}{N^2}\\\\sum _{i=1}^N \\\\operatorname{Var}(P_i).$ Using expectation values of the single-qubit polarization operator, we deduce that the variance of a single measurement of an ideal interferometer is $\\\\operatorname{Var}(P_i) = \\\\langle P_i^2 \\\\rangle - \\\\langle P_i \\\\rangle ^2 = 1 - \\\\cos ^2\\\\phi = \\\\sin ^2 \\\\phi .$ Here $\\\\langle P_i^2 \\\\rangle = 1$ because both polarization measurement outcomes square to 1, and $\\\\langle P_i \\\\rangle = -\\\\cos {\\\\phi }$ as derived in Eq.\",\n \"(REF ).\",\n \"Thus, $\\\\operatorname{Var}(P) = \\\\sin ^2 \\\\phi /N$ and $\\\\sigma _P = \\\\sin \\\\phi /\\\\sqrt{N}$ .\",\n \"Substituting back into Eq.\",\n \"(REF ), the phase sensitivity is $\\\\sigma _\\\\phi = \\\\frac{1}{\\\\sqrt{N}},$ which is equal to the standard quantum limit and is independent of $\\\\phi $ .\",\n \"Figure: Polarization measurements at φ=π/2+0.2\\\\phi = \\\\pi /2 + 0.2 (right histogram) and the corresponding inferred phases φ ˜\\\\tilde{\\\\phi } (top histogram) obtained by inverting the linear approximation (purple solid line) of the best-fitted PP vs φ\\\\phi cosine curve (black solid curve) at φ 0 =π/2\\\\phi _0 = \\\\pi /2.\",\n \"The vertical and horizontal dotted lines represent the original phase value φ 0 =π/2\\\\phi _0 = \\\\pi /2 and the corresponding polarization P(φ=π/2)P(\\\\phi = \\\\pi /2), respectively.\",\n \"The black curve is the ibmq_manila cosine fit shown earlier in Fig. .\",\n \"The data were taken from 75 sequential polarization measurements, each consisting of N=100N=100 shots.Figure REF shows how shot noise manifests in an interferometer implemented on a quantum computer.\",\n \"The central panel of the figure depicts the experimentally determined relationship between interferometer phase and polarization and a linear approximation of this curve around $\\\\phi _0 = \\\\pi /2$ .\",\n \"This initial phase and the corresponding polarization are depicted as dashed lines in the figure.\",\n \"We illustrate the sensitivity of the interferometer to a phase shift $\\\\delta \\\\phi = 0.2$ by repeatedly measuring the polarization at the output of a quantum circuit where the interferometer phase was set to $\\\\phi = \\\\pi /2 + 0.2$ .\",\n \"From each measurement of the polarization, we obtain an estimate $\\\\tilde{\\\\phi }$ using the previously determined relationship between the two variables.\",\n \"Repeatedly measuring polarization illustrates the uncertainty in each measurement due to shot noise and other technical noise sources.\",\n \"Each polarization measurement consists of $N=100$ shots.\",\n \"The histogram shown on the right margin is constructed from 75 independent measurements of probability to illustrate variability.\",\n \"The measured polarization has a standard deviation of 0.109(9), which is within error of theoretical value $\\\\sigma _P = \\\\sin {\\\\phi }/\\\\sqrt{N}$ at $\\\\phi = \\\\pi /2 + 0.2$ .\",\n \"The distribution of polarization maps to a distribution of the inferred values of $\\\\tilde{\\\\phi }$ which has a mean of $\\\\pi /2 + 0.20(1)$ and standard deviation of 0.117(10).\",\n \"The mean is consistent with the initially chosen $\\\\delta \\\\phi $ , showing that $\\\\tilde{\\\\phi }$ is an unbiased estimator for the phase shifts.\",\n \"The width of the distribution of $\\\\tilde{\\\\phi }$ is the sensitivity of the interferometer to phase shifts.\",\n \"The estimated phase sensitivity of 0.117(10) is consistent with shot noise for $N=100$ shots combined with the factor of $1/m = 1.073$ due to the reduced amplitude of the polarization curve as compared to Eq.\",\n \"(REF ).\",\n \"The model we derived in Eq.\",\n \"(REF ) for the shot noise of independent photons predicts that the phase sensitivity scales with the number of shots $N$ as $\\\\sigma _{\\\\tilde{\\\\phi }} \\\\propto 1/\\\\sqrt{N}$ .\",\n \"To verify this scaling, we repeat the procedure performed in Fig.\",\n \"REF for values of $N$ between 1 and 1024.\",\n \"We plot the measured sensitivities $\\\\sigma _{\\\\tilde{\\\\phi }}$ in Fig.\",\n \"REF .\",\n \"Shot noise corresponds to a slope of $-1/2$ on the log-log scale depicted in this figure.\",\n \"For all $N$ , the measured sensitivities are slightly above shot noise, but follow the expected scaling, with a slope of -0.506(5) on the log-log plot.\",\n \"On average, the plotted sensitivities are 11(2)% above shot noise, of which 7.3% is due to the reduced amplitude of polarization curve from Fig.\",\n \"REF .\",\n \"The remaining deviation is not statistically significant, but may be due to weak correlations between successive shots, which violate the assumption of independence required for Eq.\",\n \"(REF ).\",\n \"Overall, the results are consistent with the expected scaling for a predominantly shot noise limited device in the presence of some technical noise.\",\n \"The sensitivity of this interferometer is thus primarily limited by the number of qubits queried.\",\n \"This corresponds to noise scaling with the photons or optical power at LIGO.\",\n \"Figure: Standard deviation of inferred φ ˜\\\\tilde{\\\\phi } from polarization data taken on ibmq_manila as a function of number of qubits NN.\",\n \"The least-squares best fit line for the log-log plot is shown in solid purple.\",\n \"All data points lie slightly above the standard quantum limit (dashed line) 1/N1/\\\\sqrt{N}.\"\n ],\n [\n \"Modeling Quantum Enhanced Measurement\",\n \"LIGO now utilizes entanglement to improve sensitivity without further increasing optical power.\",\n \"Specifically, LIGO now uses light in a squeezed quantum state where photons are entangled to each other [9].\",\n \"The correlations between the photons in this state improve sensitivity to phase at the expense of amplitude uncertainty.\",\n \"Due to these correlations, the photons are not independent and can circumvent the standard quantum limit.\",\n \"This manipulation of quantum uncertainty is widely used in quantum metrology.\",\n \"Many examples of metrologically useful states can modeled using the language of quantum circuits [14].\",\n \"The simplest form of entanglement involves two qubits.\",\n \"We can create a two-qubit entangled state on a quantum computer by using a combination of $H$ , $X$ , and CNOT gates.\",\n \"Specifically, the controlled not gate (CNOT) flips a target qubit (labeled with $\\\\bigoplus $ ) based on the state of a control qubit (labeled with $\\\\bullet $ ).\",\n \"This allows the two qubits to interact, producing an entangled state that can be used as the input to an interferometer circuit in order to achieve improved sensitivity [21].\",\n \"The full circuit is $@C=0.75em @R=0.2em @!R { \\\\\\\\{ {q}_{0} : } & { {q}_{0} : } & {\\\\mathrm {\\\\left|0\\\\right\\\\rangle }} & {\\\\mathrm {H}} & {1} & {\\\\mathrm {H}} & {\\\\mathrm {R_Z}\\\\,(\\\\mathrm {\\\\phi })} & {\\\\mathrm {H}} & & & & \\\\\\\\{ {q}_{1} : } & { {q}_{1} : } & {\\\\mathrm {\\\\left|0\\\\right\\\\rangle }} & {\\\\mathrm {X}} & & {\\\\mathrm {H}} & {\\\\mathrm {R_Z}\\\\,(\\\\mathrm {\\\\phi })} & {\\\\mathrm {H}} & & & & \\\\\\\\{c:} & {c:} & {/_{_{2}}} & & & & & & {_{_{0}}} [-2] & {_{_{1}}} [-1] & & \\\\\\\\}$ To understand this circuit's dynamics, we first construct a mathematical representation of two-qubit states.\",\n \"An arbitrary two-qubit state can be expressed as a superposition of four two-qubit basis states $|\\\\psi \\\\rangle = c_0 |00\\\\rangle + c_1|01\\\\rangle + c_2|10\\\\rangle + c_3 |11\\\\rangle $ .\",\n \"The corresponding matrix notation is the $4 \\\\times 1$ vector $[c_0, c_1, c_2, c_3]^T$ .\",\n \"Here each basis state is the tensor product of single-qubit basis states for qubits $q_0$ and $q_1$ .\",\n \"For example, $|01\\\\rangle = |0\\\\rangle _0 \\\\otimes |1\\\\rangle _1$ .\",\n \"In general, states that can be represented as a tensor product $|\\\\psi \\\\rangle = |\\\\varphi _0\\\\rangle \\\\otimes |\\\\varphi _1\\\\rangle $ are not entangled.\",\n \"If a two-qubit state cannot be factorized as a tensor product of single-qubit states, the two qubits are entangled [16].\",\n \"The first three gates in the two-qubit circuit (Eq.\",\n \"(REF )) transform the initial state $|00\\\\rangle $ into an entangled state $\\\\frac{1}{\\\\sqrt{2}} (|01\\\\rangle + |10\\\\rangle )$ , known as a Fock state, to be used for quantum enhanced measurements.\",\n \"The first Hadamard gate and the $X$ gate act independently on the two qubits, preparing the product state $\\\\frac{1}{\\\\sqrt{2}}(|0\\\\rangle _0 + |1\\\\rangle _0) \\\\otimes |1\\\\rangle _1 = \\\\frac{1}{\\\\sqrt{2}} (|01\\\\rangle + |11\\\\rangle )$ .\",\n \"Next, the CNOT gate produces entanglement by flipping the target bit $q_1$ when the control bit $q_0$ is in the state $|1\\\\rangle $ .\",\n \"This operation can be represented by the $4 \\\\times 4$ matrix $\\\\text{CNOT} = \\\\begin{bmatrix}1 & 0 & 0 & 0 \\\\\\\\0 & 1 & 0 & 0 \\\\\\\\0 & 0 & 0 & 1 \\\\\\\\0 & 0 & 1 & 0 \\\\\\\\\\\\end{bmatrix}.$ Applying this matrix to the state $\\\\frac{1}{\\\\sqrt{2}}[0, 1, 0, 1]^T$ produces the Fock state $\\\\frac{1}{\\\\sqrt{2}}[0, 1, 1, 0]^T = \\\\frac{1}{\\\\sqrt{2}} (|01\\\\rangle + |10\\\\rangle )$ .\",\n \"The value of using an entangled state like the Fock state for quantum metrology can be understood visually through the Wigner representation on a spin sphere [12].\",\n \"The Wigner function is a quasi-probability distribution that maps the uncertainty of the quantum state onto the spin sphere.\",\n \"Similar to the Bloch sphere representation, single-qubit gates applied to all qubits rotate the Wigner function without changing the shape of the distribution.\",\n \"This allows for a clean representation of the behavior of each state as it evolves through the interferometer circuit.\",\n \"Figure REF shows the evolution of the two-qubit state after each gate using the Wigner distribution.\",\n \"The single-qubit Bloch spheres from Fig.\",\n \"REF are reproduced in the first row for comparison.\",\n \"The second and third row of Fig.\",\n \"REF compare the evolution of the Wigner distribution when the two qubits start in an untangled state $|00\\\\rangle $ and when they start in the Fock state $\\\\frac{1}{\\\\sqrt{2}}(|01\\\\rangle + |10\\\\rangle )$ , respectively.\",\n \"The Wigner function of the unentangled pair is a localized distribution centered about the Bloch vector, with equal uncertainty in both x and y directions.\",\n \"The width of this local distribution (dark blue region) on the unit sphere is $1/\\\\sqrt{2}$ radians [12], corresponding to the shot noise of two unentangled qubits.\",\n \"For an entangled state like the Fock state, the Wigner distribution is not necessarily symmetric and can have less uncertainty in one direction.\",\n \"The Fock state Wigner function appears as a horizontal ring centered about z=0 in the bottom left sphere in Fig.\",\n \"REF .\",\n \"The thickness of the ring, denoting the uncertainty in the z-direction, is smaller than the width of the unentangled Wigner distribution, at the expense of increased uncertainty in the azimuthal direction.\",\n \"Entanglement also leads to negative Wigner function values at the poles.\",\n \"When properly rotated during the interferometer sequence, this entangled state can exhibit increased sensitivity to the interferometer phase $\\\\phi $ due to the reduced width and increased spatial structure of Wigner distribution.\",\n \"Figure: Visualization of the quantum state over time in the two-qubit circuit using the spin Wigner distribution W(θ,φ)W(\\\\theta ,\\\\phi ).\",\n \"Successive columns depict evolution of the quantum state of the system after each gate in the interferometry sequence is applied.\",\n \"The first row reproduces the single-qubit Bloch spheres from Fig.\",\n \"for comparison.\",\n \"The second row depicts the Wigner distribution for an unentangled initial state |ψ 0 〉=|00〉|\\\\psi _0\\\\rangle = |00\\\\rangle , which is localized about the single-qubit Bloch vector.\",\n \"The third row depicts the Wigner distribution for an entangled pair of qubits initialized in the Fock state |ψ 0 〉=1 2(|01〉+|10〉)|\\\\psi _0\\\\rangle = \\\\frac{1}{\\\\sqrt{2}}(|01\\\\rangle + |10\\\\rangle ), appearing as a horizontal ring.\",\n \"The shape of the Wigner quasi-probability distribution illustrates measurement uncertainty for each quantum state and is rotated under the application of gates in the interferometer sequence in same way as the Bloch vector.To precisely model the behavior of the Fock state in the interferometer requires matrix representations of the three gates used in the interferometer sequence.\",\n \"Since these gates act independently on the two qubits, we can represent each step of the interferometer as the tensor product of two single-qubit gates.\",\n \"The interferometer's beam splitter is represented by a pair of Hadamard gates, $H \\\\otimes H = \\\\frac{1}{2}\\\\begin{bmatrix}1 & 1 & 1 & 1 \\\\\\\\1 & -1 & 1 & -1 \\\\\\\\1 & 1 & -1 & -1 \\\\\\\\1 & -1 & -1 & 1 \\\\\\\\\\\\end{bmatrix}.$ The application of this Hadamard gate transforms the Fock state $\\\\frac{1}{\\\\sqrt{2}}(|01\\\\rangle + |10\\\\rangle )$ into a cat state $\\\\frac{1}{\\\\sqrt{2}}(|00\\\\rangle - |11\\\\rangle )$ .\",\n \"Graphically this corresponds to rotating the Wigner distribution by $\\\\pi /2$ such that the ring is oriented vertically and is sensitive to azimuthal rotations due the interferometer phase.\",\n \"The phase shift portion on the interferometer is given by the product of individual qubit rotations, $R_Z(\\\\phi ) \\\\otimes R_Z(\\\\phi ) = \\\\begin{bmatrix}e^{-i\\\\phi } & 0 & 0 & 0 \\\\\\\\0 & 1 & 0 & 0 \\\\\\\\0 & 0 & 1 & 0 \\\\\\\\0 & 0 & 0 & e^{i\\\\phi } \\\\\\\\\\\\end{bmatrix}.$ The application of this gate to the cat state results in the state $|\\\\psi \\\\rangle =e^{-i\\\\phi }|00\\\\rangle - e^{i\\\\phi } |11\\\\rangle $ , introducing a phase shift of $2\\\\phi $ between the basis states.\",\n \"The state oscillates as a function of $\\\\phi $ between the two cat states $\\\\frac{1}{\\\\sqrt{2}}(|00\\\\rangle - |11\\\\rangle )$ and $\\\\frac{1}{\\\\sqrt{2}}(|00\\\\rangle + |11\\\\rangle )$ with twice the frequency at which states in the single-qubit interferometer oscillate around the Bloch sphere.\",\n \"The final pair of Hadamard gates represents the second pass through a beam splitter and maps the phase shift from the $R_Z$ gates onto the computation basis.\",\n \"The Hadamard gates map one cat state $\\\\frac{1}{\\\\sqrt{2}}(|00\\\\rangle - |11\\\\rangle )$ back to the initial Fock state, while leaving the other cat state $\\\\frac{1}{\\\\sqrt{2}}(|00\\\\rangle + |11\\\\rangle )$ unchanged.\",\n \"Thus the final state is an oscillation between the Fock state and the cat state, $|\\\\psi \\\\rangle = \\\\frac{1}{2\\\\sqrt{2}}\\\\begin{bmatrix}e^{-i\\\\phi } - e^{i\\\\phi } \\\\\\\\e^{-i\\\\phi } + e^{i\\\\phi } \\\\\\\\e^{-i\\\\phi } + e^{i\\\\phi } \\\\\\\\e^{-i\\\\phi } - e^{i\\\\phi }\\\\end{bmatrix}= \\\\frac{\\\\cos {\\\\phi }}{\\\\sqrt{2}} \\\\begin{bmatrix}0 \\\\\\\\1 \\\\\\\\1 \\\\\\\\0\\\\end{bmatrix} - \\\\frac{i\\\\sin {\\\\phi }}{\\\\sqrt{2}}\\\\begin{bmatrix}1 \\\\\\\\0 \\\\\\\\0 \\\\\\\\1\\\\end{bmatrix}.$ The Fock state is detected when the measurements of the two qubits are different, which has a total probability of $p_{01} + p_{10} = \\\\cos ^2{\\\\phi }$ .\",\n \"The cat state corresponds to both qubits being in the same state, which has a total probability of $p_{00} + p_{11} = \\\\sin ^2{\\\\phi }$ .\",\n \"To encapsulate the oscillation between the cat state and Fock state in a single number, we define a parity operator such that parity is 1 when both qubits are in the same state (e.g., cat state), and parity is -1 when the qubits are in different states (e.g., Fock state).\",\n \"In matrix form, this parity operator is $\\\\hat{\\\\Pi } = \\\\begin{bmatrix}1 & 0 & 0 & 0 \\\\\\\\0 & -1 & 0 & 0 \\\\\\\\0 & 0 & -1 & 0 \\\\\\\\0 & 0 & 0 & 1\\\\end{bmatrix}.$ The expectation value of the operator is $\\\\begin{aligned}\\\\langle \\\\Pi \\\\rangle &= \\\\langle \\\\psi |\\\\hat{\\\\Pi }| \\\\psi \\\\rangle \\\\\\\\&= (p_{00} + p_{11}) - (p_{01} + p_{10}) \\\\\\\\&= -\\\\cos {(2\\\\phi )},\\\\end{aligned}$ in close analogy to the polarization measurement for a single-qubit interferometer.\",\n \"The key difference between the expectation value of the two-qubit parity measurement (Eq.\",\n \"(REF )) and the expectation value of the average polarization of independent qubits (Eq.\",\n \"(REF )) is the frequency of oscillation when $\\\\phi $ is varied continuously.\",\n \"As illustrated by Wigner distributions in Fig.\",\n \"REF , an untangled state returns to its original orientation after a rotation of $\\\\phi = 2\\\\pi $ whereas the ring-shaped Wigner distribution for an entangled pair of qubits is indistinguishable from its initial state after a rotation by $\\\\phi =\\\\pi $ .\",\n \"The increased frequency of oscillation leads to an increased slope $m=2\\\\sin (2\\\\phi )$ of the parity vs $\\\\phi $ curve as compared to the slope $m=\\\\sin (\\\\phi )$ of the polarization curve for the single-qubit interferometer.\",\n \"This improves phase sensitivity.\",\n \"In analogy to Eq.\",\n \"(REF ), the phase sensitivity achievable by parity measurement is $\\\\sigma _{\\\\tilde{\\\\phi }} = \\\\frac{\\\\sigma _\\\\Pi }{2\\\\sin {(2\\\\phi )}},$ where $\\\\sigma _{\\\\Pi }$ is the standard deviation of the average parity.\",\n \"The standard deviation of the average parity in measurement consisting of $n$ shots of the two-qubit interferometer can be computed using the matrix form of the single-shot parity operator $\\\\hat{\\\\Pi }_i$ .\",\n \"Assuming all shots are independent, $\\\\sigma _{\\\\Pi }^2 = \\\\frac{\\\\operatorname{Var}({\\\\Pi _i})}{n} = \\\\frac{\\\\langle \\\\hat{\\\\Pi }_i^2 \\\\rangle - \\\\langle \\\\hat{\\\\Pi }_i \\\\rangle ^2}{n} = \\\\frac{\\\\sin ^2{(2\\\\phi )}}{n}.$ For a fair comparison to the single-qubit interferometer, we note that $n$ shots involve measurements of $N=2n$ qubits so the properly normalized phase sensitivity is $\\\\sigma _{\\\\tilde{\\\\phi }} = \\\\frac{1}{\\\\sqrt{2N}}.$ This is an overall improvement by a factor of $\\\\sqrt{2}$ as compared to the shot noise in an unentangled system.\"\n ],\n [\n \"Quantifying Benefits of Entanglement\",\n \"To demonstrate the improvement in sensitivity introduced by entanglement, we implemented the two-qubit interferometer circuit on the same quantum computer, ibmq_manila, that was used earlier to implement the single-qubit interferometer.\",\n \"We additionally implemented the circuit on the state vector simulator, ibmq_qasm_simulator, to provide a comparison without the effects of technical noise.\",\n \"Figure REF shows the results of parity measurements for each of the two implementations.\",\n \"Each measurement of parity $\\\\Pi = p_{00} + p_{11} - p_{01} - p_{10}$ is computed from $n = 1024$ shots of the 2-qubit interferometer circuit.\",\n \"The results show the expected sinusoidal dependence on $\\\\phi $ with a frequency twice that of the single-qubit interferometer.\",\n \"Figure: Measured parity, Π=p 00 +p 11 -p 01 -p 10 \\\\Pi = p_{00} + p_{11} - p_{01} - p_{10}, as a function of φ\\\\phi .\",\n \"Purple circles show data from the two-qubit interferometer taken on the quantum computer ibmq_manila at 13 angles between 0 and 2π2\\\\pi .\",\n \"Green squares show data from the simulator ibmq_qasm_simulator at the same angles.\",\n \"Error bars depict the standard deviation of 5 trials.\",\n \"The curves are fit to the data with amplitude and offset as free parameters.We fit the Manila and simulator data to the expected functional form $\\\\Pi = -A\\\\cos {(2\\\\phi )}+B$ with amplitude and vertical offset as free parameters.\",\n \"The simulator fit has an offset of -0.001(2) and amplitude of 0.997(3), both of which agree well with the expected values of 0 and 1.\",\n \"The Manila fit's offset of -0.007(6) is also consistent with 0, while the amplitude of 0.852(7) corresponds to a 15% reduction from the ideal value.\",\n \"This deviation in amplitude is larger than the corresponding value for the single-qubit interferometer (6.8%) because readout and gate errors in either qubit can flip overall parity, doubling the contribution from these errors.\",\n \"Additionally, the CNOT gate has an error of up to 1% [19], (varying between daily calibrations), which further contributes to the reduction in amplitude.\",\n \"Curves of parity $\\\\Pi $ vs phase $\\\\phi $ can be used to infer phase $\\\\tilde{\\\\phi }$ from measurements of the two-qubit, in analogy to the procedure illustrated for single-qubit polarization measurements in Fig.\",\n \"REF and Fig.\",\n \"REF .\",\n \"To analyze the phase sensitivity quantitatively, like we did for the one-qubit interferometer, we collected parity data on both the quantum computer, ibmq_manila, and the simulator.\",\n \"Each measurement of parity came from a trial consisting of 1 to 512 sequential shots (corresponding to $N=$ 2 to 1024 qubits), conducted with an interferometer phase $\\\\phi = \\\\pi /4$ .\",\n \"From each parity measurement, we inferred a value for the interferometer phase $\\\\tilde{\\\\phi }$ using the slope of the cosine fits from Fig.\",\n \"REF .\",\n \"We computed the standard deviation of these phase measurements $\\\\sigma _{\\\\tilde{\\\\phi }}$ using data from 600 trials.\",\n \"The resulting relationship between phase uncertainty and number of qubits contributing to each measurement is plotted in Fig.\",\n \"REF .\",\n \"The measured phase sensitivities for both ibmq_manila and simulator data are consistently below the standard quantum limit $1/\\\\sqrt{N}$ for unentangled qubits.\",\n \"Fitting the simulator data verifies that the phase sensitivity is $1/\\\\sqrt{2N}$ , as expected for $N/2$ pairs of entangled qubits.\",\n \"The fit for measurements made on the quantum computer is $(0.83 \\\\pm 0.02) N^{-0.495(6)}$ , which lies between the standard quantum limit and the limit for entangled pairs of qubits.\",\n \"Overall, the two-qubit interferometer shows an improvement of 17(2)% as compared to the standard quantum limit and 25(3)% when compared to the single-qubit interferometer implemented on the same hardware.\",\n \"This illustrates the benefits that entanglement can provide for quantum measurement, even in the smallest possible system sizes.\",\n \"Figure: Standard deviation of inferred φ ˜\\\\tilde{\\\\phi } from polarization data taken on the quantum computer Manila and the state vector simulator as a function of number of qubits NN along with their best-fit lines.\",\n \"All data points lie below the standard quantum limit (dashed line) 1/N1/\\\\sqrt{N}.\"\n ],\n [\n \"Conclusions\",\n \"We have demonstrated how entanglement between two qubits enables measurement with sensitivity below the standard quantum limit.\",\n \"Students working on these experiments gain better comprehension of quantum mechanics, quantum metrology, and entanglement.\",\n \"Running these experiments on a real quantum computer, students practice quantum programming, analyze experimental data, and assess the effects of both statistical uncertainty and technical noise sources.\",\n \"This prepares students with the tools needed to engage in modern research in quantum science and technology.\",\n \"The project presented here requires minimal prerequisites.\",\n \"The current work began as an independent student project through a STEM outreach program that pairs high school student researchers with graduate student mentors [22].\",\n \"This illustrates how the material can be accessible to motivated students for whom this is their first introduction to quantum mechanics.\",\n \"We believe this project would fit well in the undergraduate curriculum, either as a demonstration for an introductory quantum information course or as a project in a modern physics laboratory class.\",\n \"These experiments can be accomplished with minimal resources as the IBM Quantum Experience allows for free remote access.\",\n \"Additionally, students working independently for a class project could extend the techniques presented here to circuits with more than two qubits [4] in order to demonstrate different forms of entanglement and implement a variety of quantum squeezing techniques.\",\n \"We thank Stanford FAST for facilitating our collaboration and providing support for this research.\",\n \"TN acknowledges support from NSF Q-SEnSE and ESC acknowledges the support of the NSF Graduate Research Fellowship.\",\n \"We additionally thank Victoria Borish, Monika Schleier-Smith and Patrick Allamandola for useful discussions.\"\n ],\n [\n \"Author contributions\",\n \"All authors contributed equally to this project.\",\n \"CT conducted experiments with guidance from TN and ESC.\",\n \"All authors participated in the analysis of experimental data and the preparation of this manuscript.\"\n ]\n]"},"id":{"kind":"string","value":"2209.08187"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":23,"numItemsPerPage":100,"numTotalItems":1867602,"offset":2300,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MDY4NjUxNywic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUwNjkwMTE3LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.vH6NndpIBp-nftwnlUtPW3I7lGcW23-oMZCCaGrDsoVK836egPHyrYgwY1J6vBBWLkYL9GO_yUitX0ghRvP7Aw","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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"Combinatorial results on (1,2,1,2)-avoiding $GL(p,\\mathbb{C}) \\times\n GL(q,\\mathbb{C})$-orbit closures on $GL(p+q, \\mathbb{C})/B$"
],
[
"Abstract Using recent results of the second author which explicitly identify the \"$(1,2,1,2)$-avoiding\" $GL(p,\\mathbb{C}) \\times GL(q,\\mathbb{C})$-orbit closures on the flag manifold $GL(p+q,\\mathbb{C})/B$ as certain Richardson varieties, we give combinatorial criteria for determining smoothness, lci-ness, and Gorensteinness of such orbit closures.",
"(In the case of smoothness, this gives a new proof of a theorem of W.M.",
"McGovern.)",
"Going a step further, we also describe a straightforward way to compute the singular locus, the non-lci locus, and the non-Gorenstein locus of any such orbit closure.",
"We then describe a manifestly positive combinatorial formula for the Kazhdan-Lusztig-Vogan polynomial $P_{\\tau,\\gamma}(q)$ in the case where $\\gamma$ corresponds to the trivial local system on a $(1,2,1,2)$-avoiding orbit closure $Q$ and $\\tau$ corresponds to the trivial local system on any orbit $Q'$ contained in $\\overline{Q}$.",
"This combines the aforementioned result of the second author, results of A. Knutson, the first author, and A. Yong, and a formula of Lascoux and Sch\\\"{u}tzenberger which computes the ordinary (type $A$) Kazhdan-Lusztig polynomial $P_{x,w}(q)$ whenever $w \\in S_n$ is cograssmannian."
],
[
"Introduction",
"Let $G$ be a complex semisimple reductive group and $B$ a Borel subgroup of $G$ .",
"The opposite Borel subgroup $B^-$ (as well as $B$ itself) acts on the flag variety $G/B$ with finitely many orbits.",
"The closures of the orbits are known as the (opposite) Schubert varieties.",
"These orbits are indexed by the Weyl group $W$ for $G$ , so for each $w\\in W$ , we have a Schubert variety $X^w$ .",
"Especially in the case $G=\\mathrm {SL}(n,$ , frequently known as the “type A case”, the geometry of Schubert varieties has been extensively studied, both for its intrinsic interest and for its applications to the representation theory of $G$ .",
"Particularly interesting for our present purposes are results that relate the geometry of $X^w$ to the combinatorics of the indexing element $w$ using the combinatorial notion of pattern avoidance.",
"Abe and Billey have recently written an excellent survey of such results [2], of which we mention only a few here: Pattern avoidance criteria have been given in all types to determine which Schubert varieties are smooth and which are rationally smooth [28], [7], [6].",
"In type $A$ , the singular locus of a Schubert variety is completely understood in terms of patterns appearing in the Weyl group element [10], [15], [33], [24].",
"In type $A$ , criteria in terms of a generalization of pattern avoidance have been given to determine which Schubert varieties are Gorenstein [45].",
"In type $A$ , pattern avoidance criteria have been given to determine which Schubert varieties are local complete intersections (“lci\") [42].",
"Now let $\\theta $ be an involution of $G$ , and let $K=G^\\theta $ be the fixed points of the involution.",
"The pair $(G,K)$ is known as a symmetric pair.",
"The subgroup $K$ also acts with finitely many orbits on the flag variety $G/B$ [34].",
"Moreover, natural combinatorial indexing sets for the orbits have been determined in many cases.",
"The geometry of $K$ -orbits and their closures are important in the representation theory of the real forms $G_{{\\mathbb {R}}}$ of $G$ .",
"In one special case, the relationship between the geometry of the $K$ -orbit and the combinatorics of its indexing element is well understood.",
"Let our group be $G\\times G$ and our involution $\\theta $ be given by $\\theta (x,y)=(y,x)$ .",
"In this case, the symmetric pair is $(G\\times G, G)$ , and the action is the diagonal action of $G$ on $G/B\\times G/B$ .",
"The orbit closures are fiber bundles with Schubert variety fibers over the smooth base $G/B$ , so many geometric properties of these orbit closures can be understood by reference to analogous results for Schubert varieties.",
"Hence we refer to this case loosely as “the Schubert case”.",
"However, outside of the Schubert case, the interaction between the geometry of the $K$ -orbit closure and the combinatorics of the indexing set has been less well studied.",
"We know only of a few recent results in this direction: Smoothness and rational smoothness have been characterized by McGovern and McGovern–Trapa in terms of pattern avoidance for various symmetric pairs [37], [38], [36], [39].",
"Graph-theoretic criteria for rational smoothness have been given by A. Hultman [21] for pairs $(G,K)$ satisfying fairly strong hypotheses — these apply in particular to the symmetric pair $(G,K) = (GL(2n,,Sp(2n,)$ .",
"To the authors' knowledge, no explicit description of the (rationally) singular locus of an orbit closure has been given in any of these cases, nor have any combinatorial criteria been given to categorize $K$ -orbit closures with respect to more subtle singularity properties, such as Gorensteinness or lci-ness, for any symmetric pair.",
"Aside from these purely geometric questions, in the Schubert case, one can hope for combinatorial descriptions of Kazhdan–Lusztig (KL) polynomials.",
"In 1979, Kazhdan and Lusztig [25] introduced this family of polynomials $P_{v,w}(q)$ , where $v$ and $w$ are elements of a Coxeter group $W$ .",
"Defined by a recursion on $W$ or alternatively by certain axioms on elements of the Hecke algebra, these polynomials carry important information about the representation theory of both semisimple complex reductive groups and Coxeter groups.",
"Geometrically, in the case where $W$ is a Weyl group, their coefficients are dimensions of local intersection (co)homology groups of Schubert varieties, so in particular they are nonnegative.",
"(A proof for this nonnegativity that does not rely on the geometric interpretation, and hence holds generally for all Coxeter groups, was only recently given by Elias and Williamson [19].)",
"There have been numerous papers giving various combinatorial formulas for various classes of Kazhdan–Lusztig polynomials; some such papers are [31], [30], [5], [9], [23].",
"Inspired by questions in the representation theory of real reductive groups, Vogan and Lusztig–Vogan [43], [32] defined a more general family of polynomials associated to a symmetric pair $(G,K)$ , now known as Kazhdan–Lusztig–Vogan (KLV) polynomials $P_{\\tau ,\\gamma }(q)$ .",
"(Indeed, the KLV polynomials for the Schubert case are the ordinary KL polynomials.)",
"In the most general case, $\\tau $ and $\\gamma $ are pairs, each consisting of a $K$ -orbit $Q_{\\tau }$ (or $Q_{\\gamma }$ ) on $G/B$ together with a $K$ -equivariant local system on the orbit.",
"In this paper, matters are simplified by the fact that for the pair $(G,K)$ that we consider, each orbit admits only the trivial $K$ -equivariant local system.",
"Thus the reader may think of our KLV polynomials as being of the form $P_{\\tau ,\\gamma }(q)$ where $Q_{\\tau }$ and $Q_{\\gamma }$ are $K$ -orbits.",
"As with ordinary KL polynomials, the KLV polynomials can be defined by a recursion on the indexing set for local systems or alternatively by certain axioms on elements of a particular module over the Hecke algebra.",
"Like the ordinary KL-polynomials, their coefficients have a geometric interpretation as the dimensions of local intersection (co)homology groups of $K$ -orbit closures, so in particular they are non-negative.",
"However, the authors know of only one previous result [14] implying a formula for KLV polynomials for a very special case where the $K$ -orbit is a Schubert variety.",
"In this paper, we answer various combinatorial questions of the above flavor for certain $K=GL(p, \\times GL(q,$ -orbit closures on $GL(p+q,/B$ .",
"Using recent work of the second author [49] identifying a certain subset of the $K$ -orbit closures (those “whose clans avoid $(1,2,1,2)$ \" — this is explained in Section ) as certain Richardson varieties, we recover most of the results of [37] regarding pattern avoidance criteria for rational smoothness.",
"We also give combinatorial characterizations describing which orbit closures are lci and which are Gorenstein.",
"We moreover describe a diagrammatic procedure to calculate the singular locus, the non-lci locus, and the non-Gorenstein locus of a $(1,2,1,2)$ -avoiding orbit closure.",
"The remaining $K$ -orbit closures — those whose clans contain $(1,2,1,2)$ — are more difficult to study.",
"They are not Richardson varieties but are instead cut out from Richardson varieties by certain projection conditions [47].",
"(Equivalently, they can be described as the intersections of Richardson varieties and certain Hessenberg varieties.)",
"Therefore, the techniques used in this paper do not apply directly to them.",
"Nonetheless, it is natural to wonder about combinatorial characterizations of lci-ness and Gorensteinness of these more complicated orbit closures.",
"(The question of (rational) smoothness is understood; see Proposition REF .)",
"We do not give complete answers to these questions, but we do briefly describe ways to approach them computationally, giving some partial evidence and conjectures based on our own experiments.",
"Next, combining the aforementioned work of the second author, recent work of A. Knutson, the first author, and A. Yong [27] reducing local questions on Richardson varieties to similar questions on Schubert varieties, and an old result of Lascoux and Schützenberger [31] giving Kazhdan–Lusztig polynomials for cograssmannian Schubert varieties, we give explicit combinatorial formulas for the KLV polynomials associated to pairs of orbits $\\tau ,\\gamma $ , where $\\gamma $ is an orbit whose clan avoids $(1,2,1,2)$ .",
"This subsumes as a special case a result of Collingwood [14].",
"We remark briefly on a possible extension to the results of this paper.",
"The symmetric pairs $(G,K) = (Sp(2n,,GL(2n,)$ and $(SO(2n,,GL(n,)$ are like $(GL(p+q,,GL(p, \\times GL(q,)$ in that, in each case, $K$ is the Levi subgroup of a minuscule parabolic subgroup of $G$ .",
"This fact implies that a number of the $K$ -orbit closures in those cases also coincide with Richardson varieties.",
"The second author has explicitly identified the $K$ -stable Richardson varieties in these cases [48].",
"(They again correspond to “$(1,2,1,2)$ -avoiding clans.\")",
"The singular locus of minuscule Schubert varieties is described in [29], so one can in principle recover some of the pattern avoidance results of [39] for the case $(SO(2n,,GL(n,)$ and give similar results for the pair $(Sp(2n,,GL(n,)$ .",
"Additionally, the Gorenstein locus of such Schubert varieties are described in [40], so one should also be able to characterize Gorensteinness of $(1,2,1,2)$ -avoiding orbit closures in these cases.",
"Finally, KL polynomials for minuscule Schubert varieties were given by B. Boe in [11], so one can also use our methods to write down explicit formulas for some of the KLV polynomials for these symmetric pairs.",
"Finding these results will require understanding the translation between the combinatorics of (co)minuscule quotients of Weyl groups on the one hand and the combinatorics of the clan parameterization for these symmetric pairs on the other.",
"We will not attempt to make these additional translations explicit in this paper."
],
[
"Notation and conventions",
"In this paper, $G$ will denote the group $GL(n,$ .",
"We use $K$ to denote the symmetric subgroup $GL(p, \\times GL(q,$ .",
"Note that $K=G^{\\theta }$ for $\\theta = \\text{int}(I_{p,q})$ , where $ I_{p,q} :=\\begin{pmatrix}I_p & 0 \\\\0 & -I_q \\end{pmatrix} $ and where $\\text{int}(g)$ denotes conjugation by $g$ .",
"Realized in this way, $K$ is embedded in $G$ as the subgroup of block-diagonal matrices consisting of an upper-left invertible $p \\times p$ block, a lower-right invertible $q \\times q$ block, and zeros elsewhere.",
"$B$ and $B^-$ will denote the opposite Borel subgroups of $G$ consisting of upper and lower-triangular matrices, respectively, while $T = B \\cap B^-$ will be the diagonal maximal torus of $G$ .",
"The flag variety is isomorphic to $G/B$ , with the coset $gB$ corresponding under this isomorphism to the complete flag whose $i$ th subspace is the linear span of the first $i$ columns of $g$ .",
"Let $u \\in S_n$ be given as a permutation matrix in $G$ .",
"For us, the Schubert cell, denoted $X_0^u$ , will be the $B^-$ -orbit $B^- uB/B$ of the $T$ -fixed point $uB$ .",
"The corresponding Schubert variety $X^u$ is the closure $\\overline{X_0^u}$ .",
"The Schubert variety $X^u$ is a subvariety of $G/B$ of complex codimension $l(u)$ , where $l$ denotes the Coxeter length function on $S_n$ .",
"Similarly, the opposite Schubert cell $X_u^0$ will be the $B$ -orbit $BuB/B$ , while the opposite Schubert variety $X_u$ will be the closure $\\overline{X_u^0}$ .",
"The opposite Schubert variety $X_u$ is a subvariety of $G/B$ of complex dimension $l(u)$ .",
"(Note that many papers reverse our definitions of Schubert and opposite Schubert cells and varieties.)",
"Left multiplication by the longest element $w_0\\in S_n$ gives an isomorphism between the opposite Schubert variety $X_u$ and the Schubert variety $X^{w_0u}$ .",
"Also, given $u\\in S_n$ , we will sometimes refer to the point $uB/B$ simply as $u$ or as $p_u$ ."
],
[
"Combinatorial parameters for $K=GL(p, \\times GL(q,$ -orbits on {{formula:3aafc44e-af59-49e0-8c66-b5eab2a13623}}",
"The finitely many $K$ -orbits on $G/B$ are customarily indexed by $(p,q)$ -clans, as described in, for example, [35], [51], [39].",
"We now recall the details of this indexing.",
"Definition 2.1 A $(p,q)$ -clan is a string $\\gamma =(c_1, \\hdots ,c_n)$ of $n=p+q$ symbols, each of which is a $+$ , a $-$ , or a natural number.",
"The string must satisfy the following two properties: Every natural number which appears must appear exactly twice in the string.",
"The difference between the number of plus signs and the number of minus signs in the string must be $p-q$ .",
"(If $q > p$ , then there should be $q-p$ more minus signs than plus signs.)",
"We only consider such strings up to an equivalence relation saying that only the positions of matching natural numbers, rather than the actual values of the numbers, are necessary to determine the clan.",
"For instance, the clans $(1,2,1,2)$ , $(2,1,2,1)$ , and $(5,7,5,7)$ are all the same, since they all have matching natural numbers in positions 1 and 3 and also in positions 2 and 4.",
"On the other hand, $(1,2,2,1)$ is a different clan, since it has matching natural numbers in positions 1 and 4 and in positions 2 and 3.",
"A theorem of [35], elaborated upon in [51], gives an explicit bijection between the set of $(p,q)$ -clans and the set of $K$ -orbits on $G/B$ : Theorem 2.2 ([35], [51]) There is an explicit bijection between the set of $(p,q)$ -clans and the $K$ -orbits on $G/B$ .",
"For each $(p,q)$ -clan $\\tau $ , we need an explicit point of the corresponding orbit $Q_{\\tau }$ .",
"We now outline an algorithm, described in [51], which produces certain such representatives, which we call Yamamoto points of $Q_{\\tau }$ .",
"First, for each pair of matching natural numbers of $\\tau $ , we assign one of the numbers a “signature\" of $+$ , and the other a signature of $-$ .",
"We next choose a permutation $v$ of $1,\\hdots ,n$ whose one-line notation places $1,\\hdots ,p$ (in any order) in the positions of the $+$ signs and numbers assigned a signature of $+$ , and $p+1,\\hdots ,n$ (in any order) in the remaining positions.",
"Having determined such a permutation $\\sigma $ , let $F_{\\bullet } = \\left\\langle v_1, \\hdots , v_n \\right\\rangle $ to be the flag specified as follows: $ v_i ={\\left\\lbrace \\begin{array}{ll}e_{v(i)} + e_{v(j)} & \\text{ if $c_i \\in {\\mathbb {N}}$, $c_i$ has signature $+$, and $c_i = c_j$}, \\\\e_{v(i)} & \\text{ otherwise.}\\end{array}\\right.}",
"$ Then $F_{\\bullet } \\in Q_{\\tau }$ , the $K$ -orbit corresponding to the clan $\\tau $ .",
"Note that the algorithm described above allows for several choices.",
"We describe a particularly natural scheme for these choices.",
"To each pair of numbers, assign the first a signature of $+$ and the second a signature of $-$ .",
"Then choose $v$ to be the permutation whose one-line notation places the numbers $1,\\hdots ,p$ in ascending order on the positions of the $+$ signs and the first occurrences of natural numbers, and whose one-line notation places the numbers $p+1,\\hdots ,n$ , also in ascending order, on the remaining positions.",
"Definition 2.3 We call the Yamamoto point of $Q_{\\tau }$ obtained using this choice of permutation $v$ the distinguished representative of $Q_{\\tau }$ .",
"We give two examples.",
"For the $(3,3)$ -clan $\\tau =(+,+,+,-,-,-)$ , the permutation $v=123456$ , and the distinguished representative is the standard flag, $ \\left\\langle e_1,\\hdots ,e_6 \\right\\rangle .",
"$ Now, let $\\tau $ be the $(2,2)$ -clan $(1,-,+,1)$ .",
"The permutation $v$ is 1324, and the distinguished representative is $ \\left\\langle e_1 + e_4, e_3, e_2, e_4 \\right\\rangle .",
"$ Given a clan $\\tau $ , we now associate to it two Grassmannian permutations $v(\\tau )$ and $u(\\tau )$ .",
"The permutation $v(\\tau )$ is the inverse of the permutation $v$ which we have just described.",
"Explicitly, its one-line notation is formed by first listing in ascending order the positions of $\\tau $ containing a $+$ or the first occurrence of a number, then listing in ascending order the positions with a $-$ or the second occurrence of a number.",
"The permutation $u(\\tau )$ is obtained in a similar way.",
"Its one-line notation is formed by first listing in ascending order the positions of $\\tau $ which have a $+$ or the second occurrence of a number, followed by listing in ascending order the positions with a $-$ or the first occurence of a number.",
"For example, if $\\tau =(1,2,+,-,1,2)$ , then $v(\\tau ) = 123456$ , and $u(\\tau ) = 356124$ .",
"If $\\tau =(1,2,2,3,3,1)$ , then $v(\\tau ) = 124356$ and $u(\\tau ) = 356124$ .",
"Now, let $u$ be the permutation obtained from $v=v(\\tau )^{-1}$ and $\\tau $ as follows.",
"For every pair of matching natural numbers $c_i = c_j \\in {\\mathbb {N}}$ of $\\tau $ , interchange the entries of the one-line notation for $v$ in positions $i$ and $j$ .",
"Returning to the two examples above, if $\\tau =(1,2,+,-,1,2)$ , then $v=123456$ , while $u=563412$ .",
"If $\\tau =(1,2,2,3,3,1)$ , then $v=124356$ , and $u=642531$ .",
"Then we have the following easy result.",
"Lemma 2.4 Let $\\tau $ be a $(p,q)$ -clan, and let $F_{\\bullet }$ be the distinguished representative of $Q_{\\tau }$ .",
"Let $u,v$ be as just defined.",
"Then $F_{\\bullet }$ is in the $B^-$ -orbit of the $T$ -fixed point $vB/B$ and the $B$ -orbit of the $T$ -fixed point $uB/B$ .",
"In other words, $F_{\\bullet } \\in X_u^0 \\cap X_0^v$ .",
"Moreover, although $u^{-1} \\ne u(\\tau )$ in general, $u^{-1}$ is in the same left $W_K = S_p \\times S_q$ -coset as $u(\\tau )$ .",
"Let $\\tau =(\\tau _1,\\hdots ,\\tau _n)$ .",
"The flag $F_{\\bullet }$ is of the form $gB$ , where $g$ is the matrix whose columns are the vectors $v_i$ given by the Yamamoto algorithm above.",
"Evidently, $g$ is almost the permutation matrix with 1's in positions $(v(i),i)$ , except that in each column corresponding to an index $i$ such that $\\tau _i \\in {\\mathbb {N}}$ is a first occurrence, there is an extra 1 in row $v(j)$ , where $\\tau _i = \\tau _j$ .",
"Note that by our choice of $v$ , $v(j) > v(i)$ , so the 1 in position $(v(j),i)$ occurs further down in column $i$ than that in position $(v(i),i)$ .",
"Thus, using the left $B^-$ -action by downward row operations, we may eliminate these extra 1's, giving the point $vB/B$ .",
"For the second claim, we first change our matrix representative for $F_\\bullet $ (or equivalently present some of the vector spaces in $F_\\bullet $ with a different basis).",
"Using the right action of $B$ by rightward column operations, we can first eliminate the second 1 from any row consisting of two 1's.",
"There is one such row for each pair of matching natural numbers $\\tau _i = \\tau _j \\in {\\mathbb {N}}$ ($i<j$ ), namely row $u(i)$ .",
"The effect of such a column operation is to eliminate the second 1 in position $(u(i),j)$ and move it instead (up) to position $(u(j),j)$ .",
"As a result, row $u(j)$ now has two 1's, one at position $(u(j),i)$ and the other at position $(u(j),j)$ , whereas row $u(i)$ has only one 1 in position $(u(i),i)$ .",
"Now, using the left $B$ -action by upward row operations on this new representative, we can eliminate the additional 1 in position $(u(j),i)$ .",
"Doing so for all pairs $\\tau _i = \\tau _j$ gives the point $uB/B$ .",
"For the last claim, note that by construction, $u$ has a one-line notation in which $1,\\hdots ,p$ (in some order) are on the $+$ 's and second occurrences, and $p+1,\\hdots ,n$ (in some order) are on the $-$ 's and first occurrences.",
"Left-multiplying $u$ by some element of $S_p \\times S_q$ will put both sets in ascending order.",
"We then take the inverse to obtain the desired result.",
"There is a natural notion of pattern avoidance for $(p,q)$ -clans, used first by McGovern in [37].",
"We say that one clan $\\gamma $ contains another clan $\\gamma ^{\\prime }$ if there are character positions within $\\gamma $ which, when extracted from $\\gamma $ in order, give a clan equivalent to $\\gamma ^{\\prime }$ , where the equivalence is, as described above, up to permutation of the natural numbers.",
"We say that $\\gamma $ avoids the pattern $\\gamma ^{\\prime }$ if it does not contain it.",
"In particular, note that $\\gamma $ “avoids the pattern $(1,2,1,2)$ \" if any two pairs of matching natural numbers of $\\gamma $ are either nested or disjoint.",
"So, for example, $(1,1,2,2,3,3)$ avoids $(1,2,1,2)$ , but $(1,2,1,3,2,3)$ does not.",
"The first main theorem that we use in our combinatorial analysis of some of the $K$ -orbit closures is a result of [49] identifying $(1,2,1,2)$ -avoiding $K$ -orbit closures explicitly as certain Richardson varieties.",
"Associated to a $(1,2,1,2)$ -avoiding clan $\\gamma $ , we have the Grassmannian permutations (each with at most one descent at position $p$ ) $u(\\gamma )$ and $v(\\gamma )$ defined a few paragraphs ago.",
"Then we have the following theorem.",
"Theorem 2.5 ([49]) Given any $(p,q)$ -clan $\\gamma $ avoiding the pattern $(1,2,1,2)$ , let $Q_{\\gamma }$ be the associated $K$ -orbit.",
"Let $u = w_0^K u(\\gamma )^{-1}$ , and let $v = v(\\gamma )^{-1}$ , where $w_0^K$ denotes the long element of $W_K = S_p \\times S_q$ which reverses the sets $1,\\hdots ,p$ and $p+1,\\hdots ,n$ .",
"Then $\\overline{Q_{\\gamma }}$ is the Richardson variety $X_u^v = X_u \\cap X^v$ ."
],
[
"KL and KLV polynomials",
"In this section, we quickly recall the definitions — first algebraic, then geometric — of both the ordinary Kazhdan-Lusztig (KL) polynomials and the Kazhdan–Lusztig–Vogan (KLV) polynomials.",
"The latter polynomials were originally defined by Vogan [43] for a general symmetric pair $(G,K)$ .",
"We give the general definition then explicitly explain how to calculate KLV polynomials for $(GL(n,, GL(p, \\times GL(q,)$ in terms of the combinatorics of clans.",
"This case is simpler than the general case but more complicated than the cases treated by Hultman [21].",
"Our hope is that the explicit description will be helpful to combinatorialists interested in understanding these polynomials."
],
[
"Combinatorial definitions",
"First, recall the definition of the ordinary KL polynomials.",
"Given a Weyl group $W$ with simple reflections $S$ , the Hecke algebra $\\mathcal {H}_W$ is the ${\\mathbb {Z}}[q^{\\pm 1/2}]$ -algebra with basis $T_w$ for $w\\in W$ and multiplication defined by $ T_sT_w={\\left\\lbrace \\begin{array}{ll}T_{sw} & \\text{if $sw > w$} \\\\(q-1)T_w + qT_{sw} & \\text{ if $sw < w$.}\\end{array}\\right.",
"}$ The Hecke algebra has a ring involution defined by $\\overline{q^{1/2}}=q^{-1/2}$ and $\\overline{T_w}=(T_{w^{-1}})^{-1}$ .",
"We can define the $R$ -polynomials by $\\overline{T_w}=q^{-\\ell (w)}\\sum _{v}(-1)^{\\ell (v)}R_{v,w}(q)T_v;$ if we do so, then $R_{v,w}(q)$ will be polynomials in $q$ with $R_{w,w}=1$ for all $w$ and $R_{v,w}=0$ whenever $v\\lnot \\le w$ .",
"Indeed, the involution can be defined as the unique ring homomorphism satisfying $\\overline{q^{1/2}}=q^{-1/2}$ and this condition on $R_{v,w}$ .",
"In [25], Kazhdan-Lusztig showed that there exists a unique basis $\\lbrace C^\\prime _w\\rbrace _{w\\in W}$ satisfying the following: $\\overline{C^\\prime _w}=C^\\prime _w$ for all $w\\in W$ .",
"If we define $P_{x,w}(q)$ by $ C^\\prime _w=q^{-\\ell (w)/2}\\displaystyle \\sum _{x \\le w} P_{x,w}(q)T_x, $ then $P_{x,w}(q)$ is a uniquely determined polynomial in $q$ , provided that we insist that $P_{w,w}(q)=1$ , and $\\deg (P_{x,w}) \\le \\frac{1}{2}(\\ell (w)-\\ell (x)-1)$ .",
"From these facts and the base cases $C^\\prime _1=1$ and $C^\\prime _s=q^{-1/2}(1+T_s)$ for all simple reflections $s$ , one can recursively calculate the Kazhdan–Lusztig elements $C^\\prime _w$ and hence the Kazhdan-Lusztig (KL) polynomials $P_{x,w}(q)$ .",
"If $ws>w$ , then $ C^\\prime _wC^\\prime _s=C^\\prime _{ws}+\\displaystyle \\sum _{v<ws} E_v(q)C^\\prime _v, $ where $E_v(q)$ is explicitly either 0 if $vs<v$ or else the coefficient of $q^{(\\ell (w)-\\ell (v)-1)/2}$ in $P_{v,w}(q)$ .",
"However, the explicit description of $E_v(q)$ is not necessary, since we can recursively determine $E_v(q)$ purely from the degree bound (2b).",
"In particular, if the coefficients in $C^\\prime _wC^\\prime _s$ satisfy the degree bound (2b), then $E_v(q)=0$ for all $v$ and $C^\\prime _{ws}=C^\\prime _wC^\\prime _s$ .",
"KLV polynomials for a symmetric pair $(G,K)$ have a similar definition in terms of an $\\mathcal {H}_W$ -module $\\mathcal {M}_K$ .",
"Let $\\mathcal {D}$ consist of all pairs $(Q,\\delta )$ , where $Q$ is a $K$ -orbit on $G/B$ and $\\delta $ is a $K$ -equivariant local system on $Q$ .",
"Since $Q$ is determined by the local system $\\delta $ , we will write $\\delta $ to mean $(Q,\\delta )$ .",
"The module $\\mathcal {M}_K$ is free over ${\\mathbb {Z}}[q^{\\pm 1/2}]$ with basis $\\lbrace \\mathbf {T}_\\delta \\rbrace _{\\delta \\in \\mathcal {D}}$ .",
"We will not describe the action of $\\mathcal {H}_W$ on $\\mathcal {M}_K$ in general, but we later give an explicit description of this action in terms of clans when $K=GL(p, \\times GL(q,$ and $G=GL(p+q,$ .",
"On the set $\\mathcal {D}$ there is a partial order called Bruhat $\\mathcal {G}$ -order, defined in [43].",
"We indicate this order by $<$ .",
"Like Bruhat order on $W$ , Bruhat $\\mathcal {G}$ -order on $\\mathcal {D}$ is graded by a length function $\\ell $ .",
"(Bruhat $\\mathcal {G}$ -order is roughly defined by inclusion of orbits, complicated by the possibility that multiple local systems can be associated to a single orbit.",
"The length $\\ell (\\delta )$ is the dimension of the orbit associated to $\\delta $ minus the minimal dimension for all orbits.The original definitions in [43] defined length as simply the dimension of the orbit, but it is easy to see that adding a constant to all lengths has no effect as long as it is done consistently.",
"We use our definition both for simplicity and to agree with the Atlas of Lie Groups software.)",
"We can now define an involution on $\\mathcal {M}_K$ by requiring that $\\overline{h\\cdot m}=\\overline{h}\\cdot \\overline{m}$ for all $h\\in \\mathcal {H}_W$ and all $m\\in \\mathcal {M}_K$ .",
"If we define $R_{\\gamma ,\\delta }$ by $ \\overline{\\mathbf {T}_\\delta }=(-q^{-\\ell (\\delta )})\\sum (-1)^{\\ell (\\gamma )}R_{\\gamma ,\\delta }(q)\\mathbf {T}_\\delta , $ then $R_{\\gamma ,\\delta }(q)=0$ unless $\\ell (\\gamma )\\le \\ell (\\delta )$ , and $R_{\\delta ,\\delta }(q)=1$ for all $\\delta $ .",
"Given the bar involution, the KLV polynomials $P_{\\gamma ,\\delta }$ can be defined exactly as the KL polynomials are.",
"There is a unique basis $\\lbrace \\mathbf {C}^\\prime _\\delta \\rbrace _{\\delta \\in \\mathcal {D}}$ satisfying the following: $\\overline{\\mathbf {C}^{\\prime }_\\delta }=\\mathbf {C}^{\\prime }_{\\delta }$ for all $\\delta \\in \\mathcal {D}$ .",
"If we define $P_{\\gamma ,\\delta }(q)$ by $ \\mathbf {C}^{\\prime }_\\delta =q^{-\\ell (\\delta )/2}\\sum _\\tau P_{\\gamma ,\\delta }(q)\\mathbf {T}_\\delta , $ then $P_{\\gamma ,\\delta }(q)$ is a uniquely determined polynomial in $q$ , provided we insist that $P_{\\delta ,\\delta }(q)=1$ , and $\\deg (P_{\\gamma ,\\delta }) \\le \\frac{1}{2}(\\ell (\\delta )-\\ell (\\gamma )-1)$ .",
"For many groups $K$ (including $K=GL(p, \\times GL(q,$ ), one can also recursively compute $\\mathbf {C}^\\prime _\\delta $ as for KL polynomials.",
"The base cases are given by $\\mathbf {C}^\\prime _\\delta =\\mathbf {T}_\\delta $ whenever $\\delta $ is a minimal element in Bruhat $\\mathcal {G}$ -order.",
"Otherwise, for each $\\delta \\in \\mathcal {D}$ , one finds $\\tau \\in \\mathcal {D}$ with $\\tau <\\delta $ and a simple reflection $s\\in W$ such that $\\mathbf {T}_\\delta $ is the unique maximal basis element (in Bruhat $\\mathcal {G}$ -order) occuring in the expansion of $C^\\prime _s\\mathbf {C}^\\prime _\\tau $ .",
"Then one recursively computes $ C^\\prime _s\\mathbf {C}^\\prime _\\tau =\\mathbf {C}^\\prime _\\delta +\\displaystyle \\sum _{\\gamma <\\delta } E_\\gamma (q)\\mathbf {C}^\\prime _\\gamma , $ where $E_\\gamma (q)$ is explicitly either 0 or a coefficient of $P_{\\tau ,\\gamma }(q)$ depending on certain relations in Bruhat $\\mathcal {G}$ -order.",
"Again, $E_\\gamma (q)$ can be recursively determined from the degree bound (2b), and in particular, if the coefficients in $C^\\prime _s\\mathbf {C}^\\prime _\\tau $ satisfy the degree bound (2b), then $E_\\gamma (q)=0$ for all $\\gamma $ , and $\\mathbf {C}^\\prime _\\delta =C^\\prime _s\\mathbf {C}^\\prime _\\tau $ .",
"(Unfortunately, for some groups there are local systems $\\delta $ , not minimal in Bruhat $\\mathcal {G}$ -order, for which nevertheless no such $s$ and $\\tau $ exist.",
"See [18], [3] for an explanation of how to perform this calculation in that case as well as further details of this computation.",
"Unfortunately, they do not use the combinatorial language of clans.)",
"For the pair $(GL(p+q,,GL(p, \\times GL(q,)$ , as mentioned above, matters are simplified substantially by the fact that each $K$ -orbit admits only the trivial $K$ -equivariant local system.",
"Thus each element of $\\mathcal {D}$ can be thought of as simply a $K$ -orbit, and the Bruhat $\\mathcal {G}$ -order amounts to inclusion of orbits in orbit closures.",
"We remark that for the pair $(SL(p+q,,S(GL(p, \\times GL(q,)$ , the orbit set is precisely the same as for the pair $(GL(p+q,,GL(p, \\times GL(q,)$ , but here some orbits do admit non-trivial $K$ -equivariant local systems if $p=q$ .",
"In such cases, our results still give the KLV polynomials $P_{\\tau ,\\gamma }$ whenever $\\tau $ and $\\gamma $ are trivial local systems on the corresponding orbits, but there are other KLV polynomials about which we cannot say anything.",
"(Note that $K$ and the geometry of its orbits depends on the specific form of the Lie group, so $(PGL(p+q,,P(GL(p,\\times GL(q,))$ is yet another separate case with a different orbit set to which our results do not apply at all.)",
"We now describe the action of $\\mathcal {H}_W$ on $\\mathcal {M}_K$ for the pair $(GL(p+q,,GL(p, \\times GL(q,)$ by describing $T_{s_i}\\mathbf {T}_\\gamma $ for each simple transposition $s_i$ and each clan $\\gamma $ .",
"Given a clan $\\gamma =(\\gamma _1,\\hdots ,\\gamma _n)$ , let $\\gamma _i$ denote the $i$ -th entry.",
"Also, let $\\gamma \\times s_i$ denote the clan which is obtained from $\\gamma $ by switching $\\gamma _i$ and $\\gamma _{i+1}$ .",
"Finally, we need a formula for the length of a clan, denoted $\\ell (\\gamma )$ , which is given in [51] as $\\ell (\\gamma )=\\sum _{c_i=c_j\\in \\mathbb {N}, i<j} (j-i-\\#\\lbrace k\\in \\mathbb {N}\\mid c_s=c_t=k \\text{ for some } s<i<t<j\\rbrace ).$ (compact imaginary) If $\\gamma _i=\\gamma _{i+1}=+$ , or $\\gamma _i=\\gamma _{i+1}=-$ , then $T_{s_i} \\mathbf {T}_\\gamma =q \\mathbf {T}_\\gamma $ .",
"(noncompact imaginary) If $\\gamma _i$ and $\\gamma _{i+1}$ are opposite signs, then $T_{s_i}\\mathbf {T}_\\gamma =\\mathbf {T}_{\\gamma ^{\\prime }}+\\mathbf {T}_{\\gamma \\times s_i}$ , where $\\gamma ^{\\prime }$ is obtained from $\\gamma $ by changing $\\gamma _i$ and $\\gamma _{i+1}$ to an unused natural number.",
"(real) If $\\gamma _i$ and $\\gamma _{i+1}$ are mates, then $T_{s_i} \\mathbf {T}_\\gamma =(q-2)\\mathbf {T}_\\gamma +(q-1)\\mathbf {T}_{\\gamma ^{\\prime }}+(q-1)\\mathbf {T}_{\\gamma ^{\\prime \\prime }}$ , where $\\gamma ^{\\prime }$ and $\\gamma ^{\\prime \\prime }$ are obtained from $\\gamma $ by changing $\\gamma _i$ and $\\gamma _{i+1}$ to one $+$ and one $-$ in either order.",
"(complex ascent) If we are not in the above cases, then $\\ell (\\gamma \\times s_i)=\\ell (\\gamma )\\pm 1$ .",
"If $\\ell (\\gamma \\times s_i)=\\ell (\\gamma )+1$ , then $T_{s_i}\\mathbf {T}_\\gamma =\\mathbf {T}_{\\gamma \\times s_i}$ .",
"(complex descent) If $\\ell (\\gamma \\times s_i)=\\ell (\\gamma )-1$ , then $T_{s_i}\\mathbf {T}_\\gamma =q\\mathbf {T}_{\\gamma \\times s_i}+(q-1)\\mathbf {T}_\\gamma $ .",
"The labels on the cases correspond to the classification of roots for a $\\theta $ -stable torus in [43], and the multiplication rules are translations of the specific rules of [32].",
"The real and noncompact imaginary cases are “type I” in Vogan's classification; “type II” cases do not occur for $(GL(p+q,,GL(p,\\times GL(q,)$ .",
"The reader may note that the rules in the cases of a complex ascent or descent are similar to the multiplication rules in the Hecke algebra, while the others do not occur in that case.",
"We can be more precise about distinguishing complex ascents from descents.",
"By the mate of a natural number entry of a clan, we mean the other entry with the same natural number.",
"If $\\gamma _i$ is a number and $\\gamma _{i+1}$ is a sign, then $s_i$ is a complex ascent if the mate of $\\gamma _i$ is to the left and a complex descent otherwise.",
"If $\\gamma _i$ is a sign and $\\gamma _{i+1}$ is a number, then $s_i$ is a complex ascent if the mate of $\\gamma _{i+1}$ is to the right and a complex descent otherwise.",
"If $\\gamma _i$ and $\\gamma _{i+1}$ are different numbers, then $s_i$ is a complex ascent if the mate of $\\gamma _i$ occurs to the left of the mate of $\\gamma _{i+1}$ .",
"Combinatorially speaking, it would be more appropriate to consider $\\mathcal {M}_K$ as a right module rather than a left module as we have written above, since $s_i$ acts on clans by permuting positions, not entries (though it is not clear what permuting entries would mean).",
"It would also be more desirable from a geometric viewpoint (at least when considering $K$ -orbits on $G/B$ ) to consider $\\mathcal {M}_K$ as a right module, but we bow to historical convention.",
"As a simple example of our description of KLV polynomials, we now calculate $\\mathbf {C}^\\prime _{(1,2,1,2)}$ .",
"We can use the recursion with $C^\\prime _{s_2}\\mathbf {C}^\\prime _{(1,1,2,2)} = \\mathbf {C}^\\prime _{(1,2,1,2)}+\\sum E_\\gamma (q)\\mathbf {C}^\\prime _\\gamma .$ It will turn out that the coefficients of $C^\\prime _{s_2}\\mathbf {C}^\\prime _{(1,1,2,2)}$ satisfy the degree bound, so $E_\\gamma (q)=0$ for all $\\gamma $ .",
"We know that $C^\\prime _{s_2}=q^{-1/2}(T_{s_2}+T_1),$ and, since the orbit closure for $(1,1,2,2)$ is smooth (or by further recursive calculation), we have that $\\mathbf {C}^\\prime _{(1,1,2,2)}&=q^{-2/2}(\\mathbf {T}_{(1,1,2,2)}+\\mathbf {T}_{(+,-,1,1)}+\\mathbf {T}_{(-,+,1,1)}+\\mathbf {T}_{(1,1,+,-)} +\\mathbf {T}_{(1,1,-,+)}\\\\&+\\mathbf {T}_{(+,-,+,-)}+\\mathbf {T}_{(-,+,-,+)}+\\mathbf {T}_{(+,-,-,+)} +\\mathbf {T}_{(-,+,+,-)}.$ When multiplying these terms by $T_{s_2}$ , we are in the compact imaginary case for $(+,-,-,+)$ and $(-,+,+,-)$ , the noncompact imaginary case for $(-,+,-,+)$ and $(+,-,+,-)$ , and the complex ascent case for the remaining terms.",
"Hence, $T_{s_2}(\\mathbf {T}_{(+,-,-,+)}+\\mathbf {T}_{(-,+,+,-)}) &= q(\\mathbf {T}_{(+,-,-,+)}+\\mathbf {T}_{(-,+,+,-)}), \\\\T_{s_2}(\\mathbf {T}_{(-,+,-,+)}+\\mathbf {T}_{(+,-,+,-)}) &= (\\mathbf {T}_{(-,+,-,+)}+\\mathbf {T}_{(+,-,+,-)}+\\mathbf {T}_{(+,-,-,+)}+\\mathbf {T}_{(-,+,+,-)}),$ and putting the entire product together, $C^\\prime _{s_2}\\mathbf {C}^\\prime _{(1,1,2,2)}&=q^{-3/2} (\\mathbf {T}_{(1,2,1,2)}+\\mathbf {T}_{(+,1,-,1)}+\\mathbf {T}_{(-,1,+,1)}+\\mathbf {T}_{(1,+,1,-)}\\\\&+\\mathbf {T}_{(1,-,1,+)}+\\mathbf {T}_{(1,1,2,2)}+\\mathbf {T}_{(+,-,1,1)} +\\mathbf {T}_{(-,+,1,1)}+\\mathbf {T}_{(1,1,+,-)} \\\\&+\\mathbf {T}_{(1,1,-,+)}+\\mathbf {T}_{(+,-,+,-)}+\\mathbf {T}_{(-,+,-,+)}+(1+q)\\mathbf {T}_{(+,-,-,+)}+(1+q)\\mathbf {T}_{(-,+,+,-)})$ Since this expression satisfies the degree bound, it must in fact be $\\mathbf {C}^\\prime _{(1,2,1,2)}$ , and $P_{(+,-,-,+),(1,2,1,2)}=P_{(-,+,+,-),(1,2,1,2)}=1+q$ while all other KLV polynomials $P_{\\tau ,(1,2,1,2)}$ are either 1 or 0 depending on whether or not $\\tau \\le (1,2,1,2)$ in Bruhat order."
],
[
"Geometric interpretations",
"First, recall the geometric interpretation of ordinary KL polynomials due to Kazhdan and Lusztig.",
"Given a variety $X$ and a point $p\\in X$ , let $IH^i_p(X)$ denote the $i$ -th local intersection cohomology of $X$ at $p$ .",
"In principle, one can calculate this as follows.",
"From an appropriate stratification of $X$ , one constructs (as in [20]) a complex of sheaves $\\mathcal {IH}(X)$ called the intersection cohomology sheaf.",
"One construction of $\\mathcal {IH}(X)$ starts from the trivial local system on the largest stratum and extends it by certain truncations of derived pushforward (with compact support) on the derived category of sheaves on $X$ [20].",
"Therefore, the complex $\\mathcal {IH}(X)$ is also sometimes referred to as the Deligne-Goresky-MacPherson (DGM) extension of the trivial local system to $X$ .",
"One can localize this complex at the point $p$ , creating a complex of vector spaces.",
"The $i$ -th cohomology of this complex is what we call $IH^i_p(X)$ .",
"Kazhdan and Lusztig show in [26] that $ P_{v,w}(q)=\\sum _i \\dim IH^i_{vB/B}(X_w)q^{i/2}.$ An analogous result holds for KLV polynomials in the $K$ -orbit setting.",
"For the $(p,q)$ -clan $\\gamma $ , denote by $Q_{\\gamma }$ the corresponding $K$ -orbit, and denote by $Y_{\\gamma }$ the Zariski closure of $Q_{\\gamma }$ .",
"Abusing notation, let $\\gamma $ also denote the trivial local system on $Q_{\\gamma }$ , and let $IH(\\gamma )$ be the DGM extension of $\\gamma $ to $Y_{\\gamma }$ .",
"Denote by $IH^i(\\gamma )$ the $i$ th cohomology of this complex.",
"For any $(p,q)$ -clan $\\tau $ with $Q_{\\tau } \\subseteq Y_{\\gamma }$ , denote by $[\\tau :IH^i(\\gamma )]$ the composition factor multiplicity of $\\tau $ in $IH^i(\\gamma )$ (in the category of $K$ -equivariant constructible sheaves on $G/B$ ), where again we abuse notation and use $\\tau $ to denote the trivial local system on $Q_{\\tau }$ .",
"Then the KLV polynomial $P_{\\tau ,\\gamma }$ can be defined as follows [32]: $ P_{\\tau ,\\gamma }(q)=\\sum _i [\\tau :IH^i(\\gamma )] q^{i/2}.",
"$ In particular, all odd cohomology vanishes, as $P_{\\tau ,\\gamma }(q)$ is an honest polynomial in $q$ .",
"Localizing at a point $p \\in Q_{\\tau }$ , we get $ P_{\\tau ,\\gamma }(q)=\\sum _i \\dim IH^i_p(Y_{\\gamma }) q^{i/2}.",
"$ Note that the left hand side should technically be the sum of all KLV polynomials $P_{\\tau ^\\prime ,\\gamma }(q)$ where $\\tau ^\\prime $ runs over the set of all $K$ -equivariant local systems on $Q_{\\tau }$ .",
"However, as we have mentioned, in our case no non-trivial $K$ -equivariant local systems exist on any orbit.",
"Hence the KLV polynomials we consider here are actually $IH$ -Poincaré polynomials for $K$ -orbit closures, as is the case for ordinary KL polynomials and Schubert varieties."
],
[
"Combinatorial Criteria for Singularity Properties of $(1,2,1,2)$ -avoiding Orbit Closures",
"In this section, we use Theorem REF to determine combinatorially which $(1,2,1,2)$ -avoiding $K$ -orbit closures possess certain singularity properties and which do not.",
"First we need some general facts about singularity properties on Richardson varieties.",
"We say a property $\\mathcal {P}$ is local if it is determined strictly by examining the local rings at points of the variety.",
"If a property $\\mathcal {P}$ is local, we say it is open if the $\\mathcal {P}$ -locus (meaning the set of points at which $X$ has the property) is an open set.",
"Furthermore, we say a property $\\mathcal {P}$ is multiplicative if it holds on $X\\times Y$ precisely when it holds on both $X$ and $Y$ .",
"Suppose that $\\mathcal {P}$ is an open multiplicative local property of algebraic varieties.",
"For example, being smooth, being a local complete intersection (lci), and being Gorenstein are all examples of such properties.",
"Then the following result on how to determine when a Richardson variety has property $\\mathcal {P}$ is proved in [27].",
"Lemma 3.1 Let $\\mathcal {P}$ be an open multiplicative local property of algebraic varieties.",
"The Richardson variety $X_u^v$ has property $\\mathcal {P}$ if and only if the Schubert variety $X^v$ is $\\mathcal {P}$ at $u$ and the opposite Schubert variety $X_u$ is $\\mathcal {P}$ at $v$ (or equivalently, the Schubert variety $X^{w_0u}$ is $\\mathcal {P}$ at $w_0v$ ).",
"We also require another easy, generally known lemma.",
"We include its proof for lack of a suitable reference.",
"Lemma 3.2 If $\\mathcal {P}$ is an open multiplicative local property that holds for regular local rings, then the Schubert variety $X^v$ is $\\mathcal {P}$ at $u$ if and only if the Schubert variety $X^{v^{-1}}$ is $\\mathcal {P}$ at $u^{-1}$ .",
"Denote by $\\mathrm {id}$ the point $1B/B$ .",
"Consider the diagonal action of $G$ on $G/B\\times G/B$ , and consider the $G$ -orbit closure $\\mathcal {Z}_v:=\\overline{G\\cdot (\\mathrm {id}, v)}$ , with surjective projection maps $\\pi _1,\\pi _2:\\mathcal {Z}\\rightarrow G/B$ onto the first and second factors respectively.",
"The fiber $\\pi _1^{-1}(\\mathrm {id})$ is $\\mathrm {id}\\times X_v$ , and $\\pi _1$ is $G$ -equivariant, so, taking an affine neighborhood $U$ of $\\mathrm {id}$ , we have $\\pi _1^{-1}(U)=U \\times X_v$ .",
"Consider the point $(\\mathrm {id}, u)\\in \\mathcal {Z}_v$ .",
"It has an open neighborhood $U\\times V$ , where $V$ is isomorphic to an open neighborhood of $u$ in $X_v$ .",
"On the other hand, we also have that $\\mathcal {Z}_v=\\overline{G\\cdot (v^{-1}, \\mathrm {id})}$ , so $\\pi _2^{-1}(\\mathrm {id})=X_{v^{-1}}\\times \\mathrm {id}$ .",
"Taking an affine neighborhood $U^\\prime $ of $\\mathrm {id}$ , we have $\\pi _2^{-1}(U^\\prime )=X_{v^{-1}}\\times U^\\prime $ .",
"Now consider the point $(u^{-1},\\mathrm {id})\\in \\mathcal {Z}_u$ .",
"It has an open neighborhood $V^\\prime \\times U^\\prime $ , where $V^\\prime $ is isomorphic to an open neighborhood of $u^{-1}$ in $X_{v^{-1}}$ .",
"The points $(\\mathrm {id},v)$ and $(v^{-1},\\mathrm {id})$ are in the same $G$ -orbit, and $\\mathcal {Z}_u$ is $G$ -invariant, so $\\mathcal {P}$ holds on $U\\times V$ if and only if it holds on $V^\\prime \\times U^\\prime $ .",
"Since $U$ and $U^\\prime $ are smooth (since they are open subsets of $G/B$ ) and $\\mathcal {P}$ is multiplicative, $\\mathcal {P}$ holds on $V$ if and only if it holds on $V^\\prime $ .",
"Let $\\gamma $ be a $(1,2,1,2)$ -avoiding $(p,q)$ -clan, $Q_\\gamma $ the corresponding $K$ -orbit, and $Y_\\gamma =\\overline{Q_\\gamma }$ its Zariski closure.",
"By Theorem REF , $Y_\\gamma =X_u^v$ , where $u=(u(\\gamma )w_0^K)^{-1}$ and $v=v(\\gamma )^{-1}$ .",
"To determine whether $Y_\\gamma $ has property $\\mathcal {P}$ , by Lemma REF , it suffices to check whether $X^v$ is $\\mathcal {P}$ at $u$ and whether $X^{w_0 u}$ is $\\mathcal {P}$ at $w_0 v$ .",
"Now by Lemma REF , it suffices to check whether $X^{v(\\gamma )}$ is $\\mathcal {P}$ at $u(\\gamma )w_0^K$ and whether $X^{u(\\gamma )w_0^Kw_0}$ is $\\mathcal {P}$ at $v(\\gamma )w_0$ .",
"Note that, since $u(\\gamma )$ is Grassmannian, so is $u(\\gamma )w_0^Kw_0$ .",
"Hence the determination of whether $Y_\\gamma $ has property $\\mathcal {P}$ boils down to checking whether certain points of Grassmannian Schubert varieties lie in the $\\mathcal {P}$ -locus.",
"For the properties $\\mathcal {P}=$ “(rationally) smooth\", “lci\", and “Gorenstein\", the $\\mathcal {P}$ -locus of these special Schubert varieties is known.",
"Hence we are able to provide in what follows combinatorial criteria for these properties in the case of $(1,2,1,2)$ -avoiding $\\gamma $ .",
"Note that, for general permutations $w$ and $x$ , $X^{w_0ww_0}$ is isomorphic to $X^w$ by an isomorphism that takes the point $x$ to $w_0xw_0$ .",
"Hence, checking whether $X^{u(\\gamma )w_0^Kw_0}$ is $\\mathcal {P}$ at $v(\\gamma )w_0$ is equivalent to checking whether $X^{w_0u(\\gamma )w_0^K}$ is $\\mathcal {P}$ at $w_0v(\\gamma )$ .",
"By another application of Lemma REF , this is equivalent to checking if $\\mathcal {P}$ holds on $X^{v(\\gamma )}_{u(\\gamma )w_0^K}$ .",
"While this observation is not strictly necessary in what follows, we will frequently use it for brevity.",
"For the properties $\\mathcal {P}$ listed above, the results stating when a permutation $u$ is in the $\\mathcal {P}$ -locus of a Grassmannian Schubert variety $X^v$ are best described in terms of a path diagram associated to the permutations $u$ and $v$ .",
"We now describe how to draw this diagram.",
"Let $p$ be the descent of the Grassmannian permutation $v$ .",
"Start with a $p \\times q$ rectangle, and trace a lattice path from the southwest corner to the northeast, moving either one unit right or one unit up at each step.",
"At the $i$ th step, the path moves right if $v^{-1}(i) > p$ , and up if $v^{-1}(i) \\le p$ .",
"Note that this path determines a partition, namely the one whose Young diagram consists of the blocks of the $p \\times q$ grid lying weakly northwest of it.",
"However, for our purposes, it is actually the path itself we are interested in.",
"As an example, consider the Grassmannian permutation $v=1367245$ , which has a unique descent at position 4.",
"The associated path fits inside a $4 \\times 3$ rectangle as in Figure REF .",
"Figure: The path of v=1367245v=1367245If $u \\ge v$ is another permutation, then its path lies weakly southeast of that of $v$ .",
"(Note that $u$ does not have to be Grassmannian, but when $w\\in W_K$ , the paths for $u$ and $uw$ are the same, and all local properties of $X^v$ are the same at both $u$ and $uw$ .)",
"For instance, in Figure REF the paths for the two Grassmannian permutations $v=124673589 < 156892347=u$ are shown drawn in the same $5 \\times 4$ grid, with the path further southeast being that for $u$ and the one further northwest being that for $v$ .",
"Figure: The paths for v=124673589<156892347=uv=124673589 < 156892347 = uThus the two paths for $u$ and $v$ determine a chain of skew shapes, each connected to the next at a point (or perhaps by a series of line segments, if the paths for $u$ and $v$ happen to coincide over some portion of the grid).",
"We refer to those skew shapes that are not simply sequences of line segments — those that actually open and then close, bounding a region of positive area — as the components of the path diagram.",
"For each component of the path diagram, we call the portion of the path for $u$ which bounds its southeast side its bottom boundary and the portion of the path for $v$ which bounds its northwest side its top boundary.",
"For the purpose of convenience when we later recall the Lascoux-Schützenberger rule for KL-polynomials associated to Grassmannian permutations, we will prefer to draw the diagrams just described rotated clockwise $45^{\\circ }$ , and we will generally omit the portion of the $p \\times q$ grid not lying along either path, drawing only the paths for $u$ and $v$ themselves.",
"Thus the above example of $u=156892347$ and $v=124673589$ will be depicted as in Figure REF .",
"(Note that the path for $u$ now lies below the path for $v$ , so that the “bottom\" and “top\" boundaries of components are now appropriately named.)",
"Figure: The rotated path diagram for u=156892347u=156892347, v=124673589v=124673589Note that, based on the algorithm for drawing the path diagram, it is easy to see that the path diagram for $vw_0\\ge uw_0^Kw_0$ is simply the partition diagram for $u\\ge v$ flipped upside down (with $p$ and $q$ also exchanged).",
"The combinatorial translation from a $(1,2,1,2)$ -avoiding clan to a path diagram of this type is now easy to describe, using Theorem REF and the above definitions.",
"Definition 3.3 The path diagram for a $(1,2,1,2)$ -avoiding $(p,q)$ -clan $\\gamma =(c_1, \\hdots ,c_n)$ is drawn as follows: Starting at the southwest corner of a $p \\times q$ rectangle (rotated $45^{\\circ }$ ) and tracing to the northeast corner, we draw two paths, $P_1$ and $P_2$ , following these rules at step $i$ for $i=1,\\hdots ,n$ : If $c_i=+$ , both $P_1$ and $P_2$ move up; If $c_i=-$ , both $P_1$ and $P_2$ move right; If $c_i$ is the first occurrence of a natural number, then $P_1$ moves up, while $P_2$ moves right; If $c_i$ is the second occurrence of a natural number, then $P_1$ moves right, while $P_2$ moves up.",
"It is clear in the above definition that $P_1$ (the upper path) is the path for $v(\\gamma )$ , while $P_2$ (the lower path) is the path for $u(\\gamma )$ .",
"Moreover, it is clear that the components of the path diagram open at the first occurrence of a natural number which is not contained within any other matching pair of numbers and close at the second occurrence of such a number.",
"We refer to such a matching pair as outermost.",
"Example 3.4 The path diagram for the $(6,5)$ -clan $(1,+,-,+,2,+,-,-,2,+,1)$ is shown in Figure REF .",
"Figure: The rotated path diagram for γ=(1,+,-,+,2,+,-,-,2,+,1)\\gamma =(1,+,-,+,2,+,-,-,2,+,1)"
],
[
"Globally (rationally) smooth $(1,2,1,2)$ -avoiding {{formula:7a74d550-3a84-4f90-a3f9-bb3806c765c3}} -orbit closures",
"We start by determining which $(1,2,1,2)$ -avoiding orbit closures are smooth.",
"Recall that a complex variety $X$ of dimension $n$ is smooth at a point $p$ if the local ring $(\\mathcal {O}_{X,p},\\mathfrak {m},\\mathbb {k})$ is regular, meaning that $\\dim _{\\mathbb {k}} \\mathfrak {m} / \\mathfrak {m}^2$ is equal to the Krull dimension of $\\mathcal {O}_{X,p}$ .",
"A variety $X$ is simply said to be smooth if it is smooth at every point.",
"Recall also that $X$ is rationally smooth at $p$ if $ H^q(X,X \\setminus \\lbrace p\\rbrace ; {\\mathbb {Q}}) \\cong {\\left\\lbrace \\begin{array}{ll}{\\mathbb {Q}}& \\text{ if $q = 2n$} \\\\0 & \\text{ otherwise}\\end{array}\\right.",
"}$ and simply rationally smooth if it is rationally smooth at every point.",
"In general, smoothness and rational smoothness at a point are not equivalent notions, with smoothness being a strictly stronger condition.",
"However, for all points on type $A$ Schubert varieties, these conditions are known to be equivalent [17].",
"Since our checks of smoothness or rational smoothness of $(1,2,1,2)$ -avoiding orbit closures reduce to checks for the same properties on two type $A$ Schubert varieties, the two conditions amount here to the same thing.",
"Thus we simply refer to the property of interest here as “smoothness\", dropping the redundant modifier “rational\".",
"For Grassmannian permutations $u$ and $v$ , we describe how to use the path diagrams described above to decide Whether $X^v$ is globally smooth, and Whether $X^v$ is smooth at $u$ .",
"These criteria are well-known.",
"They appear explicitly in [29], but are also implicit in earlier work such as [31] (when combined with the aforementioned equivalence of smoothness and rational smoothness established in [17]) or [52].",
"We call a lattice point on the path for $v$ (the top one) an outer corner if the (unrotated) path contains both the lattice point directly south of it and the lattice point directly east of it.",
"Analogously, we call a lattice point on the path for $v$ an inner corner if the (unrotated) path contains both the lattice points directly north and directly west of it.",
"The by now classically known theorem is as follows.",
"Proposition 3.5 The Schubert variety $X_v$ is singular at $u$ if and only if there is at least one inner corner on the path for $v$ that is not on the path for $u$ or, equivalently, an inner corner on the top boundary of a component of the path diagram.",
"Note this implies that $X_v$ is singular (at some point) if and only if the path for $v$ has an inner corner within the strict interior of the $p\\times q$ rectangle.",
"For brevity, we refer to an inner corner on the path for $v$ that is not on the path for $u$ as a singular corner.",
"For example, in Figure REF , $X^v$ is singular at $u$ , with the open dot indicating the lone singular corner.",
"Remark 3.6 The above definitions of inner and outer corner given above are opposite of what is usually found in the literature.",
"This is because usually, “inner\" and “outer\" are relative to the Young diagram which lies northwest of the path.",
"However, here we are thinking of “inner\" and “outer\" relative to the interiors of the components of the path diagram for $\\gamma $ , which lie southeast of the path for $v$ .",
"Figure: X v X^v singular at uu: A singular corner on the top boundaryOn the other hand, in Figure REF , $X^v$ is smooth at $u$ .",
"Figure: X v X^v smooth at uu: No singular corners on the top boundaryRecall that, to check smoothness of a $(1,2,1,2)$ -avoiding $K$ -orbit closure $Y_\\gamma $ , or equivalently smoothness of the Richardson variety $X^{v(\\gamma )}_{u(\\gamma )w_0^K}$ , we must also check whether $X^{uw_0^Kw_0}$ is smooth at $vw_0$ .",
"However, since the path diagram for this pair is simply the one for $X^v$ at $u$ flipped upside down, we simply have to perform the same check upside down.",
"To be precise, $X^{uw_0^Kw_0}$ is singular at $vw_0$ if and only if there is an inner (meaning to the interior side of the path diagram) corner along the lower path which does not lie on the upper path or, equivalently, which lies on the bottom boundary of some component of the path diagram.",
"We also call a corner of this type a singular corner.",
"So for instance, in Figure REF , $X^v$ is smooth at $uw_0^K$ , but the Richardson variety $X_{uw_0^K}^v$ is singular because $X^{w_0uw_0^K}$ is singular at $w_0v$ .",
"The open dot in Figure REF marks the singular corner on the bottom boundary.",
"Figure: X w 0 uw 0 K X^{w_0uw_0^K} singular at w 0 vw_0v: A singular corner on the bottom boundaryGiven this pictorial characterization of smoothness of Richardson varieties of the type that we are interested in, we can now give the following pattern-avoidance criterion for smoothness of a $(1,2,1,2)$ -avoiding $K$ -orbit closure.",
"Proposition 3.7 Let $\\gamma $ be a $(1,2,1,2)$ -avoiding $(p,q)$ -clan.",
"Then $Y_\\gamma = \\overline{Q_{\\gamma }}$ is smooth if and only if $\\gamma $ avoids the patterns $(1,+,-,1)$ , $(1,-,+,1)$ , $(1,+,2,2,1)$ , $(1,-,2,2,1)$ , $(1,2,2,+,1)$ , $(1,2,2,-,1)$ , and $(1,2,2,3,3,1)$ .",
"As we have noted, singular corners occur only on either the top or bottom boundary of a component of the path diagram, so they are a result of characters of the clan occurring between an outermost pair of matching numbers.",
"More specifically, a singular corner on the top boundary of some component of the path diagram occurs if and only if there are two consecutive character positions $c_i$ and $c_{i+1}$ of $\\gamma $ between the corresponding outermost pair such that one of the following is true: $c_i$ is a $-$ , and $c_{i+1}$ is a $+$ ; $c_i$ is a $-$ , and $c_{i+1}$ is the first occurrence of a natural number; $c_i$ is the second occurrence of a natural number, and $c_{i+1}$ is a $+$ ; or $c_i$ is the second occurrence of a natural number, and $c_{i+1}$ is the first occurrence of a (different) natural number.",
"These four possibilities respectively imply that $\\gamma $ contains the pattern $(1,-,+,1)$ , $(1,-,2,2,1)$ , $(1,2,2,+,1)$ , or $(1,2,2,3,3,1)$ .",
"Furthermore, there is a singular corner on the bottom of the path diagram if and only if there are consecutive character positions $c_i$ and $c_{i+1}$ of $\\gamma $ occurring within the corresponding outermost pair and satisfying one of (1)-(4), except with inverted signs.",
"If $Y_\\gamma $ is singular, the path diagram for $v(\\gamma )$ and $u(\\gamma )$ must have a singular corner.",
"Hence $\\gamma $ must contain one of the bad patterns.",
"The other half of the proof is to show that, if $\\gamma $ contains one of the bad patterns, then its path diagram contains a singular corner.",
"Supposing that $\\gamma $ contains one of these patterns, we take the matching 1's of the pattern to be outermost.",
"Let $C$ be the component of the path diagram corresponding to this outermost pair.",
"Then one checks easily that in each case, either the bottom boundary of $C$ has an “up\" segment followed at some later point by a “right\" segment, or the top boundary of $C$ has a “right\" segment followed at some later point by an “up\" segment (or both).",
"In the former case, the bottom boundary of $C$ must change from “up\" to “right\" at some point, giving a singular corner.",
"In the latter case, the top boundary of $C$ must change from “right\" to “up\" at some point, again giving a singular corner.",
"One would naturally wonder about $(1,2,1,2)$ -containing orbit closures as well.",
"In fact, the result here is as simple as one could hope for.",
"Proposition 3.8 ([37]) If $\\gamma $ contains the pattern $(1,2,1,2)$ , then $Y_\\gamma = \\overline{Q_{\\gamma }}$ is rationally singular.",
"Proposition REF is proved in [37] using a combinatorial result of Springer [41] giving a root-theoretic necessary condition for a $K$ -orbit closure to be rationally smooth.",
"We do not have anything to add to this portion of McGovern's argument.",
"However, combining Propositions REF and REF , we have given a new proof of the main result of [37]: Theorem 3.9 ([37]) Let $\\gamma $ be a $(p,q)$ -clan.",
"The $K$ -orbit closure $\\overline{Q_{\\gamma }}$ is rationally smooth if and only if it avoids the patterns $(1,2,1,2)$ , $(1,+,-,1)$ , $(1,-,+,1)$ , $(1,+,2,2,1)$ , $(1,-,2,2,1)$ , $(1,2,2,+,1)$ , $(1,2,2,-,1)$ , and $(1,2,2,3,3,1)$ .",
"Moreover, smoothness and rational smoothness of $K$ -orbit closures are equivalent for $(GL(p+q,,GL(p,\\times GL(q,)$ .The version of this paper in the Journal of Algebra has an error in the statement of this theorem.",
"The cited version on arXiv is correct."
],
[
"The singular locus of a $(1,2,1,2)$ -avoiding {{formula:01cfa6d6-a023-4983-bfb7-94891252fa06}} -orbit closure",
"The results of [37] determine which $K$ -orbit closures are singular but do not determine where they are singular.",
"Here, we describe how to compute the singular locus of a $(1,2,1,2)$ -avoiding $K$ -orbit closure using its path diagram.",
"Let $\\gamma $ be a $(p,q)$ -clan, and let $Y_{\\gamma } = \\overline{Q_{\\gamma }}$ be the corresponding $K$ -orbit closure.",
"Let $S$ be the singular locus of $Y_{\\gamma }$ .",
"Since the left action of any element $k \\in K$ takes an open neighborhood of any point $p \\in Y_{\\gamma }$ to an isomorphic open neighborhood of the point $k \\cdot p \\in Y_{\\gamma }$ , the singular locus of $Y_{\\gamma }$ is $K$ -stable.",
"Being closed, it is hence a union of $K$ -orbit closures.",
"Thus a description of the singular locus amounts to giving the list of $K$ -orbits (or clans) whose closures are the irreducible components of $S$ .",
"Said another way, we wish to list those clans $\\tau $ such that $Y_{\\gamma }$ is singular along $Q_{\\tau }$ and such that $\\tau $ is (Bruhat) maximal with this property.",
"The following proposition, from [27] describes the singular locus of a Richardson variety in terms of the singular loci of Schubert varieties.",
"Proposition 3.10 Let $\\Sigma (X)$ denote the singular locus of a variety $X$ .",
"Then $\\Sigma (X^v_u)=(\\Sigma (X^v)\\cap X_u)\\cup (\\Sigma (X_u)\\cap X^v).$ It follows that the singular locus of a Richardson variety is a union of Richardson varieties.",
"Thus if $\\gamma $ is $(1,2,1,2)$ -avoiding, then by the previous paragraph we know that $S$ is a union of $K$ -stable Richardson varieties, or in other words, closures of $(1,2,1,2)$ -avoiding $K$ -orbits.",
"Thus in searching for those $\\tau $ described in the previous paragraph, we can restrict our attention to $(1,2,1,2)$ -avoiding clans.",
"Bruhat order on $(1,2,1,2)$ -avoiding orbit closures is determined solely by containment of path diagrams.",
"Moreover, if $\\tau < \\gamma $ are $(1,2,1,2)$ -avoiding, then by the above proposition and Proposition REF , $Y_{\\gamma }$ is singular along $Y_{\\tau }$ if and only if $\\gamma $ has a singular corner (either on the top or the bottom path) that does not lie on the path diagram for $\\tau $ .",
"Thus the path diagrams for the $\\tau $ we seek are precisely the largest ones missing a singular corner.",
"We construct these path diagrams by removing hooks from the skew diagrams bounded by the path diagram $D_{\\gamma }$ for $\\gamma $ .",
"More specifically, we have one diagram $D_{\\tau }$ for each singular corner of $D_{\\gamma }$ , formed as follows.",
"If the singular corner is on the bottom path, then we look at the corner box immediately above it and remove its hook, taking $D_{\\tau }$ to be the boundary of the resulting skew shape.",
"If the singular corner is on the top path, then we do the same for the box immediately below it.",
"Clearly, the resulting shapes are precisely those that both miss a singular corner and are maximal with respect to containment among shapes having this property.",
"Each such diagram $D_{\\tau }$ can easily be converted back to the clan $\\tau $ , as follows.",
"A path diagram does not necessarily specify a clan, but it does specify what we will call the FS-pattern for $\\tau $ , which is a sequence of $+$ 's, $-$ 's, F's, and S's, with $\\pm $ appearing wherever $\\tau $ has one of these symbols, and F (respectively, S) appearing wherever $\\tau $ has a first occurrence (respectively, a second occurrence).",
"To obtain the FS-pattern, we simply note, for each $i$ , what the two paths do at step $i$ .",
"If they both move up, character $i$ is a $+$ .",
"If they both move right, character $i$ is a $-$ .",
"If the top path moves up while the bottom path moves right, then character $i$ is an F. Finally, if the top path moves right while the bottom path moves up, then character $i$ is an S. Now, it is possible for multiple clans to have the same FS-pattern — for instance, $(1,2,1,2)$ and $(1,2,2,1)$ have the same FS-pattern $(F,F,S,S)$ , yet they are different clans.",
"However, by the above discussion, we know that the $\\tau $ we seek is $(1,2,1,2)$ -avoiding, and there is a unique $(1,2,1,2)$ -avoiding clan with a given FS-pattern.",
"To compute it, we simply move from left to right and insist that every second occurrence be matched with the most recently appearing first occurrence which does not yet have a mate.",
"Thus the FS-pattern $(F,F,S,S)$ specifies the $(1,2,1,2)$ -avoiding clan $(1,2,2,1)$ , since when we reach the first S, we insist that it be mated with the more recently appearing (and unmated) 2 rather than with the 1.",
"As another example, the FS pattern $(F,S,+,-,F,F,F,S,+,-,S,F,S,S)$ uniquely determines the $(1,2,1,2)$ -avoiding clan $(1,1,+,-,2,3,4,4,+,-,3,5,5,2)$ .",
"This discussion establishes the following.",
"Theorem 3.11 If $\\gamma $ is $(1,2,1,2)$ -avoiding, then the clans $\\tau $ computed by the above procedure index the irreducible components of the singular locus of $Y_{\\gamma }$ .",
"Example 3.12 Consider the singular $(4,4)$ -clan $\\gamma =(1,+,-,2,2,+,-,1)$ .",
"Its path diagram is pictured in Figure REF , with the four singular corners indicated and numbered.",
"(Here we draw the boxes of the skew diagram, to make it clearer what is removed to form the new skew/path diagrams.)",
"Figure: The path diagram for (1,+,-,2,2,+,-,1)(1,+,-,2,2,+,-,1)The new path diagrams for $\\tau _1$ , $\\tau _2$ , $\\tau _3$ , and $\\tau _4$ , formed by removing the hook at each of the singular corners 1, 2, 3, and 4, respectively, appear in Figure REF (read from the top left to the bottom right).",
"These determine the FS patterns $(+,+,-,F,-,+,-,S)$ , $(F,+,-,+,S,+,-,-)$ , $(F,S,-,F,+,+,-,S)$ , and $(F,+,-,-,S,+,F,S)$ , which correspond, respectively, to the clans $\\tau _1=(+,+,-,1,-,+,-,1)$ , $\\tau _2=(1,+,-,+,1,+,-,-)$ , $\\tau _3=(1,1,-,2,+,+,-,2)$ , and $\\tau _4=(1,+,-,-,1,+,2,2)$ .",
"Thus the singular locus of $Y_{\\gamma }$ is the union of the $K$ -orbit closures corresponding to those 4 clans.",
"Figure: Diagrams for the singular components of (1,+,-,2,2,+,-,1)(1,+,-,2,2,+,-,1)"
],
[
"LCI-ness",
"Recall that a local ring $R$ is a local complete intersection (lci) if there exists a regular local ring $S$ and an ideal $I$ generated by a regular sequence on $S$ such that $R \\cong S/I$ .",
"A variety or scheme $X$ is said to be lci at the point $p$ if the local ring $\\mathcal {O}_{X,p}$ of $X$ at $p$ is lci.",
"$X$ is simply said to be lci if it is lci at every point.",
"For any variety $X$ , the set of points at which $X$ is not lci is a Zariski-closed subset of $X$ .",
"As was the case with smoothness, to understand lci-ness of Richardson varieties, we must know something about the lci locus of the relevant Schubert varieties.",
"The lci locus of Grassmannian Schubert varieties is now understood by the following proposition, due to C. Darayon [16].",
"To state the result, we require one further definition.",
"Given an inner corner on a (top boundary) path diagram, we define the left (respectively right) leg length to be the number of consecutive lattice path segments on the path northwest (respectively northeast) of the corner.",
"Proposition 3.13 Let $u\\ge v$ be a Grassmannian permutations.",
"Then $X^v$ is not lci at $u$ if and only if at least one of the following hold: The path diagram for $v$ has an inner corner not on the path diagram for $u$ that has a left or right leg length greater than 1.",
"The path diagram for $v$ has two consecutive inner corners, neither of which is on the path diagram for $u$ .",
"We extend the above definition of leg lengths analogously to singular corners on both the top and bottom boundaries of path diagrams.",
"Now, combining Lemma REF and Proposition REF , we have the following: Proposition 3.14 For Grassmannian $u$ and $v$ , the Richardson variety $X_{uw_0^K}^v$ is lci if and only if both of the following hold for every component $C$ of the path diagram: Every singular corner of $C$ has both leg lengths equal to 1.",
"$C$ contains at most one singular corner on its bottom boundary and at most one singular corner on its top boundary.",
"So, for example, the $K$ -orbit $(1,+,-,+,-,1)$ is non-lci, as the lone component of its path diagram, which is shown in Figure REF , contains two singular corners along its bottom boundary.",
"Similarly, $(1,-,+,-,+,1)$ is non-lci, having two singular corners along its top boundary.",
"In addition, the path diagrams of Figures REF and REF are both diagrams of non-lci Richardson varieties, since each singular corner has one leg of length 1 but another of length 2.",
"Figure: The path diagram for the non-lci KK-orbit (1,+,-,+,-,1)(1,+,-,+,-,1)As was the case for smoothness, this pictorial requirement on the path diagram for a $(1,2,1,2)$ -avoiding clan $\\gamma $ can be translated to a pattern avoidance condition on $\\gamma $ .",
"Proposition 3.15 If $\\gamma $ is $(1,2,1,2)$ -avoiding, then $Y_\\gamma = \\overline{Q_{\\gamma }}$ is lci if and only $\\gamma $ avoids one of 35 bad patterns.",
"The bad patterns are the following, along with their negatives (the negative of a pattern being the one obtained from that pattern by inverting all signs): $(1,+,+,-,1)$ , $(1,+,-,-,1)$ , $(1,-,2,2,+,1)$ , $(1,+,+,2,2,1)$ , $(1,+,-,2,2,1)$ , $(1,2,2,-,-,1)$ , $(1,2,2,+,-,1)$ , $(1,2,+,2,-,1)$ , $(1,+,2,-,2,1)$ , $(1,+,2,3,3,2,1)$ , $(1,2,3,3,2,-,1)$ , $(1,-,2,2,3,3,1)$ , $(1,2,+,2,3,3,1)$ $(1,2,2,+,3,3,1)$ , $(1,2,2,3,-,3,1)$ , $(1,2,2,3,3,+,1)$ , $(1,2,3,3,2,4,4,1)$ , $(1,2,2,3,4,4,3,1)$ , and $(1,2,2,3,3,4,4,1)$ .",
"The possible ways for the path diagram for the clan $\\gamma $ to have a singular corner with at least one leg length not equal to 1 are as follows: Along the bottom boundary of some component, have three consecutive segments of the form “up-up-right\"; Along the bottom boundary of some component, have three consecutive segments of the form “up-right-right\"; Along the top boundary of some component, have three consecutive segments of the form “right-up-up\"; or Along the top boundary of some component, have three consecutive segments of the form “right-right-up\".",
"For the first possibility to occur, there must be three consecutive characters enclosed in an outermost matching pair such that the first two are either $+$ 's or second occurrences, and the last is either a $-$ or a first occurrence.",
"There are 8 possible sequences, and the minimal clans containing one of these sequences are $(1,+,+,-,1)$ , $(1,+,+,2,2,1)$ , $(1,2,+,2,-,1)$ , $(1,2,+,2,3,3,1)$ , $(1,2,2,+,-,1)$ , $(1,2,2,+,3,3,1)$ , $(1,2,3,3,2,-,1)$ , and $(1,2,3,3,2,4,4,1)$ .",
"For the second possibility, there must be three consecutive characters enclosed in an outermost matching pair such that the first is a $+$ or a second occurrence, and the last two are either $-$ 's or first occurrences.",
"Again, there are 8 possible sequences, with the list of minimal clans containing one of them as follows: $(1,+,-,-,1)$ , $(1,+,-,2,2,1)$ , $(1,+,2,-,2,1)$ , ($1,+,2,3,3,2,1)$ , $(1,2,2,-,-,1)$ , $(1,2,2,-,3,3,1)$ , $(1,2,2,3,-,3,1)$ , $(1,2,2,3,4,4,3,1)$ .",
"The minimal patterns for possibilities (3) and (4) above are easily seen to be the negatives of these, as was the case when checking smoothness.",
"This gives us so far 28 distinct clans.",
"Now, consider the possible ways for the path diagram to have a component with two singular corners on either its bottom boundary or its top boundary.",
"We may assume that each singular corner has equal leg lengths of 1, as otherwise we are already covered by the preceding cases.",
"Along the bottom boundary, we require a sequence of the form “up-right-up-right\" enclosed in a matching pair.",
"There are 16 possible ways this can be accomplished, choosing either a $+$ or a second occurrence for the first position, either a $-$ or a first occurrence for the second position, either a $+$ or a second occurrence for the third position, and either a $-$ or a second occurrence for the fourth position.",
"Among the 16 minimal clans containing such a string of consecutive characters within an outermost pair, only four fail to contain one of the bad patterns we have already found.",
"These are $(1,+,2,2,-,1)$ , $(1,+,2,2,3,3,1)$ , $(1,2,2,3,3,-,1)$ , and $(1,2,2,3,3,4,4,1)$ .",
"(To illustrate the previous point, the clan $(1,+,-,+,-,1)$ is a minimal clan with a sequence of the type we seek, but this clan contains the known bad patterns $(1,+,+,-,1)$ and $(1,+,-,-,1)$ , so we do not consider this to be a new bad pattern.)",
"To handle the top boundary, we simply take the negatives of the patterns causing problems along the bottom boundary, as usual.",
"This gives three new patterns.",
"Combining these 3 with the 4 listed in the previous paragraph and the first 28 patterns described above, we have argued that any non-lci clan must contain one of the 35 bad patterns.",
"Conversely, suppose that our clan $\\gamma $ contains one of the 35 patterns.",
"Then we can take the 1's of the pattern to be an outermost pair.",
"Let $C$ be the corresponding component of the path diagram for $\\gamma $ .",
"One checks easily that for each of the 35 patterns given, either the bottom boundary of $C$ contains three (not necessarily consecutive) segments of the form “up-up-right\" or of the form “up-right-right\"; or the top boundary of $C$ contains three (not necessarily consecutive) segments of the form “right-right-up\" or of the form “right-up-up\"; or both.",
"Considering the bottom boundary, there are two possibilities.",
"Either the bottom boundary changes from “up\" to “right\" at least twice, giving at least two singular corners on the bottom boundary, or it changes from “up\" to “right\" only once, in which case either the left leg (in the case “up-up-right\") or the right leg (in the case “up-right-right\") has length at least 2.",
"In either event, the path diagram must be non-lci.",
"The reasoning for the top boundary is simply that for the bottom boundary flipped upside down.",
"Remark 3.16 We can also compute the non-lci locus of a $(1,2,1,2)$ -avoiding $K$ -orbit closure, using a similar procedure to that described in Section REF .",
"For each singular corner with at least one leg length larger than 1, we get a component of the non-lci locus by removing the hook for that corner.",
"Furthermore, if some component has two adjacent singular corners (both on the top or both on the bottom) with all leg lengths 1, we also get a single component of the non-lci locus by removing the hooks for both corners simultaneously.",
"Naturally, one would again wonder about the $(1,2,1,2)$ -containing orbit closures and about combinatorial criteria for determining which ones are lci and which are not.",
"As we explain below, whether a $K$ -orbit closure is lci can be checked by computer in any given example.",
"With the help of Macaulay 2 [1] code written by the second author and A. Yong, the authors have been able to study this question experimentally.",
"Alas, the results of these experiments show that the answer here is not as simple as that of Proposition REF .",
"Indeed, some $(1,2,1,2)$ -containing orbit closures are lci, while others are not.",
"We describe the mathematics behind using Macaulay 2 [1] to check lci-ness of a $(1,2,1,2)$ -containing clan $\\gamma $ .",
"First, recall that, since the non-lci locus of $Y_{\\gamma }$ is $K$ -stable and closed, it is a union of $K$ -orbit closures.",
"Thus, if $Y_{\\gamma }$ is non-lci at some (equivalently, every) point of $Q_{\\tau }$ , and if $Q_{\\delta } \\subseteq \\overline{Q_{\\tau }}$ , then $Y_{\\gamma }$ is non-lci at every point of $Q_{\\delta }$ as well.",
"So to check whether $Y_{\\gamma }$ is globally lci, it suffices to check whether $Y_{\\gamma }$ is lci along each closed $K$ -orbit contained in it.",
"Moreover, to determine whether $Y_{\\gamma }$ is lci along a given closed $K$ -orbit, it suffices to check whether $Y_{\\gamma }$ is lci at a single point of the closed orbit.",
"The closed $K$ -orbits are parametrized by clans consisting of only $+$ 's and $-$ 's, and determining whether a given closed orbit $\\tau $ is contained in $Y_{\\gamma }$ is easy in light of [47].",
"Given such a closed orbit, we describe how to determine whether $Y_{\\gamma }$ is lci at the distinguished representative $p_v=vB/B$ of $Q_\\tau $ , which is a $T$ -fixed point whenever $Q_\\tau $ is closed.",
"(Indeed, $v$ is the permutation $v(\\tau )$ , and $p_v$ is represented by the permutation matrix having 1's in positions $(v(i),i)$ for each $i=1,\\hdots ,n$ .)",
"There exists an open affine neighborhood of $Y_{\\gamma }$ containing $p_v$ that reflects all the local structure of $Y_{\\gamma }$ near $p_v$ .",
"This open neighborhood is called the patch of $Y_{\\gamma }$ at $p_v$ (or at $\\tau $ ) in [50].",
"The patch is the reduced scheme whose underlying set is $Y_{\\gamma } \\cap vB^-B/B$ , with $vB^-B/B$ a “permuted big cell\".",
"The permuted big cell is an open affine subset of $G/B$ , and it can be coordinatized by representing its general element as a matrix with 1's in positions $(v(i),i)$ for each $i=1,\\hdots ,n$ , with 0's to the right of these 1's, and unspecialized variable entries $z_{i,j}$ in the remaining positions.",
"We view $vB^-B/B$ as $ \\mathbb {A}^{\\binom{n}{2}} \\cong \\text{Spec}(R), $ where $R = \\textbf {z}]$ , with $\\textbf {z}$ the unspecialized $z$ -variables mentioned above.",
"The point $p_v$ corresponds to the origin in $\\mathbb {A}^{\\binom{n}{2}}$ under this identification.",
"The patch can be viewed as a reduced and irreducible closed subscheme of $\\mathbb {A}^{\\binom{n}{2}}$ , defined by a prime ideal which we denote by $I_{\\gamma ,\\tau }$ .",
"As explained in [50], some “obvious\" generators for this ideal are suggested by [47].",
"These generators are known to define the patch set theoretically [50] and are conjectured [50] to be sufficient to generate all of $I_{\\gamma ,\\tau }$ .",
"Even without a proof of this conjecture, an ideal generated by these obvious equations can certainly be created in Macaulay 2 [1] in any given example, and its radicalness can be checked, for example by verifying that the generators in question form a Gröbner basis with squarefree lead terms with respect to a chosen term order.",
"Indeed, such checks were the basis of the conjecture in the first place.",
"Now, note that the patch is stable under the action of $T$ , being the intersection of two $T$ -stable subsets of $G/B$ .",
"(Any $K$ -orbit closure $Y_{\\gamma }$ is $T$ -stable since it is stable under $K$ , which contains the full maximal torus $T$ of $G$ .)",
"Furthermore, there exists a one parameter subgroup $S\\subseteq T$ such that the point $p_v$ corresponding to the origin of this affine space is an attractor for $S$ .",
"Hence the $T$ -action induces a positive ${\\mathbb {Z}}^n$ -grading on $R$ with respect to which $I_{\\gamma ,\\tau }$ is a homogeneous ideal.",
"Thus one can speak of a minimal free ${\\mathbb {Z}}^n$ -graded resolution of $I_{\\gamma ,\\tau }$ , or of $R/I_{\\gamma ,\\tau }$ .",
"Now, note that $Y_{\\gamma }$ is lci at $p_v$ if and only if the patch $\\text{Spec}(R/I_{\\gamma ,\\tau })$ is lci.",
"Since $I_{\\gamma ,\\tau }$ is homogeneous with respect to a positive grading, the maximal ideal for $p_v$ is the unique maximal graded ideal of $R$ , and hence the condition of being a local complete intersection (at $p_v$ ) is equivalent to that of being a complete intersection.",
"(See [13] as well the surrounding section for more details.)",
"This can be determined by checking whether the codimension of $R/I_{\\gamma ,\\tau }$ is equal to the minimal number of generators of $I_{\\gamma ,\\tau }$ .",
"The latter number can be computed as the first Betti number of the aforementioned minimal free resolution of $R/I_{\\gamma ,\\tau }$ .",
"Given the ideal $I_{\\gamma ,\\tau }$ , all of the aforementioned data can be computed using Macaulay 2 [1], allowing us to check lci-ness.",
"Example 3.17 Consider the $(1,2,1,2)$ -containing $(3,3)$ -clan $\\gamma =(1,2,+,-,1,2)$ .",
"The closed orbit $\\tau =(+,-,+,+,-,-)$ is contained in the orbit closure $Y_{\\gamma }$ .",
"The Macaulay 2 [1] command I := computePQPatchIdeal(\"12+-12\",3,3,\"+-++–\"), developed by the second author and A. Yong, creates the patch ideal $I_{\\gamma ,\\tau }$ described in [50], as well as giving $\\textbf {z}]$ the appropriate ${\\mathbb {Z}}^n$ -grading coming from the torus action.",
"This enables us to then perform the command resl:= prune res((ring I)⌃1/I), which computes the minimal graded free resolution, followed by betti resl.",
"From this command, we learn that the first Betti number is 2, meaning that $I_{\\gamma ,\\tau }$ is minimally generated by 2 elements.",
"On the other hand, the command codim ((ring I)/I) reveals that the patch is of codimension 2.",
"Thus the patch of $Y_{\\gamma }$ at $\\tau $ is a complete intersection, which tells us that $Y_{\\gamma }$ is lci along $Q_{\\tau }$ .",
"Similarly, one can check that $Y_{\\gamma }$ is lci along every closed orbit below it in Bruhat order, so it is in fact globally lci.",
"On the other hand, performing the same checks for $\\gamma =(1,+,2,1,2)$ at $\\tau =(+,+,-,-,+)$ reveals that $I_{\\gamma ,\\tau }$ is minimally generated by 3 elements, but the patch is again of codimension 2.",
"So in this case, $Y_{\\gamma }$ is not lci at $\\tau $ , so of course it is not globally lci.",
"The checks described in Example REF can be automated to check all $(1,2,1,2)$ -containing orbit closures along all closed orbits for a particular $p$ and $q$ .",
"This allows one to give an exhaustive list of which orbit closures are lci and which are not.",
"The authors have successfully determined the exhaustive list of non-lci $(1,2,1,2)$ -containing orbit closures for all $(p,q)$ through $p+q=8$ .",
"As it turns out, through $p+q=8$ , lci-ness is characterized by pattern avoidance, even if we allow the $(1,2,1,2)$ -containing case.",
"More precisely, for $p+q \\le 8$ , any $K$ -orbit closure, $(1,2,1,2)$ -avoiding or not, is lci if and only if it avoids the bad patterns of Proposition REF and additionally avoids the patterns $(1,+,2,1,2)$ , $(1,2,1,+,2)$ , $(1,2,1,3,2,3)$ , $(1,2,2,3,1,3)$ , $(1,2,1,3,3,2)$ , $(1,+,2,3,2,3,1)$ , $(1,2,3,2,3,+,1)$ , $(1,2,3,2,3,4,4,1)$ , $(1,2,2,3,4,3,4,1)$ , $(1,2,3,4,2,3,4,1)$ , $(1,2,3,4,3,2,4,1)$ , and $(1,2,3,4,2,4,3,1)$ , along with their negatives.",
"We do not feel confident in conjecturing that this is the complete list of bad patterns, since we have been unable to push exhaustive checks to $p+q=9$ (or even to the case $K=GL(7, \\times GL(2,$ ) and determine whether any new bad patterns occur there.",
"However, we feel that with the evidence at hand, considering the results of [42] in the analogous case of Schubert varieties, at least the following conjecture is reasonable.",
"Conjecture 3.18 LCI-ness of $K$ -orbit closures for $(GL(p+q,,GL(p,\\times GL(q,)$ is characterized by pattern avoidance."
],
[
"Gorensteinness",
"Recall that a local ring $(R, \\mathfrak {m},\\mathbb {k})$ is said to be Cohen-Macaulay if $\\text{Ext}_R^i(\\mathbb {k},R) = 0$ for $i < \\dim (R)$ .",
"$R$ is said to be Gorenstein if it is Cohen-Macaulay and, additionally, $\\text{dim}_{\\mathbb {k}}\\text{Ext}_R^{\\dim (R)}(\\mathbb {k},R) = 1$ .",
"A variety or scheme $X$ is said to be Gorenstein at the point $p$ if the local ring $\\mathcal {O}_{X,p}$ is Gorenstein.",
"$X$ is said simply to be Gorenstein if it is Gorenstein at every point.",
"Equivalently, a variety $X$ is Gorenstein if it is Cohen-Macaulay (meaning all local rings are Cohen-Macaulay) and its canonical sheaf is a line bundle.",
"To determine Gorensteinness of $(1,2,1,2)$ -avoiding $K$ -orbit closures, we need to determine Gorensteinness of Grassmannian Richardson varieties, which requires an understanding of the Gorenstein locus of a Grassmannian Schubert variety.",
"Again, this is known, thanks to the following proposition of N. Perrin [40].",
"Proposition 3.19 The Schubert variety $X^v$ is Gorenstein at $u$ if and only if every inner corner of the path of $v$ not containing a point of $u$ has equal right and left leg lengths.",
"Using Lemma REF , we can restate Perrin's criterion as follows: Proposition 3.20 Let $u$ and $v$ be Grassmannian permutations.",
"Then the Richardson variety $X_{uw_0^K}^v$ is Gorenstein if and only if the left leg length equals the right leg length for all singular corners of the path diagram.",
"As examples, the Richardson varieties whose path diagrams are given in Figures REF and REF are non-Gorenstein, since each has a singular corner with one leg length of 1 and one leg length of 2.",
"On the other hand, the Richardson variety whose path diagram is given in Figure REF (which is the closure of the $K$ -orbit $(1,+,-,+,-,1)$ ) is Gorenstein, since all three singular corners have common leg length 1.",
"Remark 3.21 The non-Gorenstein locus of a non-Gorenstein $(1,2,1,2)$ -avoiding $K$ -orbit closure is computable by the same procedure as that described in Section REF for computing the singular locus, except we should only remove hooks at non-Gorenstein corners, rather than at all singular corners.",
"Remark 3.22 It is clear that smooth orbit closures have no singular corners to check, meaning that the tests of Propositions REF and REF are passed vacuously.",
"It is also clear that an orbit closure passing the test of Proposition REF will also pass the test of Proposition REF .",
"Additionally, our described methods of computing the singular locus, the non-lci locus, and the non-Gorenstein locus makes clear that these sets are one containing the next, with the singular locus being the largest set and the non-Gorenstein locus being the smallest.",
"These simple observations reflect the general fact that “smooth $\\Rightarrow $ lci $\\Rightarrow $ Gorenstein\".",
"For a $(1,2,1,2)$ -avoiding clan $\\gamma $ , one can easily reformulate the check described by Proposition REF as a check on the clan $\\gamma $ , without reference to the path diagram.",
"The statement of the resulting criterion requires a fair amount of notation and is somewhat unpleasant to check, so we do not give it here, since it seems less useful than simply looking at the path diagram for $\\gamma $ .",
"We do offer the following example which demonstrates that, unlike the properties of smoothness and lci-ness, Gorensteinness of $(1,2,1,2)$ -avoiding orbit closures is not characterized by pattern avoidance.",
"Example 3.23 Consider first the $(1,2,1,2)$ -avoiding clan $(1,+,+,-,1)$ .",
"Its path diagram appears in Figure REF .",
"As mentioned above, this clan is non-Gorenstein.",
"Now, let $\\gamma ^{\\prime }$ be $(1,+,+,-,-,1)$ .",
"Its path diagram is given as Figure REF .",
"Figure: The Gorenstein KK-orbit (1,+,+,-,-,1)(1,+,+,-,-,1)Note that although $\\gamma ^{\\prime }$ contains the non-Gorenstein pattern $\\gamma $ , it is Gorenstein, since both legs of its lone singular corner have length 2.",
"Again, one would naturally wonder about the $(1,2,1,2)$ -containing cases as well.",
"As was the case with lci-ness, here we are able to check Gorensteinness by computer, as we briefly describe.",
"As explained in Section REF , one can check Gorensteinness of $Y_{\\gamma }$ by checking Gorensteinness along every closed orbit contained in $Y_{\\gamma }$ , and the latter is equivalent to checking Gorensteinness of the patch of $Y_{\\gamma }$ taken near a $T$ -fixed point of a given closed orbit contained in $Y_{\\gamma }$ .",
"This can be checked in Macaulay 2 [1] using the alternative definition of Gorensteinness given above, namely that a variety is Gorenstein if and only if it is Cohen-Macaulay and its canonical sheaf is a line bundle.",
"In fact, all patches are automatically Cohen-Macaulay by a general result of Brion [12].",
"(We note, though, that Macaulay 2 [1] can easily verify the Cohen-Macaulay property, if only as a sanity check.)",
"Thus the only check that need be performed is whether the rank of the canonical sheaf is 1.",
"This is easily done in Macaulay 2 [1].",
"We demonstrate with two examples.",
"Example 3.24 Consider first the $(1,2,1,2)$ -containing $(3,2)$ -clan $\\gamma =(1,2,1,+,2)$ .",
"It contains the closed orbit $\\tau =(+,-,-,+,+)$ .",
"As before, the patch ideal $I_{\\gamma ,\\tau }$ can be created using the command I := computePQPatchIdeal(\"121+2\",3,2,\"+–++\"), and a minimal free ${\\mathbb {Z}}^n$ -graded resolution computed as resl:= prune res((ring I)⌃1/I).",
"The rank of the canonical sheaf is then computed using the command rank(source(resl.dd_cod)), where cod is the codimension of $I$ , computed as before.",
"We see here that the canonical sheaf has rank 2, meaning that $Y_{\\gamma }$ is not Gorenstein at $\\tau $ and hence is not Gorenstein.",
"On the other hand, one can perform the same checks for the orbit $\\gamma =(1,2,-,1,+,2)$ at every closed orbit contained in it, and one sees that the rank of the canonical sheaf is everywhere 1, meaning that $Y_{\\gamma }$ is Gorenstein.",
"Note that the Gorenstein pattern $(1,2,-,1,+,2)$ contains the non-Gorenstein pattern $(1,2,1,+,2)$ , which shows that Gorensteinness cannot be characterized by pattern avoidance in the $(1,2,1,2)$ -containing case either.",
"Remark 3.25 Using exhaustive computer checks in the $(1,2,1,2)$ -containing cases, we have attempted to find combinatorial criteria generalizing those characterizing Gorensteinness in the $(1,2,1,2)$ -avoiding case which would apply equally well to all clans, but we have been unable to do so.",
"By analogy with the case of type $A$ Schubert varieties, it is perhaps unsurprising that smoothness and lci-ness (in the $(1,2,1,2)$ -avoiding case, and conjecturally in the $(1,2,1,2)$ -containing case) can be characterized by pattern avoidance, while Gorensteinness cannot.",
"Indeed, the pattern avoidance criterion for smoothness of type $A$ Schubert varieties due to Lakshmibai-Sandhya [28] is well-known.",
"Additionally, the first author and H. Úlfarrson [42] have recently shown that lci-ness of Schubert varieties can also be characterized by pattern avoidance.",
"However, the first author and A. Yong showed in [45] that Gorensteinness of Schubert varieties cannot be characterized by ordinary pattern avoidance.",
"A more general notion of “Bruhat-restricted pattern avoidance\" is needed.",
"This was later shown in [46] to be an example of the yet more general notion of “interval pattern avoidance\".",
"It is reasonable to wonder whether there is a similar generalization of pattern avoidance in the $K$ -orbit setting.",
"Question Is there a combinatorial notion in the $K$ -orbit setting, analogous to interval pattern avoidance in the case of Schubert varieties, which explains or “governs\" all semicontinuously stable singularity properties of $K$ -orbit closures?",
"The above question is natural not only in the case of $(G,K) = (GL(p+q,,GL(p, \\times GL(q,)$ that we are currently considering, but also for other symmetric pairs such as $(GL(2n,,Sp(2n,)$ and $(GL(n,,O(n,)$ , where combinatorial parametrizations of the orbit sets are known, and where there are reasonable corresponding notions of pattern avoidance.",
"Furthermore, Billey and Braden [8] (partially anticipated by Bergeron and Sottile [4]) give a geometric explanation for the appearance of pattern avoidance in characterizing smoothness of Schubert varieties.",
"Billey and Postnikov [6] give a uniform definition of pattern avoidance in terms of the underlying root systems that matches this geometric explanation and use it to characterize smoothness for arbitrary Schubert varieties.The paper [6] actually precedes [8], but was published later due to delays in the publication process.",
"This definition was extended to interval pattern avoidance by the first author [44].",
"We therefore also ask the following.",
"Question Is there a uniform combinatorial notion in the $K$ -orbit setting (at least applying to the case of simply connected $G$ ) that gives a geometric explanation for the appearance of pattern avoidance in smoothness results?",
"Can this notion also be extended to a notion of interval pattern avoidance?"
],
[
"Formulas for KLV Polynomials",
"In this section, we give an explicit formula for KLV polynomials $P_{\\tau ,\\gamma }(q)$ when $\\gamma $ is $(1,2,1,2)$ -avoiding.",
"A very brief sketch of the proof, which we flesh out in more detail in the coming sections, is as follows: $Y_{\\gamma }$ is the Richardson variety $X_{w_0^Ku(\\gamma )^{-1}}^{v(\\gamma )^{-1}}$ by Theorem REF , and the KLV polynomial $P_{\\tau ,\\gamma }(q)$ is the local intersection homology ($IH$ ) Poincaré polynomial for $Y_{\\gamma }$ at a point of $Q_{\\tau }$ , as we saw in Section REF .",
"By results of [27], the $IH$ Poincaré polynomial for a Richardson variety is a product of those for the individual Schubert varieties, each of which is an ordinary KL polynomial.",
"Finally, the KL polynomials in question are those for Grassmannian Schubert varieties, and an explicit formula for such KL polynomials is given explicitly by Lascoux–Schützenberger in [31]."
],
[
"The Lascoux–Schützenberger formula for cograssmannian KL polynomials",
"We start by describing the aforementioned rule of Lascoux–Schützenberger for calculating $P_{v,w}(q)$ when $w$ is cograssmannian.",
"Our account will be closer in spirit to that of [22] than to the original [31].",
"Let $w \\in S_n$ be a cograssmannian permutation, meaning one with at most one right ascent.",
"Let $p$ be the location of this ascent.",
"Just as we did with Grassmannian permutations in Section , we can associate to $w$ a lattice path in a $p \\times (n-p)$ rectangle, going from the southwest corner to the northeast, by moving up at the $i$ -th step if $w^{-1}(i) \\le p$ and moving right otherwise.",
"Note that this is the same as the path for the Grassmannian permutation $ww_0^K$ , with $w_0^K$ the long element of $W_K = S_p \\times S_{n-p}$ .",
"Following [31], we use this lattice path to associate a rooted tree $\\mathcal {T}(w)$ to $w$ .",
"First write a string of parentheses for the lattice path, replacing up steps with “(” and right steps with “)“.",
"Each matching pair of parentheses “($\\ldots $ )” will be a vertex of the tree, and vertex $V$ is a descendent of $V^{\\prime }$ if the parentheses corresponding to $V$ are nested inside the parentheses corresponding to $V^{\\prime }$ .",
"Finally, add a root vertex (not corresponding to any parentheses) which will be the ancestor of every vertex.",
"Given any permutation $x$ , we associate a path to $x$ in the same way by drawing a lattice path in the $p \\times (n-p)$ rectangle from the southwest corner to the northeast corner by the same rules.",
"(One can do this even if $x$ is neither Grassmannian nor cograssmannian.",
"This process recovers the path for $x^{\\prime }$ , where $x^{\\prime }$ is the maximal length coset representative for $xW_K$ .",
"It is well known that in this case $P_{x,w}(q)=P_{x^{\\prime },w}(q)$ .)",
"To each leaf of $\\mathcal {T}(w)$ , we assign a nonnegative integer $c$ based on $x$ , which we will call the capacity of the leaf.",
"A leaf of $\\mathcal {T}(w)$ is associated to a consecutive pair “()” of parentheses, which corresponds to an inner corner of $w$ 's path.",
"Say that this inner corner is at $(a,b)$ .",
"Assuming $x \\le w$ , so that the path for $w$ lies weakly southeast of that for $x$ , let the capacity $c$ of the corresponding leaf be the unique nonnegative integer for which $(a-c,b+c)$ is on the path for $x$ .",
"Pictorially, in our $45^{\\circ }$ -rotated path diagrams, the capacity is easy to read off as the vertical distance from an inner corner on the lower path to the upper path( note that this capacity is nonzero if and only if the inner corner is a “singular corner\" by our earlier definition).",
"As an example, let $w = 986517432$ , and let $x$ be any permutation in the same left $W_K$ coset as 764219853.",
"The rotated path diagram is given in Figure REF .",
"We reproduce it here as Figure REF with the inner corners on the bottom path indicated, as well as their vertical distances to the upper path.",
"Figure: The path diagram for w=986517432w=986517432, x=764219853x=764219853The word corresponding to $w$ is “()))(()((\".",
"The corresponding rooted tree, with leaf capacities indicated, is shown in Figure REF .",
"Figure: The rooted tree for w=986517432w=986517432, x=764219853x=764219853Now let $A_{x,w}$ be the set of edge labellings of $\\mathcal {T}(w)$ with entries in $\\mathbb {Z}_{\\ge 0}$ satisfying the following: Labels weakly increase along any path from the root to a leaf.",
"The label on any edge adjacent to a leaf does not exceed the capacity of the leaf.",
"Given $t\\in A_{x,w}$ , let $|t|$ denote the sum of the labels.",
"Then the formula of [31] is as follows: Theorem 4.1 ([31]) For any cograssmannian $w \\in S_n$ and for any $x$ , $ P_{x,w}(q)=\\sum _{t\\in A_{x,w}} q^{|t|}.",
"$ As an example, there are three valid edge labellings of the rooted tree shown in Figure REF .",
"Each of them is shown in Figure REF .",
"Thus Theorem REF tells us that $ P_{764219853,986517432}(q) = 1 + q + q^2.",
"$ Figure: Valid edge labellings for w=986517432w=986517432, x=764219853x=764219853Remark 4.2 Note that Theorem REF can also directly calculate $P_{w_0w, w_0v}(q)$ when $v$ is a Grassmannian permutation, and $w$ is arbitrary.",
"Indeed, when $v$ is a Grassmannian permutation with descent at $p$ , then $w_0v$ is a cograssmannian permutation with descent at $q=n-p$ .",
"Thus Theorem REF applies.",
"One can associate to $v$ a lattice path in the $p \\times q$ grid as described in Section .",
"We can then directly construct $\\mathcal {T}(w_0v)$ by matching parentheses the the “wrong” way, so a vertex corresponds to a matching “)$\\ldots $ (” pair, and a leaf corresponds to an inner corner of $v$ 's path.",
"Similarly, we can assign capacities to $\\mathcal {T}(w_0v)$ corresponding to $w_0w$ by constructing a path for $w$ (now southeast of that for $v$ ) and looking at the distance from each inner corner of $v$ 's path to $w$ 's path.",
"Essentially, this amounts to doing the computation for $P_{w_0w,w_0v}(q)$ (which would ordinarily be done in a $q \\times p$ rectangle) after rotating the diagrams by $180^{\\circ }$ , which is equivalent to flipping the path diagrams upside down and switching left and right.",
"As an example of this last observation, consider the case $w=156892347$ and $v=124673589$ .",
"Then the appropriate diagram for calculating $P_{w_0w, w_0v}(q)$ is that given in Figure REF , but here we consider different corners (the outer corners along the top path) and their vertical distances to the lower path, as indicated in Figure REF .",
"The word for the top path is “(()()(())\", and, matching the wrong way as described above, we obtain the rooted tree shown in Figure REF .",
"Figure: The path diagram for v=124673589v=124673589, w=156892347w=156892347Figure: The rooted tree for v=124673589v=124673589, w=156892347w=156892347One checks that this tree has 10 valid edge labellings and that Theorem REF gives the KL polynomial $ P_{954218763,986437521}(q) = 1+2q+3q^2+2q^3+q^4+q^5.",
"$"
],
[
"Statement and proof of the theorem",
"We can now give the statement and proof of our formula for the KLV polynomial $P_{\\tau ,\\gamma }(q)$ for $(1,2,1,2)$ -avoiding clans $\\gamma $ .",
"We start with the path diagram corresponding to the clan $\\gamma $ , described in Section .",
"We then add to it another path diagram for the clan $\\tau $ , constructed in the same way that the path diagram for $\\gamma $ is.",
"Namely, we start from the southwest of the $p \\times q$ grid and trace two paths, following these rules at step $i$ : If $\\tau _i=+$ , then both paths move up; If $\\tau _i=-$ , then both paths move right; If $\\tau _i$ is a first occurrence, then the bottom path moves right, while the top path moves up; and If $\\tau _i$ is a second occurrence, the bottom path moves up, while the top path moves right.",
"Note that this makes sense even if $\\tau $ is not $(1,2,1,2)$ -avoiding, and if $\\tau \\le \\gamma $ , the new path diagram for $\\tau $ fits within that for $\\gamma $ .",
"(This follows easily from [47].)",
"Note further that the top path of the new path diagram for $\\tau $ is that for the Grassmannian permutation $v(\\tau )$ , while the bottom path of the new diagram is that for the permutation $u(\\tau )$ , where $v(\\tau )$ and $u(\\tau )$ are as described in Section REF .",
"We next construct two rooted trees in the way just described in Section REF : For the first, we construct a word in `(' and `)' for the bottom path, and a rooted tree from that word by matching “()\", with leaf capacities determined by the distances from singular corners on the bottom path for $\\gamma $ to the bottom path for $\\tau $ .",
"For the second, we construct a word for the top path, and a rooted tree from that word by matching “)(\", with leaf capacities determined by the distances from singular corners on the top path for $\\gamma $ to the top path for $\\tau $ .",
"Define the set $\\mathcal {T}_{\\tau ,\\gamma }$ as the set of all pairs consisting of a valid edge labelling of the first rooted tree, combined with a valid edge labelling of the second, with valid edge labellings defined just as in Section REF .",
"For such a pair $t$ , define $|t|$ to be the sum of labels (taken over both edge labellings).",
"Then we have the following theorem.",
"Theorem 4.3 Given clans $\\tau \\le \\gamma $ in Bruhat order, with $\\gamma $ avoiding $(1,2,1,2)$ , the KLV polynomial satisfies $ P_{\\tau ,\\gamma }(q)=\\sum _{t\\in \\mathcal {T}_{\\tau ,\\gamma }} q^{|t|}.",
"$ The following Kunneth formula is given in [20]: Given two stratified pseudomanifolds $X$ and $Y$ , the intersection homology sheaf (assuming the middle perversity) is given by $ \\mathcal {IH}(X\\times Y) \\cong \\mathcal {IH}(X)\\otimes \\mathcal {IH}(Y).",
"$ Now, given points $x\\in X$ and $y\\in Y$ , localizing at $p=(x,y)\\in X\\times Y$ tells us that $ \\mathcal {IH}_{p}(X \\times Y)=\\mathcal {IH}_x(X) \\otimes \\mathcal {IH}_y(Y).",
"$ Now the usual Kunneth formula tells us (since we are working with sheaves of vector spaces, so there are no flatness issues to worry about) that $ IH^i_p(X\\times Y)\\cong \\bigoplus _{j+k=i} IH^j_x(X)\\otimes IH^k_y(Y).",
"$ We apply this to a Richardson variety $X_u^v$ , utilizing results of [27].",
"Since, for $p\\in \\mathbb {C}^n$ (or actually any smooth space), $IH^k_p(\\mathbb {C}^n)=0$ for $k>0$ and $\\dim IH^0_p(\\mathbb {C}^n)=1$ , an application of [27] gives the following: Theorem 4.4 Let $p\\in X_u^v$ , and suppose $p\\in X_y^\\circ \\cap X_\\circ ^w$ .",
"Then $ IH^i_p(X_u^v) \\cong \\bigoplus _{j+k=i} IH^j_{yB/B}(X_u)\\otimes IH^k_{wB/B}(X^v).",
"$ Consequently, $ \\sum _{i} \\dim (IH^i_p(X_u^v)) q^{i/2} = P_{w_0w,w_0v}(q)P_{y,u}(q).",
"$ Now, by Theorem REF , if $\\gamma $ avoids $(1,2,1,2)$ , $Y_\\gamma $ is the Richardson variety $X_u^v$ where $u=w_0^Ku(\\gamma )^{-1}$ , and $v = v(\\gamma )^{-1}$ .",
"By the results of Section REF , the KLV polynomial $P_{\\tau ,\\gamma }$ is the $IH$ -Poincaré polynomial for $Y_{\\gamma }$ at a point of $Q_{\\tau }$ , and Theorem REF applies to this computation directly.",
"Recall that Lemma REF tells us that the distinguished representative of $Q_{\\tau }$ lies in $X_u^0 \\cap X_0^v$ , where $v=v(\\tau )^{-1}$ , and where $u^{-1}=u(\\tau )w$ for some $w \\in W_K$ .",
"Thus Theorem REF says that $ P_{\\tau ,\\gamma }(q) = P_{w_0v, w_0v(\\gamma )^{-1}}(q) P_{u,w_0^Ku(\\gamma )^{-1}}(q) = P_{w_0v(\\tau ),w_0v(\\gamma )}(q) P_{u(\\tau )w,u(\\gamma )w_0^K}(q) = $ $ P_{w_0v(\\tau ),w_0v(\\gamma )}(q) P_{u(\\tau ),u(\\gamma )w_0^K}(q), $ where we have used the following standard facts: $P_{a,b}(q) = P_{a^{-1},b^{-1}}(q)$ for any $a,b \\in W$ ; and If $u$ is cograssmannian and $x,x^{\\prime } \\le u$ are such that $x^{\\prime } = xw$ for $w \\in W_K$ , then $P_{x^{\\prime },u}(q) = P_{x,u}(q)$ .",
"Now, recall that $u(\\gamma )$ and $v(\\gamma )$ are Grassmannian, so $u(\\gamma )w_0^K$ and $w_0v(\\gamma )$ are cograssmannian.",
"Thus each of the polynomials in the expression above is computed by Theorem REF .",
"(We use Remark REF to compute $P_{w_0v(\\tau ),w_0v(\\gamma )}(q)$ .)",
"As mentioned, the path diagram for $\\tau $ consists of the path for $v(\\tau )$ on top and the path for $u(\\tau )$ on bottom.",
"The polynomial described by our theorem is thus the appropriate product of KL polynomials.",
"Example 4.5 We compute the KLV polynomial $P_{\\tau ,\\gamma }(q)$ where $\\gamma =(1,2,+,-,+,-,2,1)$ , and $\\tau =(+,-,+,-,+,-,+,-)$ .",
"The path diagram for $\\gamma $ , with the added (dashed) path diagram for $\\tau $ , is given in Figure REF .",
"Figure: The path diagram for τ=(+,-,+,-,+,-,+,-)\\tau =(+,-,+,-,+,-,+,-), γ=(1,2,+,-,+,-,2,1)\\gamma =(1,2,+,-,+,-,2,1)The rooted trees and leaf capacities used to calculate $P_{u(\\tau ),u(\\gamma )w_0^K}(q) = P_{13572468,87536421}(q)$ and $P_{w_0v(\\tau ),w_0v(\\gamma )}(q)=P_{86427531,87645321}(q)$ are given as Figure REF .",
"Figure: The rooted trees for P τ,γ (q)P_{\\tau ,\\gamma }(q)There are 4 valid edge labellings of the first tree, and 2 of the second, giving the following two ordinary KL polynomials: $ P_{13572468,87536421}(q) = 1 + 2q + q^2, $ and $ P_{86427531,87645321}(q) = 1 + q.",
"$ Thus $ P_{\\tau ,\\gamma }(q) = (1+2q+q^2)(1+q) = 1 + 3q + 3q^2 + q^3.",
"$"
],
[
"Acknowledgements",
"We gratefully acknowledge Alexander Yong for helpful discussions, as well as for allowing our use of Macaulay 2 [1] code developed jointly by him and the second author for the purpose of furthering this project.",
"We also wish to thank Peter Trapa and Monty McGovern for their generous help with various technical details."
]
] | 1403.0363 |
[
[
"Symmetric Strong Duality for a Class of Continuous Linear Programs with\n Constant Coefficients"
],
[
"Abstract We consider Continuous Linear Programs over a continuous finite time horizon $T$, with linear cost coefficient functions and linear right hand side functions and a constant coefficient matrix, where we search for optimal solutions in the space of measures or of functions of bounded variation.",
"These models generalize the Separated Continuous Linear Programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss.",
"We present simple necessary and sufficient conditions for feasibility.",
"We formulate a symmetric dual and investigate strong duality by considering discrete time approximations.",
"We prove that under a Slater type condition there is no duality gap and there exist optimal solutions which have impulse controls at $0$ and $T$ and have piecewise constant densities in $(0,T)$.",
"Moreover, we show that under non-degeneracy assumptions all optimal solutions are of this form, and are uniquely determined over $(0,T)$."
],
[
"Introduction",
"We consider problems of the form: $&\\max & \\int _{0-}^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}dU(t) \\nonumber \\\\\\mbox{M-CLP}\\, & \\mbox{s.t.}",
"& \\qquad A\\, U(t) \\quad \\le \\beta + b t,\\quad 0 \\le t \\le T, \\\\&& U(t) \\ge 0,\\; U(t) \\mbox{ non-decreasing and right continuous on [0,T].",
"}\\nonumber $ where $A$ is a $K\\times J$ constant matrix, $\\beta ,b,\\gamma ,c$ are constant vectors of corresponding dimensions, the integrals are Lebesgue-Stieltjes, $U$ are $J$ unknown functions over the time horizon $[0,T]$ , and by convention we take $U(0-)=0$ .",
"We formulate a symmetric dual problem $&\\min & \\int _{0-}^T (\\beta +(T-t)b)^{{\\mbox{\\tiny \\bf \\sf T}}}dP(t) \\nonumber \\\\\\mbox{M-CLP$^*$}\\, & \\mbox{s.t.}",
"& \\qquad A^{{\\mbox{\\tiny \\bf \\sf T}}}P(t) \\quad \\ge \\gamma + c t,\\quad 0 \\le t \\le T, \\\\&& P(t) \\ge 0,\\; P(t) \\mbox{ non-decreasing and right continuous on [0,T].",
"}\\nonumber $ with $K$ unknown dual functions $P$ with the same convention $P(0-)=0$ .",
"It is convenient to think of dual time as running backwards, so that $P(T-t)$ corresponds to $U(t)$ .",
"The main feature to note here is that the objective as well as the left hand side of the constraints are formulated as Lebesgue-Stieltjes integrals with respect to a vector of monotone non-decreasing control function $U(t)$ , in other words our controls are in the space of measures.",
"This is in contrast to most formulations in which the objective and left hand side of the constraints are Lebesgue integrals with respect to a measurable bounded control $u(t)$ , in other words controls which are in the space of densities.",
"In particular, while in the usual formulation the left hand side of the constraints is an absolutely continuous function, our formulation allows the left hand side of the constraint to have jumps, as a result of jumps in $U(t)$ , which correspond to impulse controls.",
"Our main results in this paper include the following: We discuss how this formulation relates to and generalizes previous continuous linear programs.",
"We show weak duality and present a simple necessary and sufficient test for feasibility of M-CLP.",
"We also present a Slater type condition which is easily checked, using the same test.",
"We show that under this Slater type condition there is no duality gap between M-CLP and M-CLP$^*$ , by considering discrete time approximations.",
"We also show that in this case M-CLP and M-CLP$^*$ posses optimal solutions.",
"We further show that in that case there exist optimal solutions for which $U(t)$ and $P(t)$ have impulse controls at $0,T$ and are absolutely continuous inside $(0,T)$ , with piecewise constant densities.",
"Finally, under appropriate simple non-degeneracy assumptions we show that all optimal solutions are of this form, and that the absolutely continuous part on $(0,T)$ is uniquely determined.",
"Further research to develop a simplex-type algorithm that constructs solutions of this form is in progress.",
"We note that the question of existence of strong duality, and whether symmetric dual formulations are useful is far from simple when dealing with linear programs in infinite dimensional spaces [7], [20].",
"Our results in this paper furnish an example where indeed strong duality can hold with a symmetric dual, if a Slater type condition is satisfied."
],
[
"Background and motivation",
"Continuous linear programs were introduced by Bellman in 1953 [8], [9] to model economic processes: find a bounded measurable $u$ which $&\\max & \\int _0^T c^{{\\mbox{\\tiny \\bf \\sf T}}}(t) u(t) dt \\nonumber \\\\\\mbox{Bellman-CLP }\\quad \\quad &\\mbox{s.t.}",
"& H(t) u(t) + \\int _0^t G(s,t) u(s) ds \\le a(t), \\hspace{43.36243pt}\\\\ && u(t) \\ge 0, \\quad t \\in [0,T].",
"\\nonumber $ Where $G(s,t),H(t)$ are given matrix functions.",
"These problems were investigated by Dantzig and some of his students, to model continuous time Leontief systems, and by several other early authors [10], [11], [13], [21], [22], with many publications since, but up to date no efficient algorithms or coherent theory have emerged, and these problems are considered very hard.",
"Separated continuous linear programs (SCLP) were introduced by Anderson [1], [2] in the context of job-shop scheduling: $&\\max & \\int _0^T c(t)^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) \\, dt \\nonumber \\\\\\mbox{Anderson-SCLP }\\quad &\\mbox{s.t.}",
"& \\int _0^t G u(s) ds \\le a(t), \\hspace{122.85876pt}\\\\&& \\quad H u(t) \\qquad \\le b(t),\\nonumber \\\\ && u(t) \\ge 0, \\quad t \\in [0,T] \\nonumber .$ where $G,H$ are constant matrices, and $a(t),b(t),c(t)$ are given vector functions.",
"Some special cases of SCLP were solved by Anderson and Philpott [4], [5], and this research and related earlier work were summarized in the 1987 book of Anderson and Nash [3], which also contains many references to work on CLP up to that date.",
"Major progress in the theory of SCLP was achieved by Pullan [6], [14]–[19].",
"Pullan considered SCLP problems with $a(t)$ , $ b(t)$ and $c(t)$ piecewise analytic, and formulated a non-symmetric dual to (REF ) (here we modify Pullan's original version by letting the dual run in reversed time, as in (REF )): $&\\min & \\int _0^T a(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}d P(t) + \\nonumber \\int _0^T b(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}q(t) d t \\hspace{43.36243pt}\\\\\\mbox{Pullan-SCLP$^*$ }\\quad &\\mbox{s.t.",
"}& G^{{\\mbox{\\tiny \\bf \\sf T}}}P(t) + H^{{\\mbox{\\tiny \\bf \\sf T}}}q(t) \\ge c(T-t), \\\\&& P(t)\\ge 0,\\; P(t) \\mbox{ non-decreasing and right continuous on [0,T].}",
"\\nonumber \\\\ && q(t) \\ge 0, \\; t \\in [0,T],\\; \\nonumber $ Pullan showed that when the feasible region of $H u(t) \\le b(t)$ is bounded strong duality holds between (REF ) and (REF ).",
"In the special case that $a(t),c(t)$ are piecewise linear and $b(t)$ piecewise constant Pullan provided an infinite but convergent algorithm to solve the problems and observed that $P$ was absolutely continuous, except for atoms at the breakpoints of $a,b,c$ .",
"The results of Pullan raised several questions: Is the boundedness restriction necessary?",
"Can one formulate a symmetric dual?",
"Do solutions of the form observed by Pullan always exist?",
"More recently Weiss [24] considered the following SCLP problem $&\\max & \\int _0^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) + d^{{\\mbox{\\tiny \\bf \\sf T}}}x(t) \\,dt \\nonumber \\hspace{72.26999pt} \\\\\\mbox{SCLP} & \\mbox{s.t.}",
"& \\int _0^t G\\, u(s)\\,ds + F x(t) \\le \\alpha + a t \\\\&& \\quad \\; H u(t) \\le b \\nonumber \\\\&& \\quad x(t), u(t)\\ge 0, \\quad 0\\le t \\le T. \\nonumber $ and the symmetric dual $&\\min & \\int _0^T (\\alpha + (T-t)a)^{{\\mbox{\\tiny \\bf \\sf T}}}p(t) + b^{{\\mbox{\\tiny \\bf \\sf T}}}q(t) \\,dt \\nonumber \\hspace{65.04256pt} \\\\\\mbox{SCLP$^*$} &\\mbox{s.t.}",
"& \\int _0^t G^{{\\mbox{\\tiny \\bf \\sf T}}}\\, p(s)\\,ds + H^{{\\mbox{\\tiny \\bf \\sf T}}}q(t) \\ge \\gamma + c t \\\\&& \\quad \\; F^{{\\mbox{\\tiny \\bf \\sf T}}}p(t) \\ge d \\nonumber \\\\&& \\quad q(t), p(t)\\ge 0, \\quad 0\\le t \\le T. \\nonumber $ with constant vectors and matrices $G,F,H,\\alpha ,a,b,\\gamma ,c,d$ .",
"In contrast to previous work Weiss developed a simplex type algorithm which solves this pair of problems exactly, in a finite bounded number of steps, without using discretization.",
"The simplex type algorithm of Weiss can solve any pair of problems (REF ), (REF ) which possess optimal solutions $u(t),p(t)$ that are bounded measurable functions.",
"It produces solutions with $u(t),p(t)$ piecewise constant, and $x(t),q(t)$ continuous piecewise linear.",
"However, there exist problems for which both (REF ) and (REF ) are feasible but either (REF ) or (REF ) or both do not possess optimal solutions $u(t),p(t)$ in the space of bounded measurable functions.",
"Moreover, one can construct examples, where (REF ) possess optimal solutions in the space of bounded measurable functions, but (REF ) is infeasible.",
"Such problems cannot be solved by the algorithm of Weiss.",
"This raises the question whether they can be solved in the space of measures, and motivates our formulation of M-CLP, M-CLP$^*$ problems (REF ), (REF ).",
"Consider the SCLP problem (REF ).",
"Then the M-CLP problem with the following data: $\\begin{array}{c}A = \\left[ \\begin{array}{cccc} G & 0 & F & -F \\\\ 0 & 0 & -I & I \\\\H & I & 0 & 0 \\\\ -H & -I & 0 & 0 \\end{array} \\right] \\quad U(t) = \\left[ \\begin{array}{c} U_*(t) \\\\ U_s(t) \\\\ U^{+}(t) \\\\ U^{-}(t) \\end{array} \\right] \\quad \\beta ^* + b^*t = \\left[ \\begin{array}{c} \\alpha \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right] +\\left[ \\begin{array}{c} a \\\\ 0 \\\\ b \\\\ -b \\end{array} \\right] t, \\\\\\gamma ^* + (T-t)c^* = \\left[ \\gamma \\quad 0 \\quad \\; d \\quad -d \\right] + (T-t)\\left[ c \\quad 0 \\quad \\; 0 \\quad 0 \\right]\\end{array}$ is called the M-CLP extension of SCLP.",
"M-CLP/M-CLP$^*$ are generalizations of SCLP/SCLP$^*$ in the following sense: (i) if SCLP (REF ) and SCLP$^*$ (REF ) possess optimal solutions, then these solutions determine optimal solutions of the corresponding M-CLP/M-CLP$^*$ extensions with the same objective value.",
"(ii) If the M-CLP/M-CLP$^*$ extensions of the SCLP/SCLP$^*$ have optimal solutions with no duality gap which are absolutely continuous, then this solution determines optimal solutions of the SCLP/SCLP$^*$ , with the same objective value.",
"(iii) If SCLP is feasible and the Slater type condition holds for M-CLP/M-CLP$^*$ extensions, then the supremum of the objective of SCLP is equal to the objective value of the optimal solution of the M-CLP extension.",
"(i) Consider an optimal solution $x^*(t), u^*(t)$ of (REF ).",
"By the Structure Theorem (Theorem 3 in [24]) $x^*(t)$ is absolutely continuous and hence of bounded variation.",
"Therefore we can write $x^*(t)$ as the difference of two non-decreasing functions $x^*(t) = U^{+}(t) - U^{-}(t)$ .",
"Let $u_s(t)$ be the slacks of the constraints $Hu(t) \\le b$ , and let $U(t) = \\int _0^t u^*(t) dt$ , $U_s(t) = \\int _0^t u_s(t) dt$ .",
"Then the resulting ${\\tilde{U}}= [U_*, U_s,U^{+},U^{-}]$ satisfies the constraints of the M-CLP extension, with the same objective value.",
"A similar argument applies to an optimal solution $q^*(t), p^*(t)$ of (REF ), which determines a feasible solution ${\\tilde{P}}$ of the M-CLP$^*$ extension, which is dual to M-CLP, and has the same objective values.",
"Weak duality of M-CLP and M-CLP$^*$ (see Proposition below) then shows that these solutions are the optimal solutions of M-CLP and M-CLP$^*$ .",
"(ii) If the solution of the M-CLP extension is absolutely continuous then taking $u(t)=\\frac{dU(t)}{dt}$ and $x(t)=U^+(t)-U^-(t)$ we get a feasible solution of SCLP, with the same objective value.",
"The same holds for SCLP$^*$ , and by weak duality these are optimal solutions.",
"(iii) The proof of this part is postponed to Section , after Theorem .",
"It is not hard to see that (REF ), (REF ) generalize also Anderson and Pullan's problems (REF ), (REF ) restricted to $a(t),c(t)$ affine, and $b(t)$ constant."
],
[
"Weak duality, complementary slackness and feasibility",
"Weak duality holds for M-CLP, M-CLP$^*$ (REF ),(REF ).",
"Let $U(t),\\,P(t)$ be feasible solutions for (REF ), (REF ), and compare their objective values: $&& \\mbox{Dual objective }= \\nonumber \\\\&=& \\int _0^T \\big (\\beta +(T-t)b \\big )^{{\\mbox{\\tiny \\bf \\sf T}}}dP(t)\\nonumber \\\\&\\ge &\\int _0^T \\Big ( \\int _0^{T-t} A dU(s) \\Big )^{{\\mbox{\\tiny \\bf \\sf T}}}dP(t) \\nonumber \\\\&=& \\int _0^T \\Big ( \\int _0^{T-t} A^{{\\mbox{\\tiny \\bf \\sf T}}}dP(s) \\Big )^{{\\mbox{\\tiny \\bf \\sf T}}}dU(t) \\nonumber \\\\&\\ge & \\int _0^T \\big ( \\gamma + (T-t) c \\big )^{{\\mbox{\\tiny \\bf \\sf T}}}dU(t) \\nonumber \\hspace{57.81621pt} \\\\&& = \\mbox{Primal objective. }",
"\\nonumber $ The first inequality follows from the primal constraints at $T-t$ , and from $P(t)$ non-decreasing.",
"The equality follows by changing order of integration, using Fubini's theorem.",
"The second inequality follows from the dual constraints at $T-t$ , and from $U(t)$ non-decreasing.",
"Equality of the primal (M-CLP) and dual (M-CLP$^*$ ) objective will occur if and only if the following holds: Complementary slackness condition.",
"Let $x(t)=\\beta +bt-AU(t)$ and $q(t)= A^{{\\mbox{\\tiny \\bf \\sf T}}}P(t) - \\gamma -ct $ be the slacks in (REF ), (REF ).",
"The complementary slackness condition for M-CLP, M-CLP$^*$ is $\\int _0^T x(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}dP(t) = \\int _0^T q(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}dU(t) = 0.$ In the following propositions and theorems in this and following sections we present results for M-CLP.",
"By symmetry these results hold for M-CLP$^*$ , with the obvious modifications.",
"We present now a simple necessary and sufficient condition for feasibility.",
"This is similar to a condition derived by Wang, Zhang and Yao [23].",
"It involves the standard linear program Test-LP and its dual Test-LP$^*$ .",
"$&\\max &z = (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}+ \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U \\nonumber \\hspace{72.26999pt} \\\\\\mbox{Test-LP} &\\mbox{s.t.}",
"& A \\mathbf {u}\\le \\beta \\\\&\\mbox{} & A \\mathbf {u}+ A U \\le \\beta + bT \\nonumber \\\\&& \\quad \\mathbf {u},\\, U \\ge 0 \\nonumber $ M-CLP is feasible if and only if Test-LP is feasible.",
"(i) Sufficiency: Let $\\mathbf {u},\\, U$ be a solution of Test-LP (REF ), with slacks $x^0=\\beta -A\\mathbf {u}$ , $x^T=\\beta +bT -A\\mathbf {u}- A U$ .",
"Then $U(t)=\\mathbf {u}+ \\frac{t}{T}U,\\,0\\le t \\le T$ is a feasible solution of M-CLP (REF ), with non-negative slacks $x(t)= (1 - \\frac{t}{T}) x^0 + \\frac{t}{T} x^T$ .",
"To check this we have for $0 \\le t \\le T$ : $A U(t) + x(t) &=& A \\left(\\mathbf {u}+ \\frac{t}{T}U\\right) + \\left(1 - \\frac{t}{T}\\right) x^0 + \\frac{t}{T} x^T \\\\&=& \\frac{t}{T} \\left(A \\mathbf {u}+ A U + x^T\\right) + \\left(1 - \\frac{t}{T}\\right) \\left(A \\mathbf {u}+ x^0\\right) \\\\&=& \\beta + b t.$ (ii) Necessity: Let $U(t)$ be a feasible solution of M-CLP (REF ) with slack $x(t)\\ge 0$ .",
"Then $\\mathbf {u}=U(0),\\; U = \\int _{0^+}^T dU(t)$ with slack $x^0=x(0), x^T=x(T)$ is a feasible solution for Test-LP (REF ), as is seen immediately.",
"We use the following definition: [Slater type condition] We say that the Test-LP problem (REF ) is strictly feasible at $T$ if there exists a feasible solution $\\mathbf {u}, U$ of (REF ) and a constant $\\alpha >0$ such that $\\beta - A\\mathbf {u}\\ge \\alpha $ and $\\beta + bT - A\\mathbf {u}- AU \\ge \\alpha $ .",
"We say that M-CLP is strictly feasible at $T$ if there exists a feasible solution $U(t)$ of (REF ) and a constant $\\alpha >0$ such that $\\beta + bt - A U(t) \\ge \\alpha $ for all $t \\in [0, T]$ .",
"M-CLP is strictly feasible if and only if Test-LP is strictly feasible.",
"Simply define $\\beta ^* = \\beta - \\alpha $ and recall Theorem for problems with $\\beta $ replaced by $\\beta ^*$ ."
],
[
"Discrete time approximations and strong duality",
"In this section we consider a pair of M-CLP/M-CLP$^*$ problems which are feasible, and use time discretization to solve them approximately.",
"We prove that if M-CLP and M-CLP$^*$ are strictly feasible, then there is no duality gap and an optimal solution exists.",
"We use a discretization approach similar to [14]."
],
[
"General discretizations",
"For a partition $0=t_0 < t_1 < \\ldots < t_N=T$ we define the discretization of M-CLP to be: $&\\max &z=(\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^0 + \\sum _{i=1}^N \\left(\\gamma + \\left(T - \\frac{t_i + t_{i-1}}{2}\\right) c \\right)^{{\\mbox{\\tiny \\bf \\sf T}}}\\left(t_i - t_{i-1}\\right) u^i + \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^N \\nonumber \\\\&\\mbox{s.t.}",
"& \\displaystyle A \\mathbf {u}^0 + x^0 = \\beta \\nonumber \\\\\\mbox{dCLP$_1$} &\\mbox{} & \\displaystyle A \\mathbf {u}^0 + A \\sum _{i=1}^n \\left(t_i - t_{i-1}\\right) u^i + x^n = \\beta + b t_n \\quad \\text{for } n=1,\\dots ,N \\\\&\\mbox{} & \\displaystyle A \\mathbf {u}^0 + A \\sum _{i=1}^N \\left(t_i - t_{i-1}\\right) u^i + A \\mathbf {u}^N + {\\mathbf {x}}^N = \\beta + b T \\nonumber \\\\&& \\quad \\mathbf {u}^0,\\, u^1,\\dots ,u^N, \\mathbf {u}^N, \\,x^0, x^1,\\dots ,x^N, {\\mathbf {x}}^N \\ge 0.",
"\\nonumber $ and for the same time partition the discretization of M-CLP$^*$ is defined as: $&\\min &z=(\\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}+Tb^{{\\mbox{\\tiny \\bf \\sf T}}}) \\mathbf {p}^N + \\sum _{i=1}^N \\left(\\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}+ \\frac{t_{i} + t_{i-1}}{2} b^{{\\mbox{\\tiny \\bf \\sf T}}}\\right) \\left(t_i - t_{i-1}\\right) p^i + \\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^0 \\nonumber \\\\&\\mbox{s.t.}",
"& \\displaystyle A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N - q^N = \\gamma \\\\\\mbox{dCLP$_2$} &\\mbox{} & \\displaystyle A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N + A^{{\\mbox{\\tiny \\bf \\sf T}}}\\sum _{i=n}^N \\left(t_i - t_{i-1}\\right) p^i- q^{n-1} = \\gamma + c (T - t_{n-1}) \\quad \\text{for } n=N,\\dots ,1 \\nonumber \\\\&\\mbox{} & \\displaystyle A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N + A^{{\\mbox{\\tiny \\bf \\sf T}}}\\sum _{i=1}^N \\left(t_i - t_{i-1}\\right) p^i + A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^0 - {\\mathbf {q}}^0 = \\gamma + c T \\nonumber \\\\&& \\quad \\mathbf {p}^N,\\, p^N,\\dots ,p^1, \\mathbf {p}^0, \\,q^N, \\dots ,q^1, q^0, {\\mathbf {q}}^0 \\ge 0.",
"\\nonumber $ Note that these two problems are not dual to each other.",
"Following Pullan [14], for a partition $0=t_0 < t_1 < \\ldots < t_N=T$ and values $f(t_0),\\dots ,$ $f(t_N)$ we define the piecewise linear extension: $f_L(t) = \\left( \\frac{t_i - t}{t_i - t_{i-1}} \\right) f (t_{i-1}) + \\left( \\frac{t - t_{i-1}}{t_i - t_{i-1}}\\right) f (t_{i}) \\quad \\text{for } t \\in [t_{i-1}, t_i) \\text{ for } i = 1,\\dots ,N.$ and the piecewise constant extension: $f_C(t)= f(t_i), \\quad t \\in [ t_{i-1}, t_i ),\\;i=1,\\ldots N.$ The following proposition is an easy extension of Theorem .",
"All discretizations dCLP$_1$ (REF ) are feasible if and only if M-CLP is feasible.",
"(i) Let $U(t)$ be a feasible solution of M-CLP (REF ) with slacks $x(t)\\ge 0$ .",
"Then $\\mathbf {u}^0=U(0),\\; u^n = \\frac{1}{t_n -t_{n-1}}\\int _{t_{n-1}}^{t_n} dU(t)$ (in these integrals we take $t_0=0$ and $t_n = t_n-$ for $n=1,\\dots ,N$ ), $\\mathbf {u}^N=U(T)-U(T-)$ , and $x^0=x(0), \\; x^n=x(t_n-),\\,n=1,\\ldots ,N,\\; {\\mathbf {x}}^N=x(T)$ is a feasible solution for dCLP$_1$ (REF ).",
"To check this we have for $n=0,\\ldots ,N$ : $&& A \\mathbf {u}^0 + A \\sum _{i=1}^n \\left(t_i - t_{i-1}\\right) u^i + x^n = \\\\&& A U(0) + A \\sum _{i=1}^n \\left(t_i - t_{i-1}\\right) \\int _{t_{i-1}}^{t_i}\\frac{1}{t_i -t_{i-1}} dU(t) +x(t_n-) = A U(t_n-) + x(t_n-) =\\beta + bt_n$ (ii) Let $\\mathbf {u}^0,\\, u^1,\\dots ,u^N, \\mathbf {u}^N$ be a feasible solution of dCLP$_1$ .",
"Define $U(0)=\\mathbf {u}^0$ , let $u(t)$ be the piecewise constant extension of $u_1,\\ldots ,u_N$ , and let $U(t)=U(0)+\\int _0^t u(s) ds,\\,t\\in [0,T)$ , $U(T)=U(T-)+\\mathbf {u}^N$ .",
"Then $U(t)$ is a feasible solution of M-CLP.",
"Any feasible solution of dCLP$_1$ can be extended to a feasible solution of M-CLP with equal objective value.",
"We set $u(t)$ to be the piecewise constant extension of $u^1,\\dots ,u^N$ and take $U(t)$ to be the measure with density $u(t)$ on $(0,T)$ and impulses $U(\\lbrace 0\\rbrace ) = \\mathbf {u}^0, U(\\lbrace T\\rbrace )=\\mathbf {u}^N$ .",
"We also set $x(t)$ to be the piecewise linear extension of $x^0,\\dots ,x^N$ , and take ${\\mathbf {x}}^N$ to be the same for both problems.",
"It is immediate to see that this gives a feasible solution to M-CLP.",
"Furthermore, it is immediate to see that the objective of dCLP$_1$ equals the objective of M-CLP for this extended solution.",
"The optimal values $V$ of the various problems satisfy: $V(dCLP_1) \\le V(M\\text{-}CLP) \\le V(M\\text{-}CLP^*) \\le V(dCLP_2)$ The first and last inequalities follow from Proposition REF and the middle inequality follows from weak duality."
],
[
"Discretizations with equidistant partitions",
"Similar to Wang, Zhang and Yao [23] and to Pullan [14] we use even equidistant partitions, denoted $\\pi ^N$ which divides the interval $[0, T]$ into $N$ equal segments, each of length $2\\epsilon $ , i.e.",
"$\\epsilon = \\frac{T}{2N}$ .",
"With this partition we introduce the notations: Given a $K \\times J$ matrix $A$ we define the $N K \\times J$ matrix $A_\\Vert $ , the $N K \\times N J$ matrix $A_\\blacktriangle $ , and the $ K \\times NJ$ matrix $A_=$ as follows: $A_\\blacktriangle =\\left[ \\begin{array}{cccc}A & & & \\\\A & A & & \\\\\\dots & & & \\\\A & A & \\dots & A \\\\\\end{array} \\right], \\quad A_\\Vert =\\left[ \\begin{array}{c}A \\\\A \\\\\\dots \\\\A \\\\\\end{array} \\right],\\quad A_= = \\left[ A \\quad A \\quad \\dots \\quad A \\right].$ We define the $N$ -fold vectors, each with $N$ vector components: ${\\hat{\\beta }}= \\left[ \\begin{array}{c}\\beta \\\\\\vdots \\\\\\beta \\end{array}\\right], \\quad {\\hat{\\gamma }}= \\left[ \\begin{array}{c}\\gamma \\\\\\vdots \\\\\\gamma \\end{array}\\right]$ ${\\hat{b}}_1=\\left[ \\begin{array}{c}2b\\epsilon \\\\4b\\epsilon \\\\\\dots \\\\b T\\end{array}\\right] \\quad {\\hat{b}}_2=\\left[ \\begin{array}{c}b\\epsilon \\\\3b\\epsilon \\\\\\dots \\\\b (T -\\epsilon )\\end{array}\\right] \\quad {\\hat{c}}_1=\\left[ \\begin{array}{c}c (T -\\epsilon ) \\\\c (T -3\\epsilon ) \\\\\\dots \\\\c\\epsilon \\end{array}\\right] \\quad {\\hat{c}}_2=\\left[ \\begin{array}{c}c T \\\\c (T- 2\\epsilon ) \\\\\\dots \\\\2 c\\epsilon \\end{array}\\right]$ $\\Delta U = \\left[ \\begin{array}{c}\\Delta U^1 \\\\\\vdots \\\\\\Delta U^N\\end{array}\\right] = \\left[ \\begin{array}{c}2 u^1 \\epsilon \\\\\\vdots \\\\2 u^N \\epsilon \\end{array}\\right],\\Delta P = \\left[ \\begin{array}{c}\\Delta P^1 \\\\\\vdots \\\\\\Delta P^N\\end{array}\\right] = \\left[ \\begin{array}{c}2 p^1 \\epsilon \\\\\\vdots \\\\2 p^N \\epsilon \\end{array}\\right],{\\hat{x}}= \\left[ \\begin{array}{c}x^1 \\\\ \\vdots \\\\ x^N \\end{array}\\right],{\\hat{q}}= \\left[ \\begin{array}{c}q^0 \\\\ \\vdots \\\\ q^{N-1} \\end{array}\\right]$ Using this notation we rewrite problems (REF ), (REF ) for even equidistant partitions, as: $&\\max &z =\\displaystyle (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^0 + \\left({\\hat{\\gamma }}+{\\hat{c}}_1\\right)^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta U + \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^N \\nonumber \\hspace{43.36243pt} \\\\&\\mbox{s.t.}",
"& \\displaystyle A \\mathbf {u}^0 + x^0 = \\beta \\nonumber \\\\\\mbox{dCLP$_1(\\pi ^N)$} &\\mbox{} & \\displaystyle A _\\Vert \\mathbf {u}^0 + A_\\blacktriangle \\Delta U + {\\hat{x}}= {\\hat{\\beta }}+ {\\hat{b}}_1 \\\\&\\mbox{} & \\displaystyle A \\mathbf {u}^0 + A_= \\Delta U+ A \\mathbf {u}^N + {\\mathbf {x}}^N = \\beta + b T \\nonumber \\\\&& \\quad \\mathbf {u}^0,\\, \\Delta U, \\mathbf {u}^N, \\,x^0,{\\hat{x}}, {\\mathbf {x}}^N \\ge 0.",
"\\nonumber $ $&\\min &z =\\displaystyle (\\beta + b T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N + \\left({\\hat{\\beta }}+ {\\hat{b}}_2 \\right)^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta P + \\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^0 \\nonumber \\hspace{43.36243pt} \\\\&\\mbox{s.t.}",
"& \\displaystyle A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N - q^N = \\gamma \\nonumber \\\\\\mbox{dCLP$_2(\\pi ^N)$} &\\mbox{} & \\displaystyle A^{{\\mbox{\\tiny \\bf \\sf T}}}_\\Vert \\mathbf {p}^N + A_\\blacktriangle ^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta P - {\\hat{q}}= {\\hat{\\gamma }}+ {\\hat{c}}_2 \\\\&\\mbox{} & \\displaystyle A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N + A^{{\\mbox{\\tiny \\bf \\sf T}}}_= \\Delta P + A^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^0 - {\\mathbf {q}}^0 = \\gamma + c T \\nonumber \\\\&& \\quad \\mathbf {p}^N,\\, \\Delta P, \\mathbf {p}^0, \\,q^N, {\\hat{q}}, {\\mathbf {q}}^0 \\ge 0.",
"\\nonumber $ The reader may notice that in (REF ) we have for convenience reversed the order of variables and the order of the constraints in the middle part of the problem relative to (REF ) To quantify the discretization error for time partition $\\pi ^N$ we define a modified pair of problems mdCLP$(\\pi ^N)$ , mdCLP$^*(\\pi ^N)$ : $ \\mbox{mdCLP$(\\pi ^N)$} & \\max & z = (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^0 + \\left({\\hat{\\gamma }}+{\\hat{c}}_2 \\right)^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta U + \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^N \\nonumber \\\\&\\mbox{s.t.}",
"& \\text{Constraints of (\\ref {eqn.dCLPpi1})} \\nonumber $ $ \\mbox{mdCLP$^*(\\pi ^N)$} & \\min & z = (\\beta + b T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^N + \\left({\\hat{\\beta }}+ {\\hat{b}}_1 \\right)^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta P + \\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^0 \\nonumber \\\\& \\mbox{s.t.}",
"& \\text{Constraints of (\\ref {eqn.dCLPpi2})} \\nonumber $ We note that they are dual to each other.",
"They are both feasible if (REF ), (REF ) are feasible.",
"Moreover, since (REF ), (REF ) are (REF ), (REF ) rewritten, then problems mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ are feasible if and only if M-CLP, M-CLP$^*$ are feasible, by Proposition REF .",
"In this case mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ also posses optimal solutions.",
"Denote by $\\mathbf {u}^{0*}, \\Delta U^*,$ $ \\mathbf {u}^{N*}$ and $\\mathbf {p}^{N*}, \\Delta P^*, \\mathbf {p}^{0*}$ an optimal solution of mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ .",
"If M-CLP and M-CLP$^*$ are feasible then by solving mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ the following bounds holds: $V(\\mbox{M-CLP}^*) - V(\\mbox{M-CLP}) \\le V(\\mbox{dCLP}_2(\\pi ^N)) - V(\\mbox{dCLP}_1(\\pi ^N))\\le \\Upsilon (N) \\epsilon $ where $\\Upsilon (N) = c^{{\\mbox{\\tiny \\bf \\sf T}}}\\sum _{i=1}^N\\Delta U^{*i} - b^{{\\mbox{\\tiny \\bf \\sf T}}}\\sum _{i=1}^N \\Delta P^{*i} > 0$ Proof.",
"The first inequality follows from Proposition REF .",
"To evaluate the second inequality we note that the optimal solutions of mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ are feasible but suboptimal solutions of dCLP$_1(\\pi ^N)$ and dCLP$_2(\\pi ^N)$ .",
"Calculating the objective values of dCLP$_1(\\pi ^N)$ , dCLP$_2(\\pi ^N)$ for the solutions $\\mathbf {u}^{0*}, \\Delta U^*,$ $ \\mathbf {u}^{N*}$ , $\\mathbf {p}^{N*}, \\Delta P^*, \\mathbf {p}^{0*}$ we have: $& V(dCLP_1(\\pi ^N)) \\ge (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^{0*} + \\big ({\\hat{\\gamma }}+{\\hat{c}}_1 \\big )^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta U^* + \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^{N*}, \\nonumber \\\\& V(dCLP_2(\\pi ^N)) \\le (\\beta + b T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^{N*} + \\big ({\\hat{\\beta }}+ {\\hat{b}}_2 \\big )^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta P^* + \\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^{0*}$ On the other hand, because mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ are dual problems: $& V(\\text{mdCLP}(\\pi ^N)) = (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^{0*} + \\big ({\\hat{\\gamma }}+{\\hat{c}}_2 \\big )^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta U^* + \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^{N*} = \\nonumber \\\\& = (\\beta + b T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^{N*} + \\big ({\\hat{\\beta }}+ {\\hat{b}}_1 \\big )^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta P^* + \\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^{0*} = V(\\text{mdCLP}^*(\\pi ^N))$ Combining (REF ) and (REF ), after easy manipulations we get: $V(dCLP_2(\\pi ^N)) - V(dCLP_1(\\pi ^N)) \\le \\big ({\\hat{b}}_2 ^{{\\mbox{\\tiny \\bf \\sf T}}}- {\\hat{b}}_1 ^{{\\mbox{\\tiny \\bf \\sf T}}}\\big ) \\Delta P^* + \\big ({\\hat{c}}_2 ^{{\\mbox{\\tiny \\bf \\sf T}}}- {\\hat{c}}_1 ^{{\\mbox{\\tiny \\bf \\sf T}}}\\big ) \\Delta U^* = \\epsilon \\Upsilon (N) \\qquad $ The sequence of optimal values of the dual problems mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ has finite lower and upper bounds, $V_L,\\,V_U$ .",
"Proof.",
"We consider the single interval partition $\\pi ^1$ , where we have the problem: $&\\max &z = (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}+ (\\gamma + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}U + \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^T \\nonumber \\hspace{43.36243pt} \\\\&\\mbox{s.t.}",
"& A \\mathbf {u}+ x^0 = \\beta \\nonumber \\\\\\mbox{mdCLP}(\\pi ^1) &\\mbox{} & A \\mathbf {u}+ A U + x^t = \\beta + b T \\\\&\\mbox{} & A \\mathbf {u}+ A U + A \\mathbf {u}^T + x^T = \\beta + b T \\nonumber \\\\&& \\quad \\mathbf {u},\\, \\mathbf {u}^T, U,\\,x^0, x^t, x^T \\ge 0.",
"\\nonumber $ An optimal solution to (REF ) can be extended to a feasible solution of mdCLP$(\\pi ^N)$ as follows: $\\displaystyle \\mathbf {u}^0 = \\mathbf {u},\\; \\mathbf {u}^N = \\mathbf {u}^T,\\;\\Delta U=\\left[\\frac{U}{N},\\dots ,\\frac{U}{N}\\right],\\; {\\hat{x}}=\\left[ \\left(1 - 2 \\epsilon \\right) x^0 + 2\\epsilon x^t,\\dots , 2 \\epsilon x^0 + \\left(1 - 2 \\epsilon \\right) x^t,\\, x^t \\right],\\;$ $ x^0=x^0,\\; {\\mathbf {x}}^N = x^T$ .",
"Hence the following inequality holds: $V( \\mbox{mdCLP}(\\pi ^N)) &\\ge & (\\gamma +cT)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}+\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U+{\\hat{c}}_2^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta U+\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^T \\\\& =& (\\gamma +cT)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}+\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U+c^{{\\mbox{\\tiny \\bf \\sf T}}}\\Big (\\frac{T}{2}+\\epsilon \\Big )U+\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^T \\\\&\\ge & (\\gamma +cT)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}+\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U+\\Big ( c^+ \\frac{T}{2} - c^- T \\Big )^{{\\mbox{\\tiny \\bf \\sf T}}}U+\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^T= V_L$ where $c^+_j=\\max (c_j,0),\\,c^-_j=\\max (-c_j,0)$ , and we recall that $\\epsilon \\le \\frac{T}{2}$ .",
"Similarly, by considering the dual, an upper bound is obtained in terms of the solution of the dual test problem: $V_U = (\\beta +b T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}+\\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}P+\\Big (b^+ T - b^- \\frac{T}{2} \\Big )^{{\\mbox{\\tiny \\bf \\sf T}}}P+\\beta ^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {p}^T \\qquad $"
],
[
"Bounding the discrete solutions",
"In this section we assume that M-CLP as well as M-CLP$^*$ satisfy the Slater type condition .",
"Under this assumption we will show that all the optimal solutions of mdCLP$(\\pi ^N)$ and mdCLP$^*(\\pi ^N)$ are uniformly bounded.",
"We consider first the sequence of primal problems $\\lbrace \\text{mdCLP}(\\pi ^N)\\rbrace _{N=1}^\\infty $ .",
"We use the following notations: $&& \\lbrace \\mathbf {u}^{0\\,*(N)}, \\Delta U^{*(N)}, \\mathbf {u}^{N\\,*(N)}\\rbrace _{N=1}^\\infty , \\mbox{ are the optimal solutions}, \\\\&& u^{*(N)}(t_i)=\\Delta U^{i\\,*(N)}/2 \\epsilon , \\qquad i=1,\\ldots ,N, \\\\&& u^{*(N)}(t) \\text{ is the piecewise constant extension of the } u^{*(N)}(t_1),\\ldots ,u^{*(N)}(t_N)\\\\&& U^{*(N)}(t) = \\mathbf {u}^{0\\,*(N)}+\\int _0^t u^{*(N)}(s) ds, \\; t\\in [0,T), \\quad U^{*(N)}(T) = U^{*(N)}(T-) + \\mathbf {u}^{N\\,*(N)}.$ If M-CLP$^*$ is strictly feasible then all $J$ elements of $U^{*(N)}(T)$ have a uniform finite upper bound.",
"Proof.",
"Take any $j=1,\\ldots ,J$ , we will show that $U_j^{*(N)}(T)$ is bounded by a constant $\\Psi _j$ for all $N$ .",
"Recall that $U^{*(N)}(t)$ are non decreasing, so this bound will hold for all $U_j^{*(N)}(t),\\,t\\in [0,T]$ .",
"We choose $N_0$ large enough and corresponding $\\epsilon $ small enough so that: $\\epsilon \\le \\frac{\\alpha _1}{2} \\Rightarrow \\frac{T}{2N_0} \\le \\frac{\\alpha _1}{2} \\Rightarrow N_0 \\ge \\frac{T}{\\alpha _1}$ where $\\alpha _1$ is a small constant, to be determined later.",
"We will find a uniform bound for $U^{*(N)}(T), N\\ge N_0$ .",
"We use the following notation: $&&\\delta =\\left[ \\begin{array}{l}\\delta _1 \\\\ \\vdots \\\\\\delta _J \\end{array} \\right] \\mbox{where }\\delta _j = \\left\\lbrace \\begin{array}{ll} \\frac{\\alpha _1}{2} & \\mbox{if } c_j > 0 \\\\0 & \\mbox{if } c_j \\le 0 \\end{array} \\right., \\\\&& \\hat{c} =\\left[ \\begin{array}{l} c \\\\ \\vdots \\\\ c \\end{array} \\right] \\mbox{and }\\hat{\\delta } =\\left[ \\begin{array}{l}\\delta \\\\ \\vdots \\\\\\delta \\end{array} \\right]\\mbox{are the $N$-fold vectors of $c$'s and $\\delta $'s}$ Consider the following discrete linear optimization problem, for a discrete error bound: $&\\Psi ^{(1)}_{j}(N) = \\max & \\frac{\\alpha _1}{2} \\mathbf {u}_j^0 \\; + \\;\\frac{\\alpha _1}{2} \\sum _{i=1}^N \\Delta U_{j}^i \\; + \\; \\frac{\\alpha _1}{2} \\mathbf {u}_j^N \\nonumber \\\\\\mbox{dEBLP$(\\pi ^N)$} & \\mbox{s.t.}",
"&(\\gamma + \\delta + c T)^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^0 + ({\\hat{\\gamma }}+ \\hat{\\delta } + {\\hat{c}}_1 )^{{\\mbox{\\tiny \\bf \\sf T}}}\\Delta U + (\\gamma + \\delta )^{{\\mbox{\\tiny \\bf \\sf T}}}\\mathbf {u}^N \\ge V_L \\hspace{28.90755pt} \\\\& & \\text{Constraints of mdCLP}(\\pi ^N) \\nonumber $ One can see that ${\\hat{c}}_1 + \\hat{\\delta } \\ge {\\hat{c}}_1 + \\hat{c} \\epsilon = {\\hat{c}}_2$ and hence, by Proposition REF the first constraint of dEBLP$(\\pi ^N)$ holds for the solution $\\lbrace \\mathbf {u}^{0\\,*(N)}, \\Delta U^{*(N)}, \\mathbf {u}^{N\\,*(N)}\\rbrace $ .",
"Hence the optimal solution of mdCLP$(\\pi ^N)$ is feasible for dEBLP$(\\pi ^N)$ .",
"In particular, it follows that $\\Psi ^{(1)}_{j}(N) \\ge U_j^{*(N)}(T)$ The problem dEBLP$(\\pi ^N)$ is a discretization of the following continuous linear programming problem: $&\\Psi ^{(2)}_{j} = \\max & \\frac{\\alpha _1}{2} \\int _0^T d U_{j}(t)dt \\nonumber \\\\\\mbox{EBCLP} & \\mbox{s.t.}",
"& \\int _0^T \\left(\\gamma +\\delta + (T-t)c \\right)^{{\\mbox{\\tiny \\bf \\sf T}}}d U(t) \\ge V_L \\hspace{108.405pt} \\\\& & \\text{Constraints of M-CLP} \\nonumber $ The continuous linear program EBCLP is not formulated exactly as an M-CLP problem, the difference being that the first constraint has linearly varying coefficients rather than constant coefficients.",
"Nevertheless, one can show by similar arguments that propositions REF , REF still hold, and so for every $N$ , $U^{*(N)}(t)$ is a feasible solution of EBCLP.",
"We now have that $\\Psi ^{(2)}_j \\ge \\Psi ^{(1)}_j $ .",
"EBCLP is obviously feasible.",
"We now need to show that it is bounded.",
"We formulate the following dual problem to EBCLP: $& \\Psi ^{(3)}_{j} = \\min &\\displaystyle \\int _0^T \\left(\\beta + (T - t) b \\right)^{{\\mbox{\\tiny \\bf \\sf T}}}dP(t) - V_L P^O \\nonumber \\\\\\mbox{EBCLP$^*$} &\\mbox{s.t.}",
"&A^{{\\mbox{\\tiny \\bf \\sf T}}}P(t) \\ge (\\gamma +\\delta + c t) P^O + \\frac{\\alpha _1}{2} \\mathbf {e}^j, \\quad 0\\le t \\le T, \\\\&& P^O \\ge 0, P(t)\\ge 0, \\mbox{ non-decreasing and right continuous on }[0,T].\\nonumber $ where ${\\mathbf {e}}^{j}$ is the $j$ th unit vector.",
"It is straightforward to check that weak duality holds between problems EBCLP and EBCLP$^*$ .",
"Hence, if EBCLP$^*$ is feasible, we have $\\Psi ^{(3)}_j \\ge \\Psi ^{(2)}_j $ .",
"It remains to show that EBCLP$^*$ is feasible.",
"We now use the assumption that M-CLP$^*$ is strictly feasible.",
"Hence there exists a vector of functions $\\tilde{P}(t),\\,t\\in [0,T]$ that satisfy: $&& A^{{\\mbox{\\tiny \\bf \\sf T}}}\\tilde{P}(t) \\ge \\gamma + c t \\nonumber \\\\&& A^{{\\mbox{\\tiny \\bf \\sf T}}}\\tilde{P}(t) \\ge \\gamma + c t + \\alpha _1, \\\\&& \\quad \\tilde{P}(t) \\ge 0, \\mbox{ non-decreasing and right continuous on }[0,T].\\nonumber $ for some small enough value $\\alpha _1$ .",
"This gives us our choice for the value of $\\alpha _1$ .",
"It is now easy to check that $P^O=1$ and $\\tilde{P}(t),\\,t\\in [0,T]$ is a feasible solution of EBCLP$^*$ , indeed, for $P^O=1$ : $A^{{\\mbox{\\tiny \\bf \\sf T}}}\\tilde{P}(t) \\ge \\gamma + c t + \\alpha _1 \\ge (\\gamma +\\delta + c t) P^O + \\frac{\\alpha _1}{2} \\mathbf {e}^j, \\quad 0\\le t \\le T.$ Let $\\Psi _j^{(4)}$ be the value of the objective of EBCLP$^*$ for this solution.",
"We have: $\\Psi ^{(4)}_j \\ge \\Psi ^{(3)}_j \\ge \\Psi ^{(2)}_j \\ge \\Psi ^{(1)}_j \\ge U_j^{*(N)}(T),\\,N \\ge N_0$ .",
"Finally: $\\Psi _j = \\max \\lbrace \\Psi ^{(4)}_j, U_j^{* (N)}(T),\\,N=1,\\ldots ,N_0\\rbrace \\ge U_j^{* (N)}(T) \\mbox{ for all $N$}.",
"\\qquad $ Let $P^{*(N)}(t)$ be defined for the optimal solutions of mdCLP$^*(\\pi ^N)$ , similar to $U^{*(N)}$ .",
"A similar proof shows that if M-CLP is strictly feasible, we can construct for any $k=1,\\ldots ,K$ a bound: $\\Phi _k \\ge P_k^{*(N)}(T)$ ."
],
[
"Strong duality",
"If M-CLP and M-CLP$^*$ are strictly feasible, then both have optimal solutions, and there is no duality gap.",
"We show first that there is no duality gap.",
"In Proposition REF we have seen that $V(\\mbox{M-CLP}^*) - V(\\mbox{M-CLP}) \\le V(\\mbox{dCLP}_2(\\pi ^N)) - V(\\mbox{dCLP}_1(\\pi ^N))\\le \\Upsilon (N) \\epsilon $ where $\\Upsilon (N) = c^{{\\mbox{\\tiny \\bf \\sf T}}}U^{*(N)}(T) - b^{{\\mbox{\\tiny \\bf \\sf T}}}P^{*(N)}(T)$ In Proposition REF we saw that all components of $U^{*(N)}(T),\\,P^{*(N)}(T)$ are uniformly bounded, by quantities $\\Psi _j,\\,\\Phi _k$ .",
"We therefore have a uniform bound $\\Upsilon $ : $\\Upsilon (N) \\le \\Upsilon = \\sum _{j=1}^J c_j^+ \\Psi _j + \\sum _{k=1}^K b_k^- \\Phi _k,$ where $c^+_j=\\max (c_j,0),\\,b^-_k=\\max (-b_k,0)$ .",
"Hence, $0\\le V(\\mbox{M-CLP}^*) - V(\\mbox{M-CLP}) \\le \\epsilon \\Upsilon $ and letting $N\\rightarrow \\infty $ , so that $\\epsilon \\rightarrow 0$ , we get $V(\\mbox{M-CLP}^*) = V(\\mbox{M-CLP}) $ .",
"We next show that optimal solutions exist.",
"We saw that $U^{*(N)}(t)$ are feasible solutions for M-CLP for all $N$ .",
"$U^{*(N)}(t)$ are vectors of non-negative non-decreasing functions, and by Proposition REF they are all uniformly bounded.",
"By Helly's selection principle (Theorem 5, p. 372 in [12]), it is then possible to find a subsequence $N_m$ such that $U_j^{*(N_m)}(t)$ converge pointwise for every $t$ to a non-negative non-decreasing right continuous function $U_j(t),\\,t\\in [0,T]$ , for all $j=1,\\ldots ,J$ .",
"It is immediate to see that $U(t)$ is a feasible solution for M-CLP.",
"By Helly's convergence theorem (Theorem 4, p. 370 in [12]) by the continuity of $\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t)$ $\\lim _{N_m \\rightarrow \\infty } \\int _0^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}dU^{(N_m)}(t) =\\int _0^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}dU(t),$ but this limit equals $ V(\\text{M-CLP})$ , hence $U(t)$ is an optimal solution of M-CLP.",
"Similarly the dual problem M-CLP$^*$ has an optimal solution."
],
[
"Form of optimal solution",
"�� We now consider problems M-CLP that have an optimal primal solution $U_O(t)$ .",
"In particular, this is true if M-CLP are primal and dual strictly feasible (see Theorem REF ).",
"In this section we investigate properties of the optimal solution.",
"$c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t)$ is continuous on $(0, T)$ .",
"Assume the contrary.",
"Then, because $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t)$ is of bounded variations, it has only jump discontinuities, with left and right limits.",
"Consider a `jump' point $t_c$ with $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}^\\uparrow = c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_c +) - c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_c -) \\ne 0$ .",
"Assume first that $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}^\\uparrow > 0$ .",
"Let $t_a < t_c$ be a point such that $| c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t_c-) - c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)| < \\frac{1}{4}c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}^\\uparrow $ for all $t \\in [t_a, t_c)$ .",
"Such a point exists because $U_O$ has left and right limits at $t_c$ .",
"Consider the following solution of M-CLP: $\\tilde{U}(t)=\\left\\lbrace \\begin{array}{ll}\\displaystyle U_{O}(t_a) + \\frac{t-t_a}{t_c-t_a} \\left(U_{O}(t_c+) - U_{O}(t_a) \\right), & t \\in [ t_a, t_c ], \\\\U_{O}(t), & t \\notin [ t_a, t_c].",
"\\\\\\end{array} \\right.$ It is clear that $\\tilde{U}(t)$ is feasible.",
"Comparing the objective values for $U_{O}(t)$ and $\\tilde{U}(t)$ we obtain: $&& \\int _0^T (\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t))d{\\tilde{U}}(t) - \\int _0^T (\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t))dU_O(t) \\\\&& = \\int _{t_a}^{t_c+} c^{{\\mbox{\\tiny \\bf \\sf T}}}t dU_O(t) - \\int _{t_a}^{t_c+} c^{{\\mbox{\\tiny \\bf \\sf T}}}t d{\\tilde{U}}(t) \\\\&&= \\frac{t_c-t_a}{2} \\Big (c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O^\\uparrow + c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t_c-)+ c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t_a)\\Big )- \\int _{t_a}^{t_c-} c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t) dt \\\\&&\\ge \\frac{t_c-t_a}{2} \\Big (c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O^\\uparrow - \\big (2 \\sup _{t\\in [t_a,t_c)}c^{{\\mbox{\\tiny \\bf \\sf T}}}{U}_O(t)- c^{{\\mbox{\\tiny \\bf \\sf T}}}{U}_O(t_a) - c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t_c-)\\big )\\Big ) > 0$ where in the second equality we replace Lebesgue-Stieltjes integral by Riemann-Stieltjes integral and integrate by parts.",
"This contradicts the optimality of $U_{O}(t)$ .",
"A similar contradiction is obtained if $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}^\\uparrow < 0$ , considering a point $t_a > t_c$ .",
"We conclude that $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t)$ has no jumps, and hence is continuous on $(0,T)$ .",
"$c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t)$ is concave on $(0, T)$ .",
"Assume the contrary.",
"Then, since by Proposition $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t)$ is continuous, there exists an interval $(t_1, t_2)$ such that: $ c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t) < c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_1) + \\frac{t-t_1}{t_2-t_1} \\left( c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_2) - c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_1) \\right), \\quad t \\in (t_1, t_2)$ Consider the following solution of M-CLP: $U^*(t)=\\left\\lbrace \\begin{array}{ll}\\displaystyle U_{O}(t_1) + \\frac{t-t_1}{t_2-t_1} \\left(U_{O}(t_2) - U_{O}(t_1) \\right), & t \\in ( t_1, t_2 ), \\\\U_{O}(t), & t \\notin ( t_1, t_2).",
"\\\\\\end{array} \\right.$ It is clear that $U^*(t)$ is feasible.",
"We note also that from our assumption it follows that $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t) < c^{{\\mbox{\\tiny \\bf \\sf T}}}U^*(t)$ on $(t_1, t_2)$ .",
"Comparing objective values for $U_{O}(t)$ and $U^*(t)$ we obtain, similar to the proof of Proposition : $&& \\int _0^T (\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t))dU^*(t) - \\int _0^T (\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t))dU_O(t) \\\\&& = \\int _{t_1}^{t_2} c^{{\\mbox{\\tiny \\bf \\sf T}}}t dU_O(t) - \\int _{t_1}^{t_2} c^{{\\mbox{\\tiny \\bf \\sf T}}}t dU^*(t) \\\\&& = \\int _{t_1}^{t_2} c^{{\\mbox{\\tiny \\bf \\sf T}}}U^*(t) dt - \\int _{t_1}^{t_2} c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t) d t> 0.$ This contradicts the optimality of $U_{O}(t)$ .",
"Hence $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t)$ is concave.",
"By the Lebesgue decomposition theorem any feasible solution of M-CLP can be represented as $U(t)= U_a(t) + U_s(t)$ , where $U_a(t)$ is an absolutely continuous function and $U_s(t)$ is a singular function, including a discrete singular (`jump') part and a continuous singular part.",
"Consider an optimal solution $U_O(t)$ and its Lebesgue decomposition $U_O(t)= U_a(t) + U_s(t)$ , and let $u(t)= \\frac{dU_a(t)}{dt}$ .",
"Then the following holds: (i) $\\frac{d}{dt} c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)= c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t)$ (ii) $c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t)$ is a non-increasing function.",
"(iii) $\\int _{0-}^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}dU_O(t) = \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(T) + c^{{\\mbox{\\tiny \\bf \\sf T}}}T U_O(T-) - \\int _0^T c^{{\\mbox{\\tiny \\bf \\sf T}}}t u(t) dt$ Proof.",
"(i) By Proposition $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ is concave and hence it is absolutely continuous on the interval $(0,T)$ .",
"Therefore, by the uniqueness of the Lebesgue decomposition, $c^{{\\mbox{\\tiny \\bf \\sf T}}}(U_O(t)-U_O(0)) = c^{{\\mbox{\\tiny \\bf \\sf T}}}U_a(t)$ on this interval.",
"(ii) That $c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t)$ is non-increasing follows from the concavity of $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ .",
"(iii) Because $(\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}$ is continuous the Lebesgue-Stieltjes integral above can be replaced by the Riemann-Stieltjes integral.",
"$&& \\int _{0-}^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}dU_O(t) \\\\&& = \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(T) + \\int _{0-}^T (T-t) d c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t) \\\\&& = \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(T) + c^{{\\mbox{\\tiny \\bf \\sf T}}}T U_O(0) + \\int _{0+}^{T-} (T-t) d c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t) \\\\&& = \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(T) + c^{{\\mbox{\\tiny \\bf \\sf T}}}T U_O(0) + \\int _0^T (T-t) c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) dt \\\\&& = \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(T) + c^{{\\mbox{\\tiny \\bf \\sf T}}}T U_O(T-) - \\int _0^T c^{{\\mbox{\\tiny \\bf \\sf T}}}t u(t) dt \\qquad $ For part (iii) of the next theorem we need the following non-degeneracy assumption: Assumption The vector $c$ is in general position to the matrix $\\left[ A^{{\\mbox{\\tiny \\bf \\sf T}}}\\; I \\right]$ (it is not a linear combination of any less than $J$ columns).",
"Assume that M-CLP/M-CLP$^*$ have optimal solutions $U_O(t), P_O(t)$ with no duality gap, then: (i) $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ is piecewise linear on $(0, T)$ with a finite number of breakpoints.",
"(ii) There exists an optimal solution $U^*(t)$ that is continuous and piecewise linear on $(0, T)$ .",
"(iii) Under the non-degeneracy assumption REF , every optimal solution is of this form, and furthermore, $U_O(t)-U_O(0+)$ is unique over $(0,T)$ .",
"(i) By the Lebesgue differentiation theorem $U_O(t), P_O(t)$ can be differentiated at least almost everywhere.",
"Let $S$ be the set where $U_O(t)$ is not differentiable and $S^*$ be the set where $P_O(T-t)$ is not differentiable.",
"Let $E_0 = [0,T] \\setminus S \\cup S^*$ .",
"Then the complementary slackness condition (REF ) can be rewritten as: $\\int _{E_0} q(T-t) ^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) dt = \\int _{S \\cup S^*} q(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}dU_s(t) = \\int _{E_0} x(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}p(t) dt = \\int _{S \\cup S^*} q(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}dP_s(t) = 0$ where $u(t) = \\frac{dU_O(t)}{dt}, p(t) = \\frac{dP_O(t)}{dt}$ are the densities of $U_O(t), P_O(t)$ on $E$ and $x(t)=\\beta + bt - A U_O(t), q(t) = A^{{\\mbox{\\tiny \\bf \\sf T}}}P_O(t) - \\gamma - ct$ are slack functions.",
"Hence, we must have for every point of $E_0$ apart from another set of measure zero $S_1$ , that $q(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) = x(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}p(t) = 0$ .",
"Let $E = E_0 \\setminus S_1$ .",
"At the same time, differentiating the constraints of M-CLP/M-CLP$^*$ everywhere on the set $E$ we obtain: $Au(t) + \\dot{x}(t) = b \\qquad A^{{\\mbox{\\tiny \\bf \\sf T}}}p(t) - \\dot{q}(t) = c$ where $\\dot{x}(t), \\dot{q}(t)$ are the slopes of the corresponding slacks.",
"Note that on $E$ , $x(t) = 0 \\Rightarrow \\dot{x}(t) = 0$ and $q(t) = 0 \\Rightarrow \\dot{q}(t) = 0$ , by non-negativity.",
"Hence, for every $t \\in E$ the following holds: $\\dot{q}(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) = \\dot{x}(T-t)^{{\\mbox{\\tiny \\bf \\sf T}}}p(t) = 0$ Recall also that $U_O,P_O$ and hence also $U_a,P_a$ are non-decreasing, so: $u(t) \\ge 0, \\qquad p(t)\\ge 0.$ Consider now any point $t\\in E$ , and the values of $u(t),x(t),p(T-t),q(T-t)$ .",
"Let ${\\cal {J}}(t)$ , ${\\cal {K}}(t)$ be the indices of the non-zero components of $u(t)$ and of $p(T-t)$ respectively.",
"One can see that (REF ), (REF ), (REF ) imply that $u=u(t),\\dot{x}=\\dot{x}(t),p=p(T-t),\\dot{q}=\\dot{q}(T-t)$ are optimal solutions of the following pair of linear programming problems: $\\begin{array}{c}\\begin{array}{rcl}& \\max & c^{{\\mbox{\\tiny \\bf \\sf T}}}u \\\\&\\mbox{s.t.",
"}& A u + I \\dot{x}= b \\\\\\mbox{Rates-LP$(t)$} && u_j \\in \\mathbb {Z}\\text{ for } j \\notin {\\cal {J}}(t)\\; u_j \\in \\mathbb {P}\\text{ for } j \\in {\\cal {J}}(t) \\\\&& \\dot{x}_k \\in \\mathbb {U}\\text{ for } k \\notin {\\cal {K}}(t) \\; \\dot{x}_k \\in \\mathbb {P}\\text{ for } k \\in {\\cal {K}}(t)\\end{array} \\\\\\\\\\begin{array}{rcl}& \\min & b^{{\\mbox{\\tiny \\bf \\sf T}}}p \\\\&\\mbox{s.t.",
"}& A^{{\\mbox{\\tiny \\bf \\sf T}}}p - I \\dot{q}= c \\\\\\mbox{Rates-LP$^*(t)$} && p_k \\in \\mathbb {Z}\\text{ for } k \\notin {\\cal {K}}(t)\\; p_k \\in \\mathbb {P}\\text{ for } k \\in {\\cal {K}}(t) \\\\&& \\dot{q}_j \\in \\mathbb {U}\\text{ for } j \\notin {\\cal {J}}(t) \\; \\dot{q}_j \\in \\mathbb {P}\\text{ for } j \\in {\\cal {J}}(t)\\end{array}\\end{array}$ where by $\\mathbb {Z}, \\mathbb {P}, \\mathbb {U}$ we denote the following sign restrictions: $\\mathbb {Z}= \\lbrace 0\\rbrace $ is zero, $\\mathbb {P}= \\mathbb {R}_+$ is non-negative and $\\mathbb {U}= \\mathbb {R}$ is unrestricted.",
"Let $M$ be the finite number of subsets of indices ${\\cal {J}}(t),{\\cal {K}}(t)$ for which the dual pair of linear programs (REF ) is feasible.",
"Then it follows that the values of $c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t)$ for all $t\\in E$ must be the objective values of an optimal solution of (REF ), for one of these subsets.",
"Since by Proposition (ii) $c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t)$ is non-decreasing there must exist a partition $0=t_0 < t_1 < \\cdots < t_N=T,\\;N \\le M$ such that $c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t)$ is constant over each interval $(t_{n-1},t_n)\\cap E$ .",
"Recall that by Proposition $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ is absolutely continuous on $(0,T)$ , and hence $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t) = c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(0) + \\int _0^t c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) dt$ .",
"It follows that $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ is continuous piecewise linear on $(0,T)$ .",
"(ii) Consider the following solution of M-CLP: $U^*(t)=\\left\\lbrace \\begin{array}{ll}\\displaystyle U_{O}(t_n) + \\frac{t-t_n}{t_{n+1}-t_n} \\left(U_{O}(t_{n+1}) - U_{O}(t_n) \\right), & \\begin{array}{l}t \\in ( t_n, t_{n+1}], \\; n=0,\\dots , N-2 \\\\ \\text{and }t \\in (t_{N-1}, t_N),\\end{array} \\\\U_{O}(t), & \\; t = 0, T. \\\\\\end{array} \\right.$ where $0=t_0 < t_1 < \\dots < t_N=T$ is the time partition defined in the proof of (i).",
"Note, that $U^*(t)$ is piecewise linear and absolutely continuous on $(0,T)$ .",
"One can see that $U^*(t)$ is also a feasible solution for M-CLP.",
"Let $u^*(t) = \\frac{dU^*(t)}{dt}$ .",
"Similar to Proposition (iii) and based on its proof, we could rewrite the objective value obtained with $U^*(t)$ as follows: $\\int _{0-}^T (\\gamma + (T-t)c)^{{\\mbox{\\tiny \\bf \\sf T}}}dU^*(t) = \\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(T) + c^{{\\mbox{\\tiny \\bf \\sf T}}}T U_O(T-) - \\int _0^T c^{{\\mbox{\\tiny \\bf \\sf T}}}t u^*(t) dt$ Recall also, that by Proposition $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ is concave on $(0,T)$ and hence absolutely continuous on $(0,T)$ .",
"Then, comparing the objective values for $U_{O}(t)$ and $U^*(t)$ we obtain: $&& \\int _{0-}^T (\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t))dU_O(t) - \\int _{0-}^T (\\gamma ^{{\\mbox{\\tiny \\bf \\sf T}}}+ c^{{\\mbox{\\tiny \\bf \\sf T}}}(T-t))dU^*(t) \\\\&& = \\int _0^T c^{{\\mbox{\\tiny \\bf \\sf T}}}t u^*(t) dt - \\int _0^T c^{{\\mbox{\\tiny \\bf \\sf T}}}t u(t) dt \\\\&&= \\sum _{n=0}^{N-1} \\int _{t_n}^{t_{n+1}} t \\left(\\frac{d}{dt} \\left(c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t_n) + \\frac{t-t_n}{t_{n+1}-t_n} \\left(c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_{n+1}) - c^{{\\mbox{\\tiny \\bf \\sf T}}}U_{O}(t_n)\\right)\\right) - c^{{\\mbox{\\tiny \\bf \\sf T}}}u(t) \\right) dt = 0$ where the first equality follows from Proposition (iii) and (REF ), and the second follows from (i) of this Theorem.",
"Hence, $U^*(t)$ is an optimal solution of M-CLP.",
"(iii) Let $U_O(t)$ be any optimal solution of M-CLP.",
"As shown in (i), there is a time partition $0=t_0 < t_1 < \\dots < t_N=T$ so that $c^{{\\mbox{\\tiny \\bf \\sf T}}}U_O(t)$ is continuous piecewise linear, with constant slope in each interval, where the slopes in successive intervals are strictly decreasing.",
"Let $I_n=(t_{n-1},t_n],\\,n=1,\\ldots ,N-1,\\;I_N=(t_{N-1},t_N)$ .",
"As shown in (ii) we can construct a dual optimal solution $P^*(t)$ which is continuous piecewise linear on $(0,T)$ , with breakpoints $0=t_0 < t_1 < \\dots < t_N=T$ .",
"For this dual solution we have that $p(t) = \\frac{d P^*(t)}{dt}$ is constant on each interval $I_n$ , let $p_n=p(t),\\,t\\in I_n$ denote this vector value.",
"Then as shown in (i), $p_n$ is a solution of the dual Rates-LP$^*$ (REF ).",
"Consider now $u(t)=\\frac{dU_a(t)}{dt}$ which is defined almost everywhere on $(0,T)$ .",
"As shown in (i), $u(t)$ is an optimal solution of the primal Rates-LP (REF ), and $u(t)$ is complementary slack to $p(t)$ almost everywhere.",
"By the non-degeneracy assumption REF , $p_n$ is non-degenerate.",
"Hence, $u(t)$ is uniquely determined almost everywhere on $I_n$ , as the unique solution which is complementary slack to $p_n$ .",
"Denote this solution by $u_n,\\,n=1,\\ldots ,N$ .",
"This uniquely determines $U_a(t),\\,t\\in (0,T)$ , the absolutely continuous part of $U_O(t)$ , as the continuous piecewise linear vector of functions with slopes $u_n$ in each interval.",
"It remains to show that $U_O$ is absolutely continuous in $(0,T)$ , i.e that $U_s(T-)-U_s(0+)=0$ .",
"Assume to the contrary that in some interval $(t_{m-1}, t_{m}]$ we have $U_s(t_m)-U_s(t_{m-1})>0$ (or if $m=N$ , $U_s(T-)-U_s(t_{N-1})>0$ ).",
"Define $U^*_m(t)=\\left\\lbrace \\begin{array}{ll}\\displaystyle U^*(t) & t \\in I_m\\\\U_{O}(t), & t \\notin I_m,\\end{array} \\right.",
"\\qquad u^*_m(t)=\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{dU^*(t)}{dt} & t \\in I_m\\\\u(t), & t \\notin I_m.\\end{array} \\right.$ where $U^*(t)$ on $I_m$ is the linear interpolation as defined in proof of (ii), and $u^*_m=u^*_m(t)$ for $t\\in I_m$ is the constant slope $\\frac{U_O(t_m)-U_O(t_{m-1})}{t_m-t_{m-1}}$ on $I_m$ .",
"Similar to (ii), it follows that $U^*_m(t)$ is also an optimal solution of M-CLP.",
"Furthermore, since the solutions are identical on $t\\notin I_m$ , we must by (i) have that $c^{{\\mbox{\\tiny \\bf \\sf T}}}u_m = c^{{\\mbox{\\tiny \\bf \\sf T}}}u^*_m$ .",
"We now have, on the one hand, that: $ u^*_m = \\frac{U_{O}(t_m) - U_{O}(t_{m-1})}{t_m-t_{m-1}} = u_m + \\frac{1}{t_m-t_{m-1}} \\left(U_{s}(t_m) - U_{s}(t_{m-1})\\right) \\ne u_m$ On the other hand, as we saw before, $u^*_m(t)$ must be complementary slack to $p(t)$ almost everywhere, and hence, $u^*_m=u_m$ .",
"This contradiction proves that $U_s(T-)-U_s(0+)=0$ , and shows that $U_O(t)$ is absolutely continuous on $(0,T)$ .",
"Furthermore, $U_O(t)$ for $t\\in (0,T)$ is continuous piecewise linear, and $U_O(t)-U_O(0+)$ is uniquely determined by the partition $0=t_0 < t_1 < \\dots < t_N=T$ and the slope vectors $u_n$ .",
"This completes the proof or (iii)."
],
[
"Completion of the proof of Theorem ",
"(iii) We first show that for objective values $V$ holds $V(SCLP) \\le V(\\mbox{M-CLP})$ .",
"Consider following CLP problem: $&\\min & \\int _0^T (\\alpha + (T-t)a)^{{\\mbox{\\tiny \\bf \\sf T}}}d P(t) + \\int _0^T b^{{\\mbox{\\tiny \\bf \\sf T}}}q(t) \\,dt \\nonumber \\hspace{65.04256pt} \\\\\\mbox{DCLP} \\quad &\\mbox{s.t.}",
"& G^{{\\mbox{\\tiny \\bf \\sf T}}}\\, P(t) + H^{{\\mbox{\\tiny \\bf \\sf T}}}q(t) \\ge \\gamma + c t \\\\&& \\quad \\; F^{{\\mbox{\\tiny \\bf \\sf T}}}P(t) + P_s(t) \\ge d t \\nonumber \\\\&& \\quad \\; - F^{{\\mbox{\\tiny \\bf \\sf T}}}P(t) - P_s(t) \\ge - d t \\nonumber \\\\&& \\quad q(t)\\ge 0, \\quad P(t), P_s(t) \\mbox{ non-decreasing and right continuous on [0,T].}",
"\\nonumber $ which is a generalization of Pullans' dual for the case when $a(t),c(t)$ are affine, and $b(t)$ is constant.",
"One can see that weak duality holds between SCLP (REF ) and DCLP.",
"One can also see that weak duality holds between the M-CLP extension and DCLP.",
"But under the Slater type condition, the M-CLP/M-CLP$^*$ extensions possess primal and dual optimal solutions ${\\tilde{U}}(t), {\\tilde{P}}(t)$ with no duality gap (see Theorem REF ).",
"Now, setting $P(t) = {\\tilde{P}}_*(t),\\,P_s(t) = {\\tilde{P}}_s(t),\\, q(t) = {\\tilde{P}}^+(t) - {\\tilde{P}}^-(t)$ we obtain a feasible solution of DCLP with the same objective value.",
"This solution is optimal for DCLP by weak duality between M-CLP extension and DCLP.",
"Then, by weak duality between SCLP and DCLP $V(SCLP) \\le V(DCLP) = V(\\mbox{M-CLP})$ .",
"Now, consider $u^*(t), x^*(t)$ be a feasible solution of SCLP, where $x^*(t)$ is of bounded variation.",
"Because $G \\int _0^t u^*(s)ds$ and right hand side of SCLP are both absolutely continuous such solution could be easily found.",
"Moreover, this solution could be translated to a solution of M-CLP extension as shown in the proof of (i).",
"Consider also an additional constraint $u(t) \\le W, 0 \\le t \\le T$ , where $W \\ge \\max _{j, 0 < t < T} u^*_j(t)$ , which preserves SCLP feasibility.",
"We denote SCLP with this additional constraint as SCLP$(W)$ .",
"It is clear that $u^*(t), x^*(t)$ still be feasible for SCLP$(W)$ .",
"Let M-CLP$(W)$ be an extension of the SCLP$(W)$ .",
"One can see that M-CLP$(W)$ is nothing also as M-CLP extension of the SCLP with following additional constraints: $\\begin{array}{c}U_*(t) + U_s(t) \\le W t \\\\- U_*(t) - U_s(t) \\le - W t\\end{array}$ It is clear that M-CLP$(W)$ is feasible.",
"Moreover, one could choose $W$ big enough to preserve the Slater type condition for the M-CLP$(W)$ .",
"Furthermore, one can see that the dual M-CLP$(W)^*$ is a relaxation of the M-CLP$^*$ , and therefore the Slater type condition still holds for the M-CLP$(W)^*$ .",
"Now, consider ${\\tilde{U}}(t)$ to be an optimal solution of M-CLP$(W)$ (existence of the such solution follow from Theorem REF ).",
"One could see, that constraint (REF ) require that for this solution ${\\tilde{\\mathbf {u}}}^0_* = {\\tilde{\\mathbf {u}}}^N_* = {\\tilde{\\mathbf {u}}}^0_s = {\\tilde{\\mathbf {u}}}^N_s = 0$ , and hence this solution could be translated back to an optimal solution of SCLP$(W)$ by taking: $ u^{**} = \\frac{d {\\tilde{U}}_*}{dt} \\quad x^{**} = U^{+}(t) - U^{-}(t)$ Finally, consider a sequence $W^{(n)}= nW, n=1,\\dots $ and let ${\\tilde{U}}^{(n)}$ be a sequence of optimal solutions of M-CLP$(W^{(n)})$ , and $u^{(n)}, x^{(n)}$ be a sequence of corresponding optimal solutions of SCLP$(W^{(n)})$ .",
"It is clear that feasible region growth in $n$ , and hence sequences of objective values $V(SCLP(W^{(n)})) = V(\\mbox{M-CLP}(W^{(n)}))$ involving by corresponding solutions are increasing.",
"Moreover, ${\\tilde{U}}^{(n)}$ are vectors of non-negative non-decreasing uniformly bounded functions, which are feasible solution of M-CLP.",
"Hence, letting $n \\rightarrow \\infty $ and repeating arguments from the proof of existing optimal solution (second part of the proof of the Theorem REF ) we obtain: $\\lim _{n \\rightarrow \\infty } V(SCLP(W^{(n)})) = \\lim _{n \\rightarrow \\infty } V(\\mbox{M-CLP}(W^{(n)})) = V(\\mbox{M-CLP})$ $ \\square $"
]
] | 1403.0112 |
[
[
"Modeling Website Popularity Competition in the Attention-Activity\n Marketplace"
],
[
"Abstract How does a new startup drive the popularity of competing websites into oblivion like Facebook famously did to MySpace?",
"This question is of great interest to academics, technologists, and financial investors alike.",
"In this work we exploit the singular way in which Facebook wiped out the popularity of MySpace, Hi5, Friendster, and Multiply to guide the design of a new popularity competition model.",
"Our model provides new insights into what Nobel Laureate Herbert A. Simon called the \"marketplace of attention,\" which we recast as the attention-activity marketplace.",
"Our model design is further substantiated by user-level activity of 250,000 MySpace users obtained between 2004 and 2009.",
"The resulting model not only accurately fits the observed Daily Active Users (DAU) of Facebook and its competitors but also predicts their fate four years into the future."
],
[
"Introduction",
"Membership-based websites such as Facebook are a proven success in what the late Nobel Laureate Herbert A. Simon called “the marketplace of attention.” In a 1969 lecture [41] Simon observed that many information systems were designed as if information was scarce, when the problem is just the opposite: “[...] in an information-rich world, the wealth of information means a dearth of something else: a scarcity of whatever it is that information consumes.",
"What information consumes is rather obvious: it consumes the attention of its recipients.” In this context, Facebook, its competitors, and other analogous membership-based websites can be understood as Simon's information processing systems that speak more than they help us listen (digest information) [41].",
"Casting the website popularity competition (e.g.",
"Facebook v.s.",
"MySpace) into Simon's insightful framework, we note that membership-based websites have an extra element: user attention is converted into content through user activity, which in turn consumes the attention of other users, thus creating an attention-activity marketplace.",
"This observation inspires the following set of questions: Can the attention-activity marketplace help explain the death of MySpace, Hi5, Friendster and MultiplySection provides a brief history of the Facebook, MySpace, Hi5, Friendster and Multiply websites.?",
"Was Facebook the likely reason why MySpace, Hi5, Friendster, and Multiply popularity dwindled or did they die of “natural causes”, e.g., “users were bored”?",
"More broadly, is it possible to model the dynamics of user attention and activity as to capture the phenomenon that drives down the popularity of well established websites (e.g.",
"MySpace)?",
"Is it possible to learn the parameters of such model using widely available Daily Active Users (DAU) time series?"
],
[
"Model",
"In this work we take a positive step toward answering the above questions.",
"We use the puzzling way by which Facebook has interfered with the popularity of its competitors on July 2008 to put forth a set of desired model properties.",
"One of the key desired properties is modeling user attention as a scarce resource that must be consumed by websites or other online user activities.",
"With the captured share of attention a website engages a user in content creation (activity), which in turn captures the attention of other users.",
"Users also have other online interests aside from the website and these interests also compete for attention (a competing website or other online activities).",
"The popularity competition ensues when two websites fight for the attention of their concurrent users (the concurrent adopters of both websites).",
"Media, marketing, and word-of-mouth adoptions complement the model as the driving forces behind membership growth.",
"The resulting model is a compartmental reaction-diffusion population-level model that provides DAU popularity forecasts and offers a compelling hypothesis for the popularity competition of membership-based websites such as Facebook, MySpace, Hi5, Friendster, and Multiply.",
"Our model is an attention-activity market generalization of our previous single website model in Ribeiro [37].",
"The resulting code of the algorithm is available onlinehttp://www.cs.cmu.edu/~ribeiro/Ribeiro_F_AttComp.zip.",
"Our modeling framework reveals interesting traits in our data.",
"Websites almost universally have low barriers of adoption, as opposed to adoptions in consumer products such as smartphones and laptop computers that have a steep barrier to adopting multiple products.",
"Signing up for Facebook and the like is free and reasonably effortless.",
"The main resource consumed by these websites is our time.",
"Not coincidentally time is H.A.",
"Simon's choice of attention metric [41].",
"As long as we see value in spending our time at these websites – rather than somewhere else – Facebook, MySpace, Hi5, Friendster, and Multiply can co-exist without interfering with each other.",
"Conversely, a website that suddenly increases its attention consumption may prey on the attention of its competitors, driving them to their (popularity) death; website death can happen as a result of a negative attention-activity feedback loop (less attention$\\rightarrow $ less content$\\rightarrow $ less attention) that is a function of the size of the concurrent adopter population and the amount of attention (time) left to the competitor.",
"This observation helps explain how a Facebook website change may have suddenly (July 2008) interfered with the popularity of its competitors after a long period of non-interfering co-existence (see Figure REF ).",
"The July 2008 Facebook user behavior change is documented in Viswanath et al. [47].",
"Section offers more details about this event.",
"Last, and perhaps almost as importantly, our model showcases the possibility of predicting popularity trends of competing websites using the widely available DAU time series.",
"Using training data that includes a few months-worth of DAU data after the competition for attention starts to show a clear signal in the DAU time series, our model is able to accurately forecast the popularity of two competing websites nearly five years into the future.",
"We also show that our model principles are consistent with detailed user-level activity of 250,000 MySpace users measured between 2004 and 2009.",
"The outline of this work is as follows.",
"Section introduces the websites and datasets used in this study and presents key DAU patterns in the Facebook, MySpace, Hi5, Friendster and Multiply DAU data.",
"These patterns are used to inform a set of desired model properties that are used to guide the development of our model.",
"Section presents an overview of the related work.",
"Section introduces our model.",
"Section fits the model parameters to the data and uses the fitted model to predict the DAU years into the future.",
"Finally, Section presents our conclusions and future work.",
"Figure: (Facebook's Popularity Competition) DAU/AIP of Facebook, MySpace, Hi5, Friendester, and Multiply websites from June 2007 to February 2014.",
"Gray vertical lines show the time of the introduction of the “new Facebook”."
],
[
"Desired Model Properties",
"In what follows we draw key findings from our datasets and the existing literature to inform our model design through a set of required model properties.",
"The next section, Section , contrasts the existing literature through the lenses of these desired model properties."
],
[
"Datasets",
"In this work we use two complementary sources of data.",
"The first datasets are provided by Amazon's Alexa web analytics company totaling 32 years of DAU data.",
"The DAU time series is measured from June 2007 to January 2014 as a fraction of the total Active Internet Population (AIP) of each day.",
"We note in passing that Alexa's DAU/AIP measurements of Facebook and other websites may have a strong U.S. and Canada bias.",
"A good argument for using the DAU/AIP instead of the raw DAU value is that the DAU/AIP ameliorates seasonal effects such as school breaks and holidays.",
"But in order to simplify our notation, throughout this work we use DAU to refer to the quantity DAU/AIP.",
"As standard practice we smooth out the DAU outliers using a moving median with a 31-day DAU interval centered around each day.",
"Our second dataset records the activity of 250,000 MySpace subscribers from 2004 to early 2009, collected by Ribeiro et al. [38].",
"Table: Model properties matrixFigure REF shows the DAU time series of Facebook, MySpace, Hi5, Friendester, and Multiply.",
"It is important to note that Alexa's datasets do not include smartphone traffic.",
"Even without Facebook's smartphone data its usage reaches an impressive 45% of the AIP.",
"According to Facebook's own (unverifiable) records, adding smartphone-only users takes the DAU to 60% of the U.S. and Canada AIP [14], an extra 15% DAU of what is reported by Alexa.",
"Facebook's 60% DAU is reported to have remained stable in the last couple of years [14].",
"We are interested in the first years of the competition between Facebook, MySpace, Hi5, Friendester, and Multiply, a time when smartphone-only usage was likely small.",
"Thus, the DAU omission of smartphone traffic should not affect our analysis.",
"However, as a reference for our predictions, we include a circle in Figure *f:fb to represent the uncertainty that Facebook's smartphone-only DAU adds to our data.",
"In what follows we provide a brief overview of the websites analyzed in this work: myspace.com: MySpace was founded in 2003 and from 2005 until early 2008 MySpace was the most visited social networking website in the world.",
"In June 2006 MySpace surpassed Google as the most visited website in the United States.",
"But by April 2008 Facebook usage overtook MySpace [13].",
"facebook.com: Facebook was founded on February 4, 2004.",
"It was initially limited to students at various other universities but soon it was opened to any individual older than 13.",
"Facebook is the largest online social network in the world today.",
"Recently its IPO raised $16 billion, making it the third largest in U.S. history [13].",
"hi5.com: Founded in 2003, Hi5 is an online social network where users can share photos and play games.",
"Today, social games, virtual goods, and other premium content monetizes the website [13].",
"friendster.com: Friendster launched in 2002 as one of the first social networking sites.",
"The service allowed users to communicate with other members, share online content and media, discover new events, brands, and hobbies.",
"The site, at its peak, reached tens of millions of registered users according to CrunchBase [13].",
"multiply.com: Multiply is a mix between an e-commerce platform and a social networking website, offering sellers a combination of e-commerce and social communications tools.",
"The website ceased operations in May, 2013 [13]."
],
[
"Desired Model Properties",
"In what follows we use our data together with other measurements reported in the literature to suggest key properties that we use as guiding principles of our model of website popularity competition."
],
[
"Attention-Activity Feedback Mechanism",
"Attention is a scarce resource that must be consumed by websites.",
"With the captured share of attention a website engages users into content creation, which in turn further captures the attention of other users.",
"Users also have other interests apart from the website that also compete for their attention.",
"Our previous success in modeling the DAU times series through the attention-seeking interaction between users of successful and unsuccessful membership-based websites in Ribeiro [37] showcases the value of this property in popularity models.",
"And indeed, recent results coming out of Facebook [2] indicate that the activity of our friends on Facebook incites us to login and become active which, in turn, incites our friends to either become active or stay active.",
"The attention-activity feedback mechanism may also come about due to the marginal increase in website utility as it gains more active users, an effect known as network effect or network externality discussed at length in Farrell and Klemperer [15].",
"There are many types of network effect, but the most widely used effect in its purest form can be described in the following path-dependent cumulative return rule (see Arthur [1] for more details): higher DAU $\\rightarrow $ more advertisement revenue $\\rightarrow $ better website features $\\rightarrow $ less inactive users (increased DAU).",
"Extrapolating the above observations to all online social interactions (email, chat, OSN, blog activity) informs the first requirement of our model: The popularity of a website should be modeled by an attention-activity feedback mechanism between all user activities, both inside and outside the website of interest."
],
[
"Concurrent Adoptions",
"A key factor in modeling popularity competition lies in the interactions of concurrent adopters, users that have accounts in both competing websites.",
"The size of concurrent adopter population may be affected by factors that prevent concurrent adoptions, as follows: Once an individual joins website $a$ she will be less interested in joining other competing websites, either due to the network effect (e.g., her friends are all in $a$ ), adoption cost (e.g., her “things” are all in $a$ ), or because her “product needs” are already fulfilled by $a$ (e.g., why join two RSS news aggregators?).",
"Henceforth, we refer to this effect as the inertia effect, a force that (at least initially) opposes concurrent adoptions.",
"The inertia effect is one of the main arguments in favor of network effect [4], [15], [17], [21], [25], [30], [33], [34], [42], [43], [50] and threshold [20], [40] adoption models.",
"However, as it is often the case in social studies, the opposite explanation seems as plausible: website adopters – specially the ones that do not adopt the “leading” website as with Hi5 and Google+ adopters – are more likely to be “technology enthusiasts” than the average Internet user and, thus, also more likely to join multiple websites.",
"The case exists for the existence of concurrent adoptions in the wild, as documented by Goga et al. [19].",
"We refer to this effect as the momentum effect, where a user adopting a website signals a higher-than-average likelihood of her adopting its competitor.",
"In real world scenarios the above opposing forces likely co-exist in a population-level sense through distinct users.",
"A model of popularity competition should include a tunable parameter that covers a large spectrum of net population-level effects of concurrent adoptions, from strong inertia to strong momentum."
],
[
"Non-interfering to Sudden DAU Interference",
"In Figure REF we observe a sudden synchronous drop in the popularity of MySpace, Hi5, Friendster, and Multiply by July 2008.",
"The date coincides with the observation of a significant change in Facebook's user behavior [47].",
"Analyzing the activity of 60,000 Facebook users between September 2006 and January 2009, Viswanath et al.",
"[47] observed only one significant Facebook user activity change starting in July 20$^\\text{th}$ of 2008, which Viswanath et al.",
"points out that coincides with the time that Facebook introduced the beta-test of its “new Facebook” design [47].",
"Indeed, in July 20, 2008 Facebook introduced the “new Facebook Wall” with a radically different content-pushing (news feed) interface, offered only to “selected users” [32].",
"By September 2008 these “selected users” already amounted to 30 million [23].",
"The sudden appearance of this new feature – which could have been disproportionally adopted by “tech enthusiasts” concurrent adopters – only marginally affects Facebook's DAU (the inset in Figure *f:fb shows an insignificant DAU bump) while it shows a noticeable and seemly lasting impact on MySpace, Hi5, Friendster, and Multiply DAU time series.",
"Figure: MySpace growth ×\\times activity.Number of observed new users (bars) and active users (line) per semester from 2004 to 2008.",
"Note the bell shape in the green bars, a characteristic of adoption saturations (see Rogers ).The attention-activity marketplace provides an easy way to assess such sudden changes.",
"Facebook's growth is accompanied by a growing number of MySpace, Hi5, Friendster, and Multiply adopters that become concurrent adopters.",
"Figure REF shows that the growing concurrent adopter base does not interfere with MySpace, Hi5, Friendster, and Multiply DAU time series.",
"This happens because Facebook's attention share does not interfere with concurrent adopter attention to its competitors.",
"After July 2008 concurrent adopters find themselves spending more time on Facebook, time now taken out of the budget of attention of Facebook's competitors.",
"A significant enough reduction in attention from concurrent adopters creates a critical mass that affects content creation on these competitors, reducing the attention and content creation (activity) of other users, which in turn further reduces the attention level of concurrent adopters; if this negative attention-activity feedback crosses a particular threshold, the negative feedback loop drives the website to its death.",
"The above observation prompts the following model property: Below a given attention budget, a website may consume extra attention from its concurrent adopters without interfering with its competitors.",
"Further attention gains come at the expense of its competitors."
],
[
"Disjoint Interfering Adopters",
"Recent reports indicate that today Facebook user base (penetration) reaches 69% of the U.S. Internet population [49].",
"In contrast, MySpace, even without Facebook's competition, would in all likelihood never have reached this success.",
"By mid 2007 – when Facebook's DAU was at a mere 3% – MySpace's DAU was already stable and new adoptions in clear decline showing signs of saturation (for MySpace adoptions see green bars in Figure REF ; for the characteristics of adoption saturation see Rogers [39]).",
"Figure: MySpace hazard rate estimates for new subscribers starting in 2006, 2007 and 2008, respectively.",
"Estimates from their first year of activity.This saturation is unlikely to be due to MySpace's competition for users with Facebook.",
"The 2007 “drop-out” (hazard) rates of MySpace's new adopters were identical to that of 2006 as shown by the Kepler-Meier hazard rate estimates in Figure REF .",
"The use of the Kepler-Meier estimator is needed as our MySpace user activity data is right censored (collection stopped by January 2009).",
"It is only by 2008 that new adopters show different hazard rates for MySpace users signifying that these adopters were in average more committed to MySpace than 2006 and 2007 new adopters.",
"Thus, it is unlikely that even if MySpace was allowed to take its course without any competition it would not have reached Facebook's 69% U.S. penetration.",
"The same probably can be said about Hi5, Friendster, and Multiply although we do not have user-level data for these websites.",
"The reason behind the opposition to adopt a website – whether by principle, lack of features of broad appeal, or other factors – are transparent to the model.",
"It is only important to model that such opposition exists.",
"It is as important, however, to note that a user only interested in one website (say, Facebook) may still indirectly affect the DAU of that website's competitors (say, MySpace and Hi5).",
"To illustrate, consider Facebook users that are “opposed” to joining MySpace.",
"The activity of these users help create content on Facebook, which in turn attracts the attention of concurrent adopters.",
"Thus, the activity of “Facebook-only” users can indirectly affect the activity of “MySpace” users.",
"Hence, a model of popularity competition should include disjoint populations of adopters that, however, affect each other through the attention and activity of concurrent adopters."
],
[
"Related Work",
"Adoption models describe phenomena as diverse individuals deciding to adopt a new technology [5], [15], [16], [18], [25], [30], [36], [37] or individuals adopting a new health habit [10], [9].",
"These models have deeply influenced the study of online social network growth [3], [9], [22], [24], [27], [28], [44].",
"In the literature the work that is closest to ours is that of Prakash et al. [36].",
"Prakash et al.",
"models the popularity of two competing products (e.g., smartphones) using a generalization of the diffusion of innovation model (detailed below).",
"The model in Prakash et al.",
"does not capture users finite attention or their inertia/momentum in concurrent adoptions.",
"Moreover, the model in Prakash et al.",
"does not consider disjoint interfering adopters.",
"Beutel et al.",
"[6] extends the model of Prakash et al.",
"to include interfering adopters.",
"Table REF compares the models in the literature against the desired model properties specified in Section .",
"Our model is a generalization of the our previous single-website model in Ribeiro [37], where the popularity of membership-based websites is modeled using a population-level reaction-diffusion-decay model.",
"Our new model significantly generalizes that model by explicitly modeling user's attention, attention sharing, website competition, and disjoint adoptions in a marketplace where users must share their finite attention.",
"Note that in our generalization we make do without the spontaneous exponential decay required in Ribeiro [37] by modeling the attention grabbing influence of other online activities.",
"Another work closely related to ours is that of Cauwels and Sornette [8] which focuses on describing the evolution of the Facebook DAU.",
"Cauwels and Sornette, however, is incomplete in the sense that its time-series analysis is tailored towards Facebook's success and, thus, cannot capture DAU decays.",
"The work of Liu et al.",
"[31] models the popularity of applications inside online social networks such as Facebook.",
"More generally, regarding product adoptions can be classified as: (a) Network effect adoption models (a.k.a.",
"network externality models) [4], [15], [17], [21], [25], [30], [33], [34], [42], [43], [50], where individual rationality and adoption costs and utilities are modeled in a game-theoretic framework; (b) Threshold adoption models [20], [40], where an individual adopts if enough of his or her friends are adopters; (c) Diffusion of innovation models [5], [16], [18], [36], where adopters influence others to adopt through word-of-mouth; In the absence of fine-grained individual-level data these models provide demand forecasting at the aggregate (population) level; and finally (d) Adoption models from influence and network structure [44], [9], [3], [22], [27], [28], [24], [26], where an individual adoptions depends not only on whether his or her friends adopt but also on how these friends are connected among themselves.",
"A variety of works also consider the relationship between community growth inside an online social network (OSN) websites and their network structure [3], [27], [24].",
"These studies, however, focus on (i) the growth of communities inside the OSN (not the growth of the OSN itself) and (ii) the role of network structure disregarding whether the community is alive (active) or dead (inactive).",
"A more thorough review can be found in Ribeiro [37]."
],
[
"Model",
"Compartmental models of interacting populations have been successfully applied in mathematical biology [35] and social systems [11], [12].",
"Our model considers a large segmented user population that interact through catalytic reactions and media & marketing diffusions.",
"Reaction and diffusion processes find applications in chemistry, physics, and applied mathematics [11], [12], [35], [46], [45].",
"We choose to avoid stochastic models (which would allow us to give confidence intervals to our predictions) because very little is known about the stochastic behavior of the dynamics between inactive and active members of websites and between the latter and non-members.",
"Our model can be described as follows.",
"The user state is a tuple $(W_a,W_b) \\in \\lbrace \\emptyset ,U,A,I\\rbrace ^2$ , where $W_a$ and $W_b$ represent the state of the user with respect to competing websites $a$ and $b$ , respectively.",
"For each website a user can be in one of four states: $\\emptyset $ is a permanent state signifying that the user will never adopt the website; all remaining three states are likely transient: $U$ : the user is willing to join but she is still unaware of the website; $A$ : the user is an active member of the website, and finally $I$ : the user is an inactive member of the website."
],
[
"Catalytic Reactions & Marketing Diffusion",
"We use the following notation to mark interactions between users in distinct populations.",
"Let $S_{(W_a,W_b)}(t) \\in (0,1)$ be the fraction of the active Internet population at state $(W_a,W_b) \\in \\lbrace \\emptyset ,U,A,I\\rbrace ^2$ at time $t$ .",
"We use the notation $S_{(W_a,W_b)} \\xrightarrow{} S_{(W^\\dagger _a,W^\\dagger _b)}$ to denote a population of users in state $(W^\\prime _a,W^\\prime _b)$ acting as catalysts of users in state $(W_a,W_b)$ , inciting them to state $(W^\\dagger _a,W^\\dagger _b)$ in the next $dt$ time step, with probability $\\upsilon dt$ , $\\upsilon \\in \\mathbb {R}^+$ .",
"The above notation directly translates into the differential equations: $\\frac{dS_{(W^\\dagger _a,W^\\dagger _b)}(t) }{dt} &= \\dots + \\upsilon S_{(W^\\prime _a,W^\\prime _b)}(t) S_{(W_a,W_b)}(t) \\,, \\\\\\frac{dS_{(W_a,W_b)}(t) }{dt} &= \\dots - \\upsilon S_{(W^\\prime _a,W^\\prime _b)}(t) S_{(W_a,W_b)}(t) \\, ,$ where “$\\dots $ ” represents the contributions of other reactions and diffusions that flow into the same state.",
"We also use of other two important definitions: $(W_a,\\star )$ represents users with state $W_a$ on website $a$ and any state on website $b$ .",
"We also use $\\overline{S}_{(W_a,W_b)}(t)$ to denote the fraction of the Internet population that is not in state $(W_a,W_b)$ at time $t$ .",
"The DAU of website $a$ is given by $S_{(A,\\star )}(t) = \\sum _{k \\in \\lbrace \\emptyset , U, A, I\\rbrace } S_{(A,k)}(t)$ and website's $b$ DAU is $S_{(\\star ,A)}(t)= \\sum _{k \\in \\lbrace \\emptyset , U, A, I\\rbrace } S_{(k,A)}(t).$ The equations describing the dynamics of users of website $a$ that will never join website $b$ are as follows (a symmetric set of equations model website $b$ users that will never join website $a$ ).",
"The first reaction $S_{(U,\\emptyset )} \\xrightarrow{} S_{(A,\\emptyset )} \\quad \\textsf {\\lbrace word-of-mouth\\rbrace }$ describes the catalytic reaction at rate $\\gamma _a S_{(A,\\star )}$ that happens when an active member of website $a$ influences a unaware users $(U,\\emptyset )$ to join website $a$ , which can happen either through word-of-mouth or because of increased utility (e.g., network effects), two widely known phenomena in the specialized literature [5], [15], [39].",
"Unaware users can also join website $a$ through media & marketing campaign diffusions $S_{(U,\\emptyset )} \\xrightarrow{} S_{(A,\\emptyset )} \\quad \\textsf {\\lbrace marketing\\rbrace } \\, .$ The remaining catalytic reactions are $S_{(I,\\emptyset )} \\xrightarrow{} S_{(A,\\emptyset )} \\quad \\textsf {\\lbrace website activity\\rbrace } \\, ,$ describing the population-level influence that the content created by active users of website $a$ exert on $a$ 's inactive users, prodding them into activity (for more details on these dynamics see Ribeiro [37]); and finally $S_{(A,\\emptyset )} \\xrightarrow{} S_{(I,\\emptyset )} \\quad \\textsf {\\lbrace external activity\\rbrace } \\: ,$ modeling the influence of people doing things other than spending time on website $a$ exert on website $a$ users to also do something else."
],
[
"Joint Unaware Population Dynamics",
"We now apply the same mechanisms used above to describe the dynamics of website $a$ users that are willing to join website $b$ but are still unaware of website $b$ .",
"In what follows we only present website $a$ 's equations; the symmetric corresponding set of equations should be used to describe the dynamics from website's $b$ point of view.",
"Users unaware of website $a$ join through word-of-mouth and media & marketing diffusions: $S_{(U,U)} & \\xrightarrow{} S_{(A,U)} \\quad \\textsf {\\lbrace marketing\\rbrace } \\,,\\\\S_{(U,U)} & \\xrightarrow{} S_{(A,U)} \\quad \\textsf {\\lbrace word-of-mouth\\rbrace } \\, .$ The forces that users on website $a$ and people outside website $a$ exert on each other manifest in the following catalytic reactions: $S_{(I,U)} & \\xrightarrow{} S_{(A,U)} \\quad \\textsf {\\lbrace website activity\\rbrace }\\,, \\\\S_{(A,U)} & \\xrightarrow{} S_{(I,U)} \\quad \\textsf {\\lbrace external activity\\rbrace } \\, .$ Up until now websites $a$ and $b$ do not interfere with each other.",
"In what follows we consider the dynamic of concurrent adopters.",
"But first users must become concurrent adopters.",
"A user of website $a$ becomes a concurrent adopter of website $b$ through the following catalytic reactions and diffusions: $S_{(A,U)} &\\xrightarrow{} S_{(A,A)} \\quad \\textsf {\\lbrace marketing+inertia|momentum\\rbrace } \\,,\\\\S_{(A,U)} &\\xrightarrow{} S_{(A,A)} \\: \\textsf {\\small \\lbrace word-of-mouth+inertia|momentum\\rbrace }\\,, \\\\S_{(I,U)} &\\xrightarrow{} S_{(I,A)} \\quad \\textsf {\\lbrace marketing+inertia|momentum\\rbrace } \\,,\\\\S_{(I,U)} &\\xrightarrow{} S_{(I,A)} \\: \\textsf {\\small \\lbrace word-of-mouth+inertia|momentum\\rbrace } \\, ,$ where $\\zeta _a \\in \\mathbb {R}^+$ is a parameter that covers the spectrum of net population-level effects from inertia for $ \\zeta _a < 1$ to momentum for $\\zeta _a > 1$ .",
"In order to reduce the model complexity and also model the relative interest generated by the websites over time, we can further reduce the parameter space of our model $\\zeta _a(t) = \\zeta S_{(A,\\star )}(t)/(S_{(A,\\star )}(t)+S_{(\\star ,A)}(t))$ and $\\zeta _b = \\zeta S_{(\\star ,A)}(t)/(S_{(A,\\star )}(t)+S_{(\\star ,A)}(t))$ , where $\\zeta \\in \\mathbb {R}^+$ .",
"In what follows we cover the dynamics of concurrent adopters."
],
[
"Concurrent Adopters Dynamics",
"In our current attention-activity marketplace model concurrent adopters do not interfere with each other's inactive$\\rightarrow $ active dynamics.",
"This is because spending more time (attention) on website $a$ is assumed not to make the activity of other users (say, on website $b$ ) seem less interesting.",
"Using the fact that Facebook suffered nearly no lasting DAU effect around the July 2008 website redesign as a guiding principle (as shown by the tiny DAU bump in the inset of Figure *f:fb), we will assume that the extra time spent on website $a$ does not increase the user activity (attention-grabbing activity) rate $\\alpha _a$ , that is, $\\alpha _a$ remains unchanged in the following catalytic reactions: $S_{(I,A)} & \\xrightarrow{} S_{(A,A)} \\quad \\textsf {\\lbrace website a activity\\rbrace } \\,,\\\\S_{(A,I)} & \\xrightarrow{} S_{(A,A)} \\quad \\textsf {\\lbrace website b activity\\rbrace }\\,, \\\\S_{{(I,I)}} & \\xrightarrow{} S_{(A,I)} \\quad \\textsf {\\lbrace website a activity\\rbrace } \\,,\\\\S_{{(I,I)}} & \\xrightarrow{} S_{(I, A)} \\quad \\textsf {\\lbrace website b activity\\rbrace }\\,.$ Interestingly, according to our framework an increase in $\\alpha _a$ happens only when a new “user activity” feature is added, which our model predicts will cause an abrupt permanent change in the DAU slope.",
"For instance, in February 2009 Facebook introduced the “Like” feature [48] and, indeed, in Figure *f:fb we observe a small but sustainable sharp jump in the DAU time series.",
"Similarly, in September 2009 Facebook introduced the “tagging” feature [48] and, again, we observe another DAU sustainable sharp jump followed by a slope change.",
"In these scenarios $\\alpha _a$ becomes $\\alpha _a(t)$ , a right-continuous step function that sharply changes after a new “user activity” feature is added.",
"In order to keep model complexity down in our experiments and because these unpredictable changes are not of interest to out forecast, we consider $\\alpha _a$ as a constant in the results presented in Section ."
],
[
"Attention Sharing of Concurrent Adopters",
"In what follows we model the attention sharing behavior of users.",
"In the attention-activity marketplace the concurrent adopters are responsible for driving otherwise self-sustaining websites (see Ribeiro [37] for a precise definition of website self-sustainability) to their “unnatural” death.",
"In our model the average time a user spends on the website is latent, as we are modeling the DAU.",
"However, we can model its effect on the parameters of our model.",
"Let $B$ be the average time budget of time that a user is willing to spend at online social interactions.",
"Let $B_a$ , $B_b$ , and $B_o$ be the times that the user spends at websites $a$ , $b$ , and at other activities $o$ s.t.",
"$B_o = B - B_a - B_b$ .",
"If at time $t_0$ $B_a$ sharply increases by $\\Delta _a$ and $B_{o}$ sharply decreases by $\\Delta _a$ then $B_b$ remains constant.",
"In this scenario the DAU of website $b$ remains unchanged and websites $a$ and $b$ do not interfere with each other.",
"However, if the sharp increase in $B_a$ by $\\Delta _a$ is met with a decrease $\\Delta _b$ in $B_b$ , i.e., users are not willing to further compromise $B_{o}$ , then $\\beta _b$ also abruptly changes to follow the abrupt change in $B_b$ as $\\beta _b (1 + \\Delta _b/B_b)$ .",
"Define $\\eta _b^\\prime := \\Delta _b/B_b$ ; using the Heaviside step function, $H(t_0)$ , at time $t_0$ yields the catalytic reactions that model the attention sharing behavior of concurrent adopters: $S_{(A,A)} & \\xrightarrow{} S_{(I,A)} \\quad \\textsf {\\lbrace external activity\\rbrace } \\,, \\\\S_{(A,I)} & \\xrightarrow{} S_{{(I,I)}} \\quad \\textsf {\\lbrace external activity\\rbrace }\\,,\\\\S_{(A,A)} & \\xrightarrow{} S_{(A,I)} \\quad \\textsf {\\lbrace external activity\\rbrace }\\,, \\\\S_{(I,A)} & \\xrightarrow{} S_{{(I,I)}} \\quad \\textsf {\\lbrace external activity\\rbrace }\\, .$"
],
[
"DAU Model Fit",
"In this section we briefly introduce the challenges of learning the parameters of our model from the DAU data.",
"The DAU time series only provides information about $S_{(A,\\star )}$ and $S_{(\\star ,A)}$ .",
"The parameters introduced in Section REF need to be estimated together with the population compartment fractions.",
"The latter is what mathematical biologists call the carrying capacity of each of our four compartments, represented by the population fractions: (1) users “opposed” to both websites $a$ and $b$ : $C_{00} := S_{(\\emptyset ,\\emptyset )}$ ; (2) the users opposed to website $b$ : $C_{10} := \\sum _{k \\in \\lbrace U,A,I\\rbrace } S_{(k,\\emptyset )}$ ; (3) users opposed to website $a$ : $C_{01} := \\sum _{k \\in \\lbrace U,A,I\\rbrace } S_{(\\emptyset , k)}$ ; and (4) the fraction of concurrent adopters $C_{11} := \\sum _{k_a \\in \\lbrace U,A,I\\rbrace } \\sum _{k_b \\in \\lbrace U,A,I\\rbrace } S_{(k_a,k_b)},$ such that $C_{00} + C_{10} + C_{01} + C_{11} = 1$ .",
"Note that because the DAU time series starts June 18, 2007 and not when the websites were created, we also need to parametrize the unobservable quantities $S_{(I,\\star )}(t_{-1})$ and $S_{(\\star ,I)}(t_{-1})$ , where $t_{-1} = $ “June 18, 2007”.",
"For the remaining quantities that need to be initialized with values greater than zero due to the $t_{-1}$ start, e.g., $S_{(A,A)}(t_{-1})$ , we use independence assumptions, e.g., $S_{(A,A)}(t_{-1}) = S_{(A,\\star )}(t_{-1}) S_{(\\star ,A)}(t_{-1})$ .",
"We fit the model parameters to the DAU data using the Levenberg-Marquardt algorithm [29].",
"Our results in Section show the model fit using the first two years of DAU data to train the model, the following four months for model selection, and the remaining years as holdout data to evaluate the model predictions.",
"The Levenberg-Marquardt algorithm only finds a locally optimal solution starting from an initial parameter guess.",
"Hence, the initial guess may significantly influence the output of the algorithm.",
"Due to the large number of parameters of our model we run the Levenberg-Marquardt algorithm with multiple initial parameter guesses, choosing the fitted parameters that best fit our model selection data.",
"In order further reduce the number of parameters to be learned, we tried the options of learning $C_{11}$ or setting $C_{11} = \\min (C_{10},C_{01})$ .",
"While these two options give similar results, the option $C_{11}$ requires at least ten times the number of initializations and, in the end, $C_{11} = \\min (C_{10},C_{01})$ in all examples we tried.",
"Therefore, in the scenarios presented in Section we set $C_{11} = \\min (C_{10},C_{01})$ to speed up computations."
],
[
"Results",
"In this section we briefly introduce our results.",
"In all of our results an extra four months of DAU data is used to select the best parameter fit that does not overfit the data (model selection phase).",
"The fitted models and their predictions show great agreement with the data.",
"But the learned parameter should be interpreted carefully given that most differential equations in our model are quadratic (the ones with $\\zeta $ are cubic) and the model has a multitude of parameters and latent variables.",
"The only observable quantities (the DAUs) are the aggregates $S_{(A,\\star )}$ and $S_{(\\star ,A)}$ , the DAU data is left-censored (our DAU time series starts mid 2007 when, for instance, MySpace was already four years old), only two years of DAU data are used to fit the model, and the maximum DAU of Facebook in the training set is 20%.",
"Nevertheless, the learned parameter seem to offer interesting insights into the popularity growth of Facebook and the death of its competitors.",
"As expected, the model predicts momentum in concurrent adoptions for all websites, that is, users of MySpace, Hi5, Friendster, and Multiply were more likely than the average user to adopt Facebook."
],
[
"Facebook v.s. MySpace",
"Figure REF shows the results of the model fit using the first two years (24 months) of the DAU time series of the competition between Facebook and MySpace.",
"These 28 months (24 months for training and 4 months for the model selection phase) are shown as blue points in the plot.",
"The gray vertical line separates the training & model selection data from the remaining 50 months (4.1 years) of DAU data used to test our predictions, shown in the plot as gray points.",
"The model shows great agreement with the data, both in the training (blue line) and prediction (red line) phases.",
"The inset in Figure REF gives a closer look at the MySpace results.",
"The model – which was trained with Facebook's peaking at only $20\\%$ DAU – estimates the total (unaware + active + inactive) Facebook population at $C_{10}+C_{11} = 77\\%$ and MySpace's population at $C_{01}+C_{11} = 22\\%$ ; the model also estimates that Facebook user base grew largely due to word-of-mouth and that the July 20, 2008 change in Facebook reduced the average time spent on MySpace by 88%.",
"The model fit also estimates $\\zeta = 4$ , that is, by 2008 MySpace users were about twice as likely to join Facebook than the average Facebook adopter (recall that $\\zeta _a(t) = \\zeta S_{(\\star ,A)}(t)/(S_{(\\star ,A)}(t)+S_{(A,\\star )})(t)$ ).",
"In light of our findings, it seems that MySpace's documented “white flight” and “teen disengagement” in 2007 [7] – often anecdotally cited by the lay press as the primary reason of MySpace's death – may have had only a marginal role in MySpace's demise.",
"While in hindsight many social causes could explain MySpace's demise, they would not explain why Hi5, Friendster, and Multiply simultaneously suffered the same effect.",
"Moreover, by 2007 MySpace's DAU was stable – and our model predicts it would have remained stable in the absence of Facebook's change –, thus, making the “white flight” and “teen disengagement” 2007 hypothesis rather unlikely as the main cause of MySpace's death.",
"Figure: Model fit and predictions for the competition Facebook v.s.",
"MySpace."
],
[
"Facebook v.s. Multiply",
"Figure REF shows the results of the model fit using the first two and a half years (31 months) of the DAU time series in the competition between Facebook and Multiply.",
"The larger training data was required to achieve a better quality fit of the parameters (harder to learn on Multiply).",
"Unlike MySpace, Hi5, and Friendster, the model now predicts that Multiply survives Facebook's “attention raid”.",
"The model fit shows great agreement with the data up until May, 2013 when Multiply officially closed operations.",
"The model estimates the total (unaware + active + inactive) Facebook population at $C_{10}+C_{11} = 64\\%$ and Multiply's population at $C_{01}+C_{11} = 0.9\\%$ ; the model estimates that both Facebook and Multiply user base grew largely due to word-of-mouth.",
"The model also estimates that immediately after the July 20, 2008 event the average time spent on Multiply decreases “only” by 24%, which is why Multiply is projected to survive.",
"The model fit also estimates $\\zeta = 7.5$ , that is, Multiply users were over seven times as likely to join Facebook than the average Facebook adopter.",
"Figure: Model fit and predictions for the competition Facebook v.s.",
"Multiply."
],
[
"Facebook v.s. Hi5",
"Figure REF shows the results of the model fit using the first two years (24 months) of the DAU time series in the competition between Facebook and Hi5.",
"The model fit shows great agreement with the data, blue line shows the model fit and red line shows its prediction.",
"The model was able to capture the sharp elbow near July 20, 2008.",
"Here the model estimates the total (unaware + active + inactive) Facebook population at $C_{10}+C_{11} = 59\\%$ and Hi5's population at $C_{01}+C_{11} = 5\\%$ ; the model estimates that Hi5 user base grew largely due to media & marketing campaigns and that Facebook's growth was through word-of-mouth; the July 20, 2008 event largely reduced the average time spent on Hi5 by 95%.",
"The model fit also estimates $\\zeta = 3.7$ , that is, by 2008 Hi5 users were almost four times as likely to join Facebook than the average Facebook adopter.",
"Figure: Model fit and predictions for the competition Facebook v.s.",
"Hi5."
],
[
"Facebook v.s. Friendster",
"Figure REF shows the results of the model fit using the first two years (24 months) of the DAU time series in the competition between Facebook and Multiply.",
"The model shows good agreement with the data both in the fitting phase (blue line) and the prediction phase (red line).",
"The model estimates the total (unaware + active + inactive) Facebook population at $C_{10}+C_{11} = 59\\%$ and Multiply's population at $C_{01}+C_{11} = 3.6\\%$ ; the model estimates that both Facebook and Friendster user base grew largely due to word-of-mouth.",
"Immediately after the July 20, 2008 event the average time spent on Friendster is estimated to have decreased by nearly 99%.",
"The model fit also estimates $\\zeta = 1.8$ showing that Friendster users were almost twice as likely to join Facebook than the average Facebook adopter.",
"Figure: Model fit and predictions for the competition Facebook v.s.",
"Friendster."
],
[
"Conclusions",
"Our study sheds light onto the role of the attention-activity marketplace in the popularity (DAU time series) of membership-based websites.",
"Making use of the unique way by which Facebook affected its competitors in July, 2008, we derive a set of modeling principles that inform our proposed attention-activity model design.",
"Through a series of catalytic reactions that model user attention and activity interactions, together with media & market diffusions, we propose a model that well captures the popularity competition between websites.",
"We fit the model parameters to real-world DAU time series data and show that our model not only fits well the DAU data but can also predict its future evolution.",
"In a 1969 lecture Herbert A. Simon warned us that information systems that help us generate more content than they help us reduce our time consuming such content would exacerbate the scarcity of attention [41].",
"Our work takes a positive step towards modeling such phenomenon in Internet companies, providing insights into the connections between website popularity and user attention.",
"The model shows that two competing websites can co-exist without interfering with each other as long as users have enough attention to spare; this agrees with our data showing that before its new attention-demanding “Wall” feature Facebook did not seem to interfere with the popularity of MySpace, Hi5, Friendster, or Multiply.",
"Conversely, the model shows that websites fiercely compete when they share a sizable population of attention-starved users, and that such population can play a central role into the negative attention feedback loop that leads to the death of a website.",
"The model shows, for instance, that the popularity of a website with a large user base of tech savvy users – or novelty-driven teen users – can be easily preyed upon by a competing website, thus reducing the long-term viability of such websites.",
"Our hope is that further research in this direction will provide a better picture of the attention-activity marketplace, helping the design of information systems that do not need to overload the finite attention capacity of its users in order to survive."
],
[
"Acknowledgments",
"This work was supported by NSF grant CNS-1065133 and ARL Cooperative Agreement W911NF-09-2-0053.",
"The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied of the NSF, ARL, or the U.S. Government.",
"The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon."
]
] | 1403.0600 |
[
[
"Comment on \"Noise and Disturbance in Quantum Measurements: An\n Information-Theoretic Approach\""
],
[
"Abstract In this comment on the work of F. Buscemi, M.J.W.",
"Hall, M. Ozawa and M.M.",
"Wilde [PRL 112, 050401, 2014, arXiv:1310.6603], we point out a misrepresentation of measures of error and disturbance introduced in our recent work [PRL 111, 160405, 2013, arXiv:1306.1565] as being \"purely formal, with no operational counterparts\".",
"We also exhibit an tension in the authors' message, in that their main result is an error-disturbance relation for state-independent measures, but its importance is declared to be limited to discrete variables.",
"In contrast, we point out the separate roles played by such relations for either state-dependent or state-independent measures of error and disturbance."
],
[
"Comment on “Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach” Paul Busch [email protected] Department of Mathematics, University of York, York, United Kingdom Pekka Lahti [email protected] Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland Reinhard F. Werner [email protected] Institut für Theoretische Physik, Leibniz Universität, Hannover, Germany In their interesting paper [1], Buscemi et al derive a state-independent entropic trade-off relation for the noise (approximation error) in the measurement of one observable and the necessary disturbance imparted thereby on another observable.",
"We appreciate that this work is very much in the spirit of our recent letter [2], where we have derived two variants of error-disturbance relations for position and momentum in terms of calibration errors and worst case quadratic deviations, respectively.",
"Yet, the paper [1] contains some comments on our letter that are either misleading or incorrect.",
"Crucially, there is an internal tension in the presentation that betrays a misrepresentation of the research programme underlying both papers [1] and [2] which we feel needs correcting lest it be perpetuated by repetition.",
"In the conclusion of their work, the authors of [1] describe their results concerning state-independent entropic measurement uncertainty relations as fundamental, and we agree with this assessment.",
"However, the discussion implies that they consider the significance of their relation to be restricted to the case of discrete observables.",
"They do consider an extension to continuous observables, such as position and momentum, but immediately declare this to be “purely formal, with no operational counterparts”.",
"This verdict is then also applied to our calibration error relation.",
"Based on this “formality” claim, for which no justification is offered other than the hint that continuous observables do not have proper eigenstates, the authors conclude that “it appears that state-dependent noise-disturbance and joint-measurement relations ... may be preferable for continuous observables.” In contrast, we maintain that both approaches to formulating measurement uncertainty relations – with state independent or state dependent error measures – have their separate uses and merits, and that there is no reason why either should be limited to a certain type of observables [3].",
"State dependent error and disturbance measures are useful if, for example, one wishes to carry out information theoretic tasks in which the disturbance should be limited in the case of a specific state.",
"An error-disturbance relation would then tell us how the accuracy in a measurement must be limited.",
"By contrast, state-independent error measures, which can be defined as suitable mean or worst-case errors, are suitable as figures of merit for a measuring device.",
"The corresponding uncertainty relations describe the limitations that all possible devices are subjected to if they are to be used for jointly approximating a pair of incompatible quantities.",
"Unfortunately, the recent hype about alleged violations of Heisenberg's error-disturbance relation has distracted somewhat from appreciating the important role played by state-independent measures in quantifying this fundamental measurement limitation.",
"Moreover, it is incorrect to say that calibration error measures are without operational counterpart or content.",
"In fact, it is standard experimental practice to calibrate a measuring device by applying it to situations where the property to be measured has a fairly sharply determined, known value or distribution, and then comparing this value or distribution with the device's output distribution.",
"This is exactly what is being captured with the error measures defined in both [2] and [1].",
"These measures may be difficult to implement practically; but that does not make them void of operational meaning.",
"Incidentally, the claim of [1] that their relation $V^Q_{\\mathsf {N}}V^P_{\\mathsf {D}}\\ge \\hbar ^2/4$ is stronger than ours is logically unfounded: their $V$ quantities are defined in terms of very specific families of approximate position and momentum eigenstates while our measures take into account all possible sufficiently localized states.",
"There is therefore no direct comparison possible between our respective quantities.",
"Nevertheless it is gratifying to see that [1] attempts to strengthen our error-disturbance relation, considering that in a recent arXiv publication one of its coauthors attempted to disprove it [4], [5]."
]
] | 1403.0368 |
[
[
"Asymptotics of eigenstates of elliptic problems with mixed boundary data\n on domains tending to infinity"
],
[
"Abstract We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity.",
"We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions."
],
[
"Introduction",
"Let $\\omega $ be a bounded open set in $\\mathbb {R}^{n-1}$ .",
"For every $\\ell >0$ set $\\Omega _\\ell = (-\\ell , \\ell )\\times {}\\omega $ and write each $x\\in \\Omega _\\ell $ as $x=(x_1,X_2)$ with $X_2=(x_2,\\ldots ,x_n)$ .",
"We assume that the matrices $A(X_2)=\\begin{pmatrix}a_{11}(X_2) & A_{12}(X_2)\\\\A_{12}^t(X_2) & A_{22}(X_2)\\end{pmatrix}$ are uniformly elliptic and uniformly bounded on $\\omega $ (precise assumptions will be made in Section ).",
"The limiting behavior, when $\\ell $ goes to infinity, of the eigenvalues and eigenfunctions of the elliptic operator $-\\operatorname{div}(A(X_2)\\nabla u)$ on $\\Omega _\\ell $ with zero Dirichlet boundary conditions, was studied by Chipot and Rougirel in [7].",
"We shall recall below one of their main results that was the principal motivation for the current paper.",
"Let $\\mu {}^k$ and $\\sigma _\\ell ^k$ denote, respectively, the $k$ th eigenvalues for the problems $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A_{22}(X_2)\\nabla u)=\\mu {}u \\quad \\text{ in }\\omega ,\\\\&u=0 \\quad \\text{ on }\\partial \\omega ,\\end{aligned}\\right.$ and $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A(X_2)\\nabla u)=\\sigma u \\quad \\text{ in }\\Omega _\\ell ,\\\\&u=0 \\quad \\text{ on }\\partial \\Omega _\\ell .\\end{aligned}\\right.$ The following relation between problem (REF ) (for large $\\ell $ ) and problem (REF ) was established in [7].",
"Theorem A (Chipot-Rougirel) $\\mu {}^1 \\le \\sigma _\\ell ^1 \\le \\mu {}^{1} + \\frac{C}{\\ell ^2}\\,,$ where $C$ is a constant independent of $\\ell $ .",
"The main goal of the present article is to study the analogous problem for mixed boundary conditions, at least for $k=1$ .",
"Let us write $\\partial \\Omega _\\ell =\\Gamma _\\ell \\cup \\gamma _\\ell $ where $\\Gamma _\\ell = \\lbrace -\\ell , \\ell \\rbrace \\times {}\\omega \\text{ and } \\gamma _\\ell = (-\\ell , \\ell )\\times {}\\partial \\omega ,$ and denote by $\\lambda _\\ell ^k$ the $k$ th eigenvalue for the mixed Neumann-Dirichlet problem $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A(X_2)\\nabla u)=\\sigma u \\quad \\text{ in }\\Omega _\\ell ,\\\\&u=0 \\quad \\text{ on }\\gamma _\\ell ,\\\\& (A(X_2)\\nabla u).\\nu =0 \\quad \\text{ on }\\Gamma _\\ell .\\end{aligned}\\right.$ One of our main results establishes that $\\lim _{\\ell \\rightarrow \\infty }\\lambda _\\ell ^1$ exists, but in general it is strictly smaller than $\\mu ^1$ .",
"This “gap phenomenon” is explained by the appearance of boundary effects near $\\Gamma _\\ell $ .",
"To gain better understanding of these effects we are led to consider first the limit $\\lim _{\\ell \\rightarrow 0}\\lambda _\\ell ^1$ .",
"Asymptotic behavior of elliptic problems set on domains shrinking to zero in some directions are generally known as “Dimension Reduction\" problems and are addressed in [1], [3], [14] and in a setting particularly suitable for us, in [2] .",
"Our work establishes a somewhat surprising connection between the theory of dimension reduction (i.e., “$\\ell \\rightarrow 0$ ”) and the theory for “$\\ell \\rightarrow \\infty $ ”.",
"In order to have a more precise description of the boundary effects and to characterize the value of the limit $\\lim _{\\ell \\rightarrow \\infty }\\lambda _\\ell ^1$ , we introduce eigenvalue problems on the two semi-infinite cylinders $\\Omega _\\infty ^+=(0,\\infty )\\times \\partial \\omega $ and $\\Omega _\\infty ^-=(-\\infty ,0)\\times \\partial \\omega $ , with mixed boundary conditions.",
"Let $\\nu _\\infty ^\\pm $ denote the first eigenvalue for the operator $-\\operatorname{div}(A(X_2)\\nabla u)$ on $\\Omega _\\infty ^\\pm $ with zero boundary condition on the lateral part of the boundary $\\partial \\Omega _\\infty ^\\pm $ .",
"One might be tempted to expect that the equality $\\nu _\\infty ^+=\\nu _\\infty ^-$ always hold because of “symmetry considerations”.",
"However, as we shall see in Section , this equality is false in general.",
"Our main results are summarized in the next theorem, that combines the results of Theorem REF and Theorem REF .",
"We denote by $W_1$ the positive normalized eigenfunction corresponding to $\\mu ^1$ .",
"Main Theorem We have $\\lim _{\\ell \\rightarrow \\infty }\\lambda _\\ell ^1=\\min (\\nu _\\infty ^+,\\nu _\\infty ^-)\\,.$ If $A_{12}.\\nabla W_1\\lnot \\equiv 0\\text{ a.e.",
"on } \\omega ,$ then $\\lim _{\\ell \\rightarrow \\infty }\\lambda _\\ell ^1<\\mu ^1$ .",
"Otherwise, $\\lambda _\\ell ^1=\\mu ^1$ , $\\forall \\ell $ .",
"Many problems of the type “$\\ell \\rightarrow \\infty $ ” were studied in the past.",
"Besides the eigenvalue problem already mentioned [7], these include elliptic and parabolic equations, variational inequalities and systems, see [5], [6], [8], [9], [10], [11], [12].",
"In all these problems it is found that the limit is characterized by the solution of the corresponding problem on the section $\\omega $ .",
"We emphasize that the limiting behavior in our problem is very different.",
"The paper is organized as follows.",
"In Section we give the main definitions and notation needed in the subsequent sections.",
"In Section we illustrate the gap phenomenon in a simple model case where $\\omega =(-1,1)$ and $A$ is a $2\\times 2$ matrix with constant coefficients, namely, $A = A_\\delta =\\begin{pmatrix}1 & \\delta \\\\\\delta & 1\\end{pmatrix}$ .",
"In Section we prove the gap phenomenon for the general case.",
"In Section we prove that the limit $\\lim _{\\ell \\rightarrow \\infty }\\lambda _\\ell ^1$ exists, and identify its value using the eigenvalue problems on the semi-infinite cylinders $\\Omega _\\infty ^+$ and $\\Omega _\\infty ^-$ .",
"In Section we investigate further the problem on a semi-infinite cylinder and use it to give a more precise description of the first eigenfunction $u_\\ell $ for large $\\ell $ .",
"In the last section, Section , we address briefly two natural related problems.",
"First, we present a result on the asymptotics of the second eigenvalue $\\lambda _\\ell ^2$ as $\\ell $ goes to infinity (under some symmetry assumption on the matrix $A$ ).",
"Second, we give a partial result for the more general case of a domain becoming large in several directions."
],
[
"Preliminaries",
"For each $\\ell >0$ consider $\\Omega _\\ell = (-\\ell , \\ell ) \\times {}\\omega $ with $\\omega $ a bounded domain in $\\mathbb {R}^{n-1}$ as in the Introduction.",
"The lateral part of $\\partial \\Omega _\\ell $ and the remaining part of the cylinder (i.e., the two ends) will be denoted by $\\gamma _\\ell $ and $\\Gamma _\\ell $ , respectively.",
"Let us denote by $H^1(\\Omega _\\ell )$ and $H_0^1(\\Omega _\\ell )$ the usual spaces of functions defined by $H^1(\\Omega _\\ell ) = \\left\\lbrace v \\in L^2(\\Omega _\\ell ) |\\ \\partial _{x_i}v \\in L^2(\\Omega _\\ell ), i = 1, 2, \\ldots ,n \\right\\rbrace ,$ and $H_0^1(\\Omega _\\ell ) = \\left\\lbrace v \\in H^1(\\Omega _\\ell ) | \\ v = 0 \\text{ on} \\ \\partial \\Omega _\\ell \\right\\rbrace ,$ or in a more precise way, $H_0^1(\\Omega _\\ell )$ is the closure of $C^\\infty _c(\\Omega _\\ell )$ in $H^1(\\Omega _\\ell )$ .",
"The space $H_0^1(\\Omega _\\ell )$ is equipped with the norm $\\Vert \\nabla v \\Vert ^2_{2, \\Omega _\\ell } = \\int _{\\Omega _\\ell } |\\nabla v|^2.$ A suitable space for our problem is $V(\\Omega _\\ell ) = \\left\\lbrace v \\in H^1(\\Omega _\\ell ) \\ | \\ v = 0 \\text{ on}\\gamma _\\ell \\right\\rbrace ,$ where the boundary condition should be interpreted in the sense of traces.",
"Thanks to the Poincaré inequality, $V(\\Omega _\\ell )$ becomes an Hilbert space when equipped with the norm (REF ).",
"For later use we define the sets $\\Omega _\\ell ^+ = [0, \\ell )\\times {}\\omega \\text{ and } \\Omega _\\ell ^-= (-\\ell ,0)\\times {}\\omega ,$ We decompose $\\Gamma _\\ell $ (see (REF )) into two parts as $\\Gamma _\\ell = \\Gamma _\\ell ^+ \\cup \\Gamma _\\ell ^-$ , where $\\Gamma _\\ell ^+ = \\lbrace \\ell \\rbrace \\times \\omega ~\\text{ and }~ \\Gamma _\\ell ^- =\\lbrace -\\ell \\rbrace \\times \\omega \\,.$ Similarly, for the lateral part of $\\partial \\Omega _\\ell $ we define, $\\gamma _\\ell ^+ = (0,\\ell )\\times \\partial \\omega ~\\text{ and }~ \\gamma _\\ell ^- =(-\\ell ,0)\\times \\partial \\omega \\,.$ We shall be concerned with the operator $-\\operatorname{div}(A(X_2)\\nabla u)$ where, for each $X_2\\in \\omega $ , $\\begin{aligned}A(X_2)=\\begin{pmatrix}a_{11}(X_2) & A_{12}(X_2) \\\\A_{12}^t(X_2) & A_{22}(X_2)\\end{pmatrix}\\end{aligned}$ is a symmetric $n \\times n $ matrix, $a_{11}\\in \\mathbb {R}$ , $A_{12}$ is a $1 \\times (n-1)$ matrix and $A_{22}$ is a $(n-1)\\times (n-1)$ matrix.",
"The components of $A(X_2)$ are assumed to be bounded measurable functions on $\\omega $ and we assume the following bound $\\Vert A(X_2)\\Vert \\le C_A\\,,~~a.e.",
"\\,X_2 \\in \\omega ,$ for the Euclidean operator norm.",
"We also assume that $A(X_2)$ is uniformly elliptic and denote by $\\lambda _A$ the largest positive number for which the following inequality holds, $A(X_2)\\xi .",
"\\xi \\ge \\lambda _A |\\xi |^2\\,,~~\\forall \\xi \\in \\mathbb {R}^n, \\ a.e.",
"\\,X_2 \\in \\omega .$ The weak formulation of the eigenvalue problem (REF ) is to find $u\\in H_0^1(\\omega )\\setminus \\lbrace 0\\rbrace $ and $\\mu {}\\in \\mathbb {R}$ such that $\\int _{\\omega }(A_{22} \\nabla u).",
"\\nabla v \\,dX_2 = \\mu \\int _{\\omega }uv\\,dX_2\\,,~~\\forall v\\in H_0^1(\\omega )\\,.$ Denote by $\\mu ^1$ the first eigenvalue of the problem (REF ) with the corresponding normalized eigenfunction $W_1$ , i.e., $\\int _\\omega |W_1|^2=1$ .",
"It is well known that $\\mu {}^1$ has a variational characterization by the Rayleigh quotient: $\\mu ^1 = \\inf \\left\\lbrace \\int _{\\omega }(A_{22}(X_2)\\nabla u).\\nabla u\\big |\\,u \\in H_0^1(\\omega ) \\text{ s.t. }",
"\\int _{\\omega }u^2=1 \\right\\rbrace \\\\= \\inf _{u \\in H_0^1(\\omega )\\setminus \\lbrace 0\\rbrace } \\frac{\\int _{\\omega }(A_{22}(X_2)\\nabla u).\\nabla u }{\\int _{\\omega }u^2}.$ Moreover, $W_1$ is simple and has constant sign in $\\Omega $ (see [13]).",
"The choice of positive sign leaves us with a unique $W_1$ .",
"Similarly, the eigenvalue problem (REF ) has the following weak formulation: find $u\\in V(\\Omega _\\ell )\\setminus \\lbrace 0\\rbrace $ and a real number $\\lambda $ such that $\\int _{\\Omega _\\ell }A \\nabla u.\\nabla v \\,dx = \\lambda \\int _{\\Omega _\\ell }u v\\, dx\\,,~~\\forall v\\in V(\\Omega _\\ell ).$ It is well known, see [4], that the first eigenvalue $\\lambda _{\\ell }^1$ for (REF ) is associated with a variational characterization, $\\lambda _\\ell ^1 = \\inf \\left\\lbrace \\int _{\\Omega _\\ell }A\\nabla u.\\nabla u \\,: \\,u \\in V(\\Omega _\\ell ),\\, \\int _{\\Omega _\\ell }u^2 = 1\\right\\rbrace =\\inf _{u \\in V(\\Omega _\\ell )\\setminus \\lbrace 0\\rbrace } \\frac{\\int _{\\Omega _\\ell }A(X_2)\\nabla u.\\nabla u }{\\int _{\\Omega _\\ell }u^2}.$ It is also true, and can be proved in the same way as it is done for the corresponding Dirichlet problem, that $\\lambda _{\\ell }^1$ is simple and the corresponding eigenfunction $u_\\ell $ has constant sign in $\\Omega _\\ell $ , that we should fix as the positive sign in the sequel.",
"For some of our results we shall need to impose a certain symmetry condition on $\\omega $ and $A$ .",
"Definition 2.1 We shall say that property (S) holds if $\\omega $ is symmetric w.r.t.",
"the origin (i.e., $-\\omega =\\omega $ ) and $A(-X_2)=A(X_2)$ .",
"From the uniqueness of $u_\\ell $ we deduce easily the following symmetry result.",
"Proposition 2.1 If property (S) holds then $u_\\ell (x_1, X_2) = u_\\ell (-x_1, -X_2)$ .",
"Clearly $v_\\ell (x_1, X_2) := u_\\ell (-x_1,-X_2)$ is a positive normalized eigenfunction for $\\lambda ^1_\\ell $ , so it must be equal to $u_\\ell $ ."
],
[
"The gap phenomenon in a model problem",
"In this section we treat a two dimensional model problem in order to illustrate the main ideas behind the analysis of the general case in the next sections.",
"Throughout this section $\\omega =(-1,1)$ , $\\Omega _\\ell =(-\\ell ,\\ell )\\times (-1,1)$ , and the matrix $A$ is a constant matrix depending on the parameter $\\delta \\in [0,1)$ , namely, $A = A_\\delta =\\begin{pmatrix}1 & \\delta \\\\\\delta & 1\\end{pmatrix}.$ Clearly $A_\\delta $ satisfies all the assumptions made on $A$ in Section .",
"Since the eigenvalues of $A_\\delta $ are $1\\pm {}\\delta $ , $\\lambda _A = 1- \\delta $ (see (REF )).",
"In this section we shall denote a point in $\\mathbb {R}^2$ by $x=(x_1, x_2).$ The problem (REF ) has the following simple form $\\left\\lbrace \\begin{aligned}-&W_1^{\\prime \\prime }= \\mu {}^1 W_1 \\text{ in } (-1, 1)\\,,\\\\&W_1(-1) =W_1(1) = 0\\,.\\end{aligned}\\right.$ where $\\mu {}^1$ denotes the first eigenvalue and $W_1$ is the corresponding positive normalized eigenfunction.",
"Therefore, $\\mu {}^1=(\\frac{\\pi }{2})^2$ and $W_1(t)=\\cos (\\frac{\\pi }{2}t)$ .",
"Proposition 3.1 For $\\delta =0$ we have $\\lambda ^1_\\ell = \\mu {}^1$ for all $\\ell >0$ .",
"For $\\delta \\in (0,1)$ we have $(1-\\delta ^2)\\mu {}^1<\\lambda ^1_\\ell <\\mu {}^1,\\,\\forall \\ell >0.$ (i) Since $A_0=\\begin{pmatrix} 1 & 0\\\\0 & 1\\end{pmatrix}$ , the corresponding operator is just $-\\Delta $ , and the function $v(x_1,x_2)=W_1(x_2)$ is clearly a positive eigenfunction in (REF ) with $\\sigma =\\mu {}^1$ , for all $\\ell >0$ .",
"It follows that $\\lambda ^1_\\ell = \\mu {}^1$ as claimed.",
"(ii) Assume now that $\\delta \\in (0,1)$ .",
"Using the function $v(x_1,x_2)=W_1(x_2)$ in the Rayleigh quotient (REF ) yields the inequality $\\lambda ^1_\\ell \\le \\mu {}^1\\,.$ We claim that the inequality in (REF ) is strict as stated in (REF ).",
"Indeed, an equality would imply that the function $v$ (as defined above) is a positive eigenfunction in (REF ) for $\\sigma =\\lambda ^1_\\ell =\\mu ^1$ , and in particular, it satisfies the Neumann boundary condition $0=(A_\\delta \\nabla v).\\nu =v_{x_1}+\\delta v_{x_2}=\\delta v_{x_2}~\\text{ on}~\\Gamma _\\ell ^+=\\lbrace \\ell \\rbrace \\times (-1,1)\\,.$ But this clearly contradicts the fact that $(W_1)^{\\prime }(x_2)\\ne 0$ for $x_2\\in (-1,1)\\setminus \\lbrace 0\\rbrace $ .",
"To prove the inequality of the left in (REF ) we first notice the elementary inequality $(A_\\delta \\xi ).\\xi \\ge (1 -\\delta ^2)|\\xi _2|^2,~\\forall \\xi =(\\xi _1,\\xi _2)\\in \\mathbb {R}^2\\,.$ Indeed, (REF ) follows from the identity $(A_\\delta \\xi ).\\xi =\\xi _1^2+2\\delta \\xi _1\\xi _2+\\xi _2^2=(1-\\delta ^2)\\xi _2^2+(\\xi _1+\\delta \\xi _2)^2\\,.$ By (REF ) and(REF ) we get $\\begin{aligned}\\lambda _\\ell ^1 = \\int _{\\Omega _\\ell } (A_\\delta \\nabla u_\\ell ).\\nabla u_\\ell &\\ge (1 -\\delta ^2)\\int _{\\Omega _\\ell }|\\partial _{x_2}u_\\ell |^2 \\\\ &\\ge (1 -\\delta ^2)\\mu {}^1\\int _{\\Omega _\\ell } |u_\\ell |^2= (1 -\\delta ^2) \\mu {}^1.\\end{aligned}$ To conclude, we show that the inequality $\\lambda _\\ell ^1\\ge (1-\\delta ^2)\\mu {}^1$ is strict.",
"Indeed, equality would imply equalities in all the inequalities in (REF ), implying in particular that $u_\\ell (x_1,x_2)=W_1(x_2)$ in $\\Omega _\\ell $ .",
"It would then follow that $\\lambda _\\ell ^1=\\mu ^1$ .",
"Contradiction.",
"From now on we shall assume that $\\delta \\in (0,1)$ (the first part of Proposition REF settles completely the case $\\delta =0$ ).",
"Our main result in this section establishes the following estimate about the behavior of $\\lambda _\\ell ^1$ as $\\ell $ goes to infinity.",
"Theorem 3.1 $\\limsup _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1 < \\mu {}^1,$ for every $\\delta \\in (0,1)$ .",
"In the next section, when dealing with the general case, we shall actually see that the limit $\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1$ exists.",
"As mentioned in the Introduction, an important ingredient in the proof of Theorem REF is a study of the asymptotic behavior of $\\lambda _\\ell ^1 $ as $\\ell \\rightarrow 0$ (a dimension reduction problem).",
"Theorem 3.2 We have $\\lim _{\\ell \\rightarrow 0} \\lambda _\\ell ^1 = (1- \\delta ^2)\\mu ^1$ .",
"It suffices to consider $\\ell <1$ .",
"Fix any $\\alpha \\in (0,1)$ and let $\\rho _\\ell $ be the piecewise-linear function defined by $\\rho _\\ell (t)={\\left\\lbrace \\begin{array}{ll}\\frac{t+1}{\\ell ^\\alpha } & t\\in [-1,-1+\\ell ^\\alpha ),\\\\1 & t\\in [-1+\\ell ^\\alpha ,1-\\ell ^\\alpha ],\\\\\\frac{1-t}{\\ell ^\\alpha } & t\\in (1-\\ell ^\\alpha ,1]\\,.\\end{array}\\right.",
"}$ Consider the following test function $v_{\\ell }(x_1,\\ x_2) =W_1(x_2) -\\delta x_1W_1^{\\prime }(x_2)\\rho _{\\ell }(x_2)\\,.$ Then clearly $v_{\\ell } \\in V(\\Omega _{\\ell })$ is a valid test function.",
"From (REF ), we have $\\begin{aligned}\\lambda _{\\ell }^1 \\le \\frac{\\int _{\\Omega _{\\ell }}A_{\\delta }\\nabla v_{\\ell }.\\nabla v_{\\ell }}{\\int _{\\Omega _{\\ell }}v_{\\ell }^2} &=\\frac{\\int _{\\Omega _{\\ell }}|\\partial _{x_1}v_\\ell |^2 + \\int _{\\Omega _{\\ell }}|\\partial _{x_2}v_\\ell |^2 + 2\\delta \\int _{\\Omega _{\\ell }}\\partial _{x_1}v_\\ell \\partial _{x_2}v_\\ell }{\\int _{\\Omega _{\\ell }}v_{\\ell }^2} \\\\&= \\frac{I_1 + I_2 + I_3}{I}\\,.\\end{aligned}$ We consider each of the terms $I_1, I_2, I_3$ and $ I$ separately.",
"First, $I_1 = \\delta ^2 \\int _{\\Omega _{\\ell }}\\rho _{\\ell }^2|W_1^{\\prime }(x_2)|^2\\,dx = 2\\ell \\delta ^2 \\int _{-1}^1\\rho _{\\ell }^2|W_1^{\\prime }(x_2)|^2\\,dx_2\\,.$ Next, calculating for $I_2$ , $I_2 &= \\int _{\\Omega _{\\ell }} \\Big [W_1^{\\prime }(x_2) - \\delta x_1\\lbrace \\rho _{\\ell }W_1^{\\prime \\prime }(x_2) + W_1^{\\prime }(x_2)\\rho ^{\\prime }_\\ell (x_2)\\rbrace \\Big ]^2\\\\&= \\int _{\\Omega _{\\ell }} |W_1^{\\prime }|^2 -2\\delta \\int _{\\Omega _{\\ell }} x_1W_1^{\\prime }(x_2)\\big \\lbrace \\rho _{\\ell }W_1^{\\prime \\prime }(x_2) + W_1^{\\prime }(x_2)\\rho ^{\\prime }_\\ell (x_2)\\big \\rbrace \\\\&\\phantom{=} +\\delta ^2 \\int _{\\Omega _{\\ell }}x_1^2| \\rho _{\\ell }W_1^{\\prime \\prime }(x_2)+ W_1^{\\prime }(x_2)\\rho _\\ell ^{\\prime }(x_2)|^2.$ The integral in the middle vanishes since $\\int _{-\\ell }^{\\ell }x_1 = 0$ .",
"Hence, using $|\\rho _{\\ell }^{\\prime }| \\le \\frac{1}{\\ell ^{\\alpha }}$ and (REF ) we get $I_2=2\\ell \\mu {}^1 + \\frac{2\\delta ^2\\ell ^3}{3} \\int _{-1}^1 |\\rho _{\\ell }W_1^{\\prime \\prime } + W_1^{\\prime }\\rho ^{\\prime }_\\ell |^2\\le 2\\ell \\mu {}^1 + \\frac{2\\delta ^2\\ell ^3}{3} (C_1 + C_2\\ell ^{-2\\alpha }),$ where $C_1, C_2$ are two constants independent of $\\ell $ .",
"Next, for $I_3$ we find, $I_3 = 2\\delta \\int _{\\Omega _{\\ell }}-\\delta W_1^{\\prime }\\rho _{\\ell }\\left[W_1^{\\prime } -x_1\\delta \\left\\lbrace W_1^{\\prime }\\rho ^{\\prime }_\\ell + \\rho _{\\ell }W_1^{\\prime \\prime }\\right\\rbrace \\right]\\\\=-4\\ell \\delta ^2\\int _{-1}^1\\rho _{\\ell }|W_1^{\\prime }|^2 + 2\\delta ^3\\int _{\\Omega _\\ell }x_1 W_1^{\\prime }\\rho _{\\ell }\\left\\lbrace W_1^{\\prime }\\rho ^{\\prime }_\\ell + \\rho _{\\ell }W_1^{\\prime \\prime }\\right\\rbrace = -4\\ell \\delta ^2\\int _{-1}^1\\rho _{\\ell }|W_1^{\\prime }|^2 .$ Finally we compute the term $I$ .",
"$I = \\int _{\\Omega _{\\ell }}\\left(W_1 - \\delta x_1W_1^{\\prime }\\rho _{\\ell } \\right)^2= \\int _{\\Omega _{\\ell }}W_1^2 + \\delta ^2\\int _{\\Omega _{\\ell }} x_1^2\\rho _{\\ell }^2|W_1^{\\prime }|^2 \\\\= 2\\ell + \\frac{2\\ell ^3\\delta ^2}{3}\\int _{-1}^1\\rho _{\\ell }^2|W_1^{\\prime }|^2 \\ge 2\\ell .$ Plugging (REF )–(REF ) in (REF ) yields $\\lambda _\\ell ^1 \\le \\delta ^2 \\int _{-1}^1\\rho _{\\ell }^2|W_1^{\\prime }|^2 +\\mu {}^1 -2\\delta ^2\\int _{-1}^1\\rho _{\\ell }|W_1^{\\prime }|^2 + \\varepsilon (\\ell ),$ where $\\varepsilon (\\ell ) \\rightarrow 0$ as $\\ell \\rightarrow 0$ .",
"Since $\\rho _{\\ell } \\rightarrow 1$ pointwise, passing to the limit $\\ell \\rightarrow 0$ and using dominated convergence for the RHS of (REF ) gives $\\limsup _{\\ell \\rightarrow 0} \\lambda _{\\ell }^1 \\le (1- \\delta ^2)\\mu {}^1\\,.$ Combining (REF ) with (REF ) we obtain the result of the theorem.",
"Now we turn to the proof of Theorem REF .",
"Let $\\ell _0$ and $\\eta $ be two positive constants whose values will be determined later.",
"For $\\ell >\\ell _0+\\eta $ define $\\phi _\\ell $ by $\\phi _\\ell = {\\left\\lbrace \\begin{array}{ll}v_{\\ell _0}(x_1-\\ell +\\ell _0, x_2) & \\text{ on } (\\ell -\\ell _0, \\ell )\\times {}(-1,1) \\,,\\\\\\frac{\\left(x_1- (\\ell -\\ell _0-\\eta )\\right)W_1(x_2)}{\\eta } &\\text{ on } (\\ell -\\ell _0 -\\eta , \\ell -\\ell _0)\\times {}(-1, 1)\\,, \\\\0 & \\text{ on } \\Omega _{\\ell -\\ell _0-\\eta }\\,, \\\\\\frac{\\left(-x_1- (\\ell -\\ell _0-\\eta )\\right)W_1(x_2)}{\\eta } & \\text{ on } \\left(\\ell _0 -\\ell , -\\ell + \\ell _0 + \\eta \\right)\\times (-1, 1)\\,, \\\\v_{\\ell _0}(x_1 + \\ell -\\ell _0, x_2) & \\text{ on } (-\\ell , \\ell _0 - \\ell )\\times (-1,1)\\,,\\end{array}\\right.",
"}$ where $v_{\\ell _0}$ is given by (REF ).",
"We have $\\begin{aligned}\\int _{\\Omega _\\ell }\\phi _\\ell ^2 &= \\int _{\\Omega _\\ell \\setminus \\Omega _{\\ell -\\ell _0}}\\phi _\\ell ^2 +\\int _{\\Omega _{\\ell -\\ell _0}}\\phi _\\ell ^2 \\\\ &=\\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2+2\\left(\\int _{\\ell -\\ell _0-\\eta }^{\\ell -\\ell _0}\\frac{(x_1-\\ell + \\ell _0 + \\eta )^2}{\\eta ^2}dx_1\\right)\\left(\\int _{-1}^1W_1^2\\right)\\\\& = \\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2+ \\frac{2}{3}\\eta \\,,\\end{aligned}$ where we used the fact that $\\phi _\\ell $ is an even function in $x_1$ on $\\Omega _\\ell \\setminus \\Omega _{\\ell -\\ell _0}$ .",
"Also, $\\int _{\\Omega _\\ell }A_\\delta \\nabla \\phi _\\ell .\\nabla \\phi _\\ell = \\int _{\\Omega _{\\ell _0}}A_\\delta \\nabla v_{\\ell _0}.\\nabla v_{\\ell _0}+ \\int _{\\Omega _{\\ell -\\ell _0}}A_\\delta \\nabla \\phi _\\ell .\\nabla \\phi _\\ell \\,.$ Setting $\\mathcal {D} = \\Omega _{\\ell -\\ell _0}\\setminus \\Omega _{\\ell -\\ell _0-\\eta }$ and using the fact that $\\phi _\\ell $ is even in $\\mathcal {D}$ while $\\partial _{x_1}\\phi _\\ell $ is odd on $\\mathcal {D}$ we get $\\begin{aligned}\\int _{\\Omega _{\\ell -\\ell _0}}A_\\delta \\nabla \\phi _\\ell .\\nabla \\phi _\\ell &= \\frac{1}{\\eta ^2}\\int _{\\mathcal {D}}W_1^2+2\\delta \\int _{\\mathcal {D}}\\partial _{x_1}\\phi _\\ell \\partial _{x_2}\\phi _\\ell \\\\&\\phantom{=}+ \\frac{2}{\\eta ^2} \\int _{(\\ell -\\ell _0-\\eta ,\\ell -\\ell _0)\\times (-1,1)}|W_1^{\\prime }|^2(x_1-\\ell +\\ell _0+\\eta )^2\\\\&= \\frac{2}{\\eta }\\int _{-1}^1W_1^2 +\\frac{2\\eta }{3}\\int _{-1}^1|W_1^{\\prime }|^2= \\frac{2}{\\eta } + \\frac{2\\eta \\mu ^1}{3}\\,.\\end{aligned}$ From (REF )–(REF ) we obtain $\\lambda _\\ell ^1 \\le \\frac{\\int _{\\Omega _{\\ell _0}}A_\\delta \\nabla v_{\\ell _0}.\\nabla v_{\\ell _0} +\\frac{2}{\\eta } + \\frac{2\\eta \\mu {}^1}{3} }{\\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2 + \\frac{2}{3}\\eta }\\,.$ Noting that Theorem REF implies that $\\frac{\\int _{\\Omega _{\\ell _0}}A_{\\delta }\\nabla v_{\\ell _0}.\\nabla v_{\\ell _0}}{\\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2} = (1-\\delta ^2)\\mu {}^1 + \\varepsilon (\\ell _0)\\,,$ we obtain from (REF ) that $\\begin{aligned}\\lambda _\\ell ^1 -\\mu ^1&\\le \\frac{\\left\\lbrace (1-\\delta ^2)\\mu {}^1 +\\varepsilon (\\ell _0)\\right\\rbrace \\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2 +\\frac{2}{\\eta } +\\frac{2\\eta \\mu {}^1}{3} }{\\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2 +\\frac{2}{3}\\eta }-\\mu ^1\\\\&= \\frac{(\\varepsilon (\\ell _0)-\\delta ^2\\mu {}^1)\\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2 + \\frac{2}{\\eta } }{\\int _{\\Omega _{\\ell _0}}v_{\\ell _0}^2 + \\frac{2}{3}\\eta }\\,.\\end{aligned}$ Choosing $\\ell _0$ small enough such that $\\varepsilon (\\ell _0)-\\delta ^2\\mu {}^1 < 0$ , and then taking $\\eta $ sufficiently large, makes the RHS of (REF ) equal a negative number, say $-\\delta _0$ .",
"Hence, $\\lambda _\\ell ^1\\le \\mu ^1-\\delta _0$ for $\\ell >\\ell _0+\\eta $ , and the result follows."
],
[
"The gap phenomenon in the general case.",
"In this section we extend the results from Section to a more general framework.",
"We shall use the notation from Section and study the limit $\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1$ for $\\lambda _\\ell ^1$ given by (REF ).",
"As in Section our strategy is to study first the limit as $\\ell $ goes to 0.",
"Theorem 4.1 We have $\\lim _{\\ell \\rightarrow 0} \\lambda _\\ell ^1=\\Lambda ^1$ where $\\Lambda ^1=\\inf \\left\\lbrace \\int _{\\omega }A_{22}(X_2)\\nabla u.\\nabla u-\\frac{|A_{12}(X_2).\\nabla u|^2}{a_{11}(X_2)}:\\,u \\in H_0^1(\\omega ),\\, \\int _{\\omega }u^2=1 \\right\\rbrace .$ The reason why we find $\\Lambda ^1$ as the limiting value will be clarified by the following simple observation.",
"Let $B=\\begin{pmatrix}b_{11} & B_{12} \\\\B^t_{12} & B_{22}\\end{pmatrix}$ be a positive definite $n\\times n$ matrix and represent any vector $z$ in $\\mathbb {R}^n$ as $z=(z_1,Z_2)$ with $Z_2\\in \\mathbb {R}^{n-1}$ .",
"Then, elementary calculus shows that for any fixed $Z_2\\in \\mathbb {R}^{n-1}$ we have $\\min _{z_1\\in \\mathbb {R}} (Bz).z=(B_{22}Z_2).Z_2-\\frac{|B_{12}Z_2|^2}{b_{11}}\\,.$ Furthermore, the minimum in (REF ) is attained for $z_1=-\\frac{B_{12}Z_2}{b_{11}}\\,.$ Applying (REF ) with $B=A(X_2)$ we obtain, for any $\\ell >0$ , $\\begin{aligned}\\int _{\\Omega _\\ell } (A(X_2)\\nabla u_\\ell ).\\nabla u_\\ell &\\ge \\int _{\\Omega _\\ell }(A_{22}(X_2)\\nabla _{X_2}u_\\ell ).\\nabla _{X_2}u_\\ell -\\frac{|A_{12}(X_2)\\nabla _{X_2}u_\\ell |^2}{a_{11}(X_2)}\\\\&\\ge \\Lambda ^1\\int _{\\Omega _\\ell }u_\\ell ^2\\,.\\end{aligned}$ By (REF ) the lower-bound $\\liminf _{\\ell \\rightarrow 0} \\lambda _\\ell ^1\\ge \\Lambda ^1\\,,$ is clear.",
"We note that from the above it follows in particular that $\\Lambda ^1\\ge \\lambda _A\\cdot \\inf \\left\\lbrace \\int _{\\omega }|\\nabla u|^2:\\,u \\in H_0^1(\\omega ),\\, \\int _{\\omega }u^2=1 \\right\\rbrace .$ (see (REF )) and the infimum in (REF ) is actually a minimum, which is realized by a positive function $w_1\\in H^1_0(\\omega )$ .",
"In order to complete the proof of Theorem REF we need to establish the upper-bound part.",
"A natural generalization of the construction used in the proof of Theorem REF would be to use $v_\\ell (x)=w_1(X_2)-\\frac{\\left( A_{12}(X_2).\\nabla w_1\\right) x_1 \\rho _\\ell (X_2)}{a_{11}(X_2)}\\,,$ where $\\rho _l$ is an appropriate cut-off function.",
"However, since the coefficients of the matrix $A(X_2)$ are only assumed to be $L^\\infty $ -functions, the function on the RHS of (REF ) does not necessarily belong to $H^1$ .",
"To overcome this difficulty, we use an approximation argument, motivated by [2].",
"We apply standard mollification to define a family of functions $\\lbrace G_\\varepsilon \\rbrace _{\\varepsilon >0}\\subset C^\\infty _c(\\omega )$ satisfying $\\lim _{\\varepsilon \\rightarrow 0} G_\\varepsilon (X_2)=\\frac{A_{12}(X_2)\\cdot \\nabla w_1}{a_{11}(X_2)}~\\text{ in}L^2(\\omega )\\text{ and a.e..}$ We then define $v_\\ell ^\\varepsilon (x_1,X_2)= w_1(X_2) -G_\\varepsilon (X_2)x_1\\,.$ First notice that $\\int _{\\Omega _\\ell }|v^\\varepsilon _\\ell |^2 = \\int _{-\\ell }^\\ell \\int _{\\omega }w_1^2 - 2x_1w_1G_\\varepsilon + \\left( x_1G_\\varepsilon \\right)^2\\ge 2\\ell \\int _{\\omega }w_1^2= 2\\ell ,$ since $\\int _{-\\ell }^{\\ell }x_1\\,dx_1 = 0.$ Now $\\int _{\\Omega _\\ell }A\\nabla v_\\ell ^\\varepsilon .\\nabla v_\\ell ^\\varepsilon &= \\int _{\\Omega _\\ell } a_{11}(\\partial _{x_1} v_\\ell ^\\varepsilon )^2 + 2(A_{12}.\\nabla _{X_2}v_\\ell ^\\varepsilon ) \\partial _{x_1}v_\\ell ^\\varepsilon + (A_{22}\\nabla _{X_2} v_\\ell ^\\varepsilon ).\\nabla _{X_2} v_\\ell ^\\varepsilon \\\\&= I_1(\\varepsilon ) + I_2(\\varepsilon ) + I_3(\\varepsilon )\\,.$ For the first integral we have $I_1 (\\varepsilon )= \\int _{\\Omega _\\ell }a_{11} G_\\varepsilon ^2 = 2\\ell \\int _\\omega a_{11} G_\\varepsilon ^2\\,.$ For the second integral, $I_2(\\varepsilon ) =2\\int _{-\\ell }^\\ell \\int _{\\omega } A_{12}.\\Big \\lbrace \\nabla w_1 -x_1\\nabla G_\\varepsilon (X_2) \\Big \\rbrace \\Big \\lbrace -G_\\varepsilon (X_2)\\Big \\rbrace \\,.$ Since the integral of the term containing $x_1$ vanishes, we get $I_2 (\\varepsilon )= -4\\ell \\int _\\omega (A_{12}.",
"\\nabla w_1)G_\\varepsilon \\,.$ For the last integral we have (after dropping the term with the vanishing integral), $I_3(\\varepsilon ) = \\int _{-\\ell }^\\ell \\int _{\\omega }(A_{22}\\nabla w_1).\\nabla w_1+ x_1^2(A_{22}\\nabla G_\\varepsilon ) .\\nabla G_\\varepsilon \\\\=2\\ell \\Big \\lbrace \\int _\\omega (A_{22}\\nabla w_1).\\nabla w_1+\\frac{\\ell ^2}{3}\\int _\\omega (A_{22}\\nabla G_\\varepsilon ).\\nabla G_\\varepsilon \\Big \\rbrace \\,.$ By (REF )–(REF ) we deduce that $\\limsup _{\\ell \\rightarrow 0} \\lambda _\\ell ^1\\le \\limsup _{\\ell \\rightarrow 0} \\frac{\\int _{\\Omega _\\ell } A\\nabla v_\\ell ^\\varepsilon .\\nabla v_\\ell ^\\varepsilon }{\\int _{\\Omega _\\ell } |v_\\ell ^\\varepsilon |^2}\\le \\\\\\int _\\omega a_{11}\\left( G_\\varepsilon \\right)^2-2\\int _\\omega (A_{12}.",
"\\nabla w_1)G_\\varepsilon +\\int _\\omega \\big (A_{22}\\nabla w_1\\big ).\\nabla w_1\\,.$ Passing to the limit $\\varepsilon \\rightarrow 0$ in (REF ), using (REF ), gives $\\limsup _{\\ell \\rightarrow 0} \\lambda _\\ell ^1\\le \\int _{\\omega }(A_{22}\\nabla w_1).\\nabla w_1-\\frac{|A_{12}\\nabla w_1|^2}{a_{11}}=\\Lambda ^1\\,,$ which together with (REF ) yields the result.",
"Remark 4.1 Replacing (REF ) by $\\tilde{v}_\\ell ^\\varepsilon (x_1,X_2)= W_1(X_2) -\\tilde{G}_\\varepsilon (X_2)x_1\\,,$ where $\\tilde{G}_\\varepsilon $ is defined as in (REF ), but with $w_1$ replacing $W_1$ , and carrying out the same computation as in the last part of the proof of Theorem REF yields $\\inf _{\\varepsilon >0} \\lim _{\\ell \\rightarrow 0}\\frac{\\int _{\\Omega _\\ell } (A\\nabla \\tilde{v}_\\ell ^\\varepsilon ).\\nabla \\tilde{v}_\\ell ^\\varepsilon }{\\int _{\\Omega _\\ell }|\\tilde{v}_\\ell ^\\varepsilon |^2}=\\int _{\\omega }(A_{22}\\nabla W_1).\\nabla W_1-\\frac{|A_{12}\\nabla W_1|^2}{a_{11}}\\,.$ Our next theorem provides an analog of Theorem REF to the general case.",
"Theorem 4.2 We have $\\limsup _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1 < \\mu ^1,$ provided the following condition holds, $A_{12}.\\nabla W_1\\lnot \\equiv 0\\text{ a.e.",
"on } \\omega .$ In case (REF ) does not hold we have $\\lambda _\\ell ^1 = \\mu ^1$ for all $\\ell >0$ .",
"Remark 4.2 It is easy to construct examples where condition (REF ) doesn't hold.",
"Take for example for $\\omega $ the unit disc in $\\mathbb {R}^2$ .",
"For $A_{22}=\\begin{pmatrix} 1 & 0\\\\0 & 1\\end{pmatrix}$ , the eigenfunction $W_1$ is radially symmetric.",
"We use polar coordinates on $\\omega $ and represent each $X_2$ as $X_2=r(\\cos \\theta ,\\sin \\theta )$ .",
"Taking $a_{11}=1$ and $A_{12}(X_2)=t(-\\sin \\theta ,\\cos \\theta )$ for $|t|$ small enough (in order for the uniform ellipticity condition (REF ) to hold for the 3 by 3 matrix $A$ ) yields an example for which (REF ) doesn't hold.",
"(i) Assume first that (REF ) holds.",
"Then, $\\Lambda _1<\\mu {}^1.$ Indeed, this follows from $\\Lambda _1\\le \\int _{\\omega }A_{22}(X_2)\\nabla W_1.\\nabla W_1-\\frac{|A_{12}(X_2)\\nabla W_1|^2}{a_{11}(X_2)}<\\int _{\\omega }A_{22}(X_2)\\nabla W_1.\\nabla W_1=\\mu {}^1\\,.$ By the proof of Theorem REF there exist positive values of $\\ell _0$ and $\\varepsilon _0$ such that $\\tilde{v}_{\\ell _0}^{\\varepsilon _0}$ defined by (REF ) satisfies $\\int _{\\Omega _{\\ell _0}} A\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}.\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}<\\mu {}^1\\int _{\\Omega _{\\ell _0}}|\\tilde{v}_{\\ell _0}^{\\varepsilon _0}|^2\\,.$ Notice that $\\tilde{v}_{\\ell _0}^{\\varepsilon _0}(0,X_2)=W_1(X_2)$ .",
"Let $\\eta >0$ be a parameter whose value will be determined later.",
"For $\\ell >\\ell _0+\\eta $ define $\\phi _\\ell $ as follows, $\\phi _\\ell ={\\left\\lbrace \\begin{array}{ll}\\tilde{v}_{\\ell _0}^{\\varepsilon _0}(x_1-\\ell +\\ell _0, X_2) &\\text{ on } (\\ell -\\ell _0,\\ell )\\times \\omega \\,, \\\\\\frac{\\left(x_1- (\\ell -\\ell _0-\\eta )\\right) W_1(X_2)}{\\eta } &\\text{ on } (\\ell -\\ell _0 -\\eta , \\ell -\\ell _0)\\times \\omega \\,,\\\\0 &\\text{ on } \\Omega _{\\ell -\\ell _0-\\eta }\\,, \\\\\\frac{\\left(-x_1- (\\ell -\\ell _0-\\eta )\\right)W_1(X_2)}{\\eta } & \\text{ on } \\left(\\ell _0 -\\ell , -(\\ell -\\ell _0-\\eta )\\right)\\times \\omega \\,,\\\\\\tilde{v}_{\\ell _0}^{\\varepsilon _0}(x_1 + \\ell -\\ell _0, X_2) &\\text{ on } (-\\ell , \\ell _0 - \\ell )\\times \\omega \\,.\\end{array}\\right.",
"}$ Since $\\int _{\\Omega _\\ell \\setminus \\Omega _{\\ell -\\ell _0}}\\phi _\\ell ^2= \\int _{\\Omega _{\\ell _0}}|\\tilde{v}_{\\ell _0}^{\\varepsilon _0}|^2\\,,$ and $\\int _{\\Omega _{\\ell -\\ell _0}}\\phi _\\ell ^2=2\\left(\\int _{\\ell -\\ell _0-\\eta }^{\\ell -\\ell _0}\\frac{(x_1-\\ell + \\ell _0 + \\eta )^2}{\\eta ^2}dx_1\\right)\\left(\\int _{\\omega }W_1^2\\,dX_2\\right) = \\frac{2}{3}\\eta \\,,$ we have $\\int _{\\Omega _\\ell }\\phi _\\ell ^2 = \\int _{\\Omega _{\\ell _0}}|\\tilde{v}_{\\ell _0}^{\\varepsilon _0}|^2 + \\frac{2}{3}\\eta \\,.$ Similarly $\\int _{\\Omega _\\ell }A\\nabla \\phi _\\ell .\\nabla \\phi _\\ell = \\int _{\\Omega _{\\ell _0}}A\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}.\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}+ \\int _{\\Omega _{\\ell -\\ell _0}}A\\nabla \\phi _\\ell .\\nabla \\phi _\\ell \\, .$ Setting $D = \\Omega _{\\ell -\\ell _0}\\setminus \\Omega _{\\ell -\\ell _0-\\eta }$ and $D^{+} = (\\ell - \\ell _0 -\\eta , \\ell -\\ell _0)\\times {}\\omega $ , the last integral above can be written as $\\int _{\\Omega _{\\ell -\\ell _0}}A\\nabla \\phi _\\ell .\\nabla \\phi _\\ell = \\frac{1}{\\eta ^2}\\int _{D}a_{11}W_1^2+2\\int _{D}A_{12}.\\nabla _{X_2}\\phi _\\ell \\partial _{x_1} \\phi _\\ell \\\\+ \\frac{2}{\\eta ^2}\\int _{D^+} (x_1-\\ell +\\ell _0 +\\eta )^2A_{22}\\nabla W_1.\\nabla W_1.$ The second integral vanishes since its integrand is an odd function of $x_1$ on $D$ .",
"Therefore, $\\int _{\\Omega _{\\ell -\\ell _0}}A\\nabla \\phi _\\ell .\\nabla \\phi _\\ell = \\frac{2}{\\eta } \\int _{\\omega } a_{11} W_1^2 + \\frac{2\\eta }{3}\\int _{\\omega } A_{22}\\nabla W_1.\\nabla W_1= \\frac{2}{\\eta } \\int _{\\omega } a_{11} W_1^2 + \\frac{2\\eta \\mu {}^1}{3}\\,.$ Combining (REF ), (REF ) and (REF ) we obtain $\\lambda _\\ell ^1 \\le \\frac{\\int _{\\Omega _\\ell }A\\nabla \\phi _\\ell .\\nabla \\phi _\\ell }{\\int _{\\Omega _\\ell }\\phi _\\ell ^2}\\le \\frac{\\int _{\\Omega _{\\ell _0}}A\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}.\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0} +\\frac{2}{\\eta } \\int _{\\omega } a_{11}W_1^2 +\\frac{2}{3}\\eta \\mu {}^1}{\\int _{\\Omega _{\\ell _0}}|\\tilde{v}_{\\ell _0}^{\\varepsilon _0}|^2+ \\frac{2}{3}\\eta }\\,.$ Therefore, $\\lambda _\\ell ^1 - \\mu ^1 \\le \\frac{ \\int _{\\Omega _{\\ell _0}}A\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}.\\nabla \\tilde{v}_{\\ell _0}^{\\varepsilon _0}-\\mu ^1 \\int _{\\Omega _{\\ell _0}}|\\tilde{v}_{\\ell _0}^{\\varepsilon _0}|^2 + \\frac{2}{\\eta } \\int _{\\omega }a_{11}W_1^2}{\\int _{\\Omega _{\\ell _0}}|\\tilde{v}_{\\ell _0}^{\\varepsilon _0}|^2 + \\frac{2}{3}\\eta }.$ By (REF ) it is clear that we can fix a large enough value for $\\eta $ such that the RHS of (REF ) is negative, and the result for case (i) follows.",
"(ii) By (REF ) we have $\\Lambda _1\\le \\lambda _\\ell ^1$ for all $\\ell >0$ .",
"On the other hand, using $u(x)=W_1(X_2)$ as a test function in (REF ) gives $\\lambda _\\ell ^1\\le \\mu {}^1$ .",
"Thus we have, $\\Lambda _1\\le \\lambda _\\ell ^1\\le \\mu {}^1\\,,\\quad \\forall \\ell >0\\,.$ In view of (REF ), the result for the case where (REF ) doesn't hold would follow once we show that in this case $\\Lambda _1=\\mu ^1$ .",
"The Euler-Lagrange equation for an eigenfunction $v$ of the quadratic form in (REF ), with eigenvalue $\\lambda $ is $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A_{22}\\nabla v)+\\operatorname{div}((A_{12}.\\nabla v)A_{12}^t/{a_{11}})=\\lambda v~\\text{ in }\\omega \\,,\\\\&v=0~\\text{ on }\\partial \\omega \\,.\\end{aligned}\\right.$ Of course $v=w_1$ satisfies (REF ) with $\\lambda =\\Lambda _1$ .",
"But since we assume that (REF ) doesn't hold, $v=W_1$ is also a solution of (REF ) with $\\lambda =\\mu ^1$ .",
"However, only the first eigenvalue of the problem (REF ) can have a positive eigenfunction, so we must have $\\Lambda _1=\\mu ^1$ as claimed."
],
[
"Characterization of the limit $\\lim _{\\ell \\rightarrow \\infty } \\lambda ^1_\\ell $",
"In this section we obtain more precise results on the asymptotic behavior of the eigenfunctions$\\lbrace u_\\ell \\rbrace $ and the eigenvalues $\\lbrace \\lambda _\\ell ^1\\rbrace $ as $\\ell $ goes to infinity.",
"We shall see that when (REF ) holds, the eigenfunctions decay to zero in the bulk of the cylinder and concentration occurs near the bases of the cylinder.",
"We denote by $[x]$ the integer part of $x$ .",
"Theorem 5.1 Assume (REF ) holds.",
"Then, there exist $\\alpha \\in (0,1)$ and a positive constant $c$ such that for $\\ell >\\ell _0$ we have, for every $0<r\\le \\ell -1$ , $\\int _{\\Omega _r} u_\\ell ^2 &\\le \\alpha ^{[\\ell -r]}\\,,\\\\{and}\\int _{\\Omega _{r}} |\\nabla u_\\ell |^2&\\le c \\alpha ^{[\\ell -r]}\\,.$ Let $\\ell $ and $\\ell ^{\\prime }$ satisfy $0< \\ell ^{^{\\prime }} \\le \\ell -1$ .",
"Define $\\rho _{\\ell ^{^{\\prime }}}=\\rho _{\\ell ^{^{\\prime }}}(x_1)$ by $\\rho _{\\ell ^{^{\\prime }}}(x_1)={\\left\\lbrace \\begin{array}{ll}1 & |x_1|\\le \\ell ^{\\prime }\\,,\\\\\\ell ^{\\prime }+1-|x_1|& |x_1|\\in (\\ell ^{\\prime },\\ell ^{\\prime }+1)\\,,\\\\0 & |x_1|\\ge \\ell ^{\\prime }+1\\,.\\end{array}\\right.",
"}$ Using $v =\\rho _{\\ell ^{^{\\prime }}}^2u_\\ell \\in V(\\Omega _\\ell )$ in (REF ), we get $\\int _{\\Omega _\\ell } (A\\nabla u_\\ell ).",
"\\nabla (\\rho _{\\ell ^{^{\\prime }}}^2u_\\ell ) = \\lambda _\\ell ^1 \\int _{\\Omega _\\ell }\\rho _{\\ell ^{^{\\prime }}}^2u_\\ell ^2\\,,$ i.e., $ \\int _{\\Omega _\\ell } \\big (A \\nabla (\\rho _{\\ell ^{^{\\prime }}}u_\\ell )\\big ).\\nabla (\\rho _{\\ell ^{^{\\prime }}}u_\\ell ) -\\int _{\\Omega _\\ell }u_\\ell ^2 (A\\nabla \\rho _{\\ell ^{^{\\prime }}}).\\nabla \\rho _{\\ell ^{^{\\prime }}} = \\lambda _\\ell ^1\\int _{\\Omega _\\ell }\\rho _{\\ell ^{^{\\prime }}}^2u_\\ell ^2\\,.$ Since $\\rho _{\\ell ^{^{\\prime }}} u_\\ell \\in H_0^1(\\Omega _\\ell )$ , by the Rayleigh quotient characterization of $\\sigma _\\ell ^1$ (see (REF )) we have $\\sigma _\\ell ^1 \\int _{\\Omega _\\ell }u_\\ell ^2 \\rho _{\\ell ^{^{\\prime }}}^2 \\le \\int _{\\Omega _\\ell } A\\nabla (\\rho _{\\ell ^{^{\\prime }}}u_\\ell ).\\nabla (\\rho _{\\ell ^{^{\\prime }}}u_\\ell )\\,.$ Combining (REF )–(REF ) with (REF ) we get $\\begin{aligned}(\\sigma _\\ell ^1 -\\lambda _\\ell ^1) \\int _{\\Omega _\\ell } u_\\ell ^2 \\rho _{\\ell ^{^{\\prime }}}^2 \\le \\int _{\\Omega _\\ell }u_\\ell ^2 (A\\nabla \\rho _{\\ell ^{^{\\prime }}}).\\nabla \\rho _{\\ell ^{^{\\prime }}}&= \\int _{\\Omega _{\\ell ^{\\prime }+1}\\setminus \\Omega _{\\ell ^{\\prime }}}u_\\ell ^2 (A\\nabla \\rho _{\\ell ^{^{\\prime }}}).\\nabla \\rho _{\\ell ^{^{\\prime }}}\\\\&\\le C_A \\int _{\\Omega _{\\ell ^{\\prime }+1}\\setminus \\Omega _{\\ell ^{\\prime }}}u_\\ell ^2\\,.\\end{aligned}$ By (REF ) and (REF ) there exists $\\beta >0$ such that for $\\ell >\\ell _0$ we have $\\sigma _\\ell ^1 -\\lambda _\\ell ^1\\ge \\beta $ .",
"Therefore, from (REF ) we deduce that $(C_A+\\beta )\\int _{\\Omega _{\\ell ^{^{\\prime }}}} u_\\ell ^2 \\le C_A \\int _{\\Omega _{\\ell ^{^{\\prime }}+1}}u_\\ell ^2\\,.$ This leads to $\\int _{\\Omega _{\\ell ^{^{\\prime }}}} u_\\ell ^2 \\le \\alpha \\int _{\\Omega _{\\ell ^{^{\\prime }} + 1}} u_\\ell ^2\\,,$ with $\\alpha =\\frac{C_A}{C_A+\\beta }<1$ .",
"Applying (REF ) successively for $\\ell ^{^{\\prime }} = r, r+1, \\ldots ,r+ [\\ell -r]-1$ yields $\\int _{\\Omega _r} u_\\ell ^2 \\le \\alpha ^{[\\ell -r]} \\int _{\\Omega _\\ell }u_\\ell ^2 =\\alpha ^{[\\ell -r]} \\,.$ To prove (), we fix $r\\in (0,\\ell -2)$ and then use (REF ), with $\\ell ^{\\prime }=r$ , combined with (REF ) and (REF ), to obtain $\\lambda _A \\int _{\\Omega _r}|\\nabla u_\\ell | ^2\\le \\int _{\\Omega _\\ell } A \\nabla (\\rho _{r}u_\\ell ).\\nabla ( \\rho _{r}u_\\ell )\\\\ =\\int _{\\Omega _\\ell }u_\\ell ^2 (A\\nabla \\rho _{r}).\\nabla \\rho _{r} +\\lambda _\\ell ^1\\int _{\\Omega _\\ell } \\rho _{r}^2u_\\ell ^2\\le (C_A+\\mu ^1)\\int _{\\Omega _{r+1}}u_\\ell ^2\\,.$ Finally, () follows from (REF )–(REF ) for $r\\le \\ell -2$ .",
"Choosing a step size of $\\frac{1}{2}$ in the first part of the proof would allow $r\\le \\ell -1$ .",
"The decay of the eigenfunction in the bulk immediately implies concentration near the two ends of the cylinder.",
"Corollary 5.1 If (REF ) holds then for every $r\\in (0,\\ell -1]$ we have $\\int _{\\Omega _\\ell \\setminus \\Omega _{r}} u_\\ell ^2 \\ge 1-\\alpha ^{[\\ell -r]}~\\text{ and }~\\int _{\\Omega _\\ell \\setminus \\Omega _{\\ell -1}} \\!\\!\\!A\\nabla u_\\ell .\\nabla u_\\ell \\ge \\lambda _\\ell ^1-c_1\\alpha ^{[\\ell -r]}\\,.$ To have a more precise description of the asymptotic behavior of $\\lambda _\\ell ^1$ we introduce two variational problems on semi-infinite cylinders.",
"Set $\\Omega _{\\infty }^+ = (0, \\infty ) \\times \\omega ~\\text{ and }~ \\Omega _{\\infty }^- = (-\\infty ,0) \\times \\omega \\,,$ and denote the corresponding lateral parts of the boundary by $\\gamma _\\infty ^{+} = (0, \\infty )\\times \\partial \\omega ~\\text{ and }~ \\gamma _\\infty ^{-} = (-\\infty ,0)\\times \\partial \\omega \\,.$ Define the spaces $V(\\Omega _{\\infty }^\\pm ) := \\lbrace u\\in H^1(\\Omega _{\\infty }^\\pm )\\,: \\, u=0 \\text{ on } \\gamma _\\infty ^{\\pm } \\rbrace \\,,$ and set $\\nu _\\infty ^\\pm = \\inf _{0\\ne u \\in V(\\Omega _{\\infty }^\\pm )} \\frac{\\int _{\\Omega _\\infty ^\\pm } A\\nabla u.\\nabla u}{\\int _{\\Omega _\\infty ^\\pm } u^2}\\,.$ Remark 5.1 In case property (S) holds (see Definition REF ) we clearly have $\\nu _\\infty ^+=\\nu _\\infty ^-$ as we can use the transformation $v(x_1,X_2)\\mapsto v(-x_1,-X_2)$ to pass from a function in $V(\\Omega _{\\infty }^+ )$ to a function in $V(\\Omega _{\\infty }^-)$ (and vice versa) that has the same Rayleigh quotient.",
"In general we can only assert that $\\nu _\\infty ^-=\\tilde{\\nu }_\\infty ^+$ where $\\tilde{\\nu }_\\infty ^+$ is defined as in (REF ), but with $A$ being replaced by $\\tilde{A}$ , given by $\\begin{aligned}\\tilde{A}(X_2)=\\begin{pmatrix}a_{11}(X_2) & -A_{12}(X_2) \\\\-A_{12}^t(X_2) & A_{22}(X_2)\\end{pmatrix}\\,.\\end{aligned}$ This is easily seen by applying the transformation $v(x_1,X_2)\\mapsto v(-x_1,X_2)$ .",
"The next lemma gives the possible range of values for $\\nu _\\infty ^\\pm $ .",
"Lemma 5.1 We have $0<\\nu _\\infty ^\\pm \\le \\mu ^1\\,.$ By Remark REF it is enough to consider $\\nu _\\infty ^+$ .",
"The fact that $\\nu _\\infty ^+>0$ follows from the Poincaré inequality.",
"In order to show that $\\nu _\\infty ^+\\le \\mu ^1$ we set for each $\\varepsilon >0$ , $v_\\varepsilon (x)=e^{-\\varepsilon x_1} W_1(X_2)\\,.$ Clearly $v_\\varepsilon \\in V(\\Omega _\\infty ^+)$ and a direct computation gives $\\begin{aligned}\\int _{\\Omega _\\infty ^+} A\\nabla v_\\varepsilon .\\nabla v_\\varepsilon &=\\int _{\\Omega _\\infty ^+}\\!\\!",
"e^{-2\\varepsilon x_1}\\Big (a_{11}\\varepsilon ^2W_1^2-2\\varepsilon (A_{12}.\\nabla W_1)W_1+A_{22}\\nabla W_1.\\nabla W_1\\Big )\\\\&=(\\int _0^\\infty \\!\\!",
"e^{-2\\varepsilon x_1})\\Big (\\mu ^1+\\varepsilon ^2 \\int _\\omega a_{11}W_1^2-2\\varepsilon \\int _\\omega (A_{12}.\\nabla W_1)W_1\\Big )\\,,\\end{aligned}$ and $\\int _{\\Omega _\\infty ^+} v_\\varepsilon ^2=\\int _0^\\infty e^{-2\\varepsilon x_1}\\,\\big (=\\frac{1}{2\\varepsilon }\\big )\\,.$ By (REF )–(REF ) we obtain $\\frac{ \\int _{\\Omega _\\infty ^+} A\\nabla v_\\varepsilon .\\nabla v_\\varepsilon }{\\int _{\\Omega _\\infty ^+} v_\\varepsilon ^2}=\\mu ^1-2\\varepsilon \\int _\\omega (A_{12}.\\nabla W_1)W_1+\\varepsilon ^2 \\int _\\omega a_{11}W_1^2\\,,$ so by sending $\\varepsilon $ to 0 we deduce that $\\nu _\\infty ^+\\le \\mu ^1$ .",
"It is easy to identify $\\nu _\\infty ^\\pm $ with the limits, as $\\ell \\rightarrow \\infty $ , of certain minimization problems on $\\Omega _\\ell ^{\\pm }$ .",
"This is the content of the next lemma (see (REF ) and (REF ) for the definitions of $\\gamma _\\ell ^\\pm $ and $\\Gamma _\\ell ^\\pm $ ).",
"Lemma 5.2 We have $\\nu _\\infty ^\\pm =\\lim _{\\ell \\rightarrow \\infty }\\tilde{\\lambda }_\\ell ^{1,\\pm }\\,,$ where $\\tilde{\\lambda }_\\ell ^{1,\\pm } = \\inf \\lbrace \\int _{\\Omega _\\ell ^{\\pm }}A\\nabla u.\\nabla u \\,: \\,u\\in H^1(\\Omega _\\ell ^{\\pm }),\\, \\int _{\\Omega _\\ell ^{\\pm }}u^2 = 1, u=0 \\text{ on}\\gamma _\\ell ^{\\pm }\\cup \\Gamma _\\ell ^\\pm \\rbrace \\,.$ Remark 5.2 It is a standard fact that the infimum in (REF ) is actually attained.",
"The unique positive normalized minimizers will be denoted by $\\tilde{u}_\\ell ^\\pm $ .",
"We present the proof for $\\tilde{\\lambda }_\\ell ^{1,+}$ as the proof for $\\tilde{\\lambda }_\\ell ^{1,-}$ is completely analogous.",
"Note first that the limit $\\lim _{\\ell \\rightarrow \\infty }\\tilde{\\lambda }_\\ell ^{1,+}$ exists since the function $\\ell \\mapsto \\tilde{\\lambda }_\\ell ^{1,+}$ is non increasing.",
"Indeed, if $\\ell _1<\\ell _2$ then any admissible function in (REF ) for $\\tilde{\\lambda }_{\\ell _1}^{1,+}$ can be extended to an admissible function for $\\tilde{\\lambda }_{\\ell _2}^{1,+}$ by setting it to zero on $\\Omega _{\\ell _2}^+\\setminus \\Omega _{\\ell _1}^+$ .",
"A similar argument shows that $\\tilde{\\lambda }_\\ell ^{1,+}\\ge \\nu _\\infty ^+$ , for any $\\ell >0$ .",
"On the other hand, the density of the space $V_s(\\Omega _{\\infty }^+) = \\lbrace u\\in C^{\\infty }(\\Omega _{\\infty }^+)\\cap V(\\Omega _\\infty ^+)\\,:\\,\\exists M=M(u) > 0 \\text{ s.t. }",
"u = 0 \\text{ on } (M, \\infty )\\times \\omega \\rbrace \\,,$ in $V(\\Omega _\\infty ^+)$ implies that for each $u\\in V(\\Omega _\\infty ^+)\\setminus \\lbrace 0\\rbrace $ and any $\\varepsilon >0$ we can find an $\\ell _\\varepsilon $ and $v_\\varepsilon \\in V_s(\\Omega _{\\infty }^+)$ with $\\text{supp}(v_\\varepsilon )\\subset \\Omega _{\\ell _\\varepsilon }^{+}$ such that $\\left|\\frac{\\int _{\\Omega _\\infty ^+} (A\\nabla v_\\varepsilon ) .\\nabla v_\\varepsilon }{\\int _{\\Omega _\\infty ^+} v_\\varepsilon ^2}-\\frac{\\int _{\\Omega _\\infty ^+} (A\\nabla u).\\nabla u}{\\int _{\\Omega _\\infty ^+} u^2}\\right|\\le \\varepsilon \\,,$ and (REF ) follows (for $\\tilde{\\lambda }_\\ell ^{1,+}$ ).",
"Our next result complements the result of Theorem REF in two ways: by showing that the limit $\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1$ exists and by identifying its value.",
"Theorem 5.2 We have $\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1 =\\min (\\nu _\\infty ^+,\\nu _\\infty ^-)\\,.$ (i) We shall first show that $\\limsup _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1\\le \\min (\\nu _\\infty ^+,\\nu _\\infty ^-)\\,.$ We may assume w.l.o.g.",
"that $\\nu _\\infty ^+=\\min (\\nu _\\infty ^+,\\nu _\\infty ^-)$ .",
"Given $\\varepsilon >0$ we may find by Lemma REF an $\\ell _\\varepsilon >1/\\varepsilon $ such that $\\tilde{\\lambda }_{\\ell _\\varepsilon }^{1,+}\\le \\nu _\\infty ^++\\varepsilon $ .",
"Since $\\lambda _{\\ell /2}^1\\le \\tilde{\\lambda }_\\ell ^{1,+}$ by the definitions (REF ) and (REF ), we easily deduce (REF ).",
"(ii) We now treat the case where (REF ) holds.",
"Let $u_\\ell $ denote the positive normalized minimizer in (REF ).",
"Define $v_\\ell (x)=\\rho (x_1)u_\\ell (x)$ where $\\rho $ is given by $\\rho (x_1)={\\left\\lbrace \\begin{array}{ll}0 & x_1\\le -1\\,,\\\\1+x_1& x_1\\in (-1,0)\\,,\\\\1 & x_1\\ge 0\\,.\\end{array}\\right.",
"}$ By (REF ) and (REF ) we have $\\int _{(-1,\\ell )\\times \\omega } (A\\nabla v_\\ell ).\\nabla v_\\ell \\le \\int _{\\Omega _\\ell ^+}(A\\nabla u_\\ell ).\\nabla u_\\ell +C_A \\int _{(-1,0)\\times \\omega }|\\nabla v_\\ell |^2\\,.$ Define $w_{\\ell +1}(x_1,X_2)=v_\\ell (x_1+\\ell ,X_2)$ on $\\Omega _{\\ell +1}^-$ and notice that it is an admissible function for the infimum defining $\\tilde{\\lambda }_{\\ell +1}^{1,-}$ (see (REF )).",
"By (REF ) and (REF )–() we obtain, for some positive constant $C$ , $\\int _{\\Omega _{\\ell +1}^-} (A\\nabla w_{\\ell +1}).\\nabla w_{\\ell +1}\\le \\int _{\\Omega _\\ell ^+}(A\\nabla u_\\ell ).\\nabla u_\\ell +C\\alpha ^\\ell \\,.$ Denote $N_\\ell ^\\pm =\\int _{\\Omega _\\ell ^\\pm }(A\\nabla u_\\ell ).\\nabla u_\\ell ~\\text{ and }~ D_\\ell ^\\pm =\\int _{\\Omega _\\ell ^\\pm } |u_\\ell |^2\\,,$ so that in particular we have $N_\\ell ^++N_\\ell ^-=\\lambda _\\ell ^1~\\text{ and }~D_\\ell ^++D_\\ell ^-=1\\,.$ By (REF ) and an analogous construction on $\\Omega _{\\ell +1}^+$ we have $\\tilde{\\lambda }_{\\ell +1}^{1,-}\\le \\frac{N_\\ell ^++C\\alpha ^\\ell }{D_\\ell ^+}~\\text{ and }~\\tilde{\\lambda }_{\\ell +1}^{1,+}\\le \\frac{N_\\ell ^-+C\\alpha ^\\ell }{D_\\ell ^-}\\,.$ From (REF ) and (REF ) it follows that $\\min \\lbrace \\tilde{\\lambda }_{\\ell +1}^{1,-},\\tilde{\\lambda }_{\\ell +1}^{1,+}\\rbrace \\le D_\\ell ^+\\tilde{\\lambda }_{\\ell +1}^{1,-}+D_\\ell ^-\\tilde{\\lambda }_{\\ell +1}^{1,+}\\le \\lambda _\\ell ^1+C\\alpha ^\\ell \\,.$ Passing to the limit $\\ell \\rightarrow \\infty $ in (REF ) and using Lemma REF yields $\\min (\\nu _\\infty ^+,\\nu _\\infty ^-)\\le \\liminf _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1\\,,$ which combined with (REF ) clearly implies (REF ) (when (REF ) holds).",
"(iii) Finally, we turn to the case where (REF ) doesn't hold.",
"In this case we know already from Theorem REF that $\\lambda _\\ell ^1=\\mu ^1$ for all $\\ell $ .",
"The proof of (REF ) will be clearly completed if we show that $ \\nu _\\infty ^+=\\nu _\\infty ^-=\\mu ^1$ .",
"We shall only show that $\\nu _\\infty ^+=\\mu ^1$ as the argument for $\\nu _\\infty ^-$ is identical.",
"By Lemma REF we have $\\nu _\\infty ^+\\le \\mu ^1$ .",
"For the reverse inequality we notice that in our case, for any $u\\in V(\\Omega _\\infty ^+)$ we have, $\\int _{\\Omega _\\infty ^+} A\\nabla u.\\nabla u=\\int _{\\Omega _\\infty ^+}a_{11}u_{x_1}^2+(A_{22}\\nabla _{X_2} u).\\nabla _{X_2} u\\ge \\mu ^1\\int _{\\Omega _\\infty ^+}u^2\\,,$ implying that $\\nu _\\infty ^+\\ge \\mu ^1$ .",
"The argument of the above proof can be used to derive an additional information that will be useful in the next section.",
"Proposition 5.1 If $\\nu _\\infty ^+<\\nu _\\infty ^-$ then $\\lim _{\\ell \\rightarrow \\infty }\\int _{\\Omega _\\ell ^+}|\\nabla u_\\ell |^2+|u_\\ell |^2=0$ .",
"We use the same notation as in the proof of Theorem REF .",
"Passing to the limit $\\ell \\rightarrow \\infty $ in (REF ), using Lemma REF and (REF ) yields $\\Big (\\limsup _{\\ell \\rightarrow \\infty }D_\\ell ^+\\Big )\\nu _\\infty ^-+\\Big (1-\\limsup _{\\ell \\rightarrow \\infty }D_\\ell ^+)\\nu _\\infty ^+\\le \\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1=\\nu _\\infty ^+\\,,$ so necessarily $ \\limsup _{\\ell \\rightarrow \\infty } D_\\ell ^+=0$ .",
"Next, by (REF ) we have for $\\ell $ large, $\\frac{N_\\ell ^+}{D_\\ell ^-}+\\tilde{\\lambda }_{\\ell +1}^{1,+}-C\\alpha ^\\ell \\le \\frac{N_\\ell ^++N_\\ell ^-}{D_\\ell ^-}\\le \\frac{N_\\ell ^++N_\\ell ^-}{D_\\ell ^++D_\\ell ^-}=\\lambda _\\ell ^1\\,.$ Since in our case, $\\lim _{\\ell \\rightarrow \\infty }\\lambda _\\ell ^1=\\lim _{\\ell \\rightarrow \\infty }\\tilde{\\lambda }_{\\ell +1}^{1,+}=\\nu _\\infty ^+$ , and we know already that $\\lim _{\\ell \\rightarrow \\infty }D_\\ell ^-=1$ , we deduce from (REF ) that $\\lim _{\\ell \\rightarrow \\infty }N_\\ell ^+=0$ ."
],
[
"The problem on a semi-infinite cylinder",
"In this section we further investigate the minimization problem (REF ).",
"By Remark REF it is enough to consider $\\nu _\\infty ^+$ .",
"There are two main questions we are interested in.",
"First, we want to identify the conditions under which the infimum in (REF ) is attained.",
"Second, we would like to know when the inequality $\\nu _\\infty ^+<\\mu ^1$ hold.",
"The next proposition shows that the two questions are closely related to each other.",
"Proposition 6.1 If $\\nu _\\infty ^+<\\mu ^1 \\,,$ then $\\nu _\\infty ^+$ is attained.",
"The minimizer $\\tilde{u}^+$ is unique up to multiplication by a constant, has constant sign and satisfies $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A(X_2)\\nabla \\tilde{u}^+)=\\nu _\\infty ^+\\tilde{u}^+ \\quad \\text{ in }\\Omega _\\infty ^+\\,,\\\\&\\tilde{u}^+=0 \\text{ on } \\gamma _\\infty ^{+}\\,.\\end{aligned}\\right.$ The existence of a minimizer will be achieved by taking the limit $\\ell \\rightarrow \\infty $ of the minimizers $\\lbrace \\tilde{u}_\\ell ^+\\rbrace $ in (REF ) (see Remark REF ).",
"Since $\\lbrace \\tilde{u}_\\ell ^+\\rbrace $ is bounded in $H^1(\\Omega _\\infty ^+)$ , a subsequence $\\lbrace \\tilde{u}_{\\ell _k}^+\\rbrace $ converges weakly to some limit $\\tilde{u}^+\\in H^1(\\Omega _\\infty ^+)$ .",
"Take any $\\varphi \\in V_s(\\Omega _\\infty ^+)$ .",
"Since $\\nu _\\infty ^+=\\lim _{k\\rightarrow \\infty } {\\tilde{\\lambda }}_{\\ell _k}^{1,+}$ by Lemma REF , we can pass to the limit in the following equality, that holds for $\\ell _k>M(\\varphi )$ (see (REF )), $\\int _{\\Omega _\\infty ^+} A\\nabla \\tilde{u}_{\\ell _k}^+\\cdot \\nabla \\varphi ={\\tilde{\\lambda }}_{\\ell _k}^{1,+}\\int _{\\Omega _\\infty ^+} \\tilde{u}_{\\ell _k}^+\\varphi \\,,$ and obtain that $\\int _{\\Omega _\\infty ^+} A\\nabla \\tilde{u}^+\\cdot \\nabla \\varphi =\\nu _\\infty ^+ \\int _{\\Omega _\\infty ^+} \\tilde{u}^+\\varphi \\,.$ Since (REF ) is valid for any $\\varphi \\in V_s(\\Omega _\\infty ^+)$ , and by density also for any $\\varphi \\in V(\\Omega _\\infty ^+)$ , we obtain that $\\tilde{u}^+$ is a solution of (REF ).",
"To conclude that it is a minimizer realizing $\\nu _\\infty ^+$ in (REF ) we only need to prove that it is nontrivial, i.e., that $\\tilde{u}^+\\lnot \\equiv 0$ .",
"Actually, we are going to show that $\\int _ {\\Omega _\\infty ^+} (\\tilde{u}^+)^2=1$ and $\\tilde{u}^+>0$ .",
"For that matter we will prove decay estimates for $\\tilde{u}_\\ell ^+$ for large $x_1$ , that imply concentration near $x_1=0$ , using the same technique as the one used in the proof of Theorem REF .",
"Let $\\ell $ and $\\ell ^{\\prime }$ satisfy $0< \\ell ^{^{\\prime }} \\le \\ell -1$ .",
"Define $\\tilde{\\rho }_{\\ell ^{^{\\prime }}}=\\tilde{\\rho }_{\\ell ^{^{\\prime }}}(x_1)$ by $\\tilde{\\rho }_{\\ell ^{^{\\prime }}}(x_1)={\\left\\lbrace \\begin{array}{ll}0 & x_1\\le \\ell ^{\\prime }\\,,\\\\x_1-\\ell ^{\\prime }& x_1\\in (\\ell ^{\\prime },\\ell ^{\\prime }+1)\\,,\\\\1 & x_1\\ge \\ell ^{\\prime }+1\\,.\\end{array}\\right.",
"}$ By the Euler-Lagrange equation satisfied by $\\tilde{u}_\\ell ^+$ we have $\\int _{\\Omega ^+_\\ell } (A\\nabla \\tilde{u}_\\ell ^+).",
"\\nabla (\\tilde{\\rho }_{\\ell ^{^{\\prime }}}^2\\tilde{u}_\\ell ^+) = \\tilde{\\lambda }_\\ell ^{1,+}\\int _{\\Omega ^+_\\ell }\\tilde{\\rho }_{\\ell ^{^{\\prime }}}^2|\\tilde{u}_\\ell ^+|^2\\,.$ Repeating the argument used to derive (REF ) we obtain $\\begin{aligned}(\\sigma _{\\ell /2}^1 -\\tilde{\\lambda }_\\ell ^{1,+}) \\int _{\\Omega _\\ell ^+\\setminus \\Omega _{\\ell ^{\\prime }+1}}\\!\\!\\!|\\tilde{u}_\\ell ^+|^2 &\\le (\\sigma _{\\ell /2}^1 -\\tilde{\\lambda }_\\ell ^{1,+}) \\int _{\\Omega _\\ell ^+} |\\tilde{u}_\\ell ^+|^2 \\tilde{\\rho }_{\\ell ^{^{\\prime }}}^2 \\le \\int _{\\Omega _\\ell ^+}|\\tilde{u}_\\ell ^+|^2 (A\\nabla \\tilde{\\rho }_{\\ell ^{^{\\prime }}}).\\nabla \\tilde{\\rho }_{\\ell ^{^{\\prime }}}\\\\&=\\int _{\\Omega _{\\ell ^{\\prime }+1}^+\\setminus \\Omega _{\\ell ^{\\prime }}}\\!\\!|\\tilde{u}_\\ell ^+|^2 (A\\nabla \\tilde{\\rho }_{\\ell ^{^{\\prime }}}).\\nabla \\tilde{\\rho }_{\\ell ^{^{\\prime }}}\\le C_A \\int _{\\Omega _{\\ell ^{\\prime }+1}^+\\setminus \\Omega _{\\ell ^{\\prime }}}\\!\\!\\!|\\tilde{u}_\\ell ^+|^2\\,.\\end{aligned}$ Using (REF ) together with (REF ) and Lemma REF we deduce that there exist $\\tilde{\\ell }_0>0$ and $\\tilde{\\beta }>0$ such that for $\\ell >\\tilde{\\ell }_0$ we have $\\sigma _{\\ell /2}^1-\\tilde{\\lambda }_\\ell ^{1,+}\\ge \\tilde{\\beta }$ .",
"Therefore, we deduce from (REF ) that $\\int _{\\Omega _\\ell ^+\\setminus \\Omega _{\\ell ^{\\prime }+1}}|\\tilde{u}_\\ell ^+|^2\\le \\tilde{\\alpha }\\int _{\\Omega _\\ell ^+\\setminus \\Omega _{\\ell ^{\\prime }}}\\!\\!\\!|\\tilde{u}_\\ell ^+|^2~\\text{ with }~\\tilde{\\alpha }:=\\frac{C_A}{\\tilde{\\beta }+C_A}\\,.$ Fix any $r>1$ .",
"Applying (REF ) successively for $\\ell ^{^{\\prime }} = r-1, r-2,\\ldots ,r-[r]$ yields $\\int _{\\Omega _\\ell ^+\\setminus \\Omega _r} |\\tilde{u}_\\ell ^+|^2 \\le \\tilde{\\alpha }^{[r]}\\int _{\\Omega ^+_\\ell }|\\tilde{u}_\\ell ^+|^2 =\\tilde{\\alpha }^{[r]},~\\forall \\ell >r\\,.$ In other words, $\\int _{\\Omega _r^+} |\\tilde{u}_\\ell ^+|^2\\ge 1-\\tilde{\\alpha }^{[r]}\\,.$ Since $\\tilde{u}_{\\ell _k}\\rightarrow \\tilde{u}^+$ strongly in $L^2(\\Omega _r^+)$ , we deduce from (REF ) that $\\int _{\\Omega _r^+} (\\tilde{u}^+)^2\\ge 1-\\tilde{\\alpha }^{[r]}\\,.$ This already implies that $\\tilde{u}^+$ is a nontrivial nonnegative solution to (REF ) and therefore, a minimizer in (REF ).",
"Applying (REF ) with arbitrary large $r$ , we get that $\\int _{\\Omega _\\infty ^+} (\\tilde{u}^+)^2=1$ .",
"The uniqueness of the minimizer follows by a standard argument, using the fact that any minimizer must have a constant sign.",
"Open Problem: Is it true that (REF ) is also a necessary condition for the existence of a minimizer realizing $\\nu _\\infty ^+$ ?",
"In Theorem REF below we will show nonexistence of a minimizer when $\\nu _\\infty ^+=\\mu ^1$ , but under the additional condition (REF ).",
"The next result provides a sufficient condition for (REF ) to hold and another one for it to fail.",
"Theorem 6.1 (i) Assume that (REF ) is satisfied.",
"If the following condition holds, $\\int _\\omega (A_{12}.\\nabla W_1)W_1\\ge 0\\,,$ then (REF ) holds.",
"(ii) If $A_{12}.\\nabla W_1\\le 0\\text{ a.e.",
"in }\\omega $ then $\\nu _\\infty ^+=\\mu ^1$ .",
"Moreover, in this case there is no minimizer realizing $\\nu _\\infty ^+$ .",
"(i) Assume that (REF ) is satisfied.",
"A similar computation to the one done in the proof of Theorem REF (see also Remark REF ) shows that $\\lbrace \\tilde{v}_\\ell ^\\varepsilon \\rbrace $ given by (REF ), satisfy not only (REF ), but also $\\inf _{\\varepsilon >0} \\lim _{\\ell \\rightarrow 0}\\frac{\\int _{\\Omega _\\ell ^-} (A\\nabla \\tilde{v}_\\ell ^\\varepsilon ).\\nabla \\tilde{v}_\\ell ^\\varepsilon }{\\int _{\\Omega _\\ell ^-}|\\tilde{v}_\\ell ^\\varepsilon |^2}=\\int _{\\omega }(A_{22}\\nabla W_1).\\nabla W_1-\\frac{|A_{12}\\nabla W_1|^2}{a_{11}}\\,.$ Indeed, we only need to note that the term corresponding to the second term on the RHS of (REF ) is of the order $O(\\ell ^2)$ .",
"Hence, we can fix values of $\\ell _1$ and $\\varepsilon _1$ such that the following analog of (REF ) holds, $-\\gamma _1:=\\int _{\\Omega _{\\ell _1}^-} (A\\nabla \\tilde{v}_{\\ell _1}^{\\varepsilon _1}).\\nabla \\tilde{v}_{\\ell _1}^{\\varepsilon _1}-\\mu ^1\\int _{\\Omega _{\\ell _1}^-}|\\tilde{v}_{\\ell _1}^{\\varepsilon _1}|^2<0\\,.$ For each $\\alpha >0$ we define a test function in $V_\\infty (\\Omega _\\infty ^+)$ by $z_\\alpha (x_1,X_2)={\\left\\lbrace \\begin{array}{ll}\\tilde{v}_{\\ell _1}^{\\varepsilon _1}(x_1-\\ell _1,X_2) & x_1\\in [0,\\ell _1)\\,,\\\\W_1(X_2)e^{-\\alpha (x_1-\\ell _1)} & x_1 \\in [\\ell _1,\\infty )\\,.\\end{array}\\right.",
"}$ Above we used the fact that $\\tilde{v}_{\\ell _1}^{\\varepsilon _1}(0,X_2)=W_1(X_2)$ .",
"We have, $\\int _{\\Omega _\\infty ^+}|z_\\alpha |^2=\\int _{\\Omega _{\\ell _1}^-} |\\tilde{v}_{\\ell _1}^{\\varepsilon _1}|^2+ (\\int _0^\\infty \\!\\!e^{-2\\alpha x_1})\\int _\\omega W_1^2=\\int _{\\Omega _{\\ell _1}^-}|\\tilde{v}_{\\ell _1}^{\\varepsilon _1}|^2+\\frac{1}{2\\alpha }\\,,$ and $\\begin{aligned}\\int _{\\Omega _\\infty ^+} (A\\nabla z_\\alpha ).\\nabla z_\\alpha &=\\int _{\\Omega _{\\ell _1}^-} (A\\nabla \\tilde{v}_{\\ell _1}^{\\varepsilon _1}).\\nabla \\tilde{v}_{\\ell _1}^{\\varepsilon _1}\\\\&\\phantom{=}+ \\frac{1}{2\\alpha }\\left(\\alpha ^2\\int _\\omega a_{11}W_1^2-2\\alpha \\int _\\omega (A_{12}.\\nabla W_1)W_1+\\int _\\omega (A_{22} \\nabla W_1).\\nabla W_1\\right)\\end{aligned}$ Therefore, using (REF ) we get $\\nu _\\infty ^+-\\mu ^1\\le \\frac{ \\int _{\\Omega _\\infty ^+} A\\nabla z_\\alpha .\\nabla z_\\alpha }{\\int _{\\Omega _\\infty ^+}|z_\\alpha |^2}- \\mu ^1<\\frac{ \\frac{\\alpha }{2}\\int _\\omega a_{11}W_1^2-\\int _\\omega (A_{12}.\\nabla W_1)W_1 -\\gamma _1}{\\int _{\\Omega _{\\ell _1}^-}|v_{\\ell _1}^{\\varepsilon _1}|^2+\\frac{1}{2\\alpha }}\\,.$ Since $\\gamma _1>0$ and $\\int _\\omega (A_{12}.\\nabla W_1)W_1\\ge 0$ by (REF ), it is clear that we can choose $\\alpha $ small enough to ensure that the RHS of (REF ) is negative, completing the proof of (REF ).",
"(ii) We notice that not only $V_s(\\Omega _+^\\infty )$ is dense in $V(\\Omega _+^\\infty )$ (see (REF )), but its subspace $V_s^0(\\Omega _{\\infty }^+) = \\Big \\lbrace u\\in V_s(\\Omega _\\infty ^+)\\,:\\,\\exists \\delta =\\delta (u) > 0 \\text{ s.t. }",
"u(x) = 0 \\text{ for }\\text{dist}(x,\\gamma _\\infty ^+)\\le \\delta \\Big \\rbrace \\,,$ is dense as well.",
"By elliptic regularity and the strong maximum principle we know that $W_1$ is continuous and positive in $\\omega $ (see [15]).",
"We shall use the following version of Picone identity, $(A\\nabla u).\\nabla u-(A\\nabla v).\\nabla \\big (\\frac{u^2}{v}\\big )=A\\big (\\nabla u-\\frac{u}{v}\\nabla v\\big ).\\big (\\nabla u-\\frac{u}{v}\\nabla v\\big )\\ge 0\\,.$ Using (REF ) with any $u\\in V_s^0(\\Omega _{\\infty }^+)$ and $v=W_1$ , integrating and applying the generalized Green formula yields $\\begin{aligned}0&\\le \\int _{\\Omega _{\\infty }^+} A\\big (\\nabla u-\\frac{u}{W_1}\\nabla W_1\\big ).\\big (\\nabla u-\\frac{u}{W_1}\\nabla W_1\\big )=\\int _{\\Omega _{\\infty }^+} (A\\nabla u).\\nabla u-(A\\nabla W_1).\\nabla \\big (\\frac{u^2}{W_1}\\big )\\\\&=\\int _{\\Omega _{\\infty }^+} (A\\nabla u).\\nabla u+\\int _{\\Omega _{\\infty }^+}\\operatorname{div}(A\\nabla W_1)\\big (\\frac{u^2}{W_1}\\big )-\\int _{\\lbrace 0\\rbrace \\times \\omega }(A\\nabla W_1.\\nu )\\big (\\frac{u^2}{W_1}\\big )\\\\&=\\int _{\\Omega _{\\infty }^+} (A\\nabla u).\\nabla u-\\mu ^1u^2+\\int _{\\omega } \\Big (A_{12}.\\nabla W_1\\Big )\\frac{u^2(0,X_2)}{W_1(X_2)}\\,.\\end{aligned}$ By (REF ) and (REF ) we deduce that $0\\le \\int _{\\Omega _{\\infty }^+} A\\big (\\nabla u-\\frac{u}{W_1}\\nabla W_1\\big ).\\big (\\nabla u-\\frac{u}{W_1}\\nabla W_1\\big )\\le \\int _{\\Omega _{\\infty }^+} A\\nabla u.\\nabla u-\\mu ^1u^2\\,.$ By the density of $V_s^0(\\Omega _{\\infty }^+)$ in $V(\\Omega _{\\infty }^+)$ it follows that (REF ) holds for every $u\\in V(\\Omega _{\\infty }^+)$ , i.e., $\\nu _\\infty ^+\\ge \\mu ^1$ .",
"Finally, applying (REF ) we get that $\\nu _\\infty ^+=\\mu ^1$ .",
"To conclude, assume by negation that $\\nu _\\infty ^+$ is realized by a minimizer $u$ .",
"Then, by (REF ) we get that $\\nabla \\big (\\frac{u}{W_1}\\big )=0$ a.e., implying that $u=cW_1$ for some constant $c\\ne 0$ .",
"But this is clearly a contradiction since $W_1\\notin V(\\Omega _\\infty ^+)$ .",
"Remark 6.1 An immediate consequence of Theorem REF and Remark REF is that if (REF ) holds and $\\int _\\omega (A_{12}.\\nabla W_1)W_1= 0$ , then we have both $\\nu _\\infty ^+<\\mu ^1$ and $\\nu _\\infty ^-<\\mu ^1$ .",
"A special case is when property (S) holds.",
"Another direct consequence is that whenever (REF ) holds we have $\\min (\\nu _\\infty ^+,\\nu _\\infty ^-)<\\mu ^1$ .",
"However, this fact follows already from our previous results, by combining Theorem REF and Theorem REF .",
"Our last result provides a description of the asymptotic profile of the eigenfunctions $\\lbrace u_\\ell \\rbrace $ near the ends of the cylinder.",
"We denote by $\\tilde{u}^\\pm $ the unique positive renormalized minimizer for $\\nu _\\infty ^\\pm $ , when it exists.",
"For each $\\ell >0$ we define: $\\begin{aligned}\\tilde{u}_\\ell ^+(x_1,X_2)=u_\\ell (x_1-\\ell ,X_2)~\\text{ on}~\\Omega _\\ell ^+\\,,\\\\\\tilde{u}_\\ell ^-(x_1,X_2)=u_\\ell (x_1+\\ell ,X_2)~\\text{ on}~\\Omega _\\ell ^-\\,.\\end{aligned}$ The next theorem describes two possible scenarios that may occur: concentration near one of the ends of the cylinder, or concentration near both ends.",
"Theorem 6.2 (i) If $\\nu _\\infty ^+<\\nu _\\infty ^-$ then, for every $r>0$ , $\\tilde{u}_\\ell ^+\\rightarrow \\tilde{u}^+ \\text{ in }H^1(\\Omega _r^+) ~\\text{and }~ \\tilde{u}_\\ell ^-\\rightarrow 0 \\text{ in }H^1(\\Omega _r^-)\\,.$ (ii) If both (REF ) and property (S) hold then we have $\\tilde{u}^+(x_1,X_2)=\\tilde{u}^-(-x_1,-X_2)$ and for every $r>0$ , $\\tilde{u}_\\ell ^+\\rightarrow \\tilde{u}^+ \\text{ in }H^1(\\Omega _r^+) ~\\text{and }~ \\tilde{u}_\\ell ^-\\rightarrow \\tilde{u}^-\\text{ in }H^1(\\Omega _r^-)\\,.$ (i) The convergence of $\\lbrace \\tilde{u}_\\ell ^-\\rbrace $ to 0 in $H^1(\\Omega _r^-)$ for all $r>0$ is clear from Proposition REF , so we only need to prove the result for $\\lbrace \\tilde{u}_\\ell ^+\\rbrace $ .",
"Since $\\lbrace \\tilde{u}_\\ell ^+\\rbrace $ is bounded in $H^1(\\Omega _\\ell ^+)$ , given any sequence $\\ell _k\\rightarrow \\infty $ , we can apply a diagonal argument to $\\lbrace \\tilde{u}_{\\ell _k}^+\\rbrace $ to extract a subsequence, still denoted by $\\lbrace \\ell _k\\rbrace $ , such that $\\tilde{u}_{\\ell _k}^+$ converges weakly in $H^1(\\Omega _r^+)$ and strongly in $L^2(\\Omega _r^+)$ to some function $v^+\\in H^1(\\Omega _\\infty ^+)$ , for every $r>0$ .",
"By (REF ) and Proposition REF we have $\\int _{\\Omega _r^+} |\\tilde{u}_\\ell ^+|^2=\\int _{\\Omega _\\ell ^-\\setminus \\Omega _{\\ell -r}}\\!\\!|u_\\ell |^2=1-\\int _{\\Omega _{l-r}^-}\\!|u_\\ell |^2-\\int _{\\Omega _\\ell ^+} \\!|u_\\ell |^2\\ge 1-\\alpha ^{[r]}+o(1)\\,,$ where $o(1)$ stands for a quantity that tends to 0 when $\\ell \\rightarrow \\infty $ .",
"Passing to the limit in (REF ) with $\\ell =\\ell _k$ , yields, $\\int _{\\Omega _r^+} |v^+|^2\\ge 1-\\alpha ^{[r]}\\,,$ and since $r$ is arbitrary, we get that $\\int _{\\Omega _\\infty ^+}|v^+|^2=1$ .",
"In addition, we clearly have $\\nu _\\infty ^+=\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1\\ge \\lim _{\\ell \\rightarrow \\infty } \\int _{\\Omega _\\ell }(A\\nabla u_\\ell ).\\nabla u_\\ell \\\\\\ge \\limsup _{k\\rightarrow \\infty } \\int _{\\Omega _r^+} (A\\nabla \\tilde{u}_{\\ell _k}^+).\\nabla \\tilde{u}_{\\ell _k}^+\\ge \\int _{\\Omega _r^+} (A\\nabla v^+).\\nabla v^+\\,.$ From (REF )–(REF ) we deduce that $\\int _{\\Omega _\\infty ^+}(A\\nabla v^+).\\nabla v^+=\\nu _\\infty ^+$ , i.e., $v^+$ is a nonnegative normalized minimizer, realizing $\\nu _\\infty ^+$ in (REF ).",
"Therefore, it must coincide with $\\tilde{u}^+$ .",
"Finally, defining on $(0,\\infty )$ the function $f(r)=\\limsup _{k\\rightarrow \\infty } \\int _{\\Omega _r^+} (A\\nabla \\tilde{u}_{\\ell _k}^+).\\nabla \\tilde{u}_{\\ell _k}^+-\\int _{\\Omega _r^+} (A\\nabla \\tilde{u}^+).\\nabla \\tilde{u}^+ \\,,$ we see that on the one hand it is a nonnegative and nondecreasing function, while on the other hand $\\lim _{r\\rightarrow \\infty } f(r)=0$ .",
"Hence $f(r)\\equiv 0$ , implying the strong convergence $\\tilde{u}_{\\ell _k}\\rightarrow \\tilde{u}^+$ in $H^1(\\Omega _r^+)$ for all $r>0$ .",
"The uniqueness of the possible limit implies the the same convergence holds for the whole family $\\lbrace \\tilde{u}_{\\ell }\\rbrace $ .",
"(ii) In this case we have the symmetry relation $u_\\ell (x_1,X_2) = u_\\ell (-x_1, -X_2)$ by Proposition REF , and the same argument as in (i) gives the result.",
"Remark 6.2 Theorem REF provides a description of the profile of $u_\\ell $ near the ends of the cylinder.",
"As pointed to us by Y. Pinchover, a description of the profile of $u_\\ell $ in the bulk can be given using the characterization of positive solutions in an infinite cylinder, given in [17].",
"Indeed, setting $v_\\ell (x)=u_\\ell (x)/u_\\ell (0)$ , and employing Harnack's inequality and the boundary Harnack principle (see [16]) we obtain a subsequence $\\lbrace v_{\\ell _k}\\rbrace $ that converges uniformly on each cylinder $\\Omega _r$ , $r>0$ , to a limit $v$ .",
"The function $v$ is a positive solution on the infinite cylinder $\\Omega _\\infty =(-\\infty ,\\infty )\\times \\omega $ of $-\\operatorname{div}(A(X_2)\\nabla v)=\\lambda _\\infty v$ satisfying $v=0$ on $\\partial \\Omega _\\infty =(-\\infty ,\\infty )\\times \\partial \\omega $ , where $\\lambda _\\infty =\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1=\\min (\\nu _\\infty ^+,\\nu _\\infty ^-)$ (by Theorem REF ).",
"From [17] (that handles a much more general situation) it follows that such $v$ is a linear combination of one or two exponential solutions of the form $v_\\alpha (x)=\\Phi _\\alpha (X_2)e^{\\alpha x_1}$ .",
"In particular, when property $(S)$ holds it follows that $v$ takes the form $v(x)=g(X_2)e^{\\alpha x_1}+g(-X_2)e^{-\\alpha x_1}\\,$ for some $\\alpha >0$ , if (REF ) holds, and $v(x)=cW_1(X_2)$ if (REF ) doesn't hold."
],
[
"Some additional results ",
"So far we only studied the asymptotic behavior of the first eigenvalue $\\lambda _\\ell ^1$ and the corresponding eigenfunction $u_\\ell $ .",
"The analogous behavior of the other eigenvalues $\\lambda _\\ell ^2,\\lambda _\\ell ^3,$ etc., is also of interest.",
"For the case of Dirichlet boundary condition this was done in [7].",
"For our case of mixed boundary conditions we have the following partial result for $ \\lambda _\\ell ^2$ .",
"Theorem 7.1 If property $(S)$ holds then $\\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^2 = \\lim _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1\\,.$ Define $h_\\ell ^-$ and $h_\\ell ^+$ on $\\Omega _{\\ell }$ by $h_\\ell ^-(x) = {\\left\\lbrace \\begin{array}{ll}\\tilde{u}_\\ell ^{+}(x_1+\\ell ,X_2) &\\text{ on } \\Omega _{\\ell }^-, \\\\0 &\\text{ on } \\Omega _{\\ell }^+\\end{array}\\right.",
"}$ and $h_\\ell ^+= {\\left\\lbrace \\begin{array}{ll}\\tilde{u}_\\ell ^{-}(x_1-\\ell ,X_2) &\\text{ on } \\Omega _{\\ell }^+, \\\\0 &\\text{ on } \\Omega _{\\ell }^-,\\end{array}\\right.",
"}$ where $\\tilde{u}_\\ell ^{-}, \\tilde{u}_\\ell ^{+} $ are defined in Remark REF .",
"Set $\\mathcal {H}_\\ell = \\alpha _\\ell h_\\ell ^- + \\beta _\\ell h_\\ell ^+$ , where $\\alpha _\\ell , \\beta _\\ell $ are chosen such that $\\int _{\\Omega _{\\ell }}u_{\\ell }\\mathcal {H}_\\ell = 0~\\text{ and }~\\alpha _\\ell ^2+\\beta _\\ell ^2>0\\,.$ Such a choice is possible since we have to satisfy one linear equation in two unknowns.",
"From the Rayleigh quotient characterization of $\\lambda _\\ell ^2$ we get, since the functions $h_\\ell ^{+}$ and $h_\\ell ^{-}$ have disjoint supports, $\\lambda _{\\ell }^2 =\\min \\left\\lbrace \\frac{\\int _{\\Omega _\\ell }(A\\nabla u).\\nabla u}{\\int _{\\Omega _\\ell } u^2}\\,\\big |\\,0\\ne u \\in V(\\Omega _\\ell ),\\,\\int _{\\Omega _\\ell }uu_\\ell =0 \\right\\rbrace \\le \\frac{\\int _{\\Omega _{\\ell }}A\\nabla \\mathcal {H}_\\ell .\\nabla \\mathcal {H}_\\ell }{\\int _{\\Omega _{\\ell }}\\mathcal {H}_\\ell ^2}\\\\= \\frac{\\alpha _\\ell ^2\\int _{\\Omega _{\\ell }^{-}}(A\\nabla h_\\ell ^{-}).\\nabla h_\\ell ^{-} + \\beta _\\ell ^2 \\int _{\\Omega _{\\ell }^{+}}(A\\nabla h_\\ell ^{+}).\\nabla h_\\ell ^{+}}{\\alpha _\\ell ^2\\int _{\\Omega _{\\ell }^{-}}(h_\\ell ^{-})^2+ \\beta _\\ell ^2\\int _{\\Omega _{\\ell }^{+}}(h_\\ell ^{+})^2}=\\frac{\\alpha _\\ell ^2\\tilde{\\lambda }_\\ell ^{1,+} + \\beta _\\ell ^2\\tilde{\\lambda }_\\ell ^{1,-} }{\\alpha _\\ell ^2 + \\beta _\\ell ^2} .$ But the symmetry property $(S)$ implies, by the same proof as that of Proposition REF , that $\\tilde{u}_\\ell ^+(x_1,X_2)=\\tilde{u}_\\ell ^-(-x_1,-X_2)$ and $\\tilde{\\lambda }_\\ell ^{1,+}=\\tilde{\\lambda }_\\ell ^{1,-}$ .",
"Therefore, (REF ) implies that the RHS of (REF ) equals $\\tilde{\\lambda }_\\ell ^{1,+}$ and we obtain that $\\lambda _{\\ell }^1 < \\lambda _{\\ell }^2 \\le \\tilde{\\lambda }_\\ell ^{1,+}=\\tilde{\\lambda }_\\ell ^{1,-}\\,.$ The theorem then follows from Lemma REF and Theorem REF .",
"In the previous sections we considered the case of a cylinder which goes to infinity in one direction.",
"We now consider the more general case of a domain that tends to infinity in several directions.",
"In the rest of the paper we set $\\Omega _\\ell = (-\\ell , \\ell )^{p}\\times \\omega ,$ where $1\\le p < n$ and $\\omega $ is a bounded subset of $\\mathbb {R}^{n-p}$ .",
"The points in $\\Omega _\\ell $ are denoted by $X = (X_1,X_2) \\text{ with }X_1 = (x_1,\\ldots , x_p)\\text{ and }X_2 = (x_{p+1},\\ldots ,x_n)\\,.$ Let $A(X_2)$ be a $n \\times n$ symmetric, positive definite matrix, uniformly elliptic and uniformly bounded on $\\omega $ , as in the previous sections.",
"Now we consider the following decomposition to sub-matrices: $\\begin{aligned}A(X_2)=\\begin{pmatrix}A_{11}(X_2) & A_{12}(X_2) \\\\A_{12}^t(X_2) & A_{22}(X_2)\\end{pmatrix}\\end{aligned}$ where $A_{11}, A_{12}$ and $A_{22}$ are $p\\times p, p\\times (n-p) $ and $(n-p)\\times (n-p)$ matrices, respectively.",
"We still denote by $\\mu ^1$ and $W_1$ the first eigenvalue and the corresponding eigenfunction for the problem (REF ).",
"Let $C_i$ denote the $i$ -th row of the matrix $A_{12}$ , and denote by $B_i$ the $(n-p+1)\\times (n-p+1)$ matrix $\\begin{aligned}B_i(X_2)=\\begin{pmatrix}a_{ii}(X_2) & C_i(X_2) \\\\C_i^t(X_2) & A_{22}(X_2)\\end{pmatrix}\\end{aligned}\\,,$ for $1 \\le i \\le p$ .",
"Since the matrix $B_i$ can be viewed as a representation of the restriction of the operator associated with $A$ to the subspace of $\\mathbb {R}^n$ consisting of the vectors $v=(v_1,\\ldots ,v_n)$ satisfying $v_j=0$ for all $j$ such that $i\\ne j\\le p$ , we conclude that the matrices $B_i(X_2)$ are also uniformly elliptic for $X_2\\in \\omega $ .",
"The following eigenvalue problem is the generalization of (REF ) to our setting: $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A(X_2)\\nabla u)=\\sigma u \\quad \\text{ in }\\Omega _\\ell ,\\\\&u=0 \\quad \\text{ on } (-\\ell ,\\ell )^{p} \\times \\partial \\omega ,\\\\& (A(X_2)\\nabla u).\\nu =0 \\quad \\text{ on }\\partial (-\\ell ,\\ell )^p\\times \\omega .\\end{aligned}\\right.$ As before we denote by $\\lambda _\\ell ^1$ the first eigenvalue and by $u_\\ell $ the corresponding normalized positive eigenfunction.",
"We have the following generalization of Theorem REF .",
"Theorem 7.2 We have $\\limsup _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1 < \\mu ^1,$ provided the following condition holds, $A_{12}.\\nabla W_1\\lnot \\equiv {0}\\text{ a.e.",
"on } \\omega \\,,$ where 0 denotes the zero element in $\\mathbb {R}^p$ .",
"In case (REF ) does not hold we have $\\lambda _\\ell ^1 = \\mu ^1$ for all $\\ell >0$ .",
"Assume first that (REF ) doesn't hold.",
"Then there exists $i\\in \\lbrace 1,\\ldots ,p\\rbrace $ for which $(A_{12}\\nabla W_1)_i$ is not identically zero (a.e.)",
"on $\\omega $ .",
"It follows that the hypotheses of Theorem REF (for the case where (REF ) holds) are satisfied for the eigenvalue problem associated with the operator $-\\operatorname{div}(B_i(X_2)\\nabla v)$ on the domain $\\tilde{\\Omega }_\\ell =(-\\ell ,\\ell )\\times \\omega $ in $\\mathbb {R}^{n-p+1}$ .",
"Hence, there exist functions $\\phi _\\ell (x_1,X_2)\\in V(\\tilde{\\Omega }_\\ell ),\\,\\ell >0,$ such that $\\limsup _{\\ell \\rightarrow \\infty } \\frac{\\int _{\\tilde{\\Omega }_\\ell } (B_i(X_2)\\nabla \\phi _\\ell ).",
"\\nabla \\phi _{\\ell }}{\\int _{\\tilde{\\Omega }_\\ell } \\phi _\\ell ^2} < \\mu ^1\\,.$ Define on $\\Omega _\\ell $ , $v_\\ell (X_1,X_2) := \\phi _\\ell (x_i,X_2)$ .",
"Noting that $\\int _{\\Omega _\\ell } (A\\nabla v_\\ell ).",
"\\nabla v_\\ell = (2\\ell )^{p-1} \\int _{\\tilde{\\Omega }_\\ell } (B_i \\nabla \\phi _\\ell ).\\nabla \\phi _{\\ell } ~\\text{ and }~\\int _{\\Omega _{\\ell }}v_\\ell ^2=(2\\ell )^{p-1} \\int _{\\tilde{\\Omega }_\\ell } \\phi _\\ell ^2\\,,$ we get from (REF ) that $\\limsup _{\\ell \\rightarrow \\infty } \\lambda _\\ell ^1 \\le \\limsup _{\\ell \\rightarrow \\infty } \\frac{\\int _{\\Omega _\\ell } A\\nabla v_\\ell \\cdot \\nabla v_\\ell }{\\int _{\\Omega _{\\ell }} v_\\ell ^2} < \\mu ^1\\,.$ Assume now that (REF ) does hold.",
"Next we apply a simple generalization of an argument from Theorem REF .",
"Let $B=\\begin{pmatrix}B_{11} & B_{12} \\\\B^t_{12} & B_{22}\\end{pmatrix}$ be a positive definite $n\\times n$ matrix, where $B_{11}$ and $B_{22}$ are $p\\times p$ and $(n-p)\\times (n-p)$ matrices, respectively.",
"Represent any vector $z$ in $\\mathbb {R}^n$ as $z=(Z_1,Z_2)$ with $Z_1\\in \\mathbb {R}^{p}$ and $Z_2\\in \\mathbb {R}^{n-p}$ .",
"Then, by a similar computation to the one leading to (REF )–(REF ) we get that for any fixed $Z_2\\in \\mathbb {R}^{n-p}$ we have $\\min _{Z_1\\in \\mathbb {R}^p} (Bz).z=(B_{22}Z_2).Z_2-\\big (B_{11}^{-1}B_{12}Z_2\\big ).B_{12}Z_2\\,,$ and the minimum in (REF ) is attained for $Z_1=-B_{11}^{-1}(B_{12}Z_2)\\,.$ Applying (REF ) with $B=A(X_2)$ we obtain, for any $\\ell >0$ , $\\begin{aligned}\\int _{\\Omega _\\ell } (A(X_2)\\nabla u_\\ell ).\\nabla u_\\ell &\\ge \\int _{\\Omega _\\ell }(A_{22}\\nabla _{X_2}u_\\ell ).\\nabla _{X_2}u_\\ell -\\big (A_{11}^{-1}A_{12}\\nabla _{X_2}u_\\ell \\big ).A_{12}\\nabla _{X_2}u_\\ell \\\\&\\ge \\Lambda ^1\\int _{\\Omega _\\ell }u_\\ell ^2\\,,\\end{aligned}$ where $\\Lambda ^1$ is defined, generalizing (REF ), by $\\Lambda ^1=\\inf \\left\\lbrace \\int _{\\omega }A_{22}\\nabla u.\\nabla u-\\big (A_{11}^{-1}A_{12}\\nabla u\\big ).A_{12}\\nabla u:\\,u \\in H_0^1(\\omega ),\\, \\int _{\\omega }u^2=1 \\right\\rbrace .$ The infimum in (REF ) is attained by a unique positive function, denoted again by $w_1$ , that satisfies $\\left\\lbrace \\begin{aligned}-&\\operatorname{div}(A_{22}\\nabla w_1)+\\operatorname{div}(A_{12}^tA_{11}^{-1}A_{12}\\nabla w_1)=\\Lambda ^1w_1~\\text{ in }\\omega \\,,\\\\&w_1=0~\\text{ on }\\partial \\omega \\,.\\end{aligned}\\right.$ But if (REF ) holds, then $W_1$ is also a positive eigenfunction in (REF ), with eigenvalue $\\mu ^1$ .",
"As in the proof of Theorem REF we conclude that $\\Lambda ^1=\\mu ^1$ and the result follows from (REF ) (since clearly $\\lambda _\\ell ^1\\le \\mu ^1$ ).",
"Acknowledgements: The research of the first author was funded by the Lithuanian-Swiss cooperation programme under the project agreement No.",
"CH-SMM-01/01 and by the Swiss National Science Foundation under the contracts $\\#$ 200021-129807/1 and 200021-146620.",
"The research of the second author was supported by the Swiss National Science Foundation under the contract $\\#$ 200021-129807/1.",
"The third author thanks Yehuda Pinchover for Remark REF and for providing him the relevant references."
]
] | 1403.0094 |
[
[
"Syzygies, finite length modules, and random curves"
],
[
"Abstract We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus."
],
[
"Introduction",
"While a great deal of modern commutative algebra and algebraic geometry has taken a nonconstructive form, the theory of Gröbner bases provides an algorithmic approach.",
"Algorithms currently implemented in computer algebra systems, such as Macaulay2 and Singular, already exhibit the wide range of computational possibilities that arise from Gröbner bases [40], [15].",
"In these lectures, we focus on certain applications of Gröbner bases to syzygies and curves.",
"In Section , we use Gröbner bases to give an algorithmic proof of Hilbert's Syzygy Theorem, which bounds the length of a free resolution over a polynomial ring.",
"In Section , we prove Petri's theorem about the defining equations for canonical embeddings of curves.",
"We turn in Section to the Hartshorne–Rao module of a curve, showing by example how a module $M$ of finite length can be used to explicitly construct a curve whose Hartshorne–Rao module is $M$ .",
"Section then applies this construction to the study of the unirationality of the moduli space $\\mathfrak {M}_g$ of curves of genus $g$ ."
],
[
"Hilbert's Syzygy Theorem",
"Let $R:={\\mathbb {k}}[x_1,\\ldots ,x_n]$ be a polynomial ring in $n$ variables over a field ${\\mathbb {k}}$ .",
"A free resolution of a finitely generated $R$ -module $M$ is a complex of free modules $\\cdots \\rightarrow R^{\\beta _2}\\rightarrow R^{\\beta _1}\\rightarrow R^{\\beta _0}$ such that the following is exact: $\\cdots \\rightarrow R^{\\beta _2}\\rightarrow R^{\\beta _1}\\rightarrow R^{\\beta _0}\\rightarrow M\\rightarrow 0.$ Theorem 1.1 (Hilbert's Syzygy Theorem) Let $R={{\\mathbb {k}}[x_1, \\dots , x_n]}$ be a polynomial ring in $n$ variables over a field ${\\mathbb {k}}$ .",
"Every finitely generated $R$ -module $M$ has a finite free resolution of length at most $n$ .",
"In this section, we give an algorithmic Gröbner basis proof of Theorem REF , whose strategy is used in modern computer algebra systems like Macaulay2 and Singular for syzygy computations.",
"Gröbner bases were introduced by Gordan to provide a new proof of Hilbert's Basis Theorem [27].",
"We believe that Gordan could have given the proof of Theorem REF presented here.",
"Definition 1.2 A (global) monomial order on $R$ is a total order $>$ on the set of monomials in $R$ such that the following two statements hold: if $x^\\alpha > x^\\beta $ , then $x^\\gamma x^\\alpha > x^\\gamma x^\\beta $ for all $\\gamma \\in {\\mathbb {N}}^n$ , and $x_i > 1$ for all $i$ .",
"Given a global monomial order, the leading term of a nonzero polynomial $f = \\sum _{\\alpha } f_\\alpha x^\\alpha \\in R$ is defined to be ${\\bf {L}}(f) := f_\\beta x^\\beta ,\\quad \\text{where }x^\\beta := \\max _\\alpha \\lbrace x^\\alpha \\mid f_\\alpha \\ne 0\\rbrace .$ For convenience, set ${\\bf {L}}(0) := 0$ .",
"Theorem 1.3 (Division with Remainder) Let $>$ be a global monomial order on $R$ , and let ${f_1,\\ldots ,f_r}\\in R$ be nonzero polynomials.",
"For every $g\\in R$ , there exist uniquely determined $g_1,\\ldots , g_r \\in R$ and a remainder $h \\in R$ such that the following hold.",
"(1) We have $g=g_1f_1+\\cdots +g_rf_r + h$ .",
"(2a) No term of $g_i\\,{\\bf {L}}(f_i)$ is divisible by any ${\\bf {L}}(f_j)$ with $j<i$ .",
"(2b) No term of $h$ is a multiple of ${\\bf {L}}(f_i)$ for any $i$ .",
"The result is obvious if ${f_1,\\ldots ,f_r}$ are monomials, or more generally, if each $f_i$ has only a single nonzero term.",
"Thus there is always a unique expression $g = \\sum _{i=1}^r g^{(1)}_i\\,{\\bf {L}}(f_i) + h^{(1)},$ if we require that $g^{(1)}_1,\\ldots , g^{(1)}_r$ and $h^{(1)}$ satisfy (2a) and (2b).",
"By construction, the leading terms of the summands of the expression $\\sum _{i=1}^r g^{(1)}_i f_i+ h^{(1)}$ are distinct, and the leading term in the difference $g^{(1)}=g-(\\sum _{i=1}^r g^{(1)}_i\\,f_i + h^{(1)})$ cancels.",
"Thus ${\\bf {L}}(g^{(1)}) < {\\bf {L}}(g)$ , and we may apply induction on the number of summands in $g$ .",
"The remainder $h$ of the division of $g$ by ${f_1,\\ldots ,f_r}$ depends on the order of ${f_1,\\ldots ,f_r}$ , since the partition of the monomials in $R$ given by (2a) and (2b) depends on this order.",
"Even worse, it might not be the case that if $g \\in {\\langle } {f_1,\\ldots ,f_r}{\\rangle }$ , then $h=0$ .",
"A Gröbner basis is a system of generators for which this desirable property holds.",
"Definition 1.4 Let $I \\subset R$ be an ideal.",
"The leading ideal of $I$ (with respect to a given global monomial order) is ${\\bf {L}}(I) := {\\langle } \\,{\\bf {L}}(f) \\mid f \\in I\\, {\\rangle }.$ A finite set ${f_1,\\ldots ,f_r}$ of polynomials is a Gröbner basis when ${\\langle } {\\bf {L}}({\\langle } {f_1,\\ldots ,f_r}{\\rangle }) {\\rangle }={\\langle } \\, {\\bf {L}}(f_1),\\ldots ,{\\bf {L}}(f_r)\\,{\\rangle }.$ Gordan's proof of Hilbert's basis theorem now follows from the easier statement that monomial ideals are finitely generated.",
"In combinatorics, this result is called Dickson's Lemma [16].",
"If ${f_1,\\ldots ,f_r}$ is a Gröbner basis, then by definition, a polynomial $g$ lies in ${\\langle } {f_1,\\ldots ,f_r}{\\rangle }$ if and only if the remainder $h$ under division of $g$ by ${f_1,\\ldots ,f_r}$ is zero.",
"In particular, in this case, the remainder does not depend on the order of ${f_1,\\ldots ,f_r}$ , and the monomials $x^\\alpha \\notin {\\langle } {\\bf {L}}(f_1),\\ldots ,{\\bf {L}}(f_r) {\\rangle }$ represent a ${\\mathbb {k}}$ -vector space basis of the quotient ring $R/{\\langle } {f_1,\\ldots ,f_r}{\\rangle }$ , a fact known as Macaulay's theorem [39].",
"For these reasons, it is desirable to have a Gröbner basis on hand.",
"The algorithm that computes a Gröbner basis for an ideal is due to Buchberger [9], [10].",
"Usually, Buchberger's Criterion is formulated in terms of so-called S-pairs.",
"In the treatment below, we do not use S-pairs; instead, we focus on the partition of the monomials of $R$ induced by ${\\bf {L}}(f_1), \\ldots , {\\bf {L}}(f_r)$ via (2a) and (2b) of Theorem REF .",
"Given polynomials ${f_1,\\ldots ,f_r}$ , consider the monomial ideals $M_i := {\\langle } {\\bf {L}}(f_1),\\ldots ,{\\bf {L}}(f_{i-1}){\\rangle } : {\\bf {L}}(f_i) \\quad \\text{for \\,$i=1,\\ldots ,r$}.$ For each minimal generator $x^\\alpha $ of an $M_i$ , let $h^{(i,\\alpha )}$ denote the remainder of $x^\\alpha f_i$ divided by ${f_1,\\ldots ,f_r}$ (in this order!).",
"Theorem 1.5 (Buchberger's Criterion [10]) Let ${f_1,\\ldots ,f_r}\\in R$ be a collection of nonzero polynomials.",
"Then ${f_1,\\ldots ,f_r}$ form a Gröbner basis if and only if all of the remainders $h^{(i,\\alpha )}$ are zero.",
"We will prove this result after a few more preliminaries.",
"Example 1.6 Consider the ideal generated by the $3\\times 3$ minors of the matrix $\\begin{pmatrix}x_1 & x_2 & x_3 & x_4 & x_5 \\cr y_1 & y_2 & y_3 & y_4 & y_5 \\cr z_1 & z_2 & z_3 & z_4 & z_5 \\cr \\end{pmatrix}.$ Using the lexicographic order on ${\\mathbb {k}}[x_1,\\ldots ,z_5]$ , the leading terms of the maximal minors of this matrix and the minimal generators of the corresponding monomial ideals $M_i$ are listed in the following table.",
"$\\begin{array}{|c|l|}\\hline \\quad x_1y_2z_3\\quad &\\quad M_1=0 \\quad \\cr \\quad x_1y_2z_4\\quad & \\quad M_2={\\langle } z_3{\\rangle }\\quad \\cr \\quad x_1y_3z_4\\quad & \\quad M_3={\\langle } y_2{\\rangle }\\quad \\cr \\quad x_2y_3z_4\\quad & \\quad M_4={\\langle } x_1{\\rangle }\\quad \\cr \\quad x_1y_2z_5\\quad & \\quad M_5={\\langle } z_3,z_4{\\rangle }\\quad \\cr \\quad x_1y_3z_5\\quad & \\quad M_6={\\langle } y_2,z_4{\\rangle }\\quad \\cr \\quad x_2y_3z_5\\quad & \\quad M_7={\\langle } x_1,z_4{\\rangle }\\quad \\cr \\quad x_1y_4z_5\\quad & \\quad M_8={\\langle } y_2,y_3{\\rangle }\\quad \\cr \\quad x_2y_4z_5\\quad & \\quad M_9={\\langle } x_1,y_3{\\rangle }\\quad \\cr \\quad x_3y_4z_5\\quad & \\quad M_{10}={\\langle } x_1,x_2{\\rangle }\\quad \\cr \\hline \\end{array}$ Note that only 15 of the possible ${10 \\atopwithdelims ()2}=45$ S-pairs are needed to test Buchberger's Criterion, as stated in Theorem REF .",
"Exercise 1.7 Show that the maximal minors of the matrix in Example REF form a Gröbner basis by using the Laplace expansions of suitable $4\\times 4$ matrices.",
"In order to prove Theorems REF and REF , we now extend the notion of a monomial order to vectors of polynomials.",
"Definition 1.8 A monomial of a free module $R^r$ with basis $e_1,\\ldots ,e_r$ is an expression $x^\\alpha e_i$ .",
"A (global) monomial order on $R^r$ is a total order of the monomials of $R^r$ such that the following two statements hold: if $x^\\alpha e_i > x^\\beta e_j$ , then $x^\\gamma x^\\alpha e_i > x^\\gamma x^\\beta e_j$ for all $i,j$ and $\\gamma \\in {\\mathbb {N}}^n$ , and $x^\\alpha e_i >e_i$ for all $i$ and $\\alpha \\ne 0$ .",
"Usually, it is also the case that $x^\\alpha e_i > x^\\beta e_i$ if and only if $x^\\alpha e_j> x^\\beta e_j$ , i.e., the order on the monomials in the components induce a single monomial order on $R$ .",
"Thanks to Definition REF , we may now speak of the leading term of a vector of polynomials.",
"In this situation, the division theorem still holds.",
"Theorem 1.9 (Division with Remainder for Vectors of Polynomials) Let $>$ be a global monomial order on $R^{r_0}$ , and let $F_1,\\dots ,F_r \\in R^{r_0} $ be nonzero polynomial vectors.",
"For every $G\\in R^{r_0}$ , there exist uniquely determined $g_1,\\ldots ,g_r \\in R$ and a remainder $H \\in R^{r_0}$ such that the following hold.",
"(1) We have $G=g_1F_1+\\cdots +g_rF_r + H$ .",
"(2a) No term of $g_i\\, {\\bf {L}}(F_i)$ is a multiple of an ${\\bf {L}}(F_j)$ with $j< i$ .",
"(2b) No term of $H$ is a multiple of ${\\bf {L}}(F_i)$ for any $i$ .",
"$\\Box $ Definition 1.10 Generalizing the earlier definition, given a global monomial order on $R^r$ , the leading term of a nonzero vector of polynomials $F = ({f_1,\\ldots ,f_r})$ is defined to be the monomial ${\\bf {L}}(F) := f_{\\beta _i} x^{\\beta _i} e_i,\\quad \\text{where $x^{\\beta _i} = \\max _{\\alpha _i}\\lbrace x^{\\alpha _i}\\mid f_{\\alpha _i}x^{\\alpha _i}\\text{ is a nonzero term of } f_i \\rbrace $.",
"}$ A finite set $F_1,\\ldots ,F_s$ of vectors of polynomials in $R^r$ is a Gröbner basis when ${\\langle } {\\bf {L}}({\\langle }F_1,\\ldots ,F_s{\\rangle }) {\\rangle }= {\\langle }\\,{\\bf {L}}(F_1),\\ldots ,{\\bf {L}}(F_s)\\,{\\rangle }.$ We are now prepared to prove Theorem REF , followed by a corollary.",
"[Proof of Theorem REF (Buchberger's Criterion)] The forward direction follows by definition.",
"For the converse, assume that all remainders $h^{(i,\\alpha )}$ vanish.",
"Then for each minimal generator $x^\\alpha $ in an $M_i$ , there is an expression $x^\\alpha f_i =g_1^{(i,\\alpha )}f_1 +\\cdots + g_r^{(i,\\alpha )}f_r$ such that no term of $g^{(i,\\alpha )}_j{\\bf {L}}(f_j)$ is divisible by an ${\\bf {L}}(f_k)$ for every $k<j$ , by condition (2a) of Theorem REF .",
"(Of course, for a suitable $j<i$ , one of the terms of $g^{(i,\\alpha )}_j{\\bf {L}}(f_j)$ coincides with $x^\\alpha {\\bf {L}}(f_i)$ .",
"This is the second term in the usual S-pair description of Buchberger's Criterion.)",
"Now let $e_1,\\ldots , e_r \\in R^r$ denote the basis of the free module, and let $\\varphi :R^r\\rightarrow R$ be defined by $e_i \\mapsto f_i$ .",
"Then by (REF ), elements of the form $G^{(i,\\alpha )} :=-g^{(i,\\alpha )}_1e_1+\\cdots +(x^\\alpha -g^{(i,\\alpha )}_i)e_i+\\cdots + &(-g_r^{(i,\\alpha )})e_r$ are in the kernel of $\\phi $ .",
"In other words, the $G^{(i,\\alpha )}$ 's are syzygies between ${f_1,\\ldots ,f_r}$ .",
"We now proceed with a division with remainder in the free module $R^r$ , using the induced monomial order $>_1$ on $R^r$ defined by $\\begin{array}{ll} x^\\alpha e_i >_1 x^\\beta e_j\\iff & x^\\alpha {\\bf {L}}(f_i)> x^\\beta {\\bf {L}}(f_j) \\;\\hbox{ or }\\;\\cr & x^\\alpha {\\bf {L}}(f_i)= x^\\beta {\\bf {L}}(f_j)\\;\\hbox{ (up to a scalar)}\\;\\hbox{ with }\\;i > j.",
"\\cr \\end{array}$ With respect to this order, ${\\bf {L}}(G^{(i,\\alpha )})= x^\\alpha e_i$ because the term $cx^\\beta {\\bf {L}}(f_j)$ that cancels against $x^\\alpha {\\bf {L}}(f_i)$ in (REF ) satisfies $j<i$ , and all other terms of any $g^{(i,\\alpha )}_k {\\bf {L}}(f_k)$ are smaller.",
"Now consider an arbitrary element $g = a_1f_1+\\cdots +a_rf_r \\in {\\langle } {f_1,\\ldots ,f_r}{\\rangle }.$ We must show that ${\\bf {L}}(g) \\in {\\langle } {\\bf {L}}(f_1),\\ldots ,{\\bf {L}}(f_r) {\\rangle }$ .",
"Let $g_1e_1+\\cdots +g_re_r$ be the remainder of $a_1e_1+\\cdots +a_re_r$ divided by the collection of $G^{(i,\\alpha )}$ vectors.",
"Then $g=a_1f_1+\\cdots +a_rf_r =g_1f_1+\\cdots +g_rf_r$ because the $G^{(i,\\alpha )}$ are syzygies, and $g_1,\\ldots , g_r$ satisfy (2a) of Theorem REF when $a_1,\\ldots ,a_n$ are divided by ${f_1,\\ldots ,f_r}$ , by the definition of the $M_i$ in (REF ).",
"Therefore, the nonzero initial terms ${\\bf {L}}(g_jf_j)={\\bf {L}}(g_j){\\bf {L}}(f_j)$ are distinct and no cancellation can occur among them.",
"The proof is now complete because ${\\bf {L}}(g) := \\max _{j} \\lbrace {\\bf {L}}(g_j){\\bf {L}}(f_j) \\rbrace \\in {\\langle } {\\bf {L}}(f_1),\\ldots ,{\\bf {L}}(f_r) {\\rangle }.\\hfill $ Corollary 1.11 (Schreyer [49]) If $F_1,\\ldots ,F_{r_1} \\in R^{r_0}$ are a Gröbner basis, then the $G^{(i,\\alpha )}$ of (REF ) form a Gröbner basis of $\\ker (\\varphi _1\\colon R^{r_1} \\rightarrow R^{r_0})$ with respect to the induced monomial order $>_1$ defined in (REF ).",
"In particular, $F_1,\\ldots ,F_{r_1}$ generate the kernel of $\\varphi _1$ .",
"As mentioned in the proof of Theorem REF , the coefficients $g_1, \\ldots ,g_r$ of a remainder $g_1e_1+\\cdots +g_re_r$ resulting from division by the $G^{(i,\\alpha )}$ satisfy condition (2a) of Theorem REF when divided by ${f_1,\\ldots ,f_r}$ .",
"Hence, no cancellation can occur in the sum $g_1 {\\bf {L}}(f_1)+\\cdots +g_r {\\bf {L}}(f_r)$ , and $g_1f_1+\\cdots g_rf_r=0$ only if $g_1=\\ldots =g_r=0$ .",
"Therefore, the collection of ${\\bf {L}}(G^{(i,\\alpha )})$ generate the leading term ideal ${\\bf {L}}(\\ker \\varphi _1)$ .",
"We have reached the goal of this section, an algorithmic proof of Theorem REF .",
"[Proof of Theorem REF (Hilbert's Syzygy Theorem)] Let $M$ be a finitely generated $R$ -module with presentation $R^{r}\\overset{\\varphi }{\\longrightarrow }R^{r_0}\\longrightarrow M\\longrightarrow 0.$ Regard $\\varphi $ as a matrix and, thus, its columns as a set of generators for $\\operatorname{im}\\varphi $ .",
"Starting from these generators, compute a minimal Gröbner basis $F_1, \\ldots , F_{r_1}$ for $\\operatorname{im}\\varphi $ with respect to some global monomial $>_0$ order on $R^{r_0}$ .",
"Now consider the induced monomial order $>_1$ on $R^{r_1}$ , and let $G^{(i,\\alpha )}\\in R^{r_1}$ denote the syzygies obtained by applying Buchberger's Criterion to $F_1,\\ldots ,F_{r_1}$ .",
"By Corollary REF , the $G^{(i,\\alpha )}$ form a Gröbner basis for the kernel of the map ${\\varphi _1}\\colon R^{r_1} \\rightarrow R^{r_0}$ , so we may now repeat this process.",
"Let $\\ell $ be the maximal $k$ such that the variable $x_k$ occurs in some leading term ${\\bf {L}}(F_j)$ .",
"Sort $F_1,\\ldots ,F_{r_1}$ so that whenever $j < i$ , the exponent of $x_\\ell $ in ${\\bf {L}}(F_j)$ is less than or equal to the exponent of $x_\\ell $ in ${\\bf {L}}(F_i)$ .",
"In this way, none of the variables $x_\\ell ,\\ldots , x_n$ will occur in a leading term ${\\bf {L}}(G^{(i,\\alpha )})$ .",
"Thus the process will terminate after at most $n$ steps.",
"Note that there are a number of choices allowed in the algorithm in the proof of Theorem REF .",
"In particular, we may order each set of Gröbner basis elements as we see fit.",
"Example 1.12 Consider the ideal $I={\\langle } f_1 , \\dots ,f_5{\\rangle }\\subset R={\\mathbb {k}}[w,x,y,z]$ generated by the polynomials $f_1=w^2-xz,\\,\\,f_2=wx-yz,\\,\\,f_3=x^2-wy,\\,\\,f_4=xy-z^2,\\,\\,f_5=y^2-wz.$ To compute a finite free resolution of $M=R/I$ using the method of the proof of Theorem REF , we use the degree reverse lexicographic order on $R$ .",
"The algorithm successively produces three syzygy matrices $\\varphi _1$ , $\\varphi _2$ , and $\\varphi _3$ , which we present in a compact way as follows.",
"$\\begin{matrix}{\\bf w^2}-xz && -x & y & 0 &-z & 0 & -y^2+wz \\cr {\\bf wx}-yz&&{\\bf w} &-x &-y & 0 & z & z^2 \\cr {\\bf x^2}-wy&&-z &{\\bf w} & 0 &-y & 0 & 0 \\cr {\\bf xy}-z^2&& 0 & 0 &{\\bf w} &{\\bf x }&-y & yz \\cr {\\bf y^2}-wz&& 0 & 0 &-z &-w &{\\bf x} &{\\bf w^2} \\cr \\hline && 0 & y&-x&{\\bf w}&-z& 1 \\cr &&-y^2+wz & z^2 & -wy & yz & -w^2 &{\\bf x}\\end{matrix}$ All initial terms are printed in bold.",
"The first column of this table is the transpose of the matrix $\\varphi _1$ .",
"It contains the original generators for $I$ which, as Buchberger's Criterion shows, already form a Gröbner basis for $I$ .",
"The syzygy matrix $\\varphi _2$ resulting from the algorithm is the $5\\times 6$ matrix in the middle of our table.",
"Note that, for instance, $M_4 = {\\langle } w,x{\\rangle }$ can be read from the 4th row of $\\varphi _2$ .",
"By Corollary REF , we know that the columns of $\\varphi _2$ form a Gröbner basis for $\\ker (\\varphi _1)$ with respect to the induced monomial order on $R^5$ .",
"Buchberger's Criterion applied to these Gröbner basis elements yields a $6 \\times 2$ syzygy matrix $\\varphi _3$ , whose transpose is printed in the two bottom rows of the table above.",
"Note that there are no syzygies on the two columns of $\\varphi _3$ because the initial terms of these vectors lie with different basis vectors.",
"To summarize, we obtain a free resolution of the form $0 \\longrightarrow R^2 \\overset{\\varphi _3}{\\longrightarrow }R^6 \\overset{\\varphi _2}{\\longrightarrow }R^5 \\overset{\\varphi _1}{\\longrightarrow }R \\longrightarrow R/I \\longrightarrow 0.$ Observe that, in general, once we have the initial terms of a Gröbner basis for $I$ , we can easily compute the initial terms of the Gröbner bases for all syzygy modules, that is, all bold face entries of our table.",
"This gives us an idea on the amount of computation that will be needed to obtain the full free resolution.",
"If the polynomial ring is graded, say $R=S={\\mathbb {k}}[x_0,\\ldots ,x_n]$ is the homogeneous coordinate ring of ${\\mathbb {P}}^n$ , and $M$ is a finitely generated graded $S$ -module, then the resolution computed through the proof of Theorem REF is homogeneous as well.",
"However, this resolution is typically not minimal.",
"In Example REF , the last column of $\\varphi _2$ is in the span of the previous columns, as can be seen from the first row of $\\varphi _3^t$ .",
"Example 1.13 Recall that in Example REF , we considered the ideal $I$ of $3\\times 3$ minors of a generic $3\\times 5$ matrix over $S={\\mathbb {k}}[x_1,\\ldots ,z_5]$ with the standard grading.",
"The algorithm in the proof of Theorem REF produces a resolution of $S/I$ of the form $0 \\longrightarrow S(-5)^6 \\longrightarrow S(-4)^{15} \\longrightarrow S(-3)^{10}\\longrightarrow S \\longrightarrow S/I \\longrightarrow 0$ because $I$ is generated by 10 Gröbner basis elements, there are altogether 15 minimal generators of the $M_i$ ideals, and 6 of the monomial ideal $M_i$ have 2 generators.",
"In this case, the resolution is minimal for degree reasons.",
"Exercise 1.14 Let $I$ be a Borel-fixed monomial ideal.",
"Prove that in this case, the algorithm in the proof of Theorem REF produces a minimal free resolution of $I$ .",
"Compute the differentials explicitly and compare your result with the complex of S. Eliahou and Kervaire in [20] (see also [45])."
],
[
"Petri's Theorem",
"One of the first theoretical applications of Gröbner bases is Petri's analysis of the generators of the homogeneous ideal of a canonically embedded curve.",
"Petri was the last student of Max Noether, and he acknowledges help from Emmy Noether in his thesis.",
"As Emmy Noether was a student of Gordan, it is quite possible that Petri became aware of the concept of Gröbner bases through his communication with her, but we do not know if this was the case.",
"Let $C$ be a smooth projective curve of genus $g$ over ${\\mathbb {C}}$ .",
"Let $\\omega _1, \\ldots ,\\omega _g \\in H^0(C, \\omega _C)$ be a basis of the space of holomorphic differential forms on $C$ and consider the canonical map $\\iota : C \\rightarrow {\\mathbb {P}}^{g-1}\\quad \\text{given by}\\quad p\\mapsto [\\omega _1( p): \\cdots : \\omega _g( p)].$ The map $\\iota $ is an embedding unless $C$ is hyperelliptic.",
"We will assume that $C$ is not hyperelliptic.",
"Let $S:={\\mathbb {C}}[x_1,\\ldots ,x_g]$ be the homogeneous coordinate ring of ${\\mathbb {P}}^{g-1}$ , and let $I_C \\subset S$ be the homogeneous ideal of $C$ .",
"Theorem 2.1 (Petri's Theorem [46]) The homogeneous ideal of a canonically embedded curve is generated by quadrics unless $C$ is trigonal (i.e., there is a 3:1 holomorphic map $C \\rightarrow {\\mathbb {P}}^1$ ) or $C$ is isomorphic to a smooth plane quintic.",
"In this case, $g=6$ .",
"Petri's Theorem received much attention through the work of Mark Green [28], who formulated a conjectural generalization to higher syzygies of canonical curves in terms of the Clifford index.",
"We will not report here on the impressive progress made on this conjecture in the last two decades, but refer instead to [2], [3], [4], [29], [35], [41], [50], [51], [52], [59], [60], [61] for further reading.",
"In the cases of the exceptions in Theorem REF , also Babbage [5] observed that the ideal cannot be generated by quadrics alone.",
"If $D := p_1+\\cdots +p_d$ is a divisor of degree $d$ on $C$ , then the linear system $|\\omega _C(-D)|$ is cut out by hyperplanes through the span $\\overline{D}$ of the points $p_i \\in C \\subset {\\mathbb {P}}^{g-1}$ .",
"Thus Riemann–Roch implies that $h^0(C,{\\mathcal {O}}_C(D))= d+1- g + \\operatorname{codim}\\overline{D} = d- \\dim \\overline{D}.$ Hence the three points of a trigonal divisor span only a line, and by Bézout's theorem, we need cubic generators in the generating set of its vanishing ideal.",
"Similarly, in the second exceptional case, the 5 points of a $g^2_5$ are contained in a unique conic in the plane they span, and quadrics alone do not cut out the curve.",
"The first step of Petri's analysis builds upon a proof by Max Noether.",
"Theorem 2.2 (M. Noether [43]) A non-hyperelliptic canonical curve $C \\subset {\\mathbb {P}}^{g-1}$ is projectively normal, i.e., the maps $H^0({\\mathbb {P}}^{g-1},{\\mathcal {O}}(n))\\rightarrow H^0(C,\\omega _C^{\\otimes n})$ are surjective for every $n$ .",
"Max Noether's proof is a clever application of the base point free pencil trick.",
"This is a method which, according to Mumford, Zariski taught to all of his students.",
"Let $|D|$ be a base point free pencil on a curve, and let ${\\mathcal {L}}$ be a further line bundle on $C$ .",
"Then the Koszul complex $0 \\rightarrow \\Lambda ^2 H^0({\\mathcal {O}}_C(D)) \\otimes {\\mathcal {L}}(-D)\\rightarrow H^0({\\mathcal {O}}_C(D)) \\otimes {\\mathcal {L}}\\rightarrow {\\mathcal {L}}(D) \\rightarrow 0$ is an exact sequence.",
"To see this, note that locally, at least one section of the line bundle ${\\mathcal {O}}_C(D)$ does not vanish.",
"Thus the kernel of the multiplication map $H^0({\\mathcal {O}}_C(D)) \\otimes H^0({\\mathcal {L}}) \\rightarrow H^0({\\mathcal {L}}(D))$ is isomorphic with $H^0({\\mathcal {L}}(-D))$ .",
"(Note $\\Lambda ^2 H^0({\\mathcal {O}}_C(D)) \\cong {\\mathbb {C}}$ , as $h^0({\\mathcal {O}}_C(D))=2$ .)",
"Consider $p_1, \\ldots , p_g$ general points on $C$ and the divisor $D=p_1+\\cdots +p_{g-2}$ built from the first $g-2$ points.",
"Then the images of these points span ${\\mathbb {P}}^{g-1}$ and the span of any subset of less than $g-1$ points intersects the curve in no further points.",
"Choose a basis $\\omega _1, \\ldots , \\omega _g \\in H^0(\\omega _C)$ that is, up to scalars, dual to these points, i.e., $\\omega _i(p_j) = 0$ for $i\\ne j$ and $\\omega _i(p_i)\\ne 0$ .",
"Then $|\\omega _C(-D)|$ is a base point free pencil spanned by $\\omega _{g-1}, \\omega _g$ .",
"If we apply the base point free pencil trick to this pencil and ${\\mathcal {L}}=\\omega _C$ , then we obtain the sequence $0 \\rightarrow \\Lambda ^2 H^0\\omega _C(-D) \\otimes H^0{\\mathcal {O}}_C(D)\\rightarrow H^0\\omega _C(-D) \\otimes H^0\\omega _C\\longrightarrow ^{\\hspace{-12.80373pt}\\mu \\hspace{5.69054pt}} H^0\\omega _C^{\\otimes 2}(-D),$ and the image of $\\mu \\colon H^0(\\omega _C(-D)) \\otimes H^0(\\omega _C) \\rightarrow H^0(\\omega _C^{\\otimes 2}(-D))$ is $2g-1$ dimensional because $h^0(\\omega _C(-D))=2$ and $h^0({\\mathcal {O}}_C(D))=1$ .",
"Thus $\\mu $ in (REF ) is surjective, since $h^0(\\omega _C^{\\otimes 2}(-D))=2g-1$ holds by Riemann–Roch.",
"On the other hand, $\\omega _1^{\\otimes 2}, \\ldots ,\\omega _{g-2}^{\\otimes 2} \\in H^0(\\omega _C^{\\otimes 2})$ represent linearly independent elements of $H^0(\\omega _C^{\\otimes 2})/H^0(\\omega _C^{\\otimes 2}(-D))$ , hence represent a basis, and the map $H^0(\\omega _C) \\otimes H^0(\\omega _C)\\rightarrow H^0(\\omega _C^{\\otimes 2})$ is surjective as well.",
"This proves quadratic normality.",
"The surjectivity of the multiplication maps $H^0(\\omega _C^{\\otimes n-1})\\otimes H^0(\\omega _C)\\rightarrow H^0(\\omega _C^{\\otimes n})$ for $n\\ge 3$ is similar, but easier: $\\omega _1^{\\otimes n},\\ldots ,\\omega _{g-2}^{\\otimes n}\\in H^0(\\omega _C^{\\otimes n})$ are linearly independent modulo the codimension $g-2$ subspace $H^0(\\omega _C^{\\otimes n}(-D))$ , and the map $H^0(\\omega _C^{\\otimes n-1}) \\otimes H^0(\\omega _C(-D))\\rightarrow H^0(\\omega _C^{\\otimes n}(-D))$ is surjective simply because $H^1(\\omega _C^{\\otimes n-2}(D))=0$ for $n\\ge 3$ .",
"Corollary 2.3 The Hilbert function of the coordinate ring of a canonical curve takes values $\\hspace{82.51282pt}\\dim (S/I_C)_n = {\\left\\lbrace \\begin{array}{ll}1 & \\hbox{ if } n= 0\\\\g & \\hbox{ if } n= 1\\\\(2n-1)(g-1) &\\hbox{ if } n\\ge 2.\\hspace{64.87224pt}\\Box \\end{array}\\right.",
"}$ [Proof of Theorem REF (Petri's Theorem)] Petri's analysis begins with the map $\\mu $ in (REF ) above.",
"Choose homogeneous coordinates $x_1,\\ldots , x_g$ such that $x_i \\mapsto \\omega _i$ .",
"Since $\\omega _i\\otimes \\omega _j \\in H^0(\\omega _C^{\\otimes 2}(-D))$ for $1 \\le i <j \\le g-2$ , we find the polynomials $f_{ij} := x_ix_j -\\sum _{r=1}^{g-2}a_{ij}^r x_r - b_{ij} \\in I_C,$ where the $a^r_{ij}$ and $b_{ij}$ are linear and quadratic, respectively, in ${\\mathbb {C}}[x_{g-1},x_g]$ .",
"We may choose a monomial order such that ${\\bf {L}}(f_{ij})=x_ix_j$ .",
"Since ${g-2 \\atopwithdelims ()2}={g+1 \\atopwithdelims ()2} - (3g-3)$ , these quadrics span $(I_C)_2$ .",
"On the other hand, they do not form a Gröbner basis for $I_C$ because the $(g-2){n+1 \\atopwithdelims ()2}+(n+1)$ monomials $x_i^kx_{g-1}^\\ell x_g^m$ with $i=1, \\ldots , g-2$ and $k+\\ell +m=n$ represent a basis for $(S/{\\langle } x_ix_j\\mid 1\\le i<j\\le g-2 {\\rangle })_n$ , which is still larger.",
"We therefore need $g-3$ further cubic Gröbner basis elements.",
"To find these, Petri considers the base point free pencil trick applied to $|\\omega _C(-D)|$ and ${\\mathcal {L}}=\\omega _C^{\\otimes 2}(-D)$ .",
"The cokernel of the map $H^0(\\omega _C(-D)) \\otimes H^0(\\omega _C^{\\otimes 2}(-D))\\rightarrow H^0(\\omega _C^{\\otimes 3}(-2D))$ has dimension $h^1(\\omega _C)=1$ .",
"To find the missing element in $H^0(\\omega _C^{\\otimes 3}(-2D))$ , Petri considers the linear form $\\alpha _i=\\alpha _i(x_{g-1},x_g)$ in the pencil spanned by $x_{g-1},x_g$ that defines a tangent hyperplane to $C$ at $p_i$ .",
"Then $\\alpha _i\\omega _i^{\\otimes 2} \\in H^0(\\omega _C^{\\otimes 3}(-2D))$ because $\\omega _i^{\\otimes 2}$ vanishes quadratically at all points $p_j \\ne p_i$ , while $\\alpha _i$ vanishes doubly at $p_i$ .",
"Not all of these elements can be contained in the image of (REF ), since otherwise we would find $g-2$ further cubic Gröbner basis elements of type $\\alpha _i x_i^2 + \\hbox{ lower order terms,}$ where a lower order term is a term that is at most linear in $x_1,\\ldots ,x_{g-2}$ .",
"As this is too many, at least one of the $\\alpha _i\\omega _i^{\\otimes 2}$ spans the cokernel of the map (REF ).",
"We now argue by uniform position.",
"Since $C$ is irreducible, the behavior of $\\alpha _i\\omega _i^{\\otimes 2}$ with respect to spanning of the cokernel is the same for any general choice of points $p_1, \\ldots ,p_g$ .",
"So for general choices, each of these elements span the cokernel, and after adjusting scalars, we find that $G_{k\\ell } := \\alpha _k x_k^2 -\\alpha _\\ell x_\\ell ^2 + \\hbox{ lower order terms }$ are in $I_C$ .",
"Note that $G_{k\\ell }= - G_{\\ell k}$ and $G_{k\\ell }+G_{\\ell m} = G_{km}$ .",
"So this gives only $g-3$ further equations with leading terms $x_k^2x_{g-1}$ for $k=1,\\ldots ,g-3$ up to a scalar.",
"The last Gröbner basis element is a quartic $H$ with leading term ${\\bf {L}}(H)=x_{g-2}^3x_{g-1}$ , which we can obtain as a remainder of the Buchberger test applied to $x_{g-2}G_{k,g-2}$ .",
"There are no further Gröbner basis elements, because the quotient $S/J$ of $S$ by $J:={\\langle } x_ix_j, x_k^2x_{g-1}, x_{g-2}^3x_{g-1}\\mid 1 \\le i< j \\le g-2, 1 \\le k \\le g-3 {\\rangle }$ has the same Hilbert function as $S/I_C$ .",
"Hence ${\\bf {L}}(I_C) = J$ .",
"We now apply Buchberger's test to $x_kf_{ij}$ for a triple of distinct indices $1\\le i,j,k\\le g-2$ .",
"Division with remainder yields a syzygy $x_kf_{ij} - x_jf_{ik} + \\sum _{r\\ne k}a_{ij}^r f_{rk} - \\sum _{r \\ne j}a_{ik}^r f_{rj} + \\rho _{ijk} G_{kj} =0$ for a suitable coefficient $\\rho _{ijk} \\in {\\mathbb {C}}$ .",
"(Moreover, comparing coefficients, we find that $a_{ij}^k = \\rho _{ijk}\\alpha _k$ holds.",
"In particular, Petri's coefficients $\\rho _{ijk}$ are symmetric in $i,j,k$ , since $a_{ij}^k$ is symmetric in $i,j$ .)",
"Since $C$ is irreducible, we have that for a general choice of $p_1, \\ldots , p_g$ , either all coefficients $\\rho _{ijk} \\ne 0$ or all $\\rho _{ijk}=0$ .",
"In the first case, the cubics lie in the ideal generated by the quadrics.",
"In the second case, the $f_{ij}$ are a Gröbner basis by themselves.",
"Thus the zero locus $V(f_{ij}| 1 \\le i<j \\le g-2)$ of the quadrics $f_{ij}$ define an ideal of a scheme $X$ of dimension 2 and degree $g-2$ .",
"Since $C$ is irreducible and non-degenerate, the surface $X$ is irreducible and non-degenerate as well.",
"Thus $X \\subset {\\mathbb {P}}^{g-2}$ is a surface of minimal degree.",
"These were classified by Bertini, see, for instance, [19].",
"Either $X$ is a rational normal surface scroll, or $X$ is isomorphic to the Veronese surface ${\\mathbb {P}}^2 \\hookrightarrow {\\mathbb {P}}^5$ .",
"In the case of a scroll, the ruling on $X$ cuts out a $g^1_3$ on $C$ by Riemann–Roch.",
"In the case of the Veronese surface, the preimage of $C$ in ${\\mathbb {P}}^2$ is a plane quintic.",
"Perhaps the most surprising part of Petri's theorem is this: either $I_C$ is generated by quadrics or there are precisely $g-3$ minimal cubic generators.",
"It is a consequence of the irreducibility of $C$ that no value in between 0 and $g-3$ is possible for the number of cubic generators.",
"If we drop the assumption of irreducibility, then there are canonical curves with $1,\\ldots ,g-5$ or $g-3$ cubic generators.",
"For example, if we take a stable curve $C=C_1\\cup C_2$ with two smooth components of genus $g_i \\ge 1$ intersecting in three points, so that $C$ has genus $g=g_1+g_2+2$ , then the dualizing sheaf $\\omega _C$ is very ample and the three intersection points lie on a line by the residue theorem.",
"For general curves $C_1$ and $C_2$ of genus $g_i \\ge 3$ for $i\\in \\lbrace 1,2\\rbrace $ , the ideal $I_C$ has precisely one cubic generator, see [51].",
"However, we could not find such an example with precisely $g-4$ generators.",
"For genus $g=5$ , one cubic generator is excluded by the structure theorem of Buchsbaum–Eisenbud, and obstructions for larger $g$ are unclear to us.",
"Conjecture 2.1 Let $A=S/I$ be a graded artinian Gorenstein algebra with Hilbert function $\\lbrace 1,g-2,g-2,1\\rbrace $ .",
"Then $I$ has $0,1,\\ldots ,g-5$ or $g-3$ cubic minimal generators.",
"The veracity of this conjecture would imply the corresponding statement for reducible canonical curves because the artinian reduction $A:=S/(I_C+{\\langle } \\ell _1,\\ell _2 {\\rangle })$ of $S/I_C$ , for general linear forms $\\ell _1,\\ell _2$ , has Hilbert function $\\lbrace 1,g-2,g-2,1\\rbrace $ .",
"Petri's analysis has been treated by Mumford [42], as well as [1], [48], [57].",
"From our point of view, Gröbner bases and the use of uniform position simplify and clarify the treatment quite a bit.",
"Mumford remarks in [42] that we now have seen all curves at least once, following a claim of Petri [46].",
"We disagree with him in this point.",
"If we introduce indeterminates for all of the coefficients in Petri's equations, then the scheme defined by the condition on the coefficients that $f_{ij}, G_{kl}$ , and $H$ form a Gröbner basis can have many components [51], [38].",
"It is not clear to us how to find the component corresponding to smooth curves, much less how to find closed points on this component."
],
[
"Finite length modules and space curves",
"In the remaining part of these lectures, we report on how to find all curves in a Zariski open subset of the moduli space ${\\mathfrak {M}}_g$ of curves of genus $g$ for small $g$ .",
"In Section , we report on the known unirationality results for these moduli spaces.",
"But first, we must discuss a method to explicitly construct space curves.",
"In this section, a space curve $C \\subset {\\mathbb {P}}^3$ will be a Cohen–Macaulay subscheme of pure dimension 1; in particular, $C$ has no embedded points.",
"We denote by ${\\mathcal {I}}_C$ the ideal sheaf of $C$ and by $I_C = \\sum _{n\\in {\\mathbb {Z}}} H^0({\\mathbb {P}}^3, {\\mathcal {I}}_C(n))$ the homogeneous ideal of $C$ .",
"The goal of this section is to construct a curve $C$ of genus $g$ and degree $d$ .",
"To do so, we will use work of Rao, who showed that the construction of $C$ is equivalent to the creation of its Hartshorne–Rao module (see Theorem REF ).",
"Definition 3.1 The Hartshorne–Rao module of $C$ is the finite length module $M = M_C := \\sum _{n \\in {\\mathbb {Z}}} H^1({\\mathbb {P}}^3, {\\mathcal {I}}_C(n))\\subset \\sum _{n\\in {\\mathbb {Z}}} H^0({\\mathbb {P}}^3, {\\mathcal {O}}(n))\\cong S:={\\mathbb {k}}[x_0,..,x_3].$ The Hartshorne–Rao module measures the deviation of $C$ from being projectively normal.",
"Furthermore, $M_C$ plays an important role in liaison theory of curves in ${\\mathbb {P}}^3$ , which we briefly recall now.",
"Let $S:={\\mathbb {k}}[x_0,\\ldots ,x_3]$ and $S_C:=S/I_C$ denote the homogeneous coordinate ring of ${\\mathbb {P}}^3$ and $C \\subset {\\mathbb {P}}^3$ , respectively.",
"By the Auslander–Buchsbaum–Serre formula [17], $S_C$ has projective dimension $\\operatorname{pd}_S S_C \\le 3$ .",
"Thus its minimal free resolution has the form $0 \\leftarrow S_C \\leftarrow S \\leftarrow F_1\\leftarrow F_2 \\leftarrow F_3 \\leftarrow 0,$ with free graded modules $F_i=\\oplus S(-j)^{\\beta _{ij}}$ .",
"By the same formula in the local case, we see that the sheafified ${\\mathcal {G}}:=\\ker (\\widetilde{F}_1 \\rightarrow {\\mathcal {O}}_{{\\mathbb {P}}^3})$ is always a vector bundle, and $0\\leftarrow {\\mathcal {O}}_C \\leftarrow {\\mathcal {O}}_{{\\mathbb {P}}^3}\\leftarrow \\bigoplus _j {\\mathcal {O}}_{{\\mathbb {P}}^3}(-j)^{\\beta _{1j}}\\leftarrow {\\mathcal {G}}\\leftarrow 0$ is a resolution by locally free sheaves.",
"If $C$ is arithmetically Cohen–Macaulay, then $F_3=0$ and ${\\mathcal {G}}$ splits into a direct sum of line bundles.",
"In this case, the ideal $I_C$ is generated by the maximal minors of $F_1 \\leftarrow F_2$ by the Hilbert–Burch Theorem [34], [11], [17].",
"In general, we have $M_C \\cong \\sum _{n \\in {\\mathbb {Z}}} H^2({\\mathbb {P}}^3,{\\mathcal {G}}(n))\\quad \\text{and} \\quad \\sum _{n \\in {\\mathbb {Z}}}H^1({\\mathbb {P}}^3,{\\mathcal {G}}(n))=0.$ We explain now why curves linked by an even number of liaison steps have, up to a twist, the same Hartshorne–Rao module, thus illustrating its connection to liaison theory.",
"We will then mention Rao's Theorem, which states that the converse also holds (Theorem REF ).",
"Suppose that $f,g \\in I_C$ are homogeneous forms of degree $d$ and $e$ without common factors.",
"Let $X := V(f,g)$ denote the corresponding complete intersection, and let $C^{\\prime }$ be the residual scheme defined by the homogeneous ideal $I_{C^{\\prime }}:=(f,g):I_C$ [44].",
"The locally free resolutions of ${\\mathcal {O}}_C$ and ${\\mathcal {O}}_{C^{\\prime }}$ are closely related, as follows.",
"Applying $\\mathcal {E}xt^2(-,\\omega _{{\\mathbb {P}}^3})$ to the sequence $0 \\rightarrow {\\mathcal {I}}_{C/X} \\rightarrow {\\mathcal {O}}_X \\rightarrow {\\mathcal {O}}_C \\rightarrow 0$ gives $0 \\leftarrow \\mathcal {E}xt^2({\\mathcal {I}}_{C/X},\\omega _{{\\mathbb {P}}^3}) \\leftarrow \\omega _X \\leftarrow \\omega _C \\leftarrow 0.$ From $\\omega _X \\cong {\\mathcal {O}}_X(d+e-4)$ , we conclude that $\\mathcal {E}xt^2({\\mathcal {I}}_{C/X},{\\mathcal {O}}_{{\\mathbb {P}}^3}(-d-e))\\cong {\\mathcal {O}}_{C^{\\prime }}$ , and hence ${\\mathcal {I}}_{C^{\\prime }/X} \\cong \\omega _C(-d-e+4)$ .",
"Now the mapping cone of ${0 & {\\mathcal {O}}_C [l] & {\\mathcal {O}}_{{\\mathbb {P}}^3} [l] &\\bigoplus _j {\\mathcal {O}}_{{\\mathbb {P}}^3}(-j)^{\\beta _{1j}}[l] & {\\mathcal {G}}[l]&0 [l] \\\\0 &{\\mathcal {O}}_X [l] [u] & {\\mathcal {O}}_{{\\mathbb {P}}^3} [l] [u]_\\cong & {\\mathcal {O}}_{{\\mathbb {P}}^3}(-d) \\oplus {\\mathcal {O}}_{{\\mathbb {P}}^3}(-e) [l] [u] & {\\mathcal {O}}_{{\\mathbb {P}}^3}(-d-e) [l] [u]& 0 [l]}$ dualized with $\\operatorname{{\\mathcal {H}}om}(-,{\\mathcal {O}}_{{\\mathbb {P}}^3}(-d-e))$ gives ${0\\rightarrow {\\mathcal {O}}_{{\\mathbb {P}}^3}(-d-e) [r] [d]_\\cong &\\bigoplus _j {\\mathcal {O}}_{{\\mathbb {P}}^3}(j-d-e)^{\\beta _{1j}}[r] [d]& {\\mathcal {G}}^*(-d-e)[d] @{->>}[r]& {\\mathcal {I}}_{C^{\\prime }/X} [d] \\\\0 \\rightarrow {\\mathcal {O}}_{{\\mathbb {P}}^3}(-d-e) [r] & {\\mathcal {O}}_{{\\mathbb {P}}^3}(-e) \\oplus {\\mathcal {O}}_{{\\mathbb {P}}^3}(-d) [r] & {\\mathcal {O}}_{{\\mathbb {P}}^3} @{->>}[r] & {\\mathcal {O}}_X,}$ which yields the following locally free resolution of ${\\mathcal {O}}_{C^{\\prime }}$ : $0 \\rightarrow \\bigoplus _j {\\mathcal {O}}_{{\\mathbb {P}}^3}(j-d-e)^{\\beta _{1j}}\\rightarrow {\\mathcal {G}}^*(-d-e)\\oplus {\\mathcal {O}}_{{\\mathbb {P}}^3}(-e) \\oplus {\\mathcal {O}}_{{\\mathbb {P}}^3}(-d)\\rightarrow {\\mathcal {O}}_{{\\mathbb {P}}^3} \\rightarrow {\\mathcal {O}}_{C^{\\prime }}\\rightarrow 0.$ In particular, after truncating this complex to resolve $I_{C^{\\prime }}$ , one sees that $M_{C^{\\prime }}&:= \\sum _{n\\in {\\mathbb {Z}}} H^1({\\mathbb {P}}^3, {\\mathcal {I}}_{C^{\\prime }} (n))\\cong \\sum _{n \\in {\\mathbb {Z}}} H^1({\\mathbb {P}}^3,{\\mathcal {G}}^*(n-d-e)) \\\\& \\phantom{:}\\cong \\sum _{n\\in {\\mathbb {Z}}} H^2({\\mathbb {P}}^3,{\\mathcal {G}}(d+e-4-n))^* \\cong \\operatorname{Hom}_{\\mathbb {k}}(M_C,{\\mathbb {k}})(4-d-e).$ Thus curves that are related via an even number of liaison steps have the same Hartshorne–Rao module up to a twist.",
"Rao's famous result says that the converse is also true.",
"Theorem 3.2 (Rao's Theorem [47]) The even liaison classes of curves in ${\\mathbb {P}}^3$ are in bijection with finite length graded $S$ - modules up to twist.",
"$\\Box $ Therefore the difficulty in constructing the desired space curve $C$ (of degree $d$ and genus $g$ ) lies completely in the construction of the appropriate Hartshorne–Rao module $M=M_C$ .",
"Upon constructing $M$ , we may then obtain the desired ideal sheaf ${\\mathcal {I}}_C$ as follows.",
"Assume that we have a free $S$ -resolution of $M_C$ , $0 \\leftarrow M_C \\leftarrow F_0 \\leftarrow F_1 \\leftarrow F_2 \\leftarrow F_3 \\leftarrow F_4 \\leftarrow 0,$ with $F_i= \\bigoplus _j S(-j)^{\\beta _{ij}}$ .",
"Let ${\\mathcal {F}}:= \\widetilde{N}$ be the sheafification of $N := \\ker ( F_1 \\rightarrow F_0)$ , the second syzygy module of $M$ .",
"In this case, ${\\mathcal {F}}$ will be a vector bundle without line bundle summands such that $H_*^1({\\mathcal {F}}) \\cong H^1_* ({\\mathcal {I}}_C)$ and $H_*^2({\\mathcal {F}})=0$ .",
"Here, we have used the notation $H_*^i({\\mathcal {F}}) := \\bigoplus _n H^i({\\mathcal {F}}(n))$ .",
"If we constructed the correct Hartshorne–Rao module $M$ , then taking ${\\mathcal {L}}_1$ and ${\\mathcal {L}}_2$ to be appropriate choices of direct sums of line bundles on ${\\mathbb {P}}^3$ , a general homomorphism $\\varphi \\in \\operatorname{Hom}({\\mathcal {L}}_1,{\\mathcal {F}}\\oplus {\\mathcal {L}}_2)$ will produce the desired curve $C$ , as we will obtain ${\\mathcal {I}}_C$ as the cokernel of a map $\\varphi $ of the bundles ${ 0 [r] & {\\mathcal {L}}_1 [r]^{\\varphi \\hspace{9.95845pt}} & {\\mathcal {F}}\\oplus {\\mathcal {L}}_2 [r] &{\\mathcal {I}}_C [r] & 0 \\\\}.$ To compute the rank of ${\\mathcal {F}}$ and to choose the direct sums of line bundles ${\\mathcal {L}}_1$ and ${\\mathcal {L}}_2$ , we now make plausible assumptions about the Hilbert function of $M_C$ .",
"We illustrate this approach in the example of the construction of a smooth linearly normal curve $C$ of degree $d=11$ and genus $g=10$ .",
"Since $ 2 d > 2g-2$ , the line bundle ${\\mathcal {O}}_C(2)$ is already non-special.",
"Hence by Riemann–Roch, we have that $h^0 ({\\mathcal {O}}_C(2))=22+1-10=13$ .",
"Remark 3.3 If we assume that $C$ is a curve of maximal rank, i.e., that all maps $H^0 ({\\mathcal {O}}_{{\\mathbb {P}}^3}(n)) \\rightarrow H^0 ({\\mathcal {O}}_C(n))$ are either injective or surjective, then we can compute the Hilbert function of $M_C$ and $I_C$ .",
"Note that being of maximal rank is an open condition, so among the curves in the union ${\\mathcal {H}}_{d,g}$ of the component of the Hilbert scheme ${\\rm Hilb}_{dt+1-g}({\\mathbb {P}}^3)$ containing smooth curves, maximal rank curves form an open (and hopefully nonempty) subset.",
"There is a vast literature on the existence of maximal rank curves; see, for example, [24].",
"To gain insight into the Betti numbers of $M=M_C$ , we use Hilbert's formula for the Hilbert series: $h_M(t) = \\sum _{n\\in {\\mathbb {Z}}} \\dim M_n t^n=\\frac{ \\sum _{i=0}^3 (-1)^i \\sum _j \\beta _{ij} t^j}{(1-t)^4}.$ Since $h_{M_C}(t)= 3t^2+4t^3$ by our maximal rank assumption (Remark REF ), we have $(1-t)^4 h_{M}(t)=3t^2-8t^3+2t^4+12t^5-13t^6+4t^7,$ and thus the Betti table of $M$ must be $\\beta (M) \\ = $ Table: NO_CAPTION, if we assume that $M$ has a so called natural resolution, which means that for each degree $j$ at most one $\\beta _{ij}$ is nonzero.",
"Note that having a natural resolution is an open condition in a family of modules with constant Hilbert function.",
"Figure: With our maximal rank assumption ofRemark , this tableprovides the relevant Hilbert functions in thecase d=11d=11 and g=10g=10.Figure REF provides a detailed look at the Hilbert functions relevant to our computation.",
"From these we see that $H^0_* ({\\mathcal {O}}_C)$ and $S_C=S/I_C$ will have the following potential Betti tables, if we assume that they also have natural resolutions: $\\beta (H^0_*({\\mathcal {O}}_C)) \\ =$ Table: NO_CAPTION $\\quad $ and $\\quad \\beta (S_C) \\ =$ Table: NO_CAPTION.",
"Comparing these Betti tables, we find the following plausible choices of ${\\mathcal {F}}$ , ${\\mathcal {L}}_1$ , and ${\\mathcal {L}}_2$ : We choose ${\\mathcal {F}}:= \\widetilde{N}$ , where $N =\\ker ( \\psi : S^8(-3) \\rightarrow S^3(-2))$ is a sufficiently general $ 3\\times 8$ matrix of linear forms; in particular, $\\operatorname{rank}{\\mathcal {F}}=5$ .",
"Let ${\\mathcal {L}}_1:={\\mathcal {O}}^2(-4)\\oplus {\\mathcal {O}}^2(-5)$ and ${\\mathcal {L}}_2:=0$ .",
"The map $\\varphi \\in \\operatorname{Hom}({\\mathcal {L}}_1, {\\mathcal {F}})$ is a sufficiently general homomorphism.",
"Since the map $F_2 \\rightarrow H^0_*({\\mathcal {F}})$ is surjective, the choice of $\\varphi $ amounts to choosing an inclusion ${\\mathcal {O}}^2(-5) \\rightarrow {\\mathcal {O}}^{12}(-5)$ , i.e., a point in the Grassmannian ${\\mathbb {G}}(2,12)$ .",
"Finally, ${\\mathcal {I}}_C= {\\rm {coker}\\,}\\varphi $ .",
"It is not clear that general choices as above will necessarily yield a smooth curve.",
"If the sheaf $\\operatorname{{\\mathcal {H}}om}({\\mathcal {L}}_1,{\\mathcal {F}}\\oplus {\\mathcal {L}}_2)$ happens to be generated by its global sections $\\operatorname{Hom}({\\mathcal {L}}_1,{\\mathcal {F}}\\oplus {\\mathcal {L}}_2)$ , then a Bertini-type theorem as in [36] would apply.",
"However, since we have to take all generators of $H^0_* ({\\mathcal {F}})$ in degree 4, this is not the case.",
"On the other hand, there is no obvious reason that ${\\rm {coker}\\,}\\varphi $ should not define a smooth curve, and upon construction, it is easy to check the smoothness of such an example using a computer algebra system, e.g., Macaulay2 or Singular.",
"Doing this, we find that general choices do lead to a smooth curve.",
"Exercise 3.4 Construct examples of curves of degree and genus as prescribed in Hartshone's book [32] in Figure 18 on page 354, including those which were open cases at the time of the book's publication."
],
[
"Random curves",
"In this section, we explain how the ideas of Section lead to a computer-aided proof of the unirationality of the moduli space ${\\mathfrak {M}}_g$ of curves of genus $g$ , when $g$ is small.",
"We will illustrate this approach by example, through the case of genus $g=12$ and degree $d=13$ in Theorem REF .",
"Definition 4.1 A variety $X$ is called unirational if there exists a dominant rational map ${\\mathbb {A}}^n \\dasharrow X$ .",
"A variety $X$ is called uniruled if there exists a dominant rational map ${\\mathbb {A}}^1\\times Y \\dasharrow X$ for some variety $Y$ that does not factor through $Y$ .",
"A smooth projective variety $X$ has Kodaira dimension $\\kappa $ if the section ring $R_X:=\\sum _{n\\ge 0} H^0(X,\\omega _X^{\\otimes n})$ of pluri-canonical forms on $X$ has a Hilbert function with growth rate $h^0(\\omega _X^{\\otimes n}) \\in O(n^\\kappa )$ .",
"We say that $X$ has general type if $\\kappa = \\dim X$ , the maximal possible value.",
"Since the pluri-genera $h^0(\\omega _X^{\\otimes n})$ are birational invariants, being of general type does not depend on a choice of a smooth compactification.",
"Thus we may also speak of general type for quasi-projective varieties.",
"Unirationality and general type are on opposite ends of birational geometry.",
"If a variety is of general type, then there exists no rational curve through a general point of $X$ [37].",
"On the other hand, uniruled varieties have the pluri-canonical ring $R_X=(R_X)_0={\\mathbb {C}}$ and thus (by convention) have Kodaira dimension $\\kappa =-\\infty $ .",
"In fact, even if $X$ is unirational, then we can connect any two general points of $X$ by a rational curve.",
"We now recall results concerning the unirationality of the moduli space ${\\mathfrak {M}}_g$ .",
"There are positive results for small genus, followed by negative results for large genus.",
"Theorem 4.2 (Severi, Sernesi, Chang–Ran, Verra) The moduli space ${\\mathfrak {M}}_g$ of curves of genus $g$ is unirational for $g\\le 14$ .",
"(For $g\\le 10$ , see [56].",
"For $g=12,11,13$ , see [55], [12].",
"For $g=14$ , see [58].)",
"$\\Box $ Theorem 4.3 (Harris–Mumford, Eisenbud–Harris, Farkas [31], [18], [21], [22], [23]) The moduli space ${\\mathfrak {M}}_g$ of curves of genus $g$ is of general type for $g\\ge 24$ or $g=22$ .",
"The moduli space $M_{23}$ has Kodaira dimension $\\ge 2$ .",
"$\\Box $ We call this beautiful theorem a negative result because it says that it will be very difficult to write down explicitly a general curve of large genus.",
"Given a family of curves of genus $g\\ge 24$ that pass through a general point of ${\\mathfrak {M}}_g$ , say via an explicit system of equations with varying coefficients, none of the essential coefficients is a free parameter.",
"All of the coefficients will satisfy some complicated algebraic relations.",
"On the other hand, in unirational cases, there exists a dominant family of curves whose parameters vary freely.",
"In principle, we can compute a dominating family explicitly along with a unirationality proof.",
"In practice, this is often out of reach using current computer algebra systems; however, the following approach is feasible today in many cases.",
"By replacing each free parameter in the construction of the family by a randomly chosen value in the ground field, the computation of an explicit example is possible.",
"In particular, over a finite field ${\\mathbb {F}}$ , where it is natural to use the constant probability distribution on ${\\mathbb {F}}$ , a unirationality proof brings with it the possibility of choosing random points in ${\\mathfrak {M}}_g({\\mathbb {F}})$ , i.e., to compute a random curve.",
"These curves can then be used for further investigations of the moduli space, as well as to considerably simplify the existing unirationality proofs.",
"The advantage of using such random curves in the unirationality proof is that, with high probability, they will be smooth curves, while in a theoretical treatment, smoothness is always a delicate issue.",
"To begin this construction, we first need some information on the projective models of a general curve.",
"This is the content of Brill– Noether theory.",
"Let $W^r_d(C) :=\\lbrace L \\in \\operatorname{Pic}^d(C) \\mid h^0(C,L) \\ge r+1 \\rbrace \\subset \\operatorname{Pic}^d(C)$ denote the space of line bundles of degree $d$ on $C$ that give rise to a morphism $C \\rightarrow {\\mathbb {P}}^r$ .",
"Theorem 4.4 (Brill–Noether, Griffith–Harris, Fulton–Lazarsfeld, Gieseker) Let $C$ be a smooth projective curve of genus $g$ .",
"[7] At every point, $\\dim W^r_d(C) \\ge \\rho := g-(r+1)(g-d+r)$ .",
"[30], [25] If $\\rho \\ge 0$ , then $W^r_d(C) \\ne 0$ , and if $\\rho >0$ , then $W^r_d(C)$ is connected.",
"Further, the tangent space of $W^r_d(C)$ at a point $L \\in W^r_d(C) \\setminus W^{r+1}_d(C)$ is $T_L W^r_d(C) ={\\rm Im}\\ \\mu _L^\\perp \\subset H^1({\\mathcal {O}}_C)= T_L \\operatorname{Pic}^d(C),$ where $\\mu _L\\colon H^0(L)\\otimes H^0(\\omega _C\\otimes L^{-1})\\rightarrow H^0(\\omega _C) = H^1({\\mathcal {O}}_C)^*$ denotes the Petri map.",
"[26] If $C \\in {\\mathfrak {M}}_g$ is a general curve, then $W^r_d(C)$ is smooth of dimension $\\rho $ away from $W^{r+1}_d(C)$ .",
"More precisely, the Petri map $\\mu _L$ is injective for all $L \\in W^r_d(C)\\setminus W^{r+1}_d(C)$ .",
"$\\Box $ We now illustrate the computer-aided unirationality proof of ${\\mathfrak {M}}_g$ by example, through the case $g=12,d=13$ [54].",
"This case is not amongst those covered by Sernesi [55] or Chang–Ran [12].",
"They treated the cases $g=11,d=12$ , $g=12,d=12$ , and $g=13,d=13$ .",
"We are choosing the case $d=12,g=13$ because it illustrates well the difficulty of this construction.",
"For $g=14$ , see [58] and, for a computer aided unirationality proof, [53].",
"For a related Macaulay2 package, see [6].",
"Theorem 4.5 Let $g=12$ and $d=13$ .",
"Then ${\\rm Hilb}_{dt+1-g}({\\mathbb {P}}^3)$ has a component ${\\mathcal {H}}_{d,g}$ that is unirational and dominates the moduli space ${\\mathfrak {M}}_g$ of curves of genus $g$ .",
"This proof proceeds as follows.",
"We first compute the Hilbert function and expected syzygies of the Hartshorne–Rao module $M=H^1_*({\\mathcal {I}}_C)$ , the coordinate ring $S_C$ , and the section ring $R:=H^0_*({\\mathcal {O}}_C)$ .",
"We then use this information to choose generic matrices which realize the free resolution of $M$ .",
"Finally, we show that this construction leads to a family of curves that dominate ${\\mathfrak {M}}_{12}$ and generically contains smooth curves.",
"We first choose $r$ so that a general curve has a model of degree $d=13$ in ${\\mathbb {P}}^r$ .",
"In our case, we choose $r=3$ so that $g-d+r=2$ .",
"To compute the Hilbert function and expected syzygies of the Hartshorne–Rao module $M=H^1_*({\\mathcal {I}}_C)$ , the coordinate ring $S_C=S/I_C$ , and the section ring $R=H^0_*({\\mathcal {O}}_C)$ , we assume the open condition that $C$ has maximal rank, i.e., $H^0({\\mathbb {P}}^3,{\\mathcal {O}}(n)) \\rightarrow H^0(C, L^n)$ is of maximal rank for all $n$ , as in Remark REF .",
"In this case, $h_M(t)=5t^2+8t^3+6t^4$ , which has Hilbert numerator $h_M(t)(1-t)^4= 5t^2-12t^3+4t^4+4t^5+9t^6-16t^{10}+6t^{11}.$ If $M$ has a natural resolution, so that for each $j$ at most one $\\beta _{ij}(M)$ is nonzero, then $M$ has the Betti table $\\beta (M) \\ = $ Table: NO_CAPTION.",
"If we assume the open condition that $S_C$ and $R$ have natural syzygies as well, then their Betti tables are $\\beta (S_C) \\ = $ Table: NO_CAPTION $\\quad $ and $\\quad $ $\\beta (R) \\ = $ Table: NO_CAPTION.",
"We conclude that once we have constructed the Hartshorne–Rao module $M=M_C$ , say via its representation $0 \\leftarrow M \\leftarrow S^5(-2)\\leftarrow S^{12}(-3),$ we may choose ${\\mathcal {F}}$ to be the kernel of $0 \\leftarrow {\\mathcal {O}}^5(-2) \\leftarrow {\\mathcal {O}}^{12}(-3)\\leftarrow {\\mathcal {F}}\\leftarrow 0$ and set ${\\mathcal {L}}_1 := {\\mathcal {O}}(-4)^4 \\oplus {\\mathcal {O}}^2(-5)$ and ${\\mathcal {L}}_2:=0$ .",
"Then $C$ is determined by $M$ and the choice of a point in ${\\mathbb {G}}(2,4)$ .",
"In particular, as mentioned earlier, constructing $C$ is equivalent to constructing the finite length module $M$ with the desired syzygies.",
"If we choose the presentation matrix $\\phi $ of $M$ to be given by a general (or random) $5\\times 12$ matrix of linear forms, then its cokernel will be a module with Hilbert series $5t^2+8t^3+2t^4$ .",
"In other words, to get the right Hilbert function for $M$ , we must force 4 linear syzygies.",
"To do this, choose a general (or random) $12\\times 4$ matrix $\\psi $ of linear forms.",
"Then $\\ker (\\psi ^t\\colon S^{12}(1) \\rightarrow S^4(2))$ has at least $12\\cdot 4- 4 \\cdot 10 =8$ generators in degree 0.",
"In fact, there are precisely 8 and a general point in ${\\mathbb {G}}(5,8)$ gives rise to a $12 \\times 5$ matrix $\\varphi ^t$ of linear forms.",
"This means that $M := {\\rm {coker}\\,}(\\varphi \\colon S^12(-3)\\rightarrow S^5(-2))$ to have Hilbert series $5t^2+8t^3+6t^4$ , due to the forced 4 linear syzygies.",
"Having constructed $M$ , it remains to prove that this construction leads to a family of curves that dominates ${\\mathfrak {M}}_{12}$ .",
"To this end, we compute a random example $C$ , say over a finite prime field ${\\mathbb {F}}_p$ , and confirm its smoothness.",
"Since we may regard our computation over ${\\mathbb {F}}_p$ as the reduction modulo $p$ of a construction defined over an open part of ${{\\rm Spec}\\,}{\\mathbb {Z}}$ , semi-continuity allows us to establish the existence of a smooth example defined over ${\\mathbb {Q}}$ with the same syzygies.",
"We now consider the universal family ${\\mathfrak {W}}^r_d \\subset {\\mathfrak {P}}ic^d$ over ${\\mathfrak {M}}_g$ and a neighborhood of our example $(C,L) \\in {\\mathfrak {P}}ic^d$ .",
"Note that the codimension of ${\\mathfrak {W}}^r_d$ is at most $(r+1)(g-d+r)=4\\cdot 2 =8$ .",
"On the other hand, we claim that the Petri map $\\mu _L$ for ($C,L)$ is injective.",
"(Recall the definition of $\\mu _L$ from Theorem REF .)",
"To see this, note that the Betti numbers of $H^0_*(\\omega _C)$ correspond to the dual of the resolution of $H^0_*({\\mathcal {O}}_C)$ , so $\\beta (H^0_*(\\omega _C)) \\ = $ Table: NO_CAPTION.",
"Thus there are no linear relations among the two generators in $H^0(\\omega _C \\otimes L^{-1})$ , which means that the $\\mu _L\\colon H^0(L) \\otimes H^0(\\omega _C \\otimes L^{-1})$ is injective.",
"From this we see that $\\dim W^r_d(C)$ has dimension 4 at $(C,L)$ , and the constructed family dominates for dimension reasons.",
"The unirationality of ${\\mathfrak {M}}_{15}$ and ${\\mathfrak {M}}_{16}$ are open; however, these moduli spaces are uniruled.",
"Theorem 4.6 (Chang–Ran [13], [14], [8], [22]) The moduli space ${\\mathfrak {M}}_{15}$ is rationally connected, and ${\\mathfrak {M}}_{16}$ is uniruled.",
"$\\Box $ To explain why the unirationality in these cases is more difficult to approach using the method of Theorem REF , we conclude with a brief discussion on the space models of curves of genus $g=16$ .",
"By Brill–Noether theory, a general curve $C$ of genus 16 has finitely many models of degree $d=15$ in ${\\mathbb {P}}^3$ .",
"Again assuming the maximal rank condition of Remark REF , the Hartshorne–Rao module $M=H^1_*({\\mathcal {I}}_C)$ has Hilbert series $H_M(t)= 5t^2+10t^3+10t^4+4t^5$ and expected syzygies $\\beta (M) \\ = $ Table: NO_CAPTION.",
"The section ring $H^0_*({\\mathcal {O}}_C)$ and the coordinate ring $S_C$ have expected syzygies $\\beta (H^0 _*({\\mathcal {O}}_C)) \\ = $ Table: NO_CAPTION $\\quad $ and $\\quad $ $\\beta (S_C) \\ = $ Table: NO_CAPTION.",
"Proposition 4.7 A general curve $C$ of genus $g=16$ and degree $d=15$ in ${\\mathbb {P}}^3$ has syzygies as above.",
"In particular, the Hartshorne–Rao module $M_C$ uniquely determines $C$ .",
"Furthermore, the rational map from the component ${\\mathcal {H}}_{d,g}$ of the Hilbert scheme ${\\rm Hilb}_{15t+1-16}({\\mathbb {P}}^3)$ that dominates ${\\mathfrak {M}}_{16}$ defined by ${\\mathcal {H}}_{d,g} &\\dasharrow \\lbrace \\hbox{ 20 determinantalpoints } \\rbrace \\\\C &\\,\\,\\mapsto \\,\\,\\Gamma :=\\operatorname{supp}{\\rm {coker}\\,}( \\varphi ^t: {\\mathcal {O}}^6(-1)\\rightarrow {\\mathcal {O}}^4)$ is dominant.",
"Here $\\varphi \\colon S^4(-9) \\rightarrow S^6(-8)$ denotes the linear part of the last syzygy matrix of $M$ .",
"For the first statement, it suffices to find an example with the expected syzygies, since Betti numbers behave semi-continuously in a family of modules with constant Hilbert function.",
"We may even take a reducible example, provided that it is smoothable.",
"Consider the union $C:=E_1 \\cup E_2 \\cup E_3$ of three smooth curves of genus 2 and degree 5, such that $E_i \\cap E_j$ for $i \\ne j$ consists of 4 nodes of $C$ .",
"Then $C$ has degree $d=3\\cdot 5=15$ and genus $g=3\\cdot 2 + 4\\cdot 3-2=16$ .",
"Clearly, $C$ is smoothable as an abstract curve.",
"For general choices, it is smoothable as an embedded curve because the $g^3_{15}$ on the reducible curve is an isolated smooth point in $W^3_{15}$ (as we will see), so the smooth curves nearby have an isolated $g^3_{15}$ as well.",
"It is easy to find such a union over a finite field ${\\mathbb {F}}$ .",
"Start with the 12 intersection points $\\lbrace p_1,\\ldots ,p_4\\rbrace \\cup \\lbrace p_5,\\ldots ,p_8\\rbrace \\cup \\lbrace p_9,\\ldots ,p_{12} \\rbrace $ randomly chosen in ${\\mathbb {P}}^3({\\mathbb {F}})$ .",
"To construct $E_1$ , pick at random a quadric $Q_1$ in the pencil of quadrics through $\\lbrace p_1,\\ldots ,p_8\\rbrace $ .",
"Next, we must check if the tangent hyperplane of $Q_1$ in a point, say $p_1$ , intersects $Q_1$ in a pair of lines individually defined over ${\\mathbb {F}}$ ; this will happen about $50 \\%$ of the time.",
"Once this is true, choose one of the lines, call it $L_1$ .",
"Then $| {\\mathcal {O}}_{Q_1}(3) \\otimes {\\mathcal {O}}_{Q_1}(-L_1)|$ is a linear system of class $(3,2)$ on $Q_1 \\cong {\\mathbb {P}}^1 \\times {\\mathbb {P}}^1$ .",
"We may take $E_1$ as a general curve in this linear system that passes through $\\lbrace p_1,\\ldots ,p_8\\rbrace $ .",
"Similarly, we choose $E_2$ using $\\lbrace p_1,\\ldots ,p_4,p_9,\\ldots ,p_{12} \\rbrace $ and $E_3$ starting with $\\lbrace p_5, \\ldots ,p_{12}\\rbrace $ .",
"The union of the $E_i$ yields the desired curve $C$ , and it a straightforward computation to check that $C$ has the expected Hartshorne–Rao module and syzygies.",
"The second statement can be proved by showing that the appropriate map between tangent spaces is surjective for this example.",
"This involves computing appropriate ${\\rm {Ext}}$ -groups.",
"Define $\\overline{M} :&= {\\rm {coker}\\,}( S^6(-2)\\oplus S^6(-1)\\rightarrow S^4)\\\\& ={\\rm {Ext}}^4_S(M,S(-9))\\\\&= \\operatorname{Hom}_K(M,K)(-5)\\\\\\text{and}\\ \\ \\,N:&={\\rm {coker}\\,}( \\varphi ^t\\colon S^6(-1)\\rightarrow S^4).$ Then there is a short exact sequence $0 \\rightarrow P \\rightarrow N \\rightarrow \\overline{M} \\rightarrow 0$ of modules with Hilbert series $h_{\\overline{M}}(t) &=&4 +10t+10t^2 +5t^3\\\\h_N(t) &=& 4+10t+16t^2+20t^3+20t^4+20t^5+\\cdots \\\\h_P(t)& = & \\qquad \\qquad \\; \\;6t^2+15t^3+20t^4+20t^5+\\cdots .$ The group ${\\rm {Ext}}^1_S(\\overline{M}, \\overline{M})$ governs the deformation theory of $\\overline{M}$ (and $M$ ).",
"More details can be found in [33], for example, Theorem 2.7 applied in the affine case.",
"More precisely, the degree 0 part of this ${\\rm {Ext}}$ -group is the tangent space of homogeneous deformations of $M$ , which in turn is isomorphic to the tangent space of the Hilbert scheme in $C$ .",
"Similarly, in the given example, ${\\rm {Ext}}^1(N,N)_0$ can be identified with the tangent space to the space of twenty determinantal points.",
"Note that we have the diagram ${{\\rm {Ext}}^1_S(\\overline{M}, \\overline{M}) [r] & {\\rm {Ext}}^1_S(N,\\overline{M}) [r] & {\\rm {Ext}}^1_S(P,\\overline{M}).",
"\\\\& {\\rm {Ext}}^1_S(N,N) [u] & \\\\& {\\rm {Ext}}^1_S(N,P) [u] &\\\\ }$ In our example, computation shows that $\\nonumber \\dim {\\rm {Ext}}^1_S(\\overline{M},\\overline{M})_0&=60, \\\\\\dim {\\rm {Ext}}^1_S(N,\\overline{M})_0= \\dim {\\rm {Ext}}^1_S(N,N)_0 &= 45,\\quad \\text{and}\\\\\\dim {\\rm {Ext}}^1_S(P,\\overline{M})_0=\\dim {\\rm {Ext}}^1_S(N,P)_0&=0.$ Thus the induced map ${\\rm {Ext}}^1_S(\\overline{M}, \\overline{M})_0\\rightarrow \\dim {\\rm {Ext}}^1_S(N,N)_0$ is surjective with 15-dimensional kernel, as expected.",
"Exercise 4.8 Fill in the computational details in of the proof of Proposition REF and Theorem REF using your favorite computer algebra system, say Macaulay2 or Singular.",
"Remark 4.9 In the proof of Proposition REF , the module $P$ has syzygies $\\beta (P) \\ = $ Table: NO_CAPTION.",
"The cokernel of $\\psi ^t\\colon S^6(-1) \\rightarrow S^5$ has support on a determinantal curve $E$ of degree 15 and genus 26, which is smooth for general $C$ .",
"The points $\\Gamma $ form a divisor on $E$ with $h^0(E,{\\mathcal {O}}_E(\\Gamma ))=1$ .",
"The curves $E$ and $C$ do not intersect; in fact, we have no idea how the curve $E$ is related to $C$ , other than the fact that it can be constructed from the syzygies of $M$ .",
"It is possible that ${\\mathfrak {M}}_{16}$ is not unirational, and, even if ${\\mathfrak {M}}_{16}$ is unirational, it could be that the component of the Hilbert scheme containing $C$ is itself not unirational.",
"It is not clear to us whether it is a good idea to start with the determinantal points $\\Gamma $ in Proposition REF .",
"Perhaps entirely different purely algebraic methods might lead to a unirational construction of the modules $M$ , and we invite the reader to discover such an approach.",
"Author Addresses: Christine Berkesch Department of Mathematics, Duke University, Box 90320 Durham, NC 27708 [email protected] Frank-Olaf Schreyer Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany [email protected]"
]
] | 1403.0581 |
[
[
"Variable-Rate Linear Network Error Correction MDS Codes"
],
[
"Abstract In network communication, the source often transmits messages at several different information rates within a session.",
"How to deal with information transmission and network error correction simultaneously under different rates is introduced in this paper as a variable-rate network error correction problem.",
"Apparently, linear network error correction MDS codes are expected to be used for these different rates.",
"For this purpose, designing a linear network error correction MDS code based on the existing results for each information rate is an efficient solution.",
"In order to solve the problem more efficiently, we present the concept of variable-rate linear network error correction MDS codes, that is, these linear network error correction MDS codes of different rates have the same local encoding kernel at each internal node.",
"Further, we propose an approach to construct such a family of variable-rate network MDS codes and give an algorithm for efficient implementation.",
"This approach saves the storage space for each internal node, and resources and time for the transmission on networks.",
"Moreover, the performance of our proposed algorithm is analyzed, including the field size, the time complexity, the encoding complexity at the source node, and the decoding methods.",
"Finally, a random method is introduced for constructing variable-rate network MDS codes and we obtain a lower bound on the success probability of this random method, which shows that this probability will approach to one as the base field size goes to infinity."
],
[
"Introduction",
"Network coding allows internal nodes in a communication network to process the information received.",
"This idea was first appeared in Yeung and Zhang [1] and then developed by Ahlswede et al.",
"[2].",
"In [2], the authors showed that if coding is applied at the nodes in a network, rather than routing alone, the source node can multicast messages to all sink nodes at the theoretically maximum rate—the smallest minimum cut capacity between the source and any sink node, as the alphabet size approaches infinity.",
"Li et al.",
"[3] further indicated that linear network coding with finite alphabet size is sufficient for multicast.",
"Koetter and Médard [4] developed an algebraic characterization of network coding.",
"Although network coding has higher information rate than classical routing, Jaggi et al.",
"[5] still proposed a deterministic polynomial-time algorithm for constructing a linear network code.",
"For a detail and comprehensive discussion of network coding, refer to [6], [7], [8], [9].",
"Network coding has been studied extensively for several years under the assumption that the channels of networks are error-free.",
"Unfortunately, all kinds of errors may occur in practical network communication such as random errors, erasure errors (packet losses), errors in headers and so on.",
"In order to deal with such problems, network error correction (NEC) based on network coding was studied recently.",
"Cai and Yeung proposed the original idea of network error correction coding in their conference paper [10] and developed it in their journal papers [11][12].",
"They introduced the concept of network error correction codes as a generalization of classical error-correcting codes, and extended some important bounds in classical coding theory to network error correction coding, such as the Singleton bound, the Hamming bound, and the Gilbert-Varshamov bound.",
"Although the Singleton bound has been given by Yeung and Cai [11], Zhang[13] and Yang et al.",
"[14][15] presented the refined Singleton bound independently by the different approaches.",
"Further, the linear NEC codes satisfying this bound with equality are called linear network error correction maximum distance separable (MDS) codes, or network MDS codes for short.",
"Koetter and Kschischang [16] (see also [17]) formulated a different framework for network error correction coding when a noncoherent network model was under consideration where neither source node nor sink node was assumed to have knowledge of the channel transfer characteristic.",
"Motivated by the property that linear network coding is vector-space preserving, in their approach the source message is represented by a subspace of a fixed vector space and a basis of the subspace is injected into the network.",
"So this type of network error correction codes is called subspace codes.",
"A metric was proposed to account for the discrepancy between the transmitted and received subspaces and a coding theory based on this metric was developed.",
"For an overview of the development and some contributions in network error correction coding, refer to the survey paper [18].",
"In network communication, the source often transmits the messages at several different information rates within a session.",
"When both information transmission and network error correction are considered simultaneously, it is expected that linear network error correction MDS codes can be applied for these information rates.",
"For the problem as described above, the most efficient solution based on the existing results is that, for each information rate, design a network MDS code by constructive algorithms proposed by Yang et al.",
"[14], Guang et al.",
"[19], or others.",
"For this scheme, each node in a network has to store all local encoding kernels corresponding to different network MDS codes.",
"Hence, it takes a large amount of storage space for each node in network.",
"This also increases the complexity of the system considerably.",
"Furthermore, in transmission, the source node has to tell each non-source node which information rate is used to transmit the messages, and then each non-source node searches and uses the corresponding local encoding kernel for coding.",
"Searching and changing the local encoding kernels at each non-source node consume resources and time in the network.",
"In order to avoid these shortcomings of the above solution, we wish to construct a family of linear network error correction MDS codes with the following property: these network MDS codes with different rates have the same local encoding kernel at each non-source node.",
"In other words, for these different information rates, each non-source node can use the same local encoding kernel for coding.",
"This will save the storage space at each node and all internal nodes will not need to know which rate the source node uses to transmit messages.",
"We are partly motivated by the same problem in network coding [20], where, Fong and Yeung studied the variable-rate linear network coding with and without link failure in the case that no errors occur in the channels of networks.",
"[21] and [22] further studied different classes of variable-rate linear network codes.",
"This paper is divided into 6 sections.",
"In the next section, we first review linear network coding and linear network error correction coding, and then give some necessary notation and definitions.",
"Section iii@ is devoted to constructing variable-rate linear network error correction MDS codes and designing an algorithm for efficient implementation.",
"In this section, we give a method to construct low-dimensional linear network MDS codes from a high-dimensional one such that they have the same local encoding kernel at each non-source node.",
"In other words, we give a constructive proof to show the existence of the variable-rate network MDS codes defined in Section iii@.",
"Actually, the existence may be proved more easily by a random method as used in [23][24].",
"But the constructive approach is much more important because of its widely potential applications.",
"Furthermore, we design an algorithm for efficient implementation.",
"Section iv@ is devoted to the performance analysis of our proposed algorithm in Section iii@, including the field size, the time complexity of the algorithm, the encoding complexity at the source node, and the decoding methods.",
"In particular, we also discuss the feasibility of two algorithms proposed by Yang et al.",
"[14] for this variable-rate network error correction problem.",
"Since both algorithms design the codebook at the source node and the local encoding kernels separately, it seems likely that they might solve this variable-rate problem.",
"However, by a detailed analysis, they are either non-feasible or inefficient for solving the problem.",
"Particularly, even assuming that some certain conditions are satisfied such that Algorithm 1 in [14] can solve our problem, our proposed algorithm still has many advantages in different aspects.",
"In Section v@, a random approach for implementing variable-rate network error correction MDS codes is proposed and then we obtain a lower bound on the success probability of using the random approach to construct variable-rate network MDS codes.",
"This success probability can characterize the performance of this random method, and the obtained lower bound implies that, if the field size is sufficiently large, the random method can construct variable-rate network MDS codes with high probability close to one.",
"The last section summarizes the works done in this paper and proposes some topics for further research."
],
[
"Preliminaries",
"In the present paper, we follow [13][19] with their notation and terminology.",
"A communication network is represented as a finite acyclic directed graph $G=(V,E)$ , where $V$ and $E$ are the sets of nodes and channels of the network, respectively.",
"The node set $V$ consists of three disjoint subsets $S$ , $T$ , and $J$ , where $S$ is the set of source nodes, $T$ is the set of sink nodes, and $J=V-S-T$ is the set of internal nodes.",
"A direct edge $e=(i,j)\\in E$ stands for a channel leading from node $i$ to node $j$ .",
"Node $i$ is called the tail of $e$ and node $j$ is called the head of $e$ , denoted by $tail(e)$ and $head(e)$ , respectively.",
"Correspondingly, the channel $e$ is called an outgoing channel of $i$ and an incoming channel of $j$ .",
"For a node $i$ , define $Out(i)$ as the set of outgoing channels of $i$ and $In(i)$ as the set of incoming channels of $i$ .",
"Formally, we have $Out(i)=\\lbrace e\\in E:\\ tail(e)=i\\rbrace ,\\ \\ \\ In(i)=\\lbrace e \\in E:\\ head(e)=i\\rbrace .$ For each channel $e\\in E$ , there exists a positive number $R_e$ , say the capacity of $e$ .",
"We allow the multiple channels between two nodes and thus assume reasonably that the capacity of any channel is 1 per unit time, that is, one field element can be transmitted over a channel in one unit time.",
"A cut between node $i$ and node $j$ is a set of channels whose removal disconnects $i$ from $j$ .",
"For unit capacity channels, the capacity of a cut can be regarded as the number of channels in the cut, and the minimum of all capacities of cuts between $i$ and $j$ is called the minimum cut capacity between the two nodes.",
"A cut between node $i$ and node $j$ is called a minimum cut if its capacity achieves the minimum cut capacity between them.",
"Note that there may exist several minimum cuts between $i$ and $j$ , but the minimum cut capacity between them is determined.",
"Following the direction of the channels, there is an upstream-to-downstream order (ancestral topological order) on the channels in $E$ which is consistent with the partial order of all channels.",
"The coordinates of all vectors and rows/columns of all matrices in this paper are indexed according to this upstream-to-downstream order.",
"In particular, if $L$ is such a matrix whose column vectors are indexed by a collection $B\\subseteq E$ of channels according to an upstream-to-downstream order, then we use some symbol with subscript $e$ , $e\\in B$ , such as $l_e$ , to denote the column vector indexed by the channel $e$ , and the matrix $L$ is written as column-vector form $L=\\Big [l_e:\\ e\\in B\\Big ]$ .",
"If $L$ is a matrix whose row vectors are indexed by this collection $B$ of channels, then we use some symbol with $e$ inside a pair of brackets, such as $l(e)$ , to denote the row vector corresponding to $e$ , and the matrix $L$ is written as row-vector form $L=\\Big [ l(e):\\ e\\in B \\Big ]$ ."
],
[
"Linear Network Coding",
"In this paper, we consider single source networks, i.e., $|S|=1$ , and the unique source node is denoted by $s$ , which generates messages and transmits them to all sink nodes over the network by a linear network code.",
"The source node $s$ has no incoming channels and any sink node has no outgoing channels.",
"But we introduce the concept of imaginary incoming channels of the source node $s$ and assume that these imaginary incoming channels provide the source messages to $s$ .",
"Let the information rate be ${\\omega }$ symbols per unit time.",
"Then $s$ has ${\\omega }$ imaginary incoming channels denoted by $d_1^{\\prime },d_2^{\\prime },\\cdots ,d_{\\omega }^{\\prime }$ and let $In(s)=\\lbrace d_1^{\\prime },d_2^{\\prime },\\cdots ,d_{\\omega }^{\\prime }\\rbrace $ .",
"The source messages are ${\\omega }$ symbols ${\\bf X}=[X_1\\ X_2\\ \\cdots \\ X_{\\omega }]$ arranged in a row vector where each $X_i$ is an element of the base field $\\mathcal {F}$ .",
"Subsequently, they are assumed to be transmitted to $s$ through the ${\\omega }$ imaginary incoming channels in $In(s)$ .",
"Without loss of generality, assume that the message transmitted over the $i$ th imaginary channel is the $i$ th source message.",
"Further, at each node $i\\in V-T$ , there is an $|In(i)|\\times |Out(i)|$ matrix $K_i=[k_{d,e}]_{d\\in In(i),e\\in Out(i)}$ , called the local encoding kernel at $i$ , where $k_{d,e}\\in \\mathcal {F}$ is called the local encoding coefficient for the adjacent pair $(d,e)$ of channels.",
"We use $U_e$ to denote the message transmitted over the channel $e$ .",
"Hence, at the source node $s$ , we have $U_{d_i^{\\prime }}=X_i$ , $1\\le i \\le {\\omega }$ .",
"In general, the message $U_e$ transmitted over the channel $e\\in E$ is calculated recursively by the formulae: $U_e=\\sum _{d\\in In(tail(e))}k_{d,e}U_d.$ Furthermore, it is not difficult to see that $U_e$ is actually a linear combination of the ${\\omega }$ source symbols $X_i$ , $1\\le i\\le {\\omega }$ , that is, there is an ${\\omega }$ -dimensional column vector $f_e$ over the base field $\\mathcal {F}$ such that $U_e={\\bf X}\\cdot f_e$ (see also [6] [7]).",
"This column vector $f_e$ is called the global encoding kernel of a channel $e$ , and can be determined by the local encoding kernels as follows: $f_e=\\sum _{d\\in In(tail(e))}k_{d,e}f_d,$ with boundary condition that the vectors $f_{d_i^{\\prime }}$ , $1\\le i \\le {\\omega }$ , form the standard basis of the vector space $\\mathcal {F}^{\\omega }$ ."
],
[
"Linear Network Error Correction Coding",
"In the case that an error occurs on a channel $e$ , the output of the channel is $\\tilde{U}_e=U_e+Z_e$ , where $U_e$ is the message that should be transmitted over the channel $e$ and $Z_e\\in \\mathcal {F}$ is the error occurred in $e$ .",
"We also treat the error $Z_e$ as a message called error message.",
"Further, let the error vector be an $|E|$ -dimensional row vector ${\\bf Z}=[Z_e:\\ e\\in E]$ over the field $\\mathcal {F}$ with each component $Z_e$ representing the error occurred on the corresponding channel $e$ .",
"Firstly, we introduce the extended network as follows.",
"In the network $G=(V,E)$ , for each channel $e\\in E$ , an imaginary channel $e^{\\prime }$ is introduced, which is connected to the tail of $e$ in order to provide the error message $Z_e$ .",
"This new network $\\tilde{G}=(\\tilde{V},\\tilde{E})$ with imaginary channels is called the extended network of $G$ , where $\\tilde{V}=V$ , $\\tilde{E}=E\\cup E^{\\prime }\\cup \\lbrace d_1^{\\prime },d_2^{\\prime },\\cdots , d_{\\omega }^{\\prime }\\rbrace $ with $E^{\\prime }=\\lbrace e^{\\prime }: e\\in E\\rbrace $ .",
"Obviously, $|E^{\\prime }|=|E|$ .",
"Then a linear network code for the original network $G$ can be extended to a linear network code for the extended network $\\tilde{G}$ by setting $k_{e^{\\prime },e}=1$ and $k_{e^{\\prime },d}=0$ for all $d\\in E\\backslash \\lbrace e\\rbrace $ .",
"Note that, for each internal node $i$ in the extended network $\\tilde{G}$ , $In(i)$ only includes the real incoming channels of $i$ , that is, the imaginary channels $e^{\\prime }$ corresponding to $e\\in Out(i)$ are not in $In(i)$ .",
"But for the source node $s$ , we still define $In(s)=\\lbrace d_1^{\\prime },d_2^{\\prime },\\cdots ,d_{\\omega }^{\\prime }\\rbrace $ .",
"In order to distinguish two different types of imaginary channels, we say $d_i^{\\prime }$ , $1\\le i\\le {\\omega }$ , the imaginary message channels and $e^{\\prime }$ for $e\\in E$ the imaginary error channels.",
"Similarly, we can also define global encoding kernels $e$ for all $e\\in \\tilde{E}$ , which is an $({\\omega }+|E|)$ -dimensional column vector and the entries can be indexed by the channels in $In(s)\\cup E$ .",
"For imaginary message channels $d_i^{\\prime }$ , $1\\le i \\le {\\omega }$ , and imaginary error channels $e^{\\prime }\\in E^{\\prime }$ , let ${d_i^{\\prime }}=1_{d_i^{\\prime }}$ and ${e^{\\prime }}=1_e$ , where $1_d$ is an $({\\omega }+|E|)$ -dimensional column vector which is the indicator function of $d\\in In(s)\\cup E$ .",
"Thus, the vectors $e$ for both ${\\omega }$ imaginary message channels and $|E|$ imaginary error channels form the standard basis of vector space $\\mathcal {F}^{{\\omega }+|E|}$ .",
"For other global encoding kernels $e$ , $e\\in E$ , we have the following recursive formulae: $e=\\sum _{d\\in In(tail(e))}k_{d,e}d+1_e.$ We call $e$ the extended global encoding kernel of the channel $e$ for the original network.",
"At each sink node $t\\in T$ , the received message vector $\\tilde{U}_t\\triangleq [\\tilde{U}_e:\\ e\\in In(t)]$ and the decoding matrix $\\tilde{F}_t\\triangleq \\begin{bmatrix}e:\\ e\\in In(t)\\end{bmatrix}$ are available, and we have the following decoding equation: $\\tilde{U}_t=({\\bf X}\\ {\\bf Z})\\tilde{F}_t,$ which can be used for decoding and error correction (refer to [13][19]).",
"Similar to linear network codes [6][7], we can also define a linear network error correction code by either a local description or a global description.",
"Definition 1 Local Description of A Linear Network Error Correction Code.",
"An ${\\omega }$ -dimensional $\\mathcal {F}$ -valued linear network error correction code consists of all local encoding kernels at all internal nodes (including the source node $s$ ), i.e., $ K_i=[k_{d,e}]_{d\\in In(i), e\\in Out(i)},$ that is an $|In(i)|\\times |Out(i)|$ matrix for the node $i$ , where $k_{d,e}\\in \\mathcal {F}$ is the local encoding coefficient for the adjacent pair $(d,e)$ of channels with $d\\in In(i)$ , $e\\in Out(i)$ .",
"Global Description of A Linear Network Error Correction Code.",
"An ${\\omega }$ -dimensional $\\mathcal {F}$ -valued linear network error correction code consists of all extended global encoding kernels for all channels including imaginary message channels and imaginary error channels, which satisfy: ${d_i^{\\prime }}=1_{d_i^{\\prime }},\\ 1 \\le i \\le {\\omega }$ , and ${e^{\\prime }}=1_e$ , $e^{\\prime }\\in E^{\\prime }$ , where $1_d$ is an $({\\omega }+|E|)$ -dimensional column vector which is the indicator function of $d\\in In(s) \\cup E$ ; for other channels $e\\in E$ , $e=\\sum _{d\\in In(tail(e))}k_{d,e}d+1_e,$ where $k_{d,e}\\in \\mathcal {F}$ is the local encoding coefficient for the adjacent channel pair $(d,e)$ with $d\\in In(tail(e))$ , and again $1_e$ is an $({\\omega }+|E|)$ -dimensional column vector which is the indicator function of the channel $e\\in E$ .",
"Further, we give the following notation and definitions.",
"Definition 2 For each channel $e\\in E$ , the extended global encoding kernel $e$ is written as follows: $e=\\begin{bmatrix}f_e(d_1^{\\prime })\\\\\\vdots \\\\f_e(d_{\\omega }^{\\prime })\\\\f_e(e_1)\\\\\\vdots \\\\f_e(e_{\\mathcal {E}})\\\\\\end{bmatrix}=\\begin{bmatrix}f_e\\\\g_e\\\\\\end{bmatrix}$ where $f_e=\\begin{bmatrix}f_e(d_1^{\\prime })\\\\\\vdots \\\\f_e(d_{\\omega }^{\\prime })\\end{bmatrix}$ is an ${\\omega }$ -dimensional column vector, and $g_e=\\begin{bmatrix}f_e(e_1)\\\\\\vdots \\\\f_e(e_{\\mathcal {E}})\\\\\\end{bmatrix}$ is an $\\mathcal {E}$ -dimensional column vector with $|E|=\\mathcal {E}$ .",
"Recall that $\\tilde{F}_t=\\begin{bmatrix}e:\\ e\\in In(t)\\end{bmatrix}$ is the decoding matrix at the sink node $t\\in T$ .",
"Denote by ${\\rm row}_t(d)$ the row vector of the decoding matrix $\\tilde{F}_t$ indexed by the channel $d\\in In(s)\\cup E$ .",
"These row vectors are of dimension $|In(t)|$ .",
"Hence, $\\tilde{F}_t=\\begin{bmatrix}{\\rm row}_t(d_1^{\\prime })\\\\\\vdots \\\\{\\rm row}_t(d_{\\omega }^{\\prime })\\\\{\\rm row}_t(e_1)\\\\\\vdots \\\\{\\rm row}_t(e_{\\mathcal {E}})\\end{bmatrix}=\\begin{bmatrix}F_t\\\\G_t\\end{bmatrix}$ where $F_t=\\begin{bmatrix}{\\rm row}_t(d_1^{\\prime })\\\\\\vdots \\\\{\\rm row}_t(d_{\\omega }^{\\prime })\\end{bmatrix}$ and $G_t=\\begin{bmatrix}{\\rm row}_t(e_1)\\\\ \\vdots \\\\ {\\rm row}_t(e_{\\mathcal {E}}) \\end{bmatrix}$ are two matrices of sizes ${\\omega }\\times |In(t)|$ and $|E|\\times |In(t)|$ , respectively.",
"We use ${\\rho }$ to denote an error pattern which can be regarded as a set of channels.",
"We say that an error message vector ${\\bf Z}$ matches an error pattern ${\\rho }$ , if $Z_e=0$ for all $e\\in E\\backslash {\\rho }$ .",
"In the following, we always use ${\\bf 0}$ to denote an all zero row vector, whose dimension will always be clear from the context.",
"Definition 3 ([13]) Define $\\Delta (t,{\\rho })=\\lbrace ({\\bf 0}\\ {\\bf Z})\\tilde{F}_t={\\bf Z}\\cdot G_t:\\ {\\bf Z}\\in \\mathcal {F}^{|E|} \\mbox{ matching the error pattern }{\\rho }\\rbrace ,$ and $\\Phi (t)=\\lbrace ({\\bf X}\\ {\\bf 0})\\tilde{F}_t={\\bf X}\\cdot F_t:\\ {\\bf X}\\in \\mathcal {F}^{\\omega }\\rbrace .$ We call $\\Delta (t,{\\rho })$ and $\\Phi (t)$ the error space of the error pattern ${\\rho }$ and the message space with respect to the sink node $t$ , respectively.",
"Let $L$ be a collection of vectors in some linear space.",
"For convenience, we use $\\langle L \\rangle $ to represent the subspace spanned by vectors in $L$ .",
"Thus, we further have $\\Delta (t,{\\rho })=\\langle \\lbrace {\\rm row}_t(d):\\ d\\in {\\rho }\\rbrace \\rangle \\mbox{ and } \\Phi (t)=\\langle \\lbrace {\\rm row}_t(d):\\ d\\in In(s) \\rbrace \\rangle .$ Moreover, we give some concepts which will be used in this paper.",
"Definition 4 ([13]) We say that an error pattern ${\\rho }_1$ is dominated by another error pattern ${\\rho }_2$ with respect to a sink node $t$ , if $\\Delta (t,{\\rho }_1)\\subseteq \\Delta (t,{\\rho }_2)$ for any linear network code.",
"This relation is denoted by ${\\rho }_1\\prec _t{\\rho }_2$ .",
"Definition 5 ([13]) The rank of an error pattern ${\\rho }$ with respect to a sink node $t$ is defined by $rank_t({\\rho })=\\min \\lbrace |{\\rho }^{\\prime }|:\\ {\\rho }\\prec _t{\\rho }^{\\prime }\\rbrace ,$ where $|{\\rho }^{\\prime }|$ denotes the cardinality of the error pattern ${\\rho }^{\\prime }$ .",
"The above definition on the rank of an error pattern is abstract, and so in order to understand this concept more intuitively, we give the following proposition.",
"Proposition 1 ([19]) For an error pattern ${\\rho }$ , introduce a source node $s_{{\\rho }}$ .",
"Let ${\\rho }=\\lbrace e_1,e_2,\\cdots ,\\ e_l \\rbrace $ where $e_j\\in In(i_j)$ for $1\\le j \\le l$ and define new edges $e_j^{\\prime }=(s_{{\\rho }},i_j)$ .",
"Replace each $e_j$ by $e_j^{\\prime }$ on the network, that is, add $e_1^{\\prime },e_2^{\\prime },\\cdots ,e_l^{\\prime }$ on the network and delete $e_1,e_2,\\cdots ,e_l$ from the network.",
"Then the rank of the error pattern ${\\rho }$ with respect to a sink node $t$ in the original network is equal to the minimum cut capacity between $s_{{\\rho }}$ and $t$ .",
"Definition 6 ([13]) An ${\\omega }$ -dimensional linear network error correction code is called a regular code if for any $t\\in T$ , $\\dim (\\Phi (t))={\\omega }$ , or equivalently, ${\\mathrm {Rank}}(F_t)={\\omega }$ .",
"If the considered code is not regular, i.e., ${\\mathrm {Rank}}(F_t)<{\\omega }$ for at least one sink node $t\\in T$ , then even in the error-free case, the code is not decodable at at least one sink node $t\\in T$ , not to mention network error correction.",
"Therefore, we must consider regular codes for all information rates.",
"Definition 7 ([13]) The minimum distance of a regular linear network error correction code at sink node $t$ is defined as $d_{\\min }^{(t)}=\\min \\lbrace rank_t({\\rho }):\\ \\dim (\\Delta (t,{\\rho })\\cap \\Phi (t))>0 \\rbrace .$ Now, for linear network error correction codes, we give the refined Singleton bound as follows.",
"Proposition 2 (The Refined Singleton Bound) Let $d_{\\min }^{(t)}$ be the minimum distance of a regular linear network error correction code at a sink node $t\\in T$ .",
"Then $d_{\\min }^{(t)}\\le \\delta _t+1,$ where $\\delta _t=C_t-{\\omega }$ is called the redundancy of the sink node $t$ with $C_t$ being the minimum cut capacity between $s$ and $t$ , and ${\\omega }$ being the information rate.",
"We adopt the convention that the regular linear network error correction codes satisfying the refined Singleton bound with equality for all sink nodes are called linear network error correction maximum distance separable (MDS) codes, or network MDS codes for short."
],
[
"Variable-Rate Network Error Correction MDS Codes",
"In a single source finite acyclic communication network $G$ , assume that the source transmits the messages at several distinct rates ${\\omega }_1,{\\omega }_2,\\cdots ,{\\omega }_h$ within a session, and let ${\\omega }=\\max \\lbrace {\\omega }_1,{\\omega }_2,\\cdots ,{\\omega }_h \\rbrace $ satisfying ${\\omega }\\le \\min _{t\\in T}C_t$ to avoid triviality, where again $C_t$ is the minimum cut capacity between the source node $s$ and the sink node $t$ .",
"According to the constructive algorithm of linear network error correction codes [19], we know that an ${\\omega }$ -dimensional linear network error correction MDS code can be designed on $G$ .",
"In this section, we will show that if one ${\\omega }$ -dimensional network MDS code is given, then an $({\\omega }-1)$ -dimensional network MDS code with the same local encoding kernels at all non-source nodes can also be constructed.",
"Then a constructive algorithm is proposed.",
"By using this algorithm recursively, we can construct all ${\\omega }_i$ -dimensional $(1\\le i\\le h)$ linear network MDS codes with the same local encoding kernels at all non-source nodes.",
"First, we need several lemmas as follows.",
"Lemma 1 Let $\\lbrace e:\\ e\\in E \\rbrace $ constitute a global description of a regular linear network error correction code over a network $G$ , and $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ be an arbitrary $({\\omega }-1)$ -dimensional column vector.",
"Define the matrix $F_t^{({\\omega }-1)}(\\vec{k})=\\begin{bmatrix}I_{{\\omega }-1} & \\vec{k}\\end{bmatrix}\\cdot F_t,$ where $I_{{\\omega }-1}$ is an $({\\omega }-1)\\times ({\\omega }-1)$ identity matrix.",
"Then the row vectors of $F_t^{({\\omega }-1)}(\\vec{k})$ are still linearly independent, i.e., ${\\mathrm {Rank}}(F_t^{({\\omega }-1)}(\\vec{k}))={\\omega }-1$ .",
"For each sink node $t\\in T$ , we know $F_t=\\begin{bmatrix}{\\rm row}_t(d^{\\prime }_1)\\\\\\vdots \\\\{\\rm row}_t(d^{\\prime }_{\\omega })\\end{bmatrix}.$ Consequently, $F_t^{({\\omega }-1)}(\\vec{k})&=\\begin{bmatrix}I_{{\\omega }-1} & \\vec{k}\\end{bmatrix}\\cdot F_t\\\\&=\\begin{bmatrix}{\\rm row}_t(d^{\\prime }_1)+k_1\\cdot {\\rm row}_t(d^{\\prime }_{\\omega })\\\\{\\rm row}_t(d^{\\prime }_2)+k_2\\cdot {\\rm row}_t(d^{\\prime }_{\\omega })\\\\\\cdots \\cdots \\\\{\\rm row}_t(d^{\\prime }_{{\\omega }-1})+k_{{\\omega }-1}\\cdot {\\rm row}_t(d^{\\prime }_{\\omega })\\end{bmatrix}.$ To simply notation, let $r_i={\\rm row}_t(d^{\\prime }_i)$ for all $1\\le i \\le {\\omega }$ .",
"It follows that we only need to prove that, for any $({\\omega }-1)$ -dimensional vector $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ , the $({\\omega }-1)$ row vectors $r_1^{\\prime }\\triangleq r_1+k_1r_{\\omega }$ , $r_2^{\\prime }\\triangleq r_2+k_2r_{\\omega }$ , $\\cdots $ , $r_{{\\omega }-1}^{\\prime }\\triangleq r_{{\\omega }-1}+k_{{\\omega }-1}r_{\\omega }$ are linearly independent.",
"Conversely, suppose that $r_1^{\\prime },r_2^{\\prime },\\cdots ,r_{{\\omega }-1}^{\\prime }$ are linearly dependent.",
"This implies that there exist $({\\omega }-1)$ elements $a_1,a_2,\\cdots ,a_{{\\omega }-1}$ of $\\mathcal {F}$ , not all 0, such that $a_{1}r_{1}^{\\prime }+a_{2}r_{2}^{\\prime }+\\cdots +a_{{\\omega }-1}r_{{\\omega }-1}^{\\prime }=0,$ that is, $a_{1}r_{1}+a_{2}r_{2}+\\cdots +a_{{\\omega }-1}r_{{\\omega }-1}+(a_{1}k_{1}+a_{2}k_{2}+\\cdots +a_{{\\omega }-1}k_{{\\omega }-1})r_{{\\omega }}=0.$ Since $r_{1},r_{2},\\cdots ,r_{{\\omega }}$ are linearly independent vectors, we further have $a_{1}=a_{2}=\\cdots =a_{{\\omega }-1}=a_{1}k_{1}+a_{2}k_{2}+\\cdots +a_{{\\omega }-1}k_{{\\omega }-1}=0,$ particularly, $a_{1}=a_{2}=\\cdots =a_{{\\omega }-1}=0,$ which is a contradiction.",
"So $r_1^{\\prime },r_2^{\\prime },\\cdots ,r_{{\\omega }-1}^{\\prime }$ are linearly independent.",
"That is, for any $({\\omega }-1)$ -dimensional column vector $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ , one has ${\\mathrm {Rank}}(F_{t}^{{\\omega }-1}(\\vec{k}))={\\mathrm {Rank}}\\big (\\begin{bmatrix}r_1^{\\prime \\top }&\\cdots &r_{{\\omega }-1}^{\\prime \\top }\\end{bmatrix}^{\\top }\\big )={\\omega }-1.$ The lemma is proved.",
"Let $\\mathbf {C}_{\\omega }$ be an ${\\omega }$ -dimensional $\\mathcal {F}$ -valued regular linear network error correction code over an acyclic network $G=(V,E)$ , and $e$ represent the extended global encoding kernel of the channel $e$ for all $e\\in E$ .",
"Let $I_{{\\omega }-1}$ and $I_{\\mathcal {E}}$ denote the $({\\omega }-1)\\times ({\\omega }-1)$ and $\\mathcal {E}\\times \\mathcal {E}$ identity matrices, respectively.",
"Let $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ be an arbitrary $({\\omega }-1)$ -dimensional column vector.",
"For each non-imaginary channel $e$ , define ${e}^{({\\omega }-1)}(\\vec{k})=\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}\\cdot {e},$ where ${\\bf 0}_{a\\times b}$ represents the $a\\times b$ all-zero matrix.",
"Lemma 2 If $\\lbrace {e}:e\\in E \\rbrace $ constitutes a global description of an ${\\omega }$ -dimensional $\\mathcal {F}$ -valued regular linear network error correction code $\\mathbf {C}_{\\omega }$ over an acyclic network $G$ , then $\\lbrace {e}^{({\\omega }-1)}(\\vec{k}):e\\in E \\rbrace $ constitutes a global description of an $({\\omega }-1)$ -dimensional regular linear network error correction code for the network $G$ .",
"In particular, the local encoding kernel of this $({\\omega }-1)$ -dimensional code at each non-source node is the same as that of the original ${\\omega }$ -dimensional code $\\mathbf {C}_{\\omega }$ .",
"Let $k_{d,e}\\in \\mathcal {F}$ be the local encoding coefficient of the original ${\\omega }$ -dimensional code $\\mathbf {C}_{\\omega }$ for the adjacent pair $(d,e)$ of channels.",
"First, we show that $\\lbrace {e}^{({\\omega }-1)}(\\vec{k}):e\\in E \\rbrace $ constitutes an $({\\omega }-1)$ -dimensional linear network error correction code by demonstrating the existence of the corresponding local encoding coefficient $k_{d,e}^{({\\omega }-1)}$ for the adjacent channel pair $(d,e)$ of channels, where $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ .",
"By convention, assume that the extended global encoding kernels of the $({\\omega }-1)$ imaginary message channels are ${d^{\\prime }_1}^{({\\omega }-1)}=\\begin{bmatrix}f_{d^{\\prime }_1}^{({\\omega }-1)}\\\\{\\bf 0}_{\\mathcal {E}\\times 1} \\end{bmatrix},\\ {d^{\\prime }_2}^{({\\omega }-1)}=\\begin{bmatrix}f_{d^{\\prime }_2}^{({\\omega }-1)}\\\\{\\bf 0}_{\\mathcal {E}\\times 1} \\end{bmatrix},\\ \\cdots ,\\ {d^{\\prime }_{{\\omega }-1}}^{({\\omega }-1)}=\\begin{bmatrix}f_{d^{\\prime }_{{\\omega }-1}}^{({\\omega }-1)}\\\\{\\bf 0}_{\\mathcal {E}\\times 1} \\end{bmatrix},$ where $f_{d^{\\prime }_1}^{({\\omega }-1)},f_{d^{\\prime }_2}^{({\\omega }-1)},\\cdots ,f_{d^{\\prime }_{{\\omega }-1}}^{({\\omega }-1)}$ form the standard basis of $\\mathcal {F}^{{\\omega }-1}$ .",
"Case 1.",
"For each channel $e\\in Out(s)$ , we have ${e}^{({\\omega }-1)}(\\vec{k})&=\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}\\cdot {e}\\nonumber \\\\&=\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}\\cdot \\begin{bmatrix}f_e\\\\g_e\\end{bmatrix}\\nonumber \\\\&=\\begin{bmatrix}f_e(d^{\\prime }_1)+k_{1}f_{e}(d^{\\prime }_{{\\omega }})\\\\f_{e}(d^{\\prime }_{2})+k_{2}f_{e}(d^{\\prime }_{{\\omega }})\\\\\\cdots \\cdots \\cdots \\\\f_e(d^{\\prime }_{{\\omega }-1})+k_{{\\omega }-1}f_{e}(d^{\\prime }_{{\\omega }})\\\\g_e\\end{bmatrix}=\\begin{bmatrix}k_{d^{\\prime }_1,e}+k_{1}k_{d^{\\prime }_{\\omega },e}\\\\k_{d^{\\prime }_2,e}+k_{2}k_{d^{\\prime }_{\\omega },e}\\\\\\cdots \\cdots \\cdots \\\\k_{d^{\\prime }_{{\\omega }-1},e}+k_{{\\omega }-1}k_{d^{\\prime }_{\\omega },e}\\\\g_e\\end{bmatrix},$ where the equation (REF ) follows from $f_e(d^{\\prime }_i)=k_{d^{\\prime }_i,e}$ , the local encoding coefficient for the adjacent pair $(d_i^{\\prime }, e)$ , $1\\le i \\le {\\omega }$ .",
"Further, define $k_{d^{\\prime }_i,e}^{({\\omega }-1)}(\\vec{k})=k_{d^{\\prime }_i,e}+k_{i}k_{d^{\\prime }_{\\omega },e}$ , $i=1,2,\\cdots ,{\\omega }-1$ .",
"Thus $e^{({\\omega }-1)}(\\vec{k})=&\\sum _{d\\in In(s)}k_{d,e}^{({\\omega }-1)}(\\vec{k})\\cdot d^{({\\omega }-1)}+1_e^{({\\omega }-1)}\\\\=&\\sum _{i=1}^{{\\omega }-1}k_{d^{\\prime }_i,e}^{({\\omega }-1)}(\\vec{k})\\cdot {d^{\\prime }_i}^{({\\omega }-1)}+1_e^{({\\omega }-1)},$ where $1_e^{({\\omega }-1)}$ is an $(({\\omega }-1)+\\mathcal {E})$ -dimensional column vector which is the indicator function of the channel $e$ .",
"Case 2.",
"For other non-imaginary channels $e\\notin Out(s)$ , we know from (REF ) $e=\\sum _{d\\in In(tail(e))}k_{d,e}\\cdot d+1_e.$ Multiplying both sides by $\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}$ , together with (REF ), yields that $e^{({\\omega }-1)}(\\vec{k})=&\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}\\cdot e\\\\=&\\sum _{d\\in In(tail(e))}k_{d,e}\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}\\cdot d+\\begin{bmatrix} I_{{\\omega }-1}& \\vec{k}& {\\bf 0}_{({\\omega }-1)\\times \\mathcal {E}}\\\\{\\bf 0}_{\\mathcal {E}\\times ({\\omega }-1)}&{\\bf 0}_{\\mathcal {E}\\times 1}&I_{\\mathcal {E}}\\end{bmatrix}\\cdot 1_e\\\\=&\\sum _{d\\in In(tail(e))}k_{d,e}\\cdot d^{({\\omega }-1)}(\\vec{k})+1_e^{({\\omega }-1)},$ which leads to $k_{d,e}^{({\\omega }-1)}(\\vec{k})=k_{d,e}$ for all adjacent pairs $(d,e)$ of channels $d,e\\in E$ .",
"Combining the above two cases, $\\lbrace f_e^{({\\omega }-1)}(\\vec{k}):e\\in E \\rbrace $ consists of all the extended global encoding kernels of an $({\\omega }-1)$ -dimensional linear network error correction code, and for each adjacent pair $(d,e)$ of channels $d,e\\in E$ , $k_{d,e}$ is also the local encoding coefficient of this $({\\omega }-1)$ -dimensional code.",
"Applying Lemma REF and the fact that the ${\\omega }$ -dimensional linear network error correction code $\\mathbf {C}_{\\omega }$ is regular, $\\lbrace e^{({\\omega }-1)}(\\vec{k}):e\\in E \\rbrace $ also constitutes an $({\\omega }-1)$ -dimensional regular linear network error correction code.",
"This completes the proof.",
"Moreover, we need the following lemma, which gives three equivalent relations on the minimum distance.",
"Lemma 3 ([19]) For the minimum distance of a regular linear network error correction code at every sink node $t$ , we have the following equalities: $d_{\\min }^{(t)}&=\\min \\lbrace rank_t({\\rho }):\\ \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace \\\\&=\\min \\lbrace |{\\rho }|:\\ \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace \\\\&=\\min \\lbrace \\dim (\\Delta (t,{\\rho })):\\ \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace .$ For a network MDS code, define a set of error patterns for each sink node $t\\in T$ : $Q(t)=\\Big \\lbrace \\mbox{error\\;pattern\\;}{\\rho }: \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\mbox{ and } |{\\rho }|=d_{\\min }^{(t)} \\Big \\rbrace .$ Lemma 4 For an ${\\omega }$ -dimensional network MDS code on $G$ , we have for any sink node $t\\in T$ and any error pattern ${\\rho }\\in Q(t)$ , $\\dim (\\Delta (t,{\\rho })\\cap \\Phi (t))=1.$ For the given network MDS code, we know $d_{\\min }^{(t)}=C_t-{\\omega }+1={\\delta _t}+1$ for each sink node $t\\in T$ .",
"This means that $Q(t)=\\Big \\lbrace \\mbox{error\\;pattern\\;} {\\rho }: \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\mbox{ and } |{\\rho }|={\\delta _t}+1 \\Big \\rbrace .$ For any ${\\rho }\\in Q(t)$ , $|{\\rho }|={\\delta _t}+1$ implies that $\\dim (\\Delta (t,{\\rho }))\\le |{\\rho }|={\\delta _t}+1 .$ On the other hand, by Lemma REF and the definition of network MDS codes, it is readily seen that $d_{\\min }^{(t)}=\\min \\lbrace \\dim (\\Delta (t,{\\rho }^{\\prime })): \\Delta (t,{\\rho }^{\\prime })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace ={\\delta _t}+1.$ Together with $\\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace $ , it follows that $\\dim (\\Delta (t,{\\rho }))\\ge d_{\\min }^{(t)}={\\delta _t}+1 .$ Combining the inequalities (REF ) and (REF ), one has $\\dim (\\Delta (t,{\\rho }))={\\delta _t}+1=|{\\rho }|.$ For simplicity, let $d=d_{\\min }^{(t)}={\\delta _t}+1$ , ${\\rho }=\\lbrace e_1,e_2,\\cdots , e_d\\rbrace $ and $r_i\\triangleq {\\rm row}_t(e_i)$ , $i=1,2,\\cdots ,d$ .",
"Hence, $r_1,r_2,\\cdots ,r_d$ are $d$ linearly independent vectors since $\\dim (\\Delta (t,{\\rho }))=|{\\rho }|=d$ from (REF ) and $\\Delta (t,{\\rho })=\\langle \\lbrace r_i:\\ 1\\le i \\le d \\rbrace \\rangle $ .",
"Suppose that $\\dim (\\Delta (t,{\\rho })\\cap \\Phi (t))\\ge 2$ and then let $\\vec{l_1},\\ \\vec{l_2}$ be two linearly independent vectors in the vector space $\\Delta (t,{\\rho })\\cap \\Phi (t)$ .",
"Then there exist $a_1,a_2,\\cdots ,a_d$ in $\\mathcal {F}$ , not all 0, and $b_1,b_2,\\cdots ,b_d$ in $\\mathcal {F}$ , not all 0, such that ${\\left\\lbrace \\begin{array}{ll}\\vec{l_1}=a_1r_1+a_2r_2+\\cdots +a_dr_d\\\\\\vec{l_2}=b_1r_1+b_2r_2+\\cdots +b_dr_d.\\end{array}\\right.",
"}$ Further, for all $i=1,2,\\cdots ,d$ , we claim that either $a_i$ or $b_i$ is zero.",
"Assume the contrary, that is, there exists some $i\\ (1\\le i \\le d)$ such that $a_i\\ne 0$ , $b_i\\ne 0$ .",
"If so, we have $a_i\\vec{l_2}-b_i\\vec{l_1}\\in \\Delta (t,{\\rho }\\backslash \\lbrace e_i\\rbrace )\\cap \\Phi (t)$ and $a_i\\vec{l_2}-b_i\\vec{l_1}\\ne {\\bf 0}$ because of the linear independence between $\\vec{l_1}$ and $\\vec{l_2}$ , which means that $\\Delta (t,{\\rho }\\backslash \\lbrace e_i\\rbrace )\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace $ .",
"Hence, $d_{\\min }^{(t)}=&\\min \\lbrace \\dim (\\Delta (t,{\\rho }^{\\prime })): \\Delta (t,{\\rho }^{\\prime })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace \\\\\\le &\\dim (\\Delta (t,{\\rho }\\backslash \\lbrace e_i\\rbrace ))={\\delta _t},$ which is a contradiction to $d_{\\min }^{(t)}={\\delta _t}+1$ .",
"Now, we can say that for all $i=1,2,\\cdots ,d$ , either $a_i=0$ or $b_i=0$ .",
"Without loss of generality, assume $a_1\\ne 0$ and $b_1=0$ .",
"That is, the non-zero vector $\\vec{l_2}&=b_1r_1+b_2r_2+\\cdots +b_dr_d\\\\&=b_2r_2+\\cdots +b_dr_d\\in \\Delta (t,{\\rho }\\backslash \\lbrace e_1\\rbrace ),$ which, together with $\\vec{l_2}\\in \\Delta (t,{\\rho })\\cap \\Phi (t)$ , leads to ${\\bf 0}\\ne \\vec{l_2}\\in \\Phi (t)\\cap \\Delta (t,{\\rho }\\backslash \\lbrace e_1\\rbrace ).$ It also follows that $d_{\\min }^{(t)}&=\\min \\lbrace \\dim (\\Delta (t,{\\rho }^{\\prime })):\\Phi (t)\\cap \\Delta (t,{\\rho }^{\\prime })\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace \\\\&\\le \\dim (\\Delta (t,{\\rho }\\backslash \\lbrace e_1\\rbrace ))={\\delta _t}.$ This also violates the condition $d_{\\min }^{(t)}={\\delta _t}+1$ .",
"Therefore, we have shown that $\\dim (\\Delta (t,{\\rho })\\cap \\Phi (t))=1$ for any ${\\rho }\\in Q(t)$ .",
"This completes the proof.",
"Lemma 5 For an acyclic network $G$ , an ${\\omega }$ -dimensional $\\mathcal {F}$ -valued linear network MDS code with field size $|\\mathcal {F}|>\\sum _{t\\in T}|Q(t)|$ is given.",
"Then there exists an $({\\omega }-1)$ -dimensional column vector $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top } \\in \\mathcal {F}^{{\\omega }-1}$ such that $\\dim (\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k}))=0$ for each sink node $t\\in T$ and each error pattern ${\\rho }\\in Q(t)$ , where $\\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\langle \\lbrace {\\rm row}_t(d^{\\prime }_i)+k_i\\cdot {\\rm row}_t(d^{\\prime }_{\\omega }): 1\\le i \\le {\\omega }-1 \\rbrace \\rangle .$ First, we show that, when a fixed sink node $t\\in T$ and a fixed ${\\rho }\\in Q(t)$ are under consideration, there exists an $({\\omega }-1)$ -dimensional column vector $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}$ such that $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\lbrace {\\bf 0}\\rbrace $ .",
"Conversely, suppose that for any $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ , $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})\\ne \\lbrace {\\bf 0}\\rbrace .$ Clearly, $\\Phi ^{({\\omega }-1)}(t,\\vec{k})\\subseteq \\Phi (t)$ , which shows that $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})\\subseteq \\Delta (t,{\\rho })\\cap \\Phi (t).$ Using formulae (REF ), (REF ) and $\\dim (\\Delta (t,{\\rho })\\cap \\Phi (t))=1$ from Lemma REF , we have $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\Delta (t,{\\rho })\\cap \\Phi (t).$ To simply notation, again let $r_i={\\rm row}_t(d^{\\prime }_i)$ , $1\\le i \\le {\\omega }$ and $r_j^{\\prime }=r_j+k_jr_{\\omega }$ , $1\\le j \\le {\\omega }-1$ .",
"Then $r_1,r_2,\\cdots ,r_{\\omega }$ form a basis of vector space $\\Phi (t)$ , and $r_1^{\\prime },r_2^{\\prime },\\cdots ,r_{{\\omega }-1}^{\\prime }$ form a basis of vector space $\\Phi ^{({\\omega }-1)}(t,\\vec{k})$ since $r_1^{\\prime },r_2^{\\prime },\\cdots ,r_{{\\omega }-1}^{\\prime }$ are linearly independent from Lemma REF .",
"Let $\\vec{l}$ be a non-zero vector in $\\Delta (t,{\\rho })\\cap \\Phi (t)$ .",
"Then there exist unique elements $a_1,a_2,\\cdots ,a_{\\omega }\\in \\mathcal {F}$ , not all 0, such that $\\vec{l}=a_1r_1+a_2r_2+\\cdots +a_{{\\omega }-1}r_{{\\omega }-1}+a_{{\\omega }}r_{\\omega }.$ Moreover, it is certain that $\\vec{l}\\in \\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})$ .",
"This means that there also exist unique elements $b_1,b_2,\\cdots ,b_{{\\omega }-1}\\in \\mathcal {F}$ such that $\\vec{l}=b_1r_1^{\\prime }+b_2r_2^{\\prime }+\\cdots +b_{{\\omega }-1}r_{{\\omega }-1}^{\\prime }.$ Hence, $\\vec{l}=&b_1(r_1+k_1r_{\\omega })+b_2(r_2+k_2r_{\\omega })+\\cdots +b_{{\\omega }-1}(r_{{\\omega }-1}+k_{{\\omega }-1}r_{\\omega })\\nonumber \\\\=&b_1r_1+b_2r_2+\\cdots +b_{{\\omega }-1}r_{{\\omega }-1}+(b_1k_1+b_2k_2+\\cdots +b_{{\\omega }-1}k_{{\\omega }-1})r_{\\omega }.$ Due to both representations (REF ) and (REF ) of $\\vec{l}$ , one has $a_i=b_i$ for $1\\le i\\le {\\omega }-1$ and $a_{\\omega }&=b_1k_1+b_2k_2+\\cdots +b_{{\\omega }-1}k_{{\\omega }-1}\\\\&=a_1k_1+a_2k_2+\\cdots +a_{{\\omega }-1}k_{{\\omega }-1}.$ This implies that, for any $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ , it always follows $a_{\\omega }=a_1k_1+a_2k_2+\\cdots +a_{{\\omega }-1}k_{{\\omega }-1},$ which is obviously impossible.",
"Therefore, there exists an $({\\omega }-1)$ -dimensional column vector $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}$ such that $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\lbrace {\\bf 0}\\rbrace $ .",
"Furthermore, consider the following set: $K(t,{\\rho })=\\Big \\lbrace \\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}: \\sum _{i=1}^{{\\omega }-1}a_ik_i=a_{\\omega }\\Big \\rbrace .$ It is not hard to see $|K(t,{\\rho })|=|\\mathcal {F}|^{{\\omega }-2}$ , and, because $0\\le \\dim (\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k}))\\le 1$ , $\\dim (\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k}))=0$ for any $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}\\backslash K(t,{\\rho })$ .",
"Thus, $K(t,{\\rho })$ can be rewritten as the following equivalent form: $K(t,{\\rho })=\\Big \\lbrace \\vec{k}\\in \\mathcal {F}^{{\\omega }-1}: \\Delta (t,{\\rho })\\cap \\Phi (t)=\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k}) \\Big \\rbrace .$ As a result, for any $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho })$ , we have $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\lbrace {\\bf 0}\\rbrace $ for any $t\\in T$ and ${\\rho }\\in Q(t)$ .",
"At last, we show that $|\\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho })|>0$ under the condition $|\\mathcal {F}|>\\sum _{t\\in T}|Q(t)|$ .",
"This follows because $\\begin{split}&\\big |\\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho })\\big |\\\\=&\\big |\\mathcal {F}^{{\\omega }-1}\\big |-\\big |\\mathcal {F}^{{\\omega }-1}\\cap [\\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho })]\\big |\\\\=&\\big |\\mathcal {F}\\big |^{{\\omega }-1}-\\big |\\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}\\mathcal {F}^{{\\omega }-1}\\cap K(t,{\\rho })\\big |\\\\\\ge &\\big |\\mathcal {F}\\big |^{{\\omega }-1}-\\sum _{t\\in T}\\sum _{{\\rho }\\in Q(t)}\\big |\\mathcal {F}^{{\\omega }-1}\\cap K(t,{\\rho })\\big |\\\\=&\\big |\\mathcal {F}\\big |^{{\\omega }-1}-\\big |\\mathcal {F}\\big |^{{\\omega }-2}\\sum _{t\\in T}\\big |Q(t)\\big |\\\\=&\\big |\\mathcal {F}\\big |^{{\\omega }-2}\\big [\\big |\\mathcal {F}\\big |-\\sum _{t\\in T}\\big |Q(t)\\big |\\big ]\\\\>&0.\\end{split}$ The lemma is proved.",
"Under the support of the above five lemmas, we can give the main theorem below.",
"Theorem 6 Let $\\mathbf {C}_{\\omega }$ be an ${\\omega }$ -dimensional $\\mathcal {F}$ -valued network MDS code.",
"If the size of the base field $\\mathcal {F}$ satisfies $|\\mathcal {F}|>\\sum _{t\\in T}|Q(t)|$ , then there exists an $({\\omega }-1)$ -dimensional $\\mathcal {F}$ -valued network MDS code for this network $G$ with the same local encoding kernels at all non-source nodes as that of $\\mathbf {C}_{\\omega }$ .",
"For the given network MDS code on an acyclic network $G$ , Lemmas REF and REF imply that there exists an $({\\omega }-1)$ -dimensional column vector $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}$ such that $\\lbrace {e}^{({\\omega }-1)}(\\vec{k}):\\ e\\in E\\rbrace $ is the set of all extended global encoding kernels of an $({\\omega }-1)$ -dimensional regular linear network error correction code, and $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\lbrace {\\bf 0}\\rbrace $ for any $t\\in T$ and any error pattern ${\\rho }\\in Q(t)$ .",
"On the other hand, by the definition of network MDS codes and Lemma REF , we know $\\Delta (t,{\\rho })\\cap \\Phi (t)=\\lbrace {\\bf 0}\\rbrace $ for all error patterns ${\\rho }$ with $|{\\rho }|<{\\delta _t}+1$ .",
"Hence, for any error pattern ${\\rho }$ satisfying $|{\\rho }|<{\\delta _t}+1$ , or $|{\\rho }|={\\delta _t}+1$ but $\\Delta (t,{\\rho })\\cap \\Phi (t)=\\lbrace {\\bf 0}\\rbrace $ , one has that $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\lbrace {\\bf 0}\\rbrace $ since $\\Phi ^{({\\omega }-1)}(t,\\vec{k})\\subseteq \\Phi (t)$ .",
"Combining the above, for any $t\\in T$ and any error pattern ${\\rho }$ with $|{\\rho }|\\le {\\delta _t}+1$ , it always follows $\\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})=\\lbrace {\\bf 0}\\rbrace ,$ which implies that $d_{\\min }^{(t,{\\omega }-1)}\\triangleq \\min \\lbrace |{\\rho }|: \\Delta (t,{\\rho })\\cap \\Phi ^{({\\omega }-1)}(t,\\vec{k})\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace \\ge {\\delta _t}+2=C_t-({\\omega }-1)+1.$ On the other hand, the refined Singleton bound on linear network error correction codes (Proposition REF ) indicates that, for each sink node $t\\in T$ , $d_{\\min }^{(t,{\\omega }-1)}\\le C_t-({\\omega }-1)+1={\\delta _t}+2.$ Thus, $d_{\\min }^{(t,{\\omega }-1)}=C_t-({\\omega }-1)+1={\\delta _t}+2$ .",
"That is, $\\lbrace {e}^{({\\omega }-1)}(\\vec{k}):e\\in E\\rbrace $ constitutes a global description of an $({\\omega }-1)$ -dimensional network MDS code on the network $G$ , which completes the proof.",
"Again let $\\mathbf {C}_{\\omega }$ be an ${\\omega }$ -dimensional $\\mathcal {F}$ -valued network MDS code over an acyclic network $G$ .",
"Using the above constructive method recursively, if the field size $|\\mathcal {F}|$ is big enough, then, for any information rate ${\\omega }^{\\prime }\\le {\\omega }$ , it is feasible to construct an ${\\omega }^{\\prime }$ -dimensional $\\mathcal {F}$ -valued network MDS code over the network $G$ satisfying the condition that the local encoding kernels of this ${\\omega }^{\\prime }$ -dimensional network MDS code at all internal nodes are the same as that of the original ${\\omega }$ -dimensional network MDS code.",
"These network MDS codes with the same local encoding kernels at all internal nodes are called a family of variable-rate network MDS codes.",
"By [19], it follows that if $|\\mathcal {F}|\\ge \\sum _{t\\in T}|R_t({\\delta _t})|$ where $R_t({\\delta _t})=\\lbrace \\mbox{error pattern}\\ {\\rho }:\\ |{\\rho }|=rank_t({\\rho })={\\delta _t}\\rbrace $ and ${\\delta _t}=C_t-{\\omega }$ , we can construct an ${\\omega }$ -dimensional network MDS code $\\mathbf {C}_{\\omega }$ .",
"By Theorem REF , if $|\\mathcal {F}|>\\sum _{t\\in T}|Q(t)|$ , where recall that $Q(t)=\\Big \\lbrace \\mbox{error\\;pattern\\;}{\\rho }: \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\mbox{ and } |{\\rho }|=d_{\\min }^{(t)} \\Big \\rbrace ,$ we can construct an $({\\omega }-1)$ -dimensional network MDS code $\\mathbf {C}_{{\\omega }-1}$ with the same local encoding kernels at all internal nodes as that of $\\mathbf {C}_{\\omega }$ .",
"Subsequently, for any error pattern ${\\rho }$ with $rank_t({\\rho })<{\\delta _t}+1$ , we have $\\Delta (t,{\\rho })\\cap \\Phi (t)=\\lbrace {\\bf 0}\\rbrace $ because of the definition of the minimum distance $d_{\\min }^{(t)}\\triangleq \\min \\lbrace rank_t({\\rho }):\\ \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\rbrace $ and the refined Singleton bound (Proposition REF ) that $d_{\\min }^{(t)}\\le C_t-{\\omega }+1={\\delta _t}+1$ .",
"Consequently, we have $Q(t)&=\\Big \\lbrace {\\rho }\\subseteq E: \\Delta (t,{\\rho })\\cap \\Phi (t)\\ne \\lbrace {\\bf 0}\\rbrace \\mbox{ and } rank_t({\\rho })=|{\\rho }|={\\delta _t}+1 \\Big \\rbrace \\\\&\\subseteq \\Big \\lbrace {\\rho }\\subseteq E: rank_t({\\rho })=|{\\rho }|={\\delta _t}+1\\Big \\rbrace \\\\&=R_t({\\delta _t}+1)\\triangleq R_t({\\delta _t}^{({\\omega }-1)}),$ where ${\\delta _t}^{({\\omega }-1)}=C_t-({\\omega }-1)={\\delta _t}+1$ .",
"Therefore, if the base field size $|\\mathcal {F}|>\\max \\Big \\lbrace \\sum _{t\\in T}|R_t({\\delta _t})|,\\ \\sum _{t\\in T}|R_t({\\delta _t}+1)| \\Big \\rbrace ,$ then, applying our approach, we can construct two variable-rate network MDS codes with respective information rates ${\\omega }$ and ${\\omega }-1$ , that is, the constructed $({\\omega }-1)$ -dimensional and ${\\omega }$ -dimensional network MDS codes $\\mathbf {C}_{{\\omega }-1}$ and $\\mathbf {C}_{{\\omega }}$ have the same local encoding kernels at all internal nodes.",
"Recursively, if the field size satisfies $|\\mathcal {F}|>\\max \\Big \\lbrace \\sum _{t\\in T}|R_t({\\delta _t})|,\\sum _{t\\in T}|R_t({\\delta _t}+1)|,\\cdots , \\sum _{t\\in T}|R_t({\\delta _t}+{\\omega }-1)| \\Big \\rbrace ,$ or equivalently, $|\\mathcal {F}|>\\max \\Big \\lbrace \\sum _{t\\in T}|R_t(C_t-{\\omega })|,\\sum _{t\\in T}|R_t(C_t-({\\omega }-1))|,\\cdots , \\sum _{t\\in T}|R_t(C_t-1)| \\Big \\rbrace ,$ we can construct all ${\\omega }^{\\prime }$ -dimensional $(1\\le {\\omega }^{\\prime }\\le {\\omega })$ network MDS codes having the same local encoding kernel at each internal node.",
"Therefore, we have the following theorem.",
"Theorem 7 For a single source multicast acyclic network $G$ , if the size of the base field satisfies $|\\mathcal {F}|>\\max _{0\\le i \\le {\\omega }-1 }\\sum _{t\\in T}\\big |R_t({\\delta _t}+i)\\big |=\\max _{0\\le i \\le {\\omega }-1 }\\sum _{t\\in T}\\big |R_t(C_t-{\\omega }+i)\\big |,$ then we can construct a family of variable-rate $\\mathcal {F}$ -valued network MDS codes of dimensions $1,2,\\cdots ,{\\omega }$ .",
"Further, we have for each ${\\omega }^{\\prime }$ , $\\sum _{t\\in T}\\big |R_t(C_t-{\\omega }^{\\prime })\\big |\\le \\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-{\\omega }^{\\prime }}$ from [19].",
"Thus, if $|\\mathcal {F}|>\\max \\Big \\lbrace \\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}},\\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}+1},\\cdots ,\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-1}\\Big \\rbrace ,$ we are more able to construct a family of variable-rate network MDS codes of dimensions $1,2,\\cdots , {\\omega }$ , which have the same local encoding kernel at each internal node.",
"This result can be described by the following corollary.",
"Corollary 8 For a single source multicast network $G$ , if the size of the base field satisfies $|\\mathcal {F}|>\\max _{1\\le i\\le {\\omega }}\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-i}$ , then we can construct a family of variable-rate $\\mathcal {F}$ -valued network MDS codes of dimensions $1,2,\\cdots ,{\\omega }$ .",
"Remark 9 Generally speaking, in most communication networks, $C_t\\le \\lfloor \\frac{|E|}{2}\\rfloor $ for any sink node $t\\in T$ .",
"Therefore, $\\max _{1\\le i\\le {\\omega }} \\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-i} =\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-1}$ .",
"This shows that we can construct a family of variable-rate network MDS codes provided $|\\mathcal {F}|>\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-1}$ .",
"Now, we can give an algorithm for constructing a family of variable-rate network MDS codes based on our discussion above.",
"Step 1: Construct an ${\\omega }$ -dimensional network MDS code $\\mathbf {C}_{\\omega }$ by Algorithm 1 in [19]; Step 2: Choose an $({\\omega }-1)$ -dimensional column vector $\\vec{k}=[k_1\\ k_2\\ \\cdots \\ k_{{\\omega }-1}]^{\\top }\\in \\mathcal {F}^{{\\omega }-1}$ such that $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho }),$ where $K(t,{\\rho })$ is a collection of $({\\omega }-1)$ -dimensional $\\mathcal {F}$ -valued column vectors as defined in Lemma REF .",
"Step 3: $\\lbrace {({\\omega }-1)}_e(\\vec{k}): e\\in E \\rbrace $ constitutes an $({\\omega }-1)$ -dimensional $\\mathcal {F}$ -valued network MDS code with the same local encoding kernels at all internal nodes as that of $\\mathbf {C}_{\\omega }$ .",
"Using this algorithm recursively, we can construct a family of variable-rate network MDS codes of dimensions $1,2,\\cdots ,{\\omega }$ .",
"Remark 10 For the proposed variable-rate network error correction problem, we have to simultaneously consider the information transmission and network error correction, or equivalently, the regular property and MDS property of the codes.",
"If we assume that all channels are error-free, that is, only information transmission is under the consideration, our constructive algorithm degenerates into an algorithm to construct variable-rate linear network codes presented in [20] since [20] can be regarded as a special case of Lemma 1 in the present paper.",
"Further, together with other conditions such as Lemma 3 and a similar result, Lemma 5, in [20], it will become the algorithm of Fong and Yeung for constructing variable-rate linear broadcast and static linear broadcast network codes.",
"Now, we give a simple example to show how to construct an $({\\omega }-1)$ -dimensional network MDS code from an ${\\omega }$ -dimensional one satisfying that both network MDS codes have the same local encoding kernels at all internal nodes by applying the above algorithm.",
"Example 1 Let $G$ be a network with $C_{t_1}=C_{t_2}=3$ as showed by Fig.",
"REF , and let ${\\omega }=2$ .",
"Figure: The network GG with C t 1 =C t 2 =3C_{t_1}=C_{t_2}=3.For simplicity, for all $d^{\\prime }_i \\in In(s),\\ e_j\\in Out(s)$ , denote by $k_{d^{\\prime }_i,j}$ the local encoding coefficient of the adjacent channel pair $(d^{\\prime }_i,e_j)$ ; and for $e_i,e_j\\in E$ with $tail(e_j)=head(e_i)$ , denote by $k_{i,j}$ the local encoding coefficient of the adjacent channel pair $(e_i,e_j)$ .",
"Let the base field $\\mathcal {F}$ be $\\mathbb {F}_3$ , and let $k_{d^{\\prime }_1,3}=k_{d^{\\prime }_2,2}=k_{d^{\\prime }_2,5}=0$ and $k_{d^{\\prime }_1,1}=k_{d^{\\prime }_1,2}=k_{d^{\\prime }_1,4}=k_{d^{\\prime }_1,5}=k_{d^{\\prime }_2,1}=k_{d^{\\prime }_2,3}=k_{d^{\\prime }_2,4}=k_{3,6}=k_{3,7}=1.$ Then the extended global encoding kernels of all channels are ${d^{\\prime }_1}=\\begin{bmatrix}1&0&0&0&0&0&0&0&0\\end{bmatrix}^\\top ,\\qquad {d^{\\prime }_2}=\\begin{bmatrix}0&1&0&0&0&0&0&0&0\\end{bmatrix}^\\top ,$ ${e_1}=\\begin{bmatrix}1&1&1&0&0&0&0&0&0\\end{bmatrix}^\\top ,\\qquad {e_2}=\\begin{bmatrix}1&0&0&1&0&0&0&0&0\\end{bmatrix}^\\top ,$ ${e_3}=\\begin{bmatrix}0&1&0&0&1&0&0&0&0\\end{bmatrix}^\\top ,\\qquad {e_4}=\\begin{bmatrix}1&1&0&0&0&1&0&0&0\\end{bmatrix}^\\top ,$ ${e_5}=\\begin{bmatrix}1&0&0&0&0&0&1&0&0\\end{bmatrix}^\\top ,\\qquad {e_6}=\\begin{bmatrix}0&1&0&0&1&0&0&1&0\\end{bmatrix}^\\top ,$ ${e_7}=\\begin{bmatrix}0&1&0&0&1&0&0&0&1\\end{bmatrix}^\\top .$ The decoding matrices at sink nodes $t_1$ and $t_2$ are given respectively by $\\tilde{F}_{t_1}=\\begin{bmatrix}1&1&0\\\\1&0&1\\\\1&0&0\\\\0&1&0\\\\0&0&1\\\\0&0&0\\\\0&0&0\\\\0&0&1\\\\0&0&0\\end{bmatrix},\\ \\mbox{ and }\\tilde{F}_{t_2}=\\begin{bmatrix}1&1&0\\\\1&0&1\\\\0&0&0\\\\0&0&0\\\\0&0&1\\\\1&0&0\\\\0&1&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}.$ By checking the row vector of $\\tilde{F}_{t_1}$ (respectively, $\\tilde{F}_{t_2}$ ), we can see that the intersections of all one-dimensional error spaces with the message space are $\\lbrace {\\bf 0}\\rbrace $ .",
"This implies that the minimum distance of this code at $t_1$ (respectively, $t_2$ ) is 2.",
"This shows that $\\lbrace e: e\\in E \\rbrace $ constitutes a global description of a two-dimensional $\\mathbb {F}_3$ -valued network MDS code over the network $G$ .",
"Further we can choose an one-dimensional $\\mathbb {F}_3$ -valued column vector $\\vec{k}=k=1$ .",
"Then after a simple calculation, we have $k^{({\\omega }-1)}_{d^{\\prime }_1,1}(\\vec{k})=k^{({\\omega }-1)}_{d^{\\prime }_1,4}(\\vec{k})=2,$ $k^{({\\omega }-1)}_{d^{\\prime }_1,2}(\\vec{k})=k^{({\\omega }-1)}_{d^{\\prime }_1,3}(\\vec{k})=k^{({\\omega }-1)}_{d^{\\prime }_1,5}(\\vec{k})=1,$ $k^{({\\omega }-1)}_{3,6}(\\vec{k})=k_{3,6}=1, \\mbox{ and } k^{({\\omega }-1)}_{3,7}(\\vec{k})=k_{3,7}=1,$ and the $({\\omega }-1)$ -dimensional decoding matrices are $\\tilde{F}^{({\\omega }-1)}_{t_1}(\\vec{k})=\\begin{bmatrix}2&1&1\\\\1&0&0\\\\0&1&0\\\\0&0&1\\\\0&0&0\\\\0&0&0\\\\0&0&1\\\\0&0&0\\end{bmatrix}, \\mbox{ and }\\tilde{F}^{({\\omega }-1)}_{t_2}(\\vec{k})=\\begin{bmatrix}2&1&1\\\\0&0&0\\\\0&0&0\\\\0&0&1\\\\1&0&0\\\\0&1&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}.$ Further, by checking the row vectors of $\\tilde{F}^{({\\omega }-1)}_{t_1}(\\vec{k})$ (respectively, $\\tilde{F}^{({\\omega }-1)}_{t_2}(\\vec{k})$ ), we can see that the intersections of all two-dimensional error spaces with the message space are $\\lbrace {\\bf 0}\\rbrace $ .",
"This implies that the minimum distance of this code at $t_1$ (respectively, $t_2$ ) is 3.",
"Therefore, $\\lbrace {({\\omega }-1)}_e(\\vec{k}): e\\in E \\rbrace $ constitutes an one-dimensional $\\mathbb {F}_3$ -valued network MDS code and the local encoding kernels at all internal nodes are the same as that of $\\lbrace e:e\\in E \\rbrace $ over the network $G$ ."
],
[
"Performance Analysis",
"In this section, we will focus on the performance of our proposed algorithm for constructing variable-rate network MDS codes in different aspects including the field size, the time complexity of the algorithm, the encoding complexity at the source node, and the decoding methods.",
"First, recall that Yang et al.",
"[14] proposed two algorithms for constructing network MDS codes and both of them design the codebook at the source node and local encoding kernels separately.",
"The first one needs to find a codebook based on a given set of local encoding kernels, and the second one needs to find a set of local encoding kernels based on a given classical error-correcting code at the source node satisfying a certain minimum distance requirement as the codebook.",
"Hence, it seems likely that these two algorithms might solve this variable-rate network error correction problem.",
"However, by a detailed analysis below, they are either non-feasible or inefficient for solving the problem.",
"To be specific, for the second one, the design of the set of local encoding kernels is based on a given classical error-correcting code, say $\\mathcal {C}$ , at the source node, so the local encoding kernels are different for the distinct classical error-correcting codes with distinct information rates at the source node.",
"Mathematically, for each updating channel $e$ , where $e$ is the edge appended to the graph at the $i$ th iteration, $i>0$ , let ${\\bf k}_e=\\begin{bmatrix} k_{d,e}: & d\\in E^{i-1}\\end{bmatrix}^\\top $ be an $(|Out(s)|+i-1)$ -dimensional column vector consisting of all local encoding coefficients $k_{d,e}$ for the channels $d,e$ , where $k_{d,e}=0$ if $d$ and $e$ are not adjacent, and $E^{i-1}$ is the set of channels in the $(i-1)$ -th subnetwork $G^{i-1}$ of $G$ .",
"Note that ${\\bf k}_e$ has to be chosen to satisfy the following feasible condition, that is, $\\left(F_t^i(X,-Z^i)\\right)^{\\backslash \\mathcal {L}}\\ne {\\bf 0}$ for all combinations of C1) $t\\in T$ ; C2) $\\mathcal {L}\\subset \\lbrace 1,2,\\cdots ,r_t\\rbrace $ with $0\\le |\\mathcal {L}| \\le d_t-1$ ; C3) nonzero $ X\\in \\mathcal {C}\\subseteq \\mathcal {F}^{|Out(s)|}$ ; C4) error vector $Z^i$ with $w_H(Z^i)\\le d_t-1-|\\mathcal {L}|$ ; where $F_t^i(X,-Z^i)$ represents the output of the channels in $In(t)$ for input $X\\in \\mathcal {C}$ and error vector $-Z^i$ , in the $i$ th subnetwork $G^i$ corresponding to the $i$ th iteration, the designed rank of the matrix $F_{s,t}$ to be introduced in (REF ) below is $r_t$ and the designed minimum distance is $d_t$ for each sink $t\\in T$ (refer to [14][15] for more details).",
"So it is easily seen that ${\\bf k}_e$ depends on some initial parameters including the given algebraic code $\\mathcal {C}$ , the minimum distance $d_t$ , the rank $r_t$ of the matrix $F_{s,t}$ , and so on, which further depend on the information rate ${\\omega }$ .",
"This implies that it is impossible to use Yang et al.s' Algorithm 2 to construct variable-rate network MDS codes.",
"The first algorithm needs to find a codebook at the source node after a set of local encoding kernels is given.",
"Thus, it seems likely that it is feasible to design variable-rate network MDS codes to solve this variable-rate problem.",
"However, for the first algorithm, as described by Yang et al.",
"[14], they just give a method to find the proper codebook at the source node and the part of constructing local encoding kernels makes use of the existing Jaggi et al.s' algorithm [5] directly.",
"Actually, the way of Jaggi et al.s' algorithm to obtain linear network codes is to construct global encoding kernels for all channels one by one from the source node $s$ to each sink $t\\in T$ , including all outgoing channels of the source node.",
"In other words, it designs the matrix $M\\triangleq \\begin{bmatrix}f_e:& e\\in E\\end{bmatrix}$ , where $f_e$ is the global encoding kernel of channel $e$ .",
"Further, each sink node $t\\in T$ can use the corresponding decoding matrix: $M_t\\triangleq MA_{In(t)}^\\top =\\begin{bmatrix} f_e: & e\\in In(t) \\end{bmatrix},$ where we use $A_{{\\rho }}$ to denote a $|{\\rho }|\\times |E|$ matrix with ${\\rho }$ being a collection of channels, to be specific, $A_{{\\rho }}=[A_{d,e}]_{d\\in {\\rho },e\\in E}$ satisfying $A_{d,e}={\\left\\lbrace \\begin{array}{ll}1, & d=e,\\\\0, & \\text{otherwise.}\\end{array}\\right.",
"}$ Particularly, $A_{In(t)}=[A_{d,e}]_{d\\in In(t),e\\in E}$ and $A_{Out(s)}=[A_{d,e}]_{d\\in Out(s),e\\in E}$ .",
"Thus, by [4] (also see [6][7]), it is not difficult to obtain that $M_t=M\\cdot A_{In(t)}^\\top =K_s\\cdot A_{Out(s)}\\cdot (I-K)^{-1}\\cdot A_{In(t)}^\\top = K_s\\cdot F_{s,t},$ where $F_{s,t}\\triangleq A_{Out(s)}\\cdot (I-K)^{-1}\\cdot A_{In(t)}^\\top ;$ $K=[k_{d,e}]_{d\\in E,e\\in E}$ is the system transfer matrix (also called one-step transformation matrix) of size $|E|\\times |E|$ with $k_{d,e}$ being the local encoding coefficient for the adjacent pair $(d,e)$ of channels, and $k_{d,e}=0$ otherwise; $K_s=[k_{d,e}]_{d\\in In(s),e\\in Out(s)}$ is the local encoding kernel at the source node, and $I$ represents an $|E|\\times |E|$ identity matrix.",
"Recall that, Yang et al.s' Algorithm 1 first needs to construct a set of local encoding kernels satisfying ${\\mathrm {Rank}}(F_{s,t})=r_t$ for each sink node $t\\in T$ .",
"But, together with the equality (REF ), it seems that it is not feasible to apply Jaggi et al.s' algorithm directly.",
"To be specific, by Jaggi et al.s' algorithm, one obtains decoding matrices $M_t$ for all sink nodes $t\\in T$ .",
"But only from $M_t$ , it is difficult to find a matrix $K_s$ such that $M_t=K_s\\cdot F_{s,t}$ and $F_{s,t}$ satisfies ${\\mathrm {Rank}}(F_{s,t})=r_t$ for each sink node $t\\in T$ .",
"So in order to apply Yang et al.s' Algorithm 1 to solve the variable-rate problem it is necessary to design a new algorithm or modify Jaggi et al.s' algorithm to achieve the above requirements, that is, construct local encoding kernels at all internal nodes such that ${\\mathrm {Rank}}(F_{s,t})=r_t$ for each $t\\in T$ .",
"We believe that modifying Jaggi et al.s' algorithm supposedly makes sense.",
"Furthermore, even assuming that all local encoding kernels at internal nodes satisfying the condition that ${\\mathrm {Rank}}(F_{s,t})=r_t$ for each sink node $t\\in T$ are given, our proposed algorithm still has many advantages in different aspects such as the size of base finite field, the time complexity of the algorithms, the encoding complexity at the source node, and the decoding algorithms.",
"In the following, we show the detailed discussion in order to characterize the performance analysis of our algorithms."
],
[
"Field Size",
"From [19], we have known that the required field size of our algorithm for constructing a network MDS code is smaller (in some cases much smaller) than that of Yang et al.s' algorithms.",
"If the variable-rate network MDS coding is considered simultaneously, the required field size of our algorithm is still smaller (also in some cases much smaller) than that of Yang et al.s' algorithms.",
"Without loss of generality, we consider two variable-rate network MDS codes with respective information rates ${\\omega }$ and ${\\omega }-1$ .",
"As stated in the last section, we have obtained that if the base field size: $|\\mathcal {F}|>\\max \\Big \\lbrace \\sum _{t\\in T}|R_t({\\delta _t})|,\\ \\sum _{t\\in T}|R_t({\\delta _t}+1)| \\Big \\rbrace ,$ then, applying our algorithm, we can construct two variable-rate network MDS codes with respective information rates ${\\omega }$ and ${\\omega }-1$ , that is, the constructed $({\\omega }-1)$ -dimensional and ${\\omega }$ -dimensional network MDS codes $\\mathbf {C}_{{\\omega }-1}$ and $\\mathbf {C}_{{\\omega }}$ have the same local encoding kernels at all internal nodes.",
"Particularly, if ${\\delta _t}+1\\le \\lceil C_t/2 \\rceil $ , then we have $|R_t({\\delta _t})|\\le |R_t({\\delta _t}+1)|$ from [19], which means that the field size satisfying $|\\mathcal {F}|>\\sum _{t\\in T}|R_t({\\delta _t}+1)|$ is enough.",
"In fact, notice that the field size satisfying $|\\mathcal {F}|>\\max \\Big \\lbrace \\sum _{t\\in T}|R_t({\\delta _t})|,\\ \\sum _{t\\in T}|Q(t)| \\Big \\rbrace $ is enough for constructing such two network MDS codes.",
"Together with $|Q(t)|\\le |R_t({\\delta _t}+1)|$ and $|R_t({\\delta _t})|\\le |R_t({\\delta _t}+1)|$ , it follows $\\max \\Big \\lbrace \\sum _{t\\in T}|R_t({\\delta _t})|,\\ \\sum _{t\\in T}|Q(t)| \\Big \\rbrace \\le \\sum _{t\\in T}|R_t({\\delta _t}+1)|.$ This implies that the left hand side of the above inequality is big enough for the required field size for the existence of $({\\omega }-1)$ -dimensional network MDS codes, which usually is smaller than the previous result $\\sum _{t\\in T}|R_t({\\delta _t}+1)|$ proposed in [19].",
"On the other hand, from Theorem 10 in [14], in order to construct ${\\omega }$ -dimensional network MDS code, the required filed size is not less than $\\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}}$ .",
"Further, for constructing an $({\\omega }-1)$ -dimensional network MDS code with the same local encoding kernel at the internal nodes, the required field size is not less than $\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-({\\omega }-1)}=\\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}+1}.$ Combining the above, the required base field size of Yang et al.s' Algorithm 1 satisfies: $|\\mathcal {F}|>\\max \\left\\lbrace \\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}},\\ \\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}+1} \\right\\rbrace .$ In particular, if ${\\delta _t}+1\\le \\lfloor |E|/2 \\rfloor $ , then we deduce $|\\mathcal {F}|>\\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}+1}$ .",
"In addition, Lemma 6 in [19] shows that $\\sum _{t\\in T}|R_t({\\delta _t})|\\le \\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}}$ and $\\sum _{t\\in T}|R_t({\\delta _t}+1)|\\le \\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}+1},$ which indicates that our algorithm needs smaller field size than Yang et al.s'.",
"Example 2 Let $G$ be a combination network [6][7] with parameters $N=6$ and $k=4$ .",
"To be specific, $G$ is a single source multicast network with $N=6$ internal nodes, where there is one and only one channel from the source node $s$ to each internal node, and arbitrary $k=4$ internal nodes are connective with one and only one sink node, which implies that there are totally ${6\\atopwithdelims ()4}=15$ sink nodes.",
"Thus, for $G$ , we know that $|J|=6$ , $|T|={6 \\atopwithdelims ()4}=15$ , and $|E|=6+4\\times {6\\atopwithdelims ()4}=66$ .",
"It is evident that the minimum cut capacity $C_t$ between $s$ and any sink node $t$ is 4.",
"For example, Fig.",
"REF shows a combination network with $N=3,k=2$ .",
"Figure: Combination Network with N=3,k=2N=3,k=2.Furthermore, let the information rates be ${\\omega }=2$ and ${\\omega }=1$ , and thus ${\\delta _t}^{({\\omega }=2)}=C_t-2=2$ and ${\\delta _t}^{({\\omega }=1)}=C_t-1=3$ , respectively.",
"Therefore, for each sink node $t\\in T$ , one has $|R_t({\\delta _t}^{({\\omega }=2)})|&=|R_t(2)|=2^2\\cdot {4 \\atopwithdelims ()2}=24,\\\\|R_t({\\delta _t}^{({\\omega }=1)})|&=|R_t(1)|=2^3\\cdot {4 \\atopwithdelims ()3}=32,$ which further leads to $\\sum _{t\\in T}|R_t({\\delta _t}^{({\\omega }=2)})|&=15\\times 24=360,\\\\\\sum _{t\\in T}|R_t({\\delta _t}^{({\\omega }=1)})|&=15\\times 32=480.$ So the field size satisfying $|\\mathcal {F}|>480$ is enough for our proposed algorithm.",
"On the other hand, we further have $\\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}^{({\\omega }=2)}}=\\sum _{t\\in T}{66 \\atopwithdelims ()2}=32175,$ and $\\sum _{t\\in T}{|E| \\atopwithdelims (){\\delta _t}^{({\\omega }=1)}}=\\sum _{t\\in T}{66 \\atopwithdelims ()3}=45760.$ Thus, Yang et al.s' algorithm needs the field size satisfying $|\\mathcal {F}|>45760$ ."
],
[
"Time Complexity of Algorithms",
"Below we will discuss the time complexity of constructive algorithms.",
"Similarly, we still consider one representative case constructing two variable-rate network MDS codes with respective dimensions ${\\omega }$ and ${\\omega }-1$ .",
"As discussed in [19], the time complexity of our used algorithm for constructing an ${\\omega }$ -dimensional network MDS code is smaller than that of Yang et al.s' two algorithms by using either the random analysis method or the deterministic analysis method.",
"In the following, we further discuss the time complexity of constructing a variable-rate $({\\omega }-1)$ -dimensional network MDS code.",
"Note that the key for constructing such an $({\\omega }-1)$ -dimensional network MDS code from a given ${\\omega }$ -dimensional network MDS code is to choose a proper $({\\omega }-1)$ -dimensional $\\mathcal {F}$ -valued vector $\\vec{k}$ , which has to satisfy the condition (REF ).",
"Therefore, from [5] and [14], the time complexity of our algorithm to construct such an $({\\omega }-1)$ -dimensional network MDS code from a given ${\\omega }$ -dimensional network MDS code is at most $\\mathcal {O}\\left( ({\\omega }-1)^3\\sum _{t\\in T}|Q(t)|+({\\omega }-1)\\left[ \\sum _{t\\in T}|Q(t)| \\right]^2 \\right).$ Note that it is also the encoding time complexity at the source node by using our algorithm.",
"In the following, we consider Yang et al.s' Algorithm 1.",
"First, assume that all local encoding kernels at internal nodes are given and satisfy ${\\mathrm {Rank}}(F_{s,t})=C_t$ , where again recall that $C_t$ is the minimum cut capacity between the source node $s$ and the sink node $t$ , and $F_{s,t}=A_{Out(s)}\\cdot (I-K)^{-1}\\cdot A_{In(t)}^\\top $ .",
"By [14], if we want to construct a network MDS code with the information rate ${\\omega }$ , we have to derive ${\\omega }$ $n_s$ -dimensional vectors ${\\bf g}_1,{\\bf g}_2,\\cdots ,{\\bf g}_{\\omega }$ in turn satisfying: ${\\bf g}_1&\\notin \\Delta _t({\\bf 0}, C_t-{\\omega }),\\\\{\\bf g}_i&\\notin \\Delta _t({\\bf 0}, C_t-{\\omega })+\\langle {\\bf g}_1,{\\bf g}_2,\\cdots ,{\\bf g}_{i-1} \\rangle ,$ for each $i$ , $2\\le i \\le {\\omega }$ , where $n_s$ is the number of outgoing channels of the source node $s$ , i.e., $n_s=|Out(s)|$ , and $\\Delta _t({\\bf 0}, C_t-{\\omega })=\\lbrace {\\bf g}:\\ {\\bf g}\\in \\mathcal {F}^{n_s} \\mbox{ satisfying } \\min \\lbrace w_H({\\bf Z}):\\ {\\bf Z}\\in \\mathcal {F}^{|E|} \\mbox{ such that } {\\bf g}F_{s,t}={\\bf Z}G_t \\rbrace \\le C_t-{\\omega }\\rbrace $ with $w_H(\\cdot )$ representing the Hamming weight.",
"Further, when an $({\\omega }-1)$ -dimensional network MDS code with the same local encoding kernels is constructed, we similarly derive $({\\omega }-1)$ $n_s$ -dimensional vectors ${\\bf g}_1^{\\prime },{\\bf g}_2^{\\prime },\\cdots ,{\\bf g}_{({\\omega }-1)}^{\\prime }$ in turn according to the same way.",
"However, it is necessary to notice that $\\Delta _t({\\bf 0}, C_t-{\\omega })\\subseteq \\Delta _t({\\bf 0}, C_t-{\\omega }+1),$ which implies that ${\\bf g}_1,{\\bf g}_2,\\cdots ,{\\bf g}_{\\omega }$ for ${\\omega }$ -dimensional network MDS code may be useless for deriving ${\\bf g}_1^{\\prime },{\\bf g}_2^{\\prime },\\cdots ,{\\bf g}_{({\\omega }-1)}^{\\prime }$ .",
"In other words, it has to repeat the same procedure to choose the proper vectors ${\\bf g}_1^{\\prime },{\\bf g}_2^{\\prime },\\cdots ,{\\bf g}_{({\\omega }-1)}^{\\prime }$ .",
"This evidently increases the complexity.",
"From [14], based on the given local encoding kernels at all internal nodes, the time complexity of Yang et al.s' Algorithm 1 for constructing such an $({\\omega }-1)$ -dimensional network MDS code is $\\mathcal {O}\\left( ({\\omega }-1)n_s^3\\sum _{t\\in T}{|E|\\atopwithdelims (){\\delta _t}+1}+({\\omega }-1)n_s\\left[ \\sum _{t\\in T}{|E|\\atopwithdelims (){\\delta _t}+1} \\right]^2 \\right),$ which is also the encoding time complexity at the source node by using Yang et al.s' algorithm.",
"Since $|Q(t)|\\le |R_t({\\delta _t}+1)|\\le {|E| \\atopwithdelims (){\\delta _t}+1}$ for each sink node $t\\in T$ , it is easily seen that $&({\\omega }-1)^3\\sum _{t\\in T}\\big |Q(t)\\big |+({\\omega }-1)\\left[ \\sum _{t\\in T}\\big |Q(t)\\big | \\right]^2\\\\<& ({\\omega }-1)n_s^3\\sum _{t\\in T}{|E|\\atopwithdelims (){\\delta _t}+1}+({\\omega }-1)n_s\\left[ \\sum _{t\\in T}{|E|\\atopwithdelims (){\\delta _t}+1} \\right]^2,$ and, in general cases, the former is much smaller than the later.",
"In view of the above discussion, the total time complexity of our algorithm for constructing variable-rate network MDS codes is also smaller than that of Yang et al.s' algorithm.",
"Particularly, for our algorithm, the encoding time complexity at the source node is smaller (in general much smaller) than that of Yang et al.s' algorithm.",
"This time complexity is important, in particular, when the local encoding kernels at all internal nodes are fixed.",
"In addition, during the analysis of time complexity of these algorithms, it is assumed that any arithmetic in the base finite field is $\\mathcal {O}(1)$ regardless of the finite field.",
"Actually, it is well-known that the cost of arithmetic in small field is smaller than that in a bigger one, together with the above conclusion that the size of the base field used in our algorithm is smaller than that of others, which implies that the time complexity of the proposed algorithm can be reduced further."
],
[
"Decoding Algorithms",
"In [14], Yang et al.",
"just gave two decoding principles by using the concept of the minimum weight (refer to Definitions 2 and 3 in [14]), which are similar to the minimum distance decoding principle.",
"This minimum distance decoding problem (usually called nearest codeword problem) is known to be NP-hard for classical linear codes which can be regarded as special linear network codes.",
"Moreover, as mentioned in [13] and [19], our algorithm can make use of the better and faster decoding algorithms proposed by Zhang, Yan, and Balli in a series of papers [13], [25], and [18] such as the brute force decoding algorithm and, particularly, the statistical decoding algorithm.",
"For the case of decoding in packet networks [25] and [18], where all messages such as $X_i$ $(1\\le i \\le {\\omega })$ , $Z_e$ , and $\\tilde{U}_e$ $(e\\in E)$ are column vectors over the base field $\\mathcal {F}$ , all message scalar components in a packet share the same extended global encoding kernel, and the decoding principle is applied to each message scalar component of the packets, our algorithm has more advantages on decoding network error correction codes beyond the error correction capability, even beyond the minimum distance."
],
[
"Random Variable-Rate Network MDS Codes",
"At present, as described in [15][14], there are roughly two classes of network error correction coding.",
"One class is called coherent network error correction if the sink nodes know the network topology as well as the network codes used in transmission.",
"Otherwise, the network error correction without this assumption is called noncoherent network error correction.",
"Actually, coherent and noncoherent transmissions for network coding are analogous to the coherent and noncoherent transmissions for multiple antenna channels in wireless communication.",
"When using the deterministic construction of linear network codes such as [3] and [5], the network transmission is usually regarded as coherent, and when using random network coding such as [26] and [23], the network transmission is usually considered to be noncoherent.",
"Here the main idea of random network coding is that when a node (maybe the source node $s$ ) receives the messages from its all incoming channels, for each outgoing channel, it randomly and uniformly picks the encoding coefficients from the base field $\\mathcal {F}$ , uses them to encode the received messages, and transmits the encoded messages over the outgoing channel.",
"In other words, the local coding coefficients $k_{d,e}$ are independently and uniformly distributed random variables taking values in the base field $\\mathcal {F}$ .",
"However, it is possible to use noncoherent transmission for deterministicly constructed linear network codes and use coherent transmission for randomly constructed linear network codes.",
"When the noncoherent network error correction is under consideration, for the problem discussed in this paper, the deterministic constructive algorithm may not be used since the network topology is unknown.",
"So the above random method is also applied to noncoherent network error correction, and the linear network error correction codes constructed by this method are called random linear network error correction codes.",
"Further, we obtain the following result.",
"Theorem 11 Consider noncoherent network error correction coding on a single source multicast network $G$ .",
"Using random method to construct two variable-rate network MDS codes with respective dimensions ${\\omega }$ and ${\\omega }-1$ , then the success probability $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})$ for constructing such two codes satisfies: $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})\\ge \\left[1-\\frac{\\sum _{t\\in T}|Q(t)|}{|\\mathcal {F}|}\\right]\\cdot \\left[1-\\frac{\\sum _{t\\in T}|R_t({\\delta _t})|}{|\\mathcal {F}|-1}\\right]^{|J|+1},$ where again ${\\delta _t}=C_t-{\\omega }$ , and $J$ is the set of internal nodes in $G$ .",
"This further indicates two variable-rate network MDS codes with respective dimensions ${\\omega }$ and ${\\omega }-1$ can be constructed with high probability close to one by random method, if the size of the base field $\\mathcal {F}$ is sufficiently large.",
"By [19], we know the probability $Pr(\\mathbf {C}_{\\omega })$ that ${\\omega }$ -dimensional network MDS codes are constructed by the random method is lower bounded by: $Pr(\\mathbf {C}_{{\\omega }})\\ge \\left[1-\\frac{\\sum _{t\\in T}|R_t({\\delta _t})|}{|\\mathcal {F}|-1}\\right]^{|J|+1}.$ Together with $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})=Pr(\\mathbf {C}_{{\\omega }})Pr(\\mathbf {C}_{{\\omega }-1}|\\mathbf {C}_{{\\omega }})$ , it suffices to take the probability $Pr(\\mathbf {C}_{{\\omega }-1}|\\mathbf {C}_{{\\omega }})$ into account.",
"We randomly and uniformly pick an $({\\omega }-1)$ -dimensional column vector $\\vec{k}$ from $\\mathcal {F}^{{\\omega }-1}$ , i.e., $\\vec{k}$ is a uniformly distributed random vector taking values in $\\mathcal {F}^{{\\omega }-1}$ .",
"By Lemma REF , it follows that, if $\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho }),$ then $\\lbrace {e}^{{\\omega }-1}(\\vec{k}):\\ e\\in E\\rbrace $ constitutes an $({\\omega }-1)$ -dimensional network MDS code and its local encoding kernel at each non-source node are the same as that of $\\mathbf {C}_{\\omega }$ .",
"Thus, we will focus on the probability $Pr(\\mathbf {C}_{{\\omega }-1}|\\mathbf {C}_{{\\omega }})\\ge P(\\vec{k})\\triangleq Pr(\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho })).$ It is not difficult to obtain $P(\\vec{k})=&Pr(\\vec{k}\\in \\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho }))\\nonumber \\\\=&\\frac{|\\mathcal {F}^{{\\omega }-1}\\backslash \\cup _{t\\in T}\\cup _{{\\rho }\\in Q(t)}K(t,{\\rho })|}{|\\mathcal {F}|^{{\\omega }-1}}\\nonumber \\\\\\ge &\\frac{|\\mathcal {F}|^{{\\omega }-1}-\\sum _{t\\in T}\\sum _{{\\rho }\\in Q(t)}|\\mathcal {F}^{{\\omega }-1}\\cap K(t,{\\rho })|}{|\\mathcal {F}|^{{\\omega }-1}}\\nonumber \\\\=&1-\\frac{\\sum _{t\\in T}|Q(t)|}{|\\mathcal {F}|}.$ Combining the inequalities (REF ), (REF ) and (REF ), one obtains a lower bound on the success probability: $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})\\ge \\left[1-\\frac{\\sum _{t\\in T}|R_t({\\delta _t})|}{|\\mathcal {F}|-1}\\right]^{|J|+1}\\cdot \\left[1-\\frac{\\sum _{t\\in T}|Q(t)|}{|\\mathcal {F}|}\\right].$ Furthermore, it is not difficult to see that $Pr(\\mathbf {C}_{{\\omega }})\\rightarrow 1$ and $P(\\vec{k})\\rightarrow 1$ as $|\\mathcal {F}|\\rightarrow \\infty $ from (REF ) and (REF ), respectively.",
"Therefore, for sufficiently large base field $\\mathcal {F}$ , an ${\\omega }$ -dimensional and an $({\\omega }-1)$ -dimensional network MDS codes with the same local encoding kernel at each non-source node can be constructed by random method with high probability close to one.",
"This accomplishes the proof.",
"Together with Corollary REF , the above theorem leads to the following corollary immediately.",
"Corollary 12 Using random method to construct two variable-rate network MDS codes with respective dimensions ${\\omega }$ and ${\\omega }-1$ , then the success probability $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})$ for constructing such two codes satisfies: $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})\\ge \\left[1-\\frac{\\sum _{t\\in T}{ |E|\\atopwithdelims ()C_t-{\\omega }+1}}{|\\mathcal {F}|}\\right]\\cdot \\left[1-\\frac{\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-{\\omega }}}{|\\mathcal {F}|-1}\\right]^{|J|+1},$ and further for general cases with $C_t\\le \\lfloor \\frac{|E|}{2}\\rfloor $ for all sink nodes $t\\in T$ , $Pr(\\mathbf {C}_{{\\omega }}\\cap \\mathbf {C}_{{\\omega }-1})\\ge \\left[1-\\frac{\\sum _{t\\in T}{|E| \\atopwithdelims ()C_t-{\\omega }+1}}{|\\mathcal {F}|-1}\\right]^{|J|+2},$ where again $J$ is the set of internal nodes in $G$ .",
"For constructing a family of variable-rate network MDS codes by the random method, we similarly have the following corollary.",
"Corollary 13 A family of variable-rate network MDS codes can be constructed with high probability close to one by the random method, if the size of the base field $\\mathcal {F}$ is sufficiently large.",
"In [20], the authors proposed a further research problem that is the performance analysis of randomly designed codes for variable-rate linear network coding.",
"Actually, the discussions in this section analyze the performance of randomly designed network MDS codes for our variable-rate network error correction problem, which is more complicated than variable-rate linear network coding problem.",
"Therefore, our analysis method also can be applied to characterize the performance of randomly designed variable-rate linear network codes."
],
[
"Conclusion",
"In network communication, the source often transmits the messages at several different information rates within a session.",
"When both information transmission and network error correction are under consideration, linear network error correction MDS codes are expected to be used for these different rates.",
"In this paper, we propose a more efficient scheme for this purpose than using the known algorithms to construct network MDS code for each rate.",
"In addition, these network MDS codes designed by the proposed scheme have the same local encoding kernels at all internal nodes.",
"This saves the storage space for each internal node and resources and time for the transmission.",
"Some interesting problems in this direction remain open.",
"For instance, we can also consider a family of variable-rate general linear network error correction codes with certain error correction capacity instead of network MDS codes, partly because the field size required by general linear network error correction codes is smaller than that of network MDS codes."
]
] | 1403.0214 |
[
[
"Equilibrating dynamics in quenched Bose gases: characterizing multiple\n time regimes"
],
[
"Abstract We address the physics of equilibration in ultracold atomic gases following a quench of the interaction parameter.",
"We focus on the momentum distribution of the excitations, $n_{\\mathbf k}$, and observe that larger ${\\mathbf k}$ modes will equilibrate faster, as has been claimed in recent experimental work.",
"We identify three time regimes.",
"At short times $n_{\\mathbf k}$ exhibits oscillations; these are damped out at intermediate times where the system appears to be in a false-equilibrium.",
"Finally, at longer times, full equilibration occurs.",
"This false-equilibrium is associated with the necessarily slower relaxation of the condensate which sufficiently high ${\\mathbf k}$-states (of the excitation response) will then quasi-adiabatically follow.",
"Our work bears on the recent literature focus on interaction quench experiments.",
"We take issue with the fact that theories to date assume that the oscillatory regime is adequate for addressing experiments."
],
[
"Supplemental materials",
"We give here the technical details of our bath approach to the dynamics of a quench Bose gas that was developed in Ref. Rancon2013b.",
"Following Ref.",
"Rancon2013b, we compute the momentum distribution $n_k(t)$ .",
"Based on Bogoliubov theory, we consider the full Hamiltonian in the absence of a trap to be given by $\\hat{H}(g;n_0)=\\hat{H}_{{\\rm Bog}}(g;n_0) +\\hat{H}_{{\\rm bath}} +\\hat{H}_{\\rm {coup}}$ where $\\hat{H}_{{\\rm Bog}}(g;n_0) &=& \\sum _k \\big [\\hat{\\psi }^\\dag _k (\\epsilon _k+gn_0) \\hat{\\psi }_k + \\frac{gn_0}{2} \\hat{\\psi }_k\\hat{\\psi }_{-k} \\nonumber \\\\&+& \\frac{gn_0}{2} \\hat{\\psi }^\\dag _k\\hat{\\psi }^\\dag _{-k}\\big ] \\nonumber \\\\\\hat{H}_{{\\rm bath}}&=& \\sum _{i,k} \\big [\\omega _{i,k}\\hat{W}^\\dag _{i,k}\\hat{W}_{i,k}+\\nu _{i,k}\\hat{V}^\\dag _{i,k}\\hat{V}_{i,k}\\big ] \\nonumber \\\\\\hat{H}_{\\rm {coup}}&=& \\sum _{i,k} \\big [\\eta _{i,k}^* \\hat{W}^\\dag _{i,k} \\hat{\\psi }_k +\\zeta _{i,k} \\hat{V}^\\dag _{i,-k} \\hat{\\psi }^\\dag _k+ h.c.\\big ] \\nonumber $ where $\\hat{\\psi }^{(\\dag )}_k$ annihilates (creates) an atom with momentum $k\\ne 0$ Here $n_0$ is the condensate density and $g$ is the interaction strength.",
"The bath is characterized by two kinds of bosonic modes, $\\hat{W}^{(\\dag )}_{i,k}$ and $\\hat{V}^{(\\dag )}_{i,k}$ , which allow for a well-behaved spectral function [23], [13].",
"The dynamics of the system after an interaction quench from $g_i$ to $g_f$ is described by $i\\partial _t \\hat{\\psi }_k(t)=\\big [\\hat{\\psi }_k(t),\\hat{H}\\big (g_f,n_0(t)\\big )\\big ]$ , etc., where we have allowed a time dependent condensate.",
"One can solve the equation for the bath operators which in turn gives $\\begin{split}i\\partial _t\\hat{\\psi }_k(t) =& \\omega _k(t) \\hat{\\psi }_k(t)+ g_f n_0(t) \\hat{\\psi }^\\dag _{-k}(t)+\\hat{D}_k(t)\\\\&-i \\int _{t_0}^{t}ds\\,\\gamma _k(t-s)\\hat{\\psi }_k(s) ,\\\\i\\partial _t\\hat{\\psi }^\\dag _{-k}(t) =& -\\omega _k(t) \\hat{\\psi }^\\dag _{-k}(t)- g_f n_0(t) \\hat{\\psi }_{k}(t)-\\hat{D}^\\dag _{-k}(t)\\\\&-i \\int _{t_0}^{t}ds\\,\\gamma _k(t-s)\\hat{\\psi }^\\dag _{-k}(s) .\\end{split}$ Here, $\\omega _k(t) =\\epsilon _k+g_fn_0(t)$ , $\\hat{D}_k(t)=\\sum _j \\eta _{j,k} e^{-i\\omega _{j,k}t}\\hat{W}_{j,k}(0)+\\sum _j \\zeta _{j,k} e^{i\\nu _{j,k}t}\\hat{V}^\\dag _{j,k}(0)$ and $\\gamma _k(t) =\\int _\\omega \\Sigma _2(k,\\omega ) e^{-i\\omega t}$ with $\\int _\\omega =\\int d\\omega /(2\\pi )$ .",
"We define the spectral function of the bath $\\Sigma _2(k,\\omega )=2\\pi \\sum _j \\Big [|\\eta _{j,k}|^2 \\delta (\\omega -\\omega _{j,k})- |\\zeta _{j,k}|^2\\delta (\\omega +\\nu _{j,k})\\Big ]$ .",
"In the following, we will use an Ohmic bath $\\Sigma _2(k,\\omega )=2\\Gamma _k \\omega f(\\omega /\\Omega )$ where $f(\\omega /\\Omega )$ is an even function that regularizes the high-energy behavior with cut-off $\\Omega $ [21].",
"Note that in this framework, the bath parameter $\\Gamma _k$ is independent of the temperature of the bath (it is only related to the microscopic coupling between the bath and the system).",
"Note that $\\hat{D}_k(t)$ plays the role of a random force operator and $\\gamma _k(t)$ reflects the damping.",
"The relaxation to equilibrium will be insured by the satisfaction of the fluctuation-dissipation relation $\\Big [\\hat{D}_k(t),\\hat{D}^\\dag _k(s)\\Big ]=\\gamma _k(t-s).$ The equations of motion (REF ) can be formally solved by introducing a matrix Green's function $M_k(t,s)= \\begin{pmatrix}M_{1,k}(t,s) && M_{2,k}(t,s)\\\\M_{3,k}(t,s) && M_{4,k}(t,s)\\end{pmatrix},$ where $\\begin{split}i\\partial _t M_k(t,s)=& \\begin{pmatrix}\\omega _k(t)-i\\gamma * && g_f n_0(t)\\\\-g_f n_0(t) && \\omega _k(t)-i\\gamma *\\end{pmatrix}M_k(t,s),\\end{split}$ and $\\gamma _k*f(t,s)= \\int _0^t du\\,\\gamma _k(t-u) \\, f(u,s)$ for any function $f(t,s)$ .",
"The initial condition is given by $M_k(t,t)=-i\\large {1}$ .",
"One readily shows that $M_{4,k}^*(t,s)=M_{1,k}(t,s)$ $M_{3,k}^*(t,s)=M_{2,k}(t,s)$ .",
"The formal solution of Eq.",
"(REF ) can be written as $\\begin{pmatrix}\\hat{\\psi }_k(t)\\\\\\hat{\\psi }^\\dag _{-k}(t)\\end{pmatrix}&=& M_k(t,0)\\begin{pmatrix}i\\hat{\\psi }_{k,0}\\\\ i\\hat{\\psi }^\\dag _{-k,0}\\end{pmatrix} \\nonumber \\\\&+&\\int _0^{t}ds M_k (t,s)\\begin{pmatrix}\\hat{D}_k(s)\\\\ -\\hat{D}^\\dag _{-k}(s)\\end{pmatrix}.",
"$ Eq.",
"(REF ) is a generalization of the time-dependent Bogoliubov-de Gennes equation, that includes both dissipation and equilibration.",
"For a time-dependent condensate, $M_k(t,s)$ depends on two times separately, whereas it is a function of $t-s$ if the condensate is constant, as was studied in Ref.",
"Rancon2013b.",
"Here one has to solve the time evolution matrix numerically when the time dependence of the condensate is specified (see main text).",
"To compute an observable such as the momentum distribution $n_k=\\langle \\hat{\\psi }^\\dag _{k}(t)\\hat{\\psi }_k(t) \\rangle $ , one has to specify the initial state of the system, through the initial correlation functions $\\langle i\\hat{\\psi }^{(\\dag )}_{\\pm k,0}\\hat{\\psi }^{(\\dag )}_{\\pm k,0} \\rangle $ , $\\langle \\hat{\\psi }^{(\\dag )}_{\\pm k,0}\\hat{W}^{(\\dag )}_{j,\\pm k}(0) \\rangle $ , etc.",
"In order to simplify both the discussion and the numerical calculations, we will assume that at $t=0$ , the system is an ideal Bose gas ($n_0(t=0^-)=n_{0,i}$ and $g_i=0$ ) that does not interact with the bath, leading to the simplification that all cross-correlation functions such as $\\langle \\hat{\\psi }_{k,0} \\hat{W}_{j,k}(0) \\rangle $ vanish.",
"We will furthermore assume that $\\Gamma _k=\\Gamma $ is momentum independent.",
"For simplicity, the bath is assumed to be at zero temperature.",
"Then, the momentum distribution is given by $\\begin{split}n_k(t)=&-M_{k,3}(t,0)M_{k,2}(t,0)-\\int _0^t ds\\,\\int _0^t du\\,{\\cal D}_k(s-u) \\\\& \\Big [M_{k,1}(t,u)M_{k,4}(t,s)+M_{k,2}(t,u)M_{k,3}(t,s)\\Big ],\\end{split}$ where ${\\cal D}_k(s-u)=\\langle \\hat{D}^\\dag _k(s)\\hat{D}_k(u) \\rangle =\\langle \\hat{D}_k(s)\\hat{D}^\\dag _k(u) \\rangle $ is given by ${\\cal D}_k(s-u)=\\int _{\\omega <0} \\Sigma _2(k,\\omega ) e^{i\\omega (s-u)}$ .",
"In the limit of vanishing system-bath coupling, we obtain the standard Bogoliubov results [8]."
]
] | 1403.0141 |
[
[
"Coordinated Direct and Relay Schemes for Two-Hop Communication in VANETS"
],
[
"Abstract In order to accommodate increasing need and offer communication with high performance, both vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) communications are exploited.",
"The advantages of static nodes and vehicular nodes are combined to achieve an optimal routing scheme.",
"In this paper, we consider the communications between a static node and the vehicular nodes moving in an adjacent area of it.",
"The adjacent area is defined as the zone where a vehicular can communicate with the static node within maximum two hops.",
"We only consider single-hop and two-hop transmissions because these transmissions can be considered as building blocks to construct transmissions with a higher number of hops.",
"Different cases in which an uplink or a downlink for the two-hop user combined with an uplink or a downlink for the single-hop user correspond to different CDR schemes.",
"Using side information to intentionally cancel the interference, Network Coding (NC), CDR, overhearing and multi-way schemes aggregate communications flows in order to increase the performance of the network.",
"We apply the mentioned schemes to a V2I network and propose novel schemes to optimally arrange and combine the transmissions."
],
[
"Introduction",
"Vehicular ad hoc networks (VANETs) have attracted a great attention in research community due to their huge benefits in safety and entertaining applications [1].",
"In order to accommodate increasing need and offer communication with high performance, both vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) communications are exploited [2].",
"The advantages of static nodes and vehicular nodes are combined to achieve an optimal routing scheme [3].",
"Considering a two-tier network as such, a message from a vehicular source to its destination may be routed via a certain static node.",
"This static node may store the message until it has a good link to a vehicular node on the way to the destination [4].",
"This static node can also be connected to the backbone network through a backhaul link using an orthogonal channel from over which the message will be forwarded.",
"In another example, a certain message destined to a vehicular node which is nearby that static node can be routed from the source via this static node.",
"To a certain extent, a static node together with its surrounding vehicular nodes can be considered as a small cell.",
"In this paper, we consider the communications between a static node and the vehicular nodes moving in an adjacent area of it.",
"The adjacent area is defined as the zone, as shown in Fig.",
"REF , where a vehicular can communicate with the static node within maximum two hops.",
"We only consider single-hop and two-hop transmissions because these transmissions can be considered as building blocks to construct transmissions with a higher number of hops.",
"Assume at a certain time, we have only a downlink for node U$_1$ and an uplink for node U$_4$ .",
"This means that transmissions 1, 2, 3 and 8 are to be conducted.",
"A conventional scheme will conduct all of them in different time slots therefore they do not interfere with each other.",
"In an advanced scheme, two or more transmissions may be conducted simultaneously in order to use less time slots therefore increase the spectrum efficiency.",
"If we only consider transmissions 1, 2 and 8 and try to combine these flows, we have a Coordinated Direct and Relay (CDR) scheme.",
"This CDR scheme combined with transmission 3 can accommodate the required downlink and uplink for U$_1$ and U$_4$ respectively.",
"Different cases in which an uplink or a downlink for the two-hop user combined with an uplink or a downlink for the single-hop user correspond to different CDR schemes which are described in details in [5].",
"Figure: We consider vehicular users which can communicate with the SN in at most two hops.If at a certain time, there are an uplink and a downlink for two users using the same relay node (e.g.",
"U$_2$ and U$_3$ in Fig.",
"REF ) respectively.",
"The transmissions can be combined in an overhearing scheme [6], [7].",
"On the other hand, if there are two-hop users using different relay nodes: one has an uplink and one has a downlink (e.g.",
"U$_2$ and U$_5$ ), we have a multi-way scheme [8], [9], [10].",
"If we use the CDR, overhearing and multi-way schemes as described above to respective cases instead of the corresponding conventional schemes, several time slots are saved and a significant improvement is gained.",
"Using side information to intentionally cancel the interference, Network Coding (NC), CDR, overhearing and multi-way schemes aggregate communications flows in order to increase the performance of the network.",
"Let us regard these types of schemes as multi-flow schemes.",
"So far, multi-flow schemes have been proposed and analyzed in terms of spectrum efficiency [5] and diversity [11].",
"In those works, the nodes are assumed to be static in the whole scheme which may last for several time slots.",
"For a vehicular network, the positions of the nodes in different time slots are different.",
"This may make the performance of the network lower than that of a non-multi-flow based network.",
"In this paper, we apply the mentioned schemes to a V2I network and propose novel schemes to optimally arrange and combine the transmissions."
],
[
"System Model",
"We consider a static nodes (SN) and $n$ adjacent vehicular nodes U$_i$ , $i\\in \\lbrace 1, 2, ..., n\\rbrace $ .",
"We assume that all considered traffic from/to the vehicular nodes goes through the SN as an intermediate node on the way to the destination.",
"The SN therefore can be considered as a Base Station of a cell in cellular networks.",
"The vehicular nodes with distances to the SN smaller than $R$ can directly communicate with the SN while the vehicular nodes outside the circle with radius $R$ cannot directly communicate with the SN, due to the negligible magnitude of the channel between it and the SN, and have to rely on vehicular nodes inside the circle as relays.",
"Consider a map with horizontal and vertical streets equally separated with distance $d_o$ .",
"At time $t$ , vehicular node $i$ moves with velocity $v_i$ to the direction $d_i$ , where $d_i = 1, 2, 3, 4$ correspond to East, North, West, South.",
"At an intersection, it turns left, goes straight or turns right with probabilities $\\frac{1 - p_o}{2}$ , $p_o$ and $\\frac{1 - p_o}{2}$ respectively.",
"Assume that at the beginning of a scheme which lasts for a few time slots, all vehicular nodes in the map reports their expected positions in the whole scheme to the SN.",
"The positions of a vehicle in a few time slots can be calculated.",
"All transmissions are in one frequency with a normalized bandwidth of 1 Hz.",
"All nodes are single-antenna and half-duplexed.",
"Each of the complex channels is reciprocal, known at the receivers.",
"The receivers here include overhearing receivers as well as receivers of the second-hop transmission of an Amplify-and-Forward (AF) relaying communication.",
"Each vehicular node requests an uplink or a downlink transmission to the SN.",
"We assume that the data to/from each user is infinitely backlogged so that there are always data to transmit as in many works regarding downlink [12] and Two-way Relaying optimization [13] and scheduling [14], [15].",
"Thus the achievable rate for a user at a certain time is equal to the information theoretic capacity, i.e.",
"${\\rm C}(\\gamma ) = \\log _2(1 + \\gamma )$ , where $\\gamma $ denotes an instantaneous received SINR at the receiver.",
"We use the following notation, with a slight abuse: $x_i$ may denote a packet or a single symbol, and it will be clear from the context.",
"For example, the packet that the SN wants to send to a user is denoted by $x$ ; but if we want to express the signal received, then we use expressions of type $y = hx + z$ , where $y$ , $x$ and $z$ denote symbols (received, sent and noise respectively).",
"We assume perfect power control i.e.",
"the transmit power is selected so as the received power at the aimed receiver is at a fixed level of $P$ [16].",
"The principle is also applied to the case of AF or DF transmission.",
"The received power at the relay and at the final receiver is also $P$ .",
"The considered scheme is divided into $2n$ time slots so that each two of them can be assigned to each user.",
"If it is a relayed user, each time slot is used for each of two hop transmissions.",
"If it is a direct user, only one of the two slots assigned to it is used.",
"The number of total slots used is $n_T = 2n_1 + n_2$ , where $n_1$ and $n_2$ are the numbers of relayed and direct users respectively.",
"Note that $n_T$ depends on the way the transmissions of all users are scheduled because a user which is relayed in a certain slot can be direct in another time slot because it moves toward the SN."
],
[
"Single-Flow Scheme",
"The single-flow scheme is a conventional scheme in which, a direct communication requires one time slot while a relay communication requires two slots.",
"The purpose is to schedule the transmissions so that the performance is maximized considering a fixed consumed energy.",
"For a direct user, it is optimal when it is near the SN.",
"For a relayed user, it should be when the relay moves in a good direction e.g.",
"the relay moves from the relayed user toward the SN when there is an uplink.",
"Optimality of a user does not mean optimality of the whole network therefore we will find the optimal transmission scheduling by trying all the permutations of all transmissions to see which scheduling gives the highest performance.",
"Assigning time slots $2i - 1$ and $2i$ to user $i$ , we have totally $2n$ slots as in Fig.",
"REF a.",
"If user $i$ is a relayed user, slots $2i - 1$ and $2i$ are used for first-hop and second-hop transmissions respectively.",
"First, from the list of the slots in the first row, all permutations of the slots are listed.",
"There are totally $(2n)!$ permutations.",
"The permutations in which slot $2i$ appears before slot $2i - 1$ is invalid and therefore crossed out.",
"Fig.",
"REF b shows one of the valid permutation.",
"In a permutation, the two slots assigned to each user is checked.",
"If the distance from a user to the SN is smaller than $R$ ($d(t) \\le R$ ) in only one of the two slots, that slot is used for the user.",
"For example, Fig.",
"REF c shows that between slots 3 and 4 of user 2, slot 4 is worse and slot 3 is chosen.",
"For user 3, slot 4 is chosen rather than slot 6.",
"If in both slots $d(t) \\le R$ , the slot when it is closer to the SN is selected.",
"If in both slots $d(t) > R$ , the two slots are used for the first-hop and second-hop transmissions respectively as slots 1 and 2 of user 1, slots 7 and 8 of user 4.",
"The rate for each user is calculated.",
"The sum–rate for all users in one permutation is calculated.",
"The permutation with the highest sum–rate is selected."
],
[
"Multi-Flow Scheme",
"After the transmissions are fixed in the single-flow scheme, we still can combine transmissions to decrease the time slots used in order to increase the spectrum efficiency.",
"The combination is performed based on the optimal permutation of the single-flow scheme therefore the energy consumptions of the single-flow and multi-flow schemes are almost the same.",
"The advanced schemes we applied here include CDR, overhearing and multi-way schemes."
],
[
"CDR schemes",
"First, we look for a direct user and a relayed user which can be combined in two certain time slots.",
"In one slot, the direct transmission and a relayed transmission (can be the first or the second hop depending on which CDR scheme is used) are conducted simultaneously.",
"Let us call this one the CDR simultaneous slot as slots 3 and 7 in Fig.",
"REF d. In the other slot called CDR single slot, the other relayed transmissions is conducted as slot 8 in the figure.",
"Second, we look for a time slot which can host slots 3 and 7 by step by step move slots 3 and 7 along the row and check if it fits (the slot must be empty and the direct user is still direct in the slot) and calculate the performance metric in that case.",
"The case with the highest performance is selected."
],
[
"Description of Individual Schemes",
"In this section, we present single-flow (conventional) and multi-flow (CDR) individual schemes in two relaying modes: AF and DF.",
"An individual scheme is presented for two users: one relayed user (denoted as user 1) and one direct user (user 2).",
"The vehicular node acts as a relay is denoted as R as shown in Fig.",
"REF .",
"The whole composite scheme is a multiplexing of $\\frac{n}{2}$ individual schemes.",
"Figure: Single-flow (conventional) and multi-flow (CDR 1) individual schemes.The signals for user 1 and 2 are $x_1$ and $x_2$ respectively.",
"We denote the received signal, noise and channel in time slot 1, 2 and 3 as $y,z,h$ , $y^{\\prime },z^{\\prime },h^{\\prime }$ , $y^{\\prime \\prime },z^{\\prime \\prime },h^{\\prime \\prime }$ respectively.",
"Denote $_i\\mathrm {SNR}^j_k$ as the SNR when signal $x_k$ is decoded at node $i$ in decoding option $j$ ."
],
[
"Non-CDR Schemes",
"The scheme is conducted in three equal time slots.",
"In time slot 1, the SN transmits $x_1$ towards the relay.",
"The transmit power is set to $\\frac{P}{|h_1|^2}$ so as to compensate the SN-to-relay channel.",
"Because channel between the SN and user 1 has a negligible magnitude, user 1 does not receive the signal from the SN.",
"The relay receives $y_R = \\frac{h_1\\sqrt{P}}{|h_1|}x_1 + z_R,$ and scales with a power factor $\\alpha = \\frac{1}{P + \\sigma ^2}.$ In time slot 2, the relay transmits the amplified version of the received signal in time slot 1 towards user 1.",
"User 1 receives $\\begin{array}{lll}y_1^{\\prime } &=& \\frac{h_2^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}y_R + z_1^{\\prime }\\\\&=&\\frac{h_2^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}\\frac{h_1\\sqrt{P}}{|h_1|}x_1 + \\frac{h_2^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}z_R + z_1^{\\prime }\\end{array}$ and decode with SNR $_1\\mathrm {SNR}_1= \\frac{\\alpha P^2}{\\alpha P\\sigma ^2 + \\sigma ^2} = \\frac{\\gamma _o^2}{2\\gamma _o + 1}.$ In time slot 3, user 2 transmits $x_2$ to the SN.",
"The SN receives $y^{\\prime \\prime }_S = \\frac{h^{\\prime \\prime }_3\\sqrt{P}}{|h^{\\prime \\prime }_3|}x_1 + z^{\\prime \\prime }_S,$ and decodes with SNR $_S\\mathrm {SNR}_2= \\gamma _o.$ Finally, we have the rates for two users $\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_1\\mathrm {SNR}_1\\right)\\\\R_2 \\le {\\rm C}\\left(_S\\mathrm {SNR}_2\\right)\\end{array}\\right.$"
],
[
"CDR Schemes",
"The scheme is conducted in two equal time slots.",
"In time slot 1, the SN transmits $x_1$ towards the relay.",
"The transmit power is set to $\\frac{P}{|h_1|^2}$ so as to compensate the SN-to-relay channel.",
"Because channel between the SN and user 1 has a negligible magnitude, user 1 does not receive the signal from the SN.",
"The relay receives $y_R = \\frac{h_1\\sqrt{P}}{|h_1|}x_1 + z_R,$ and scales with a power factor $\\alpha = \\frac{1}{P + \\sigma ^2}.$ In time slot 2, the relay transmits the amplified version of the received signal in time slot 1 towards user 1 and user 2 transmits $x_2$ towards the SN simultaneously.",
"The transmit power of the first transmission is set to $\\frac{\\alpha P}{|h_2|^2}$ so that user 1 receives the signal with power of $P$ .",
"User 1 and the SN respectively receive $\\begin{array}{lll}y_1^{\\prime } &=& \\frac{h_2^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}y_R + \\frac{h_4^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_1^{\\prime }\\\\&=&\\frac{h_2^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}\\frac{h_1\\sqrt{P}}{|h_1|}x_1 + \\frac{h_2^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}z_R + \\frac{h_4^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_1^{\\prime },\\end{array}$ $\\begin{array}{lll}y_F^{\\prime } &=& \\frac{h_1^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}y_R + \\frac{h_3^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_F^{\\prime }\\\\&=&\\frac{h_1^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}\\frac{h_1\\sqrt{P}}{|h_1|}x_1 + \\frac{h_1^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}z_R + \\frac{h_3^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_F^{\\prime }.\\end{array}$ Because the SN has the information about $x_1$ and the channels, the contribution of $x_1$ in $y_F^{\\prime }$ is cancelled $\\begin{array}{lll}\\tilde{y}_S^{\\prime } = \\frac{h_1^{\\prime }\\sqrt{\\alpha P}}{|h_2^{\\prime }|}z_R + \\frac{h_3^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_F^{\\prime }.\\end{array}$ At user 1, there are two options to decode Option 1: User 1 decodes $x_1$ treating $x_2$ as noise with SNR, using MMSE [17], [18], $_1\\mathrm {SNR}^1_1= \\frac{\\alpha P^2}{\\frac{|h_4^{\\prime }|^2}{|h_3^{\\prime }|^2}P + \\alpha P\\sigma ^2 + \\sigma ^2} = \\frac{\\gamma _o^2}{\\left(\\frac{\\gamma _4^{\\prime }}{\\gamma _3^{\\prime }}\\gamma _o + 1\\right)(\\gamma _o + 1) + \\gamma _o}.$ Option 2: User 1 decodes $x_2$ treating $x_1$ as noise with SNR $_1\\mathrm {SNR}^2_2 = \\frac{\\frac{|h_4^{\\prime }|^2}{|h_3^{\\prime }|^2}P}{\\alpha P^2 + \\alpha P\\sigma ^2 + \\sigma ^2} = \\frac{\\gamma _4^{\\prime }\\gamma _o}{\\gamma _3^{\\prime }(\\gamma _o + 1)},$ cancels the contribution of $x_2$ in $\\tilde{y}_1$ and decodes $x_1$ with SNR $_1\\mathrm {SNR}^2_1 = \\frac{\\alpha P^2}{\\alpha P\\sigma ^2 + \\sigma ^2} = \\frac{\\gamma _o^2}{2\\gamma _o + 1}.$ This option corresponds to the case when the contribution of $x_2$ in $\\tilde{y}_1$ is higher than that of $x_1$ .",
"In the opposite case, option 1 is appropriate.",
"In both options, the SN decodes $x_2$ from $\\tilde{y}_S$ with $_S\\mathrm {SNR}_2 = \\frac{P}{\\frac{|h_1^{\\prime }|^2\\alpha P}{|h_2^{\\prime }|^2}\\sigma ^2 + \\sigma ^2} = \\frac{\\gamma _o}{\\frac{\\gamma _1^{\\prime }}{\\gamma _2^{\\prime }} + \\gamma _o + 1}.$ In summary, we have two options $\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_1\\mathrm {SNR}_1^1\\right)\\\\R_2 \\le {\\rm C}\\left(_S\\mathrm {SNR}_2\\right)\\end{array}\\right.\\mbox{or}\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_1\\mathrm {SNR}_1^2\\right)\\\\R_2 \\le {\\rm C}\\left(\\min \\left(_1\\mathrm {SNR}_2^2, _S\\mathrm {SNR}_2\\right)\\right)\\end{array}\\right.$"
],
[
"Non-CDR Schemes",
"The scheme is conducted in three equal time slots.",
"In time slot 1, the SN transmits $x_1$ towards the relay.",
"The transmit power is set to $\\frac{P}{|h_1|^2}$ so as to compensate the SN-to-relay channel.",
"Because channel between the SN and user 1 has a negligible magnitude, user 1 does not receive the signal from the SN.",
"The relay receives $y_R = \\frac{h_1\\sqrt{P}}{|h_1|}x_1 + z_R,$ and scales decodes $x_1$ with SNR $_R\\mathrm {SNR}_1 = \\frac{P}{\\sigma ^2} = \\gamma _o$ , transmits $x_1$ in time slot 2 towards user 1.",
"User 1 receives and decodes with SNR $_1\\mathrm {SNR}_1 = \\gamma _o$ .",
"In time slot 3, user 2 transmits $x_2$ to the SN.",
"The SN receives and decodes with SNR $_S\\mathrm {SNR}_1 = \\gamma _o$ .",
"Finally, we have the rates for two users $\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}(\\gamma _o)\\\\R_2 \\le {\\rm C}(\\gamma _o).\\end{array}\\right.$"
],
[
"CDR Schemes",
"In time slot 1, the relay receives $y_R = \\frac{h_1\\sqrt{P}}{|h_1|}x_1 + z_R,$ and decodes $x_1$ with SNR $_1\\mathrm {SNR}_1= \\gamma _o.$ In time slot 2, the relay transmits $x_1$ and user 2 transmits $x_2$ simultaneously.",
"User 1 and the SN respectively receive $y_1^{\\prime } = \\frac{h_2^{\\prime }\\sqrt{P}}{|h_2^{\\prime }|}x_1 + \\frac{h_4^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_1^{\\prime },$ $y_F^{\\prime } = \\frac{h_1^{\\prime }\\sqrt{P}}{|h_2^{\\prime }|}x_1 + \\frac{h_3^{\\prime }\\sqrt{P}}{|h_3^{\\prime }|}x_2 + z_1^{\\prime },$ Option 1: User 1 decodes $x_1$ treating $x_2$ as noise with SNR $_1\\mathrm {SNR}^1_1= \\frac{\\gamma _o}{\\frac{\\gamma _4^{\\prime }}{\\gamma _3^{\\prime }}\\gamma _o + 1}.$ Option 2: User 1 decodes $x_2$ treating $x_1$ as noise with SNR $_1\\mathrm {SNR}^2_2 = \\frac{\\gamma _4^{\\prime }\\gamma _o}{\\gamma _3^{\\prime }(\\gamma _o + 1)},$ cancels the contribution of $x_2$ in $\\tilde{y}_1$ and decodes $x_1$ with SNR $_1\\mathrm {SNR}^2_1 = \\gamma _o.$ In summary, we have two options $\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(\\min \\left(_1\\mathrm {SNR}_1^1, _R\\mathrm {SNR}_1\\right)\\right)\\\\R_2 \\le {\\rm C}\\left(_S\\mathrm {SNR}_2\\right)\\end{array}\\right.\\mbox{or}\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(\\min \\left(_1\\mathrm {SNR}_1^2, _R\\mathrm {SNR}_1\\right)\\right)\\\\R_2 \\le {\\rm C}\\left(\\min \\left(_1\\mathrm {SNR}_2^2, _S\\mathrm {SNR}_2\\right)\\right)\\end{array}\\right.$ Figure: Sum–rate versus P σ 2 \\frac{P}{\\sigma ^2}.Figure: Sum–rate versus go-straight ratio p o p_o (%).Monto Carlo simulation is conducted for a map with parallel horizontal and vertical streets equally separated with $d_o = 30$ m. Other parameters include maximum one-hop distance $R = 65$ m, $n = 4$ users.",
"All vehicular nodes move with the same velocity $v = 10m/$ time slot.",
"[Proof of Proposition]"
],
[
"$S_2$",
" Option 1: $_2\\mathrm {SNR}^1_1=\\frac{\\xi \\gamma _{11} + \\gamma _{12} + \\gamma _b}{\\xi \\gamma _{21} + \\gamma _{22} + \\xi }.$ $_2\\mathrm {SNR}^1_2=\\gamma _{21} + \\frac{\\gamma _{22}}{\\xi }.$ Option 2: $_2\\mathrm {SNR}^2_2=\\frac{\\xi \\gamma _{21} + \\gamma _{22} + \\gamma _b}{\\xi \\gamma _{11} + \\gamma _{12} + \\xi }$ $\\begin{array}{lll}\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(\\min \\left(_S\\mathrm {SNR}_1, _2\\mathrm {SNR}_1^1 \\right)\\right)\\\\R_2 \\le {\\rm C}\\left(_2\\mathrm {SNR}_2^1\\right)\\end{array}\\right.&\\mbox{or}&\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_S\\mathrm {SNR}_1\\right)\\\\R_2 \\le {\\rm C}\\left(_2\\mathrm {SNR}_2^2\\right)\\end{array}\\right.\\end{array}$"
],
[
"$S_3$",
" Option 1: $_2\\mathrm {SNR}^1_2=\\frac{\\xi \\gamma _{21} + \\gamma _{22} + \\gamma _b}{\\xi \\gamma _{11} + \\gamma _{12} + \\xi }$ Option 2: $_2\\mathrm {SNR}^2_1=\\frac{\\xi \\gamma _{11} + \\gamma _{12} + \\gamma _b}{\\xi \\gamma _{21} + \\gamma _{22} + \\xi }.$ $_2\\mathrm {SNR}^2_2=\\frac{\\gamma _{22}}{\\xi }.$ $\\begin{array}{lll}\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_1\\mathrm {SNR}_1\\right)\\\\R_2 \\le {\\rm C}\\left(_2\\mathrm {SNR}_2^1\\right)\\end{array}\\right.&\\mbox{or}&\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(\\min \\left(_1\\mathrm {SNR}_1, _2\\mathrm {SNR}_1^2 \\right)\\right)\\\\R_2 \\le {\\rm C}\\left(_2\\mathrm {SNR}_2^2\\right)\\end{array}\\right.\\end{array}$"
],
[
"$S_4$",
" Option 1: $_S\\mathrm {SNR}^1_1=\\frac{\\xi \\gamma _{11} + \\gamma _{12} + \\gamma _b}{\\xi \\gamma _{21} + \\gamma _{22} + \\xi }.$ $_S\\mathrm {SNR}^1_2=\\gamma _{21} + \\frac{\\gamma _{22}}{\\xi }.$ Option 2: $_S\\mathrm {SNR}^2_2=\\frac{\\xi \\gamma _{21} + \\gamma _{22} + \\gamma _b}{\\xi \\gamma _{11} + \\gamma _{12} + \\xi }$ $_S\\mathrm {SNR}^2_1=\\gamma _{11} + \\frac{\\gamma _{12}}{\\xi }.$ $\\begin{array}{lll}\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_S\\mathrm {SNR}_1^1\\right)\\\\R_2 \\le {\\rm C}\\left(_S\\mathrm {SNR}_2^1\\right)\\end{array}\\right.&\\mbox{or}&\\left\\lbrace \\begin{array}{l}R_1 \\le {\\rm C}\\left(_S\\mathrm {SNR}_1^2\\right)\\\\R_2 \\le {\\rm C}\\left(_S\\mathrm {SNR}_2^2\\right)\\end{array}\\right.\\end{array}$"
]
] | 1403.0173 |
[
[
"The spectral theorem for unitary operators based on the $S$-spectrum"
],
[
"Abstract The quaternionic spectral theorem has already been considered in the literature, see e.g.",
"[22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem.",
"In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum.",
"In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices.",
"The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus.",
"The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum.",
"The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem."
],
[
"Introduction",
"One of the main motivations to study spectral theory of linear operators in the quaternionic setting is due to the fact that Birkhoff and von Neumann, see [12], showed that there are essentially two possible settings in which to write the Schrödinger equation: one may use complex-valued functions or one may use quaternion-valued functions.",
"Since then, many efforts have been made by several authors, see [1], [20], [22], [26], to develop a quaternionic version of quantum mechanics.",
"Fundamental tools in this framework are the theory of quaternionic groups and semigroups on quaternionic Banach spaces which have been studied only recently in the papers [3], [14], [25] using the notion of $S$ -spectrum and of $S$ -resolvent operator as well as the spectral theorem, which is the main result of this paper.",
"To fully understand the aim of this work, we start by recalling some basic facts in complex spectral theory.",
"Let $A$ be a linear operator acting on a complex Banach space $X$ , and let $\\sigma (A)$ and $\\rho (A)$ be the spectrum and the resolvent sets of $A$ , respectively.",
"One of the most natural ways to associate to a linear operator $A$ the linear operator $f(A)$ is to use the Cauchy formula for holomorphic functions $f(A)=\\frac{1}{2\\pi i}\\int _{\\partial \\Omega } (\\lambda I-A)^{-1} f(\\lambda ) d\\lambda ,$ where $\\partial \\Omega $ is a smooth closed curve that belongs to the resolvent set of $A$ and $f$ a holomorphic function on an open set $\\Omega $ which contains the spectrum of $A$ .",
"This holomorphic functional calculus is known as Riesz-Dunford functional calculus, see [18].",
"To any linear operator $A$ , it is possible to associate the notion of spectral measures, which can be written explicitly using the Riesz-projectors, as described below.",
"A subset of $\\sigma (A)$ that is open and closed in the relative topology of $\\sigma (A)$ is called a spectral set.",
"The spectral sets form a Boolean algebra and with each spectral set $\\sigma $ one can associate the projector operator $P(\\sigma )=\\frac{1}{2\\pi i}\\int _{C_\\sigma } (\\lambda I-A)^{-1}d\\lambda $ where $C_\\sigma $ is a smooth closed curve belonging to the resolvent set $\\rho (A)$ , such that $C_\\sigma $ surrounds $\\sigma $ but no other points of the spectrum.",
"A spectral measure in the complex Banach space $X$ is then a homomorphic map of the Boolean algebra of the sets into the Boolean algebra of projection operators in $X$ which has the additional property that it maps the unit in its domain into the identity operator in its range.",
"As is well known, the spectrum $\\sigma (A)$ appearing in the definition of the Riesz-projectors $P(\\sigma )$ is the same spectrum on which is supported the spectral measure $E(\\lambda )$ appearing in the spectral theorem for normal linear operators in a complex Hilbert space.",
"Precisely, for a normal linear operator $B$ on a Hilbert space, given a continuous function $g$ on the spectrum $\\sigma (B)$ , we have $g(B)=\\int _{\\sigma (B)}g(\\lambda ) dE(\\lambda ).$ In the quaternionic setting, before the introduction of the $S$ -spectrum, see [17], two spectral problems were considered.",
"We discuss the case of a right linear quaternionic operator (the case of a left linear operator being similar) $T:\\mathcal {V}\\rightarrow \\mathcal {V}$ acting on a quaternionic two sided Banach space $\\mathcal {V}$ , that is $T(w_1\\alpha +w_2\\beta )=T(w_1)\\alpha +T(w_2)\\beta $ , for $\\alpha ,\\beta \\in \\mathbb {H}$ and $w_1$ , $w_2\\in \\mathcal {V}$ .",
"The symbol $\\mathcal {B}^R(\\mathcal {V})$ denotes the Banach space of bounded right linear operators.",
"The left spectrum $\\sigma _L(T)$ of $T$ is related to the resolvent operator $(s\\mathcal {I}-T)^{-1}$ that is $\\sigma _L(T)=\\lbrace s\\in \\mathbb {H}\\ \\ :\\ \\ s\\mathcal {I}-T\\ \\ \\ {\\rm is\\ not\\ invertible\\ in\\ }\\mathcal {B}^R(\\mathcal {V}) \\rbrace ,$ where the notation $s\\mathcal {I}$ in $\\mathcal {B}^R(\\mathcal {V})$ means that $(s\\mathcal {I} )(v)=sv$ .",
"The right spectrum $\\sigma _R(T)$ of $T$ is associated with the right eigenvalue problem, i.e.",
"the search for nonzero vectors satisfying $T(v)=vs$ .",
"It is important to note that if $s$ is an eigenvalue, then all quaternions belonging to the sphere $r^{-1}s r$ , $r\\in \\mathbb {H}\\setminus \\lbrace 0\\rbrace $ , are also eigenvalues.",
"But observe that the operator $ \\mathcal {I}s -T$ associated to the right eigenvalue problem is not linear, so it is not clear what is the resolvent operator to be considered.",
"A natural notion of spectrum that arises in the definition of the quaternionic functional calculus is the one of $S$ -spectrum.",
"In the case of matrices, the $S$ -spectrum coincides with the set of right eigenvalues; in the general case of a linear operator, the point $S$ -spectrum coincides with the set of right eigenvalues.",
"In the literature there are several papers on the quaternionic spectral theorem, see e.g.",
"[22], [32], however the notion of spectrum in use is not made clear.",
"Recently, there has been a resurgence of interest in this topic, see [24], where the authors prove the spectral theorem, based on the $S$ -spectrum, for compact normal operators on a quaternionic Hilbert space.",
"In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$ -spectrum, which is then realized to be the correct notion of spectrum for the quaternionic spectral theory of unitary operators.",
"The $S$ -spectrum, see [17], is defined as $\\sigma _S(T)=\\lbrace s\\in \\mathbb {H}\\ \\ :\\ \\ T^2-2 {\\rm Re}(s)T+|s|^2\\mathcal {I}\\ \\ \\ {\\rm is\\ not\\ invertible}\\rbrace ,$ while the $S$ -resolvent set is $\\rho _S(T):= \\mathbb {H}\\setminus \\sigma _S(T)$ where $s=s_0+s_1i+s_2j+s_3k$ is a quaternion, $i$ , $j$ and $k$ are the imaginary units of the quaternion $s$ , ${\\rm Re}(s)=s_0$ is the real part and the norm $|s|$ is such that $|s|^2=s_0^2+s_1^2+s_2^2+s_3^2$ .",
"Due to the noncommutativity of the quaternions, there are two resolvent operators associated with a quaternionic linear operator: the left and the right $S$ -resolvent operators which are defined as $S_L^{-1}(s,T):=-(T^2-2{\\rm Re}(s) T+|s|^2\\mathcal {I})^{-1}(T-\\overline{s}\\mathcal {I}),\\ \\ \\ s \\in \\rho _S(T)$ and $S_R^{-1}(s,T):=-(T-\\overline{s}\\mathcal {I})(T^2-2{\\rm Re}(s) T+|s|^2\\mathcal {I})^{-1},\\ \\ \\ s \\in \\rho _S(T),$ respectively.",
"Using the notion of $S$ -spectrum and the notion of slice hyperholomorphic functions, see Section 4, we can define the quaternionic functional calculus, see [15], [16], [17].",
"We point out that the $S$ -resolvent operators are also used in Schur analysis in the realization of Schur functions in the slice hyperholomorphic setting see [6], [7], [8] and [2], [10] for the classical case.",
"To set the framework in which we will work, we give some preliminaries.",
"Consider the complex plane $\\mathbb {C}_I:=\\mathbb {R}+I\\mathbb {R}$ , for $I\\in \\mathbb {S}$ , where $\\mathbb {S}$ is the unit sphere of purely imaginary quaternions.",
"Observe that $\\mathbb {C}_I$ can be identified with a complex plane since $I^2=-1$ for every $I\\in \\mathbb {S}$ .",
"Let $\\Omega \\subset \\mathbb {H}$ be a suitable domain that contains the $S$ -spectrum of $T$ .",
"We define the quaternionic functional calculus for left slice hyperholomorphic functions $f:\\Omega \\rightarrow \\mathbb {H}$ as $f(T)={{1}\\over {2\\pi }} \\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)} S_L^{-1} (s,T)\\ ds_I \\ f(s),$ where $ds_I=- ds I$ ; for right slice hyperholomorphic functions, we define $f(T)={{1}\\over {2\\pi }} \\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)} \\ f(s)\\ ds_I \\ S_R^{-1} (s,T).$ These definitions are well posed since the integrals depend neither on the open set $\\Omega $ nor on the complex plane $\\mathbb {C}_I$ .",
"Using a similar idea, we define the projection operators which will provide the link between the spectral theorem and the $S$ -spectrum.",
"Our proofs will make use of a quaternionic version of Herglotz's theorem proved in the recent paper [5].",
"This theorem will be the starting point to prove the quaternionic spectral theorem for unitary operators, in analogy with the classical case.",
"We have proved that if $U$ is a unitary operator acting on a quaternionic Hilbert space $\\mathcal {H}$ , then, for $x$ , $y\\in \\mathcal {H}$ , there exists a spectral measure $E$ defined on the Borel sets of $[0,2\\pi )$ such that for every slice continuous function $f\\in \\mathcal {S}(\\sigma _S(U))$ , we have $\\langle f(U) x, y \\rangle = \\int _0^{2\\pi } f(e^{I t}) \\langle dE(t)x, y \\rangle , \\quad \\quad x,y, \\in \\mathcal {H}.$ Moreover, for $t$ belonging to the Borel sets of $[0,2\\pi )$ , the measures $\\nu _{x,y}(t) = \\langle E(t)x, y \\rangle , \\quad \\quad x,y \\in \\mathcal {H},$ are related to the $S$ -spectrum of $U$ by the quaternionic Riesz projectors through the relation $\\mathcal {P}(\\sigma ^0_S(U))=E(t_1)-E(t_0),$ where $\\sigma ^0_S(U)$ is the spectral set in the unit circle in $\\mathbb {C}_I$ delimited by the angles $t_0$ , $t_1$ .",
"The plan of the paper is the following.",
"In Section 2, we introduce the quaternionic Riesz projectors.",
"Section 3 contains the proof of the main result of the paper, namely the spectral theorem for the unitary operatrs.",
"In Section 4, we discuss the relation with the $S$ -spectrum."
],
[
"Quaternionic Riesz projectors",
"In the following we denote by $\\mathcal {B}(\\mathcal {V})$ the space of bounded quaternionic linear operators on the left or on the right, since the results of this section hold in both cases.",
"The classical Riesz projectors are a powerful tool in spectral analysis and the study of such projectors is based on the resolvent equation.",
"Recently, in the paper [4], it has been shown that there exists a $S$ -resolvent equation but in the quaternionic setting it involves both the $S$ -resolvent operators.",
"Precisely we have: Theorem 2.1 (The $S$ -resolvent equation) Let $T\\in \\mathcal {B}(\\mathcal {V})$ and let $s$ and $p\\in \\rho _S(T)$ .",
"Then we have $\\small S_R^{-1}(s,T)S_L^{-1}(p,T)=((S_R^{-1}(s,T)-S_L^{-1}(p,T))p-\\overline{s}(S_R^{-1}(s,T)-S_L^{-1}(p,T)))(p^2-2s_0p+|s|^2)^{-1},$ but also $\\small S_R^{-1}(s,T)S_L^{-1}(p,T)=(s^2-2p_0s+|p|^2)^{-1}(s(S_R^{-1}(s,T)-S_L^{-1}(p,T))-(S_R^{-1}(s,T)-S_L^{-1}(p,T))\\overline{p} ).$ The quaternionic functional calculus is defined on the class of slice hyperholomorphic functions $f: \\Omega \\subseteq \\mathbb {H} \\rightarrow \\mathbb {H}$ .",
"Such functions have a Cauchy formula, that works on specific domains which are called axially symmetric slice domain.",
"On this Cauchy formula is based the quaternionic functional calculus.",
"If we consider an element $I$ in the unit sphere of purely imaginary quaternions $\\mathbb {S}=\\lbrace q=ix_1+jx_2+kx_3\\ {\\rm such \\ that}\\ x_1^2+x_2^2+x_3^2=1\\rbrace $ then $I^2=-1$ , and for this reason the elements of $\\mathbb {S}$ are also called imaginary units.",
"Note that $\\mathbb {S}$ is a 2-dimensional sphere in $\\mathbb {R}^4$ .",
"Given a nonreal quaternion $p=x_0+{\\rm Im} (p)=x_0+I |{\\rm Im} (p)|$ , $I={\\rm Im} (p)/|{\\rm Im} (p)|\\in \\mathbb {S}$ , we can associate to it the 2-dimensional sphere defined by $[p]=\\lbrace x_0+I|{\\rm Im} (p)|\\ :\\ I\\in \\mathbb {S}\\rbrace .$ For any fixed $I\\in \\mathbb {S}$ , the set $\\mathbb {C}_I=\\lbrace u+Iv\\ :\\ u,v\\in \\mathbb {R}\\rbrace $ can be identified with the complex plane $\\mathbb {C}$ .",
"Definition 2.2 (Axially symmetric slice domain) Let $\\Omega $ be a domain in $\\mathbb {H}$ .",
"We say that $\\Omega $ is a slice domain (s-domain for short) if $\\Omega \\cap \\mathbb {R}$ is non empty and if $\\Omega \\cap \\mathbb {C}_I$ is a domain in $\\mathbb {C}_I$ for all $I \\in \\mathbb {S}$ .",
"We say that $\\Omega $ is axially symmetric if, for all $q \\in \\Omega $ , the 2-sphere $[q]$ is contained in $\\Omega $ .",
"Definition 2.3 An axially symmetric set $\\sigma \\subseteq \\sigma _S(T)$ which is both open and closed in $\\sigma _S(T)$ in its relative topology, is called a S-spectral set.",
"For sake of simplicity we will call it a spectral set.",
"The definition of a S-spectral set is suggested by the symmetry properties of the $S$ -spectrum.",
"In fact, if $p\\in \\sigma _S(T)$ , then all of the elements of the 2-sphere $[p]$ are contained in $\\sigma _S(T)$ .",
"Definition 2.4 Let $T$ be a quaternionic linear operator on a quaternionic Banach space $\\mathcal {V}$ .",
"Denote by $\\Omega _\\sigma $ an axially symmetric s-domain that contains the spectral set $\\sigma $ but not any other points of the $S$ -spectrum.",
"Suppose that the Jordan curves $\\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I)$ belong to the $S$ -resolvent set $\\rho _S(T)$ , for every $I\\in \\mathbb {S}$ .",
"We define the family $\\mathcal {P}(\\sigma )$ of quaternionic operators, depending on the spectral sets $\\sigma $ , as $\\mathcal {P}(\\sigma )=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I)}S_L^{-1}(s,T)ds_I.$ The operators $\\mathcal {P}(\\sigma )$ are called (quaternionic) Riesz projectors.",
"Remark 2.5 The definition of $\\mathcal {P}(\\sigma )$ can be given using the right $S$ -resolvent operator $S_R^{-1}(s,T)$ , that is $\\mathcal {P}(\\sigma )=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I)}ds_IS_R^{-1}(s,T).$ Using the left $S$ -resolvent operator we define the Riesz projectors associated with the $S$ -spectrum.",
"In [4] we proved that $\\mathcal {P}(\\sigma )$ is a projector and that it commutes with $T$ .",
"The following lemma will be useful in the sequel.",
"Lemma 2.6 Let $B\\in \\mathcal {B}(\\mathcal {V})$ and let $\\Omega $ be an axially symmetric s-domain.",
"If $p\\in \\Omega $ , then $\\frac{1}{2\\pi }\\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)}ds_I(\\overline{s}B-Bp)(p^2-2s_0p+|s|^2)^{-1}=B.$ Moreover, if $s\\in \\Omega $ , then $\\frac{1}{2\\pi }\\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)}(\\overline{s}B-Bp)(p^2-2s_0p+|s|^2)^{-1}dp_I=-B.$ It follows the same lines of the proof of Lemma 3.18 in [4].",
"Theorem 2.7 Let $T$ be a quaternionic linear operator.",
"Then the family of operators $\\mathcal {P}(\\sigma )$ has the following properties: (i) $(\\mathcal {P}(\\sigma ))^2=\\mathcal {P}(\\sigma ){\\rm ;}$ (ii) $T\\mathcal {P}(\\sigma )=\\mathcal {P}(\\sigma )T{\\rm ;}$ (iii) $\\mathcal {P}(\\sigma _S(T))=\\mathcal {I}{\\rm ;}$ (iv) $\\mathcal {P}(\\emptyset )=0{\\rm ;}$ (v) $\\mathcal {P}(\\sigma \\cup \\delta )=\\mathcal {P}(\\sigma )+\\mathcal {P}(\\delta ){\\rm ;} \\ \\ \\ \\sigma \\cap \\delta =\\emptyset ,$ (vi) $\\mathcal {P}(\\sigma \\cap \\delta )=\\mathcal {P}(\\sigma )\\mathcal {P}(\\delta ).$ Properties (i) and (ii) are proved in Theorem 3.19 in [4].",
"Property (iii) follows from the quaternionic functional calculus since $T^m=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)}S_L^{-1}(s,T)ds_I \\; s^m, \\ \\ \\ m\\in \\mathbb {N}_0$ for $\\sigma _S(T) \\subset \\Omega $ , which for $m=0$ gives $\\mathcal {I}=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)}S_L^{-1}(s,T)ds_I.$ Property (iv) is a consequence of the functional calculus as well.",
"Property (v) follows from $\\begin{split}\\mathcal {P}(\\sigma \\cup \\delta )&=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _{\\sigma \\cup \\delta }\\cap \\mathbb {C}_I)}S_L^{-1}(s,T)ds_I\\\\&=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I)}S_L^{-1}(s,T)ds_I+\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _\\delta \\cap \\mathbb {C}_I)}S_L^{-1}(s,T)ds_I\\\\&=\\mathcal {P}(\\sigma )+\\mathcal {P}(\\delta ).\\end{split}$ To prove (vi), assume that $\\sigma \\cap \\delta \\ne \\emptyset $ and consider $\\begin{split}\\mathcal {P}(\\sigma )\\mathcal {P}(\\delta )&=\\frac{1}{(2\\pi )^2 }\\int _{\\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I)}ds_IS_R^{-1}(s,T)\\int _{\\partial (\\Omega _\\delta \\cap \\mathbb {C}_I)}S_L^{-1}(p,T)dp_I\\\\&=\\frac{1}{(2\\pi )^2 }\\int _{ \\partial ( \\Omega _\\sigma \\cap \\mathbb {C}_I) } ds_I \\int _{ \\partial (\\Omega _\\delta \\cap \\mathbb {C}_I) }[S_R^{-1}(s,T)-S_L^{-1}(p,T)]p(p^2-2s_0p+|s|^2)^{-1}dp_I\\\\&-\\frac{1}{(2\\pi )^2 }\\int _{ \\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I) } ds_I\\int _{ \\partial ( \\Omega _\\delta \\cap \\mathbb {C}_I) }\\overline{s}[S_R^{-1}(s,T)-S_L^{-1}(p,T)](p^2-2s_0p+|s|^2)^{-1}dp_I,\\end{split}$ where we have used the $S$ -resolvent equation (see Theorem REF ).",
"We rewrite the above relation as $\\begin{split}\\mathcal {P}(\\sigma )\\mathcal {P}(\\delta ) &=-\\frac{1}{(2\\pi )^2 }\\int _{ \\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I) } ds_I \\int _{ \\partial (\\Omega _\\delta \\cap \\mathbb {C}_I) }[\\overline{s}S_R^{-1}(s,T)-S_R^{-1}(s,T)p](p^2-2s_0p+|s|^2)^{-1}dp_I\\\\&+\\frac{1}{(2\\pi )^2 }\\int _{ \\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I) } ds_I\\int _{ \\partial ( \\Omega _\\delta \\cap \\mathbb {C}_I) }[\\overline{s}S_L^{-1}(p,T)-S_L^{-1}(p,T)p](p^2-2s_0p+|s|^2)^{-1}dp_I\\\\& := \\mathcal {J}_1+\\mathcal {J}_2.\\end{split}$ Now thanks to Lemma REF and Remark REF we have $\\begin{split}\\mathcal {J}_1&=-\\frac{1}{(2\\pi )^2 }\\int _{ \\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I) } ds_I \\int _{ \\partial ( \\Omega _\\delta \\cap \\mathbb {C}_I) }[\\overline{s}S_R^{-1}(s,T)-S_R^{-1}(s,T)p](p^2-2s_0p+|s|^2)^{-1}dp_I\\\\&=\\frac{1}{2\\pi }\\int _{ \\partial (\\Omega _\\sigma \\cap \\mathbb {C}_I) } ds_IS_R^{-1}(s,T),\\ \\ {\\rm for}\\ \\ \\ s\\in \\Omega _\\delta \\cap \\mathbb {C}_I\\\\&=\\frac{1}{2\\pi }\\int _{ \\partial ( \\Omega _\\sigma \\cap \\mathbb {C}_I) } S_L^{-1}(s,T) ds_I,\\ \\ {\\rm for}\\ \\ \\ s\\in \\Omega _\\delta \\cap \\mathbb {C}_I\\end{split}$ while $\\mathcal {J}_1=0$ when $s\\notin \\Omega _\\delta \\cap \\mathbb {C}_I$ since $\\int _{ \\partial ( \\Omega _\\delta \\cap \\mathbb {C}_I) }[\\overline{s}S_R^{-1}(s,T)-S_R^{-1}(s,T)p](p^2-2s_0p+|s|^2)^{-1}dp_I=0.$ Similarly, one can show that $\\begin{split}\\mathcal {J}_2=\\frac{1}{2\\pi }\\int _{ \\partial ( \\Omega _\\delta \\cap \\mathbb {C}_I) }S_L^{-1}(p,T)dp_I,\\ \\ \\ \\ {\\rm for}\\ \\ \\ p\\in \\Omega _\\sigma \\cap \\mathbb {C}_I\\end{split}$ while $\\mathcal {J}_2=0$ when $p\\notin \\Omega _\\sigma \\cap \\mathbb {C}_I$ .",
"The integrals $\\mathcal {J}_1$ , $\\mathcal {J}_2$ are either both zero or both nonzero, so with a change of variable we get $\\mathcal {J}_1+\\mathcal {J}_2=\\frac{1}{2\\pi }\\int _{ \\partial ( \\Omega _{\\sigma \\cap \\delta } \\cap \\mathbb {C}_I) }S_L^{-1}(r ,T) dr _I=\\mathcal {P}(\\sigma \\cap \\delta ).$ From now on we will always work in quaternionic Hilbert spaces, so we will recall some definitions.",
"Let $\\mathcal {H}$ be a right linear quaternionic Hilbert space with an $\\mathbb {H}$ -valued inner product $\\langle \\cdot , \\cdot \\rangle $ which satisfies $\\langle x, y \\rangle = \\overline{ \\langle y , x \\rangle }{\\rm ;}$ $\\langle x, x \\rangle \\ge 0 \\quad {\\rm and} \\quad \\Vert x \\Vert ^2 := \\langle x,x \\rangle = 0 \\Longleftrightarrow x = 0{\\rm ;}$ $\\langle x \\alpha + y \\beta , z \\rangle = \\langle x, z \\rangle \\alpha + \\langle y, z \\rangle \\beta {\\rm ;}$ $\\langle x, y \\alpha + z \\beta \\rangle = \\overline{\\alpha } \\langle x, z \\rangle + \\overline{\\beta } \\langle x, z \\rangle ,$ for all $x,y,z \\in \\mathcal {H}$ and $\\alpha ,\\beta \\in \\mathbb {H}$ .",
"Any right linear quaternionic Hilbert space can be made also a left linear space, by fixing an Hilbert basis, see [23], Section 3.1.",
"We call an operator $A$ from the right quaternionic Hilbert space $\\mathcal {H}_1$ , with inner product $\\langle \\cdot , \\cdot \\rangle _1$ , to another right quaternionic Hilbert space $\\mathcal {H}_2$ , with inner product $\\langle \\cdot , \\cdot \\rangle _2$ , right linear if $A(x\\alpha + y \\beta ) = (Ax)\\alpha + (Ay)\\beta ,$ for all $x,y$ in the domain of $A$ and $\\alpha ,\\beta \\in \\mathbb {H}$ .",
"We call an operator $A$ bounded if $\\Vert A \\Vert := \\sup _{ \\Vert x \\Vert \\le 1 } \\Vert A x \\Vert < \\infty .$ Corresponding to any bounded right linear operator $A: \\mathcal {H}_1 \\rightarrow \\mathcal {H}_2$ there exists a unique bounded right linear operator $A^*: \\mathcal {H}_2 \\rightarrow \\mathcal {H}_1$ such that $\\langle A x, y \\rangle _2 = \\langle x, A^* y \\rangle _1,$ and $\\Vert A \\Vert = \\Vert A^* \\Vert $ (see Proposition 6.2 in [11]).",
"Let $\\mathcal {H}$ be a right quaternionic Hilbert space with inner product $\\langle \\cdot , \\cdot \\rangle $ .",
"We call a right linear operator $U: \\mathcal {H}\\rightarrow \\mathcal {H}$ unitary if $\\langle U^* U x, y \\rangle = \\langle x,y \\rangle ,\\quad \\quad {\\rm for}\\; {\\rm all} \\; x,y \\in \\mathcal {H},$ or, equivalently, $U^{-1} = U^*$ .",
"Theorem 2.8 Let $\\mathcal {H}$ be a right linear quaternionic Hilbert space and let $U$ be a unitary operator on $\\mathcal {H}$ .",
"Then the $S$ -spectrum of $U$ belongs to the unit sphere of the quaternions.",
"See Theorem 4.8 in [23].",
"By $ \\mathbf {B}([0,2\\pi ))$ we denote the Borel sets in $[0,2\\pi )$ .",
"Lemma 2.9 Let $x,y \\in \\mathcal {H}$ and let $\\mathcal {P}(\\sigma )$ be the projector associated with the unitary operator $U$ .",
"We define $m_{x,y}(\\sigma ):= \\langle \\mathcal {P}(\\sigma )x,y \\rangle ,\\ \\ \\ \\ x, \\, y\\in \\mathcal {H},\\ \\ \\ \\sigma \\in \\mathbf {B}([0,2\\pi )).$ Then the $\\mathbb {H}$ -valued measures $m_{x,y}$ defined on $\\mathbf {B}([0, 2\\pi ))$ enjoy the following properties (i) $m_{x \\alpha +y \\beta ,z} = m_{x,z}\\alpha + m_{y,z} \\beta $ ; (ii) $m_{x, y\\alpha + z \\beta } = \\overline{\\alpha } m_{x,y} + \\overline{\\beta } m_{x,z}$ ; (iii) $m_{x,y}([0, 2\\pi )) \\le \\Vert x \\Vert \\Vert y \\Vert $ , where $x,y,z \\in \\mathcal {H}$ and $\\alpha ,\\beta \\in \\mathbb {H}$ .",
"Properties (i) and (ii) follow from the properties of the quaternionic scalar product, while (iii) follows from Property (iii) in Theorem REF and the Cauchy-Schwarz inequality (see Lemma 5.6 in [11])."
],
[
"The spectral theorem for quaternionic unitary operators",
"We recall some classical results and also their quaternionic analogs which will be useful to prove a spectral theorem for quaternionic unitary operators.",
"Theorem 3.1 (Herglotz's theorem) The function $n \\mapsto r(n)$ from $\\mathbb {Z}$ into $\\mathbb {C}^{s \\times s}$ is positive definite if and only if there exists a unique $\\mathbb {C}^{s \\times s}$ -valued measure $\\mu $ on $[0, 2\\pi )$ such that $r(n) = \\int _0^{2\\pi } e^{i n t} d\\mu (t), \\quad n \\in \\mathbb {Z}.$ Given $P \\in \\mathbb {H}^{s \\times s}$ , there exist unique $P_1, P_2 \\in \\mathbb {C}^{s \\times s}$ such that $P = P_1 + P_2 j$ .",
"Recall the bijective homomorphism $\\chi : \\mathbb {H}^{s \\times s} \\rightarrow \\mathbb {C}^{2s \\times 2s}$ given by $\\chi \\hspace{1.42262pt} P = \\begin{pmatrix} P_1 & P_2 \\\\ - \\overline{P}_2 & \\overline{P}_1 \\end{pmatrix} \\quad {\\rm where} \\; P = P_1 + P_2 j ,$ Definition 3.2 Given a $\\mathbb {H}^{s \\times s}$ -valued measure $\\nu $ , we may always write $\\nu = \\nu _1 + \\nu _2 j$ , where $\\nu _1$ and $\\nu _2$ are uniquely determined $\\mathbb {C}^{s \\times s}$ -valued measures.",
"We call a measure $d\\nu $ on $[0, 2\\pi )$ $q$ -positive if the $\\mathbb {C}^{2s \\times 2s}$ -valued measure $\\mu = \\begin{pmatrix} \\nu _1 & \\nu _2 \\\\ \\nu ^*_2 & \\nu _3 \\end{pmatrix}, \\quad {\\rm where}\\; \\nu _3(t) = \\nu _1(2\\pi - t),\\;\\; t \\in [0, 2\\pi )$ is positive and, in addition, $\\nu _2(t) = -\\nu _2(2\\pi -t)^T, \\quad t \\in [0, 2\\pi ).$ Remark 3.3 If $\\nu $ is $q$ -positive, then $\\nu = \\nu _1 + \\nu _2 j$ , where $\\nu _1$ is a uniquely determined positive $\\mathbb {C}^{s \\times s}$ -valued measure and $\\nu _2$ is a uniquely determined $\\mathbb {C}^{s \\times s}$ -valued measure.",
"Remark 3.4 If $r = (r(n))_{n \\in \\mathbb {Z}}$ is a $\\mathbb {H}^{s \\times s}$ -valued sequence on $\\mathbb {Z}$ such that $r(n) = \\int _0^{2\\pi } e^{i n t} d\\nu (t),$ where $d\\nu $ is a $q$ -positive measure, then $r$ is Hermitian, i.e., $r(-n)^* = r(n)$ .",
"The following result has been proved in [5].",
"Theorem 3.5 (Herglotz's theorem for the quaternions) The function $n \\mapsto r(n)$ from $\\mathbb {Z}$ into $\\mathbb {H}^{s \\times s}$ is positive definite if and only if there exists a unique $q$ -positive measure $\\nu $ on $[0, 2\\pi )$ such that $r(n) = \\int _0^{2\\pi } e^{i n t} d\\nu (t), \\quad n \\in \\mathbb {Z}.$ Remark 3.6 For every $I \\in \\mathbb {S}$ , there exists $J \\in \\mathbb {S}$ so that $I J = -J I$ .",
"Thus, $\\mathbb {H}= \\mathbb {C}_I \\oplus \\mathbb {C}_I J$ and we may rewrite (REF ) as $r(n) = \\int _0^{2\\pi } e^{I n t} d\\nu (t), \\quad n \\in \\mathbb {Z},$ where $\\nu = \\nu _1 + \\nu _2 J$ is a $q$ -positive measure (in the sense that $\\mu = \\begin{pmatrix} \\nu _1 & \\nu _2 \\\\ \\nu _2^* & \\nu _3 \\end{pmatrix}$ is positive).",
"Here $\\nu _3(t) = \\nu _1(2\\pi - t)$ .",
"For our purpose the scalar case will be important.",
"Lemma 3.7 If $U$ is a unitary operator on $\\mathcal {H}$ , then $r_x = (r_x(n))_{n \\in \\mathbb {Z}}$ , where $r_x(n) = \\langle U^n x, x \\rangle $ for $x \\in \\mathcal {H}$ , is an $\\mathbb {H}$ -valued positive definite sequence.",
"If $\\lbrace p_0, \\ldots , p_N \\rbrace \\subset \\mathbb {H}$ , then $\\sum _{m,n=0}^N \\bar{p}_m r_x(n - m) p_n =& \\; \\sum _{m,n=0}^N \\bar{p}_m \\langle U^{n-m} x, x \\rangle p_n \\\\=& \\; \\sum _{m,n=0}^N \\langle U^{n-m} x p_n, x p_m \\rangle \\\\=& \\; \\sum _{m,n=0}^N \\langle U^n x p_n, U^m x p_m \\rangle \\\\=& \\; \\langle \\sum _{n=0}^N U^n x p_n, \\sum _{m=0}^N U^m x p_m \\rangle \\\\=& \\; \\left\\Vert \\sum _{n=0}^N U^n x p_n \\right\\Vert ^2 \\ge 0.$ Thus, $r_x$ is a positive definite $\\mathbb {H}$ -valued sequence.",
"Let $r_x$ be as in Lemma REF .",
"It follows from Theorem REF that there exists a unique $q$ -positive measure $d\\nu _x$ such that $r_x(n) = \\langle U^n x, x \\rangle = \\int _0^{2\\pi } e^{i n t } d\\nu _x(t), \\quad \\quad n \\in \\mathbb {Z}.$ One can check that $4 \\langle U^n x, y \\rangle =& \\; \\langle U^n (x+y), x+y \\rangle - \\langle U^n (x-y), x-y \\rangle + i \\langle U^n(x+yi), x+yi \\rangle \\nonumber \\\\& \\; \\; - i \\langle U^n (x - y i), x - yi \\rangle + i \\langle U^n (x - y j), x - y j \\rangle k - i \\langle U^n (x + y j) , x+ y j \\rangle k \\nonumber \\\\& \\; \\; + \\langle U^n(x+yk), x+yk \\rangle k - \\langle U^n(x-yk), x-yk \\rangle k $ and hence if we let $4 \\nu _{x,y} :=& \\; \\nu _{x+y} - \\nu _{x-y} + i \\nu _{x+yi} - i \\nu _{x-yi} + i \\nu _{x-yj} k - i \\nu _{x+yj} k \\nonumber \\\\& \\; \\; + \\nu _{x+yk} k - \\nu _{x-yk} k, $ then $\\langle U^n x, y \\rangle = \\int _0^{2\\pi } e^{i n t } d\\nu _{x,y}(t), \\quad \\quad x,y \\in \\mathcal {H}\\quad {\\rm and }\\quad n \\in \\mathbb {Z}.$ Theorem 3.8 The $\\mathbb {H}$ -valued measures $\\nu _{x,y}$ defined on $\\mathbf {B}([0, 2\\pi ))$ enjoy the following properties: (i) $\\nu _{x \\alpha +y \\beta ,z} = \\nu _{x,z}\\alpha + \\nu _{y,z} \\beta ,\\ \\ \\alpha , \\beta \\in \\mathbb {H}$ ; (ii) $\\nu _{x, y\\alpha + z \\beta } = \\bar{\\alpha } \\nu _{x,y} +\\bar{\\beta } \\nu _{x,z},\\ \\ \\alpha , \\beta \\in \\mathbb {C}_i$ ; (iii) $\\nu _{x,y}([0, 2\\pi )) \\le \\Vert x \\Vert \\Vert y \\Vert $ , where $x,y,z \\in \\mathcal {H}$ and $\\alpha ,\\beta \\in \\mathbb {H}$ .",
"It follows from (REF ) that $\\int _0^{2\\pi } e^{i n t} d\\nu _{x\\alpha + y \\beta , z}(t) =& \\; \\langle U^n x, z \\rangle \\alpha + \\langle U^n y, z \\rangle \\beta \\\\=& \\; \\int _0^{2\\pi } e^{i n t} (d \\nu _{x,z}(t)\\alpha + d\\nu _{y,z}(t) \\beta ), \\quad \\quad n \\in \\mathbb {Z}.$ Use the uniqueness in Theorem REF to conclude that $\\nu _{x\\alpha + y \\beta , z}(t) = \\nu _{x,z}(t)\\alpha + \\nu _{y,z}(t) \\beta $ and hence we have proved (i).",
"Property (ii) is proved in a similar fashion, observing that $\\bar{\\alpha }$ , $\\bar{\\beta }$ commute with $e^{int}$ .",
"If $n = 0$ in (REF ), then $\\langle x , y \\rangle = \\int _0^{2\\pi } d\\nu _{x,y}(t) = \\nu _{x,y}([0, 2\\pi ))$ and thus we can use an analog of the Cauchy-Schwarz inequality (see Lemma 5.6 in [11]) to obtain $\\nu _{x,y}([0, 2\\pi )) \\le \\Vert x \\Vert \\Vert y \\Vert $ and hence we have proved (iii).",
"Remark 3.9 Contrary to the classical complex Hilbert space setting, $\\nu _{x,y}$ need not equal ${\\bar{\\nu }_{y,x}}$ .",
"It follows from statements (i), (ii) and (iii) in Theorem REF that $\\phi (x) = \\nu _{x,y}(\\sigma )$ , where $y \\in \\mathcal {H}$ and $\\sigma \\in \\mathbf {B}([0, 2\\pi ))$ are fixed, is a continuous right linear functional.",
"It follows from an analog of the Riesz representation theorem (see Theorem 6.1 in [11] or Theorem 7.6 in [13]) that corresponding to any $x \\in \\mathcal {H}$ , there exists a uniquely determined vector $w \\in \\mathcal {H}$ such that $\\phi (x) = \\langle x, w \\rangle ,$ i.e.",
"$\\nu _{x,y}(\\sigma ) = \\langle x, w \\rangle $ .",
"Use (i) and (ii) in Theorem REF to deduce that $w = E(\\sigma )^* y$ .",
"The uniqueness of $E$ follows readily from the construction.",
"Thus, we have $\\nu _{x,y}(\\sigma ) = \\langle E(\\sigma )x, y \\rangle , \\quad \\quad x,y \\in \\mathcal {H}\\quad {\\rm and} \\quad \\sigma \\in \\mathbf {B}([0,2\\pi )),$ whence $\\langle U^n x, y \\rangle = \\int _0^{2\\pi } e^{ i n t} \\langle dE(t) x, y \\rangle .$ To prove the main properties of the operator $E$ we need a uniqueness results on quaternionic measures which is a corollary of the following: Theorem 3.10 Let $\\mu $ and $\\nu $ be $\\mathbb {C}$ -valued measures on $[0, 2\\pi )$ .",
"If $r(n) = \\int _0^{2\\pi } e^{i n t} d\\mu (t) = \\int _0^{2\\pi } e^{i n t} d\\nu (t), \\quad n \\in \\mathbb {Z},$ then $\\mu = \\nu $ .",
"See, e.g., Theorem 1.9.5 in [29].",
"Theorem 3.11 Let $\\mu $ and $\\nu $ be $\\mathbb {H}$ -valued measures on $[0, 2\\pi )$ .",
"If $r(n) = \\int _0^{2\\pi } e^{i n t} d\\mu (t) = \\int _0^{2\\pi } e^{i n t} d\\nu (t), \\quad n \\in \\mathbb {Z},$ then $\\mu = \\nu $ .",
"Write $r(n) = r_1(n) + r_2(n)j$ , $\\mu = \\mu _1 + \\mu _2j$ and $\\nu = \\nu _1 + \\nu _2 j$ , where $r_1(n), r_2(n) \\in \\mathbb {C}$ and $\\mu _1, \\mu _2, \\nu _1, \\nu _2$ are $\\mathbb {C}$ -valued measures on $[0, 2\\pi )$ .",
"It follows from (REF ) that $r_1(n) = \\int _0^{2\\pi } e^{i n t} d\\mu _1(t) = \\int _0^{2\\pi } e^{i n t} d\\nu _1(t), \\quad n \\in \\mathbb {Z}$ and $r_2(n) = \\int _0^{2\\pi } e^{i n t} d\\mu _2(t) = \\int _0^{2\\pi } e^{i n t} d\\nu _2(t), \\quad n \\in \\mathbb {Z}.$ Use Theorem REF to conclude that $\\mu _1 = \\nu _1$ , $\\mu _2 = \\nu _2$ and hence that $\\mu = \\nu $ .",
"Theorem 3.12 The operator $E$ given in (REF ) enjoys the following properties: (i) $\\Vert E(\\sigma ) \\Vert \\le 1$ ; (ii) $E(\\emptyset ) = 0$ and $E([0, 2\\pi )) = I_{\\mathcal {H}}$ ; (iii) If $\\sigma \\cap \\tau = \\emptyset $ , then $E( \\sigma \\cup \\tau ) = E(\\sigma ) + E(\\tau )$ ; (iv) $E(\\sigma \\cap \\tau ) = E(\\sigma ) E(\\tau )$ ; (v) $E(\\sigma )^2 = E(\\sigma )$ ; (vi) $E(\\sigma )$ commutes with $U$ for all $\\sigma \\in \\mathbf {B}([0, 2\\pi ))$ .",
"Use (REF ) with $y = E(\\sigma ) x$ and (iii) in Theorem (REF ) to obtain $\\Vert E(\\sigma ) x \\Vert ^2 \\le \\Vert x \\Vert \\Vert E(\\sigma ) x \\Vert ,$ whence we have shown (i).",
"The first part of property (ii) follows directly from the fact that $\\nu _{x,y}(\\emptyset ) = 0$ .",
"The last part follows from (REF ) when $n = 0$ .",
"Statement (iii) follows easily from the additivity of the measure $\\nu _{x,y}$ .",
"We will now prove property (iv).",
"It follows from (REF ) that $\\langle U^{n+m} x, y \\rangle =& \\; \\int _0^{2\\pi } e^{i n t} e^{i m t} \\langle dE(t)x, y \\rangle \\\\=& \\; \\langle U^n (U^m x), y \\rangle \\\\=& \\; \\int _0^{2\\pi } e^{i n t} d\\langle E(t) U^m x, y \\rangle .$ Using the uniqueness in Theorem REF we obtain $e^{i m t} d\\langle E(t) x, y \\rangle = \\langle dE(t) U^m x, y \\rangle $ and hence, denoting by $\\mathbf {1}_\\sigma $ the characteristic function of the set $\\sigma $ , we have $\\int _0^{2\\pi } \\mathbf {1}_\\sigma (t) e^{i m t} \\langle dE(t) x, y \\rangle = \\langle E(\\sigma ) U^m x , y \\rangle .$ But $\\int _0^{2\\pi } \\mathbf {1}_\\sigma (t) e^{i m t} \\langle dE(t) x, y \\rangle = \\langle U^k x, E(\\sigma )^* y \\rangle = \\int _0^{2\\pi } e^{i m t} d\\langle E(t) x, E(\\sigma )^* y \\rangle .$ Using the uniqueness in Theorem REF once more we get $\\mathbf {1}_\\sigma (t) d\\langle E(t)x, y \\rangle = \\langle dE(t) x, E(\\sigma )^* y \\rangle $ and hence $\\int _0^{2\\pi } \\mathbf {1}_\\tau (t) \\mathbf {1}_\\sigma (t) \\langle dE(t) x, y \\rangle = \\langle E(t)x, E(\\sigma )^* y \\rangle $ and thus $ \\langle E( \\sigma \\cap \\tau ) x, y \\rangle = \\langle E(\\sigma ) E(\\tau ) x, y \\rangle .$ Property (v) is obtained from (iv) by letting $\\sigma = \\tau $ .",
"Finally, since $U$ is unitary one can check that $\\langle U(x \\pm U^* y), x \\pm U^* y \\rangle = \\langle U(Ux \\pm y), Ux \\pm y \\rangle $ and hence from (REF ) and the uniqueness in Theorem REF we obtain $\\nu _{x \\pm U^*y} = \\nu _{Ux \\pm y}$ .",
"Similarly, $\\nu _{x \\pm U^* y i} = \\nu _{Ux \\pm yi}$ $\\nu _{x\\pm U^* y j} = \\nu _{U x \\pm yj }$ and $\\nu _{x \\pm U^* y k} = \\nu _{Ux \\pm yk}.$ It follows from (REF ) that $\\nu _{x, U^* y} = \\nu _{Ux, y}.$ Now use (REF ) to obtain $\\langle E(\\sigma ) x, U^* y \\rangle = \\langle E(\\sigma ) U x, y \\rangle ,$ i.e., $ \\langle U E(\\sigma ) x, y \\rangle = \\langle E(\\sigma ) U x , y \\rangle ,\\quad \\quad x,y \\in \\mathcal {H}.$ Given any quaternionic Hilbert space $\\mathcal {H}$ , there exists a subspace $\\mathcal {M} \\subset \\mathcal {H}$ on $\\mathbb {C}$ so that for any $x \\in \\mathcal {H}$ we have $x = x_1 + x_2 j, \\quad x_1,x_2 \\in \\mathcal {M}.$ Theorem 3.13 Let $U$ be a unitary operator on a quaternionic Hilbert space $\\mathcal {H}$ and let $E$ be the corresponding operator given by (REF ).",
"$E$ is self-adjoint if and only if $U: \\mathcal {M} \\rightarrow \\mathcal {M}$ , where $\\mathcal {M}$ is as above.",
"If $E = E^*$ , then it follows from (REF ) that $\\nu _{x,y} = \\bar{\\nu }_{y,x}$ for all $x,y \\in \\mathcal {H}$ .",
"In particular, we get $\\nu _{x,x} = \\bar{\\nu }_{x,x}$ , i.e.",
"$\\nu _x = \\bar{\\nu }_x, \\quad x \\in \\mathcal {H}.$ Since $\\nu _x$ is a $q$ -positive measure we may write $\\nu _x = \\alpha _x + \\beta _x j$ , where $\\alpha _x$ is a positive Borel measure on $[0, 2\\pi )$ and $\\beta _x$ is a complex Borel measure on $[0, 2\\pi )$ .",
"It follows from (REF ) that $\\beta _x = - \\beta _x,$ i.e.",
"$\\beta _x = 0$ .",
"Thus, we may make use of the spectral theorem for unitary operators on a complex Hilbert space (see, e.g., Section 31.7 in [27]) to deduce that $U: \\mathcal {M} \\rightarrow \\mathcal {M}$ .",
"Conversely, if $U: \\mathcal {M} \\rightarrow \\mathcal {M}$ , then the spectral theorem for unitary operators on a complex Hilbert space yields that $E = E^*$ .",
"If $U: \\mathbb {H}^n \\rightarrow \\mathbb {H}^n$ is unitary, then (REF ) and Theorem REF assert that $U = \\sum _{a=1}^n e^{i \\theta _a} P_a,$ where $\\theta _1, \\ldots , \\theta _n \\in [0, 2\\pi )$ and $P_1, \\ldots , P_n$ are oblique projections (i.e.",
"$(P_a)^2 = P_a$ but $(P_a)^*$ need not equal $P_a$ ).",
"Corollary 6.2 in Zhang [33] asserts, in particular, the existence of $V: \\mathbb {H}^n \\rightarrow \\mathbb {H}^n$ which is unitary and $\\theta _1, \\ldots , \\theta _n \\in [0, 2\\pi )$ so that $U = V^* {\\rm diag}(e^{i \\theta _1}, \\ldots , e^{i \\theta _n}) V.$ In the following remark we will explain how (REF ) and (REF ) are consistent.",
"Remark 3.14 Let $U: \\mathbb {H}^n \\rightarrow \\mathbb {H}^n$ be unitary.",
"Let $V$ and $\\theta _1, \\ldots , \\theta _n$ be as above.",
"If we let $e_a = (0, \\ldots , 0, 1, 0, \\ldots , 0)^T \\in \\mathbb {H}^n$ , where the 1 is the $a$ -th position, then we can rewrite (REF ) as $U = \\sum _{a=1}^n V^* e^{i \\theta _a} e_a e_a^* V.$ Note that $V^* e^{i \\theta _a} e_a e_a^* V = e^{i \\theta _a} V^* e_a e_a^* V$ if and only if $V: \\mathbb {C}^n \\rightarrow \\mathbb {C}^n$ .",
"In this case $U: \\mathbb {C}^n \\rightarrow \\mathbb {C}^n$ and $U = \\sum _{a=1}^n e^{i \\theta _a} P_a,$ where $P_a$ denotes the orthogonal projection given by $V^* e^{i \\theta _a} e_a e_a^* V$ .",
"Remark 3.15 Observe that in the proof of the spectral theorem for $U^n$ we have taken the imaginary units $i$ , $j$ , $k$ for the quaternions and we have determined spectral measures $\\langle dE(t) x, y \\rangle $ that are supported on the unit circle in $\\mathbb {C}_i$ .",
"In the case one uses other orthogonal units $I$ , $J$ and $K\\in \\mathbb {S}$ to represent quaternions, then the spectral measures are supported on the unit circle in $\\mathbb {C}_I$ .",
"Observe that (REF ) provides a vehicle to define a functional calculus for unitary operators on a quaternionic Hilbert space.",
"For a continuous $\\mathbb {H}$ -valued function $f$ on the unit circle, which will be approximated by the polynomials $\\sum _{k}e^{i k t} a_k $ .",
"We will consider a subclass of continuous quaternionic-valued functions defined as follows, see [23]: Definition 3.16 The quaternionic linear space of slice continuous functions on an axially symmetric subset $\\Omega $ of $\\mathbb {H}$ , denoted by $\\mathcal {S}(\\Omega )$ consists of functions of the form $f(u+Iv)=\\alpha (u,v)+I\\beta (u,v)$ where $\\alpha ,\\beta $ are quaternionic valued functions such that $\\alpha (x,y)=\\alpha (u,-v)$ , $\\beta (u,v)=-\\beta (u,-v)$ and $\\alpha $ , $\\beta $ are continuous functions.",
"When $\\alpha ,\\beta $ are real valued we say that the continuous slice function is intrinsic.",
"The subspace of intrinsic continuous slice functions is denoted by $\\mathcal {S}_{\\mathbb {R}}(\\Omega )$ .",
"It is important to note that any polynomial of the form $P(u+Iv)=\\sum _{k=0}^n (u+Iv)^n a_n$ , $a_n\\in \\mathbb {H}$ is a slice continuous function in the whole $\\mathbb {H}$ .",
"A trigonometric polynomial of the form $P(e^{It})=\\sum _{m=-n}^n e^{Imt} a_m$ is a slice continuous function on $\\partial \\mathbb {B}$ , where $\\mathbb {B}$ denotes the unit ball of quaternions.",
"Let us now denote by $\\mathcal {PS}(\\sigma _S(T))$ the set of slice continuous functions $f(u+Iv)=\\alpha (u,v)+I\\beta (u,v)$ where $\\alpha $ , $\\beta $ are polynomials in the variables $u,v$ .",
"In the sequel we will work on the complex plane $\\mathbb {C}_I$ and we denote by $\\mathbb {T}_I$ the boundary of $\\mathbb {B}\\cap \\mathbb {C}_I$ .",
"Any other choice of an imaginary unit in the unit sphere $\\mathbb {S}$ will provide an analogous result.",
"Remark 3.17 For every $I \\in \\mathbb {S}$ , there exists $J \\in \\mathbb {S}$ so that $I J = -J I$ .",
"Bearing in mind Remark REF , we can construct $\\nu _{x,y}^{(J)}$ so that (REF ) can also be written as $\\langle U^n x, y \\rangle = \\int _0^{2\\pi } e^{I n t } d\\nu ^{ (J)}_{x,y}(t), \\quad \\quad x,y \\in \\mathcal {H}\\quad {\\rm and }\\quad n \\in \\mathbb {Z}.$ Consequently, (REF ) can be written as $\\langle U^n x, y \\rangle = \\int _0^{2\\pi } e^{i n t} \\langle E_J(t) x, y \\rangle ,$ where $E_J$ is given by $\\nu _{x,y}^{(J)}(\\sigma ) = \\langle E_J(\\sigma )x, y \\rangle , \\quad x,y \\in \\mathcal {H} \\quad {\\rm and} \\quad \\sigma \\in {\\rm B}(\\mathbb {T}_I).$ Moreover, $E_J$ satisfy properties (i)-(v) listed in Theorem REF .",
"Theorem 3.18 (The spectral theorem for quaternionic unitary operators) Let $U$ be an unitary operator on a right linear quaternionic Hilbert space $\\mathcal {H}$ .",
"Let $I, J \\in \\mathbb {S}$ , $I$ orthogonal to $J$ .",
"Then there exists a unique spectral measure $E_J$ defined on the Borel sets of $\\mathbb {T}_I$ such that for every slice continuous function $f\\in \\mathcal {S}(\\sigma _S(U))$ , we have $f(U) = \\int _0^{2\\pi } f(e^{I t}) dE_J(t).$ Let us consider a polynomial $P(t) = \\sum _{m=-n }^n e^{I m t}a_m$ defined on $\\mathbb {T}_I$ .",
"Then using (REF ) we have $\\langle U^m x, y \\rangle = \\int _0^{2\\pi } e^{Im t} \\langle dE_J(t)x, y \\rangle \\quad \\quad x,y, \\in \\mathcal {H}.$ By linearity, we can define $\\langle P(U) x, y \\rangle = \\int _0^{2\\pi } P(e^{I t}) \\langle dE_J(t)x, y \\rangle ,\\quad \\quad x,y, \\in \\mathcal {H}.$ The map $\\Psi :\\ \\mathcal {PS}(\\sigma _S(U))\\rightarrow \\mathcal {H}$ defined by $\\psi _U(P) = P(U)$ is $\\mathbb {R}$ -linear.",
"By fixing a basis for $\\mathbb {H}$ , e.g.",
"the basis $1,i,j,k$ , each slice continuous function $f$ can be decomposed using intrinsic function, i.e.",
"$f=f_0+f_1i+f_2j+f_3k$ with $f_\\ell \\in \\mathcal {S}_{\\mathbb {R}}(\\sigma _S(U))$ , $\\ell =0,\\ldots ,3$ , see [23].",
"For these functions the spectral mapping theorem holds, thus $f_\\ell (\\sigma _S(U))=\\sigma _S(f_\\ell (U))$ and so $\\Vert f_\\ell (U)\\Vert =\\Vert f_\\ell \\Vert _\\infty $ , see [23].",
"The map $\\psi $ is continuous and so there exists $C>0$ , that does not depend on $\\ell $ , such that $\\Vert P(U) \\Vert _{\\mathcal {H}} \\le C \\max _{ t \\in \\sigma _S(U) } |P(t) |.$ A slice continuous function $f\\in \\mathcal {S}(\\sigma _S(U))$ is defined on an axially symmetric subset $K\\subseteq \\mathbb {T}$ and thus it can be written as a function $f(e^{It})=\\alpha (\\cos t,\\sin t) +I \\beta (\\cos t,\\sin t)$ .",
"By fixing a basis of $\\mathbb {H}$ , e.g.",
"$1,i,j,k$ , $f$ can be decomposed into four slice continuous intrinsic functions $f_\\ell (\\cos t,\\sin t)=\\alpha _\\ell (\\cos t,\\sin t) +I \\beta _\\ell (\\cos t,\\sin t)$ , $\\ell =0,\\ldots ,3$ , such that $f= f_0+f_1i+f_2j+f_3k$ .",
"By the Weierstrass approximation theorem for trigonometric polynomials, see, e.g., Theorem 8.15 in [28], each function $f_\\ell $ can be approximated by a sequence of polynomials $\\tilde{R}_{\\ell n} = \\tilde{a}_{\\ell n} (\\cos t,\\sin t) +I \\tilde{b}_{\\ell n} (\\cos t,\\sin t),$ $\\ell =0,\\ldots ,3$ which tend uniformly to $f_\\ell $ .",
"These polynomials do not necessarily belong to the class of the continuous slice functions since $\\tilde{a}_{\\ell n}, \\tilde{b}_{\\ell n}$ do not satisfy, in general, the even and odd conditions in Definition REF .",
"However, by setting $a_{\\ell n}(u,v)= \\frac{1}{2}(\\tilde{a}_{\\ell n}(u,v)+\\tilde{a}_{\\ell n}(u,-v)),$ $b_{\\ell n}(u,v)= \\frac{1}{2}(\\tilde{b}_{\\ell n}(u,-v)-\\tilde{b}_{\\ell n}(u,v))$ we obtain that the sequence of polynomials $a_{\\ell n}+Ib_{\\ell n}$ still converges to $f_\\ell $ , $\\ell =0,\\ldots , 3$ .",
"By putting $R_{\\ell n} = a_{\\ell n} (\\cos t,\\sin t) +I b_{\\ell n} (\\cos t,\\sin t)$ , $\\ell =0,\\ldots ,3$ and $R_n=R_{0n}+R_{1n}i+R_{2n}j+R_{3n}k$ we have a sequence of slice continuous polynomials $R_n(e^{It})$ converging to $f(e^{It})$ uniformly on $\\mathbb {R}$ .",
"By the previous discussion, $\\lbrace R_n(U)\\rbrace $ is a Cauchy sequence in the space of bounded linear operators since $\\Vert R_n(U)-R_m(U) \\Vert \\le C \\max _{ t \\in \\sigma _S(U) } |R_n(t)-R_m(t) |,$ so as ${R_n(U)}$ has a limit which we denote by $f(U)$ .",
"Remark 3.19 Fix $I \\in \\mathbb {S}$ .",
"It is worth pointing out that $f(u+Iv) = (u+Iv)^{-1}$ is an intrinsic function on $\\mathbb {C}_{I} \\cap \\partial \\mathbb {B}$ , where $\\partial \\mathbb {B} = \\lbrace q \\in \\mathbb {H}: |q| = 1 \\rbrace $ , since $f(u+Iv) = \\frac{u}{u^2+v^2} + \\left(\\frac{-v}{u^2+v^2}\\right)J.$ Thus, using Theorem REF , we may write $U^{-1} = \\int _0^{2\\pi } e^{-It} d E_J(t)$ and $U = \\int _0^{2\\pi } e^{It} dE_J(t).$"
],
[
"The $S$ -spectrum and the spectral theorem ",
"We now want to show that the spectral theorem is based on the $S$ -spectrum.",
"We will be in need of the Cauchy formula for slice hyperholomorphic functions, see [17] for more details.",
"Definition 4.1 (Cauchy kernels) We define the (left) Cauchy kernel, for $q\\notin [s]$ , by $S_L^{-1}(s,q):=-(q^2-2q{\\rm Re}(s)+|s|^2)^{-1}(q-\\bar{s}).$ We define the right Cauchy kernel, for $q\\notin [s]$ , by $S^{-1}_R(s,q):=-(q-\\bar{s})(q^2-2{\\rm Re}(s)q+|s|^2)^{-1}.$ Theorem 4.2 Let $\\Omega \\subseteq \\mathbb {H}$ be an axially symmetric s-domain such that $\\partial (\\Omega \\cap \\mathbb {C}_I)$ is union of a finite number of continuously differentiable Jordan curves, for every $I\\in \\mathbb {S}$ .",
"Let $f$ be a slice hyperholomorphic function on an open set containing $ \\overline{\\Omega }$ and, for any $I\\in \\mathbb {S}$ , set $ds_I=-Ids$ .",
"Then for every $q=u+I_qv\\in \\Omega $ we have: $f(q)=\\frac{1}{2 \\pi }\\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)} S_L(s,q) ds_I f(s).$ Moreover, the value of the integral depends neither on $\\Omega $ nor on the imaginary unit $I\\in \\mathbb {S}$ .",
"If $f$ is a right slice regular function on a set that contains $\\overline{\\Omega }$ , then $f(q)=\\frac{1}{2 \\pi }\\int _{\\partial (\\Omega \\cap \\mathbb {C}_I)} f(s)ds_I S_R^{-1}(s,q).$ Moreover, the value of the integral depends neither on $\\Omega $ nor on the imaginary unit $I\\in \\mathbb {S}$ .",
"We conclude the paper with the following result, based on the Cauchy formula, that shows the relation between the spectral measures and the $S$ -spectrum.",
"Theorem 4.3 Fix $I, J \\in \\mathbb {S}$ , with $I$ orthogonal to $J$ .",
"Let $U$ be an unitary operator on a right linear quaternionic Hilbert space $\\mathcal {H}$ and let $E(t)=E_J(t)$ be its spectral measure.",
"Assume that $\\sigma ^0_S(U)\\cap \\mathbb {C}_I$ is contained in the arc of the unit circle in $\\mathbb {C}_I$ with endpoints $t_0$ and $t_1$ .",
"Then $\\mathcal {P}(\\sigma ^0_S(U))=E(t_1)-E(t_0).$ The spectral theorem implies that the operator $S_R^{-1}(s,U)$ can be written as $S_R^{-1}(s,U)=\\int _0^{2\\pi }S_R^{-1}(e^{It},s) dE(t).$ The Riesz projector is given by $\\mathcal {P}(\\sigma ^0_S(U))=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _0\\cap \\mathbb {C}_I)} ds_I S_R^{-1}(s,U)$ where $\\Omega _0$ is an open set containing $\\sigma ^0_S(U)$ and such that $\\partial (\\Omega _0\\cap \\mathbb {C}_I)$ is a smooth closed curve in $\\mathbb {C}_I$ .",
"Now we write $\\mathcal {P}(\\sigma ^0_S(U))=\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _0\\cap \\mathbb {C}_I)} ds_I \\Big (\\int _0^{2\\pi }S_R^{-1}(e^{It},s) dE(t) \\Big )$ and using the Fubini theorem we get $\\mathcal {P}(\\sigma ^0_S(U))=\\int _0^{2\\pi } \\Big ( \\frac{1}{2\\pi }\\int _{\\partial (\\Omega _0\\cap \\mathbb {C}_I)} ds_I S_R^{-1}(e^{It},s) \\Big ) dE(t).$ It follows from the Cauchy formula that $\\frac{1}{2\\pi }\\int _{\\partial (\\Omega _0\\cap \\mathbb {C}_I)} ds_I S_R^{-1}(e^{It},s)=\\mathbf {1}_{[t_0,t_1]},$ where $\\mathbf {1}_{[t_0,t_1]}$ is the characteristic function of the set $[t_0,t_1]$ , and so we get the statement, since $\\mathcal {P}(\\sigma ^0_S(U))=\\int _0^{2\\pi } \\mathbf {1}_{[t_0,t_1]}dE(t)=E(t_1)-E(t_2).$"
]
] | 1403.0175 |
[
[
"On some varieties associated with trees"
],
[
"Abstract This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras.",
"Their definition involves a canonical coloring of vertices of trees into three colors.",
"These varieties are proved to be smooth and to admit sometimes free actions of algebraic tori.",
"Some results are obtained on their number of points over finite fields and on their cohomology."
],
[
"Introduction",
"The theory of cluster algebras, introduced by S. Fomin and A. Zelevinsky around 2000 [9], [10], was motivated initially by the study of total positivity in Lie groups and canonical bases in quantum groups.",
"It has since then developed rapidly in many directions, among which one can cite (for example) triangulated categories [3], triangulations of surfaces [8] and Poisson geometry [11], [13].",
"Because cluster algebras are commutative algebras endowed with more structure, it is natural to study them from the point of view of algebraic geometry.",
"The geometric study of cluster algebras has nevertheless been mostly concentrated on aspects related to Poisson geometry or symplectic geometry.",
"The appearance of the known cluster structure on coordinate rings of grassmannians in a physical context [1] has raised recently the interest in the computation of integrals on the varieties associated with cluster algebras.",
"The natural context for this is of course the cohomology ring.",
"The present article aims to study some varieties closely related to the spectrum of cluster algebras, and their cohomology rings.",
"General cluster algebras are defined using a quiver or a skew-symmetric matrix.",
"For our purposes, one needs as a starting point a presentation by generators and relations of the cluster algebras.",
"This is available for cluster algebras with an acyclic quiver [2] and in a few other cases (see for example [17]).",
"The choice has been made here to restrict to a still smaller class, namely cluster algebras with a quiver which is a tree, in the hope that the answers may be simpler in that case, and also because all finite Dynkin diagrams are trees.",
"Cluster algebras come with a subalgebra generated by so-called frozen (or coefficient) variables, which are invertible elements.",
"This corresponds to a morphism from the spectrum of the cluster algebra to an algebraic torus.",
"At the start of this work, our intention was to study both the fibers of this map and the spectrum in full.",
"Later it turned out that it is possible (for cluster algebras associated with trees) to define more general varieties.",
"Cohomology and number of points on similar varieties have been considered in some previous works [12], [16], [6].",
"Some results of these articles will be recalled when necessary.",
"The article is organized as follows.",
"In the first section, one recalls a canonical tri-coloring of the vertices of trees, originally defined in [4], [7], [21] and not so well-known.",
"This coloring is closely connected to matchings and independent sets in the trees.",
"It will be used in an intensive way in the rest of the article, as it enters in the very definition of the varieties under study.",
"One introduces the notion of red-green components of a tree, and defines an important integer invariant, the dimension of a tree.",
"The second section is devoted to the definition of the varieties.",
"This is rather involved, and the definition itself only appears after a long preparation.",
"One first considers a very general family of varieties, depending one many invertible parameters.",
"By considering these varieties as objects in a groupoid, one can reduce this family to a much smaller one, with less parameters.",
"One proves that every variety in the big family is isomorphic to a variety in the small family.",
"One also introduces an explicit condition of genericity.",
"Then everything is ready for the definition, which involves making an independent choice for every red-green component of the tree.",
"The third section is devoted to some geometric properties of these varieties.",
"One proves by induction that all these varieties are smooth, by finding explicit coverings by products of varieties of the same type and algebraic tori.",
"One next shows that some of these varieties are endowed with a free torus action, which turns them into principal torus bundles.",
"The fourth section turns to the study of the number of points over finite fields.",
"One shows by induction that the number of points is a polynomial in the cardinality $q$ of the finite field.",
"This is done by finding an appropriate decomposition into pieces isomorphic to products of varieties of the same type and algebraic tori.",
"One then gives formulas for some classical trees, including Dynkin diagrams.",
"One also obtains (Prop.",
"REF ) a general decomposition as a disjoint union of products of tori and affine spaces (indexed by independent sets), which allows to compute the Euler characteristic.",
"The three next sections (5,6 and 7) deal with some computations regarding the cohomology rings.",
"Section 5 is a very short reminder about known results about differential forms on varieties associated with cluster algebras, and about the general theory of (mixed) Hodge structure on the cohomology ring of algebraic varieties.",
"Section 6 deals with some examples of trees, namely linear trees (the case of which forms a useful building stone) and some trees of shape $H$ with no parameters.",
"Section 7 is about varieties where parameters have been given a generic value.",
"Our results about cohomology are rather partial, restricted to special cases, but there does not seem to be any simple general answer.",
"The prominent missing case is in type $\\mathbb {A}$ with an odd number of vertices, where one proposes a conjecture.",
"The appendix A presents a simple algorithm for the computation of the canonical coloring of trees.",
"This algorithm is not needed in the rest of the article.",
"Let us finish this introduction by a few side remarks.",
"Another interesting question which has not been considered here is the study of the real points of the same varieties, and their cohomology.",
"This is probably also rather complicated, but certainly worth looking at.",
"There seems to be some kind of vague analogy between the counting-points polynomials considered here and the characteristic polynomials of bipartite Coxeter elements (cf [15] and [20]), namely the general look and feel of these two families of polynomials are similar in various points (including some relations to Pisot and Salem numbers).",
"At the end of section 1.2 of [5], one can find some speculations about the idea of “quadratic spectra” for graphs, that would be an analog of the usual spectrum but related to quadratic equations instead of linear equations.",
"Maybe one can argue that the cluster varieties considered here and their counting-point polynomials are a good candidate for such a quadratic spectrum (even if they involve polynomial relations of arbitrary degree).",
"This work has been supported by the ANR program CARMA (ANR-12-BS01-0017-02)."
],
[
"Combinatorics of trees",
"In this article, a tree is a finite connected and simply-connected graph.",
"A leaf is a vertex with at most one neighbor.",
"A forest is a disjoint union of trees."
],
[
"Canonical red-orange-green coloring of trees",
"In this section, one recalls a canonical coloring of the vertices of all trees, using the colors red, orange and green.",
"This coloring has first appeared in an article by J. Zito [21] and has been studied independently later by S. Coulomb and M. Bauer in [7], [4].",
"Figure: Canonical coloring: {1,2} are orange, {4,6} green and {3,5,7} redLet us consider a tree $T$ .",
"A vertex cover of $T$ is a subset $S$ of the vertices of $T$ such that every edge of $T$ has at least one end in $S$ .",
"A minimum vertex cover of $T$ is a vertex cover of minimal cardinality among all vertex covers of $T$ .",
"Let us use this notion to color the vertices of $T$ according to the following rule: a vertex $v$ is green if $v$ is present in all minimum vertex covers, orange if $v$ is present in some but not all minimum vertex covers, red if $v$ is present in no minimum vertex covers.",
"The colors have been chosen to match this definition with traffic lights colors.",
"For the tree of figure REF , the minimum vertex covers are made of the two green vertices $\\lbrace 4,6\\rbrace $ and one of the two orange vertices $\\lbrace 1,2\\rbrace $ .",
"Remark 1.1 By taking the complementary subset, there is a bijection between minimum vertex covers, and sets of non-adjacent vertices of maximal cardinality (maximum independent sets, also called maximum stable sets).",
"This coloring is also related to maximum matchings of $T$ .",
"A matching of $T$ is a set $D$ of edges of $T$ , such that every vertex belongs to at most one element of $D$ .",
"The elements of $D$ will be called dominoes.",
"A maximum matching is a matching of maximal cardinality among all matchings of $T$ .",
"Then, a vertex $v$ is green if $v$ is present in all maximum matchings, in several different dominoes.",
"orange if $v$ is present in all maximum matchings, always in the same domino.",
"red if $v$ is absent in some maximum matchings.",
"The proof of the equivalence of these two descriptions of the coloring can be found in [4].",
"For the tree of figure REF , the maximum matchings are made of three dominoes, one of them being the edge between the two orange vertices $\\lbrace 1,2\\rbrace $ .",
"Proposition 1.2 The orange vertices are matched in pairs by the unique domino in which they are contained in any maximum matching.",
"In maximum matchings, green vertices are matched with red vertices in several different ways.",
"Proof.",
"This is proved in [4].",
"This coloring has a third equivalent description, also given in [4].",
"It is the unique coloring of the vertices such that the induced forest on orange vertices has a perfect matching, every green vertex has at least two red neighbors, every red vertex has only green neighbors.",
"It follows from this description that the coloring is stable by any of the following operations: taking the induced forest on orange vertices, taking the induced forest on the union of red and green vertices, removing a matched pair of orange vertices, removing a green vertex.",
"An algorithm to compute the coloring is presented in appendix .",
"Figure: A typical exampleFurther properties of the coloring Let us first state a corollary of the third description of the coloring.",
"Lemma 1.3 A tree admits a perfect matching if and only if all vertices are orange.",
"Proof.",
"If all vertices are orange, there is a perfect matching by the first condition in the third description.",
"If the tree has a perfect matching, letting all the vertices be orange gives a coloring which satisfies all the required conditions, and therefore is the correct one by uniqueness.",
"Note that the maximum matching is unique for these trees.",
"We will call them orange trees.",
"They are also known as perfect trees or matched trees [19].",
"Let $T$ be a tree.",
"The red-green components of $T$ are the connected components of the graph defined by keeping only the edges of $T$ with one red end and one green end.",
"Every red-green component is a tree, which is moreover bipartite with only red leaves.",
"In these trees, every green vertex has valency at least two.",
"This kind of trees has been considered under the name of $bc$ -trees in the study of blocks and cut-vertices of graphs, see for example [14].",
"Even trees with no orange vertex can have several such components, because there can be edges with two green ends, and these edges are not kept in the red-green components.",
"By the third description of the coloring, the coloring is stable by taking a red-green component.",
"A tree which is equal to its only red-green component will be called a red-green tree.",
"Lemma 1.4 The set of maximum matchings of a tree is in bijection with the product of the sets of maximum matchings of its red-green components.",
"Proof.",
"The dominoes are fixed on the set of orange vertices, and cannot connect two distinct red-green components by proposition REF .",
"Therefore, one can choose a maximum matching independently on every red-green component.",
"Let us denote by $r(T), o(T)$ and $g(T)$ the number of red, orange and green vertices in the coloring of $T$ .",
"Let us call dimension of a tree $T$ the quantity $\\dim (T) = r(T) - g(T).$ Remark 1.5 The dimension of $T$ is also the dimension of the kernel of the adjacency matrix of $T$ , see [4].",
"Lemma 1.6 The dimension of $T$ is the number of vertices not covered by dominoes in any maximum matching.",
"Proof.",
"By the precise description of maximum matchings given in proposition REF , the number of dominoes in a maximum matching is $g(T)+o(T)/2$ .",
"The number of covered vertices is therefore $2g(T)+o(T)$ .",
"The statement follows.",
"Lemma 1.7 The dimension of $T$ is the sum of the dimensions of the red-green components of $T$ .",
"Every red-green component has dimension at least 1.",
"Proof.",
"The formula (REF ) for the dimension does not depend on orange vertices, and is clearly additive on red-green components.",
"Let $T$ be a red-green tree.",
"The Euler characteristic is given by $\\chi (T) = 1 = r(T) + g(T) - e(T),$ where $e(T)$ is the number of edges of $T$ .",
"On the other hand, $e(T) \\ge 2 g(T),$ because every green vertex has at least two red neighbors.",
"Lemma 1.8 Let $T$ be a red-green tree.",
"Let $F$ be the forest obtained by removing one red vertex of $T$ .",
"Then $\\dim (F) = \\dim (T) -1$ .",
"Proof.",
"Removing the vertex makes a big difference in the colorings of $F$ and $T$ .",
"The coloring of $F$ can be obtained from the restriction of the coloring of $T$ by some avalanche of orange vertices, as follows.",
"At start, the restriction of the coloring of $T$ gives a bad coloring of $F$ , where some green vertices $v$ may have exactly one red neighbor.",
"If not, then the coloring is the canonical one.",
"Otherwise, one can turn every such vertex $v$ and its unique red neighbor into an orange domino.",
"Doing that may create a certain number of green vertices with exactly one red neighbor.",
"For each of them, replace it and its unique red neighbor by a domino.",
"Repeat this as long as there is some green vertex with exactly one red neighbor.",
"This must stop at some point, because we work in a finite union of trees.",
"At the end of this avalanche of orange dominoes, one has obtained a canonical coloring of $F$ .",
"This construction implies that the dimension of $F$ is the dimension of $T$ minus 1, because it only involves turning pairs (green vertex, red vertex) into orange dominoes.",
"Lemma 1.9 Let $T$ be a red-green tree and $u-v$ be any edge of $T$ .",
"Let $F$ be the forest induced from $T$ by removing the vertices $u$ and $v$ .",
"Then the dimension of $T$ is the sum of the dimensions of the trees in $F$ .",
"Proof.",
"Assume that $u$ is green and $v$ is red.",
"Let $S_1,\\dots ,S_k$ be the trees in $F$ attached to $u$ and let $T_1,\\dots ,T_\\ell $ be the trees in $F$ attached to $v$ .",
"Then the coloring of every $S_i$ is just obtained by restriction, because it still satisfies the third description of the canonical coloring.",
"On the other hand, let us denote by $\\widehat{T}_j$ the tree obtained from $T_j$ by adding back the red vertex $v$ .",
"Then the coloring of every $\\widehat{T}_j$ is just obtained by restriction, because it still satisfies the third description of the canonical coloring.",
"By the definition (REF ) of the dimension, one therefore finds that $\\dim (T) = \\sum _i \\dim (S_i) + \\sum _j (\\dim (\\widehat{T}_j)-1).$ By lemma REF , this is equal to the expected result.",
"Lemma 1.10 Let $T$ be a tree.",
"Let $u-v$ be a red-green edge of $T$ .",
"There exists a maximum matching of $T$ containing $u-v$ .",
"Proof.",
"One can assume that $T$ is a red-green component, as maximum matchings of different red-green components are independent.",
"One can take maximum matchings of the connected components of the forest $F$ induced from $T$ by removing $u$ and $v$ .",
"From lemma REF and lemma REF , the number of vertices not covered on $F$ is the dimension of $T$ .",
"Therefore, adding the domino $u-v$ gives a maximum matching of $T$ .",
"Lemma 1.11 Let $T$ be a red-green tree and let $v$ be a leaf of $T$ .",
"There exists a maximum matching of $T$ where the vertices which are not covered are leaves.",
"Moreover, unless $T$ is reduced to the single vertex $v$ , one can find such a matching where $v$ is in a domino.",
"Proof.",
"By induction on the size of the tree $T$ .",
"This is true for the tree with 1 vertex.",
"Let us call $u$ the green neighbor of the red leaf $v$ .",
"The induced forest $F$ defined as $T\\setminus \\lbrace u,v\\rbrace $ is made of red-green trees, whose sum of dimensions is the dimension of $T$ by lemma REF .",
"By induction, one can find a maximum matching of $F$ such that vertices which are not covered are leaves of $F$ .",
"Moreover, one can choose this matching such that the vertices which are not covered are in fact leaves of $T$ .",
"One then obtains by adding the domino $u-v$ a maximum matching of $T$ with all the required properties.",
"Lemma 1.12 Let $T$ be a tree and let $v$ be a red vertex of $T$ .",
"There exists a maximum matching of $T$ not containing $v$ .",
"Proof.",
"Otherwise, one would get a contradiction with the characterization of the canonical coloring.",
"Lemma 1.13 The trees obtained by removing a leaf in an orange tree are exactly the trees of dimension 1.",
"They have exactly one red-green component.",
"Proof.",
"Let us pick an orange tree $T$ and a leaf $v$ with adjacent vertex $w$ .",
"Removing the leaf $v$ gives a tree $T\\setminus \\lbrace v\\rbrace $ with a matching covering all vertices but $w$ .",
"This is clearly a maximum matching, hence $T\\setminus \\lbrace v\\rbrace $ has dimension 1 by lemma REF .",
"Conversely, consider a tree $T^{\\prime }$ of dimension 1.",
"It has exactly one red-green component, as every red-green component contributes at least 1 to the dimension by lemma REF .",
"This red-green component has dimension 1.",
"By lemma REF , one can find a maximum matching of $T^{\\prime }$ missing only one leaf $w$ .",
"Adding a vertex $v$ attached to $w$ gives a tree with a perfect matching, i.e.",
"an orange tree.",
"We will call the trees of dimension 1 unimodal trees.",
"Remark 1.14 The classical Dynkin diagrams are simple examples of trees: Type $\\mathbb {A}_n$ : orange for even $n$ , unimodal for odd $n$ , Type $\\mathbb {D}_n$ : unimodal for odd $n$ , Type $\\mathbb {E}_n$ : orange for $n=6, 8$ , unimodal for $n=7$ .",
"The type $\\mathbb {D}_n$ with $n$ even has dimension 2.",
"Affine algebraic varieties Using the coloring of the previous section, one can define several affine algebraic varieties attached to a tree $T$ and some auxiliary choices.",
"These varieties are closely related to cluster algebras.",
"First, let us consider the system of equations $x_i x^{\\prime }_i = 1 + \\alpha _i \\prod _{i-j} x_j$ for all vertices $i$ of $T$ , where the product runs over vertices $j$ adjacent to $i$ .",
"Here $x_i$ and $x^{\\prime }_i$ are called cluster variables, and $\\alpha _i$ are called coefficient variables.",
"By a special case of [2], this system is a presentation of the cluster algebra associated with the quiver given by a bipartite orientation of $T$ , with one frozen vertex attached to every vertex of $T$ (in such a way that all vertices of $T$ remain sources or sinks).",
"In the context of cluster algebras, the equations (REF ) are called exchange relations.",
"We will be interested here in considering the $\\alpha _i$ as parameters, and letting them either vary in some well-chosen families or take fixed generic values (and even a mix of these two possibilities), so that the resulting space is smooth.",
"Jumping around a groupoid Let us denote by $X_T(\\alpha )$ the algebraic scheme defined by fixing some invertible values for all coefficient variables $\\alpha _i$ .",
"Recall the following lemma ([6]).",
"Lemma 2.1 Let $u-v$ be an edge of $T$ .",
"Let $\\beta $ be defined by $ \\beta _w = \\alpha _w/\\alpha _u$ if $w$ is a neighbor of $v$ (in particular $ \\beta _u = 1$ ) and $\\beta _w = \\alpha _w$ otherwise.",
"Then $X_T(\\alpha )$ and $X_T(\\beta )$ are isomorphic, by the change of variables $x_v=\\alpha _u x_v$ and $x^{\\prime }_v=x^{\\prime }_v/\\alpha _u$ .",
"One may say that the coefficient $\\alpha _u$ has jumped away from $u$ over $v$ and its inverse has got spread over all other neighbors of $v$ .",
"When $v$ has $u$ as only neighbor, the coefficient $\\alpha _u$ just disappears from the equations.",
"From now on, we will only admit the following kinds of jumps: a red vertex over one of its green neighbors, a green vertex over one of its red neighbors, an orange vertex over its matched orange neighbor.",
"Let us now define a groupoid $G_T$ with objects the schemes $X_T(\\alpha )$ indexed by invertible values of the parameters $\\alpha $ , and isomorphisms $X_T(\\alpha ) \\simeq X_T(\\bar{\\alpha })$ of the shape ${\\left\\lbrace \\begin{array}{ll}\\bar{x}_i &\\mapsto \\lambda _i x_i,\\\\\\bar{x}^{\\prime }_i &\\mapsto x^{\\prime }_i/\\lambda _i,\\end{array}\\right.",
"}$ where $\\lambda _i$ are some invertible elements.",
"The parameters are then related by ${\\alpha }_i = \\bar{\\alpha }_i \\prod _{i - j}\\lambda _j .$ Note that every jump corresponds to an isomorphism in the groupoid $G_T$ .",
"Proposition 2.2 For every maximum matching $M$ and given parameters $\\alpha $ , there exists unique parameters $\\beta $ (given by monic Laurent monomials in $\\alpha $ ) such that the function $\\beta $ is 1 except on the set of red vertices not covered by $M$ .",
"$X_T(\\alpha )$ is isomorphic to $X_T(\\beta )$ by a sequence of jumps.",
"Moreover, (a) the function $\\beta $ only depends on the values of $\\alpha $ on the red vertices of $T$ , (b) the values of $\\beta $ on a red-green component are Laurent monomials in the values of $\\alpha $ on the same red-green component.",
"Proof.",
"Let us first prove the existence of such parameters $\\beta $ .",
"The main idea is to iterate lemma REF by jumping over dominoes of $M$ .",
"Let us define an auxiliary oriented graph $\\mathcal {G}$ as follows: the vertices of $\\mathcal {G}$ are the vertices of $T$ , and there is an edge $u \\rightarrow w$ in $\\mathcal {G}$ if $u-v$ is a domino in $M$ and $v-w$ is another edge in $T$ .",
"With this notation, if there are edges starting from $u$ in $\\mathcal {G}$ , one can use lemma REF (by jumping over $v$ ) to turn the coefficient $\\beta _u$ into 1 and replace the coefficients $\\beta _w$ by $\\beta _w / \\beta _u$ , for all vertices at the end of an arrow $u \\rightarrow w$ .",
"One can see that the graph $\\mathcal {G}$ has no oriented cycle, otherwise there would be a cycle in $T$ made of concatenated dominoes.",
"Moreover, edges in the graph $\\mathcal {G}$ can only go from green to green, from orange to orange or green, or start from red.",
"Then one can do these jumps starting from the sources in $\\mathcal {G}$ and then proceeding along any linear extension of the partial order defined by $\\mathcal {G}$ .",
"At the end of this process, all vertices covered by dominoes have coefficient 1.",
"There only remains coefficients on the red vertices not covered by the maximum matching $M$ .",
"This proves the existence of the required parameters $\\beta $ .",
"The fact that the coefficients $\\beta _j$ are products of coefficients $\\alpha _i$ and their inverses is immediate from the definition of jumping.",
"Let us now prove uniqueness.",
"Assume there are two such sets of parameters $\\beta $ and $\\bar{\\beta }$ .",
"Let $x$ and $\\bar{x}$ be the coordinates on the isomorphic $X_T(\\beta )$ and $X_T(\\bar{\\beta })$ .",
"Let us first prove that any isomorphism in the groupoid $G_T$ from $X_T(\\beta )$ to $X_T(\\bar{\\beta })$ maps $\\bar{x}_j$ to $x_j$ for every green vertex $j$ .",
"This is done by induction using the auxiliary graph $\\mathcal {G}$ , starting with the green vertices that do not have any outgoing edge in $\\mathcal {G}$ .",
"For every green vertex, one just has to consider the equation (REF ) for the unique red vertex that is in the same domino in $M$ .",
"Using then the equation (REF ) for all red vertices $i$ not covered by $M$ , one obtains that $\\beta _i =\\bar{\\beta }_i$ .",
"This proves uniqueness.",
"For the statement $(a)$ , consider what happens to the coefficient attached to an orange or a green vertex $u$ .",
"By proposition REF , the domino containing $u$ must be orange or green-red.",
"The coefficient can therefore only jump to green or orange vertices.",
"So they must disappear at some point, because only red vertices bear coefficients at the end of the process.",
"Similarly for the statement $(b)$ , consider the coefficient attached to a red vertex $u$ .",
"Again by proposition REF , the domino containing $u$ must be red-green.",
"The coefficient can only jump to red vertices in the same red-green component, or to orange and green vertices.",
"As the coefficients on orange or green vertices will disappear by the previous point, coefficients can only stay within a given red-green component.",
"Recall that the dimension $\\dim (T)$ of $T$ is (by lemma REF ) the number of red vertices that are not covered in any maximum matching of $T$ .",
"Proposition REF justifies this terminology, as this gives the number of independent parameters for the varieties $X_T(\\alpha )$ (inside the groupoid $G_T$ ).",
"Remark 2.3 In the particular case when the tree $T$ is orange, all $X_T(\\alpha )$ are isomorphic.",
"By proposition REF , in order to study all isomorphism classes of such varieties, one can restrict oneself to attach parameters only to red vertices not covered by a maximum matching $M$ .",
"For a maximum matching $M$ of $T$ , let us define a scheme $X^M_T(\\alpha )$ by the set of equations (REF ), where $\\alpha _i$ are invertible fixed parameters, equal to 1 if $i$ is covered by $M$ .",
"Given two matchings $M$ and $M^{\\prime }$ , one can always find by Proposition REF a sequence of jumps that provides an isomorphism in $G_T$ between $X^M_T(\\alpha )$ and $X^{M^{\\prime }}_T(\\beta )$ , where the parameters $\\beta $ are uniquely determined Laurent monomials in $\\alpha $ .",
"Let us consider now the automorphism group $\\operatorname{Aut}(X^M_T(\\alpha ))$ of the object $X^M_T(\\alpha )$ in the groupoid $G_T$ .",
"Proposition 2.4 The automorphism group $\\operatorname{Aut}(X^M_T(\\alpha ))$ is an algebraic torus isomorphic to $\\mathbb {G}_m^{\\dim (T)}$ .",
"If $(\\lambda _i)_{i\\in T}$ is an element of $\\operatorname{Aut}(X^M_T(\\alpha ))$ , then $\\lambda _i = 1$ on green and orange vertices of $T$ .",
"Proof.",
"Let us consider an automorphism in $G_T$ given by invertible elements $\\lambda _i$ .",
"The condition that the equation (REF ) for the vertex $i$ is preserved is $\\prod _{j - i} \\lambda _j = 1.$ This just means that the $\\lambda _i$ belongs to the kernel of the adjacency matrix of $T$ (seen as an endomorphism of $\\mathbb {G}_m^T$ ).",
"Looking at the induced linear equations on the tangent space at one, one can deduce from remark REF that the dimension of $\\operatorname{Aut}(X^M_T(\\alpha ))$ is $\\dim (T)$ .",
"By the same argument (using induction on the auxiliary graph $\\mathcal {G}$ ) as in the uniqueness step of the proof of Prop.",
"REF , every automorphism fixes $x_j$ for every green vertex $j$ .",
"By a similar argument (starting with orange vertices attached to green vertices in the auxiliary graph $\\mathcal {G}$ ), one can then prove that every automorphism fixes $x_j$ for every orange vertex $j$ .",
"There remains to show that $\\operatorname{Aut}(X^M_T(\\alpha ))$ is connected.",
"Let us prove that, given any choice for the values of $\\lambda _i$ for $i\\notin M$ , there is a unique element of $\\operatorname{Aut}(X^M_T(\\alpha ))$ extending this choice.",
"This is once again done by induction using the auxiliary graph $\\mathcal {G}$ .",
"Let us consider a red vertex $j$ that is pointing in $\\mathcal {G}$ only toward vertices with known $\\lambda $ .",
"Then there is a unique way to fix the value $\\lambda _j$ such that (REF ) holds for the green vertex $i$ in the domino of $j$ .",
"This proves that the kernel is isomorphic to $\\mathbb {G}_m^{\\dim (T)}$ .",
"Note that the torus $\\operatorname{Aut}(X^M_T(\\alpha ))$ and its action on $X^M_T(\\alpha )$ do not depend on $\\alpha $ .",
"This action therefore extends to varieties defined as the union of $X^M_T(\\alpha )$ over some family of parameters $\\alpha $ .",
"The torus $\\operatorname{Aut}(X^M_T(\\alpha ))$ can be written as a product of several tori, indexed by the red-green components.",
"Every factor acts only on the red vertices inside a fixed red-green component.",
"This factorization will be useful later to describe free actions on some varieties.",
"Genericity A non-empty set $S$ of red vertices in a red-green component $C$ is called an admissible set if every green vertex in $C$ has either 0 or 2 neighbors in $S$ .",
"Lemma 2.5 Given a red vertex $u$ in $C$ , there is an admissible set containing $u$ .",
"Proof.",
"One can build an admissible set $S$ starting from $\\lbrace u\\rbrace $ by repeated addition of red vertices.",
"If there is a green vertex $v$ with exactly one red neighbor in $S$ , then add to $S$ one of the other red neighbors of $v$ .",
"Repeat until the set $S$ is admissible.",
"Let us now introduce an explicit genericity condition on the parameters attached to a given red-green component $C$ .",
"For every admissible set $S$ of red vertices of $C$ , the alternating product $\\prod _{i \\in S} \\alpha _i^{\\pm } \\ne (-1) ^{\\#S},$ where any two red vertices sharing a common green neighbor have opposite powers in the left hand side.",
"Lemma 2.6 The genericity condition is preserved under jumping moves.",
"Proof.",
"Indeed, consider the jumping move from a red vertex $u$ over a green vertex $v$ .",
"The coefficients of all red neighbors of $v$ are divided by $\\alpha _u$ .",
"Let $S$ be an admissible set.",
"If the vertex $v$ has no neighbor in $S$ , nothing is changed in the genericity condition for $S$ .",
"Otherwise, the vertex $v$ has two neighbors in $S$ .",
"Then two terms are changed in the left-hand side of (REF ), both being divided by $\\alpha _u$ .",
"But they appear with opposite powers, hence the product is not changed.",
"The two other kinds of jumping moves (green over red and orange over orange) do not change the parameters of red vertices.",
"Definition of the varieties Let us now carefully define the varieties that will be studied in the rest of the article.",
"Let us fix a tree $T$ , a choice function $\\varphi $ from the set of red-green components of $T$ to the set $\\lbrace \\mathtt {generic}, \\mathtt {versal}\\rbrace $ and a maximum matching $M$ of $T$ .",
"For every red-green component $C$ such that $\\varphi (C)$ is $\\mathtt {generic}$ , let us fix for every vertex $u$ of $C$ not covered by the maximum matching $M$ , an invertible value $\\alpha _u$ .",
"To this data, one associates a scheme $X^{\\varphi ,M}_{T, \\alpha }$ as follows.",
"The variables are $x_i$ and $x^{\\prime }_i$ for all vertices of $T$ , $\\alpha _i$ for all vertices not covered by the matching $M$ in the red-green components $C$ of $T$ such that $\\varphi (C)$ is $\\mathtt {versal}$ .",
"The equations are the system of equations (REF ), all variables $\\alpha _i$ are invertible.",
"In fact, there is no true dependency on the matching $M$ .",
"Let us consider two maximum matchings $M$ and $M^{\\prime }$ .",
"Using proposition REF , one can find an isomorphism between $X^{\\varphi ,M}_{T,\\alpha }$ for arbitrary invertible parameters $\\alpha $ and $X^{\\varphi ,M^{\\prime }}_{T,\\beta }$ for parameters $\\beta $ depending on the parameters $\\alpha $ .",
"One will therefore forget the matching and use the notation $X^\\varphi _T$ from now on, keeping the parameters $\\alpha $ implicit as well.",
"Moreover, by lemma REF , if the genericity condition (REF ) holds for the parameters $\\alpha $ with respect to one matching $M$ , they will also hold for the corresponding parameters $\\beta $ for another matching $M^{\\prime }$ .",
"One can therefore impose that the genericity condition (REF ) holds for all $\\mathtt {generic}$ red-green components of $T$ .",
"This will always be assumed from now on.",
"Let us summarize this lengthy definition.",
"Once the tree $T$ is chosen, one picks a maximum matching $M$ of $T$ .",
"Any choice of matching will lead to isomorphic varieties.",
"One then decides for every red-green component of $T$ either to take the union over all invertible parameters or to fix some generic parameters.",
"One will use the simplified notation $X_T$ for orange trees, as there is then no choice to be made for the function $\\varphi $ .",
"One will also use the notations $X^\\mathtt {generic}_T$ and $X^\\mathtt {versal}_T$ when the function $\\varphi $ is constant.",
"Remark 2.7 One can as well consider forests instead of trees in the definition of the varieties $X^\\varphi _T$ , but then everything factors according to the connected components.",
"This possibility will be used implicitly in the rest of the article.",
"Let us introduce the notation $U(x)$ for the open set defined by $x\\ne 0$ .",
"Lemma 2.8 If $a-b$ is an edge in a tree $T$ , then the two open sets $U(x_a)$ and $U(x_b)$ cover the variety $X^\\varphi _T$ .",
"Proof.",
"This follows from the exchange relation $x_a x^{\\prime }_a = 1 + \\alpha _a x_b y,$ where $y$ is some product of other cluster variables.",
"Remark 2.9 When removing red vertices or green vertices in a tree $T$ , some red-green components may split into several red-green components.",
"One can then define a function $\\widehat{\\varphi }$ on the new set of red-green components, whose value on a red green component $C$ is the value of $\\varphi $ in the unique red-green component of $T$ containing $C$ .",
"Abusing notation, one will denote this induced function $\\widehat{\\varphi }$ simply by $\\varphi $ .",
"Smoothness and free actions Theorem 3.1 For every choice of $\\varphi $ , the variety $X^\\varphi _T$ is smooth.",
"Proof.",
"The proof is by induction on the size of the tree $T$ .",
"For the tree with only one vertex, the only equation is $x x^{\\prime } = 1 + \\alpha .$ In the $\\mathtt {generic}$ case when $\\alpha $ is considered to have a fixed value, different from $-1$ by the genericity condition (REF ), the variety is isomorphic to the punctured affine line $\\mathbb {G}_m$ and is therefore smooth.",
"In the $\\mathtt {versal}$ case when $\\alpha $ is considered to be a variable and assumed to be invertible, the variety is an open set in the variety defined by (REF ) where $\\alpha $ is not assumed to be invertible.",
"This last variety is isomorphic to the affine plane $A_{2}$ , hence smooth.",
"The rest of the proof by induction is organized as follows.",
"One first considers the case when the tree has at least one red-green component, and treat separately the case when there is a red-green component which is $\\mathtt {generic}$ and the case when there is one which is $\\mathtt {versal}$ .",
"Otherwise, the tree is orange.",
"These three cases are done in the next three subsections.",
"Let us first state a few useful lemmas.",
"Lemma 3.2 If one variable $x_i$ is assumed to be non-zero, then one can get rid of the associated variable $x^{\\prime }_i$ and of the equation (REF ) of index $i$ .",
"Proof.",
"Indeed, one can just use the equation to eliminate $x^{\\prime }_i$ .",
"Lemma 3.3 If one variable $x_i$ is assumed to be zero, then $x^{\\prime }_i$ becomes a free variable and the equation (REF ) of index $i$ reduces to $-1 = \\alpha _i \\prod _{i-j} x_j.$ Let us now introduce a useful variant of the varieties $X_T^\\varphi $ .",
"Let $v$ be a vertex of $T$ .",
"Let $X^\\varphi _T[v]$ be defined just as $X_T^\\varphi $ , but with one more invertible variable $\\gamma _v$ attached to the vertex $v$ as a coefficient (playing the same role as $\\alpha _v$ in the equations).",
"This variable defines a morphism $\\gamma _v$ from $X^\\varphi _T[v]$ to $\\mathbb {G}_m$ .",
"Lemma 3.4 If $v$ is an orange or green vertex, then $X^\\varphi _T[v]$ is isomorphic as a variety over $\\mathbb {G}_m$ to $X_T^\\varphi \\times \\mathbb {G}_m$ endowed with the projection to the second factor.",
"Proof.",
"By proposition REF and its proof, one can find an isomorphism in the groupoid $G_T$ between $X^\\varphi _T$ and $X^\\varphi _T[v]$ that only changes the coordinates $x_i$ for orange and green vertices.",
"More precisely, using the auxiliary oriented graph $\\mathcal {G}$ , one can find a sequence of jumps (corresponding to edges in $\\mathcal {G}$ starting with a green or orange vertex) that makes the coefficient $\\gamma _v$ disappear from the equations.",
"The isomorphism associated with this sequence of jumps is multiplying the variables $x_i$ by monic Laurent monomials in the parameter $\\gamma _v$ , hence defines an isomorphism over $\\mathbb {G}_m$ .",
"Lemma 3.5 If $v$ is a red vertex in a versal red-green component $C$ , then $X^\\varphi _T[v]$ is isomorphic as a variety over $\\mathbb {G}_m$ to $X_T^\\varphi \\times \\mathbb {G}_m$ endowed with the projection to the second factor.",
"Proof.",
"If the red vertex $v$ is not covered by the matching $M$ chosen to define $X^\\varphi _T$ , then one has two coefficient variables $\\alpha _v$ and $\\gamma _v$ attached to the vertex $v$ .",
"By the simple change of coordinates $\\alpha _v := \\alpha _v \\gamma _v$ and $\\gamma _v := \\gamma _v$ , one gets the expected isomorphism.",
"Assume now that that red vertex $v$ is covered by the matching $M$ .",
"By proposition REF and its proof, one can find an isomorphism in the groupoid $G_T$ between $X^\\varphi _T[v]$ and a variety $X^{\\varphi ,M}_{T,\\beta }$ that only changes the coordinates $x_i$ for orange and green vertices and for red vertices in the red-green component $C$ .",
"More precisely, using the auxiliary oriented graph $\\mathcal {G}$ , one can find a sequence of jumps that moves the coefficient $\\gamma _v$ towards the red vertices in $C$ not covered by the matching.",
"At the end, every new coefficient $\\beta _i$ is the product of $\\alpha _i$ by a Laurent monomial in $\\gamma _v$ .",
"The isomorphism associated with this sequence of jumps is multiplying the variables $x_i$ by monic Laurent monomials in the parameter $\\gamma _v$ , hence defines an isomorphism over $\\mathbb {G}_m$ .",
"One can then compose this isomorphism with a relabeling of the coefficients $\\alpha _i := \\beta _i$ in order to get the expected isomorphism, still defined over $\\mathbb {G}_m$ , between $X^\\varphi _T[v]$ and $X_T^\\varphi \\times \\mathbb {G}_m$ .",
"One could say that the coefficient $\\gamma _v$ can be detached from $T$ in these cases.",
"This will be used frequently in the rest of the article.",
"Trees with a generic component One assumes now that $T$ has at least two vertices and a $\\mathtt {generic}$ component $C$ .",
"Let us pick an admissible set $S$ of red vertices in $C$ , as defined in §REF .",
"Lemma 3.6 The open sets $U(x_i)$ for $i \\in S$ form a covering of $X^\\varphi _T$ .",
"Proof.",
"Indeed, the complement of their union is the set where all variables $x_i$ for $i \\in S$ vanish.",
"This implies that $\\alpha _i \\prod _{j-i} x_j = -1$ for every $i$ in $S$ .",
"Taking the alternating product of these equalities gives $\\prod _{i \\in S} \\alpha _i^{\\pm } = (-1)^{\\# S},$ because for every green vertex $j$ attached by an edge to some element of $S$ , the cluster variable $x_j$ appears exactly twice by definition of admissible sets, hence disappears in the alternating product.",
"But the equation (REF ) is incompatible with the genericity condition (REF ).",
"Let us now show that the open sets $U(x_i)$ are smooth.",
"Let $F$ be the forest $ T \\setminus \\lbrace i\\rbrace $ .",
"In the forest $F$ , the coloring is changed only on the red-green component containing $i$ , where an avalanche of orange dominoes can take place when removing $i$ .",
"The red-green component $C$ is therefore split into a number of red-green components.",
"Let us moreover introduce a function $\\varphi $ on $F$ , which is $\\mathtt {generic}$ on every red-green component coming from $C$ , and unchanged on all other red-green components.",
"Lemma 3.7 The open set $U(x_i)$ is isomorphic to $\\mathbb {G}_m\\times X^\\varphi _F$ .",
"Proof.",
"The condition that $x_i$ is not zero allows one to get rid of the variable $x^{\\prime }_i$ by using the equation (REF ) of index $i$ .",
"What remains are the equations for the forest $F = T \\setminus \\lbrace i\\rbrace $ , where now $x_i$ is treated as a parameter attached to all neighbors of $i$ in $T$ .",
"Because all neighbors of $i$ in $T$ are green, they become either green or orange in $F$ .",
"It follows from lemma REF that one can, without changing the variety, consider instead that the parameter $x_i$ is not attached to any vertex of $F$ .",
"Let us check that the genericity condition still holds on all $\\mathtt {generic}$ red-green components.",
"If the component $D$ does not come from the splitting of $C$ , then the genericity conditions are unchanged on this red-green component.",
"Otherwise, let us choose an admissible set in $D$ .",
"It was then already an admissible set in $C$ , by inspection of what happens during the avalanche of orange dominoes.",
"Therefore the genericity condition for $D$ is inherited from that for $C$ .",
"One has therefore obtained an isomorphism $U(x_i) \\simeq \\mathbb {G}_m\\times X^\\varphi _F,$ which is smooth by induction.",
"Therefore $X^\\varphi _T$ is also smooth.",
"Trees with a versal component One assumes now that $T$ has at least two vertices, and has a $\\mathtt {versal}$ component $C$ .",
"Let us choose a red leaf $v$ in this component.",
"By proposition REF , one can find a maximum matching $M$ not containing $v$ .",
"Therefore there is a coefficient variable $\\alpha _v$ .",
"Let $u$ be the green vertex adjacent to $v$ .",
"By lemma REF , the two open sets $U(x_u)$ and $U(x_v)$ cover $X^\\varphi _T$ .",
"Let us first prove that $U(x_v)$ is smooth.",
"Let $T^{\\prime }$ be the tree $T\\setminus \\lbrace v\\rbrace $ .",
"The coloring of $T^{\\prime }$ is obtained from $T$ by an avalanche of orange dominoes.",
"The dimension of $T^{\\prime }$ is $\\dim (T) - 1$ .",
"The avalanche may split the red-green component of $T$ containing $v$ into several components.",
"Let $\\varphi $ be the function which maps all these new components to the $\\mathtt {versal}$ condition, and unchanged condition on all the other red-green components.",
"Lemma 3.8 The open set $U(x_v)$ is isomorphic to $\\mathbb {G}_m^2 \\times X^\\varphi _{T^{\\prime }}$ .",
"Proof.",
"Assuming that $x_v$ is not zero allows one to get rid of the variable $x^{\\prime }_v$ by using (REF ) with index $v$ .",
"The coefficient variable $\\alpha _v$ also disappears from the equations: this gives one factor $\\mathbb {G}_m$ .",
"Then the variable $x_v$ is seen as a coefficient attached to the vertex $u$ in $T^{\\prime }$ , which is either green or orange.",
"The coefficient can therefore be detached by lemma REF , and one obtains a factor isomorphic to $\\mathbb {G}_m\\times X^\\varphi _{T^{\\prime }}$ .",
"Therefore $U(x_v)$ is smooth by induction.",
"Let us now prove that $U(x_u)$ is smooth.",
"Let us choose instead a matching $M$ containing the domino $u-v$ , thanks to lemma REF .",
"This amounts to go through an isomorphism in the groupoid $G_T$ , hence preserves the open set $U(x_u)$ .",
"Let $F$ be the forest $T\\setminus \\lbrace u\\rbrace $ .",
"Because $u$ is green, the coloring of $F$ is obtained from that of $T$ by restriction and the dimension of $F$ is $\\dim (T) + 1$ .",
"Let $v, T_1, \\dots , T_k$ be the connected components of the forest $F$ .",
"By removing the domino $u-v$ , one can restrict the matching $M$ to a matching of the forest $F$ .",
"The red-green component of $T$ containing $u$ splits into several red-green components in $F$ , one of them being the vertex $v$ .",
"One takes the $\\mathtt {versal}$ condition on all of these red-green components of $F$ , and unchanged condition on all the other red-green components.",
"Lemma 3.9 The open set $U(x_u)$ is isomorphic to $X^{\\mathtt {versal}}_{\\lbrace v\\rbrace } \\times \\prod _{j=1}^k X^\\varphi _{T_j},$ where the first component is the vertex $v$ with coefficient variable $x_u$ .",
"Proof.",
"Setting $x_u\\ne 0$ in the equations allows to get rid of the variable $x^{\\prime }_u$ .",
"The result can be described as a fiber product over $\\mathbb {G}_m$ , where the same coefficient variable $x_u$ is attached to every connected component of $F$ at a red vertex in a versal red-green component.",
"By repeated use of lemma REF on all connected components (but not on the isolated vertex $v$ ), one finds that the open set $U(x_u)$ is isomorphic to the product $X^{\\mathtt {versal}}_{v} \\times \\prod _{j=1}^k X^\\varphi _{T_j},$ where the first component is the vertex $v$ with coefficient $x_u$ .",
"Therefore $U(X_u)$ is smooth by induction, and hence $X^\\varphi _T$ is also smooth.",
"Orange trees Let us now assume that $T$ is an orange tree and let us choose one domino $u-v$ in the perfect matching of $T$ .",
"By lemma REF , the two open sets $U(x_u)$ and $U(x_v)$ cover the variety $X_T$ .",
"By symmetry between $u$ and $v$ , it is enough to prove that $U(x_u)$ is smooth.",
"Let $T_1, \\dots , T_k$ be the trees attached to $u$ in $T\\setminus \\lbrace v\\rbrace $ .",
"The $T_i$ are clearly orange trees.",
"Let $R$ be the connected component of $v$ in $T\\setminus \\lbrace u\\rbrace $ .",
"The tree $R$ is obtained by removing a leaf in an orange tree, hence (by lemma REF ) has dimension 1 and a unique red-green component.",
"Moreover, $R$ has a maximum matching avoiding only $v$ and the vertex $v$ is red in the coloring of $R$ .",
"Lemma 3.10 The open set $U(x_u)$ is isomorphic to the product of the varieties $X_{T_i}$ and the variety $X^\\mathtt {versal}_R$ .",
"Proof.",
"Assuming that $x_u$ is not zero allows to eliminate the variable $x^{\\prime }_u$ and the equation (REF ) of index $u$ .",
"There remains the equations for the union of $R$ and the $T_i$ , with $x_u$ considered as a parameter attached to all of them at the former neighbors of $u$ .",
"Because the trees $T_i$ are orange, one can consider instead (by lemma REF ) that the parameter $x_u$ is only attached to the vertex $v$ of $R$ .",
"This proves that the open set $U(x_u)$ is isomorphic to the product of the varieties $X_{T_i}$ and the variety $X^\\mathtt {versal}_R$ .",
"By induction, this proves that $U(x_u)$ is smooth.",
"Therefore $X_T$ is smooth too.",
"Torus actions Let $T$ be a tree and let $\\varphi $ be a choice in $\\lbrace \\mathtt {generic},\\mathtt {versal}\\rbrace $ for every red-green component of $T$ .",
"Let us also choose a maximum matching $M$ of $T$ .",
"One can deduce from proposition REF and the remarks following it that there is an action of an algebraic torus of dimension $\\dim (T)$ on $X^\\varphi _T$ , and that this torus (and its action) can be written as a product over red-green components $C$ of tori $\\Lambda ^C_T$ .",
"Let us define a smaller torus $\\Lambda ^\\varphi _T$ acting on $X^\\varphi _T$ as the product of $\\Lambda ^C_T$ over all $\\mathtt {generic}$ red-green components of $T$ .",
"Let us call the rank of $(T, \\varphi )$ and denote by $\\operatorname{rk}(T,\\varphi )$ the sum of the dimensions of the generic red-green components of $T$ .",
"This is the dimension of $\\Lambda ^\\varphi _T$ .",
"Proposition 3.11 If $\\varphi (C)$ is generic, the action of $\\Lambda ^C_T$ on $X^\\varphi _T$ is free.",
"Proof.",
"Let us assume that there is a non-trivial element $\\lambda =(\\lambda _i)_i$ of $\\Lambda ^C_T$ that fixes a point $(x_i)_i$ in $X^\\varphi _T$ .",
"Let $i$ be a red vertex in $C$ such that $\\lambda _i \\ne 1$ .",
"For every green neighbor $j$ of $i$ , one can find another red vertex $k$ incident to $j$ such that $\\lambda _k \\ne 1$ , because of (REF ).",
"Iterating this process, one can build an admissible set $S$ (as defined in §REF ), such that $\\lambda _s \\ne 1$ for every $s \\in S$ .",
"Because $\\lambda $ fixes the given point, one then has $x_s = 0$ for every $s \\in S$ .",
"But this is impossible by Lemma REF .",
"Corollary 3.12 There is on $X^\\varphi _T$ a free action by a torus $\\Lambda ^\\varphi _T$ of dimension the rank $\\operatorname{rk}(T,\\varphi )$ .",
"This gives $X^\\varphi _T$ the structure of a principal bundle with structure group $\\Lambda ^\\varphi _T$ .",
"As one will see later, this bundle is not trivial in general (i.e.",
"not a product), as can be seen from our results for the cohomology already in type $\\mathbb {A}_3$ .",
"Number of points over finite fields and Euler characteristic Let us denote by $N^\\varphi _T(q)$ the number of points on $X^\\varphi _T$ over the finite field $\\mathbb {F}_q$ .",
"When the tree is orange, one will use the shorthand notation $N_T$ .",
"When the function $\\varphi $ is constant, one will use the notations $N^\\mathtt {versal}_T$ and $N^\\mathtt {generic}_T$ .",
"Proposition 4.1 The numbers $N^\\varphi _T(q)$ are monic polynomials in $q$ of degree $\\dim X_T^\\varphi $ .",
"Proof.",
"The proof is by induction on the size of the tree.",
"For the tree with one vertex, the number of points is $q-1$ in the $\\mathtt {generic}$ case and $q^2 - q + 1$ in the $\\mathtt {versal}$ case, by the description given at the beginning of the proof of theorem REF .",
"Then either the tree has a red-green component, which can be $\\mathtt {generic}$ or $\\mathtt {versal}$ , or it is an orange tree.",
"The proof is decomposed into the three following geometric decomposition lemmas, or rather into their obvious corollaries on the number of points over finite fields.",
"Let $T$ be a tree and $v$ be a red leaf in a red-green component $C$ of $T$ .",
"Let $u$ be the neighbor of $v$ .",
"Removing the vertex $v$ creates an orange avalanche and may separate the red-green component $C$ into several ones.",
"Let $\\varphi $ be the induced genericity condition (as defined in Remark REF ).",
"Let $F$ be the forest $T\\setminus \\lbrace u, v\\rbrace $ .",
"The component $C$ may also split into several red-green components in $F$ .",
"Let $\\varphi $ be the induced genericity condition.",
"Let us consider now the case of a generic red-green component $C$ .",
"Lemma 4.2 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_mX^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"In the case where $x_v \\ne 0$ , one uses lemma REF .",
"This gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u= -1$ is attached to some red vertices of $F$ , as a coefficient.",
"One has to check that the genericity condition still holds on every connected component of $F$ .",
"Let $S$ be an admissible set in one of these components.",
"Either $S$ was already an admissible set in $T$ , and then the genericity condition still holds, or it contains exactly one of the neighbors of $u$ in $T$ .",
"In this case, one can extend $S$ by adding $v$ to form an admissible set in $T$ .",
"The genericity condition for $S \\sqcup \\lbrace v\\rbrace $ in $T$ implies the condition for $S$ , because of the additional $-1$ coefficient attached to $S$ in $F$ .",
"Keeping the same notations, let us consider now the case of a versal red-green component $C$ .",
"Lemma 4.3 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_m^2 X^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"If $x_v \\ne 0$ , using lemma REF gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u = -1$ is attached to red vertices of $F$ .",
"By lemma REF , it can be detached, and this just gives the expected second term.",
"Let $T$ be an orange tree and $u-v$ be a domino in $T$ .",
"Let $(T_{u,i})_i$ (resp.",
"$(T_{v,j})_j$ ) be the connected components of $T\\setminus \\lbrace u,v\\rbrace $ that were attached to $u$ (resp.",
"to $v$ ).",
"All these trees are orange.",
"Let us denote by $S_{u,i}$ and $S_{v,j}$ the forests obtained from them by removing the vertex that was linked to $u$ or $v$ .",
"These forests are unimodal, in the sense that they have one unimodal connected component, all the other connected components being orange.",
"Lemma 4.4 In this situation, one has $X_T = \\mathbb {G}_m^2 \\prod _i X_{T_{u,i}} \\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X^\\mathtt {versal}_{S_{u,i}}\\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X_{T_{u,i}}\\prod _j X^\\mathtt {versal}_{S_{v,j}} .$ Proof.",
"Because the open sets $U(x_u)$ and $U(x_v)$ are a covering by lemma REF , one can cut the variety $X_T$ into three pieces: either both $x_u$ and $x_v$ are not zero, or exactly one of them is zero.",
"If both are not zero, then one obtains the product of $\\mathbb {G}_m^2$ (with coordinates $x_u$ and $x_v$ ) with the product of the varieties attached to the $T_{u,i}$ and the $T_{v,j}$ .",
"Indeed, one first get that $x_u$ becomes a parameter attached to all trees $T_{u,i}$ and $x_v$ becomes a parameter attached to all trees $T_{v,j}$ .",
"But these trees are orange, so $x_u$ and $x_v$ can be detached by lemma REF .",
"This gives the first term.",
"If $x_u$ is zero and $x_v$ is not zero, then there is a free variable $x^{\\prime }_u$ and the variable $x_v$ is determined by the variables attached to the vertices of the trees $T_{u,i}$ linked to $u$ , which must be non-zero.",
"One obtains therefore a versal condition on each forest $S_{u,i}$ .",
"For the trees $T_{v,j}$ , the coefficient $x_v$ is attached to all of them, but because they are orange it can be detached.",
"This gives the second term.",
"The third term is the same after exchanging $u$ and $v$ .",
"Reciprocal property Recall from §REF that the rank $\\operatorname{rk}(T,\\varphi )$ of the pair $(T,\\varphi )$ formed by a tree $T$ and a choice function $\\varphi $ is the sum of the dimensions of the $\\mathtt {generic}$ red-green components of $T$ .",
"Proposition 4.5 The polynomial $N^\\varphi _T(q)$ is divisible par $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ .",
"Proof.",
"This follows from the existence of the free action obtained in corollary REF .",
"Let us refine this slightly.",
"Proposition 4.6 The polynomial $N_T^\\varphi $ can be written as $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ times a reciprocal polynomial.",
"Proof.",
"By induction.",
"This is true for the tree with one vertex.",
"One just has to look carefully at the decompositions given in the three lemmas that were used to prove polynomiality by induction.",
"For lemma REF , let $D$ be the rank for $T$ .",
"Then the rank is $D-1$ for $T \\setminus \\lbrace v\\rbrace $ and $D$ for $F$ .",
"Using the additional factor $q-1$ coming from $\\mathbb {G}_m$ , there is a common factor $(q-1)^D$ to all terms involved.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is the same in all terms involved.",
"One uses that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is 0 in all terms involved, as there is no generic red-green component.",
"One uses again that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 pieces ensures that the reciprocal property holds.",
"Enumeration and coincidences In the following remarks, one will describe trees by their numbers in the tables at the end of [5] and by their graph6 string (which is a standard format for graphs).",
"Remark 4.7 One can find distinct orange trees with the same enumerating polynomial.",
"This happens first for trees with 8 vertices.",
"The trees 2.188 (graph6 'IhGGOC@?G') and 2.189 (graph6 'IhC_GCA?G') have the same polynomial, as well as the trees 2.172 (graph6 'IhGGOCA?G') and 2.174 (graph6 'IhGH?C@?G').",
"The number of different polynomials for orange trees with $2n$ vertices is the sequence $1, 1, 2, 5, 13, 41, 138, \\dots $ whereas the number of orange trees is $1, 1, 2, 5, 15, 49, 180, \\dots $ Remark 4.8 For unimodal trees with $\\mathtt {versal}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of trees with 9 vertices, numbered 2.83 (graph6 'HhCGOCA') and 2.85 (graph6 'HhGGGG@').",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 6, 19, 65, \\dots $ whereas the number of unimodal trees is $1, 1, 2, 6, 20, 76, 313, 1361, \\dots $ Remark 4.9 For unimodal trees with $\\mathtt {generic}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of the Dynkin diagrams $\\mathbb {A}_7$ and $\\mathbb {E}_7$ .",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 5, 13, 46, 168, \\dots $ Linear trees Let us denote by $\\mathbb {A}_n$ the linear tree with $n$ vertices.",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) node[fill=orange!20] 3 – (3,0) node[fill=orange!20] 4 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=red!20] 1 – (1,0) node[fill=green!20]2 – (2,0) node[fill=red!20] 3 – (3,0) node[fill=green!20] 4 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; One can check that $\\mathbb {A}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.10 The number of points on varieties attached to $\\mathbb {A}_n$ is given by $N_{\\mathbb {A}_n} = \\frac{q^{n+2} - 1}{q^2 -1}$ if $n$ is even and by $N_{\\mathbb {A}_n}^\\mathtt {versal}= \\frac{q^{n+2} + 1}{q + 1} \\quad \\text{and} \\quad N_{\\mathbb {A}_n}^\\mathtt {generic}= \\frac{(q^{(n+1)/2} - 1)(q^{(n+3)/2} - 1)}{q^2 -1}$ if $n$ is odd.",
"Proof.",
"This follows easily by induction from lemmas REF , REF and REF .",
"Trees of type $\\mathbb {D}$ Let us denote by $\\mathbb {D}_n$ the tree with $n$ vertices associated with the Dynkin diagram of type $\\mathbb {D}$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=orange!20] 4 – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=red!20] 4 – (3,0) node[fill=green!20] 5 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; One can check that $\\mathbb {D}_n$ is unimodal if $n$ is odd and has dimension 2 if $n$ is even.",
"Proposition 4.11 The number of points on varieties attached to $\\mathbb {D}_n$ is given by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}+q^3-q+1}{q + 1} \\quad \\text{and} \\quad N^\\mathtt {generic}_{\\mathbb {D}_n} = (q^{n/2} - 1)^2$ if $n$ is even and by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}-q^3+q-1}{q^2-1} \\quad \\text{and} \\quad N_{\\mathbb {D}_n}^\\mathtt {generic}= q^n-1$ if $n$ is odd.",
"Proof.",
"This is easily deduced from the type $\\mathbb {A}$ case, using REF , REF applied to a red leaf on a short branch.",
"Trees of type $\\mathbb {E}$ Let us consider now a family of trees containing the Dynkin diagrams of type $\\mathbb {E}$ .",
"The tree $\\mathbb {E}_n$ is the tree with one triple point and branches of size 1, 2 and $n-4$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=red!20] 5 – (4,0) node[fill=green!20] 6 – (5,0) node[fill=red!20]7; (2,1) node[fill=red!20] 3 – (2,0) node[fill=green!20] 4; One can check that $\\mathbb {E}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.12 The number of points on varieties attached to $\\mathbb {E}_n$ is given by $N_{\\mathbb {E}_n} = (q^2 - q + 1)\\frac{q^{n-1} - 1}{q-1}$ if $n$ is even and by $N_{\\mathbb {E}_n}^\\mathtt {versal}= (q^2 - q + 1)(1+q^{n-1}) $ and $N_{\\mathbb {E}_n}^\\mathtt {generic}= \\frac{q^{n+1}-q^{n}+q^{n-1}-q^{(n+3)/2}-q^{(n-1)/2}+q^{2}-q+1}{q-1}$ if $n$ is odd.",
"Proof.",
"In the even case, one uses lemma REF applied to the domino on the short branch, and the known type $\\mathbb {A}$ cases.",
"In the odd case, one uses lemmas REF and REF applied to the red leaf on the short branch, and the known type $\\mathbb {A}$ cases.",
"Orange trees and unimodal trees Let us now describe a recursion involving only the polynomials for orange trees and versal unimodal trees.",
"Let $T$ be an orange tree and $v$ be a leaf of $T$ .",
"Let $T^{\\prime }$ be the unimodal tree $T \\setminus \\lbrace v\\rbrace $ and let $F$ be the orange forest obtained from $T$ by removing the domino $u - v$ containing $v$ .",
"Lemma 4.13 There is a decomposition $X_T = X^{\\mathtt {versal}}_{T^{\\prime }} \\sqcup A_{1} X_F.$ Proof.",
"This decomposition is made according to the value of $x_v$ .",
"If $x_v = 0$ , then one has a free parameter $x^{\\prime }_v$ , which gives the factor $A_{1}$ .",
"One also has $x_u = -1$ and one can get rid of $x^{\\prime }_u$ .",
"The value $-1$ is attached as a coefficient to some orange vertices of $F$ , but one can detach this coefficient by lemma REF .",
"There remains the equations for $X_F$ .",
"If $x_v \\ne 0$ , one can use lemma REF .",
"In the special case of a leaf, this gives an isomorphism with $X^\\mathtt {versal}_{T^{\\prime }}$ .",
"One can use lemma REF to compute the enumerating polynomials for orange trees and versal unimodal trees only, by the following algorithm.",
"Step 0: if the tree $T$ is of type $\\mathbb {A}_n$ with $n$ even, use the known value from (REF ) in proposition REF .",
"Step 1: if the tree $T$ is orange, find a leaf $v$ whose branch has minimal length.",
"Here the branch is the longest sequence of vertices of valency 2 starting at the unique neighbor of the leaf (it could be empty).",
"Then use lemma REF applied to the leaf $v$ to compute $N_T$ .",
"Step 2: if the tree $T$ is unimodal, find a red leaf $w$ whose branch has maximal length.",
"Adding a vertex $v$ at the end of this branch gives an orange tree $T^{\\prime }$ .",
"Then use lemma REF (backwards) applied to the tree $T^{\\prime }$ and its leaf $v$ to compute $N_T$ .",
"This will work because each step either shorten the shortest branch or add some vertex to the longest branch.",
"This makes sure that the tree become more and more linear, and that at some point one is reduced to the initial step.",
"This is a decreasing induction on the number of points of valency at least 3 and the length of the longest branch.",
"Remark 4.14 For orange trees, one can use instead in this algorithm the lemma REF , maybe choosing a domino close to the center of the tree for a better complexity.",
"Euler characteristic and independent sets Let us denote by $\\operatorname{vc}(T)$ the number of minimum vertex covers of $T$ .",
"This is also the number of maximum independent sets.",
"Let us now describe a decomposition of the versal varieties according to independent sets (not necessarily maximal).",
"If $S$ is a subset of the vertices of $T$ , one can define $W_T(S)$ as the set of points in $X_T^\\mathtt {versal}$ where $x_u = 0 & \\quad \\text{if } u\\in S,\\\\x_u \\ne 0 & \\quad \\text{if } u\\notin S.$ The sets $W_T(S)$ are obviously disjoint in $X_T^\\mathtt {versal}$ .",
"Lemma 4.15 If the set $W_T(S)$ is not empty, then $S$ is an independent set in $T$ .",
"Proof.",
"This follows from lemma REF .",
"Proposition 4.16 Let $S$ be an independent set in $T$ .",
"There is an isomorphism $W_T(S) \\simeq (\\mathbb {G}_m)^{t + \\dim (T)- 2s} \\times (A_{1})^{s},$ where $t$ is the size of $T$ and $s$ the size of $S$ .",
"Proof.",
"Let us fix a maximum matching $M$ of $T$ .",
"For every $u$ not in $S$ , one can use the hypothesis $x_u \\ne 0$ to get rid of $x^{\\prime }_u$ and of the equation of index $u$ .",
"There remains only the equations of index $v$ for $v \\in S$ .",
"Because $x_v=0$ when $v \\in S$ , the variables $x^{\\prime }_v$ for $v \\in S$ do no longer appear in the equations, hence they are free.",
"This gives the factor $(A_{1})^s$ .",
"Then there remains $s$ equations of the general shape $-1 = \\alpha _i \\prod _{j - i} x_j,\\qquad \\mathrm {(E_i)}$ involving the $t-s$ invertible variables $x_u$ and the $\\dim (T)$ coefficient variables $\\alpha _i$ .",
"The factor $\\alpha _i$ is present in this equation only if the vertex $i$ is not covered by the chosen maximum matching $M$ .",
"One will use the following auxiliary graph $\\widehat{T}$ .",
"The vertices are the vertices of $T$ and new vertices $Z_i$ indexed by coefficient variables $\\alpha _i$ for $i \\notin M$ .",
"The edges of $\\widehat{T}$ are edges of $T$ and new edges between the vertex $Z_i$ and the vertex $i$ for every $i \\notin M$ .",
"Clearly, this graph is still a tree and admits a perfect matching $\\widehat{M}$ , by adding dominoes $i-Z_i$ to the matching $M$ .",
"Because $S$ is an independent set in $T$ , there is at most one element of $S$ in every edge of $\\widehat{T}$ .",
"Let us orient every edge containing an element of $S$ towards this element if the edge is a domino and in the other way otherwise.",
"This defines a partial order on the vertices of $\\widehat{T}$ , decreasing along the chosen orientation of edges.",
"Consider now the equation REF associated with a vertex $i \\in S$ .",
"There is a unique domino $i-j$ in $\\widehat{T}$ containing $i$ .",
"The equation can then be used to express the variable $x_j$ in terms of variables of lower index in the partial order.",
"One can therefore eliminate one variable for every equation.",
"At the end, one obtains an algebraic torus whose dimension is the difference between the number $ t -s +\\dim (T)$ of initial variables and the number $s$ of equations.",
"Corollary 4.17 The Euler characteristic of $X_T^\\mathtt {versal}$ is $\\operatorname{vc}(T)$ .",
"Proof.",
"Every set $W_T(S)$ contributes either 0 or 1 to the Euler characteristic.",
"It contributes by 1 if and only if the exponent $t+ \\dim (T)- 2s$ is zero.",
"This exponent can be expressed as $(r(T) + o(T) + g(T)) + (r(T) - g(T)) - 2 s.$ It is therefore zero if and only if $s = r(T) + o(T)/2$ , which is the size of the maximum independent sets in $T$ .",
"Of course, one can also use Proposition REF to give a formula for the number of points $N^\\mathtt {versal}_T$ as a sum over independent sets.",
"Corollary 4.18 The value at $q=1$ of the polynomial $N^\\mathtt {versal}_T$ is the number $\\operatorname{vc}(T)$ of maximum independent sets of $T$ .",
"Cohomology: general setting and results This section first describes some differential forms that are always present in the varieties under study, and then very briefly recalls the results one needs about (mixed) Hodge structures.",
"For a general reference about mixed Hodge structures, see for example [18].",
"Weil-Petersson two-form Let $T$ be a tree and let $S$ be a subset of $T$ .",
"Consider the augmented tree $T+S$ obtained by adding a new edge out of every vertex in $S$ , and endow this tree with a bipartite orientation, where every vertex is either a sink or a source.",
"As a variant of the definition of the variety $X_T^\\varphi $ , one can define a variety $X(T+S)$ attached to this data, with invertible variables associated to the new vertices, playing the role of coefficients in the equations (as the $\\alpha $ do).",
"Let $\\omega _i$ denote $d \\log (x_i)$ .",
"The following lemma has been proved by Greg Muller in [16] in a more general context.",
"Lemma 5.1 The differential form $\\operatorname{WP}= \\sum _{i \\rightarrow j} \\omega _i \\omega _j,$ where the sum is running over edges of $T+S$ , is an algebraic differential form on the variety $X(T+S)$ .",
"Proof.",
"Let us prove that it has no pole.",
"Let us fix $i$ .",
"To study the possible pole along $x_i = 0$ , it is enough to look at the sum $\\sum _{j \\leftrightarrow i} \\omega _i \\omega _j$ restricted to edges containing $i$ .",
"By the relation $x_i x^{\\prime }_i = 1 + \\prod _{j\\leftrightarrow i} x_j$ , one has $x_i d x^{\\prime }_i + x^{\\prime }_i d x_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j,$ and therefore $x_i dx^{\\prime }_i dx_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j d x_i.$ This implies $dx^{\\prime }_i dx_i / \\prod _{k\\leftrightarrow i} x_k = \\sum _{j\\leftrightarrow i} \\omega _j \\omega _i,$ where the left-hand side has clearly no pole at $x_i$ .",
"Note that $\\operatorname{WP}$ stands here for Weil-Petersson.",
"Abusing notations, one will use the same symbol $\\operatorname{WP}$ to denote these differential forms on different varieties.",
"The ambient variety should be clear from the context.",
"Hodge structures We will use the notation $\\mathbb {Q}(-i)$ to denote a one dimensional vector space over $\\mathbb {Q}$ endowed with a pure Hodge structure of Tate type, of weight $2i$ and type $(i,i)$ .",
"The tensor product of $\\mathbb {Q}(-i)$ and $\\mathbb {Q}(-j)$ is $\\mathbb {Q}(-i-j)$ .",
"Recall that the cohomology of $\\mathbb {G}_m$ has an Hodge structure described by $\\mathsf {H}^k(\\mathbb {G}_m) = \\mathbb {Q}(-k)$ for $0 \\le k \\le 1$ .",
"There is no morphism between pure Hodge structures of distinct weights.",
"The Künneth isomorphism is compatible with the Hodge structures.",
"The Mayer-Vietoris long exact sequence is an exact sequence of Hodge structures.",
"Cohomology: orange and versal cases This section deals with the cohomology, in several cases where either varieties do not depend on parameters, or versal conditions are assumed on all parameters.",
"The first part is devoted to linear trees; the results there can then be used as building blocks.",
"Linear trees $\\mathbb {A}$ Let $\\mathbb {A}_n$ be the linear tree with $n$ vertices numbered from 1 to $n$ .",
"As seen in §REF , this is an orange tree if $n$ is even, and an unimodal tree otherwise.",
"Some of the results of this section were already obtained in [6] using instead the cohomology with compact supports.",
"Cohomology of some auxiliary varieties for $\\mathbb {A}$ Let us introduce three varieties $X_n$ , $Y_n$ and $Z_n$ with dimensions $n, n+1$ and $n+1$ .",
"The variety $Z_n$ is defined by variables $x_1,\\dots ,x_n$ , $x^{\\prime }_1,\\dots ,x^{\\prime }_n$ and $\\alpha $ such that $x_1 x^{\\prime }_1 &= 1 + \\alpha x_2 ,\\\\x_i x^{\\prime }_i &= 1 + x_{i-1} x_{i+1}, \\\\x_n x^{\\prime }_n &= 1 + x_{n-1}.$ The variety $Y_n$ is the open set in $Z_n$ where $\\alpha $ is invertible.",
"The variety $X_n$ is the closed set in $Y_n$ where $\\alpha $ is fixed to a generic invertible value (where generic means distinct from $(-1)^{(n+1)/2}$ if $n$ is odd).",
"In our general notations, $Y_n$ is $X_{\\mathbb {A}_n}^\\mathtt {versal}$ and $X_n$ is $X_{\\mathbb {A}_n}^\\mathtt {generic}$ .",
"Let us first describe the variety $Z_n$ .",
"Proposition 6.1 There exists an isomorphism between $Z_n$ and the affine space $A_{n+1}$ .",
"Proof.",
"This has been proved in [6].",
"Therefore, the cohomology of $Z_n$ is known for all $n$ : $\\mathsf {H}^k(Z_n) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}1 &\\text{ if } k=0,\\\\0 &\\text{ if } k>0.\\end{array}\\right.",
"}$ The Hodge structure on $\\mathsf {H}^0(Z_n)$ is $\\mathbb {Q}(0)$ .",
"Let us now compute the cohomology of $Y_n$ by induction.",
"This uses the Mayer-Vietoris long exact sequence for the covering of $Z_n$ by the two open sets $U(x_1)$ and $U(\\alpha )$ .",
"First, let us note that $U(\\alpha ) \\simeq Y_n $ by definition.",
"Next, one finds that $U(x_1) \\simeq A_{1} Y_{n-1}$ .",
"Indeed one can eliminate $x^{\\prime }_1$ using the first equation.",
"Then $\\alpha $ becomes a free variable, and there remains the equations for $Y_{n-1}$ , with $x_1$ now playing the role of $\\alpha $ .",
"Last, the intersection $U(\\alpha ) \\cap U(x_1)$ is isomorphic to $ \\mathbb {G}_mY_{n-1}$ , by the same argument.",
"Let us write $\\omega _{\\alpha }$ for $d \\log (\\alpha )$ .",
"Proposition 6.2 The cohomology ring of $Y_n$ has the following description: $\\mathsf {H}^k(Y_n) = \\mathbb {Q}(-k)$ for $0 \\le k \\le n+1$ .",
"It has a basis given by powers of $\\operatorname{WP}$ in even degrees and by powers of $\\operatorname{WP}$ times $\\omega _{\\alpha }$ in odd degrees.",
"It is generated by the 1-form $\\omega _{\\alpha }$ and the 2-form $\\operatorname{WP}$ .",
"Proof.",
"Because of the vanishing of $\\mathsf {H}^k(Z_n)$ for $k>0$ , the Mayer-Vietoris long exact sequence gives short exact sequences $0 \\rightarrow \\mathsf {H}^0(Z_n) \\rightarrow \\mathsf {H}^0(Y_n)\\oplus \\mathsf {H}^0(U(x_1)) \\rightarrow \\mathsf {H}^0(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ and $0 \\rightarrow \\mathsf {H}^k(Y_n)\\oplus \\mathsf {H}^k(U(x_1)) \\rightarrow \\mathsf {H}^k(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ for every $k > 0$ .",
"This determines by induction the Hodge structure of the cohomology of $Y_{n}$ .",
"Let us now proceed to the expected basis.",
"One already knows that $\\operatorname{WP}$ and $\\omega _{\\alpha }$ are indeed algebraic differential forms on $Y_{n}$ .",
"By the short exact sequences above, one can check that for $k>0$ the union of the expected basis of $\\mathsf {H}^k(Y_n)$ with the known basis of $\\mathsf {H}^k(U(x_1))$ is mapped to a basis of $\\mathsf {H}^k(U(\\alpha ) \\cap U(x_1))$ .",
"This implies the statement.",
"Cohomology for $\\mathbb {A}_n$ with even $n$ Let us now consider the linear tree $\\mathbb {A}_n$ for even $n$ , and compute the cohomology of $X_n$ .",
"Proposition 6.3 The Hodge structure of the cohomology of $X_n$ is $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for all even $k$ between 0 and $n$ , and 0 otherwise.",
"A basis is given by powers of $\\operatorname{WP}$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ .",
"Proof.",
"This follows from the known cohomology of $Y_n$ and the Künneth theorem applied to the isomorphism $Y_n \\simeq X_n \\mathbb {G}_m$ given by lemma REF .",
"The Künneth theorem gives immediately the Hodge structure.",
"For the basis, it is enough to recall that the $\\mathbb {G}_m$ factor is given by the value of $\\alpha $ , and to check that fixing the value $\\alpha = 1$ maps $\\operatorname{WP}$ (for $Y_n$ ) to $\\operatorname{WP}$ (for $X_n$ ).",
"Cohomology for orange trees of shape $H$ 0.7 [scale=0.7] patterns,decorations.pathreplacing every node=[draw,shape=circle,very thick,fill=white] (0,0) node ... – (1,0) node ... – (3,0) node ... – (4,0) node ...; (0,1) node ... – (1,1) node ... – (3,1) node ... – (4,1) node ...; (2,1) node $a$ – (2,0) node $b$ ; every node=[draw=none,fill=none]; [thick,decoration= brace, raise=0.3cm ,decorate] (-0.3,1) node – (1.3,1) node ; [thick,decoration= brace, raise=0.3cm ,decorate] (2.7,1) node – (4.3,1) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (-0.3,0) node – (1.3,0) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (2.7,0) node – (4.3,0) node ; t (0.5,-1) $m$ ; t (0.5,2) $k$ ; t (3.5,-1) $n$ ; t (3.5,2) $\\ell $ ; Let us denote by $H_{k,\\ell ,m,n}$ the tree described as two chains joined by an edge, such that by removing the joining edge and its extremities $a$ and $b$ , one gets two chains of lengths $k$ and $\\ell $ on the $a$ side (top) and two chains of lengths $m$ and $n$ on the $b$ side (bottom).",
"We assume now that $H_{k,\\ell ,m,n}$ is an orange tree.",
"It implies that either $k, \\ell , m$ and $n$ are even if the middle edge is an orange domino, or that (without loss of generality) $k$ and $m$ are odd and $l$ and $n$ are even otherwise.",
"Then one can compute the cohomology of $H_{k,\\ell ,m,n}$ using the Mayer-Vietoris long exact sequence for the open covering by $U(x_a)$ and $U(x_b)$ .",
"When the middle edge is an orange domino, one has $\\begin{aligned}U(x_a) &\\simeq X_k X_\\ell Y_{m+n+1},\\\\U(x_b) &\\simeq Y_{k+\\ell +1} X_m X_n ,\\\\U(x_a) \\cap U(x_b) &\\simeq (\\mathbb {G}_m)^2 X_k X_\\ell X_m X_n.\\end{aligned}$ When the middle edge is not an orange domino, one finds instead $\\begin{aligned}U(x_a) &\\simeq Y_k X_\\ell X_{m+n+1},\\\\U(x_b) &\\simeq X_{k+\\ell +1} Y_m X_n,\\\\U(x_a) \\cap U(x_b) &\\simeq Y_k X_\\ell Y_m X_n.\\end{aligned}$ Let us introduce some notations: call $K,L,M,N$ the subsets of vertices corresponding to the four branches of $H$ (i.e.",
"the connected components of $H \\setminus \\lbrace a,b\\rbrace $ ).",
"Let us denote by $W_S$ the Weil-Petersson 2-form associated with a subset $S$ of the vertices of $H$ .",
"For conciseness, one will use shortcuts such as $W_{KaL}$ or $W_{MabN}$ .",
"Note that there holds $\\omega _a W_{aL} = \\omega _a W_{L}$ and other similar simplifications, by the definition (REF ) of these forms.",
"Let us now describe generators and bases of the cohomology of the open sets $ U(x_a)$ , $U(x_b)$ and $U(x_b)\\cap U(x_b)$ .",
"This can be computed using the isomorphisms (REF ), (REF ) and the known cohomology of varieties $X$ and $Y$ .",
"It turns out that the result does not depend on whether or not the middle edge $a-b$ is an orange domino.",
"The cohomology of $U(x_a)$ is generated by $\\omega _a$ , $W_{Ka}$ , $W_{aL}$ and $W_{MabN}$ .",
"A basis is given by $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B} \\quad \\text{and}\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{MbN}^{B},$ where $0 \\le \\kappa \\le k/2$ , $0 \\le \\lambda \\le l/2$ and $0 \\le B \\le (m+n+2)/2$ (left) or $0 \\le B \\le (m+n)/2$ (right).",
"Similarly, the cohomology of $U(x_b)$ is generated by $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ and $W_{KabL}$ .",
"A basis is given by $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A} \\quad \\text{and}\\quad \\omega _b W_{M}^{\\mu } W_{N}^{\\nu } W_{KaL}^{A},$ where $0 \\le \\mu \\le m/2$ , $0 \\le \\nu \\le n/2$ and $0 \\le A \\le (k+l+2)/2$ (left) or $0 \\le A \\le (k+l)/2$ (right).",
"The cohomology of $U(x_b)\\cap U(x_b)$ is generated by $\\omega _a$ , $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ , $W_{Ka}$ and $W_{aL}$ .",
"A basis is given by $\\begin{aligned}W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\\\\\omega _a \\omega _b W_{K}^{\\kappa } W_{L}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }\\quad \\text{and}\\quad \\omega _b W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu },\\end{aligned}$ with the same conditions as above on $\\kappa ,\\lambda ,\\mu $ and $\\nu $ .",
"There is a bigrading corresponding to the top and bottom parts of the $H$ shape.",
"Every differential form involved in the bases just described is a sum of products of $\\omega _i$ .",
"The bidegree of a monomial in the $\\omega _i$ is the pair (number of $\\omega _i$ where $i$ is in the top row, number of $\\omega _i$ where $i$ is in the bottom row).",
"Among the various Weil-Petersson forms involved, only the differential forms $W_{KabL}$ and $W_{MabN}$ are not homogeneous for the bidegree, but have terms in bidegrees $(2,0)$ and $(1,1)$ (resp.",
"$(0,2)$ and $(1,1)$ ).",
"One needs now to compute explicitly the following maps in the Mayer-Vietoris long exact sequence: $\\mathsf {H}^i(U(x_a))\\oplus \\mathsf {H}^i(U(x_b)) \\stackrel{f_i}{\\longrightarrow } \\mathsf {H}^i(U(x_a) \\cap U(x_b)).$ Because one has bases of all these spaces, this is a matter of matrices.",
"For odd degree $i$ , let us show that the differential is injective.",
"Because in this case all basis elements (given by right columns of (REF ), (REF ) and (REF )) are homogeneous for the bigrading, one can separate the cases of bidegree congruent to $(0,1)$ and to $(1,0)$ modulo $(2,2)$ .",
"Let us give details only for the first possibility, the other case being similar after exchanging top and bottom of $H$ .",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_b))$ is given by $\\omega _{b}W_{KaL}^{A}W_{M}^{\\mu } W_{N}^{\\nu }$ with $i =1+2A+2\\mu +2\\nu $ .",
"The corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a))$ is zero.",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a) \\cap U(x_b))$ is given by $\\omega _{b}W_{Ka}^{\\kappa }W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }$ with $i = 1+2\\kappa +2\\lambda +2\\mu +2\\nu $ .",
"But $W_{KaL}^{A}$ can be written as a linear combination of $W_{Ka}^{\\kappa }W_{aL}^{\\lambda }$ with $\\kappa + \\lambda = A$ .",
"Therefore the basis elements are mapped to linear combinations with disjoint supports.",
"It follows that the map $f_i$ is injective.",
"Let us now turn to even degrees.",
"Proposition 6.4 For even degree $2i$ , the kernel of the differential $f_{2i}$ has dimension 1, spanned by the $i^{th}$ power of the form $\\operatorname{WP}$ .",
"Proof.",
"First note that one can define an injective map $\\Delta $ from the space $\\mathsf {H}^{2i}(U(x_a) \\cap U(x_b))$ to the space $D_i$ spanned by all products of $i$ 2-forms of the shape $\\omega _{s}\\omega _{t}$ for $s-t$ an edge of the tree (always written in the order given by a fixed alternating orientation of the tree).",
"Indeed, both terms in the left column of (REF ) can be written as linear combinations of such products.",
"The injectivity holds because distinct elements in this part of the basis are mapped to linear combinations with disjoint supports.",
"To recover a basis element $B$ from any monomial in its image by $\\Delta $ , first count in $\\Delta (B)$ if the number of $\\omega _k$ in the top row is odd or even.",
"This tells if the basis elements $B$ contains $\\omega _a\\omega _b$ or not.",
"Then it is easy to recover the exponents $(\\kappa , \\lambda , \\mu , \\nu )$ defining $B$ by counting in $\\Delta (B)$ how many $\\omega _k$ there are in the different parts of the tree.",
"To prove the statement of the proposition, it is therefore enough to compute the kernel of the composite map $\\Delta \\circ f_{2i}$ .",
"It turns out that the matrix of this composite map has a nice description.",
"First, every monomial $d$ made of $i$ 2-forms $\\omega _{s}\\omega _{t}$ as above appears in exactly two images, the image of a form $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B}$ and the image of a form $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A}$ (with opposite signs).",
"Let us denote these two forms by $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ .",
"On the other hand, the image of every basis element is the sum of several monomials (at least one), with constant sign.",
"Let us pick an element $z$ of the kernel of $f_{2i}$ .",
"Then for every monomial $d$ in $D_i$ , the coefficients of $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ in $z$ must be the same.",
"One can make a graph with vertices given by all forms in the basis, and edges corresponding to the relations $\\mathsf {F}_a(d)-\\mathsf {F}_b(d)$ for all monomials $d$ .",
"By a combinatorial argument, one can check that this graph is connected.",
"For this, one just has to show that one can go from any monomial $d$ to any monomial $d^{\\prime }$ , using two kinds of moves: replace $d$ by another monomial appearing in the same $\\mathsf {F}_a(d)$ , or replace $d$ by another monomial appearing in the same $\\mathsf {F}_b(d)$ .",
"This is not difficult once translated in terms of dominoes, and details are left to the reader.",
"From the connectedness of this graph, one deduces that the kernel is spanned by the sum of all basis elements of $\\mathsf {H}^{2i}(U(x_a))\\oplus \\mathsf {H}^{2i}(U(x_b))$ , which is just $(\\operatorname{WP}^i,\\operatorname{WP}^i)$ .",
"This proposition and the injectivity in the case of odd degree allow to give a description of the weights of the Hodge structure on the cohomology.",
"This can easily be made explicit, but one will not do that here.",
"There would remain to find explicit expressions for the cohomology classes coming from the co-image of the differentials $f_i$ .",
"In the case of the Dynkin diagrams $\\mathbb {E}_6$ and $\\mathbb {E}_8$ , one can go further and compute explicit representatives of the cohomology classes.",
"By the general proof, the cohomology for $\\mathbb {E}_6$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-6),$ where the $\\mathbb {Q}(-i)$ with $i$ even correspond to the powers of $\\operatorname{WP}$ .",
"Using the connection homomorphism in the long exact sequence, one finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ .",
"Similarly, the cohomology for $\\mathbb {E}_8$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-5)\\oplus \\mathbb {Q}(-6) \\mid 0 \\mid \\mathbb {Q}(-8),$ where the even $\\mathbb {Q}(-i)$ are the powers of $\\operatorname{WP}$ .",
"One finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ , and its product by $\\operatorname{WP}$ corresponds to $\\mathbb {Q}(-5)$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6 – (5,0) node[fill=orange!20] 7 – (6,0) node[fill=orange!20] 8; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; Cohomology: generic cases This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers.",
"Cohomology for $\\mathbb {A}$ odd and generic Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Affine algebraic varieties",
"Using the coloring of the previous section, one can define several affine algebraic varieties attached to a tree $T$ and some auxiliary choices.",
"These varieties are closely related to cluster algebras.",
"First, let us consider the system of equations $x_i x^{\\prime }_i = 1 + \\alpha _i \\prod _{i-j} x_j$ for all vertices $i$ of $T$ , where the product runs over vertices $j$ adjacent to $i$ .",
"Here $x_i$ and $x^{\\prime }_i$ are called cluster variables, and $\\alpha _i$ are called coefficient variables.",
"By a special case of [2], this system is a presentation of the cluster algebra associated with the quiver given by a bipartite orientation of $T$ , with one frozen vertex attached to every vertex of $T$ (in such a way that all vertices of $T$ remain sources or sinks).",
"In the context of cluster algebras, the equations (REF ) are called exchange relations.",
"We will be interested here in considering the $\\alpha _i$ as parameters, and letting them either vary in some well-chosen families or take fixed generic values (and even a mix of these two possibilities), so that the resulting space is smooth."
],
[
"Jumping around a groupoid",
"Let us denote by $X_T(\\alpha )$ the algebraic scheme defined by fixing some invertible values for all coefficient variables $\\alpha _i$ .",
"Recall the following lemma ([6]).",
"Lemma 2.1 Let $u-v$ be an edge of $T$ .",
"Let $\\beta $ be defined by $ \\beta _w = \\alpha _w/\\alpha _u$ if $w$ is a neighbor of $v$ (in particular $ \\beta _u = 1$ ) and $\\beta _w = \\alpha _w$ otherwise.",
"Then $X_T(\\alpha )$ and $X_T(\\beta )$ are isomorphic, by the change of variables $x_v=\\alpha _u x_v$ and $x^{\\prime }_v=x^{\\prime }_v/\\alpha _u$ .",
"One may say that the coefficient $\\alpha _u$ has jumped away from $u$ over $v$ and its inverse has got spread over all other neighbors of $v$ .",
"When $v$ has $u$ as only neighbor, the coefficient $\\alpha _u$ just disappears from the equations.",
"From now on, we will only admit the following kinds of jumps: a red vertex over one of its green neighbors, a green vertex over one of its red neighbors, an orange vertex over its matched orange neighbor.",
"Let us now define a groupoid $G_T$ with objects the schemes $X_T(\\alpha )$ indexed by invertible values of the parameters $\\alpha $ , and isomorphisms $X_T(\\alpha ) \\simeq X_T(\\bar{\\alpha })$ of the shape ${\\left\\lbrace \\begin{array}{ll}\\bar{x}_i &\\mapsto \\lambda _i x_i,\\\\\\bar{x}^{\\prime }_i &\\mapsto x^{\\prime }_i/\\lambda _i,\\end{array}\\right.",
"}$ where $\\lambda _i$ are some invertible elements.",
"The parameters are then related by ${\\alpha }_i = \\bar{\\alpha }_i \\prod _{i - j}\\lambda _j .$ Note that every jump corresponds to an isomorphism in the groupoid $G_T$ .",
"Proposition 2.2 For every maximum matching $M$ and given parameters $\\alpha $ , there exists unique parameters $\\beta $ (given by monic Laurent monomials in $\\alpha $ ) such that the function $\\beta $ is 1 except on the set of red vertices not covered by $M$ .",
"$X_T(\\alpha )$ is isomorphic to $X_T(\\beta )$ by a sequence of jumps.",
"Moreover, (a) the function $\\beta $ only depends on the values of $\\alpha $ on the red vertices of $T$ , (b) the values of $\\beta $ on a red-green component are Laurent monomials in the values of $\\alpha $ on the same red-green component.",
"Proof.",
"Let us first prove the existence of such parameters $\\beta $ .",
"The main idea is to iterate lemma REF by jumping over dominoes of $M$ .",
"Let us define an auxiliary oriented graph $\\mathcal {G}$ as follows: the vertices of $\\mathcal {G}$ are the vertices of $T$ , and there is an edge $u \\rightarrow w$ in $\\mathcal {G}$ if $u-v$ is a domino in $M$ and $v-w$ is another edge in $T$ .",
"With this notation, if there are edges starting from $u$ in $\\mathcal {G}$ , one can use lemma REF (by jumping over $v$ ) to turn the coefficient $\\beta _u$ into 1 and replace the coefficients $\\beta _w$ by $\\beta _w / \\beta _u$ , for all vertices at the end of an arrow $u \\rightarrow w$ .",
"One can see that the graph $\\mathcal {G}$ has no oriented cycle, otherwise there would be a cycle in $T$ made of concatenated dominoes.",
"Moreover, edges in the graph $\\mathcal {G}$ can only go from green to green, from orange to orange or green, or start from red.",
"Then one can do these jumps starting from the sources in $\\mathcal {G}$ and then proceeding along any linear extension of the partial order defined by $\\mathcal {G}$ .",
"At the end of this process, all vertices covered by dominoes have coefficient 1.",
"There only remains coefficients on the red vertices not covered by the maximum matching $M$ .",
"This proves the existence of the required parameters $\\beta $ .",
"The fact that the coefficients $\\beta _j$ are products of coefficients $\\alpha _i$ and their inverses is immediate from the definition of jumping.",
"Let us now prove uniqueness.",
"Assume there are two such sets of parameters $\\beta $ and $\\bar{\\beta }$ .",
"Let $x$ and $\\bar{x}$ be the coordinates on the isomorphic $X_T(\\beta )$ and $X_T(\\bar{\\beta })$ .",
"Let us first prove that any isomorphism in the groupoid $G_T$ from $X_T(\\beta )$ to $X_T(\\bar{\\beta })$ maps $\\bar{x}_j$ to $x_j$ for every green vertex $j$ .",
"This is done by induction using the auxiliary graph $\\mathcal {G}$ , starting with the green vertices that do not have any outgoing edge in $\\mathcal {G}$ .",
"For every green vertex, one just has to consider the equation (REF ) for the unique red vertex that is in the same domino in $M$ .",
"Using then the equation (REF ) for all red vertices $i$ not covered by $M$ , one obtains that $\\beta _i =\\bar{\\beta }_i$ .",
"This proves uniqueness.",
"For the statement $(a)$ , consider what happens to the coefficient attached to an orange or a green vertex $u$ .",
"By proposition REF , the domino containing $u$ must be orange or green-red.",
"The coefficient can therefore only jump to green or orange vertices.",
"So they must disappear at some point, because only red vertices bear coefficients at the end of the process.",
"Similarly for the statement $(b)$ , consider the coefficient attached to a red vertex $u$ .",
"Again by proposition REF , the domino containing $u$ must be red-green.",
"The coefficient can only jump to red vertices in the same red-green component, or to orange and green vertices.",
"As the coefficients on orange or green vertices will disappear by the previous point, coefficients can only stay within a given red-green component.",
"Recall that the dimension $\\dim (T)$ of $T$ is (by lemma REF ) the number of red vertices that are not covered in any maximum matching of $T$ .",
"Proposition REF justifies this terminology, as this gives the number of independent parameters for the varieties $X_T(\\alpha )$ (inside the groupoid $G_T$ ).",
"Remark 2.3 In the particular case when the tree $T$ is orange, all $X_T(\\alpha )$ are isomorphic.",
"By proposition REF , in order to study all isomorphism classes of such varieties, one can restrict oneself to attach parameters only to red vertices not covered by a maximum matching $M$ .",
"For a maximum matching $M$ of $T$ , let us define a scheme $X^M_T(\\alpha )$ by the set of equations (REF ), where $\\alpha _i$ are invertible fixed parameters, equal to 1 if $i$ is covered by $M$ .",
"Given two matchings $M$ and $M^{\\prime }$ , one can always find by Proposition REF a sequence of jumps that provides an isomorphism in $G_T$ between $X^M_T(\\alpha )$ and $X^{M^{\\prime }}_T(\\beta )$ , where the parameters $\\beta $ are uniquely determined Laurent monomials in $\\alpha $ .",
"Let us consider now the automorphism group $\\operatorname{Aut}(X^M_T(\\alpha ))$ of the object $X^M_T(\\alpha )$ in the groupoid $G_T$ .",
"Proposition 2.4 The automorphism group $\\operatorname{Aut}(X^M_T(\\alpha ))$ is an algebraic torus isomorphic to $\\mathbb {G}_m^{\\dim (T)}$ .",
"If $(\\lambda _i)_{i\\in T}$ is an element of $\\operatorname{Aut}(X^M_T(\\alpha ))$ , then $\\lambda _i = 1$ on green and orange vertices of $T$ .",
"Proof.",
"Let us consider an automorphism in $G_T$ given by invertible elements $\\lambda _i$ .",
"The condition that the equation (REF ) for the vertex $i$ is preserved is $\\prod _{j - i} \\lambda _j = 1.$ This just means that the $\\lambda _i$ belongs to the kernel of the adjacency matrix of $T$ (seen as an endomorphism of $\\mathbb {G}_m^T$ ).",
"Looking at the induced linear equations on the tangent space at one, one can deduce from remark REF that the dimension of $\\operatorname{Aut}(X^M_T(\\alpha ))$ is $\\dim (T)$ .",
"By the same argument (using induction on the auxiliary graph $\\mathcal {G}$ ) as in the uniqueness step of the proof of Prop.",
"REF , every automorphism fixes $x_j$ for every green vertex $j$ .",
"By a similar argument (starting with orange vertices attached to green vertices in the auxiliary graph $\\mathcal {G}$ ), one can then prove that every automorphism fixes $x_j$ for every orange vertex $j$ .",
"There remains to show that $\\operatorname{Aut}(X^M_T(\\alpha ))$ is connected.",
"Let us prove that, given any choice for the values of $\\lambda _i$ for $i\\notin M$ , there is a unique element of $\\operatorname{Aut}(X^M_T(\\alpha ))$ extending this choice.",
"This is once again done by induction using the auxiliary graph $\\mathcal {G}$ .",
"Let us consider a red vertex $j$ that is pointing in $\\mathcal {G}$ only toward vertices with known $\\lambda $ .",
"Then there is a unique way to fix the value $\\lambda _j$ such that (REF ) holds for the green vertex $i$ in the domino of $j$ .",
"This proves that the kernel is isomorphic to $\\mathbb {G}_m^{\\dim (T)}$ .",
"Note that the torus $\\operatorname{Aut}(X^M_T(\\alpha ))$ and its action on $X^M_T(\\alpha )$ do not depend on $\\alpha $ .",
"This action therefore extends to varieties defined as the union of $X^M_T(\\alpha )$ over some family of parameters $\\alpha $ .",
"The torus $\\operatorname{Aut}(X^M_T(\\alpha ))$ can be written as a product of several tori, indexed by the red-green components.",
"Every factor acts only on the red vertices inside a fixed red-green component.",
"This factorization will be useful later to describe free actions on some varieties.",
"Genericity A non-empty set $S$ of red vertices in a red-green component $C$ is called an admissible set if every green vertex in $C$ has either 0 or 2 neighbors in $S$ .",
"Lemma 2.5 Given a red vertex $u$ in $C$ , there is an admissible set containing $u$ .",
"Proof.",
"One can build an admissible set $S$ starting from $\\lbrace u\\rbrace $ by repeated addition of red vertices.",
"If there is a green vertex $v$ with exactly one red neighbor in $S$ , then add to $S$ one of the other red neighbors of $v$ .",
"Repeat until the set $S$ is admissible.",
"Let us now introduce an explicit genericity condition on the parameters attached to a given red-green component $C$ .",
"For every admissible set $S$ of red vertices of $C$ , the alternating product $\\prod _{i \\in S} \\alpha _i^{\\pm } \\ne (-1) ^{\\#S},$ where any two red vertices sharing a common green neighbor have opposite powers in the left hand side.",
"Lemma 2.6 The genericity condition is preserved under jumping moves.",
"Proof.",
"Indeed, consider the jumping move from a red vertex $u$ over a green vertex $v$ .",
"The coefficients of all red neighbors of $v$ are divided by $\\alpha _u$ .",
"Let $S$ be an admissible set.",
"If the vertex $v$ has no neighbor in $S$ , nothing is changed in the genericity condition for $S$ .",
"Otherwise, the vertex $v$ has two neighbors in $S$ .",
"Then two terms are changed in the left-hand side of (REF ), both being divided by $\\alpha _u$ .",
"But they appear with opposite powers, hence the product is not changed.",
"The two other kinds of jumping moves (green over red and orange over orange) do not change the parameters of red vertices.",
"Definition of the varieties Let us now carefully define the varieties that will be studied in the rest of the article.",
"Let us fix a tree $T$ , a choice function $\\varphi $ from the set of red-green components of $T$ to the set $\\lbrace \\mathtt {generic}, \\mathtt {versal}\\rbrace $ and a maximum matching $M$ of $T$ .",
"For every red-green component $C$ such that $\\varphi (C)$ is $\\mathtt {generic}$ , let us fix for every vertex $u$ of $C$ not covered by the maximum matching $M$ , an invertible value $\\alpha _u$ .",
"To this data, one associates a scheme $X^{\\varphi ,M}_{T, \\alpha }$ as follows.",
"The variables are $x_i$ and $x^{\\prime }_i$ for all vertices of $T$ , $\\alpha _i$ for all vertices not covered by the matching $M$ in the red-green components $C$ of $T$ such that $\\varphi (C)$ is $\\mathtt {versal}$ .",
"The equations are the system of equations (REF ), all variables $\\alpha _i$ are invertible.",
"In fact, there is no true dependency on the matching $M$ .",
"Let us consider two maximum matchings $M$ and $M^{\\prime }$ .",
"Using proposition REF , one can find an isomorphism between $X^{\\varphi ,M}_{T,\\alpha }$ for arbitrary invertible parameters $\\alpha $ and $X^{\\varphi ,M^{\\prime }}_{T,\\beta }$ for parameters $\\beta $ depending on the parameters $\\alpha $ .",
"One will therefore forget the matching and use the notation $X^\\varphi _T$ from now on, keeping the parameters $\\alpha $ implicit as well.",
"Moreover, by lemma REF , if the genericity condition (REF ) holds for the parameters $\\alpha $ with respect to one matching $M$ , they will also hold for the corresponding parameters $\\beta $ for another matching $M^{\\prime }$ .",
"One can therefore impose that the genericity condition (REF ) holds for all $\\mathtt {generic}$ red-green components of $T$ .",
"This will always be assumed from now on.",
"Let us summarize this lengthy definition.",
"Once the tree $T$ is chosen, one picks a maximum matching $M$ of $T$ .",
"Any choice of matching will lead to isomorphic varieties.",
"One then decides for every red-green component of $T$ either to take the union over all invertible parameters or to fix some generic parameters.",
"One will use the simplified notation $X_T$ for orange trees, as there is then no choice to be made for the function $\\varphi $ .",
"One will also use the notations $X^\\mathtt {generic}_T$ and $X^\\mathtt {versal}_T$ when the function $\\varphi $ is constant.",
"Remark 2.7 One can as well consider forests instead of trees in the definition of the varieties $X^\\varphi _T$ , but then everything factors according to the connected components.",
"This possibility will be used implicitly in the rest of the article.",
"Let us introduce the notation $U(x)$ for the open set defined by $x\\ne 0$ .",
"Lemma 2.8 If $a-b$ is an edge in a tree $T$ , then the two open sets $U(x_a)$ and $U(x_b)$ cover the variety $X^\\varphi _T$ .",
"Proof.",
"This follows from the exchange relation $x_a x^{\\prime }_a = 1 + \\alpha _a x_b y,$ where $y$ is some product of other cluster variables.",
"Remark 2.9 When removing red vertices or green vertices in a tree $T$ , some red-green components may split into several red-green components.",
"One can then define a function $\\widehat{\\varphi }$ on the new set of red-green components, whose value on a red green component $C$ is the value of $\\varphi $ in the unique red-green component of $T$ containing $C$ .",
"Abusing notation, one will denote this induced function $\\widehat{\\varphi }$ simply by $\\varphi $ .",
"Smoothness and free actions Theorem 3.1 For every choice of $\\varphi $ , the variety $X^\\varphi _T$ is smooth.",
"Proof.",
"The proof is by induction on the size of the tree $T$ .",
"For the tree with only one vertex, the only equation is $x x^{\\prime } = 1 + \\alpha .$ In the $\\mathtt {generic}$ case when $\\alpha $ is considered to have a fixed value, different from $-1$ by the genericity condition (REF ), the variety is isomorphic to the punctured affine line $\\mathbb {G}_m$ and is therefore smooth.",
"In the $\\mathtt {versal}$ case when $\\alpha $ is considered to be a variable and assumed to be invertible, the variety is an open set in the variety defined by (REF ) where $\\alpha $ is not assumed to be invertible.",
"This last variety is isomorphic to the affine plane $A_{2}$ , hence smooth.",
"The rest of the proof by induction is organized as follows.",
"One first considers the case when the tree has at least one red-green component, and treat separately the case when there is a red-green component which is $\\mathtt {generic}$ and the case when there is one which is $\\mathtt {versal}$ .",
"Otherwise, the tree is orange.",
"These three cases are done in the next three subsections.",
"Let us first state a few useful lemmas.",
"Lemma 3.2 If one variable $x_i$ is assumed to be non-zero, then one can get rid of the associated variable $x^{\\prime }_i$ and of the equation (REF ) of index $i$ .",
"Proof.",
"Indeed, one can just use the equation to eliminate $x^{\\prime }_i$ .",
"Lemma 3.3 If one variable $x_i$ is assumed to be zero, then $x^{\\prime }_i$ becomes a free variable and the equation (REF ) of index $i$ reduces to $-1 = \\alpha _i \\prod _{i-j} x_j.$ Let us now introduce a useful variant of the varieties $X_T^\\varphi $ .",
"Let $v$ be a vertex of $T$ .",
"Let $X^\\varphi _T[v]$ be defined just as $X_T^\\varphi $ , but with one more invertible variable $\\gamma _v$ attached to the vertex $v$ as a coefficient (playing the same role as $\\alpha _v$ in the equations).",
"This variable defines a morphism $\\gamma _v$ from $X^\\varphi _T[v]$ to $\\mathbb {G}_m$ .",
"Lemma 3.4 If $v$ is an orange or green vertex, then $X^\\varphi _T[v]$ is isomorphic as a variety over $\\mathbb {G}_m$ to $X_T^\\varphi \\times \\mathbb {G}_m$ endowed with the projection to the second factor.",
"Proof.",
"By proposition REF and its proof, one can find an isomorphism in the groupoid $G_T$ between $X^\\varphi _T$ and $X^\\varphi _T[v]$ that only changes the coordinates $x_i$ for orange and green vertices.",
"More precisely, using the auxiliary oriented graph $\\mathcal {G}$ , one can find a sequence of jumps (corresponding to edges in $\\mathcal {G}$ starting with a green or orange vertex) that makes the coefficient $\\gamma _v$ disappear from the equations.",
"The isomorphism associated with this sequence of jumps is multiplying the variables $x_i$ by monic Laurent monomials in the parameter $\\gamma _v$ , hence defines an isomorphism over $\\mathbb {G}_m$ .",
"Lemma 3.5 If $v$ is a red vertex in a versal red-green component $C$ , then $X^\\varphi _T[v]$ is isomorphic as a variety over $\\mathbb {G}_m$ to $X_T^\\varphi \\times \\mathbb {G}_m$ endowed with the projection to the second factor.",
"Proof.",
"If the red vertex $v$ is not covered by the matching $M$ chosen to define $X^\\varphi _T$ , then one has two coefficient variables $\\alpha _v$ and $\\gamma _v$ attached to the vertex $v$ .",
"By the simple change of coordinates $\\alpha _v := \\alpha _v \\gamma _v$ and $\\gamma _v := \\gamma _v$ , one gets the expected isomorphism.",
"Assume now that that red vertex $v$ is covered by the matching $M$ .",
"By proposition REF and its proof, one can find an isomorphism in the groupoid $G_T$ between $X^\\varphi _T[v]$ and a variety $X^{\\varphi ,M}_{T,\\beta }$ that only changes the coordinates $x_i$ for orange and green vertices and for red vertices in the red-green component $C$ .",
"More precisely, using the auxiliary oriented graph $\\mathcal {G}$ , one can find a sequence of jumps that moves the coefficient $\\gamma _v$ towards the red vertices in $C$ not covered by the matching.",
"At the end, every new coefficient $\\beta _i$ is the product of $\\alpha _i$ by a Laurent monomial in $\\gamma _v$ .",
"The isomorphism associated with this sequence of jumps is multiplying the variables $x_i$ by monic Laurent monomials in the parameter $\\gamma _v$ , hence defines an isomorphism over $\\mathbb {G}_m$ .",
"One can then compose this isomorphism with a relabeling of the coefficients $\\alpha _i := \\beta _i$ in order to get the expected isomorphism, still defined over $\\mathbb {G}_m$ , between $X^\\varphi _T[v]$ and $X_T^\\varphi \\times \\mathbb {G}_m$ .",
"One could say that the coefficient $\\gamma _v$ can be detached from $T$ in these cases.",
"This will be used frequently in the rest of the article.",
"Trees with a generic component One assumes now that $T$ has at least two vertices and a $\\mathtt {generic}$ component $C$ .",
"Let us pick an admissible set $S$ of red vertices in $C$ , as defined in §REF .",
"Lemma 3.6 The open sets $U(x_i)$ for $i \\in S$ form a covering of $X^\\varphi _T$ .",
"Proof.",
"Indeed, the complement of their union is the set where all variables $x_i$ for $i \\in S$ vanish.",
"This implies that $\\alpha _i \\prod _{j-i} x_j = -1$ for every $i$ in $S$ .",
"Taking the alternating product of these equalities gives $\\prod _{i \\in S} \\alpha _i^{\\pm } = (-1)^{\\# S},$ because for every green vertex $j$ attached by an edge to some element of $S$ , the cluster variable $x_j$ appears exactly twice by definition of admissible sets, hence disappears in the alternating product.",
"But the equation (REF ) is incompatible with the genericity condition (REF ).",
"Let us now show that the open sets $U(x_i)$ are smooth.",
"Let $F$ be the forest $ T \\setminus \\lbrace i\\rbrace $ .",
"In the forest $F$ , the coloring is changed only on the red-green component containing $i$ , where an avalanche of orange dominoes can take place when removing $i$ .",
"The red-green component $C$ is therefore split into a number of red-green components.",
"Let us moreover introduce a function $\\varphi $ on $F$ , which is $\\mathtt {generic}$ on every red-green component coming from $C$ , and unchanged on all other red-green components.",
"Lemma 3.7 The open set $U(x_i)$ is isomorphic to $\\mathbb {G}_m\\times X^\\varphi _F$ .",
"Proof.",
"The condition that $x_i$ is not zero allows one to get rid of the variable $x^{\\prime }_i$ by using the equation (REF ) of index $i$ .",
"What remains are the equations for the forest $F = T \\setminus \\lbrace i\\rbrace $ , where now $x_i$ is treated as a parameter attached to all neighbors of $i$ in $T$ .",
"Because all neighbors of $i$ in $T$ are green, they become either green or orange in $F$ .",
"It follows from lemma REF that one can, without changing the variety, consider instead that the parameter $x_i$ is not attached to any vertex of $F$ .",
"Let us check that the genericity condition still holds on all $\\mathtt {generic}$ red-green components.",
"If the component $D$ does not come from the splitting of $C$ , then the genericity conditions are unchanged on this red-green component.",
"Otherwise, let us choose an admissible set in $D$ .",
"It was then already an admissible set in $C$ , by inspection of what happens during the avalanche of orange dominoes.",
"Therefore the genericity condition for $D$ is inherited from that for $C$ .",
"One has therefore obtained an isomorphism $U(x_i) \\simeq \\mathbb {G}_m\\times X^\\varphi _F,$ which is smooth by induction.",
"Therefore $X^\\varphi _T$ is also smooth.",
"Trees with a versal component One assumes now that $T$ has at least two vertices, and has a $\\mathtt {versal}$ component $C$ .",
"Let us choose a red leaf $v$ in this component.",
"By proposition REF , one can find a maximum matching $M$ not containing $v$ .",
"Therefore there is a coefficient variable $\\alpha _v$ .",
"Let $u$ be the green vertex adjacent to $v$ .",
"By lemma REF , the two open sets $U(x_u)$ and $U(x_v)$ cover $X^\\varphi _T$ .",
"Let us first prove that $U(x_v)$ is smooth.",
"Let $T^{\\prime }$ be the tree $T\\setminus \\lbrace v\\rbrace $ .",
"The coloring of $T^{\\prime }$ is obtained from $T$ by an avalanche of orange dominoes.",
"The dimension of $T^{\\prime }$ is $\\dim (T) - 1$ .",
"The avalanche may split the red-green component of $T$ containing $v$ into several components.",
"Let $\\varphi $ be the function which maps all these new components to the $\\mathtt {versal}$ condition, and unchanged condition on all the other red-green components.",
"Lemma 3.8 The open set $U(x_v)$ is isomorphic to $\\mathbb {G}_m^2 \\times X^\\varphi _{T^{\\prime }}$ .",
"Proof.",
"Assuming that $x_v$ is not zero allows one to get rid of the variable $x^{\\prime }_v$ by using (REF ) with index $v$ .",
"The coefficient variable $\\alpha _v$ also disappears from the equations: this gives one factor $\\mathbb {G}_m$ .",
"Then the variable $x_v$ is seen as a coefficient attached to the vertex $u$ in $T^{\\prime }$ , which is either green or orange.",
"The coefficient can therefore be detached by lemma REF , and one obtains a factor isomorphic to $\\mathbb {G}_m\\times X^\\varphi _{T^{\\prime }}$ .",
"Therefore $U(x_v)$ is smooth by induction.",
"Let us now prove that $U(x_u)$ is smooth.",
"Let us choose instead a matching $M$ containing the domino $u-v$ , thanks to lemma REF .",
"This amounts to go through an isomorphism in the groupoid $G_T$ , hence preserves the open set $U(x_u)$ .",
"Let $F$ be the forest $T\\setminus \\lbrace u\\rbrace $ .",
"Because $u$ is green, the coloring of $F$ is obtained from that of $T$ by restriction and the dimension of $F$ is $\\dim (T) + 1$ .",
"Let $v, T_1, \\dots , T_k$ be the connected components of the forest $F$ .",
"By removing the domino $u-v$ , one can restrict the matching $M$ to a matching of the forest $F$ .",
"The red-green component of $T$ containing $u$ splits into several red-green components in $F$ , one of them being the vertex $v$ .",
"One takes the $\\mathtt {versal}$ condition on all of these red-green components of $F$ , and unchanged condition on all the other red-green components.",
"Lemma 3.9 The open set $U(x_u)$ is isomorphic to $X^{\\mathtt {versal}}_{\\lbrace v\\rbrace } \\times \\prod _{j=1}^k X^\\varphi _{T_j},$ where the first component is the vertex $v$ with coefficient variable $x_u$ .",
"Proof.",
"Setting $x_u\\ne 0$ in the equations allows to get rid of the variable $x^{\\prime }_u$ .",
"The result can be described as a fiber product over $\\mathbb {G}_m$ , where the same coefficient variable $x_u$ is attached to every connected component of $F$ at a red vertex in a versal red-green component.",
"By repeated use of lemma REF on all connected components (but not on the isolated vertex $v$ ), one finds that the open set $U(x_u)$ is isomorphic to the product $X^{\\mathtt {versal}}_{v} \\times \\prod _{j=1}^k X^\\varphi _{T_j},$ where the first component is the vertex $v$ with coefficient $x_u$ .",
"Therefore $U(X_u)$ is smooth by induction, and hence $X^\\varphi _T$ is also smooth.",
"Orange trees Let us now assume that $T$ is an orange tree and let us choose one domino $u-v$ in the perfect matching of $T$ .",
"By lemma REF , the two open sets $U(x_u)$ and $U(x_v)$ cover the variety $X_T$ .",
"By symmetry between $u$ and $v$ , it is enough to prove that $U(x_u)$ is smooth.",
"Let $T_1, \\dots , T_k$ be the trees attached to $u$ in $T\\setminus \\lbrace v\\rbrace $ .",
"The $T_i$ are clearly orange trees.",
"Let $R$ be the connected component of $v$ in $T\\setminus \\lbrace u\\rbrace $ .",
"The tree $R$ is obtained by removing a leaf in an orange tree, hence (by lemma REF ) has dimension 1 and a unique red-green component.",
"Moreover, $R$ has a maximum matching avoiding only $v$ and the vertex $v$ is red in the coloring of $R$ .",
"Lemma 3.10 The open set $U(x_u)$ is isomorphic to the product of the varieties $X_{T_i}$ and the variety $X^\\mathtt {versal}_R$ .",
"Proof.",
"Assuming that $x_u$ is not zero allows to eliminate the variable $x^{\\prime }_u$ and the equation (REF ) of index $u$ .",
"There remains the equations for the union of $R$ and the $T_i$ , with $x_u$ considered as a parameter attached to all of them at the former neighbors of $u$ .",
"Because the trees $T_i$ are orange, one can consider instead (by lemma REF ) that the parameter $x_u$ is only attached to the vertex $v$ of $R$ .",
"This proves that the open set $U(x_u)$ is isomorphic to the product of the varieties $X_{T_i}$ and the variety $X^\\mathtt {versal}_R$ .",
"By induction, this proves that $U(x_u)$ is smooth.",
"Therefore $X_T$ is smooth too.",
"Torus actions Let $T$ be a tree and let $\\varphi $ be a choice in $\\lbrace \\mathtt {generic},\\mathtt {versal}\\rbrace $ for every red-green component of $T$ .",
"Let us also choose a maximum matching $M$ of $T$ .",
"One can deduce from proposition REF and the remarks following it that there is an action of an algebraic torus of dimension $\\dim (T)$ on $X^\\varphi _T$ , and that this torus (and its action) can be written as a product over red-green components $C$ of tori $\\Lambda ^C_T$ .",
"Let us define a smaller torus $\\Lambda ^\\varphi _T$ acting on $X^\\varphi _T$ as the product of $\\Lambda ^C_T$ over all $\\mathtt {generic}$ red-green components of $T$ .",
"Let us call the rank of $(T, \\varphi )$ and denote by $\\operatorname{rk}(T,\\varphi )$ the sum of the dimensions of the generic red-green components of $T$ .",
"This is the dimension of $\\Lambda ^\\varphi _T$ .",
"Proposition 3.11 If $\\varphi (C)$ is generic, the action of $\\Lambda ^C_T$ on $X^\\varphi _T$ is free.",
"Proof.",
"Let us assume that there is a non-trivial element $\\lambda =(\\lambda _i)_i$ of $\\Lambda ^C_T$ that fixes a point $(x_i)_i$ in $X^\\varphi _T$ .",
"Let $i$ be a red vertex in $C$ such that $\\lambda _i \\ne 1$ .",
"For every green neighbor $j$ of $i$ , one can find another red vertex $k$ incident to $j$ such that $\\lambda _k \\ne 1$ , because of (REF ).",
"Iterating this process, one can build an admissible set $S$ (as defined in §REF ), such that $\\lambda _s \\ne 1$ for every $s \\in S$ .",
"Because $\\lambda $ fixes the given point, one then has $x_s = 0$ for every $s \\in S$ .",
"But this is impossible by Lemma REF .",
"Corollary 3.12 There is on $X^\\varphi _T$ a free action by a torus $\\Lambda ^\\varphi _T$ of dimension the rank $\\operatorname{rk}(T,\\varphi )$ .",
"This gives $X^\\varphi _T$ the structure of a principal bundle with structure group $\\Lambda ^\\varphi _T$ .",
"As one will see later, this bundle is not trivial in general (i.e.",
"not a product), as can be seen from our results for the cohomology already in type $\\mathbb {A}_3$ .",
"Number of points over finite fields and Euler characteristic Let us denote by $N^\\varphi _T(q)$ the number of points on $X^\\varphi _T$ over the finite field $\\mathbb {F}_q$ .",
"When the tree is orange, one will use the shorthand notation $N_T$ .",
"When the function $\\varphi $ is constant, one will use the notations $N^\\mathtt {versal}_T$ and $N^\\mathtt {generic}_T$ .",
"Proposition 4.1 The numbers $N^\\varphi _T(q)$ are monic polynomials in $q$ of degree $\\dim X_T^\\varphi $ .",
"Proof.",
"The proof is by induction on the size of the tree.",
"For the tree with one vertex, the number of points is $q-1$ in the $\\mathtt {generic}$ case and $q^2 - q + 1$ in the $\\mathtt {versal}$ case, by the description given at the beginning of the proof of theorem REF .",
"Then either the tree has a red-green component, which can be $\\mathtt {generic}$ or $\\mathtt {versal}$ , or it is an orange tree.",
"The proof is decomposed into the three following geometric decomposition lemmas, or rather into their obvious corollaries on the number of points over finite fields.",
"Let $T$ be a tree and $v$ be a red leaf in a red-green component $C$ of $T$ .",
"Let $u$ be the neighbor of $v$ .",
"Removing the vertex $v$ creates an orange avalanche and may separate the red-green component $C$ into several ones.",
"Let $\\varphi $ be the induced genericity condition (as defined in Remark REF ).",
"Let $F$ be the forest $T\\setminus \\lbrace u, v\\rbrace $ .",
"The component $C$ may also split into several red-green components in $F$ .",
"Let $\\varphi $ be the induced genericity condition.",
"Let us consider now the case of a generic red-green component $C$ .",
"Lemma 4.2 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_mX^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"In the case where $x_v \\ne 0$ , one uses lemma REF .",
"This gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u= -1$ is attached to some red vertices of $F$ , as a coefficient.",
"One has to check that the genericity condition still holds on every connected component of $F$ .",
"Let $S$ be an admissible set in one of these components.",
"Either $S$ was already an admissible set in $T$ , and then the genericity condition still holds, or it contains exactly one of the neighbors of $u$ in $T$ .",
"In this case, one can extend $S$ by adding $v$ to form an admissible set in $T$ .",
"The genericity condition for $S \\sqcup \\lbrace v\\rbrace $ in $T$ implies the condition for $S$ , because of the additional $-1$ coefficient attached to $S$ in $F$ .",
"Keeping the same notations, let us consider now the case of a versal red-green component $C$ .",
"Lemma 4.3 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_m^2 X^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"If $x_v \\ne 0$ , using lemma REF gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u = -1$ is attached to red vertices of $F$ .",
"By lemma REF , it can be detached, and this just gives the expected second term.",
"Let $T$ be an orange tree and $u-v$ be a domino in $T$ .",
"Let $(T_{u,i})_i$ (resp.",
"$(T_{v,j})_j$ ) be the connected components of $T\\setminus \\lbrace u,v\\rbrace $ that were attached to $u$ (resp.",
"to $v$ ).",
"All these trees are orange.",
"Let us denote by $S_{u,i}$ and $S_{v,j}$ the forests obtained from them by removing the vertex that was linked to $u$ or $v$ .",
"These forests are unimodal, in the sense that they have one unimodal connected component, all the other connected components being orange.",
"Lemma 4.4 In this situation, one has $X_T = \\mathbb {G}_m^2 \\prod _i X_{T_{u,i}} \\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X^\\mathtt {versal}_{S_{u,i}}\\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X_{T_{u,i}}\\prod _j X^\\mathtt {versal}_{S_{v,j}} .$ Proof.",
"Because the open sets $U(x_u)$ and $U(x_v)$ are a covering by lemma REF , one can cut the variety $X_T$ into three pieces: either both $x_u$ and $x_v$ are not zero, or exactly one of them is zero.",
"If both are not zero, then one obtains the product of $\\mathbb {G}_m^2$ (with coordinates $x_u$ and $x_v$ ) with the product of the varieties attached to the $T_{u,i}$ and the $T_{v,j}$ .",
"Indeed, one first get that $x_u$ becomes a parameter attached to all trees $T_{u,i}$ and $x_v$ becomes a parameter attached to all trees $T_{v,j}$ .",
"But these trees are orange, so $x_u$ and $x_v$ can be detached by lemma REF .",
"This gives the first term.",
"If $x_u$ is zero and $x_v$ is not zero, then there is a free variable $x^{\\prime }_u$ and the variable $x_v$ is determined by the variables attached to the vertices of the trees $T_{u,i}$ linked to $u$ , which must be non-zero.",
"One obtains therefore a versal condition on each forest $S_{u,i}$ .",
"For the trees $T_{v,j}$ , the coefficient $x_v$ is attached to all of them, but because they are orange it can be detached.",
"This gives the second term.",
"The third term is the same after exchanging $u$ and $v$ .",
"Reciprocal property Recall from §REF that the rank $\\operatorname{rk}(T,\\varphi )$ of the pair $(T,\\varphi )$ formed by a tree $T$ and a choice function $\\varphi $ is the sum of the dimensions of the $\\mathtt {generic}$ red-green components of $T$ .",
"Proposition 4.5 The polynomial $N^\\varphi _T(q)$ is divisible par $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ .",
"Proof.",
"This follows from the existence of the free action obtained in corollary REF .",
"Let us refine this slightly.",
"Proposition 4.6 The polynomial $N_T^\\varphi $ can be written as $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ times a reciprocal polynomial.",
"Proof.",
"By induction.",
"This is true for the tree with one vertex.",
"One just has to look carefully at the decompositions given in the three lemmas that were used to prove polynomiality by induction.",
"For lemma REF , let $D$ be the rank for $T$ .",
"Then the rank is $D-1$ for $T \\setminus \\lbrace v\\rbrace $ and $D$ for $F$ .",
"Using the additional factor $q-1$ coming from $\\mathbb {G}_m$ , there is a common factor $(q-1)^D$ to all terms involved.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is the same in all terms involved.",
"One uses that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is 0 in all terms involved, as there is no generic red-green component.",
"One uses again that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 pieces ensures that the reciprocal property holds.",
"Enumeration and coincidences In the following remarks, one will describe trees by their numbers in the tables at the end of [5] and by their graph6 string (which is a standard format for graphs).",
"Remark 4.7 One can find distinct orange trees with the same enumerating polynomial.",
"This happens first for trees with 8 vertices.",
"The trees 2.188 (graph6 'IhGGOC@?G') and 2.189 (graph6 'IhC_GCA?G') have the same polynomial, as well as the trees 2.172 (graph6 'IhGGOCA?G') and 2.174 (graph6 'IhGH?C@?G').",
"The number of different polynomials for orange trees with $2n$ vertices is the sequence $1, 1, 2, 5, 13, 41, 138, \\dots $ whereas the number of orange trees is $1, 1, 2, 5, 15, 49, 180, \\dots $ Remark 4.8 For unimodal trees with $\\mathtt {versal}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of trees with 9 vertices, numbered 2.83 (graph6 'HhCGOCA') and 2.85 (graph6 'HhGGGG@').",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 6, 19, 65, \\dots $ whereas the number of unimodal trees is $1, 1, 2, 6, 20, 76, 313, 1361, \\dots $ Remark 4.9 For unimodal trees with $\\mathtt {generic}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of the Dynkin diagrams $\\mathbb {A}_7$ and $\\mathbb {E}_7$ .",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 5, 13, 46, 168, \\dots $ Linear trees Let us denote by $\\mathbb {A}_n$ the linear tree with $n$ vertices.",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) node[fill=orange!20] 3 – (3,0) node[fill=orange!20] 4 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=red!20] 1 – (1,0) node[fill=green!20]2 – (2,0) node[fill=red!20] 3 – (3,0) node[fill=green!20] 4 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; One can check that $\\mathbb {A}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.10 The number of points on varieties attached to $\\mathbb {A}_n$ is given by $N_{\\mathbb {A}_n} = \\frac{q^{n+2} - 1}{q^2 -1}$ if $n$ is even and by $N_{\\mathbb {A}_n}^\\mathtt {versal}= \\frac{q^{n+2} + 1}{q + 1} \\quad \\text{and} \\quad N_{\\mathbb {A}_n}^\\mathtt {generic}= \\frac{(q^{(n+1)/2} - 1)(q^{(n+3)/2} - 1)}{q^2 -1}$ if $n$ is odd.",
"Proof.",
"This follows easily by induction from lemmas REF , REF and REF .",
"Trees of type $\\mathbb {D}$ Let us denote by $\\mathbb {D}_n$ the tree with $n$ vertices associated with the Dynkin diagram of type $\\mathbb {D}$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=orange!20] 4 – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=red!20] 4 – (3,0) node[fill=green!20] 5 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; One can check that $\\mathbb {D}_n$ is unimodal if $n$ is odd and has dimension 2 if $n$ is even.",
"Proposition 4.11 The number of points on varieties attached to $\\mathbb {D}_n$ is given by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}+q^3-q+1}{q + 1} \\quad \\text{and} \\quad N^\\mathtt {generic}_{\\mathbb {D}_n} = (q^{n/2} - 1)^2$ if $n$ is even and by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}-q^3+q-1}{q^2-1} \\quad \\text{and} \\quad N_{\\mathbb {D}_n}^\\mathtt {generic}= q^n-1$ if $n$ is odd.",
"Proof.",
"This is easily deduced from the type $\\mathbb {A}$ case, using REF , REF applied to a red leaf on a short branch.",
"Trees of type $\\mathbb {E}$ Let us consider now a family of trees containing the Dynkin diagrams of type $\\mathbb {E}$ .",
"The tree $\\mathbb {E}_n$ is the tree with one triple point and branches of size 1, 2 and $n-4$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=red!20] 5 – (4,0) node[fill=green!20] 6 – (5,0) node[fill=red!20]7; (2,1) node[fill=red!20] 3 – (2,0) node[fill=green!20] 4; One can check that $\\mathbb {E}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.12 The number of points on varieties attached to $\\mathbb {E}_n$ is given by $N_{\\mathbb {E}_n} = (q^2 - q + 1)\\frac{q^{n-1} - 1}{q-1}$ if $n$ is even and by $N_{\\mathbb {E}_n}^\\mathtt {versal}= (q^2 - q + 1)(1+q^{n-1}) $ and $N_{\\mathbb {E}_n}^\\mathtt {generic}= \\frac{q^{n+1}-q^{n}+q^{n-1}-q^{(n+3)/2}-q^{(n-1)/2}+q^{2}-q+1}{q-1}$ if $n$ is odd.",
"Proof.",
"In the even case, one uses lemma REF applied to the domino on the short branch, and the known type $\\mathbb {A}$ cases.",
"In the odd case, one uses lemmas REF and REF applied to the red leaf on the short branch, and the known type $\\mathbb {A}$ cases.",
"Orange trees and unimodal trees Let us now describe a recursion involving only the polynomials for orange trees and versal unimodal trees.",
"Let $T$ be an orange tree and $v$ be a leaf of $T$ .",
"Let $T^{\\prime }$ be the unimodal tree $T \\setminus \\lbrace v\\rbrace $ and let $F$ be the orange forest obtained from $T$ by removing the domino $u - v$ containing $v$ .",
"Lemma 4.13 There is a decomposition $X_T = X^{\\mathtt {versal}}_{T^{\\prime }} \\sqcup A_{1} X_F.$ Proof.",
"This decomposition is made according to the value of $x_v$ .",
"If $x_v = 0$ , then one has a free parameter $x^{\\prime }_v$ , which gives the factor $A_{1}$ .",
"One also has $x_u = -1$ and one can get rid of $x^{\\prime }_u$ .",
"The value $-1$ is attached as a coefficient to some orange vertices of $F$ , but one can detach this coefficient by lemma REF .",
"There remains the equations for $X_F$ .",
"If $x_v \\ne 0$ , one can use lemma REF .",
"In the special case of a leaf, this gives an isomorphism with $X^\\mathtt {versal}_{T^{\\prime }}$ .",
"One can use lemma REF to compute the enumerating polynomials for orange trees and versal unimodal trees only, by the following algorithm.",
"Step 0: if the tree $T$ is of type $\\mathbb {A}_n$ with $n$ even, use the known value from (REF ) in proposition REF .",
"Step 1: if the tree $T$ is orange, find a leaf $v$ whose branch has minimal length.",
"Here the branch is the longest sequence of vertices of valency 2 starting at the unique neighbor of the leaf (it could be empty).",
"Then use lemma REF applied to the leaf $v$ to compute $N_T$ .",
"Step 2: if the tree $T$ is unimodal, find a red leaf $w$ whose branch has maximal length.",
"Adding a vertex $v$ at the end of this branch gives an orange tree $T^{\\prime }$ .",
"Then use lemma REF (backwards) applied to the tree $T^{\\prime }$ and its leaf $v$ to compute $N_T$ .",
"This will work because each step either shorten the shortest branch or add some vertex to the longest branch.",
"This makes sure that the tree become more and more linear, and that at some point one is reduced to the initial step.",
"This is a decreasing induction on the number of points of valency at least 3 and the length of the longest branch.",
"Remark 4.14 For orange trees, one can use instead in this algorithm the lemma REF , maybe choosing a domino close to the center of the tree for a better complexity.",
"Euler characteristic and independent sets Let us denote by $\\operatorname{vc}(T)$ the number of minimum vertex covers of $T$ .",
"This is also the number of maximum independent sets.",
"Let us now describe a decomposition of the versal varieties according to independent sets (not necessarily maximal).",
"If $S$ is a subset of the vertices of $T$ , one can define $W_T(S)$ as the set of points in $X_T^\\mathtt {versal}$ where $x_u = 0 & \\quad \\text{if } u\\in S,\\\\x_u \\ne 0 & \\quad \\text{if } u\\notin S.$ The sets $W_T(S)$ are obviously disjoint in $X_T^\\mathtt {versal}$ .",
"Lemma 4.15 If the set $W_T(S)$ is not empty, then $S$ is an independent set in $T$ .",
"Proof.",
"This follows from lemma REF .",
"Proposition 4.16 Let $S$ be an independent set in $T$ .",
"There is an isomorphism $W_T(S) \\simeq (\\mathbb {G}_m)^{t + \\dim (T)- 2s} \\times (A_{1})^{s},$ where $t$ is the size of $T$ and $s$ the size of $S$ .",
"Proof.",
"Let us fix a maximum matching $M$ of $T$ .",
"For every $u$ not in $S$ , one can use the hypothesis $x_u \\ne 0$ to get rid of $x^{\\prime }_u$ and of the equation of index $u$ .",
"There remains only the equations of index $v$ for $v \\in S$ .",
"Because $x_v=0$ when $v \\in S$ , the variables $x^{\\prime }_v$ for $v \\in S$ do no longer appear in the equations, hence they are free.",
"This gives the factor $(A_{1})^s$ .",
"Then there remains $s$ equations of the general shape $-1 = \\alpha _i \\prod _{j - i} x_j,\\qquad \\mathrm {(E_i)}$ involving the $t-s$ invertible variables $x_u$ and the $\\dim (T)$ coefficient variables $\\alpha _i$ .",
"The factor $\\alpha _i$ is present in this equation only if the vertex $i$ is not covered by the chosen maximum matching $M$ .",
"One will use the following auxiliary graph $\\widehat{T}$ .",
"The vertices are the vertices of $T$ and new vertices $Z_i$ indexed by coefficient variables $\\alpha _i$ for $i \\notin M$ .",
"The edges of $\\widehat{T}$ are edges of $T$ and new edges between the vertex $Z_i$ and the vertex $i$ for every $i \\notin M$ .",
"Clearly, this graph is still a tree and admits a perfect matching $\\widehat{M}$ , by adding dominoes $i-Z_i$ to the matching $M$ .",
"Because $S$ is an independent set in $T$ , there is at most one element of $S$ in every edge of $\\widehat{T}$ .",
"Let us orient every edge containing an element of $S$ towards this element if the edge is a domino and in the other way otherwise.",
"This defines a partial order on the vertices of $\\widehat{T}$ , decreasing along the chosen orientation of edges.",
"Consider now the equation REF associated with a vertex $i \\in S$ .",
"There is a unique domino $i-j$ in $\\widehat{T}$ containing $i$ .",
"The equation can then be used to express the variable $x_j$ in terms of variables of lower index in the partial order.",
"One can therefore eliminate one variable for every equation.",
"At the end, one obtains an algebraic torus whose dimension is the difference between the number $ t -s +\\dim (T)$ of initial variables and the number $s$ of equations.",
"Corollary 4.17 The Euler characteristic of $X_T^\\mathtt {versal}$ is $\\operatorname{vc}(T)$ .",
"Proof.",
"Every set $W_T(S)$ contributes either 0 or 1 to the Euler characteristic.",
"It contributes by 1 if and only if the exponent $t+ \\dim (T)- 2s$ is zero.",
"This exponent can be expressed as $(r(T) + o(T) + g(T)) + (r(T) - g(T)) - 2 s.$ It is therefore zero if and only if $s = r(T) + o(T)/2$ , which is the size of the maximum independent sets in $T$ .",
"Of course, one can also use Proposition REF to give a formula for the number of points $N^\\mathtt {versal}_T$ as a sum over independent sets.",
"Corollary 4.18 The value at $q=1$ of the polynomial $N^\\mathtt {versal}_T$ is the number $\\operatorname{vc}(T)$ of maximum independent sets of $T$ .",
"Cohomology: general setting and results This section first describes some differential forms that are always present in the varieties under study, and then very briefly recalls the results one needs about (mixed) Hodge structures.",
"For a general reference about mixed Hodge structures, see for example [18].",
"Weil-Petersson two-form Let $T$ be a tree and let $S$ be a subset of $T$ .",
"Consider the augmented tree $T+S$ obtained by adding a new edge out of every vertex in $S$ , and endow this tree with a bipartite orientation, where every vertex is either a sink or a source.",
"As a variant of the definition of the variety $X_T^\\varphi $ , one can define a variety $X(T+S)$ attached to this data, with invertible variables associated to the new vertices, playing the role of coefficients in the equations (as the $\\alpha $ do).",
"Let $\\omega _i$ denote $d \\log (x_i)$ .",
"The following lemma has been proved by Greg Muller in [16] in a more general context.",
"Lemma 5.1 The differential form $\\operatorname{WP}= \\sum _{i \\rightarrow j} \\omega _i \\omega _j,$ where the sum is running over edges of $T+S$ , is an algebraic differential form on the variety $X(T+S)$ .",
"Proof.",
"Let us prove that it has no pole.",
"Let us fix $i$ .",
"To study the possible pole along $x_i = 0$ , it is enough to look at the sum $\\sum _{j \\leftrightarrow i} \\omega _i \\omega _j$ restricted to edges containing $i$ .",
"By the relation $x_i x^{\\prime }_i = 1 + \\prod _{j\\leftrightarrow i} x_j$ , one has $x_i d x^{\\prime }_i + x^{\\prime }_i d x_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j,$ and therefore $x_i dx^{\\prime }_i dx_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j d x_i.$ This implies $dx^{\\prime }_i dx_i / \\prod _{k\\leftrightarrow i} x_k = \\sum _{j\\leftrightarrow i} \\omega _j \\omega _i,$ where the left-hand side has clearly no pole at $x_i$ .",
"Note that $\\operatorname{WP}$ stands here for Weil-Petersson.",
"Abusing notations, one will use the same symbol $\\operatorname{WP}$ to denote these differential forms on different varieties.",
"The ambient variety should be clear from the context.",
"Hodge structures We will use the notation $\\mathbb {Q}(-i)$ to denote a one dimensional vector space over $\\mathbb {Q}$ endowed with a pure Hodge structure of Tate type, of weight $2i$ and type $(i,i)$ .",
"The tensor product of $\\mathbb {Q}(-i)$ and $\\mathbb {Q}(-j)$ is $\\mathbb {Q}(-i-j)$ .",
"Recall that the cohomology of $\\mathbb {G}_m$ has an Hodge structure described by $\\mathsf {H}^k(\\mathbb {G}_m) = \\mathbb {Q}(-k)$ for $0 \\le k \\le 1$ .",
"There is no morphism between pure Hodge structures of distinct weights.",
"The Künneth isomorphism is compatible with the Hodge structures.",
"The Mayer-Vietoris long exact sequence is an exact sequence of Hodge structures.",
"Cohomology: orange and versal cases This section deals with the cohomology, in several cases where either varieties do not depend on parameters, or versal conditions are assumed on all parameters.",
"The first part is devoted to linear trees; the results there can then be used as building blocks.",
"Linear trees $\\mathbb {A}$ Let $\\mathbb {A}_n$ be the linear tree with $n$ vertices numbered from 1 to $n$ .",
"As seen in §REF , this is an orange tree if $n$ is even, and an unimodal tree otherwise.",
"Some of the results of this section were already obtained in [6] using instead the cohomology with compact supports.",
"Cohomology of some auxiliary varieties for $\\mathbb {A}$ Let us introduce three varieties $X_n$ , $Y_n$ and $Z_n$ with dimensions $n, n+1$ and $n+1$ .",
"The variety $Z_n$ is defined by variables $x_1,\\dots ,x_n$ , $x^{\\prime }_1,\\dots ,x^{\\prime }_n$ and $\\alpha $ such that $x_1 x^{\\prime }_1 &= 1 + \\alpha x_2 ,\\\\x_i x^{\\prime }_i &= 1 + x_{i-1} x_{i+1}, \\\\x_n x^{\\prime }_n &= 1 + x_{n-1}.$ The variety $Y_n$ is the open set in $Z_n$ where $\\alpha $ is invertible.",
"The variety $X_n$ is the closed set in $Y_n$ where $\\alpha $ is fixed to a generic invertible value (where generic means distinct from $(-1)^{(n+1)/2}$ if $n$ is odd).",
"In our general notations, $Y_n$ is $X_{\\mathbb {A}_n}^\\mathtt {versal}$ and $X_n$ is $X_{\\mathbb {A}_n}^\\mathtt {generic}$ .",
"Let us first describe the variety $Z_n$ .",
"Proposition 6.1 There exists an isomorphism between $Z_n$ and the affine space $A_{n+1}$ .",
"Proof.",
"This has been proved in [6].",
"Therefore, the cohomology of $Z_n$ is known for all $n$ : $\\mathsf {H}^k(Z_n) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}1 &\\text{ if } k=0,\\\\0 &\\text{ if } k>0.\\end{array}\\right.",
"}$ The Hodge structure on $\\mathsf {H}^0(Z_n)$ is $\\mathbb {Q}(0)$ .",
"Let us now compute the cohomology of $Y_n$ by induction.",
"This uses the Mayer-Vietoris long exact sequence for the covering of $Z_n$ by the two open sets $U(x_1)$ and $U(\\alpha )$ .",
"First, let us note that $U(\\alpha ) \\simeq Y_n $ by definition.",
"Next, one finds that $U(x_1) \\simeq A_{1} Y_{n-1}$ .",
"Indeed one can eliminate $x^{\\prime }_1$ using the first equation.",
"Then $\\alpha $ becomes a free variable, and there remains the equations for $Y_{n-1}$ , with $x_1$ now playing the role of $\\alpha $ .",
"Last, the intersection $U(\\alpha ) \\cap U(x_1)$ is isomorphic to $ \\mathbb {G}_mY_{n-1}$ , by the same argument.",
"Let us write $\\omega _{\\alpha }$ for $d \\log (\\alpha )$ .",
"Proposition 6.2 The cohomology ring of $Y_n$ has the following description: $\\mathsf {H}^k(Y_n) = \\mathbb {Q}(-k)$ for $0 \\le k \\le n+1$ .",
"It has a basis given by powers of $\\operatorname{WP}$ in even degrees and by powers of $\\operatorname{WP}$ times $\\omega _{\\alpha }$ in odd degrees.",
"It is generated by the 1-form $\\omega _{\\alpha }$ and the 2-form $\\operatorname{WP}$ .",
"Proof.",
"Because of the vanishing of $\\mathsf {H}^k(Z_n)$ for $k>0$ , the Mayer-Vietoris long exact sequence gives short exact sequences $0 \\rightarrow \\mathsf {H}^0(Z_n) \\rightarrow \\mathsf {H}^0(Y_n)\\oplus \\mathsf {H}^0(U(x_1)) \\rightarrow \\mathsf {H}^0(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ and $0 \\rightarrow \\mathsf {H}^k(Y_n)\\oplus \\mathsf {H}^k(U(x_1)) \\rightarrow \\mathsf {H}^k(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ for every $k > 0$ .",
"This determines by induction the Hodge structure of the cohomology of $Y_{n}$ .",
"Let us now proceed to the expected basis.",
"One already knows that $\\operatorname{WP}$ and $\\omega _{\\alpha }$ are indeed algebraic differential forms on $Y_{n}$ .",
"By the short exact sequences above, one can check that for $k>0$ the union of the expected basis of $\\mathsf {H}^k(Y_n)$ with the known basis of $\\mathsf {H}^k(U(x_1))$ is mapped to a basis of $\\mathsf {H}^k(U(\\alpha ) \\cap U(x_1))$ .",
"This implies the statement.",
"Cohomology for $\\mathbb {A}_n$ with even $n$ Let us now consider the linear tree $\\mathbb {A}_n$ for even $n$ , and compute the cohomology of $X_n$ .",
"Proposition 6.3 The Hodge structure of the cohomology of $X_n$ is $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for all even $k$ between 0 and $n$ , and 0 otherwise.",
"A basis is given by powers of $\\operatorname{WP}$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ .",
"Proof.",
"This follows from the known cohomology of $Y_n$ and the Künneth theorem applied to the isomorphism $Y_n \\simeq X_n \\mathbb {G}_m$ given by lemma REF .",
"The Künneth theorem gives immediately the Hodge structure.",
"For the basis, it is enough to recall that the $\\mathbb {G}_m$ factor is given by the value of $\\alpha $ , and to check that fixing the value $\\alpha = 1$ maps $\\operatorname{WP}$ (for $Y_n$ ) to $\\operatorname{WP}$ (for $X_n$ ).",
"Cohomology for orange trees of shape $H$ 0.7 [scale=0.7] patterns,decorations.pathreplacing every node=[draw,shape=circle,very thick,fill=white] (0,0) node ... – (1,0) node ... – (3,0) node ... – (4,0) node ...; (0,1) node ... – (1,1) node ... – (3,1) node ... – (4,1) node ...; (2,1) node $a$ – (2,0) node $b$ ; every node=[draw=none,fill=none]; [thick,decoration= brace, raise=0.3cm ,decorate] (-0.3,1) node – (1.3,1) node ; [thick,decoration= brace, raise=0.3cm ,decorate] (2.7,1) node – (4.3,1) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (-0.3,0) node – (1.3,0) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (2.7,0) node – (4.3,0) node ; t (0.5,-1) $m$ ; t (0.5,2) $k$ ; t (3.5,-1) $n$ ; t (3.5,2) $\\ell $ ; Let us denote by $H_{k,\\ell ,m,n}$ the tree described as two chains joined by an edge, such that by removing the joining edge and its extremities $a$ and $b$ , one gets two chains of lengths $k$ and $\\ell $ on the $a$ side (top) and two chains of lengths $m$ and $n$ on the $b$ side (bottom).",
"We assume now that $H_{k,\\ell ,m,n}$ is an orange tree.",
"It implies that either $k, \\ell , m$ and $n$ are even if the middle edge is an orange domino, or that (without loss of generality) $k$ and $m$ are odd and $l$ and $n$ are even otherwise.",
"Then one can compute the cohomology of $H_{k,\\ell ,m,n}$ using the Mayer-Vietoris long exact sequence for the open covering by $U(x_a)$ and $U(x_b)$ .",
"When the middle edge is an orange domino, one has $\\begin{aligned}U(x_a) &\\simeq X_k X_\\ell Y_{m+n+1},\\\\U(x_b) &\\simeq Y_{k+\\ell +1} X_m X_n ,\\\\U(x_a) \\cap U(x_b) &\\simeq (\\mathbb {G}_m)^2 X_k X_\\ell X_m X_n.\\end{aligned}$ When the middle edge is not an orange domino, one finds instead $\\begin{aligned}U(x_a) &\\simeq Y_k X_\\ell X_{m+n+1},\\\\U(x_b) &\\simeq X_{k+\\ell +1} Y_m X_n,\\\\U(x_a) \\cap U(x_b) &\\simeq Y_k X_\\ell Y_m X_n.\\end{aligned}$ Let us introduce some notations: call $K,L,M,N$ the subsets of vertices corresponding to the four branches of $H$ (i.e.",
"the connected components of $H \\setminus \\lbrace a,b\\rbrace $ ).",
"Let us denote by $W_S$ the Weil-Petersson 2-form associated with a subset $S$ of the vertices of $H$ .",
"For conciseness, one will use shortcuts such as $W_{KaL}$ or $W_{MabN}$ .",
"Note that there holds $\\omega _a W_{aL} = \\omega _a W_{L}$ and other similar simplifications, by the definition (REF ) of these forms.",
"Let us now describe generators and bases of the cohomology of the open sets $ U(x_a)$ , $U(x_b)$ and $U(x_b)\\cap U(x_b)$ .",
"This can be computed using the isomorphisms (REF ), (REF ) and the known cohomology of varieties $X$ and $Y$ .",
"It turns out that the result does not depend on whether or not the middle edge $a-b$ is an orange domino.",
"The cohomology of $U(x_a)$ is generated by $\\omega _a$ , $W_{Ka}$ , $W_{aL}$ and $W_{MabN}$ .",
"A basis is given by $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B} \\quad \\text{and}\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{MbN}^{B},$ where $0 \\le \\kappa \\le k/2$ , $0 \\le \\lambda \\le l/2$ and $0 \\le B \\le (m+n+2)/2$ (left) or $0 \\le B \\le (m+n)/2$ (right).",
"Similarly, the cohomology of $U(x_b)$ is generated by $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ and $W_{KabL}$ .",
"A basis is given by $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A} \\quad \\text{and}\\quad \\omega _b W_{M}^{\\mu } W_{N}^{\\nu } W_{KaL}^{A},$ where $0 \\le \\mu \\le m/2$ , $0 \\le \\nu \\le n/2$ and $0 \\le A \\le (k+l+2)/2$ (left) or $0 \\le A \\le (k+l)/2$ (right).",
"The cohomology of $U(x_b)\\cap U(x_b)$ is generated by $\\omega _a$ , $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ , $W_{Ka}$ and $W_{aL}$ .",
"A basis is given by $\\begin{aligned}W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\\\\\omega _a \\omega _b W_{K}^{\\kappa } W_{L}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }\\quad \\text{and}\\quad \\omega _b W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu },\\end{aligned}$ with the same conditions as above on $\\kappa ,\\lambda ,\\mu $ and $\\nu $ .",
"There is a bigrading corresponding to the top and bottom parts of the $H$ shape.",
"Every differential form involved in the bases just described is a sum of products of $\\omega _i$ .",
"The bidegree of a monomial in the $\\omega _i$ is the pair (number of $\\omega _i$ where $i$ is in the top row, number of $\\omega _i$ where $i$ is in the bottom row).",
"Among the various Weil-Petersson forms involved, only the differential forms $W_{KabL}$ and $W_{MabN}$ are not homogeneous for the bidegree, but have terms in bidegrees $(2,0)$ and $(1,1)$ (resp.",
"$(0,2)$ and $(1,1)$ ).",
"One needs now to compute explicitly the following maps in the Mayer-Vietoris long exact sequence: $\\mathsf {H}^i(U(x_a))\\oplus \\mathsf {H}^i(U(x_b)) \\stackrel{f_i}{\\longrightarrow } \\mathsf {H}^i(U(x_a) \\cap U(x_b)).$ Because one has bases of all these spaces, this is a matter of matrices.",
"For odd degree $i$ , let us show that the differential is injective.",
"Because in this case all basis elements (given by right columns of (REF ), (REF ) and (REF )) are homogeneous for the bigrading, one can separate the cases of bidegree congruent to $(0,1)$ and to $(1,0)$ modulo $(2,2)$ .",
"Let us give details only for the first possibility, the other case being similar after exchanging top and bottom of $H$ .",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_b))$ is given by $\\omega _{b}W_{KaL}^{A}W_{M}^{\\mu } W_{N}^{\\nu }$ with $i =1+2A+2\\mu +2\\nu $ .",
"The corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a))$ is zero.",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a) \\cap U(x_b))$ is given by $\\omega _{b}W_{Ka}^{\\kappa }W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }$ with $i = 1+2\\kappa +2\\lambda +2\\mu +2\\nu $ .",
"But $W_{KaL}^{A}$ can be written as a linear combination of $W_{Ka}^{\\kappa }W_{aL}^{\\lambda }$ with $\\kappa + \\lambda = A$ .",
"Therefore the basis elements are mapped to linear combinations with disjoint supports.",
"It follows that the map $f_i$ is injective.",
"Let us now turn to even degrees.",
"Proposition 6.4 For even degree $2i$ , the kernel of the differential $f_{2i}$ has dimension 1, spanned by the $i^{th}$ power of the form $\\operatorname{WP}$ .",
"Proof.",
"First note that one can define an injective map $\\Delta $ from the space $\\mathsf {H}^{2i}(U(x_a) \\cap U(x_b))$ to the space $D_i$ spanned by all products of $i$ 2-forms of the shape $\\omega _{s}\\omega _{t}$ for $s-t$ an edge of the tree (always written in the order given by a fixed alternating orientation of the tree).",
"Indeed, both terms in the left column of (REF ) can be written as linear combinations of such products.",
"The injectivity holds because distinct elements in this part of the basis are mapped to linear combinations with disjoint supports.",
"To recover a basis element $B$ from any monomial in its image by $\\Delta $ , first count in $\\Delta (B)$ if the number of $\\omega _k$ in the top row is odd or even.",
"This tells if the basis elements $B$ contains $\\omega _a\\omega _b$ or not.",
"Then it is easy to recover the exponents $(\\kappa , \\lambda , \\mu , \\nu )$ defining $B$ by counting in $\\Delta (B)$ how many $\\omega _k$ there are in the different parts of the tree.",
"To prove the statement of the proposition, it is therefore enough to compute the kernel of the composite map $\\Delta \\circ f_{2i}$ .",
"It turns out that the matrix of this composite map has a nice description.",
"First, every monomial $d$ made of $i$ 2-forms $\\omega _{s}\\omega _{t}$ as above appears in exactly two images, the image of a form $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B}$ and the image of a form $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A}$ (with opposite signs).",
"Let us denote these two forms by $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ .",
"On the other hand, the image of every basis element is the sum of several monomials (at least one), with constant sign.",
"Let us pick an element $z$ of the kernel of $f_{2i}$ .",
"Then for every monomial $d$ in $D_i$ , the coefficients of $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ in $z$ must be the same.",
"One can make a graph with vertices given by all forms in the basis, and edges corresponding to the relations $\\mathsf {F}_a(d)-\\mathsf {F}_b(d)$ for all monomials $d$ .",
"By a combinatorial argument, one can check that this graph is connected.",
"For this, one just has to show that one can go from any monomial $d$ to any monomial $d^{\\prime }$ , using two kinds of moves: replace $d$ by another monomial appearing in the same $\\mathsf {F}_a(d)$ , or replace $d$ by another monomial appearing in the same $\\mathsf {F}_b(d)$ .",
"This is not difficult once translated in terms of dominoes, and details are left to the reader.",
"From the connectedness of this graph, one deduces that the kernel is spanned by the sum of all basis elements of $\\mathsf {H}^{2i}(U(x_a))\\oplus \\mathsf {H}^{2i}(U(x_b))$ , which is just $(\\operatorname{WP}^i,\\operatorname{WP}^i)$ .",
"This proposition and the injectivity in the case of odd degree allow to give a description of the weights of the Hodge structure on the cohomology.",
"This can easily be made explicit, but one will not do that here.",
"There would remain to find explicit expressions for the cohomology classes coming from the co-image of the differentials $f_i$ .",
"In the case of the Dynkin diagrams $\\mathbb {E}_6$ and $\\mathbb {E}_8$ , one can go further and compute explicit representatives of the cohomology classes.",
"By the general proof, the cohomology for $\\mathbb {E}_6$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-6),$ where the $\\mathbb {Q}(-i)$ with $i$ even correspond to the powers of $\\operatorname{WP}$ .",
"Using the connection homomorphism in the long exact sequence, one finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ .",
"Similarly, the cohomology for $\\mathbb {E}_8$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-5)\\oplus \\mathbb {Q}(-6) \\mid 0 \\mid \\mathbb {Q}(-8),$ where the even $\\mathbb {Q}(-i)$ are the powers of $\\operatorname{WP}$ .",
"One finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ , and its product by $\\operatorname{WP}$ corresponds to $\\mathbb {Q}(-5)$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6 – (5,0) node[fill=orange!20] 7 – (6,0) node[fill=orange!20] 8; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; Cohomology: generic cases This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers.",
"Cohomology for $\\mathbb {A}$ odd and generic Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Smoothness and free actions",
"Theorem 3.1 For every choice of $\\varphi $ , the variety $X^\\varphi _T$ is smooth.",
"Proof.",
"The proof is by induction on the size of the tree $T$ .",
"For the tree with only one vertex, the only equation is $x x^{\\prime } = 1 + \\alpha .$ In the $\\mathtt {generic}$ case when $\\alpha $ is considered to have a fixed value, different from $-1$ by the genericity condition (REF ), the variety is isomorphic to the punctured affine line $\\mathbb {G}_m$ and is therefore smooth.",
"In the $\\mathtt {versal}$ case when $\\alpha $ is considered to be a variable and assumed to be invertible, the variety is an open set in the variety defined by (REF ) where $\\alpha $ is not assumed to be invertible.",
"This last variety is isomorphic to the affine plane $A_{2}$ , hence smooth.",
"The rest of the proof by induction is organized as follows.",
"One first considers the case when the tree has at least one red-green component, and treat separately the case when there is a red-green component which is $\\mathtt {generic}$ and the case when there is one which is $\\mathtt {versal}$ .",
"Otherwise, the tree is orange.",
"These three cases are done in the next three subsections.",
"Let us first state a few useful lemmas.",
"Lemma 3.2 If one variable $x_i$ is assumed to be non-zero, then one can get rid of the associated variable $x^{\\prime }_i$ and of the equation (REF ) of index $i$ .",
"Proof.",
"Indeed, one can just use the equation to eliminate $x^{\\prime }_i$ .",
"Lemma 3.3 If one variable $x_i$ is assumed to be zero, then $x^{\\prime }_i$ becomes a free variable and the equation (REF ) of index $i$ reduces to $-1 = \\alpha _i \\prod _{i-j} x_j.$ Let us now introduce a useful variant of the varieties $X_T^\\varphi $ .",
"Let $v$ be a vertex of $T$ .",
"Let $X^\\varphi _T[v]$ be defined just as $X_T^\\varphi $ , but with one more invertible variable $\\gamma _v$ attached to the vertex $v$ as a coefficient (playing the same role as $\\alpha _v$ in the equations).",
"This variable defines a morphism $\\gamma _v$ from $X^\\varphi _T[v]$ to $\\mathbb {G}_m$ .",
"Lemma 3.4 If $v$ is an orange or green vertex, then $X^\\varphi _T[v]$ is isomorphic as a variety over $\\mathbb {G}_m$ to $X_T^\\varphi \\times \\mathbb {G}_m$ endowed with the projection to the second factor.",
"Proof.",
"By proposition REF and its proof, one can find an isomorphism in the groupoid $G_T$ between $X^\\varphi _T$ and $X^\\varphi _T[v]$ that only changes the coordinates $x_i$ for orange and green vertices.",
"More precisely, using the auxiliary oriented graph $\\mathcal {G}$ , one can find a sequence of jumps (corresponding to edges in $\\mathcal {G}$ starting with a green or orange vertex) that makes the coefficient $\\gamma _v$ disappear from the equations.",
"The isomorphism associated with this sequence of jumps is multiplying the variables $x_i$ by monic Laurent monomials in the parameter $\\gamma _v$ , hence defines an isomorphism over $\\mathbb {G}_m$ .",
"Lemma 3.5 If $v$ is a red vertex in a versal red-green component $C$ , then $X^\\varphi _T[v]$ is isomorphic as a variety over $\\mathbb {G}_m$ to $X_T^\\varphi \\times \\mathbb {G}_m$ endowed with the projection to the second factor.",
"Proof.",
"If the red vertex $v$ is not covered by the matching $M$ chosen to define $X^\\varphi _T$ , then one has two coefficient variables $\\alpha _v$ and $\\gamma _v$ attached to the vertex $v$ .",
"By the simple change of coordinates $\\alpha _v := \\alpha _v \\gamma _v$ and $\\gamma _v := \\gamma _v$ , one gets the expected isomorphism.",
"Assume now that that red vertex $v$ is covered by the matching $M$ .",
"By proposition REF and its proof, one can find an isomorphism in the groupoid $G_T$ between $X^\\varphi _T[v]$ and a variety $X^{\\varphi ,M}_{T,\\beta }$ that only changes the coordinates $x_i$ for orange and green vertices and for red vertices in the red-green component $C$ .",
"More precisely, using the auxiliary oriented graph $\\mathcal {G}$ , one can find a sequence of jumps that moves the coefficient $\\gamma _v$ towards the red vertices in $C$ not covered by the matching.",
"At the end, every new coefficient $\\beta _i$ is the product of $\\alpha _i$ by a Laurent monomial in $\\gamma _v$ .",
"The isomorphism associated with this sequence of jumps is multiplying the variables $x_i$ by monic Laurent monomials in the parameter $\\gamma _v$ , hence defines an isomorphism over $\\mathbb {G}_m$ .",
"One can then compose this isomorphism with a relabeling of the coefficients $\\alpha _i := \\beta _i$ in order to get the expected isomorphism, still defined over $\\mathbb {G}_m$ , between $X^\\varphi _T[v]$ and $X_T^\\varphi \\times \\mathbb {G}_m$ .",
"One could say that the coefficient $\\gamma _v$ can be detached from $T$ in these cases.",
"This will be used frequently in the rest of the article.",
"Trees with a generic component One assumes now that $T$ has at least two vertices and a $\\mathtt {generic}$ component $C$ .",
"Let us pick an admissible set $S$ of red vertices in $C$ , as defined in §REF .",
"Lemma 3.6 The open sets $U(x_i)$ for $i \\in S$ form a covering of $X^\\varphi _T$ .",
"Proof.",
"Indeed, the complement of their union is the set where all variables $x_i$ for $i \\in S$ vanish.",
"This implies that $\\alpha _i \\prod _{j-i} x_j = -1$ for every $i$ in $S$ .",
"Taking the alternating product of these equalities gives $\\prod _{i \\in S} \\alpha _i^{\\pm } = (-1)^{\\# S},$ because for every green vertex $j$ attached by an edge to some element of $S$ , the cluster variable $x_j$ appears exactly twice by definition of admissible sets, hence disappears in the alternating product.",
"But the equation (REF ) is incompatible with the genericity condition (REF ).",
"Let us now show that the open sets $U(x_i)$ are smooth.",
"Let $F$ be the forest $ T \\setminus \\lbrace i\\rbrace $ .",
"In the forest $F$ , the coloring is changed only on the red-green component containing $i$ , where an avalanche of orange dominoes can take place when removing $i$ .",
"The red-green component $C$ is therefore split into a number of red-green components.",
"Let us moreover introduce a function $\\varphi $ on $F$ , which is $\\mathtt {generic}$ on every red-green component coming from $C$ , and unchanged on all other red-green components.",
"Lemma 3.7 The open set $U(x_i)$ is isomorphic to $\\mathbb {G}_m\\times X^\\varphi _F$ .",
"Proof.",
"The condition that $x_i$ is not zero allows one to get rid of the variable $x^{\\prime }_i$ by using the equation (REF ) of index $i$ .",
"What remains are the equations for the forest $F = T \\setminus \\lbrace i\\rbrace $ , where now $x_i$ is treated as a parameter attached to all neighbors of $i$ in $T$ .",
"Because all neighbors of $i$ in $T$ are green, they become either green or orange in $F$ .",
"It follows from lemma REF that one can, without changing the variety, consider instead that the parameter $x_i$ is not attached to any vertex of $F$ .",
"Let us check that the genericity condition still holds on all $\\mathtt {generic}$ red-green components.",
"If the component $D$ does not come from the splitting of $C$ , then the genericity conditions are unchanged on this red-green component.",
"Otherwise, let us choose an admissible set in $D$ .",
"It was then already an admissible set in $C$ , by inspection of what happens during the avalanche of orange dominoes.",
"Therefore the genericity condition for $D$ is inherited from that for $C$ .",
"One has therefore obtained an isomorphism $U(x_i) \\simeq \\mathbb {G}_m\\times X^\\varphi _F,$ which is smooth by induction.",
"Therefore $X^\\varphi _T$ is also smooth.",
"Trees with a versal component One assumes now that $T$ has at least two vertices, and has a $\\mathtt {versal}$ component $C$ .",
"Let us choose a red leaf $v$ in this component.",
"By proposition REF , one can find a maximum matching $M$ not containing $v$ .",
"Therefore there is a coefficient variable $\\alpha _v$ .",
"Let $u$ be the green vertex adjacent to $v$ .",
"By lemma REF , the two open sets $U(x_u)$ and $U(x_v)$ cover $X^\\varphi _T$ .",
"Let us first prove that $U(x_v)$ is smooth.",
"Let $T^{\\prime }$ be the tree $T\\setminus \\lbrace v\\rbrace $ .",
"The coloring of $T^{\\prime }$ is obtained from $T$ by an avalanche of orange dominoes.",
"The dimension of $T^{\\prime }$ is $\\dim (T) - 1$ .",
"The avalanche may split the red-green component of $T$ containing $v$ into several components.",
"Let $\\varphi $ be the function which maps all these new components to the $\\mathtt {versal}$ condition, and unchanged condition on all the other red-green components.",
"Lemma 3.8 The open set $U(x_v)$ is isomorphic to $\\mathbb {G}_m^2 \\times X^\\varphi _{T^{\\prime }}$ .",
"Proof.",
"Assuming that $x_v$ is not zero allows one to get rid of the variable $x^{\\prime }_v$ by using (REF ) with index $v$ .",
"The coefficient variable $\\alpha _v$ also disappears from the equations: this gives one factor $\\mathbb {G}_m$ .",
"Then the variable $x_v$ is seen as a coefficient attached to the vertex $u$ in $T^{\\prime }$ , which is either green or orange.",
"The coefficient can therefore be detached by lemma REF , and one obtains a factor isomorphic to $\\mathbb {G}_m\\times X^\\varphi _{T^{\\prime }}$ .",
"Therefore $U(x_v)$ is smooth by induction.",
"Let us now prove that $U(x_u)$ is smooth.",
"Let us choose instead a matching $M$ containing the domino $u-v$ , thanks to lemma REF .",
"This amounts to go through an isomorphism in the groupoid $G_T$ , hence preserves the open set $U(x_u)$ .",
"Let $F$ be the forest $T\\setminus \\lbrace u\\rbrace $ .",
"Because $u$ is green, the coloring of $F$ is obtained from that of $T$ by restriction and the dimension of $F$ is $\\dim (T) + 1$ .",
"Let $v, T_1, \\dots , T_k$ be the connected components of the forest $F$ .",
"By removing the domino $u-v$ , one can restrict the matching $M$ to a matching of the forest $F$ .",
"The red-green component of $T$ containing $u$ splits into several red-green components in $F$ , one of them being the vertex $v$ .",
"One takes the $\\mathtt {versal}$ condition on all of these red-green components of $F$ , and unchanged condition on all the other red-green components.",
"Lemma 3.9 The open set $U(x_u)$ is isomorphic to $X^{\\mathtt {versal}}_{\\lbrace v\\rbrace } \\times \\prod _{j=1}^k X^\\varphi _{T_j},$ where the first component is the vertex $v$ with coefficient variable $x_u$ .",
"Proof.",
"Setting $x_u\\ne 0$ in the equations allows to get rid of the variable $x^{\\prime }_u$ .",
"The result can be described as a fiber product over $\\mathbb {G}_m$ , where the same coefficient variable $x_u$ is attached to every connected component of $F$ at a red vertex in a versal red-green component.",
"By repeated use of lemma REF on all connected components (but not on the isolated vertex $v$ ), one finds that the open set $U(x_u)$ is isomorphic to the product $X^{\\mathtt {versal}}_{v} \\times \\prod _{j=1}^k X^\\varphi _{T_j},$ where the first component is the vertex $v$ with coefficient $x_u$ .",
"Therefore $U(X_u)$ is smooth by induction, and hence $X^\\varphi _T$ is also smooth.",
"Orange trees Let us now assume that $T$ is an orange tree and let us choose one domino $u-v$ in the perfect matching of $T$ .",
"By lemma REF , the two open sets $U(x_u)$ and $U(x_v)$ cover the variety $X_T$ .",
"By symmetry between $u$ and $v$ , it is enough to prove that $U(x_u)$ is smooth.",
"Let $T_1, \\dots , T_k$ be the trees attached to $u$ in $T\\setminus \\lbrace v\\rbrace $ .",
"The $T_i$ are clearly orange trees.",
"Let $R$ be the connected component of $v$ in $T\\setminus \\lbrace u\\rbrace $ .",
"The tree $R$ is obtained by removing a leaf in an orange tree, hence (by lemma REF ) has dimension 1 and a unique red-green component.",
"Moreover, $R$ has a maximum matching avoiding only $v$ and the vertex $v$ is red in the coloring of $R$ .",
"Lemma 3.10 The open set $U(x_u)$ is isomorphic to the product of the varieties $X_{T_i}$ and the variety $X^\\mathtt {versal}_R$ .",
"Proof.",
"Assuming that $x_u$ is not zero allows to eliminate the variable $x^{\\prime }_u$ and the equation (REF ) of index $u$ .",
"There remains the equations for the union of $R$ and the $T_i$ , with $x_u$ considered as a parameter attached to all of them at the former neighbors of $u$ .",
"Because the trees $T_i$ are orange, one can consider instead (by lemma REF ) that the parameter $x_u$ is only attached to the vertex $v$ of $R$ .",
"This proves that the open set $U(x_u)$ is isomorphic to the product of the varieties $X_{T_i}$ and the variety $X^\\mathtt {versal}_R$ .",
"By induction, this proves that $U(x_u)$ is smooth.",
"Therefore $X_T$ is smooth too.",
"Torus actions Let $T$ be a tree and let $\\varphi $ be a choice in $\\lbrace \\mathtt {generic},\\mathtt {versal}\\rbrace $ for every red-green component of $T$ .",
"Let us also choose a maximum matching $M$ of $T$ .",
"One can deduce from proposition REF and the remarks following it that there is an action of an algebraic torus of dimension $\\dim (T)$ on $X^\\varphi _T$ , and that this torus (and its action) can be written as a product over red-green components $C$ of tori $\\Lambda ^C_T$ .",
"Let us define a smaller torus $\\Lambda ^\\varphi _T$ acting on $X^\\varphi _T$ as the product of $\\Lambda ^C_T$ over all $\\mathtt {generic}$ red-green components of $T$ .",
"Let us call the rank of $(T, \\varphi )$ and denote by $\\operatorname{rk}(T,\\varphi )$ the sum of the dimensions of the generic red-green components of $T$ .",
"This is the dimension of $\\Lambda ^\\varphi _T$ .",
"Proposition 3.11 If $\\varphi (C)$ is generic, the action of $\\Lambda ^C_T$ on $X^\\varphi _T$ is free.",
"Proof.",
"Let us assume that there is a non-trivial element $\\lambda =(\\lambda _i)_i$ of $\\Lambda ^C_T$ that fixes a point $(x_i)_i$ in $X^\\varphi _T$ .",
"Let $i$ be a red vertex in $C$ such that $\\lambda _i \\ne 1$ .",
"For every green neighbor $j$ of $i$ , one can find another red vertex $k$ incident to $j$ such that $\\lambda _k \\ne 1$ , because of (REF ).",
"Iterating this process, one can build an admissible set $S$ (as defined in §REF ), such that $\\lambda _s \\ne 1$ for every $s \\in S$ .",
"Because $\\lambda $ fixes the given point, one then has $x_s = 0$ for every $s \\in S$ .",
"But this is impossible by Lemma REF .",
"Corollary 3.12 There is on $X^\\varphi _T$ a free action by a torus $\\Lambda ^\\varphi _T$ of dimension the rank $\\operatorname{rk}(T,\\varphi )$ .",
"This gives $X^\\varphi _T$ the structure of a principal bundle with structure group $\\Lambda ^\\varphi _T$ .",
"As one will see later, this bundle is not trivial in general (i.e.",
"not a product), as can be seen from our results for the cohomology already in type $\\mathbb {A}_3$ .",
"Number of points over finite fields and Euler characteristic Let us denote by $N^\\varphi _T(q)$ the number of points on $X^\\varphi _T$ over the finite field $\\mathbb {F}_q$ .",
"When the tree is orange, one will use the shorthand notation $N_T$ .",
"When the function $\\varphi $ is constant, one will use the notations $N^\\mathtt {versal}_T$ and $N^\\mathtt {generic}_T$ .",
"Proposition 4.1 The numbers $N^\\varphi _T(q)$ are monic polynomials in $q$ of degree $\\dim X_T^\\varphi $ .",
"Proof.",
"The proof is by induction on the size of the tree.",
"For the tree with one vertex, the number of points is $q-1$ in the $\\mathtt {generic}$ case and $q^2 - q + 1$ in the $\\mathtt {versal}$ case, by the description given at the beginning of the proof of theorem REF .",
"Then either the tree has a red-green component, which can be $\\mathtt {generic}$ or $\\mathtt {versal}$ , or it is an orange tree.",
"The proof is decomposed into the three following geometric decomposition lemmas, or rather into their obvious corollaries on the number of points over finite fields.",
"Let $T$ be a tree and $v$ be a red leaf in a red-green component $C$ of $T$ .",
"Let $u$ be the neighbor of $v$ .",
"Removing the vertex $v$ creates an orange avalanche and may separate the red-green component $C$ into several ones.",
"Let $\\varphi $ be the induced genericity condition (as defined in Remark REF ).",
"Let $F$ be the forest $T\\setminus \\lbrace u, v\\rbrace $ .",
"The component $C$ may also split into several red-green components in $F$ .",
"Let $\\varphi $ be the induced genericity condition.",
"Let us consider now the case of a generic red-green component $C$ .",
"Lemma 4.2 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_mX^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"In the case where $x_v \\ne 0$ , one uses lemma REF .",
"This gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u= -1$ is attached to some red vertices of $F$ , as a coefficient.",
"One has to check that the genericity condition still holds on every connected component of $F$ .",
"Let $S$ be an admissible set in one of these components.",
"Either $S$ was already an admissible set in $T$ , and then the genericity condition still holds, or it contains exactly one of the neighbors of $u$ in $T$ .",
"In this case, one can extend $S$ by adding $v$ to form an admissible set in $T$ .",
"The genericity condition for $S \\sqcup \\lbrace v\\rbrace $ in $T$ implies the condition for $S$ , because of the additional $-1$ coefficient attached to $S$ in $F$ .",
"Keeping the same notations, let us consider now the case of a versal red-green component $C$ .",
"Lemma 4.3 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_m^2 X^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"If $x_v \\ne 0$ , using lemma REF gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u = -1$ is attached to red vertices of $F$ .",
"By lemma REF , it can be detached, and this just gives the expected second term.",
"Let $T$ be an orange tree and $u-v$ be a domino in $T$ .",
"Let $(T_{u,i})_i$ (resp.",
"$(T_{v,j})_j$ ) be the connected components of $T\\setminus \\lbrace u,v\\rbrace $ that were attached to $u$ (resp.",
"to $v$ ).",
"All these trees are orange.",
"Let us denote by $S_{u,i}$ and $S_{v,j}$ the forests obtained from them by removing the vertex that was linked to $u$ or $v$ .",
"These forests are unimodal, in the sense that they have one unimodal connected component, all the other connected components being orange.",
"Lemma 4.4 In this situation, one has $X_T = \\mathbb {G}_m^2 \\prod _i X_{T_{u,i}} \\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X^\\mathtt {versal}_{S_{u,i}}\\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X_{T_{u,i}}\\prod _j X^\\mathtt {versal}_{S_{v,j}} .$ Proof.",
"Because the open sets $U(x_u)$ and $U(x_v)$ are a covering by lemma REF , one can cut the variety $X_T$ into three pieces: either both $x_u$ and $x_v$ are not zero, or exactly one of them is zero.",
"If both are not zero, then one obtains the product of $\\mathbb {G}_m^2$ (with coordinates $x_u$ and $x_v$ ) with the product of the varieties attached to the $T_{u,i}$ and the $T_{v,j}$ .",
"Indeed, one first get that $x_u$ becomes a parameter attached to all trees $T_{u,i}$ and $x_v$ becomes a parameter attached to all trees $T_{v,j}$ .",
"But these trees are orange, so $x_u$ and $x_v$ can be detached by lemma REF .",
"This gives the first term.",
"If $x_u$ is zero and $x_v$ is not zero, then there is a free variable $x^{\\prime }_u$ and the variable $x_v$ is determined by the variables attached to the vertices of the trees $T_{u,i}$ linked to $u$ , which must be non-zero.",
"One obtains therefore a versal condition on each forest $S_{u,i}$ .",
"For the trees $T_{v,j}$ , the coefficient $x_v$ is attached to all of them, but because they are orange it can be detached.",
"This gives the second term.",
"The third term is the same after exchanging $u$ and $v$ .",
"Reciprocal property Recall from §REF that the rank $\\operatorname{rk}(T,\\varphi )$ of the pair $(T,\\varphi )$ formed by a tree $T$ and a choice function $\\varphi $ is the sum of the dimensions of the $\\mathtt {generic}$ red-green components of $T$ .",
"Proposition 4.5 The polynomial $N^\\varphi _T(q)$ is divisible par $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ .",
"Proof.",
"This follows from the existence of the free action obtained in corollary REF .",
"Let us refine this slightly.",
"Proposition 4.6 The polynomial $N_T^\\varphi $ can be written as $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ times a reciprocal polynomial.",
"Proof.",
"By induction.",
"This is true for the tree with one vertex.",
"One just has to look carefully at the decompositions given in the three lemmas that were used to prove polynomiality by induction.",
"For lemma REF , let $D$ be the rank for $T$ .",
"Then the rank is $D-1$ for $T \\setminus \\lbrace v\\rbrace $ and $D$ for $F$ .",
"Using the additional factor $q-1$ coming from $\\mathbb {G}_m$ , there is a common factor $(q-1)^D$ to all terms involved.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is the same in all terms involved.",
"One uses that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is 0 in all terms involved, as there is no generic red-green component.",
"One uses again that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 pieces ensures that the reciprocal property holds.",
"Enumeration and coincidences In the following remarks, one will describe trees by their numbers in the tables at the end of [5] and by their graph6 string (which is a standard format for graphs).",
"Remark 4.7 One can find distinct orange trees with the same enumerating polynomial.",
"This happens first for trees with 8 vertices.",
"The trees 2.188 (graph6 'IhGGOC@?G') and 2.189 (graph6 'IhC_GCA?G') have the same polynomial, as well as the trees 2.172 (graph6 'IhGGOCA?G') and 2.174 (graph6 'IhGH?C@?G').",
"The number of different polynomials for orange trees with $2n$ vertices is the sequence $1, 1, 2, 5, 13, 41, 138, \\dots $ whereas the number of orange trees is $1, 1, 2, 5, 15, 49, 180, \\dots $ Remark 4.8 For unimodal trees with $\\mathtt {versal}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of trees with 9 vertices, numbered 2.83 (graph6 'HhCGOCA') and 2.85 (graph6 'HhGGGG@').",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 6, 19, 65, \\dots $ whereas the number of unimodal trees is $1, 1, 2, 6, 20, 76, 313, 1361, \\dots $ Remark 4.9 For unimodal trees with $\\mathtt {generic}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of the Dynkin diagrams $\\mathbb {A}_7$ and $\\mathbb {E}_7$ .",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 5, 13, 46, 168, \\dots $ Linear trees Let us denote by $\\mathbb {A}_n$ the linear tree with $n$ vertices.",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) node[fill=orange!20] 3 – (3,0) node[fill=orange!20] 4 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=red!20] 1 – (1,0) node[fill=green!20]2 – (2,0) node[fill=red!20] 3 – (3,0) node[fill=green!20] 4 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; One can check that $\\mathbb {A}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.10 The number of points on varieties attached to $\\mathbb {A}_n$ is given by $N_{\\mathbb {A}_n} = \\frac{q^{n+2} - 1}{q^2 -1}$ if $n$ is even and by $N_{\\mathbb {A}_n}^\\mathtt {versal}= \\frac{q^{n+2} + 1}{q + 1} \\quad \\text{and} \\quad N_{\\mathbb {A}_n}^\\mathtt {generic}= \\frac{(q^{(n+1)/2} - 1)(q^{(n+3)/2} - 1)}{q^2 -1}$ if $n$ is odd.",
"Proof.",
"This follows easily by induction from lemmas REF , REF and REF .",
"Trees of type $\\mathbb {D}$ Let us denote by $\\mathbb {D}_n$ the tree with $n$ vertices associated with the Dynkin diagram of type $\\mathbb {D}$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=orange!20] 4 – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=red!20] 4 – (3,0) node[fill=green!20] 5 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; One can check that $\\mathbb {D}_n$ is unimodal if $n$ is odd and has dimension 2 if $n$ is even.",
"Proposition 4.11 The number of points on varieties attached to $\\mathbb {D}_n$ is given by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}+q^3-q+1}{q + 1} \\quad \\text{and} \\quad N^\\mathtt {generic}_{\\mathbb {D}_n} = (q^{n/2} - 1)^2$ if $n$ is even and by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}-q^3+q-1}{q^2-1} \\quad \\text{and} \\quad N_{\\mathbb {D}_n}^\\mathtt {generic}= q^n-1$ if $n$ is odd.",
"Proof.",
"This is easily deduced from the type $\\mathbb {A}$ case, using REF , REF applied to a red leaf on a short branch.",
"Trees of type $\\mathbb {E}$ Let us consider now a family of trees containing the Dynkin diagrams of type $\\mathbb {E}$ .",
"The tree $\\mathbb {E}_n$ is the tree with one triple point and branches of size 1, 2 and $n-4$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=red!20] 5 – (4,0) node[fill=green!20] 6 – (5,0) node[fill=red!20]7; (2,1) node[fill=red!20] 3 – (2,0) node[fill=green!20] 4; One can check that $\\mathbb {E}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.12 The number of points on varieties attached to $\\mathbb {E}_n$ is given by $N_{\\mathbb {E}_n} = (q^2 - q + 1)\\frac{q^{n-1} - 1}{q-1}$ if $n$ is even and by $N_{\\mathbb {E}_n}^\\mathtt {versal}= (q^2 - q + 1)(1+q^{n-1}) $ and $N_{\\mathbb {E}_n}^\\mathtt {generic}= \\frac{q^{n+1}-q^{n}+q^{n-1}-q^{(n+3)/2}-q^{(n-1)/2}+q^{2}-q+1}{q-1}$ if $n$ is odd.",
"Proof.",
"In the even case, one uses lemma REF applied to the domino on the short branch, and the known type $\\mathbb {A}$ cases.",
"In the odd case, one uses lemmas REF and REF applied to the red leaf on the short branch, and the known type $\\mathbb {A}$ cases.",
"Orange trees and unimodal trees Let us now describe a recursion involving only the polynomials for orange trees and versal unimodal trees.",
"Let $T$ be an orange tree and $v$ be a leaf of $T$ .",
"Let $T^{\\prime }$ be the unimodal tree $T \\setminus \\lbrace v\\rbrace $ and let $F$ be the orange forest obtained from $T$ by removing the domino $u - v$ containing $v$ .",
"Lemma 4.13 There is a decomposition $X_T = X^{\\mathtt {versal}}_{T^{\\prime }} \\sqcup A_{1} X_F.$ Proof.",
"This decomposition is made according to the value of $x_v$ .",
"If $x_v = 0$ , then one has a free parameter $x^{\\prime }_v$ , which gives the factor $A_{1}$ .",
"One also has $x_u = -1$ and one can get rid of $x^{\\prime }_u$ .",
"The value $-1$ is attached as a coefficient to some orange vertices of $F$ , but one can detach this coefficient by lemma REF .",
"There remains the equations for $X_F$ .",
"If $x_v \\ne 0$ , one can use lemma REF .",
"In the special case of a leaf, this gives an isomorphism with $X^\\mathtt {versal}_{T^{\\prime }}$ .",
"One can use lemma REF to compute the enumerating polynomials for orange trees and versal unimodal trees only, by the following algorithm.",
"Step 0: if the tree $T$ is of type $\\mathbb {A}_n$ with $n$ even, use the known value from (REF ) in proposition REF .",
"Step 1: if the tree $T$ is orange, find a leaf $v$ whose branch has minimal length.",
"Here the branch is the longest sequence of vertices of valency 2 starting at the unique neighbor of the leaf (it could be empty).",
"Then use lemma REF applied to the leaf $v$ to compute $N_T$ .",
"Step 2: if the tree $T$ is unimodal, find a red leaf $w$ whose branch has maximal length.",
"Adding a vertex $v$ at the end of this branch gives an orange tree $T^{\\prime }$ .",
"Then use lemma REF (backwards) applied to the tree $T^{\\prime }$ and its leaf $v$ to compute $N_T$ .",
"This will work because each step either shorten the shortest branch or add some vertex to the longest branch.",
"This makes sure that the tree become more and more linear, and that at some point one is reduced to the initial step.",
"This is a decreasing induction on the number of points of valency at least 3 and the length of the longest branch.",
"Remark 4.14 For orange trees, one can use instead in this algorithm the lemma REF , maybe choosing a domino close to the center of the tree for a better complexity.",
"Euler characteristic and independent sets Let us denote by $\\operatorname{vc}(T)$ the number of minimum vertex covers of $T$ .",
"This is also the number of maximum independent sets.",
"Let us now describe a decomposition of the versal varieties according to independent sets (not necessarily maximal).",
"If $S$ is a subset of the vertices of $T$ , one can define $W_T(S)$ as the set of points in $X_T^\\mathtt {versal}$ where $x_u = 0 & \\quad \\text{if } u\\in S,\\\\x_u \\ne 0 & \\quad \\text{if } u\\notin S.$ The sets $W_T(S)$ are obviously disjoint in $X_T^\\mathtt {versal}$ .",
"Lemma 4.15 If the set $W_T(S)$ is not empty, then $S$ is an independent set in $T$ .",
"Proof.",
"This follows from lemma REF .",
"Proposition 4.16 Let $S$ be an independent set in $T$ .",
"There is an isomorphism $W_T(S) \\simeq (\\mathbb {G}_m)^{t + \\dim (T)- 2s} \\times (A_{1})^{s},$ where $t$ is the size of $T$ and $s$ the size of $S$ .",
"Proof.",
"Let us fix a maximum matching $M$ of $T$ .",
"For every $u$ not in $S$ , one can use the hypothesis $x_u \\ne 0$ to get rid of $x^{\\prime }_u$ and of the equation of index $u$ .",
"There remains only the equations of index $v$ for $v \\in S$ .",
"Because $x_v=0$ when $v \\in S$ , the variables $x^{\\prime }_v$ for $v \\in S$ do no longer appear in the equations, hence they are free.",
"This gives the factor $(A_{1})^s$ .",
"Then there remains $s$ equations of the general shape $-1 = \\alpha _i \\prod _{j - i} x_j,\\qquad \\mathrm {(E_i)}$ involving the $t-s$ invertible variables $x_u$ and the $\\dim (T)$ coefficient variables $\\alpha _i$ .",
"The factor $\\alpha _i$ is present in this equation only if the vertex $i$ is not covered by the chosen maximum matching $M$ .",
"One will use the following auxiliary graph $\\widehat{T}$ .",
"The vertices are the vertices of $T$ and new vertices $Z_i$ indexed by coefficient variables $\\alpha _i$ for $i \\notin M$ .",
"The edges of $\\widehat{T}$ are edges of $T$ and new edges between the vertex $Z_i$ and the vertex $i$ for every $i \\notin M$ .",
"Clearly, this graph is still a tree and admits a perfect matching $\\widehat{M}$ , by adding dominoes $i-Z_i$ to the matching $M$ .",
"Because $S$ is an independent set in $T$ , there is at most one element of $S$ in every edge of $\\widehat{T}$ .",
"Let us orient every edge containing an element of $S$ towards this element if the edge is a domino and in the other way otherwise.",
"This defines a partial order on the vertices of $\\widehat{T}$ , decreasing along the chosen orientation of edges.",
"Consider now the equation REF associated with a vertex $i \\in S$ .",
"There is a unique domino $i-j$ in $\\widehat{T}$ containing $i$ .",
"The equation can then be used to express the variable $x_j$ in terms of variables of lower index in the partial order.",
"One can therefore eliminate one variable for every equation.",
"At the end, one obtains an algebraic torus whose dimension is the difference between the number $ t -s +\\dim (T)$ of initial variables and the number $s$ of equations.",
"Corollary 4.17 The Euler characteristic of $X_T^\\mathtt {versal}$ is $\\operatorname{vc}(T)$ .",
"Proof.",
"Every set $W_T(S)$ contributes either 0 or 1 to the Euler characteristic.",
"It contributes by 1 if and only if the exponent $t+ \\dim (T)- 2s$ is zero.",
"This exponent can be expressed as $(r(T) + o(T) + g(T)) + (r(T) - g(T)) - 2 s.$ It is therefore zero if and only if $s = r(T) + o(T)/2$ , which is the size of the maximum independent sets in $T$ .",
"Of course, one can also use Proposition REF to give a formula for the number of points $N^\\mathtt {versal}_T$ as a sum over independent sets.",
"Corollary 4.18 The value at $q=1$ of the polynomial $N^\\mathtt {versal}_T$ is the number $\\operatorname{vc}(T)$ of maximum independent sets of $T$ .",
"Cohomology: general setting and results This section first describes some differential forms that are always present in the varieties under study, and then very briefly recalls the results one needs about (mixed) Hodge structures.",
"For a general reference about mixed Hodge structures, see for example [18].",
"Weil-Petersson two-form Let $T$ be a tree and let $S$ be a subset of $T$ .",
"Consider the augmented tree $T+S$ obtained by adding a new edge out of every vertex in $S$ , and endow this tree with a bipartite orientation, where every vertex is either a sink or a source.",
"As a variant of the definition of the variety $X_T^\\varphi $ , one can define a variety $X(T+S)$ attached to this data, with invertible variables associated to the new vertices, playing the role of coefficients in the equations (as the $\\alpha $ do).",
"Let $\\omega _i$ denote $d \\log (x_i)$ .",
"The following lemma has been proved by Greg Muller in [16] in a more general context.",
"Lemma 5.1 The differential form $\\operatorname{WP}= \\sum _{i \\rightarrow j} \\omega _i \\omega _j,$ where the sum is running over edges of $T+S$ , is an algebraic differential form on the variety $X(T+S)$ .",
"Proof.",
"Let us prove that it has no pole.",
"Let us fix $i$ .",
"To study the possible pole along $x_i = 0$ , it is enough to look at the sum $\\sum _{j \\leftrightarrow i} \\omega _i \\omega _j$ restricted to edges containing $i$ .",
"By the relation $x_i x^{\\prime }_i = 1 + \\prod _{j\\leftrightarrow i} x_j$ , one has $x_i d x^{\\prime }_i + x^{\\prime }_i d x_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j,$ and therefore $x_i dx^{\\prime }_i dx_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j d x_i.$ This implies $dx^{\\prime }_i dx_i / \\prod _{k\\leftrightarrow i} x_k = \\sum _{j\\leftrightarrow i} \\omega _j \\omega _i,$ where the left-hand side has clearly no pole at $x_i$ .",
"Note that $\\operatorname{WP}$ stands here for Weil-Petersson.",
"Abusing notations, one will use the same symbol $\\operatorname{WP}$ to denote these differential forms on different varieties.",
"The ambient variety should be clear from the context.",
"Hodge structures We will use the notation $\\mathbb {Q}(-i)$ to denote a one dimensional vector space over $\\mathbb {Q}$ endowed with a pure Hodge structure of Tate type, of weight $2i$ and type $(i,i)$ .",
"The tensor product of $\\mathbb {Q}(-i)$ and $\\mathbb {Q}(-j)$ is $\\mathbb {Q}(-i-j)$ .",
"Recall that the cohomology of $\\mathbb {G}_m$ has an Hodge structure described by $\\mathsf {H}^k(\\mathbb {G}_m) = \\mathbb {Q}(-k)$ for $0 \\le k \\le 1$ .",
"There is no morphism between pure Hodge structures of distinct weights.",
"The Künneth isomorphism is compatible with the Hodge structures.",
"The Mayer-Vietoris long exact sequence is an exact sequence of Hodge structures.",
"Cohomology: orange and versal cases This section deals with the cohomology, in several cases where either varieties do not depend on parameters, or versal conditions are assumed on all parameters.",
"The first part is devoted to linear trees; the results there can then be used as building blocks.",
"Linear trees $\\mathbb {A}$ Let $\\mathbb {A}_n$ be the linear tree with $n$ vertices numbered from 1 to $n$ .",
"As seen in §REF , this is an orange tree if $n$ is even, and an unimodal tree otherwise.",
"Some of the results of this section were already obtained in [6] using instead the cohomology with compact supports.",
"Cohomology of some auxiliary varieties for $\\mathbb {A}$ Let us introduce three varieties $X_n$ , $Y_n$ and $Z_n$ with dimensions $n, n+1$ and $n+1$ .",
"The variety $Z_n$ is defined by variables $x_1,\\dots ,x_n$ , $x^{\\prime }_1,\\dots ,x^{\\prime }_n$ and $\\alpha $ such that $x_1 x^{\\prime }_1 &= 1 + \\alpha x_2 ,\\\\x_i x^{\\prime }_i &= 1 + x_{i-1} x_{i+1}, \\\\x_n x^{\\prime }_n &= 1 + x_{n-1}.$ The variety $Y_n$ is the open set in $Z_n$ where $\\alpha $ is invertible.",
"The variety $X_n$ is the closed set in $Y_n$ where $\\alpha $ is fixed to a generic invertible value (where generic means distinct from $(-1)^{(n+1)/2}$ if $n$ is odd).",
"In our general notations, $Y_n$ is $X_{\\mathbb {A}_n}^\\mathtt {versal}$ and $X_n$ is $X_{\\mathbb {A}_n}^\\mathtt {generic}$ .",
"Let us first describe the variety $Z_n$ .",
"Proposition 6.1 There exists an isomorphism between $Z_n$ and the affine space $A_{n+1}$ .",
"Proof.",
"This has been proved in [6].",
"Therefore, the cohomology of $Z_n$ is known for all $n$ : $\\mathsf {H}^k(Z_n) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}1 &\\text{ if } k=0,\\\\0 &\\text{ if } k>0.\\end{array}\\right.",
"}$ The Hodge structure on $\\mathsf {H}^0(Z_n)$ is $\\mathbb {Q}(0)$ .",
"Let us now compute the cohomology of $Y_n$ by induction.",
"This uses the Mayer-Vietoris long exact sequence for the covering of $Z_n$ by the two open sets $U(x_1)$ and $U(\\alpha )$ .",
"First, let us note that $U(\\alpha ) \\simeq Y_n $ by definition.",
"Next, one finds that $U(x_1) \\simeq A_{1} Y_{n-1}$ .",
"Indeed one can eliminate $x^{\\prime }_1$ using the first equation.",
"Then $\\alpha $ becomes a free variable, and there remains the equations for $Y_{n-1}$ , with $x_1$ now playing the role of $\\alpha $ .",
"Last, the intersection $U(\\alpha ) \\cap U(x_1)$ is isomorphic to $ \\mathbb {G}_mY_{n-1}$ , by the same argument.",
"Let us write $\\omega _{\\alpha }$ for $d \\log (\\alpha )$ .",
"Proposition 6.2 The cohomology ring of $Y_n$ has the following description: $\\mathsf {H}^k(Y_n) = \\mathbb {Q}(-k)$ for $0 \\le k \\le n+1$ .",
"It has a basis given by powers of $\\operatorname{WP}$ in even degrees and by powers of $\\operatorname{WP}$ times $\\omega _{\\alpha }$ in odd degrees.",
"It is generated by the 1-form $\\omega _{\\alpha }$ and the 2-form $\\operatorname{WP}$ .",
"Proof.",
"Because of the vanishing of $\\mathsf {H}^k(Z_n)$ for $k>0$ , the Mayer-Vietoris long exact sequence gives short exact sequences $0 \\rightarrow \\mathsf {H}^0(Z_n) \\rightarrow \\mathsf {H}^0(Y_n)\\oplus \\mathsf {H}^0(U(x_1)) \\rightarrow \\mathsf {H}^0(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ and $0 \\rightarrow \\mathsf {H}^k(Y_n)\\oplus \\mathsf {H}^k(U(x_1)) \\rightarrow \\mathsf {H}^k(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ for every $k > 0$ .",
"This determines by induction the Hodge structure of the cohomology of $Y_{n}$ .",
"Let us now proceed to the expected basis.",
"One already knows that $\\operatorname{WP}$ and $\\omega _{\\alpha }$ are indeed algebraic differential forms on $Y_{n}$ .",
"By the short exact sequences above, one can check that for $k>0$ the union of the expected basis of $\\mathsf {H}^k(Y_n)$ with the known basis of $\\mathsf {H}^k(U(x_1))$ is mapped to a basis of $\\mathsf {H}^k(U(\\alpha ) \\cap U(x_1))$ .",
"This implies the statement.",
"Cohomology for $\\mathbb {A}_n$ with even $n$ Let us now consider the linear tree $\\mathbb {A}_n$ for even $n$ , and compute the cohomology of $X_n$ .",
"Proposition 6.3 The Hodge structure of the cohomology of $X_n$ is $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for all even $k$ between 0 and $n$ , and 0 otherwise.",
"A basis is given by powers of $\\operatorname{WP}$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ .",
"Proof.",
"This follows from the known cohomology of $Y_n$ and the Künneth theorem applied to the isomorphism $Y_n \\simeq X_n \\mathbb {G}_m$ given by lemma REF .",
"The Künneth theorem gives immediately the Hodge structure.",
"For the basis, it is enough to recall that the $\\mathbb {G}_m$ factor is given by the value of $\\alpha $ , and to check that fixing the value $\\alpha = 1$ maps $\\operatorname{WP}$ (for $Y_n$ ) to $\\operatorname{WP}$ (for $X_n$ ).",
"Cohomology for orange trees of shape $H$ 0.7 [scale=0.7] patterns,decorations.pathreplacing every node=[draw,shape=circle,very thick,fill=white] (0,0) node ... – (1,0) node ... – (3,0) node ... – (4,0) node ...; (0,1) node ... – (1,1) node ... – (3,1) node ... – (4,1) node ...; (2,1) node $a$ – (2,0) node $b$ ; every node=[draw=none,fill=none]; [thick,decoration= brace, raise=0.3cm ,decorate] (-0.3,1) node – (1.3,1) node ; [thick,decoration= brace, raise=0.3cm ,decorate] (2.7,1) node – (4.3,1) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (-0.3,0) node – (1.3,0) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (2.7,0) node – (4.3,0) node ; t (0.5,-1) $m$ ; t (0.5,2) $k$ ; t (3.5,-1) $n$ ; t (3.5,2) $\\ell $ ; Let us denote by $H_{k,\\ell ,m,n}$ the tree described as two chains joined by an edge, such that by removing the joining edge and its extremities $a$ and $b$ , one gets two chains of lengths $k$ and $\\ell $ on the $a$ side (top) and two chains of lengths $m$ and $n$ on the $b$ side (bottom).",
"We assume now that $H_{k,\\ell ,m,n}$ is an orange tree.",
"It implies that either $k, \\ell , m$ and $n$ are even if the middle edge is an orange domino, or that (without loss of generality) $k$ and $m$ are odd and $l$ and $n$ are even otherwise.",
"Then one can compute the cohomology of $H_{k,\\ell ,m,n}$ using the Mayer-Vietoris long exact sequence for the open covering by $U(x_a)$ and $U(x_b)$ .",
"When the middle edge is an orange domino, one has $\\begin{aligned}U(x_a) &\\simeq X_k X_\\ell Y_{m+n+1},\\\\U(x_b) &\\simeq Y_{k+\\ell +1} X_m X_n ,\\\\U(x_a) \\cap U(x_b) &\\simeq (\\mathbb {G}_m)^2 X_k X_\\ell X_m X_n.\\end{aligned}$ When the middle edge is not an orange domino, one finds instead $\\begin{aligned}U(x_a) &\\simeq Y_k X_\\ell X_{m+n+1},\\\\U(x_b) &\\simeq X_{k+\\ell +1} Y_m X_n,\\\\U(x_a) \\cap U(x_b) &\\simeq Y_k X_\\ell Y_m X_n.\\end{aligned}$ Let us introduce some notations: call $K,L,M,N$ the subsets of vertices corresponding to the four branches of $H$ (i.e.",
"the connected components of $H \\setminus \\lbrace a,b\\rbrace $ ).",
"Let us denote by $W_S$ the Weil-Petersson 2-form associated with a subset $S$ of the vertices of $H$ .",
"For conciseness, one will use shortcuts such as $W_{KaL}$ or $W_{MabN}$ .",
"Note that there holds $\\omega _a W_{aL} = \\omega _a W_{L}$ and other similar simplifications, by the definition (REF ) of these forms.",
"Let us now describe generators and bases of the cohomology of the open sets $ U(x_a)$ , $U(x_b)$ and $U(x_b)\\cap U(x_b)$ .",
"This can be computed using the isomorphisms (REF ), (REF ) and the known cohomology of varieties $X$ and $Y$ .",
"It turns out that the result does not depend on whether or not the middle edge $a-b$ is an orange domino.",
"The cohomology of $U(x_a)$ is generated by $\\omega _a$ , $W_{Ka}$ , $W_{aL}$ and $W_{MabN}$ .",
"A basis is given by $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B} \\quad \\text{and}\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{MbN}^{B},$ where $0 \\le \\kappa \\le k/2$ , $0 \\le \\lambda \\le l/2$ and $0 \\le B \\le (m+n+2)/2$ (left) or $0 \\le B \\le (m+n)/2$ (right).",
"Similarly, the cohomology of $U(x_b)$ is generated by $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ and $W_{KabL}$ .",
"A basis is given by $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A} \\quad \\text{and}\\quad \\omega _b W_{M}^{\\mu } W_{N}^{\\nu } W_{KaL}^{A},$ where $0 \\le \\mu \\le m/2$ , $0 \\le \\nu \\le n/2$ and $0 \\le A \\le (k+l+2)/2$ (left) or $0 \\le A \\le (k+l)/2$ (right).",
"The cohomology of $U(x_b)\\cap U(x_b)$ is generated by $\\omega _a$ , $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ , $W_{Ka}$ and $W_{aL}$ .",
"A basis is given by $\\begin{aligned}W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\\\\\omega _a \\omega _b W_{K}^{\\kappa } W_{L}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }\\quad \\text{and}\\quad \\omega _b W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu },\\end{aligned}$ with the same conditions as above on $\\kappa ,\\lambda ,\\mu $ and $\\nu $ .",
"There is a bigrading corresponding to the top and bottom parts of the $H$ shape.",
"Every differential form involved in the bases just described is a sum of products of $\\omega _i$ .",
"The bidegree of a monomial in the $\\omega _i$ is the pair (number of $\\omega _i$ where $i$ is in the top row, number of $\\omega _i$ where $i$ is in the bottom row).",
"Among the various Weil-Petersson forms involved, only the differential forms $W_{KabL}$ and $W_{MabN}$ are not homogeneous for the bidegree, but have terms in bidegrees $(2,0)$ and $(1,1)$ (resp.",
"$(0,2)$ and $(1,1)$ ).",
"One needs now to compute explicitly the following maps in the Mayer-Vietoris long exact sequence: $\\mathsf {H}^i(U(x_a))\\oplus \\mathsf {H}^i(U(x_b)) \\stackrel{f_i}{\\longrightarrow } \\mathsf {H}^i(U(x_a) \\cap U(x_b)).$ Because one has bases of all these spaces, this is a matter of matrices.",
"For odd degree $i$ , let us show that the differential is injective.",
"Because in this case all basis elements (given by right columns of (REF ), (REF ) and (REF )) are homogeneous for the bigrading, one can separate the cases of bidegree congruent to $(0,1)$ and to $(1,0)$ modulo $(2,2)$ .",
"Let us give details only for the first possibility, the other case being similar after exchanging top and bottom of $H$ .",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_b))$ is given by $\\omega _{b}W_{KaL}^{A}W_{M}^{\\mu } W_{N}^{\\nu }$ with $i =1+2A+2\\mu +2\\nu $ .",
"The corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a))$ is zero.",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a) \\cap U(x_b))$ is given by $\\omega _{b}W_{Ka}^{\\kappa }W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }$ with $i = 1+2\\kappa +2\\lambda +2\\mu +2\\nu $ .",
"But $W_{KaL}^{A}$ can be written as a linear combination of $W_{Ka}^{\\kappa }W_{aL}^{\\lambda }$ with $\\kappa + \\lambda = A$ .",
"Therefore the basis elements are mapped to linear combinations with disjoint supports.",
"It follows that the map $f_i$ is injective.",
"Let us now turn to even degrees.",
"Proposition 6.4 For even degree $2i$ , the kernel of the differential $f_{2i}$ has dimension 1, spanned by the $i^{th}$ power of the form $\\operatorname{WP}$ .",
"Proof.",
"First note that one can define an injective map $\\Delta $ from the space $\\mathsf {H}^{2i}(U(x_a) \\cap U(x_b))$ to the space $D_i$ spanned by all products of $i$ 2-forms of the shape $\\omega _{s}\\omega _{t}$ for $s-t$ an edge of the tree (always written in the order given by a fixed alternating orientation of the tree).",
"Indeed, both terms in the left column of (REF ) can be written as linear combinations of such products.",
"The injectivity holds because distinct elements in this part of the basis are mapped to linear combinations with disjoint supports.",
"To recover a basis element $B$ from any monomial in its image by $\\Delta $ , first count in $\\Delta (B)$ if the number of $\\omega _k$ in the top row is odd or even.",
"This tells if the basis elements $B$ contains $\\omega _a\\omega _b$ or not.",
"Then it is easy to recover the exponents $(\\kappa , \\lambda , \\mu , \\nu )$ defining $B$ by counting in $\\Delta (B)$ how many $\\omega _k$ there are in the different parts of the tree.",
"To prove the statement of the proposition, it is therefore enough to compute the kernel of the composite map $\\Delta \\circ f_{2i}$ .",
"It turns out that the matrix of this composite map has a nice description.",
"First, every monomial $d$ made of $i$ 2-forms $\\omega _{s}\\omega _{t}$ as above appears in exactly two images, the image of a form $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B}$ and the image of a form $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A}$ (with opposite signs).",
"Let us denote these two forms by $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ .",
"On the other hand, the image of every basis element is the sum of several monomials (at least one), with constant sign.",
"Let us pick an element $z$ of the kernel of $f_{2i}$ .",
"Then for every monomial $d$ in $D_i$ , the coefficients of $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ in $z$ must be the same.",
"One can make a graph with vertices given by all forms in the basis, and edges corresponding to the relations $\\mathsf {F}_a(d)-\\mathsf {F}_b(d)$ for all monomials $d$ .",
"By a combinatorial argument, one can check that this graph is connected.",
"For this, one just has to show that one can go from any monomial $d$ to any monomial $d^{\\prime }$ , using two kinds of moves: replace $d$ by another monomial appearing in the same $\\mathsf {F}_a(d)$ , or replace $d$ by another monomial appearing in the same $\\mathsf {F}_b(d)$ .",
"This is not difficult once translated in terms of dominoes, and details are left to the reader.",
"From the connectedness of this graph, one deduces that the kernel is spanned by the sum of all basis elements of $\\mathsf {H}^{2i}(U(x_a))\\oplus \\mathsf {H}^{2i}(U(x_b))$ , which is just $(\\operatorname{WP}^i,\\operatorname{WP}^i)$ .",
"This proposition and the injectivity in the case of odd degree allow to give a description of the weights of the Hodge structure on the cohomology.",
"This can easily be made explicit, but one will not do that here.",
"There would remain to find explicit expressions for the cohomology classes coming from the co-image of the differentials $f_i$ .",
"In the case of the Dynkin diagrams $\\mathbb {E}_6$ and $\\mathbb {E}_8$ , one can go further and compute explicit representatives of the cohomology classes.",
"By the general proof, the cohomology for $\\mathbb {E}_6$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-6),$ where the $\\mathbb {Q}(-i)$ with $i$ even correspond to the powers of $\\operatorname{WP}$ .",
"Using the connection homomorphism in the long exact sequence, one finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ .",
"Similarly, the cohomology for $\\mathbb {E}_8$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-5)\\oplus \\mathbb {Q}(-6) \\mid 0 \\mid \\mathbb {Q}(-8),$ where the even $\\mathbb {Q}(-i)$ are the powers of $\\operatorname{WP}$ .",
"One finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ , and its product by $\\operatorname{WP}$ corresponds to $\\mathbb {Q}(-5)$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6 – (5,0) node[fill=orange!20] 7 – (6,0) node[fill=orange!20] 8; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; Cohomology: generic cases This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers.",
"Cohomology for $\\mathbb {A}$ odd and generic Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Number of points over finite fields and Euler characteristic",
"Let us denote by $N^\\varphi _T(q)$ the number of points on $X^\\varphi _T$ over the finite field $\\mathbb {F}_q$ .",
"When the tree is orange, one will use the shorthand notation $N_T$ .",
"When the function $\\varphi $ is constant, one will use the notations $N^\\mathtt {versal}_T$ and $N^\\mathtt {generic}_T$ .",
"Proposition 4.1 The numbers $N^\\varphi _T(q)$ are monic polynomials in $q$ of degree $\\dim X_T^\\varphi $ .",
"Proof.",
"The proof is by induction on the size of the tree.",
"For the tree with one vertex, the number of points is $q-1$ in the $\\mathtt {generic}$ case and $q^2 - q + 1$ in the $\\mathtt {versal}$ case, by the description given at the beginning of the proof of theorem REF .",
"Then either the tree has a red-green component, which can be $\\mathtt {generic}$ or $\\mathtt {versal}$ , or it is an orange tree.",
"The proof is decomposed into the three following geometric decomposition lemmas, or rather into their obvious corollaries on the number of points over finite fields.",
"Let $T$ be a tree and $v$ be a red leaf in a red-green component $C$ of $T$ .",
"Let $u$ be the neighbor of $v$ .",
"Removing the vertex $v$ creates an orange avalanche and may separate the red-green component $C$ into several ones.",
"Let $\\varphi $ be the induced genericity condition (as defined in Remark REF ).",
"Let $F$ be the forest $T\\setminus \\lbrace u, v\\rbrace $ .",
"The component $C$ may also split into several red-green components in $F$ .",
"Let $\\varphi $ be the induced genericity condition.",
"Let us consider now the case of a generic red-green component $C$ .",
"Lemma 4.2 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_mX^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"In the case where $x_v \\ne 0$ , one uses lemma REF .",
"This gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u= -1$ is attached to some red vertices of $F$ , as a coefficient.",
"One has to check that the genericity condition still holds on every connected component of $F$ .",
"Let $S$ be an admissible set in one of these components.",
"Either $S$ was already an admissible set in $T$ , and then the genericity condition still holds, or it contains exactly one of the neighbors of $u$ in $T$ .",
"In this case, one can extend $S$ by adding $v$ to form an admissible set in $T$ .",
"The genericity condition for $S \\sqcup \\lbrace v\\rbrace $ in $T$ implies the condition for $S$ , because of the additional $-1$ coefficient attached to $S$ in $F$ .",
"Keeping the same notations, let us consider now the case of a versal red-green component $C$ .",
"Lemma 4.3 In this situation, the variety $X^\\varphi _T$ can be decomposed as $X^\\varphi _T = \\mathbb {G}_m^2 X^\\varphi _{T\\setminus \\lbrace v\\rbrace } \\sqcup A_{1} X^\\varphi _F.$ Proof.",
"Either $x_v$ is not zero or $x_v$ is zero.",
"This will give the required disjoint union.",
"If $x_v \\ne 0$ , using lemma REF gives the first term of the right hand side.",
"Let us pick a maximum matching $M$ of $T$ containing $v$ .",
"This is possible by lemma REF .",
"This does not change the open set $U(x_v)$ and its complement, up to isomorphism.",
"Assume now that $x_v$ is zero.",
"Then $x^{\\prime }_v$ is a free variable, and $x_u$ is equal to $-1$ , because there are no coefficients on $v$ .",
"One then gets rid of $x^{\\prime }_u$ .",
"The coloring of the forest $F$ is by restriction of the coloring of $T$ .",
"Therefore the parameter $x_u = -1$ is attached to red vertices of $F$ .",
"By lemma REF , it can be detached, and this just gives the expected second term.",
"Let $T$ be an orange tree and $u-v$ be a domino in $T$ .",
"Let $(T_{u,i})_i$ (resp.",
"$(T_{v,j})_j$ ) be the connected components of $T\\setminus \\lbrace u,v\\rbrace $ that were attached to $u$ (resp.",
"to $v$ ).",
"All these trees are orange.",
"Let us denote by $S_{u,i}$ and $S_{v,j}$ the forests obtained from them by removing the vertex that was linked to $u$ or $v$ .",
"These forests are unimodal, in the sense that they have one unimodal connected component, all the other connected components being orange.",
"Lemma 4.4 In this situation, one has $X_T = \\mathbb {G}_m^2 \\prod _i X_{T_{u,i}} \\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X^\\mathtt {versal}_{S_{u,i}}\\prod _j X_{T_{v,j}} \\sqcup A_{1} \\prod _i X_{T_{u,i}}\\prod _j X^\\mathtt {versal}_{S_{v,j}} .$ Proof.",
"Because the open sets $U(x_u)$ and $U(x_v)$ are a covering by lemma REF , one can cut the variety $X_T$ into three pieces: either both $x_u$ and $x_v$ are not zero, or exactly one of them is zero.",
"If both are not zero, then one obtains the product of $\\mathbb {G}_m^2$ (with coordinates $x_u$ and $x_v$ ) with the product of the varieties attached to the $T_{u,i}$ and the $T_{v,j}$ .",
"Indeed, one first get that $x_u$ becomes a parameter attached to all trees $T_{u,i}$ and $x_v$ becomes a parameter attached to all trees $T_{v,j}$ .",
"But these trees are orange, so $x_u$ and $x_v$ can be detached by lemma REF .",
"This gives the first term.",
"If $x_u$ is zero and $x_v$ is not zero, then there is a free variable $x^{\\prime }_u$ and the variable $x_v$ is determined by the variables attached to the vertices of the trees $T_{u,i}$ linked to $u$ , which must be non-zero.",
"One obtains therefore a versal condition on each forest $S_{u,i}$ .",
"For the trees $T_{v,j}$ , the coefficient $x_v$ is attached to all of them, but because they are orange it can be detached.",
"This gives the second term.",
"The third term is the same after exchanging $u$ and $v$ .",
"Reciprocal property Recall from §REF that the rank $\\operatorname{rk}(T,\\varphi )$ of the pair $(T,\\varphi )$ formed by a tree $T$ and a choice function $\\varphi $ is the sum of the dimensions of the $\\mathtt {generic}$ red-green components of $T$ .",
"Proposition 4.5 The polynomial $N^\\varphi _T(q)$ is divisible par $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ .",
"Proof.",
"This follows from the existence of the free action obtained in corollary REF .",
"Let us refine this slightly.",
"Proposition 4.6 The polynomial $N_T^\\varphi $ can be written as $(q-1)^{\\operatorname{rk}(T,\\varphi )}$ times a reciprocal polynomial.",
"Proof.",
"By induction.",
"This is true for the tree with one vertex.",
"One just has to look carefully at the decompositions given in the three lemmas that were used to prove polynomiality by induction.",
"For lemma REF , let $D$ be the rank for $T$ .",
"Then the rank is $D-1$ for $T \\setminus \\lbrace v\\rbrace $ and $D$ for $F$ .",
"Using the additional factor $q-1$ coming from $\\mathbb {G}_m$ , there is a common factor $(q-1)^D$ to all terms involved.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is the same in all terms involved.",
"One uses that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 piece ensures that the reciprocal property holds.",
"For lemma REF , the rank $D$ is 0 in all terms involved, as there is no generic red-green component.",
"One uses again that $(q-1)^2$ is reciprocal.",
"The factor $A_{1}$ in the codimension 1 pieces ensures that the reciprocal property holds.",
"Enumeration and coincidences In the following remarks, one will describe trees by their numbers in the tables at the end of [5] and by their graph6 string (which is a standard format for graphs).",
"Remark 4.7 One can find distinct orange trees with the same enumerating polynomial.",
"This happens first for trees with 8 vertices.",
"The trees 2.188 (graph6 'IhGGOC@?G') and 2.189 (graph6 'IhC_GCA?G') have the same polynomial, as well as the trees 2.172 (graph6 'IhGGOCA?G') and 2.174 (graph6 'IhGH?C@?G').",
"The number of different polynomials for orange trees with $2n$ vertices is the sequence $1, 1, 2, 5, 13, 41, 138, \\dots $ whereas the number of orange trees is $1, 1, 2, 5, 15, 49, 180, \\dots $ Remark 4.8 For unimodal trees with $\\mathtt {versal}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of trees with 9 vertices, numbered 2.83 (graph6 'HhCGOCA') and 2.85 (graph6 'HhGGGG@').",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 6, 19, 65, \\dots $ whereas the number of unimodal trees is $1, 1, 2, 6, 20, 76, 313, 1361, \\dots $ Remark 4.9 For unimodal trees with $\\mathtt {generic}$ condition, one can also find pairs with the same enumerating polynomials.",
"The smallest one is made of the Dynkin diagrams $\\mathbb {A}_7$ and $\\mathbb {E}_7$ .",
"The number of different polynomials for unimodal trees with $2n+1$ vertices is the sequence $1, 1, 2, 5, 13, 46, 168, \\dots $ Linear trees Let us denote by $\\mathbb {A}_n$ the linear tree with $n$ vertices.",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) node[fill=orange!20] 3 – (3,0) node[fill=orange!20] 4 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=red!20] 1 – (1,0) node[fill=green!20]2 – (2,0) node[fill=red!20] 3 – (3,0) node[fill=green!20] 4 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; One can check that $\\mathbb {A}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.10 The number of points on varieties attached to $\\mathbb {A}_n$ is given by $N_{\\mathbb {A}_n} = \\frac{q^{n+2} - 1}{q^2 -1}$ if $n$ is even and by $N_{\\mathbb {A}_n}^\\mathtt {versal}= \\frac{q^{n+2} + 1}{q + 1} \\quad \\text{and} \\quad N_{\\mathbb {A}_n}^\\mathtt {generic}= \\frac{(q^{(n+1)/2} - 1)(q^{(n+3)/2} - 1)}{q^2 -1}$ if $n$ is odd.",
"Proof.",
"This follows easily by induction from lemmas REF , REF and REF .",
"Trees of type $\\mathbb {D}$ Let us denote by $\\mathbb {D}_n$ the tree with $n$ vertices associated with the Dynkin diagram of type $\\mathbb {D}$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=orange!20] 4 – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] ... – (5,0) node[fill=orange!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0.3,0.7) node[fill=red!20] 1 – (1,0) – (2,0) node[fill=red!20] 4 – (3,0) node[fill=green!20] 5 – (4,0) node[fill=red!20] ... – (5,0) node[fill=green!20]... – (6,0) node[fill=red!20]$n$ ; (0.3,-0.7) node[fill=red!20] 2 – (1,0) node[fill=green!20] 3; One can check that $\\mathbb {D}_n$ is unimodal if $n$ is odd and has dimension 2 if $n$ is even.",
"Proposition 4.11 The number of points on varieties attached to $\\mathbb {D}_n$ is given by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}+q^3-q+1}{q + 1} \\quad \\text{and} \\quad N^\\mathtt {generic}_{\\mathbb {D}_n} = (q^{n/2} - 1)^2$ if $n$ is even and by $N_{\\mathbb {D}_n}^\\mathtt {versal}= \\frac{q^{n+3}-q^{n+2}+q^{n}-q^3+q-1}{q^2-1} \\quad \\text{and} \\quad N_{\\mathbb {D}_n}^\\mathtt {generic}= q^n-1$ if $n$ is odd.",
"Proof.",
"This is easily deduced from the type $\\mathbb {A}$ case, using REF , REF applied to a red leaf on a short branch.",
"Trees of type $\\mathbb {E}$ Let us consider now a family of trees containing the Dynkin diagrams of type $\\mathbb {E}$ .",
"The tree $\\mathbb {E}_n$ is the tree with one triple point and branches of size 1, 2 and $n-4$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=red!20] 5 – (4,0) node[fill=green!20] 6 – (5,0) node[fill=red!20]7; (2,1) node[fill=red!20] 3 – (2,0) node[fill=green!20] 4; One can check that $\\mathbb {E}_n$ is orange if $n$ is even and unimodal if $n$ is odd.",
"Proposition 4.12 The number of points on varieties attached to $\\mathbb {E}_n$ is given by $N_{\\mathbb {E}_n} = (q^2 - q + 1)\\frac{q^{n-1} - 1}{q-1}$ if $n$ is even and by $N_{\\mathbb {E}_n}^\\mathtt {versal}= (q^2 - q + 1)(1+q^{n-1}) $ and $N_{\\mathbb {E}_n}^\\mathtt {generic}= \\frac{q^{n+1}-q^{n}+q^{n-1}-q^{(n+3)/2}-q^{(n-1)/2}+q^{2}-q+1}{q-1}$ if $n$ is odd.",
"Proof.",
"In the even case, one uses lemma REF applied to the domino on the short branch, and the known type $\\mathbb {A}$ cases.",
"In the odd case, one uses lemmas REF and REF applied to the red leaf on the short branch, and the known type $\\mathbb {A}$ cases.",
"Orange trees and unimodal trees Let us now describe a recursion involving only the polynomials for orange trees and versal unimodal trees.",
"Let $T$ be an orange tree and $v$ be a leaf of $T$ .",
"Let $T^{\\prime }$ be the unimodal tree $T \\setminus \\lbrace v\\rbrace $ and let $F$ be the orange forest obtained from $T$ by removing the domino $u - v$ containing $v$ .",
"Lemma 4.13 There is a decomposition $X_T = X^{\\mathtt {versal}}_{T^{\\prime }} \\sqcup A_{1} X_F.$ Proof.",
"This decomposition is made according to the value of $x_v$ .",
"If $x_v = 0$ , then one has a free parameter $x^{\\prime }_v$ , which gives the factor $A_{1}$ .",
"One also has $x_u = -1$ and one can get rid of $x^{\\prime }_u$ .",
"The value $-1$ is attached as a coefficient to some orange vertices of $F$ , but one can detach this coefficient by lemma REF .",
"There remains the equations for $X_F$ .",
"If $x_v \\ne 0$ , one can use lemma REF .",
"In the special case of a leaf, this gives an isomorphism with $X^\\mathtt {versal}_{T^{\\prime }}$ .",
"One can use lemma REF to compute the enumerating polynomials for orange trees and versal unimodal trees only, by the following algorithm.",
"Step 0: if the tree $T$ is of type $\\mathbb {A}_n$ with $n$ even, use the known value from (REF ) in proposition REF .",
"Step 1: if the tree $T$ is orange, find a leaf $v$ whose branch has minimal length.",
"Here the branch is the longest sequence of vertices of valency 2 starting at the unique neighbor of the leaf (it could be empty).",
"Then use lemma REF applied to the leaf $v$ to compute $N_T$ .",
"Step 2: if the tree $T$ is unimodal, find a red leaf $w$ whose branch has maximal length.",
"Adding a vertex $v$ at the end of this branch gives an orange tree $T^{\\prime }$ .",
"Then use lemma REF (backwards) applied to the tree $T^{\\prime }$ and its leaf $v$ to compute $N_T$ .",
"This will work because each step either shorten the shortest branch or add some vertex to the longest branch.",
"This makes sure that the tree become more and more linear, and that at some point one is reduced to the initial step.",
"This is a decreasing induction on the number of points of valency at least 3 and the length of the longest branch.",
"Remark 4.14 For orange trees, one can use instead in this algorithm the lemma REF , maybe choosing a domino close to the center of the tree for a better complexity.",
"Euler characteristic and independent sets Let us denote by $\\operatorname{vc}(T)$ the number of minimum vertex covers of $T$ .",
"This is also the number of maximum independent sets.",
"Let us now describe a decomposition of the versal varieties according to independent sets (not necessarily maximal).",
"If $S$ is a subset of the vertices of $T$ , one can define $W_T(S)$ as the set of points in $X_T^\\mathtt {versal}$ where $x_u = 0 & \\quad \\text{if } u\\in S,\\\\x_u \\ne 0 & \\quad \\text{if } u\\notin S.$ The sets $W_T(S)$ are obviously disjoint in $X_T^\\mathtt {versal}$ .",
"Lemma 4.15 If the set $W_T(S)$ is not empty, then $S$ is an independent set in $T$ .",
"Proof.",
"This follows from lemma REF .",
"Proposition 4.16 Let $S$ be an independent set in $T$ .",
"There is an isomorphism $W_T(S) \\simeq (\\mathbb {G}_m)^{t + \\dim (T)- 2s} \\times (A_{1})^{s},$ where $t$ is the size of $T$ and $s$ the size of $S$ .",
"Proof.",
"Let us fix a maximum matching $M$ of $T$ .",
"For every $u$ not in $S$ , one can use the hypothesis $x_u \\ne 0$ to get rid of $x^{\\prime }_u$ and of the equation of index $u$ .",
"There remains only the equations of index $v$ for $v \\in S$ .",
"Because $x_v=0$ when $v \\in S$ , the variables $x^{\\prime }_v$ for $v \\in S$ do no longer appear in the equations, hence they are free.",
"This gives the factor $(A_{1})^s$ .",
"Then there remains $s$ equations of the general shape $-1 = \\alpha _i \\prod _{j - i} x_j,\\qquad \\mathrm {(E_i)}$ involving the $t-s$ invertible variables $x_u$ and the $\\dim (T)$ coefficient variables $\\alpha _i$ .",
"The factor $\\alpha _i$ is present in this equation only if the vertex $i$ is not covered by the chosen maximum matching $M$ .",
"One will use the following auxiliary graph $\\widehat{T}$ .",
"The vertices are the vertices of $T$ and new vertices $Z_i$ indexed by coefficient variables $\\alpha _i$ for $i \\notin M$ .",
"The edges of $\\widehat{T}$ are edges of $T$ and new edges between the vertex $Z_i$ and the vertex $i$ for every $i \\notin M$ .",
"Clearly, this graph is still a tree and admits a perfect matching $\\widehat{M}$ , by adding dominoes $i-Z_i$ to the matching $M$ .",
"Because $S$ is an independent set in $T$ , there is at most one element of $S$ in every edge of $\\widehat{T}$ .",
"Let us orient every edge containing an element of $S$ towards this element if the edge is a domino and in the other way otherwise.",
"This defines a partial order on the vertices of $\\widehat{T}$ , decreasing along the chosen orientation of edges.",
"Consider now the equation REF associated with a vertex $i \\in S$ .",
"There is a unique domino $i-j$ in $\\widehat{T}$ containing $i$ .",
"The equation can then be used to express the variable $x_j$ in terms of variables of lower index in the partial order.",
"One can therefore eliminate one variable for every equation.",
"At the end, one obtains an algebraic torus whose dimension is the difference between the number $ t -s +\\dim (T)$ of initial variables and the number $s$ of equations.",
"Corollary 4.17 The Euler characteristic of $X_T^\\mathtt {versal}$ is $\\operatorname{vc}(T)$ .",
"Proof.",
"Every set $W_T(S)$ contributes either 0 or 1 to the Euler characteristic.",
"It contributes by 1 if and only if the exponent $t+ \\dim (T)- 2s$ is zero.",
"This exponent can be expressed as $(r(T) + o(T) + g(T)) + (r(T) - g(T)) - 2 s.$ It is therefore zero if and only if $s = r(T) + o(T)/2$ , which is the size of the maximum independent sets in $T$ .",
"Of course, one can also use Proposition REF to give a formula for the number of points $N^\\mathtt {versal}_T$ as a sum over independent sets.",
"Corollary 4.18 The value at $q=1$ of the polynomial $N^\\mathtt {versal}_T$ is the number $\\operatorname{vc}(T)$ of maximum independent sets of $T$ .",
"Cohomology: general setting and results This section first describes some differential forms that are always present in the varieties under study, and then very briefly recalls the results one needs about (mixed) Hodge structures.",
"For a general reference about mixed Hodge structures, see for example [18].",
"Weil-Petersson two-form Let $T$ be a tree and let $S$ be a subset of $T$ .",
"Consider the augmented tree $T+S$ obtained by adding a new edge out of every vertex in $S$ , and endow this tree with a bipartite orientation, where every vertex is either a sink or a source.",
"As a variant of the definition of the variety $X_T^\\varphi $ , one can define a variety $X(T+S)$ attached to this data, with invertible variables associated to the new vertices, playing the role of coefficients in the equations (as the $\\alpha $ do).",
"Let $\\omega _i$ denote $d \\log (x_i)$ .",
"The following lemma has been proved by Greg Muller in [16] in a more general context.",
"Lemma 5.1 The differential form $\\operatorname{WP}= \\sum _{i \\rightarrow j} \\omega _i \\omega _j,$ where the sum is running over edges of $T+S$ , is an algebraic differential form on the variety $X(T+S)$ .",
"Proof.",
"Let us prove that it has no pole.",
"Let us fix $i$ .",
"To study the possible pole along $x_i = 0$ , it is enough to look at the sum $\\sum _{j \\leftrightarrow i} \\omega _i \\omega _j$ restricted to edges containing $i$ .",
"By the relation $x_i x^{\\prime }_i = 1 + \\prod _{j\\leftrightarrow i} x_j$ , one has $x_i d x^{\\prime }_i + x^{\\prime }_i d x_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j,$ and therefore $x_i dx^{\\prime }_i dx_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j d x_i.$ This implies $dx^{\\prime }_i dx_i / \\prod _{k\\leftrightarrow i} x_k = \\sum _{j\\leftrightarrow i} \\omega _j \\omega _i,$ where the left-hand side has clearly no pole at $x_i$ .",
"Note that $\\operatorname{WP}$ stands here for Weil-Petersson.",
"Abusing notations, one will use the same symbol $\\operatorname{WP}$ to denote these differential forms on different varieties.",
"The ambient variety should be clear from the context.",
"Hodge structures We will use the notation $\\mathbb {Q}(-i)$ to denote a one dimensional vector space over $\\mathbb {Q}$ endowed with a pure Hodge structure of Tate type, of weight $2i$ and type $(i,i)$ .",
"The tensor product of $\\mathbb {Q}(-i)$ and $\\mathbb {Q}(-j)$ is $\\mathbb {Q}(-i-j)$ .",
"Recall that the cohomology of $\\mathbb {G}_m$ has an Hodge structure described by $\\mathsf {H}^k(\\mathbb {G}_m) = \\mathbb {Q}(-k)$ for $0 \\le k \\le 1$ .",
"There is no morphism between pure Hodge structures of distinct weights.",
"The Künneth isomorphism is compatible with the Hodge structures.",
"The Mayer-Vietoris long exact sequence is an exact sequence of Hodge structures.",
"Cohomology: orange and versal cases This section deals with the cohomology, in several cases where either varieties do not depend on parameters, or versal conditions are assumed on all parameters.",
"The first part is devoted to linear trees; the results there can then be used as building blocks.",
"Linear trees $\\mathbb {A}$ Let $\\mathbb {A}_n$ be the linear tree with $n$ vertices numbered from 1 to $n$ .",
"As seen in §REF , this is an orange tree if $n$ is even, and an unimodal tree otherwise.",
"Some of the results of this section were already obtained in [6] using instead the cohomology with compact supports.",
"Cohomology of some auxiliary varieties for $\\mathbb {A}$ Let us introduce three varieties $X_n$ , $Y_n$ and $Z_n$ with dimensions $n, n+1$ and $n+1$ .",
"The variety $Z_n$ is defined by variables $x_1,\\dots ,x_n$ , $x^{\\prime }_1,\\dots ,x^{\\prime }_n$ and $\\alpha $ such that $x_1 x^{\\prime }_1 &= 1 + \\alpha x_2 ,\\\\x_i x^{\\prime }_i &= 1 + x_{i-1} x_{i+1}, \\\\x_n x^{\\prime }_n &= 1 + x_{n-1}.$ The variety $Y_n$ is the open set in $Z_n$ where $\\alpha $ is invertible.",
"The variety $X_n$ is the closed set in $Y_n$ where $\\alpha $ is fixed to a generic invertible value (where generic means distinct from $(-1)^{(n+1)/2}$ if $n$ is odd).",
"In our general notations, $Y_n$ is $X_{\\mathbb {A}_n}^\\mathtt {versal}$ and $X_n$ is $X_{\\mathbb {A}_n}^\\mathtt {generic}$ .",
"Let us first describe the variety $Z_n$ .",
"Proposition 6.1 There exists an isomorphism between $Z_n$ and the affine space $A_{n+1}$ .",
"Proof.",
"This has been proved in [6].",
"Therefore, the cohomology of $Z_n$ is known for all $n$ : $\\mathsf {H}^k(Z_n) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}1 &\\text{ if } k=0,\\\\0 &\\text{ if } k>0.\\end{array}\\right.",
"}$ The Hodge structure on $\\mathsf {H}^0(Z_n)$ is $\\mathbb {Q}(0)$ .",
"Let us now compute the cohomology of $Y_n$ by induction.",
"This uses the Mayer-Vietoris long exact sequence for the covering of $Z_n$ by the two open sets $U(x_1)$ and $U(\\alpha )$ .",
"First, let us note that $U(\\alpha ) \\simeq Y_n $ by definition.",
"Next, one finds that $U(x_1) \\simeq A_{1} Y_{n-1}$ .",
"Indeed one can eliminate $x^{\\prime }_1$ using the first equation.",
"Then $\\alpha $ becomes a free variable, and there remains the equations for $Y_{n-1}$ , with $x_1$ now playing the role of $\\alpha $ .",
"Last, the intersection $U(\\alpha ) \\cap U(x_1)$ is isomorphic to $ \\mathbb {G}_mY_{n-1}$ , by the same argument.",
"Let us write $\\omega _{\\alpha }$ for $d \\log (\\alpha )$ .",
"Proposition 6.2 The cohomology ring of $Y_n$ has the following description: $\\mathsf {H}^k(Y_n) = \\mathbb {Q}(-k)$ for $0 \\le k \\le n+1$ .",
"It has a basis given by powers of $\\operatorname{WP}$ in even degrees and by powers of $\\operatorname{WP}$ times $\\omega _{\\alpha }$ in odd degrees.",
"It is generated by the 1-form $\\omega _{\\alpha }$ and the 2-form $\\operatorname{WP}$ .",
"Proof.",
"Because of the vanishing of $\\mathsf {H}^k(Z_n)$ for $k>0$ , the Mayer-Vietoris long exact sequence gives short exact sequences $0 \\rightarrow \\mathsf {H}^0(Z_n) \\rightarrow \\mathsf {H}^0(Y_n)\\oplus \\mathsf {H}^0(U(x_1)) \\rightarrow \\mathsf {H}^0(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ and $0 \\rightarrow \\mathsf {H}^k(Y_n)\\oplus \\mathsf {H}^k(U(x_1)) \\rightarrow \\mathsf {H}^k(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ for every $k > 0$ .",
"This determines by induction the Hodge structure of the cohomology of $Y_{n}$ .",
"Let us now proceed to the expected basis.",
"One already knows that $\\operatorname{WP}$ and $\\omega _{\\alpha }$ are indeed algebraic differential forms on $Y_{n}$ .",
"By the short exact sequences above, one can check that for $k>0$ the union of the expected basis of $\\mathsf {H}^k(Y_n)$ with the known basis of $\\mathsf {H}^k(U(x_1))$ is mapped to a basis of $\\mathsf {H}^k(U(\\alpha ) \\cap U(x_1))$ .",
"This implies the statement.",
"Cohomology for $\\mathbb {A}_n$ with even $n$ Let us now consider the linear tree $\\mathbb {A}_n$ for even $n$ , and compute the cohomology of $X_n$ .",
"Proposition 6.3 The Hodge structure of the cohomology of $X_n$ is $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for all even $k$ between 0 and $n$ , and 0 otherwise.",
"A basis is given by powers of $\\operatorname{WP}$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ .",
"Proof.",
"This follows from the known cohomology of $Y_n$ and the Künneth theorem applied to the isomorphism $Y_n \\simeq X_n \\mathbb {G}_m$ given by lemma REF .",
"The Künneth theorem gives immediately the Hodge structure.",
"For the basis, it is enough to recall that the $\\mathbb {G}_m$ factor is given by the value of $\\alpha $ , and to check that fixing the value $\\alpha = 1$ maps $\\operatorname{WP}$ (for $Y_n$ ) to $\\operatorname{WP}$ (for $X_n$ ).",
"Cohomology for orange trees of shape $H$ 0.7 [scale=0.7] patterns,decorations.pathreplacing every node=[draw,shape=circle,very thick,fill=white] (0,0) node ... – (1,0) node ... – (3,0) node ... – (4,0) node ...; (0,1) node ... – (1,1) node ... – (3,1) node ... – (4,1) node ...; (2,1) node $a$ – (2,0) node $b$ ; every node=[draw=none,fill=none]; [thick,decoration= brace, raise=0.3cm ,decorate] (-0.3,1) node – (1.3,1) node ; [thick,decoration= brace, raise=0.3cm ,decorate] (2.7,1) node – (4.3,1) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (-0.3,0) node – (1.3,0) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (2.7,0) node – (4.3,0) node ; t (0.5,-1) $m$ ; t (0.5,2) $k$ ; t (3.5,-1) $n$ ; t (3.5,2) $\\ell $ ; Let us denote by $H_{k,\\ell ,m,n}$ the tree described as two chains joined by an edge, such that by removing the joining edge and its extremities $a$ and $b$ , one gets two chains of lengths $k$ and $\\ell $ on the $a$ side (top) and two chains of lengths $m$ and $n$ on the $b$ side (bottom).",
"We assume now that $H_{k,\\ell ,m,n}$ is an orange tree.",
"It implies that either $k, \\ell , m$ and $n$ are even if the middle edge is an orange domino, or that (without loss of generality) $k$ and $m$ are odd and $l$ and $n$ are even otherwise.",
"Then one can compute the cohomology of $H_{k,\\ell ,m,n}$ using the Mayer-Vietoris long exact sequence for the open covering by $U(x_a)$ and $U(x_b)$ .",
"When the middle edge is an orange domino, one has $\\begin{aligned}U(x_a) &\\simeq X_k X_\\ell Y_{m+n+1},\\\\U(x_b) &\\simeq Y_{k+\\ell +1} X_m X_n ,\\\\U(x_a) \\cap U(x_b) &\\simeq (\\mathbb {G}_m)^2 X_k X_\\ell X_m X_n.\\end{aligned}$ When the middle edge is not an orange domino, one finds instead $\\begin{aligned}U(x_a) &\\simeq Y_k X_\\ell X_{m+n+1},\\\\U(x_b) &\\simeq X_{k+\\ell +1} Y_m X_n,\\\\U(x_a) \\cap U(x_b) &\\simeq Y_k X_\\ell Y_m X_n.\\end{aligned}$ Let us introduce some notations: call $K,L,M,N$ the subsets of vertices corresponding to the four branches of $H$ (i.e.",
"the connected components of $H \\setminus \\lbrace a,b\\rbrace $ ).",
"Let us denote by $W_S$ the Weil-Petersson 2-form associated with a subset $S$ of the vertices of $H$ .",
"For conciseness, one will use shortcuts such as $W_{KaL}$ or $W_{MabN}$ .",
"Note that there holds $\\omega _a W_{aL} = \\omega _a W_{L}$ and other similar simplifications, by the definition (REF ) of these forms.",
"Let us now describe generators and bases of the cohomology of the open sets $ U(x_a)$ , $U(x_b)$ and $U(x_b)\\cap U(x_b)$ .",
"This can be computed using the isomorphisms (REF ), (REF ) and the known cohomology of varieties $X$ and $Y$ .",
"It turns out that the result does not depend on whether or not the middle edge $a-b$ is an orange domino.",
"The cohomology of $U(x_a)$ is generated by $\\omega _a$ , $W_{Ka}$ , $W_{aL}$ and $W_{MabN}$ .",
"A basis is given by $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B} \\quad \\text{and}\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{MbN}^{B},$ where $0 \\le \\kappa \\le k/2$ , $0 \\le \\lambda \\le l/2$ and $0 \\le B \\le (m+n+2)/2$ (left) or $0 \\le B \\le (m+n)/2$ (right).",
"Similarly, the cohomology of $U(x_b)$ is generated by $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ and $W_{KabL}$ .",
"A basis is given by $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A} \\quad \\text{and}\\quad \\omega _b W_{M}^{\\mu } W_{N}^{\\nu } W_{KaL}^{A},$ where $0 \\le \\mu \\le m/2$ , $0 \\le \\nu \\le n/2$ and $0 \\le A \\le (k+l+2)/2$ (left) or $0 \\le A \\le (k+l)/2$ (right).",
"The cohomology of $U(x_b)\\cap U(x_b)$ is generated by $\\omega _a$ , $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ , $W_{Ka}$ and $W_{aL}$ .",
"A basis is given by $\\begin{aligned}W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\\\\\omega _a \\omega _b W_{K}^{\\kappa } W_{L}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }\\quad \\text{and}\\quad \\omega _b W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu },\\end{aligned}$ with the same conditions as above on $\\kappa ,\\lambda ,\\mu $ and $\\nu $ .",
"There is a bigrading corresponding to the top and bottom parts of the $H$ shape.",
"Every differential form involved in the bases just described is a sum of products of $\\omega _i$ .",
"The bidegree of a monomial in the $\\omega _i$ is the pair (number of $\\omega _i$ where $i$ is in the top row, number of $\\omega _i$ where $i$ is in the bottom row).",
"Among the various Weil-Petersson forms involved, only the differential forms $W_{KabL}$ and $W_{MabN}$ are not homogeneous for the bidegree, but have terms in bidegrees $(2,0)$ and $(1,1)$ (resp.",
"$(0,2)$ and $(1,1)$ ).",
"One needs now to compute explicitly the following maps in the Mayer-Vietoris long exact sequence: $\\mathsf {H}^i(U(x_a))\\oplus \\mathsf {H}^i(U(x_b)) \\stackrel{f_i}{\\longrightarrow } \\mathsf {H}^i(U(x_a) \\cap U(x_b)).$ Because one has bases of all these spaces, this is a matter of matrices.",
"For odd degree $i$ , let us show that the differential is injective.",
"Because in this case all basis elements (given by right columns of (REF ), (REF ) and (REF )) are homogeneous for the bigrading, one can separate the cases of bidegree congruent to $(0,1)$ and to $(1,0)$ modulo $(2,2)$ .",
"Let us give details only for the first possibility, the other case being similar after exchanging top and bottom of $H$ .",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_b))$ is given by $\\omega _{b}W_{KaL}^{A}W_{M}^{\\mu } W_{N}^{\\nu }$ with $i =1+2A+2\\mu +2\\nu $ .",
"The corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a))$ is zero.",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a) \\cap U(x_b))$ is given by $\\omega _{b}W_{Ka}^{\\kappa }W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }$ with $i = 1+2\\kappa +2\\lambda +2\\mu +2\\nu $ .",
"But $W_{KaL}^{A}$ can be written as a linear combination of $W_{Ka}^{\\kappa }W_{aL}^{\\lambda }$ with $\\kappa + \\lambda = A$ .",
"Therefore the basis elements are mapped to linear combinations with disjoint supports.",
"It follows that the map $f_i$ is injective.",
"Let us now turn to even degrees.",
"Proposition 6.4 For even degree $2i$ , the kernel of the differential $f_{2i}$ has dimension 1, spanned by the $i^{th}$ power of the form $\\operatorname{WP}$ .",
"Proof.",
"First note that one can define an injective map $\\Delta $ from the space $\\mathsf {H}^{2i}(U(x_a) \\cap U(x_b))$ to the space $D_i$ spanned by all products of $i$ 2-forms of the shape $\\omega _{s}\\omega _{t}$ for $s-t$ an edge of the tree (always written in the order given by a fixed alternating orientation of the tree).",
"Indeed, both terms in the left column of (REF ) can be written as linear combinations of such products.",
"The injectivity holds because distinct elements in this part of the basis are mapped to linear combinations with disjoint supports.",
"To recover a basis element $B$ from any monomial in its image by $\\Delta $ , first count in $\\Delta (B)$ if the number of $\\omega _k$ in the top row is odd or even.",
"This tells if the basis elements $B$ contains $\\omega _a\\omega _b$ or not.",
"Then it is easy to recover the exponents $(\\kappa , \\lambda , \\mu , \\nu )$ defining $B$ by counting in $\\Delta (B)$ how many $\\omega _k$ there are in the different parts of the tree.",
"To prove the statement of the proposition, it is therefore enough to compute the kernel of the composite map $\\Delta \\circ f_{2i}$ .",
"It turns out that the matrix of this composite map has a nice description.",
"First, every monomial $d$ made of $i$ 2-forms $\\omega _{s}\\omega _{t}$ as above appears in exactly two images, the image of a form $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B}$ and the image of a form $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A}$ (with opposite signs).",
"Let us denote these two forms by $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ .",
"On the other hand, the image of every basis element is the sum of several monomials (at least one), with constant sign.",
"Let us pick an element $z$ of the kernel of $f_{2i}$ .",
"Then for every monomial $d$ in $D_i$ , the coefficients of $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ in $z$ must be the same.",
"One can make a graph with vertices given by all forms in the basis, and edges corresponding to the relations $\\mathsf {F}_a(d)-\\mathsf {F}_b(d)$ for all monomials $d$ .",
"By a combinatorial argument, one can check that this graph is connected.",
"For this, one just has to show that one can go from any monomial $d$ to any monomial $d^{\\prime }$ , using two kinds of moves: replace $d$ by another monomial appearing in the same $\\mathsf {F}_a(d)$ , or replace $d$ by another monomial appearing in the same $\\mathsf {F}_b(d)$ .",
"This is not difficult once translated in terms of dominoes, and details are left to the reader.",
"From the connectedness of this graph, one deduces that the kernel is spanned by the sum of all basis elements of $\\mathsf {H}^{2i}(U(x_a))\\oplus \\mathsf {H}^{2i}(U(x_b))$ , which is just $(\\operatorname{WP}^i,\\operatorname{WP}^i)$ .",
"This proposition and the injectivity in the case of odd degree allow to give a description of the weights of the Hodge structure on the cohomology.",
"This can easily be made explicit, but one will not do that here.",
"There would remain to find explicit expressions for the cohomology classes coming from the co-image of the differentials $f_i$ .",
"In the case of the Dynkin diagrams $\\mathbb {E}_6$ and $\\mathbb {E}_8$ , one can go further and compute explicit representatives of the cohomology classes.",
"By the general proof, the cohomology for $\\mathbb {E}_6$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-6),$ where the $\\mathbb {Q}(-i)$ with $i$ even correspond to the powers of $\\operatorname{WP}$ .",
"Using the connection homomorphism in the long exact sequence, one finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ .",
"Similarly, the cohomology for $\\mathbb {E}_8$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-5)\\oplus \\mathbb {Q}(-6) \\mid 0 \\mid \\mathbb {Q}(-8),$ where the even $\\mathbb {Q}(-i)$ are the powers of $\\operatorname{WP}$ .",
"One finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ , and its product by $\\operatorname{WP}$ corresponds to $\\mathbb {Q}(-5)$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6 – (5,0) node[fill=orange!20] 7 – (6,0) node[fill=orange!20] 8; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; Cohomology: generic cases This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers.",
"Cohomology for $\\mathbb {A}$ odd and generic Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Cohomology: general setting and results",
"This section first describes some differential forms that are always present in the varieties under study, and then very briefly recalls the results one needs about (mixed) Hodge structures.",
"For a general reference about mixed Hodge structures, see for example [18]."
],
[
"Weil-Petersson two-form",
"Let $T$ be a tree and let $S$ be a subset of $T$ .",
"Consider the augmented tree $T+S$ obtained by adding a new edge out of every vertex in $S$ , and endow this tree with a bipartite orientation, where every vertex is either a sink or a source.",
"As a variant of the definition of the variety $X_T^\\varphi $ , one can define a variety $X(T+S)$ attached to this data, with invertible variables associated to the new vertices, playing the role of coefficients in the equations (as the $\\alpha $ do).",
"Let $\\omega _i$ denote $d \\log (x_i)$ .",
"The following lemma has been proved by Greg Muller in [16] in a more general context.",
"Lemma 5.1 The differential form $\\operatorname{WP}= \\sum _{i \\rightarrow j} \\omega _i \\omega _j,$ where the sum is running over edges of $T+S$ , is an algebraic differential form on the variety $X(T+S)$ .",
"Proof.",
"Let us prove that it has no pole.",
"Let us fix $i$ .",
"To study the possible pole along $x_i = 0$ , it is enough to look at the sum $\\sum _{j \\leftrightarrow i} \\omega _i \\omega _j$ restricted to edges containing $i$ .",
"By the relation $x_i x^{\\prime }_i = 1 + \\prod _{j\\leftrightarrow i} x_j$ , one has $x_i d x^{\\prime }_i + x^{\\prime }_i d x_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j,$ and therefore $x_i dx^{\\prime }_i dx_i = \\sum _{j\\leftrightarrow i} \\left(\\prod _{{k \\ne j}\\atop {k\\leftrightarrow i}} x_k\\right) d x_j d x_i.$ This implies $dx^{\\prime }_i dx_i / \\prod _{k\\leftrightarrow i} x_k = \\sum _{j\\leftrightarrow i} \\omega _j \\omega _i,$ where the left-hand side has clearly no pole at $x_i$ .",
"Note that $\\operatorname{WP}$ stands here for Weil-Petersson.",
"Abusing notations, one will use the same symbol $\\operatorname{WP}$ to denote these differential forms on different varieties.",
"The ambient variety should be clear from the context.",
"Hodge structures We will use the notation $\\mathbb {Q}(-i)$ to denote a one dimensional vector space over $\\mathbb {Q}$ endowed with a pure Hodge structure of Tate type, of weight $2i$ and type $(i,i)$ .",
"The tensor product of $\\mathbb {Q}(-i)$ and $\\mathbb {Q}(-j)$ is $\\mathbb {Q}(-i-j)$ .",
"Recall that the cohomology of $\\mathbb {G}_m$ has an Hodge structure described by $\\mathsf {H}^k(\\mathbb {G}_m) = \\mathbb {Q}(-k)$ for $0 \\le k \\le 1$ .",
"There is no morphism between pure Hodge structures of distinct weights.",
"The Künneth isomorphism is compatible with the Hodge structures.",
"The Mayer-Vietoris long exact sequence is an exact sequence of Hodge structures.",
"Cohomology: orange and versal cases This section deals with the cohomology, in several cases where either varieties do not depend on parameters, or versal conditions are assumed on all parameters.",
"The first part is devoted to linear trees; the results there can then be used as building blocks.",
"Linear trees $\\mathbb {A}$ Let $\\mathbb {A}_n$ be the linear tree with $n$ vertices numbered from 1 to $n$ .",
"As seen in §REF , this is an orange tree if $n$ is even, and an unimodal tree otherwise.",
"Some of the results of this section were already obtained in [6] using instead the cohomology with compact supports.",
"Cohomology of some auxiliary varieties for $\\mathbb {A}$ Let us introduce three varieties $X_n$ , $Y_n$ and $Z_n$ with dimensions $n, n+1$ and $n+1$ .",
"The variety $Z_n$ is defined by variables $x_1,\\dots ,x_n$ , $x^{\\prime }_1,\\dots ,x^{\\prime }_n$ and $\\alpha $ such that $x_1 x^{\\prime }_1 &= 1 + \\alpha x_2 ,\\\\x_i x^{\\prime }_i &= 1 + x_{i-1} x_{i+1}, \\\\x_n x^{\\prime }_n &= 1 + x_{n-1}.$ The variety $Y_n$ is the open set in $Z_n$ where $\\alpha $ is invertible.",
"The variety $X_n$ is the closed set in $Y_n$ where $\\alpha $ is fixed to a generic invertible value (where generic means distinct from $(-1)^{(n+1)/2}$ if $n$ is odd).",
"In our general notations, $Y_n$ is $X_{\\mathbb {A}_n}^\\mathtt {versal}$ and $X_n$ is $X_{\\mathbb {A}_n}^\\mathtt {generic}$ .",
"Let us first describe the variety $Z_n$ .",
"Proposition 6.1 There exists an isomorphism between $Z_n$ and the affine space $A_{n+1}$ .",
"Proof.",
"This has been proved in [6].",
"Therefore, the cohomology of $Z_n$ is known for all $n$ : $\\mathsf {H}^k(Z_n) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}1 &\\text{ if } k=0,\\\\0 &\\text{ if } k>0.\\end{array}\\right.",
"}$ The Hodge structure on $\\mathsf {H}^0(Z_n)$ is $\\mathbb {Q}(0)$ .",
"Let us now compute the cohomology of $Y_n$ by induction.",
"This uses the Mayer-Vietoris long exact sequence for the covering of $Z_n$ by the two open sets $U(x_1)$ and $U(\\alpha )$ .",
"First, let us note that $U(\\alpha ) \\simeq Y_n $ by definition.",
"Next, one finds that $U(x_1) \\simeq A_{1} Y_{n-1}$ .",
"Indeed one can eliminate $x^{\\prime }_1$ using the first equation.",
"Then $\\alpha $ becomes a free variable, and there remains the equations for $Y_{n-1}$ , with $x_1$ now playing the role of $\\alpha $ .",
"Last, the intersection $U(\\alpha ) \\cap U(x_1)$ is isomorphic to $ \\mathbb {G}_mY_{n-1}$ , by the same argument.",
"Let us write $\\omega _{\\alpha }$ for $d \\log (\\alpha )$ .",
"Proposition 6.2 The cohomology ring of $Y_n$ has the following description: $\\mathsf {H}^k(Y_n) = \\mathbb {Q}(-k)$ for $0 \\le k \\le n+1$ .",
"It has a basis given by powers of $\\operatorname{WP}$ in even degrees and by powers of $\\operatorname{WP}$ times $\\omega _{\\alpha }$ in odd degrees.",
"It is generated by the 1-form $\\omega _{\\alpha }$ and the 2-form $\\operatorname{WP}$ .",
"Proof.",
"Because of the vanishing of $\\mathsf {H}^k(Z_n)$ for $k>0$ , the Mayer-Vietoris long exact sequence gives short exact sequences $0 \\rightarrow \\mathsf {H}^0(Z_n) \\rightarrow \\mathsf {H}^0(Y_n)\\oplus \\mathsf {H}^0(U(x_1)) \\rightarrow \\mathsf {H}^0(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ and $0 \\rightarrow \\mathsf {H}^k(Y_n)\\oplus \\mathsf {H}^k(U(x_1)) \\rightarrow \\mathsf {H}^k(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ for every $k > 0$ .",
"This determines by induction the Hodge structure of the cohomology of $Y_{n}$ .",
"Let us now proceed to the expected basis.",
"One already knows that $\\operatorname{WP}$ and $\\omega _{\\alpha }$ are indeed algebraic differential forms on $Y_{n}$ .",
"By the short exact sequences above, one can check that for $k>0$ the union of the expected basis of $\\mathsf {H}^k(Y_n)$ with the known basis of $\\mathsf {H}^k(U(x_1))$ is mapped to a basis of $\\mathsf {H}^k(U(\\alpha ) \\cap U(x_1))$ .",
"This implies the statement.",
"Cohomology for $\\mathbb {A}_n$ with even $n$ Let us now consider the linear tree $\\mathbb {A}_n$ for even $n$ , and compute the cohomology of $X_n$ .",
"Proposition 6.3 The Hodge structure of the cohomology of $X_n$ is $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for all even $k$ between 0 and $n$ , and 0 otherwise.",
"A basis is given by powers of $\\operatorname{WP}$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ .",
"Proof.",
"This follows from the known cohomology of $Y_n$ and the Künneth theorem applied to the isomorphism $Y_n \\simeq X_n \\mathbb {G}_m$ given by lemma REF .",
"The Künneth theorem gives immediately the Hodge structure.",
"For the basis, it is enough to recall that the $\\mathbb {G}_m$ factor is given by the value of $\\alpha $ , and to check that fixing the value $\\alpha = 1$ maps $\\operatorname{WP}$ (for $Y_n$ ) to $\\operatorname{WP}$ (for $X_n$ ).",
"Cohomology for orange trees of shape $H$ 0.7 [scale=0.7] patterns,decorations.pathreplacing every node=[draw,shape=circle,very thick,fill=white] (0,0) node ... – (1,0) node ... – (3,0) node ... – (4,0) node ...; (0,1) node ... – (1,1) node ... – (3,1) node ... – (4,1) node ...; (2,1) node $a$ – (2,0) node $b$ ; every node=[draw=none,fill=none]; [thick,decoration= brace, raise=0.3cm ,decorate] (-0.3,1) node – (1.3,1) node ; [thick,decoration= brace, raise=0.3cm ,decorate] (2.7,1) node – (4.3,1) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (-0.3,0) node – (1.3,0) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (2.7,0) node – (4.3,0) node ; t (0.5,-1) $m$ ; t (0.5,2) $k$ ; t (3.5,-1) $n$ ; t (3.5,2) $\\ell $ ; Let us denote by $H_{k,\\ell ,m,n}$ the tree described as two chains joined by an edge, such that by removing the joining edge and its extremities $a$ and $b$ , one gets two chains of lengths $k$ and $\\ell $ on the $a$ side (top) and two chains of lengths $m$ and $n$ on the $b$ side (bottom).",
"We assume now that $H_{k,\\ell ,m,n}$ is an orange tree.",
"It implies that either $k, \\ell , m$ and $n$ are even if the middle edge is an orange domino, or that (without loss of generality) $k$ and $m$ are odd and $l$ and $n$ are even otherwise.",
"Then one can compute the cohomology of $H_{k,\\ell ,m,n}$ using the Mayer-Vietoris long exact sequence for the open covering by $U(x_a)$ and $U(x_b)$ .",
"When the middle edge is an orange domino, one has $\\begin{aligned}U(x_a) &\\simeq X_k X_\\ell Y_{m+n+1},\\\\U(x_b) &\\simeq Y_{k+\\ell +1} X_m X_n ,\\\\U(x_a) \\cap U(x_b) &\\simeq (\\mathbb {G}_m)^2 X_k X_\\ell X_m X_n.\\end{aligned}$ When the middle edge is not an orange domino, one finds instead $\\begin{aligned}U(x_a) &\\simeq Y_k X_\\ell X_{m+n+1},\\\\U(x_b) &\\simeq X_{k+\\ell +1} Y_m X_n,\\\\U(x_a) \\cap U(x_b) &\\simeq Y_k X_\\ell Y_m X_n.\\end{aligned}$ Let us introduce some notations: call $K,L,M,N$ the subsets of vertices corresponding to the four branches of $H$ (i.e.",
"the connected components of $H \\setminus \\lbrace a,b\\rbrace $ ).",
"Let us denote by $W_S$ the Weil-Petersson 2-form associated with a subset $S$ of the vertices of $H$ .",
"For conciseness, one will use shortcuts such as $W_{KaL}$ or $W_{MabN}$ .",
"Note that there holds $\\omega _a W_{aL} = \\omega _a W_{L}$ and other similar simplifications, by the definition (REF ) of these forms.",
"Let us now describe generators and bases of the cohomology of the open sets $ U(x_a)$ , $U(x_b)$ and $U(x_b)\\cap U(x_b)$ .",
"This can be computed using the isomorphisms (REF ), (REF ) and the known cohomology of varieties $X$ and $Y$ .",
"It turns out that the result does not depend on whether or not the middle edge $a-b$ is an orange domino.",
"The cohomology of $U(x_a)$ is generated by $\\omega _a$ , $W_{Ka}$ , $W_{aL}$ and $W_{MabN}$ .",
"A basis is given by $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B} \\quad \\text{and}\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{MbN}^{B},$ where $0 \\le \\kappa \\le k/2$ , $0 \\le \\lambda \\le l/2$ and $0 \\le B \\le (m+n+2)/2$ (left) or $0 \\le B \\le (m+n)/2$ (right).",
"Similarly, the cohomology of $U(x_b)$ is generated by $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ and $W_{KabL}$ .",
"A basis is given by $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A} \\quad \\text{and}\\quad \\omega _b W_{M}^{\\mu } W_{N}^{\\nu } W_{KaL}^{A},$ where $0 \\le \\mu \\le m/2$ , $0 \\le \\nu \\le n/2$ and $0 \\le A \\le (k+l+2)/2$ (left) or $0 \\le A \\le (k+l)/2$ (right).",
"The cohomology of $U(x_b)\\cap U(x_b)$ is generated by $\\omega _a$ , $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ , $W_{Ka}$ and $W_{aL}$ .",
"A basis is given by $\\begin{aligned}W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\\\\\omega _a \\omega _b W_{K}^{\\kappa } W_{L}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }\\quad \\text{and}\\quad \\omega _b W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu },\\end{aligned}$ with the same conditions as above on $\\kappa ,\\lambda ,\\mu $ and $\\nu $ .",
"There is a bigrading corresponding to the top and bottom parts of the $H$ shape.",
"Every differential form involved in the bases just described is a sum of products of $\\omega _i$ .",
"The bidegree of a monomial in the $\\omega _i$ is the pair (number of $\\omega _i$ where $i$ is in the top row, number of $\\omega _i$ where $i$ is in the bottom row).",
"Among the various Weil-Petersson forms involved, only the differential forms $W_{KabL}$ and $W_{MabN}$ are not homogeneous for the bidegree, but have terms in bidegrees $(2,0)$ and $(1,1)$ (resp.",
"$(0,2)$ and $(1,1)$ ).",
"One needs now to compute explicitly the following maps in the Mayer-Vietoris long exact sequence: $\\mathsf {H}^i(U(x_a))\\oplus \\mathsf {H}^i(U(x_b)) \\stackrel{f_i}{\\longrightarrow } \\mathsf {H}^i(U(x_a) \\cap U(x_b)).$ Because one has bases of all these spaces, this is a matter of matrices.",
"For odd degree $i$ , let us show that the differential is injective.",
"Because in this case all basis elements (given by right columns of (REF ), (REF ) and (REF )) are homogeneous for the bigrading, one can separate the cases of bidegree congruent to $(0,1)$ and to $(1,0)$ modulo $(2,2)$ .",
"Let us give details only for the first possibility, the other case being similar after exchanging top and bottom of $H$ .",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_b))$ is given by $\\omega _{b}W_{KaL}^{A}W_{M}^{\\mu } W_{N}^{\\nu }$ with $i =1+2A+2\\mu +2\\nu $ .",
"The corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a))$ is zero.",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a) \\cap U(x_b))$ is given by $\\omega _{b}W_{Ka}^{\\kappa }W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }$ with $i = 1+2\\kappa +2\\lambda +2\\mu +2\\nu $ .",
"But $W_{KaL}^{A}$ can be written as a linear combination of $W_{Ka}^{\\kappa }W_{aL}^{\\lambda }$ with $\\kappa + \\lambda = A$ .",
"Therefore the basis elements are mapped to linear combinations with disjoint supports.",
"It follows that the map $f_i$ is injective.",
"Let us now turn to even degrees.",
"Proposition 6.4 For even degree $2i$ , the kernel of the differential $f_{2i}$ has dimension 1, spanned by the $i^{th}$ power of the form $\\operatorname{WP}$ .",
"Proof.",
"First note that one can define an injective map $\\Delta $ from the space $\\mathsf {H}^{2i}(U(x_a) \\cap U(x_b))$ to the space $D_i$ spanned by all products of $i$ 2-forms of the shape $\\omega _{s}\\omega _{t}$ for $s-t$ an edge of the tree (always written in the order given by a fixed alternating orientation of the tree).",
"Indeed, both terms in the left column of (REF ) can be written as linear combinations of such products.",
"The injectivity holds because distinct elements in this part of the basis are mapped to linear combinations with disjoint supports.",
"To recover a basis element $B$ from any monomial in its image by $\\Delta $ , first count in $\\Delta (B)$ if the number of $\\omega _k$ in the top row is odd or even.",
"This tells if the basis elements $B$ contains $\\omega _a\\omega _b$ or not.",
"Then it is easy to recover the exponents $(\\kappa , \\lambda , \\mu , \\nu )$ defining $B$ by counting in $\\Delta (B)$ how many $\\omega _k$ there are in the different parts of the tree.",
"To prove the statement of the proposition, it is therefore enough to compute the kernel of the composite map $\\Delta \\circ f_{2i}$ .",
"It turns out that the matrix of this composite map has a nice description.",
"First, every monomial $d$ made of $i$ 2-forms $\\omega _{s}\\omega _{t}$ as above appears in exactly two images, the image of a form $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B}$ and the image of a form $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A}$ (with opposite signs).",
"Let us denote these two forms by $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ .",
"On the other hand, the image of every basis element is the sum of several monomials (at least one), with constant sign.",
"Let us pick an element $z$ of the kernel of $f_{2i}$ .",
"Then for every monomial $d$ in $D_i$ , the coefficients of $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ in $z$ must be the same.",
"One can make a graph with vertices given by all forms in the basis, and edges corresponding to the relations $\\mathsf {F}_a(d)-\\mathsf {F}_b(d)$ for all monomials $d$ .",
"By a combinatorial argument, one can check that this graph is connected.",
"For this, one just has to show that one can go from any monomial $d$ to any monomial $d^{\\prime }$ , using two kinds of moves: replace $d$ by another monomial appearing in the same $\\mathsf {F}_a(d)$ , or replace $d$ by another monomial appearing in the same $\\mathsf {F}_b(d)$ .",
"This is not difficult once translated in terms of dominoes, and details are left to the reader.",
"From the connectedness of this graph, one deduces that the kernel is spanned by the sum of all basis elements of $\\mathsf {H}^{2i}(U(x_a))\\oplus \\mathsf {H}^{2i}(U(x_b))$ , which is just $(\\operatorname{WP}^i,\\operatorname{WP}^i)$ .",
"This proposition and the injectivity in the case of odd degree allow to give a description of the weights of the Hodge structure on the cohomology.",
"This can easily be made explicit, but one will not do that here.",
"There would remain to find explicit expressions for the cohomology classes coming from the co-image of the differentials $f_i$ .",
"In the case of the Dynkin diagrams $\\mathbb {E}_6$ and $\\mathbb {E}_8$ , one can go further and compute explicit representatives of the cohomology classes.",
"By the general proof, the cohomology for $\\mathbb {E}_6$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-6),$ where the $\\mathbb {Q}(-i)$ with $i$ even correspond to the powers of $\\operatorname{WP}$ .",
"Using the connection homomorphism in the long exact sequence, one finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ .",
"Similarly, the cohomology for $\\mathbb {E}_8$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-5)\\oplus \\mathbb {Q}(-6) \\mid 0 \\mid \\mathbb {Q}(-8),$ where the even $\\mathbb {Q}(-i)$ are the powers of $\\operatorname{WP}$ .",
"One finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ , and its product by $\\operatorname{WP}$ corresponds to $\\mathbb {Q}(-5)$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6 – (5,0) node[fill=orange!20] 7 – (6,0) node[fill=orange!20] 8; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; Cohomology: generic cases This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers.",
"Cohomology for $\\mathbb {A}$ odd and generic Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Cohomology: orange and versal cases",
"This section deals with the cohomology, in several cases where either varieties do not depend on parameters, or versal conditions are assumed on all parameters.",
"The first part is devoted to linear trees; the results there can then be used as building blocks."
],
[
"Linear trees $\\mathbb {A}$",
"Let $\\mathbb {A}_n$ be the linear tree with $n$ vertices numbered from 1 to $n$ .",
"As seen in §REF , this is an orange tree if $n$ is even, and an unimodal tree otherwise.",
"Some of the results of this section were already obtained in [6] using instead the cohomology with compact supports."
],
[
"Cohomology of some auxiliary varieties for $\\mathbb {A}$",
"Let us introduce three varieties $X_n$ , $Y_n$ and $Z_n$ with dimensions $n, n+1$ and $n+1$ .",
"The variety $Z_n$ is defined by variables $x_1,\\dots ,x_n$ , $x^{\\prime }_1,\\dots ,x^{\\prime }_n$ and $\\alpha $ such that $x_1 x^{\\prime }_1 &= 1 + \\alpha x_2 ,\\\\x_i x^{\\prime }_i &= 1 + x_{i-1} x_{i+1}, \\\\x_n x^{\\prime }_n &= 1 + x_{n-1}.$ The variety $Y_n$ is the open set in $Z_n$ where $\\alpha $ is invertible.",
"The variety $X_n$ is the closed set in $Y_n$ where $\\alpha $ is fixed to a generic invertible value (where generic means distinct from $(-1)^{(n+1)/2}$ if $n$ is odd).",
"In our general notations, $Y_n$ is $X_{\\mathbb {A}_n}^\\mathtt {versal}$ and $X_n$ is $X_{\\mathbb {A}_n}^\\mathtt {generic}$ .",
"Let us first describe the variety $Z_n$ .",
"Proposition 6.1 There exists an isomorphism between $Z_n$ and the affine space $A_{n+1}$ .",
"Proof.",
"This has been proved in [6].",
"Therefore, the cohomology of $Z_n$ is known for all $n$ : $\\mathsf {H}^k(Z_n) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}1 &\\text{ if } k=0,\\\\0 &\\text{ if } k>0.\\end{array}\\right.",
"}$ The Hodge structure on $\\mathsf {H}^0(Z_n)$ is $\\mathbb {Q}(0)$ .",
"Let us now compute the cohomology of $Y_n$ by induction.",
"This uses the Mayer-Vietoris long exact sequence for the covering of $Z_n$ by the two open sets $U(x_1)$ and $U(\\alpha )$ .",
"First, let us note that $U(\\alpha ) \\simeq Y_n $ by definition.",
"Next, one finds that $U(x_1) \\simeq A_{1} Y_{n-1}$ .",
"Indeed one can eliminate $x^{\\prime }_1$ using the first equation.",
"Then $\\alpha $ becomes a free variable, and there remains the equations for $Y_{n-1}$ , with $x_1$ now playing the role of $\\alpha $ .",
"Last, the intersection $U(\\alpha ) \\cap U(x_1)$ is isomorphic to $ \\mathbb {G}_mY_{n-1}$ , by the same argument.",
"Let us write $\\omega _{\\alpha }$ for $d \\log (\\alpha )$ .",
"Proposition 6.2 The cohomology ring of $Y_n$ has the following description: $\\mathsf {H}^k(Y_n) = \\mathbb {Q}(-k)$ for $0 \\le k \\le n+1$ .",
"It has a basis given by powers of $\\operatorname{WP}$ in even degrees and by powers of $\\operatorname{WP}$ times $\\omega _{\\alpha }$ in odd degrees.",
"It is generated by the 1-form $\\omega _{\\alpha }$ and the 2-form $\\operatorname{WP}$ .",
"Proof.",
"Because of the vanishing of $\\mathsf {H}^k(Z_n)$ for $k>0$ , the Mayer-Vietoris long exact sequence gives short exact sequences $0 \\rightarrow \\mathsf {H}^0(Z_n) \\rightarrow \\mathsf {H}^0(Y_n)\\oplus \\mathsf {H}^0(U(x_1)) \\rightarrow \\mathsf {H}^0(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ and $0 \\rightarrow \\mathsf {H}^k(Y_n)\\oplus \\mathsf {H}^k(U(x_1)) \\rightarrow \\mathsf {H}^k(U(\\alpha ) \\cap U(x_1)) \\rightarrow 0,$ for every $k > 0$ .",
"This determines by induction the Hodge structure of the cohomology of $Y_{n}$ .",
"Let us now proceed to the expected basis.",
"One already knows that $\\operatorname{WP}$ and $\\omega _{\\alpha }$ are indeed algebraic differential forms on $Y_{n}$ .",
"By the short exact sequences above, one can check that for $k>0$ the union of the expected basis of $\\mathsf {H}^k(Y_n)$ with the known basis of $\\mathsf {H}^k(U(x_1))$ is mapped to a basis of $\\mathsf {H}^k(U(\\alpha ) \\cap U(x_1))$ .",
"This implies the statement.",
"Cohomology for $\\mathbb {A}_n$ with even $n$ Let us now consider the linear tree $\\mathbb {A}_n$ for even $n$ , and compute the cohomology of $X_n$ .",
"Proposition 6.3 The Hodge structure of the cohomology of $X_n$ is $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for all even $k$ between 0 and $n$ , and 0 otherwise.",
"A basis is given by powers of $\\operatorname{WP}$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ .",
"Proof.",
"This follows from the known cohomology of $Y_n$ and the Künneth theorem applied to the isomorphism $Y_n \\simeq X_n \\mathbb {G}_m$ given by lemma REF .",
"The Künneth theorem gives immediately the Hodge structure.",
"For the basis, it is enough to recall that the $\\mathbb {G}_m$ factor is given by the value of $\\alpha $ , and to check that fixing the value $\\alpha = 1$ maps $\\operatorname{WP}$ (for $Y_n$ ) to $\\operatorname{WP}$ (for $X_n$ ).",
"Cohomology for orange trees of shape $H$ 0.7 [scale=0.7] patterns,decorations.pathreplacing every node=[draw,shape=circle,very thick,fill=white] (0,0) node ... – (1,0) node ... – (3,0) node ... – (4,0) node ...; (0,1) node ... – (1,1) node ... – (3,1) node ... – (4,1) node ...; (2,1) node $a$ – (2,0) node $b$ ; every node=[draw=none,fill=none]; [thick,decoration= brace, raise=0.3cm ,decorate] (-0.3,1) node – (1.3,1) node ; [thick,decoration= brace, raise=0.3cm ,decorate] (2.7,1) node – (4.3,1) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (-0.3,0) node – (1.3,0) node ; [thick,decoration= brace, mirror, raise=0.3cm ,decorate] (2.7,0) node – (4.3,0) node ; t (0.5,-1) $m$ ; t (0.5,2) $k$ ; t (3.5,-1) $n$ ; t (3.5,2) $\\ell $ ; Let us denote by $H_{k,\\ell ,m,n}$ the tree described as two chains joined by an edge, such that by removing the joining edge and its extremities $a$ and $b$ , one gets two chains of lengths $k$ and $\\ell $ on the $a$ side (top) and two chains of lengths $m$ and $n$ on the $b$ side (bottom).",
"We assume now that $H_{k,\\ell ,m,n}$ is an orange tree.",
"It implies that either $k, \\ell , m$ and $n$ are even if the middle edge is an orange domino, or that (without loss of generality) $k$ and $m$ are odd and $l$ and $n$ are even otherwise.",
"Then one can compute the cohomology of $H_{k,\\ell ,m,n}$ using the Mayer-Vietoris long exact sequence for the open covering by $U(x_a)$ and $U(x_b)$ .",
"When the middle edge is an orange domino, one has $\\begin{aligned}U(x_a) &\\simeq X_k X_\\ell Y_{m+n+1},\\\\U(x_b) &\\simeq Y_{k+\\ell +1} X_m X_n ,\\\\U(x_a) \\cap U(x_b) &\\simeq (\\mathbb {G}_m)^2 X_k X_\\ell X_m X_n.\\end{aligned}$ When the middle edge is not an orange domino, one finds instead $\\begin{aligned}U(x_a) &\\simeq Y_k X_\\ell X_{m+n+1},\\\\U(x_b) &\\simeq X_{k+\\ell +1} Y_m X_n,\\\\U(x_a) \\cap U(x_b) &\\simeq Y_k X_\\ell Y_m X_n.\\end{aligned}$ Let us introduce some notations: call $K,L,M,N$ the subsets of vertices corresponding to the four branches of $H$ (i.e.",
"the connected components of $H \\setminus \\lbrace a,b\\rbrace $ ).",
"Let us denote by $W_S$ the Weil-Petersson 2-form associated with a subset $S$ of the vertices of $H$ .",
"For conciseness, one will use shortcuts such as $W_{KaL}$ or $W_{MabN}$ .",
"Note that there holds $\\omega _a W_{aL} = \\omega _a W_{L}$ and other similar simplifications, by the definition (REF ) of these forms.",
"Let us now describe generators and bases of the cohomology of the open sets $ U(x_a)$ , $U(x_b)$ and $U(x_b)\\cap U(x_b)$ .",
"This can be computed using the isomorphisms (REF ), (REF ) and the known cohomology of varieties $X$ and $Y$ .",
"It turns out that the result does not depend on whether or not the middle edge $a-b$ is an orange domino.",
"The cohomology of $U(x_a)$ is generated by $\\omega _a$ , $W_{Ka}$ , $W_{aL}$ and $W_{MabN}$ .",
"A basis is given by $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B} \\quad \\text{and}\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{MbN}^{B},$ where $0 \\le \\kappa \\le k/2$ , $0 \\le \\lambda \\le l/2$ and $0 \\le B \\le (m+n+2)/2$ (left) or $0 \\le B \\le (m+n)/2$ (right).",
"Similarly, the cohomology of $U(x_b)$ is generated by $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ and $W_{KabL}$ .",
"A basis is given by $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A} \\quad \\text{and}\\quad \\omega _b W_{M}^{\\mu } W_{N}^{\\nu } W_{KaL}^{A},$ where $0 \\le \\mu \\le m/2$ , $0 \\le \\nu \\le n/2$ and $0 \\le A \\le (k+l+2)/2$ (left) or $0 \\le A \\le (k+l)/2$ (right).",
"The cohomology of $U(x_b)\\cap U(x_b)$ is generated by $\\omega _a$ , $\\omega _b$ , $W_{Mb}$ , $W_{bN}$ , $W_{Ka}$ and $W_{aL}$ .",
"A basis is given by $\\begin{aligned}W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\quad \\omega _a W_{K}^{\\kappa } W_{L}^{\\lambda } W_{Mb}^{\\mu } W_{bN}^{\\nu },\\\\\\omega _a \\omega _b W_{K}^{\\kappa } W_{L}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }\\quad \\text{and}\\quad \\omega _b W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu },\\end{aligned}$ with the same conditions as above on $\\kappa ,\\lambda ,\\mu $ and $\\nu $ .",
"There is a bigrading corresponding to the top and bottom parts of the $H$ shape.",
"Every differential form involved in the bases just described is a sum of products of $\\omega _i$ .",
"The bidegree of a monomial in the $\\omega _i$ is the pair (number of $\\omega _i$ where $i$ is in the top row, number of $\\omega _i$ where $i$ is in the bottom row).",
"Among the various Weil-Petersson forms involved, only the differential forms $W_{KabL}$ and $W_{MabN}$ are not homogeneous for the bidegree, but have terms in bidegrees $(2,0)$ and $(1,1)$ (resp.",
"$(0,2)$ and $(1,1)$ ).",
"One needs now to compute explicitly the following maps in the Mayer-Vietoris long exact sequence: $\\mathsf {H}^i(U(x_a))\\oplus \\mathsf {H}^i(U(x_b)) \\stackrel{f_i}{\\longrightarrow } \\mathsf {H}^i(U(x_a) \\cap U(x_b)).$ Because one has bases of all these spaces, this is a matter of matrices.",
"For odd degree $i$ , let us show that the differential is injective.",
"Because in this case all basis elements (given by right columns of (REF ), (REF ) and (REF )) are homogeneous for the bigrading, one can separate the cases of bidegree congruent to $(0,1)$ and to $(1,0)$ modulo $(2,2)$ .",
"Let us give details only for the first possibility, the other case being similar after exchanging top and bottom of $H$ .",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_b))$ is given by $\\omega _{b}W_{KaL}^{A}W_{M}^{\\mu } W_{N}^{\\nu }$ with $i =1+2A+2\\mu +2\\nu $ .",
"The corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a))$ is zero.",
"The basis of the corresponding bihomogeneous subspace of $\\mathsf {H}^i(U(x_a) \\cap U(x_b))$ is given by $\\omega _{b}W_{Ka}^{\\kappa }W_{aL}^{\\lambda } W_{M}^{\\mu } W_{N}^{\\nu }$ with $i = 1+2\\kappa +2\\lambda +2\\mu +2\\nu $ .",
"But $W_{KaL}^{A}$ can be written as a linear combination of $W_{Ka}^{\\kappa }W_{aL}^{\\lambda }$ with $\\kappa + \\lambda = A$ .",
"Therefore the basis elements are mapped to linear combinations with disjoint supports.",
"It follows that the map $f_i$ is injective.",
"Let us now turn to even degrees.",
"Proposition 6.4 For even degree $2i$ , the kernel of the differential $f_{2i}$ has dimension 1, spanned by the $i^{th}$ power of the form $\\operatorname{WP}$ .",
"Proof.",
"First note that one can define an injective map $\\Delta $ from the space $\\mathsf {H}^{2i}(U(x_a) \\cap U(x_b))$ to the space $D_i$ spanned by all products of $i$ 2-forms of the shape $\\omega _{s}\\omega _{t}$ for $s-t$ an edge of the tree (always written in the order given by a fixed alternating orientation of the tree).",
"Indeed, both terms in the left column of (REF ) can be written as linear combinations of such products.",
"The injectivity holds because distinct elements in this part of the basis are mapped to linear combinations with disjoint supports.",
"To recover a basis element $B$ from any monomial in its image by $\\Delta $ , first count in $\\Delta (B)$ if the number of $\\omega _k$ in the top row is odd or even.",
"This tells if the basis elements $B$ contains $\\omega _a\\omega _b$ or not.",
"Then it is easy to recover the exponents $(\\kappa , \\lambda , \\mu , \\nu )$ defining $B$ by counting in $\\Delta (B)$ how many $\\omega _k$ there are in the different parts of the tree.",
"To prove the statement of the proposition, it is therefore enough to compute the kernel of the composite map $\\Delta \\circ f_{2i}$ .",
"It turns out that the matrix of this composite map has a nice description.",
"First, every monomial $d$ made of $i$ 2-forms $\\omega _{s}\\omega _{t}$ as above appears in exactly two images, the image of a form $W_{Ka}^{\\kappa } W_{aL}^{\\lambda } W_{MabN}^{B}$ and the image of a form $W_{Mb}^{\\mu } W_{bN}^{\\nu } W_{KabL}^{A}$ (with opposite signs).",
"Let us denote these two forms by $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ .",
"On the other hand, the image of every basis element is the sum of several monomials (at least one), with constant sign.",
"Let us pick an element $z$ of the kernel of $f_{2i}$ .",
"Then for every monomial $d$ in $D_i$ , the coefficients of $\\mathsf {F}_a(d)$ and $\\mathsf {F}_b(d)$ in $z$ must be the same.",
"One can make a graph with vertices given by all forms in the basis, and edges corresponding to the relations $\\mathsf {F}_a(d)-\\mathsf {F}_b(d)$ for all monomials $d$ .",
"By a combinatorial argument, one can check that this graph is connected.",
"For this, one just has to show that one can go from any monomial $d$ to any monomial $d^{\\prime }$ , using two kinds of moves: replace $d$ by another monomial appearing in the same $\\mathsf {F}_a(d)$ , or replace $d$ by another monomial appearing in the same $\\mathsf {F}_b(d)$ .",
"This is not difficult once translated in terms of dominoes, and details are left to the reader.",
"From the connectedness of this graph, one deduces that the kernel is spanned by the sum of all basis elements of $\\mathsf {H}^{2i}(U(x_a))\\oplus \\mathsf {H}^{2i}(U(x_b))$ , which is just $(\\operatorname{WP}^i,\\operatorname{WP}^i)$ .",
"This proposition and the injectivity in the case of odd degree allow to give a description of the weights of the Hodge structure on the cohomology.",
"This can easily be made explicit, but one will not do that here.",
"There would remain to find explicit expressions for the cohomology classes coming from the co-image of the differentials $f_i$ .",
"In the case of the Dynkin diagrams $\\mathbb {E}_6$ and $\\mathbb {E}_8$ , one can go further and compute explicit representatives of the cohomology classes.",
"By the general proof, the cohomology for $\\mathbb {E}_6$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-6),$ where the $\\mathbb {Q}(-i)$ with $i$ even correspond to the powers of $\\operatorname{WP}$ .",
"Using the connection homomorphism in the long exact sequence, one finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ .",
"Similarly, the cohomology for $\\mathbb {E}_8$ is described by $\\mathbb {Q}(0) \\mid 0 \\mid \\mathbb {Q}(-2) \\mid 0 \\mid \\mathbb {Q}(-3)\\oplus \\mathbb {Q}(-4) \\mid 0 \\mid \\mathbb {Q}(-5)\\oplus \\mathbb {Q}(-6) \\mid 0 \\mid \\mathbb {Q}(-8),$ where the even $\\mathbb {Q}(-i)$ are the powers of $\\operatorname{WP}$ .",
"One finds that the form $dx_2 dx_3 dx_5 \\omega _4$ corresponds to $\\mathbb {Q}(-3)$ , and its product by $\\operatorname{WP}$ corresponds to $\\mathbb {Q}(-5)$ .",
"0.7 [scale=0.7] every node=[draw,shape=circle,very thick,fill=white] (0,0) node[fill=orange!20] 1 – (1,0) node[fill=orange!20] 2 – (2,0) – (3,0) node[fill=orange!20] 5 – (4,0) node[fill=orange!20] 6 – (5,0) node[fill=orange!20] 7 – (6,0) node[fill=orange!20] 8; (2,1) node[fill=orange!20] 3 – (2,0) node[fill=orange!20] 4; Cohomology: generic cases This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers.",
"Cohomology for $\\mathbb {A}$ odd and generic Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Cohomology: generic cases",
"This section contains one conjecture and one result in some specific cases about the cohomology of generic fibers."
],
[
"Cohomology for $\\mathbb {A}$ odd and generic",
"Let us now consider the linear tree $\\mathbb {A}_n$ for odd $n$ , which is unimodal.",
"In this section, one proposes a conjectural description for the cohomology of the variety $X_{\\mathbb {A}_n}^\\mathtt {generic}$ (which is also denoted $X_n$ in §REF ).",
"Conjecture 7.1 The Hodge structure on the cohomology of $X_n$ is given by $\\mathsf {H}^k(X_n) = \\mathbb {Q}(-k)$ for even $k$ in $0 \\le k \\le (n-1)$ , and $\\mathsf {H}^{n}(X_n) = \\oplus _{i=(n+1)/2}^{n} \\mathbb {Q}(-i).$ The cohomology ring has a basis given by all powers $\\operatorname{WP}^i$ for $0 \\le i \\le (n-1)/2$ and by a basis of $\\mathsf {H}^{n}(X_n)$ .",
"The cohomology ring is generated by $\\operatorname{WP}$ in degree 2 and by the elements of $\\mathsf {H}^{n}(X_n)$ in degree $n$ .",
"One approach for this computation would be using the covering of $X_n$ by the $(n+1)/2$ open sets $U(x_i)$ ($i$ odd) given by Lemma REF .",
"One can then consider the spectral sequence for this covering (where $d_1$ is the deRham differential and $d_2$ is the Cech differential).",
"The intersection of open sets in this covering have a simple description: they are products $\\mathbb {G}_m$ times two varieties of the type $X_k$ with $k$ even, times some varieties of type $Y_k$ with $k$ odd.",
"Lemma 7.2 This spectral sequence degenerates at $E_2$ .",
"Proof.",
"This follows from the purity of the Hodge structure on the cohomology of the open sets in the covering.",
"It would therefore be enough to understand the behavior of the Cech differential acting on the cohomology groups of the open sets.",
"This is still a rather intricate question.",
"The conjecture has been checked by computer for $n \\le 11$ .",
"Maybe one should look for a better approach.",
"Remark 7.3 To give an explicit description of the generators of the top cohomology group seems to be an interesting problem.",
"Cohomology for $\\mathbb {D}$ odd and generic Let us now consider the tree $\\mathbb {D}_n$ for odd $n$ , which is unimodal.",
"Our aim is to compute the cohomology of the variety $X_{\\mathbb {D}_n}^\\mathtt {generic}$ .",
"One will assume that the generic parameter $\\alpha $ is attached to the vertex 1, where 1 and 2 are the two red vertices on the short branches.",
"By Lemma REF , one has a covering by $U(x_1)$ and $U(x_2)$ .",
"One will use the Mayer-Vietoris long exact sequence for this covering.",
"One has $U(x_1) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_2) &\\simeq \\mathbb {G}_mX_{n-1},\\\\U(x_1) \\cap U(x_2) & \\simeq \\mathbb {G}_mY_{n-2}.$ Given the known explicit description of the cohomology rings of $X_{n-1}$ and $Y_{n-2}$ , one can write very explicitly the long exact sequence.",
"First note that the Hodge structure of $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ for $0\\le k \\le n$ .",
"Similarly, the Hodge structure of $\\mathsf {H}^k(U(x_1) \\cap U(x_2))$ is $2\\,\\mathbb {Q}(-k)$ , unless $k=0$ or $n$ where it is $\\mathbb {Q}(-k)$ .",
"Using the known basis of the cohomology, one can describe the map $\\rho _k$ from $\\mathsf {H}^k(U(x_1))\\oplus \\mathsf {H}^k(U(x_2))$ to $\\mathsf {H}^k(U(x_1)\\cap U(x_2))$ .",
"One can see that this map has rank 1 if $k$ is even.",
"One can also check that it is an isomorphism if $k$ is odd, unless $k=n$ where it has rank 1.",
"It follows that the Hodge structure on $\\mathsf {H}^k(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by ${\\left\\lbrace \\begin{array}{ll}\\mathbb {Q}(-k) \\quad & \\text{if} \\quad k\\equiv 0\\, (\\operatorname{mod}2),\\\\\\mathbb {Q}(-k+1) \\quad &\\text{if} \\quad k \\equiv 1\\, (\\operatorname{mod}2),\\, k \\notin \\lbrace 1,n\\rbrace \\\\\\mathbb {Q}(-n+1)\\oplus \\mathbb {Q}(-n) \\quad &\\text{if} \\quad k=n.\\end{array}\\right.",
"}$ Moreover, it also follows from the explicit knowledge of the long exact sequence that the classes in even cohomological degree are just the powers of the 2-form $\\operatorname{WP}$ .",
"One can also see that the Hodge structure $\\mathbb {Q}(-n)$ in cohomological degree $n$ is given by the differential form $\\Lambda _{i=1}^{n} \\omega _i$ .",
"There remains to understand the even Hodge structures present in odd cohomological degrees.",
"By a small diagram chase, and using the formula $\\frac{1-\\alpha }{x_1 x_2} = \\frac{x^{\\prime }_1}{x_2} - \\alpha \\frac{x^{\\prime }_2}{x_1},$ one finds that a basis of the $\\mathbb {Q}(-2)$ part of $\\mathsf {H}^3(X_{\\mathbb {D}_n}^\\mathtt {generic})$ is given by the differential form $dx_3 \\omega _1 \\omega _2.$ Moreover, a similar computation shows that products of this form by powers of $\\operatorname{WP}$ give a basis for the even Hodge structures in odd cohomological degrees.",
"The cohomology ring is therefore generated by one generator in each degree 2, 3 and $n$ (of Hodge type $\\mathbb {Q}(-2)$ , $\\mathbb {Q}(-2)$ and $\\mathbb {Q}(-n)$ ).",
"Algorithm for the canonical coloring of trees Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
],
[
"Algorithm for the canonical coloring of trees",
"Let us now describe an algorithm to find the red-orange-green coloring.",
"Let $T$ be a tree.",
"At start, all vertices are considered to be red.",
"Then, one changes the colors according to the following rule: If a vertex $v$ has exactly one red neighbor $w$ , this red neighbor becomes green.",
"If moreover $v$ is green, then one puts a domino on the edge $v-w$ .",
"One repeats the previous step until no color can change.",
"Then one colors in orange the green vertices that do not have a red neighbor.",
"One gets in that way a coloring of the tree with green, orange and red vertices, together with a collection of dominoes.",
"Proposition A.1 This algorithm defines the same coloring as in section .",
"Moreover the dominoes obtained are those that are present in all maximum matchings.",
"Proof.",
"At the end of step 3, one has obtained a tree with red and green vertices, with the property that every vertex has either no red neighbor or at least two red neighbors.",
"Let us prove that a red vertex can not have at least two red neighbors.",
"Assume that there is such a vertex $v_1$ .",
"Let $v_2$ be one of its red neighbors.",
"Then $v_2$ must also have at least two red neighbors.",
"Hence one can find another red neighbor $v_3$ of $v_2$ .",
"Going on in this way, and because $T$ is a tree, one can build an infinite sequence of red vertices, which is absurd.",
"So, after step 3, one has three kinds of vertices: red vertices (they have only green neighbors), green vertices with no red neighbors and green vertices with at least two red neighbors.",
"It follows that after step 4, one has the following situation: red vertices with only green neighbors, green vertices with at least two red neighbors, and orange vertices with no red neighbors.",
"Using the third characterization of the coloring, it just remains to prove that the induced forest on orange vertices has a perfect matching.",
"This matching is provided by the set of dominoes computed by the algorithm.",
"When a domino is introduced, both its vertices are green.",
"We need a lemma.",
"Lemma A.2 During the algorithm, the configuration $\\colorbox {red!20}{u} - \\colorbox {green!20}{v} - \\colorbox {green!20}{w}$ where $u$ is red and $v-w$ is a domino, does not appear.",
"Proof.",
"Let us assume the contrary, and let $u-v-w$ be such a configuration.",
"Because $v$ still has a red neighbor, the domino $v-w$ must have been created by turning green the vertex $v$ as the last red neighbor of the green vertex $w$ .",
"Let us go back to this previous step of this algorithm, where $u$ and $v$ are red, $w$ is green with $v$ as only red neighbor.",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}$ So $w$ must have another neighbor $z$ , such that $w$ has turned green as the last red neighbor of $z$ .",
"$\\colorbox {red!20}{u} - \\colorbox {red!20}{v} - \\colorbox {green!20}{w}- \\colorbox {green!20}{z}$ One can assume, by changing maybe the order in which the algorithm has been performed, that $z$ has turned green before $w$ .",
"This is because trees are bipartite, and the algorithm can be run independently on the two parts of the bipartition.",
"Therefore, $w$ has turned green as the last red neighbor of the green vertex $z$ , and hence belongs to a domino $w-z$ .",
"Hence one has found a configuration $v-w-z$ similar to the initial one: $\\colorbox {red!20}{v} - \\colorbox {green!20}{w} - \\colorbox {green!20}{z}.$ This can be iterated to provide an infinite sequence of vertices.",
"This is absurd.",
"It follows from the lemma that once a domino is created, its vertices do not have any red neighbors.",
"Therefore they will be orange at the end.",
"This also implies that the dominoes are disjoint, because the creation of a domino takes a red vertex with only green neighbors and a green vertex with exactly one red neighbor, and produces a pair of green vertices with only green neighbors.",
"Therefore a vertex can only enter once in a domino.",
"Moreover, every orange vertex $v$ is in a domino.",
"This is because green vertices surrounded only by green vertices can only be introduced during the creation of a domino.",
"Remark A.3 From the previous proof, one can see that one can modify the algorithm as follows: when creating a new domino, color in orange its two vertices, and forget step 4."
]
] | 1403.0540 |
[
[
"Vertically symmetric alternating sign matrices and a multivariate\n Laurent polynomial identity"
],
[
"Abstract In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely.",
"Computer experiments suggest that the numbers appearing in a conjecture concerning the number of vertically symmetric alternating sign matrices with respect to the position of the first 1 in the second row of the matrix establish the solution of a linear equation system similar to the one for the ordinary refined ASM numbers.",
"In this paper we show how our attempt to prove this fact naturally leads to a more general conjectural multivariate Laurent polynomial identity.",
"Remarkably, in contrast to the ordinary refined ASM numbers, we need to extend the combinatorial interpretation of the numbers to parameters which are not contained in the combinatorial admissible domain.",
"Some partial results towards proving the conjectured multivariate Laurent polynomial identity and additional motivation why to study it are presented as well."
],
[
"Introduction",
"An Alternating Sign Matrix (ASM) is a square matrix with entries in $\\lbrace 0,1,-1\\rbrace $ where in each row and column the non-zero entries alternate in sign and sum up to 1.",
"Combinatorialists are especially fond of these objects since they discovered that ASMs belong to the class of objects which possess a simple closed enumeration formula while at the same time no easy proof of this formula is known.",
"Mills, Robbins and Rumsey [11] introduced ASMs in the course of generalizing the determinant and conjectured that the number of $n \\times n$ ASMs is given by $\\prod _{j=0}^{n-1} \\frac{(3j+1)!}{(n+j)!",
"}.$ More than ten years later, Zeilberger [14] finally proved their conjecture.",
"Soon after, Kuperberg [9] gave another, shorter proof which makes use of a connection to statistical physics where ASMs have appeared before in an equivalent form as a model for plane square ice (six vertex model).",
"Subsequently, it turned out that also many symmetry classes of ASMs can be enumerated by a simple product formula; a majority of the cases were dealt with in [10].",
"A standard tool to prove these results are determinantal expressions for the partition function of the six vertex model.",
"A beautiful account on the history of ASMs is provided by Bressoud [2].",
"Since an ASM has precisely one 1 in its first row, it is natural to ask for the number of ASMs where this 1 is in a prescribed column.",
"Indeed, it turned out that also this refined enumeration leads to a simple product formula [15].",
"Hence, it is also interesting to explore refined enumerations of symmetry classes of ASMs.",
"The task of this paper is to present our attempt to prove the first author's conjecture [5] on a refined enumeration of vertically symmetric alternating sign matrices.",
"While we are not yet able to complete our proof, we are able to show how it naturally leads to a conjecture on a much more general multivariate Laurent polynomial identity.",
"Moreover, we present some partial results concerning this conjecture and additional motivation why it is interesting to study the conjecture.",
"A Vertically Symmetric Alternating Sign Matrix (VSASM) is an ASM which is invariant under reflection with respect to the vertical symmetry axis.",
"For instance, $\\begin{pmatrix}0 & 0 & 1 & 0 & 0 \\\\1 & 0 & -1 & 0 & 1 \\\\0 & 0 & 1 & 0 & 0 \\\\0 & 1 & -1 & 1 & 0 \\\\0 & 0 & 1 & 0 & 0 \\\\\\end{pmatrix}$ is a VSASM.",
"Since the first row of an ASM contains a unique 1, it follows that VSASMs can only exist for odd dimensions.",
"Moreover, the alternating sign condition and symmetry imply that no 0 can occur in the middle column.",
"Thus, the middle column of a VSASM has to be $(1,-1,1,\\ldots ,-1,1)^T$ .",
"The fact that the unique 1 of the first row is always in the middle column implies that the refined enumeration with respect to the first row is trivial.",
"However, it follows that the second row contains precisely two 1s and one $-1$ .",
"Therefore, a possible refined enumeration of VSASMs is with respect to the unique 1 in the second row that is situated left of the middle column.",
"Let $B_{n,i}$ denote the number of $(2n+1)\\times (2n+1)$ -VSASMs where the first 1 in the second row is in column $i$ .",
"In [5], the first author conjectured that $B_{n,i}= \\frac{\\binom{2n+i-2}{2n-1} \\binom{4n-i-1}{2n-1}}{\\binom{4n-2}{2n-1}}\\prod _{j=1}^{n-1} \\frac{(3j-1)(2j-1)!(6j-3)!}{(4j-2)!(4j-1)!",
"}, \\quad i=1,\\ldots ,n.$ Let us remark that another possible refined enumeration is the one with respect to the first column's unique 1.",
"Let $B_{n,i}^*$ denote the number of VSASMs of size $2n+1$ where the first column's unique 1 is located in row $i$ .",
"In [13], A. Razumov and Y. Stroganov showed that $B_{n,i}^* = \\prod _{j=1}^{n-1} \\frac{(3j-1)(2j-1)!(6j-3)!}{(4j-2)!(4j-1)!}",
"\\sum _{r=1}^{i-1}(-1)^{i+r-1}\\frac{\\binom{2n+r-2}{2n-1} \\binom{4n-r-1}{2n-1}}{\\binom{4n-2}{2n-1}}, \\quad i=1,\\ldots ,2n+1.$ Interestingly, the conjectured formula (REF ) would also imply a particularly simple linear relation between the two refined enumerations, namely $B_{n,i} = B_{n,i}^* + B_{n,i+1}^*, \\quad i=1,\\ldots ,n.$ To give a bijective proof of this relation is an open problem.",
"Such a proof would also imply (REF ).",
"Our approach is similar to the one used in the proof of the Refined Alternating Sign Matrix Theorem provided by the first author in [4].",
"We summarize some relevant facts from there: Let $A_{n,i}$ denote the number of $n \\times n$ ASMs where the unique 1 in the first row is in column $i$ .",
"It was shown that $(A_{n,i})_{1 \\le i \\le n}$ is a solution of the following linear equation system (LES): $\\begin{aligned}A_{n,i} & = \\sum _{j=i}^{n} \\binom{2n-i-1}{j-i} (-1)^{j+n} A_{n,j}, &\\quad i=1,\\ldots ,n, \\\\A_{n,i} &= A_{n,n+1-i}, &\\quad i=1,\\ldots ,n.\\end{aligned}$ Moreover it was proven that the solution space of this system is one-dimensional.",
"The LES together with the recursion $A_{n,1} = \\sum _{i=1}^{n-1} A_{n-1,i}$ enabled the first author to prove the formula for $A_{n,i}$ by induction with respect to $n$ .",
"The research presented in this paper started after observing that the numbers $B_{n,i}$ seem to be a solution of a similar LES: $\\begin{aligned}B_{n,n-i} & = \\sum _{j=i}^{n-1} \\binom{3n-i-2}{j-i} (-1)^{j+n+1} B_{n,n-j}, &\\quad i=-n,-n+1,\\ldots ,n-1, \\\\B_{n,n-i} & = B_{n,n+i+1}, &\\quad i=-n,-n+1,\\ldots ,n-1.\\end{aligned}$ Here we have to be a bit more precise: $B_{n,i}$ is not yet defined if $i=n+1,n+2,\\ldots ,2n$ .",
"However, if we use for the moment (REF ) to define $B_{n,i}$ for all $i \\in \\mathbb {Z}$ , basic hypergeometric manipulations (in fact, only the Chu-Vandermonde summation is involved) imply that $(B_{n,i})_{1 \\le i \\le 2n}$ is a solution of (REF ); in Proposition REF we show that the solution space of this LES is also one-dimensional.",
"Coming back to the combinatorial definition of $B_{n,i}$ , the goal of this paper is to show how to naturally extend the combinatorial interpretation of $B_{n,i}$ to $i=n+1,\\ldots ,2n$ and to present a conjecture of a completely different flavor which, once it is proven, implies that the numbers are a solution of the LES.",
"The identity analogous to (REF ) is $B_{n,1} = \\sum _{i=1}^{n-1} B_{n-1,i}.$ The Chu-Vandermonde summation implies that also the numbers on the right-hand side of (REF ) fulfill this identity, and, once the conjecture presented next is proven, (REF ) also follows by induction with respect to $n$ .",
"In order to be able to formulate the conjecture, we recall that the unnormalized symmetrizer $\\operatorname{\\mathbf {Sym}}$ is defined as $\\operatorname{\\mathbf {Sym}}p(x_1,\\ldots ,x_n) := \\sum \\limits _{\\sigma \\in {\\mathcal {S}}_{n}} p(x_{\\sigma (1)},\\ldots ,x_{\\sigma (n)})$ .",
"Conjecture 1.1 For integers $s,t \\ge 0$ , consider the following rational function in $z_1,\\ldots ,z_{s+t-1}$ $P_{s,t}(z_1,\\ldots ,z_{s+t-1}):=\\prod _{i=1}^{s} z_i^{2s-2i-t+1} (1-z_i^{-1})^{i-1}\\prod _{i=s+1}^{s+t-1} z_i^{2i-2s-t} (1-z_i^{-1})^s \\prod _{1 \\le p < q \\le s+t-1}\\frac{1 - z_p + z_p z_q}{z_q - z_p}$ and let $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) := \\operatorname{\\mathbf {Sym}}P_{s,t}(z_1,\\ldots ,z_{s+t-1})$ .",
"If $s \\le t$ , then $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) = R_{s,t}(z_1,\\ldots ,z_{i-1},z_i^{-1},z_{i+1},\\ldots ,z_{s+t-1})$ for all $i \\in \\lbrace 1,2,\\ldots ,s+t-1\\rbrace $ .",
"Note that in fact the following more general statement seems to be true: if $s \\le t$ , then $ R_{s,t}(z_1,\\ldots ,z_{s+t-1})$ is a linear combination of expressions of the form $\\prod \\limits _{j=1}^{s+t-1} [(z_j -1)(1-z_j^{-1})]^{i_j}$ , $i_j \\ge 0$ , where the coefficients are non-negative integers.",
"Moreover, it should be mentioned that it is easy to see that $R_{s,t}(z_1,\\ldots ,z_{s+t-1})$ is in fact a Laurent polynomial: Observe that $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) =\\frac{\\operatorname{\\mathbf {ASym}}\\left(P_{s,t}(z_1,\\ldots ,z_{s+t-1}) \\prod \\limits _{1 \\le i < j \\le s+t-1} (z_j-z_i) \\right)}{\\prod \\limits _{1 \\le i < j \\le s+t-1} (z_j - z_i)}$ with the unnormalized antisymmetrizer $\\operatorname{\\mathbf {ASym}}p(x_1,\\ldots ,x_n) := \\sum \\limits _{\\sigma \\in {\\mathcal {S}}_{n}} \\operatorname{sgn}\\sigma \\,p(x_{\\sigma (1)},\\ldots ,x_{\\sigma (n)})$ .",
"The assertion follows since $P_{s,t}(z_1,\\ldots ,z_{s+t-1}) \\prod \\limits _{1 \\le i < j \\le s+t-1} (z_j-z_i)$ is a Laurent polynomial and every antisymmetric Laurent polynomial is divisible by $\\prod \\limits _{1 \\le i < j \\le s+t-1} (z_j - z_i)$ .",
"We will prove the following two theorems.",
"Theorem 1.2 Let $R_{s,t}(z_1,\\ldots ,z_{s+t-1})$ be as in Conjecture REF .",
"If $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) = R_{s,t}(z_1^{-1},\\ldots ,z_{s+t-1}^{-1})$ for all $1 \\le s \\le t$ , then (REF ) is fulfilled.",
"Theorem 1.3 Let $R_{s,t}(z_1,\\ldots ,z_{s+t-1})$ be as in Conjecture REF .",
"Suppose $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) = R_{s,t}(z_1^{-1},\\ldots ,z_{s+t-1}^{-1})$ if $t=s$ and $t=s+1$ , $s \\ge 1$ .",
"Then (REF ) holds for all $s,t$ with $1 \\le s \\le t$ .",
"While we believe that (REF ) should probably be attacked with the six vertex model approach (although we have not tried), we also think that the more general Conjecture REF is interesting in its own right, given the fact that it only involves very elementary mathematical objects such as rational functions and the symmetric group.",
"The paper is organized as follows.",
"We start by showing that the solution space of (REF ) is one-dimensional.",
"Then we provide a first expression for $B_{n,i}$ and present linear equation systems that generalize the system in the first line of (REF ) and the system in the first line of (REF ) when restricting to non-negative $i$ in the latter.",
"Next we use the expression for $B_{n,i}$ to extend the combinatorial interpretation to $i=n+1,n+2,\\ldots ,2n$ and also extend the linear equation system to negative integers $i$ accordingly.",
"In Section , we justify the choice of certain constants that are involved in this extension.",
"Afterwards we present a first conjecture implying (REF ).",
"Finally, we are able to prove Theorem REF .",
"The proof of Theorem REF is given in Section .",
"It is independent of the rest of the paper and, at least for our taste, quite elegant.",
"We would love to see a proof of Conjecture REF which is possibly along these lines.",
"We conclude with some remarks concerning the special $s=0$ in Conjecture REF , also providing additional motivation why it is of interest to study these symmetrized functions."
],
[
"The solution space of (",
"The goal of this section is the proof of the proposition below.",
"Let us remark that we use the following extension of the binomial coefficient in this paper $\\binom{x}{j}:={\\left\\lbrace \\begin{array}{ll}\\frac{x(x-1)\\cdots (x-j+1)}{j!}",
"& \\text{if } j \\ge 0, \\\\0 & \\text{if } j < 0,\\end{array}\\right.",
"}$ where $x \\in \\mathbb {C}$ and $j \\in \\mathbb {Z}$ .",
"Proposition 2.1 For fixed $n \\ge 1$ , the solution space of the following LES $Y_{n,i} &= \\sum _{j=i}^{n-1} \\binom{3n-i-2}{j-i} (-1)^{j+n+1} Y_{n,j}, &i=-n,-n+1,\\ldots ,n-1, \\\\Y_{n,i} &= Y_{n,-i-1}, &i=-n,-n+1\\ldots ,n-1, \\\\$ in the variables $(Y_{n,i})_{-n \\le i \\le n-1}$ is one-dimensional.",
"As mentioned before, the numbers on the right-hand side of (REF ) are defined for all $i \\in \\mathbb {Z}$ and establish a solution after replacing $i$ by $n-i$ .",
"This implies that the solution space is at least one-dimensional.",
"Since $Y_{n,i} = \\sum _{j=-n}^{n-1} \\binom{3n-i-2}{j-i} (-1)^{j+n+1} Y_{n,-j-1} = \\sum _{j=-n}^{n-1} \\binom{3n-i-2}{-j-i-1} (-1)^{j+n} Y_{n,j}$ it suffices to show that the 1-eigenspace of $\\left( \\binom{3n-i-2}{-j-i-1}(-1)^{j+n} \\right)_{-n \\le i,j \\le n-1}$ is 1-dimensional.",
"So, we have to show that $\\operatorname{rk}\\left( \\binom{4n-i-1}{2n-i-j+1}(-1)^{j+1} -\\delta _{i,j} \\right)_{1 \\le i,j \\le 2n} = 2n-1.$ After removing the first row and column and multiplying each row with $-1$ , we are done as soon as we show that $\\det \\left( \\binom{4n-i-1}{2n-i-j+1}(-1)^{j}+\\delta _{i,j} \\right)_{2 \\le i,j \\le 2n} \\ne 0.$ If $n=1$ , this can be checked directly.",
"Otherwise, it was shown in [4] that $\\det \\left( \\binom{2m-i-1}{m-i-j+1}(-1)^{j}+\\delta _{i,j} \\right)_{2 \\le i,j \\le m} = \\det \\left( \\binom{i+j}{j-1}+\\delta _{i,j} \\right)_{1 \\le i,j \\le m-2}$ when $m \\ge 3$ , whereby the last determinant counts descending plane partitions with no part greater than $m-1$ , see [1].",
"However, this number is given by (REF ) if we set $n=m-1$ there."
],
[
"Monotone triangles and an expression for $B_{n,i}$",
"A Monotone Triangle (MT) of size $n$ is a triangular array of integers $(a_{i,j})_{1 \\le j \\le i \\le n}$ , often arranged as follows $\\begin{array}{ccccccc}&&& a_{1,1} \\\\&& a_{2,1} && a_{2,2} \\\\& \\rotatebox {75}{\\ddots } &&&& \\ddots \\\\a_{n,1} &&\\cdots & &\\cdots && a_{n,n}\\end{array},$ with strict increase along rows, i.e.",
"$a_{i,j} < a_{i,j+1}$ , and weak increase along North-East- and South-East-diagonals, i.e.",
"$a_{i+1,j} \\le a_{i,j} \\le a_{i+1,j+1}$ .",
"It is well-known [11] that MTs with $n$ rows and bottom row $(1,2\\ldots ,n)$ are in one-to-one correspondence with ASMs of size $n$ : the $i$ -th row of the MT contains an entry $j$ if the first $i$ rows of the $j$ -th column in the corresponding ASM sum up to 1.",
"In order to see that $(2n+1) \\times (2n+1)$ VSASMs correspond to MTs with bottom row $(2,4,\\ldots ,2n)$ , rotate the VSASM by 90 degrees.",
"The $(n+1)$ -st row of the rotated VSASM is $(1,-1,1,\\ldots ,-1,1)$ .",
"From the definition of ASMs, it follows that the vector of partial column sums of the first $n$ rows is $(0,1,0,\\ldots ,1,0)$ in this case, i.e.",
"the $n$ -th row of the corresponding MT is $(2,4,\\ldots ,2n)$ .",
"Since the rotated VSASM is uniquely determined by its first $n$ rows, this establishes a one-to-one correspondence between VSASMs of size $2n+1$ and MTs with bottom row $(2,4,\\ldots ,2n)$ .",
"An example of the upper part of a rotated VSASM and its corresponding MT is depicted in Figure REF .",
"The refined enumeration of VSASMs directly translates into a refined enumeration of MTs with bottom row $(2,4,\\ldots ,2n)$ : from the correspondence it follows that $B_{n,i}$ counts MTs with bottom row $(2,4,\\ldots ,2n)$ and exactly $n+1-i$ entries equal to 2 in the left-most North-East-diagonal (see Figure REF ).",
"Figure: Upper part of a rotated VSASM and its corresponding MonotoneTriangle.The problem of counting MTs with fixed bottom row $(k_1,\\ldots ,k_n)$ was considered in [3].",
"For each $n\\ge 1$ , an explicit polynomial $\\alpha (n;k_1,\\ldots ,k_n)$ of degree $n-1$ in each of the $n$ variables $k_1,\\ldots ,k_n$ was provided such that the evaluation at strictly increasing integers $k_1 < k_2 < \\cdots < k_n$ is equal to the number of MTs with fixed bottom row $(k_1,\\ldots ,k_n)$ – for instance $\\alpha (3;1,2,3)=7$ .",
"In [7], it was described how to use the polynomial $\\alpha (n;k_1,\\ldots ,k_n)$ to compute the number of MTs with given bottom row and a certain number of fixed entries in the left-most NE-diagonal: Let $E_x f(x) &:= f(x+1), \\\\\\Delta _x f(x) &:= (E_x - \\operatorname{id}) f(x) = f(x+1)-f(x), \\\\\\delta _x f(x) &:= (\\operatorname{id}- E_x^{-1}) f(x) = f(x)-f(x-1)$ denote the shift operator and the difference operators.",
"Suppose $k_1 \\le k_2 < \\cdots < k_n$ and $i \\ge 0$ , then $(-1)^i \\Delta _{k_1}^i \\alpha (n;k_1,\\ldots ,k_n)$ is the number of MTs with bottom row $(k_1-1,k_2,\\ldots ,k_n)$ where precisely $i+1$ entries in the left-most NE-diagonal are equal to $k_1-1$ (see Figure REF ).",
"There exists an analogous result for the right-most SE-diagonal: if $k_1 < \\cdots < k_{n-1} \\le k_n$ , then $\\delta _{k_n}^i \\alpha (n;k_1,\\ldots ,k_n)$ is the number of MTs where precisely $i+1$ entries in the right-most SE-diagonal are equal to $k_n+1$ (see Figure REF ).",
"Figure: δ k n i α(n;k 1 ,...,k n )\\delta _{k_n}^i \\alpha (n;k_1,\\ldots ,k_n)This implies the following formula $B_{n,n-i}=(-1)^{i} \\Delta _{k_1}^{i} \\alpha (n; k_1,4,6,\\ldots ,2n)|_{k_1=3}.$ Let us generalize this by defining $C_{n,i}^{(d)}:=(-1)^{i}\\Delta _{k_1}^{i}\\alpha (n;k_1,2d,3d,\\ldots ,nd)|_{k_1=d+1}, \\quad d \\in \\mathbb {Z},\\; i \\ge 0,$ which is for $d \\ge 1$ the number of MTs with bottom row $(d,2d,3d,\\ldots ,nd)$ and exactly $i+1$ entries equal to $d$ in the left-most NE-diagonal.",
"If $d=2$ , we obtain $B_{n,n-i}$ , and it is also not hard to see that we obtain the ordinary refined enumeration numbers $A_{n,i+1}$ if $d=1$ .",
"Next we prove that the numbers $C_{n,i}^{(d)}$ fulfill a certain LES.",
"For $d=1$ , this proves the first line of (REF ), while for $d=2$ it proves the first line of (REF ) for non-negative $i$ .",
"Proposition 3.1 For fixed $n,d \\ge 1$ the numbers $(C_{n,i}^{(d)})_{0 \\le i \\le n-1}$ satisfy the following LES $C_{n,i}^{(d)}=\\sum _{j=i}^{n-1} \\binom{n(d+1)-i-2}{j-i} (-1)^{j+n+1} C_{n,j}^{(d)}, \\quad i=0,\\ldots ,n-1.$ The main ingredients of the proof are the identities $\\alpha (n;k_1,k_2,\\ldots ,k_n) &= (-1)^{n-1} \\alpha (n; k_2,k_3,\\ldots ,k_n,k_1-n),\\\\\\alpha (n;k_1,k_2,\\ldots ,k_n) &= \\alpha (n;k_1+c,k_2+c,\\ldots ,k_n+c), \\quad c \\in \\mathbb {Z}.$ A proof of the first identity was given in [4].",
"The second identity is obvious by combinatorial arguments if $k_1 < k_2 < \\cdots < k_n$ and is therefore also true as identity satisfied by the polynomial.",
"Together with $\\Delta _x =E_x \\delta _x$ , $E^{-1}_x=(\\operatorname{id}- \\delta _x)$ and the Binomial Theorem we obtain $C_{n,i}^{(d)}&= (-1)^{i}\\Delta _{k_1}^{i}\\alpha (n;k_1,2d,3d,\\ldots ,nd)|_{k_1=d+1} \\\\&= (-1)^{i+n+1} \\Delta _{k_1}^{i} \\alpha (n;2d,3d,\\ldots ,nd,k_1-n)|_{k_1=d+1} \\\\&= (-1)^{i+n+1} E_{k_1}^{-n-nd+i+2} \\delta _{k_1}^{i}\\alpha (n;2d,3d,\\ldots ,nd,k_1+d)|_{k_1=nd-1} \\\\&= (-1)^{i+n+1} (\\operatorname{id}-\\delta _{k_1})^{n(d+1)-i-2} \\delta _{k_1}^{i}\\alpha (n;d,2d,\\ldots ,(n-1)d,k_1)|_{k_1=nd-1} \\\\&= \\sum _{j \\ge 0} \\binom{n(d+1)-i-2}{j}(-1)^{i+j+n+1} \\delta _{k_1}^{i+j}\\alpha (n;d,2d,\\ldots ,(n-1)d,k_1)|_{k_1=nd-1} \\\\&= \\sum _{j \\ge i} \\binom{n(d+1)-i-2}{j-i}(-1)^{j+n+1} \\delta _{k_1}^{j}\\alpha (n;d,2d,\\ldots ,(n-1)d,k_1)|_{k_1=nd-1}.",
"\\\\$ Since applying the $\\delta $ -operator to a polynomial decreases its degree, and $\\alpha (n;k_1,\\ldots ,k_n)$ is a polynomial of degree $n-1$ in each $k_i$ , it follows that the summands of the last sum are zero whenever $j\\ge n$ .",
"So, it remains to show that $C_{n,j}^{(d)} = \\delta _{k_1}^{j} \\alpha (n;d,2d,\\ldots ,(n-1)d,k_1)|_{k_1=nd-1}.$ From the discussion preceding the proposition we know that the right-hand side of (REF ) is the number of MTs with bottom row $(d,2d,\\ldots ,nd)$ and exactly $j+1$ entries equal to $nd$ in the right-most SE-diagonal.",
"Replacing each entry $x$ of the MT by $(n+1)d-x$ and reflecting it along the vertical symmetry axis gives a one-to-one correspondence with the objects counted by $C_{n,j}^{(d)}$ ."
],
[
"The numbers $C_{n,i}^{(d)}$ for {{formula:9595c5c5-e830-47eb-831c-679d7db1a569}}",
"In order to prove (REF ), it remains to extend the definition of $C_{n,i}^{(2)}$ to $i=-n,\\ldots ,-1$ in such a way that both the symmetry $C_{n,i}^{(2)} = C_{n,-i-1}^{(2)}$ and the first line of (REF ) is satisfied for negative $i$ .",
"Note that the definition of $C_{n,i}^{(2)}$ contains the operator $\\Delta _{k_1}^i$ which is per se only defined for $i \\ge 0$ .",
"The difference operator is (in discrete analogy to differentiation) only invertible up to an additive constant.",
"This motivates the following definitions of right inverse difference operators: Given a polynomial $p: \\mathbb {Z} \\rightarrow \\mathbb {C}$ , we define the right inverse difference operators as ${^z \\Delta ^{-1}_x} p(x) := - \\sum _{x^{\\prime }=x}^{z} p(x^{\\prime }) \\qquad \\text{ and } \\qquad {^z \\delta ^{-1}_x} p(x) := \\sum _{x^{\\prime }=z}^{x} p(x^{\\prime })$ where $x, z \\in \\mathbb {Z}$ and the following extended definition of summation $\\sum _{i=a}^b f(i) := {\\left\\lbrace \\begin{array}{ll} 0, & \\quad b=a-1, \\\\ -\\sum \\limits _{i =b+1}^{a-1} f(i), & \\quad b+1 \\le a-1, \\end{array}\\right.",
"}$ is used.",
"The motivation for the extended definition is that it preserves polynomiality: suppose $p(i)$ is a polynomial in $i$ then $(a,b) \\mapsto \\sum \\limits _{i=a}^{b} p(i)$ is a polynomial function on $\\mathbb {Z}^2$ .",
"The following identities can be easily checked.",
"Proposition 4.1 Let $z \\in \\mathbb {Z}$ and $p: \\mathbb {Z} \\rightarrow \\mathbb {C}$ a function.",
"Then $\\Delta _x \\, {^z \\Delta ^{-1}_x} = \\operatorname{id}$ and ${^z \\Delta ^{-1}_x} \\Delta _x p(x) = p(x) - p(z+1)$ , $\\delta _x \\, {^z \\delta ^{-1}_x} = \\operatorname{id}$ and ${^z \\delta ^{-1}_x} \\delta _x p(x) = p(x) - p(z-1)$ , $\\Delta _x = E_x \\delta _x$ and ${^z \\Delta ^{-1}_x} = E_x^{-1} E_z \\, {^z \\delta ^{-1}_x}$ , $\\Delta _y \\, {^z \\Delta ^{-1}_x} = {^z \\Delta ^{-1}_x} \\Delta _y$ and $\\delta _y \\, {^z \\Delta ^{-1}_x} = {^z \\Delta ^{-1}_x} \\delta _y$ for $y \\ne x,z$ .",
"Now we are in the position to define higher negative powers of the difference operators: For $i < 0$ and $\\mathbf {z}=(z_i,z_{i+1},\\ldots ,z_{-1}) \\in \\mathbb {Z}^{-i}$ we let ${^\\mathbf {z} \\Delta ^{i}_x} &:= {^{z_i} \\Delta ^{-1}_x} \\, {^{z_{i+1}} \\Delta ^{-1}_x} \\dots {^{z_{-1}} \\Delta ^{-1}_x}, \\\\{^\\mathbf {z} \\delta ^{i}_x} &:= {^{z_i} \\delta ^{-1}_x} \\, {^{z_{i+1}} \\delta ^{-1}_x} \\dots {^{z_{-1}} \\delta ^{-1}_x}.$ After observing that ${^z} \\delta ^{-1}_x E^{-1}_x = E^{-1}_x E^{-1}_z {^z} \\delta ^{-1}_x$ we can deduce the following generalization of Proposition REF (REF ) inductively: ${^\\mathbf {z} \\Delta ^{i}_x} =E_x^{i} E_{z_i}^{i+2} E_{z_{i+1}}^{i+3} \\dots E_{z_{-1}}^{1}{^\\mathbf {z} \\delta ^{i}_x}.$ The right inverse difference operator allows us to naturally extend the definition of $C_{n,i}^{(d)}$ : First, let us fix a sequence of integers $\\mathbf {x}=(x_j)_{j < 0}$ and set $\\mathbf {x}_i=(x_i,x_{i+1},\\ldots ,x_{-1})$ for $i<0$ .",
"We define $C_{n,i}^{(d)} :={\\left\\lbrace \\begin{array}{ll}\\left.",
"(-1)^{i}\\Delta _{k_1}^{i} \\alpha (n;k_1,2d,3d,\\ldots ,nd)\\right|_{k_1=d+1}, & i=0,\\ldots ,n-1, \\\\\\left.",
"(-1)^{i} \\,\\, {^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,2d,3d,\\ldots ,nd)\\right|_{k_1=d+1}, & i=-n,\\ldots ,-1.\\end{array}\\right.",
"}$ We detail on the choice of $\\mathbf {x}$ in Section .",
"If $d \\ge 1$ , it is possible to give a rather natural combinatorial interpretation of $C_{n,i}^{(d)}$ also for negative $i$ which is based on previous work of the authors.",
"It is of no importance for the rest of the paper, however, it provides a nice intuition: One can show that for non-negative $i$ , the quantity $C_{n,i}^{(d)}$ counts partial MT where we cut off the bottom $i$ elements of the left-most NE-diagonal, prescribe the entry $d+1$ in position $i+1$ of the NE-diagonal and the entries $2d, 3d, \\ldots , n d$ in the bottom row of the remaining array (see Figure REF ); in fact, in the exceptional case of $d=1$ we do not require that the bottom element 2 of the truncated left-most NE-diagonal is strictly smaller than its right neighbor.",
"From (REF ) it follows that applying the inverse difference operator has the opposite effect of prolonging the left-most NE-diagonal: if $i < 0$ , the quantity $C_{n,i}^{(d)}$ is the signed enumeration of arrays of the shape as depicted in Figure REF subject to the following conditions: For the elements in the prolonged NE-diagonal including the entry left of the entry $2d$ , we require the following: Suppose $e$ is such an element and $l$ is its SW-neighbor and $r$ its SE-neighbor: if $l \\le r$ , then $l \\le e \\le r$ ; otherwise $r < e < l$ .",
"In the latter case, the element contributes a $-1$ sign.",
"Inside the triangle, we follow the rules of Generalized Monotone Triangles as presented in [12].",
"The total sign is the product of the sign of the Generalized Monotone Triangle and the signs of the elements in the prolonged NE-diagonal.",
"Figure: C n,i (d) C_{n,i}^{(d)} for i<0i < 0."
],
[
"Extending the LES to negative $i$",
"The purpose of this section is the extension of the LES in Proposition REF to negative $i$ .",
"This is accomplished with the help of the following lemma which shows that certain identities for $\\Delta _{k_1}^i \\alpha (n;k_1,\\ldots ,k_n)$ , $i \\ge 0$ , carry over into the world of inverse difference operators.",
"Lemma 5.1 Let $n,d\\ge 1$ .",
"Suppose $i \\ge 0$ .",
"Then $\\left.",
"(-1)^{i} \\,\\, {\\Delta _{k_1}^{i}} \\alpha (n;k_1,2d,3d,\\ldots ,n d) \\right|_{k_1=d+1} \\\\= \\left.",
"{\\delta _{k_n}^{i}} \\alpha (n;d,2d,\\ldots ,(n-1) d,k_n) \\right|_{k_n=nd-1}.$ Suppose $i < 0$ , and let $\\mathbf {x}_i=(x_i,\\ldots ,x_{-1})$ and $\\mathbf {y}_i=(y_i,\\ldots ,y_{-1})$ satisfy the relation $y_j = (n+1)d - x_j$ for all $j$ .",
"Then (see Figure REF ) $\\left.",
"(-1)^{i} \\,\\, {^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,2d,3d,\\ldots ,n d) \\right|_{k_1=d+1} \\\\= \\left.",
"{^{\\mathbf {y}_i} \\delta _{k_n}^{i}} \\alpha (n;d,2d,\\ldots ,(n-1) d,k_n) \\right|_{k_n=nd-1}.$ Suppose $i \\ge 0$ .",
"Then $\\Delta ^i_{k_1} \\alpha (n;k_1,\\ldots ,k_n) = (-1)^{n-1} E^{i-n}_{k_1} \\delta ^i_{k_1} \\alpha (n;k_2,\\ldots ,k_n,k_1).$ Suppose $i < 0$ , and let $\\mathbf {x}_i=(x_i,\\ldots ,x_{-1})$ and $\\mathbf {y}_i=(y_i,\\ldots ,y_{-1})$ satisfy the relation $y_j = x_j+j-n+2$ for all $j$ .",
"Then ${{^{\\mathbf {x}_i}}\\Delta ^i_{k_1}} \\alpha (n;k_1,\\ldots ,k_n) = (-1)^{n-1} E^{i-n}_{k_1} \\,\\, {^{\\mathbf {y}_i} \\delta ^i_{k_1}} \\alpha (n;k_2,\\ldots ,k_n,k_1).$ Figure: Symmetry of inverse difference operators if y j =(n+1)d-x j y_j = (n+1)d - x_j.For the first part we refer to (REF ).",
"Concerning the second part, we actually show the following more general statement: if $r=(n+1)d-l$ and $i \\le 0$ , then $\\left.",
"(-1)^{i} \\,\\, {^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,2d,3d,\\ldots ,n d) \\right|_{k_1=l} \\\\= \\left.",
"{^{\\mathbf {y}_i} \\delta _{k_n}^{i}} \\alpha (n;d,2d,\\ldots ,(n-1) d,k_n) \\right|_{k_n=r}.$ We use induction with respect to $i$ ; the case $i=0$ is covered by the first part (${^{\\mathbf {x}_0} \\Delta _{k_1}^{0}}=\\operatorname{id}={^{\\mathbf {y}_0} \\delta _{k_n}^{0}}$ ).",
"If $i<0$ , then, by the definitions of the right inverse operators and the induction hypothesis, we have $\\left.",
"(-1)^{i} \\,\\, {^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,2d,3d,\\ldots ,n d) \\right|_{k_1=l}&= \\sum _{k^{\\prime }_1=l}^{x_i} (-1)^{i+1} \\,\\, {^{\\mathbf {x}_{i+1}} \\Delta _{k^{\\prime }_1}^{i+1}} \\alpha (n;k^{\\prime }_1,2d,3d,\\ldots ,n d) \\\\&= \\sum _{k^{\\prime }_1=l}^{x_i} \\left.",
"{^{\\mathbf {y}_{i+1}} \\delta _{k^{\\prime }_n}^{i+1}} \\alpha (n;d,2d,\\ldots ,(n-1) d,k^{\\prime }_n) \\right|_{k^{\\prime }_n=(n+1)d-k^{\\prime }_1} \\\\&= \\sum _{k^{\\prime }_n=(n+1)d-x_i}^{(n+1)d-l} {^{\\mathbf {y}_{i+1}} \\delta _{k^{\\prime }_n}^{i+1}} \\alpha (n;d,2d,\\ldots ,(n-1) d,k^{\\prime }_n).$ The last expression is equal to the right-hand side of the claimed identity.",
"The third part follows from (REF ) and Proposition REF (REF ).",
"The last part is shown by induction with respect to $i$ ; in fact $i=0$ can be chosen to be the initial case of the induction.",
"If $i<0$ , then the induction hypothesis and (REF ) imply ${{^{\\mathbf {x}_i}}\\Delta ^i_{k_1}} \\alpha (n;k_1,\\ldots ,k_n) &= - \\sum _{l_1=k_1}^{x_i}{{^{\\mathbf {x}_{i+1}}}\\Delta ^{i+1}_{l_1}} \\alpha (n;l_1,k_2,\\ldots ,k_n) \\\\& = - \\sum _{l_1=k_1}^{x_i} (-1)^{n-1} E^{i+1-n}_{l_1} \\,\\, {^{\\mathbf {y}_{i+1}}\\delta ^{i+1}_{l_1}} \\alpha (n;k_2,\\ldots ,k_n,l_1) \\\\& = \\sum _{l_1=x_i+i-n+2}^{k_1+i-n} (-1)^{n-1} \\, {^{\\mathbf {y}_{i+1}}\\delta ^{i+1}_{l_1}} \\alpha (n;k_2,\\ldots ,k_n,l_1).$ The last expression is obviously equal to the right-hand side of the identity in the lemma.",
"Now we are in the position to generalize Proposition REF .",
"Proposition 5.2 Let $n,d \\ge 1$ .",
"For $i<0$ , let $\\mathbf {x}_i, \\mathbf {z}_i \\in \\mathbb {Z}^{-i}$ with $z_j =(n+2)(d+1)- x_j-j-4$ and define $D^{(d)}_{n,i} :={\\left\\lbrace \\begin{array}{ll}\\left.",
"(-1)^{i}\\Delta _{k_1}^{i} \\alpha (n;k_1,2d,3d,\\ldots ,nd)\\right|_{k_1=d+1}, & i=0,\\ldots ,n-1, \\\\\\left.",
"(-1)^{i} \\,\\, {^{\\mathbf {z}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,2d,3d,\\ldots ,nd)\\right|_{k_1=d+1}, & i=-n,\\ldots ,-1.\\end{array}\\right.",
"}$ Then $C_{n,i}^{(d)} = \\sum _{j=i}^{n-1}\\binom{n(d+1)-i-2}{j-i} (-1)^{j+n+1} D^{(d)}_{n,j} .$ holds for all $i=-n,\\ldots ,n-1$ .",
"To simplify notation let us define ${^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} := \\Delta _{k_1}^{i}$ for $i \\ge 0$ .",
"Since the definition of $C_{n,i}^{(d)}$ and $D_{n,i}^{(d)}$ only differ in the choice of constants, the fact that the system of linear equations is satisfied for $i=0,\\ldots ,n-1$ is Proposition REF .",
"For $i=-n,\\ldots ,-1$ first note that, by Lemma REF , () and $E_x^{-1}\\, {^{z}}\\delta ^{-1}_x = {^{z+1}} \\delta ^{-1}_x E_x^{-1}$ , we have $C^{(d)}_{n,i} = \\left.",
"(-1)^{n-1+i} E^{i-n}_{k_1} \\,\\,{^{\\mathbf {y}_i} \\delta ^i_{k_1}} \\alpha (n;d,2d,\\ldots ,(n-1) d, k_1) \\right|_{k_1=1}$ where $\\mathbf {y}_i= (y_i,\\ldots ,y_{-1})$ with $y_j = x_j+j+2-n-d$ .",
"This is furthermore equal to $\\left.",
"(-1)^{n-1+i} E^{i-n (d+1) +2}_{k_1} \\,\\,{^{\\mathbf {y}_i} \\delta ^i_{k_1}} \\alpha (n;d,2d,\\ldots ,(n-1) d, k_1) \\right|_{k_1=n d -1}.$ Now we use $E^{i-n(d+1)+2}_{k_1} = (\\operatorname{id}- \\delta _{k_1})^{n(d+1)-i-2} = \\sum _{j=0}^{n(d+1)-i-2} \\binom{n(d+1)-i-2}{j} (-1)^j \\delta ^{j}_{k_1}$ and Proposition REF (REF ) to obtain $\\sum _{j=0}^{n(d+1)-i-2} \\binom{n(d+1)-i-2}{j} (-1)^{n-1+i+j} \\,\\, \\left.",
"{^{\\mathbf {y}_{i+j}} \\delta ^{i+j}_{k_1}} \\alpha (n;d,2d,\\ldots ,(n-1) d, k_1) \\right|_{k_1=n d -1}.$ Since the (ordinary) difference operator applied to a polynomial decreases the degree, the upper summation limit can be changed to $n-1-i$ .",
"Together with Lemma REF this transforms into $\\sum _{j=i}^{n-1} & \\binom{n(d+1)-i-2}{j-i} (-1)^{n-1+j} \\,\\, \\left.",
"{^{\\mathbf {y}_{j}} \\delta ^{j}_{k_1}} \\alpha (n;d,2d,\\ldots ,(n-1) d, k_1) \\right|_{k_1=n d -1} \\\\&= \\sum _{j=i}^{n-1} \\binom{n(d+1)-i-2}{j-i} (-1)^{n-1} \\,\\, \\left.",
"{^{\\mathbf {z}_{j}} \\Delta ^{j}_{k_1}} \\alpha (n;k_1,2d,3d,\\ldots ,n d) \\right|_{k_1=d+1}.$ Now it remains to find an integer sequence $(x_j)_{j<0}$ such that $C_{n,i}^{(2)} = C_{n,-i-1}^{(2)}$ and $C_{n,i}^{(2)} = D_{n,i}^{(2)}$ for negative $i$ ."
],
[
"How to choose the sequence $\\mathbf {x}=(x_j)_{j < 0}$",
"In the section, it is shown that $C_{n,i}^{(2)} = C_{n,-i-1}^{(2)}$ if we choose $\\mathbf {x}=(x_j)_{j < 0}$ with $x_j=-2j+1$ , $j<0$ .",
"This can be deduced from the following more general result.",
"Proposition 6.1 Let $x_j=-2j+1$ , $j <0$ , and set $\\mathbf {x}_i = (x_i,x_{i+1},\\ldots ,x_{-1})$ for all $i<0$ .",
"Suppose $p:\\mathbb {Z} \\rightarrow \\mathbb {C}$ and let $ c_i :={\\left\\lbrace \\begin{array}{ll}\\left.",
"(-1)^{i} \\Delta _y^{i} p(y) \\right|_{y=3}, & i \\ge 0, \\\\\\left.",
"(-1)^{i} \\,\\, {^{\\mathbf {x}_i} \\Delta ^{i}_y} p(y) \\right|_{y=3}, & i < 0, \\\\\\end{array}\\right.",
"}$ for $i \\in \\mathbb {Z}$ .",
"Then the numbers satisfy the symmetry $c_i = c_{-i-1}$ .",
"We may assume $i \\ge 0$ .",
"Then $c_i = \\left.",
"(-1)^{i} (E_y - \\operatorname{id})^{i} p(y) \\right|_{y=3} = \\sum _{d_1=3}^{i+3} \\binom{i}{d_1-3} (-1)^{d_1+1} p(d_1),$ and $c_{-i-1} = (-1)^{i+1} \\,\\,{^{\\mathbf {x}_{-i-1}} \\Delta _{y}^{-i-1}} \\left.",
"p(y) \\right|_{y=3}= \\sum _{d_{i+1}=3}^{2i+3} \\sum _{d_{i} = d_{i+1}}^{2i+1} \\cdots \\sum _{d_2 = d_3}^5 \\sum _{d_1=d_2}^3 p(d_1).$ The situation is illustrated in Figure REF .",
"According to (REF ), the iterated sum is the signed summation of $(d_1,d_2,\\ldots ,d_{i+1}) \\in \\mathbb {Z}^{i+1}$ subject to the following restrictions: We have $3 \\le d_{i+1} \\le 2i+3$ , and for $1 \\le j \\le i$ the restrictions are $\\begin{aligned}d_{j+1} \\le d_j \\le 2j+1 \\qquad & \\text{if } d_{j+1} \\le 2j+1, \\\\d_{j+1} > d_j > 2j+1 \\qquad & \\text{if } d_{j+1} > 2j+1.\\end{aligned}$ Note that there is no admissible $(d_1,d_2,\\ldots ,d_{i+1})$ with $d_{j+1}=2j+2$ .",
"The sign of $(d_1,d_2,\\ldots ,d_{i+1})$ is computed as $(-1)^{\\# \\lbrace 1 \\le j \\le i :\\, d_{j} > 2j +1\\rbrace }$ .",
"Figure: Combinatorial interpretation of () if p(y)=α(n;y,4,6,...,2n)p(y)=\\alpha (n;y,4,6,\\ldots ,2n).The proof now proceeds by showing that the signed enumeration of $(d_1,\\ldots ,d_{i+1})$ with fixed $d_1$ is just $\\binom{i}{d_1-3} (-1)^{d_1+1}$ .",
"The reversed sequence $(d_{i+1},d_{i},\\ldots ,d_1)$ is weakly increasing as long as we are in the first case of (REF ).",
"However, once we switch from Case 1 to Case 2, the sequence is strictly decreasing afterwards, because $d_{j+1} > 2j+1$ implies $d_{j} > 2j+1 > 2j-1$ .",
"Thus, the sequence splits into two parts: there exists an $l$ , $0 \\le l \\le i$ , with $3 \\le d_{i+1} \\le d_i \\le \\ldots \\le d_{l+1} > d_l > \\ldots > d_1.$ Moreover, it is not hard to see that (REF ) implies $d_{l+1}=2l+3$ and $d_l=2l+2$ .",
"The sign of the sequence is $(-1)^l$ .",
"Thus it suffices to count the following two types of sequences.",
"$3 \\le d_{i+1} \\le d_{i} \\le \\cdots \\le d_{l+2} \\le d_{l+1} = 2l+3$ .",
"$d_{l} = 2l+2 > d_{l-1} > \\cdots > d_2 > d_1 > 3$ and $d_k > 2k+1$ for $1 < k \\le l-1$ ; $d_1$ fixed.",
"For the first type, this is accomplished by the binomial coefficient $\\binom{i+l}{i-l}$ .",
"If $l \\ge 1$ , then the sequences in (REF ) are prefixes of Dyck paths in disguise: to see this, consider prefixes of Dyck paths starting in $(0,0)$ with $a$ steps of type $(1,1)$ and $b$ steps of type $(1,-1)$ .",
"Such a partial Dyck path is uniquely determined by the $x$ -coordinates of its $(1,1)$ -steps.",
"If $p_i$ denotes the position of the $i$ -th $(1,1)$ -step, then the coordinates correspond to such a partial Dyck path if and only if $0 = p_1 < p_2 < \\cdots < p_a < a+b \\quad \\text{and} \\quad p_k < 2k-1.$ In order to obtain (REF ) set $a \\mapsto l-1$ , $b \\mapsto l+3-d_1$ and $p_k \\mapsto 2l+2-d_{l-k+1}$ .",
"By the reflection principle, the number of prefixes of Dyck paths is $\\binom{a+b}{b} \\frac{a+1-b}{a+1}= \\binom{2l+2-d_1}{l+3-d_1} \\frac{d_1-3}{l}.$ If $l=0$ , then $d_1=d_2=\\ldots =d_{i+1}=3$ and this is the only case where $d_1=3$ .",
"Put together, we see that the coefficient of $p(d_1)$ in (REF ) is $\\sum _{l=1}^{i} (-1)^l \\binom{i+l}{i-l} \\binom{2l+2-d_1}{l+3-d_1} \\frac{d_1-3}{l}$ if $d_1 \\ge 4$ .",
"Using standard tools to prove hypergeometric identities, it is not hard to see that this is equal to $\\binom{i}{d_1-3} (-1)^{d_1+1}$ if $d_1 \\ge 4$ and $i \\ge 0$ .",
"For instance, C. Krattenthaler's mathematica package HYP [8] can be applied as follows: After converting the sum into hypergeometric notation, one applies contiguous relation C16.",
"Next we use transformation rule T4306, before it is possible to apply summation rule S2101 which is the Chu-Vandermonde summation.",
"In the following, we let $\\mathbf {x}=(x_j)_{j < 0}$ with $x_j=-2j+1$ and $\\mathbf {z}=(z_j)_{j < 0}$ with $z_j =(n+2)(d+1)+j-5$ .",
"Recall that $\\mathbf {x}$ is crucial in the definition of $C_{n,i}^{(d)}$ , see (REF ), while $\\mathbf {z}$ appears in the definition of $D^{(d)}_{n,i} $ , see (REF ).",
"To complete the proof of (REF ), it remains to show $C^{(2)}_{n,i} = D^{(2)}_{n,i}$ for $i=-n,-n+1,\\ldots ,-1$ , since Proposition REF and Proposition REF then imply that the numbers $C^{(2)}_{n,i}$ , $i=-n,-n+1,\\ldots ,n-1$ , are a solution of the LES (REF ).",
"The situation is depicted in Figure REF .",
"Figure: Combinatorial interpretation of the open problem ().When trying to proceed as in the proof of Proposition REF one eventually ends up with having to show that the refined VSASM numbers $B_{n,i}$ satisfy a different system of linear equations: $\\sum _{j=0}^{n-1} \\left( \\binom{3n-i-2}{i+j+1}-\\binom{3n-i-2}{i-j} \\right) (-1)^j B_{n,n-j} = 0, \\quad i=0,1,\\ldots ,n-1.$ While computer experiments indicate that this LES uniquely determines $(B_{n,1},\\ldots ,B_{n,n})$ up to a multiplicative constant for all $n \\ge 1$ , it is not clear at all how to derive that the refined VSASM numbers satisfy (REF ).",
"We therefore try a different approach in tackling (REF ).",
"The task of the rest of the paper is to show that (REF ) follows from a more general multivariate Laurent polynomial identity and present partial results towards proving the latter."
],
[
"A first conjecture implying (",
"We start this section by showing that the application of the right inverse difference operator ${^{z} \\Delta }^{-1}_{k_1}$ to $\\alpha (n;k_1,\\ldots ,k_n)$ can be replaced by the application of a bunch of ordinary difference operators to $\\alpha (n+1;k_1,z, k_2, \\ldots ,k_{n})$ .",
"Some preparation that already appeared in [3] is needed: The definition of MTs implies (see Figure REF ) that the polynomials $\\alpha (n;k_1,\\ldots ,k_n)$ satisfy the recursion $\\alpha (n;k_1,\\ldots ,k_n)= \\sum _{\\begin{array}{c}(l_1,\\ldots ,l_{n-1})\\in \\mathbb {Z}^{n-1}, \\\\k_1 \\le l_1 \\le k_2 \\le l_2 \\le \\cdots \\le k_{n-1} \\le l_{n-1} \\le k_n,\\\\ l_i < l_{i+1}\\end{array}} \\alpha (n-1; l_1,\\ldots ,l_{n-1}),$ whenever $k_1 < k_2 < \\cdots < k_n$ , $k_i \\in \\mathbb {Z}$ .",
"Figure: Bottom and penultimate row of a Monotone Triangle.In fact, one can define a summation operator $\\sum \\limits _{(l_1,\\ldots ,l_{n-1})}^{(k_1,\\ldots ,k_n)}$ such that $\\alpha (n;k_1,\\ldots ,k_n) = \\sum _{(l_1,\\ldots ,l_{n-1})}^{(k_1,\\ldots ,k_n)} \\alpha (n-1;l_1,\\ldots ,l_{n-1})$ for all $(k_1,\\ldots ,k_n) \\in \\mathbb {Z}^n$ .",
"The postulation that the summation operator should extend (REF ) motivates the recursive definition $\\sum _{(l_1,\\ldots ,l_{n-1})}^{(k_1,\\ldots ,k_n)} A(l_1,\\ldots ,l_{n-1}) :=&\\sum _{(l_1,\\ldots ,l_{n-2})}^{(k_1,\\ldots ,k_{n-1})}\\sum _{l_{n-1} = k_{n-1}+1}^{k_n} A(l_1,\\ldots ,l_{n-2},l_{n-1}) \\\\&+\\sum _{(l_1,\\ldots ,l_{n-2})}^{(k_1,\\ldots ,k_{n-2},k_{n-1}-1)}A(l_1,\\ldots ,l_{n-2},k_{n-1}), \\quad n \\ge 2$ with $\\sum \\limits _{()}^{(k_1)}:=\\operatorname{id}$ .",
"Recall the extended definition of the sum over intervals (REF ) to make sense of this definition for all $(k_1,\\ldots ,k_n) \\in \\mathbb {Z}^n$ .",
"One can show that this definition ensures that the summation operator preserves polynomiality, i.e.",
"$(k_1,\\ldots ,k_n) \\;\\mapsto \\; \\sum _{(l_1,\\ldots ,l_{n-1})}^{(k_1,\\ldots ,k_n)} A(l_1,\\ldots ,l_{n-1})$ is a polynomial function on $\\mathbb {Z}^n$ whenever $A(l_1,\\ldots ,l_{n-1})$ is a polynomial.",
"Since a polynomial in $(k_1,\\ldots ,k_n)$ is uniquely determined by its evaluations at $k_1 < k_2 < \\cdots < k_n$ , we may also use any other recursive description of the summation operator as long as it is based on the extended definition of ordinary sums (REF ) and specializes to (REF ) whenever $k_1 < k_2 < \\cdots < k_n$ .",
"So, we can also use the recursive definition $\\sum _{(l_1,\\ldots ,l_{n-1})}^{(k_1,\\ldots ,k_n)} A(l_1,\\ldots ,l_{n-1}) =& \\sum _{(l_2,\\ldots ,l_{n-1})}^{(k_2,\\ldots ,k_{n})} \\sum _{l_{1} =k_1}^{k_2-1} A(l_1,l_2,\\ldots ,l_{n-1}) \\\\&+\\sum _{(l_2,\\ldots ,l_{n-1})}^{(k_2+1,k_3,\\ldots ,k_n)}A(k_2,l_2,\\ldots ,l_{n-1}), \\quad n \\ge 2.$ Lemma 7.1 Let $i<0$ and $\\mathbf {x}_i \\in \\mathbb {Z}^{-i}$ .",
"Then ${^{\\mathbf {x}_i}\\Delta ^{i}_{k_j}} \\alpha (n;k_1,\\ldots ,k_n)=(-1)^{i j} \\\\\\times \\Delta _{k_1}^{-i} \\ldots \\Delta _{k_{j-1}}^{-i} \\delta ^{0}_{x_{i}} \\delta ^1_{x_{i+1}} \\dots \\delta _{x_{-1}}^{-i-1} \\delta ^{-i}_{k_{j+1}} \\dots \\delta ^{-i}_{k_n} \\alpha (n-i;k_1,\\ldots ,k_j,x_{i},x_{i+1},\\ldots ,x_{-1},k_{j+1},\\ldots ,k_n)$ and ${^{\\mathbf {x}_i} \\delta _{k_j}^i} \\alpha (n;k_1,\\ldots ,k_n)= (-1)^{(j-1) i + \\binom{-i}{2}} \\\\\\times \\Delta _{k_1}^{-i} \\dots \\Delta _{k_{j-1}}^{-i}\\Delta _{x_{-1}}^{-i-1} \\Delta _{x_{-2}}^{-i-2} \\dots \\Delta _{x_{i}}^{0}\\delta ^{-i}_{k_{j+1}} \\dots \\delta ^{-i}_{k_{n}} \\alpha (n-i;k_1,\\ldots ,k_{j-1},x_{-1},x_{-2},\\ldots ,x_{i},k_{j},\\ldots ,k_n).$ Informally, the lemma follows from the following two facts: The quantity ${^{\\mathbf {x}_i}\\Delta ^{i}_{k_j}} \\alpha (n;k_1,\\ldots ,k_n)$ can be interpreted as the signed enumeration of Monotone Triangle structures of the shape as depicted in Figure REF where the $j$ -th NE-diagonal has been prolonged.",
"Similarly, for ${^{\\mathbf {x}_i}\\delta ^{i}_{k_j}} \\alpha (n;k_1,\\ldots ,k_n)$ , where the shape is depicted in Figure REF and the $j$ -th SE-diagonal has been prolonged.",
"The application of the $(-\\Delta )$ -operator truncates left NE-diagonals, while the $\\delta $ -operator truncates right SE-diagonals.",
"This idea first appeared in [7].",
"Figure: 𝐱 i Δ k j i α(n;k 1 ,...,k n ){^{\\mathbf {x}_i}\\Delta ^{i}_{k_j}} \\alpha (n;k_1,\\ldots ,k_n)Figure: 𝐱 i δ k j i α(n;k 1 ,...,k n ){^{\\mathbf {x}_i}\\delta ^{i}_{k_j}} \\alpha (n;k_1,\\ldots ,k_n)Formally, let us prove the first identity by induction with respect to $i$ .",
"First note that (REF ) and (REF ) imply $(-1)^j \\Delta _{k_1} \\dots \\Delta _{k_{j-1}} & \\delta _{k_{j+1}} \\delta _{k_{j+2}} \\dots \\delta _{k_n}\\sum _{(l_1,\\ldots ,l_n)}^{(k_1,\\ldots ,k_{j-1},k_j,x,k_{j+1},\\ldots ,k_n)} A(l_1,\\ldots ,l_n) \\\\&= - \\sum _{(l_j)}^{(k_j,x)} A(k_1,\\ldots ,k_{j-1},l_j,k_{j+1},\\ldots ,k_n) = {^x}\\Delta _{k_j}^{-1} A(k_1,k_2,\\ldots ,k_n).$ Together with (REF ) the base case $i=-1$ follows.",
"For the inductive step $i<-1$ , apply the induction hypothesis, (REF ), (REF ) and Proposition REF (REF ) to obtain $& {^{\\mathbf {x}_i}\\Delta _{k_j}^i} \\alpha (n;k_1,\\ldots ,k_n) \\\\&= {^{x_i}\\Delta _{k_j}^{-1}} (-1)^{(i+1)j} \\Delta _{k_1}^{-i-1} \\ldots \\Delta _{k_{j-1}}^{-i-1} \\delta ^{0}_{x_{i+1}} \\delta ^1_{x_{i+2}} \\dots \\delta _{x_{-1}}^{-i-2} \\delta ^{-i-1}_{k_{j+1}} \\dots \\delta ^{-i-1}_{k_n} \\\\& \\qquad \\alpha (n-i-1;k_1,\\ldots ,k_j,x_{i+1},x_{i+2},\\ldots ,x_{-1},k_{j+1},\\ldots ,k_n) \\\\&= (-1)^{i j} \\Delta _{k_1}^{-i} \\ldots \\Delta _{k_{j-1}}^{-i} \\delta ^{1}_{x_{i+1}} \\delta ^2_{x_{i+2}} \\dots \\delta _{x_{-1}}^{-i-1} \\delta ^{-i}_{k_{j+1}} \\dots \\delta ^{-i}_{k_n} \\\\&\\qquad \\sum _{(l_1,\\ldots ,l_{j},y_{i+1},\\ldots ,y_{-1},l_{j+1},\\ldots , l_{n})}^{(k_1,\\ldots ,k_j,x_i,x_{i+1},\\ldots ,x_{-1},k_{j+1},\\ldots ,k_n)} \\alpha (n-i-1;l_1,\\ldots ,l_j,y_{i+1},y_{i+2},\\ldots ,y_{-1},l_{j+1},\\ldots ,l_n) \\\\&=(-1)^{i j} \\Delta _{k_1}^{-i} \\ldots \\Delta _{k_{j-1}}^{-i} \\delta ^{1}_{x_{i+1}} \\dots \\delta _{x_{-1}}^{-i-1} \\delta ^{-i}_{k_{j+1}} \\dots \\delta ^{-i}_{k_n} \\alpha (n-i;k_1,\\ldots ,k_j,x_{i},x_{i+1},\\ldots ,x_{-1},k_{j+1},\\ldots ,k_n).$ The second identity can be shown analogously.",
"The sign is again obtained by taking the total number of applications of the $\\Delta $ -operator into account.",
"In the following, we let $V_{x,y}:=E^{-1}_x + E_y - E^{-1}_x E_y$ and $S_{x,y}f(x,y) := f(y,x)$ .",
"In [3] it was shown that $(\\operatorname{id}+ E_{k_{i+1}} E^{-1}_{k_i} S_{k_i,k_{i+1}}) V_{k_i,k_{i+1}} \\alpha (n;k_1,\\ldots , k_n) = 0$ for $1 \\le i \\le n-1$ .",
"This property together with the fact that the degree of $\\alpha (n;k_1,\\ldots , k_n)$ in each $k_i$ is $n-1$ determines the polynomial up to a constant.",
"Next we present a conjecture on general polynomials with property (REF ); the goal of the current section is to show that this conjecture implies (REF ).",
"Conjecture 7.2 Let $1 \\le s \\le t$ and $a(k_1,\\ldots ,k_{s+t-1})$ be a polynomial in $(k_1,\\ldots ,k_{s+t-1})$ with $(\\operatorname{id}+ E_{k_{i+1}} E^{-1}_{k_i} S_{k_i,k_{i+1}}) V_{k_i,k_{i+1}} a(k_1,\\ldots ,k_{s+t-1}) = 0$ for $1 \\le i \\le s+t-2$ .",
"Then $\\prod _{i=1}^{s} E^{2s+3-2i}_{y_i} \\delta ^{i-1}_{y_i} \\prod _{i=2}^{t} E_{k_i}^{2 i} \\delta ^s_{k_i}a(y_1,\\ldots ,y_s,k_2,\\ldots ,k_t) \\\\ =\\prod _{i=2}^t E^{2i}_{k_i} (- \\Delta _{k_i})^s \\prod _{i=1}^s E^{2t+3-2i}_{y_i} (- \\Delta _{y_i})^{s-i}a(k_2,\\ldots ,k_t,y_1,\\ldots ,y_s)$ if $y_1=y_2=\\ldots =y_s=k_2=k_3=\\ldots =k_t$ .",
"Proposition 7.3 Let $\\mathbf {x}=(-2j+1)_{j<0}$ and $\\mathbf {z}=(3n+j+1)_{j<0}$ .",
"Under the assumption that Conjecture REF is true, it follows for all $-n \\le i \\le -1$ that $\\left.",
"{^{\\mathbf {x}_i} \\Delta _{k_1}^{i}}\\alpha (n;k_1,4,6,\\ldots ,2n)\\right|_{k_1=3n+2+i}=0$ , ${^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,4,6,\\ldots ,2n) = {^{\\mathbf {z}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,4,6,\\ldots ,2n)$ ; in particular $C^{(2)}_{n,i} = D^{(2)}_{n,i}$ .",
"According to Lemma REF we have ${^{\\mathbf {x}_i} \\Delta _{k_1}^{i}} \\alpha (n;k_1,4,6,\\ldots ,2n) \\\\= \\left.",
"(-1)^i \\prod _{j=i}^{-1} E^{-2j+1}_{y_j} \\delta ^{j-i}_{y_j} \\prod _{j=2}^{n} E^{2 j}_{k_j} \\delta ^{-i}_{k_j}\\alpha (n-i;k_1,y_i,y_{i+1},\\ldots ,y_{-1},k_2,\\ldots ,k_n)\\right|_{{{\\scriptstyle \\begin{matrix} (y_i,y_{i+1},\\ldots ,y_{-1})=0, \\\\ (k_2,\\ldots ,k_n)=0 \\end{matrix}}}}.$ We set $\\overline{y}_j=y_{i+j-1}$ and $s=-i$ to obtain $\\left.",
"(-1)^s \\prod _{j=1}^{s} E^{2s+3-2j}_{\\overline{y}_j} \\delta ^{j-1}_{\\overline{y}_j}\\prod _{j=2}^{n} E^{2 j}_{k_j} \\delta ^{s}_{k_j}\\alpha (n+s;k_1,\\overline{y}_1,\\overline{y}_2,\\ldots ,\\overline{y}_s,k_2,\\ldots ,k_n) \\right|_{{{\\scriptstyle \\begin{matrix} (\\overline{y}_1,\\ldots ,\\overline{y}_s)=0, \\\\ (k_2,\\ldots ,k_n)=0 \\end{matrix}}}}.$ By our assumption that Conjecture REF is true, this is equal to $\\left.",
"(-1)^s \\prod _{j=2}^n E^{2j}_{k_j} (- \\Delta _{k_j})^s \\prod _{j=1}^s E^{2n+3-2j}_{\\overline{y}_j} (- \\Delta _{\\overline{y}_j})^{s-j}\\alpha (n+s;k_1,k_2,\\ldots ,k_n,\\overline{y}_1,\\ldots ,\\overline{y}_s) \\right|_{{{\\scriptstyle \\begin{matrix} (\\overline{y}_1,\\ldots ,\\overline{y}_s)=0, \\\\ (k_2,\\ldots ,k_n)=0 \\end{matrix}}}}.$ Now we use (REF ) and () to obtain $(-1)^{n+1} \\prod _{j=2}^n E^{2j+n+s}_{k_j} (- \\Delta _{k_j})^s \\prod _{j=1}^sE^{3n+3-2j+s}_{\\overline{y}_j} (- \\Delta _{\\overline{y}_j})^{s-j} \\\\\\left.",
"\\alpha (n+s;k_2,\\ldots ,k_n,\\overline{y}_1,\\ldots ,\\overline{y}_s,k_1)\\vphantom{\\prod _{j=2}^n}\\right|_{{{\\scriptstyle \\begin{matrix} (\\overline{y}_1,\\ldots ,\\overline{y}_s)=0, \\\\ (k_2,\\ldots ,k_n)=0 \\end{matrix}}}}.$ According to Lemma REF , this is $(-1)^{n+1} \\;{^{\\mathbf {w}_i} \\delta _{k_1}^i}\\alpha (n;4+n-i,6+n-i,\\ldots ,3n-i,k_1) $ where $\\mathbf {w}_i=(3n+3+i,3n+5+i,\\ldots ,3n+1-i)$ .",
"Setting $k_1=3n+2+i$ , the first assertion now follows since ${^{x+1} \\delta _{x}^{-1}} p(x)=0$ .",
"For the second assertion we use induction with respect to $i$ .",
"In the base case $i=-1$ note that the two sides differ by $\\left.",
"{^{3n}\\Delta _{k_1}^{-1}}\\alpha (n;k_1,4,6,\\ldots ,2n)\\right|_{k_1=4}$ .",
"By (REF ) this is equal to $-\\left.",
"{^{3}\\Delta _{k_1}^{-1}}\\alpha (n;k_1,4,6,\\ldots ,2n)\\right|_{k_1=3n+1},$ which vanishes due to the first assertion.",
"For $i < -1$ observe that ${^{\\mathbf {x}_i} \\Delta ^i_{k_1}} & \\alpha (n;k_1,4,6,\\ldots ,2n) \\\\&= {^{-2i+1} \\Delta _{k_1}^{-1}} \\;\\; {^{\\mathbf {x}_{i+1}} \\Delta ^{i+1}_{k_1}}\\alpha (n;k_1,4,6,\\ldots ,2n) \\\\&= - \\sum _{l_1=k_1}^{-2i+1} {^{\\mathbf {x}_{i+1}} \\Delta ^{i+1}_{l_1}}\\alpha (n;l_1,4,6,\\ldots ,2n) \\\\&=- \\sum _{l_1=k_1}^{3n+1+i} {^{\\mathbf {z}_{i+1}} \\Delta ^{i+1}_{l_1}}\\alpha (n;l_1,4,6,\\ldots ,2n)+\\sum _{l_1=-2i+2}^{3n+1+i} {^{\\mathbf {x}_{i+1}}\\Delta ^{i+1}_{l_1}} \\alpha (n;l_1,4,6,\\ldots ,2n),$ where we have used the induction hypothesis in the first sum.",
"Now the first sum is equal to the right-hand side in the second assertion, while the second sum is by (REF ) just the expression in the first assertion and thus vanishes."
],
[
"Proof of Theorem ",
"Let $p(x_1,\\ldots ,x_n)$ be a function in $(x_1,\\ldots ,x_n)$ and $T \\subseteq {\\mathcal {S}}_n$ a subset of the symmetric group.",
"We define $(T p)(x_1,\\ldots ,x_n) :=\\sum _{\\sigma \\in T} \\operatorname{sgn}\\sigma \\, p(x_{\\sigma (1)},\\ldots ,x_{\\sigma (n)}).$ If $T=\\lbrace \\sigma \\rbrace $ , then we write $( T p)(x_1,\\ldots ,x_n)=(\\sigma p)(x_1,\\ldots ,x_n)$ .",
"Observe that $\\operatorname{\\mathbf {ASym}}$ as defined in the introduction satisfies $\\operatorname{\\mathbf {ASym}}p(x_1,\\ldots ,x_n) = ({\\mathcal {S}}_n p)(x_1,\\ldots ,x_n)$ .",
"A function is said to be antisymmetric if $(\\sigma p)(x_1,\\ldots ,x_n) = \\operatorname{sgn}\\sigma \\cdot p(x_1,\\ldots ,x_n)$ for all $\\sigma \\in {\\mathcal {S}}_n$ .",
"We need a couple of auxiliary results.",
"Lemma 8.1 Let $a(z_1,\\ldots ,z_n)$ be a polynomial in $(z_1,\\ldots ,z_n)$ with $(\\operatorname{id}+ E_{z_{i+1}} E^{-1}_{z_i} S_{z_i,z_{i+1}}) V_{z_i,z_{i+1}} a(z_1,\\ldots ,z_{n}) = 0$ for $1 \\le i \\le n-1$ .",
"Then there exists an antisymmetric polynomial $b(z_1,\\ldots ,z_n)$ with $a(z_1,\\ldots ,z_{n}) = \\prod _{1 \\le p < q \\le n} W_{z_q,z_p} b(z_1,\\ldots ,z_n)$ where $W_{x,y} := E_x V_{x,y} = \\operatorname{id}- E_y + E_x E_y$ .",
"By assumption, we have $S_{z_i,z_{i+1}} W_{z_i,z_{i+1}} a(\\mathbf {z}) = E_{z_{i+1}} S_{z_i,z_{i+1}}V_{z_i,z_{i+1}} a(\\mathbf {z}) = -E_{z_i} V_{z_i,z_{i+1}}a(\\mathbf {z}) =-W_{z_i,z_{i+1}}a(\\mathbf {z}).$ This implies that $c(z_1,\\ldots ,z_{n}) := \\prod _{1 \\le p < q \\le n} W_{z_p,z_q} a(z_1,\\ldots ,z_{n})$ is an antisymmetric polynomial.",
"Now observe that $W_{x,y}=\\operatorname{id}+ E_y \\Delta _x$ is invertible on $\\mathbb {C}[x,y]$ , to be more concrete $W^{-1}_{x,y} = \\sum \\limits _{i=0}^{\\infty } (-1)^i E_y^{i} \\Delta ^{i}_x$ .",
"Hence, $b(z_1,\\ldots ,z_{n}) := \\prod \\limits _{1 \\le p \\ne q \\le n} W^{-1}_{z_p,z_q}c(z_1,\\ldots ,z_{n})$ is an antisymmetric polynomial with $a(z_1,\\ldots ,z_{n}) = \\prod \\limits _{1 \\le p < q \\le n} W_{z_q,z_p} b(z_1,\\ldots ,z_{n})$ .",
"Lemma 8.2 Suppose $\\operatorname{Op}(x_1,\\ldots ,x_n)$ is a Laurent polynomial and $a(z_1,\\ldots ,z_n)$ is an antisymmetric function.",
"If there exists a non-empty subset $T$ of ${\\mathcal {S}}_{n}$ with $(T \\operatorname{Op})(x_1,\\ldots ,x_n)=0$ , then $\\left.",
"\\left( \\operatorname{Op}(E_{z_1},\\ldots ,E_{z_n}) a(z_1,\\ldots ,z_n) \\right) \\right|_{z_1=z_2=\\ldots =z_n}=0.$ First observe that the antisymmetry of $a(z_1,\\ldots ,z_n)$ implies $(T^{\\prime } a)(z_1,\\ldots ,z_n) = \\sum _{\\sigma \\in T^{\\prime }} \\operatorname{sgn}\\sigma a(z_{\\sigma (1)},\\ldots ,z_{\\sigma (n)}) = |T^{\\prime }| a(z_1,\\ldots ,z_n).$ for any subset $T^{\\prime } \\subseteq {\\mathcal {S}}_n$ .",
"Letting $\\operatorname{Op}(x_1,\\ldots ,x_n)= \\sum _{(i_1,\\ldots ,i_n) \\in \\mathbb {Z}^n} c_{i_1,\\ldots ,i_n} x_1^{i_1} x_2^{i_2} \\cdots x_n^{i_n},$ we observe that $& \\left.",
"\\left( \\operatorname{Op}(E_{z_1},\\ldots ,E_{z_n}) a(z_1,\\ldots ,z_n) \\right)\\right|_{(z_1,\\ldots ,z_n)=(d,\\ldots ,d)} \\\\&\\qquad = \\sum _{(i_1,\\ldots ,i_n) \\in \\mathbb {Z}^n}c_{i_1,\\ldots ,i_n} a(i_1+d,\\ldots ,i_n+d)= \\frac{1}{|T|} \\sum _{(i_1,\\ldots ,i_n)\\in \\mathbb {Z}^n}c_{i_1,\\ldots ,i_n} (T^{-1} a)(i_1+d,\\ldots ,i_n+d)$ with $T^{-1} = \\lbrace \\sigma ^{-1} | \\sigma \\in T\\rbrace $ , since $(i_1,\\ldots ,i_n) \\mapsto a(i_1+d,\\ldots ,i_n+d)$ is also an antisymmetric function.",
"This is equal to $& \\frac{1}{|T|} \\sum _{(i_1,\\ldots ,i_n)\\in \\mathbb {Z}^n} c_{i_1,\\ldots ,i_n}\\sum _{\\sigma \\in T} \\operatorname{sgn}\\sigma \\left.",
"E^{i_{\\sigma ^{-1}(1)}}_{z_1} \\dots E^{i_{\\sigma ^{-1}(n)}}_{z_n} a(z_1,\\ldots ,z_n) \\right|_{(z_1,\\ldots ,z_n)=(d,\\ldots ,d)} \\\\&\\qquad = \\frac{1}{|T|} \\sum _{(i_1,\\ldots ,i_n)\\in \\mathbb {Z}^n} c_{i_1,\\ldots ,i_n}\\sum _{\\sigma \\in T} \\operatorname{sgn}\\sigma \\left.",
"E^{i_{1}}_{z_{\\sigma (1)}} \\dots E^{i_{n}}_{z_{\\sigma (n)}}a(z_1,\\ldots ,z_n) \\right|_{(z_1,\\ldots ,z_n)=(d,\\ldots ,d)} \\\\&\\qquad = \\left.",
"\\frac{1}{|T|} \\left[(T \\operatorname{Op})(E_{z_1},\\ldots ,E_{z_n}) \\right]a(z_1,\\ldots ,z_n) \\right|_{(z_1,\\ldots ,z_n)=(d,\\ldots ,d)} = 0.$ Now we are in the position to prove Theorem REF .",
"In order to prove (REF ), it suffices to show that Conjecture REF holds under the theorem's assumptions.",
"We set $\\overline{\\operatorname{Op}}(z_1,\\ldots ,z_{s+t-1}) &:= \\prod _{i=1}^{s} z_i^{2s+3-2i} (1-z_i^{-1})^{i-1}\\prod _{i=s+1}^{s+t-1} z_i^{2i-2s+2} (1-z_i^{-1})^s \\\\&\\qquad - \\prod _{i=1}^{t-1} z_i^{2i+2} (1-z_i)^{s} \\prod _{i=t}^{s+t-1} z_i^{4t+1-2i}(1-z_i)^{s+t-1-i}$ and observe that the claim of Conjecture REF is that $\\overline{\\operatorname{Op}}(E_{z_1},\\ldots ,E_{z_{s+t-1}}) a(z_1,\\ldots ,z_{s+t-1})$ vanishes if $z_1=\\ldots =z_{s+t-1}$ .",
"According to Lemma REF , there exists an antisymmetric polynomial $b(z_1,\\ldots ,z_{s+t-1})$ with $a(z_1,\\ldots ,z_{s+t-1}) = \\prod _{1 \\le p < q \\le s+t-1} W_{z_q,z_p} b(z_1,\\ldots ,z_{s+t-1}).$ Thus, let us deduce that $\\operatorname{Op}(E_{z_1},\\ldots ,E_{z_{s+t-1}}) b(z_1,\\ldots ,z_{s+t-1})=0$ if $z_1=\\ldots =z_{s+t-1}$ where $\\operatorname{Op}(z_1,\\ldots ,z_{s+t-1}):= \\overline{\\operatorname{Op}}(z_1,\\ldots ,z_{s+t-1}) \\prod _{1 \\le p < q \\le s+t-1} (1-z_p + z_p z_q) \\prod _{i=1}^{s+t-1} z_i^{-2-t}.$ Now, Lemma REF implies that it suffices to show $\\operatorname{\\mathbf {ASym}}\\operatorname{Op}(z_1,\\ldots ,z_{s+t-1}) = 0$ .",
"Observe that $\\operatorname{Op}(z_1,\\ldots ,z_{s+t-1}) = \\overline{P}_{s,t}(z_1,\\ldots ,z_{s+t-1}) - \\overline{P}_{s,t}(z_{s+t-1}^{-1},\\ldots ,z_1^{-1}) \\prod _{i=1}^{s+t-1} z_i^{s+t-2}$ where $\\overline{P}_{s,t}(z_1,\\ldots ,z_{s+t-1}) = P_{s,t}(z_1,\\ldots ,z_{s+t-1}) \\prod \\limits _{1 \\le i < j \\le s+t-1} (z_j - z_i)$ and $P_{s,t}(z_1,\\ldots ,z_{s+t-1})$ is as defined in Conjecture REF .",
"Furthermore, $\\operatorname{\\mathbf {ASym}}\\operatorname{Op}(z_1,\\ldots ,z_{s+t-1}) &= R_{s,t}(z_1,\\ldots ,z_{s+t-1}) \\prod _{1 \\le i < j \\le s+t-1} (z_j-z_i) \\\\& \\quad -R_{s,t}(z_{s+t-1}^{-1},\\ldots ,z_1^{-1}) \\prod _{1 \\le i < j \\le s+t-1} (z_{s+t-j}^{-1}-z_{s+t-i}^{-1}) \\prod _{i=1}^{s+t-1} z_i^{s+t-2}$ where $R_{s,t}(z_1,\\ldots ,z_{s+t-1})$ is also defined in Conjecture REF .",
"Since $R_{s,t}(z_1,\\ldots ,z_{s+t-1})$ is symmetric we have that $\\operatorname{\\mathbf {ASym}}\\operatorname{Op}(z_1,\\ldots ,z_{s+t-1}) = 0$ follows once it is shown that $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) = R_{s,t}(z_1^{-1},\\ldots ,z_{s+t-1}^{-1})$ ."
],
[
"Proof of Theorem ",
"For integers $s,t \\ge 1$ , we define the following two rational functions: $S_{s,t}(z;z_1,\\ldots ,z_{s+t-2})&:= z^{2s-t-1} \\prod _{i=1}^{s+t-2} \\frac{(1-z+ z_i z)(1-z_i^{-1})}{(z_i-z)}, \\\\T_{s,t}(z;z_1,\\ldots ,z_{s+t-2}) &:= (1-z^{-1})^s z^{t-2} \\prod _{i=1}^{s+t-2} \\frac{1-z_i + z_i z}{(z-z_i) z_i}.$ Based on these two functions, we define two operators on functions $f$ in $s+t-2$ variables that transform them into functions in $(z_1,\\ldots ,z_{s+t-1})$ : $\\operatorname{PS}_{s,t}[f]&:= S_{s,t}(z_{1};z_2,\\ldots ,z_{s+t-1}) \\cdot f(z_2,\\ldots ,z_{s+t-1}), \\\\\\operatorname{PT}_{s,t}[f]&:=T_{s,t}(z_{s+t-1};z_1,\\ldots ,z_{s+t-2}) \\cdot f(z_1,\\ldots ,z_{s+t-2}).$ The definitions are motivated by the fact that $P_{s,t}(z_1,\\ldots ,z_{s+t-1})$ as defined in Conjecture REF satisfies the two recursions $P_{s,t}=\\operatorname{PS}_{s,t}[P_{s-1,t}] \\quad \\text{ and } \\quad P_{s,t} = \\operatorname{PT}_{s,t}[P_{s,t-1}].$ We also need the following two related operators, which are again defined on functions $f$ in $s+t-2$ variables: $\\operatorname{QS}_{s,t}[f]&:= S_{s,t}(z_{s+t-1}^{-1};z_{s+t-2}^{-1},z_{s+t-3}^{-1},\\ldots ,z_1^{-1}) \\cdot f(z_1,\\ldots ,z_{s+t-2}), \\\\\\operatorname{QT}_{s,t}[f]&:=T_{s,t}(z_{1}^{-1};z_{s+t-1}^{-1},z_{s+t-2}^{-1},\\ldots ,z_2^{-1}) \\cdot f(z_2,\\ldots ,z_{s+t-1}).$ Note that if we set $Q_{s,t}(z_1,\\ldots ,z_{s+t-1}):=P_{s,t}(z_{s+t-1}^{-1},\\ldots ,z_1^{-1})$ , then $Q_{s,t}=\\operatorname{QS}_{s,t}[Q_{s-1,t}] \\quad \\text{ and } \\quad Q_{s,t} = \\operatorname{QT}_{s,t}[Q_{s,t-1}].$ From the definitions, one can deduce the following commutation properties; the proof is straightforward and left to the reader.",
"Lemma 9.1 Let $s,t$ be positive integers.",
"If $s, t \\ge 1$ , then $\\operatorname{PS}_{s,t} \\circ \\operatorname{PT}_{s-1,t} = \\operatorname{PT}_{s,t} \\circ \\operatorname{PS}_{s,t-1}$ and $\\operatorname{QS}_{s,t} \\circ \\operatorname{QT}_{s-1,t} = \\operatorname{QT}_{s,t} \\circ \\operatorname{QS}_{s,t-1}$ .",
"If $t \\ge 2$ , then $\\operatorname{PT}_{s,t} \\circ \\operatorname{QT}_{s,t-1} = \\operatorname{QT}_{s,t} \\circ \\operatorname{PT}_{s,t-1}$ .",
"Moreover, we need the following identities, which follow from the fact that $S_{s,t}(z;z_1,\\ldots ,z_{s+t-2})$ and $T_{s,t}(z;z_1,\\ldots ,z_{s+t-2})$ are symmetric in $z_1,\\ldots ,z_{s+t-2}$ (the symbol $\\widehat{z_i}$ indicates that $z_i$ is missing from the argument): $\\begin{aligned}\\operatorname{\\mathbf {Sym}}\\operatorname{PS}_{s,t}[f]&= \\sum _{i=1}^{s+t-1} S_{s,t}(z_{i};z_1,\\ldots ,\\widehat{z_i},\\ldots ,z_{s+t-1}) \\operatorname{\\mathbf {Sym}}f(z_1,\\ldots ,\\widehat{z_i},\\ldots ,z_{s+t-1}), \\\\\\operatorname{\\mathbf {Sym}}\\operatorname{PT}_{s,t}[f] &=\\sum _{i=1}^{s+t-1} T_{s,t}(z_{i};z_1,\\ldots ,\\widehat{z_i},\\ldots ,z_{s+t-1}) \\operatorname{\\mathbf {Sym}}f(z_1,\\ldots ,\\widehat{z_i},\\ldots ,z_{s+t-1}), \\\\\\operatorname{\\mathbf {Sym}}\\operatorname{QS}_{s,t}[f]&= \\sum _{i=1}^{s+t-1} S_{s,t}(z_{i}^{-1};z_1^{-1},\\ldots ,\\widehat{z_i^{-1}},\\ldots , z_{s+t-1}^{-1}) \\operatorname{\\mathbf {Sym}}f(z_1,\\ldots ,\\widehat{z_i},\\ldots , z_{s+t-1}), \\\\\\operatorname{\\mathbf {Sym}}\\operatorname{QT}_{s,t}[f]&=\\sum _{i=1}^{s+t-1} T_{s,t}(z_{i}^{-1};z_1^{-1},\\ldots ,\\widehat{z_i^{-1}},\\ldots , z_{s+t-1}^{-1}) \\operatorname{\\mathbf {Sym}}f(z_1,\\ldots ,\\widehat{z_i},\\ldots , z_{s+t-1}).\\end{aligned}$ We consider words $w$ over the alphabet ${\\mathcal {A}}:=\\lbrace \\operatorname{PS}, \\operatorname{PT}, \\operatorname{QS}, \\operatorname{QT}\\rbrace $ and let $|w|_S$ denote the number of occurrences of $\\operatorname{PS}$ and $\\operatorname{QS}$ in the word and $|w|_T$ denote the number of occurrences of $\\operatorname{PT}$ and $\\operatorname{QT}$ .",
"It is instructive to interpret these words as labelled lattice paths with starting point in the origin, step set $\\lbrace (1,0),(0,1)\\rbrace $ and labels $P, Q$ .",
"The letters $\\operatorname{PS}$ and $\\operatorname{QS}$ correspond to $(1,0)$ -steps labelled with $P$ and $Q$ , respectively, while the letters $\\operatorname{PT}$ and $\\operatorname{QT}$ correspond to $(0,1)$ -steps.",
"With this interpretation, $(|w|_S,|w|_T)$ is the endpoint of the path (see Figure REF ).",
"Figure: Labelled lattice path corresponding to w=(PT,PS,QT,PT,QS,QT)w=(\\operatorname{PT},\\operatorname{PS},\\operatorname{QT},\\operatorname{PT},\\operatorname{QS},\\operatorname{QT}).To every word $w$ of length $n$ , we assign a rational function $F_{w}(z_1,\\ldots ,z_{n+1})$ as follows: If $w$ is the empty word, then $F_{w}(z_1):=1$ .",
"Otherwise, if $L \\in {\\mathcal {A}}$ and $w$ is a word over ${\\mathcal {A}}$ , we set $F_{w L}:= L_{|w L |_S+1,|w L|_T+1} [F_{w}].$ For example, the rational function assigned to $w$ in Figure REF is $F_{w}(z_1,\\ldots ,z_7) = \\operatorname{QT}_{3,5} \\circ \\operatorname{QS}_{3,4} \\circ \\operatorname{PT}_{2,4} \\circ \\operatorname{QT}_{2,3} \\circ \\operatorname{PS}_{2,2} \\circ \\operatorname{PT}_{1,2}[1].$ In this context, Lemma REF has the following meaning: on the one hand, we may swap two consecutive steps with the same label, and, one the other hand, we may swap two consecutive $(0,1)$ -steps without changing the corresponding rational functions.",
"For example, the rational functions corresponding to the words in Figure REF and Figure REF coincide.",
"Figure: Labelled lattice path corresponding to w ˜=(PT,PS,PT,QT,QT,QS)\\tilde{w}=(\\operatorname{PT},\\operatorname{PS},\\operatorname{PT},\\operatorname{QT},\\operatorname{QT},\\operatorname{QS}).We assume $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) = R_{s,t}(z_1^{-1},\\ldots ,z_{s+t-1}^{-1})$ if $t=s$ and $t=s+1$ .",
"We show the following more general statement: Suppose $w_1, w_2$ are two words over ${\\mathcal {A}}$ with $|w_1|_S=|w_2|_S$ and $ |w_1|_T=|w_2|_T$ , and every prefix $w^{\\prime }_i$ of $w_i$ fulfills $|w^{\\prime }_i|_S \\le |w^{\\prime }_i|_T$ , $i=1,2$ .",
"(In the lattice paths language this means that $w_1$ and $w_2$ are both prefixes of Dyck paths sharing the same endpoint; there is no restriction on the labels $P$ and $Q$ .)",
"Then $\\operatorname{\\mathbf {Sym}}F_{w_1} = \\operatorname{\\mathbf {Sym}}F_{w_2}.$ The assertion of the theorem then follows since $F_w = P_{|w|_S+1,|w|_T+1}$ if $w$ is a word over $\\lbrace \\operatorname{PS}, \\operatorname{PT}\\rbrace $ and $F_w = Q_{|w|_S+1,|w|_T+1}$ if $w$ is a word over $\\lbrace \\operatorname{QS},\\operatorname{QT}\\rbrace $ , and therefore $R_{s,t}(z_1,\\ldots ,z_{s+t-1}) &= \\operatorname{\\mathbf {Sym}}P_{s,t}(z_1,\\ldots ,z_{s+t-1}) = \\operatorname{\\mathbf {Sym}}Q_{s,t}(z_1,\\ldots ,z_{s+t-1})\\\\&= \\operatorname{\\mathbf {Sym}}P_{s,t}(z_{s+t-1}^{-1},\\ldots ,z_1^{-1})= R_{s,t}(z_1^{-1},\\ldots ,z_{s+t-1}^{-1}).$ The proof is by induction with respect to the length of the words; there is nothing to prove if the words are empty.",
"Otherwise let $w_1, w_2$ be two words over ${\\mathcal {A}}$ with $|w_1|_S=|w_2|_S=:s-1$ and $ |w_1|_T=|w_2|_T=:t-1$ , and every prefix $w^{\\prime }_i$ of $w_i$ fulfills $|w^{\\prime }_i|_S \\le |w^{\\prime }_i|_T$ , $i=1,2$ .",
"Note that the induction hypothesis and (REF ) imply that $\\operatorname{\\mathbf {Sym}}F_{w_i}$ only depends on the last letter of $w_i$ (and on $s$ and $t$ of course).",
"Thus the assertion follows if the last letters of $w_1$ and $w_2$ coincide; we assume that they differ in the following.",
"If $s=t$ , then the assumption on the prefixes implies that the last letters of $w_1$ and $w_2$ are in $\\lbrace \\operatorname{PS},\\operatorname{QS}\\rbrace $ .",
"W.l.o.g.",
"we assume $w_1= w_1^{\\prime } \\operatorname{PS}$ and $w_2 = w_2^{\\prime } \\operatorname{QS}$ .",
"By the induction hypothesis and (REF ), we have $\\operatorname{\\mathbf {Sym}}F_{w_1} = \\operatorname{\\mathbf {Sym}}P_{s,s}$ and $\\operatorname{\\mathbf {Sym}}F_{w_2} = \\operatorname{\\mathbf {Sym}}Q_{s,s}$ .",
"The assertion now follows from (REF ), since $\\operatorname{\\mathbf {Sym}}P_{s,s}(z_1,\\ldots ,z_{2s-1})=R_{s,s}(z_1,\\ldots ,z_{2s-1})$ and $\\operatorname{\\mathbf {Sym}}Q_{s,s}(z_1,\\ldots ,z_{2s-1})=R_{s,s}(z_1^{-1},\\ldots ,z_{2s-1}^{-1})$ .",
"If $s < t$ , we show that we may assume that the last letters of $w_1$ and $w_2$ are in $\\lbrace \\operatorname{PT}, \\operatorname{QT}\\rbrace $ : if this is not true for the last letter $L_1$ of $w_i$ , we may at least assume by the induction hypothesis and (REF ) that the penultimate letter $L_2$ is in $\\lbrace \\operatorname{PT},\\operatorname{QT}\\rbrace $ ; to be more precise, we require $L_2=\\operatorname{PT}$ if $L_1=\\operatorname{PS}$ and $L_2=\\operatorname{QT}$ if $L_1=\\operatorname{QS}$ ; now, according to Lemma REF , we can interchange the last and the penultimate letter in this case.",
"If $t=s+1$ , then (REF ) now follows from (REF ) in a similar fashion as in the case when $s=t$ .",
"If $s+1 < t$ , we may assume w.l.o.g.",
"that the last letter of $w_1$ is $\\operatorname{PT}$ and the last letter of $w_2$ is $\\operatorname{QT}$ .",
"By the induction hypothesis and (REF ), we may assume that the penultimate letter of $w_1$ is $\\operatorname{QT}$ .",
"According to Lemma REF , we can interchange the last and the penultimate letter of $w_1$ and the assertion follows also in this case."
],
[
"Some remarks on the case $s=0$ in Conjecture ",
"If $s=0$ in Conjecture REF , then the rational function simplifies to $\\prod _{1 \\le i < j \\le n} \\frac{z_i^{-1} + z_j -1}{1-z_i z_j^{-1}}$ where $n=t-1$ .",
"This raises the question of whether there are also other rational functions $T(x,y)$ such that symmetrizing $\\prod \\limits _{1 \\le i < j \\le n} T(z_i,z_j)$ leads to a Laurent polynomial that is invariant under replacing $z_i$ by $z_i^{-1}$ .",
"Computer experiments suggest that this is the case for $T(x,y) = \\frac{[a(x^{-1}+y)+c][b(x+y^{-1})+c]}{1-x y^{-1}} + a b x^{-1} y +d$ where $a,b,c,d \\in \\mathbb {C}$ .",
"(Since $T(x,y)=T(y^{-1},x^{-1})$ it is obvious that the symmetrized function is invariant under replacing all $z_i$ simultaneously by $z_i^{-1}$ .)",
"In case $a=0$ it can be shown with a degree argument that symmetrizing leads to a function that does not depend on $z_1,z_2,\\ldots ,z_n$ .",
"(In fact, this is also true for $\\prod \\limits _{1 \\le i < j \\le n} \\frac{A z_i z_j+ B z_i + C z_j + D}{z_j - z_i}$ , and our case is obtained by specializing $A=b c,B=-d,C=c^2+d,D=bc$ .)",
"In case $T(x,y)=\\frac{x^{-1}+y}{1-x y^{-1}}$ (which is obtained from the above function by setting $b=d=0$ then dividing by $c$ and setting $a=1, c=0$ afterwards) this is also easy to see, since the symmetrized function can be computed explicitly as follows: $\\operatorname{\\mathbf {Sym}}& \\prod _{1 \\le i < j \\le n} \\frac{z_i^{-1} + z_j}{1-z_i z_j^{-1}}= \\operatorname{\\mathbf {Sym}}\\prod _{1 \\le i < j \\le n} \\frac{z_i^{-1} z_j (1+z_i z_j)}{z_j - z_i} \\\\&= \\prod _{1 \\le i < j \\le n} (1 + z_i z_j) \\prod _{i=1}^{n} z_i^{-n+1}\\operatorname{\\mathbf {Sym}}\\frac{\\prod _{i=1}^{n} z_i^{2i-2}}{\\prod _{1 \\le i < j \\le n} (z_j - z_i)} \\\\&= \\prod _{1 \\le i < j \\le n} (1 + z_i z_j) \\prod _{i=1}^{n} z_i^{-n+1} \\frac{\\det _{1 \\le i, j \\le n} ((z_i^{2})^{j-1})}{\\prod _{1 \\le i < j \\le n} (z_j - z_i)} \\\\&= \\prod _{1 \\le i < j \\le n} (1 + z_i z_j) \\prod _{i=1}^{n} z_i^{-n+1} \\prod _{1 \\le i < j \\le n} \\frac{z_j^{2} - z_i^{2}}{z_j-z_i} = \\prod _{1 \\le i < j \\le n} (1 + z_i z_j) (z_i + z_j) \\prod _{i=1}^{n} z_i^{-n+1}.$ We come back to (REF ).",
"In our computer experiments we observed that if we specialize $z_1=z_2=\\ldots =z_n=1$ in the symmetrized function then we obtain the number of $(2n+1) \\times (2n+1)$ Vertically Symmetric Alternating Sign Matrices.",
"Next we aim to prove a generalization of this.",
"For this purpose we consider the following slight generalization of $\\alpha (n;k_1,\\ldots ,k_n)$ for non-negative integers $m$ : $\\alpha _m(n;k_1,\\ldots ,k_n)= \\prod _{1 \\le p < q \\le n} (\\operatorname{id}+ E_{k_p} E_{k_q} + (X-2) E_{k_p}) \\det _{1 \\le i, j \\le n} \\left( \\binom{k_i}{j-1 + m \\, \\delta _{j,n} } \\right)$ In [3] it was shown that $\\alpha _0(n;k_1,\\ldots ,k_n)=\\alpha (n;k_1,\\ldots ,k_n)$ if $X=1$ .",
"For $k_1 \\le t \\le k_n$ , choose $c_m \\in \\mathbb {C}$ , almost all of them zero, such that the polynomial $\\sum _{m=0}^{\\infty } c_m \\binom{x}{m}$ is 1 if $x=t$ and 0 if $x \\in \\lbrace k_1,k_1+1,\\ldots ,k_n\\rbrace \\setminus \\lbrace t\\rbrace $ .",
"In [6] it was shown that $\\sum _{m=0}^{\\infty } c_m \\alpha _{m}(n;k_1,k_2,\\ldots ,k_n)$ is the generating function ($X$ is the variable) of Monotone Triangles $(a_{i,j})_{1 \\le j \\le i \\le n}$ with bottom row $(k_1,k_2,\\ldots ,k_n)$ and top entry $t$ with respect to the occurrences of the “local pattern” $a_{i+1,j} < a_{i,j} < a_{i+1,j+1}$ .",
"In fact, these patterns correspond to the $-1$ s in the corresponding Alternating Sign Matrix if $(k_1,\\ldots ,k_n)=(1,2,\\ldots ,n)$ .",
"Proposition 10.1 Fix integers $k_1,k_2,\\ldots ,k_n$ and a non-negative integer $m$ , and define $Q(z_1,\\ldots ,z_n) := \\operatorname{\\mathbf {Sym}}\\left( \\prod \\limits _{i=1}^{n} z_i^{k_i} \\prod \\limits _{1 \\le i < j \\le n} \\frac{1 + z_i z_j + (X-2) z_i}{z_j-z_i} \\right).$ Then $\\alpha _m(n;k_1,\\ldots ,k_n) = \\left.",
"Q(1,1,\\ldots ,1,E_{l}) \\binom{l}{m}\\right|_{l=0}.$ We set $P(z_1,\\ldots ,z_n)= \\prod \\limits _{i=1}^{n} z_i^{k_i} \\prod \\limits _{1 \\le i < j \\le n} ( 1 + z_i z_j + (X-2) z_i)$ .",
"Then $\\alpha _m(n;k_1,\\ldots ,k_n)&= \\left.",
"E_{l_1}^{k_1} \\cdots E_{l_n}^{k_n} \\alpha _{m}(n;l_1,\\ldots ,l_n) \\right|_{l_1=\\ldots =l_n=0} \\\\ &=\\left.",
"P(E_{l_1},\\ldots ,E_{l_n}) \\sum _{\\sigma \\in {\\mathcal {S}}_n} \\operatorname{sgn}\\sigma \\binom{l_{\\sigma (1)}}{0} \\ldots \\binom{l_{\\sigma (n-1)}}{n-2}\\binom{l_{\\sigma (n)}}{n-1+m} \\right|_{l_1=\\ldots =l_n=0}.$ With $P(z_1,\\ldots ,z_n) = \\sum _{i_1,i_2,\\ldots ,i_n} p_{i_1,i_2,\\ldots ,i_n} z_1^{i_1} z_2^{i_2} \\cdots z_n^{i_n}$ , this is equal to $\\sum _{\\sigma \\in \\mathcal {S}_n, i_1,i_2,\\ldots ,i_n} \\operatorname{sgn}\\sigma \\, p_{i_1,i_2,\\ldots ,i_n} \\binom{i_{\\sigma (1)}}{0}\\ldots \\binom{i_{\\sigma (n-1)}}{n-2}\\binom{i_{\\sigma (n)}}{n-1+m} \\\\= \\left.",
"\\sum _{\\sigma \\in \\mathcal {S}_n, i_1,i_2,\\ldots ,i_n} \\operatorname{sgn}\\sigma \\,p_{i_1,i_2,\\ldots ,i_n} E_{l_1}^{i_{\\sigma (1)}} \\ldots E_{l_n}^{i_{\\sigma (n)}}\\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0}\\\\= \\left.",
"\\sum _{\\sigma \\in \\mathcal {S}_n, i_1,i_2,\\ldots ,i_n} \\operatorname{sgn}\\sigma \\,p_{i_1,i_2,\\ldots ,i_n} E_{l_{\\sigma ^{-1}(1)}}^{i_1} \\ldots E_{l_{\\sigma ^{-1}(n)}}^{i_n}\\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0}\\\\= \\left.",
"\\operatorname{\\mathbf {ASym}}P(E_{l_1},\\ldots ,E_{l_n}) \\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0}.$ By definition, $\\operatorname{\\mathbf {ASym}}P(z_1,\\ldots ,z_n) = Q(z_1,\\ldots ,z_n) \\prod _{1 \\le i < j \\le n} (z_j-z_i).$ Now we can conclude that $&\\alpha _m(k_1,\\ldots ,k_n) = \\left.",
"Q(E_{l_1},E_{l_2},\\ldots ,E_{l_n}) \\prod _{1 \\le i < j \\le n} (E_{l_j} - E_{l_i}) \\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0} \\\\&\\quad =\\left.",
"Q(E_{l_1},E_{l_2},\\ldots ,E_{l_n}) \\prod _{1 \\le i < j \\le n} (\\Delta _{l_j} - \\Delta _{l_i}) \\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\ldots \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0} \\\\&\\quad =\\left.",
"Q(E_{l_1},E_{l_2},\\ldots ,E_{l_n}) \\det _{1 \\le i,j \\le n} \\left( \\Delta _{l_i}^{j-1} \\right) \\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\ldots \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0} \\\\&\\quad =\\left.",
"Q(E_{l_1},E_{l_2},\\ldots ,E_{l_n}) \\sum _{\\sigma \\in \\mathcal {S}_n} \\operatorname{sgn}\\sigma \\Delta _{l_1}^{\\sigma (1)-1} \\ldots \\Delta _{l_n}^{\\sigma (n)-1} \\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\binom{l_n}{n-1+m} \\right|_{l_1=\\ldots =l_n=0} \\\\&\\quad =\\left.",
"Q(1,1,\\ldots ,1,E_{l_n}) \\binom{l_n}{m} \\right|_{l_n=0},$ since $\\Delta _{l_1}^{\\sigma (1)-1} \\ldots \\Delta _{l_n}^{\\sigma (n)-1} \\binom{l_1}{0} \\ldots \\binom{l_{n-1}}{n-2} \\binom{l_n}{n-1+m}=0$ except when $\\sigma =\\operatorname{id}$ .",
"Corollary 10.2 Let $k_1,k_2,\\ldots ,k_n,m$ and $Q(z_1,\\ldots ,z_n)$ be as in Proposition REF .",
"Then the coefficient of $z^t X^k$ in $Q(1,1,\\ldots ,1,z)$ is the number of Monotone Triangles $(a_{i,j})_{1 \\le j \\le i \\le n}$ with bottom row $k_1,k_2,\\ldots ,k_n$ , top entry $t$ and $k$ occurrences of the local pattern $a_{i+1,j} < a_{i,j} < a_{i+1,j+1}$ .",
"We fix $t$ and observe that the combination of (REF ) and Proposition REF implies that $\\sum _{m \\ge 0} c_m \\left.",
"Q(1,1,\\ldots ,1,E_l) \\binom{l}{m} \\right|_{l=0}$ is the generating function described after (REF ).",
"Now, if we suppose $Q(1,1,\\ldots ,1,z) = \\sum _{s,k} b_{s,k} z^s X^k,$ then we see that this generating function is equal to $\\sum _{m \\ge 0} c_m \\left.",
"\\sum _{s,k} b_{s,k} E_l^s X^k \\binom{l}{m} \\right|_{l=0}= \\sum _{m \\ge 0} c_m \\sum _{s,k} b_{s,k} X^k \\binom{s}{m}= \\sum _{s,k} b_{s,k} X^k \\sum _{m \\ge 0} c_m \\binom{s}{m} \\\\= \\sum _{s,k} b_{s,k} X^k \\delta _{s,t} = \\sum _{k} b_{t,k} X^k,$ where the third equality follows from the choice of the coefficients $c_m$ .",
"A short calculation shows that $\\operatorname{\\mathbf {Sym}}$ applied to (REF ) is equal to $\\prod \\limits _{i=1}^{n} z_i^{-n+1} Q(z_1,\\ldots ,z_n)$ if we set $k_i=2(i-1)$ and $X=1$ in $Q(z_1,\\ldots ,z_n)$ .",
"If we also specialize $z_1=\\dots =z_{n-1}=1$ , then Conjecture REF implies $Q(1,\\ldots ,1,z)= z^{2n-2} Q(1,\\ldots ,1,z^{-1})$ .",
"However, by Corollary REF , this is just the trivial fact that the number of Monotone Triangles with bottom row $(0,2,4,\\ldots ,2n-2)$ and top entry $t$ is equal to the number of Monotone Triangles with bottom row $(0,2,4,\\ldots ,2n-2)$ and top entry $2n-2-t$ , or, equivalently, that the number of $(2n+1) \\times (2n+1)$ Vertically Symmetric Alternating Sign Matrices with a 1 in position $(t,1)$ equals the number of $(2n+1) \\times (2n+1)$ Vertically Symmetric Alternating Sign Matrices with a 1 in position $(2n+1-t,1)$ .",
"So in the special case $s=0$ , Conjecture REF is a generalization of this obvious symmetry.",
"Finally, we want to remark that the symmetrized functions under consideration in Proposition REF can easily be computed recursively.",
"For instance, considering the case of Vertically Symmetric Alternating Sign Matrices, let $\\operatorname{VSASM}(X;z_1,\\ldots ,z_n) = \\operatorname{\\mathbf {Sym}}\\prod _{i=1}^{n} z_i^{2i-2}\\prod _{1 \\le i < j \\le n} \\frac{1+z_i (X-2) + z_i z_j}{z_j-z_i}.$ Then $\\operatorname{VSASM}(X;z_1,\\ldots ,z_n)=\\sum _{j=1}^{n} z_j^{2n-2} \\prod _{1 \\le i \\le n, i \\ne j}\\frac{1+ z_i (X-2) + z_i z_j}{z_j - z_i} \\operatorname{VSASM}(X;z_1,\\ldots ,\\widehat{z_j},\\ldots ,z_n).$ Similarly, in the case of ordinary Alternating Sign Matrices, let $\\operatorname{ASM}(X;z_1,\\ldots ,z_n) = \\operatorname{\\mathbf {Sym}}\\prod _{i=1}^{n} z_i^{i-1}\\prod _{1 \\le i < j \\le n} \\frac{1+z_i (X-2) + z_i z_j}{z_j-z_i}.$ Then $\\operatorname{ASM}(X;z_1,\\ldots ,z_n)=\\sum _{j=1}^{n} z_j^{n-1} \\prod _{1 \\le i \\le n, i \\ne j}\\frac{1+ z_i (X-2) + z_i z_j}{z_j - z_i} \\operatorname{ASM}(X;z_1,\\ldots ,\\widehat{z_j},\\ldots ,z_n).$ Let us conclude by mentioning that in order to reprove the formula for the number of Vertically Symmetric Alternating Sign Matrices of given size, it would suffice to compute $\\operatorname{VSASM}(1;1,1,\\ldots ,1)$ , while the ordinary Alternating Sign Matrix Theorem is equivalent to computing $\\operatorname{ASM}(1;1,1,\\ldots ,1)$ ."
]
] | 1403.0535 |
[
[
"CCD BVRI and 2MASS Photometry of the Poorly Studied Open Cluster NGC\n 6631"
],
[
"Abstract Here we have obtained the {\\it BVRI CCD} photometry down to a limiting magnitude of $V \\sim$ 20 for the southern poorly studied open cluster NGC 6631.",
"It is observed from the {\\it 1.88 m} Telescope of Kottamia Observatory in Egypt.",
"About 3300 stars have been observed in an area of $\\sim 10^{\\prime} \\times 10^{\\prime}$ around the cluster center.",
"The main photometric parameters have been estimated and compared with the results that determined for the cluster using {\\it JHKs 2MASS} photometric database.",
"The cluster's diameter is estimated to be 10 arcmin; the reddening E(B-V)= 0.68 $\\pm$ 0.10 mag, E(J-H)= 0.21 $\\pm$ 0.10 mag, the true modulus (m-M)$_{o}$= 12.16 $\\pm$ 0.10 mag, which corresponds to a distance of 2700 $\\pm$125 pc and age of 500 $\\pm$ 50 Myr."
],
[
"Observations and Data Reductions",
"The BVRI CCD photometric observations of the star cluster NGC 6631 were obtained during June 12 to 14, 2012, using the Newtonian focus scale 22.53 arcsec per mm of the 1.88 m Reflector Telescope at Kottamia Observatory in Egypt.",
"Fig.",
"1 represents the image of the cluster in I band.",
"The characteristics of the CCD camera are listed in Table (1), while the observation log is given in Table (2).",
"The images were bias subtracted and flat-fielded using standard procedures in IRAF and the photometry was done using IRAF/DAOPHOT (Stetson, 1987, 1992).",
"The standard stars field SA 110 230 (Landolt, 1992) were observed for standardization and the APPHOT photometry was used to derive the observed magnitudes.",
"Extinction coefficients and zero points were determined to standardize the data.",
"Fig.",
"2 shows the magnitude versus error for the BVRI bands from the DAOPHOT photometry.",
"The total number of stars in the B, V, R and I bands are 1259, 1806, 3043 and 3250 respectively.",
"The limits of errors in our final photometry are 0.363, 0.068, 0.034 and 0.094 respectively.",
"On the other hand, the near-IR JHK data are taken from the digital Two Micron All Sky Survey (2MASS) of Skrutskie et al.",
"(2006).",
"It is uniformly scanning the entire Sky in three IR bands J (1.25 $\\mu $ m), H (1.65 $\\mu $ m) and Ks (2.17 $\\mu $ m).",
"Data extraction has been performed at a preliminary radius of 10 arcmin using the known tool of VizieR for 2MASShttp://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=II/246 database.",
"A cutoff of photometric completeness limit at $J \\ge 16.5$ mag is applied to the data to avoid the over-sampling (cf.",
"Bonatto et al.",
"2004).",
"Also, for photometric quality, stars with errors in J, H and Ks bigger than 0.20 mag have been excluded (cf.",
"Tadross 2011 and references therein).",
"Figure: The CCD I image of open star cluster NGC 6631 as observed by 1.88 m Kottamia Telescope of Egypt.",
"North is up, East on the left.Figure: The BVRI errors of the observed magnitudes for the stars of NGC 6631.Table: The characteristics of CCD camera used in the observations.Table: The Log of Optical Photometric CCD Observations."
],
[
"The Radial Density Profile of the cluster",
"To establish the radial density profile (RDP) of NGC 6631, we counted the stars of the cluster (taken from 2MASS database) within concentric shells in equal incremental steps of 0.1 arcmin from the cluster center.",
"We repeated this process for $0.1<r\\le 0.2$ up to 10 arcmin, i.e.",
"the stellar density is derived out to the preliminary radius of the cluster.",
"The stars of the next steps should be subtracted from the later ones, so that we obtained only the amount of the stars within the relevant shell's area, not a cumulative count.",
"Finally, we divided the star counts in each shell to the area of that shell those stars belong to.",
"The density uncertainties in each shell were calculated using Poisson noise statistics.",
"Fig.",
"3 shows the RDP for NGC 6631 to the maximum angular separation of 5 arcmin where the decay becomes asymptotically at that point.",
"Applying the empirical King model (1966), where it parameterizes the density function $\\rho (r)$ as: $\\rho (r)=f_{bg}+\\frac{f_{0}}{1+(r/r_{c})^{2}}$ where $f_{bg}$ , $f_{0}$ and $r_{c}$ are background, central star density and the core radius of the cluster respectively.",
"In this context, $f_{bg} \\sim $ 36 stars per arcmin$^{2}$ , $f_{0}$ = 42 stars per arcmin$^{2}$ , and $r_{c}$ = 0.59 arcmin.",
"According to the next section, consequently, the radial diameter of the cluster is determined to be 7.85 pc.",
"Figure: The radial density distribution of the stars in NGC 6631.",
"The decay of the density reaches a value of ρ=37\\rho =37 stars/arcmin 2 ^{2} at 5.0 arcmin, where the decay becomes asymptotic.",
"The curved solid line represents the fitting of King (1966) model.",
"Error bars are determined from sampling statistics (1/N1/\\sqrt{N} where N is the number of stars used in the density estimation at that point).",
"The background field density f bg ∼f_{bg} \\sim 36 stars per arcmin 2 ^{2}.",
"The core radius r c r_{c} = 0.59 arcmin."
],
[
"Color-Magnitude Diagrams",
"The Color-Magnitude Diagrams (CMDs) of the observed stars: V$\\sim $ (B–V), V$\\sim $ (V–I), V$\\sim $ (V–R), R$\\sim $ (R–I), and of the obtained JHK-2MASS: J$\\sim $ (J–H) and K$\\sim $ (J–K) are constructed for the cluster.",
"The theoretical isochrones of Padovahttp://stev.oapd.inaf.it/cgi-bin/cmd that computed by Marigo et al.",
"(2008) are used in fitting processes.",
"Several solar isochrones (Z $\\sim $ 0.02) of different ages have been applied to the CMDs of NGC 6631.",
"The best fittings for BVRI diagrams are obtained at distance modulus of 14.20 $\\pm $ 0.10 mag, age of 500 $\\pm $ 50 Myr, and reddening of 0.68, 1.00, 0.54, and 0.47 $\\pm $ 0.10 mag respectively, from left to right as shown in Fig.",
"4.",
"On the other hand, the fittings for JHK-2MASS diagrams are obtained at distance modulus of 12.75 $\\pm $ 0.10 mag, age of 500 $\\pm $ 50 Myr, and reddening of 0.21 and 0.33 $\\pm $ 0.10 mag, from left to right as shown in Fig.",
"5.",
"The resulting total visual absorption is taken from the ratio $A_{v}/E(B-V)$ = 3.1, following Garcia et al.",
"(1988).",
"JHK-2MASS data has been corrected for interstellar reddening using the coefficient ratios $\\frac{A_{J}}{A_{V}}=0.276$ and $\\frac{A_{H}}{A_{V}}=0.176$ , which were derived from absorption rations in Schlegel et al.",
"(1998), while the ratio $\\frac{A_{K_s}}{A_{V}}=0.118$ was derived from Dutra et al.",
"(2002).",
"Applying the calculations of Fiorucci & Munari (2003) for the color excess of 2MASS photometric system; we ended up with the following results: $\\frac{E_{J-H}}{E_{B-V}}=0.309\\pm 0.130$ , $\\frac{E_{J-K_s}}{E_{B-V}}=0.485\\pm 0.150$ , where R$_{V}=\\frac{A_{V}}{E_{B-V}}= 3.1$ .",
"Also, we can de-reddened the distance modulus using these formulae: $\\frac{A_{J}}{E_{B-V}}$ = 0.887, $\\frac{A_{K_s}}{E_{B-V}}$ = 0.322.",
"Therefore, the true distance modulus is calculated to be $(V-M_{v})_{o} =12.16 \\pm 0.10$ mag, corresponding to a distance of $2700\\pm 125$ pc.",
"After estimating the cluster's distance from the Sun, $R_{\\odot }$ , the distance from the galactic center ($R_{g}$ ), the projected distances on the galactic plane from the Sun ($X_{\\odot }~\\&~Y_{\\odot }$ ) and the distance from the galactic plane ($Z_{\\odot }$ ) are estimated to be 6000, –2545, 900 and –8.95 pc respectively.",
"Figure: Theoretical BVRI-isochrones fit to the observed CMDs of NGC 6631.",
"The distance modulus is found to be 14.20 mag, and the color excesses are found to be (from left to right) 0.68, 1.00, 0.54 and 0.47 mag respectively.Figure: Theoretical JHK-isochrones fit to the obtained CMDs of NGC 6631.",
"The distance modulus is found to be 12.75 mag, and the color excesses are found to be (from left to right) 0.21 and 0.33 mag respectively."
],
[
"The Mass Function of NGC 6631",
"It is difficult to determine the membership of the cluster using only the stellar RDP.",
"The stellar membership is found more precisely for those stars are closer to the cluster's center and in the same time very near to the main-sequence (MS) in CMDs.",
"These MS stars are very important in determining the luminosity and mass functions of the investigated cluster.",
"The number of stars per luminosity interval, or in other words, the number of stars in each magnitude bin, gives us what so-called the luminosity function (LF) of the cluster.",
"In order to estimate the LF of NGC 6631, we count the observed stars in terms of absolute magnitude after applying the distance modulus.",
"The magnitude bin intervals are selected to include a reasonable number of stars in each bin and for the best possible statistics of the luminosity and mass functions.",
"From LF, we can infer that the massive bright stars seem to be centrally concentrated more than the low masses and fainter ones (Montgomery et al.",
"1993).",
"The LF and the mass function (MF) are correlated to each other according the known Mass-luminosity relation.",
"The accurate determination of both of them (LF & MF) suffers from some problems e.g.",
"the contamination of field stars; the observed incompleteness at low-luminosity (or low-mass) stars; and mass segregation, which may affect even poorly populated, relatively young clusters (Scalo 1998).",
"On the other hand, the properties and evolution of a star are closely related to its mass, so the determination of the initial mass function (IMF) is needed, that is an important diagnostic tool for studying large quantities of star clusters.",
"IMF is an empirical relation that describes the mass distribution (a histogram of stellar masses) of a population of stars in terms of their theoretical initial mass (the mass they were formed with).",
"The IMF is defined in terms of a power law as follows: $\\frac{dN}{dM} \\propto M^{-\\alpha }$ where $\\frac{dN}{dM}$ is the number of stars of mass interval (M:M+dM), and $\\alpha $ is a dimensionless exponent.",
"The IMF for massive stars ($>$ 1 $M_{\\odot }$ ) has been studied and well established by Salpeter (1955), where $\\alpha $ = 2.35.",
"This form of Salpeter shows that the number of stars in each mass range decreases rapidly with increasing mass.",
"Fig.",
"6 shows that the BVRI and JHK mass functions of NGC 6631, where the slopes of the two MFs close to Salpeter's value.",
"The right panel of Fig.",
"6 seems to complete the left panel.",
"The mean slope of the mass function taken to be 2.3$\\pm $ 0.05.",
"Figure: The BVRI and JHK mass functions of NGC 6631.",
"The slopes of the two panels are close to Salpeter's value, see Sec.",
"5."
],
[
"Conclusion",
"The open star cluster NGC 6631 has been observed using BVRI pass-band of the 1.88 m Kottamia Telescope of Egypt.",
"The main astrophysical properties of the cluster have been estimated and confirmed by the JHK 2MASS bass-band data.",
"It is noted that the determination of the cluster radius made by the uniformity of 2MASS database allow us to obtain reliable data on the projected distribution of stars for large extensions to the clusters' halos.",
"However, a comparison between the results of the present work with those of Ram Sagar (2001) is given in Table 3.",
"Table: Comparison between the present and previous studies.acknowledgements This paper is a part of the project No.",
"STDF-1335; funded by Science & Technology Development Fund (STDF) under the Egyptian Ministry for Scientific Research.",
"The project team expresses their deep appreciation to the administrators of STDF and its organization.",
"REFERENCES Alter, G. et al.",
"1970, Catalogue of star clusters and associations, 2nd ed., Akademiai Kiado, Budapest Bonatto, Ch., Bica, E., Girardi, L. 2004, A&A, 415, 571 Dutra, C., Santiago, B., Bica, E. 2002, A&A, 381, 219 Fiorucci, M., Munari, U.",
"2003, A&A, 401, 781 Garcia, B., Clari$\\acute{a}$ , J., Levato, H. 1988, Ap&SS, 143, 377 King, I.",
"1966, AJ, 71, 64 Landolt, A. U.",
"1992, AJ, 104, 340 Lyngå, G., Palous, J.",
"1987, Astro.",
"Astrophys., 188, 35 Marigo, P., et al.",
"2008, A&A, 482, 883 Montgomery, K.A., Marschall, L.A., Janes, K.A.",
"1993, AJ, 106, 181 Ram Sagar, et al.",
"2001, Bull.",
"Astr.",
"Soc.",
"India, 29, 519 Ruprecht, J.",
"1966, Bulletin of the Astronomical Institute of Czechoslovakia, 17, 33 Salpeter, E. 1955, ApJ, 121, 161 Scalo, J.",
"1998, ASPC, 142, 201 Schlegel, D., et al.",
"1998, ApJ, 500, 525 Skrutskie, M., et al.",
"2006, AJ, 131, 1163 Stetson, P.B.",
"1987, PASP, 99, 191 Stetson, P.B.",
"1992, IAU col. 136, 291 Tadross, A. L. 2011, JKAS, 44, 1"
]
] | 1403.0546 |
[
[
"Ultrafast Dynamics of Massive Dirac Fermions in Bilayer Graphene"
],
[
"Abstract Bilayer graphene is a highly promising material for electronic and optoelectronic applications since it is supporting massive Dirac fermions with a tuneable band gap.",
"However, no consistent picture of the gap's effect on the optical and transport behavior has emerged so far, and it has been proposed that the insulating nature of the gap could be compromised by unavoidable structural defects, by topological in-gap states, or that the electronic structure could be altogether changed by many-body effects.",
"Here we directly follow the excited carriers in bilayer graphene on a femtosecond time scale, using ultrafast time- and angle-resolved photoemission.",
"We find a behavior consistent with a single-particle band gap.",
"Compared to monolayer graphene, the existence of this band gap leads to an increased carrier lifetime in the minimum of the lowest conduction band.",
"This is in sharp contrast to the second sub-state of the conduction band, in which the excited electrons decay through fast, phonon-assisted inter-band transitions."
],
[
"Ultrafast Dynamics of Massive Dirac Fermions in Bilayer Graphene Søren Ulstrup Department of Physics and Astronomy, Interdisciplinary Nanoscience Center (iNANO), Aarhus University, 8000 Aarhus C, Denmark Jens Christian Johannsen Institute of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland Federico Cilento Sincrotrone Trieste, 34149 Trieste, Italy Jill A. Miwa Department of Physics and Astronomy, Interdisciplinary Nanoscience Center (iNANO), Aarhus University, 8000 Aarhus C, Denmark Alberto Crepaldi Sincrotrone Trieste, 34149 Trieste, Italy Michele Zacchigna IOM-CNR Laboratorio TASC, Area Science Park, 34012 Trieste, Italy Cephise Cacho Richard Chapman Emma Springate Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom Samir Mammadov Felix Fromm Christian Raidel Thomas Seyller Institut für Physik, Technische Universität Chemnitz, 09126 Chemnitz, Germany Fulvio Parmigiani Sincrotrone Trieste, 34149 Trieste, Italy Department of Physics, University of Trieste, 34127 Trieste, Italy Marco Grioni Institute of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland Phil D. C. King SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY16 9SS, United Kingdom Philip Hofmann Department of Physics and Astronomy, Interdisciplinary Nanoscience Center (iNANO), Aarhus University, 8000 Aarhus C, Denmark [][email protected] Bilayer graphene is a highly promising material for electronic and optoelectronic applications since it is supporting massive Dirac fermions with a tuneable band gap.",
"However, no consistent picture of the gap's effect on the optical and transport behavior has emerged so far, and it has been proposed that the insulating nature of the gap could be compromised by unavoidable structural defects, by topological in-gap states, or that the electronic structure could be altogether changed by many-body effects.",
"Here we directly follow the excited carriers in bilayer graphene on a femtosecond time scale, using ultrafast time- and angle-resolved photoemission.",
"We find a behavior consistent with a single-particle band gap.",
"Compared to monolayer graphene, the existence of this band gap leads to an increased carrier lifetime in the minimum of the lowest conduction band.",
"This is in sharp contrast to the second sub-state of the conduction band, in which the excited electrons decay through fast, phonon-assisted inter-band transitions.",
"78.67.Wj, 78.47.jh, 79.60.-i, 81.05.ue The lack of a band gap is the most important obstacle to using graphene in electronic devices but this can be elegantly solved in bilayer graphene (BLG) when an asymmetry between the layers is induced by a transverse electric field [1], [2], [3].",
"The promising properties of the thereby induced massive Dirac particles have been intensively explored for the development of semiconducting devices with tuneable band gaps [4], [5], [6] and for efficient photodetectors extending to the THz regime [7], [8], [9], [10].",
"However, the quantitative transport properties of BLG are inconsistent with the simple scenario of a small band gap semiconductor [5] and different hypotheses for this have been given, including broken-symmetry ground states [11] or topological edge state effects [12].",
"Recently, a study of BLG by angle-resolved photoemission spectroscopy (ARPES) has revealed the presence of electronic states throughout the gap, arising from intrinsic AA-stacked domains [13], something that might be expected to short-circuit the gap of BLG.",
"While static ARPES results provide crucial information about the spectral function of BLG, time-resolved spectroscopy is needed to complement the transport studies.",
"Time- and angle-resolved photoemission (TR-ARPES) experiments near the Dirac cone have only recently become technically feasible [14], [15] due to the high photon energies needed to reach the $\\bar{K}$ point in the Brillouin zone.",
"Here we report TR-ARPES measurements carried out using a Ti:sapphire amplified laser system with a repetition rate of 1 kHz.",
"This provided ultrafast infrared pulses with a wavelength of 785 nm, a full width at half-maximum (FWHM) duration of 30 fs, and an energy per pulse of 12 mJ.",
"A part of the laser energy was applied for high harmonic generation of extreme ultraviolet pulses in a pulsed jet of argon gas.",
"A time-preserving monochromator was used to single-out the 13th harmonic with a photon energy of 21 eV.",
"The remaining laser energy was utilized to drive an optical parametric amplifier (HE-Topas), which can provide tuneable pump pulses from the UV to the mid-infrared.",
"We used the second harmonic of the signal to produce 30 fs pulses at 1.55 eV.",
"Both beams were polarized perpendicular to the scattering plane ($s$ -polarized) The data were acquired by sweeping 45 time delay points.",
"Such a cycle was repeated between 700 and 1000 times in order to obtain satisfactory statistics in the data, amounting to total acquisition times between 8 and 14 h per dataset.",
"The experiments were carried out at the Artemis facility at the Central Laser Facility, Rutherford Appleton Laboratory[16].",
"Figure: (Color online) Ultrafast ARPES measurements of bilayer graphene: (a) Normalized photoemission intensity integrated above the Fermi level.",
"Vertical dashed lines mark the times of the dispersion snapshots in (g)-(j).",
"(b)-(e) Diagrams of ultrafast processes and relaxation dynamics involving optical pumping (straight arrows), electron scattering (curled arrows), optical (blue wiggled arrows) and acoustic (green wiggled arrows) phonon scattering.",
"Filled (open) circles signify electrons (holes).",
"(f) Spectrum before the arrival of the pump pulse with the inset showing the acquisition direction for all data (Γ ¯-K ¯\\bar{\\Gamma }-\\bar{K}).",
"(g)-(j) ARPES difference spectra obtained by subtracting the spectrum in (f) from the spectrum taken at the given time.",
"A pump laser fluence of F=1.0F=1.0 mJ/cm 2 ^2 was used.",
"The dashed parabolic bands are the result of a tight binding calculation, and have been added as a guide to the eye.Samples of high quality, so-called quasi-free-standing mono- and bilayer graphene were prepared ex situ by thermal decomposition of SiC substrates followed by hydrogen intercalation to decouple the graphene layers from the substrate.",
"The method is described in detail in Ref.",
"[17].",
"Both samples are hole doped with carrier concentration on the order of $5\\times 10^{12}$ cm$^{-2}$ , which places the Dirac point 240 meV above the Fermi level in the monolayer sample, while in the bilayer the center of the band gap is 180 meV above the Fermi level.",
"Both samples were transferred through air into the TR-ARPES ultrahigh vacuum end station, where they were cleaned by annealing to 550 K in order to remove adsorbed impurities.",
"The samples were held at 90 K throughout the experiment.",
"Fig.",
"REF (a) presents the integrated intensity of photo-induced excited states above the Fermi level during and immediately after the pumping phase.",
"The pump signal induces direct transitions from the two parabolic valence bands to their conduction band counterparts.",
"This pumping phase is characterized by a fast rising edge (see Fig.",
"REF (a)) that establishes the time resolution of our experiment to be 40 fs.",
"The excited electrons quickly thermalize, consistent with rapid Auger-like processes [18], [19], [14] as depicted in Fig.",
"REF (b).",
"This leaves a dense transient population of hot electrons in both conduction bands as sketched in Fig.",
"REF (c).",
"The electrons then quickly cascade down to the region around the band gap, thereby depleting the conduction bands by emission of phonons (Fig.",
"REF (d)).",
"This process continues (Fig.",
"REF (e)) until all of the excited electrons have recombined with the remaining holes.",
"Snapshots of these events are presented in Figs.",
"REF (g)-(j), displaying the ARPES difference spectra between the equilibrium signal before the arrival of the pump (Fig.",
"REF (f)) and the signal at the given time delay.",
"The bare band dispersion lines obtained by a tight binding calculation (see Ref.",
"[1] and Supplementary Material) are plotted on top of the difference spectra to match the intensity with the band structure.",
"The two bilayer bands are clearly discerned in the raw data as well as in the valence band part of the difference spectra.",
"Note that we do not observe discrete spikes in the intensity distribution at the energies corresponding to the direct transition, not even on the midpoint of the rising edge at $t=0$ fs (see Fig.",
"REF (g)), but rather an even intensity distribution along the bands.",
"This implies that the initial thermalization of the hot electrons by electron-electron scattering occurs on timescales less than 40 fs, too fast to be resolved here.",
"Once the peak distribution is reached at $t=40$ fs (see Fig.",
"REF (h)), the phonon scattering processes lead to a fast decrease of intensity as observed in Figs.",
"REF (i)-(j).",
"Within the considered time window, the described behavior is qualitatively identical for the laser pump fluence of $F=$ 1.0 mJ/cm$^2$ used for acquiring the data in Fig.",
"REF and for another data set with $F=3.5$ mJ/cm$^2$ (see Supplementary Material for the high fluence data.).",
"However, with the higher fluence a larger density of photoexcited electrons is created, which allows us to study a larger population of hot electrons in the two conduction bands.",
"A quantitative study of this is introduced in Fig.",
"REF .",
"Here we compare the time-dependent experimental data with simulations where we populate the conduction bands thermally by multiplying a model spectral function with a single high temperature Fermi-Dirac (FD) function (see Supplementary Material for details on the simulations).",
"Figure: (Color online) Disentangling the dynamics in the bilayer bands: (a), (e) Photoemission intensity integrated between the dashed lines in (b) and (f) as a function of binding energy.",
"Markers correspond to data points at the given time delay, and lines are integrated intensity from simulated photoemission data with the given temperatures.",
"(b), (f) Experimental photoemssion data at the peak excitation signal at time t=40t = 40 fs.",
"(c), (g) Simulated data at the given temperature taking the experimental resolution into account.",
"(d), (h) Simulated data without resolution broadening.",
"The vertical dashed lines in (a) and (e) and the horizontal dashed lines in (d) and (h) mark the onset energy of the two conduction bands.",
"The data in (a)-(d) correspond to a fluence of 1.0 mJ/cm 2 ^2 while the data in (e)-(h) correspond to 3.5 mJ/cm 2 ^2.We extract the combined statistical distribution of the carriers in the four bands at a given time delay by integrating the photoemission intensity in between the dashed lines in Figs.",
"REF (b) and REF (f) [20].",
"Interestingly, the integrated distribution curves (IDCs) of the transient spectra presented in Figs.",
"REF (a) and REF (e) do not appear to be merely described by a featureless hot electron FD tail, but a clear shoulder can be discerned in the raw data.",
"The extent of this feature is a function of time and of fluence.",
"As it is not possible to extract the temperature from the data by a fit to a simple FD function, we do so by matching the observed photoemission intensity to a simulation, consisting of the calculated spectral function that is multiplied by a hot FD function and broadened by the experimental resolution.",
"Note that the intensity variation between the bands and within the bands originate from the photoemission matrix elements that are accounted for in the simulated spectral functions.",
"These matrix elements are assumed to be energy- and band-dependent but independent of the fluence or time, as described further in the Supplementary Material.",
"We find that the electrons are heated from the initial temperature of 90 K, which is the sample temperature during the experiment, to reach temperatures of 2900 K and 4200 K for low and high fluence, respectively.",
"A remarkable agreement between experimental and simulated data is obtained at both low and high fluence.",
"It is a central result that the complex shape of the IDCs from the peak signal at $t = 40$ fs and onwards in time can be described by this simple simulation.",
"The observed shoulder in the IDCs always appears at the same energy which coincides with the onset of the inner conduction band, as seen by comparing the IDCs to the non-broadened BLG spectral function via the dashed lines in Fig.",
"REF .",
"The shoulder can therefore be ascribed to the accumulation of electrons in the upper conduction band, confirming the consistency of the calculated spectral function with the data.",
"The fact that all the IDCs can be fitted with this simple model also suggests that the excited carriers in both conduction bands are in a quasi-equilibrium state within our time resolution.",
"Figure: (Color online) Comparing the relaxation dynamics of massive and massless Dirac Fermions: (a) Spectrum before the arrival of the pump pulse, and (b)-(d) difference spectra at selected time delays for BLG.",
"(f)-(i) Corresponding data for MLG.",
"Dispersion lines from tight binding models have been added as guide to the eye.",
"(e), (j) Normalized photoemission intensity binned over the boxes in (b)-(d) for (e) BLG and (g)-(i) for (j) MLG.",
"The time traces have been labelled with a number according to the box numbers, and the given time constants are results of single exponential fits to the corresponding trace.",
"The error bar on these fits amounts to ±20\\pm 20 fs.",
"The fluence is 0.6 mJ/cm 2 ^2 for all data presented here.The model presented in Fig.",
"REF integrates over all the sub-bands in the system and can thus neither be used to study possible differences in the sub-bands, nor describe lifetime effects that do not follow a FD distribution.",
"A more detailed analysis and a direct comparison to the situation in MLG is therefore shown in Fig.",
"REF .",
"For both materials, the Dirac point imposes a bottleneck for relaxation processes, but the key-consequence of a true band gap in BLG would be to impose a serious constriction for the decay of hot electrons, e.g.",
"by scattering with low-energy acoustic phonons.",
"To avoid any complications from high fluence effects, as described in the supplementary material, we consider both MLG and BLG at a low fluence of $0.6$ mJ/cm$^2$ .",
"The equilibrium spectra for BLG and MLG at this fluence are shown in Figs.",
"REF (a) and 3(f), and the difference spectra from $t=40$ fs to long time delays of $t=2000$ fs are shown in Figs.",
"REF (b)-(d) and 3(g)-(i), respectively.",
"Again we show the tight binding bands as a guide to the eye.",
"The timescales of the relaxation processes are directly extracted by analysing the intensity binned in small boxes following the dispersion, as shown in the difference spectra in Fig.",
"REF .",
"We emphasize that this analysis, contrary to the simple model in Fig.",
"2, provides access to the spectrally resolved dynamics at specified binding energies [21], capturing dynamics that would otherwise be integrated out using the model.",
"The time-dependent intensity for all boxes is shown in Figs.",
"REF (e) and 3(j) and we find that it can be described by a simple exponential decay, apart from box 1, the box of the conduction band minimum in both samples (Note that the location of box 1 has been defined such that it is at equal distance from the center of the gap and the Dirac point in BLG and MLG, respectively).",
"We shall discuss the intensity in this spectral region later.",
"The electron population in the upper conduction band in BLG is tracked by boxes 5 and 6 in Fig.",
"REF (e) and we find a fast decay with a time constant of 300 and 200 fs, respectively.",
"For box 6, this is expected because the electrons can cascade down to the minimum of the upper conduction band.",
"For box 5, the fast decay can only be explained by phonon-induced inter-band scattering to the lower conduction band.",
"A similarly fast decay as for boxes 5 and 6 is found for corresponding boxes in the lower conduction band (boxes 3-4), suggesting that the inter-band scattering is sufficiently effective to avoid any electron accumulation near the upper conduction band edge.",
"Indeed, this is also responsible for the fact that the simple FD fit works in Fig.",
"REF .",
"The decay times for the corresponding energies in MLG are also similar.",
"The situation is more complicated for box 1 in both materials where the intensity decay cannot be fit to a single exponential.",
"The time-dependent intensity of box 1 is shown in Fig.",
"REF up to a measured delay time of 4 ps.",
"For this spectral region, the very bottom of the conduction band, pronounced differences for BLG and MLG are found.",
"Both data sets can be well-described by a double-exponential decay where an initial fast decay with a decay time of 650 fs (400 fs) for BLG (MLG) is followed by a slow decay with 10,000 fs (3,500 fs).",
"Indeed, the slow decay for BLG is so slow that a precise determination of the decay constant is not possible within our experimental time window and the value of 10,000 fs merely sets a lower limit for the long decay time.",
"The pronounced difference between the two materials can actually already be seen in the raw data in Figs.",
"REF (d) and REF (i).",
"In BLG there is still intensity from hot electrons in box 1 while in MLG the whole spectrum is fully relaxed.",
"Figure: (Color online) Gap dependent relaxation times: (a) Long time traces of the integrated intensity in box 1 in Fig.",
"3 for both BLG and MLG.",
"Lines correspond to double-exponential function fits with the given time constants.",
"(b), (c) Sketch of the relaxation dynamics involving hot electrons (red spheres), holes (yellow spheres) and phonon emission (wiggled arrows) around the Dirac point for (b) massive Dirac Fermions and (c) massless Dirac Fermions.The true difference in the hot electron carrier dynamics thus occurs mainly at the Dirac point - an experimental verification that is only possible using this direct momentum-resolved technique.",
"This result is consistent with the naive expectation of a substantially longer decay time in the presence of a gap.",
"Without a gap, hot electrons can cascade down continuously via acoustic phonon scattering whereas the presence of the gap requires relaxations by optical phonons, giving rise to a bottleneck for the decay.",
"The efficiency of a relaxation across the gap might be further restricted by the role of the pseudospin in BLG [1].",
"An intuitive picture of these differences between BLG and MLG is depicted in Figs.",
"REF (b) and (c).",
"In view of the disputed character of the electronic structure of BLG and the role of possible in-gap states, we should also point out that the spectral function fits and the data analysis in Fig.",
"REF are perfectly consistent with the expected single-particle tight-binding band structure.",
"The possibility of in-gap states still exists, especially that of gap-crossing states in AA-stacked domains, since these were identified for the same kind of BLG samples used here [13].",
"In order to obtain a quantitative estimate of the impact these defects have on the lifetime of the carriers, a calculation of the carrier lifetime for the perfect crystalline system would be required.",
"Nevertheless, the results shown here reveal that, even in the presence of such gap states, the band gap leads to a markedly enhanced lifetime over MLG.",
"We gratefully acknowledge financial support from the VILLUM foundation, The Danish Council for Independent Research / Technology and Production Sciences, the Lundbeck Foundation, the Swiss National Science Foundation (NSF), EPSRC, The Royal Society and the Italian Ministry of University and Research (Grants No.",
"FIRBRBAP045JF2 and No.",
"FIRB-RBAP06AWK3).",
"Access to the Artemis Facility was funded by STFC.",
"Work in Erlangen and Chemnitz was supported by the European Union through the project ConceptGraphene, and by the German Research Foundation in the framework of the SPP 1459 Graphene."
]
] | 1403.0122 |
[
[
"Detection of distinct power spectra in soft and hard X-ray bands in the\n hard state of GRS 1915+105"
],
[
"Abstract The well-known black hole X-ray binary GRS 1915+105 is a unique source in the sense that it cannot be classified within the standard picture of black hole binary states.",
"In this work we study archival XMM-Newton observations taken between 2003 and 2004 of the \\c{hi} variability class of GRS 1915+105, which corresponds to the hard state in the standard black hole X-ray binary state classification.",
"The crucial point of our study is that by using XMM-Newton data we can access the variability below 3 keV, an energy range that is not covered with RXTE.",
"We focus on the study of the power spectral shape in the soft and hard X-ray band, in light of our work done with Swift on MAXI J1659-152.",
"In the hard band (above 2.5 keV) power density spectra consist of band-limited noise and quasi-periodic oscillations, corresponding to the power spectral shape seen in the hard or intermediate state, while in the soft band the averaged power density spectrum is consistent with a power-law noise, corresponding to the power spectral shape usually seen in the soft state.",
"The coexisting of two different power spectral shapes in the soft and hard band, where the soft band power spectrum is dominated by a power-law noise, is consistent with MAXI J1659-152, and confirms the energy dependence of power spectral states.",
"Our additional spectral analysis shows that the disc component does contribute to the soft band flux.",
"These findings support that the observed black hole power spectral state depends on which spectral component we are looking at, which implies that power spectral analysis is probably a more sensitive method than spectral modeling to trace the emergence of the disc component in the hard or intermediate state."
],
[
"Introduction",
"The known population of low-mass black hole X-ray binaries mainly consists of transient sources, that can be studied only during outburst, as they are too faint to detect their variability reliably with present X-ray instruments during quiescence [14].",
"The outbursts begin and end in the so-called low-hard state (LHS) and in between there is normally a transition to the high-soft state (HSS).",
"All these states show characteristic timing and spectral properties.",
"In the LHS the rms variability is larger than ten per cent and the power spectrum shows one or more band-limited noise (BLN) components and sometimes a specific timing feature named type-C quasi-periodic oscillations (QPOs) can be observed [5].",
"The power spectrum of the HSS is well-described by a power-law noise (PLN) component, sometimes with a break around 10 Hz, and the rms variability is at a few percent [15].",
"In a recent study [39], we showed an energy dependence of the power spectra in the black hole candidate MAXI J1659-152.",
"The source was unique in several respects.",
"The spectral evolution was slow and the source was at a high latitude, allowing us to study the emergence of the soft disc component and how the soft component came in just before the transition to the soft state.",
"We investigated energy and power density spectra of Swift/XRT (0.3 – 2 keV) and RXTE/PCA (2 – 60 keV) observations that covered the outburst rise from the LHS to the HSS.",
"During the LHS the power density spectra in the 0.01 to 20 Hz range can be well described by BLN and QPOs in both, the soft and the hard, X-ray bands.",
"With the onset of the hard intermediate state, which coincided with a disc fraction exceeding $\\sim $ 30% in the 0.3 – 2.0 keV rangeThis fraction has to be taken with caution as it is obtained from fitting a disc blackbody plus power-law model to the data, without taken into account that the power-law model should have a low energy cutoff to mimic Comptonization., this changed dramatically.",
"While BLN and QPOs are still present above 2 keV, below 2 keV the power spectra are now dominated by PLN, as commonly observed in the HSS.",
"This suggests that the photons responsible for the BLN and the QPO origin from the innermost hot flow subjected to Comptonization, while the photons responsible for the PLN can be related to the optically thick disc.",
"Furthermore, we tried to constrain cut-off energies for the PLN and BLN plus QPO components, investigating contributions of each component in the 2 – 4 keV XRT band and at PCA energies below 5 keV.",
"Based on these investigations we could constrain the cut-offs to occur in the 2.8 – 3.5 keV range.",
"The well known black hole low mass X-ray binary GRS 1915+105 [13] was initially discovered by the WATCH instrument on-board GRANAT in 1992 [8].",
"Since then, GRS 1915+105 has been observed densely at different wavelength ranges.",
"A systematic monitoring in the X-rays revealed a rich pattern of variability on all time scales.",
"[3] identified 12 classes of variability and showed that, though complex, the behaviour of GRS 1915+105 can be understood as transitions between three basic spectral states A, B, C. Despite its many distinct accretion states, GRS 1915+105 appears similar to other black hole binaries [29], [35].",
"The closest analogue to the conventional canonical “low hard\" state in other X-ray binaries is the $\\chi $ variability class, that is found exclusively in the C state [27].",
"In this state low frequency QPOs are present [9], which energy spectrum consists with that of the hard component [20].",
"Correlations of the centroid frequency with the power-law index [36] and with the inner-disc radius [32], [33] have been conducted and it was shown that the centroid frequency correlates positively with the flux of the disc component [25].",
"For completeness, we would like to mention that with IGR J17091-3624, a second source is known, that shows variability similar to the variability classes observed in GRS 1915+105 [1], [38], [7], [26], [28].",
"In this paper we make use of archival XMM-Newton observations to study the properties of power density spectra in different energy bands.",
"Specifically, we intend to study the power spectra of the soft disc component in the energy range below $\\sim $ 2 keV and to check if the soft band variability in the $\\chi $ class is indeed not only different from that in the hard band but also shows a power-law noise component similar to that in the soft state.",
"XMM-Newton observed GRS 1915+105 several times in 2003, 2004 [21], and 2007.",
"All but one of these observations are taken with the pn detector in burst mode.",
"This mode was chosen because of the high source flux of GRS 1915+105.",
"The only observation which has been taken with EPIC/pn in Timing mode (2004 April 17) suffers from frequent telemetry drop-outs and can hence not be used in our timing study [21].",
"For our study we selected five observations of GRS 1915+105 being in the $\\chi $ variability class.",
"Details on these observations can be found in Table REF .",
"We used the standard SAS (version 13.0.0) tools to filter and extract pn event files, paying particular attention to extract the list of photons not randomized in time.",
"For our timing study we selected the longest, continuous exposure available in each observation (see table REF ), i. e. the longest available standard good time interval.",
"As all observations are taken in burst mode, we selected photons from a 15 column wide strip in RAWX centered on the column with the highest count rate, and we impose RAWY $<$ 150 to avoid direct illumination by the source.",
"We selected single and double events (PATTERN$<=$ 4).",
"We made use of the SAS task epatplot to investigate whether the observations are affected by pile-up.",
"As there is a clear deviation from the theoretical predictions at energies below 1.5 keV, we focused our investigations on energies above 1.5 keV, where the observed pattern distributions follow the theoretical predictions.",
"Applying this selection we only exclude 3 – 4 % of the source photons, as GRS 1915+105 is highly absorbed at energies below 1.5 keV [21].",
"All values given in this paper are 1 $\\sigma $ values.",
"Table: Details of XMM-Newton observationsTable: Details of RXTE observationsFigure: Power density spectra of all five XMM-Newton observations in the 4.5 – 8 keV band.",
"The best fit model is indicated by a solid line and the individual components are given as dashed lines.Figure: Shown are rms spectra of the QPO of the five XMM-Newton χ\\chi variability class observations investigated in this study (frequency range: 1.8×10 -3 1.8\\times 10^{-3} – 34 Hz).",
"For both observations taken in 2004, the rms spectrum of the main QPO, as well as its upper harmonic (indicated by an \"X\") is given.",
"For comparison the rms spectra of the main QPO and its upper harmonic derived from RXTE data are indicated by gray triangles and squares.We extracted power density spectra (PDS) for each observation in four energy bands: 1.5 – 2.5 keV, 2.5 – 3.5 keV, 3.5 – 4.5 keV, and 4.5 – 8.0 keV.",
"These four energy bands contain about 15%, 20%, 20%, 35 – 40% of the source photons, respectively.",
"We investigated the noise level at frequencies above 30 Hz and found that it is consistent with 2, as expected for Poissonian noise [40].",
"We subtracted the contribution due to Poissonian noise, normalised the PDS according to [18] and converted to square fractional rms [2].",
"As in [39] the PDS were fitted with models composed of a power-law noise, zero-centered Lorentzians for BLN components, and Lorentzians for QPOs."
],
[
"Spectral investigation",
"We also extracted energy spectra from all XMM-Newton observations, following the procedure to extract spectra from XMM-Newton burst mode data outlined in [17].",
"As for the timing study we selected source photons from a 15 column wide strip in RAWX centered on the column with the highest count rate, including single and double events.",
"Of course, no energy selection has been applied here, and we used RAWY $<$ 140 as suggested by [17].",
"Background spectra have been extracted from columns 3 to 5.",
"Energy spectra obtained form XMM-Newton EPIC-pn fast-readout modes are known to be affected by gain shift due to Charge-transfer inefficiency (CTI) which leads to an apparent shift of the instrumental edges visible at low energies.",
"This shift can be corrected by applying the SAS task epfast to the data.",
"However, epfast is likely unsuited to do CTI corrections at higher energies at present, as it applies an energy-independent correction, which leads to an over-correction at higher energies.",
"This leads to a striking difference between the RXTE/PCA and the epfast-corrected XMM-Newton/pn spectra.",
"In contrast, the unmodified XMM-Newton spectrum is quite similar to the RXTE/PCA spectrum.",
"This deviation of the spectral shape above $\\sim $ 6 keV due to the application of epfast has been already reported for simultaneous XMM-Newton and BeppoSAX data [37].",
"For three of the XMM-Newton observations (Obs.",
"B, C, and E) RXTE data taken on the same day are available.",
"Details on the observation with the longest exposure taken on the same day as an XMM-Newton observation are given in table REF .",
"For the remaining two XMM-Newton observations (Obs.",
"A and D) the RXTE observation located closest in time is two days away.",
"We refrain from including these observations in our study."
],
[
"Timing analysis",
"The variability study of the RXTE observations is based on data from the Proportional Counter Array (PCA).",
"We computed power density spectra (PDS) for each observation following the procedure outlined in [4].",
"PDS production has been limited to the PCA channel band 0 – 35 (2 – 15 keV) and used 16 second long stretches of Event mode data.",
"As for the XMM-Newton data, we subtracted the contribution due to Poissonian noise [40], normalised the PDS according to [18] and converted to square fractional rms [2].",
"Figure: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION figurePower density spectra of the three RXTE observations in the 4.9 – 14.8 keV band.",
"The best fit model is indicated by a solid line and the individual components are given as dashed lines.",
"The letter in the upper right corner indicates the corresponding XMM-Newton observation."
],
[
"Spectral investigation",
"We used the PCA Standard 2 mode (STD2), which covers the 2–60 keV energy range with 129 channels for the spectral analysis.",
"The standard RXTE software within heasoft V. 6.13 was used to extract background and dead-time corrected energy spectra for each observation, following [34].",
"Solely Proportional Counter Unit 2 from the PCA was used since only this unit was on during all the observations.",
"To account for residual uncertainties in the instrument calibration a systematic error of 0.6 per cent was added to the PCA spectraA detailed discussion on PCA calibration issues can be found at: http://www.universe.nasa.gov/xrays/programs/rxte/pca/doc/rmf/pcarmf-11.7/."
],
[
"Power density spectra above 2.5 keV",
"We fitted the PDS using a model consisting of a zero centered Lorentzian for the BLN.",
"In the XMM-Newton 4.5 – 8 keV band QPOs are clearly visible in each observation (see Fig.",
"REF , and table REF ), which have been fitted with a Lorentzian centered on the QPO frequency.",
"The centroid frequencies lie in a range of 0.6 to 1.3 Hz.",
"The fit statistics with this model is of $\\chi ^2/\\nu = $ 28/26, 30/25, 16/21, 17/27, 23/25 for Obs.",
"A–E, respectively.",
"In Obs.",
"A and B there is an excess at low frequencies.",
"Fitting it with a power-law yields a non-significant component and a change in the fractional rms of $\\sim $ 0.3 % in Obs.",
"B.",
"Adding a power law component in Obs.",
"A affects the fit of the whole spectrum and leads to a decrease of the overall variability by $\\sim $ 3.0 %.",
"The feature observed in the PDS of the 4.5 – 8 keV band (BLN and QPOs) are also detected in the 2.5 – 3.5 keV and 3.5 – 4.5 keV bands, and the model used for the highest energy band gives also decent fits of the PDS in these bands.",
"In these energy ranges no additional power-law component is needed in Obs.",
"A and B.",
"The parameters obtained for the BLN and QPO in all three energy bands above 2.5 keV are given in table REF .",
"In Fig.",
"REF rms spectra of all five XMM-Newton observations and for comparison the rms spectra of the RXTE observations are shown.",
"The rms spectra of the main QPO obtained form RXTE data show a monotonic increase with energy and a flattening towards higher energies [31].",
"In Obs.",
"A, C, and E the rms variability is highest in the 3.5 – 4.5 keV band.",
"While the rms values obtained from XMM-Newton and RXTE in Obs.",
"B are consistent within errors, the XMM-Newton values in Obs.",
"E are systematically higher than those obtained from RXTE data.",
"The discrepancy of the XMM-Newton and RXTE rms values in the 3.5 – 4.5 keV band in Obs.",
"C is most likely related to the poor determinability of the contribution of the BLN component in this energy band.",
"In Obs.",
"D we observe a monotonic increase in the rms variability of the main QPO peak with energy.",
"We also noticed that in the 2.5 – 3.5 keV band the QPO peak is broader than in the 4.5 – 8 keV band.",
"Inspired by the results obtained for the PDS below 2.5 keV (see Sect.",
"REF ), we tried to fit the PDS in the 2.5 – 3.5 keV band using a model, where we substitute the zero centered Lorentzian for the BLN with a power-law component.",
"With this substitution the Lorentzian competent to fit the upper harmonic is no longer needed.",
"The rms variability of the upper harmonic in Obs.",
"D and E is lowest in the 3.5 – 4.5 keV band, and a similar behaviour is observed in Obs.",
"B.",
"For all observations the centroid QPO frequency stays constant within errors in all bands.",
"In addition, we investigated the PDS of the three RXTE observations in the 4.9 to 14.8 keV range unsung the same frequency range as for the XMM-Newton observations ($1.8\\times 10^{-3}$ – 34 Hz).",
"The PDS are shown in Fig.",
"REF , and their overall shape agrees with the one found from XMM-Newton data.",
"In the RXTE data, an upper harmonic is present in all three observations (parameters are given in table REF ), although it is less prominent in the first two observations than in the observation corresponding to Obs.",
"E, where the upper harmonic is seen with XMM-Newton.",
"In the third RXTE observation an additional power-law component was needed to obtain an acceptable fit at the lowest frequencies.",
"All three RXTE observations require the addition of a second upper harmonic to obtain acceptable fits.",
"Table: Variability parameters at energies above 2.5 keV"
],
[
"Power density spectra below 2.5 keV",
"For the lowest energy band (1.5 – 2.5 keV), we had to average the PDS of all five observations to obtain decent fit statistics.",
"Assuming a power-law noise, we obtained a decent fit, without statistical need of a zero centered Lorentzian component, with $\\chi ^2/\\nu = $ 16/14 and a power-law index of $0.7_{-0.1}^{+0.2}$ .",
"Futhermore, the QPO, seen clearly at energies above 2.5 keV, seems not to be present in this energy band.",
"The averaged PDS of the 1.5 – 2.5 keV band is shown in Fig.",
"REF together with the averaged PDS of the 1.5 – 8 keV band.",
"Adding a Lorentzian component with the centroid frequency (1.37 Hz) and width fixed at the values obtained from a fit to the averaged PDS in the 1.5 – 8 keV band, we obtain a 1 $\\sigma $ upper limit for the QPO amplitude of 7.2 % rms.",
"This value lies a little bit below the QPO amplitude value at soft energies obtained from [31] for a similar centroid frequency; $\\sim $ 8 – 9 % rms.",
"However, the upper limit we obtained from the XMM-Newton data is not very stringent.",
"We also tried to fit the PDS with a zero centered Lorentzian instead of the power-law, which results in a $\\chi ^2/\\nu $ of 11/14.",
"We find a break frequency of the Lorentzian of 0.45$^{+0.65}_{-0.21}$ Hz, which is clearly below the break frequency of the Lorentzian in the 1.5 – 8 keV band (3.35$^{+0.52}_{-0.49}$ Hz).",
"Figure: Energy spectra and residuals of all five XMM-Newton observations.",
"For Obs.",
"B, C, and E the simultaneous RXTE/PCA data are included in the fit.",
"The best fit model as well as the contribution of the disc blackbody emission and of the power law with reflection component are indicated by solid lines.Table: Selected spectral parameters"
],
[
"Spectral results",
"Furthermore, the presence of a power-law component in the PDS below 2.5 keV, which is commonly observed in the HSS where the energy spectrum is dominated by emission form the accretion disc, suggests that a disc component should be present in the energy spectra, as inferred from the study of MAXI J1659-152 [39].",
"This is in contrast to the results presented in [21], where only a reflected power-law component (but no direct disc emission) was need to obtain acceptable fits, using solely XMM-Newton data.",
"Hence, we fitted combined XMM-Newton/EPIC-pn+RXTE/PCA spectra within isis V. 1.6.2 [16] in the 0.5 – 10 keV and 5 – 60 keV range.",
"We fitted the spectra with the model used in [21], consisting of a power law with reflection component (refsch in Xspec) modified by cold absorption, several emission features, and an additional component to model the 1 keV excess.",
"Using an unmodified XMM-Newton spectrum plus an RXTE/PCA spectrum or an epfast-corrected XMM-Newton spectrum plus an RXTE/PCA spectrum, the features present in the data do not allow us to obtain a reduced $\\chi ^2$ below 2 (see Sect.",
"REF ).",
"To get formally acceptable fits we applied epfast to the EPIC-pn spectrum and ignored energies above 6 keV in this spectrum, while the RXTE/PCA spectrum was used in the 5 – 60 keV range.",
"For Obs.",
"A and D where no simultaneous RXTE data are available XMM-Newton data are used up to 10 keV.",
"We grouped the XMM-Newton data to contain at least 20 channels per bin.",
"Both XMM-Newton and RXTE data are grouped to have a signal-to-noise ratio larger than three per bin.",
"The energy spectra and residuals of all five observations with their best fit model are shown in Fig.",
"REF .The $\\chi ^2/\\nu $ are 98/95, 202/130, 172/123, 137/95, and 150/126 for Obs.",
"A, B, C, D, and E, respectively.",
"The fold energy obtained form fits of combined XMM-Newton and RXTE data are outside the energy range covered by XMM-Newton (in the range of 11 – 12 keV for Obs.",
"B and C and $\\sim $ 14 keV for Obs.",
"E).",
"That is why we fixed the fold energy in Obs.",
"A and D at 11 and 14 keV, respectively.",
"The obtained photon index, inner disc temperature and inner disc radius are given in tabel REF .",
"The disc component always peaks around 1.5 – 2.5 keV.",
"In most observations the flux from the disc component in the 1.5 – 2.5 keV band makes up about 20 – 30 % of the total flux, while in the 2.5 – 3.5 keV band the contribution of the disc component reduces to a few per cent.",
"This finding puts additional support on the presence of a power-law component in the PDS below 2.5 keV, as we found a power-law noise in MAXI J1659-152 at a disc fraction exceeding $\\sim $ 30 % in the 0.3 – 2 keV band [39].",
"It is worth noting that an extension of the power-law spectral component to soft energies is not physical.",
"Using a simple power-law spectral component in the spectral fit would actually underestimate the disc component.",
"We used Obs.",
"E, that has the longest exposure of all three XMM-Newton observations with simultaneous RXTE data, to verify that in all three cases – unmodified XMM-Newton spectrum plus RXTE/PCA spectrum, epfast-corrected XMM-Newton spectrum plus RXTE/PCA spectrum, and epfast-corrected XMM-Newton spectrum below 6 keV plus RXTE/PCA spectrum above 5 keV – the addition of a multicolor disc blackbody component leads to a significant improvement of the fit."
],
[
"Discussion & Conclusion",
"We studied studied archival XMM-Newton data of GRS 1915+105 during its $\\chi $ variability class obtained in 2003 and 2004.",
"The focus of our study was put on an investigation of the power spectral shape in different X-ray energy bands, in the light of our work done with Swift on MAXI J1659-152 [39].",
"We found that while the PDS at energies above 2.5 keV is dominated by BLN plus QPO, corresponding to the power spectral shape seen in the hard or intermediate state, the PDS in the energy range between 1.5 and 2.5 keV shows power-law noise, which corresponds to the power spectral shape usually seen in the soft state.",
"A similar existence of two distinct power spectral states, BLN plus QPO above 2 keV and PLN below 2 keV, has been found for MAXI J1659–152 in its hard intermediate state [39].",
"Our result that the PDS of the $\\chi $ class of GRS 1915+105 shows a similar energy dependence as the hard intermediate state in MAXI J1659–152 fits well into the known connection of the $\\chi $ variability class in GRS 1915+105 with the intermediate state (preferentially close to the hard state) in other black hole X-ray binaries [29], [27].",
"In the study of MAXI J1659-152, [39] have found that the power-law noise in the soft band seems to have a cut-off at or below the QPO and BLN break frequency.",
"The XMM-Newton data of GRS 1915+105 do not allow us to determine if there is such a cut-off.",
"Studying RXTE PDS in individual energy bands, only those of Obs.",
"3 (the one corresponding to XMM-Newton Obs.",
"E) show at energies below 4.5 keV a clear deviation from a BLN plus QPO shape at frequencies below $\\sim $ 0.2 Hz that can be described by a power-law component.",
"The frequency at which the deviation from the BLN plus QPO shape occurs decreases with increasing energy.",
"However, this finding does not allow to draw conclusions on a possible cut-off in the 1.5 – 2.5 keV band, as the band covered by RXTE is too broad (up to 4.5 keV; and data with a higher energy resolution are not available) and the presence of an additional power-law noise is only found in one out of three RXTE observations.",
"In summary, the observations of GRS 1915+105 show similar energy-dependent power spectral states as in MAXI J1659-152, which means that two different power spectral shapes are coexisting in the hard and soft band simultaneously.",
"In the soft band, which is contributed by emission from the thermal disc component, not only the variability amplitude is lower, as has been known before, but the power spectral shape is of a power-law shape.",
"Such an energy dependence reveals a geometry in which the photons in the soft energy band and in the hard energy band come from different locations in the system, i. e. the cold optically thick accretion disc and the region of Comptonization of hot electrons (being it either an optically thin hot corona [12] or a jet flow [19]), respectively.",
"The inner radius at which the cold disc component ends would be determined by future accurate measurements of the power-law noise in the soft band.",
"Notice that the radius at which the cold disc ends may be not the radius at which the cold disc terminates [11], [22], [10], since a hot flow or corona would cover the innermost cold disc so the radius one determines from the cut-off frequency of the power-law noise would correspond to the radius to which the hot flow or corona extends [6], [23], [24], [30].",
"In conclusion, the energy dependence of the power spectral state found supports the idea that the observed power spectral state depends on which spectral component (and thus the geometrical location – disc or corona) we are looking at, and that a multi-wavelength picture of power spectra in black hole X-ray binaries is needed.",
"The important consequence of such an energy-dependent picture of black hole power spectral state is that power spectral analysis in the soft X-ray energy band is more sensitive to the emergence of the disc component in the hard or intermediate state than energy spectral analysis, which in many circumstances is model dependent."
],
[
"Acknowledgments",
"We would like to thank Tomaso Belloni, Masaru Matsuoka, Phil Kaaret, Mike Nowak, and Deepto Chakrabarty for comments and useful discussions.",
"This work was supported by the National Natural Science Foundation of China under grant No.",
"11073043, 11333005, and 11350110498, by Strategic Priority Research Program \"The Emergence of Cosmological Structures\" under Grant No.",
"XDB09000000 and the XTP project under Grant No.",
"XDA04060604, by the Shanghai Astronomical Observatory Key Project and by the Chinese Academy of Sciences Fellowship for Young International Scientists Grant."
]
] | 1403.0591 |
[
[
"Exhausting the Information: Novel Bayesian Combination of Photometric\n Redshift PDFs"
],
[
"Abstract The estimation and utilization of photometric redshift probability density functions (photo-$z$ PDFs) has become increasingly important over the last few years and currently there exist a wide variety of algorithms to compute photo-$z$'s, each with their own strengths and weaknesses.",
"In this paper, we present a novel and efficient Bayesian framework that combines the results from different photo-$z$ techniques into a more powerful and robust estimate by maximizing the information from the photometric data.",
"To demonstrate this we use a supervised machine learning technique based on random forest, an unsupervised method based on self-organizing maps, and a standard template fitting method but can be easily extend to other existing techniques.",
"We use data from the DEEP2 and the SDSS surveys to explore different methods for combining the predictions from these techniques.",
"By using different performance metrics, we demonstrate that we can improve the accuracy of our final photo-$z$ estimate over the best input technique, that the fraction of outliers is reduced, and that the identification of outliers is significantly improved when we apply a Na\\\"{\\i}ve Bayes Classifier to this combined information.",
"Our more robust and accurate photo-$z$ PDFs will allow even more precise cosmological constraints to be made by using current and future photometric surveys.",
"These improvements are crucial as we move to analyze photometric data that push to or even past the limits of the available training data, which will be the case with the Large Synoptic Survey Telescope."
],
[
"Introduction",
"Spectroscopic galaxy surveys have played an important role in understanding the origin, composition, and evolution of our Universe.",
"Surveys like the Sloan Digital Sky Survey (SDSS; [63]), WiggleZ [27], and BOSS [24] have imposed important constraints on the allowed parameter values of the standard cosmological model [55], [7], [57].",
"However, spectroscopic measurements are considerable more expensive to obtain than photometric data, they are more likely to suffer from selection effects, and they provide much smaller galaxy samples per unit telescope time.",
"As a consequence, current ongoing and future galaxy surveys like the Dark Energy Survey (DEShttp://www.darkenergysurvey.org/) and the Large Synoptic Survey Telescope (LSSThttp://www.lsst.org/lsst/) are pure photometric surveys.",
"These surveys will enable cosmological measurements on galaxy samples that are currently at least a hundred times larger than comparable spectroscopic samples, that have relatively simple and uniform selection functions, that extend to fainter flux limits and larger angular scales, thereby probing much larger cosmic volumes and will photometrically detect galaxies that are too faint to be spectroscopically observed.",
"With the growth of these large photometric surveys, the estimation of galaxy redshifts by using multi band photometry has grown significantly over the last two decades.",
"As a result, a variety of different algorithms for estimating photo-$z$ 's based on statistical techniques have been developed [39], [1], [58].",
"Over the last several years, particular attention has been focused on techniques that compute a full probability density function (PDF) for each galaxy in the sample.",
"A photo-$z$ PDF contains more information than a single photo-$z$ estimate, and the use of photo-$z$ PDFs has been shown to improve the accuracy of cosmological measurements [46], [49], [41].",
"Photo-$z$ techniques can be broadly divided into two categories: spectral energy distribution (SED) fitting, and training based algorithms.",
"Template fitting approaches [6], [8], [30], [40], [3] estimate photo-$z$ s by finding the best match between the observed set of magnitudes or colors, and the synthetic magnitudes or colors taken from the suite of templates that are sampled across the expected redshift range of the photometric observations.",
"This method is often preferred over empirical techniques as they can be applied without obtaining a high-quality spectroscopic training sample.",
"However, these techniques do require a representative sample of template galaxy spectra, and they are not exempt from uncertainties due to measurement errors on the survey filter transmission curves, mismatches when fitting the observed magnitudes or colors to template SEDs, and color–redshift degeneracies.",
"The use of training data that include known redshifts can also improve these predictions [40], [50].",
"On the other hand, machine learning methods have been shown to have similar or even better performance [18], [12] when the spectroscopic training sample is populated by representative galaxies from the photometric sample.",
"Machine learning methods have the advantage that it is easier to include extra information, such as galaxy profiles, concentrations, or different modeled magnitudes within the algorithm.",
"However, they are only reliable within the limits of the training data, and one must exercise sufficient caution when extrapolating these algorithms.",
"These techniques can be sub-categorized into supervised and unsupervised machine learning approaches.",
"For supervised techniques [19], [11], [18], [61], [4], [45], [32], [34], [12], the input attributes (e.g., magnitudes or colors) are provided along with the desired output (e.g., redshift).",
"This training information is directly used by the algorithm during the learning process.",
"In this case, the redshift information from the training set supervises the learning process and decisions are made by using this information.",
"On the other hand, unsupervised machine learning photo-$z$ techniques [33], [62], [14] are less common as they do not use the desired output value (e.g., redshifts from the spectroscopic sample) during the training process.",
"Only the input attributes are processed during the training, leaving aside the redshift information until the evaluation phase.",
"Given the importance of these photo-$z$ PDFs, there is a present demand to compute them as efficiently and accurately as possible.",
"Additional requirements include the need to understand the impact of systematics from the spectroscopic sample on the estimation of these PDFs [53], [20], [21], and to maximally reduce the fraction of catastrophic outliers [35].",
"Considerable effort has, therefore, been put into both the development of different techniques and the exploration of new approaches in order to maximize the efficacy of photo-$z$ PDF estimation.",
"Yet, the combination of multiple, independent photo-$z$ PDF techniques has remained under explored [13], [22].",
"In this paper we extend our previous exploratory work in combining machine learning techniques with template fitting methods [13] to explicitly address this issue by presenting a novel Bayesian framework to combine and fully exploit different photo-$z$ PDF techniques.",
"In particular, we show that the combination of a standard template fitting technique with both a supervised and an unsupervised machine learning method can improve the overall accuracy over any individual method.",
"We also demonstrate how this combined approach can both reduce the number of outliers and improve the identification of catastrophic outliers when compared to the individual techniques.",
"Finally, we show that this methodology can be easily extended to include additional, independent techniques and that we can maximize the complex information contained within a photometric galaxy sample.",
"This paper is organized as follows.",
"In Section 2 we present the algorithms used in this work to generate the individual photo-$z$ PDF estimates and we provide a brief description on their individual functionality.",
"We describe, in Section 3, the different Bayesian approaches by which different photo-$z$ techniques are combined.",
"Section 4 introduces the data sets employed to test this Bayesian approach taken from the SDSS and DEEP2 surveys.",
"In Section 5 we present the main results of our combination approach and compare these results to those from the individual photo-$z$ PDF methods.",
"In Section 6 we discuss the application of a Naïve Bayes combination technique for outlier detection.",
"In Section 7 we conclude with a summary of our main points and a more general discussion of this new approach."
],
[
"Photo-z methods",
"To develop and test our combination framework, we consider three, distinct photo-$z$ PDF estimation techniques; we briefly discuss each one of them in this section.",
"We make the reasonable assumption that these three techniques are independent in their nature where two of these methods implement machine learning algorithms.",
"The first method is a supervised machine learning technique we have published called TPZ [12], which uses prediction trees and a random forest to produce probability density functions.",
"The second method is an unsupervised technique we have published called SOM$z$ [14], which uses self organizing maps (SOM) and a random atlas to produce a probability density function.",
"We have recently incorporated these two implementations into a new, publicly available and growing photo-$z$ PDF prediction framework called MLZhttp://lcdm.astro.illinois.edu/code/mlz.html (Machine Learning for photo-Z).",
"The third method is a Bayesian template fitting technique based on BPZ [6], which fits spectral energy density templates from a preselected library to an observed set of measured flux values.",
"Taken together, these three methods span the three standard published approaches in computing photo-$z$ s in the literature.",
"Any new method would, very likely, be functionally similar to one of these three methods; therefore, any of these three methods could in principle be replaced by a similar method to avoid redundancy.",
"This can be most easily demonstrated for template fitting methods, where an additional set of photo-$z$ estimations can be utilized by adopting a different template library .",
"In this particular case, the underlying code is essentially unchanged, but the photo-$z$ results will change as different spectral libraries are adopted.",
"Figure: Left: A simplified example of a binary prediction tree plotted radially, taken from CB13.",
"The initial node is close to the center of the figure; each node is subdivided and the splitting process terminates when a pre-defined stopping criterion is reached.",
"Individual colors represent a unique variable (e.g., a magnitude like gg or rr, or a color like g-rg - r) used to split an individual node.",
"Each leaf node provides a specific prediction based on the information contained within that terminal node (gray triangles in the figure).",
"The subpanel highlights a specific branch of the tree at higher resolution for additional clarity.",
"Right: A schematic representation of a self organized map, taken from CB14.",
"The training set of nn galaxies is mapped onto a two-dimensional lattice of KK neurons that are represented by vectors containing the weights for each input attribute.",
"Note that the galaxies and the weight vectors are of the same dimension mm, and that one neuron can represent morethanone training galaxy.",
"The colors used in the map encode the target property from the galaxies grouped within that cell."
],
[
"TPZ",
"TPZ (CB13) is a parallel, supervised algorithm that uses prediction trees and random forest techniques [10], [9] to produce photo-$z$ PDFs and ancillary information for a sample of galaxies.",
"Among the different non-linear methods that are used to compute photometric redshifts, prediction trees and random forests are one of the simplest yet most accurate techniques.",
"Furthermore, they have been shown to be one of the most accurate algorithms for low as well as high multi-dimensional data [16].",
"Prediction trees are built by asking a sequence of questions that recursively split the data into two branches until a terminal leaf is created that meets a pre-defined stopping criterion (e.g., a minimum leaf size or a maximum rms within that leaf).",
"The small region bounding the data in the terminal leaf node represents a specific subsample of the entire data that all share similar characteristics.",
"A comprehensive predictive model is applied to the data within each leaf that enables predictions to be rapidly computed in situations where many variables might exist that possibly interact in a nonlinear manner, which is often the case with photo-$z$ estimation.",
"A visualization of an example tree generated by TPZ is shown in the left panel of Figure REF .",
"In this figure, the plotting colors represent the magnitudes (or source colors) in which the data are recursively divided.",
"In practice, however, the prediction trees are generally both denser and deeper than the sample tree shown in the Figure.",
"To compute photo-$z$ PDFs in this study, we have used regression trees, which are a specific type of prediction trees.",
"Regression trees are built by first starting with a single node that encompasses the entire data, and subsequently splitting the data within a node recursively into two branches along the dimension that provides the most information about the desired output.",
"The procedure used to select the optimal split dimension is based on the minimization of the sum of the squared errors, which for a specific ${\\rm node}$ is given by $S({\\rm node}) = \\sum \\limits _{m \\in values(M)} \\sum \\limits _{i \\in m} (z_i - \\hat{z}_m)^2$ where $m$ are the possible values (bins) of the dimension $M$ , $z_i$ are the values of the target variable on each branch, and $\\hat{z}_m$ is the specific prediction model used.",
"In the case of the arithmetic mean, for example, we would have that $\\hat{z}_m = \\frac{1}{n_m}\\sum _{i \\in m} z_i$ , where $n_m$ are the members on branch $m$ .",
"This allows us to rewrite Equation REF as $S({\\rm node}) = \\sum \\limits _{m \\in values(M)} n_m V_m$ where $V_m$ is the variance of the estimator $\\hat{z}_m$ .",
"At each node in our tree, we scan all dimensions to identify the split point that minimizes the function $S({\\rm node})$ .",
"We choose the dimension that minimizes $S({\\rm node})$ as the splitting direction, and this process is recursively repeated until either a predefined threshold in $S({\\rm node})$ is reached or any new child nodes would contain less than the predefined minimum leaf size.",
"When constructed, each terminal leaf within the prediction tree contains spectroscopic data with different redshift values; the final prediction value for a given leaf node is determined from a regression model that covers these spectroscopic data.",
"The simplest model is to simply return the mean value of the set of spectroscopic training redshifts contained within the leaf node, which provides a single estimate of a continuous variable.",
"Alternatively, all of the spectroscopic training redshifts can be retained and subsequently combined with data from the matching leaf nodes in other prediction trees to form an aggregate, final prediction.",
"We create bootstrap samples from the input training data by sampling repeatedly from the magnitude using the magnitude errors.",
"We use these bootstrap samples to construct multiple, uncorrelated prediction trees whose individual predictions are aggregated to construct a photo-$z$ PDF for each individual galaxy by using a technique called a random forest.",
"We also use a cross validation technique called Out-of-Bag [10] within TPZ to provide extra information about the galaxy sample.",
"This information includes an unbiased estimation of the errors and a ranking of the relative importance of the individual input attributes used for the prediction.",
"This extra information can prove extremely valuable when calibrating the algorithm, when deciding what attributes to incorporate in the construction of the forest, and when combining this approach with other techniques.",
"TPZ has been tested extensively on different datasets, including the SDSS, DEEP2, and DES.",
"In all tests, TPZ has performed comparable to if not better than other machine learning approaches.",
"When high quality training data are available, TPZ has been shown to actually outperform other comparable techniques, both training and template based.",
"[12] provides a more detailed discussion of the TPZ algorithm and its application to different datasets."
],
[
"SOM$z$",
"A Self Organized Map (SOM): [43], [44] is an unsupervised, artificial neural network algorithm that is capable of projecting high-dimensional input data onto a low-dimensional map through a process of competitive learning.",
"In our case, the high dimensional input data can be galaxy magnitudes, colors, or some other photometric attributes, and two dimensions are generally sufficient for the output map.",
"A SOM differs from other neural network based-algorithms in that a SOM is unsupervised (the redshift information is not used during training), there are no hidden layers and therefore no extra parameters, and it produces a direct mapping between the training set and the output network.",
"In fact, a SOM can be viewed as a non-linear generalization of a principal component analysis (PCA).",
"The key characteristic of the self organization is that it retains the topology of the input training set, revealing correlations between inputs that are not obvious.",
"The method is unsupervised since the user is not required to specify the desired output during the creation of the low-dimensional map, as the mapping of the components from the input vectors is a natural outcome of the competitive learning process.",
"Another important characteristic of a SOM when applied to photo-$z$ estimation is the creation of a structured ordering of the spectroscopic training data, since similar galaxies in the training sample are mapped to neighboring neural nodes in the trained feature map (CB14).",
"We demonstrate the construction of a self-organizing map in the right-hand panel of Figure REF .",
"During this phase, each node on the two-dimensional map is represented by weight vectors of the same dimension as the number of attributes used to create the map itself.",
"In an iterative process, each galaxy in the input sample is individually used to correct these weight vectors.",
"This correction is determined so that the specific neuron (or node), which at a given moment best represents the input galaxy, is modified along with the weight vectors of that node's neighboring neurons.",
"As a result, this sector within the map becomes a better representation of the current input galaxy.",
"This process is repeated for every galaxy in the train sample, and this entire process is repeated for several iterations.",
"Eventually the SOM converges to its final form where the training data is separated into groups of similar features, which is illustrated in Figure REF by the different cell colors within the output map.",
"The result of this direct mapping procedure is an approximation of the galaxy training probability density function, and the map itself can be considered a simplified representation of the full attribute space of the input galaxy sample.",
"Building on our experience in creating TPZ, we have developed a similar approach, named SOM$z$ (CB14), where prediction trees are replaced by SOMs to create what we called a random atlas.",
"The random atlas is constructed from multiple maps that are each constructed from different bootstrap samples selected from the input training data by perturbing the input attributes using their measured error, where each one of these maps are built using a random subsample of the attribute space.",
"The multiple, uncorrelated maps are aggregated to generate a photo-$z$ PDF, in a similar manner as described earlier for the random forest.",
"As described previously, our SOM implementation not only updates the best-matching node but also the topologically closest nodes to it.",
"This functionality ensures that the entire region surrounding the best-matching node is identified as being similar to the current input galaxy.",
"As a result, similar nodes within the map are co-located, which naturally mimics how the input galaxies that have similar properties tend to be co-located in the higher dimensional input parameter space.",
"We apply this procedure iteratively to all input galaxies, which are processed randomly during each iteration to avoid any biases that might arise if galaxies are processed in a specific order.",
"When running SOM$z$, there are few different parameters that must be determined, including the map resolution (i.e., the number of pixels in the map), the number of iterations required to build the map, and, most importantly, the underlying two-dimensional topology used for the maps.",
"In this paper we follow the guidelines we presented in CB14 for these parameters, and use a spherical topology for the map, which are constructed by using HEALPIX [36], where each pixel in our maps has the same area.",
"This topology was shown to be more accurate in many cases when compared to other topologies like a rectangular or hexagonal grid.",
"In addition, a spherical topology has natural periodic boundary conditions which avoids possible edge effects.",
"In analogy with TPZ, we use cross validation, or OOB data, to estimate unbiased errors and to determine the relative importance of the different input attributes for this technique.",
"These are both key pieces of information that will be used during the combination process, as we need to ensure that the same process is uniformly applied to each photo-$z$ estimation technique.",
"By doing this, we will enable a robust analysis of the final results from the combination of the different techniques.",
"[14] (CB14) provides a complete description of the SOM$z$ implementation, the performance of this technique when applied to real data, and an exploration of specific parameter configurations."
],
[
"Template fitting approach",
"Using spectral templates to estimate galaxy photo-$z$ s from broadband photometry has a long history [5]; and this approach is, not surprisingly, one of the most utilized techniques.",
"A primary advantage of this technique is the fact that a training sample is not required, thus this approach can be considered unsupervised.",
"On the other hand, this technique has the disadvantage that a complete and representative library of spectral energy distributions (SEDs) are required.",
"Thus any incompleteness in our knowledge of the template SEDs that fully span the input galaxy photometry will lead to inaccuracies or misestimates in the computation of a galaxy photo-$z$ .",
"A number of different groups have published template fitting photo-$z$ estimation methods, all of which are roughly similar in nature.",
"In this work, we have modified and parallelized one of the most popular, publicly available template fitting algorithms, BPZ [6].",
"BPZ uses Bayesian inference to quantify the relative probability that each template matches the galaxy input photometry and determines a photo-$z$ PDF by computing the posterior probability that a given galaxy is at a particular redshift.",
"We can write this probability as $P(z\\mid \\mathbf {x})$ for a specific template $t$ , where $\\mathbf {x}$ represents a given set of magnitudes (or colors).",
"If the identification of a specific template is not required, we can later marginalize over the entire set of templates $\\mathbf {T}$ .",
"By using Bayes theorem, we have: $P(z \\mid \\mathbf {x}) = \\sum \\limits _{t \\in \\mathbf {T}} P(z,t\\mid \\mathbf {x}) \\propto \\sum \\limits _{t \\in \\mathbf {T}} \\mathcal {L}(\\mathbf {x} \\mid z,t) P(z,t) .$ $\\mathcal {L} (\\mathbf {x} \\mid z,t) $ is the likelihood that, for a given redshift $z$ and spectral template $t$ , a specific galaxy has the set of magnitudes (or colors) $\\mathbf {x}$ .",
"$P(z,t)$ is the prior probability of a specific galaxy is at redshift $z$ and has spectral type $t$ , this prior probability can be computed from a spectroscopic sample if one is available.",
"The photo-$z$ PDF is, therefore, either the posterior probability, if a prior is used, or the likelihood itself if no prior is used.",
"This last point arises since the likelihood only depends on the collection of template SEDs; and, if this collection is representative of the overall galaxy sample, the likelihood can be used by itself as a photo-$z$ PDF even without a spectroscopic training sample.",
"Figure: An Elliptical galaxy spectrum at z=0 and redshifted to z = 0.4 overlaid by the eight photometric filters from the DEEP2 galaxy survey (3 from the original survey and ugrizugriz from a matched catalog ).The use of a prior in a Bayesian analysis, however, is recommended.",
"In this case, the prior probability can be computed directly from physical assumptions, from an empirical function calibrated by using a spectroscopic training sample [6], or from an empirical function calibrated by using machine learning techniques [13].",
"For example, [6] propose the following function for a single magnitude $m_0$ : $P(z,t\\mid m_0) = P(t\\mid m_0)P(z\\mid t,m_0) \\\\\\propto f_T e^{-k_t (m-m_0)} \\times z^{\\alpha _t} \\exp \\left( -\\left[\\frac{z}{z_{mt}(m)} \\right]^{\\alpha _t}\\right).$ where $z_{mt}(m) = z_0t + k_{mt} (m-m_0)$ .",
"The five parameters of this function: $f_T$ , $m_0$ , $\\alpha _t$ , $z_{mt}$ , and $k_{mt}$ can be constrained either by using direct fitting routines, or by using Markov Chain Monte Carlo methods to sample these parameters.",
"These five parameters are dependent on the template $t$ and can be quantified independently.",
"For additional details on the underlying Bayesian approach, we refer the reader to the original paper by [6].",
"As the goal of a template fitting method is to minimize the difference between observed and theoretical magnitudes (or colors), this approach is heavily dependent on both the library of galaxy SED templates that are used for the computation and the accuracy of the transmission functions for the filters used for particular survey.",
"SED libraries are generally built from a base set of SED templates.",
"These base templates broadly cover the Elliptical, Spiral, and Irregular categories, and a template library can be constructed by interpolating between the base spectral templates to create new spectra.",
"One of the most widely used set of base templates are the four CWW spectra [17], which include an Elliptical, an Sba, an Sbb, and an Irregular galaxy template.",
"When extending an analysis to higher redshift, these temples are often augmented with two star bursting galaxy templates published by [42].",
"One additional effect some template approaches consider is the presence of interstellar dust, which will introduce artificial reddening.",
"Once the library of galaxy SED templates has been constructed, the templates are convolved with the transmission functions for a particular survey to generate synthetic magnitudes as a function of redshift for each galaxy template.",
"For the most accurate results, these transmission functions should include the effects of the Earth's atmosphere (if the observations are ground-based), as well as all telescope and instrument effects.",
"This convolution process is demonstrated visually in Figure REF , which presents an example Elliptical galaxy spectral template at redshift zero and at a redshift 0.4.",
"Overplotted on this figure is the filter set ($B$ , $R$ , and $I$ ) used by the DEEP2 survey, which is the data analyzed in this paper, along with the five extra filters: $u, g, r, i, z$ presented in the DEEP2 photometry catalog compiled by [47].",
"We now turn our attention to the different methods with which we can combine distinct photo-$z$ PDF estimation techniques [13].",
"In the statistics and machine learning communities, this topic is known as ensemble learning [56].",
"Recently, [22] have demonstrated that, on average, an improved photo-$z$ estimate can be realized by combining the results from multiple template fitting methods.",
"In this section, we build on this previous work to identify how Bayesian techniques can be used to construct a combined photo-$z$ PDF estimator.",
"We can frame the problem mathematically by writing the set of photo-$z$ PDFs for a given galaxy as a set of models $\\mathbf {M}$ , where each individual model $M_k$ (e.g., TPZ, SOM$z$, or modified BPZ) provides a distinct photo-$z$ PDF or posterior probability.",
"A photo-$z$ PDF can be written as $P(z \\mid \\mathbf {x}, \\mathbf {D}, M_k)$ , where $\\mathbf {x}$ is the set of magnitudes or colors (note that without loss of generality we can use other attributes in this process) used to make the prediction and $\\mathbf {D}$ corresponds to the training set which consists of $N_d$ galaxies.",
"We can also abbreviate this photo-$z$ PDF as $P_k(z)$ .",
"These photo-$z$ PDFs are each subject to the following constraint: $\\int _{z_1}^{z_2} P_k(z) dz = 1$ for every model $M_k$ , where $z_1$ and $z_2$ are the lower and upper limits, respectively, for the redshift range spanned by the galaxy sample.",
"In the following subsections, we introduce different methods to aggregate these photo-$z$ PDFs and show the results of these different methods in §.",
"Given the variety of photo-$z$ PDF estimation methods we are using (i.e., supervised, unsupervised, and model-based), we fully expect the relative performance of the individual techniques to vary across the parameter space spanned by the data.",
"For example, supervised methods should perform the best in areas populated by high quality training data, while unsupervised or model-based methods should perform better where we have little or no training data.",
"As a result, we can bin a specific subspace of our multi-dimensional parameter space and apply an individual combination method to each bin separately.",
"This technique is demonstrated later in more detail with the Bayesian Model Averaging method (although it is more generally applicable)."
],
[
"Weighted Average",
"The simplest approach to combine different photo-$z$ PDF techniques is to simply add the individual PDFs and renormalize the sum.",
"In this case the final photo-$z$ PDF is given by: $P(z \\mid \\mathbf {x},\\mathbf {M})=\\sum \\limits _{k}P(z \\mid \\mathbf {x},M_k) .$ We can improve on this simple approach by including weights in the previous equation: $P(z \\mid \\mathbf {x},\\mathbf {M})=\\sum \\limits _{k} \\omega _k P(z \\mid \\mathbf {x},M_k) .$ These weights, $\\omega _k$ , can be estimated for each input method by using the cross validation or OOB data, or from an intrinsic characteristic of the photo-$z$ PDF, such as $zConf$ that we introduced in CB13.",
"In this work we use three weight schemes in addition to the uniform case:"
],
[
"PDF shape weights",
"In this case, $\\omega _k$ is given by the the $zConf$ parameter, which is similar to the odds parameter presented in [6] $zConf$ is defined as the integrated probability between $z_{\\rm phot} \\pm \\sigma _{k}(1+z_{\\rm phot})$ , where $z_{\\rm phot}$ is a single estimated value for the photo-$z$ PDF.",
"This single photo-$z$ estimate can be either the mean or the mode of the photo-$z$ PDF.",
"Likewise, we can estimate $\\sigma _k$ for each input method either by using the OOB data, by selecting a constant value across all input methods, or by selecting these values separately so that all photo-$z$ PDFs have the same cumulative $zConf$ distributions.",
"$zConf$ quantifies the sharpness of the PDF and can take values from zero to one.",
"In CB13 and CB14, we demonstrated that there is a correlation between this value and the accuracy of the overall photo-$z$ .",
"Specifically, we observed that, on average, galaxies with higher $zConf$ have more accurate photo-$z$ PDFs than galaxies with lower $zConf$ values.",
"An alternative method to compute the values of $\\omega _k$ is to use the cross-validation data to first determine the weight values that minimize the difference between $z_{\\rm phot}$ and $z_{\\rm spec}$ ; and, second to apply these best fit values to the test data.",
"This method seeks the optimal linear combination of each individual PDF, thus it allows the values of $\\omega _k$ to be negative.",
"After the combination is completed, we renormalize according to Equation REF .",
"This method can be applied to a binned sub-sample to take advantages of the performance of each method in different areas of the attribute space.",
"As mentioned, when the input, multi-dimensional data have been binned (c.f.",
"Figure REF ), we can use the cross-validation data to select only one model from among all available input models to only be used with the test data located within that specific bin.",
"Since we are allowed to only select one input model, this will result in an assigned weight value of one for the chosen model and zero otherwise, however the chosen model is allowed to vary between bins.",
"The primary disadvantage of these simple, additive models is that incorrect estimates for the errors for the selected input model can bias the final result.",
"On the one hand, if a technique has underestimated errors, the final result will be biased towards this one input method.",
"On the other hand, overestimation of the errors will bias the final result away from this particular method.",
"One approach to address this issue, as discussed by [22], is to either smooth or sharpen the photo-$z$ PDFs estimated by each method by using the OOB data until their error distributions are approximately Gaussian with unit variance.",
"We can generalize this approach to transform a photo-$z$ PDF as $P_k(z) = P_k(z)^{\\alpha _k}$ , where we adjust the value of $\\alpha _k$ by using either the cross validation data when errors are over estimated or use a Gaussian smoothing filter when they are under estimated."
],
[
"Bayesian Model Averaging",
"Bayesian Model Averaging (BMA) is an ensemble technique that combines different models within a Bayesian framework.",
"BMA accounts for any uncertainty in the correctness of a given model by integrating over the model space and weighting each model by the estimated probability of being the correct model.",
"As a result, BMA acts as a model selection procedure that handles the uncertainty in selecting the best model by using a combination of models instead.",
"This is because BMA considers the uncertainty in selecting the best model while working under the assumption that only one model is actually the best [48].",
"BMA has been used for astrophysical problems [37], [60], [25] in, for example, the determination of cosmological parameters and variable star classification [54].",
"When using BMA, the training data are used to characterize each of the models that will be combined.",
"For each galaxy, the final PDF, $P(z \\mid \\mathbf {x}, \\mathbf {D}, \\mathbf {M})$ , is given by: $P(z \\mid \\mathbf {x},\\mathbf {D},\\mathbf {M}) = \\sum \\limits _{k}P(z \\mid \\mathbf {x},M_k)P(M_k \\mid \\mathbf {D}) .$ $P(M_k \\mid \\mathbf {D})$ is the probability of the model $M_k$ given the training data $\\mathbf {D}$ , which can be viewed as a simple, model dependent weighting scheme.",
"This probability can be computed by using Bayes' Theorem: $P(M_k \\mid \\mathbf {D}) = \\frac{P(M_k)}{P(\\mathbf {D})} P(\\mathbf {D} \\mid M_k) \\\\\\propto P(M_k) \\prod \\limits _{i=1}^{N_d} P(d_i \\mid M_k) .$ We have omitted the $P(\\mathbf {D})$ term as it is merely a normalization factor and we use the same data for all models.",
"$d_i$ is the $i^{\\textrm {th}}$ element from the training data $\\mathbf {D}$ , which are assumed to be independent.",
"For each model, we assign the value $\\epsilon _k$ as an average error for the estimation process.",
"$\\epsilon _k$ can be computed as the fraction $N^{(b)}_k/N_d$ , where $N^{(b)}_k$ is the number of galaxies considered to be misestimated or bad for the particular photo-$z$ PDF method $k$ .",
"To quantify when a specific galaxy is a bad prediction we compute $N^{(b)}_{k,i} =\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if } \\int _{z_s-\\delta _z}^{z_s+\\delta _z} P(z \\mid \\mathbf {x},d_i) dz \\le \\pi _z ,\\\\0 & \\mbox{otherwise} .\\end{array}\\right.$ In this equation, $z_s$ is the spectroscopic redshift for the $i^{\\textrm {th}}$ training set galaxy.",
"The first parameter, $\\delta _z$ , controls the width of a window centered on $z_s$ within which we accumulate photo-$z$ probability for the $i^{\\textrm {th}}$ training galaxy around the true redshift.",
"The second parameter, $\\pi _z$ , is the minimum probability within this window for which we consider the model prediction to be good.",
"We find that $\\pi _z = 0.5$ and $\\delta _z = 0.05$ provides a good discriminant between good and bad photo-$z$ model estimates.",
"Given the individual good/bad predictions for each training set galaxy, we can compute the total number of bad predictions, $N^{(b)}_k$ , by summing over the individual predictions, $N^{(b)}_{k,i}$ , for the entire training data, $\\mathbf {D}$ .",
"The total number of good prediction will naturally be $N_d-N^{(b)}_k$ .",
"As a result, we can rewrite Equation REF : $P(M_k \\mid \\mathbf {D}) \\propto P(M_k) (1-\\epsilon _k)^{N_d-N^{(b)}_k}(\\epsilon _k)^{N_k^{(b)}},$ where $P(M_k)$ is the probability of each model $k$ , which we can assume to be unity for all models.",
"Therefore, the final PDF for each galaxy is given by $P(z \\mid \\mathbf {x},\\mathbf {D}, \\mathbf {M}) \\propto \\sum \\limits _{k}P(z \\mid \\mathbf {x},M_k) P(M_k) \\times \\\\(1-\\epsilon _k)^{N_d-N^{(b)}_k}(\\epsilon _k)^{N_k^{(b)}} .$ We applied the BMA technique to individual bins within the multi-dimensional parameter space occupied by a given data set.",
"We demonstrate this binned BMA technique in Figure REF , where we use a Self Organized Map to project our entire input parameter space to a two-dimensional map.",
"In this manner, all magnitudes or colors are used to form the binned regions within which the parameters of the ensemble learning approach can vary.",
"After computing photo-$z$ PDFs for all galaxies with each method, we use BMA to determine the relative weights for these input techniques within each bin; we can visualize these weights as different colors across the two-dimensional map, as shown in Figure REF .",
"This figure graphically displays how the accuracy of each photo-$z$ PDF estimation varies across the parameter space, and thus how the different weights themselves vary."
],
[
"Bayesian Model Combination",
"As discussed, Bayesian Model Averaging tries to select the best model among the ones introduced to the algorithm.",
"Alternatively, we can modify BMA to produce an more optimal model combination technique [48] known as Bayesian Model Combination (BMC).",
"With BMC, instead of directly combining the three different photo-$z$ PDF estimates as was the case with BMA, the Bayesian process is used to explore different combinations of the individual photo-$z$ PDF techniques.",
"Thus, an ensemble of different photo-$z$ PDF combinations are generated and we directly compare different model combinations.",
"As a simple example, we could first generate hundreds different random weights for all three of our photo-$z$ PDF estimation techniques, and second use these to compute hundreds of new sets of PDFs by computing a simple weighted average by using Equation REF .",
"Finally, we could apply BMA to this PDF ensemble to determine the final PDF.",
"In this case, we could write Equation REF : $P(z \\mid \\mathbf {x},\\mathbf {D},\\mathbf {M}, \\mathbf {E}) = \\sum \\limits _{e \\in \\mathbf {E}}P(z \\mid \\mathbf {x},\\mathbf {M},e)P(e \\mid \\mathbf {D}) ,$ where $e$ is an element from the set $\\mathbf {E}$ of these hundreds combined models.",
"Here we need to compute the performance of each combination $e$ and apply the BMA formulation, shown in Equations REF and REF , to those models by using the model $e$ instead of $M_k$ , i.e., $P(e \\mid \\mathbf {D}) \\propto P(e) \\prod \\limits _{i=1}^{N_d} P(d_i \\mid e) .$ Fundamentally, with BMC we are marginalizing over the uncertainty in the correct model combination, where in BMA we marginalized over the uncertainty in identifying the correct model from the entire ensemble.",
"The number of model combinations $\\mathbf {E}$ is, in principle, infinite, and in practice can be very large.",
"To overcome this, we can use sampling techniques over a reasonable, finite number of models.",
"Naively we might use randomly generated weights, however, this approach can be costly to fully span the allowed range of weights and convergence towards a satisfactory solution might be slow.",
"Thus, instead of assigning weights randomly or using incremental steps within a regular grid, we sample the weights from a Dirichlet distribution where the concentration parameters are modified until they converge to stable values.",
"We require that the set of weights, $w_k$ , for each of the three models, $M_k$ , satisfy $\\sum w_k = 1$ and also $w_k > 0$ .",
"For a concentration parameter $\\alpha $ of the same dimension as $\\mathbf {w}$ , we have that the probability distribution for $\\mathbf {w}$ is given by: $P(\\mathbf {w}) \\sim \\mathcal {D}{\\rm ir}(\\alpha ) = \\frac{\\Gamma (\\sum _k \\alpha _k)}{\\prod _k \\Gamma (\\alpha _k)}\\prod \\limits _k w_k^{\\alpha _k -1} ,$ where $\\mathcal {D}{\\rm ir}(\\alpha )$ is the Dirichlet distribution, $\\Gamma (\\alpha _k)$ is the gamma function and $k$ are the base models, which in this paper are TPZ, SOM$z$, and our modified BPZ.",
"In order to generate a set $\\mathbf {E}$ of combined models, we first set $\\alpha _k$ to unity for all values of $k$ .",
"Second, we sample from this distribution $n_s$ times ($n_s$ is a fixed number, generally between 2 and 5, which we fixed at 3) to get a set of $n_s$ weights and $n_s$ new model combinations.",
"Next, we compute $P(e \\mid D)$ by using Equations REF and REF for each model in the set of $n_s$ models.",
"We, temporarily, select the best model among the set $n_s$ , i.e, the one with highest $P(e \\mid \\mathbf {D})$ , and update the $\\alpha _k$ parameters by simply adding the weights from the corresponding model to the current values of $\\alpha $ , $\\alpha ^{(t+1)} = \\alpha ^{t} + \\max _{\\mathbf {w}_e \\in n_s} P(e \\mid \\mathbf {D})$ where $t$ is just a symbolic reference to the fact that $\\alpha $ is being updated every 3 steps.",
"We use the latest values for $\\alpha $ to continue the sampling process to obtain the next set $n_s$ of model combinations.",
"As a result, we continually (by adding $n_s$ new models at each step) extend our set of model combinations $\\mathbf {E}$ .",
"As the chain of models in this set is constructed iteratively, the process can be terminated either when a predefined number of model combinations has been reached or when new model combinations have started to converge.",
"This process behaves similarly to a Markov Chain Monte Carlo process, and we have an analogous phase to the burn in step, where we can omit some number of model combinations at the start of our set $\\mathbf {E}$ of model combinations.",
"Thus, our final photo-$z$ PDF prediction is the application of BMA over the remaining elements in $\\mathbf {E}$ , we have set for this work the size of $E$ to be 800.",
"Finally, we note that, as was the case with BMA, we can develop a binned version of our BMC technique, where we develop different model combinations for different region of the magnitude (color) space by using a SOM."
],
[
"Hierarchical Bayes",
"A Hierarchical Bayesian (HB) method provides a different approach to combine the individual photo-$z$ PDFs.",
"In a manner similar to BMA, we include the uncertainty that a given photo-$z$ PDF for a specific galaxy might be incorrectly predicted as a set of nuisance parameters over which we later marginalize.",
"Adopting our previous notation, we follow a similar approach to [29] and [22], and we write the photo-$z$ PDF for an individual galaxy for each base method $k$ : $P(z \\mid \\mathbf {x},\\mathbf {D},M_k, \\theta _k) = \\sum \\limits _j P(z \\mid \\mathbf {x}, \\mathbf {D}, M_k, \\theta _{k j}) \\times \\\\P(\\theta _{k j} \\mid \\mathbf {D}, M_k) ,$ where we have introduced the hyperparameter $\\theta _{k}$ , a nuisance parameter that characterizes our uncertainty in the prior distribution of model $k$ .",
"The parameter $\\theta _{k}$ can be quantified in different forms, but essentially is the misclassification probability of the $k^{\\textrm {th}}$ method.",
"Thus, we quantify this mis-prediction probability with $P(\\theta _k)$ ; and we drop the dependence on $\\mathbf {x}$ , the measured galaxy attributes, as it does not directly affect the parameter $\\theta _k$ .",
"Since we will marginalize over $\\theta $ , we keep the term $\\mathbf {D}$ as we can use the training data to place limits on $\\theta _k$ by using the cross-validation data.",
"We note that these probabilities are subject to: $\\sum \\limits _{j} P(\\theta _{k j} \\mid \\mathbf {D}, M_k) = 1 .$ If we consider the case where galaxies are predicted correctly or are outliers, $j$ is a binary state.",
"In this model, if we assume that $\\gamma _k$ is the fraction of galaxies that are mispredictions or are labeled as outliers for method $k$ , we have: $P(\\theta _{k 0} \\mid \\mathbf {D}, M_k) = \\gamma _k$ and $P(\\theta _{k 1} \\mid \\mathbf {D}, M_k) = (1 - \\gamma _k)$ .",
"In this case, Equation REF becomes: $P(z \\mid \\mathbf {x},\\mathbf {D},M_k, \\theta _k) = P_{def}(z \\mid M_k, \\theta _k)\\gamma _k + \\\\P(z \\mid \\mathbf {x}, \\mathbf {D}, M_k, \\theta _k)(1-\\gamma _k) ,$ where $P_{def}(z \\mid M_k, \\theta _k)$ is the default PDF that should be used for the $k^{\\textrm {th}}$ method when the original PDF for that method has been determined to be mis-predicted or wrong.",
"In the second term, we use the original PDF for the method $k$ , which is multiplied by the fraction of well predicted objects $1-\\gamma _k$ .",
"The final PDF after we combine the different photo-$z$ PDFs from our base methods in the HB approach is given by: $P(z \\mid \\mathbf {x},\\mathbf {D},\\theta ) = \\prod \\limits _k P(z \\mid \\mathbf {x},\\mathbf {D},M_k, \\theta _k)^{1/\\beta } .$ Here, following [22], we have introduced an extra parameter $\\beta $ , which is a constant value that quantifies the degree of covariance between the different base methods.",
"$\\beta =1$ corresponds to complete independence between the base methods, while $\\beta =3$ (or, more generally, the total number of methods) would correspond to full covariance between them.",
"We can compute $\\beta $ from the OOB sample in such way the final error distribution follows a normal distribution with zero mean and unit variance, as we have done in this paper.",
"Alternatively, we can marginalize over all possibles values of $\\beta $ when no cross validation data is available and we can integrate over the uncertainty of this parameter.",
"Finally, by marginalizing over $\\theta $ we have our final PDF: $P(z \\mid \\mathbf {x}, \\mathbf {D} )$ , or simply $P(z)$ given by: $P(z) = \\int _0^1 P(z \\mid \\mathbf {x},\\mathbf {D},\\theta )P(\\theta )d\\theta ,$ where $P(\\theta )$ is a constant which in the simple case is equal to unity.",
"If OOB data is available, we can narrow down the range of allowed values for $\\theta $ (or effectively $\\gamma _k$ ), so we can set up a limited range for $\\gamma _k$ based on the performance of each method $k$ on this data.",
"In this case, $P(\\theta )$ will act as a top-hat window function.",
"In any case, the final $P(z)$ is subject to Equation REF .",
"As discussed before, we can either apply the HB approach to the entire data set, or we can partition the input space and apply the HB approach independently to the binned regions of the parameter space.",
"Table: The photo-zz PDF combination methods, their weights and abbreviations presented in this paper.To explore different configurations and to demonstrate the capabilities and the efficacy of these photo-$z$ combination techniques, we follow the approach we presented in CB13 and CB14, but in this paper we restrict our analysis to data obtained by the Deep Extragalactic Evolutionary Probe (DEEP) survey and the Sloan Digital Sky Survey (SDSS).",
"In the rest of this section we provide a summary of these data and detail how we extracted the data sets from these surveys that we use in the analysis presented in §."
],
[
"Deep Extragalactic Evolutionary Probe",
"The DEEP survey is a multi-phase, deep spectroscopic survey that was performed with the Keck telescope.",
"Phase I used the Low Resolution Imaging Spectrometer (LIRS) instrument [52], while phase II used the DEep Imaging Multi-Object Spectrograph (DEIMOS) [28].",
"The DEEP2 Galaxy Redshift Survey is a magnitude limited spectroscopic survey of objects with $R_{AB} < 24.1$ [23], [51].",
"The survey includes photometry in three bands from the Canada-France-Hawaii Telescope (CFHT) 12K: $B$ , $R$ , and $I$ and it was recently extended by cross-matching the data to other photometric data sets.",
"In this work, we use Data Release 4 [47], the latest DEEP2 release that includes secure and accurate spectroscopy for over 38,000 sources.",
"The original input photometry for the sources in this catalog was supplemented by using two $u$ , $g$ , $r$ , $i$ , and $z$ surveys: the Canada-France-Hawaii Legacy Survey [38], and the SDSS.",
"For additional details about the photometric extension of the DEEP2 catalog, see [47].",
"To use the DEEP2 data with our implementation, we have selected sources with secure redshifts (ZQUALITY$\\ge 3$ ), which were securely classified as galaxies, have no bad flags, and have full photometry.",
"Even though the filter responses are similar, the $u$ , $g$ , $r$ , $i$ , and $z$ photometry originates from two different surveys and are thus not identical.",
"We therefore only present the results from those galaxies that lie within field 1 that have CFHTLS photometry.",
"Furthermore, we have corrected these observed magnitudes by using the extinction maps from [59].",
"In the end, this leaves us with a total of 10,210 galaxies each with eight band photometry and redshifts.",
"From this data set, we randomly select 5,000 galaxies for training and hold the remainder out for testing.",
"The computation of photo-$z$ PDFs was completed by using the magnitudes in the bands $B$ , $R$ , $I$ , $u$ , $g$ , $r$ , $i$ , and $z$ and their corresponding colors $B-R$ , $R-I$ , $u - g$ , $g - r$ , $r - i$ , and $i - z$ , providing a total of fourteen dimensions."
],
[
"Sloan Digital Sky Survey",
"The Sloan Digital Sky Survey [63] phases I, II and III conducted a photometric survey in the optical bands: $u$ , $g$ , $r$ , $i$ , $z$ that covered more than 14,000 square degrees, more then one-quarter of the entire sky.",
"The resultant photometric catalog contains photometry for over $10^8$ galaxies, making the SDSS one of the largest sky surveys ever completed.",
"The SDSS also conducted a spectroscopic survey of targets selected from the SDSS photometric catalog.",
"In this paper, we use a subset of the spectroscopic data contained within the Data Release 10 catalog [2], which includes over two million spectra of galaxies and quasars which include those taken as apart as the Baryonic Oscillation Spectroscopic Survey (BOSS) program [24].",
"Specifically, we selected galaxies by using the online CasJobs websitehttp://skyserver.sdss3.org/CasJobs/ and the following query from the DR10 data base: SELECT spec.specObjID, gal.dered_u, gal.dered_g, gal.dered_r, gal.dered_i, gal.dered_z, gal.err_u, gal.err_g, gal.err_r, gal.err_i, gal.err_z, spec.z AS zs INTO mydb.DR10_spec_clean_phot FROM SpecObj AS spec JOIN Galaxy AS gal ON spec.specobjid = gal.specobjid, PhotoObj AS phot WHERE spec.class = `GALAXY' -- Spectroscopic class -- (GALAXY, QSO, or STAR) AND gal.objId = phot.ObjID AND phot.CLEAN=1 -- Clean photometry flag -- (1=clean, 0=unclean) AND spec.zWarning = 0 -- Bitmask of warning -- vaules; 0 means all -- is well We also removed some additional bad photometric observations, ensured the redshift values were positive, and compute colors for the final catalog, which contains 1,147,397 galaxies.",
"The spectroscopic data range from $z \\approx 0.02$ up to $z \\approx 0.8$ ; the full spectroscopic redshift distribution of these galaxies is shown in the gray shaded histogram presented in Figure REF .",
"These data are dominated by the Main Galaxy Sample (MGS) at low redshifts, with mean redshift of $z \\sim 0.1$ , and by luminous red galaxies (LRG) at higher redshifts, with mean redshift of $z \\sim 0.5$ .",
"From this sample, we randomly selected 50,000 galaxies for training and hold the remaining 1,097,397 for testing.",
"This training set corresponds to approximately 4.5% of the test set.",
"We note that this is a blind test, as the testing data are not used in any way to train or calibrate the algorithms.",
"Of all the measured attributes in the SDSS photometric catalog, we have only used the nine dimensions corresponding to the five galaxy, extinction corrected, model magnitudes and the four colors derived from these five magnitudes: $u$ , $g$ , $r$ , $i$ , $z$ , $u - g$ , $g - r$ , $r - i$ , and $i - z$ ."
],
[
"results/discussion",
"We now turn to the actual application of the ensemble learning approaches described in § to the data introduced in §.",
"We present the seven combination methodologies we use in this section in Table REF , which also includes an abbreviated name that we will use to refer to a specific technique.",
"We follow a similar approach to CB14 in order to compare different combination methods, and define the bias to be $\\Delta z^{\\prime } = |z_{\\rm phot}-z_{\\rm spec}|/(1+z_{\\rm spec})$ .",
"We also present the standard metrics we use to compare the performance of the different combination techniques in Table REF .",
"As shown in this table, we define five metrics to address the bias and the variance of the results (the first five rows) and we present three values to characterize the outlier fraction.",
"We also use the $KS$ metric, which represents the results of a Kolmogorov–Smirnov test that quantifies the likelihood that the predicted photo-$z$ distribution and the spectroscopic redshift distribution $N(z)$ are drawn from the same underlying population.",
"This metric provides a single, robust value to compare both distributions that does not depend on how the results are binned by redshift, and it is defined as the maximum distance between both empirical distributions.",
"To determine this statistic, we compute the empirical cumulative distribution function (ECDF) for both distributions.",
"For the spectroscopic sample, the ECDF is defined as: $F_{\\rm spec} (z) = \\sum \\limits _{i=1}^{N} \\Omega _{z_{\\rm spec}^i < z}$ where N is the number of galaxies in the redshift sample, and $\\Omega _{z_{\\rm spec}^i < z} = {\\left\\lbrace \\begin{array}{ll} 1, & \\mbox{if } z_{{\\rm spec},i} < z \\\\ 0 , & \\mbox{otherwise } \\end{array}\\right.",
"}$ The ECDF for the photo-$z$ distribution is simply the accumulation of the probability presented in the photo-$z$ PDF.",
"The summation is carried out over all galaxies in the sample.",
"Given the ECDF for both the photo-$z$ and spectroscopic distributions, we compute the KS statistic as: ${\\rm KS} = \\max _z \\left(\\vert \\vert F_{\\rm phot} (z) - F_{\\rm spec} (z)\\vert \\vert \\right)$ Thus, as the KS statistic decreases, the two distributions become more similar.",
"All of the metrics listed in Table REF are positive and characterized by the fact that lower metric values indicate a more accurate photo-$z$ PDF.",
"In CB14 we defined a new, meta-statistic called $I$ -score (symbolically represented by $I_{\\Delta z^{\\prime }}$ ) that provides a single statistic to simplify the comparison of different photo-$z$ techniques.",
"To compute this metric, we first normalize each set of metrics across all different photo-$z$ estimation techniques so that we are not biased by different dynamic ranges.",
"Thus, for example, we first compute the mean and standard deviation for $<\\Delta z^{\\prime }>$ for each combination technique, and subsequently rescale all individual $<\\Delta z^{\\prime }>$ values so that this set of values has zero mean and unit variance.",
"We continue this process for all nine statistics listed in Table REF , and compute their weighted sum to obtain the total $I$ -score: $I_{\\Delta z^{\\prime }} = \\sum w_i M_i,$ where $M_i$ is the rescaled metric and weight value for metric $i$ out of the nine available.",
"For simplicity, we use equal weights in the remainder of this paper (and thus the $I$ -score is simply the average of the nine rescaled metrics for each technique).",
"As a result, the photo-$z$ method (or parameter configuration) with the lowest $I$ -score will be the optimal estimation technique.",
"On the other hand, if we were looking for the technique or the specific parameter configuration with, for instance, the lower outlier fraction, we could assign higher weights accordingly to select the best technique.",
"In this way, we can efficiently select the best method or configuration for specific research requirement.",
"Table: The definition of the metrics used to compare different photo-zz combination methods."
],
[
"Cross validation data",
"In CB13, we introduced OOB data and demonstrated its use as a cross-validation data set that provided error quantification and overall performance similar to what could be expected when applying an algorithm directly to the test data set.",
"When building a tree with TPZ or a map with SOM$z$, a fraction of the overall training data, usually one-third, is extracted and not used during the tree/map construction process.",
"The resultant tree/map is subsequently applied to this unused data to make a photo-$z$ prediction, and this process is repeated for every tree/map.",
"These photo-$z$ predications are aggregated for each galaxy to make a photo-$z$ PDF; and by construction a galaxy can never be used to train any tree/map that is subsequently used to predict that galaxy's photo-$z$ .",
"Thus, as long as the OOB data remains similar to the final testing data, the OOB data provide results that will be similar to the final test data results and can be used to guide expectations when applied blindly to other data.",
"As an illustration of this process, Figure REF compares the photometric (as computed by using SOM$z$) and spectroscopic redshifts for galaxies in the training (5,000 in total) and testing (5,210) samples as selected from field 1 of the DEEP2 data set.",
"As shown in this Figure, the performance on both the OOB and the testing data are visually similar and there is no indication of overfitting.",
"In addition, general features in the result, like the spread of the data or the slight tilt of the distribution of points relative to the diagonal, are observed in both samples.",
"A similar conclusion is observed with the SDSS data, as shown in Figure REF where the photometric (as computed by using TPZ) and spectroscopic redshifts for 50,000 galaxies from the training set are compared to 50,000 randomly selected galaxies from the test set.",
"Both distributions show similar behavior and global trends, thus we conclude that, as expected, the OOB data can be used to predict the performance of an PDF combination algorithm on real data.",
"Figure: A comparison of the photometric (computed by using SOMzz) and spectroscopic redshifts for training set (left) and test set (right) galaxies from field 1 of the DEEP2 survey.Figure: A comparison of the photometric (computed by using TPZ) and the spectroscopic redshift from the SDSS-DR10 for the 50,000 training set galaxies (left) and 50,000 galaxies randomly subsampled from the 1,097,397 galaxies in the test set (right).Another method to contrast the results from these data is to compute the correlation between each of the three photo-$z$ estimation techniques discussed earlier as a function of redshift.",
"For this, we use the photo-$z$ PDFs for all galaxies, and we calculate the Pearson correlation coefficient $R_{ik}$ within each redshift bin.",
"Even if the three input methods are completely independent, we should expect a positive correlation between them if their predictions are similar.",
"In fact, we desire a positive correlation (but not necessarily a perfect correlation) between the techniques as this will indicate the different techniques are all performing well.",
"We present the Pearson correlation coefficient for the three photo-$z$ PDF estimation techniques for the DEEP2 data (top panel) and the SDSS data (bottom panel) in Figure REF .",
"In this figure we display these correlation coefficient computed from the cross-validation (OOB) data (dashed line) and the test data (solid line).",
"The global agreement between these lines further demonstrates the importance of the OOB data as a predictor of the performance of a given technique.",
"This figure also demonstrates a tighter correlation between the two machine learning algorithms than between any machine learning algorithm and the template technique, which is not surprising given the similarities in the methods.",
"While not shown, the shape of the covariance matrices resemble the spectroscopic $N(z)$ distributions presented in Figures REF and REF .",
"We conclude that this is expected since a larger number of galaxies can naturally produce a greater chance for divergent photo-$z$ estimates.",
"Figure: The Pearson correlation coefficient between the individual photo-zz PDF estimation methods as a function of redshift for the DEEP2 (top) and SDSS (bottom) data.",
"The coefficients measured from the cross-validation (OOB) data (dashed line) and from the test data (solid line) are nearly identical, indicating the utility of the OOB data in predicting the performance of an algorithm on blind test data.",
"Note that a positive correlation is beneficial since this measures the relative performance of different techniques in predicting redshifts.As mentioned previously, a concern when combining photo-$z$ PDFs from different methods is to reduce the likelihood of being biased by methods that might under- or overestimate their errors.",
"To further demonstrate the importance of the cross-validation data, we compare the normalized error distribution between the cross-validation (OOB) and test data in Figure REF for both DEEP2 (top panel) and SDSS (bottom panel) data, where the photo-$z$ PDFs were generated by TPZ .",
"In both cases, the two curves are nearly identical, and we confirmed the same result with both SOM$z$ and BPZ.",
"Thus we can use the OOB data error estimate to rescale the PDF for the test data by using the results computed from the OOB data.",
"Figure: The normalized error distributions for galaxies in DEEP2 (top) and SDSS (bottom).",
"The error distribution computed from the test data is shown in red, while the error distribution for the cross-validation (OOB data) is shown in black.",
"The excellent agreement highlights the importance of the OOB data in predicting the results of blind test data predictions."
],
[
"Photo-$z$ PDF Combination for DEEP2",
"To combine the three photo-$z$ PDF techniques discussed in §, we employ a binning strategy to allow different method combinations to be used in different parts of parameter space.",
"We first create a two dimensional, $10 \\times 10$ SOM representation of the full 14-dimensional space (eight magnitudes and six colors, note that we do not compute a color between the two different photometric input surveys) by using a rectangular topology to facilitate visualization.",
"With this map we can perform an analysis of all galaxies that lie within the same cell, in a similar process to that described in CB14, but now instead of predicting a photo-$z$ , we are computing the optimal model combination.",
"We apply all seven combination methods presented in Table REF to all galaxies within each cell by using the OOB data that are also contained within the same cell.",
"We note that the ${\\rm WA}_{\\rm flat}$ and ${\\rm WA}_{\\rm shape}$ methods do not depend on this binning, and can, therefore, be used without OOB data.",
"We also could employ the ${\\rm HB}$ approach without using this map, but in this case we would need to define $P_{def}(z \\mid M_k, \\theta _k)$ and perform the marginalization over the entire range of $\\theta _k$ without any prior on this value.",
"Figure: A comparison of the average performance for the three individual photo-zz PDF estimation methods and the seven different photo-zz PDF combination approaches for five different metrics as defined in Table for the DEEP2 data.",
"The horizontal dashed line indicates the best result for a given statistic among the three individual methods (note, BPZ is not always shown at the provided scale), and the shaded area separates the individual methods from the combined approaches.",
"All values are presented in Table .Table: A summary of the performance results for the three individual methods and the seven different photo-zz PDF combination methods as applied to the DEEP2 data, no magnitude cut was applied during the training phase.",
"The bold entries highlight the best value within each column to aid in the interpretation of the table (c.f.",
"Figure ).We present a summary of the results obtained by applying the seven different combination techniques to all the galaxies within the DEEP2 data in Table REF .",
"The bold entries in this Table highlight the best technique for any particular metric.",
"The first three rows in this Table show the individual photo-$z$ PDF estimation techniques, of which TPZ generally performs the best and is thus shown in the first row as the benchmark.",
"This Table also clearly indicates that the seven different combination techniques generally have a similar performance, and, as shown in the last four rows, often perform better than TPZ.",
"We observe that the last four methods: ${\\rm WA_{\\rm fit}}$ , ${\\rm BMA}$ , ${\\rm BMC}$ , and ${\\rm HB}$ all use the binned model combination approach, and thus can take advantage of the different performance characteristics of individual codes.",
"In this case, ${\\rm BMC}$ provides the best performance as measured by the $I$ -score $I_{\\Delta z^{\\prime }}$ , the bias $<\\Delta z^{\\prime }>$ , the scatter $\\sigma _{\\Delta z^{\\prime }}$ , and the outlier fraction ${\\rm out}_{0.1}$ .",
"Overall, the differences are close to 5% for many of the metrics, which, while small, are still significant since these are averaged metrics over the full test galaxy sample.",
"In Figure REF , we present a visual comparison between the ten different photo-$z$ estimation techniques for five different metrics: bias, scatter, outlier fraction, KS test, and the $I$ -score.",
"In each panel, the horizontal dashed line shows the best value from the individual photo-$z$ PDF estimation methods and the shaded area separates the individual from the combined methods.",
"This Figure demonstrates that the Bayesian modeling techniques provide better performance than the best individual method over all five metrics, and also that by employing the binning scheme to optimize the combination approach we achieve better performance than for the best individual technique.",
"We compare the actual photo-$z$ PDF for a single galaxy selected from the DEEP2 survey as estimated by the three individual techniques with the photo-$z$ PDF estimated by the ${\\rm BMC}$ method in Figure REF .",
"This Figure clearly shows how the re-normalized combined PDF from the three individual photo-$z$ PDF estimation techniques has been improved as the ${\\rm BMC}$ result is closer to the true galaxy redshift, shown by the vertical line.",
"These combination techniques identify which individual method works best in different cells, and can use that information to either weight the individual photo-$z$ PDFs accordingly, or in the case of ${\\rm BMC}$ to marginalize over the uncertainty in the correct weights to produce the best combination.",
"Figure: An comparison between the three individual photo-zz PDF estimation techniques and a combined PDF computed by using BMC {\\rm BMC} and Equation for a single example galaxy taken from the DEEP2.",
"The vertical line indicates the true source redshift.Figure: A two-dimensional SOM showing the relative weights for the BMA combination scheme applied to the three individual methods for the DEEP2 field 1 data (TPZ is top left, BPZ is top right, and SOMzz is bottom left).",
"In each panel, the color map indicates the value of the weight relative to the other cells in the map.",
"The bottom right panel shows the same cells colored by the mean RR-band magnitude for the cross validation galaxies.We apply a SOM to the DEEP2 field 1 data in order to construct a two-dimensional, binned combination of the three individual photo-$z$ PDF estimation methods.",
"We use this SOM to determine the weights for the three individual methods for each cell, and present the results in Figure REF when using the BMA approach as it is easy to interpret.",
"We also show the mean DEEP2 $R$ -band magnitude for all galaxies in a given cell in the lower right panel, which clearly indicates the ability of the SOM to preserve relationships between galaxies when projecting from the higher dimensional space to the two-dimensional map.",
"Of course, the SOM mapping is a non-linear representation of all magnitudes and colors, thus the DEEP2 $R$ -band map should only be used to provide guidance.",
"In the three weight maps, a redder color indicates a higher weight, or equivalently that the corresponding method performs better in that region.",
"These weight maps demonstrate the variation in the performance of the individual techniques across the two-dimensional parameter space defined by the SOM.",
"For example, BPZ performs the best, as expected, in the upper left corner of the map, which is approximately where the faintest galaxies, at least in the DEEP2 $R$ -band, are stored.",
"On the other hand, TPZ performs better in the lower sections of the map, which approximates to brighter DEEP2 $R$ -band magnitudes.",
"Interestingly, SOM$z$ performs relatively better in the upper middle of the map, which corresponds to the middle range $21 R 23$ .",
"The overall variation in weights across the map reflects the performance differences between the individual methods, which are exploited by the combination algorithms in order to identify the optimal combined performance.",
"We can also compare the global performance of the ${\\rm BMC}$ method with the three individual photo-$z$ PDF methods as a function of the spectroscopic redshift as shown in Figure REF .",
"In this Figure, the photometric redshifts are the computed as the mean of each PDF, and the median is shown as black points along with the tenth and ninetieth percentiles as vertical error bars, enclosing 80% of the distribution on each redshift bin.",
"The performance of the ${\\rm BMC}$ method is generally more accurate, resulting in a tighter distribution that suffers fewer outliers when compared to the benchmark TPZ method.",
"Interestingly, the SOM$z$ performance is similar to TPZ, while BPZ is worse, with wider spread and several discontinuities.",
"Nevertheless, the combined method still uses BPZ, as shown in the weight maps, as appropriate to generate an overall improved performance, especially for the faintest galaxies as discussed previously.",
"We note, however, that the number counts in the last few bins are very low for the DEEP2 training and testing sets as shown in Figure REF .",
"Therefore, although on average BPZ has better performance statistics over those bins (with large error bars), the photo-$z$ results remain subject to Poissonian fluctuations (which is important when constructing a SOM to subdivide the galaxies when applying the combination models), thus the BMC results do not emphasize the BPZ results in the highest redshift bins.",
"Figure: A comparison of the photometric and the spectroscopic redshifts for all DEEP2 field1 galaxies.",
"From left to right, the comparison is for the TPZ, SOMzz, BPZ, and the BMCBMC techniques.The black dots are the median values of z phot z_{\\rm phot} and the errors bars correspond to the tenth and ninetieth percentiles within a given spectroscopic redshift bin of width Δz=0.1\\Delta z = 0.1Of all of the ten different metrics presented in Table REF , only the $KS$ test does not show a marked improvement over the benchmark TPZ method.",
"This metric does not depend on the redshift binning and it is computed by using the stacked PDF for each method.",
"As a result, this metric is expected to be less sensitive to a combination approach, since stacking the PDF smooths out little discrepancies between the models.",
"After integrating over a large number of galaxies PDFs, the individual methods will not differ significantly from one another and the final $N(z)$ distribution will resemble the one from the benchmark method.",
"Figure REF shows the final $N(z)$ produced by stacking the PDFs from the ${\\rm BMC}$ technique for galaxies from the DEEP2 (in solid black) and the corresponding DEEP2 spectroscopic $N(z)$ for the same galaxies (in gray).",
"As also seen in CB13 and CB14 for TPZ and SOM$z$ respectively, both distributions match exceedingly well.",
"Figure: Top panel: The N(z)N(z) for the DEEP2 sample computed directly from the spectroscopic redshifts (gray) and by stacking the photo-zz PDF estimates from the BMCBMC method (black).",
"Bottom Panel: The absolute difference between these two N(z)N(z) distributions."
],
[
"Photo-$z$ PDF Combination for the SDSS",
"We now change our focus to the analysis of the SDSS galaxy sample, which consists of 1,097,397 galaxies taken from the SDSS-DR10 data; we now retain 50,000 galaxies for training purposes.",
"We apply the same three photo-$z$ PDF estimation methods and seven different combination methods.",
"We construct a SOM-defined, $10 \\times 10$ two-dimensional map to subdivide the multi-dimensional magnitude and color space by using a rectangular topology to facilitate visualization.",
"As before, we use cross-validation data to identify the best set of model parameters within each individual cell in our two-dimensional map.",
"As shown in Figures REF and REF , the photo-$z$ PDFs computed by using the cross-validation and testing data sets are comparable and unbiased.",
"We present in Table REF the same ten metrics for each method, and in bold we highlight the best method for each metric.",
"Overall, the results obtained for this data set are remarkable, especially for the outlier fraction and the dispersion.",
"We once again treat TPZ as the benchmark method; but note that, interestingly enough, in two cases, including the $KS$ metric, TPZ does provide the best result.",
"In addition, both ${\\rm BMA}$ and ${\\rm BMC}$ have very similar results, with the latter being slightly better.",
"After these two models, ${\\rm WA_{\\rm shape}}$ , which is OOB data independent, shows good performance, especially when looking at the $I_{\\Delta z^{\\prime }}$ score.",
"For any given individual metric, however, it does not perform better than other combination methods.",
"For this data, BPZ provides good results; thus we expect that the set of template described in §REF are a good representation of the galaxies in the SDSS photometric data.",
"In particular, this seems true of the LRGs that dominate this sample for $z 0.3$ .",
"Table: A summary of the performance results for the three individual methods and the seven different photo-zz PDF combination methods as applied to the SDSS-DR10 data, with no magnitude cut applied to the training data set.",
"The bold entries highlight the best value within each column to aid in the interpretation of the table (c.f.",
"Figure ).We present the performance of the three individual and seven combination methods when applied to the SDSS data for five of the most common metrics in Figure REF .",
"As was the case with the DEEP2 data, the Bayesian combination methods provide good performance.",
"We also see the same variation in the $KS$ metric, especially when comparing the combination methods to TPZ.",
"However, TPZ is not always the best performer among the individual techniques, for example SOM$z$ displays the best performance as measured by $\\sigma _{\\Delta z^{\\prime }}$ and ${\\rm out}_{0.1}$ .",
"As we discussed in CB14, SOM$z$ performs quite well when using a spherical topology; in the current application to the SDSS data, we have used a random atlas containing 300 maps that use spherical topology each with 3072 total cells.",
"Interestingly, the ${\\rm WA_{\\rm oracle}}$ method, which selects the best method within each binned cell, often selects the SOM$z$ result as we can infer from Figure REF .",
"Although in general the oracle combination method is not the best possible combination, as shown by the overall performance of the ${\\rm BMA}$ and ${\\rm BMC}$ combination methods on this data.",
"Figure: A comparison of the average performance for the three individual photo-zz PDF estimation methods and the seven different photo-zz PDF combination approaches for five different metrics as defined in Table for the SDSS data.",
"The horizontal dashed line indicates the best result for a given statistic among the three individual methods, and the shaded area separates the individual methods from the combined approaches.",
"All values are presented in Table .We also display the SOM-defined, $10 \\times 10$ two-dimensional map used to determine the weights for the three individual methods for each cell in Figure REF .",
"In this map, we identify galaxies within the OOB and test data to determine the parameters for the combination models.",
"One of the benefits of using an unsupervised learning method for this mapping is that we can use any property from the galaxies within this map to construct a representation, such as the mean SDSS $r$ -band magnitude map shown in the bottom right panel of Figure REF .",
"In this panel the brighter galaxies are generally on the right while the fainter galaxies are on the left, even though all five magnitudes and four colors were used to construct the SOM-defined, two-dimensional map.",
"The weighting for the three individual methods show interesting patterns, and TPZ and SOM$z$ seem complimentary in that TPZ is weighted most strongly at fainter $r$ -band magnitudes (the left side of the map) while SOM$z$ is weighted most strongly at brighter $r$ -band magnitudes (the right side of the map).",
"This result is most likely an artifact from the bi-modality of the training data, which is dominated at low redshift by the SDSS main galaxy sample and at high redshifts by the SDSS-III LRG sample.",
"At brighter magnitudes and lower redshifts, the SOM$z$ approach where a high-dimensional space is projected to two-dimensions does a better job of maintaining complex relationships within the data.",
"At fainter magnitudes and higher redshifts, however, the data are dominated by the homogeneous LRG sample.",
"The TPZ approach performs better for this sample, since the high-dimensional space is recursively sub-divided by TPZ to maximize the information gain, which may only require one or two dimensions.",
"Figure: A two-dimensional SOM showing the relative weights for the BMA combination scheme applied to the three individual methods for the SDSS data (TPZ is top left, BPZ is top right, and SOMzz is bottom left).",
"In each panel, the color map indicates the value of the weight relative to the other cells in the map.",
"The bottom right panel shows the same cells colored by the mean SDSS rr-band magnitude for the cross validation galaxies.Figure: A comparison of the photometric and the spectroscopic redshifts for all SDSS galaxies.",
"From left to right, the comparison is for the TPZ, SOMzz, BPZ, and the BMCBMC techniques.The black dots are the median values of z phot z_{\\rm phot} and the errors bars correspond to the tenth and ninetieth percentiles within a given spectroscopic redshift bin of width Δz=0.05\\Delta z = 0.05Another interesting observation from these weight maps is that BPZ performs well over much of the parameter space, with a particular strong weighting in a narrow vertical band on the extreme left of the map and again in the center of the map.",
"Given the nature of the input galaxy sample, it seems reasonable to expect that these areas of the map are dominated by Elliptical galaxies.",
"Another interesting observation is that there are six cells in the second column from the left that all have the same value in each weight map (pink for TPZ, white for BPZ, and light blue for SOM$z$).",
"These cells are primarily empty, i.e., they contain weights and training data but they lack test galaxies and thus have a constant value, which illustrates how strongly the galaxies (i.e., MGS or LRG) are concentrated in this SOM-defined, two-dimensional topology.",
"The number of galaxies, either for training or testing, within each cell can vary significantly, which is simply due to the fact that we used a fixed number of cells (in this case 100) to represent the higher dimensional space when fewer cells would have been sufficient.",
"However, the empty cells do not affect the performance of the photo-$z$ combination methods, they are simply not used during the analysis.",
"It is the fact that these individual methods perform differently across these cells that makes the combination approach a powerful technique to maximally extract information from the available data.",
"We next provide a comparison between the photo-$z$ PDFs computed by the three individual techniques and the ${\\rm BMC}$ technique and the SDSS spectroscopic redshift for all 1,097,397 galaxies in Figure REF .",
"The first observation from the figure is the bi-modality of the sample, which is the result of the two primary sub-populations (i.e., MGS and LRGs).",
"Overall, the results are quite good with a very tight correlation, especially in areas of high source density areas.",
"The main exception is at the highest redshifts where there is a slight underestimation; and, as seen before, we can observe how these different approaches provide similar results, which are therefore correlated, while still differing in other areas where one method may outperform the others.",
"The most right panel is the ${\\rm BMC}$ which shows a slightly tighter distribution in comparison to the others.",
"Finally, in Figure REF we present the galaxy redshift distribution for both the spectroscopic sample (in gray) and the photometric redshift distribution, computed by stacking the individual galaxy PDFs (in black).",
"This Figure highlights that the underestimation of the photo-$z$ at high redshifts in Figure REF coincides with the strong decline in the number of galaxies after $z = 0.75$ .",
"More importantly, however, this $N(z)$ figure shows the excellent agreement between the photometric and spectroscopic galaxy redshift distributions.",
"Given the fact that the SDSS galaxy sample contains two distinct populations, this agreement is remarkable.",
"Figure: Top panel: The N(z)N(z) computed directly from the spectroscopic redshifts (gray) and by stacking the photo-zz PDF estimates from the BMCBMC method (black).",
"Bottom Panel: The absolute difference between these two N(z)N(z) distributions."
],
[
"Outliers identification",
"As we have discussed previously, aggregating information from multiple photo-$z$ PDFs estimation techniques can improve the overall photo-$z$ solution.",
"In this section, however, we explore how this information can be combined to improve the identification of outliers within the test data.",
"In particular, we attempt to use all possible information in order to identify these objects, from the shape of each photo-$z$ PDF as computed by all individual methods to the differences in their predicted photo-$z$ .",
"We adopt a Naïve Bayes Classifier (NBC) [64] to identify these two groups, a technique that has found widespread adoption to identify spam email messages.",
"The advantage of this approach is that it is easy to implement, is fast and efficient for large dimensional data, and can be very competitive with other classifiers [26], [31].",
"Let $\\theta $ be the set of $N_{\\theta }$ parameters, $\\theta _i$ , we will use to identify the outliers.",
"By using the Bayes Theorem, we can compute the probability for an object to be an outlier, given $\\theta $ as: $P({\\rm out} \\mid \\theta ) = \\frac{P({\\rm out}) P(\\theta \\mid {\\rm out})}{P(\\theta )}$ where the evidence, $P(\\theta )$ is given by $P(\\theta ) = P(\\theta \\mid {\\rm out}) + P(\\theta \\mid {\\rm in})$ and out refers to outliers and in refers to inliers, the only two classes we identify in this analysis.",
"The Naïve Bayes Classifier assumes that all $\\theta _i$ variables are independent, even if their independence is weak or even if there is a strong dependence between any of them.",
"Each variable provides information about these two classes, and this information can be combined to make a stronger classifier [64].",
"For instance, in CB13 we showed that outliers tend to have a broader (larger values of $zConf$ ) and multi-peaked PDFs, and herein we treat these values as independent data even though multi-peaked PDFs are indeed generally broader.",
"By using this assumption, we can write: $P(\\theta \\mid {\\rm out}) = P(\\theta _1, \\theta _2, \\dots , \\theta _{N_\\theta } \\mid {\\rm out}) = \\prod \\limits _{i=1}^{N_\\theta } P(\\theta _i \\mid {\\rm out})$ and similarly, $P(\\theta \\mid {\\rm in}) = \\prod \\limits _{i=1}^{N_\\theta } P(\\theta _i \\mid {\\rm in})$ We can now rewrite Equation REF : $P({\\rm out} \\mid \\theta ) = \\frac{ P({\\rm out}) \\prod P(\\theta _i \\mid {\\rm out})}{\\prod P(\\theta _i \\mid {\\rm out}) + \\prod P(\\theta _i \\mid {\\rm in}) } ,$ which is similar to the method used by [35], who demonstrated the potential of this approach to identify photo-$z$ outliers.",
"Here, however, we use a different set of variables that are generated for all three individual photo-$z$ PDF methods.",
"In our case we use $N_{peak}$ , the number of peaks in each photo-$z$ PDF; $r_{peak}$ , the logarithm of the ratio between the height of the first peak and the height of the second peak; $z_{mean}$ , the mean of each photo-$z$ PDF; $z_{mode}$ , the mode of each PDF;$zConf$ , measured with respect to the mean and the mode of the photo-$z$ PDF; and the difference in the photo-$z$ , as enumerated by the mean and the mode between each of the three methods.",
"Thus, we have six metrics computed individually for each of our three photo-$z$ PDF estimation techniques, and an additional six metrics for the difference in photo-$z$ mean and mode between the three techniques.",
"As a result, we have a total of twenty-four metrics, to which we can add the input data for each survey.",
"We, therefore, have a total of thirty-eight variables for the DEEP2 survey, while for the SDSS we have a total of thirty-three variables to use for outlier detection.",
"For convenience, we rescale each of these variables to lie between zero and one.",
"$P(\\theta _i \\mid {\\rm in})$ and $P(\\theta _i \\mid {\\rm out})$ are evaluated by using the OOB or cross-validation data, which we have shown can reliably predict the results on the test data.",
"Once computed, these distributions are evaluated for the test data, where $ P({\\rm out} \\mid \\theta ) $ is evaluated separately for each galaxy in the test data.",
"Figure: The normalized distributions of four of the set of thirty-eight (rescaled) θ\\theta variables from the DEEP2 data that are used for outlier detection.",
"The variables are binned as outliers (black line histograms) or inliers (gray histogram).",
"From the top left and following in a clockwise direction: N peak N_{peak}, the number of peaks in the TPZ PDF; zConfzConf, as computed from TPZ, the RR-band magnitude, and the difference between the photo-zz computed by using the mean of the TPZ and BPZ PDFs.Figure: The normalized distributions of four of the set of thirty-three (rescaled) θ\\theta variables from the SDSS data that are used for outlier detection.",
"The variables are binned as outliers (black line histograms) or inliers (gray histogram).",
"From the top left and following in a clockwise direction: r peak r_{peak}, the logarithmic ratio of the first two peaks in the TPZ PDF; zConfzConf, as computed from SOMzz, the SDSS zz-band magnitude, and the difference between the photo-zz computed by using the mode of the SOMzz and BPZ PDFs.Figure REF presents the normalized distributions of four rescaled variables (i.e., $\\theta _i$ ) taken from the DEEP2 test data.",
"Note that the inlier and outlier distributions are normalized to have unit area, thus these distributions illustrate how these two populations differ and not how the relative numbers between the inlier and outlier populations vary.",
"The four variables shown in this Figure include the number of peaks in the TPZ PDFs, $zConf$ computed by TPZ, the $R$ -band magnitude, and the difference between the mean of the TPZ and BPZ photo-$z$ PDFs.",
"In just these four distributions, there is clear separation between the galaxies labeled as outliers (black line) and inliers (gray shaded area), where the outlier identification metrics are defined by using Table REF .",
"In particular, for this Figure we use ${\\rm out}_{0.1}$ , i.e., galaxies for which $\\Delta z^{\\prime } > 0.1$ .",
"While not shown, a similar result is seen for the other distributions.",
"The result that outliers and inliers follow distinct distributions is what makes this a powerful approach.",
"In effect, all information is assumed to be independent, and when combined allows an efficient identification of catastrophic outliers.",
"We see a similar trend in Figure REF , but now for galaxies in the SDSS test data.",
"In this Figure, we have selected four different rescaled variables; namely, the logarithmic ratio between the first and the second peaks of the TPZ PDF (note that if the PDF has one peak, we fix this value to be four), the $zConf$ computed from SOM$z$, the SDSS $z$ -band magnitude, and the difference between the mode of the SOM$z$ and BPZ photo-$z$ PDFs.",
"Once again, this Figure highlights that in each of these distributions there is a separation between the outliers and inliers, and that in combination we obtain an even better discriminant between these two classes.",
"By using Equation REF , we can combine the values of all of the rescaled variables (i.e., $\\theta _i$ ) to compute $P({\\rm out} \\mid \\theta )$ for each galaxy in the DEEP2 and SDSS, both for the OOB and the test data.",
"We present these $P({\\rm out} \\mid \\theta )$ distributions for the DEEP2 in Figure REF and for the SDSS in Figure REF .",
"Both Figures are similar, showing a clear separation between the outliers and inliers in both data sets.",
"The probability ranges between zero and one, and the outliers are generally concentrated near one, while the inliers are concentrated near zero.",
"While some mis-classifications remain, the contamination has been greatly reduced, meaning we can successfully identify a majority of the outlier population.",
"Lastly, while there are a few galaxies with probabilities lying somewhere between zero and one, these distributions are highly bimodal, which reinforces the belief that this method provides a remarkably good discriminant between these two populations.",
"Figure: The count distribution of P( out ∣θ)P({\\rm out} \\mid \\theta ) for the DEEP2 OOB data (top) and test data (bottom) showing both the outliers (orange) and inliers (gray).Figure: The count distribution of P( out ∣θ)P({\\rm out} \\mid \\theta ) for the SDSS OOB data (top) and test data (bottom) showing both the outliers (orange) and inliers (gray ).Once again, in both Figures REF and REF , the OOB and test data distributions show strong similarities.",
"As a result, we can expect that any cut we make on the OOB data will produce similar results in the test data, allowing us to make a robust classification of outliers in potentially blind test data.",
"To quantify this, we show in Table REF the effects of selecting outliers by using this NBC approach and by using the $zConf$ approach we initially presented in CB13 for the DEEP2 data.",
"To simplify the comparison, we first select inlier galaxies by using the $P({\\rm out} \\mid \\theta )$ to cut the test data sample, and subsequently choosing those galaxies in the test data that have the highest $zConf$ so that we have the same number of galaxies selected via both techniques.",
"Table: The effect of removing outliers from the DEEP2 test data on several, select performance metrics by using the Naïve Bayes Classifier and the zConfzConf cut approach.",
"The two techniques are applied to ensure equal numbers of galaxies are selected, which is indicated by the Fraction column.The information in this Table demonstrates that the NBC approach produces a sample of galaxies that have a smaller spread in $\\Delta z^{\\prime }$ along with a smaller number of outliers than the $zConf$ method, which was previously shown to be beneficial in this regard (CB13).",
"We interpret this result as suggesting that a $zConf$ cut can potentially remove good galaxies whose photo-$z$ PDF happens top be broad, while retaining some bad galaxies that have a well-localized photo-$z$ PDF.",
"By using a Naïve Bayes approach, we collect all information from photo-$z$ PDFs predicted by using different, semi-independent methods, allowing a more robust discriminant between outliers and inliers.",
"Finally, we notice that as always there is a trade-off between completeness, whereby we try to retain as many good galaxies, and contamination, whereby we try to minimize the inclusion of bad galaxies.",
"The final choice in this conflict should be determined by the scientific application, but by producing a probabilistic value, subsequent researchers can make these cuts more easily.",
"We performed a similar analysis on the SDSS galaxy sample and present the results in Table REF .",
"As was the case with the DEEP2 galaxies, we see that the NBC approach once again does better in identifying outliers within the sample, as the NBC cuts have a smaller scatter and the fraction of remaining outliers is remarkably small.",
"We also notice that the mean bias is similar between the two approaches, but the number of outliers, defined as $\\Delta z^{\\prime } > 0.1$ , is significantly reduced when we adopt the Bayesian approach.",
"This is yet another piece of evidence supporting the benefits of aggregating information to make decisions.",
"Table: The effect of removing outliers from the SDSS test data on several, select performance metrics by using the Naïve Bayes Classifier and the zConfzConf cut approach.",
"The two techniques are applied to ensure equal numbers of galaxies are selected, which is indicated by the Fraction column.We can also test how the definition of an outlier affects this approach.",
"Previously we identified an outlier as a galaxy that had $\\Delta z^{\\prime } > 0.1$ ; but for the purpose of this test, we apply a much more restrictive cut of $\\Delta z^{\\prime } > 0.05$ .",
"We apply the NBC cut and produce a matched sample by imposing a $zConf$ cut to both the DEEP2 and the SDSS galaxies, presenting the information in Table REF .",
"We find, once again, that even for this more restrictive approach we produce a cleaner catalog (of the same size) as compared to using only the $zConf$ parameter.",
"Interestingly, even after removing almost 30% of the galaxies from the DEEP2 galaxy sample, we still have over a 10% outlier contamination.",
"On the other hand, this tight cut applied to the SDSS galaxies produces a very small contamination of $\\sim $ 2%, for both methods, albeit the NBC approach is still slightly better.",
"Table: The effect of removing outliers, defined as Δz ' >0.05\\Delta z^{\\prime } > 0.05, from the DEEP2 and SDSS test data on several, select performance metrics by using the Naïve Bayes Classifier and the zConfzConf cut approach.",
"For each data set, the two techniques are applied to ensure equal numbers of galaxies are selected, which is indicated by the Fraction column.While producing galaxy samples that are less affected by outliers than competing techniques, the NBC approach has an additional advantage in that it can easily be extended to other variables and to other photo-$z$ algorithms.",
"In effect, any information that might increase the efficacy of outlier identification can be included in order to improve this discriminant while still maximizing the overall galaxy sample size."
],
[
"Conclusions",
"We have presented and analyzed different techniques for combining photo-$z$ PDF estimations on galaxy samples from the DEEP2 and SDSS projects.",
"In particular, we use three independent photo-$z$ PDF estimation methods: TPZ, a supervised machine learning technique based on prediction trees and a random forest; SOM$z$, an unsupervised machine learning approach based on self organizing maps and a random atlas; and BPZ, a standard template-fitting method that we have slightly modified to parallelize the implementation.",
"Both TPZ and SOM$z$ are currently available within a new software package entitled MLZhttp://lcdm.astro.illinois.edu/code/mlz.html.",
"We developed seven different combination methods that employ ensemble learning with cross-validation data to maximize the information extracted.",
"Of these seven methods, four employ a weighted average where the weights can either be selected to be uniform across the input methods, to be determined from the shape of the photo-$z$ PDF (e.g., by using the $zConf$ parameter), to be determined by an oracle estimator where one (ideally the best) method is preferentially selected, and where the weights are obtained by a fitting procedure applied to the OOB data.",
"Three of the combination methods were Bayesian techniques: Bayesian Model Averaging (BMA), Bayesian Model Combination (BMC), and Hierarchical Bayes (HB).",
"We expect the individual photo-$z$ PDF estimation techniques to perform differently across the parameter space spanned by our galaxy samples; for example, template-fitting techniques are expected to work better at higher redshifts than machine learning methods, which perform optimally when provided high-quality, representative training data.",
"Thus we construct a two-dimensional, $10 \\times 10$ self-organizing map (SOM) to subdivide the high-dimensional parameter space occupied by the galaxy samples.",
"We apply different photo-$z$ PDF estimation techniques within each cell in this map, since each cell should contain galaxies with similar properties.",
"A visual inspection of these maps indicates that the two machine learning methods: TPZ and SOM$z$ are generally complementary, and that in combination with a model based technique such as BPZ we are able to maximize the coverage of this multidimensional space efficiently.",
"We also verified that by using the OOB data, as introduced in CB13, we can an obtain an accurate, unbiased and honest estimation of the performance of a photo-$z$ PDF estimation technique on the test data.",
"We also computed the correlation coefficient and the error distribution and showed they also behave similarly for the cross-validation (i.e., the OOB data) and the test data.",
"These computations are extremely important when combining photo-$z$ PDF techniques as we can learn from the OOB data the optimal parameters needed for a specific ensemble learning approach, and thereby maximize the performance of that combination technique when applied to blind test data.",
"Overall, we found that the BMA and BMC are the best photo-$z$ PDF combination techniques as they have better performance metrics when compared to the individual photo-$z$ PDF estimation techniques, especially when unbiased cross-validation data is available.",
"This result is true for both the DEEP2 and the SDSS data.",
"When OOB data is not available, we can instead use the $zConf$ parameter as a weight for each method after first renormalizing the individual photo-$z$ PDFs.",
"We can also use the Hierarchical Bayes method to combine these predictions, which we demonstrated can also lead to better results.",
"Within this Bayesian Framework, we also developed a novel, Naïve Bayesian Classifier (NBC) that efficiently identifies outliers within the galaxy sample.",
"The approach we present gathers all available information from the different photo-$z$ PDF estimation techniques regarding the shape of the PDF, the location of the mean and mode, and the magnitudes and colors, which are all naively assumed to be independent, in order to compute a Bayesian posterior probability that a certain galaxy is an outlier.",
"The distribution of these probabilities for an entire galaxy sample indicate that this is a very powerful method to separate outliers from inliers (i.e., good galaxies), and we further demonstrated that this approach can produce a more accurate and cleaner sample of galaxies than competing techniques, such as the use of the $zConf$ parameter.",
"An important takeaway point is that all information provided by the catalogs and the photo-$z$ PDF methods, no matter how redundant the information might appear, helps in building this discriminant probability.",
"Given the probabilistic nature of this computation, the final application of this technique can be chosen to maximize the scientific utility of the resulting galaxy data for a specific application.",
"The computational cost to apply these Bayesian models to galaxy samples will depend directly on the size of the data set, the number of photo-$z$ estimation techniques used, and the resolution of the given photo-$z$ PDFs.",
"In [15] we demonstrate how a sparse basis representation can reduce the storage significantly and that manipulation of these PDFs can be improved within the bases framework thereby reducing computational costs.",
"We plan to adopt this representation framework to compute the combination models, which will allow fast and accurate combination of multiple photo-$z$ PDFs.",
"Finally, we have demonstrated that even when a photo-$z$ PDF technique is very accurate, we can still make improvements by extracting additional information about the distribution of galaxies in the higher dimensional parameter space and the individual performance of the photo-$z$ PDF algorithms.",
"There are currently a large number of published algorithms to compute photo-$z$ 's, many of which also compute photo-$z$ PDFs.",
"Even if their performance is similar, these techniques will all have their own advantages and disadvantages.",
"Thus we believe the combination of different techniques is the future of photo-$z$ research, and we expect additional research to be forthcoming in this area.",
"Overall, the combination of photo-$z$ PDFs is a powerful, new approach that can be easily extended to incorporate new techniques in order to generate a meta-predictor that accelerate our knowledge and understanding of the Universe."
],
[
"Acknowledgements",
"The authors thank the referee for a careful reading of the manuscript and for comments that improved this work.",
"RJB and MCK acknowledge support from the National Science Foundation Grant No.",
"AST-1313415.",
"MCK has been supported by the Computational Science and Engineering (CSE) fellowship at the University of Illinois at Urbana-Champaign.",
"RJB has been supported in part by the Institute for Advanced Computing Applications and Technologies faculty fellowship at the University of Illinois.",
"The authors gratefully acknowledge the use of the parallel computing resource provided by the Computational Science and Engineering Program at the University of Illinois.",
"The CSE computing resource, provided as part of the Taub cluster, is devoted to high performance computing in engineering and science.",
"This work also used resources from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.",
"Funding for the DEEP2 Galaxy Redshift Survey has been provided by NSF grants AST-95-09298, AST-0071048, AST-0507428, and AST-0507483 as well as NASA LTSA grant NNG04GC89G.",
"Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.",
"The SDSS-III web site is http://www.sdss3.org/.",
"SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University."
]
] | 1403.0044 |
[
[
"One-dimensional potential for image-potential states on graphene"
],
[
"Abstract In the framework of dielectric theory the static non-local self-energy of an electron near an ultra-thin polarizable layer has been calculated and applied to study binding energies of image-states near free-standing graphene.",
"The corresponding series of eigenvalues and eigenfunctions have been obtained by solving numerically the one-dimensional Schr{\\\"o}dinger equation.",
"Image-potential-state wave functions accumulate most of their probability outside the slab.",
"We find that a Random Phase Approximation (RPA) for the non-local dielectric function yields a superior description for the potential inside the slab, but a simple Fermi-Thomas theory can be used to get a reasonable quasi-analytical approximation to the full RPA result that can be computed very economically.",
"Binding energies of the image-potential states follow a pattern close to the Rydberg series for a perfect metal with the addition of intermediate states due to the added symmetry of the potential.",
"The formalism only requires a minimal set of free parameters; the slab width and the electronic density.",
"The theoretical calculations are compared to experimental results for work function and image-potential states obtained by two-photon photoemission."
],
[
"Introduction",
"Graphene layers display a number of interesting properties and potential applications owing to the linearly-dispersing bands found near the $\\overline{\\mathrm {K}}$ point in the Brillouin zone [1], [2].",
"However, in order to get a complete characterization of graphene other regions in the Brillouin zone need to be considered.",
"In particular, unoccupied states in the vicinity of the $\\overline{\\Gamma }$ point can play a significant role in the transport of currents[3].",
"Indeed, it is well known the importance of conduction band minima valleys located around $\\overline{\\Gamma }$ in ballistic electron emission processes, where currents are injected on a substrate under voltage-dependent matching restrictions (e.g.",
"$k_{\\parallel }$ -conservation[4]) that are relevant to design field emission transistors[5].",
"On the other hand, the transport of heat also involves other regions of the Brillouin Zone apart from the $\\overline{\\mathrm {K}}$ point.",
"Recently it has been shown how the thermal conductivity of few layers of graphene supported on silicon dioxide depend crucially on flexural modes[6] that are sensitive to the electronic structure around $\\overline{\\Gamma }$[7].",
"Therefore, the dielectric response of very few layers of graphene is a key physical element to effectively design devices based on graphene.",
"The dielectric response is directly related, and therefore can be investigated, by trapping electrons in the region of unoccupied states between the Fermi level and the vacuum level.",
"These states, bound by their self-induced long-range image potential, are called image-potential states [8].",
"The experimental [9], [10], and theoretical [11] study of image-potential states constitutes an ideal probe to better understand the properties of graphene layers.",
"The image force is a non-local effect asymptotically dominated by correlation effects [12].",
"In order to study the infinite Rydberg series arising from the image potential one needs to compute an effective one-dimensional potential, $\\Phi (z)$ , representing the real part of the quasi-static self-energy for an external probe charge.",
"This self-induced potential is a continuous function spanning from inside the material, where it represents the exchange and correlation energy, to the vacuum region, where it should have the correct Coulomb-like asymptotic behavior $\\propto -\\frac{1}{4z}$ .",
"Such a potential can only be obtained from a non-local spatial formalism, since a local approach results in a correlation potential decaying exponentially in the vacuum region, following the density behavior outside the solid [13].",
"For a self-consistent first-principles theory such a non-local functional dependence requires costly numerical calculations [14].",
"Therefore, it is useful and natural to search for simpler ways to obtain such an effective potential, which is the basic ingredient needed to understand the physics of image-potential states bound by an ultra-thin polarizable layer like graphene.",
"The simplest of these alternatives is to introduce a set of fitting parameters to continuously join solutions valid either inside or outside the solid.",
"This point of view has been taken, e. g. by Silkin et al.",
"to study image-potential states in free-standing graphene, joining a function with the correct classical asymptotic behavior outside to a first-principles calculation inside a graphene layer [11].",
"Such ab-initio calculations depends on choosing a model for the exchange and correlation potential; Local Density Approximation (LDA) has been employed by Silkin et al.",
"[11].",
"Furthermore, it needs setting a very large unit cell to minimize effects between charged periodic images (i. e. 85 Å vacuum separator was used to reach convergence, implying a large number of plane-waves and a serious computational effort).",
"Finally, such a calculation needs to be supplemented by a few adjustable parameters, e. g. the choosing of a matching point to join the inside potential to the asymptotic classical potential.",
"In this paper we analyze an alternative that makes use of a minimal free parameter set and makes a simple, flexible and accurate basis for interpreting experimental results.",
"We use a well-known model for the reflection of electromagnetic fields at the surface (infinite barrier specular model [15]), coupled with a non-local static dielectric response so the desired self-energy can be obtained [16].",
"For the electrodynamics model, two free parameters are introduced, i. e. the electronic polarizability of the thin slab, which is determined by the Fermi-Thomas screening wavelength inside the slab (electron density of the material), and a geometrical dimension given by the layer thickness.",
"This approach leads in a natural way to a potential with proper physical features: it is continuous and finite over the full spatial domain and it has the right asymptotic behavior towards the vacuum region.",
"Recently, Ghaznavi et al.",
"[17] have studied the non-linear screening of a external charge near a doped graphene layer by solving a Fermi-Thomas model via a non-linear integral equation.",
"The non-linear image potential shows changes up to $0.2$ eV with respect to the classical image potential, and supports the use of RPA dielectric response for graphene.",
"In the case of a graphene layer laying on a metallic support another free parameter may be introduced in this model: the wave-function penetration in the material.",
"Image-potential states are supported in many metallic surfaces for energies between the vacuum level and the Fermi level due to the existence of surface band gaps that prevent the penetration of the wave function towards the bulk and allows the existence of bound states [8].",
"Therefore, penetration of wave functions is determined by the band structure of particular materials and surface orientations in a way that cannot be incorporated in our model, except by including a free parameter that globally determines this penetration.",
"Here we shall first discuss two limiting cases: 0 or 100% penetration of the graphene layer, and then the continuous evolution from one limit to the other.",
"Since for image-potential states we are interested in regions in reciprocal space with $\\vec{k}$ near $\\overline{\\Gamma }$ and energies between the vacuum level and the Fermi energy, it is well justified to model graphene as a polarizable electron gas with a quadratic energy dispersion, as seen from the relevant bands for graphite and graphene[18], [19], [20], [11].",
"A connection between the interlayer states and the image potential states on graphite, graphene and other carbon-based materials has been established by different groups, e.g.",
"Silkin et al.",
"[11], Feng et al.",
"[21], and Hu et al.[22].",
"These authors have pointed out that such states exist universally on two-dimensional materials.",
"The static dielectric response, $\\epsilon (\\vec{k})$ , has been modeled by a Random Phase Approximation (RPA), and by its small $k$ expansion, the Fermi-Thomas Approximation (FT) [23].",
"While the RPA yields a more accurate description of excitation in the material, introducing the FT allows us to write the potentials as analytic expressions or quasi-analytical ones which merely depend on a final numerical step involving the simple integration of a function decaying quickly for large values of the argument.",
"A single parameter, the screening constant related to the density of states at the Fermi level, $k_\\mathrm {FT} \\propto \\frac{\\partial n_0}{\\partial \\mu }$ , fixes both the scales for energies and lengths facilitating the rescaling of results for different materials and sizes (atomic units are used throughout the paper, except where explicitly stated otherwise).",
"Taking graphite as a model (2 g/cm$^{3}$ , 2s$^2$ 2p$^2$ ), a typical value for graphene is $k_\\mathrm {FT} \\approx 1$ ($r_s \\approx 2.5$ ), although its precise value should depend on factors like doping, external potentials, etc; this is accommodated in our results through the simple aforementioned scaling with $k_\\mathrm {FT}$ .",
"The other parameter used to characterize a thin slab is its width, $2d$ .",
"For a single atom thick layer of graphene a reasonable value for $d$ , should be related to the spatial extension of $\\pi $ carbon orbitals, $d \\approx 1$ a.u.",
"The value of this parameter turns out not to be critical for this work because wave functions spread over regions much larger than $d$ .",
"The theoretical results are compared to experimental results for graphene on various substrates.",
"The presence of the substrate leads to a charge transfer from or to the graphene which can be described also as doping.",
"The resulting work function change due to graphene has been modeled by Giovanetti et al.",
"[24], [25].",
"We find a good agreement with the experimental data.",
"The doping changes the available screening charge and leads in turn to a change in the energy of the image-potential states.",
"The experimental data from two-photon photoemission agree with the calculated dependence.",
"For an external probe charge near a slab ($Q=1$ ) we seek the potential acting on $Q$ by the polarization charges induced in the medium by $Q$ itself.",
"This is obtained by computing the total potential, and subtracting the charge's own naked potential.",
"To ensure the proper boundary conditions, and according to the specular reflection model at the surface, auxiliary pseudo-media are introduced for the polarizable slab and the vacuum that reduce the calculation to matching solutions obtained in different regions of space for homogeneous media everywhere [12].",
"Details of the calculation for the thin slab are given in .",
"The resulting potential depends on a number of integrals that include the dielectric response of the system: for the RPA these are computed numerically.",
"On the other hand, within the FT approximation an expression that only depends on a single numerical integration can be obtained, $\\Phi _\\mathrm {FT}(z > d)=-\\frac{k_\\mathrm {FT}^2}{2}\\int _{0}^{\\infty }\\, d \\kappa \\,\\frac{e^{-2 \\kappa z} }{ \\left(\\chi +\\kappa \\coth {\\left[\\chi d \\right]}\\right)\\left(\\chi +\\kappa \\tanh \\left[\\chi d \\right]\\right)}$ $\\Phi _\\mathrm {FT}(0<z \\le d)=$ $\\int _{0}^{\\infty } \\, d \\kappa \\bigg \\lbrace \\frac{\\chi +e^{4 \\chi d}\\left(\\kappa e^{2 \\chi z }+\\kappa -\\chi \\right)}{2 \\left(e^{4 \\chi d}-1\\right) \\chi }+\\frac{\\kappa }{2 \\chi }\\bigg [-\\frac{2 \\kappa \\left((\\chi +\\kappa )e^{2 (2 d+z)\\chi }+(\\chi -\\kappa )\\right)}{(2 \\kappa ^2+k_\\mathrm {FT}^2)\\left(e^{4 d \\chi }-1\\right) +2 \\kappa \\chi \\left(e^{4 d \\chi }+1\\right)}+$ $+\\frac{e^{-2 \\chi z}+1}{e^{4 \\chi d}-1}-\\frac{e^{-2 \\chi (d+z) }\\left(1+e^{2 \\chi z}\\right) \\kappa \\left(\\kappa +\\chi \\left(e^{2 (d+z) \\chi }+\\cosh \\left[2 \\chi d \\right]\\right)\\mathrm {csch} \\,\\left[2 \\chi d \\right]\\right)}{2 \\kappa \\chi \\cosh \\left[2 \\chi d \\right]+\\left(2 \\kappa ^2+k_\\mathrm {FT}^2\\right) \\sinh \\left[2 \\chi d \\right]}\\bigg ]\\bigg \\rbrace $ where $\\chi =\\sqrt{\\kappa ^2+k_\\mathrm {FT}^2}$ .",
"This is an useful expression that can be computed very efficiently.",
"We shall see that for the purpose of computing the energy levels of image-potential states it makes an excellent approximation to the more costly RPA calculation.",
"In figure REF we show the potential for a slab occupying the region $-d \\le z \\le d$ ; both in the FT approximation (black continuous line), and in the RPA one (black dots).",
"In the region determining the Rydberg series ($|z|\\ge d$ ), both approaches yield similar values and agree with the correct asymptotic power-law.",
"Near the center of the slab, FT overestimates the interaction over RPA by about 30-40%, $\\frac{\\Phi _{RPA}}{\\Phi _\\mathrm {FT}} \\bigg |_{z=0} \\approx 0.82-\\frac{r_s}{12.5}\\quad ; \\quad 2 \\le r_s \\le 6$ a difference that is reflected mainly in the lowest state (node-less) that has a significant weight in the central part of the slab where the difference between RPA and FT is larger.",
"This state is located outside the window between the vacuum level and the Fermi energy and is not part of the image-potential states series.",
"For $d\\gg \\frac{1}{k_\\mathrm {FT}}$ the $\\kappa $ integral in Equations (REF ) and (REF ) can be evaluated analytically to obtain the following useful particular values (see ): $\\Phi _\\mathrm {FT}(z=d)=-\\frac{k_\\mathrm {FT}}{3}$ , and $\\Phi _\\mathrm {FT}(z=0)=-\\frac{k_\\mathrm {FT}}{2}$ .",
"We remark that the latter value corresponds to the Coulomb hole [26].",
"Furthermore, for $z > \\frac{1}{k_\\mathrm {FT}}$ (vacuum region) the potential is well approximated by a classical law, $\\frac{1}{4(z-z_0)}$ , corrected by an image plane, $z_0$ .",
"The value of $z_0$ can be obtained by expanding the integrals for $\\kappa \\approx 0$ , which fixes the position of the image plane in this model: $z_0=-\\frac{1}{k_\\mathrm {FT}}$ (dashed blue line in figure REF , see ).",
"Finally, Equation (3) suggests a simple procedure to find a good approximant to the RPA potential.",
"Given a physically representative value for $k_{FT}(r_s)$ , one can find another value, $k_{FT}^{*}$ , such that $\\Phi _{RPA}(k_{FT},d=0) \\approx \\Phi _{RPA}(k_{FT}^{*},d=0)$ .",
"For the purpose of obtaining energy eigenvalues of $\\Phi _{RPA}(k_{FT},z)$ , $\\Phi _{FT}(k_{FT}^{*},z)$ is a good approximation to the full RPA potential that can be computed quick and easy."
],
[
"Eigenvalues",
"We solve numerically the Schrödinger equation [27], [28] to compute the eigenvalues and eigenfunctions corresponding to the RPA and FT model potential described above.",
"Quite generally, bound states can be unequivocally labeled by the number of nodes $n$ , with energies increasing as the number of nodes increases.",
"Furthermore, as long as the potential is symmetric, eigenfunctions have either even or odd parity, for even or odd number of nodes, $n$ .",
"Table: Binding energies, E n E_{n} (eV), measured with respect to the vacuum leveland labeled according tothe number of nodes in wave functions.Top three rows:Results for RPA and Fermi-Thomas (Eq.",
")are given for a free-standing graphene ultra-thin layerand compared with similar results obtained byfrom an ab-initio LDA calculation matched to anasymptotic expression for the image potential by Silkin et al.",
"(states are labelled accordingly totheir ordering in the image-potential series of unoccupied stateslying above the Fermi energy and below the vacuum level).The effect of a repulsive barrier located at z≈-10z \\approx -10 Å ispresented under R b=-20 R_{b=-20} (notice that in this case symmetryis broken and the parity is no longer a good quantum number).For comparison, experimental values measured on Gr/Ir are quoted,and compared with the eigenvalues for the classical Whittaker's problem.Finally, we show the effect of approaching the repulsive barrier to z≈0.5z \\approx 0.5 Å,R b=1 R_{b=1},that can also be computed by introducing an appropriate quantum defect (δ=0.2\\delta =0.2).To guide the eye, we highlight in bold face numbers that can be compared across differentcalculations or experiments and have been given an accompanying interpretationin the text.We show in figure REF the first five eigenvalues for $\\Phi (z)$ ; dashed (red) and dotted (blue) horizontal lines for even and odd parities.",
"It is worth noticing the structure of this series: there is an isolated eigenvalue (the lowest one), while the remaining states cluster near the vacuum level.",
"For standard densities (e. g. $1 < r_s < 10$ ) this first level appears below $\\sim -4.5$ eV; an estimate for the work function in graphene (thick line).",
"Therefore, this $n=0$ eigenstate does not fit in the standard definition for a Rydberg state, necessarily located between the vacuum and Fermi levels to be observable in a standard experiment.",
"Figure: Bound states energies, -E n -E_{n} (eV), for thehydrogenic series supported by the potential of theultra-thin slab in figure (FT: triangles up, blue; RPA: triangles down, red).These are compared withexperimental values forGr/Ir (squares, brown) ,and Whittaker series (circles, cyan).Energies have been labeled on the abscissa by thequantum numbers nncorresponding to wave functions from the symmetrical potential(Whittaker quantum numbers have been accordingly transformed to n=2m-1n=2 m - 1,m=1,2,3,⋯m=1, 2, 3, \\dots ).In table REF we compare eigenenergies calculated for the RPA and FT models with similar theoretical values obtained from a formalism based in Density Functional Theory[11], and with experimental values reported for Gr/Ir [9].",
"We have also included, as a reference, the limiting case of the Rydberg series for a perfect metal $E_{m+1}=-\\frac{1}{32 (m+1)^2} ; \\, m= 0, 1, 2, ...$ , where $m$ refers to the number of nodes for each state (note that these wave functions only extend to $z>0$ half-space, and that the zero at the origin is not counted as a node since it derives from the boundary conditions).",
"Figure REF shows in a log-log scale the scaling law for the first few members of the series.",
"We remark that for the purpose of computing binding energies for image-potential states the FT model with a suitable value for $k_{FT}^{*}$ yields values of the same quality as the RPA at a much lower computational cost.",
"For the potential created by free standing graphene Silkin et al.",
"[11] have computed binding energies for image states by matching an ab-initio Density Functional Theory (DFT) model for the total energy to an asymptotic expression for the image potential beyond some distance, $z_0$ .",
"A pseudopotential with valence electrons 2s2 2p2 and a pseudized core for 1s2 electrons has been used.",
"In order to compare results from these different theoretical approaches we align the states attending to their energies in the same order as they appear in the image series.",
"This labeling of states is straightforward and unambiguous, and most importantly it facilitates comparison between different theories and with experimental values.",
"On the other hand, theoretical eigenvalues can be classified by looking at the properties of their eigenfunctions, notably the number of nodes and for a symmetric potential the parity.",
"Eigenvalues corresponding to a symmetric potential like the one plotted in Fig.",
"1 are easily classified in the same order as they appear, $n$ , by the number of nodes (n), and its parity ($+$ for even $n$ , and $-$ for odd $n$ ).",
"However, parity might become not a good quantum number in the presence of defects or a perturbing potential modifying the symmetry of the potential (e.g., a supporting substrate for the graphene layer), while the ordering dictated by the number of nodes is related to the orthonormality of wavefunctions and makes a robust labeling scheme.",
"Within this ordering, the first image-potential state is simply the first unoccupied one to lie above the Fermi energy and below the vacuum level determined by the work function, $W$ .",
"In the RPA/FT model we are introducing in this paper the first image-potential state has been labeled $n=1$ in table REF , its wavefunction has one node and it is antisymmetric, $E(1)_{RPA}=-.68$ eV.",
"In the DFT model, the first image-potential state we find is $E(1)_{DFT}=-1.29$ eV, and its wavefunction has two nodes and it is symmetric (labeled in ref.",
"[11] as $1^{+}$ ).",
"The number of nodes on this wavefunction has been correlated in ref[11] with the occupied $\\sigma $ and $\\pi $ states below and the required orthonormality condition between them.",
"The discrepancy between the number of nodes and parity on wave functions from these two theoretical approaches is not a fundamental one.",
"Presumably a pseudopotential including a different core, like a simpler one constructed with only 2p2 electrons, or a more complete one including the 1s2 electrons, would alter the number of nodes in the first unoccupied state because the orthonormality condition to the low lying states would be different.",
"Similarly, in our RPA model one could imagine an scenario where the value of $k_{FT}$ is so small as to make the $n=0$ state the first unoccupied state, or inversely, so large as to get the $n=1$ state below the Fermi level to make the $n=2$ state the first member of the series of unoccupied states.",
"Therefore, we conclude that the more convenient way to compare different image series from different theories, or with experiments, is the ordering attending their energy, because it does not depend on the fine details of the model being used.",
"This criterion brings good agreement between DFT and RPA/FT, and with experiments, which is remarkable taking into account how different are both theories, and something to be highlighted.",
"Perhaps the exception is the first value predicted by DFT, $-1.29$ eV, that is a bit too low.",
"We do not take this small discrepancy as serious since both theories involve a number of approximations and parameters (the pseudopotential and the exchange-correlation potential for DFT; $k_{FT}$ and $d$ for RPA), and it is well known the difficulties in DFT to get accurate values for empty states (e.g., band gaps) without including a more sophisticated description for electron correlation, which is the physical effect responsible for the asymptotic behavior, and one of the good assets in the RPA/FT formalism."
],
[
"Eigenfunctions",
"The similarity of the values found for the antisymmetric (odd, $n^{-}$ ) members of the series of states for $\\Phi (z)$ and the classical Rydberg series is striking, and merits some attention (third row in table REF ).",
"Such a similarity can be understood by looking at the corresponding wave functions (figure REF ).",
"While wave functions for Rydberg states are spatially located mostly in the vacuum region, the symmetric members of the series have $\\psi (z=0) \\ne 0$ at the origin of the slab (figure REF , continuous line in upper-left panel).",
"This is most conspicuous for the ground state wave function $\\psi _{0}$ that is more alike to the ground state of a harmonic oscillator fitted to the bottom of the potential well ($\\Phi (z)\\approx -0.5 + 0.13 z^2$ a.u., $E^{\\prime }_0=-6.2$ eV), than to states in the one-dimensional hydrogen-like series for a semi-infinite metal, that are assumed to go to zero at the image plane.",
"Figure: First four eigenfunctions for the potential displayedas a black continuous line infigure (FT).Eigenvalues are in eV and referred to the vacuum level.For comparison, Whittaker wave functions (n=1,3n=1, 3, dashed),and the fitted harmonic oscillator wave function(n=0n=0, dotted) are shown.Boundary conditions force all Whittaker wave functions to go to zero at the origin, a condition that in the case of a symmetric well can only be fulfilled by odd wave functions.",
"Moreover, if $n^{-}$ and $m$ give the number of nodes for odd wave functions for the symmetric potential, and the Rydberg one respectively, we can make a one-to-one correspondence, $\\frac{n^{-}-1}{2}=m$ , that simply tells us that both sets of wave functions have the same number of nodes if $n^{-}$ is divided by two (only half-space) and the node at the origin is discounted.",
"To compare with wavefunctions corresponding to Whittaker's classical problem defined only over half the space one might use Silkin's et al.",
"labelling for image states: a quantum number related to nodes of wavefunctions for the symmetrical problem only on half of the space supplied with its parity to take into account that antisymmetric wavefunctions add an extra node to the count.",
"From a physical point of view, we can envisage two relevant limits: A free standing slab producing a symmetric potential with states labeled by the number of nodes and their parity, and a slab on a substrate where a particular surface gap may prevent penetration of wave functions inside the material leaving only half the space accessible for image-potential states.",
"In this later scenario, parity would not be an useful quantum number.",
"It is reasonable to describe the case of a graphene layer grown and supported on a particular substrate as having a place somewhere in between these two limits depending on details like the interaction between the graphene layer and the support, the electronic surface structure of the combined system, work function, etc.",
"In what follows, we shall assume that all these details can be taken into account in the simplest terms by a single free parameter giving the amount of penetration of wave functions.",
"This electrodynamics formalism is not intended to describe these details, but it can bring useful quantitative information on the observed levels, and as a consequence to further conceptual understanding.",
"Table: Expectation valuez ¯\\overline{z} (Å)for Whittaker and RPA wave functions(FT results are very similar to RPA ones).This formalism predicts, for a free-standing graphene layer, the appearance of new states associated with the even parity (e. g., eigenvalues between $-.44$ and $-.36$ eV for $n=2$ , and between $-.15$ and $-.14$ eV for $n=4$ in Figure REF ).",
"These new states have been obtained numerically and fit well into the classical scaling law proportional to $n^{-2}$ for $n$ up to 7.",
"Obviously, the symmetric potential can be perturbed by external ones (e. g. the cases of graphene on a support), and these states would be affected accordingly."
],
[
"Modeling the substrate",
"So far we have discussed a model that effectively represents a free-standing graphene layer.",
"For those cases where the graphene layer has been deposited on a metallic support the substrate is expected to manifest itself in two main physical ways: (i) The wave-functions may be constrained to be outside some spatial region where the substrate enforces an electronic gap, and (ii) as a consequence of the interaction between the layer and the support, some charge may be transferred to/from the slab, modifying the density of states at the Fermi level, i. e., the value of $k_\\mathrm {FT}$ .",
"To assess how sensitive the eigenvalues are to the penetration of wave functions into the material we have added to $\\Phi (z)$ a repulsive term modeled as an exponential wall, $R_{b}(z)=e^{-a (z-b)}$ .",
"Since we are only interested in creating a decaying state inside the support, we fix the parameter $a$ to a large value, $a= 20$ a.u., akin to the infinite hard-wall limit; its effective role is to expel states from the $z<b$ region, ensuring the exponential decay of wave functions inside an electronic band gap.",
"The resulting potential for the repulsive barrier located near the slab surface ($b=1$ a.u., green dashed line in figure REF ) is similar to the classical series with an image plane, $\\frac{1}{4(z+z_0)}$ , and can be easily solved by introducing a quantum defect in Rydberg formula (compare energies for the same number of nodes in the eigenfunctions for $R_{b=1}$ and Whittaker series with quantum defect $\\delta =.2$ in table REF ).",
"The barrier, on the other hand, can be introduced below the surface, mimicking the effect of an electronic band gap due to a supporting substrate.",
"The evolution of the first few eigenvalues with the position of the barrier have been shown in the two limits in table REF .",
"As long as the barrier is located far away from the ultra-thin slab (e. g. $b \\le 20$ a.u.)",
"we get values reminiscent from the original members for the unperturbed symmetrical potential.",
"On the other limit, a barrier located just on the surface very much reminds of the classical solution.",
"For large $m$ the eigenvalues are determined by the potential in the vacuum region and the existence of such a barrier distorts less and less the states as they approach the vacuum level.",
"Table REF gives the expectation mean values in Å, $\\overline{z}=\\int _{0}^{\\infty } \\psi (z) \\, z \\, \\psi (z) \\, dz$ , for Whittaker wave functions compared with the ones obtained for the RPA or FT potentials (e. g. (REF )).",
"These values compare well with each other, which reflect the manifest similarity between wave functions commented on figure REF , and show how the important region for the potential moves quickly away from the surface as $m$ grows.",
"The fact that $\\overline{z} \\gg d$ for image-potential states with $n > 1$ implies that wave functions are quite insensitive to the potential inside or near the layer and they are mostly influenced by the asymptotic region where FT and RPA are equivalent.",
"This suggests that higher $k$ -corrections to the dielectric function arising from the random phase approximation are not very important, at least for $n > 1$ states.",
"Taking away the first level, largely affected by the details near the bottom of the potential, the rest of the series is only modified by a percentage comparable to differences found in table REF between similar entries.",
"Figure: Solid lines (red): Energy of n=1,2,n=1, 2, and 3 image-potential statesas a function of work function, W=k FT 6W=\\frac{k_\\mathrm {FT}}{6} (eV).Large open circles (blue) show experimental results, , .For Ru the data are plotted by filled circles (green) fortwo different work functions: 4.004.00 and 4.244.24 eV.Data for graphite have been taken from.The effect of doping may be explored by looking at the $k_\\mathrm {FT}$ dependence of the energies of the image series.",
"The important parameter of the dielectric model is the charge density of the graphene layer which can be related to the doping level and the work function.",
"This is summarized in the current formalism via a single parameter, the Fermi-Thomas wave-vector $k_\\mathrm {FT}$ .",
"While $k_\\mathrm {FT}$ cannot be easily extracted in our formalism from doping levels of graphene, we can compare to experiments by exploiting the linear dependence predicted by this theory between the screening wave-vector and the work function of the thin slab.",
"To compare with available experimental results it suffices to establish a connection between $k_\\mathrm {FT}$ and a relevant energy scale, e. g. the distance between the vacuum level to the Fermi level, i. e. the work function, $W$ .",
"To this end, we note that within our approach the work function for a semi-infinite surface, or a slab with $d>1$ Å, is $-\\frac{k_\\mathrm {FT}}{2}$ in the Fermi-Thomas model, and $\\approx -\\frac{k_\\mathrm {FT}}{3}$ in the RPA one.",
"We can see that RPA, so far our best approach, overestimates $W$ when compared with experimental values by about a factor $\\approx 2$ .",
"The linear dependence between $W$ and $k_\\mathrm {FT}$ , on the other hand, is solidly anchored in the theory, having its origin in the net attractive interaction between the external electrons and the polarization charges created inside the polarizable material.",
"Therefore, $W$ should be proportional to $ \\alpha k_\\mathrm {FT}$ , albeit the proportionality constant should be corrected to compare with the experimental value.",
"We take $\\alpha =\\frac{1}{6}$ , that corresponds to the empirical work function for graphene, $W_0=4.5$ eV.",
"Therefore, the empirical dependence $W=\\frac{k_\\mathrm {FT}}{6}$ allows us to translate $k_\\mathrm {FT}$ into an experimental energy scale, and to set up an energy origin at the value $W_0$ .",
"Figure REF shows the results for the first three members of the series by straight lines.",
"The experimental data will be discussed in section .",
"Figure: Scaling of the slope of ∂E n (W)/∂W\\partial E_n(W)/\\partial Wcomputed for the first three image-potential states.An interesting feature of this result is the fact that the higher member of the series are less and less affected by changes in $k_\\mathrm {FT}$ ($W$ and/or $\\rho (E_F)$ ).",
"This is clearly seen in Figure REF where the slope of $\\partial E_n(W)/\\partial W$ decreases for the higher $n$ values.",
"In Figure REF we plot this result, and show that it is well fitted by the empirical function $\\frac{1}{6 n^{2}}$ .",
"Since $k_\\mathrm {FT}$ is linearly proportional to the density of states at the Fermi level, $\\rho (E_F)$ , this gives us the empirical scaling law expected for the different terms of the image-potential-state series with doping."
],
[
"Experimental results",
"The theoretical results of the preceding section can be tested experimentally by two-photon photoemission (2PPE).",
"This technique is able to measure image-potential states with high accuracy and results for various graphene-covered surfaces have been reported [9], [10], [29], [30].",
"Photoelectron spectroscopy can also provide precise values for the work function of the surfaces which is correlated to the doping level and charge density of the graphene layer.",
"We first discuss the work function before turning to the image-potential states."
],
[
"Work function",
"The amount of charge transfer from the substrate to the graphene layer is determined by the work function difference between substrate and graphene.",
"This situation has been modeled with a simple capacitor model by Giovanetti et al.",
"and validated by comparison to results from calculations for various surfaces [24], [25].",
"The curve plotted in figure REF shows the calculated work function of the graphene-covered surfaces versus the work function of the clean substrates for the capacitor model.",
"The calculated values plotted by green open squares fit the curve quite well [24], [25].",
"The available experimental data are shown by blue open circles [9], [10], [29], [30], [32], [33].",
"The size of the circles represents approximately typical experimental error bars.",
"The curve was fitted to the experimental results for the noble-metal surfaces [29] and monolayer graphene on SiC [30].",
"The fit was done with the work function of graphene of 4.50 eV compared to 4.48 eV in the original work [24], [25].",
"The chemical shift was reduced from 0.90 eV to 0.89 eV.",
"Overall the fit of this work describes also the calculated values (green open squares in figure REF ) very well.",
"The surfaces included in the fit all have a graphene-metal distance around 3.4 Å [34], [35], [36], [37], [38].",
"The surfaces of Ni, Ru, and Pd were not included in the fit and shown by blue dashed circles.",
"They have shorter graphene-metal distances [24], [25], [39] and would require a larger chemical shift for a satisfactory description.",
"The capacitor model provides a meaningful measure to compare different surfaces with comparable graphene-metal distances via the work function.",
"In addition, the work function for graphene-covered surfaces can be estimated from the work function of the substrate at least for weakly coupled graphene.",
"Figure: Work function of graphene-covered surfaces versus the work function of the clean substrates (blue open circles).",
"The solid curve shows the results of a capacitor model which match the calculated results plotted by green open squares , .The energies of the Dirac point (red filled circles) follow the same curvewith an energy scale shifted 4.5 eV relative to the vacuum level.The charges transferred between the substrate and the graphene layer originate from the Dirac cone on the graphene side.",
"These might be electron or holes depending on the doping of the graphene.",
"The shift of the work function due to the charge transfer can therefore be directly related to the energy of the Dirac point relative to the Fermi energy.",
"Zero is obtained for a work function of 4.5 eV.",
"The values are taken from the literature [36], [40], [41], [42], [43] and are plotted in figure REF as filled red circles.",
"The agreement is perfect for SiC.",
"but for the other surfaces larger deviations are observed.",
"One has to keep in mind that for the noble-metal substrates the graphene layer is p-doped and the Dirac point cannot be observed directly in photoemission spectroscopy.",
"Its energy is extrapolated from the dispersion of the Dirac cone below the Fermi energy.",
"The experimental results are in good agreement with the calculations, which show the constant difference between work function and energy of Dirac point as predicted by calculations [24], [25].",
"This has been found also for different modifications of graphene on SiC [30]."
],
[
"Binding energies",
"The important parameter of the dielectric model is the charge density of the graphene layer which can be related to the doping level and work function.",
"The energies of the image-potential states as a function of work function have been plotted as red solid lines in figure REF .",
"Large open circles (blue) show the available experimental values [9], [29], [30].",
"The symbol size represents typical experimental uncertainties.",
"The data points for graphite were taken from[31] which observed the lowest three image-potential states.",
"Other groups reported binding energies of $0.85\\pm 0.10$ eV for the $n=1$ image-potential state[44], [45].",
"The experimental values are slightly above the calculated lines, but the slope agrees fairly well.",
"The agreement could be improved by using a smaller value for the parameter $\\alpha $ relating work function and Fermi-Thomas wave vector $W=\\alpha k_\\mathrm {FT}$ .",
"The data point for the $n=1$ state of SiC lies significantly too high.",
"This might be an effect of different screening properties of the dielectric substrate or residual binding to the buffer layer.",
"The accuracy and work function range of the experimental data for the noble-metal surfaces is not sufficient to determine the slope independently.",
"Therefore we only conclude that the experimental data are in reasonable agreement with the predictions of the dielectric model calculations.",
"Finally, we discuss the interpretation of experimental data measured on Gr/Ru [10].",
"Three bound states have been measured with respect to the Fermi energy (all energies in eV): $3.44$ (1'), $3.59$ (1) and $3.82$ (2).",
"The work function was measured as 4.24 eV.",
"On the corrugated graphene on Ru(0001) surface also lower areas exist which have a work function of 4.00 eV.",
"On all other surfaces only the $n=1$ and $n=3$ image-potential states have been found.",
"It is therefore worthwhile to check whether on Ru the additional state might be the $n=2$ image-potential state.",
"The data are plotted by green solid circles with error bars in figure REF for the two different values of the work function.",
"For a work function of 4.00 eV the experimental binding energies agree reasonably well with the calculated lines.",
"This assignment would also be compatible with the observed monotonous decrease of the lifetime with binding energy [10].",
"However, we cannot rule out the consistent interpretation based on different local work functions on the corrugated graphene on Ru as proposed by Armbrust et al.",
"[10]."
],
[
"Conclusions",
"Using standard models for the dielectric response and the reflection of electromagnetic waves at a surface we have computed the static self-energy for an ultra-thin slab mimicking a graphene layer.",
"The self-induced potential goes continuously from the exchange and correlation energy inside the material to the classical asymptotic image potential in the vacuum.",
"For the purpose of obtaining image-potential states binding energies we find that FT makes an excellent and convenient approximation to the accurate RPA.",
"Eigenvalues and eigenfunctions have been compared with Whittaker classical series and recent experiments on Gr/Ir.",
"A free standing graphene ultra-thin layer produces a spatially symmetric self-energy that induces an image potential series with even and odd states.",
"The odd members of the series show a remarkable resemblance to the solution of Schrödinger equation for the classical image potential (Whittaker wave functions).",
"On the other hand, even wave functions arise as new states that differ from Whittaker in several key respects, e. g. their non-zero density probability at the origin.",
"While the qualitative aspects of the image series supported by a thin slab can be described in terms of quite general physical properties, the detailed quantitative values depend crucially on the penetration of wave functions into the substrate and the thin graphene film supported on it.",
"We have considered the limiting cases of full and none penetration.",
"For the case of films weakly interacting with a support and wave functions penetrating well inside the system some new states may consequently appear in between the classical ones, that can be traced back to the even states in a free-standing slab.",
"In cases where the interaction is strong and the surface electronic structure prevents the penetration of wave-functions a behavior more similar to a standard metallic surface is expected.",
"We notice that the formalism used here can be easily generalized to include the effect of surface plasmons via a frequency-dependent dielectric function [46].",
"The experimental results for graphene on various substrates compare well to theoretical predictions.",
"The measured work function change due to graphene agree with the capacitor model of Giovanetti et al.",
"[24], [25].",
"This opens the possibility to predict the work function of graphene-covered surfaces from the substrate work function.",
"The work function difference from an isolated graphene sheet is related to the doping of the graphene layer and determines in turn the available screening charge.",
"The resulting change in the energy of the image-potential states calculated with the theoretical model is in agreement with the experimental data from two-photon photoemission.",
"This work has been financed by the Governments of Spain (MAT2011-26534, and FIS2010-19609-C01-01), and the Basque Country (IT-756-13).",
"Computing resources provided by the CTI-CSIC are gratefully acknowledged."
],
[
"Electrodynamics of a slab characterized by a width $2 d$ \nand {{formula:a50fdf3e-f951-4ecf-9c1d-ae283e3094da}}",
"We are interested in the quasi-static self-energy potential by an external charge near a slab characterized by a dielectric function, $\\epsilon (k)$ .",
"The dynamical problem, however, is solved exactly in the same way by introducing a k and w-dependent response function $\\epsilon (k,w)$ .",
"The problem is solved independently for two homogeneous systems (pseudo-vacuum and pseudo-medium), with a set of fictitious charges, $\\sigma $ , to reproduce the real fields in the regions for the vacuum and the material respectively [12].",
"Notice that the electromagnetic field propagator in vacuum, $\\frac{4 \\pi }{k^2}$ , corresponds to three spatial dimensions, $\\vec{k} =(\\vec{\\kappa },q)$ .",
"Symmetric and antisymmetric solutions are analyzed separately and combined to yield a solution for the most general case.",
"The extra fictitious charges are finally removed from the solutions by using matching conditions.",
"Since a free standing graphene slab is symmetric with respect to the middle plane, matching conditions on only one surface need to be considered.",
"Figure: Q inside the ultra-thin slab (0≤z 1 ≤d0 \\le z_1 \\le d, region II).The problem is decomposed as a superpositionof symmetric (S, upper part)and antisymmetric (A, lower part) configurations.The slab polarization is characterized by ϵ(k →)\\epsilon (\\vec{k}),and its width, 2d2d,and it is symmetric around O 1 _1 (z=0z=0).Specular reflection conditions for the fieldsare imposed at one of the two equivalent surfaces (O 2 _{2} z=dz=d).The self-energy of the external probe charge, $Q$ is obtained by computing the total potential in each region, and subtracting the bare potential due to $Q$ : $\\Phi (z, z_1)= \\frac{1}{2}\\int \\frac{d^2 k_{\\parallel }}{(2 \\pi )^2} \\frac{dq}{2 \\pi }e^{i q z}\\lbrace \\phi ^{S}(k) + \\phi ^{A}(k) - \\frac{4 \\pi Q}{k^2} e^{-i q z_1}\\rbrace $ where $\\phi ^{S,A}(k)$ must be separately obtained for Q inside/outside the slab.",
"The case where Q is inside the material is schematically shown in figure REF ."
],
[
"Q inside the slab (II and III, $-d \\le z \\le d$ )",
"An schematic distribution for the pseudo-charges when the external Q is inside the slab is given in figure REF for the antisymmetric case.",
"The symmetric configuration follows easily by substituting the factors $(-1)^n$ in the sums by $(+1)^n$ .",
"Continuity of the perpendicular component of the displacement field results in $\\sigma ^{V}=-\\sigma ^{M}$ for the pseudo-vacuum, $V$ , and the pseudo-medium, $M$ , for both the symmetric and antisymmetric cases.",
"Figure: Schematics for the extendedpseudo-vacuum and pseudo-medium with charges distributionappropriate for theantisymmetric case.Upper panel: QQ inside the slab.Lower panel: QQ in the vacuum region.Therefore, the total potential for the symmetric (antisymmetric) cases, are: $\\phi ^{V}(k) =-\\frac{4 \\pi }{k^2}\\sigma _{\\kappa }$ $\\hspace*{-25.60747pt}\\phi ^{M}(k) =\\frac{4 \\pi }{k^2 \\epsilon (k)}\\left(\\sigma _{\\kappa }\\sum _{n=-\\infty }^{\\infty } (\\pm )^n e^{i q 2 d n}+Q\\sum _{n=0}^{\\infty } (\\pm )^n\\left(e^{i q (z_1 + 2 d n)} \\pm e^{i q (2 d + z_1 + 2 d n)}\\right)\\right)$ The surface charges, $\\sigma ^{S}$ and $\\sigma ^{A}$ , are separately obtained from the condition: $\\Phi ^{V}(z=d^{+})=\\Phi ^{M}(z=d^{-})$ .",
"Convergence of the different series in these expressions is guaranteed by the zeros of the dielectric function, i. e. the normal modes dressed by the interaction in the medium [12].",
"For the simple Thomas-Fermi dielectric function we have, $k^2 \\epsilon _\\mathrm {FT}(k) = (q^2 + \\kappa ^2 + k_\\mathrm {FT}^2) =(q+i\\chi )(q-i\\chi )$ , and the integration over the perpendicular moment, $q$ , can be performed to obtain analytic or semi-analytic expressions that are given below."
],
[
"Q inside the vacuum (I and IV, $\\mid z \\mid \\ge d$ )",
"A similar procedure yields for the case of Q in the vacuum region: $\\phi ^{V}(k) =\\frac{4 \\pi }{k^2}\\lbrace -\\sigma _{\\kappa }+\\frac{Q}{2} \\left(e^{-i q z_1} -e^{i q z_1} \\right)\\rbrace $ $\\phi ^{M}(k) =\\frac{4 \\pi }{k^2 \\epsilon (k)}\\sigma _{\\kappa }\\sum _{n=-\\infty }^{\\infty } (\\pm 1)^n e^{i q 2 d n}$ Equation (REF ) yields a numerical procedure that allows to obtain the self-induced potential.",
"Using the FT approximation, quasi-explicit expressions that only depend on a single numerical integration, have been obtained and given in (REF ).",
"We remark this is an excellent approximation to the full RPA result for the purpose of computing binding energies of image-potential states."
],
[
"Useful analytical results in the\nFermi-Thomas approximation.",
"The semi-infinite system ($d \\rightarrow \\infty $ ), along with the use of Fermi-Thomas dielectric function, brings some simplifications to the expression for the potential that can be exploited to compute exact values at the surface, well inside the material, and in the asymptotic vacuum region.",
"For convenience, in this appendix we move the origin to the surface ($z=d$ )."
],
[
"Induced potential in the vacuum region.",
"Using the equations in we obtain the induced potential in the vacuum (now $z>0$ ): $\\Phi (z>0) =\\frac{1}{2} \\int _{0}^{\\infty } d \\kappa e^{- \\kappa 2 z}[1 - \\frac{2}{1+\\frac{\\kappa }{\\pi } \\int \\frac{d q}{k^2 \\epsilon (k)}}]$ that can be further simplified by the use of $k^2 \\epsilon (k)=k^2 \\epsilon _\\mathrm {FT}(k)=\\kappa ^2 +q^2 + k^2_\\mathrm {FT}$ : $=\\frac{1}{2} \\int _{0}^{\\infty } d \\kappa e^{- \\kappa 2 z}\\frac{\\kappa -\\chi }{\\kappa +\\chi } =$ $-\\frac{1}{12 z^3}\\lbrace 6 ~ {\\bf _{0} \\tilde{F}_{1}}\\left(;-1;-k_\\mathrm {FT}^2z^2\\right)-3 \\pi k_\\mathrm {FT}^2 z^2 {\\bf H}_2(2 k_\\mathrm {FT}z)+k_\\mathrm {FT}^2 z^2 (4 k_\\mathrm {FT} z+3)+$ $+\\left(3-6k_\\mathrm {FT}^2 z^2 \\log (k_\\mathrm {FT} z)\\right) J_0(2 k_\\mathrm {FT}z)+6 k_\\mathrm {FT} z (\\log (k_\\mathrm {FT})+\\log (z)+1) J_1(2k_\\mathrm {FT} z)+3\\rbrace $ where ${\\bf _{0} \\tilde{F}_{1}}(;b;z)$ is the regularized confluent hypergeometric function, ${\\bf H}_2(z)$ is the Struve function, and $J_{n}(z)$ are Bessel functions of the first kind."
],
[
"Asymptotic behaviour and position of the image plane.",
"We can study the asymptotic behavior in the long wave-length region ($\\kappa \\approx 0$ ) by expanding the fraction in the integrand in Eq.",
"REF , $\\frac{\\kappa -\\chi }{\\kappa +\\chi } \\approx -1 + \\frac{2 \\kappa }{k_\\mathrm {FT}} + O(\\kappa ^2)$ , to obtain: $\\Phi (z) \\approx -\\frac{1}{4z} + \\frac{1}{4 k_\\mathrm {FT} z^2}$ On the other hand, we can expand the classical image potential around $z=\\infty $ $-\\frac{1}{4(z-z_0)} \\approx -\\frac{1}{4z} -\\frac{z_0}{4z^2} + O(\\frac{1}{z^3})$ which allows us to identify the position of the image plane by direct comparison of both expressions, $z_0 = -\\frac{1}{k_\\mathrm {FT}}$ .",
"This asymptotic expression, corrected by the position of the image plane obtained above, makes an excellent approximation to the full potential either in the vacuum region outside the slab or in the middle of a vacuum gap, the case discussed below."
],
[
"Particular values for $z=0$ and {{formula:e4358a24-02de-4baf-9fee-350660bc71bd}} .",
"For $z=0$ ($d\\gg \\frac{1}{k_\\mathrm {FT}}$ ) we have: $\\Phi (z=0) =\\frac{1}{2} \\int _{0}^{\\infty }e^{- \\kappa 2 z}\\frac{\\kappa -\\chi }{\\kappa +\\chi } d \\kappa = -\\frac{k_\\mathrm {FT}}{3}$ For $z=-d$ , $\\hspace*{-25.60747pt}\\Phi (z=-d) =\\frac{1}{2} \\int \\frac{d^{3} \\vec{k}}{(2 \\pi )^{3}} \\frac{4 \\pi }{k^2}\\left( \\frac{1}{\\epsilon _\\mathrm {FT}(k)}-1\\right)=\\frac{k_\\mathrm {FT}^2}{2} \\int _{0}^{\\infty } d \\kappa \\frac{1}{\\kappa \\chi + \\chi ^2} = -\\frac{k_\\mathrm {FT}}{2}$"
],
[
"Potential in a vacuum gap.",
"It is handy to apply the same techniques to the inverse problem: the potential in a vacuum gap between two semi-infinite media at $z< \\pm d$ .",
"The theoretical procedure proceeds along similar lines we have analyzed in this paper, and we simply give the Fermi-Thomas result here: $\\Phi (z>d)=-\\frac{k_\\mathrm {FT}}{2}-\\frac{e^{2 k_\\mathrm {FT} (d-z)}}{4 (d-z)} -$ $\\hspace*{-5.69054pt}-\\int _{0}^{\\infty } d \\kappa \\ \\frac{ \\kappa ^2 e^{2 (d-z) \\sqrt{\\kappa ^2+k_\\mathrm {FT}^2}}\\left(\\kappa +\\sqrt{\\kappa ^2+k_\\mathrm {FT}^2} \\coth [2 d \\kappa ]\\right)}{\\sqrt{\\kappa ^2+k_\\mathrm {FT}^2}\\left(\\kappa +\\sqrt{\\kappa ^2+k_\\mathrm {FT}^2} \\coth [d \\kappa ]\\right)\\left(\\kappa +\\sqrt{\\kappa ^2+k_\\mathrm {FT}^2} \\tanh [d \\kappa ]\\right)}$ and, $\\Phi (-d \\le z \\le d)=$ $\\int _{0}^{\\infty } d \\kappa \\ e^{-2 \\kappa z} \\ \\frac{-k_\\mathrm {FT}^2-e^{4 \\kappa z} k_\\mathrm {FT}^2+e^{2 \\kappa (-d+z)} \\left(2 k_\\mathrm {FT}^2+4 \\kappa \\left(\\kappa -\\sqrt{\\kappa ^2+k_\\mathrm {FT}^2}\\right)\\right)}{4 \\left(2 \\kappa \\sqrt{\\kappa ^2+k_\\mathrm {FT}^2}\\cosh [2d \\kappa ]+\\left(2 \\kappa ^2+k_\\mathrm {FT}^2\\right)\\sinh [2 d \\kappa ]\\right)}$ This potential has been plotted in figure REF , where it is compared with: (i) the classical value [46], [47], $\\Phi (|z|\\le d)=-\\frac{2 \\gamma -\\Psi ^{(0)}\\left(\\frac{1}{2} +\\frac{z}{2 d}\\right)-\\Psi ^{(0)}\\left(\\frac{1}{2} -\\frac{z}{2 d}\\right)}{8 d}$ ($\\Psi ^{0}(z)$ is the digamma function, and $\\gamma =-0.577$ is the Euler constant), (ii) the classical value corrected by an image plane determined by an expansion to the full non-local potential calculated in the Fermi-Thomas approximation, and (iii) the independent non-local potential for each of the two surfaces delimiting the vacuum gap.",
"Figure: Self-induced potential, Φ(z)\\Phi (z), of a unit test charge at different positionsin a vacuum gap of width2 a.u., and inside the surrounding metals (k FT =1k_\\mathrm {FT}=1 a.u.",
").Continuous line (black): non-local potential for the gap (Fermi-Thomas).Dashed line (black): non-local potential for each surface in the gapconsidered as independent systems (Fermi-Thomas).Dashed line (red): classical result taking into account all theclassical images and counter-images .Dashed line (blue): semi-classical result adding the contributionsof all the classical images, but corrected by an image planeat z 0 =±1 k FT z_0=\\pm \\frac{1}{k_\\mathrm {FT}} (valid in the vacuum region only)."
]
] | 1403.0391 |
[
[
"Micro to macro models for income distribution in the absence and in the\n presence of tax evasion"
],
[
"Abstract We investigate the effect of tax evasion on the income distribution and the inequality index of a society through a kinetic model described by a set of nonlinear ordinary differential equations.",
"The model allows to compute the global outcome of binary and multiple microscopic interactions between individuals.",
"When evasion occurs, both individuals involved in a binary interaction take advantage of it, while the rest of the society is deprived of a part of the planned redistribution.",
"In general, the effect of evasion on the income distribution is to decrease the population of the middle classes and increase that of the poor and rich classes.",
"We study the dependence of the Gini index on several parameters (mainly taxation rates and evasion rates), also in the case when the evasion rate increases proportionally to a taxation rate which is perceived by citizens as unfair.",
"Finally, we evaluate the relative probability of class advancement of individuals due to direct interactions and welfare provisions, and some typical temporal rates of convergence of the income distribution to its equilibrium state."
],
[
"Introduction",
"The rise of inequalities in income and wealth, the implementation of different tax policies and the effects of tax evasion constitute important socioeconomic questions for most countries.",
"Especially in times of economic crisis, like the current one, such issues become of major concern and are the object of frequent studies and debates.",
"Involving a large number of interacting agents, as well as a multiplicity of aspects and levels, this matter certainly falls within the realm of the science of complex systems.",
"We think that also mathematics can contribute to some extent to the analysis of these problems; for example, it can help to understand the micro-processes and mechanisms which lead to certain collective patterns.",
"Through modelling and simulations, made possible by the power of modern computers, mathematics allows the exploration of several possible scenarios.",
"Thus, in conjunction with the expertise from economics, political economics and other disciplines, and suitably supported by empirical data, mathematical models could in some cases even suggest concrete policies.",
"Basically motivated by this belief, we consider in this paper some microscopic models of taxation and redistribution in a closed market society, both in the absence and in the presence of tax evasion.",
"These models are constructed within a general framework which was first introduced in [2] and then further investigated in [3], [4].",
"They are formulated as systems of nonlinear ordinary differential equations.",
"More precisely, the systems expressing them consist of a number $n$ of equations equal to the number of income classes in which one divides a population.",
"The $j$ -th equation (with $j = 1, ... , n$ ) describes the variation in time of the fraction, say $x_j$ , of individuals belonging to the $j$ -th class.",
"The vector $(x_1, ... , x_n)$ represents the discrete income distribution over the population, whose size is supposed to remain constant.",
"According to the findings of [2], [3], [4], in correspondence to any value $\\mu $ of the total income (which is a conserved quantity too) a stationary income distribution exists, which is the asymptotic trend of all initial distributions with total income $\\mu $ .",
"In [3], [4] it was also shown that, if the number $n$ of classes is large enough ($n$ was taken to be equal to 25 in those papers) and if the value $\\mu $ is compatible with initial distributions having the majority of individuals in the lower income classes, then for models constructed with suitable parameters (and those discussed here are of this type) the asymptotic income distributions exhibit fat tails with Pareto power-law behaviour like the real world distributions.",
"The main novelty with respect to the models explored in [2], [3], [4] is that we treat here also cases in which tax evasion occurs.",
"This addition is not irrelevant.",
"Indeed, the illegal practice of tax evasion affects probably all societies, causing the “loss” of huge amounts of money, which could be employed towards social and economic policies.",
"We are especially interested in the differences of the asymptotic income distributions in cases of tax compliance and in cases with tax evasion.",
"Below, we investigate these differences and we examine how quantities and indicators like the Gini index, the tax revenue and the probability of class promotion due to welfare change in the various cases.",
"In our approach the aggregate behaviour of a system, represented by the observable income distribution curves, emerges from the complex of interactions which take place between single heterogeneous individuals.",
"The underlying behaviour- and interaction-based perspective differs intrinsically from the traditional viewpoint of mainstream economics, whose cornerstones are the assumption of a representative agent and the rational choice theory.",
"The interaction-based paradigm began to take shape during the last decades and it counts among its pioneers various exponents of the economics community, e.g.",
"T. Schelling, A. Kirman, B.W.",
"Arthur, and M. Gallegati, see e.g.",
"[1], [10], [13], [17].",
"The tool kit of researchers adopting this perspective typically includes agent-based computational simulations and complex networks.",
"For example, questions related to tax evasion have been investigated via agent-based models in [6], [11], [20].",
"In these papers the focus is on the effect of interactions among behaviourally different agent types (honest, imitative, tax evaders and so on) on the changes in individual behaviour patterns.",
"An experimental approach to such kind of questions has been developed and described in [15].",
"On another side, starting in the mid-1990s a branch of physics denoted econophysics The term econophysics was coined by H.E.",
"Stanley.",
"has been developed, which explores the dynamical behaviour of economic and financial markets by means of methods and tools originally developed in statistical mechanics and in gas kinetic theory (see e.g.",
"in this connection [7], [8], [9], [14], [18], [19]).",
"In econophysics the phenomenon of tax evasion has been described (e.g.",
"in [20]) through an analogy with the Ising model, which is an array, typically 2-dimensional, of �spin� variables $s_{ij}$ that interact with their nearest neighbors and can only assume the values $\\pm 1$ .",
"In the analogy each spin represents a citizen, which can be either in the �tax compliant� state $+ 1$ or in the �tax evader� state $- 1$ and can undergo transitions from $+ 1$ to $- 1$ due to imitation and from $- 1$ to $+ 1$ due to tax audits.",
"Through numerical simulations or approximations typical of statistical mechanics it is possible to compute the average $\\langle \\sum _{i,j} s_{ij} \\rangle $ , directly related to the total evaders/compliant rate, as a function of several global or local parameters.",
"This approach is helpful for the analysis of evasion phenomena in relation to local interaction and external controls, but not for studying the effect of evasion on the income distribution as we do here.",
"The paper is organized as follows.",
"In the next section we sketch some models, which were constructed and analysed in [2], [3], [4], and we recall some of their features as established in these papers.",
"In Section we incorporate into these models the tax evasion phenomenon and we discuss some first results concerning the asymptotic stationary income distributions which are found in the absence and in the presence of tax evasion.",
"We then explore in Section the case in which to an increase of tax rates a proportional increase of evasion corresponds.",
"The fifth and the sixth section are devoted respectively to an in-depth analysis of some interesting quantities characteristic of the stationary solutions and of the relative times of convergence.",
"Some summarizing comments are contained in Section ."
],
[
"The tax compliance case",
"We shortly review in this section a family of models regarding a tax compliance case, and we recall their main features as established in [2], [3], [4].",
"Imagine dividing a population of individuals into a finite number $n$ of classes, each one characterized by its average income, the average incomes being the positive numbers $r_1< r_2 < ... <\\ r_n$ .",
"We refer to [2] for a detailed illustration of the stylized micro scale mechanism we have in mind.",
"Here, we just recall that also the part of the government (which of course plays a role in connection with the taxation system) can be described through monetary exchanges between pairs of individuals, and we emphasise that consequently two kinds of interactions may take place: the so called direct ones, between an $h$ -individual and a $k$ -individual, occurring when the first one pays the second one, and the indirect ones, between the $h$ -individual and every $j$ -individual with $j \\ne n$ , occurring on the occasion of the direct $h$ -$k$ interaction.",
"The indirect interactions represent the transactions corresponding to the payment of taxes and to the benefit of the redistribution.",
"In short, and we are referring here to a tax compliance case, in correspondence to any direct $h$ -$k$ interaction, if $S$ (with $S < (r_{i+1} - r_{i})$ for all $i = 1, ..., n$ ) denotes the amount of money that the $h$ -individual should pay to the $k$ -one, the overall effect of payment, taxation and redistribution is that of an $h$ -individual paying a quantity $S \\, (1 - \\tau )$ to a $k$ -individual and paying as well a quantity $S \\, \\tau $ , which is divided among all $j$ -individuals for $j \\ne n$ .",
"Individuals of the $n$ -th class cannot receive money.",
"Otherwise, they would possibly advance to a higher class.",
"And this is not permitted in the present context.",
"The quantity $\\tau = \\tau _k$ , which is assumed to depend on the class to which the earning individual belongs, corresponds to the taxation rate of the $k$ -th class.",
"A suitable framework towards modelling taxation and redistribution processes relative to such a population was shown in [2] to be provided by the system of $n$ nonlinear differential equations ${{d x_i} \\over {d t}} =\\sum _{h=1}^n \\sum _{k=1}^n {\\Big (} C_{hk}^i + T_{[hk]}^i(x) {\\Big )}x_h x_k - x_i \\sum _{k=1}^n x_k \\, ,\\qquad i= 1\\, ,... \\, ,n \\, .$ Here, $x_i(t)$ with $x_i : {\\bf R} \\rightarrow [0,+\\infty )$ denotes the fraction at time $t$ of individuals belonging to the $i$ -th class; the coefficients $C_{hk}^i \\in [0,+\\infty )$ , satisfying $\\sum _{i=1}^n C_{hk}^i = 1$ for any fixed $h$ and $k$ , represent transition probability densities due to the direct interactions (more precisely, $C_{hk}^i $ expresses the probability density that an individual of the $h$ -th class will belong to the $i$ -th class after a direct interaction with an individual of the $k$ -th class), and the functions $T_{[hk]}^i : {\\bf R}^n \\rightarrow {\\bf R}$ , continuous and satisfying $\\sum _{i=1}^n T_{[hk]}^i(x) = 0$ for any fixed $h$ , $k$ and $x \\in {\\bf R}^n$ , represent transition variation densities due to the direct interactions (more precisely, $T_{[hk]}^i$ expresses the variation density in the $i$ -th class due to an interaction between an individual of the $h$ -th class with an individual of the $k$ -th class).",
"The system $(\\ref {evolution eq eta = 1})$ accounts for the fact that any direct or indirect interaction possibly causes a slight increase or decrease of the income of individuals.",
"To choose a particular family of models, by specifying the values of the parameters $C_{hk}^i$ and the functions $T_{[hk]}^i(x)$ , we first define certain coefficients $p_{h,k}$ for $h, k = 1, ... , n$ .",
"These have the function of expressing the probability that in an encounter between an $h$ -individual and a $k$ -individual, the one who pays is the $h$ -individual.",
"Since also the possibility that none of the two pays has to be taken into account, the requirement which the $p_{h,k}$ must satisfy is that $0 \\le p_{h,k} \\le 1$ and $p_{h,k} + p_{k,h} \\le 1$ .",
"We take $p_{h,k} = \\min \\lbrace r_h,r_k\\rbrace /{4 r_n} \\, ,$ with the exception of the terms $p_{j,j} = {r_j}/{2 r_n}$ for $j = 2, ..., n-1$ , $p_{h,1} = {r_1}/{2 r_n}$ for $h = 2, ..., n$ , $p_{n,k} = {r_k}/{2 r_n}$ for $k = 1, ..., n-1$ , $p_{1,k} = 0$ for $k = 1, ..., n$ and $p_{hn} = 0$ for $h = 1, ..., n$ .",
"This choice, among others, was proposed and discussed in [4].",
"Our choice (see [2], [4] for details) for the coefficients $C_{hk}^i$ and the functions $T_{[hk]}^i(x)$ is reported next.",
"The only possibly nonzero elements among the $C_{hk}^i$ are: $C_{i+1,k}^{i} & =& p_{i+1,k} \\, \\frac{S \\, (1-\\tau _k) }{r_{i+1} - r_{i}} \\, ,\\nonumber \\\\C_{i,k}^i & =& 1 - \\, p_{k,i} \\, \\frac{S \\, (1-\\tau _i)}{r_{i+1} - r_{i}}- \\, p_{i,k} \\, \\frac{S \\, (1-\\tau _k)}{r_{i} - r_{i-1}} \\, ,\\nonumber \\\\C_{i-1,k}^i & =& p_{k,i-1} \\, \\frac{S \\, (1-\\tau _{i-1})}{r_{i} -r_{i-1}} \\, .$ We stress that the expression for $C_{i+1,k}^{i}$ in $(\\ref {choiceforC})$ holds true for $i \\le n-1$ and $k\\le n-1$ ; the second addendum of the expression for $C_{i,k}^i$ is effectively present only provided $i \\le n-1$ and $k \\ge 2$ , while its third addendum is present only provided $i \\ge 2$ and $k \\le n-1$ ; the expression for $C_{i-1,k}^i$ holds true for $i \\ge 2$ and $k\\ge 2$ .",
"Following [2], we take the functions $T_{[hk]}^i(x)$ as $T_{[hk]}^i(x) & = & \\frac{p_{h,k} \\, S \\, \\tau _k}{\\sum _{j=1}^{n} x_{j}} {\\bigg (} \\frac{x_{i-1}}{(r_i - r_{i-1})} - \\frac{x_{i}}{(r_{i+1} - r_{i})} {\\bigg )} \\nonumber \\\\\\ & + & p_{h,k} \\, S \\, \\tau _k \\,{\\bigg (}\\frac{\\delta _{h,i+1}}{r_h - r_{i}} \\, - \\, \\frac{\\delta _{h,i}}{r_h - r_{i-1}}{\\bigg )}\\, \\frac{\\sum _{j=1}^{n-1} x_{j}}{\\sum _{j=1}^{n} x_{j}} \\, ,$ with $\\delta _{h,k}$ denoting the Kronecker delta.",
"In the r.h.s.",
"of $(\\ref {choiceforT})$ , $h >1$ and the terms involving the index $i-1$ [respectively, $i+1$ ] are effectively present only provided $i-1 \\ge 1$ [respectively, $i+1 \\le n$ ].",
"We point out that, due to the bound on the income of individuals in the $n$ -th class, in the model under consideration, the effective amount of money representing taxes, which is paid in correspondence to a payment of $S (1 - \\tau (k))$ and is then redistributed is $S \\, \\tau (k) \\,({\\sum _{j=1}^{n-1} x_{j}})/{(\\sum _{j=1}^{n} x_{j}})$ instead of $S \\, \\tau (k)$ .",
"To fix ideas, we take $S = 1$ , $r_j = 10 \\, j \\, ,$ and $\\tau _j = \\tau _{min} + \\frac{j - 1}{n-1} \\, (\\tau _{max} - \\tau _{min}) \\, ,$ for $j = 1, ... , n$ .",
"Still, the value of $\\tau _{min}$ and $\\tau _{max}$ have to be fixed.",
"Hence, the equations $(\\ref {evolution eq eta = 1})$ describe a family of models rather than a single model.",
"They are well beyond analytical solutions.",
"But, relevant facts can be understood through simulations.",
"We notice that, to run simulations, we take $n=25$ .",
"It is worthwhile observing that the choices of the $C_{hk}^i$ and the $T_{[hk]}^i(x)$ in $(\\ref {choiceforC})$ and $(\\ref {choiceforT})$ respectively are quite natural.",
"In contrast, the choice of the $p_{h,k}$ is an arbitrary one.",
"We make it, because it seems to be reasonable and it guarantees some heterogeneity in the saving propensity across classes.",
"The following properties have been found to hold true in [2], [3], [4].",
"Precisely, the first two have been analytically proved; the remaining ones are in fact suggested by a large number of simulations.",
"$\\bullet $ In correspondence to any initial condition $x_0 = (x_{01} , \\ldots , x_{0n})$ , for which $x_{0i} \\ge 0$ for all $i = 1, ... , n$ and $\\sum _{i=1}^n x_{0i} = 1$ , a unique solution $x(t) = (x_1(t),\\ldots ,x_n(t))$ of $(\\ref {evolution eq eta = 1})$ exists, which is defined for all $t \\in [0,+\\infty )$ , satisfies $x(0) = x_0$ and also $x_{i}(t) \\ge 0 \\ \\hbox{for} \\ i = 1, ... , n \\ \\hbox{and} \\ \\sum _{i=1}^n x_{i}(t) = 1 \\ \\hbox{for all} \\ t \\ge 0 \\, .$ In particular, this implies that the expressions of the $T_{[hk]}^i(x)$ in (REF ) simplify and the right hand sides of system $(\\ref {evolution eq eta = 1})$ are polynomials of degree 3, see [2].",
"$\\bullet $ The scalar function $\\mu (x)=\\sum _{i=1}^n r_i x_i$ , expressing the global (and mean) income, is a first integral for the system $(\\ref {evolution eq eta = 1})$ , see [2].",
"$\\bullet $ For any fixed value $\\mu \\in [r_1,r_n]$ , an equilibrium of $(\\ref {evolution eq eta = 1})$ exists, to which all solutions of $(\\ref {evolution eq eta = 1})$ , whose initial conditions $x_0 = (x_{01} , \\ldots , x_{0n})$ satisfy $x_{0i} \\ge 0$ for all $i = 1, ... , n$ , $\\sum _{i=1}^n x_{0i} = 1$ , and $\\sum _{i=1}^n r_i x_{0i} = \\mu $ tend asymptotically as $t \\rightarrow +\\infty $ .",
"In other words, a one-parameter family of asymptotic stationary distributions exists, the parameter being the total income value, [2], [3], [4].",
"$\\bullet $ The profile of the asymptotic stationary distribution depends on the difference between the maximum and the minimum tax rates, $\\tau _{max} - \\tau _{min}$ .",
"Specifically, if this difference is enlarged, while all other data are kept unchanged, an increase of the fraction of individuals belonging to the middle classes (to the detriment of those in the poorest and richest classes) can be detected at the asymptotic equilibrium, [2], [3].",
"$\\bullet $ The asymptotic stationary distributions corresponding to suitable values of the total income $\\mu $ exhibit tails which have a power-law decreasing behaviour.",
"Such a property has been observed in real world income distributions since the work by Pareto, [16].",
"The condition on $\\mu $ is related to the fact that the total income cannot be too high if a tail is expected in the asymptotic distribution.",
"In practice, in the admissible distributions of the population, the majority of individuals have to be concentrated in lower income classes.",
"Anyway, this occurs quite naturally in the real world, [3], [4]."
],
[
"Incorporating tax evasion into the model",
"In this section we incorporate in the model the occurrence of a partial tax evasion.",
"Our first natural inspection is then devoted to comparing the asymptotic income distributions which evolve from identical initial conditions in the absence and in the presence of this illegal practice.",
"Preliminar results in this direction have been reported in [5].",
"The case we model is that one in which, the effect of tax evasion in a trade corresponds to an advantage for both the individual who is receiving money and that one who is paying.",
"(Of course, other cases of tax evasion could be considered).",
"Such cases happen e.g.",
"in connection with value added taxes.",
"Indeed, the payment of this kind of tax relies upon invoices and receipts.",
"A way how tax evasion may take place may be described as follows.",
"The individual who receives the money (in our scheme, a $k$ -individual), be he an entrepreneur, a professional, a trader or similar, colludes with the individual who is paying him (a $h$ -individual), offering a discount on condition that this will not require any invoice or receipt.",
"In this way, the $h$ -individual has the advantage of the discount, and the $k$ -individual will conceal his gain from the tax return.",
"In such a case e.g.",
"taxes called in Italy, our country, I.V.A.",
"and I.R.P.E.F.",
"are evaded.",
"In order to keep into account, at least to some extent, such a behaviour, we recall that according to Section , in the case of tax compliance, when an $h$ -individual is supposed to pay an amount of money $S$ to a $k$ -individual, what equivalently happens is that the $h$ -individual pays a quantity $S \\, (1 - \\tau _k)$ to the $k$ -individual and he pays a quantity $S \\, \\tau _k$ to the government.",
"We take now $\\theta _k \\le \\tau _k$ .",
"To fix ideas, we take $\\theta _k = (1-q) \\, \\tau _k \\, ,\\qquad \\hbox{with} \\ 0 \\le q \\le 1 \\, .$ We write the scaling coefficient in (REF ) as $1 - q$ so as to have that the absence of evasion corresponds to $q = 0$ and a total evasion corresponds to $q = 1$ .",
"The effect of a partial tax evasion can be produced provided e.g.",
"we postulate that, when an $h$ -individual is supposed to pay an amount of money $S$ to a $k$ -individual, as a matter of fact the $h$ -individual pays a quantity $S \\, (1 - (\\tau _k + \\theta _k)/2)$ to the $k$ -individual and he pays a quantity $S \\, \\theta _k$ to the government.",
"If $q > 0$ , then the $h$ -individual pays less than he should, the $k$ -individual gains in the end more than he would have done in the tax compliant situation and the government collects less than it should.",
"Summarizing, in the presence of tax evasion the evolution equations are given by $(\\ref {evolution eq eta = 1})$ , with the $C_{hk}^i$ and the $T_{[hk]}^i(x)$ as in $(\\ref {choiceforC})$ and $(\\ref {choiceforT})$ , where however $S \\, (1 - \\tau _k)$ is replaced by $S \\, (1 - (\\tau _k + \\theta _k)/2)$ and $S \\, \\tau _k$ is replaced by $S \\, \\theta _k$ for any $k = 1, ... , n$ .",
"As described in [5], we ran several simulations, with different values of $q$ , relative to pairs of cases with the same initial conditions, the first one without and the other one with tax evasion.",
"The simulations systematically show that the effect produced by tax evasion is an increment in the number of individuals belonging to the poorest and to the richest classes at the detriment of the middle ones.",
"Moreover, examining the percentage in each class of the variation of the number of individuals, shows that, when passing from the straight to the dishonest case, the increasing effect in the high income classes is larger for greater income, while in the low income classes it is larger for lower income.",
"Correspondingly, those who benefit from tax evasion are individuals of the richest classes; and the richer they are, the most they benefit.",
"In contrast, things are getting worse for individuals with low income: many of them even pass to the lowest income class.",
"The four panels in the Figure REF illustrate the typical output.",
"Figure: In the first row, the panel on the left displays an asymptotic distribution in a case of tax compliance, whilethe panel on the right displays the asymptotic distribution for the same initial condition in the presence of tax evasion(τ min =30%\\tau _{min} = 30 \\%, τ max =45%\\tau _{max}=45 \\%, q=1/3q = 1/3);in the second row,the histograms in the panel on the left express the difference between the fraction of individuals in each class in the two cases with tax evasionand with tax compliance; the histograms in the right panel represent the percentual variation in each classof the fraction of individuals when passing from the tax compliance case to the one with tax evasion.The histograms are scaled differently on different pictures.Also the Gini index $G$ , which provides a measure of the income inequality, turns out to be larger when tax evasion is present.",
"This index takes values in $[0,1]$ , with 0 representing complete equality and 1 the maximal inequality.",
"For the example in the Figure REF , it is approximately equal to $0.383$ in the tax compliance case and $0.410$ in the tax evasion case.",
"If the evasion rate is increased e.g.",
"to $q = 2/3$ , it is approximately equal to $0.444$ , and so on.",
"We recall that the definition of $G$ involves the Lorenz curve, which plots the cumulative percentage of the total income of a population (on the $y$ axis) earned by the bottom percentage of individuals (on the $x$ axis).",
"Specifically, $G$ is the ratio $A_1/A_2$ of the area $A_1$ between the Lorenz curve and the line of perfect equality (the line at 45 degrees) and the total area $A_2$ under the line of perfect equality.",
"In our simulations we estimated the Gini index by calculating the area under the Lorenz curve as a sum of areas of trapezia."
],
[
"Proportional increase of tax rates and evasion",
"The extent of tax evasion in a society depends on several factors, like for instance the strength of moral values, imitation phenomena, frequency of the tax audits, strictness of the penalties etc.",
"A further important factor which influences the decision of a citizen to attempt some form of evasion is his or her perception of the fairness and rationality of the taxation scheme, and the “control” that he or she can exert on the final destination and good use of the money collected through taxation, [15].",
"In our model we can simulate quite easily a situation related to the perception of fairness, namely a situation in which the evasion rate increases in response to a sharp increase of the maximum tax rate $\\tau _{max}$ .",
"Notice that, in view of $(\\ref {progressivetaxrates})$ , increasing $\\tau _{max}$ produces an increase of all tax rates $\\tau _k$ but $\\tau _{min}$ .",
"For simplicity we shall suppose that the evasion rate $q$ varies in the same way for all income classes, but it is straightforward to turn to the general case of class-specific evasion rates $q_k$ .",
"As a first illustrative example, suppose to keep $\\tau _{min}$ fixed, $\\tau _{min}$ = 20 %, while varying $\\tau _{max}$ and $q$ according to Table REF .",
"For each couple of values we compute the equilibrium distribution, its Gini index and also the tax revenue or government budget $W_{tot}$ (compare Section REF ).",
"Our purpose is to see if there exists an “optimal” value of $\\tau _{max}$ which allows, even in the presence of evasion, to minimize the inequality, while keeping at the same time the government budget in a reasonable range.",
"Table: Example of a situation in which tax evasion increases in response to an increase of the maximum tax rate τ max \\tau _{max}.Note that q=0q=0 corresponds to honest behaviour and q=1q=1 to total evasion.",
"Both τ max \\tau _{max} and qqare expressend in percentual form.",
"α\\alpha is an integer index for reference to the graphs.",
"The ratio Δq/Δτ max \\Delta q / \\Delta \\tau _{max} is equal to 1.From the point of view of real economics this representation may still look quite naive, but at the mathematical level it is far from trivial: we are dealing with a manifold of equilibrium solutions of 25 coupled non-linear equations which depend on two variable parameters (plus, of course, on the total income and on the other parameters which define the choice of a specific model within the entire “family”).",
"Let us consider for simplicity only situations where $\\tau _{max}$ and $q$ are varied keeping the ratio of their variation constant.",
"In Table REF , for instance, the ratio is 1.",
"The resulting plot of the Gini index $G = G(\\tau _{max})$ (Fig.",
"REF ) is decreasing: this means that, even though evasion increases with increasing taxation, the total effect is always a decrease of inequality (accompanied by sizeable variations in the total government budget).",
"Figure: Behaviour of the Gini index GG and the government budget W tot W_{tot} as a function of τ max \\tau _{max},for ratio Δq/Δτ max \\Delta q / \\Delta \\tau _{max} equal to 1 (compare Table ).",
"The function G(τ max )G(\\tau _{max}) is decreasing.For graphical reasons the value of W tot W_{tot} has been multiplied by 100 here and in Fig.",
".Figure: Typical population variation for each income class due to the occurrence of evasion, in cases of variable qqand fixed τ max \\tau _{max} (left panel) and in cases of fixed Δq/Δτ max \\Delta q / \\Delta \\tau _{max} (right panel).In the case of the right panel, the middle class is split in two, and the super-rich decrease.If we look in detail at the population variations for each income class with respect to the case without evasion, we note a curious phenomenon, which did not occur in the solutions with constant $\\tau _{max}$ .",
"When $\\tau _{max}$ grows, at first the number of the poor and rich increases, while the middle class shrinks, as it happens when there is evasion for fixed $\\tau _{max}$ (Fig.",
"REF , left panel); at some point, however, for certain values of the ratio $\\Delta q / \\Delta \\tau _{max}$ , the super-rich begin to decrease and the middle class is split in two sections with different trends: a middle-rich section, increasing, and a middle-poor section, decreasing (Fig.",
"REF , right panel).",
"If, however, we choose a markedly different value for the ratio $\\Delta q / \\Delta \\tau _{max}$ , the behaviour of the Gini index is altered quite radically, and a minimum appears in the graph of $G(\\tau _{max})$ (Fig.",
"REF ).",
"The minimum is clearly visible, for instance, when $\\Delta q / \\Delta \\tau _{max} = 2$ , i.e.",
"when the evasion grows twice as fast in response to higher tax rates, compared to the case of Table REF .",
"It is natural to ask for which value of the $\\Delta q / \\Delta \\tau _{max}$ ratio the minimum begins to appear.",
"The numerical solutions indicate that this happens for $\\Delta q / \\Delta \\tau _{max} \\simeq 1.1$ .",
"Of course, this value does not have an absolute meaning, since it still depends on the $p_{h,k}$ parameters of the model, on $\\tau _{min}$ and on the initial value arbitrarily chosen for $q$ in the variations (here, $q = 0,2$ ).",
"Yet the existence of “phases” with different behaviour in the parameter space is quite clear.",
"Figure: Behaviour of the Gini index GG and the government budget W tot W_{tot} as a function of τ max \\tau _{max},for ratio Δq/Δτ max \\Delta q / \\Delta \\tau _{max} equal to 2.",
"The function G(τ max )G(\\tau _{max}) has a minimum.",
"This corresponds toan “optimal” value of τ max \\tau _{max} which allows, even in the presence of evasion, to minimize the inequalitywhile keeping at the same time the government budget in a reasonable range."
],
[
"Further “dynamical” quantities characteristic of the equilibrium",
"The equilibrium income distribution resulting from our model in correspondence to certain given parameters is completely described by the histogram of the equilibrium class populations $\\left\\lbrace \\hat{x}_1,...,\\hat{x}_{25} \\right\\rbrace $ .",
"This histogram can be compared to real statistical data or to suitable fit functions, like the Gibbs function, the log-normal function or the Kaniadakis function (see [12] and [4] and references therein), also in order to check for the possible presence of “fat” power-law tails.",
"Some integral indices of the income distribution are usually computed in dependence of the model parameters and allow a quick comparison to real data.",
"We have analyzed in Section for instance, the dependence of the Gini inequality index $G$ on the variations of the evasion parameter $q$ and then in Section the dependence of $G$ of simultaneous variations of $\\tau _{max}$ and $q$ .",
"Further integral indices which are usually computed for the distribution functions of statistical physics are the average $\\langle x \\rangle $ and the variance $\\sigma _x^2$ .",
"In our case, however, the average income is not meaningful because it is fixed by the initial conditions, and the variance does not appear to be of special interest.",
"There are, nevertheless, some other interesting and peculiar integral quantities which characterize the equilibrium state of our model and depend on the choice of its parameters.",
"Such quantities cannot be computed only from the asymptotic stationary distribution function, but depend on the underlying dynamical structure of the model.",
"In order to define them, let us first recall that the equilibrium state is, like in any kinetic model, the result of a dynamical equilibrium: the number of individuals belonging to a certain class remains constant in time when the total rate of individuals leaving that class is equal in absolute value to the total rate of individuals arriving from other classes.",
"Each single rate contains in turn contributions due to direct interaction and to taxation and redistribution."
],
[
"The tax revenue",
"The simplest dynamical integral quantity is the total amount of tax collected in the unit time and redistributed as welfare provisions.",
"It is often called tax revenue or government budget and it is given by $W_{tot} = \\sum _{h=1}^{n} \\, \\sum _{k=1}^{n} \\,\\sum _{j=1}^{n - 1} \\, S \\, p_{hk} \\, \\theta _k \\, {{\\hat{x}}_j}{{\\hat{x}}_h}{{\\hat{x}}_k} \\, ,$ where ${\\hat{x}}_i$ is the class population at equilibrium.",
"On one hand, it is then interesting to compare this budget with the total amount of the private-sector direct exchanges (see Section REF ).",
"On the other hand, it is also important, to keep track of the variations of the government budget when different taxations schemes are simulated.",
"We have seen, for instance, that by increasing the gap $\\tau _{max} - \\tau _{min}$ between the maximum and minimum tax rates one typically obtains an income distribution where the middle classes are more populated and the Gini inequality index is smaller.",
"This does not automatically imply, however, that the total amount of the collected taxes also increases; $W_{tot}$ might as well stay constant or decrease, and this would have significant consequences for the public sector.",
"Another typical case where one should be careful to keep the government budget under control while simulating changes of taxation is that of an increase of evasion following the increase of the maximum tax rate (Section ).",
"Supposing that some amount of evasion is inevitable in practice, we have been looking for values of the evasion-taxation variation ratio $\\Delta q / \\Delta \\tau _{max}$ which yield a minimum $G$ index.",
"In correspondence to such values, one can in principle obtain a situation of relatively low inequality even in the presence of evasion.",
"But what about the government budget?",
"Will it still be sufficient to keep the state administration and welfare working, without the need for major cuts?",
"If this is not the case, then one should conclude that social equality and evasion are incompatible and that the government must in any case enforce tax compliance by introducing further audits, fines etc.",
"Consider for instance the case of the Figure REF ."
],
[
"Relative probability of class promotion due to welfare or direct interaction",
"The probability of class promotion per unit time following the interaction of an individual with others is given by the ratio between the money gained in the interaction and the income difference of the classes.",
"This probability has therefore various contributions; some represent the money gained in direct binary interactions and others the money gained in indirect interactions due to taxation and welfare redistribution, represented in our model by terms of degree 3 in the population densities $x_i$ .",
"It is interesting to compute the ratio of these two contributions.",
"Suppose, for instance, that for a certain class the total promotion probability per unit time at equilibrium is 0.15, of which 0.1 is due to direct interactions and 0.05 to indirect interactions.",
"We can conclude that each individual of that class receives on the average, in the unit time, a certain amount of money from direct economic interactions, and half as much in the form of government welfare.",
"If with different model parameters the ratio was, say, 1/10 instead of 1/2, then we could conclude that with those parameters the model represents a more “liberistic” society, and so on.",
"We can also sum the probabilities over all classes, before computing that ratio.",
"In that way we obtain a figure referred to the entire society, namely the ratio between the total amount collected and redistributed by the government through welfare schemes and the total amount of direct exchanges between individuals.",
"In conditions of dynamical equilibrium, the probability $P_{i,welfare}$ of class promotion due to welfare of an individual of the class $i$ , for $1 \\le i \\le n - 1$ , is ${P_{i,welfare}} = \\frac{S}{{{r_{i + 1}} - {r_i}}} \\, \\sum _{h=1}^{n} \\, \\sum _{k=1}^{n} \\, {{p_{h,k}}{\\theta _k}{{\\hat{x}}_h}{{\\hat{x}}_k}} \\, ,$ where ${\\hat{x}}_i$ is the class population at equilibrium.",
"The probability $P_{i,exchanges}$ of class promotion due to direct exchanges is obtained from the appropriate terms of the matrix $C^i_{hk}$ , i.e.",
"is ${P_{i,exchanges}} = \\frac{S}{{{r_{i + 1}} - {r_i}}} \\,\\sum _{k=1}^{n} \\, {{p_{k,i}}\\left( 1-\\frac{\\theta _i+\\tau _i}{2} \\right){{\\hat{x}}_k}} \\, .$ Figure: An exampleof the ratio 𝑅=P i,welfare /P i,exchanges {\\it R} = {P_{i,welfare}}/{P_{i,exchanges}}, giving the relative probability foran individual of the class ii to be promoted to the upper class due to welfare provisions or direct exchanges.",
"(Here τ min =20\\tau _{min}=20, τ max =50\\tau _{max}=50, q=0.3q=0.3; the coefficients p h,k p_{h,k} are proportional to min(r i ,r k )\\min (r_i,r_k)).The mentioned ratio ${P_{i,welfare}}/{P_{i,exchanges}}$ , computed in a special case, is represented in the Figure REF .",
"From this graph we may deduce that for the very poor the welfare is an important factor of social promotion, while in the middle classes it becomes less important, in comparison to direct exchanges.",
"What is quite surprising, is that the ratio stays almost constant when we pass to the rich classes, although one might expect welfare provisions to be quite irrelevant for the class advancement of the super-rich.",
"A possible explanation is that in our kinetic model the direct exchanges involving the super-rich are quite rare, while all the tax money is constantly redistributed to everybody except to individuals of the $n$ -th class (not represented in Figure REF ).",
"A more realistic version might take into account the fact that welfare benefits are usually not accessible to the super-rich."
],
[
"The time evolution scale",
"In our model the equilibrium income distribution is obtained from the numerical solutions of the differential equations at large times.",
"The convergence of the solution to its equilibrium value is apparent from the numerical values of the $x_i(t)$ and from their temporal graphs.",
"In general, the various class populations converge to their equilibrium values in different times, depending on “how far” they were from those values at the beginning.",
"This is clear from the simultaneous graph in time of all the $x_i$ components (Fig.",
"REF ), where some of their lines can be seen crossing each other at different instants.",
"Figure: Time evolution of the class populations x i x_i from “extreme” initial conditions where all individualsare in the same income class, the class nr.",
"7.",
"(Total income μ=70\\mu =70; τ min =30\\tau _{min}=30, τ max =50\\tau _{max}=50, q=0.5q=0.5,coefficients p h,k p_{h,k} proportional to min(r i ,r k )\\min (r_i,r_k).",
")Figure: Example of time evolution for a society already at equilibrium which adapts to a moderate change in the taxation rate.We have taken the equilibrium state of Fig.",
"as new initial condition and changed τ min \\tau _{min} to 10 and τ min \\tau _{min} to 70,all the other parameters being the same.In order to obtain an estimate of the temporal convergence which is uniform with respect to the various income classes, one can consider a vector norm applied to the difference between the configuration at time $t$ and the configuration at time $t+\\xi $ , being $\\xi $ an arbitrary fixed delay; i.e.",
"one can consider the function $F_\\xi (t) = \\left[ \\sum _{i=1}^{25} \\left( x_i(t)-x_i(t+\\xi ) \\right)^2 \\right]^{1/2} \\, .$ This reminds of the Cauchy convergence criterum for discrete successions.",
"The plot of $ F_\\xi (t)$ converges quickly to zero as $t \\rightarrow \\infty $ (compare Fig.",
"REF ).",
"The plot of $\\ln \\, (F_\\xi (t))$ in function of $t$ is linear (Fig.",
"REF ), thus showing that the convergence of $ F_\\xi (t)$ to zero is exponential.",
"In correspondence of any small $\\varepsilon $ one can find a convergence time $T$ such that $F_\\xi (t) \\le \\varepsilon $ .",
"This time also depends on $\\xi $ and does not have any special significance in itself, but it allows to compare situations with slow and quick convergence.",
"For instance, one of the longest convergence times, which we could consider as a reference time for our system, is the one obtained with the “artificial” initial conditions where all individuals are in the same income class (Fig.",
"REF ).",
"If we could actually “reset” a real society in this way and record the subsequent evolution, the convergence time would be likely of the order of years.",
"This can be compared with the time it takes for a society already at equilibrium to adapt to a moderate change in the taxation rate (Fig.",
"REF ).",
"Figure: Convergence norm F ξ (t)F_\\xi (t) computed with ξ=100\\xi =100 for the two examples of slow evolution (left panel) and fast evolution (right panel) shown respectivelyin Figs.",
", .",
"At the same time, the norm in the right panel is approximately ten times smaller than in the left panel.",
"Due to the exponential behaviour,if we fix a threshold ε\\varepsilon , the time TT necessary to reach it is approximatelytwice as large for the case in the left panel, compared to that in the right panel; for instance,for ε=10 -4 \\varepsilon =10^{-4} one has T a ≃11000T_a \\simeq 11000, T b ≃5500T_b \\simeq 5500.Figure: Logarithm of the norm F ξ (t)F_\\xi (t) in the case of slow and fast evolution in Fig.",
", left and right panel.The exponential convergence of the norm is apparent."
],
[
"Conclusion",
"In this paper a microscopic model for the complex of monetary exchanges, taxation and redistribution in a closed society is investigated, both in the tax compliance case and in the presence of tax evasion.",
"The focus is on the effects of the tax evasion phenomenon on the income distribution over the population.",
"Various comparisons between the situations without and with tax evasion are made through a direct inspection of the asymptotic income profiles, and by means of indicators like the Gini index, the tax revenue and the welfare-induced probability of class promotion.",
"In a nutshell, this stylised model supports the belief that a fair fiscal policy and a honest behaviour of the population individuals play a decisive role towards the overcoming of social inequalities."
]
] | 1403.0015 |
[
[
"On-flow and strong solutions to Killing-type equations"
],
[
"Abstract If we impose infinitesimal invariance up to a boundary term of the action functional for Lagrangian ordinary differential equations, we are led to Killing-type equations, which are related to first integrals through Noether theorem.",
"We review the \"on-flow\" and \"strong\" interpretations of the Killing-type equation, and for each we detail the complete explicit structure of the solution set in terms of the associated first integral.",
"We give examples that reappraise the usefulness of the \"on-flow\" solutions.",
"Finally, we describe an equivalent alternative approach to variational invariance."
],
[
"Introduction",
"Suppose we are given a smooth Lagrangian function $L(t,q,\\dot{q})$ , with $t\\in \\mathbb {R}$ , $q,\\dot{q}\\in \\mathbb {R}^n$ .",
"The variational principle for Lagrangian dynamics posits that $\\delta \\int _{t_1}^{t_2}L\\bigl (t,q(t),\\dot{q}(t)\\bigr )dt=0,$ which is equivalent to the Euler-Lagrange equation $\\frac{d}{dt}\\partial _{\\dot{q}}L\\bigl (t,q(t),\\dot{q}(t)\\bigr )-\\partial _{q}L\\bigl (t,q(t),\\dot{q}(t)\\bigr )=0.$ We will assume that the Euler-Lagrange equation can be put into normal form $\\ddot{q}=\\Lambda (t,q,\\dot{q}),$ Following closely the notation of Sarlet and Cantrijn (except that $\\dot{q}$ is an independent variable from the outset), we consider an infinitesimal transformation in the $(t,q)$ space given by $\\bar{t}=t+\\varepsilon \\tau (t,q,\\dot{q}),\\qquad \\bar{q}=q+\\varepsilon \\xi (t,q,\\dot{q}).$ This transformation is said to leave the action integral (infinitesimally) invariant up to boundary terms (using the nomenclature recommended by P.G.L.",
"Leach), if a function $f(t,q,\\dot{q})$ exists, such that for the given smooth curve $t\\mapsto q(t)$ we have $\\int _{\\bar{t}_1}^{\\bar{t}_2}L\\Bigl (\\bar{t},\\bar{q}(\\bar{t}),\\frac{d\\bar{q}}{d\\bar{t}}(\\bar{t})\\Bigr )d\\bar{t}=\\int _{t_1}^{t_2}L\\bigl (t,q(t),\\dot{q}(t)\\bigr )dt+{}\\\\+\\varepsilon \\int _{t_1}^{t_2}\\frac{df}{dt}\\bigl (t,q(t),\\dot{q}(t)\\bigr )dt+O(\\varepsilon ^2),$ which is equivalent to the following Killing-type equation for ODEs: $\\tau \\partial _t L+\\partial _q L\\cdot \\xi +\\partial _{\\dot{q}}L\\cdot \\bigl (\\dot{\\xi }-\\dot{q}\\dot{\\tau }\\bigr )+L\\dot{\\tau }=\\dot{f}.$ This is the same formula as Sarlet and Cantrijn , formula (9) p. 471.",
"We will write $\\partial _{t}, \\partial _{q}, \\partial _{\\dot{q}}$ for the partial derivative and gradients, $\\dot{x}$ for the total time derivative of the function $x$ , and $x\\cdot y$ will denote the ordinary scalar product of $x,y\\in \\mathbb {R}^n$ .",
"Noether's theorem states that equation (REF ) is a sufficient condition so that the function $N=f-L\\tau -\\partial _{\\dot{q}}L\\cdot \\bigl (\\xi -\\dot{q}\\tau \\bigr )$ () be a constant of motion for the solutions to the Lagrange equation (REF ).",
"The function $L(t,q,\\dot{q})$ will be given, and we solve Killing-type equation (REF ) for the triple $(\\tau ,\\xi ,f)$ .",
"The equation in the terse form (REF ) is open to at least three interpretations that we know of, differing on what the independent variables are and on how to treat the $\\ddot{q}$ terms.",
"The most restrictive approach is when we seek $\\tau ,\\xi , f$ as functions of $(t,q)$ only: ($\\tau (t,q),\\xi (t,q),f(t,q)$ ).",
"Since in this case $\\ddot{q}$ does not appear, the independent variables are $t,q,\\dot{q}$ only, and we want the equation to hold identically.",
"We will be concerned with this approach mainly in Section .",
"If we instead allow full dependence of $\\tau ,\\xi ,f$ on $\\dot{q}$ , the expanded-out form of the total time derivatives $\\dot{\\xi },\\dot{\\tau },\\dot{f}$ will contain $\\ddot{q}$ .",
"We distinguish two alternatives: Strong form: we treat $\\ddot{q}$ as just another independent variable and require the equation to hold for all $t,q,\\dot{q},\\ddot{q}$ .",
"This way the Hamiltonian action functional will be infinitesimally invariant along all smooth trajectories $q(t)$ .",
"The strong form was introduced by Djukic , and studied also by Kobussen .",
"Since the equation depends linearly on $\\ddot{q}$ , it is equivalent to a system of $n+1$ equations that do not contain $\\ddot{q}$ , but we will not exploit this fact in the sequel.",
"On-flow form: we replace every occurrence of $\\ddot{q}$ with $\\Lambda (t,q,\\dot{q})$ from the normalized Lagrange equation (REF ), and require the resulting equation to hold for all $t,q,\\dot{q}$ .",
"This way the Hamiltonian action functional will be infinitesimally invariant as in (REF ) only along the Lagrangian motions $q(t)$ , a condition which is however enough for the expression (REF ) to be a first integral.",
"In Sections and of this work we give a full description of the structure of the solution sets of the Killing-like equation, in the above senses, assuming that we are given both $L$ and the first integral $N$ .",
"This problem is also known as “reverse Noether theorem”.",
"The main results of Sections and , specially for the strong form, are basically contained already in Sarlet and Cantrijn's 1981 paper , where they are deduced as a corollaries of the theory of $d\\theta $ -symmetries.",
"Here we make a direct derivation, which we hope will be helpful to some readers.",
"In Section we illustrate the general theory with examples.",
"In Subsection REF we re-examine the free particle's notorious “non-noetherian” symmetries that were “noetherized” by P.G.L.",
"Leach by substituting a new Lagrangian for the usual one.",
"We will see what we can say about those symmetries from the point of view of on-flow and strong solutions without switching Lagrangian.",
"In Subsection REF we search for classes of superintegrable systems among the ones of the form $\\ddot{x}=-G(x)$ , $\\ddot{y}=-G^{\\prime }(x)y$ , by making a suitable ansatz on the solution triple $(\\tau ,\\xi ,f)$ in the on-flow form.",
"For the sake of clarity we will stop short of pursuing the method beyond known territory.",
"The on-flow form of Killing-like equation was dismissed by most authors, the reason being that it has too many solution.",
"Our point is that a large solution set can work to our advantage when we do not know exactly the form of the system and of the first integral, because an ansatz has more chances of catching a solution when the solutions are plenty.",
"Subsection REF is devoted to the Laplace-Runge-Lenz vector conservation of the classical Kepler problem.",
"We review some known formulas for solutions to Killing-like equation in either on-flow or strong interpretation, and propose some new ones in dimension 3, that feel simpler to us.",
"In Section we return to an alternative way of performing time change in the action integral, that we proposed in a recent paper .",
"We show how this approach leads to a different, but equivalent, Killing-like equation and to some formulas that match closely with formulas written in a 1972 paper by Candotti, Palmieri and Vitale."
],
[
"Structure of the solution sets",
"Equation (REF ) is referred to as “Killing-type” because of the important particular case when the Lagrangian function is a quadratic form in the $\\dot{q}$ variable: $L=\\frac{1}{2}\\dot{q}\\cdot A(q)\\dot{q}$ , with $A(q)$ symmetric $n\\times n$ non-singular matrix.",
"The Lagrange equation (REF ) reduces to the equation of geodesics.",
"Equation (REF ) with $\\tau \\equiv 0$ , $f\\equiv 0$ , and $\\xi (q)$ as a function of $q$ only, becomes $\\partial _q L\\cdot \\xi (q)+ \\partial _{\\dot{q}}L\\cdot \\xi ^{\\prime }(q)\\dot{q}=0$ , which is quadratic homogeneous in $\\dot{q}$ : if we equate to zero the coefficients, we get the well-known Killing equations of Differential Geometry.",
"The first integral (REF ) simplifies to $-\\partial _{\\dot{q}} L\\cdot \\xi (q)=-A(q)\\dot{q}\\cdot \\xi (q)=-\\dot{q} \\cdot A(q) \\xi (q)$ .",
"Back to the general Killing-type equation (REF ), we will tacitly assume that $L\\in C^3$ and that the following usual regularity condition is satisfied: $\\det g\\ne 0\\qquad \\text{where }g:=\\partial ^2_{\\dot{q},\\dot{q}}L(t,q,\\dot{q}).$ The notation $\\partial ^2_{\\dot{q},\\dot{q}}L$ means the Hessian matrix of the second derivatives of $L$ with respect to $\\dot{q}$ .",
"This will ensure that the Lagrange equation can indeed be put into normal form (REF ), and that there is existence and uniqueness of the solutions to the Cauchy problems.",
"The following result was basically found by Lutzky , who uses it to argue that it is not useful to allow $f$ to depend on $\\dot{q}$ .",
"Theorem 1 (General solution of the on-flow equation) Let $L(t,q,\\dot{q})$ be a Lagrangian function.",
"Suppose that the Lagrange equation has the $C^1$ first integral $N(t,q,\\dot{q})$ .",
"Then a triple $(\\tau ,\\xi ,f)$ is a solution of the on-flow version of the Killing-type equation with $N$ as associated first integral if and only if $f=\\tau L+N+\\partial _{\\dot{q}}L\\cdot (\\xi -\\tau \\dot{q}).$ Equation (REF ) is simply a rearrangement of formula (REF ).",
"If the triple $(\\tau ,\\xi ,f)$ is a solution associated with $N$ then it must satisfy (REF ).",
"Conversely, suppose that the triple satisfies (REF ) and let us check that it is an on-flow solution.",
"Taking the time derivative of $f=\\tau L+N+ \\partial _{\\dot{q}} L\\cdot (\\xi -\\tau \\dot{q})$ along a solution of Lagrange equation and replacing into Killing-type equation (REF ) $\\tau \\partial _t L+\\partial _q L\\cdot \\xi +\\partial _{\\dot{q}}L\\cdot \\bigl (\\dot{\\xi }-\\dot{q}\\dot{\\tau }\\bigr )+L\\dot{\\tau }=\\\\=\\dot{f}=\\dot{\\tau }L+\\tau \\dot{L}+0+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}} L\\Bigr )\\cdot (\\xi -\\tau \\dot{q})+\\partial _{\\dot{q}} L\\cdot (\\dot{\\xi }-\\dot{\\tau }\\dot{q}-\\tau \\ddot{q})$ Canceling out the common terms and using Lagrange equation we get $\\tau \\partial _t L+\\partial _q L\\cdot \\xi =\\tau \\dot{L}+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}} L\\Bigr )\\cdot (\\xi -\\tau \\dot{q})+\\partial _{\\dot{q}} L\\cdot (-\\tau \\ddot{q}).$ Using Lagrange equation (REF ) this becomes $\\tau \\partial _t L+\\partial _q L\\cdot \\xi =\\tau \\dot{L}+\\partial _{q} L\\cdot (\\xi -\\tau \\dot{q})-\\tau \\partial _{\\dot{q}} L\\cdot \\ddot{q}.$ Canceling out and rearranging we get $\\tau \\bigl (\\partial _t L+\\partial _{q} L\\cdot q+\\partial _{\\dot{q}} L\\cdot \\ddot{q}\\bigr )=\\tau \\dot{L},$ which is simply the chain rule.",
"From formula (REF ) we can express $\\tau $ as a function of $f,\\xi $ , at least when $L-\\partial _{\\dot{q}}L\\cdot \\dot{q}\\ne 0$ .",
"Of particular interest are the solutions with $f=0$ : Corollary 1 (Simplest on-flow solution with $f=0$ ) Let $L$ be a Lagrangian function.",
"Suppose that the Lagrange equation has the $C^1$ first integral $N$ .",
"Then the on-flow version of the Killing-type equation (REF ) is satisfied by the triple $\\tau =-\\frac{N}{L},\\qquad \\xi =-\\frac{N}{L}\\dot{q},\\qquad f\\equiv 0.$ The corresponding first integral (REF ) is precisely $N$ .",
"Around points where $L=0$ we can take $\\tau =-N/(L+c)$ , $\\xi =- N/ (L+c)$ , for a constant $c\\ne 0$ .",
"Corollary 2 (More general on-flow solution with $f=0$ ) Let $L(t,q,\\dot{q})$ be a Lagrangian function.",
"Suppose that the Lagrange equation has the $C^1$ first integral $N(t,q,\\dot{q})$ .",
"Let $R(t,q,\\dot{q})$ be an arbitrary smooth function with values in $\\mathbb {R}^n$ .",
"Then the on-flow version of the Killing-type equation (REF ) is satisfied by the triple $\\tau (t,q,\\dot{q})=-\\frac{N+\\partial _{\\dot{q}}L\\cdot R}{L},\\qquad \\xi (t,q,\\dot{q})=R-\\dot{q}\\frac{N+\\partial _{\\dot{q}}L\\cdot R}{L},\\qquad f\\equiv 0.$ The corresponding first integral (REF ) is precisely $N$ .",
"The solutions of the strong Killing equation were given indirectly by Sarlet and Cantrijn as a consequence of a result on $d\\theta $ -symmetries.",
"Here we give a direct formulation and proof.",
"Theorem 2 (General solution of the strong equation) Let $L(t,q,\\dot{q})$ be a Lagrangian function.",
"Suppose that the Lagrange equation has the $C^2$ first integral $N(t,q,\\dot{q})$ .",
"Then a triple $(\\tau ,\\xi ,f)$ is a solution of the strong version of the Killing-type equation with $N$ as associated first integral if and only if $\\xi = \\tau \\dot{q}-g^{-1}\\partial _{\\dot{q}}N,\\\\f=\\tau L+N-\\partial _{\\dot{q}}L\\cdot g^{-1}\\partial _{\\dot{q}}N,$ where $g=\\partial ^2_{\\dot{q},\\dot{q}}L$ is the Hessian matrix as in (REF ).",
"Let us establish some formulas first.",
"The function $\\Lambda $ appearing in the normal form of Lagrange equation (REF ) can be made explicit this way: $\\Lambda \\equiv g^{-1}\\bigl (\\partial _{q}L-\\partial ^2_{\\dot{q},t}L-\\partial ^2_{\\dot{q},q}L\\;\\dot{q}\\bigr ).$ For any smooth $q(t)$ , not necessarily a Lagrangian motion, the following relations holds: $\\partial _qL-\\frac{d}{dt}\\partial _{\\dot{q}}L=g(\\Lambda -\\ddot{q}),\\\\\\dot{L}=\\partial _tL+\\partial _qL\\cdot \\dot{q}+\\partial _{\\dot{q}}L\\cdot \\ddot{q}$ Since $N$ is a first integral, again for any smooth $q(t)$ we have $\\begin{split}\\dot{N}={}&\\partial _tN+\\partial _qN\\cdot \\dot{q}+\\partial _{\\dot{q}}N\\cdot \\ddot{q}=\\\\={}&\\underbrace{\\partial _tN+\\partial _qN\\cdot \\dot{q}+\\partial _{\\dot{q}}N\\cdot \\Lambda }_{=0}+\\partial _{\\dot{q}}N\\cdot (\\ddot{q}-\\Lambda )=\\\\={}&\\partial _{\\dot{q}}N\\cdot (\\ddot{q}-\\Lambda ).\\end{split}$ A solution of the strong form with $N$ as associated first integral must in particular satisfy relation (REF ).",
"Suppose that the triple $(\\tau ,\\xi ,f)$ satisfies (REF ) and let us impose that it solves the strong version of Killing equation.",
"The total time derivative of $f$ along a generic smooth $q(t)$ is: $\\dot{f}={}&\\frac{d}{dt}\\bigl (\\tau L+N+\\partial _{\\dot{q}}L\\cdot (\\xi -\\tau \\dot{q})\\bigr )=\\\\={}&\\dot{\\tau }L+\\tau \\dot{L}+\\dot{N}+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}}L\\Bigr )\\cdot (\\xi -\\tau \\dot{q})+\\partial _{\\dot{q}}L\\cdot (\\dot{\\xi }-\\dot{\\tau }\\dot{q}-\\tau \\ddot{q}).$ Equating this with the left-hand side of the strong Killing-like equation we get $\\tau \\partial _t L+\\partial _q L\\cdot \\xi +\\partial _{\\dot{q}}L\\cdot \\bigl (\\dot{\\xi }-\\dot{q}\\dot{\\tau }\\bigr )+L\\tau =\\\\=\\dot{\\tau }L+\\tau \\dot{L}+\\dot{N}+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}}L\\Bigr )\\cdot (\\xi -\\tau \\dot{q})+\\partial _{\\dot{q}}L\\cdot (\\dot{\\xi }-\\dot{\\tau }\\dot{q}-\\tau \\ddot{q})$ which simplifies immediately to $\\tau \\partial _t L+\\partial _q L\\cdot \\xi =\\tau \\dot{L}+\\dot{N}+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}}L\\Bigr )\\cdot (\\xi -\\tau \\dot{q})-\\tau \\partial _{\\dot{q}}L\\cdot \\ddot{q}.$ Using () to replace $\\dot{L}$ it becomes $\\tau \\partial _t L+\\partial _q L\\cdot \\xi =(\\partial _tL+\\partial _qL\\cdot \\dot{q})\\tau +\\dot{N}+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}}L\\Bigr )\\cdot (\\xi -\\tau \\dot{q}),$ which further simplifies to $\\partial _q L\\cdot \\xi =\\tau \\partial _qL\\cdot \\dot{q}+\\dot{N}+\\Bigl (\\frac{d}{dt}\\partial _{\\dot{q}}L\\Bigr )\\cdot (\\xi -\\tau \\dot{q}),$ which can be rearranged to $\\Bigl (\\partial _q L-\\frac{d}{dt}\\partial _{\\dot{q}}L\\Bigr )\\cdot (\\xi -\\tau \\dot{q})=\\dot{N}.$ Using (REF ) and (REF ), formula (REF ) becomes $\\bigl (g(g-\\ddot{q})\\bigr )\\cdot (\\xi -\\tau \\dot{q})=\\partial _{\\dot{q}}N(\\ddot{q}-\\Lambda )=-\\partial _{\\dot{q}}N\\cdot (\\Lambda -\\ddot{q}).$ Since the hessian matrix $g$ is symmetric, this becomes $\\bigl (g(\\xi -\\tau \\dot{q})\\bigr )\\cdot (\\Lambda -\\ddot{q})=-\\partial _{\\dot{q}}N\\cdot (\\Lambda -\\ddot{q}).$ Finally, since $\\ddot{q}$ is arbitrary, we conclude that $g(\\xi -\\tau \\dot{q})=-\\partial _{\\dot{q}}N,$ which is equivalent to (REF ).",
"Equation () is simply a consequence of (REF ) and (REF ).",
"A direct proof of the “if” part of Theorem REF can be found in a paper by Boccaletti and Pucacco .",
"If we know a solution to the Killing-type equation, either on-flow or strong, we can easily generate infinitely many others, parameterized by an arbitrary function: Corollary 3 (Multiplicity for both on-flow and strong equation) Let $L(t,q,\\dot{q})$ be a Lagrangian function.",
"Suppose that the triple $\\bigl (\\tau (t,q,\\dot{q}), \\xi (t,q,\\dot{q}),f(t,q,\\dot{q}))$ satisfies the Killing-type equation (REF ) in either the strong or the on-flow version.",
"Take an arbitrary smooth function $h(t,q,\\dot{q})$ .",
"Then also the following triple $\\tilde{\\tau }=\\tau +\\frac{h-f}{L},\\qquad \\tilde{\\xi }=\\xi +\\dot{q}\\,\\frac{h-f}{L},\\qquad \\tilde{f}=h$ satisfies the Killing-type equation of the same form.",
"The corresponding first integral (REF ) is the same.",
"If we assume that any of the equations (REF ), () and (REF ) holds for the triple $(\\tau ,\\xi ,f)$ , a simple replacement shows that the equation holds also for $(\\tilde{\\tau },\\tilde{\\xi },\\tilde{f})$ .",
"Within the family of solutions given by Theorem REF there is always one with trivial (i.e., zero) time change and another one with trivial boundary term.",
"This simple fact was already established in a more general setting (including, for example, nonlocal constants of motion) and different notations by the authors .",
"Corollary 4 (Trivializing either time-change or gauge) Let $L(t,q,\\dot{q})$ be a Lagrangian function.",
"Suppose that the triple $\\bigl (\\tau (t,q,\\dot{q}), \\xi (t,q,\\dot{q}),f(t,q,\\dot{q}))$ satisfies the Killing-type equation (REF ) in either the strong or the on-flow form.",
"Then also the following two triples are solutions: $(0,\\;\\xi -\\dot{q}\\tau ,\\;f-L\\tau ),\\qquad \\Bigl (\\tau -\\frac{f}{L},\\;\\xi -\\dot{q}\\,\\frac{f}{L},\\;0\\Bigr ).$ The corresponding first integrals are the same.",
"Simply take either $h=f-L\\tau $ or $h=f$ in Corollary REF ."
],
[
"Strong solutions independent of $\\dot{q}$",
"Given $\\xi $ and $\\tau $ that do not depend on $\\dot{q}$ there is a simple necessary condition for them to be part of a solution triple $(\\tau ,\\xi ,f)$ of the strong form of Killing-like equation, regardless of $N$ .",
"Proposition 1 Suppose that $(\\tau ,\\xi ,f)$ solves the strong form of Killing equation, and that $\\xi (t,q)$ and $\\tau (t,q)$ do not depend on $\\dot{q}$ .",
"Then $f$ does not depend on $\\dot{q}$ either.",
"Moreover, the left-hand side of the Killing-like equation $\\tau \\partial _t L+\\partial _q L\\cdot \\xi +\\partial _{\\dot{q}}L\\cdot \\bigl (\\dot{\\xi }-\\dot{q}\\dot{\\tau }\\bigr )+L\\dot{\\tau }$ after replacing with the given $L(t,q,\\dot{q}),\\xi (t,q),\\tau (t,q)$ , depends linearly on $\\dot{q}$ .",
"Starting from formula (REF ), which holds in the strong case too, $f=\\tau L+N+\\partial _{\\dot{q}}L\\cdot (\\xi -\\tau \\dot{q})$ and taking the gradient with respect to $\\dot{q}$ we get $\\begin{split}\\partial _{\\dot{q}}f={}&\\partial _{\\dot{q}}N+\\partial _{\\dot{q}}\\bigl (\\tau L+\\partial _{\\dot{q}}L\\cdot (\\xi -\\tau \\dot{q})\\bigr )=\\\\={}&-g(\\xi -\\tau \\dot{q})+\\partial _{\\dot{q}}\\bigl (\\tau L+\\partial _{\\dot{q}}L\\cdot (\\xi -\\tau \\dot{q})\\bigr ),\\end{split}$ where we have used the replacement $\\partial _{\\dot{q}}N=-g(\\xi -\\tau \\dot{q})$ , which is a rearrangement of (REF ).",
"Using now the assumption that $\\xi ,\\tau $ do not depend on $\\dot{q}$ we can carry on the calculation $\\begin{split}\\partial _{\\dot{q}}f={}&-g(\\xi -\\tau \\dot{q})+\\tau \\partial _{\\dot{q}}L+g (\\xi -\\tau \\dot{q})+\\partial _{\\dot{q}}L(-\\tau )\\equiv 0.\\end{split}$ We deduce that $f$ does not depend on $\\dot{q}$ either.",
"Hence $\\dot{f}$ is linear in $\\dot{q}$ .",
"We conclude that expression (REF ), which is identically equal to $\\dot{f}$ , must be linear in $\\dot{q}$ too.",
"Only some first integrals $N$ can be deduced from a triple $(\\tau (t,q),\\xi (t,q),f(t,q))$ independent of $\\dot{q}$ .",
"Proposition 2 A first integral $N(t,q,\\dot{q})$ can be deduced from a triple $(\\tau ,\\xi , f)$ that does not depend on $\\dot{q}$ if and only if $g^{-1} \\partial _{ \\dot{q}}N=a(t,q)+b(t,q)\\dot{q}$ , where $a(t,q)$ is vector-valued and $b(t,q)$ is scalar-valued.",
"If $N$ can be deduced from $\\tau (t,q),\\xi (t,q),f(t,q)$ , then from Theorem REF , formula (REF ), $g^{-1}\\partial _{\\dot{q}}N= -\\xi (t,q)+\\tau (t,q)\\dot{q}$ .",
"Conversely, if $g^{-1} \\partial _{ \\dot{q}}N=a(t,q)+b(t,q)\\dot{q}$ , we can choose $\\xi =-a$ , $\\tau =b$ and $f$ given by equation (), so that the triple $(\\tau ,\\xi , f)$ is a solution of Killing-like equation in the strong sense (Theorem REF ).",
"Finally, $f$ does not depend on $\\dot{q}$ because of Proposition REF ."
],
[
"The free particle",
"In one of his papers , Leach argues that Lie point symmetries that are usually called “nonnoetherian” are indeed fully Noetherian, provided that we switch from the obvious Lagrangian to some other Lagrangian which retains the same equations of motion.",
"The point is illustrated with the example of the free particle in one dimension: the equation of motion is $\\ddot{q}=0$ , whose Lie symmetries $\\xi \\partial _q +\\tau \\partial _t$ are an 8-dimensional space (Table REF ).",
"If we examine these symmetries in the Noetherian sense together with the “natural” Lagrangian $L=\\dot{q}^2/2$ , we see that only five of them can be completed to a Noetherian triple $(\\tau ,\\xi ,f)$ .",
"Let us see what we can say about the three remaining “nonnoetherian” symmetries from the point of view of the strong and on-flow solutions, without resorting to a different Lagrangian.",
"Table: Lie and Noether symmetries of the free particleFor the Lie symmetries $\\Gamma _6,\\Gamma _7,\\Gamma _8$ the Killing-like equation becomes respectively $\\dot{q}^2=\\dot{f}_6,\\qquad -\\frac{\\dot{q}^3}{2}=\\dot{f}_7,\\qquad \\frac{3q\\dot{q}^2-t\\dot{q}^3}{2}=\\dot{f}_8.$ These equations have no solution in $f$ in the strong sense, because the left-hand sides are not linear in $\\dot{q}$ (Proposition REF ).",
"As for the on-flow version of the Killing-like equation, given any arbitrary couple $(\\tau ,\\xi )$ and a first integral $N$ , formula (REF ) of Theorem REF immediately gives a boundary term $f$ that completes to a solution triple.",
"Specifically: $\\text{for }\\Gamma _6\\qquad f_6=q\\dot{q}+N\\\\\\text{for }\\Gamma _7\\qquad f_7=-\\frac{q\\dot{q}^2}{2}+N\\\\\\text{for }\\Gamma _8\\qquad f_8=\\frac{1}{2}(2q-t\\dot{q})q\\dot{q}+N.$ Of course, these boundary terms depend on $\\dot{q}$ .",
"With solutions in the on-flow solutions we can recover all first integral of the system, i.e., all function of the form $k(\\dot{q},q-t\\dot{q})$ .",
"Using Proposition REF , since $g^{-1}=1$ , we can say that with solutions $(\\tau (t,q),\\xi (t,q),f(t,q))$ in the strong sense we cannot obtain first integrals that are not quadratic in $\\dot{q}$ , for example $(q-t\\dot{q})^3$ ."
],
[
"Superintegrable systems related to isochrony",
"The authors are familiar with with the following system of two scalar differential equations $\\ddot{x}=-G(x),\\quad \\ddot{y}=-G^{\\prime }(x)y,$ which are the Lagrange equations of the Lagrangian $L(t,q,\\dot{q})=\\dot{x}\\,\\dot{y}-G(x)y,\\qquad \\text{where }q=\\binom{x}{y},\\ \\dot{q}=\\binom{\\dot{x}}{\\dot{y}}$ This system is rich with first integrals.",
"One is $N_1=\\dot{x}\\dot{y}+G(x)y$ .",
"Since $g^{-1}\\partial _{\\dot{q}}N_1=\\dot{q}$ , following Proposition REF we can deduce $N_1$ from the following triple independent of $\\dot{x},\\dot{y}$ : $\\tau =1,\\quad \\xi =0,\\quad f=0,$ which is a solution in the strong sense.",
"Another first integral is $N_2=\\dot{x}^2/2+\\int G(x)dx$ .",
"Since $g^{-1}\\partial _{\\dot{q}}N_2=({\\begin{matrix}0&0\\\\ -1&0 \\end{matrix}})\\dot{q}$ and because of Proposition REF , to deduce $N_2$ we must accept dependence on $\\dot{q}$ .",
"According to Theorem REF , all solutions in the strong sense are given by an arbitrary $\\tau $ and $\\xi =\\tau \\dot{q}-g^{-1}\\partial _{\\dot{q}}N_2=\\binom{\\tau \\dot{x}}{\\tau \\dot{y}-\\dot{x}},\\\\f=\\tau L+N_2-\\partial _{\\dot{q}}g^{-1}\\partial _{\\dot{q}}N_2=\\tau \\bigl (\\dot{x}\\dot{y}-G(x)y\\bigr )-\\frac{1}{2}\\dot{x}^2+\\int G(x)dx$ The special feature of the system (REF ) is that for some classes of function $G$ the system has a third, independent, first integral.",
"One way to detect some of these superintegrable system is by making a plausible ansatz on the triple $(\\tau ,\\xi ,f)$ and solving the Killing-like equation for $G$ as an additional unknown function.",
"We think it is preferable to use the on-flow version of the equation, simply because it has so many more solution, heightening the chances that the ansatz may catch one.",
"Our starting ansatz is $\\tau \\equiv 0,\\qquad \\xi =\\binom{h(x,\\dot{x})}{0}.$ In keeping with the on-flow version, the first and second total time derivatives of $h$ will take the Lagrange equations into account: $\\dot{h}=\\dot{x}\\partial _x h+\\ddot{x}\\partial _{\\dot{x}}h=\\dot{x}\\partial _x h-G(x)\\partial _{\\dot{x}}h,\\qquad \\ddot{h}=\\dot{x}\\partial _x \\dot{h}-G(x)\\partial _{\\dot{x}}\\dot{h}$ The Killing-like equation becomes $-yG^{\\prime }(x)h+\\dot{y}\\dot{h}=\\dot{f}.$ With the further ansatz that $f=y\\dot{h}$ the equation is $-yG^{\\prime }(x)h+\\dot{y}\\dot{h}=\\dot{y}\\dot{h}+y\\ddot{h}$ which simplifies to $-G^{\\prime }(x)h=\\ddot{h}.$ We can make a third ansatz by setting $h$ to be a polynomial in $\\dot{x}$ of the form $h=\\alpha (x)+\\beta (x)\\dot{x}^2$ .",
"Replacing into (REF ) we obtain a polynomial of degree 4 in $\\dot{x}$ equated to 0: $\\beta ^{\\prime \\prime }(x)\\dot{x}^4 +\\bigl (\\alpha ^{\\prime \\prime }(x)-\\beta (x)G^{\\prime }(x)-5 G(x)\\beta ^{\\prime }(x)\\bigr )\\dot{x}^2+{}\\\\+2 G(x)^2\\beta (x)-G(x)\\alpha ^{\\prime }(x)+\\alpha (x) G^{\\prime }(x)=0.$ The coefficient of $\\dot{x}^4$ is $\\beta ^{\\prime \\prime }(x)$ , which must be 0.",
"Let us simply take $\\beta (x)\\equiv x$ .",
"Equating the coefficient of $\\dot{x}^0$ in (REF ) to 0 we get $\\alpha (x) G^{\\prime }(x)-G(x)\\alpha ^{\\prime }(x)+2 x G(x)^2=0,$ which can be solved for $\\alpha $ as $\\alpha (x)=(c+x^2)G(x)$ .",
"Replacing into the coefficient of $\\dot{x}^2$ we get the second order linear equation in $G$ $(c+x^2) G^{\\prime \\prime }(x)+3 x G^{\\prime }(x)-3 G(x)=0,$ whose linear space of real solutions around $x=0$ is generated by $G(x)=x$ and by $\\frac{1}{x^3}\\quad \\text{if }c=0,\\qquad \\frac{c+2x^2}{\\sqrt{c+x^2}}\\quad \\text{if }c>0,\\qquad \\frac{-c-2x^2}{\\sqrt{-c-x^2}}\\quad \\text{if }c<0,$ If we take any $G$ in this space, and set $h=(c+x^2)G(x)+x\\dot{x}^2$ , the triple given by equations (REF ) and (REF ) is an on-flow solution to Killing-like equation, and the associated first integral is $\\begin{split}N_3={}&f-L\\tau -\\partial _{\\dot{q}}L\\cdot \\bigl (\\xi -\\tau \\dot{q}\\bigr )=\\\\={}&y\\dot{h}-\\partial _{\\dot{q}}L\\cdot \\xi =\\\\={}&(c+x^2)G^{\\prime }(x)\\dot{x}y-(c+x^2) G(x)\\dot{y}-x\\dot{x}^2 \\dot{y}+\\dot{x}^3 y.\\end{split}$ It can be verified that the three first integrals $N_1,N_2,N_3$ are functionally independent.",
"The triple $(\\tau ,\\xi ,f)$ that we have found is not a solution in the strong sense, since $\\xi -(\\tau \\dot{q}-g^{-1}\\partial _{\\dot{q}}N_3)=(0, (c+x^2) G^{\\prime }(x)y+(3 \\dot{x} y-2 x\\dot{y})\\dot{x} )$ does not vanish identically.",
"Now that we know the expression of the first integral $N_3$ we can construct the solution triples $(\\tau ,\\Xi ,f)$ of the Killing-like equation in the strong sense: $\\tau =T,\\qquad \\Xi =\\binom{(c+x^2) G(x)+(T +x \\dot{x})\\dot{x}}{-y(c+x^2) G^{\\prime }(x)-3 \\dot{x}^2 y+2 x \\dot{x} \\dot{y}+T \\dot{y}},\\\\f=(T +2 x \\dot{x})\\dot{x} \\dot{y}-2 \\dot{x}^3 y-G(x)T y.$ Summing up, we have used the on-flow version of the Killing-like equation to detect a class of superintegrable systems: Proposition 3 The Lagrangian system given by equations (REF ) and (REF ) is superintegrable whenever the function $G$ satisfies equation (REF ).",
"This class of systems with the parameter $c>0$ overlaps with the one that was found by the second author using a totally different line of reasoning.",
"The on-flow method that we have illustrated here was pushed further (albeit with a different language) by the two authors , using with the more general ansatz $h=\\alpha (x)+\\beta (x)\\dot{x}^2+\\gamma (x)\\dot{x}^4$ .",
"We expect that a larger class of superintegrable systems can readily be found by increasing the degree of $h$ with respect to $\\dot{x}$ ."
],
[
"The Laplace-Runge-Lenz vector for Kepler's problem",
"Consider the Lagrangian function and Lagrange equation of Kepler's problem in dimension 3 $L(t,\\vec{r}, \\vec{v})=\\frac{1}{2} \\Vert \\vec{v}\\Vert ^2+\\frac{\\mu }{\\Vert \\vec{r}\\Vert },\\quad \\vec{r}\\in \\mathbb {R}^3\\setminus \\lbrace 0\\rbrace ,\\\\\\ddot{\\vec{r}}=-\\frac{\\mu }{\\Vert \\vec{r}\\Vert ^3}\\,\\vec{r}\\,.$ Here we depart from the $q,\\dot{q}$ notation and use $\\vec{r},\\vec{v}$ instead, as done in common introductory mechanics textbooks.",
"The vector product “$\\times $ ” for 3-dimensional vectors will allow more compact formulas than what we get in the otherwise equivalent 2-dimensional treatment we gave in an earlier paper .",
"Besides energy and angular momentum, the Kepler system has the LRL vector first integral $\\vec{A}:= \\vec{v}\\times (\\vec{r}\\times \\vec{v})-\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\,\\vec{r}.$ Fix an arbitrary vector $\\vec{u}\\in \\mathbb {R}^3$ and consider the scalar first integral $N:=-\\vec{u}\\cdot \\vec{A}.$ If we check the condition of Proposition REF we see that $N$ cannot be obtained from a triple $(\\tau ,\\xi ,f)$ which is independent of $\\vec{v}$ .",
"Let us see what we can do with either on-flow or strong solutions involving $\\vec{v}$ .",
"Theorem REF gives us so many on-flow solutions that we may be choosy and aim for subjectively simple, elegant formulas.",
"One that is simple enough is $\\tau _0=\\frac{\\vec{u}\\cdot \\vec{v}\\times (\\vec{r}\\times \\vec{v})}{L},\\qquad {\\vec{\\xi }}_0=\\frac{\\vec{u}\\cdot \\vec{v}\\times (\\vec{r}\\times \\vec{v})}{L}\\vec{v},\\qquad f_0=\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{r}\\cdot \\vec{u}.$ Levy-Leblond proposed the following one, without explanation as to how he came up with the formula: $\\tau _L=0,\\qquad {\\vec{\\xi }}_L=-\\frac{1}{2}\\partial _{\\vec{v}}N=(\\vec{r}\\cdot \\vec{u})\\vec{v}-\\frac{1}{2}(\\vec{v}\\cdot \\vec{u})\\vec{r}-\\frac{1}{2}(\\vec{v}\\cdot \\vec{r})\\vec{u},\\\\f_L=\\tau _L L-N+\\partial _{\\dot{q}}L\\cdot (\\xi _L-\\tau _L\\dot{q})=\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{r}\\cdot \\vec{u}=f_0,$ To find different solutions with trivial first order time variation $\\tau \\equiv 0$ , we write the Killing-type equation (REF ) within the current setting: $\\partial _{\\vec{r}} L\\cdot \\vec{\\xi }+\\partial _{\\vec{v}}L\\cdot \\dot{\\vec{\\xi }}=\\dot{f},$ and we impose that the first order space variation $\\vec{\\xi }$ be such that $\\partial _{\\vec{v}}L\\cdot \\vec{\\xi }=\\vec{u}\\cdot \\vec{v}\\times (\\vec{r}\\times \\vec{v}).$ Our favourite way to satisfy this condition is ${\\vec{\\xi }}_Z=(\\vec{r}\\times \\vec{v})\\times \\vec{u}=(\\vec{r}\\cdot \\vec{u})\\vec{v}-(\\vec{v}\\cdot \\vec{u})\\vec{r}.$ This choice is not the only possible: $\\vec{\\xi }_L$ satisfies the same condition: $\\partial _{\\vec{v}}L\\cdot \\vec{\\xi }_{L}=\\vec{v}\\cdot \\vec{\\xi }_{L}=\\vec{u}\\cdot \\vec{v}\\times (\\vec{r}\\times \\vec{v})=\\Vert \\vec{v}\\Vert ^2\\vec{u}\\cdot \\vec{r}-(\\vec{v}\\cdot \\vec{r})(\\vec{v}\\cdot \\vec{u}).$ Using Lagrange equation () we have ${\\dot{\\vec{\\xi }}}_Z=\\bigl (\\partial _{\\vec{r}}{\\vec{\\xi }}_Z\\bigr ) \\vec{v}+\\partial _{\\vec{v}}{\\vec{\\xi }}_Z\\,\\Bigl (-\\frac{\\mu }{\\Vert \\vec{r}\\Vert ^3}\\, \\vec{r}\\Bigr )=\\vec{0}.$ Using equation (REF ), the left-hand side of equation (REF ) becomes $\\partial _{\\vec{r}} L\\cdot {\\vec{\\xi }}_Z+\\partial _{\\vec{}v}L\\cdot {\\dot{\\vec{\\xi }}}_Z={}&\\partial _{\\vec{r}} L\\cdot {\\vec{\\xi }}_Z+\\partial _{\\vec{v}}L\\cdot \\vec{0}=-\\frac{\\mu }{\\Vert \\vec{r}\\Vert ^3}\\vec{r}\\cdot (\\vec{r}\\times \\vec{v})\\times \\vec{u}=\\\\={}&-\\frac{\\mu }{\\Vert \\vec{r}\\Vert ^3}\\vec{r}\\times (\\vec{r}\\times \\vec{v})\\cdot \\vec{u}=\\\\={}&-\\frac{\\mu }{\\Vert \\vec{r}\\Vert ^3}\\,\\vec{u}\\cdot \\bigl (\\vec{r}(\\vec{r}\\cdot \\vec{v})-\\vec{v}\\Vert \\vec{r}\\Vert ^2\\bigr )=\\vec{v}\\cdot \\partial _{\\vec{r}}\\Bigl (\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{r}\\cdot \\vec{u}\\Bigr )=\\\\={}&\\frac{d}{dt}\\Bigl (\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{r}\\cdot \\vec{u}\\Bigr )=\\dot{f}_0.$ We can complete the solution triple as follows: $\\tau _Z\\equiv 0,\\qquad {\\vec{\\xi }}_Z=(\\vec{r}\\times \\vec{v})\\times \\vec{u},\\qquad f_Z=f_0=\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\,\\vec{r}\\cdot \\vec{u}.$ The resulting first integral of formula (REF ) is what we expected: $f_Z-\\partial _{\\dot{\\vec{}}v}L\\cdot {\\vec{\\xi }}_Z={}&\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{r}\\cdot \\vec{u}-\\vec{v}\\cdot (\\vec{r}\\times \\vec{v})\\times \\vec{u}=\\\\={}&-\\Bigl (\\vec{v}\\times (\\vec{r}\\times \\vec{v})-\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{r}\\Bigr )\\cdot \\vec{u}=N.$ The triple (REF ) belongs to the family of Theorem REF , as can be checked by direct computation.",
"Corollary REF , formula (REF ), applied to the triple (REF ), gives a whole family of solution triples, depending on an arbitrary function $h(t,\\vec{r},\\vec{v})$ : $\\tau =\\frac{1}{L}\\Bigl (h-\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{u}\\cdot \\vec{r}\\Bigr ),\\quad \\vec{\\Xi }= (\\vec{r}\\times \\vec{v})\\times \\vec{u}+\\frac{1}{L}\\Bigl (h-\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\vec{u}\\cdot \\vec{r}\\Bigr )\\vec{v},\\quad f=h.$ As in Corollary REF , the choice $h\\equiv 0$ will trivialize the boundary term.",
"Using a computer algebra system the reader can directly check all these solutions to the Killing-type equation, independently of the theorems in Section .",
"For example here is some simple code written for Wolfram Mathematica that implements the solution triple (REF ) and then checks that the on-flow Killing-type equation is satisfied and that the first integral is the LRL vector: (*Defining the variables*) r = {r1, r2, r3}; v = {v1, v2, v3}; L = v.v/2 + mu/Sqrt[r.r]; A = Cross[v, Cross[r, v]] - mu*r/Sqrt[r.r]; u = {u1, u2, u3}; arbitrary = h[t, r1, r2, r3, v1, v2, v3]; f0 = mu*(u.r)/Sqrt[r.r]; T = (arbitrary - f0)/L; Xi = Cross[Cross[r, v], u] + T*v; f = arbitrary; (*the time dot derivatives are on-flow*) rDotDot = -mu*(u.r)*r/(r.r)^(3/2); Tdot = D[T, t] + D[T, {r}].v + D[T, {v}].rDotDot; XiDot = D[Xi, t] + D[Xi, {r}].v + D[Xi, {v}].rDotDot; fDot = D[f, t] + D[f, {r}].v + D[f, {v}].rDotDot; (*checking on-flow Killing-type equation*) Simplify[ D[L, t]*T + D[L, {r}].Xi + D[L, {v}].",
"(XiDot - v*Tdot) + L*Tdot == fDot] (*checking LRL vector as first integral*) Simplify[ f - L*T - D[L, {v}].",
"(Xi - T*v) == -A.u] Upon evaluation, the code gives True and True in an instant.",
"Solutions of the Killing-type equation in the strong sense are fewer, and there is less freedom to simplify formulas.",
"The explicit triples given by Sarlet and Cantrijn , and by the authors are for dimension 2.",
"Boccaletti an Pucacco deduce their solution, also in dimension 2, by assuming $\\tau \\equiv 0$ and $\\xi $ to be a bilinear function of $q,\\dot{q}$ and then working out the coefficients.",
"Here we contribute a triple written for dimension 3, where the vector cross product again leads to elegant formulas for the solution $(\\tau ,\\Xi ,f)$ , and where we incorporate the multiplicity Corollary REF : $\\vec{b}:=-\\vec{u} \\bigl (\\vec{r}\\cdot \\vec{v}\\bigr )-\\vec{r}\\bigl (\\vec{v}\\cdot \\vec{u}\\bigr )+\\vec{v}\\bigl (\\vec{u}\\cdot \\vec{r}\\bigr ),\\\\\\tau =\\frac{1}{L}\\biggl (h-\\vec{u}\\cdot \\Bigl (\\vec{v}\\times \\bigl (\\vec{r}\\times \\vec{v}\\bigr )+\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\, \\vec{r}\\Bigr )\\biggr ),\\\\\\vec{\\Xi }=\\frac{1}{L}\\Bigl (h\\,\\vec{v}+\\frac{1}{2}\\vec{v}\\times \\bigl (\\vec{b}\\times \\vec{v}\\bigr )+\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\,\\vec{b}\\Bigr ),\\qquad f=h,$ where $\\vec{u}\\in \\mathbb {R}^3$ is an arbitrary parameter vector, as in the previous section.",
"The first integral associated to the triple through Noether's theorem (REF ) is the same as before: $L\\tau +\\partial _{\\vec{v}}L\\cdot \\bigl (\\vec{\\Xi }-\\tau \\vec{v}\\bigr )=-\\vec{u}\\cdot \\Bigl (\\vec{v}\\times \\bigl (\\vec{r}\\times \\vec{v}\\bigr )-\\frac{\\mu }{\\Vert \\vec{r}\\Vert }\\,\\vec{r}\\Bigr )=-\\vec{u}\\cdot \\vec{A}.$ Again we provide below some Mathematica code that implements the solution triple and checks that it solves the Killing-type equation in the strong sense, and that it gives the Laplace-Runge-Lenz first integral.",
"(*Defining the variables*) r = {r1, r2, r3}; v = {v1, v2, v3}; L = v.v/2 + mu/Sqrt[r.r]; A = Cross[v, Cross[r, v]] - mu*r/Sqrt[r.r]; u = {u1, u2, u3}; arbitrary = h[t, r1, r2, r3, v1, v2, v3]; T = (arbitrary - u.",
"(Cross[v, Cross[r, v]] + mu*r/Sqrt[r.r]))/L; b = -u*(r.v) - r*(u.v) + v*(r.u); Xi = (arbitrary*v + Cross[v, Cross[b, v]]/2 + mu*b/Sqrt[r.r])/L; f = arbitrary; (*the time dot derivatives are generic, not on-flow*) rDotDot = {a1, a2, a3}; Tdot = D[T, t] + D[T, {r}].v + D[T, {v}].rDotDot; XiDot = D[Xi, t] + D[Xi, {r}].v + D[Xi, {v}].rDotDot; fDot = D[f, t] + D[f, {r}].v + D[f, {v}].rDotDot; (*checking Killing-type equation*) Simplify[ D[L, t]*T + D[L, {r}].Xi + D[L, {v}].",
"(XiDot - v*Tdot) + L*Tdot == fDot] (*checking LRL vector as first integral*) Simplify[ f - L*T - D[L, {v}].",
"(Xi - T*v) == -A.u]\\end{verbatim} \\noindent The evaluation gives \\verb!True!, as expected.",
"%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\section{Killing-like equation for a different time\\\\ change} \\label{differentTimeChangeSection} Infinitesimal invariance up to boundary terms usually refers to the dependence of the quantity \\begin{equation}\\label{actionWithInverseTimeChange} \\int_{\\bar t_1}^{\\bar t_2}L\\Bigl(\\bar t,\\bar q(\\bar t), \\frac{d\\bar q}{d\\bar t}(\\bar t)\\Bigr)d\\bar t \\end{equation} with respect to $\\varepsilon$, as in equation~\\eqref{infinitesimalInvariance} of Section~\\ref{introduction}, where again \\begin{equation}\\label{timeAndSpaceChange} \\bar t_\\varepsilon(t)=t+\\varepsilon \\tau\\bigl(t,q(t),\\dot q(t)\\bigr),\\qquad \\bar q_\\varepsilon(t)=q+\\varepsilon \\xi\\bigl(t,q(t),\\dot q(t)\\bigr).",
"\\end{equation} In an earlier paper on Noether's theorem~\\cite{GorniZampieri} we proposed, among other things, a generalization of infinitesimal invariance that leads to nonlocal constants of motion, and also, more to the point here, that infinitesimal invariance with time change can be based on the following expression \\begin{equation}\\label{actionWithNewTimeChange} \\int_{\\bar t_\\varepsilon(t_1)}^{ \\bar t_\\varepsilon(t_2)}L\\Bigl(t,\\bar q_\\varepsilon(t), \\frac{d\\bar q_\\varepsilon}{dt}(t)\\Bigr)dt.",
"\\end{equation} instead of~\\eqref{actionWithInverseTimeChange} \\cite[Sec.~4]{GorniZampieri}. What is different is that the time derivative of $\\bar q_\\varepsilon$ and the integration in~\\eqref{actionWithNewTimeChange} is made by respect to the original time~$t$, whilst in Section~\\ref{introduction} the derivative was made with respect to the transformed time~$\\bar t$.",
"We will say that the transformation~\\eqref{timeAndSpaceChange} leaves the action integral \\emph{alternatively}-invariant up to boundary terms if a function $f(t,q,\\dot q)$ exists, such that for all $t_1,t_2$ we have \\begin{multline}\\label{infinitesimalInvarianceAlternative} \\int_{\\bar t_\\varepsilon(t_1)}^{ \\bar t_\\varepsilon(t_2)}L\\Bigl(t,\\bar q_\\varepsilon(t), \\frac{d\\bar q_\\varepsilon}{dt}(t)\\Bigr)dt+{}\\\\ +\\varepsilon\\int_{t_1}^{t_2} \\frac{df}{dt}\\bigl(t,q(t),\\dot q(t)\\bigr)dt +O(\\varepsilon^2) \\quad\\text{as }\\varepsilon\\to0.",
"\\end{multline} To translate this condition into a differential equation, we take the integral \\begin{equation} \\int_{\\bar t_\\varepsilon(t_1)}^{ \\bar t_\\varepsilon(t_2)}L\\Bigl(t,\\bar q_\\varepsilon(t), \\frac{d\\bar q_\\varepsilon}{dt}(t)\\Bigr)dt \\end{equation} and replace the $t$ variable with $\\bar t_\\varepsilon(t)$.",
"The integral becomes with fixed extrema $t_1,t_2$: \\begin{equation} \\int_{t_1}^{t_2}L\\bigl(\\bar t_\\varepsilon(t), \\bar q_\\varepsilon(\\bar t_\\varepsilon(t)), \\dot{\\bar q}(\\bar t_\\varepsilon(t))\\bigr) \\dot{\\bar t}_\\varepsilon(t)\\,dt.",
"\\end{equation} The first-order expansion of this expression as $\\varepsilon\\to0$ is \\begin{multline} \\int_{t_1}^{t_2}L\\bigl(t,q(t),\\dot q(t)\\bigr)dt+{}\\\\ +\\varepsilon\\int_{t_1}^{t_2} \\bigl(\\tau\\partial_t L +\\partial_q L\\cdot (\\xi+\\tau\\dot q) +\\partial_{\\dot q}L\\cdot(\\dot\\xi+\\tau\\ddot q) +L\\dot\\tau\\bigr)dt +O(\\varepsilon^2).",
"\\end{multline} Hence the alternative invariance~\\eqref{infinitesimalInvarianceAlternative} is equivalent to the following \\emph{alternative} Killing-type equation for ODEs: \\begin{equation}\\label{Killing-typeAlternative} \\tau\\partial_t L +\\partial_q L\\cdot (\\xi+\\tau\\dot q) +\\partial_{\\dot q}L\\cdot(\\dot\\xi+\\tau\\ddot q) +L\\dot\\tau= \\dot f. \\end{equation} The commonly made assumption that $\\tau,\\xi,f$ only depend on~$(t,q)$ eliminated~$\\ddot q$ for the standard Killing-type equation~\\eqref{Killing-type}, collapsing the on-flow and the strong interpretation. The alternative equation~\\eqref{Killing-typeAlternative} does not lend itself to this simplification, so that we are forced to take position on how to understand~$\\ddot q$: either as an independent $n$-dimensional variable (strong form), or as a shorthand for the $\\Lambda(t,q,\\dot q)$ of the Lagrange equation~\\eqref{normalformLagrange} (on-flow form).",
"If a triple $(\\tau,\\xi,f)$ solves the alternative equation~\\eqref{Killing-typeAlternative} in either sense, then the following function is a first integral for the Lagrangian system: \\begin{equation}\\label{firstintegralAlternative} N=f-L\\tau-\\partial_{\\dot q}L\\cdot\\xi.",
"\\end{equation} The expression is different from the corresponding formula~\\eqref{firstintegral} for the standard Killing-like equation. Compare however our formula~\\eqref{firstintegralAlternative} with formula~(I.11) from Candotti, Palmieri and Vitale~\\cite{Candotti}.",
"There is a simple correspondence between the solutions to the standard and the alternative Killing-type equations, either in the strong or in the on-flow interpretation: if $(\\tau,\\xi,f)$ solves the alternative version~\\eqref{Killing-typeAlternative} then $(\\tau,\\xi+\\tau\\dot q,f)$ solves the standard~\\eqref{Killing-type}. Conversely, if $(\\tau,\\xi,f)$ solves the standard~\\eqref{Killing-type} then $(\\tau,\\xi-\\tau\\dot q,f)$ solves the alternative~\\eqref{Killing-typeAlternative}. This is basically Theorem~8 of the previous paper~\\cite{GorniZampieri}, except that the ``alternative'' tag is used in the opposite sense.",
"Given a first integral~$N$, the associated general solution for the strong interpretation of the standard Killing-type equation are given by Theorem~\\ref{familyOfSolutionsWithoutLagrange}.",
"The equivalent statement for the solutions to the alternative Killing-type equation is simply obtained by replacing equations~(\\ref{strongSolutionCondition}--\\ref{strongSolutionGauge}) with \\begin{gather}\\label{strongSolutionConditionAlternative} \\xi= -g^{-1}\\partial_{\\dot q}N,\\\\ f=\\tau L+N-\\partial_{\\dot q}L\\cdot g^{-1}\\partial_{\\dot q}N. \\label{strongSolutionGaugeAlternative} \\end{gather} If we choose $\\tau$ so as to get trivial $f\\equiv0$, we obtain the alternative strong solution \\begin{equation} \\label{strongSolutionConditionAlternativeWithTrivialGauge} \\tau=-\\frac{1}{L}\\Bigl(N- \\partial_{\\dot q}L\\cdot g^{-1}\\partial_{\\dot q}N\\Bigr),\\qquad \\xi= -g^{-1}\\partial_{\\dot q}N\\qquad f=0.",
"\\end{equation} Compare with equation~(I.17) from Candotti, Palmieri and Vitale~\\cite{Candotti}.",
"%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\begin{thebibliography}{10} \\frenchspacing \\bibitem{Boccaletti} Boccaletti, D., Pucacco, G.: Theory of Orbits 1: Integrable systems and non-perturbative methods.",
"Springer (1996).",
"\\bibitem{BoccalettiPucacco} Boccaletti, D., Pucacco, G.: Killing equations in classical mechanics. Il Nuovo Cimento~\\textbf{112 B}, No.~2--3, 181--212 (1997).",
"\\bibitem{Candotti} Candotti, E., Palmieri, C., Vitale B.: On the Inversion of Noether's Theorem in Classical Dynamical Systems.",
"Am. J.",
"Phys.~\\textbf{40}, 424--429 (1972) \\bibitem{Djukic} Djukic, Dj. S.",
": A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians.",
"Internat. J. Non-Linear Mech.~\\textbf{8}, 479--488 (1973).",
"\\bibitem{GorniZampieri} Gorni, G., Zampieri, G.: Revisiting Noether's theorem on constants of motion.",
"Journal of Nonlinear Mathematical Physics~\\textbf{21}, No.~1, 43--73 (2014).",
"\\bibitem{Kobussen} Kobussen, J.A.: On a systematic search for integrals of motion.",
"Helv. Phys.",
"Acta~\\textbf{53}, 183--200 (1980).",
"\\bibitem{Leach2} Leach P.G.L.: Lie symmetries and Noether symmetries. Applicable Analysis and Discrete Mathematics~\\textbf{6}, 238--246 (2012).",
"\\bibitem{Leblond} L\\'evy-Leblond, J.: Conservation laws for gauge-variant Lagrangians in classical mechanics.",
"Am. J. Phys.~\\textbf{39}, 502--506 (1971).",
"\\bibitem{Lutzky} M. Lutzky, Dynamical symmetries and conserved quantities. J.~Phys. A \\textbf{2}, 973--981 (1979).",
"\\bibitem{SarletCantrijn} Sarlet, W., Cantrijn, F.: Generalizations of Noether's theorem in classical mechanics.",
"SIAM Review~\\textbf{23}, No.~4, 467--494 (1981).",
"\\bibitem{Zampieri} Zampieri, G.: Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony. \\emph{Comm. Math. Phys.}~\\textbf{303}, 73--87 (2011).",
"\\end{thebibliography} \\end{document} %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%"
]
] | 1403.0506 |
[
[
"Magnons and a two-component spin gap in FeV2O4"
],
[
"Abstract The spinel vanadates have become a model family for exploring orbital order on the frustrated pyrochlore lattice, and recent debate has focused on the symmetry of local crystal fields at the cation sites.",
"Here, we present neutron scattering measurements of the magnetic excitation spectrum in $\\mathrm{FeV_2O_4}$, a recent example of a ferrimagnetic spinel vanadate which is available in single crystal form.",
"We report the existence of two emergent magnon modes at low temperatures, which draw strong parallels with the closely related material, $\\mathrm{MnV_2O_4}$.",
"We were able to reproduce the essential elements of both the magnetic ordering pattern and the dispersion of the inelastic modes with semi- classical spin wave calculations, using a minimal model that implies a sizeable single-ion anisotropy on the vanadium sublattice.",
"Taking into account the direction of ordered spins, we associate this anisotropy with the large trigonal distortion of $\\mathrm{VO_6}$ octahedra, previously observed via neutron powder diffraction measurements.",
"We further report on the spin gap, which is an order-of-magnitude larger than that observed in $\\mathrm{MnV_2O_4}$.",
"By looking at the overall temperature dependence, we were able to show that the gap magnitude is largely associated with the ferro-orbital order known to exist on the iron sublattice, but the contribution to the gap from the vanadium sublattice is in fact comparable to what is reported in the Mn compound.",
"This reinforces the conclusion that the spin canting transition is associated with the ordering of vanadium orbitals in this system, and closer analysis indicates closely related physics underlying orbital transitions in $\\mathrm{FeV_2O_4}$ and $\\mathrm{MnV_2O_4}$."
],
[
"Introduction",
"The spinel vanadates ($\\mathrm {AV_2O_4}$ ) are an important model family for the study of orbital order, and notable for the frustrated pyrochlore network of spin and orbitally-active vanadium cations therein[1].",
"The major outstanding question in the study of these compounds is the nature of the orbital ordered state at low temperatures, and scattering studies are playing an important role in determining the relative importance of sub-dominate interaction and crystal field terms in the magnetic Hamiltonian.",
"Here, we present inelastic neutron scattering data on $\\mathrm {FeV_2O_4}$ which contribute to this important conversation.",
"$\\mathrm {FeV_2O_4}$ is a ferrimagnetic spinel, characterized by two orbitally active cation sites, and shown by previous diffraction studies to have three structural and two magnetic transitions[2], [3], [4].",
"We report on the existence of two inelastic spin-wave modes in the low temperature magnetic phases whose dispersions are well described by a model that assumes significant trigonal crystal fields at the vanadium site.",
"Our data further show that the spin gap has a temperature dependence that reflects both magnetic transitions, and suggests that a full description of magnetism in this material requires consideration of the orbital order which exists on both cation sites.",
"Spinel vanadates with divalent transition metal atoms on the A-site are Mott insulators, and the octahedrally coordinated vanadium ($V^{3+}$ ) cations on the pyrochlore sublattice have S=1 spin and orbital triplet degrees-of-freedom in the ideal cubic phase.",
"Nearly without exception, each exhibits a cubic-to-tetragonal structural transition with decreasing temperature, which partially lifts the orbital degeneracy and alters magnetic properties.",
"In materials with non-magnetic A-site cations ($A^{2+} \\in (Zn^{2+}, Cd^{2+}, Mg^{2+})$ ), the cubic-tetragonal transition precedes the onset of $\\mathbf {Q}= 2\\pi (0,0,1)$ antiferromagnetic order at lower temperature[1], [5].",
"The ferrimagnetic materials $\\mathrm {MnV_2O_4}$ and $\\mathrm {FeV_2O_4}$ are observed to have successive collinear Neel and canted antiferromagnetic transitions[6], [7], [8], [9], [3], where the spin canting is coincident with the onset of the low-temperature tetragonal structure.",
"Though generally associated with orbital order, the nature of these structural transitions and the exact configuration of electron orbitals at low temperatures are issues of active interest.",
"Discussions of orbital order have largely focused on the relative importance of two proposed patterns: a form of antiferro-orbital order (AFO) containing an alternating pattern of real orbitals in the $ab$ -plane [10], [11], and a form of ferro-orbital order (FOO) where each site is occupied by a complex superposition of orbitals[12].",
"The two patterns are associated with slightly different tetragonal space groups, with the FOO pattern containing an additional glide plane symmetry (space group $I4_1/amd$ , rather than $I4_1/a$ ), but both were derived assuming a tetragonally distorted cubic crystal field environment.",
"Recent developments, however, have indicated that a full understanding of the orbital ground state might require a proper treatment of the trigonal crystal fields which exist in the spinel structure when the fractional coordinate, $x$ , deviates from its ideal value of 0.25[13].",
"Trigonal fields have been invoked to explain neutron scattering data in $\\mathrm {MgV_2O_4}$[5] and the existence of a high field transition in $\\mathrm {CdV_2O_4}$[14].",
"First-principles calculations have indicated that trigonal crystal fields play a defining role for the low-temperature states of $\\mathrm {MnV_2O_4}$[15] and $\\mathrm {FeV_2O_4}$[16].",
"The “quantum $120^{\\circ }$ model” of Chern $\\textit {et al.",
"}$[17] was able to explain the experimental picture surrounding $\\mathrm {MnV_2O_4}$ by assuming that the trigonal crystal fields were a dominant contribution to the magnetic Hamiltonian.",
"Our own neutron powder diffraction (NPD) work subsequently demonstrated that the observed magnetic ground state of $\\mathrm {FeV_2O_4}$ is consistent with the predictions of the quantum $120^{\\circ }$ model in the strong spin-orbit coupling limit[3].",
"The current study follows up on our initial neutron powder diffraction study of $\\mathrm {FeV_2O_4}$ , and presents inelastic neutron scattering data on single crystals.",
"$\\mathrm {FeV_2O_4}$ is quite unique among the spinel vanadates, in that it has an orbital doublet degree-of-freedom on the A-site $Fe^{2+}$ cation, in addition to the orbitally active vanadium cation on the spinel `B'-site.",
"It is observed to have three distinct structural transitions, evolving from high-temperature cubic (HTC) to high-temperature tetragonal (HTT) to face-centered orthorhombic (FCO) to higher symmetry low-temperature tetragonal (LTT) structure with decreasing temperature[2], [3], [4].",
"The HTT-FCO and FCO-LTT structural transitions are further associated with the onset of collinear and canted spin structures, respectively, involving both of the cation sites[3], [18].",
"The lowest temperature canted phase has additionally been shown to exhibit a net ferroelectric moment, which can be manipulated with moderate applied magnetic field[19], [20].",
"The physics underlying these transitions has been argued mostly from analogy to other spinel systems.",
"The highest temperature structural transition is thought to be the result of a Jahn-Teller transition and the onset of ferro-orbital order on the $Fe^{2+}$ site, similar to $\\mathrm {FeCr_2O_4}$[21], [22], [23], [24] and consistent with the distortion of the local $FeO_4$ tetrahedra[2], [3], [4].",
"The HTT-FCO transition is thought to be driven by antiferromagnetic exchange between the two cation sites, consistent with spin-only chromates[23] and $\\mathrm {MnV_2O_4}$[8].",
"In analogy to $\\mathrm {MnV_2O_4}$ , we have also argued that the lowest transition can be understood to result from the onset of orbital order on the vanadium sublattice[3] and pointed to the predictions of the quantum 120$^\\circ $ model[17].",
"In this article, we further this discussion by presenting neutron scattering measurements of spin-wave spectra in a large single crystal of $\\mathrm {FeV_2O_4}$ .",
"Two low energy modes are identified, with strong parallels to observed modes in $\\mathrm {MnV_2O_4}$[8], [9], but fit to a model which more appropriately includes a local $< 111 >$ single-ion anisotropy to encompass trigonal fields on the vanadium sites.",
"Using these fits, we argue that both [001] anisotropy on the iron site and the $<111>$ anisotropy on the vanadium sites are essential for a proper description of the low-temperature physics.",
"Implications for the low temperature orbital order and the parallels to $\\mathrm {MnV_2O_4}$ are discussed."
],
[
"Experimental Methods",
"Single crystals of $\\mathrm {FeV_2O_4}$ were grown by the float zone method, as described elsewhere[3].",
"Crystals were characterized first by heat capacity, using a Quantum Design Physical Property Measurement System at the National High Magnetic Field Laboratory in Tallahassee.",
"The resulting data were previously published in the Supplementary Materials of Ref.",
"macdougall12, and are presented again in Figure REF .",
"Bulk magnetization was measured using a Vibrating Sample Magnetometer at the University of Illinois at Urbana-Champaign, with applied fields of H = 500Oe along the cubic (001) direction, and plotted with the heat capacity data for direct comparison.",
"Neutron scattering experiments were performed using instruments at both the Spallation Neutron Source (SNS) and the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory.",
"Spin waves were first measured using the SEQUOIA Fine Resolution Chopper Spectrometer at the SNS[25], [26].",
"The majority of measurements used $E_i$ =55meV neutrons and the coarse chopper on SEQUOIA[26], giving energy resolution of approximately 3 meV at the elastic line.",
"One 3g crystal was mounted in a closed cycle refrigerator with the pseudo-cubic (HK0) plane horizontal, and spectra at three different temperatures (T = 4K, 85K and 120K) were built from measurements taken at 0.5$^\\circ $ steps over a 100$^\\circ $ range using the Mantid framework[27].",
"Background scattering from the cryostat and sample can were measured separately and subtracted from the data.",
"Plots of SEQUOIA data were made using the Horace software package[28].",
"Further measurements to explore the temperature dependence of the spin gap were performed at the HFIR, using the HB3 and CTAX triple-axis (TA) spectrometers.",
"Both sets of measurements made use of the same crystal explored with SEQUOIA, oriented in the same scattering plane.",
"For the lowest temperatures, the large energy transfers involved demanded the use of thermal neutrons.",
"The HB3 spectrometer was employed, with a PG002 monochromator and analyzer, 48$^{\\prime }$ -40$^{\\prime }$ -40$^{\\prime }$ -120$^{\\prime }$ collimation and $E_f$ =14.7meV neutrons.",
"Higher order contamination was removed with two PG filters.",
"Finer resolution measurements were performed at temperatures near the upper magnetic transitions, using the CTAX cold TA spectrometer, with guide-open-80$^{\\prime }$ -open collimation.",
"The energy of the scattered neutrons was fixed at $E_f$ = 5 meV.",
"Higher order contamination was removed by a cooled Be filter placed between the sample and analyzer."
],
[
"Results and Discussion",
"Characterization and elastic neutron scattering data shown in Fig.",
"REF largely confirm the temperature and nature of the phase transitions reported by our previous NPD work[3].",
"Peaks in the heat capacity data (Fig.",
"REF (a)) identify phase transitions at 138 K, 107 K and 60 K, and can be immediately associated with previously identified Jahn-Teller ($T_c$ ), the collinear N$\\mathrm {\\acute{e}}$ el ($T_{N1}$ ) and the spin canting ($T_{N2}$ ) transitions, respectively.",
"Magnetization versus temperature data for the current single crystal sample are plotted in the same panel, measured in both field-cooled (solid line) and zero-field-cooled (dashed line) configurations.",
"As with measurements on powders[3], [19], the field-cooled curve reveals an increase in net magnetization at the 107 K and 60 K transitions, reflecting the onset of collinear and canted ferrimagnetism.",
"The divergence of ZFC and FC lines below $T_{N1}$ can be associated with ferrimagnetic domain formation.",
"Fig.",
"REF (b) shows scans of elastic neutron scattering intensity across the cubic (400) Bragg position.",
"Temperatures are representative of the four distinct structural phases, identified above, and the peak profiles exhibit an evolution in-line with the known cubic-HTT-FCO-LTT sequence of transitions.",
"The single cubic peak at T = 200 K splits at 120 K, reflecting the cubic-tetragonal transition at $T_{c}$ = 140 K. The scan at T = 90 K shows a shift in scattering intensity from the high-angle to low angle peak, with the latter peak broader than resolution and best described by two Gaussians, as expected for the FCO phase.",
"The T = 5 K scan is again described by two peaks, with intensity distributed between them with opposite sense to what is observed at T = 120 K. As with our NPD study[3], and confirmed by a subsequent single-crystal x-ray study[4], there is no indication of further structural transitions below $T_{N2}$ = 60 K. The existence of a magnetic transition at $T_{N1}$ = 110K is confirmed by the emergence of spin wave excitations and the temperature dependence of the spin gap, as laid out below.",
"Fig.",
"REF (c) further shows temperature dependence of elastic scattering at the (220) Bragg position, as determined by scans of neutron energy with constant-Q (e.g.",
"Inset, Fig.",
"REF (a)).",
"The intensity of this peak reflects the ordered moment on the iron sublattice, and serves as an order parameter for the Neel antiferromagnetic state.",
"The intensity of the Bragg peak at the (200) position was measured via radial scans in the elastic channel, and is plotted in Fig.",
"REF (d).",
"The (200) Bragg peak reflects a breaking of a local glide plane symmetry preserved in the collinear spin state, and its intensity acts as an order parameter for spin canting.",
"Data in Fig.",
"REF (c) and (d) were fitted to the function $I(T) = I_0*(1-\\frac{T}{T_N})^{2\\beta } + const,$ in the temperature range $\\frac{T}{T_N} > 0.75$ , to extract approximate values for critical temperatures and exponents.",
"The extracted critical exponents, $\\beta _1 = 0.16 \\pm 0.06$ and $\\beta _2 = 0.32 \\pm 0.03$ , are broadly consistent with Ising transitions in two and three dimensions, respectively, though more detailed measurements would be required to comment further.",
"The fitted transition temperatures, $T_{N1} = 107.5 K \\pm 0.1 K$ and $T_{N2} = 61.1 K \\pm 0.5 K$ , in-line with our earlier estimates from heat capacity and NPD.",
"A new feature revealed by the single-crystal measurements is the significant elastic intensity about the (200) position which is seen to persist to temperatures well above $T_{N2}$ , and decrease monotonically with warming.",
"In this temperature region, the (200) peaks are broader in Q than instrument resolution and can be associated with the existence of short-range spin canting correlations within the collinear Neel state.",
"It is important to note that there is no change in the (400) Bragg intensity over the same temperature range, and so this effect cannot be simply associated with scattering from this structural peak with imperfectly filtered $\\lambda /2$ neutrons.",
"Spin canting is symmetrically allowed in the Fddd spacegroup of the FCO phase, and the observed short range correlations above the ordering transition is similar to what is seen in other frustrated geometries.",
"Figure: (a)Heat capacity and magnetization data from the single crystal sample from the current neutron scattering study.",
"Peaks in heat capacity identify transitions at 138 K, 107 K, and 60 K. Magnetization was measured under field-cooled (upper/blue line) and zero-field-cooled (lower/red line) conditions, with field oriented along the cubic (001) direction.",
"(b) Variation of the structural (400) Bragg peak, as measured by elastic neutron scattering.",
"Temperatures were chosen in each of the previously identified phases.",
"Solid lines are fits to Gaussian curves.",
"(c) Elastic scattering intensity of the cubic (220) position, reflecting the square magnitude of ordered moments on the Fe 2+ Fe^{2+} sublattice and (d) (200), reflecting the canting component of spins on the V 3+ V^{3+} sublattice.",
"Data in panels (c) and (d) were measured using the HB3 and CTAX triple-axis spectrometers, respectively.",
"Solid lines represent fits to Eq.",
"and are discussed in the main text.Main inelastic results from the SEQUOIA chopper spectrometer are summarized in Figures REF and REF .",
"At the lowest temperatures, we observe two distinct dispersive modes with energies below 25 meV, and interpret them as magnons of the ferrimagnetic ordered state.",
"The dispersion of these modes is plotted along five different symmetry directions in Fig.",
"REF (a)-(d) and Fig.",
"REF (a).",
"The dominant mode is roughly twenty times as intense as the “weak” mode, and has shape and bandwidth reminiscent of the lowest energy mode reported by Chung et al.",
"for $\\mathrm {MnV_2O_4}$[9].",
"The minima coincide with the zone-centers of the diamond sublattice, consistent with the previous claim that this excitation is associated with the motion of A-site spins.",
"The second mode is more difficult to discern in the contour plots of Fig.",
"REF , but plots of neutron scattering intensity versus (H00) (Fig.",
"REF (e)) and energy transfer (Fig.",
"REF (f)) indicate that the weak mode has minima (maxima) where the strong mode has maxima (minima), with an equally large spin gap.",
"These statements are confirmed below via triple-axis measurements.",
"A full theoretical construction to describe magnetic excitations in $\\mathrm {FeV_2O_4}$ must take into account the local ionic levels of both $Fe^{2+}$ and $V^{3+}$ cations[29].",
"Here, we show instead that the essential elements of the observed spin excitations at low temperatures are captured by a minimal spin-wave model, with the inclusion of appropriate single-ion anisotropy terms.",
"Dispersion data were fit using semiclassical spin-wave calculations, assuming an ideal cubic structure and the Hamiltonian $\\mathcal {H} = \\sum \\limits _{< i,j >} J_{i,j} \\mathbf {S_i} \\cdot \\mathbf {S_j} + \\sum \\limits _i D_i (\\mathbf {S_i} \\cdot \\mathbf {\\hat{n}_i})^2$ and the results are shown as solid lines in Fig.",
"REF .",
"Here, the sums are over all spins, including both cation sites, and interactions are truncated beyond nearest neighbors for the $Fe^{2+}$ sites (nearest neighbor Fe-V interactions) and next-nearest neighbors for the $V^{3+}$ sites (nearest neighbor V-V and nearest neighbor Fe-V interactions).",
"The single-ion anisotropies are constrained to be along the (001) direction for the iron site, consistent with conclusions from x-ray diffraction and magnetization[2], and along the local $<111>$ directions for the vanadium sites, consistent with an important role for the trigonal crystal field.",
"The exchange parameter, $J_{i,j}$ is also fitted differently when describing V-V pairs in ($J_{BB}$ ) or out of ($J^{^{\\prime }}_{BB}$ ) the tetragonal a-b plane.",
"The canting angle of spins in the ground state was determined self-consistently through fits of the inelastic data, and found to be within error equal to the value inferred from NPD[3].",
"Further fit parameters are listed in Table 1.",
"The spin-wave model explains the dispersion of the two observed modes in all directions, and identifies them as having a majority contribution from the motion of iron cations.",
"The model predicts the existence of four additional modes with primarily vanadium character, not discernable in our inelastic neutron scattering data below energy transfers of 35 meV.",
"One possible explanation for their absence is that they exist at energies above our measurement range; as shown in the Supplementary Information[30], spin-wave curves associated with the parameters in Table 1 indeed predict the four remaining modes reside in the energy range 30 - 60 meV.",
"However, we further note that the ordered moment on the vanadium site is roughly 5 times weaker than the order iron moment[3], and thus the missing modes are expected to be much less intense than those reported here.",
"Table: Exchange and anisotropy parameters inferred from fits of TOF neutron scattering data collected at T = 5 K to a semiclassical spin-wave model.Of the parameters listed in Table 1, the exchange parameter $J_{AB}$ is most tightly constrained by the available data, but there was significant freedom in chosing $J_{BB}$ and $J^{^{\\prime }}_{BB}$ .",
"The given values provide a good description of the current data, but further constraints on $J_{BB}$ and $J^{^{\\prime }}_{BB}$ will likely require measurements of the four remaining modes.",
"It is interesting to note, however, that even a good description of all dispersion curves was possible only if we allowed the parameter $J^{^{\\prime }}_{BB}$ , describing the exchange between V-V nearest neighbor pairs in the (101) direction, to be ferromagnetic.",
"An antiferromagnetic $J_{BB}$ and ferromagnetic $J^{^{\\prime }}_{BB}$ are in fact perfectly consistent with the 2-in-2-out spin structure stabilized in $\\mathrm {FeV_2O_4}$ .",
"Importantly, both anisotropy parameters $D^{[001]}_{A}$ and $D^{<111>}_{B}$ are predicted to contribute to the spin gap.",
"The best fit value for $D^{<111>}_{B}$ is large and negative.",
"A large, negative anisotropy favoring spins in the $<111>$ directions is consistent with the assumptions underlying the quantum 120$^{\\circ }$ model of Chern et al.",
"[17] In Fig.",
"REF , we show a representative contour plot of the low-energy excitation spectrum at three temperatures, T = 5 K, 85 K and 125 K, corresponding to measurements in the canted spin, the collinear ferrimagnetic, and the paramagnetic states, respectively.",
"One can see that the overall shape and bandwidth of the dominant mode is unchanged as the system evolves from the canted to the collinear spin state, with only a loss of scattering intensity resulting from a smaller ordered moment size at 85 K. This is as expected, as the spins on the iron sublattice are unaffected at the canting transition.",
"Above $T_{N1}$ , the modes vanish entirely, confirming they are magnetic in origin.",
"The most notable change with temperature is the magnitude of the spin gap, which is $\\Delta \\sim 9 meV$ in the LTT phase but barely distinguishable from zero in Fig.",
"REF (d) at T = 85 K. Figure: (a)-(d) Contour plots of neutron scattering intensity in several energy-momentum transfer planes, showing the Q-dependence of magnon excitations along several different symmetric directions in reciprocal space.",
"Especially notable are the dispersive magnetic excitations which arise about points (2 0 0), (4 0 0) and symmetric equivalents.",
"(e) A plot of scattering intensity versus (H 0 0) in the energy range E = 8-16 meV, revealing peaks at (-4 0 0) and (±\\pm 2 0 0).",
"(f) Plot of scattering intensity versus energy transfer at (2 2 0) and (2 0 0).",
"In all plots, we use the high-temperature cubic basis to index the reciprocal lattice cell.Figure: (a)-(c) Plots of inelastic neutron scattering intensity along the line (2 H 0) in reciprocal space, as measured with the SEQUOIA spectrometer.",
"Shown are data taken at T = 4 K (a), 85 K (b) and 125 K (c).",
"Solid lines are spin-wave fits described in the main text.",
"Curves in (b) are shifted by 5 meV.",
"(d) Plots of neutron scattering intensity versus energy transfer for all three temperatures at the point (2 2 0).To explore the temperature dependence of the magnetic states more fully, SEQUOIA results were supplemented by targeted triple-axis measurements over the range 5 K $<$ T $<$ 120 K. Figure REF (a) shows a comparison of scattering intensity at the (220) reciprocal lattice position versus energy transfer well-below and above the upper magnetic transition at $T_{N1}$ = 110K.",
"The choice of (220) was dictated by the minimum of the dominant band in Fig.",
"REF .",
"The inset is plotted on a logarithmic scale, and reveals significant quasi-elastic scattering intensity at the higher temperature which collapses into the gapped magnon modes below the ordering transitions.",
"The modes themselves are shown more clearly on a linear scale in the main panel.",
"Similar base temperature scans are shown in Figure REF (b) at the (200) position- the location of the lowest-Q minimum of the weak mode in Fig.",
"REF - and at (240)- a symmetrically equivalent position with larger dynamic range on the TA spectrometer.",
"Scans in both panels indicate multiple magnetic excitations below 25 meV.",
"The dominant and weak magnon modes identified with SEQUOIA are confirmed in Fig.",
"REF (a), where the greater intensity branch has a gap 8.9 meV $\\pm $ 0.1 meV and the lower intensity branch appears much broader and is located around 23 meV.",
"Both peaks are also present at (240) with the relative intensities reversed.",
"In addition, the scans seem to indicate that an excitation exists with E$\\approx $ 12 meV.",
"An equivalent excitation is seen at all measured Q, with an intensity that decreases with increasing scattering angle.",
"As shown in Fig.",
"REF (c), this peak also persists to temperatures as high as 120 K. The origin of this peak is a subject for future investigation.",
"To track the temperature dependence of the spin gap, we took scans at the (220) position with limited energy transfer and temperature increasing by 5 K increments from 5 K to 120 K. Select scans taken using a thermal TA instrument are shown in Fig.",
"REF (c), and using a cold TA in Fig.",
"REF (d).",
"Solid lines represent the results of fitting to a series of Lorentzian peaks convolved with instrument resolution, with the energy of the 12 meV excitation held constant below T = 100 K to stabilize the fitting.",
"The fitted gap is plotted as a function of temperature in Fig.",
"REF .",
"Results from the two datasets are consistent in the temperature region overlap, and the overall gap function is seen to rise up in a smooth, mean-field like way from the upper N$\\mathrm {\\acute{e}}$ el temperature.",
"The temperature dependence was characterized using both single and double order-parameter functions, with the best fits shown as dashed and solid lines, respectively.",
"The single OP fit gives $T_{N1}$ = 107.7 K $\\pm $ 0.5 K, within error equal to other measurements of critical temperature.",
"However, a far superior description of the data is provided by the two-OP fit, which yields $T_{N1}$ = 107.0 K $\\pm $ 0.5 K and $T_{N2}$ = 60.3 K $\\pm $ 0.7 K. These values independently confirm the critical temperatures obtained from heat capacity, NPD and elastic neutron measurements.",
"The increase of the gap at the canting transition, $T_{N2}$ , is reminiscent of the opening of a gap from zero in $\\mathrm {MnV_2O_4}$ and reinforces the notion that this temperature can be associated with the ordering of orbitals on the vanadium sublattice.",
"It is natural to ascribe the second, higher-temperature contribution to the gap function to the ferro-orbital order that exists on the iron sublattice below T = 140 K. Thus, the two-OP gap function presented in Fig.",
"REF indicates that orbital order on both cations sites is playing a defining role in this material.",
"Figure: (a) Constant-Q scans on the HB3 triple-axis spectrometer at the cubic (2 2 0) position.",
"Shown are scans at temperatures above and well below the ordering temperatures in FeV 2 O 4 \\mathrm {FeV_2O_4} on logarithmic (inset) and linear (main panel) scales.",
"(b) Constant-Q scans at the (2 0 0) and (2 4 0) positions, which are symmetrically equivalent.",
"(c)(d) show the temperature evolution of the constant-Q scans at (2 2 0), as measured with the HB3 and CTAX spectrometers, respectively.Figure: Temperature dependence of the excitation gap at the cubic (2 2 0) position in FeV 2 O 4 \\mathrm {FeV_2O_4}.",
"Included are data from the HB3 (diamonds) and CTAX (circles) spectrometers.",
"Lines are fits to power-law temperature dependence, assuming one (dashed) or two (solid) order-parameters.",
"The latter was a better description of the data, and the parameters from that fit are shown explicitly.The most distinguishing feature of the inelastic spectrum of $\\mathrm {FeV_2O_4}$ is the order-of-magnitude larger spin-gap over other spinel vanadates.",
"However, the data in Fig.",
"REF imply that this is primarily due to the orbital order on the A-site, combined with spin-orbit coupling and the large ordered iron moment, and has little to do with the physics of vanadium cations.",
"Other distinguishing features of $\\mathrm {FeV_2O_4}$ , including the (0 0 1) easy-axis observed via magnetization and associated magnetostrictive effects observed with x-ray diffraction[2], can also be explained by a strong single ion anisotropy for the spin on the A-site, as made clear by first principles calculations[16].",
"When focusing instead on the contribution to the gap from the vanadium sublattice, the magnitude and temperature dependence is comparable to what is reported for $\\mathrm {MnV_2O_4}$[8].",
"We have noted in the past[3] that the elastic ordering pattern of spins in $\\mathrm {FeV_2O_4}$ at low temperatures is consistent with the predictions of the quantum 120$^\\circ $ model[17] of Chern et al.",
"in the strong spin-orbit coupling limit.",
"In this model, developed to explain observations on $\\mathrm {MnV_2O_4}$ , the in-plane direction of vanadium spins in the canted state is set by a competition between orbital exchange interactions and coupling to local trigonal distortions, which prefer spins to point along primary cubic axes and local $<11>$ spin directions, respectively.",
"Intuitively then, the statement that $\\mathrm {FeV_2O_4}$ lies in the strong spin-orbit coupling limit is equivalent to highlighting the dominant role of trigonal distortions, consistent with the success of our spin-wave model above.",
"In the paper of Chern, it was concluded that $\\mathrm {MnV_2O_4}$ lies in the opposite, strong orbital exchange limit.",
"This is surprising in the current context, as $\\mathrm {MnV_2O_4}$ has both a larger trigonal distortion than $\\mathrm {FeV_2O_4}$ and a larger spin-gap arising from the vanadium sublattice.",
"A possible resolution comes from noting that the classification of $\\mathrm {MnV_2O_4}$ as being in the strong-exchange limit was based primarily on the directions for vanadium spins stated in the NPD paper of Garlea et al.[8].",
"However, the unpolarized NPD data of Garlea et al.",
"were unable to uniquely determine in-plane spin directions[31], and so the authors were careful to say that the decision to constrain the in-plane portion of the ordered spins to point along primary axes was an assumption of the structure refinement[8].",
"As a further point of interest, one recent neutron diffraction study on single crystals of $\\mathrm {MnV_2O_4}$[32] largely confirms the structure of Garlea et al., but reports a slightly smaller canting angle of 54.9$^\\circ $ for that material, nearly identical to the present case.",
"Thus, the spin and orbital structures of $\\mathrm {MnV_2O_4}$ and $\\mathrm {FeV_2O_4}$ may be more closely related than previously thought.",
"A strong counter-argument to this conclusion is the different tetragonal space groups reported for $\\mathrm {MnV_2O_4}$ and $\\mathrm {FeV_2O_4}$ .",
"In fact, the $I4_1/amd$ space group reported for $\\mathrm {FeV_2O_4}$[2], [3] contains a glide-plane symmetry that is expected to be broken in the quantum 120$^\\circ $ model, and normal mode analysis of recent x-ray diffraction data suggests fundamentally different orbitally ordered states for the Mn and Fe compounds[4].",
"The trigonal distortion in $\\mathrm {FeV_2O_4}$ is large, but smaller than in $\\mathrm {MnV_2O_4}$ .",
"We suggest that the underlying assumption of a dominant (i.e.",
"infinite) trigonal distortion may be violated in the case of $\\mathrm {FeV_2O_4}$ .",
"Detailed measurements of the inelastic spectrum of the two compounds, and an analysis which incorporates the full orbital and spin degrees-of-freedom in the $t_{2g}$ manifold[29] may be key to a complete understanding of these systems."
],
[
"Summary and Conclusions",
"In conclusion, we have examined single crystal samples of $\\mathrm {FeV_2O_4}$ with elastic and inelastic neutron scattering.",
"The sequence of structural and magnetic phase transitions identified in powders are confirmed by heat capacity, magnetization and elastic neutron scattering on crystals.",
"Inelastic neutron scattering measurements reveal two magnon branches below the ferrimagnetic transition at 110 K, and a semiclassical spin wave analysis was shown to be successful if single ion anisotropies are assumed to be in the (0 0 1) direction for the iron sublattice and the local $<111>$ directions on the vanadium sublattice.",
"The spin-gap was measured to be 8.9 $\\pm $ 0.1 meV, mostly resulting from the ferro-orbital order on the A-site sublattice.",
"The small increase in the gap from the vanadium sublattice is comparable to other materials, and consistent with the identification of the canting transition at 60 K with an orbital ordering transition on the B-site.",
"Similarities in the measured inelastic spectra and the local direction of ordered spins suggest that the physics describing $\\mathrm {MnV_2O_4}$ and $\\mathrm {FeV_2O_4}$ systems are closely related.",
"Reconciling the different crystallographic space groups in the two materials may require future measurements of the inelastic spectrum."
],
[
"Acknowledgements",
"Authors would like to acknowledge valuable discussions with S. Hahn and R. Fishman at ORNL.",
"Research at the Spallation Neutron Source and the High Flux Isotope Reactor was sponsored by the U. S. Department of Energy, Office of Basic Energy Sciences, Scientific User Facilities Division.",
"G.J.M and I.B.",
"are further supported by U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-07ER46453.",
"H.D.Z.",
"thanks the support of the JDRD program of the University of Tennessee."
]
] | 1403.0269 |
[
[
"On the existence of topological hairy black holes in $\\mathfrak{su}(N)$\n EYM theory with a negative cosmological constant"
],
[
"Abstract We investigate the existence of black hole solutions of four dimensional $\\mathfrak{su}(N)$ EYM theory with a negative cosmological constant.",
"Our analysis differs from previous works in that we generalise the field equations to certain non-spherically symmetric spacetimes.",
"We prove the existence of non-trivial solutions for any integer $N$, with $N-1$ gauge degrees of freedom.",
"Specifically, we prove two results: existence of solutions for fixed values of the initial parameters and as $|\\Lambda|\\rightarrow\\infty$, and existence of solutions for any $\\Lambda<0$ in some neighbourhood of existing trivial solutions.",
"In both cases we can prove the existence of `nodeless' solutions, i.e.",
"such that all gauge field functions have no zeroes; this fact is of interest as we anticipate that some of them may be stable."
],
[
"Introduction",
"Until relatively recently, the question of classifying black holes seemed resolved, with Birkhoff's theorem [10] and uniqueness theorems of Israel's for Schwarzschild [25] and Reissner-Nördstrom [26] black holes.",
"This lead to Wheeler's so-called `no-hair' conjecture [43], which placed limits of the kinds of black holes one could expect in spherically symmetric space (and specifically the very small number of parameters which must entirely specify the solution).",
"However, in the 1980s and since, more and more examples have been found which show that the conjecture is violated in the strictest terms, beginning with the discovery of black hole and soliton solutions in 4D flat spacetime; in the case of scalar hair by Bekenstein and others (e.g.",
"[8], [7], [15]), and for $\\mathfrak {su}(2)$ hair by Bizon, Bartnik, McKinnon and others (e.g.",
"[12], [2], [52], [33]).",
"Since then, many other directions of research in this area have been pursued (see for instance [54] for a review.)",
"Introducing a non-zero cosmological constant is known to yield very interesting results, due to the differing geometry it bestows upon the manifold.",
"In asymptotically flat space, soliton and black hole solutions of $\\mathfrak {su}(2)$ EYM theory are known to exist only at discrete points in the parameter space.",
"All such solutions will have a gauge field function $\\omega $ with at least one zero, the number of unstable modes of the solution being directly proportional to the number of zeroes of $\\omega $ ($2k$ unstable modes for $k$ zeroes [34], [51], [40]).",
"However, for anti-de Sitter (adS) space (i.e.",
"a negative cosmological constant), it changes completely; we get solutions existing in continuous ranges of the parameters governing the dynamic variables [56], [14], [13], [19], and at least some of them possess gauge fields with no zeroes (so-called `nodeless' solutions).",
"It can be shown that some of the solutions with non-zero gauge fields everywhere are stable under linear perturbations [56], [14], [13], [47], [58], [5], provided that $|\\Lambda |$ is large enough.",
"(We shall leave the question of $\\Lambda >0$ entirely, except to say that for a comprehensive classification of constructed asymptotically de Sitter spacetimes, see [18].)",
"Recent research has expanded upon this, and the existence has been established of asymptotically anti-de Sitter soliton and black hole solutions to 4D $\\mathfrak {su}(N)$ EYM equations [57], [3], [4].",
"Purely magnetic solutions have been found which are described by $N-1$ gauge field functions $\\omega _j$ ($j=1,...,N-1$ ).",
"The most recent results we have [6] show that there exist genuinely non-trivial solutions for any $N$ where the gauge field functions have no zeroes, and that for $|\\Lambda |$ sufficiently large, at least some of these are stable [4], [5].",
"These hairy black holes in adS, despite being indistinguishable from Reissner-Nördstrom black holes asymptotically, therefore require an extra $N-1$ parameters to describe them, and this number is obviously unbounded (though still countable) as $N\\rightarrow \\infty $ .",
"For more detail, see the recent work [48], in which there do seem to be global charges characterising at least some of the stable $\\mathfrak {su}(N)$ solutions.",
"Some work from van der Bij and Radu [9] (among others) has suggested a whole new direction in which to take these `hairy' black holes.",
"Their work is on topological black holes – that is, black holes that possess isometries other than the usual requirement of spherical symmetry.",
"This will be expanded upon later, but briefly, the requirement is no longer that the metric and gauge potential are spherically symmetric (i.e.",
"endow the manifold with gauge fields that are invariant under an action of $SU(2)$ by principal bundle automorphisms); it is relaxed to encompass other isometry groups such that the Lie group of isometries is the structure group of one of the three surfaces of constant curvature – i.e., the sphere, the plane, or a hyperbolic surface (with the possibilities of event horizons of non-zero genus).",
"Several authors have also considered various other cases of topological black holes, including those set in higher dimensions and with event horizons of genus $g\\ge 1$ (e.g.",
"[20], [39], [44], [50], [35], [37], [11], [36], [21]).",
"This work [9] yielded interesting results for $\\mathfrak {su}(2)$ : the authors found topological black hole solutions numerically; noted that the Yang-Mills field equation could be cast in a form in which it was obvious that all solutions would be nodeless; and demonstrated their stability under linear spherically symmetric perturbations for $|\\Lambda |$ sufficiently large.",
"It is the aim of this paper to build on their successes (though using largely different methods) by generalising their results to $\\mathfrak {su}(N)$ ; and thus to prove that, for $|\\Lambda |\\rightarrow \\infty $ , solutions exist for arbitrary values of the initial parameters of the field variables at the event horizon which are regular everywhere and which are analytic in their parameters, and that genuine (i.e.",
"non-trivial) solutions to the 4D $\\mathfrak {su}(N)$ EYM field equations with a negative cosmological constant can be found in a neighbourhood of known (trivial) solutions and, for $k\\ne 1$ , these solutions will be nodeless i.e.",
"will possess gauge field functions with no zeroes.",
"The results for $k=1$ (which correspond to spherical symmetry) have been adequately explored in [6], and so will largely be ignored except as comparison to the new results we obtain for $k\\ne 1$ .",
"Also, we note that here we use a time-honoured `shooting' method to verify the existence of solutions, though this is not the only method: recent work by Nolan and Winstanley [44] uses an argument involving Banach spaces to prove the existence of dyons and dyonic black holes in spherically symmetric $\\mathfrak {su}(2)$ theory.",
"The outline of this paper is as follows.",
"First, we present the ansätze, field equations and boundary conditions for 4D $\\mathfrak {su}(N)$ EYM theory with a negative cosmological constant.",
"We use the work of Wang [55] and Künzle [31] to derive a new, more general $\\mathfrak {su}(N)$ -invariant gauge potential which applies to the topologies we are interested in.",
"Some trivial solutions to the field equations are found, including Reissner-Nördstrom-type and embedded $\\mathfrak {su}(2)$ solutions, both of which are vital to the final existence argument.",
"Next we prove a series of propositions: we demonstrate that solutions exist locally nearby the singular points of the field equations (that is, $r=r_h$ and $r\\rightarrow \\infty $ ); and that solutions which are regular in some small interval within $r_h<r<\\infty $ can continue to be integrated outwards to the asymptotic regime provided the metric function $\\mu >0$ .",
"We note that such solutions will be analytic in their initial values at the event horizon.",
"We then use these propositions to prove a theorem which states that topological $\\mathfrak {su}(N)$ solutions will always exist in open sets, and that solutions in a sufficiently small neighbourhood of each other will possess the same number of gauge field zeroes.",
"Finally we establish our two main results: that non-trivial solutions to these field equations exist both for fixed values of the boundary conditions $\\omega _{j,h}$ and for $|\\Lambda |\\rightarrow \\infty $ , and also in some small neighbourhood of Reissner-Nördstrom-type solutions for any fixed $\\Lambda <0$ .",
"The conclusions are presented at the end along with an indication of future directions this work may take."
],
[
"2D spaces of constant curvature",
"In this section we give a brief review of the three different topological cases of 2-dimensional spaces of constant Gaussian curvature $K$ (i.e.",
"spherical, planar and hyperboloidal), including a discussion of the metrics and the Lie groups which describe the symmetries of the spaces."
],
[
"Metrics and isometry groups: 3 different cases",
"We begin with a manifold $M$ endowed with a metric, possessing some symmetries of its own; and wish to endow it with fields possessing some other symmetry group $G$ .",
"We can consider it as a principal fibre bundle, i.e.",
"with base space $M$ and $G$ -valued fibres; and endow that bundle with a connection which possesses the property that it is invariant under any isometric action (automorphism) performed on it.",
"In the case of spherical symmetry, such an action would be a rotation around a general axis.",
"It can be noted that each such action can be mapped to the (continuous) rotation group $U(1)$ – as such, this is associated with the $\\mathfrak {u}(1)$ subalgebra.",
"In the case of a sphere, the collection of all these actions form orbit surfaces which in the trivial case are clearly spheres $S^2$ , which regularly foliate the 4-manifold.",
"The whole group of isometries is isomorphic to $SU(2)$ .",
"In the case of a general space of constant curvature, we find that the $U(1)$ subgroup is still present in $LG(2)$ (the general name we are giving to the Lie group of isometries of the 2D constant-curvature manifold in question), and since the collection of all possible actions of $LG(2)$ regularly foliates the 4-manifold in question, then the isometries are classified by all possible actions of $LG(2)$ on a principal $G$ -bundle.",
"In order to generalise the above statements, and the arguments that follow it in the original paper by Künzle [31], to $k\\ne 1$ , we will need appropriate replacements for the subspace $S^2$ and the Lie group of isometries $SU(2)$ .",
"Then we will have a way forward when it comes to constructing the gauge potential, in order to give the gauge fields the correct symmetry.",
"Hence, in what shortly follows we shall briefly explain what we mean by surfaces of constant curvature, categorising them by the sign of the Gaussian curvature, and showing the appropriate topology and Lie algebra we use in each case.",
"As is well known, the metric for the sphere of unit radius is given by $d\\Omega _1^2 = d\\theta ^2+\\sin ^2\\theta d\\phi ^2.$ The symmetry group is also well known, as the Lie group $SU(2)$ , with the associated Lie algebra $\\mathfrak {su}(2)$Henceforth Lie groups will be denoted by Latin script, and their associated algebras by Germanic script..",
"The most common representation of this algebra is to use the well-known Pauli matrices, i.e.",
": $\\begin{array}{ccc}\\sigma _1=\\left( \\begin{array}{cc}0 & 1 \\\\1 & 0 \\end{array}\\right) &\\sigma _2=\\left(\\begin{array}{cc}0 & -i \\\\i & 0 \\end{array}\\right) &\\sigma _3=\\left(\\begin{array}{cc}1 & 0 \\\\0 & -1 \\end{array}\\right),\\end{array}$ to obtain the three generators.",
"It should be noted that the Lie group $SU(2)$ forms a double cover of the Lie group $SO(3)$ ; thus, the three generators of the algebra $\\mathfrak {su}(2)$ can be mapped to the three (infinitesimal) basis vectors for ordinary 3-dimensional Euclidean spatial rotations.",
"If we let $K_j = \\frac{-i\\sigma _j}{2}$ with $j\\in \\lbrace 1,2,3\\rbrace $ , then the $K_j$ obey the following commutation relations: $\\begin{array}{ccccc}[K_1,K_2]=K_3 & \\quad &[K_2,K_3]=K_1 & \\quad &[K_3,K_1]=K_2.\\end{array}$ As well as using the Pauli matrices as a basis, we can use the following definitions of infinitesimal differential operators to satisfy the commutators (REF ): $\\begin{split}K_1&\\mapsto \\frac{\\partial }{\\partial \\alpha }=z\\frac{\\partial }{\\partial y} - y\\frac{\\partial }{\\partial z},\\\\K_2&\\mapsto \\frac{\\partial }{\\partial \\beta }=x\\frac{\\partial }{\\partial z} -z\\frac{\\partial }{\\partial x},\\\\K_3&\\mapsto \\frac{\\partial }{\\partial \\phi }=y\\frac{\\partial }{\\partial x} - x\\frac{\\partial }{\\partial y};\\end{split}$ where $\\partial /\\partial \\alpha $ , $\\partial /\\partial \\beta $ and $\\partial /\\partial \\phi $ can be mapped to ordinary $SO(3)$ rotations about the $x$ , $y$ and $z$ axes."
],
[
"$k=0$ : The plane",
"The metric for a 2D plane can be given as the following: $d\\Omega _0^2 = d\\theta ^2+\\theta ^2d\\phi ^2.$ The isometry group for the 2D plane is just the ordinary 2D Euclidean Lie group $E(2)$ , with associated Lie algebra $\\mathfrak {e}(2)$ .",
"The Lie group $E(2)$ has three generators: two spatial translations and one rotation.",
"The basis for this is most helpfully taken as infinitesimal translations in the $x$ and the $y$ directions, which for later convenience we call $K_1$ and $K_2$ ; and the rotation in the $xy$ plane, for which we use the name $K_3$ .",
"The commutation relations for $\\mathfrak {e}(2)$ are then: $\\begin{array}{ccccc}[K_1,K_2]=0, & \\quad &[K_2,K_3]=K_1, & \\quad &[K_3,K_1]=K_2.\\end{array}$ Unfortunately, the isometries of $\\mathbb {R}^2$ cannot be given explicitly in terms of matrices under multiplication – the closest we can get is the set of transforms $\\begin{array}{rl}\\left(\\begin{array}{c}x\\\\y \\end{array}\\right)\\mapsto & \\left(\\begin{array}{c}x+a\\\\y\\\\ \\end{array}\\right)\\\\\\left(\\begin{array}{c}x\\\\y \\end{array}\\right)\\mapsto & \\left(\\begin{array}{c}x\\\\y+b\\\\ \\end{array}\\right)\\\\\\left(\\begin{array}{c}x\\\\y \\end{array}\\right)\\mapsto & R_\\theta \\left(\\begin{array}{c}x\\\\y \\end{array}\\right)=\\left(\\begin{array}{cc}\\sin \\theta & \\cos \\theta \\\\\\cos \\theta & -\\sin \\theta \\end{array}\\right)\\left(\\begin{array}{c}x\\\\y \\end{array}\\right)\\end{array}$ for a translation in the plane by the vector $\\left(\\begin{array}{c}a\\\\b \\end{array}\\right)$ and a rotation by an angle $\\theta $ .",
"So the best representation for this algebra can be given in terms of the following correspondence to differential operators: $\\begin{split}K_1&\\mapsto \\frac{\\partial }{\\partial x},\\\\K_2&\\mapsto \\frac{\\partial }{\\partial y},\\\\K_3&\\mapsto \\frac{\\partial }{\\partial \\phi }=y\\frac{\\partial }{\\partial x} - x\\frac{\\partial }{\\partial y}.\\end{split}$ These are just the ordinary killing vectors for 2-dimensional Euclidean space."
],
[
"$k=-1$ : The hyperboloid",
"The analogous negatively curved space is endowed with the metric $d\\Omega _{-1}^2 = d\\theta ^2+\\sinh ^2\\theta d\\phi ^2,$ where $d\\Omega _{-1}^2$ is regarded as the metric on the sphere of unit imaginary radius, i.e.",
"the unit hyperboloid.",
"The symmetry group on the space can be shown to be the Lie group $SU(1,1)$ , associated with the Lie algebra $\\mathfrak {su}(1,1)$ .",
"The Lie algebra $\\mathfrak {su}(1,1)$ again has 3 infinitesimal generators which we call $K_j$ for $j\\in \\lbrace 1,2,3\\rbrace $ ; and can be cast in the form where they have the following commutation relations (see e.g.",
"[46], [1]): $\\begin{array}{ccccc}[K_1,K_2]=-K_3,& \\quad &[K_2,K_3]=K_1,& \\quad &[K_3,K_1]=K_2.\\end{array}$ A representation for this can be given in terms of the following matrices: $\\begin{array}{c}K_1\\mapsto -\\frac{i}{2}\\left( \\begin{array}{cc}0 & -i \\\\-i & 0 \\end{array}\\right),\\\\K_2\\mapsto -\\frac{i}{2}\\left(\\begin{array}{cc}0 & -1 \\\\1 & 0 \\end{array}\\right),\\\\K_3\\mapsto -\\frac{i}{2}\\left(\\begin{array}{cc}1 & 0 \\\\0 & -1 \\end{array}\\right).\\end{array}$ Note that $K_3$ is identical to the definition of $K_3$ for the spherical case – this is no coincidence and will be discussed below."
],
[
"Topological considerations",
"In all 3 cases, giving the appropriate symmetry ($SU(2)$ , $E(2)$ or $SU(1,1)$ ) to the 4D manifold requires that they be considered regularly foliated by 2D surfaces of constant curvature (which we shall call $\\Sigma _k^2$ ) that are appropriate to the case in question, endowing them with the topology $\\mathbb {R}^2\\times \\Sigma _k^2$ .",
"In the case of the sphere (for $k=1$ ), it is clear that $\\Sigma _1^2=S^2$ , and it can be shown that $S^2\\simeq SU(2)/U(1)$ .",
"$SU(2)$ as a manifold is topologically the 3-sphere $S^3$ , and the quotient is taken using $U(1)$ (which is the circle $S^1$ ).",
"To form this quotient we can use the fact that the Abelian subgroup of diagonal matrices is isomorphic to the circle group $U(1)$ , so we let the generator $K_3 = (-i/2)\\sigma _3$ span the $\\mathfrak {u}(1)$ subalgebra – the reason for this identification is to simplify the later derivation of the gauge potential.",
"In the case of the flat plane $\\Sigma _0^2=\\mathbb {R}^2$ (for $k=0$ ), the quotient works slightly differently: since $\\mathbb {R}^2$ is a normal subgroup of $E(2)$ but $O(2)$ is not, we find that $O(2)\\simeq E(2)/\\mathbb {R}^2$ (through the usual map that associates to an affine transformation a linear transformation of the tangent vector space associated to the affine space).",
"Again though, $U(1)\\le O(2)\\le E(2)$ , and we let the generator $K_3$ (in this case, the rotation generator) span the $\\mathfrak {u}(1)$ subalgebra.",
"With this identification the remaining two generators (the translations) span the $\\mathbb {R}^2$ plane.",
"Using the geometrical construction given above, it can be seen how this continues over from the spherical case, with the $z$ -axis rotation generator being $K_3$ in both cases.",
"One can even consider the above spherical case, and simply let the radius of the sphere go to infinity; in that case, the rotation about the $x$ - and $y$ -axes become the $y$ and $x$ translation generators respectively and the rotation generator stays as it is.",
"For the hyperbolic space $H^2\\equiv \\Sigma _{-1}^2$ , and $H^2\\simeq SU(1,1)/U(1)$ .",
"Thus, although it is harder to visualise, we have essentially a similar situation as for $k=1$ : one of the generators ($K_3$ ) is of an ordinary rotation type, and it is this generator that we use to span the subalgebra $\\mathfrak {u}(1)$ in the quotient.",
"The other two generators will be hyperbolic rotations on the space $\\Sigma _{-1}^2$ ."
],
[
"The general case",
"It can be seen from the previous subsection that we could express this information in a much simpler way, exploiting the three values we have chosen for $k$ .",
"Examining (REF , REF , REF ), the metric in all the above cases can be expressed as $d\\Omega _k^2 = d\\theta ^2+f_k^2(\\theta )d\\phi ^2$ where $f_k(\\theta ) = \\left\\lbrace \\begin{array}{ll}\\sin \\theta & \\mbox{for } k=1 \\\\\\theta & \\mbox{for } k=0 \\\\\\sinh \\theta & \\mbox{for } k=-1 \\end{array}\\right.$ Examining (REF , REF , REF ), we can also express the commutation relations of the three Lie algebras in the following succinct way: $\\begin{array}{ccccc}[K_1,K_2]=kK_3 & \\quad &[K_2,K_3]=K_1 & \\quad &[K_3,K_1]=K_2\\end{array}$ with the algebra (which we call $\\mathfrak {lg}(2)$ in general) being $\\mathfrak {su}(2)$ for $k=1$ , $\\mathfrak {e}(2)$ for $k=0$ and $\\mathfrak {su}(1,1)$ for $k=-1$ ; and where the three generators of the group are the three $K_j$ s. It can be noted that in each of the three cases above, $K_3$ in particular takes a similar form: it has an identical representation each time in the form of differential operators, and thus represents in each case a rotation which can easily be arranged about the same axis (essentially $z$ ).",
"The significance of this in later sections is linked to the presence of the factor of $k$ which accompanies the $K_3$ generator above.",
"Finally, we can also give the topology of the foliated hypersurfaces $\\Sigma _k^2$ as $\\Sigma _k^2\\simeq LG(2)/U(1)$ for $k = \\pm 1$ and $O(2)\\simeq E(2)/\\Sigma _k^2$ for $k=0$ ; and the topology of the 4D manifolds themselves as $\\mathbb {R}^2\\times \\Sigma _k^2;$ where $LG(2)$ is the Lie group in question, and where it is understood that the generator $K_3$ (as noted above) plays a privileged role in forming this quotient.",
"These facts, and equations (REF , REF ) will be useful when later we derive the gauge potential using a proof analogous to that of Künzle [31], and when we derive the field equations themselves.",
"In this section we give the general mathematical background we shall need to describe and model topological black holes.",
"The discussion proper will begin with a detailed outline of our metric and gauge potential ansätze.",
"This will include a full derivation of the gauge potential in this case, since this is a fairly technical procedure.",
"These ansätze will then be used to derive the field equations and boundary conditions for the topological black hole solutions.",
"Finally, we describe some trivial embedded solutions of the theory (in analogy with [6])."
],
[
"Ansätze",
"The action used for the four-dimensional $\\mathfrak {su}(N)$ EYM theory with a negative cosmological constant is $S_{EYM}=\\frac{1}{2}\\int d^{4}x\\sqrt{-g}[R-2\\Lambda -\\mbox{Tr}F_{\\mu \\nu }F^{\\mu \\nu }],$ where $R$ is the Ricci scalar of the geometry and $\\Lambda $ is the cosmological constant.",
"Throughout the paper the metric has signature $(-, +, +, +)$ and we use units in which $4\\pi G=1=c$ .",
"In this paper, we focus on $\\Lambda <0$ .",
"Varying the action gives the field equations $\\begin{split}-2T_{\\mu \\nu }&=R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R + \\Lambda g_{\\mu \\nu },\\\\0&=\\nabla _\\lambda F^\\lambda _{\\:\\:\\mu }+[A_\\lambda ,F^\\lambda _{\\:\\:\\mu }]\\\\\\end{split}$ where the YM stress-energy tensor is $T_{\\mu \\nu }=\\mbox{Tr}F_{\\mu \\lambda }F_\\nu ^\\lambda -\\frac{1}{4}g_{\\mu \\nu }\\mbox{Tr}F_{\\lambda \\sigma }F^{\\lambda \\sigma },$ and where it should be noted that `Tr' represents the trace in the Lie algebra sense, rather than the ordinary matrix trace.",
"In equations (REF , REF ) we have employed the usual Einstein summation convention where it is understood that summation occurs over repeated indices.",
"However where appropriate, summations will be shown explicitly.",
"In this paper we are interested in static, topological black hole solutions of the field equations (REF ), specifically for spaces regularly foliated by 2D (spacelike) hypersurfaces of constant and unit- or zero-magnitude Gaussian curvature $k$ , and hence (following the last section) we write the metric in standard Schwarzschild co-ordinates as $ds^2=-\\mu S^2dt^2+\\mu ^{-1}dr^2+r^2d\\theta ^2+r^2f^2_k(\\theta )d\\phi ^2,$ where $\\mu $ and $S$ depend on $r$ alone.",
"For convenience, we may take $\\mu (r)=k-\\frac{2m(r)}{r}-\\frac{\\Lambda r^2}{3}.$ Note the presence of our modified Gaussian curvature constant $k$ .",
"We emphasise again that in our case, $\\Lambda <0$ ; and that the $k=1$ case has been thoroughly investigated in our previous work [6], and so will be passed over more quickly and mainly used for comparison to the other cases.",
"The most general gauge potential we used in the spherically symmetric case was [31]: $A=\\mathcal {A}\\,dt+\\mathcal {B}\\,dr+\\frac{1}{2}(C-C^H)d\\theta -\\frac{i}{2}[(C+C^H)\\sin \\theta +D\\cos \\theta ]d\\phi ,$ where $\\mathcal {A},\\mathcal {B},C$ and $D$ are all ($N\\times N$ ) matrices and $C^H$ is the Hermitian conjugate of $C$ ; the matrices $\\mathcal {A}$ and $\\mathcal {B}$ are purely imaginary, diagonal and traceless and depend only on $r$ ; the matrix $C$ (which also depends solely on $r$ ) is upper-triangular, with non-zero entries only immediately above the diagonal, i.e.",
": $C_{j,j+1}=\\omega _j(r)e^{i\\gamma _j(r)}$ for $j=1,...,N-1$ ; and $D$ is the constant matrix $D=\\mbox{Diag}(N-1, N-3,...,-N+3, -N+1).$ However in the cases $k=0, -1$ we must alter (REF ) slightly to take into account the different geometry.",
"The new gauge potential becomes $A=\\mathcal {A}\\,dt+\\mathcal {B}\\,dr+\\frac{1}{2}(C-C^H)d\\theta -\\frac{i}{2}\\left[(C+C^H)f_k(\\theta )+D\\frac{df_k(\\theta )}{d\\theta }\\right]d\\phi ,$ with the same definitions for the matrices as above, and including $f_k(\\theta )$ (from (REF )) and its first derivative.",
"We shall now demonstrate that this is an appropriate choice of potential, given the symmetry considerations, by using an argument analogous to one due to Künzle in [31]."
],
[
"Deriving the ansatz for the gauge potential",
"The gauge potential for any system is never unique (see e.g.",
"[31]).",
"By definition it is invariant under the action of some group, so it always has freedom under transformations within that group (automorphisms); therefore it is merely a case of finding one invariant under actions of the requisite symmetry group that fulfills the appropriate gauge constraints – we shall show that (REF ) is such a potential.",
"The procedure to find all such irreducible representations can be found in [23].",
"Following on from section , we now present the derivation of the gauge potential in the case of general $k$ .",
"The similarity in the derivations for the three values of $k$ is such that we can talk very generally about the Lie groups/algebras we are using, as long as we are using one of the three we have specified; so we shall talk about the Lie algebra $\\mathfrak {lg}(2)$ , realising that this will either mean $\\mathfrak {su}(2)$ , $\\mathfrak {e}(2)$ or $\\mathfrak {su}(1,1)$ as appropriate.",
"What we wish to do is find the possible $LG(2)$ -invariant connections on an $SU(N)$ principal bundle $P$ over our spacetime manifold $M$ ; i.e.",
"find a gauge potential that is invariant under an action of $LG(2)$ by principal bundle automorphisms.",
"According to Wang's theorem [55] (see also [30]), the $LG(2)$ -invariant connections on $P$ are in one-to-one correspondence with the linear maps $\\mathfrak {lg}(2)\\rightarrow \\mathfrak {su}(N)$ satisfying $\\Pi (X) = \\lambda ^\\prime (X)$ for $X\\in \\mathfrak {u}(1)$ , $\\lambda $ being the homomorphism $\\lambda : U(1)\\rightarrow SU(N)$ and $\\lambda ^\\prime $ the induced map of the Lie algebras.",
"Also, we have $\\Pi \\circ ad_z = ad_{\\lambda (z)} \\circ \\Pi \\quad \\forall z\\in U(1).$ Now, because $SU(2)$ and $SU(1,1)$ have nice succinct matrix representations, but $E(2)$ does not, that means that we have to treat the cases for $k=\\pm 1$ very slightly differently to the case $k=0$ , but the differences are largely technical.",
"In practice, all it really means is that we keep track of which representation we are using and exactly what is meant by `commutator brackets' when they appear – whether they represent the obvious matrix commutator, or are the more abstract commutator relationships possessed by any Lie algebra.",
"Following [31], let $K_l\\quad (l\\in \\lbrace 1,2,3\\rbrace )$ , defined in terms of the general generators referred to in (REF ), be a basis for $\\mathfrak {lg}(2)$ (i.e., for $k=1$ let $K_l = -(i/2)\\sigma _l$ ; for $k=-1$ let them be the matrices in (REF ); and for $k=0$ let them be the more abstract differential operators (REF )).",
"In particular, let the rotation generator $K_3$ in each case span the $\\mathfrak {u}(1)$ subalgebra.",
"Note that this associates $K_3$ with the matrix $-\\frac{i}{2}\\left(\\begin{array}{cc}1 & 0\\\\0 & -1 \\end{array}\\right)$ for the cases of $k=\\pm 1$ , and for the case of $k=0$ , with the differential operator stated in (REF ) $y\\frac{\\partial }{\\partial x} - x\\frac{\\partial }{\\partial y}.$ Now we need a map embedding $U(1)$ isomorphically in $LG(2)$ .",
"As was stated in [31], the map $z\\mapsto \\left( \\begin{array}{cc}z & 0 \\\\0 & z^{-1}\\\\\\end{array} \\right)$ embeds $U(1)$ isomorphically in $SU(2)$ .",
"Hence, the homomorphisms described by (REF ) are explicitly given as $\\lambda (z)=\\mbox{diag}(z^{q_1},...,z^{q_N})$ (for sets of $N$ integers $q_i$ such that $\\sum ^N_i{q_i}=0$ , conjugate homomorphisms being described by different orderings of the $q_i$ s); and the induced map of Lie algebras is given as $\\lambda ^\\prime _q=-(i/2)D_{q_i}$ where $D_{q_i}=\\mbox{diag}(q_1,q_2,...,q_N).$ Obviously we now need to produce analogous maps for $k\\ne 1$ .",
"The reason it works in the $k=1$ case is that we can express a general element of $SU(2)$ as the matrix $A=\\left( \\begin{array}{cc}a & b \\\\-b^* & a^*\\\\\\end{array} \\right),$ for some choice of $a,b\\in \\mathbb {C}$ for which $\\mbox{det A} = |a|^2+|b|^2=1$ ; and then if we let $a=e^{i\\theta }=z$ and $b=0$ , we find that the matrix specified in (REF ) is indeed in $SU(2)$ , and it reduces infinitesimally to the rotation generator $K_3$ .",
"It is for a similar reason that the same map (REF ) can also be used in the case $k=-1$ , since a general element of $SU(1,1)$ can be expressed as $B=\\left( \\begin{array}{cc}a & b \\\\b^* & a^*\\\\\\end{array} \\right),$ for some choice of $a,b\\in \\mathbb {C}$ for which $\\mbox{det B}=|a|^2-|b|^2=1$ ; and then if we again let $a=e^{i\\theta }=z$ and $b=0$ , we find that the matrix specified in (REF ) is also in $SU(1,1)$ , and again it reduces infinitesimally to $K_3$ .",
"For $k=0$ , it is slightly different, since we are working with $K_3$ as a differential operator.",
"However we are still identifying $K_3$ with the rotation generator of the space.",
"So, given that $z\\in U(1)$ , we instead simply use the map $z\\mapsto R_\\theta ,$ where $R_\\theta $ refers to the rotation generator of $E(2)$ (expressed explicitly in (REF )).",
"This is a map that embeds $U(1)$ isomorphically in $E(2)$ , so that once again $\\lambda (z)=\\mbox{diag}(z^{q_1},...,z^{q_N})$ and $\\lambda ^\\prime _q=-(i/2)D_{q_i}$ , as in (REF ).",
"We note that infinitesimally, $R_\\theta $ can be expressed as the differential operator (REF ), which we associate again with the generator $K_3$ , since it obeys the correct commutation laws (REF ).",
"Defining $\\Pi _q\\equiv \\Pi (K_q)$ , then $\\Pi _3=-(i/2)D_{q_i}$ and infinitesimally (i.e.",
"where $ad_z:z\\mapsto [z,\\quad ]$ ) (REF ) becomes $\\Pi ([K_3,K_l])=-(i/2)[D_{q_i},\\Pi _l]$ where in the second step we have let $z\\mapsto K_3$ since this generator is spanning the $\\mathfrak {u}(1)$ subalgebra, and hence we replace the map $\\lambda $ with the induced map $\\lambda ^{\\prime }$ ; and where the square bracketed expression on the right hand sides (but not the left) are the ordinary matrix commutatorsOnce again we must note a minor technicality, due to the differing values of $k$ which necessitates a slightly different treatment of the map embedding $U(1)$ in $SU(2)$ .",
"For $k=\\pm 1$ , the square brackets on the left of (REF ), i.e.",
"the abstract commutator relations, become the ordinary matrix commutator in the case that the representation used is the matrix representation given in terms of either the Pauli matrices (for $k=1$ ) or the matrices shown in (REF ) (for $k=-1$ ).",
"For $k=0$ however, the square brackets refer to the more abstract commutator, as defined in (REF ), acting on the differential expressions (REF )..) Using the commutation relations (REF ), we find that $\\begin{array}{ccc}\\Pi _1=(i/2)[D_{q_i},\\Pi _2], & \\quad &\\Pi _2=-(i/2)[D_{q_i},\\Pi _1].\\end{array}$ Substituting the second of those expressions into the first gives us $\\Pi _1=(1/4)[D_{q_i},[D_{q_i},\\Pi _1]],$ from which it is easy to see that $((q_i-q_j)^2-4)\\Pi _{1,ij}=0$ .",
"Given this, and since the $\\Pi $ are traceless and anti-Hermitian (and assuming that $q_i \\ge q_j$ for $i < j$ ), we can deduce that $\\begin{array}{cc}\\Pi _1=\\frac{1}{2}(C-C^H), & \\Pi _2=-\\frac{i}{2}(C+C^H),\\\\\\end{array}$ where $C$ is (as previously described) an upper triangular complex $N\\times N$ matrix with Hermitian conjugate $C^H$ and where $C_{ij}\\ne 0$ iff $q_i = q_j+2.$ According to the second part of Wang's theorem, the curvature 2-form of this invariant connection on the two-dimensional subspace is given by $\\langle \\tilde{X}\\wedge \\tilde{Y}/\\Omega \\rangle =[\\Pi (X),\\Pi (Y)] - \\Pi ([X,Y])$ where $X, Y\\in SU(N)$ , and $\\tilde{X}$ and $\\tilde{Y}$ are the corresponding generators of $P$ – since $\\Omega =d\\tilde{A}+\\tilde{A}\\wedge \\tilde{A}$ here refers purely to the angular part of the gauge potential 1-form $\\tilde{A}$ , then $\\langle \\tilde{X}\\wedge \\tilde{Y}/\\Omega \\rangle $ represents the curvature 2-form on the bundle itself restricted to the angular part.",
"Therefore, we find $\\langle \\tilde{K_1}\\wedge \\tilde{K_2}/\\Omega \\rangle =[\\Pi _1,\\Pi _2] - \\Pi \\left([K_1,K_2]\\right).$ Substituting in the forms for $\\Pi _1$ and $\\Pi _2$ given in (REF ), and noting from (REF ) that (since $\\Pi $ is a linear map) $\\Pi ([K_1,K_2]) = -(i/2)kD_{q_i}$ we find that the curvature is given as $\\tilde{F}\\equiv \\langle \\tilde{K_1}\\wedge \\tilde{K_2}/\\Omega \\rangle =-(i/2)([C,C^H]-kD)$ with other components vanishing.",
"However, since the curvature is a tensorial form it must be the pullback of a scalar multiple of the area element of the 2-space we are working with (sphere, plane or hyperboloid), which we have already described metrically using the function $f_k(\\theta )$ .",
"That is, the angular component of the full 4D curvature must be of the form $\\tilde{F}f_k(\\theta )d\\theta \\wedge d\\phi $ (in the $(\\theta ,\\phi )$ co-ordinate system appropriate to the space in question.)",
"This gives us the equation $d\\tilde{A}+\\frac{1}{2}[\\tilde{A}, \\tilde{A}]=\\tilde{F}f_k(\\theta )d\\theta \\wedge d\\phi .$ Now we are ready to construct the angular part of the connection, i.e.",
"the local potential $\\tilde{A}=A_\\theta d\\theta + A_\\phi d\\phi $ , an $\\mathfrak {su}(N)$ valued one-form on the two dimensional subspace.",
"We can easily check that (REF ) is satisfied by the potential $\\begin{array}{rlcrl}A_\\theta &=\\frac{1}{2}(C-C^H), & \\quad &A_\\phi &=-\\frac{i}{2}\\left[ (C+C^H)f_k(\\theta )+D\\frac{df_k}{d\\theta }\\right].\\end{array}$ A helpful relation in showing this is $\\frac{d^2f_k(\\theta )}{d\\theta ^2}=-kf_k(\\theta )\\quad \\forall k\\in \\lbrace -1,0,1\\rbrace ;$ note that the presence of the $\\frac{df_k}{d\\theta }$ term is due to the fact that the curvature must be the pullback of a tensorial form and hence we end up with a logarithmic derivative, evident when we take the $f_k(\\theta )$ out of the square brackets in $A_\\phi $ .",
"Finally, we augment the potential with $t$ and $r$ components, which we may do easily due to the product topology of the whole 4-manifold (REF ), such that $\\begin{array}{l}A_t\\equiv \\mathcal {A}\\,dt,\\\\A_r\\equiv \\mathcal {B}\\,dr,\\end{array}$ where the forms of $\\mathcal {A}$ and $\\mathcal {B}$ are given in the next subsection.",
"Hence we have confirmed our ansatz for the gauge potential (REF ).",
"As we noted, the gauge potential is never unique since there are degrees of freedom left over in the structure equations – there are many other possible forms, including Witten's [59] – but it is enough that we have found a manifestly $SU(N)$ invariant gauge potential that satisfies the structural constraint (REF ).",
"Another approach that can be taken is to use the symmetry equations for the Lie algebra in question, as in [23]."
],
[
"Deriving the field equations",
"We now outline a way of constructing an irreducible representation of $\\mathfrak {su}(N)$ [31] with reference to the previously derived gauge potential (REF ).",
"We first consider $\\mathcal {A}$ and $\\mathcal {B}$ to be general imaginary diagonal and traceless matrices which can be written in the form $\\begin{array}{cc}\\left(\\mathcal {A}\\right)_{jj}=\\frac{i}{2}\\alpha _{j}(r),\\quad & \\left(\\mathcal {B}\\right)_{jj}=\\frac{i}{2}\\beta _{j}(r)\\\\\\end{array}$ for real functions $\\alpha _j(r)$ , $\\beta _j(r)$ .",
"We also recall that the form of the diagonal matrix $D$ is specified uniquely, given the equation (REF ), i.e.",
"$\\left(D\\right)_{jj}=N+1-2j.$ The form of the matrices $C$ (and hence $C^H$ ) were also determined in the previous section (REF , REF ).",
"Now, we can simplify the field equations considerably: For purely magnetic solutions, we must set $\\alpha _j=0$ (as opposed to the cases where dyons are considered – see for instance [14], [13], [44]).",
"Due to gauge freedom, we may set $\\beta _j=0$ as well.",
"Finally, one of the Yang-Mills equations has the immediate solution $\\gamma _j=0$ .",
"The last item also gives us a final form for $C$ – the only non-zero elements, indexed by $(j, j+1)$ , are $(C)_{j,j+1}=\\omega _j.$ The details of this derivation are given in [31], [32].",
"The Yang-Mills equations with a cosmological constant thus take the form: $r^2\\mu \\omega ^{\\prime \\prime }_{j}+\\left(2m-2r^3 p_{\\theta ,k}-\\frac{2\\Lambda r^3}{3}\\right)\\omega ^{\\prime }_{j}+W_{k,j}\\omega _j=0$ where $\\begin{split}p_{\\theta ,k}&=\\frac{1}{4r^4}\\sum ^N_{j=1}\\left[\\left(\\omega ^2_j-\\omega ^2_{j-1}-k(N+1-2j)\\right)^2\\right],\\\\W_{k,j}&=k-\\omega ^2_j+\\frac{1}{2}\\left(\\omega ^2_{j-1}+\\omega ^2_{j+1}\\right),\\end{split}$ and the Einstein equations take the form $m^{\\prime } =\\mu G+r^2p_{\\theta ,k}$ and $\\Delta ^{\\prime } = \\frac{S^{\\prime }}{S}=\\frac{2G}{r},$ where $G=\\sum ^{N-1}_{j=1}\\omega _j^{\\prime 2}.\\\\$ It can readily be checked that these reduce in the correct limit – see equations (6, 7, 8) in [9].",
"We can note several things.",
"From now on, we assume that all the $\\omega _j(r)$ are in general non-zero so that we are dealing with genuinely $\\mathfrak {su}(N)$ solutions (see, for example, [53], [27], [28], [29] for the consequences of violating this assumption in the $\\Lambda =0$ case).",
"For later convenience, we have also introduced $S\\equiv e^\\Delta $ .",
"We define $\\omega _0\\equiv \\omega _N\\equiv 0$ , so that (REF , REF , REF ) reduce to the $\\mathfrak {su}(2)$ equations in the correct limit (see below).",
"As in the purely $k=1$ case, there are two symmetries respected by the ansätze.",
"Firstly the mapping $\\omega _j\\rightarrow -\\omega _j$ is an invariant mapping separately for each $j$ , and secondly the substitution $j\\rightarrow N-j$ for all $j$ is a symmetry.",
"Finally we note that, since the Einstein equation for $S$ decouples from the rest, it can be integrated separately once we know the character of the metric function $\\mu $ and the gauge field functions $\\omega _j$ , to give the solution $S=e^{\\Delta }=\\exp \\left(\\int ^r_c\\frac{2G}{r}dr\\right)$ for some arbitrary constant $c$ .",
"So the $N$ functions on which we will concentrate are $\\mu $ and the $\\omega _j$ s. For completeness, we shall show what forms these equations take for the simplest non-trivial case, $\\mathfrak {su}(2)$ .",
"In this case, there is one gauge field function which we call $\\omega $ , and the following functions simplify considerably: $\\begin{split}p_{\\theta ,k}&=\\frac{(k-\\omega ^2)^2}{4r^4},\\\\G&=\\omega ^{\\prime 2}.\\end{split}$ This means the field equations (REF ) become the single equation $r^2\\mu \\omega ^{\\prime \\prime }+\\left(2m-2r^3p_{\\theta ,k}-\\frac{\\Lambda r^3}{3}\\right)\\omega ^{\\prime }+(k-\\omega ^2)\\omega =0,$ and the field equations (REF , REF ) become $\\begin{split}m^{\\prime } &=\\mu \\omega ^{\\prime 2}+\\frac{(k-\\omega ^2)^2}{4r^2},\\\\\\Delta ^{\\prime } &= \\frac{S^{\\prime }}{S}=\\frac{2G}{r}=\\frac{2\\omega ^{\\prime 2}}{r}.\\end{split}$ The existence of exact solutions to these $\\mathfrak {su}(2)$ equations will be necessary later in our constructive proof of the existence of non-trivial solutions to the $\\mathfrak {su}(N)$ equations."
],
[
"Boundary conditions",
"We are interested in black hole solutions to the field equations (REF , REF , REF ), but these equations are singular at the event horizon $r=r_h$ and as $r\\rightarrow \\infty $ .",
"(Since $\\Lambda <0$ , there is no cosmological horizon to consider.)",
"A first step is to derive the boundary conditions at these singular points.",
"Local existence has been proven already in $\\mathfrak {su}(N)$ EYM theory for the cases $\\Lambda =0$ [45], [32] and for $\\Lambda <0$ with spherical symmetry ($k=1$ ) [6]; our intention is to extend these results to $k\\ne 1$ .",
"We are ignoring solitons, the particle-like solutions to the equations which possess no event horizon and so continue to be regular in the limit $r\\rightarrow 0$ .",
"(See [6], [2], [54], [17], [3], [19], [49] among others, for a discussion of these in the literature.)",
"The Ricci curvature scalar $R = g_{\\mu \\nu }R^{\\mu \\nu }$ (for $R^{\\mu \\nu }$ the Ricci tensor) is complicated, but always involves the terms: $R=-4\\Lambda +\\frac{2(k-1)}{r^2}+...$ where the ellipses refer to terms involving $m$ , $S$ and their derivatives.",
"Hence, only for $k=1$ is the curvature non-singular at $r=0$ .",
"For $k\\ne 1$ , at $r=0$ the Riemann scalar has an essential singularity, hence there is no such thing as globally regular solutions in these cases.",
"Hence, we shall only need to pay attention to (exterior) black hole solutions, meaning that our two singular points of interest are the event horizon $r=r_h$ and the asymptotic region $r\\rightarrow \\infty $ ."
],
[
"Event horizon",
"We assume that the black hole solutions have a regular, non-extremal event horizon at $r=r_h$ ; i.e.",
"for which $\\mu (r_h)$ has a single zero.",
"This fixes the value of $m_h\\equiv m(r_h)$ as: $2m_h=kr_h-\\frac{\\Lambda r_h^3}{3}.$ We assume that the field variables $m(r)$ , $S(r)$ and $\\omega _j(r)$ have regular Taylor expansions about $r_h$ , i.e.",
"that for $r\\approx r_h$ , $\\begin{split}m(r)&=m(r_h)+m^{\\prime }(r_h)(r-r_h)+O(r-r_h)^2,\\\\S(r)&=S(r_h)+S^{\\prime }(r_h)(r-r_h)+O(r-r_h)^2,\\\\\\omega _j(r)&=\\omega _j(r_j)+\\omega ^{\\prime }_j(r_h)(r-r_h)+O(r-r_h)^2.\\\\\\end{split}$ Letting $\\mu (r_h)=0$ in the field equation (REF ) gives us the following boundary conditions for $\\omega ^{\\prime }_j(r_h)$ : $\\omega ^{\\prime }_j(r_h)=\\frac{W_{k,j}(r_h)\\omega _j(r_h)}{2m(r_h)-2r^3_hp_{\\theta ,k}(r_h)-\\frac{2\\Lambda r_h^3}{3}}.$ So for fixed $\\Lambda $ the expansions (REF ) are determined entirely by the $N+1$ quantities given by $r_h$ , $\\omega _j(r_h)\\equiv \\omega _{j,h}$ and $S(r_h)$ .",
"Setting $\\mu (r_h)=0$ in the field equation (REF ), we can show that for a non-extremal event horizon we require $2m^{\\prime }(r_h)=2r^2_hp_{\\theta ,k}(r_h)<k-\\Lambda r_h^2,$ which weakly constrains the possible values of $\\omega _j$ near the horizon.",
"It also follows from (REF ) that for $k=-1$ , we have a minimum event horizon radius given by the constraint $r_h>\\sqrt{\\frac{-1}{2p_{\\theta ,k}(r_h)+\\Lambda }}.$ Since the field equations are invariant under $\\omega _j\\mapsto -\\omega _j$ , we may consider $\\omega _{j,h}>0$ (without loss of generality)."
],
[
"Infinity",
"At infinity, we expect that the black hole will approach the topological analogue to adS space, i.e.",
"that $\\mu (r)\\rightarrow k-\\frac{\\Lambda r^2}{3}$ as $r\\rightarrow \\infty $ .",
"We assume that all the field variables have regular expansions in $r^{-1}$ as $r$ approaches infinity: $\\begin{array}{rclcrcl}m(r)&=&M+O(r^{-1}),&\\qquad &S(r)&=&1+O(r^{-1}),\\\\\\omega _j(r)&=&\\omega _{j,\\infty }+O(r^{-1}).&&&&\\\\\\end{array}$ Just as in the spherically symmetric case with $\\Lambda <0$ , there are no a priori constraints on the values of $\\omega _{j,\\infty }$ , so that in general, the adS topological black holes will carry a global magnetic charge [48]."
],
[
"Trivial solutions",
"A key part of our proof of the existence of non-trivial $\\mathfrak {su}(N)$ solutions is the analyticity in the initial conditions of the field variables, so that we may find solutions with initial conditions in a neighbourhood of those producing known solutions.",
"Thus, it is important that we can find some `trivial' known solutions.",
"The field equations (REF , REF , REF ) are non-linear coupled equations, but they possess a number of analytic, trivial solutions.",
"These arise by letting the functions $\\omega _j(r)$ be identical to a constant.",
"This produces as a constraint $W_{k,j}\\omega _j=0.$"
],
[
"Reissner-Nordström-adS (RNadS)",
"Equation (REF ) clearly has the solution $\\omega _j(r)\\equiv 0$ for all $j$ , which gives the RNadS black hole with metric function $\\mu (r)=k-\\frac{2M}{r}+\\frac{Q^2}{r^2}-\\frac{\\Lambda r^2}{3}$ where the magnetic charge $Q$ is defined as $Q^2=\\frac{k^2}{6}N(N+1)(N-1).$ This solution exists for all three values of $k$ .",
"Note that the magnetically charged RNadS black hole is only a solution of the field equations if the charge is exactly this value.",
"Note also that for $k=0$ there is a difference.",
"In that case, $Q=0$ , and hence in the $\\mathfrak {su}(N)$ case with planar symmetry, the solution is more similar to the Schwarzchild-adS solution, discussed next."
],
[
"Schwarzchild-adS (SadS)",
"Writing (REF ) out in full and letting $\\Omega _j\\equiv \\omega _j^2$ turns the $W_{k,j}=0$ into a system of linear equations, with the solution $\\Omega _j=kj(N-j).$ For $k=1$ , we have $\\Omega _j=\\omega ^2_j=j(N-j)$ , so that $\\omega _j=\\pm \\sqrt{j(N-j)},$ for all $j$ , which agrees with previous work [6].",
"This gives us $m(r)=M=\\mbox{ constant.",
"}$ For the case $k=-1$ , it can be immediately seen that for (REF ) to be correct, $\\omega _j$ will have to be imaginary.",
"Thus in this case, all we have is RNadS-type solutions (due to $Q\\ne 0$ ) and the only trivial embedded solutions are found by setting $\\omega _j\\equiv 0$ for all $j$ .",
"For $k=0$ however, it is obvious that $\\omega _j\\equiv 0$ , which coincides with the $Q=0$ case above, hence the only type of solution we have is the SadS-type solution."
],
[
"$\\mathfrak {su}(2)$ embedded solutions",
"We now demonstrate that it is possible to embed any solution of the $\\mathfrak {su}(2)$ field equations as an $\\mathfrak {su}(N)$ solution by a simple rescaling of variables.",
"Note that this is a fairly trivial step from the similar $\\Lambda =0$ , $k=1$ case discussed in [32], [41] and the $\\Lambda <0$ , $k=1$ case discussed in [6].",
"Proposition 1 Any $\\mathfrak {su}(2)$ solution can be rescaled and embedded as an $\\mathfrak {su}(N)$ solution.",
"Proof We begin with the field equations (REF , REF , REF ).",
"We attempt to rescale them with the following definitions: $\\begin{array}{lll}\\tilde{N}\\equiv \\frac{1}{6}N(N-1)(N+1), & R\\equiv \\tilde{N}^{-\\frac{1}{2}}r, & \\tilde{\\Lambda }\\equiv \\tilde{N}\\Lambda , \\\\\\omega _j\\mapsto \\sqrt{j(N-j)}\\omega , & \\tilde{m}\\equiv \\tilde{N}^{-\\frac{1}{2}}m. & \\\\\\end{array}$ This rescaling leads to the following equations: $\\begin{array}{rl}0&=R^2\\mu \\left(\\displaystyle {\\frac{d^2\\omega }{dR^2}}\\right)+\\left(2\\tilde{m}-\\displaystyle {\\frac{2\\tilde{\\Lambda }R^3}{3}}-2R^3\\tilde{p}_{\\theta ,k}\\right)\\left(\\displaystyle {\\frac{d\\omega }{dR}}\\right)+(k-\\omega ^2)\\omega \\\\&\\\\\\displaystyle {\\frac{d\\tilde{m}}{dR}}&=\\mu \\left(\\displaystyle {\\frac{d\\omega }{dR}}\\right)^2+R^2\\tilde{p}_{\\theta ,k}\\\\&\\\\\\displaystyle {\\frac{1}{S}}\\frac{dS}{dR}&=\\displaystyle {\\frac{2}{R}}\\left(\\displaystyle {\\frac{d\\omega }{dR}}\\right)^2\\\\\\end{array}$ with $\\tilde{p}_{\\theta ,k}\\equiv \\displaystyle {\\frac{(\\omega ^2-k)^2}{2R^4}}.$ It can be checked that these are exactly the same equations as the $\\mathfrak {su}$ (2) field equations with general $k$ (REF , REF ).$\\Box $ It is easy to see that the boundary conditions (REF , REF ) reduce to the topological $\\mathfrak {su}(2)$ boundary equations given in [9].",
"We have therefore proven that any $\\mathfrak {su}(2)$ topological black hole solution can be embedded into $\\mathfrak {su}(N)$ EYM to yield an asymptotically topological anti-de-Sitter black hole.",
"One final fact of note is that these embedded solutions will have the same number of nodes as the original $\\mathfrak {su}(2)$ solution, since if $\\omega (r_0)=0$ for some $r_0$ , then so will $\\tilde{\\omega }_j(r_0)=0$ , according to (REF ).",
"This fact will be of importance in the proof of our second result (Theorem REF )."
],
[
"Existence of non-trivial $\\mathfrak {su}(N)$ black hole solutions",
"Our goal is to prove that we can find solutions throughout the whole range of the spacetime; in other words, solutions which can be proven to exist at the event horizon, and can be integrated out arbitrarily far into the asymptotic regime.",
"The way we do this is by proving that solutions exist locally at the event horizon and locally as $r\\rightarrow \\infty $ ; then we prove that given any solution which remains regular in a specified range, we can continue to integrate that solution outwards, right into the asymptotic regime.",
"We use these results to argue that locally regular solutions can be `patched together' to form global black hole solutions.",
"We note existence has already been proven for $\\Lambda =0$ case in $\\mathfrak {su}(N)$ EYM [41] and with $\\Lambda < 0$ and $k=1$ [6]."
],
[
"Local existence of solutions",
"In this section, following e.g.",
"[16], we shall make use of an established theorem of differential equations [22] to demonstrate the local existence of solutions at the event horizon and near infinity.",
"We begin by stating the theorem: Theorem 2 [16] Consider a system of differential equations for $n+m$ functions $\\mathbf {a}=(a_1,a_2,\\ldots ,a_n)$ and $\\mathbf {b}=(b_1,b_2,\\ldots ,b_m)$ of the form $ \\begin{split}x\\frac{da_i}{dx}&=x^{p_i}f_i(x,\\mathbf {a},\\mathbf {b}),\\\\x\\frac{db_i}{dx}&=-\\lambda _ib_i+x^{q_i}g_i(x,\\mathbf {a},\\mathbf {b})\\\\\\end{split}$ with constants $\\lambda _i>0$ and integers $p_i,q_i\\ge 1$ and let $\\mathcal {C}$ be an open subset of $\\mathbb {R}^n$ such that the functions $f_i$ and $g_i$ are analytic in a neighbourhood of $x=0$ , $\\mathbf {a}=\\mathbf {c}$ , $\\mathbf {v}=\\mathbf {0}$ , for all $\\mathbf {c}\\in \\mathcal {C}$ .",
"Then there exists an $n$ -parameter family of solutions of the system such that $\\begin{matrix}&a_i(x)=c_i+O(x^{p_i}),&b_i=O(x^{q_i}),\\end{matrix}$ where $a_i(x)$ and $b_i(x)$ are defined for $\\mathbf {c}\\in \\mathcal {C}$ , $|x|<x_0(\\mathbf {c})$ and are analytic in $x$ and $\\mathbf {c}$ .",
"This theorem allows us to parameterize the family of solutions near a singular point of a set of ordinary differential equations.",
"We need to take each singular point in turn (here, $r=r_h$ and $r\\rightarrow \\infty $ ) and change variables so that the field equations are in the form required by the theorem.",
"After that, it is elementary to verify the forms we have chosen for our expansions of the field variables near the singular points (REF , REF ).",
"Now we turn our attention to the field equations at $r=r_h$ .",
"We assume the existence of a non-degenerate event horizon, so that $\\mu (r_h)=0$ but $\\mu ^{\\prime }(r_h)>0$ is finite.",
"We begin by proving a proposition analogous to that proven for $\\mathfrak {su}(N)$ , $k=1$ in [6].",
"Proposition 3 There exists an N-parameter family of local solutions of the field equations near $r=r_h$ analytic in $r_h$ , $\\omega _{j,h}$ and $\\rho =r-r_h$ such that $\\begin{split}\\mu (r_h+\\rho )&=\\mu ^{\\prime }(r_h)+O(\\rho ),\\\\\\omega _j(r_h+\\rho )&=\\omega _{j,h}+\\omega ^{\\prime }(r_h)\\rho +O(\\rho ^2),\\\\\\end{split}$ where $\\mu ^{\\prime }(r_h)$ and $\\omega ^{\\prime }(r_h)$ are functions of the $\\omega _{j,h}$ .",
"Proof Following [16], [41], [45], [6], let us define a new independent variable $x=r-r_h$ , and define some new dependent variables: $\\rho \\equiv r,\\quad \\quad \\lambda \\equiv \\frac{\\mu }{\\rho },\\quad \\quad \\psi _j\\equiv \\omega _j,\\quad \\quad \\xi _j\\equiv \\frac{\\mu \\omega ^{\\prime }_j}{\\rho }=\\lambda \\omega _j^{\\prime }.\\\\$ The field equations take the form: $\\begin{array}{lll}x\\displaystyle {\\frac{d\\rho }{dx}}=x, & x\\displaystyle {\\frac{d\\psi _j}{dx}}=\\displaystyle {\\frac{x\\xi _j}{\\lambda }}, & x\\displaystyle {\\frac{d\\lambda }{dx}}=-\\lambda +xH_\\lambda +F_\\lambda ,\\\\&&\\\\x\\displaystyle {\\frac{d\\Lambda }{dx}}=0, & x\\displaystyle {\\frac{dS}{dx}}=\\displaystyle {\\frac{2x}{\\rho }}GS, & x\\displaystyle {\\frac{d\\xi _j}{dx}}=-\\xi _j+xH_{\\xi _j}+F_{\\xi _j};\\end{array}$ where $\\begin{split}F_\\lambda \\equiv &\\frac{1}{\\rho }\\left(k-\\frac{2}{\\rho ^2}\\mathcal {P}-\\Lambda \\rho ^2\\right),\\\\H_\\lambda \\equiv &-\\frac{\\lambda }{x}(1+2\\mathcal {G}),\\\\F_{\\xi _j}\\equiv &-\\frac{W_{k,j}\\psi _j}{\\rho ^2},\\\\H_{\\xi _j}\\equiv &x\\frac{H_\\lambda \\xi _j}{\\lambda };\\\\\\end{split}$ and we note that $\\begin{split}W_{k,j}=&k-\\psi ^2_j+\\frac{1}{2}(\\psi ^2_{j-1}+\\psi ^2_{j+1});\\end{split}$ and define $\\begin{split}\\mathcal {P}\\equiv & r^4p_{\\theta ,k}=\\frac{1}{4}\\sum ^N_{j=1}\\left[\\left(\\psi ^2_j-\\psi ^2_{j-1}-k(N+1-2j)\\right)^2\\right],\\\\\\mathcal {G}\\equiv & \\frac{1}{\\lambda ^2}\\sum ^{N-1}_{j=1}\\xi _j^2.\\end{split}$ So the functions $F_\\lambda $ , $F_{\\xi _j}$ , $H_\\lambda $ and $H_{\\xi _j}$ are polynomials in $1/\\rho $ , $1/\\lambda $ , $\\rho $ , $\\lambda $ , and $\\Lambda $ .",
"The equations (REF ) are not yet in the required form, so next we let $\\begin{array}{cc}\\tilde{\\xi _j}\\equiv \\xi _j-F_{\\xi _j}, & \\tilde{\\lambda }\\equiv \\lambda -F_\\lambda .\\end{array}$ This makes the two non-conforming equations (for $xd\\xi _j/dx$ and $xd\\lambda /dx$ ) take the form: $\\begin{split}x\\frac{d\\tilde{\\xi _j}}{dx}&\\equiv -\\tilde{\\xi _j}+xG_{\\xi _j},\\\\x\\frac{d\\tilde{\\lambda }}{dx}&\\equiv -\\tilde{\\lambda }+x G_\\lambda ,\\\\\\end{split}$ in which $G_\\lambda $ and $G_{\\xi _j}$ are polynomials given by $G_{\\xi _j}\\equiv H_{\\xi _j}-\\frac{d F_{\\xi _j}}{dx},\\quad \\quad G_{\\lambda }\\equiv H_{\\lambda }-\\frac{d F_\\lambda }{dx}.$ The polynomials $G_\\lambda $ and $G_{\\xi _j}$ are lengthy; suffice it to say, it can be checked that both are polynomials in $1/\\rho $ , $1/\\lambda $ , $\\rho $ , $\\lambda $ , $\\tilde{\\lambda }$ , $\\tilde{\\xi _j}$ and $\\Lambda $ .",
"Hence, due to Theorem REF , there then exist solutions to the equations of the form $\\begin{array}{lll}\\rho =r_h+O(x), & \\psi _j=\\omega _{j,h}+O(x), & \\tilde{\\lambda }=O(x), \\\\\\tilde{\\xi _j}=O(x), & S=S(r_h)+O(x); & \\end{array}$ with $\\rho $ , $\\tilde{\\lambda }$ , $\\psi _j$ and $\\tilde{\\xi _j}$ all analytic in $x$ , $r_h$ , $\\omega _j(r_h)$ , $\\Lambda $ and $S(r_h)$ .",
"Transforming back to our original variables gives us the correct behaviour and analyticity.$\\Box $ We have thus proven existence of solutions to the field equations for a black hole in some neighbourhood of the event horizon $r=r_h$ , satisfying the boundary conditions (REF )."
],
[
"Local existence of solutions at infinity",
"Local existence at infinity for asymptotically adS spherically symmetric black holes has been proven in [6], and the proof extends almost directly to the topological case.",
"We note that, as in the adS spherically symmetric case, it is relatively easy to prove existence and we need only go to first order in the field variables, unlike in the asymptotically flat case where higher order terms were needed and the analysis was more involved [45], [32].",
"We also note that for adS space, the field equations are not actually singular as $r\\rightarrow \\infty $ , but it is convenient to use this method anyway – the above theorem does not require the boundary points to be singular, but it can account for that if needed.",
"Again we prove a proposition analogous to one in [56].",
"Proposition 4 There exists an $2N$ -parameter family of local solutions of the field equations near $r=\\infty $ , analytic in $\\Lambda $ , $\\omega _{j,\\infty }$ , $M$ and $r^{-1}$ such that $\\begin{split}\\mu (r)&=k-\\frac{2M}{r}-\\frac{\\Lambda r^2}{3}+O\\left(\\frac{1}{r^2}\\right),\\\\\\omega _j(r)&=\\omega _{j,\\infty }-\\frac{c_j}{r}+O\\left(\\frac{1}{r^2}\\right).\\\\\\end{split}$ Proof We introduce new variables, following [16], [32], [6]: $\\begin{matrix}&x=r^{-1},&\\psi _j=\\omega _j,&\\xi _j=r^2\\omega ^{\\prime }_j,&\\lambda =r\\left(k-\\mu -\\frac{\\Lambda r^2}{3}\\right)\\equiv 2m.&\\end{matrix}$ Then the field equations take the form: $\\begin{array}{lll}x\\displaystyle {\\frac{d\\lambda }{dx}}=-xf_\\lambda , & x\\displaystyle {\\frac{d\\psi _j}{dx}}=-x\\xi _j, &x\\displaystyle {\\frac{d\\xi _j}{dx}}=xg_{\\xi _{j}},\\\\&&\\\\x\\displaystyle {\\frac{dS}{dx}}=x^4f_S, & x\\displaystyle {\\frac{d\\Lambda }{dx}}=0; &\\end{array}$ where $\\begin{array}{rl}f_S\\equiv & S\\sum _{j=1}^{N-1}\\xi _j^2,\\\\f_\\lambda \\equiv &2\\mu x^2\\sum ^{N-1}_{j=1}\\xi ^2_j+\\frac{1}{2}\\sum ^N_{j=1}\\left(\\psi _j^2-\\psi _{j-1}^2-k(N+1-2j)\\right)^2,\\\\g_{\\xi _j}\\equiv & -\\frac{W_{j,k}\\psi _j}{\\mu x^2}+\\frac{1}{\\mu }\\left(\\lambda -\\frac{2\\mathcal {P}}{x^3}\\right)\\\\\\end{array}$ (where we have used the forms from (REF , REF )); and since 1/$\\mu $ is at least of order $x^2$ as $x\\rightarrow 0$ , it can be observed that all of these polynomials are non-singular as $x\\rightarrow 0$ .",
"Therefore we have proven local existence of solutions at infinity, and Theorem REF confirms that the functions exhibit the required behaviour near infinity (REF ).",
"We also note that solutions are analytic in $x$ , $\\Lambda $ , $M$ , $\\omega _{j,\\infty }$ and $S_\\infty $ .",
"Also, by rescaling the time co-ordinate in the metric we can fix $S_\\infty =1$ so that the spacetime is asymptotically `topological adS' as we have described it – this satisfies the boundary conditions (REF ), and the field variables are analytic in $M$ , $\\omega _{j,\\infty }$ , $c_j$ , $r$ and $\\Lambda $ .$\\Box $ Our strategy for proving the existence of genuinely $\\mathfrak {su}(N)$ solutions involves showing that given initial conditions near the singular point at the event horizon $r=r_h$ , we can regularly integrate the field equations out to infinity.",
"To be specific, we wish to show that as long as $\\mu >0$ (required for the spacetime to be regular), then any solution that is regular up to a certain $r$ value can be extended for larger values of $r$ .",
"To do this we use the field equations to prove the following lemma, which has been proven for the spherically symmetric cases of $\\mathfrak {su}(2)$ with $\\Lambda =0$ [16], of $\\mathfrak {su}(N)$ with $\\Lambda =0$ [41], and of $\\mathfrak {su}(N)$ with $\\Lambda <0$ and $k=1$ [6].",
"Lemma 5 As long as $\\mu >0$ all field variables are regular functions of $r$ .",
"To prove this lemma we shall take some interval $I=[r_0,r_1)$ (such that $r_h<r_0<r_1$ ), assume the solution is regular in this interval and that $\\mu (r)>0$ on the interval $\\bar{I}=[r_0,r_1]$ , and then show that this implies regularity at $r=r_1$ as well.",
"In other words, as long as $\\mu >0$ , if we start off with arbitrary initial conditions outside the event horizon (i.e.",
"a `piece' of a regular solution) then we can continue to integrate it regularly outwards into the asymptotic region.",
"This method is adapted from a similar one used in [16].",
"Proof If $\\mu (r_1)>0$ , then by integrating the equation for $m^{\\prime }$ in (REF ), we obtain $kr_1-\\frac{\\Lambda r_1^3}{3}>2m(r_1)\\ge 2\\int ^{r_1}_{r_0}\\sum _{j=1}^{N-1}\\mu \\omega ^{\\prime 2}_j dr.$ Since $\\mu $ must have a minimum in $\\bar{I}$ , we define $\\mu _{min}\\equiv \\inf (\\mu :r\\in \\bar{I})>0.$ Therefore: $2\\mu _{min}\\int ^{r_1}_{r_0}\\sum _{j=1}^{N-1}\\omega ^{\\prime 2}_j dr\\le kr_1-\\frac{\\Lambda r_1^3}{3}$ or $\\int ^{r_1}_{r_0}\\sum _{j=1}^{N-1}\\omega ^{\\prime 2}_jdr\\le \\frac{kr_1-\\frac{\\Lambda r_1^3}{3}}{2\\mu _{min}}.$ The LHS is bounded above by the RHS, which we notice is constrained to be positive – this is because of the constraint (REF ), where we remember we require $m_h>0$ , and because the mass function is monotonic ($m^{\\prime }(r)>0\\:\\: \\forall r$ ), which together means that $m(r)>0\\:\\: \\forall r$ .",
"We can also see that since the LHS is a sum of squared (i.e.",
"positive) terms in $\\omega ^{\\prime }_j$ , each term must be bounded below by zero.",
"Hence $G$ is bounded and so is $2G/r$ (and thus $\\Delta ^{\\prime }$ ), so direct integration gives us $\\Delta $ (or $S$ ) bounded.",
"Using the Cauchy-Schwarz inequality and performing an integration gives us $\\begin{split}\\int ^{r_1}_{r_0}\\sum _{j=1}^{N-1}\\omega ^{\\prime 2}_jdr&\\ge \\frac{1}{r_1-r_0}\\sum _{j=1}^{N-1}(\\omega _j(r_1)-\\omega _j(r_0))^2.\\end{split}$ From here we see that each $\\omega _j(r_1)$ is finite.",
"Finally, we take the Yang-Mills equations, which can be rewritten in the form $\\left(\\mu S\\omega _j^{\\prime }\\right)^{\\prime }=-\\frac{SW_{k,j}\\omega _j}{r^2}.$ Let $\\mu \\omega ^{\\prime }_j\\equiv y$ .",
"Then we can write the above equation as $(Sy)^{\\prime }=-\\frac{SW_{k,j}\\omega _j}{r^2}.$ From earlier, we can say that $S(r)$ is finite for all $r\\in \\bar{I}$ .",
"Now we have $S(r_1)y(r_1)-S(r_0)y(r_0)=-\\int ^{r_1}_{r_0}S\\frac{W_{k,j}\\omega _j}{r^2}dr.$ We can now apply the Cauchy-Schwarz inequality again to obtain $\\left(\\int ^{r_1}_{r_0}S\\frac{W_{k,j}\\omega _j}{r^2}dr\\right)^2\\le \\int ^{r_1}_{r_0}S^2dr\\int ^{r_1}_{r_0}\\frac{W^2_{kj}\\omega ^2_j}{r^4}dr.$ Now, both $S(r)$ and $\\frac{W_{k,j}\\omega _j}{r^2}$ are finite on the interval $[r_0,r_1]$ , therefore the integrals on the RHS are finite, therefore the LHS is bounded.",
"Therefore, from (REF ) we get $(Sy)(r_1)$ finite, therefore $\\mu (r_1)S(r_1)\\omega ^{\\prime }_j(r_1)$ is finite, and since $\\mu (r_1)>0$ , we finally have $\\omega ^{\\prime }_j(r_1)$ is finite.",
"Therefore we have proven that if $\\mu >0$ , then given some initial conditions (either on or outside the event horizon, so that $\\mu >0$ ) to begin with, we can continue to regularly integrate out to obtain a solution.$\\Box $ We now have all the pieces we need to `patch together' solutions which exist and are regular throughout the range of the spacetime.",
"The exact nature of this patching together will be discussed at the end.",
"We now briefly turn our attention to the behaviour of the equations in the asymptotic limit."
],
[
"Asymptotic behaviour of solutions in adS space",
"The main reason for the abundance of black hole solutions in the $\\Lambda <0$ case (as opposed to the $\\Lambda =0$ case) is the behaviour of the field equations in the asymptotic limit $r\\rightarrow \\infty $ .",
"This section closely follows our work in [6].",
"As $r\\rightarrow \\infty $ , the Yang-Mills equations (REF ) approximate to: $r^2\\left(-\\frac{\\Lambda r^4}{3}\\right)\\omega ^{\\prime \\prime }_j-\\frac{2\\Lambda r^3}{3}\\omega ^{\\prime }_j+W_{k,j}\\omega _j=0.$ We attempt to make the equations autonomous.",
"First we introduce a new variable $\\tau $ [6] such that $\\tau =\\sqrt{-\\frac{3}{\\Lambda }}\\frac{1}{r},$ which reduces the equations (REF ) down to $\\ddot{\\omega _j}+W_{k,j}\\omega _j=0.$ This tells us that the system has a critical point when $\\left(k-\\omega ^2_j+\\frac{1}{2}\\left(\\omega _{j+1}^2+\\omega _{j-1}^2\\right)\\right)\\omega _j=0,\\qquad j=1\\ldots N-1.$ This constraint is identical to (REF ), so we already know the solutions (given by (REF ).",
"For $k=1$ , the solutions are $\\begin{matrix}\\omega _j=0 & \\mbox{ or } & \\omega _j=\\pm \\sqrt{j(N-j)}.\\end{matrix}$ For $k\\ne 1$ in general $\\mathfrak {su}(N)$ , the only critical point is at $\\omega _j=0$ .",
"However, as in previous works (e.g.",
"[6], [56]), it is not the nature of the critical points that is important, so much as the choice of variable (REF ) we used to make the equations autonomous.",
"In the $\\Lambda =0$ case, an appropriate choice of radial parameter was $\\tau \\propto -\\log r$ (see e.g.",
"[16]), which goes to infinity as $r$ goes to infinity, so that the system had to reach the critical point (as it had to go `all the way along' its trajectory in the phase space.)",
"In our case, however, we use $\\tau \\propto 1/r$ as being more appropriate.",
"As our radial parameter $r$ goes to infinity, the trajectory on the phase space gets shorter and shorter, so it does not have to reach the critical point of the system at infinity.",
"To put it another way, a solution which corresponds to one within some interval $r\\in [r_1,\\infty )$ (for some large value of $r_1$ ) will be transformed to one in the interval $(0, \\tau _1]$ , and $\\tau _1$ is arbitrarily small for large $r_1$ .",
"This also explains why the values of the $\\omega _j$ s are allowed to be arbitrary at infinity, rather than being constrained to continue along a trajectory until reaching a limit defined by the position of the critical points."
],
[
"Existence of non-trivial $\\mathfrak {su}(N)$ solutions",
"Now have all of the pieces we need (Propositions REF and REF and Lemma REF ) to establish the existence of non-trivial $\\mathfrak {su}(N)$ solutions to the 4D topological EYM field equations.",
"The first theorem we shall prove (Theorem REF ) is one of the two main results of our paper: it establishes the existence of nodeless topological $\\mathfrak {su}(N)$ solutions in the regime $|\\Lambda |\\rightarrow \\infty $ , for any fixed $r_h$ (respecting possible minimum bounds on $r_h$ if $k=-1$ ) and $\\omega _{j,h}$ .",
"Note that Theorem REF is very much in line with the similar proof given in [6] for $k=1$ .",
"The next proposition proves the existence of nodeless topological $\\mathfrak {su}(N)$ solutions in some sufficiently small neighbourhood of an existing such solution – i.e.",
"that these solutions exist in open sets – for any $\\Lambda <0$ .",
"Finally, we use this proposition and the existence of trivial solutions to the field equations to prove the existence of non-trivial, nodeless topological $\\mathfrak {su}(N)$ solutions for $\\Lambda <0$ ."
],
[
"Existence of nodeless topological $\\mathfrak {su}(N)$ solutions for sufficiently large {{formula:9c79444c-9058-4c4b-b109-6b25b27fe786}}",
"For $k=1$ it was demonstrated numerically in [3] that for any value of $N$ , if $|\\Lambda |$ was large enough, all of the solutions were such that the gauge field functions were nodeless; and in [9] it was shown for $\\mathfrak {su}(2)$ that if $k\\ne 1$ , the gauge field is monotonically increasing and therefore nodeless, for a positive $\\omega _h$ .",
"However, it may be observed, as in [6], that as $N$ increases (for fixed $\\Lambda $ ), the region in which solutions may be found grows ever smaller.",
"We now show that for any $N$ , for fixed initial parameters at the event horizon, and for $|\\Lambda |\\rightarrow \\infty $ , corresponding solutions to the field equations exist with gauge field functions non-zero everywhere.",
"It can be noted that the situation here is very similar indeed to both the $\\mathfrak {su}(2)$ [56] and $\\mathfrak {su}(N)$ [6] spherically symmetric cases for $\\Lambda <0$ , for which the analogue of the following proposition has already been proven; so we will only sketch out the derivation here.",
"Theorem 6 For fixed $r_h$ and $\\omega _{j,h}$ , and for $|\\Lambda |$ sufficiently large, there exists a black hole solution of the $\\mathfrak {su}(N)$ EYM field equations such that all the gauge field functions $\\omega _j(r)$ have no zeroes.",
"Proof First, a qualification: if $k = -1$ , and for $\\Lambda <0$ , we find that there is a minimum bound on $r_h$ given by $m(r)$ being monotonic, i.e.",
"$r_h>\\sqrt{\\frac{2m^{\\prime }_h+1}{-\\Lambda }},$ with the RHS manifestly positive.",
"However, it should be noted that if $|\\Lambda |\\rightarrow \\infty $ , this minimum radius $r_h\\rightarrow 0$ .",
"We note that for fixed $r_h$ and $\\omega _{j,h}$ , the constraint (REF ) for a regular event horizon is satisfied for all sufficiently large $|\\Lambda |$ .",
"As in [56], [6], it is helpful to define a length scale $\\ell $ such that $\\ell ^2=-3/\\Lambda $ , and then new variables $\\tilde{m}$ and $\\tilde{\\mu }$ , which will be finite as $|\\Lambda |\\rightarrow \\infty $ , $\\ell \\rightarrow 0$ : $\\tilde{m}=m\\ell ^2,\\qquad \\tilde{\\mu }=k\\ell ^2-\\frac{2\\tilde{m}}{r}+r^2.$ The field equations then take the form $\\begin{split}\\tilde{m}^{\\prime }&=\\left(k\\ell ^2-\\frac{2\\tilde{m}}{r}+r^2\\right)G+\\ell ^2r^2p_{\\theta ,k},\\\\0&=r^2\\left(k\\ell ^2-\\frac{2\\tilde{m}}{r}+r^2\\right)\\omega ^{\\prime \\prime }_j+\\left[2\\tilde{m}-2\\ell ^2r^3p_{\\theta ,k}+2r^3\\right]\\omega ^{\\prime }_j+\\ell ^2W_{k,j}\\omega _j.\\\\\\end{split}$ In the limit $\\ell \\rightarrow 0$ , these equations simplify considerably and have the unique solution $\\tilde{m}(r)=\\tilde{m}(r_h)=\\frac{1}{2}r_h^3,\\qquad S(r)=1,\\qquad \\omega _j(r)=\\omega _j(r_h).$ However we would also like to extend these results to the case of $\\ell $ arbitrarily small (and not just $\\ell =0$ ) by analyticity.",
"We continue along the lines of [6].",
"We can use the change of variables $\\tilde{\\lambda }=\\ell ^2\\lambda ;$ then the equations (REF ) are unchanged except for $x\\frac{d\\ell }{dx}=0$ instead of $x\\frac{d\\Lambda }{dx}=0$ , and for $x\\frac{d\\tilde{\\lambda }}{dx}$ , we get: $x\\frac{d\\tilde{\\lambda }}{dx}=-\\tilde{\\lambda }+xH_{\\tilde{\\lambda }}+F_{\\tilde{\\lambda }},$ where $H_{\\tilde{\\lambda }}=-\\frac{\\tilde{\\lambda }}{\\rho }(1+2G),\\qquad F_{\\tilde{\\lambda }}=\\frac{k\\ell ^2}{\\rho }+3\\rho -\\frac{2\\ell ^2}{\\rho ^3}\\mathcal {P}.$ We can note by looking at equations (REF ) and (REF ) that the only appearance of the constant $k$ is as a `coefficient' of terms in $\\ell ^2$ .",
"Therefore as $\\ell \\rightarrow 0$ we expect the influence of $k$ to be removed completely from the analysis, and the results are then the same as in the spherically symmetric case: the field equations are all regular as $\\ell \\rightarrow 0$ , and the solutions are analytic in $r_h$ , $\\omega _{j,h}$ and $\\ell $ .",
"Near $r=\\infty $ , exactly the same argument used to attain Proposition REF shows we have solutions for $\\ell $ small, and such solutions will be analytic in the field variables.",
"Therefore, we choose some values for $r_h$ and $\\omega _{j,h}$ and fix some $r_1 \\gg r_h$ .",
"When we vary $\\ell $ we see that solutions exist for arbitrarily small $\\ell $ for which all of the gauge field functions have no zeroes in the interval $[r_h,r_1]$ .",
"Finally if $r_1$ is large enough we can use the previously discussed fact that in the asymptotic region we must use a parameter like $\\tau \\propto 1/r$ to show that as $r\\rightarrow \\infty $ , the solutions will remain regular and nodeless.$\\Box $"
],
[
"Existence of nodeless topological $\\mathfrak {su}(N)$ solutions for fixed {{formula:9918a590-7b51-440f-b695-6a33df2c7afb}} in a neighbourhood of trivial solutions",
"We now show that solutions exist in another regime.",
"First, we prove a powerful proposition: that if we assume the existence of an $\\mathfrak {su}(N)$ black hole solution, then we can definitely find other such solutions for any $\\Lambda <0$ in some sufficiently small neighbourhood of it, which possess the same number of nodes.",
"That is, $\\mathfrak {su}(N)$ solutions exist in open sets which all have the same number of nodes.",
"Proposition 7 Assume we have an existing topological $\\mathfrak {su}(N)$ solution of the field equations, with the gauge field functions all nodeless for $k\\ne 1$ , and the initial gauge field values given by $\\omega _{1,h}$ , $\\omega _{2,h}$ , ... , $\\omega _{N-1,h}$ .",
"Then all initial gauge field values in a neighbourhood of these values will give an $\\mathfrak {su}(N)$ solution to the field equations in which all the gauge field functions are nodeless.",
"Proof Assume we know of a non-trivial nodeless topological $\\mathfrak {su}(N)$ black hole solution of the field equations with event horizon radius $r_h$ and initial gauge field values at the event horizon $\\omega _{j,h}$ .",
"As we pointed out in Section REF we are only interested in black holes as solitons can be shown not to exist for $k\\ne 1$ .",
"So using these initial conditions and the field equations (REF , REF , REF ), we can integrate out and get a solution that is regular all the way to infinity.",
"For the rest of this argument we assume that $|\\Lambda |$ and $r_h$ are fixed and that the gauge field functions $\\omega _j$ are all nodeless.",
"From the local existence theorems we proved (Propositions REF , REF ), we know that for any set of initial values of the gauge field $\\omega _{1,h}$ , ..., $\\omega _{N-1,h}$ , there are solutions locally near the event horizon, and that the solutions are analytic in their choice of initial values.",
"For the existing $\\mathfrak {su}(N)$ solution, it must be true that $\\mu (r)>0$ on the interval $[r_h,\\infty )$ .",
"So, by analyticity of solutions, the nearby $\\mathfrak {su}(N)$ solution with initial gauge field values close to those of the existing solution, will all have $\\mu (r)>0$ for all $r\\in [r_h,r_c]$ for some $r_c$ .",
"By Lemma REF , these nearby solutions will also be regular on $[r_h,r_c]$ .",
"Now take some $r_1\\gg r_h$ such that for the existing solution, $m(r_1)$ satisfies $m(r_1)/r_1\\ll 1$ ; and let $\\tilde{\\omega }_{j,h}$ be a different set of initial gauge field values (for another $\\mathfrak {su}(N)$ solution) in some sufficiently small neighbourhood of the $\\omega _{j,h}$ s. By the analyticity argument above, these nearby $\\mathfrak {su}(N)$ solutions evolved from $\\tilde{\\omega }_{j,h}$ will be regular in the interval $[r_h,r_1]$ .",
"That is, in this interval, it is again true that $\\mu >0$ and all the gauge field functions are nodeless.",
"Furthermore, it will still be the case at $r_1$ that $m(r_1)/r_1\\ll 1$ for these nearby $\\mathfrak {su}(N)$ solutions as well as the existing $\\mathfrak {su}(N)$ solution.",
"Because of this, and because $r_1\\gg r_h$ , we are able to consider the solutions in the asymptotic regime.",
"Provided $r_1$ is sufficiently large, the solution (parameterised proportional to $1/r$ ) will not move very far along the phase space trajectory as $r$ increases from $r_1$ towards infinity.",
"Therefore $m(r)/r$ will continue to be small and the asymptotic regime will continue to be valid.$\\Box $ Lastly, we will use Proposition REF to prove the second of our main results – the existence of genuinely non-trivial $\\mathfrak {su}(N)$ solutions, for any fixed $\\Lambda <0$ , in some neighbourhood of certain trivial solutions (see Section REF ).",
"Theorem 8 There exist non-trivial black hole solutions to the $\\mathfrak {su}(N)$ topological EYM equations (REF , REF , REF ), regular throughout the range $r_h<r<\\infty $ .",
"These solutions are analytic in the parameters $r_h$ , $\\omega _{j,h}$ and $m(r_h)$ , and are nodeless.",
"Proof From Section REF , it is clear that there exist RNadS-type trivial solutions to the topological $\\mathfrak {su}(2)$ equations.",
"From Proposition REF , we can therefore infer the existence of nearby $\\mathfrak {su}(2)$ solutions which are non-trivial (i.e.",
"whose gauge field functions are not identically zero).",
"Furthermore, from van der Bij and Radu [9] we see that that with $\\mathfrak {su}(2)$ black holes for $k\\ne 1$ , $\\omega $ (which we can assume without loss of generality is positive at the event horizon) is monotonically increasing, so it can never equal zero; hence all these $\\mathfrak {su}(2)$ solutions will be nodeless.",
"Therefore, we are also able to take one such nodeless $\\mathfrak {su}(2)$ solution and embed it as a topological $\\mathfrak {su}(N)$ solution (remembering that this solution will similarly be nodeless, due to to (REF )).",
"Finally, using Proposition REF once again, we may infer the existence of genuinely non-trivial and nodeless $\\mathfrak {su}(N)$ solutions in some sufficiently small neighbourhood of this embedded nodeless $\\mathfrak {su}(N)$ solution.$\\Box $"
],
[
"Conclusion",
"The aim of this paper was to prove the existence of genuine (i.e.",
"non-trivial and non-embedded) solutions of the 4D $\\mathfrak {su}(N)$ EYM topological field equations.",
"We began by justifying the topological metric ansatz we used.",
"Then, we described the field equations and the gauge potential ansatz, taking a detour to prove its validity (i.e.",
"that it is invariant under an action of $\\mathfrak {su}(N)$ by principal bundle automorphisms), adapting a theorem of Künzle's [31].",
"We then considered the boundary conditions at the important singular points: the event horizon ($r=r_h$ ) and asymptotically ($r\\rightarrow \\infty $ ).",
"We note that we considered only purely magnetic gauge fields, for which we had $N-1$ gauge field functions $\\omega _j$ .",
"We also derived some trivial solutions to these equations, and described how any $\\mathfrak {su}(2)$ solution can be embedded in $\\mathfrak {su}(N)$ to yield a solution by a scaling of variables.",
"The field equations in question are singular at the event horizon and asymptotically, thus we began by proving local existence of solutions in the neighbourhood of these two singular regimes of the equations, using a method following [6], [16], [45], [56].",
"We also proved a lemma which states that any solution which regular in some small interval can continue to be integrated using the field equations to produce a solution which remains regular arbitrarily far from the event horizon, as long as $\\mu (r)$ remains positive.",
"We then used these to prove one of our two main results: that for fixed values of the field variables at the event horizon ($r_h$ , $\\omega _{j,h}$ , $m_h$ ), then as $|\\Lambda |\\rightarrow \\infty $ we can demonstrate the existence of solutions to the $\\mathfrak {su}(N)$ EYM field equations.",
"Next, we proved a proposition stating that given the existence of any nodeless topological $\\mathfrak {su}(N)$ solution, we can always find solutions arbitrarily close that possess the same number of nodes.",
"Finally, we used the existence of known trivial solutions, and this proposition, to demonstrate the existence of non-trivial nodeless solutions to the field equations.",
"Our main two results, therefore, prove the existence (in the stated regimes) of four-dimensional, topological solutions to $\\mathfrak {su}(N)$ EYM field equations with a negative cosmological constant, with $N-1$ gauge degrees of freedom as in the spherically symmetric case ($k=1$ ), previously examined in [6].",
"There are several very productive directions in which this research could next be taken.",
"The most obvious and pressing issue is that of the linear stability (under non-spherically symmetric perturbations, for £$k\\ne 1$ ) of the solutions we found: as we noted, it has previously been shown in [34], [51], [40] that the number of unstable modes of a solution is related to the number of nodes the gauge fields possess, and we have proven the existence (in certain regimes) of nodeless black hole solutions.",
"Given that we were successfully able to generalise previous work (for black holes in $\\mathfrak {su}(N)$ and $\\Lambda <0$ ) to include spaces whose topologies give the symmetries other than spherical, it is natural to ask what other sorts of solutions we could generalise in this way; and as we said in the introduction, there is no shortage of work that we may possibly generalise.",
"We mentioned the work of Mann on topological higher-dimensional black holes [39] – again, perhaps this work could be extended to $\\mathfrak {su}(N)$ .",
"Finally, as in [6], since there is no theoretical upper limit on $N$ , neither is there on the amount of `hair' we may give a topological black hole, though an interesting area of exploration might be $\\mathfrak {su}(\\infty )$ : there is some evidence for existence of solutions to these [42] but more work is required, and this work could extend into the inclusion of alternate topologies.",
"Results from this may or may not be significant, especially in view of the adS/CFT (conformal field theory) correspondence (see [38], among others) and the fact that it has been conjectured that there are observables in the dual CFT which are sensitive to the presence of black hole hair (see [24] for a discussion of non-Abelian solutions in the context of the adS/CFT correspondence.)",
"The author would like to thank Prof. E. Winstanley for many useful conversations and moral support; and Prof. H. P. Künzle for a very useful email exchange."
]
] | 1403.0171 |
[
[
"Asymmetric Flexural-gravity Lumps in Nonuniform Media"
],
[
"Abstract Here we show that asymmetric fully-localized flexural-gravity lumps can propagate on the surface of an inviscid and irrotational fluid covered by a variable-thickness elastic material, provided that the thickness varies only in one direction and has a local minimum.",
"We derive and present equations governing the evolution of the envelope of flexural-gravity wave packets allowing the flexing material to have small variations in the transverse (to propagation) direction.",
"We show that the governing equation belongs to the general family of Davey-Stewartson equations, but with an extra term in the surface evolution equation that accounts for the variable thickness of the elastic cover.",
"We then use an iterative Newton-Raphson scheme, with a numerical continuation procedure via Lagrange interpolation, in a search to find fully-localized solutions of this system of equations.",
"We show that if the elastic sheet thickness has (at least) a local minimum, flexural-gravity lumps can propagate near the minimum thickness, and in general have an asymmetric bell-shape in the transverse to the propagation direction.",
"In applied physics, flexural-gravity waves describe for instance propagation of waves over the ice-covered bodies of water.",
"Ice is seldom uniform, nor is the seafloor, and in fact near the boundaries (ice-edges, shorelines) they typically vary only in one direction (toward to edge), and are uniform in the transverse direction.",
"This research suggests that fully localized waves are not restricted to constant ice-thickness/water-depth areas and can exist under much broader conditions.",
"Presented results may have implications in experimental generation and observation of flexural-gravity (as well as capillary-gravity) lumps."
],
[
"Introduction",
"Fully localized solitary waves (and wavepackets) are of importance in applied sciences as they can transport mass, momentum and energy over long distances.",
"It is known today that gravity waves cannot admit fully-localized waves[1], [2], but in the presence of surface-tension or flexural-rigidity on the water surface such localized structures may exist, , , [6], [7].",
"These structures are formed as a result of a careful balance between nonlinearity and the dispersion.",
"Of interest in this paper, is the fully-localized wave packets of flexural-gravity systems.",
"Equations governing the evolution of flexural-gravity wave packets are described by the Davey-Stewartson equation [4], [8].",
"Davey-Stewartson (DS) equation [9] is a two-dimensional extension of Nonlinear Schrödinger (NLS) equation[10], [11].",
"Compared to two-dimensional Nonlinear Schrödinger (2DNLS) equation, DS further includes the coupling with an auxiliary field (which is the mean current in the context of hydrodynamics).",
"In contrast to one-dimensional NLS, 2DNLS is not integrable and does not admit localized solutions.",
"The coupling with the mean field allows finite-depth DS equation to be integrable (under certain conditions), and to admit fully-localized solutions.",
"In the limit of infinite depth DS equation reduces to 2DNLS.",
"While DS equation has been studied extensively due to its importance in different areas of science [12], [13], [14], [15], [16], investigation of its fully-localized solutions is a relatively young field of research.",
"Two families of such solutions have so far been discovered: Dromions and Lumps.",
"Dromions are fully-localized surface structures with exponentially decaying tails that form at the intersection of mean-flow line-solitary tracks [17], [18], [3], [19], [20], [21].",
"Lumps on the other hand have algebraically decaying tails but both the surface elevation and the mean field are fully-localized [22], [23].",
"Lump solution is not restricted to DS, but several other systems including Kadomtsev-Petviashvili equation [24], [25], Benney-Luke equation [5] and full Euler equation [26], [27] also admit lump solutions.",
"Flexural-gravity waves in two-dimension have been the subject of considerable interest from various standpoints [28], [29], [30], [31], [32].",
"In three-dimensional few existing research endeavors are mainly focused on the problem of moving loads on the ice [33], [34], [35], [36], [37], [38], [39].",
"The first consideration of fully-localized waves in three-dimension, to our knowledge, has been made recently where it has been shown that in a constant-depth water, wave packets of flexural-gravity waves may admit three-dimensional fully-localized structures in the form of lumps [8], [40] and dromions [4], [41].",
"Here we consider weakly nonlinear flexural-gravity wave packets propagating over a (relatively) thin but variable-thickness elastic cover that lies on the surface of an inviscid and incompressible fluid.",
"A perturbation scheme is used to derive the evolution equation for the envelope of waves, and it is shown that the governing equation belongs to the general family of Davey-Stewartson [9], [8], [4], but with an extra term (linear in the surface elevation) whose coefficient is a function of thickness variations.",
"We consider the Elliptic-Elliptic subset of this equation for which, in the absence of thickness variation, lump solution exists.",
"We then look for fully-localized lump solutions for a variety of thickness variation functions with the help of Lagrange interpolation of the lump shape combined with Newton-Raphson iteration scheme to solve the equation through a numerical continuation procedure without prescribed boundary conditions.",
"We show that lump solution over a transversely variable thicknesses exists if the thickness function has a local minimum.",
"These lumps, however, have an asymmetric shape in the transverse to the propagation direction with a steeper slope on the side that the gradient of thickness variation is higher.",
"For a constant thickness and variable depth, the governing equation is similar in the form, but different in the coefficient.",
"In this case lump solutions exist if the depth perturbation has a maximum, or in other words if the water depth has a minimum.",
"In the nature and in practice, media through which waves propagate is usually non-uniform.",
"Results presented here show that fully-localized solutions are not limited to perfectly uniform environments and in fact can exist and propagate in media with variable properties."
],
[
"Governing Equation",
"Consider an incompressible, inviscid and homogeneous fluid of density $\\rho _f$ and constant depth $h$ bounded on top by a thin sheet of an elastic material (such as ice) with the density $\\rho _{i}$ and spatially variable (small) thickness $L(x,y)$ .",
"We define a Cartesian coordinate system with the $x,y$ -axes on the mean bottom and $z$ -axis positive upward.",
"Assuming that flow is irrotational, a potential function $\\phi $ can be defined such that $\\vec{\\nabla }\\phi \\equiv \\vec{u}$ where $\\vec{u}=\\vec{u}(x,y,z,t)$ is the Eulerian velocity of the flow field.",
"If $\\eta =\\eta (x,y,t)$ denotes the elevation of the free surface from the mean water level, the governing equations read xx+yy+zz=0, 0<z<h+(x,y) z=t+xx+yy, z=h+(x,y) t+12(x2+y2+z2)+g= -[H4 + 2 Hx2x+2Hy2y+2 H2 .",
".-(1-)(Hxxyy-2Hxyxy+Hyyxx)+Rtt], z=h+(x,y) z=0, z=0 where $H=EL^3/12\\rho (1-\\nu ^2),~R=\\rho _{i}L/\\rho _{f}$ [42].",
"We define the following dimensionless variables $&&x^*,y^*=\\frac{x,y}{\\lambda },~~~z^*=\\frac{z}{h},~~~\\eta ^*=\\frac{\\eta }{a},~~~t^*=\\frac{t\\sqrt{gh}}{\\lambda }\\nonumber \\\\&&\\phi ^*=\\frac{\\phi h}{\\lambda a \\sqrt{gh}},~H^*=\\frac{H}{g\\lambda ^4},~~~R^*=\\frac{R h}{\\lambda ^2},~\\epsilon =\\frac{a}{h},~\\delta =\\frac{h}{\\lambda },$ where $\\lambda $ is the characteristic wavelength of the carrier wave and $a$ is the characteristic amplitude.",
"Using definitions Eq.",
"(REF ), the dimensionless form of the governing equation Eq.",
"(), after dropping asterisks for notational simplicity, becomes zz+2(xx+yy)=0, 0z1+, z=2(t+xx+yy), z=1+, t+12(x2+y2+12 z2)+= -[H4 + 2 Hx2x+2Hy2y+2 H2 .",
".-(1-)(Hxxyy-2Hxyxy+Hyyxx)+Rtt], z=1+9112, z=0, z=0.988 We are interested in weakly nonlinear harmonic waves of wavenumber $k$ with slowly varying amplitudes in both $x,y$ directions.",
"To achieve this solution we assume $\\epsilon \\ll O(1)$ but leaving $\\delta $ to be arbitrary.",
"Permanent-form fully-localized structures, for example lumps, are not expected to exist if the medium (e.g.",
"thickness of the elastic cover) changes in the direction of the propagation (in this paper we take the direction of the motion along the $x$ -axis).",
"In fact, if the medium has a constant mean but with random perturbations, we expect that the waves attenuate over time, as they propagate, via the so-called localization mechanism [43], [44].",
"Therefore here we assume that the thickness of the elastic cover only changes in the transverse (i.e.",
"$y$ ) direction and also very slowly, that is, $L(x,y)=L(y)=L_0[1+\\epsilon ^2 f(y)]$ , where $L_0$ is the mean thickness and $f(y)$ is a continuous function that specifies how thickness varies about the mean and satisfies $f(y)\\sim O(1),~f^{\\prime }(y)\\sim O(\\epsilon )$ .",
"Under these assumptions, the dynamic boundary condition Eq.",
"(), correct to the order $\\epsilon ^2$ , becomes $\\phi _{t}+\\frac{1}{2}\\epsilon (\\phi _{x}^{2}+\\phi _{y}^{2}+\\frac{1}{\\delta ^2}\\phi _{z}^{2})+\\eta =-H_0[1+3\\epsilon ^2 f(y)]\\nabla ^{4}\\eta -R_0[1+\\epsilon ^2 f(y)]\\eta _{tt}.$ where $H_0=EL_0^3/12\\rho (1-\\nu ^2)g\\lambda ^4$ and $R_0=\\rho _{i}L_0h/\\rho \\lambda ^2$ .",
"We further define the following different scale variables $\\xi =x-c_pt,~~~\\zeta =\\epsilon (x-c_gt),~~~Y=\\epsilon y,~~~\\tau =\\epsilon ^2 t,$ where $c_p(k),c_g(k)$ are respectively the phase and group velocity of the carrier wave.",
"Note that $c_p(k),c_g(k)$ are unknown speeds at this stage, but leading order (i.e.",
"linear) analysis proves $c_p(k)$ to be the phase velocity and first order analysis shows that $c_g(k)$ must in fact be the group velocity of the wave system.",
"We assume that the solution to the governing equations can be expressed by a convergent asymptotic series in terms of our small parameter $\\epsilon $ .",
"In terms of new variables Eq.",
"(REF ) we suggest the series solution in the form $&&\\phi (\\xi ,\\zeta ,Y,z,\\tau )=f_0(\\zeta ,Y,\\tau )+\\sum _{n=0}^{\\infty }\\epsilon ^n\\left\\lbrace \\sum _{m=0}^{n+1} F_{nm}(z,\\zeta ,Y,\\tau )E^m+{\\rm c.c.}",
"\\right\\rbrace ,\\hspace{28.45274pt}\\\\&&\\eta (\\xi ,\\zeta ,Y,\\tau )=\\sum _{n=0}^{\\infty }\\epsilon ^n\\left\\lbrace \\sum _{m=0}^{n+1} A_{nm}(\\zeta ,Y,\\tau )E^m+{\\rm c.c.}",
"\\right\\rbrace ,\\hspace{17.07182pt}$ where $E=\\exp (ik\\xi )$ and $A_{00}=0$ .",
"Leading order (i.e.",
"linear) analysis provides the expression for the dispersion relation $c_{p}^2=\\frac{(1+\\tilde{H})\\tanh \\delta k}{\\delta k+\\tilde{R}\\tanh \\delta k}.$ where $\\tilde{H}=H_0k^4$ and $\\tilde{R}=R_0k^2$ .",
"If the perturbation analysis perused to the second order of nonlinearity, equations governing the evolution of the envelope $A_{01}(\\zeta ,Y,\\tau )$ and the mean field $f_{0}(\\zeta ,Y,\\tau )$ obtain in the following form [45], [46] f0+f0YY=-|A0|2 930a iA0+A0+A0YY=(1|A0|2+2f0+3)A0930b in which =1-cg2, =12[2k cp+(2k2cp2cg)(1-2)]0 =”2, =cg2k='2k0 1=k34, 2=k[1+2k2cpcg(1-2)-2R22(k +R)]0 3= f(y)2(3H-cp2R)kk(1+H-cp2R)+cp2kk+2Rcp2k where $\\Gamma =p/q$ and q= (R+k)3[(R+k)(-3+12H)+k(1+H)(3-2)] p = a+b+c(1-2)+d(1-2)+e(1-2)2+f(1-2)2+g(1-2)3 a = (52H2+44H-8)4k4+(48H2+36H-12)R22k2 c = (-104H+8-112H2)4k4+(36-144H2-108H)R22k2 b = (100H2+80H-20)R3k3, d= (32-176H-208H2)R3k3 e = (-42H-63H2-28H3-7)4k4+(72H-24+96H2)R22k2 f = (-30H-24H3-51H2-3)R3k3, g = (-2H3-6H2-6H-2)4k4in which $\\sigma =\\tanh \\delta k$ .",
"Note that $f(y)$ is now in the coefficient $\\nu _3$ .",
"Eq.",
"() is a more general form of Davey-Stewartson[9] (also known as Benney-Roskes-Davey-Stewartson equation [47], [8]).",
"In the special case of Eq.",
"() when the thickness of the top elastic layer does not vary, i.e.",
"$f(y)=0$ , and if its inertia can be neglected, i.e.",
"$R_0=0$ , then we recover Eq.",
"(2.10) in Ref.",
"M.RezaAlam2012 or Eqs.",
"(15),(16) in Ref.",
"Milewski2013.",
"The effect of inertia appears in all coefficients in Eq.",
"().",
"We finally note that the effect of variable thickness appears only in the new term $\\nu _3A_0$ in Eq.",
"().",
"In order for underlying assumptions of Eq.",
"() to be valid the function $f(y)$ must satisfy $|f(y)|=O(1)$ , $|f^{\\prime }(y)|=O(\\epsilon )$ ."
],
[
"Numerical Method",
"To cast Eq.",
"() in a more commonly-used form we define $v=-f_{0\\zeta }+g|A_0|^2$ , where $g=-\\beta /\\alpha $ and for simplicity of notations replace $A_0,\\zeta ,Y$ with $u,x,y$ .",
"Eq.",
"() turns into vxx+vyy=g|u|2yy iu+uxx+uyy+2uv+|u|2u-3u=0 where $\\gamma =-\\nu _{1}-\\nu _{2}g$ .",
"Depending on whether $\\gamma >0$ or $\\gamma <0$ Davey-Stewartson equation is known as, respectively, focusing or defocusing.",
"Likewise signs of $\\alpha $ and $\\vartheta $ determine whether Eq.",
"() is elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic or hyperbolic-hyperbolic (the latter is not possible in the context of water waves[48]).",
"Figure REF (a) compares the area of focusing and defocusing of the governing equation Eq.",
"() for $\\hat{R}\\equiv \\tilde{R}/(k\\delta )^2=$ 0, 0.05.",
"The new variable $\\hat{R}$ in physical space is simply the ratio of the mass of the elastic sheet to the mass of the water underneath it, and therefore is independent of the wavelength.",
"Whether Eq.",
"() is focusing or defocusing is a function of both $k\\delta $ and $\\hat{H}\\equiv \\tilde{H}/(k\\delta )^4$ and this functionality is shown in Figure REF (a).",
"Areas for which the governing Davey-Stewartson is elliptic-elliptic is overlain on top of boundaries of focusing/defocusing curves in Figure REF (b).",
"The case of $\\hat{R}$ =0 is shown by gray area, and for $\\hat{R}$ =0.05 by hatched area.",
"Here we only consider focusing elliptic-elliptic Davey-Stewartson equation which is known to admit localized solitary waves[48].",
"We further assume that the amplitude has a periodic phase with an unknown frequency $\\Omega $ , i.e.",
"$u=r(x,y) e^{i\\Omega \\tau }.$ By implementing the following re-scaling $\\bar{r}=r \\Omega ^{-1/2}\\gamma ^{1/2},~\\bar{v}=\\frac{\\gamma }{g\\Omega }v,~\\bar{x}=x\\sqrt{\\frac{\\Omega }{\\vartheta }},~\\bar{y}=y\\sqrt{\\frac{\\Omega }{\\mu }}\\nonumber $ After dropping bars Eq.",
"() becomes -1vxx+vyy=|r|2yy941 -r+rxx+ryy+2g-1r v+r3-3-1r=0942.",
"Figure: (a) Focusing (denoted by “+\") and defocusing (denoted by “-\") areas of Davey-Stewartson equation with the effect of inertia of the elastic sheet (R ^=0.05\\hat{R}=0.05, dashed-line) and without the effect of inertia (R ^=0\\hat{R}=0, solid lines).",
"The choice of R ^=0.05\\hat{R}=0.05 is made so that the figure shows clearly direction of departure from the ground state of R ^=0\\hat{R}=0.",
"(b) Areas of elliptic-elliptic Davey-Stewartson for R ^=0.05\\hat{R}=0.05 (hatched area) and R ^\\hat{R}=0 (gray area).The objective here is to look for lumps solutions of Eq.",
"() particularly when $f(y)$ is nonzero and non-constant (note that $f(y)$ is hidden in $\\nu _3$ ).",
"The numerical procedure is as follows: we first consider Eq.",
"() with $\\nu _2=\\nu _3=0$ , i.e., $-r+r_{xx}+r_{yy}+r^3=0$ .",
"This equation can be solved numerically using shooting method under the condition that $r(x,y)$ is axisymmetric and vanishes at infinity.",
"It has countably many solutions, and the one that is always positive and asymptotically goes to zero as $r\\rightarrow \\infty $ (the so-called ground state) will be considered here as a first guess for solving Eq.",
"(), .",
"With the ground state solution in hand, we aim at solving Eqs.",
"() and () together including $\\nu _2$ but keeping $\\nu _3=0$ .",
"It is to be noted that Eq.",
"() with $\\nu _3=0$ is invariant under the parity transformation (i.e.",
"when $x \\rightarrow -x$ , likewise for $y$ ).",
"Although this by itself does not guarantee that the solution is parity invariant, we restrict our attention to solutions of Eq.",
"() (with $\\nu _3=0$ ) that are symmetric about both $x$ - and $y$ -axis , .",
"This assumption later on will be relaxed in the $y$ -direction in a search for asymmetric lumps.",
"We express $r(x,y)$ and $v(x,y)$ in terms of Lagrangian polynomials whose coefficients are found via the iterative Newton-Raphson method.",
"Since lumps have algebraically decaying tails the transformation $x,y=Lp,q/\\sqrt{1-(p,q)^2}$ is used to map physical domain ($-\\infty , \\infty $ ) to a finite computational domain(-1,1).",
"The effect of the right-hand side of Eq.",
"() is included via numerical continuation (i.e.",
"by slowly increasing an artificial coefficient in front of that term from zero to one [49]).",
"To find lump solutions of Eq.",
"() including $\\nu _3$ term, we start from the symmetric lump of the previous step and implement another level of numerical continuation on the $\\nu _3$ term.",
"Note that at this stage symmetricity in $y$ -direction is relaxed."
],
[
"Results and Discussion",
"We show here that a variable-thickness elastic sheet, with the thickness variation function $f(y)$ , overlying the surface of an inviscid and irrational fluid can admit asymmetric flexural-gravity lumps.",
"We first consider a case with the thickness variation function $f(y)=5\\text{exp}(-y^2)\\sin y$ .",
"Chosen parameters are chosen as $\\delta k$ =2.8, $\\hat{R}$ =0.01, $\\hat{H}^{\\frac{1}{4}}$ =0.19, $\\Omega $ =1/3, and domain transformation variable $L$ =5.",
"The number of grid points in the simulation domain is set at 24$\\times $ 96.",
"Note that computational domain is half of the full domain.",
"Therefore this choice of grid size implies that the full domain is 48$\\times $ 96.",
"Chebyshev nodes are used as grid points to reduce the effect of Runge phenomenon.",
"Targeted computational error is set to $1e-12$ .",
"It is to be noted that for the same parameters different choices of $L$ result in different lump solutions.",
"This is not unexpected as, for instance, analytical dromion solution to Davey-Stewartson equation has several free parameters and for every set of parameters an infinite number of dromions can be found , .",
"There are, however, limitations on the choice of this number because for smaller $L$ we get a finer mesh close to the center of the domain of computation and coarser grid farther away and vice versa.",
"Therefore care must be taken in a proper choice of scaling parameter $L$ .",
"Figures REF (a)-(c) show the thickness variation function $f(y)$ (Figure REF (c)) along with central cross-sections of $r,v$ , i.e.",
"$r(x=0,y)$ (Figure REF (a)) and $v(x=0,y)$ (Figure REF (b)).",
"To highlight that the lump is asymmetric, the mirror image of each side about the peak of the lump is plotted by dashed-lines on the other side of the peak.",
"In the absence of the thickness variation (i.e.",
"when $f(y)$ =0), the peak is at y=0.",
"With the current form of $f(y)$ the peak is slipped away from $y$ =0 toward the trough of the thickness variation function.",
"A close examination of the Figures REF (a)-(c) shows that the peak of the lump is not exactly at the trough but slightly to the left (this is further highlighted in the next example).",
"The lump profile is leaned forward toward the steeper side of $f(y)$ , i.e., it is steeper on the side where $f(y)$ is steeper.",
"To further highlight the effect of steepness of the profile on the shape of the lumps we consider a piecewise linear thickness variation function given by (see Figure REF (f)) $f(y)={\\left\\lbrace \\begin{array}{ll}-y-1 & -1 <y< 0 \\\\\\frac{1}{5}y-1& 0 <y< 5 \\\\0 & \\text{otherwise.}\\end{array}\\right.",
"}$ Note that Eq.",
"(REF ) has only one trough (minimum) and no distinguished crest (maximum).",
"With the choice of parameters the same as in the previous case, the lump surface and associated mean current are shown in Figures REF (d)-(e).",
"Clearly the lump and the mean current have a steeper slope on the side that $f(y)$ is steeper.",
"And the crest of the lump is formed away from the trough of the thickness on its right where the slope is milder.",
"The trends are similar to the previous case but further highlighted.",
"Note that strictly speaking a thickness variation function must be at least two times differentiable (i.e.",
"$f(y)\\in \\mathcal {C}^2$ , c.f.",
"Eq.",
"()), but nevertheless no derivative of $f(y)$ appears in the final form of governing equation Eq.",
"().",
"Our numerical experiments show that the obtained asymmetric shape for the lump is quite robust to the perturbations to the thickness variation function.",
"The last case of our interest is if $f(y)$ has a crest, but no trough, e.g.",
"mirror of $f(y)$ shown in Figure REF (f) about the horizontal axis.",
"In this case our iterative algorithm does not converge to a lump solution.",
"Iteration of the shape of the initial profile shows that the peak of the lump slips away from the crest of the topography (perhaps in a search for a trough), but continues to move to the right or left and never reaches a convergence.",
"Figure: Cross sections of asymmetric flexural-gravity lumps propagating over a transversely uneven elastic sheet.",
"(a),(b) cross sections r(x=0,y)r(x=0,y), v(x=0,y)v(x=0,y) for an asymmetric lump propagating over an elastic sheet with the thickness variation function f(y)=5exp(-y 2 )sinyf(y)=5\\exp (-y^2) \\sin y in (c).",
"Dashed lines show the mirror of the right/left side profile (about the peak) on the left/right side and are presented to highlight the asymmetry of the shape.",
"(d),(e) cross sections for the thickness variation function f(y)f(y) in (f)(as is shown in Eq.",
"()).",
"Parameters used here are δk=2.8,R ^=0.01,H ^ 1 4 =0.19\\delta k=2.8,~\\hat{R}=0.01, \\hat{H}^{\\frac{1}{4}}=0.19, Ω=1/3\\Omega =1/3 and L=5L=5.One important feature of the governing equation Eq.",
"() in the presence of a non-zero thickness variation function is that the frequency of the periodic phase of the amplitude $\\Omega $ appears explicitly in the equation (c.f.",
"Eq.",
"(), last term).",
"In other words if $f(y)$ =0, by the rescaling introduced $\\Omega $ disappears from the governing equation Eq.",
"(), but if $f(y)\\ne $ 0 there is no rescaling that can remove $\\Omega $ .",
"Therefore $\\Omega $ appears as a free parameter in the governing equation.",
"In the previous two examples (Figures REF (a)-(f)), we set $\\Omega $ =1/3.",
"Figure REF shows the effect of $\\Omega $ on the shape of the lump obtained ($f(y)$ and other parameters are the same as those in Figures REF (a)-(c)).",
"Clearly with the increase in $\\Omega $ the height of the lump increases but the wavelength is almost unchanged.",
"In the case presented in Figure REF , for $\\Omega <$ 0.24 the lump height vanished and we obtained a trivial solution of a constant mean current.",
"Note for $\\Omega =0.25$ , the mean current also has hump-like profile but its amplitude is of O($0.01$ ).",
"Figure: Cross sections of asymmetric flexural-gravity lumps for different oscillation frequencies Ω\\Omega (introduced in Eq.",
"()).",
"(a),(b) respectively show cross sections r(x=0,y)r(x=0,y), v(x=0,y)v(x=0,y).",
"Parameters used here are δk=2.8,R ^=0.01,H ^ 1 4 =0.19\\delta k=2.8,~\\hat{R}=0.01, \\hat{H}^{\\frac{1}{4}}=0.19, L=5L=5.The effect of the amplitude of the thickness variation is higher on the lump height than the mean current.",
"Figure REF compares lumps obtained for $f(y)=c\\exp (-y^2) \\sin y$ and for four choices of $c$ =0, 0.1 ,1 and 2.5.",
"The case $c$ =0 corresponds to a uniform thickness function (i.e.",
"$f(y)$ =0) that is known to yield symmetric lumps.",
"Once variation function is introduced (i.e.",
"$c>$ 0) the peak equilibrium location changes, but the peak stays at the same $y$ for different values of $c$ .",
"Further increase in the $c$ decreases both the height and mean current.",
"Effect of the $c$ is small on the lump height ($\\sim $ 15% when $c$ changes by a factor of 25) and the mean current ($\\sim $ 40%).",
"Clearly the mean current is affected more than the lump height by the height of the thickness anomaly .",
"For flexural-gravity waves, variable medium can be a result of variations in the water depth as well.",
"Governing equation for flexural-gravity waves over a variable bottom can be derived via a similar procedure followed §2 and the final equation is in the same form as in Eq.",
"() with different coefficient $\\nu _3$ (see Appendix).",
"Similar procedure can be followed to derive the governing equation for Capillary-Gravity waves in the presence of bottom variations.",
"Capillary-gravity lumps excited by a localized moving pressure have been observed in the laboratory[50], [51], [52].",
"Our results suggest that carefully architected variable mediums may be used in laboratory experiments to trap lumps, or restrict their motion along desired paths.",
"Figure: Cross sections of asymmetric flexural-gravity lumps for different amplitudes of f(y)f(y).",
"(a),(b) show cross sections r(x=0,y)r(x=0,y), v(x=0,y)v(x=0,y) respectively.",
"Parameters used here are δk=2.8,R ^=0.01,H ^ 1 4 =0.19\\delta k=2.8,~\\hat{R}=0.01, \\hat{H}^{\\frac{1}{4}}=0.19, L=5L=5."
],
[
"Conclusion",
"Here we showed that asymmetric flexural-gravity lumps can exist on a surface of an inviscid and irrotational fluid covered by a (transversely) variable-thickness elastic material.",
"Assuming that the variation in the thickness of the overlying elastic sheet is small, we derived, via perturbation expansion, the governing equation for the envelope of wavepackets in a flexural-gravity wave system.",
"The governing equation is in the form of classical Davey-Stewartson equation but with an additional term in the surface evolution equation that accounts for the variation in the thickness.",
"In order to find fully-localized solitary waves, we then carried out a continuation procedure by Lagrange interpolation, combined with Newton-Raphson iteration scheme.",
"We showed that the peak of the asymmetric lump forms near a local minimum of the elastic sheet thickness.",
"Also, in contrast to lumps propagating over a flat sheet, asymmetric lumps over a non-uniform thicknesses can only exist for frequencies greater than a minimum frequency.",
"In practical applications the medium is often not perfectly uniform.",
"In fact in many cases the variation is only in one spatial direction (e.g.",
"sloping shoreline, sloping edge of the ice).",
"Possibility of existence of lumps in these systems suggests that these fully-localized solitary waves may exist more widely than expected before.",
"It also suggests where lumps may be more often expected: for instance, contours of constant water depth over a sloping beach (that has a local minimum along the beach slope) in ice-covered waters are loci of lumps.",
"Whether these loci can act as attractors for wandering lumps is an interesting subject worth further research.",
"*"
],
[
"VARIABLE TOPOGRAPHY",
"Here we present the governing equation for evolution of wavepackets of flexural-gravity waves, assuming that the thickness of the elastic sheet is constant, but the seabed has small variations in the transverse (to propagation) direction.",
"We consider that the seabed is given by a mean bottom at the depth $h$ , with small perturbations $b(y)/h\\sim O(\\epsilon ^2)$ .",
"Under this assumption, governing equations are the same as Eq.",
"() except the bottom boundary condition in Eq.",
"() now is $\\phi _z=\\phi _yb_y, ~~~z=b(y)\\nonumber $ We define a normalized bottom perturbation $b^*=bh/a^2$ .",
"Now using scaling variables we introduced before in Eq.",
"(REF ), the dimensionless bottom boundary condition Eq.",
"(), after dropping asterisks, turns into $\\phi _z=\\epsilon ^2\\delta ^2\\phi _yb_y, ~~~z=\\epsilon ^2 b(y)\\nonumber $ By substituting this into the governing equations and following the similar procedure we arrive at Eq.",
"(), we in fact arrive at the same form equation with just a different coefficient $\\nu _3$ which is now $\\nu _{3}=-b(y)\\frac{\\delta k \\omega [-\\delta c_{p}^{2} k \\sinh \\delta k+\\cosh \\delta k(\\tilde{H}+1-\\tilde{R}c_{p}^{2})]}{\\sinh \\delta k(\\tilde{H}+1-\\tilde{R}c_{p}^{2})+\\delta c_{p}^{2}k \\cosh \\delta k+2\\tilde{R}c_{p}^{2}\\sinh \\delta k}\\nonumber $ It can be shown that the sign of $\\nu _3/b(y)$ is decided by the sign of $-(1+\\tilde{H})$ , which is always negative.",
"In the case of variable thickness the sign of $\\nu _3/f(y)$ is decided by $\\tilde{R}\\tanh \\delta k(2\\tilde{H}-1)+3\\tilde{H}\\delta k$ .",
"If effect of inertia is small (i.e.",
"$\\tilde{R}\\approx 0$ ), the sign of $\\nu _3/f(y)$ is always positive.",
"Since in the case of variable thickness lumps are formed near the local minima of the thickness variation function $f(y)$ , we then conclude that in the case of variable seabed, lumps are formed near the local maxima of the seabed, i.e.",
"where the water depth is minimum.",
"Extension of these formula to the capillary-gravity waves over non-uniform water depth is also obtained, but is similar to the above case, and hence is not presented here."
]
] | 1403.0043 |
[
[
"Energy Harvesting Cooperative Networks: Is the Max-Min Criterion Still\n Diversity-Optimal?"
],
[
"Abstract This paper considers a general energy harvesting cooperative network with M source-destination (SD) pairs and one relay, where the relay schedules only m user pairs for transmissions.",
"For the special case of m = 1, the addressed scheduling problem is equivalent to relay selection for the scenario with one SD pair and M relays.",
"In conventional cooperative networks, the max-min selection criterion has been recognized as a diversity-optimal strategy for relay selection and user scheduling.",
"The main contribution of this paper is to show that the use of the max-min criterion will result in loss of diversity gains in energy harvesting cooperative networks.",
"Particularly when only a single user is scheduled, analytical results are developed to demonstrate that the diversity gain achieved by the max-min criterion is only (M+1)/2, much less than the maximal diversity gain M. The max-min criterion suffers this diversity loss because it does not reflect the fact that the source-relay channels are more important than the relay-destination channels in energy harvesting networks.",
"Motivated by this fact, a few user scheduling approaches tailored to energy harvesting networks are developed and their performance is analyzed.",
"Simulation results are provided to demonstrate the accuracy of the developed analytical results and facilitate the performance comparison."
],
[
"Introduction",
"Simultaneous wireless information and power transfer (SWIPT) has recently received a lot of attention.",
"Compared to conventional energy harvesting techniques, SWIPT can be used even if wireless nodes do not have access to external energy sources, such as solar and winder power.",
"The key idea of SWIPT is to collect energy from radio frequency (RF) signals, and this new concept of energy harvesting was first proposed in [1] and [2].",
"Particularly by assuming that the receiver has the capability to carry out energy harvesting and information decoding at the same time, the tradeoff between information rate and harvested energy has been characterized in [1] and [2].",
"Motivated by the difficulty of designing a circuit performing both energy harvesting and signal detection simultaneously, a practical receiver architecture has been developed in [3], where two receiver strategies, power splitting and time sharing, have been proposed and their performance have been analyzed.",
"The concept of SWIPT was initially studied in simple scenarios with one source-destination pair, where the use of co-channel interference for energy harvesting was considered in [4] and the combination of multiple-input multiple-output (MIMO) technologies with SWIPT was investigated in [5].",
"SWIPT has been recently applied to various important communication scenarios more complicated than the case with one source-destination pair.",
"For example, in [6] the application of SWIPT to multiple access channels has been considered, where a few solutions for system throughput maximization have been proposed.",
"Broadcasting scenarios have been considered in [7] and [8], where one transmitter is to serve two types of users, energy receivers and information receivers, simultaneously.",
"In [9] the joint design of uplink information transfer and downlink energy transfer has been considered, where sophisticated algorithms for energy beamforming, power allocation and throughput maximization have been proposed.",
"The idea of SWIPT has also been applied to wireless cognitive radio systems, where opportunistic energy harvesting from RF signals has been studied in [10].",
"The application of SWIPT to cooperative networks is important since the lifetime of the relay batteries can be extended by efficiently using the energy harvested from the relay observations.",
"In [11] a greedy switching approach between data decoding and energy harvesting has been proposed for the case with one source-destination pair and one relay.",
"In [12] the outage performance achieved by amplified-and-forward (AF) relaying protocols has been developed, and the use of decode-and-forward (DF) strategies has been investigated in multi-user energy harvesting cooperative networks [13].",
"Relay selection has been studied in a broadcasting scenario where energy harvesting was carried out at the destinations, instead of relays [14].",
"The impact of the random locations of wireless nodes on the path loss and the outage performance has been characterized by applying stochastic geometry in [15].",
"In conventional cooperative networks, the max-min criterion has been recognized as a diversity-optimal selection strategy [16], [17], [18].",
"Take a DF cooperative network with one source-destination pair and $M$ relays as an example.",
"Provided that the $i$ -th relay is used, the capacity of a DF relay channel is $\\min \\lbrace \\log (1+\\rho |h_i|^2), \\log (1+\\rho |g_i|^2)\\rbrace $ , where $\\rho $ is the transmit signal-to-noise ratio (SNR), $h_i$ is the channel gain between the source and the relay, and $g_i$ is the channel gain between the relay and the destination.",
"Obviously the max-min criterion, i.e.",
"$\\max \\lbrace \\min \\lbrace |h_i|^2,|g_i|^2\\rbrace , 1\\le i \\le M\\rbrace $ , is capacity optimal and can achieve the maximal diversity gain, $M$ .",
"But is this conclusion still valid when energy harvesting relays are used?",
"The main contribution of this paper is to characterize the performance of the max-min selection criterion in energy harvesting cooperative networks.",
"We first construct a general framework of energy harvesting cooperative networks, where $M$ pairs of sources and destinations communicate with each other via a relay.",
"Among the $M$ user pairs, the relay will schedule $m$ of them to transmit.",
"It is important to point out that the problem of relay selection for the scenario with one source-destination pair is a special case of the formulated framework by setting $m=1$ .",
"When only a single user is scheduled, the exact expression for the outage probability achieved by the max-min criterion is developed by carefully grouping the possible outage events and then applying order statistics.",
"Based on this obtained expression, asymptotic studies of the outage probability are carried out to show that the diversity gain achieved by the max-min criterion is only $\\frac{M+1}{2}$ , much less than the full diversity gain, $M$ .",
"The reason for this loss of the diversity gain is that the max-min criterion treats the source-relay channels and the relay-destination channels equally.",
"However, when an energy harvesting relay is used, it is important to observe that the source-relay channels become more important.",
"For example, the source-relay channels impact not only the reception reliability at the relay, but also the relay transmission power.",
"Recognizing this fact, a few modified user scheduling approaches are developed, which is the second contribution of this paper.",
"Particularly for the case of $m=1$ , an efficient user scheduling approach is proposed, and analytical results are developed to demonstrate that this approach can achieve the maximal diversity gain.",
"This approach can be extended to the case of $m>1$ , by applying exhaustive search.",
"A greedy user scheduling approach is also developed by assuming that the relay always has data to be sent to all the destinations.",
"The use of this greedy approach yields closed-form expressions for the outage probability and diversity order, which can be used as an upper bound for the other approaches.",
"Simulation results are also provided to demonstrate the accuracy of the developed analytical results and facilitate the performance comparison among the addressed user scheduling approaches."
],
[
"System Model",
"Consider a cooperative communication scenario with $M$ source-destination pairs and one energy harvesting relay.",
"The $M$ users compete for the wireless medium, and the relay will schedule $m$ user pairs over $2m$ time slots, $0\\le m \\le M$ .",
"All the channels are assumed to be independent and identically (i.i.d.)",
"quasi-static Rayleigh fading, and this indoor slow fading model is valid for many applications of wireless energy transfer, such as wireless body area networks and smart homes [14] and [15].",
"In Section , the impact of the path loss and the random locations of the users on the outage performance will be studiedNote that when the users are randomly deployed, the effective channel gains, i.e.",
"the combinations of Rayleigh fading and large scale path loss, can be still approximated as independent and identically exponentially distributed variables [15]..",
"It is assumed that the relay has access to global channel state information (CSI), which is important for the relay to carry out user scheduling.",
"During the $j$ -th time slot, consider that the $i$ -th user pair is scheduled to transmit its message $s_i$ , where the details for user scheduling will be provided in the next two sections.",
"The power splitting strategy will be used at the DF relay.",
"Particularly the relay will first direct the observation flow to the detection circuit, and then to the energy harvesting circuit if there is any energy left after successful detection [3] and [13].",
"Therefore the observation at the relay is given by $y_{ri} = \\sqrt{P(1-\\theta _i)}h_is_i+n_{ri},$ where $\\theta _i$ is the power splitting factor, $P$ is the transmission power at the source, $h_i$ denotes the channel gain from the $i$ -th source to the relay, and $n_{ri}$ denotes the additive white Gaussian noise.",
"As discussed in [13], the optimal value of $\\theta _i$ for a DF relay is $\\max \\left\\lbrace 1-\\frac{\\epsilon }{|h_i|^2},0\\right\\rbrace $ , the maximal value of $\\theta _i$ constrained by successful detection at the relay, where $\\epsilon =\\frac{2^{2R}-1}{P}$ and $R$ denotes the targeted data rate.",
"The power obtained at the relay after carrying out energy harvesting from the $i$ -th user pair is given by $P_{ri} = \\eta P\\left[|h_i|^2-\\epsilon \\right]^+,$ where $\\eta $ denotes the energy harvesting coefficient, and $[x]^+$ denotes $\\max \\lbrace x,0\\rbrace $ .",
"At the $(m+j)$ -th time slot, the relay forwards $s_i$ to the $i$ -th destination, and the receive SNR at this destination is given by $SNR_i = P_{i} |g_i|^2,$ where $P_{i}$ denotes the relay transmission power allocated to the $i$ -th destination, and $g_i$ denotes the channel gain between the relay and the $i$ -th destination.",
"Note that $P_{ri}$ is not necessarily equal to $P_i$ , depending on the used relay strategy, as discussed in the following sections."
],
[
"User scheduling based on the max-min criterion",
"In this section, the performance achieved by the user scheduling strategy based on the max-min criterion is studied.",
"Particularly we will focus on the case that the relay selects only one user pair, i.e.",
"$m=1$ , and more discussions about the case with $m>1$ will be provided in the next section.",
"Note that the scenario addressed in this section can be shown mathematically the same as the problem of relay selection for the case with one source-destination pair and $M$ relays.",
"Therefore the results obtained for the addressed scheduling problem will be also applicable to the max-min relay selection cases.",
"Since only one user pair is scheduled, the energy harvested from the $i$ -th source will be used to power the relay transmission to the $i$ -th destination, i.e.",
"$P_{i}=P_{ri}$ .",
"The max-min user scheduling strategy can be described as follows: The relay first finds out the worst link of each user pair.",
"Denote $z_i= \\min \\lbrace |h_i|^2, |g_i|^2\\rbrace $ .",
"The user pair with the strongest worst link is selected, i.e.",
"the $i^*$ -th user pair is selected because $i^*=\\arg \\max \\left\\lbrace z_1, \\ldots , z_M\\right\\rbrace $ .",
"Provided that the relay can decode the $i^*$ -th source's message correctly, the SNR at the corresponding destination is given by $SNR_{i^*} = \\eta P\\left(|h_{i^*}|^2-\\epsilon \\right) |g_{i^*}|^2.$"
],
[
"Performance evaluation",
"The outage probability achieved by the max-min based scheduling scheme can be written as follows: $\\mathrm {P}_{o}\\triangleq \\mathrm {P}\\left(|h_{i^*}|^2<\\epsilon \\right) + \\mathrm {P}\\left((|h_{i^*}|^2-\\epsilon )|g_{i^*}|^2<\\epsilon _1, |h_{i^*}|^2>\\epsilon \\right),$ where $\\epsilon _1=\\frac{\\epsilon }{\\eta }$ .",
"Although the outage probability achieved by the max-min criterion is shown in a simple term as in (REF ), it is challenging to evaluate this probability.",
"The reason is that the use of the scheduling strategy has changed the statistical property of the channel gains.",
"For example, $|h_{i^*}|^2$ is no longer exponentially distributed.",
"The density function of $\\min \\lbrace |h_{i^*}|^2,|g_{i^*}|^2\\rbrace $ can be found by using order statistics, and the key step is to restructure the expression of the outage probability shown in (REF ) into a form to which the density function of $\\min \\lbrace |h_{i^*}|^2,|g_{i^*}|^2\\rbrace $ can be applied.",
"In the following theorem, the exact expression for the outage probability achieved by the max-min scheme is provided.",
"$\\mathbf {Theorem}$ 1 When a single user is scheduled, the outage probability achieved by the max-min user scheduling strategy is given by $\\mathrm {P}_o&=& \\frac{e^{-\\epsilon }}{2}\\sum _{i=0}^{M}{M \\atopwithdelims ()i} \\frac{(-1)^i}{2i-1}\\left(1-e^{-(2i-1)\\epsilon }\\right) \\\\\\nonumber &&+M \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-\\epsilon }-e^{-(2i+2)\\epsilon }}{2i+1} +\\frac{e^{-(2i+2)\\epsilon }-e^{-(2i+2)\\epsilon _0}}{2i+2} - e^{-\\epsilon } \\beta (\\epsilon _0,i) \\right)\\\\ \\nonumber &&+ \\frac{\\left(1-e^{-2\\epsilon }\\right)^M}{2}+ M\\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-2(i+1)\\epsilon }-e^{-2(i+1)\\epsilon _0}}{2(i+1)} - e^{-(2i+1)\\epsilon }\\beta (\\epsilon _0-\\epsilon , i) \\right),$ where $\\beta (y,i)=\\int ^{\\epsilon _0}_{0}e^{-(2i+1)y-\\frac{\\epsilon _1}{y}} dy$ , and $\\epsilon _0=\\frac{\\epsilon +\\sqrt{\\epsilon ^2+4\\epsilon _1}}{2}$ .",
"See the appendix.",
"The expression shown in (REF ) can be used to numerically evaluate the outage probability achieved by the max-min scheduling approach, as shown in Section .",
"In addition, it can also be used for the analysis of the diversity gain achieved by the max-min approach, as shown in the following theorem.",
"$\\mathbf {Theorem}$ 2 When a single user pair is scheduled, the diversity order achieved by the max-min user scheduling approach is $\\frac{M+1}{2}$ .",
"See the appendix.",
"For the addressed topology, there are $M$ independent pathes given $M$ user pairs, which means that the maximal diversity gain is $M$ .",
"And Theorem REF indicates that the max-min scheduling approach cannot achieve this maximum diversity.",
"As a benchmark scheme, recall a conventional cooperative network that has the same topology as the one described in Section .",
"Without loss generality, let $P_i=P$ , i.e.",
"the relay transmission power is the same as the source power.",
"It can be easily verified that the max-min approach can achieve the optimal diversity gain, $M$ , as shown in the following.",
"The outage probability achieved by the max-min approach is $\\nonumber \\mathrm {P}_o&=&\\mathrm {P}(|h_{i^*}|^2<\\epsilon )+\\mathrm {P}(|g_{i^*}|^2<\\epsilon ,|h_{i^*}|^2>\\epsilon ) \\\\ \\nonumber &=&\\mathrm {P}(|h_{i^*}|^2<\\epsilon ,|g_{i^*}|^2>\\epsilon )+\\mathrm {P}(|h_{i^*}|^2<\\epsilon ,|g_{i^*}|^2<\\epsilon )+\\mathrm {P}(|g_{i^*}|^2<\\epsilon ,|h_{i^*}|^2>\\epsilon )\\\\ &=& \\mathrm {P}(\\min \\lbrace |h_{i^*}|^2,|g_{i^*}|^2\\rbrace <\\epsilon )\\rightarrow \\epsilon ^M,$ where the last step is obtained by using the probability density function (pdf) shown in (REF ) and applying the high SNR approximation.",
"Comparing (REF ) to (REF ), one can observe that the performance of the max-min scheduling approach in two system setups is significantly different, and new efficient user scheduling strategies are needed for energy harvesting cooperative networks."
],
[
"Scheduling a single user pair",
"A straightforward approach of user scheduling for the energy harvesting scenario is described as follows: Construct a subset of user pairs containing all the destinations whose source information can be decoded correctly at the relay.",
"Denote this subset as $\\mathcal {S}\\triangleq \\lbrace i\\in \\mathcal {S}: |h_i|^2\\ge \\epsilon \\rbrace $ .",
"Select a destination from $\\mathcal {S}$ to minimize the outage probability of the relay transmission.",
"Denote the index of the selected user by $i^*$ , i.e.",
"$i^*=\\arg \\max \\lbrace (|h_i|^2-\\epsilon )|g_i|^2, i\\in \\mathcal {S}\\rbrace $ .",
"The outage probability achieved by this user scheduling strategy can be expressed as follows: $\\mathrm {P}_o &\\triangleq & \\mathrm {P}\\left( |S|=0 \\right)+\\mathrm {P}\\left( (|h_{i^*}|^2-\\epsilon )|g_{i^*}|^2 <\\epsilon _1,|\\mathcal {S}|>0\\right) \\\\ \\nonumber &=&\\mathrm {P}\\left( |S|=0 \\right)+\\sum ^{M}_{n=1}\\underset{T_1}{\\underbrace{\\mathrm {P}\\left( (|h_{i^*}|^2-\\epsilon )|g_{i^*}|^2 <\\epsilon _1||\\mathcal {S}|=n\\right)}}\\mathrm {P}(|\\mathcal {S}|=n),$ where $|\\mathcal {S}|$ denotes the cardinality of the set.",
"Denote $x_i=|h_i|^2$ , and order $x_i$ as $x_{(1)}\\le \\cdots \\le x_{(M)}$ .",
"The probability of $\\mathrm {P}(|\\mathcal {S}|=n)$ can be calculated as follows: $\\mathrm {P}(|\\mathcal {S}|=n) &=& \\mathrm {P}(x_{(M-n)}<\\epsilon , x_{(M-n+1)}>\\epsilon )\\\\ \\nonumber &=& \\frac{M!}{(M-n)!n!}",
"\\left(1-e^{-\\epsilon }\\right)^{M-n} e^{-n\\epsilon },$ for $0\\le n \\le M$ , where the last equation is obtained by applying the joint pdf of $x_{(M-n)}$ and $x_{(N-n+1)}$ [19] and [20].",
"On the other hand $T_1$ can be simply expressed as follows: $T_1 &=& \\left[\\mathrm {P}\\left( (x_i-\\epsilon )y_i <\\epsilon _1, |i\\in \\mathcal {S}, |\\mathcal {S}|=n\\right)\\right]^{n},$ where $y_i=|g_i|^2$ .",
"In the following we first consider the case of $n\\ge 1$ .",
"The conditions of $T_1$ , $i\\in \\mathcal {S}$ and $ |\\mathcal {S}|=n$ , imply $x\\ge \\epsilon $ , which means that the conditional CDF of $x_i$ is given by $F_{x_i|i\\in \\mathcal {S}, |\\mathcal {S}|\\ge 1}(x) = \\frac{e^{-\\epsilon }-e^{-x}}{e^{-\\epsilon }},$ for $x\\ge \\epsilon $ .",
"The two conditions, $i\\in \\mathcal {S}$ and $|\\mathcal {S}|=n$ , do not affect $y_i$ which is still exponentially distributed.",
"Therefore the factor $T_1$ can be calculated as follows: $T_1 &=& \\left(\\mathcal {E}_{y}\\left(\\frac{e^{-\\epsilon }-e^{-\\frac{\\epsilon _1}{y}-\\epsilon }}{e^{-\\epsilon }}\\right)\\right)^n\\\\ \\nonumber &=& \\left(1 - 2\\sqrt{\\epsilon _1}\\mathbf {K}_{1}\\left(2\\sqrt{\\epsilon _1}\\right)\\right)^n,$ where $\\mathbf {K}_n(\\cdots )$ denotes the modified Bessel function of the second kind.",
"Recall that $x\\mathbf {K}_1(x)\\approx 1+\\frac{x^2}{2}\\ln \\frac{x}{2}$ for $x\\rightarrow 0$ , [13], which means $T_1\\approx \\epsilon \\ln \\frac{1}{\\epsilon }$ .",
"The overall outage probability can be approximated as follows: $\\mathrm {P}_o &=&\\left(1-e^{-\\epsilon }\\right)^M +\\sum ^{M}_{n=1}\\left(1 - 2\\sqrt{\\epsilon _1}\\mathbf {K}_{1}\\left(2\\sqrt{\\epsilon _1}\\right)\\right)^n \\frac{M!}{(M-n)!n!}",
"\\left(1-e^{-\\epsilon }\\right)^{M-n} e^{-n\\epsilon }\\\\ \\nonumber &\\approx & \\epsilon ^M +\\sum ^{M}_{n=1} \\epsilon ^n \\left(\\ln \\frac{1}{\\epsilon }\\right)^n \\frac{M!}{(M-n)!n!}",
"\\epsilon ^{M-n}.$ When $\\epsilon \\rightarrow 0$ , it is straightforward to show $\\frac{\\log \\mathrm {P}_o}{\\log \\epsilon }\\rightarrow M$ , which results in the following lemma.",
"$\\mathbf {Lemma}$ 1 The proposed user scheduling strategy can achieve the full diversity gain $M$ .",
"Compared to the maxi-min based approach, the proposed scheduling strategy can achieve a larger diversity gain.",
"The reason for this performance improvement is that the source-relay channels have been given a more important role for use scheduling, compared to the relay-destination channels.",
"Particularly the source-relay channels have been considered when forming $\\mathcal {S}$ and also selecting the best user from the set, whereas the relay-destination channels affect only the second step."
],
[
"Scheduling $m$ user pairs",
"The approach proposed in the previous subsection can be extended to the case of scheduling $m$ user pairs, as described in the following.",
"Construct a subset of user pairs, $\\mathcal {S}$ , as defined in Section REF .",
"Find all possible combinations of the users in $ \\mathcal {S} $ , denoted by $\\lbrace \\pi _1, \\cdots , \\pi _{{|\\mathcal {S}| \\atopwithdelims ()\\min \\lbrace m,|\\mathcal {S}|\\rbrace }}\\rbrace $ , where each set contains $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ users, i.e.",
"$\\pi _i=\\lbrace \\pi _i(1),\\ldots , \\pi _i(\\min \\lbrace m,|\\mathcal {S}|\\rbrace )\\rbrace $ .",
"For each possible combination, $\\pi _i$ , $1\\le i \\le {|\\mathcal {S}| \\atopwithdelims ()\\min \\lbrace m,|\\mathcal {S}|\\rbrace }$ Calculate the accumulated power obtained from energy harvesting, $\\sum _{j=1}^{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }P_{r\\pi _i(j)}$ .",
"Distribute the overall power among $m$ destinations equally, i.e.",
"$P_i=\\frac{\\sum _{j=1}^{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }P_{r\\pi _i(j)}}{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }$ .",
"Find the worst outage performance among the $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ users in $\\pi _i$ , denoted by $\\mathrm {P}_{o,\\pi _i}$ .",
"Select the combination which minimize the worst user outage performance, i.e.",
"$i^*=\\arg \\min \\lbrace \\mathrm {P}_{o,\\pi _1}, \\cdots , \\mathrm {P}_{o,\\pi _{{|\\mathcal {S}| \\atopwithdelims ()\\min \\lbrace m,|\\mathcal {S}|\\rbrace }}}\\rbrace $ .",
"This scheduling approach is to exhaustively search all possible combinations of the $|\\mathcal {S}|$ user pairs, and one combination will be selected if it can minimize the outage probability for the worst user case.",
"Provided that there is a large number of users to be scheduled, the complexity of this exhaustive search scheme can be infeasible due to the large number of the possible combinations.",
"Note that in this paper, we consider only the equal power allocation strategy, whereas other power allocation strategies, such as the sequential water filling scheme proposed in [13], can also be applied.",
"It is difficult to analyze the performance achieved by the exhaustive search approach, since the channel gains from different combinations might be correlated.",
"Instead, we will propose a greedy approach which is applicable to delay tolerant networks, and also serves as an upper bound for the system performance."
],
[
"Greedy user scheduling approach",
"First order all the source-relay channels and the relay-destination channels, i.e.",
"$|h_{(1)}|^2\\le \\ldots \\le |h_{(M)}|^2$ and $|g_{(1)}|^2\\le \\ldots \\le |g_{(M)}|^2$ .",
"The greedy user scheduling approach can be described as follows: Construct a subset of user pairs, $\\mathcal {S}$ , as defined in Section REF .",
"Schedule $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ sources with the best source-relay channel conditions during the first $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ time slots, i.e.",
"the $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ sources with the following channels, $|h_{(M-\\min \\lbrace m,|\\mathcal {S}|\\rbrace +1)}|^2\\le \\ldots \\le |h_{(M)}|^2$ .",
"Calculate the accumulated power obtained from energy harvesting, $\\sum _{j=1}^{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }P_{r (M-j+1)}$ .",
"Schedule $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ destinations with the best relay-destination channel conditions during the second $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ time slots, i.e.",
"the $\\min \\lbrace m,|\\mathcal {S}|\\rbrace $ destinations with the following channels, $|g_{(M-\\min \\lbrace m,|\\mathcal {S}|\\rbrace +1)}|^2\\le \\ldots \\le |g_{(M)}|^2$ , with equally allocated transmission power, denoted by $P_{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }=\\frac{\\sum _{j=1}^{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }P_{r (M-j+1)}}{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }$ .",
"Note that the scheduled destinations are not necessarily the partners of the scheduled sources, so this greedy approach assumes that the relay always has data to be transmitted to all the destinations.",
"Based on the above strategy description, the outage probability at the $i$ -th best destination, $1\\le i \\le \\min \\lbrace m, |\\mathcal {S}|\\rbrace $ , can be written as follows: $\\mathrm {P}_{oi}&\\triangleq & \\mathrm {P}\\left(|\\mathcal {S}|=0\\right) +\\sum ^{M}_{n=1}\\mathrm {P}\\left(\\left.P_{\\min \\lbrace m,|\\mathcal {S}|\\rbrace }|g_{(M-i+1)}|^2<(2^{2R}-1)\\right||\\mathcal {S}|=n\\right) \\mathrm {P}\\left(|\\mathcal {S}|=n\\right) .$ And the following lemma provides the exact expression of the above outage probability.",
"$\\mathbf {Lemma}$ 2 The outage probability achieved by the greedy user scheduling approach is given by: $\\mathrm {P}_{oi}&\\triangleq & \\mathrm {P}\\left(|\\mathcal {S}|=0\\right) +\\sum ^{m}_{n=1} T_2\\mathrm {P}\\left(|\\mathcal {S}|=n\\right)+\\sum ^{M}_{n=m+1}T_3\\mathrm {P}\\left(|\\mathcal {S}|=n\\right),$ where $\\mathrm {P}(|\\mathcal {S}|=n)$ is defined in (REF ), $T_2= i{M \\atopwithdelims ()i}\\sum ^{M-i}_{k=0} {M-i \\atopwithdelims ()k} \\frac{ (-1)^k}{k+i} \\left(1- \\frac{2\\left((k+i)n\\epsilon _1\\right)^{\\frac{n}{2}}}{(n-1)!",
"}\\mathbf {K}_n\\left(2\\sqrt{(k+i)n\\epsilon _1}\\right) \\right)$ , $T_3=\\frac{M!}{(M-i)!(i-1)!",
"}\\sum ^{M-i}_{l=0}{M-i \\atopwithdelims ()l} \\frac{(-1)^l}{l+i} \\left( 1- T_4\\right)$ , $T_4=\\sum ^{n-m-1}_{k=0} d_{m,k}\\left(\\sum ^{m}_{j=1}\\frac{2a_{j,k}\\left(m\\epsilon _1(l+i)\\right)^{\\frac{j}{2}} \\mathbf {K}_{j}\\left(2\\sqrt{m\\epsilon _1(l+i)}\\right) }{(j-1)!}",
"\\right.$ $ \\left.",
"+2b_k \\sqrt{\\frac{m\\epsilon _1(l+i)}{1+\\frac{k+1}{m}}} \\mathbf {K}_1\\left(2\\sqrt{\\epsilon _1(l+i)\\left(m+ k+1 \\right)}\\right)\\right)$ , $d_{m,k}=\\frac{n!",
"}{(n-m-1)!m!m}{n-m-1 \\atopwithdelims ()k} (-1)^k$ , $b_k=(-1)^m\\frac{m^{m}}{(k+1)^{m}}$ , and $a_{j,k}= \\frac{(-1)^{m-j}m^{m-j+1}}{(k+1)^{m-j+1}}$ .",
"See the appendix.",
"Although the outage probability expression in Lemma REF can be used for numerical studies, this form is quite complicated and cannot be used for analyzing diversity gains.",
"For the special case of $m=1$ , asymptotic studies can be carried out and the achievable diversity gain can be obtained, as shown in the following lemma.",
"$\\mathbf {Lemma}$ 3 When scheduling only a single user pair, i.e.",
"$m=1$ , the diversity gain achieved by the greedy user scheduling approach is $M$ .",
"See the appendix.",
"The fact that the greedy user scheduling approach can achieve the full diversity gain is not surprising, since the greedy approach outperforms the diversity-optimal one described in Section REF ."
],
[
"Numerical Results",
"In this section, computer simulations will be carried out to evaluate the performance of the user scheduling approaches addressed in this paper.",
"To simplify clarifications, we term the user scheduling approaches described in Section REF , REF and REF as “Approach I\", “Approach II\", and “Approach III\", respectively.",
"We first focus on the scenario where only a single user is scheduled.",
"In Fig.",
"REF the accuracy of the developed analytical results about the outage probability shown in Theorem REF , (REF ), and Lemma REF , is verified by using simulation results, where the targeted data rate is $R=4$ bits per channel use (BPCU), and the energy harvesting efficiency coefficient is $\\eta =1$ .",
"As can be seen from the figure, the developed analytical results match the simulation results exactly.",
"In Fig.",
"REF the outage probabilities achieved by different user scheduling approaches are examined with more details, where analytical results are used to generate the figure.",
"As a benchmark, the scheme with a random selected user is also shown in the figure, and its outage performance is the worst among all the scheduling approaches.",
"On the other hand, Approach III, the greedy user scheduling approach, can achieve the best outage performance.",
"The max-min scheduling approach can outperform random relaying, since its diversity gain can be improved when more users join in the competition, as shown in Theorem REF .",
"However, it will result in some performance loss compared to Approach I and Approach III, since it cannot achieve the full diversity gain, as indicated in Theorem REF .",
"Figure: Analytical results vs computer simulations.",
"Only one user pair is scheduled, η=1\\eta =1.",
"The targeted data rate is R=4R=4 BPCU.Figure: Comparison of various user scheduling approaches.",
"Only one user pair is scheduled.",
"η=1\\eta =1.",
"The targeted data rate is R=2R=2 BPCU.Since the main focus of this paper is to study the performance of the max-min user scheduling approach, Fig.",
"REF is provided in order to closely examine the diversity order achieved by this approach.",
"Particularly the analytical results developed in Theorem REF are used to generate the curves of outage probabilities.",
"To clearly demonstrate achievable diversity gains, auxiliary lines with the diversity order of $\\frac{M+1}{2}$ are also shown as a benchmark.",
"As can be seen from the figure, the outage probability curves for the max-min approach are always parallel to the benchmarking curves.",
"Recall that the diversity order is indicated by the slope of an outage probability curve.",
"Therefore Fig.",
"REF confirms that the diversity order achieved by the max-min approach is $\\frac{M+1}{2}$ , as indicated by Theorem REF .",
"The reason for this loss of diversity gains is that the max-min approach treats the source-relay channels and the relay-destination channels equally important when user scheduling is carried out.",
"However, when an energy harvesting relay is used, the source-relay channels become more important, since they affect not only the transmission reliability during the first phase, but also the transmission power for the second phase.",
"Figure: Verification of the diversity order for the max-min scheduling approach.",
"Only one user pair is scheduled.",
"η=1\\eta =1.",
"The targeted data rate is R=2R=2 BPCU.Figure: Comparison of various user scheduling approaches.",
"The total number of user pairs is M=10M=10, η=1\\eta =1 and two user pairs are scheduled, m=2m=2.Figure: Analytical results vs computer simulations.",
"The total number of user pairs is M=6M=6, η=1\\eta =1 and three user pairs are scheduled, m=3m=3.In Figs.",
"REF and REF we will focus on the scenario when multiple user pairs are scheduled.",
"Particularly, in Fig.",
"REF we compare the outage performance achieved by the three schemes, the max-min approach and the two approaches proposed in Section .",
"The total number of the user pairs is $M=10$ and two user pairs will be scheduled.",
"Since the scheduled users experience different outage performance, in the figure we show the outage performance for the user with the strongest SNR and also the user with the weakest SNR.",
"As can be observed from the figure, Approach III, the greedy user scheduling approach, can achieve the best outage performance, and the max-min approach achieves the worst performance.",
"But it is worthy to point out that Approach II outperforms the max-min approach at a price of high computational complexity, since Approach II needs to enumerate all possible combinations of the user pairs.",
"In Fig.",
"REF , we evaluate the accuracy of the analytical results developed in Lemma REF , by comparing the outage probability calculated using (REF ) to computer simulations.",
"The total number of the user pairs is $M=6$ and three user pairs will be scheduled.",
"As can be observed from the figure, the developed analytical results match the computer simulations exactly.",
"Finally we present some simulation results when $\\eta <1$ and the large scale path loss is considered.",
"Particularly consider a disk with the relay at its center and its diameter as 4 meters.",
"The $M$ pairs of sources and destinations are uniformly deployed in this disc, and the used path loss exponent is 2.",
"In Fig.",
"REF and Fig.",
"REF , the performance of the user scheduling approaches for the cases of $m=1$ and $m=2$ are shown, respectively.",
"As can be seen from Fig.",
"REF , the use of the user scheduling approaches can improve the system performance compared to the random relaying scheme.",
"Another observation from both figures is that, among all the opportunistic scheduling approaches, the max-min approach achieves the worst performance, and the greedy approach outperforms the other user scheduling approaches, which is consistent to the previous figures.",
"Figure: Comparison of various user scheduling approaches.",
"η=0.5\\eta =0.5.",
"The total number of user pairs is M=6M=6, and one user pair is scheduled, m=1m=1.Figure: Comparison of various user scheduling approaches.",
"η=0.5\\eta =0.5.",
"The total number of user pairs is M=6M=6, and two user pairs are scheduled, m=3m=3.",
"The targeted data rate is R=2R=2 BPCU."
],
[
"Conclusions",
"In this paper, we considered an energy harvesting cooperative network with $M$ source-destination pairs and one relay, where the relay schedules only $m$ user pairs for transmissions.",
"It is important to point out that for the special case of $m=1$ , the addressed scheduling problem is the same as relay selection for the scenario with one source-destination pair and $M$ relays.",
"The main contribution of this paper is to show that the use of the max-min criterion will result in loss of diversity gains, when an energy harvesting relay is employed.",
"Particularly when only one user is scheduled, analytical results have been developed to demonstrate that the diversity gain achieved by the max-min criterion is only $\\frac{M+1}{2}$ , much less than the maximal diversity gain $M$ .",
"Motivated by this performance loss, a few user scheduling approaches tailored to energy harvesting networks have been proposed and their performance is analyzed.",
"Simulation results have been provided to demonstrate the accuracy of the developed analytical results and facilitate the performance comparison.",
"When developing user scheduling approaches, only reception reliability is considered, and it is assumed that the network is delay tolerant.",
"It is a promising future direction to study how to achieve a balanced tradeoff between reception reliability and user delay.",
"Proof of Theorem REF : To simplify notation, define $x=|h_{i^*}|^2$ and $y=|g_{i^*}|^2$ , and the outage probability in (REF ) can be expressed as follows: $\\mathrm {P}_{o}\\triangleq \\mathrm {P}\\left(x<\\epsilon \\right) + \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon \\right).$ The scheduling strategy has changed the statistical property of $x$ and $y$ , but the density function of $\\min \\lbrace x,y\\rbrace $ can be found simply by applying order statistics.",
"To use such a density function, we need to first rewrite the outage probability as follows: $\\mathrm {P}_{o}&=&\\mathrm {P}\\left(x<\\epsilon ,x>y\\right) + \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon , x>y \\right)\\\\ \\nonumber &&+\\mathrm {P}\\left(x<\\epsilon ,x<y\\right) + \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon , x<y \\right).$ Converting the joint probabilities to conditional probabilities, the outage probability is given by $\\mathrm {P}_{o}&=&\\mathrm {P}\\left(x<\\epsilon |x>y\\right)\\mathrm {P}(x>y) + \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon | x>y \\right)\\mathrm {P}(x>y)\\\\ \\nonumber &&+\\mathrm {P}\\left(x<\\epsilon |x<y\\right)\\mathrm {P}(x<y) + \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon |x<y \\right)\\mathrm {P}(x<y).$ Since the incoming and outgoing channels at the relay are independent and identically distributed, we have $\\mathrm {P}(x>y)=\\mathrm {P}(x<y)=\\frac{1}{2}$ .",
"Consequently the outage probability can be expressed as in the following form: $\\mathrm {P}_{o}&=&\\frac{1}{2}\\underset{Q_1}{\\underbrace{\\mathcal {E}_{y|x>y}\\left\\lbrace \\mathrm {P}\\left(x<\\epsilon |x>y\\right)\\right\\rbrace }} + \\frac{1}{2}\\underset{Q_2}{\\underbrace{\\mathcal {E}_{y|x>y}\\left\\lbrace \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon | x>y \\right)\\right\\rbrace }}\\\\ \\nonumber &&+\\frac{1}{2}\\underset{Q_3}{\\underbrace{\\mathcal {E}_{x|x<y}\\left\\lbrace \\mathrm {P}\\left(x<\\epsilon |x<y\\right)\\right\\rbrace }} + \\frac{1}{2}\\underset{Q_4}{\\underbrace{\\mathcal {E}_{y|x>y}\\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon |x<y \\right)}},$ where $\\mathcal {E}\\lbrace \\cdot \\rbrace $ denotes the expectation operation.",
"The rationale to have the above expression is following.",
"Take $Q_1$ as an example.",
"$Q_1$ can be calculated in two steps.",
"The first step is to calculate $Q_1$ by treating $y$ as a constant and using the condition $x>y$ .",
"The second step is to calculate the expectation of the probability by using the density function of $y$ .",
"Since $x>y$ , $y=\\min \\lbrace x,y\\rbrace $ , and the density function of $y$ can be found easily.",
"In the following the four terms $Q_i$ will be evaluated individually.",
"Calculating $Q_1$ We start from the calculation of $Q_1$ , the first terms in (REF ).",
"In particular, $Q_1$ can be expressed as follows: $Q_1&=& \\int ^{\\epsilon }_{0}\\int ^{\\epsilon }_{y}f_{x|x>y,y}(x)dx f_{y|x>y}(y)dy,$ where $f_{x|x>y,y}(x)$ is the pdf of $x$ conditioned on a fixed $y$ and $x>y$ , and $f_{y|x>y}(y)$ is the pdf of $y$ also conditioned on $x>y$ .",
"To find the two conditional pdfs, we first define $x_i=|h_i|^2$ and $y_i=|g_i|^2$ , $z_i=\\min \\lbrace x_i,y_i\\rbrace $ , and $z=\\min \\lbrace z_i, 1\\le i \\le M\\rbrace $ .",
"From order statistics [19], the pdf of $z_i $ is $f_{z_i}(z)=2e^{-2z}$ , and the pdf of $z $ can be found as follows: $f_{z}(z) = 2Me^{-2z}\\left(1-e^{-2z}\\right)^{M-1}.$ Conditioned on $x>y$ , the pdf of $y$ is the same as $z$ , i.e.",
"$f_{y|x>y}(y)=f_z(y)$ .",
"On the other hand, conditioned on a fixed $y$ and $x>y$ , the cumulative distribution function (CDF) of $x$ can be found as follows: $F_{x|x>y,y}(x) = \\frac{e^{-y}-e^{-x}}{e^{-y}},$ where the factor $e^{-y}$ at the denominator is to ensure $F_{x|x>y}(x)\\rightarrow 1$ when $x\\rightarrow \\infty $ .",
"By using the obtained conditional pdfs, $Q_1$ can be calculated as follows: $Q_1&=& \\int ^{\\epsilon }_{0}\\int ^{\\epsilon }_{z}f_{x|x>y}(x)dx f_{z}(z)dy\\\\ \\nonumber &=& e^{-\\epsilon }\\int ^{\\epsilon }_{0} \\left(1-e^{-2z} \\right)^{M}e^{z}dz.$ By applying binomial expansions, we obtain the following: $Q_1 &=& e^{-\\epsilon }\\sum _{i=0}^{M}{M \\atopwithdelims ()i} \\frac{(-1)^i}{2i-1}\\left(1-e^{-(2i-1)\\epsilon }\\right).$ Calculating $Q_2$ Recall that $Q_2=\\mathcal {E}_{y|x>y}\\left\\lbrace \\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon | x>y \\right)\\right\\rbrace $ .",
"The conditional density functions, $f_{x|x>y,y}(x)$ and $f_{y|x>y}(y)$ , obtained in (REF ) and (REF ) can be used again.",
"An important step to calculate $Q_2$ is to determine the domain of integration.",
"The constrains, $x>y$ and $x<\\frac{\\epsilon _1}{y}+\\epsilon $ , imply that $y<\\frac{\\epsilon _1}{y}+\\epsilon $ .",
"Together with the additional constraint, $x>\\epsilon $ , the integration domain for $Q_2$ is given by $\\left\\lbrace \\begin{array}{ll}y<x <\\frac{\\epsilon _1}{y}+\\epsilon , & if \\quad \\epsilon <y<\\epsilon _0 \\\\ \\epsilon <x <\\frac{\\epsilon _1}{y}+\\epsilon , & if \\quad 0\\le y <\\epsilon \\end{array} \\right.,$ where $\\epsilon _0\\triangleq \\frac{\\epsilon +\\sqrt{\\epsilon ^2+4\\epsilon _1}}{2}$ is the positive root of $y^2-\\epsilon y-\\epsilon =0$ , due to the constraint $y<\\frac{\\epsilon _1}{y}+\\epsilon $ .",
"With the obtained integration domain, $Q_2$ can be rewritten as follows: $Q_2 &=& \\int ^{\\epsilon }_{0}\\int ^{\\epsilon +\\frac{\\epsilon _1}{y}}_{\\epsilon }f_{x|x>y}(x)dx f_{y|x>y}(y)dy\\\\ \\nonumber &&+ \\int ^{\\epsilon _0}_{\\epsilon }\\int ^{\\epsilon +\\frac{\\epsilon _1}{y}}_{y}f_{x|x>y}(x)dx f_{y|x>y}(y)dy\\\\ \\nonumber &=&\\underset{Q_{21}}{\\underbrace{\\int ^{\\epsilon }_{0} \\left(\\frac{e^{-\\epsilon } - e^{-\\epsilon -\\frac{\\epsilon _1}{y}}}{e^{-y}}\\right)f_{y|x>y}(y)dy}}\\\\ \\nonumber &&+\\underset{Q_{22}}{\\underbrace{\\int ^{\\epsilon _0}_{\\epsilon } \\left(\\frac{e^{-y} - e^{-\\epsilon -\\frac{\\epsilon _1}{y}}}{e^{-y}}\\right)f_{y|x>y}(y)dy}}.$ Now applying the conditional pdf of $y$ , the first factor, $Q_{21}$ , in the above equation can be expressed as follows: $Q_{21} &=& 2Me^{-\\epsilon } \\int ^{\\epsilon }_{0} \\left(1-e^{-\\frac{\\epsilon _1}{y}}\\right)e^{-y}\\left(1-e^{-2y}\\right)^{M-1}dy\\\\ \\nonumber &=& 2Me^{-\\epsilon } \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\left(\\frac{1-e^{-(2i+1)\\epsilon }}{2i+1} - \\int ^{\\epsilon }_{0}e^{-(2i+1)y-\\frac{\\epsilon _1}{y}} dy \\right).$ Similarly the factor $Q_{22}$ can be calculated as follows: $Q_{22} &=& 2M \\int ^{\\epsilon _0}_{\\epsilon } \\left(e^{-y}- e^{-\\epsilon -\\frac{\\epsilon _1}{y}}\\right)e^{-y}\\left(1-e^{-2y}\\right)^{M-1}dy\\\\ \\nonumber &=& 2M \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-(2i+2)\\epsilon }-e^{-(2i+2)\\epsilon _0}}{2i+2} - e^{-\\epsilon } \\int ^{\\epsilon _0}_{\\epsilon }e^{-(2i+1)y-\\frac{\\epsilon _1}{y}} dy \\right).$ By combining (REF ) and (REF ), the factor $Q_2$ can be expressed as follows: $Q_{2} &=& 2M \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-\\epsilon }-e^{-(2i+2)\\epsilon }}{2i+1} +\\frac{e^{-(2i+2)\\epsilon }-e^{-(2i+2)\\epsilon _0}}{2i+2} \\right.",
"\\\\ \\nonumber &&\\left.- e^{-\\epsilon } \\int ^{\\epsilon _0}_{0}e^{-(2i+1)y-\\frac{\\epsilon _1}{y}} dy \\right).$ Calculating $Q_4$ Recall that $Q_4=\\mathcal {E}_{y|x>y}\\mathrm {P}\\left((x-\\epsilon )y<\\epsilon _1, x>\\epsilon |x<y \\right)$ .",
"Again it is important to determine the integration domain of $Q_4$ .",
"Particularly, the integral constraints, $y< \\frac{\\epsilon _1}{x-\\epsilon }$ , $x>\\epsilon $ and $x<y$ , imply the inegration domain of $x<y< \\frac{\\epsilon _1}{x-\\epsilon }$ and $\\epsilon <x<\\epsilon _0$ , where the inequality of $x<\\epsilon _0$ is due to $ x<\\frac{\\epsilon _1}{x-\\epsilon }$ , i.e.",
"$x^2-\\epsilon x-\\epsilon _1<0$ .",
"By applying the obtained integration domain, $Q_4$ is calculated as follows: $Q_4 &=& \\int ^{\\epsilon _0}_{\\epsilon }\\int ^{ \\frac{\\epsilon _1}{x-\\epsilon }}_{x}f_{y|x<y,x}(y)dy f_{x|x<y}(x)dx\\\\ \\nonumber &=&\\int ^{\\epsilon _0}_{\\epsilon } \\left(\\frac{e^{-x} - e^{-\\frac{\\epsilon _1}{x-\\epsilon }}}{e^{-x}}\\right)f_{x|x<y}(x)dx,$ where the last equation follows from the symmetry of incoming and outgoing channels, i.e.",
"$f_{y|x<y,x}(y)=f_{x|y>x,y}(x)$ .",
"Similarly we have $f_{x|x<y}(x)=f_{y|x>y}(y)$ , which yields the following expression of $Q_4$ : $Q_4 &=& 2M\\int ^{\\epsilon _0}_{\\epsilon } \\left(1-e^{x-\\frac{\\epsilon _1}{x-\\epsilon }}\\right)e^{-2x} \\left(1-e^{-2x}\\right)^{M-1}dx \\\\ \\nonumber &=& 2M\\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-2(i+1)\\epsilon }-e^{-2(i+1)\\epsilon _0}}{2(i+1)} - \\int ^{\\epsilon _0}_{\\epsilon } e^{-(2i+1)x-\\frac{\\epsilon _1}{x-\\epsilon }}dx\\right).$ On the other hand, $Q_3$ can be easily calculated as $Q_3= F_z(\\epsilon ) = \\left(1-e^{-2\\epsilon }\\right)^M $ , where $F_z(z)$ is the CDF corresponding to the pdf in (REF ).",
"Therefore, the overall outage probability can be expressed as $\\nonumber \\mathrm {P}_o&=& \\frac{e^{-\\epsilon }}{2}\\sum _{i=0}^{M}{M \\atopwithdelims ()i} \\frac{(-1)^i}{2i-1}\\left(1-e^{-(2i-1)\\epsilon }\\right) \\\\\\nonumber &&+M \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-\\epsilon }-e^{-(2i+2)\\epsilon }}{2i+1} +\\frac{e^{-(2i+2)\\epsilon }-e^{-(2i+2)\\epsilon _0}}{2i+2} - e^{-\\epsilon } \\int ^{\\epsilon _0}_{0}e^{-(2i+1)y-\\frac{\\epsilon _1}{y}} dy \\right)\\\\ \\nonumber &&+ \\frac{\\left(1-e^{-2\\epsilon }\\right)^M}{2}+ M\\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i \\left(\\frac{e^{-2(i+1)\\epsilon }-e^{-2(i+1)\\epsilon _0}}{2(i+1)} - e^{-(2i+1)\\epsilon }\\int ^{\\epsilon _0-\\epsilon }_{0} e^{-(2i+1)x-\\frac{\\epsilon _1}{x }}dx\\right),$ and the proof of the theorem is completed.",
"$\\blacksquare $ Proof of Theorem REF : To simplify the analytical development, we let $\\eta =1$ , which means $\\epsilon _1=\\epsilon $ .",
"Note that this simplification has no impact on the developed analytical results, since the diversity order is obtained at high SNR.",
"As shown in (REF ), the outage probability can be expressed as $\\mathrm {P}_o=\\frac{1}{2}\\sum ^{4}_{l=1}Q_l$ .",
"In the following the asymptotic study for the four terms will be carried out individually.",
"Asymptotic study of $Q_1$ The aim of the asymptotic study is to convert $Q_1$ in a form of $t\\epsilon ^d$ , where $t$ should be a constant, not a function of $\\epsilon $ , and $d$ will be used to determine the diversity order.",
"By applying series expansion of exponential functions, $Q_1$ , the first term in (REF ), can be expressed as follows: $Q_1 &=& e^{-\\epsilon }\\sum _{i=0}^{M}{M \\atopwithdelims ()i} \\frac{(-1)^i}{2i-1}\\left(1-\\sum _{k=0}^{\\infty }\\frac{(-1)^k (2i-1)^k \\epsilon ^k}{k!}",
"\\right)\\\\ \\nonumber &=&-e^{-\\epsilon }\\sum _{k=1}^{\\infty }\\frac{ (-1)^k \\epsilon ^k}{k!}",
"\\sum _{i=0}^{M}{M \\atopwithdelims ()i} (-1)^i (2i-1)^{k-1}.$ Compared to the expression of $Q_1$ in (REF ), the above form is more complicated, but facilitates the asymptotic studies as shown in the following.",
"Recall the following two properties about the sums of binomial coefficients: [21] $\\sum ^{M}_{i=0}(-1)^i {M \\atopwithdelims ()i} i^{j}=0,$ for $0\\le j\\le (M-1)$ , and $\\sum ^{M}_{i=0}(-1)^i {M \\atopwithdelims ()i} i^{M}=(-1)^MM!.$ These prosperities are useful to remove the terms at the order of $\\epsilon ^{d}$ , $d<\\frac{M+1}{2}$ , from $Q_1$ , as described in the following.",
"To make the above properties applicable, we rewrite $Q_1$ as follows: $Q_1&=&-e^{-\\epsilon }\\sum _{k=1}^{\\infty }\\frac{ (-1)^k \\epsilon ^k}{k!}",
"\\sum _{i=0}^{M}{M \\atopwithdelims ()i} (-1)^i \\sum ^{k-1}_{j=0}{k-1 \\atopwithdelims ()j}(-1)^{k-1-j}2^ji^j.$ All the terms with $i^j$ for $j\\le (M-1)$ can be removed because of (REF ).",
"At high SNR, i.e.",
"$\\epsilon \\rightarrow 0$ , all the factors with $\\epsilon ^{k}$ for $k\\ge (M+2)$ can be also ignored.",
"So the dominant factor of $Q_1$ will be the one at the order of $\\epsilon ^{M+1}$ .",
"By applying (REF ), $Q_1$ can be approximated as follows: $Q_1&\\approx & -\\frac{ (-1)^{M+1} \\epsilon ^{M+1}}{(M+1)!}",
"\\sum _{i=0}^{M}{M \\atopwithdelims ()i} (-1)^i 2^Mi^M\\\\ \\nonumber &=&-\\frac{ (-1)^{M+1} \\epsilon ^{M+1}}{(M+1)!}",
"2^M (-1)^MM!",
"= \\frac{2^M\\epsilon ^{M+1}}{M+1}.$ Therefore the first factor of the outage probability expression in (REF ) is at the order of $\\epsilon ^{M+1}$ .",
"Asymptotic study of $Q_2$ The approximation of $Q_2$ is more difficult than that of $Q_1$ , since $Q_2$ contains an integral which cannot be expressed analytically.",
"As shown in (REF ), $Q_2$ can be re-written as follows: $\\nonumber Q_{2} &=&2M\\underset{\\tilde{Q}_{21}}{\\underbrace{ \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\frac{e^{-\\epsilon }-e^{-(2i+2)\\epsilon }}{2i+1}}} +2M\\underset{\\tilde{Q}_{22}}{\\underbrace{\\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\frac{e^{-(2i+2)\\epsilon }-e^{-(2i+2)\\epsilon _0}}{2i+2}}} \\\\ && -2Me^{-\\epsilon } \\underset{\\tilde{Q}_{23}}{\\underbrace{\\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\int ^{\\epsilon _0}_{0}e^{-(2i+1)y-\\frac{\\epsilon }{y}} dy }}.$ Again by applying the properties in (REF ) and (REF ), $ \\tilde{Q}_{21}$ can be approximated as follows: $\\tilde{Q}_{21}&=&e^{-\\epsilon }\\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^{i+1} \\sum ^{\\infty }_{k=1} \\frac{(-1)^k}{k!",
"}\\epsilon ^k \\sum ^{k-1}_{j=0} {k \\atopwithdelims ()j} 2^j i^j \\\\ \\nonumber &\\approx &e^{-\\epsilon } \\frac{2^{M-1}\\epsilon ^M}{M}.$ Similarly the factor $ \\tilde{Q}_{22}$ can be approximated as follows: $\\tilde{Q}_{22} &=&\\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\left(\\sum ^{\\infty }_{k=1}\\frac{(-1)^k}{k!",
"}2^{k-1}\\left(\\epsilon ^{k}-\\epsilon _0^k\\right) \\sum ^{k-1}_{j=0}{k-1 \\atopwithdelims ()j} i^j\\right)\\\\ \\nonumber &\\approx & \\frac{(-1)^M}{M!",
"}2^{M-1}\\left(\\epsilon ^{M}-\\epsilon _0^M\\right) (-1)^{M-1}(M-1)!",
"= \\frac{2^{M-1}\\left(\\epsilon _0^M-\\epsilon ^M\\right)}{M}.$ Different from $\\tilde{Q}_{21}$ and $\\tilde{Q}_{22}$ , it is difficult to directly find the the closed form of the asymptotic expression for the term $\\tilde{Q}_{23}$ .",
"Instead, we will first develop the upper and lower bounds on $\\tilde{Q}_{23}$ and then show that they converge at high SNR.",
"Observe that for the integral of $\\tilde{Q}_{23}$ , $ \\int ^{\\epsilon _0}_{0}e^{-(2i+1)y-\\frac{\\epsilon }{y}} dy $ , the range of $y$ is from 0 to $\\epsilon _0$ , so $y\\rightarrow 0$ at high SNR.",
"Therefore the term in the integral, $e^{-(2i+1)y}$ , can be approximated at high SNR.",
"This observation motivates us to rewrite $\\tilde{Q}_{23}$ as follows: $\\tilde{Q}_{23}&=& \\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i \\int ^{\\epsilon _0}_{0}\\left(\\sum ^{\\infty }_{k=0} \\frac{(-1)^k(2i+1)^ky^k}{k!}",
"\\right)e^{-\\frac{\\epsilon }{y}} dy \\\\ \\nonumber &=& \\int ^{\\epsilon _0}_{0}\\left(\\sum ^{\\infty }_{k=0} \\frac{(-1)^ky^k}{k!}",
"\\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i (2i+1)^k\\right) e^{-\\frac{\\epsilon }{y}} dy.$ By using the properties in (REF ) and (REF ), $\\tilde{Q}_{23}$ can be approximated as follows: $\\tilde{Q}_{23} &\\approx &\\int ^{\\epsilon _0}_{0}\\left( \\frac{(-1)^{M-1}y^{M-1}}{(M-1)!}",
"2^{M-1} (-1)^{M-1}(M-1)!",
"\\right) e^{-\\frac{\\epsilon }{y}} dy\\\\ \\nonumber &=& \\int ^{\\epsilon _0}_{0} 2^{M-1} y^{M-1}e^{-\\frac{\\epsilon }{y}} dy\\triangleq \\bar{Q}_{23},$ where the approximation follows from the fact that $0\\le y\\le \\epsilon _o$ and $\\epsilon _0\\rightarrow 0$ at high SNR.",
"To obtain the upper and lower bounds on $\\bar{Q}_{23}$ , the use of the inequalities for exponential functions yields the following: $1-\\frac{\\epsilon }{y}\\le e^{-\\frac{\\epsilon }{y}}\\le \\frac{1}{1+\\frac{\\epsilon }{y}},$ for $0\\le y\\le \\epsilon _0$ .",
"Now the upper bound of $Q_{23}$ can be computed as follows: $\\bar{Q}_{23} &\\le &2^{M-1}\\int ^{\\epsilon _0}_{0} y^{M-1}\\frac{1}{1+\\frac{\\epsilon }{y}} dy\\\\ \\nonumber &= &2^{M-1}\\sum ^{M}_{i=1}{M \\atopwithdelims ()i} (-1)^{M-i}\\epsilon ^{M-i}\\frac{(\\epsilon _0+\\epsilon )^i-\\epsilon ^i}{i} +2^{M-1}(-1)^M\\epsilon ^M\\ln \\frac{\\epsilon _0+\\epsilon }{\\epsilon }.$ At high SNR, $\\epsilon \\rightarrow 0$ , and $\\epsilon _0\\rightarrow \\epsilon ^{\\frac{1}{2}}$ .",
"Therefore the dominant factors in the upper bound of $\\bar{Q}_{23}$ are the terms with $i=M$ and $i=M-1$ , which means $\\bar{Q}_{23} &\\le &2^{M-1} \\left( \\frac{(\\epsilon _0+\\epsilon )^M-\\epsilon ^M}{M} -\\epsilon \\frac{M(\\epsilon _0+\\epsilon )^{M-1}-M\\epsilon ^{M-1}}{M-1} \\right).$ Combining (REF ), (REF ) and (REF ), $Q_2$ can be lower bounded as follows: $Q_2 &=& 2M\\tilde{Q}_{21}+2M\\tilde{Q}_{22} -2Me^{-\\epsilon }\\tilde{Q}_{23}\\\\ \\nonumber &\\ge & 2^{M}\\epsilon ^M+ 2^{M}\\left(\\epsilon _0^M-\\epsilon ^M\\right)- 2^{M} (\\epsilon _0+\\epsilon )^M+2^M\\epsilon ^M\\\\ \\nonumber &&+2^{M} \\epsilon \\frac{M^2(\\epsilon _0+\\epsilon )^{M-1}}{M-1}-2^{M} \\epsilon \\frac{M^2\\epsilon ^{M-1}}{M-1}\\\\ \\nonumber &\\underset{(a)}{\\approx }& 2^M\\left(\\epsilon _0^M - \\epsilon _0^M-M\\epsilon \\epsilon _0^{M-1} +\\epsilon \\frac{M^2\\epsilon _0^{M-1}}{M-1} \\right) \\\\ \\nonumber &=& \\frac{2^M}{M-1}\\epsilon \\epsilon _0^{M-1}\\rightarrow \\epsilon ^{\\frac{M+1}{2}},$ where $(a)$ is obtained by keeping only the terms at the order of $\\epsilon _0^M$ and $\\epsilon \\epsilon _0^{M-1}$ .",
"The lower bound of $\\bar{Q}_{23}$ can be obtained as follows: $\\bar{Q}_{23} &\\ge &2^{M-1}\\int ^{\\epsilon _0}_{0} y^{M-1}\\left(1-\\frac{\\epsilon }{y}\\right) dy\\\\ \\nonumber &= &2^{M-1}\\left(\\frac{\\epsilon _0^M}{M}-\\epsilon \\frac{\\epsilon _0^{M-1}}{M-1}\\right).$ Combining (REF ) with (REF ) and (REF ), the upper bound of $Q_2$ can be asymptotically shown in the following: $Q_2 &=& 2M\\tilde{Q}_{21}+2M\\tilde{Q}_{22} -2Me^{-\\epsilon }\\tilde{Q}_{23}\\\\ \\nonumber &\\le & 2^{M}\\epsilon ^M+ 2^{M}\\left(\\epsilon _0^M-\\epsilon ^M\\right)- 2^{M} \\left(\\epsilon _0^M -\\epsilon \\frac{M\\epsilon _0^{M-1}}{M-1}\\right)\\\\ \\nonumber &=& \\frac{M2^M}{M-1}\\epsilon \\epsilon _0^{M-1}\\rightarrow \\epsilon ^{\\frac{M+1}{2}}.$ As can be observed from (REF ) and (REF ), the upper and lower bounds converge at high SNR, which implies $Q_2 \\rightarrow \\epsilon ^{\\frac{M+1}{2}}.$ Asymptotic study of $Q_4$ First rewrite $Q_4$ in the following expression: $Q_4&=&2M\\underset{\\tilde{Q}_{41}}{\\underbrace{\\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i \\frac{e^{-2(i+1)\\epsilon }-e^{-2(i+1)\\epsilon _0}}{2(i+1)} }}\\\\ \\nonumber &&- 2M\\underset{\\tilde{Q}_{42}}{\\underbrace{\\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i e^{-(2i+1)\\epsilon }\\int ^{\\epsilon _0-\\epsilon }_{0} e^{-(2i+1)x-\\frac{\\epsilon }{x }}dx }}.$ Comparing (REF ) to (REF ), we observe that $\\tilde{Q}_{41} $ is the same as $\\tilde{Q}_{22} $ , and therefore can be approximated similarly as follows: $\\tilde{Q}_{41} = \\tilde{Q}_{22} \\approx \\frac{2^{M-1}\\left(\\epsilon _0^M-\\epsilon ^M\\right)}{M}.$ Similar to $\\tilde{Q}_{23} $ , the term $\\tilde{Q}_{42}$ also contains an integral whose analytical closed-form expression cannot be found.",
"Following the previous steps, we can first use the series expansion of $e^{-(2i+1)(\\epsilon +x)}$ to get the following: $\\tilde{Q}_{42}&=& \\int ^{\\epsilon _0-\\epsilon }_{0} \\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i \\sum ^{\\infty }_{k=0}\\frac{(-1)^k}{k!}",
"(2i+1)^k(\\epsilon +x)^k e^{ -\\frac{\\epsilon }{x }}dx\\\\ \\nonumber &=& \\int ^{\\epsilon _0-\\epsilon }_{0} \\sum ^{\\infty }_{k=0}\\frac{(-1)^k}{k!}",
"\\sum ^{M-1}_{i=0} {M-1\\atopwithdelims ()i} (-1)^i \\left(\\sum ^{k}_{j=0}{k \\atopwithdelims ()j} 2^j i^j\\right)(\\epsilon +x)^k e^{ -\\frac{\\epsilon }{x }}dx.$ And by using the properties in (REF ) and (REF ), we obtain $\\tilde{Q}_{42}&\\approx & \\int ^{\\epsilon _0-\\epsilon }_{0} \\frac{(-1)^{M-1}}{(M-1)!}",
"\\left( 2^{M-1} (-1)^{M-1}(M-1)!\\right)(\\epsilon +x)^{M-1} e^{ -\\frac{\\epsilon }{x }}dx\\\\ \\nonumber &=& \\int ^{\\epsilon _0-\\epsilon }_{0} 2^{M-1} (\\epsilon +x)^{M-1} e^{ -\\frac{\\epsilon }{x }}dx\\triangleq \\bar{Q}_{42}.$ Again applying the upper bound of exponential functions, we have $\\bar{Q}_{42}&\\le & \\int ^{\\epsilon _0-\\epsilon }_{0} 2^{M-1} (\\epsilon +x)^{M-1} \\frac{1}{1+\\frac{\\epsilon }{x }}dx\\\\ \\nonumber &=&2^{M-1} \\sum ^{M-2}_{i=0} {M-2 \\atopwithdelims ()i} \\epsilon ^{M-2-i}\\frac{(\\epsilon _0-\\epsilon )^{i+2}}{i+2} \\\\ \\nonumber &\\approx & 2^{M-1}\\frac{(\\epsilon _0-\\epsilon )^{M}}{M}\\rightarrow \\epsilon ^{-\\frac{M+1}{2}}.$ By subsisting this upper bound to the expression of $Q_4$ , the lower bound of $Q_4$ is given by $Q_4&\\ge & 2^{M}\\left(\\epsilon _0^M-\\epsilon ^M\\right) - 2^{M} (\\epsilon _0-\\epsilon )^{M} \\approx 2^MM \\epsilon \\epsilon _0^{M-1}.$ On the other hand, the lower bound of $\\tilde{Q}_{42}$ can be expressed as follows: $\\bar{Q}_{42}&\\ge & \\int ^{\\epsilon _0-\\epsilon }_{0} 2^{M-1} x^{M-1} \\left(1-\\frac{\\epsilon }{x }\\right)dx \\\\ \\nonumber &=& 2^{M-1} \\left(\\frac{(\\epsilon _0-\\epsilon )^M}{M} -\\epsilon \\frac{(\\epsilon _0-\\epsilon )^{M-1}}{M-1} \\right).$ Therefore the upper bound of $Q_4$ can be shown as follows: $Q_4&\\le & 2^{M}\\left(\\epsilon _0^M-\\epsilon ^M\\right) - 2^{M} \\left( (\\epsilon _0-\\epsilon )^M -\\epsilon \\frac{M(\\epsilon _0-\\epsilon )^{M-1}}{M-1} \\right) \\\\ \\nonumber &\\approx & 2^{M} \\epsilon _0^M - 2^{M} \\left( \\epsilon _0^M-M\\epsilon \\epsilon _0^{M-1} - \\frac{M\\epsilon \\epsilon _0^{M-1}}{M-1} \\right) \\\\\\nonumber &=& \\frac{M^22^M}{M-1}\\epsilon \\epsilon _0^{M-1}\\rightarrow \\epsilon ^{\\frac{M+1}{2}},$ where the approximation is carried out by keeping only the terms at $\\epsilon _0^M$ and $\\epsilon _0^{M-1}\\epsilon $ .",
"Combining (REF ) and (REF ), one can observe that the upper and lower bounds converge at high SNR, and the following conclusion can be obtained $Q_4 \\rightarrow \\epsilon ^{\\frac{M+1}{2}}.$ Applying the series expansion of exponential functions, $Q_3$ can be simply approximated as $Q_3 \\approx 2^M\\epsilon ^M$ .",
"Therefore the asymptotic expression for the overall outage probability can be obtained as follows: $\\mathrm {P}_o &=& \\frac{1}{2}\\sum ^{4}_{i=1}Q_4 \\\\ \\nonumber &\\rightarrow & \\epsilon ^{M+1} +2\\epsilon ^{\\frac{M+1}{2}}+\\epsilon ^M\\rightarrow \\epsilon ^{\\frac{M+1}{2}},$ and the proof of the theorem is completed.",
"$\\blacksquare $ Proof of Lemma REF : Based on the equal power allocation strategy, the power allocated to each destination is given by $ \\left\\lbrace \\begin{array}{ll} \\frac{1}{m}\\sum ^{m}_{i=1}P\\eta \\left(x_{(M-i+1)}-\\epsilon \\right) & if \\quad M\\ge n\\ge m \\\\\\frac{1}{n}\\sum ^{n}_{i=1}P\\eta \\left(x_{(M-i+1)}-\\epsilon \\right) & if \\quad 1\\le n\\le m-1 \\end{array}\\right..$ Therefore the overall outage probability will be $\\mathrm {P}_{oi}&\\triangleq & \\mathrm {P}\\left(|\\mathcal {S}|=0\\right) +\\sum ^{m}_{n=1}\\underset{T_2}{\\underbrace{\\mathrm {P}\\left(\\left.|g_{(M-i+1)}|^2\\sum ^{n}_{j=1} \\left(x_{(M-j+1)}-\\epsilon \\right) <n\\epsilon _1\\right||\\mathcal {S}|=n\\right)}} \\mathrm {P}\\left(|\\mathcal {S}|=n\\right) \\\\ \\nonumber &&+\\sum ^{M}_{n=m+1}\\underset{T_3}{\\underbrace{\\mathrm {P}\\left(\\left.|g_{(M-i+1)}|^2\\sum ^{m}_{j=1} \\left(x_{(M-j+1)}-\\epsilon \\right) <m\\epsilon _1\\right||\\mathcal {S}|=n\\right)}} \\mathrm {P}\\left(|\\mathcal {S}|=n\\right).$ $T_3$ can be first rewritten as follows: $T_3 &=& \\mathrm {P}\\left(\\left.y_{(M-i+1)}\\alpha _m <m\\epsilon _1\\right||\\mathcal {S}|=n\\right),$ where $\\alpha _m=\\sum ^{m}_{j=1} \\left(x_{(M-j+1)}-\\epsilon \\right)$ and $m\\le n\\le M$ .",
"The condition of $T_3$ implies that there are $n$ , $n> m$ , sources whose information can be decoded by the relay and $m$ of the $n$ users will be scheduled.",
"Therefore the conditional pdf of $\\alpha _m$ will be the same as that of $\\sum ^{m}_{j=1} \\left(\\tilde{x}_{(n-i+1)}-\\epsilon \\right)$ , where $\\tilde{x}_{(i)}$ are from the parents $\\tilde{x}_i$ , and $\\tilde{x}_i$ , $1\\le i \\le n$ , are i.i.d.",
"exponentially variables with the constraint $\\tilde{x}_i>\\epsilon $ .",
"It is straightforward to verify that the CDF of $\\tilde{x}_i$ conditioned on $\\tilde{x}_i>\\epsilon $ is $ F_{\\tilde{x}_i}(x) = \\frac{e^{-\\epsilon }-e^{-x}}{e^{-\\epsilon }}$ .",
"Consequently $w_i\\triangleq (\\tilde{x}_i-\\epsilon )$ is simply another exponential variable.",
"Therefore the pdf of $\\alpha _m$ is the same as the pdf of $w\\triangleq \\sum ^{m}_{j=1}w_{(n-j+1)}$ , the sum of $m$ largest order statistics chosen from $n$ i.i.d exponential variables.",
"Following the steps in [22], [23], the pdf of $w$ is given by $f_{w}(w) = \\sum ^{n-m-1}_{k=0} d_{m,k}\\left(\\sum ^{m}_{j=1}\\frac{a_{j,k}e^{-w}w^{j-1}}{(j-1)!}",
"+b_k e^{-\\left(1+\\frac{k+1}{m}\\right)w}\\right).$ From [19], the pdf of $y_{(M-i+1)}$ is $f_{y_{(M-i+1)}}(y) = \\frac{M!}{(M-i)!(i-1)!",
"}e^{-iy} \\left(1-e^{-y}\\right)^{M-i}$ .",
"So $T_3$ can be calculated as follows: $T_3 &=&\\int ^{\\infty }_{0} f_w(w) \\int ^{\\frac{m\\epsilon _1}{w}}_{0} f_{y_{(M-i+1)}}(y)dydw\\\\ \\nonumber &=&\\frac{M!}{(M-i)!(i-1)!",
"}\\sum ^{M-i}_{l=0}{M-i \\atopwithdelims ()l} \\frac{(-1)^l}{l+i}\\int ^{\\infty }_{0} f_w(w) \\left( 1-e^{-\\frac{m\\epsilon _1(l+i)}{w}} \\right)dw\\\\ \\nonumber &=&\\frac{M!}{(M-i)!(i-1)!",
"}\\sum ^{M-i}_{l=0}{M-i \\atopwithdelims ()l} \\frac{(-1)^l}{l+i} \\left( 1-\\underset{T4}{\\underbrace{\\int ^{\\infty }_{0} f_w(w)e^{-\\frac{m\\epsilon _1(l+i)}{w}} dw}}\\right).$ The integral in the above equation can be calculated as follows: $T_4&=& \\sum ^{n-m-1}_{k=0} d_{m,k}\\left(\\sum ^{m}_{j=1}\\frac{a_{j,k}\\int ^{\\infty }_{0}e^{-w}w^{j-1}e^{-\\frac{m\\epsilon _1(l+i)}{w}}dw}{(j-1)!}",
"+b_k \\int ^{\\infty }_{0} e^{-\\left(1+\\frac{k+1}{m}\\right)w}e^{-\\frac{m\\epsilon _1(l+i)}{w}} dw \\right).$ With some straightforward manipulations, $T_4$ can be further simplified as shown in the lemma.",
"$T_2$ can be first recalculated as follows: $T_2 &=& \\mathrm {P}\\left(\\left.|g_{(M-i+1)}|^2\\alpha _n <n\\epsilon _1\\right||\\mathcal {S}|=n\\right),$ where $\\alpha _n=\\sum ^{n}_{j=1} \\left(x_{(M-j+1)}-\\epsilon \\right)$ .",
"Different to $\\alpha _m$ in (REF ), the pdf of $\\alpha _n$ can be found simply as in the following.",
"The condition of $T_2$ implies that there are $n$ sources whose information can be decoded by the relay and all these users will be scheduled.",
"Therefore the conditional pdf of $\\alpha _n$ will be the same as that of $\\sum ^{n}_{j=1} \\left(\\tilde{x}_i-\\epsilon \\right)$ .",
"Following the same arguments as previously, $(\\tilde{x}_i-\\epsilon )$ is simply an exponential variable, which means $\\alpha _n$ is Chi-square distributed, i.e.",
"$f_{\\alpha _n}(z) = \\frac{e^{-x}x^{n-1}}{(n-1)!",
"}$ .",
"Therefore $T_2$ can be calculated as follows: $T_2 &=& \\int ^{\\infty }_{0}\\frac{e^{-z}z^{n-1}}{(n-1)!",
"}\\int ^{\\frac{n\\epsilon _1}{z}}_{0} \\frac{M!}{(M-i)!(i-1)!",
"}e^{-ix} \\left(1-e^{-x}\\right)^{M-i}dydz\\\\ \\nonumber &=&\\frac{M!}{(M-i)!(i-1)!(n-1)!",
"}\\sum ^{M-i}_{k=0} {M-i \\atopwithdelims ()k} \\frac{ (-1)^k}{k+i} \\int ^{\\infty }_{0} \\left(e^{-z}z^{n-1}- z^{n-1}e^{-z-\\frac{(k+i)n\\epsilon _1}{z}} \\right) dz.$ Combining (REF ), (REF ) and (REF ), and also with some algebraic manipulation, the outage probability shown in the lemma can be obtained.",
"The proof of the lemma is completed.",
"$\\blacksquare $ Proof of Lemma REF : When $m=1$ , the overall outage probability can be simplified as follows: $\\mathrm {P}_{oi}&\\triangleq & \\mathrm {P}\\left(|\\mathcal {S}|=0\\right) +\\sum ^{M}_{n=1}T_3\\mathrm {P}\\left(|\\mathcal {S}|=n\\right) .$ The condition that only one user pair will be scheduled can also help to simplify the expression of $T_3$ as follows: $T_3&=&n\\int ^{\\infty }_{0} e^{-y} \\left(1-e^{-y}\\right)^{n-1}\\int ^{\\frac{\\epsilon _1}{y}}_{0} d\\left(1-e^{-z}\\right)^{M}dy\\\\ \\nonumber &=&n\\sum ^{M}_{k=0} {M \\atopwithdelims ()k} (-1)^k \\sum ^{n-1}_{i=0} {n-1 \\atopwithdelims ()i} (-1)^i 2 \\sqrt{\\frac{k\\epsilon _1}{i+1}} \\mathbf {K}_1\\left(2\\sqrt{(i+1)k\\epsilon _1}\\right),$ where the first equation follows from the density function of the largest order statistics.",
"Recall the series representation of the Bessel function as follows: $x\\mathbf {K}_1(x) &=&1+ x \\mathbf {I}_1(x)\\left(\\ln \\frac{x}{2} +\\mathbf {C}\\right) -\\frac{1}{2} \\sum ^{\\infty }_{l=0}\\frac{\\left(\\frac{x}{2}\\right)^{2l+1}x}{l!(l+2)!",
"}\\left( \\sum ^{l}_{k=1}\\frac{1}{k} +\\sum ^{l+2}_{k=1}\\frac{1}{k} \\right)\\\\ \\nonumber &\\approx & 1+\\sum ^{\\infty }_{q=1}\\kappa _qx^{2q}\\ln x,$ for $x\\rightarrow 0$ , where $\\kappa _q$ is the constant coefficient associated to $x^{2q}\\ln x$ .",
"Note that the terms of $x^{2q}$ have been ignored since they are dominated by the terms of $x^{2q}\\ln x$ when $x\\rightarrow 0$ .",
"It is also worthy to point out that the exact value of $\\kappa _q$ has no effect to diversity gains.",
"By applying the above approximation, we can rewrite $T_3$ as follows: $\\nonumber T_3 &\\approx &n\\sum ^{M}_{k=0} {M \\atopwithdelims ()k} (-1)^k \\sum ^{n-1}_{i=0} {n-1 \\atopwithdelims ()i} \\frac{(-1)^i}{i+1} \\left(1+ \\sum ^{\\infty }_{q=1} \\frac{\\kappa _q}{2 }\\phi _{i,k}^{q}\\ln \\phi _{i,k} \\right),$ where $\\phi _{i,k}=4 (i+1)k\\epsilon _1$ .",
"We first focus on the case of $n=M$ .",
"Since $\\sum ^{M-1}_{k=0}{M-1 \\atopwithdelims ()k} (-1)^k =0$ , we have $\\nonumber M\\sum ^{M}_{k=0} {M \\atopwithdelims ()k} (-1)^k \\sum ^{M-1}_{i=0} {M-1 \\atopwithdelims ()i} \\frac{(-1)^i}{i+1} \\cdot 1 =0.$ To show that the terms at the order of $\\epsilon ^q \\ln \\epsilon $ , $1\\le q\\le (M-1)$ , are zero, we first observe the following: $\\phi _{i,k}^{q}\\ln \\phi _{i,k}&=&4^q (i+1)^qk^q\\epsilon _1^q \\ln \\left[4 (i+1)k\\epsilon _1\\right]\\\\ \\nonumber &=&\\underset{T_5}{\\underbrace{ 4^q (i+1)^qk^q\\epsilon _1^q \\ln \\left[4 (i+1)\\epsilon _1\\right]}} +\\underset{T_6}{\\underbrace{ 4^q (i+1)^qk^q\\epsilon _1^q \\ln k }}.$ By using the above separated expression, we can show that $M\\sum ^{M}_{k=0} {M \\atopwithdelims ()k} (-1)^k \\sum ^{M-1}_{i=0} {M-1 \\atopwithdelims ()i} \\frac{(-1)^i}{i+1}T_5=0,$ since $\\sum ^{M}_{k=0}{M \\atopwithdelims ()k} (-1)^k k^q =0$ , $1\\le q\\le (M-1)$ , and $M\\sum ^{M}_{k=0} {M \\atopwithdelims ()k} (-1)^k \\sum ^{M-1}_{i=0} {M-1 \\atopwithdelims ()i} \\frac{(-1)^i}{i+1}T_6=0,$ since $\\sum ^{M-1}_{i=0}{M-1 \\atopwithdelims ()i} (-1)^i i^{q-1} =0$ , $1\\le q\\le (M-1)$ .",
"Therefore the term at the order of $\\epsilon ^{M-1} \\ln \\epsilon $ will be removed from $T_3$ , and the overall outage probability can be expressed as $\\nonumber T_3 &\\approx &M\\sum ^{M}_{k=0} {M \\atopwithdelims ()k} (-1)^k \\sum ^{M-1}_{i=0} {M-1 \\atopwithdelims ()i} \\frac{(-1)^i}{i+1} \\left(1+ \\sum ^{\\infty }_{q=M} \\frac{\\kappa _q}{2 }\\phi _{i,k}^{q}\\ln \\phi _{i,k} \\right).$ Therefore the dominant factor is at the order of $\\epsilon ^M\\ln \\epsilon $ .",
"Similarly the dominant factors for $T_3$ , $1\\le n<M$ , is $\\epsilon ^n \\ln \\epsilon $ .",
"Substituting this result into (REF ) and also using the fact that $\\mathrm {P}\\left(|\\mathcal {S}|=n\\right)\\rightarrow \\epsilon ^{M-n}$ , the diversity gain of the overall outage probability will be $M$ .",
"And the proof of the lemma is completed.",
"$\\blacksquare $"
]
] | 1403.0354 |
[
[
"Azimuthal anisotropies of reconstructed jets in Pb+Pb collisions at\n $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV in a multiphase transport model"
],
[
"Abstract Azimuthal anisotropies of reconstructed jets [$v_{n}^{jet} (n=2, 3)$] have been investigated in Pb+Pb collisions at the center of mass energy $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV within a framework of a multiphase transport (AMPT) model.",
"The $v_{2}^{jet}$ is in good agreement with the recent ATLAS data.",
"However, the $v_{3}^{jet}$ shows a smaller magnitude than $v_{2}^{jet}$, and approaches zero at a larger transverse momentum.",
"It is attributed to the path-length dependence in which the jet energy loss fraction depends on the azimuthal angles with respect to different orders of event planes.",
"The ratio $v_{n}^{jet}/\\varepsilon_{n}$ increases from peripheral to noncentral collisions, and $v_{n}^{jet}$ increases with the initial spatial asymmetry ($\\varepsilon_{n}$) for a given centrality bin.",
"These behaviors indicate that the $v_{n}^{jet}$ is produced by the strong interactions between jet and the partonic medium with different initial geometry shapes.",
"Therefore, azimuthal anisotropies of reconstructed jet are proposed as a good probe to study the initial spatial fluctuations, which are expected to provide constraints on the path-length dependence of jet quenching models."
],
[
"Introduction",
"A deconfined quark-gluon plasma (QGP) could be created in the early state of high-energy heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC) [1], [2].",
"A jet, produced by initial hard processes, is an important probe to understand the properties of the QGP, since it losses its energy when it passes through the hot partonic medium [3].",
"This phenomenon, so-called jet quenching, has been confirmed by many experimental observations.",
"For example, the nuclear modification factor $R_{AA}$ shows a strong suppression at high transverse momentum $p_{T}$ in central A+A collisions at RHIC [4] and LHC [5] energies.",
"The measured elliptic anisotropy (or elliptic “flow\") $v_{2}$ of final hadrons remains positive above $\\sim $ 10 GeV/$c$ in A+A collisions at the RHIC [6] and LHC [7] energies, which discloses a path-length dependence of jet quenching [8].",
"Besides these above jet measurements based on high-$p_T$ leading hadrons, the recent LHC measurements on fully reconstructed jets provide a comprehensive characterization of jet quenching.",
"For instance, a larger dijet $p_{T}$ asymmetry has been observed in central Pb+Pb collisions than in p+p collisions at the LHC energy [9], [10], which is thought to be additional direct evidence of jet energy loss in the QGP, as important as the disappearance of the away-side peak in dihadron azimuthal correlation in central Au+Au collisions at the top RHIC energy [11].",
"The data on the elliptic anisotropy of reconstructed jets are recently released by the ATLAS Collaboration, which show nonzero $v_{2}$ values for the $p_{T}$ range from 45 to 160 GeV/$c$ for all centrality bins in Pb+Pb collisions [12], [13].",
"It is consistent with a path-length dependence of jet energy loss with respect to the reaction plane.",
"The elliptic anisotropy of reconstructed jets can be theoretically reproduced by the JEWEL model within a perturbative framework for jet evolution in a QGP medium [14].",
"The recent studies of higher orders of harmonic flow, especially for triangular flow $v_{3}$ , have deepened our understanding of many aspects of high energy heavy-ion collisions [15], [16], [17].",
"It would be interesting to study the third order of anisotropy $v_{3}$ of reconstructed jets, as it serves as the jet response to the initial geometry triangularity which could provide a greater constraint on jet quenching models.",
"The conversion efficiency $v_{n}^{jet}/\\varepsilon _{n}$ , the ratio of jet $v_{n}$ over the initial spatial eccentricity, is also an important observable to learn about how the energy loss of reconstructed jets depends the initial geometry asymmetry.",
"As heavy-ion collisions are dynamical evolutions, it is also necessary for understanding the whole jet quenching picture to study how reconstructed jets evolve dynamically during different evolution stages.",
"In this work, the elliptic anisotropy $v_{2}$ and triangular anisotropy $v_{3}$ of reconstructed jets are investigated in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV within a multiphase transport (AMPT) model, which includes both dynamical evolutions of partonic and hadronic phases.",
"In the remaining part of this paper, we refer to “jet” as a fully reconstructed jet for simplicity unless otherwise noted.",
"We find that the AMPT model can well describe the experimental results about jet $v_{2}$ .",
"Jet $v_{n}$ (n=2 and 3) arises owing to the strong interactions between jets and a partonic matter with different geometrical asymmetries.",
"The other final state interactions, such as hadronization (via coalescence) and hadronic rescatterings, show little impact on the measured jet $v_{n}$ .",
"We do observe that the jet energy loss fraction is dependent on the azimuthal angle with respect to the different orders of event plane.",
"We find that jet $v_{n}$ is sensitive to the spatial eccentricity ($\\varepsilon _{n}$ ) of initial parton distribution.",
"We further propose azimuthal anisotropies of reconstructed jets as a good probe to study the initial spatial fluctuations, and expect that jet $v_{n}$ provides constraints on the path-length dependence of jet quenching models."
],
[
"The AMPT Model",
"The AMPT model with the string melting mechanism is utilized in this work [18].",
"It consists of four main stages of high-energy heavy-ion collisions: the initial condition, parton cascade, hadronization, and hadronic rescatterings.",
"In order to increase the simulation efficiency of jets with $p_{T} > 45$ GeV/$c$ , a dijet of $p_{T}\\sim $ 40 GeV/$c$ is triggered in the initial condition based on the HIJING model [19], [20].",
"The high-$p_{T}$ primary partons evolve into jet showers full of lower virtuality partons through initial- and final- state QCD radiations.",
"In the string melting mechanism, all excited strings and jets are fragmented into hadrons according to the Lund string fragmentation [21].",
"Then these hadrons are converted to quarks according to the flavor and spin structures of their valence quarks.",
"After the melting process, the jet parton showers are converted into clusters of on-shell constituent quarks and anti-quarks, and a plasma of on-shell constituent quarks and anti-quarks is also formed.",
"Next, Zhang's parton cascade (ZPC) model [22] automatically simulates all possible elastic partonic interactions among the medium quarks and jet shower quarks, but without including inelastic parton interactions or further radiations at present.",
"When the quarks freeze out, they are recombined into medium hadrons or jet shower hadrons via a simple coalescence model which combines two nearest quarks into a meson and three nearest quarks into a baryon.",
"The final-state hadronic interactions, including elastic and inelastic hadronic scatterings and resonance decays, can be described by a relativistic hadronic transport (ART) model [23].",
"For more details on the AMPT model, we refer the reader to Ref. [18].",
"Recently, the AMPT model with a partonic interaction cross section of 1.5 mb has successfully given many qualitative descriptions of the experimental results about pseudorapidity and $p_{T}$ distributions [24], harmonic flows [25], [26], and reconstructed jet observables, including $\\gamma $ -jet $p_{T}$ imbalance [27], dijet $p_{T}$ asymmetry [28], jet fragmentation function [29] and jet shape [30] in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV.",
"Consistently with the previous studies, a partonic interaction cross section, 1.5 mb, is kept to simulate Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV in this work."
],
[
"Jet Reconstruction",
"To fully reconstruct jets, our kinematic cuts are chosen to be the same as in the ATLAS experiment [12], [13].",
"An anti-$k_{t}$ algorithm from the standard Fastjet package is used to reconstruct full jets [31], in which the jet cone size $R$ is set to be 0.2.",
"A pseudorapidity strip of width $\\Delta \\eta $ =1.0 centered on the jet position, with two highest-energy jets excluded, is used to estimate the background (“average energy per jet area\"), which is subtracted from the reconstructed jet energy in Pb+Pb collisions.",
"Only jets within a mid-rapidity range of $|\\eta |<2$ are considered in our analysis."
],
[
"Results and Discussions",
"The path-length dependence of jet energy loss can be characterized by jet $v_{2}$ , i.e., $v_{2}^{jet}$ =$\\left\\langle cos2(\\phi ^{jet}-\\Psi _{RP}) \\right\\rangle $ , where $\\phi ^{jet}$ is the azimuthal angle of the jet and $\\Psi _{RP}$ is the azimuthal angle of the reaction plane formed by the impact parameter $b$ and the beam direction which is fixed to $\\Psi _{RP}$ =0 in our AMPT simulations.",
"Fig.",
"REF (a)-(d) show the comparison of $v_{2}^{jet}$ as functions of the number of participant nucleons ($N_{part}$ ) between the AMPT results and the ATLAS experimental data for different jet $p_{T}$ bins in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ = 2.76 TeV.",
"The AMPT results give qualitative trends similar to the experimental data, but slightly overestimate the magnitudes.",
"Figure: (Color online) v n jet v_{n}^{jet} (n= 2 and 3) as functions of N part N_{part} for jet p T p_{T} bins of 45<p T <6045< p_{T} <60 GeV/cc (a) and 60<p T <8060< p_{T} <80 GeV/cc (b) in Pb+Pb collisions at s NN \\sqrt{s_{_{\\rm NN}}} = 2.76 TeV, where open triangles represent v 2 jet v_{2}^{jet} with respect to Ψ 2 r \\Psi _{2}^{r}, open circles represent v 2 jet v_{2}^{jet} with respect to Ψ RP \\Psi _{RP} = 0, open squares represent v 3 jet v_{3}^{jet} with respect to Ψ 3 r \\Psi _{3}^{r} and solid circles represent the ATLAS experimental data , .",
"Some points are slightly shifted along the xx axis for better representation.It is well known that the odd orders of harmonic flows can arise from the initial geometry fluctuations through final state interactions [15].",
"On the other hand, the even orders of harmonic flows are also affected if considering the initial geometry fluctuations [32].",
"To calculate the n-th Fourier coefficient $v_{n}$ , the n-th event plane $\\Psi _{n}^{r}$ can be defined as $ \\Psi _{n}^{r}=\\frac{1}{n}\\left[arctan\\frac{\\left\\langle r^{n}sin(n\\varphi )\\right\\rangle }{\\left\\langle r^{n}cos(n\\varphi )\\right\\rangle }+\\pi \\right],$ where $r$ and $\\varphi $ are the coordinate position and azimuthal angle of each parton in the AMPT initial state and the average $\\langle \\cdots \\rangle $ denotes density weighting.",
"Then the n-th harmonic coefficient of jets, $v_{n}^{jet}$ , can be obtained by the following equation: $ v_{n}^{jet}=\\left\\langle cos \\left[ n(\\phi ^{jet}-\\Psi _{n}^{r}) \\right] \\right\\rangle .$ Note that the $v_{n}^{jet}$ definition is the same as that for a single hadron; however, $v_{n}^{jet}$ is expected to have smaller bias because the reconstructed jet has kinematic properties that are more closely related to those of the parent partons [12], [13].",
"Jet $v_{2}$ and $v_{3}$ as functions of $N_{part}$ for two typical $p_{T}$ bins of $45< p_{T} <60$ GeV/$c$ and $60< p_{T} <80$ GeV/$c$ , calculated by Eqs.",
"(REF ) and (REF ) and denoted as $v_{2}^{jet}\\lbrace \\Psi _{2}^{r}\\rbrace $ and $v_{3}^{jet}\\lbrace \\Psi _{3}^{r}\\rbrace $ , are shown in Figs.",
"REF (a) and (b), respectively.",
"$v_{2}^{jet}\\lbrace \\Psi _{2}^{r}\\rbrace $ (open triangles) is consistent with the previous jet $v_{2}$ calculations of $v_{2}^{jet}\\lbrace \\Psi _{RP}=0\\rbrace $ (open circles), though it has a little higher magnitudes due to the initial fluctuation contribution [32].",
"For jet $v_{3}$ , it is smaller than jet $v_{2}$ .",
"By comparing jet $v_{3}$ between two different jet $p_{T}$ bins, jet $v_{3}$ tends to vanish with increasing jet $p_{T}$ .",
"Figure: (Color online) The AMPT results on v 2 jet v_{2}^{jet} (a) and v 3 jet v_{3}^{jet} (b) as functions of N part N_{part} for the jet p T p_{T} bin of 45<p T <6045< p_{T} <60 GeV/cc for different evolution stages in Pb+Pb collisions at s NN \\sqrt{s_{_{\\rm NN}}} = 2.76 TeV.",
"Some points are slightly shifted along the xx axis for better representation.Since heavy-ion collisions are dynamical evolutions which involve many important evolution stages, it is important to investigate the stage evolution of jet $v_{n}$ .",
"Figs.",
"REF (a) and (b) display jet $v_{2}$ and $v_{3}$ for the $p_{T}$ bin of $45< p_{T} <60$ GeV/$c$ at different evolution stages in Pb+Pb collisions from the AMPT simulations, respectively.",
"The jet $v_{n}$ is nearly zero in the initial state.",
"However, jet $v_{n}$ arises from the process of parton cascade, which indicates jet $v_{n}$ is generated owing to the strong interactions between jet and the partonic medium.",
"On the other hand, the processes of hadronization via coalescence and final hadronic rescatterings have little impact on jet $v_{n}$ .",
"Figure: The AMPT results on the jet energy loss fraction, Δp T /p T \\Delta p_{T}/p{_{T}}, as functions of Δφ\\Delta \\phi = φ jet -Ψ n r \\phi ^{jet}-\\Psi _{n}^{r} [n=2 (solid circles) and 3 (open circles)] for the jet p T p_{T} bins of 45<p T <6045< p_{T} <60 GeV/cc (a) and 60<p T <8060< p_{T} <80 GeV/cc (b) in the centrality bin of 20-30% in Pb+Pb collisions at s NN \\sqrt{s_{_{\\rm NN}}}=2.76 TeV.Figs.",
"REF (a) and (b) show the AMPT results for the averaged jet energy loss fraction $\\Delta p_{T}/p_{T}$ =$(p_{T}^{jet, initial}-p_{T}^{jet final})/p_{T}^{jet, initial}$ as functions of the relative azimuthal angle $\\Delta \\phi =\\phi ^{jet}-\\Psi _{n}^{r}$ for two jet $p_{T}$ bins of $45< p_{T} <60$ GeV/$c$ and $60< p_{T} <80$ GeV/$c$ in their first azimuth periods for the centrality bin of 20-30% in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ =2.76 TeV, respectively.",
"Jets lose more energy at $\\Delta \\phi \\sim \\pi /2$ with respect to the second order of the event plane or $\\Delta \\phi \\sim \\pi /3$ with respect to the third order of the event plane.",
"It can be reasonably understood because jets transverse a longer path length through the medium in the direction of $\\Delta \\phi \\sim \\pi /2$ or $\\Delta \\phi \\sim \\pi /3$ for an elliptic or triangle shape profile, which is consistent with the path-length effect of jet energy loss [8].",
"Figure: The AMPT results for v n jet /ε n v_{n}^{jet}/\\varepsilon _{n} [n=2 (solid circles) and 3 (open circles)] as functions of N part N_{part} for the jet p T p_{T} bin of 45<p T <6045< p_{T} <60 GeV/cc in Pb+Pb collisions at s NN \\sqrt{s_{_{\\rm NN}}}=2.76 TeV.",
"Some points are slightly shifted along the xx axis for better representation.Figure: The AMPT results for v n jet v_{n}^{jet} as functions of ε n \\varepsilon _{n} [n=2 (solid circles) and 3 (open circles)] for the jet p T p_{T} bin of 45<p T <6045< p_{T} <60 GeV/cc in the centrality bin of 20-30% in Pb+Pb collisions at s NN \\sqrt{s_{_{\\rm NN}}}=2.76 TeV.",
"Some points are slightly shifted along the xx axis for better representation.The conversion efficiency ($v_{n}/\\varepsilon _{n}$ ) has been used as an important observable to understand the collective flow phenomena in high-energy heavy-ion collisions [32], [15].",
"Similarly, $v_{n}^{jet}/\\varepsilon _{n}$ could disclose how jet quenching depends on the initial geometry shape.",
"To calculate the n-th order eccentricity $\\varepsilon _{n}$ , we use the definition as follows, $ \\varepsilon _{n}=\\frac{\\sqrt{{\\left\\langle r^{n}sin(n\\varphi )\\right\\rangle }^{2}+{\\left\\langle r^{n}cos(n\\varphi ) \\right\\rangle }^{2}}}{\\left\\langle r^{n} \\right\\rangle },$ according to the information about the coordinate space of initial partons.",
"Fig.",
"REF shows the AMPT results for $v_{n}^{jet}/\\varepsilon _{n}$ as functions of $N_{part}$ for the jet $p_{T}$ bin of $45< p_{T} <60$ GeV/$c$ in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ =2.76 TeV.",
"The $v_{n}^{jet}/\\varepsilon _{n}$ increases with $N_{part}$ except in the most central centrality bin where jet $v_{n}$ is close zero, which reveals that azimuthal anisotropies of jets are more easily formed in more central collisions owing to the larger jet energy loss in a denser partonic matter.",
"Fig.",
"REF presents jet $v_{n}$ as functions of the eccentricity $\\varepsilon _{n}$ for the jet $p_{T}$ bin of $45< p_{T} <60$ GeV/$c$ in a selected centrality bin of 20-30% in Pb+Pb collisions.",
"It is shown that the final jet $v_{n}$ increases with the initial spatial eccentricity or triangularity, which indicates that jet azimuthal anisotropies are produced by the interactions between jets and the partonic medium with different asymmetrical geometry shapes."
],
[
"CONCLUSION",
"In conclusion, azimuthal anisotropies of reconstructed jets have been investigated in Pb+Pb collisions at $\\sqrt{s_{_{\\rm NN}}}$ =2.76 TeV within a framework of a multiphase transport (AMPT) model.",
"The model gives a qualitative description about the measured $v_{2}$ of reconstructed jets for the $p_{T}$ range from 45 to 160 GeV/$c$ .",
"We predict that $v_{3}$ of reconstructed jets, which has a smaller magnitude than its $v_{2}$ , approaches zero with increasing jet $p_{T}$ .",
"It can be attributed to the dependence of the jet energy loss fraction on the azimuthal angles with respect to the different orders of event planes.",
"The dynamical stage evolution of reconstructed jets discloses that jet $v_{n}$ mostly arises from a strong parton cascade process with little effect from the final stages such as hadronization and hadronic rescatterings.",
"The ratio $v_{n}^{jet}/\\varepsilon _{n}$ increases with $N_{part}$ in non-central Pb+Pb collisions, furthermore, jet $v_{n}$ increases with the initial spatial asymmetry ($\\varepsilon _{n}$ ) for a given centrality bin.",
"These behaviors indicate that jet $v_{n}$ is produced by the strong interactions between the jet and the partonic medium with different initial asymmetrical geometry shapes.",
"Therefore, the azimuthal anisotropies of reconstructed jets can be utilized as a good probe to study the initial spatial asymmetry, and imposes constraints on the path-length dependence of jet quenching models."
],
[
"ACKNOWLEDGMENTS",
"This work was supported by the Major State Basic Research Development Program in China under Contract No.",
"2014CB845404, the NSFC of China under Projects No.",
"11175232, No.",
"11035009, and No.",
"11375251, the Knowledge Innovation Program of CAS under Grant No.",
"KJCX2-EW-N01, the Youth Innovation Promotion Association of CAS, the project sponsored by SRF for ROCS, SEM, the CCNU-QLPL Innovation Fund under Grant No.",
"QLPL2011P01, and the “Shanghai Pujiang Program\" under Grant No.",
"13PJ1410600."
]
] | 1403.0328 |
[
[
"Momentum Distribution of Near-Zero-Energy Photoelectrons in the\n Strong-Field Tunneling Ionization in the Long Wavelength Limit"
],
[
"Abstract We investigate the ionization dynamics of Argon atoms irradiated by an ultrashort intense laser of a wavelength up to 3100 nm, addressing the momentum distribution of the photoelectrons with near-zero-energy.",
"We find a surprising accumulation in the momentum distribution corresponding to meV energy and a \\textquotedblleft V\"-like structure at the slightly larger transverse momenta.",
"Semiclassical simulations indicate the crucial role of the Coulomb attraction between the escaping electron and the remaining ion at extremely large distance.",
"Tracing back classical trajectories, we find the tunneling electrons born in a certain window of the field phase and transverse velocity are responsible for the striking accumulation.",
"Our theoretical results are consistent with recent meV-resolved high-precision measurements."
],
[
"Momentum Distribution of Near-Zero-Energy Photoelectrons in the Strong-Field Tunneling Ionization in the Long Wavelength Limit Q.",
"Z. Xia National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China D. F. Ye National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China L. B. Fu [email protected] National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China X. Y. Han National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China J. Liu [email protected] National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100084, China We investigate the ionization dynamics of Argon atoms irradiated by an ultrashort intense laser of a wavelength up to 3100 nm, addressing the momentum distribution of the photoelectrons with near-zero-energy.",
"We find a surprising accumulation in the momentum distribution corresponding to meV energy and a “V\"-like structure at the slightly larger transverse momenta.",
"Semiclassical simulations indicate the crucial role of the Coulomb attraction between the escaping electron and the remaining ion at extremely large distance.",
"Tracing back classical trajectories, we find the tunneling electrons born in a certain window of the field phase and transverse velocity are responsible for the striking accumulation.",
"Our theoretical results are consistent with recent meV-resolved high-precision measurements.",
"32.30.-r,32.80.Fb,33.60.-q,31.15.Ar Introduction—The above-threshold ionization (ATI) phenomenon of atoms exposed to a strong field has attracted sustaining attention for decades since it was first discovered in 1979 [1].",
"One of the most pronounced features of the ATI in the long-wavelength limit, is the high-energy photoelectron spectrum plateau extending up to 10$U_{P}$ ($U_{P}=I/4\\omega ^{2}$ , denotes the ponderomotive energy, where $I$ is the laser intensity and $\\omega $ the frequency in atomic units) [2].",
"The underlying mechanism has been attributed to the tunneled electron's multiple return and rescattering by its parent ion [5], [6], [3], [4], [7].",
"Recently, some unexpected low-energy structures (LES) of ATI [8], [9], [10] were observed in the tunneling regime, initially at several eV and then at lower energies of less than 1eV, triggering a new surge of attention of ATI.",
"Although from all aspects, including quantum and semiclassical, numerous theoretical investigations [8], [9], [10], [11], [17], [12], [13], [14], [15], [16] of the underlying physics of the LES were reported, the controversies on the surprising structure have been continuing.",
"In the above experiments, the photoelectrons are detected only in a small angle around the laser polarization direction by a time-of-flight spectrometer.",
"Because the LES is subtle and sensitive, it is difficult to disentangle the origin of the various findings without large solid angle measurements including momentum information with high resolution.",
"Recent experimental work by J. Dura et al [18] steps forward in this direction.",
"In the experiment, with a specifically developed ultrafast mid-IR light source of 3100 nm in the combination with a 3D reaction microscope, the strong-field dynamics is explored in three-dimensional momentum space down to meV electron energies with an unprecedented precision [18].",
"Instead of structures on the eV level, an apparent meV electron accumulation in the ATI spectrum and a striking momentum distribution for the near-zero-energy electron are observed.",
"Nevertheless, the physics underlying the meV electron distribution is unsettled that calls for an investigation from theoretical side urgently.",
"In this Letter, stimulated by the recent experiment and attempting to resolve the controversies on LES, we theoretically investigate the ionization dynamics of Argon atoms by intense laser fields in the deep tunneling regime of $\\gamma \\ll 1$ (Keldysh parameter $\\gamma =\\sqrt{I_{p}/2U_{p}}$ , where $I_{p}$ is the ionization potential), with special emphasis on addressing the low momentum distribution of the near-zero-energy photoelectrons.",
"Our study is facilitated by an improved semiclassical rescattering model that includes the Coulomb attraction and trajectory interference and can precisely produce the momenta of the near-zero-energy photoelectrons regardless of the Coulomb long-range tail.",
"Our theory accounts the surprising accumulation around near-zero momenta corresponding to meV energies and predicts a “V\"-like structure at slightly larger transverse momenta.",
"We identify the roles of the Coulomb attraction and trajectory interference, by tracing back the trajectories of soft and chaotic scattering, respectively.",
"Our work provides profound insight into the meV low-energy ATI mechanism, and combined with the recent high-precision experimental results can help unraveling the debate about the LES.",
"Model—In the semiclassical model, the atomic ionization consists two essential physical processes, i.e., an electron tunnels through the Coulomb field that has been dramatically suppressed by the laser field, and the released electron is driven by laser field to scatter with its parent ion [6].",
"The tunneled electrons (released at a distance $r_{0}$ from the ion) have initially zero longitudinal velocity and a Gaussian transverse velocity distribution.",
"Each trajectory is weighed by the ADK ionization rate $\\varpi (t_{0},v_{\\perp }^{i})=\\varpi _{0}(t_{0})\\varpi _{1}(v_{\\perp }^{i})$ [19], where $\\varpi _{1}(v_{\\perp }^{i})=(2\\sqrt{2I_{p}}v_{\\perp }^{i}/|\\varepsilon (t_{0})|)\\exp [-\\sqrt{2I_{p}}(v_{\\perp }^{i})^{2}/|\\varepsilon (t_{0})|]$ is the distribution of initial transverse velocity, and $\\varpi _{0}(t_{0})=|\\varepsilon (t_{0})|^{(1-2/\\sqrt{2I_{p}})}\\exp [-2(2I_{p})^{3/2}/|3\\varepsilon (t_{0})|]$ , depending on the field phase $\\omega t_{0}$ at the instant of tunneling as well as on the ionization potential $I_{p}$ .",
"In the post-tunneling process, the electron evolution in the combined oscillating laser field and Coulomb field is traced via the classical Newtonian equation $d^{2}\\vec{r}/dt^{2}=-\\vec{r}/r^{3}-\\vec{\\varepsilon }(t)$ .",
"The physical quantities can then be calculated through weighted averaging over the ensemble of trajectories corresponding to diverse initial laser phases and transverse velocities at tunneling.",
"We have made simulations for Ar atoms with $I_{p}=0.583$ a.u..",
"The laser parameters are chosen as $\\varepsilon _{0}=0.053$ a.u.",
"and $\\omega =0.0147$ a.u.",
"($\\lambda =3100$ nm) to match the experiment [18].",
"Thus, the Keldysh parameter $\\gamma =0.3$ .",
"The pulse envelope is half-trapezoidal, constant for the first six cycles and ramped off within the last six cycles.",
"After the laser pulse is over, the instantaneous positions and momenta of the emitted electrons are recorded.",
"However, the instantaneous momenta are not equal to the asymptotic ones (i.e., $r \\rightarrow \\infty $ ) collected by a detector due to the Coulomb long-range tail.",
"To precisely reproduce the near-zero momentum distribution, we need to extract the asymptotic momenta from the instantaneous positions and momenta.",
"The Coulomb two-body system has two conserved vectors: the angular momentum $\\vec{M}=\\vec{r}\\times \\vec{p}$ and the Laplace-Runge-Lenz vector $\\vec{A}=\\vec{p}\\times \\vec{M}-\\vec{r}/r$ .",
"Using the conserved quantities and after some coordinate rotations, we can then obtain the asymptotic momenta of the emitted electrons.",
"Figure: (Color online) (a) Momentum distribution calculated from the modelwithout considering the Coulomb attraction in rescattering; (b) Results fromthe model with considering the Coulomb attraction; (c) The results withconsidering both Coulomb field and trajectory interference; (d) Momentumspectrum from experiment cited from for comparison.Figure: (Color online)(a)-(h)The parallel momentum distribution and theenergy distribution counted from Fig.",
"1(b), with respect to the four areaswithin the momentum map of Fig. (d).",
"(a) and (e) correspond tothe area IV; (b) and (f) correspond to the area III; (c) and (g) to the areaII; and (d) and (h) to the area I.",
"Red curves denote the results of thesemiclassical simulation with the Coulomb field, while the blue onesrepresent the simulation results without the Coulomb field.",
"(i) and (j) aresimulation results from screening potentials: (i) Low energy distributionwith respect to screening parameters; (j) meV photoelectron yields vs. thescreening parameters.The above classical trajectory evolution, however, ignore the quantum interference totally.",
"To retrieve the interference effect, we assign a phase to each trajectory.",
"The phase $S$ is determined by the integral $S(t_{0},v_{\\perp }^{i})=-i\\int _{t_{0}}^{t_{f}}[\\frac{1}{2}\\vec{v}^{2}(t)-\\frac{1}{r(t)}+I_{p}]dt$ [17], here $\\vec{v}(t)$ and $r(t)$ is the solution of the Newton equation with the initial condition $t_{0}$ , $v_{\\perp }^{i}$ and tunneling position $r_{0}$ .",
"Then, the transition amplitude from the initial state to the continuum state with the asymptotic momentum $(p_{\\parallel }, p_{\\perp })$ can be calculated as $M(p_{\\parallel }, p_{\\perp })=\\sum _{t_{0},v_{\\perp }^{i}}\\sqrt{\\varpi (t_{0},v_{\\perp }^{i})}\\exp {[S(t_{0},v_{\\perp }^{i})]}$ , where the summation includes all the trajectories leading to the same final momentum $(p_{\\parallel }, p_{\\perp }) $ .",
"The momentum spectra can be obtained from $|M(p_{\\parallel }, p_{\\perp })|^{2}$ .",
"Here, we only consider the electrons released within the first cycle $(\\omega t_{0}\\in [0,2\\pi ])$ .",
"In our simulation, more than 5 million trajectories are calculated and the convergence of the results has been tested by increasing the number of trajectories.",
"Figure: (Color online) The dependence of final kinetic energy on tunnelingtime and initial transverse velocity around peak field.",
"The data are plottedin different colors with respect to different electron energy ranges.",
"Theblank areas contribute to high-energy electrons with E>0.5E>0.5eV, which arenot in the focus of the present discussion.Momentum Distribution of Low-Energy Electrons— Fig.",
"REF (a) (b) and (c) show our model calculations on the momentum distribution spectra of low-energy electrons in the momentum ranges $p_{\\parallel } \\in (-0.4,0.4)$ and $p_{\\perp } \\in (0,0.5)$ , indicating the prominent roles of both the Coulomb potential and trajectory interference.",
"We can first see the important role of the Coulomb attraction by comparing Fig.",
"1(a) and (b).",
"In Fig.",
"1(a), we artificially remove the Coulomb potential in the post-tunneling scattering.",
"Here, the momentum spectrum exhibits a simple “sandwich\"-like structure, with the dense distribution in the central belt only reflecting the Gaussian type distribution of initial transverse velocities.",
"In this case, we can not see any accumulation near zero momentum.",
"However, in the presence of the Coulomb potential (see Fig.",
"1 (b)), we can obviously observe the accumulation near zero momentum as well as a “V\"-like structure at the slightly larger transverse momenta.",
"Figure: (Color online) Three typical trajectories leading to meV energy andtheir corresponding temporal evolutions of the momentum and energy.",
"(a) (d)and (g), the soft forward scattering trajectory; (b) (e) and (h), the softbackward scattering; (c) (f) and (i), the chaotic scattering.The role of the trajectory interference becomes obvious when comparing Fig.",
"1 (c) and (b): The dense belt structure broadens and the accumulation near origin fades out a little, in better agreement with the experiment.",
"Fig.",
"1(c) also shows fine vertical interference contrasts.",
"To achieve deeper insight into the origin of the striking structure in the momentum distribution and address its relation to the energy spectrum, we take average the transverse momentum and generate energy distributions with respect to distinct transverse momentum regimes, i.e., accumulation regime (I), V-structure regime (II), and belt regime (III).",
"Symbol IV represents the sum over the above three regimes.",
"In our energy statistics, the energy interval is chosen as $0.4$ meV, consistent with the experimental resolution.",
"The results are shown in Fig.",
"2.",
"From Fig.",
"2 (f) to (h), we see the Coulomb effects increase and become very significant at small transverse velocities.",
"The envelope of the total energy spectrum (e) shows a prominent hump around 0.5 eV and then extends to meV energies or even less.",
"The hump is also apparent in experiment [18].",
"Nevertheless, it has nothing to do with the Coulomb attraction, is just due to momentum space density compression when transformed to energy distribution.",
"An estimation of the hump center is given by $\\partial \\varpi (t_{0},v_{\\perp }^{i})/\\partial t_{0}=0$ and $\\partial \\varpi (t_{0},v_{\\perp }^{i})/\\partial v_{\\perp }^{i}=0$ .",
"In the long-wavelength limit of $\\omega \\rightarrow 0$ , we analytically obtain an electron kinetic energy of $\\varepsilon _{0}/4\\sqrt{2I_{p}}$ , here $\\varepsilon _{0}$ is the maximum field strength.",
"Substituting the atom and laser parameters, it gives $0.33$ eV, in close agreement with the results of the simulation.",
"In contrast to the experiment, below the hump, our simulation exhibits a plateau structure in energy spectrum that spreads down the regime of $10^{-4}$ eV or less.",
"The meV electrons are closely related to the Coulomb attraction.",
"In particular, we find that the long-range Coulomb attraction between the electron and ion plays a crucial role, in contrast to the Coulomb focusing effect [20] that is significant only when the electron is closer to ion.",
"We have replaced the Coulomb potential by a Yukawa type potential of $-exp[-\\lambda r]/r$ , to screen the Coulomb long-range tail (see Fig.",
"2 (i) and (j) ).",
"We find surprisingly that, even with a very small screening parameter of $\\lambda = 0.01$ , the meV electron yield decreases rapidly, analogous to the case where the Coulomb potential is completely absent.",
"Only with much smaller screening parameters of $0.001$ or less, the meV electron accumulation can recover.",
"The above observation unambiguously indicates the crucial role of the Coulomb attraction between the escaping electron and ion at the extremely long distance ($\\gg 100$ a.u.",
"), and provides the strong evidence that the highly excited Rydberg states are involved in the meV electron dynamics.",
"We have also calculated the meV electron yields (i. e. energy less than 0.01 eV) with respect to the screening parameters in Fig.",
"2 (j), which mainly exhibits a logarithm feature.",
"The singularity stemmed from the Coulomb long tail also emerges in heavy ion impact ionization with manifesting a sharp cusp-like peak at zero transverse momentum [21].",
"Recently, the cutoff of Coulomb potential at the distance of a few atomic units is found to affect the momentum spectra of the electrons with eV energy in multiphoton regime [22], [23].",
"Here, we find that the tiny Coulomb tail at distance much larger than 100 a.u.",
"can significantly help producing the meV photoelectrons and lead to the striking distribution in momentum spectrum.",
"The striking meV electron accumulation is found to be dependent on laser wavelength.",
"We have extended our simulations to shorter wavelength of 800 nm and find that apparent accumulation around near-zero momenta become less visible.",
"This is due to the stabilization of the high-excited Rydberg states that are deeply involved in the meV photoelectron dynamics.",
"When field frequency is much larger than the Rydberg orbit frequency, the electrons that are pumped into the Rydberg states become stabilized against ionization [24], [25].",
"This effect can reduce the meV electron accumulation.",
"Actually, we have simply estimated the most possible energy of the electrons that are pumped by the laser field.",
"According to the zero origin of the energy, we can obtain a critical Keldysh parameter of $\\gamma _{c}=(\\sqrt{I_{p}}/2\\sqrt{2})^{1/2}$ , below which the most possible electron energy shifts to a positive value indicating the emergence of the hump structure similar to Fig.",
"2(e).",
"We therefore predict that the surprising meV electron accumulation around near-zero momenta should be universal in the deep tunneling regime of $\\gamma <\\gamma _{c}$ .",
"This is the case for the present experiment [18], where $\\gamma =0.3$ and the critical Keldysh parameter for Ar atom is around 0.5.",
"Classical Trajectory Analysis of the source of the meV electrons—In our semiclassical model, the emitted electron's energy is determined by the tunneling time and initial perpendicular velocity.",
"In Fig.",
"REF , we show the dependence of the final kinetic energy on the tunneling time and initial transverse velocity around peak field.",
"The meV electrons originate from the red area that consists of a regular arc region and scattered irregular regions below the arc.",
"The irregular zone is self-similar and has fractal properties [6], [26], [27], [28].",
"The trajectories originated from this region might experience multiple returns to the ion and the final energies is very sensitive to the initial conditions.",
"The chaotic multiple rescatterings by the Coulomb field might lead to extremely high energy electrons that are responsible for the well-known ATI plateau structure [6].",
"It can also result in extremely low-energy electrons, as shown by the scattered red dots below the arc region.",
"Besides these chaotic trajectories, we find the electrons with meV energy usually experience soft scattering and then move forward or backward [17], [12].",
"We plot such three kinds of typical trajectories in Fig.",
"REF (a), (b) and (c), respectively.",
"Their initial conditions correspond to the “stars\" in Fig.",
"3 .",
"Temporal evolution of the momenta and energy for each trajectory is shown in Fig.",
"REF (d-f) and (g-i), respectively.",
"Since the electrons oscillate in the laser field, here we use the compensated energy [29] calculated via the canonical momentum instead of the kinetic energy.",
"Fig.",
"4 (a) (g) and (b) (h) indicate that, some electrons originated in a certain window of laser phase and initial transverse velocity can tunnel into Rydberg states without ionization.",
"They locate far away from the parent ion and are weakly bounded by the ion's Coulomb attraction.",
"They subsequently experience very soft rescattering during which they can only acquire limited field energy to be pumped into the continuum with meV energy.",
"During the process, the Coulomb potential attracts the electron that reduces the electron's tunneling momentum to zero showing a kind of “friction\" effect (see Fig.",
"4 (d) and (e)).",
"While for the chaotic trajectory, the electrons experience multiple scatterings with ion and \"occasionally\" emit with meV energy.",
"Our model calculation indicates both chaotic and soft (forward or backward) recattering trajectories are the source of meV photoelectrons and the ratio of the two kinds events (i.e., chaotic events vs. soft rescattering events) is about $1/4$ .",
"In summary, we have investigated the dynamics of meV ATI photoelectrons from theoretical side for the first time.",
"Our simulations indicate that the meV electron generation is subtle and attributed to the extremely long-range Coulomb tail, while it is also of universality in the deep tunneling regime.",
"Our theoretical results account for the recent high-precision ATI experiments and some predictions are given that calls for the verification from further experiments.",
"Since the meV energy corresponds to terahertz, the present results can also have implications in the generation of terahertz radiation [30].",
"This work is supported by the National Fundamental Research Program of China (Contact No 2011CB921503, 2013CBA01502, and 2013CB83410), the NNSF of China (Contact Nos.",
"11274051, 11374040, 11078001,10933001)."
]
] | 1403.0414 |
[
[
"Spin and charge thermopower of resonant tunneling diodes"
],
[
"Abstract We investigate thermoelectric effects in quantum well systems.",
"Using the scattering approach for coherent conductors, we calculate the thermocurrent and thermopower both in the spin-degenerate case and in the presence of giant Zeeman splitting due to magnetic interactions in the quantum well.",
"We find that the thermoelectric current at linear response is maximal when the well level is aligned with the Fermi energy and is robust against thermal variations.",
"Furthermore, our results show a spin voltage generation in response to the applied thermal bias, giving rise to large spin Seebeck effects tunable with external magnetic fields, quantum well tailoring and background temperature."
],
[
"Spin and charge thermopower of resonant tunneling diodes Javier H. Nicolau Institute for Cross-Disciplinary Physics and Complex Systems IFISC (UIB-CSIC), E-07122 Palma de Mallorca, Spain David Sánchez Institute for Cross-Disciplinary Physics and Complex Systems IFISC (UIB-CSIC), E-07122 Palma de Mallorca, Spain We investigate thermoelectric effects in quantum well systems.",
"Using the scattering approach for coherent conductors, we calculate the thermocurrent and thermopower both in the spin-degenerate case and in the presence of giant Zeeman splitting due to magnetic interactions in the quantum well.",
"We find that the thermoelectric current at linear response is maximal when the well level is aligned with the Fermi energy and is robust against thermal variations.",
"Furthermore, our results show a spin voltage generation in response to the applied thermal bias, giving rise to large spin Seebeck effects tunable with external magnetic fields, quantum well tailoring and background temperature.",
"72.20.Pa, 85.75.Mm, 75.50.Pp Resonant tunneling diodes are versatile devices that further enable investigations in fundamental physics.",
"[1] Current responses in the GHz regime have been demonstrated with double-barrier heterojunctions,[2], [3] which also show bistability [4] and super-Poissonian noise [5], [6] arising from their intrinsic nonlinearities.",
"These systems are also useful in discussions on coherent versus sequential scattering processes.",
"[7], [8] Very recently, tunnel diodes have been used as detectors of hypersonic wave packets.",
"[9] Temperature effects dramatically alter the behavior of resonant tunneling devices.",
"The peak-to-valley ratio in the current–voltage characteristics quickly decreases as temperature increases.",
"[2], [10] Hole transport has been shown to be quite sensitive to thermal variations.",
"[11] Furthermore, temperature can tune the transition from static domains to self-sustained oscillations in multiple-quantum-well structures.",
"[12], [13] However, these works consider a fixed background temperature common to both the sample and the leads.",
"More interesting is the generation of thermovoltages in response to temperature gradients applied to the attached reservoirs (the Seebeck effect).",
"[14] Recent works suggest that significant improvements of the heat-to-energy conversion efficiency can be obtained with low dimensional systems in general [15], [16], [17] and with quantum-well tunnel devices in particular.",
"[18], [19] Therefore, it is important to investigate the thermopower properties of a resonant-tunneling double-barrier system, which have been little explored up to date.",
"Our findings reveal a thermocurrent peak when the quantum well level is aligned with the leads' Fermi energy.",
"This is in stark contrast with the typical quantum dot behavior, for which the thermoelectric conductance vanishes at the electron-hole symmetry point.",
"[20], [21], [22] We attribute this difference to the crucial contribution from the transversal energy degrees of freedom in tunnel diodes.",
"Furthermore, we consider giant Zeeman effects arising from diluted magnetic impurities [23], [24] and find significant values of the spin bias voltage created from temperature differences (the spin Seebeck effect)[25], [26] and highly tunable with the well level position or the base temperature.",
"Consider a semiconductor heterostructure with two potential barriers and a quantum well sandwiched between them, as sketched in Fig.",
"REF .",
"Quite generally, the energy levels are spin split due to an external magnetic field to be specified below.",
"The mean electrochemical potential at lead $\\alpha =L,R$ is given by $\\mu _{\\alpha }=(\\mu _{\\alpha \\uparrow }+\\mu _{\\alpha \\downarrow })/2$ and the bias voltage between the two contacts is $V=(\\mu _L-\\mu _R)/e$ .",
"We denote with $E_F$ the common Fermi energy.",
"In the quantum well, we consider a single subband $E_\\sigma =E_\\perp +E_0+\\sigma h/2$ only because in tunnel diodes with narrow wells the level spacing is quite large (e.g., 550–750 meV in Ref. bon85).",
"Here, $E_\\perp $ is the energy associated to the lateral modes perpendicular to the current direction, $E_0$ is the level position measured from the well bottom, $\\sigma =+$ ($-$ ) for spins $\\uparrow $ ($\\downarrow $ ) with the spin quantization axis taken along the magnetic field direction, and $h$ is the giant Zeeman splitting of the order of 10 meV for small fields around 1 T.[23] This splitting arises from the combined effect of an in-plane magnetic field and diluted magnetic impurities present in the quantum well.",
"We remark that for the same field strengths, the spin splittings in the nonmagnetic leads are negligible.",
"Therefore, possible spin biases will emerge from the application of thermal gradients, as shown below.",
"Figure: (Color online) Sketch of a double-barrier tunnel device.",
"Transport occurs along the zz direction.E 0 E_0 is the spin-split level in the quantum well coupledto left (LL) and right (RR) reservoirs with spin-dependent chemical potentials μ\\mu and two different temperatures, T L T_L and T R T_R.Within the scattering approach, the electronic current per spin is given by $I_\\sigma =\\frac{e}{h}\\int {\\cal T}_\\sigma (E)\\left[f_L(E)-f_R(E)\\right]dE,$ where ${\\cal T}_\\sigma (E)$ is the transmission function for a carrier with spin $\\sigma $ and energy $E=E_\\perp +E_z$ , and $f_{\\alpha }\\left(E\\right)=1/1+e^{\\left(E-\\mu _{\\alpha \\sigma }\\right)/k_B T_{\\alpha }}$ is the Fermi-Dirac function for left and right contacts with temperature $T_{\\alpha }$ .",
"Neglecting interfacial roughness effects, the lateral momentum is conserved during tunneling and the transmission thus depends only on the energy parallel to the current direction, $E_z$ .",
"We integrate Eq.",
"(REF ) over the transversal modes and find I=C dEz T(Ez) [TL (1+e(L - Ez )/kB TL).",
"- .",
"TR (1+e(R - Ez )/kB TR)] , where ${\\cal C}=em^* Ak_B/4\\pi ^2 \\hbar ^3$ with $m^*$ the carrier effective mass and $A$ the device cross sectional area.",
"The total current is $I=I_++I_-$ .",
"Consider for the moment the spin-degenerate case ($h=0$ ).",
"We focus on the linear response regime because Seebeck nonlinearities appear only for transmission line shapes which depend strongly on energy.",
"[27], [28], [29], [30] Hence, the current can be linearized as $I=GV+L\\Delta T$ , where $\\Delta T=T_L-T_R$ is the temperature difference between the two contacts and the transport coefficients are G=2 eCkB dEz T(Ez) f(Ez) , L= 2 C dEz T(Ez) [Ez-EFkB T f(Ez) + (1+e(EF-Ez )/kB T)] , where $T=(T_L+T_R)/2$ is the base temperature.",
"Importantly, the thermopower $S=-(V/\\Delta T)_{I=0}=L/G$ is independent of $m^*$ and $A$ .",
"Equations (REF ) and (REF ) are completely general for elastic transport and arbitrary transmission functions.",
"For definiteness, we consider the Breit-Wigner approximation and model the transmission as ${\\cal T}(E_z)=\\Gamma _L \\Gamma _R/[(E_z-E_0)^2+(\\Gamma _L +\\Gamma _R)^2]$ , where $\\Gamma _\\alpha $ is the level broadening due to coupling to lead $\\alpha =L,R$ .",
"Without loss of generality, we consider symmetric barriers, $\\Gamma _L=\\Gamma _R=\\Gamma /2$ .",
"Figure REF shows the thermoelectric conductance $L$ (inset) as a function of the relative position of the energy level in the quantum well.",
"Strikingly, $L$ reaches a maximum when $E_0$ is aligned with $E_F$ irrespectively of the temperature $T$ .",
"The background temperature enhances the peak broadening.",
"This sharply differs with a double-barrier tunnel system in effectively one dimension (a quantum dot), for which the thermoelectric conductance vanishes at the particle-hole symmetry point and reaches a maximum (minimum) when the $E_0-E_F$ is of the order of $\\Gamma $ ($-\\Gamma $ ).",
"[20], [21], [22] In our case, $L$ is always positive, implying that the current direction is completely determined by the sign of the temperature difference.",
"In other words, electrons are always transported from the hot to the cold side at $V=0$ .",
"In contrast, molecular junctions and quantum dots can exhibit flow against the thermal gradients when electrons below $E_F$ (holes) are dominant.",
"[31] This is not possible for a quasi-three dimensional tunnel diode because energy is distributed not only along the transport direction but also among lateral momenta in the perpendicular plane.",
"Figure: (Color online) Seebeck coefficient SS and thermoelectric conductance LL (inset) for a resonant tunneling diode as a function of the quantum well energy level E 0 E_0.",
"LL is normalized to L 0 =em * Ak B Γ/2π 2 ℏ 3 L_0=em^* A k_B \\Gamma /2\\pi ^2 \\hbar ^3 .A maximum is found at E 0 =E F E_0=E_F independently of k B T/Γk_B T/\\Gamma .",
"Both SS and LL peakshave a width proportional to k B T/Γk_B T/\\Gamma .Accordingly, the thermopower $S=L/G$ attains positive values only, as shown in the main panel of Fig.",
"REF .",
"Therefore, the generated thermovoltage always counteracts the applied temperature difference and its sign cannot be changed with tuning the well below or above the Fermi energy.",
"We find that $S$ reaches values as large as $0.5$ mV/K for quantum levels far beyond $E_F$ .",
"The Seebeck coefficient is quite sensitive to modifications of the base temperature: the maximum position shifts to higher energies and the peak quickly broadens.",
"When a typical value is used ($\\Gamma =1$ meV),[24] $T$ changes in Fig.",
"REF from 46 K to 186 K. Of course, for these temperature values one would expect contributions from inelastic scattering due, e.g., to interaction with phonons which are neglected in our model.",
"Nevertheless, our results suggest a significant change in $S$ that should be observable at moderate temperatures.",
"In fact, we find that the thermopower maximum increases proportionally to $ T^{0.33}$ for $\\Gamma =1$ meV and its position as a function of $E_0$ is approximately given by $E_F$ when $T\\lesssim O \\!",
"\\left(10^1\\right)$ K and by $7.5k_B T$ when $T\\gtrsim O \\!",
"\\left(10^2\\right)$ K. We now turn to spintronic effects.",
"Recently, a thermal gradient applied to a metallic ferromagnet was shown to generate a spin voltage (which produces a spin current) detected from a spin Hall effect signal.",
"[25], [26] This discovery has motivated the study of related phenomena in phase-coherent systems.",
"[32], [33], [34] Here, we consider a magnetically doped quantum well which exhibits a large spin splitting (denoted with $h$ ) in the presence of low magnetic fields.",
"[23] In the linear response regime, the total current is $I=\\left(L_{\\uparrow }+L_{\\downarrow }\\right) \\Delta T + \\left(G_{\\uparrow }+G_{\\downarrow }\\right) V + \\left(G_{\\uparrow }-G_{\\downarrow }\\right) V_S/2$ while the electronic spin current is $I_S=\\left(L_{\\uparrow }+L_{\\downarrow }\\right) \\Delta T + \\left(G_{\\uparrow }+G_{\\downarrow }\\right) V + \\left(G_{\\uparrow }-G_{\\downarrow }\\right) V_S/2$ , where $V_S=\\left[\\left(\\mu _{L \\uparrow }-\\mu _{L \\downarrow }\\right)-\\left(\\mu _{R \\uparrow }-\\mu _{R \\downarrow }\\right)\\right]/e$ represents the spin voltage.",
"The spin-dependent responses can be obtained from Eqs.",
"(REF ) and (REF ) substituting $E_0$ with $E_0+\\sigma h/2$ in the transmission Lorentzian function.",
"As a consequence, the Seebeck coefficient becomes split as $h$ is of the order of the thermopower peak (not shown here).",
"More interesting is the spin thermopower $S_S$ since it determines the possibility to create a spin voltage from a temperature bias only: $S_S=-\\left.\\frac{V_S}{\\Delta T}\\right|_{I=I_S=0}=\\left(\\frac{L_{\\uparrow }}{G_{\\uparrow }}-\\frac{L_{\\downarrow }}{G_{\\downarrow }}\\right)\\,.$ We note in passing that an electric (charge) voltage $V=-(L_\\uparrow /G_\\uparrow +L_\\downarrow /G_\\downarrow )\\Delta T/2$ is to be applied in order the conditions $I=0$ and $I_S=0$ to be fulfilled.",
"An experimentally more accessible alternative considers $I=0$ and $V=0$ but it then results in a nonzero spin current.",
"Figure: (Color online) Spin Seebeck coefficient for a magnetic tunnel diode as a function of E 0 E_0 for (a) fixed temperature k B T/Γ=4k_B T/\\Gamma = 4 and different spin splittings hh and (b) fixed splitting h/Γ=75h/\\Gamma =75 and different temperatures.In Fig.",
"REF we plot the spin Seebeck coefficient for different magnetic fields (upper panel) and different temperatures (lower panel).",
"We can observe that unlike the charge thermopower, $S_S$ can be positive or negative depending on the relative position of $E_0$ with respect to $E_F$ .",
"Therefore, the spin bias due to a temperature difference ($V_S=-S_S \\Delta T$ ) can change its sign if we modify the position of $E_0$ using, e.g., nearby gates, doping or growing techniques.",
"This is not possible with charge degrees of freedom only (cf.",
"Fig.",
"REF ).",
"At low temperature, Fig.",
"REF shows that the spin thermopower is manifestly positive (negative) for well level positions below (above) the Fermi energy.",
"The peak separation grows when $h$ increases, as expected.",
"In Fig.",
"REF (b) we analyze the influence of temperature for a fixed splitting $h$ .",
"As the base temperature increases, the peak positions shift to higher energies but the maximum value of $S_S$ stays roughly constant.",
"In fact, for moderate temperatures ($T=35\\Gamma /k_B$ ) there is a wide range of energy levels around which the spin Seebeck coefficient attains a sizeable value (around $0.2$ mV/K).",
"Our numerical simulations reveal a maximum of $|S_S|$ at $T\\simeq 50K$ for the experimental values $\\Gamma =1$ meV, $E_F=50$ meV and $h=35$ meV.",
"[23] To sum up, we have analyzed the thermoelectric properties of magnetic and nonmagnetic resonant tunneling diodes using the scattering approach.",
"We have found that in the absence of magnetic fields the thermoelectric peak conductance occurs when the quantum well level is aligned with the reservoirs' Fermi energy, and this effect persists when temperature changes.",
"In magnetically doped quantum wells, the spin thermopower can be tuned with an external field and reaches significant values even if the background temperature increases.",
"Therefore, our work suggests that quantum well systems are quite promising in developing substantial spin Seebeck effects at large output powers.",
"Electron-electron interactions will not qualitatively alter our conclusions since these interactions are effectively screened in large-area heterostructures, although further work should take into account the role of phonons and disorder in the intermediate temperature range.",
"This work has been supported by MINECO under grant No.",
"FIS2011-23526."
]
] | 1403.0484 |
[
[
"Optimization problems involving the first Dirichlet eigenvalue and the\n torsional rigidity"
],
[
"Abstract We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue."
],
[
"Introduction",
"A shape optimization problem can be written in the very general form $\\min \\big \\lbrace F(\\Omega )\\ :\\ \\Omega \\in \\mathcal {A}\\big \\rbrace ,$ where $\\mathcal {A}$ is a class of admissible domains and $F$ is a cost functional defined on $\\mathcal {A}$ .",
"We consider in the present paper the case where the cost functional $F$ is related to the solution of an elliptic equation and involves the spectrum of the related elliptic operator.",
"We speak in this case of spectral optimization problems.",
"Shape optimization problems of spectral type have been widely considered in the literature; we mention for instance the papers [14], [18], [17], [20], [21], [22], [23], [30], and we refer to the books [16], [27], [28], and to the survey papers [2], [19], [26], where the reader can find a complete list of references and details.",
"In the present paper we restrict ourselves for simplicity to the Laplace operator $-\\Delta $ with Dirichlet boundary conditions.",
"Furthermore we shall assume that the admissible domains $\\Omega $ are a priori contained in a given bounded domain $D\\subset \\mathbb {R}^d$ .",
"This assumption greatly simplifies several existence results that otherwise would require additional considerations in terms of concentration-compactness arguments [14], [32].",
"The most natural constraint to consider on the class of admissible domains is an inequality on their Lebesgue measure.",
"Our admissible class $\\mathcal {A}$ is then $\\mathcal {A}=\\big \\lbrace \\Omega \\subset D\\ :\\ |\\Omega |\\le 1\\big \\rbrace .$ Other kinds of constraints are also possible, but we concentrate here to the one above, referring the reader interested in possible variants to the books and papers quoted above.",
"The following two classes of cost functionals are the main ones considered in the literature.",
"Integral functionals.",
"Given a right-hand side $f\\in L^2(D)$ , for every $\\Omega \\in \\mathcal {A}$ let $u_\\Omega $ be the unique solution of the elliptic PDE $-\\Delta u=f\\hbox{ in }\\Omega ,\\qquad u\\in H^1_0(\\Omega ).$ The integral cost functionals are of the form $F(\\Omega )=\\int _\\Omega j\\big (x,u_\\Omega (x),\\nabla u_\\Omega (x)\\big )\\,dx,$ where $j$ is a suitable integrand that we assume convex in the gradient variable.",
"We also assume that $j$ is bounded from below by $j(x,s,z)\\ge -a(x)-c|s|^2,$ with $a\\in L^1(D)$ and $c$ smaller than the first Dirichlet eigenvalue of the Laplace operator $-\\Delta $ in $D$ .",
"For instance, the energy $\\mathcal {E}_f(\\Omega )$ defined by $\\mathcal {E}_f(\\Omega )=\\inf \\left\\lbrace \\int _D\\Big (\\frac{1}{2}|\\nabla u|^2-f(x)u\\Big )\\,dx\\ :\\ u\\in H^1_0(\\Omega )\\right\\rbrace ,$ belongs to this class since, integrating by parts its Euler-Lagrange equation, we have that $\\mathcal {E}_f(\\Omega )=-\\frac{1}{2}\\int _D f(x)u_\\Omega \\,dx,$ which corresponds to the integral functional above with $j(x,s,z)=-\\frac{1}{2} f(x)s.$ The case $f=1$ is particularly interesting for our purposes.",
"We denote by $w_\\Omega $ the torsion function, that is the solution of the PDE $-\\Delta u=1\\hbox{ in }\\Omega ,\\qquad u\\in H^1_0(\\Omega ),$ and by the torsional rigidity $T(\\Omega )$ the $L_1$ norm of $w_{\\Omega }$ , $T(\\Omega )=\\int _\\Omega w_\\Omega \\,dx=-2\\mathcal {E}_1(\\Omega ).$ Spectral functionals.",
"For every admissible domain $\\Omega \\in \\mathcal {A}$ we consider the spectrum $\\Lambda (\\Omega )$ of the Laplace operator $-\\Delta $ on $H^1_0(\\Omega )$ .",
"Since $\\Omega $ has a finite measure, the operator $-\\Delta $ has a compact resolvent and so its spectrum $\\Lambda (\\Omega )$ is discrete: $\\Lambda (\\Omega )=\\big (\\lambda _1(\\Omega ),\\lambda _2(\\Omega ),\\dots \\big ),$ where $\\lambda _k(\\Omega )$ are the eigenvalues counted with their multiplicity.",
"The spectral cost functionals we may consider are of the form $F(\\Omega )=\\Phi \\big (\\Lambda (\\Omega )\\big ),$ for a suitable function $\\Phi :\\mathbb {R}^\\mathbb {N}\\rightarrow \\overline{\\mathbb {R}}$ .",
"For instance, taking $\\Phi (\\Lambda )=\\lambda _k(\\Omega )$ we obtain $F(\\Omega )=\\lambda _k(\\Omega ).$ We take the torsional rigidity $T(\\Omega )$ and the first eigenvalue $\\lambda _1(\\Omega )$ as prototypes of the two classes above and we concentrate our attention on cost functionals that depend on both of them.",
"We note that, by the maximum principle, when $\\Omega $ increases $T(\\Omega )$ increases, while $\\lambda _1(\\Omega )$ decreases."
],
[
"Statement of the problem",
"The optimization problems we want to consider are of the form $ \\min \\left\\lbrace \\Phi \\big (\\lambda _1(\\Omega ),T(\\Omega )\\big )\\ :\\ \\Omega \\subset D,\\ |\\Omega |\\le 1\\right\\rbrace ,$ where we have normalized the constraint on the Lebesgue measure of $\\Omega $ , and where $\\Phi $ is a given continuous (or lower semi-continuous) and non-negative function.",
"In the rest of this paper we often take for simplicity $D=\\mathbb {R}^d$ , even if most of the results are valid in the general case.",
"For instance, taking $\\Phi (a,b)=ka+b$ with $k$ a fixed positive constant, the quantity we aim to minimize becomes $k\\lambda _1(\\Omega )+T(\\Omega )\\qquad \\hbox{with $\\Omega \\subset D,$ and }|\\Omega |\\le 1.$ Remark 2.1 If the function $\\Phi (a,b)$ is increasing with respect to $a$ and decreasing with respect to $b$ , then the cost functional $F(\\Omega )=\\Phi \\big (\\lambda _1(\\Omega ),T(\\Omega )\\big )$ turns out to be decreasing with respect to the set inclusion.",
"Since both the torsional rigidity and the first eigenvalue are $\\gamma $ -continuous functionals and the function $\\Phi $ is assumed lower semi-continuous, we can apply the existence result of [21], which provides the existence of an optimal domain.",
"In general, if the function $\\Phi $ does not verify the monotonicity property of Remark REF , then the existence of an optimal domain is an open problem, and the aim of this paper is to discuss this issue.",
"For simplicity of the presentation we limit ourselves to the two-dimensional case $d=2$ .",
"The case of general $d$ does not present particular difficulties but requires the use of several $d-$ dependent exponents.",
"Remark 2.2 The following facts are well known.",
"i) If $B$ is a disk in $\\mathbb {R}^2$ we have $T(B)=\\frac{1}{8\\pi }|B|^2.$ ii) If $j_{0,1}\\approx 2.405$ is the first positive zero of the Bessel functions $J_0(x)$ and $B$ is a disk of $\\mathbb {R}^2$ we have $\\lambda _1(B)=\\frac{\\pi }{|B|}j^2_{0,1}.$ iii) The torsional rigidity $T(\\Omega )$ scales as $T(t\\Omega )=t^4T(\\Omega ),\\qquad \\forall t>0.$ iv) The first eigenvalue $\\lambda _1(\\Omega )$ scales as $\\lambda _1(t\\Omega )=t^{-2}\\lambda _1(\\Omega ),\\qquad \\forall t>0.$ v) For every domain $\\Omega $ of $\\mathbb {R}^2$ and any disk $B$ we have $|\\Omega |^{-2}T(\\Omega )\\le |B|^{-2}T(B)=\\frac{1}{8\\pi }.$ vi) For every domain $\\Omega $ of $\\mathbb {R}^2$ and any disk $B$ we have (Faber-Krahn inequality) $|\\Omega |\\lambda _1(\\Omega )\\ge |B|\\lambda _1(B)=\\pi j^2_{0,1}.$ vii) A more delicate inequality is the so-called Kohler-Jobin inequality (see [29], [11]): for any domain $\\Omega $ of $\\mathbb {R}^2$ and any disk $B$ we have $\\lambda ^2_1(\\Omega )T(\\Omega )\\ge \\lambda ^2_1(B)T(B)=\\frac{\\pi }{8}j^4_{0,1}.$ This had been previously conjectured by G. Pólya and G.Szegö [31].",
"We recall the following inequality, well known for planar regions (Section 5.4 in [31]), between torsional rigidity and first eigenvalue.",
"Proposition 2.3 For every domain $\\Omega \\subset \\mathbb {R}^d$ we have $\\lambda _1(\\Omega )T(\\Omega )\\le |\\Omega |.$ By definition, $\\lambda _1(\\Omega )$ is the infimum of the Rayleigh quotient $\\int _\\Omega |\\nabla u|^2\\,dx\\bigg /\\int _\\Omega u^2\\,dx\\qquad \\hbox{over all }u\\in H^1_0(\\Omega ),\\ u\\ne 0.$ Taking as $u$ the torsion function $w_\\Omega $ , we have $\\lambda _1(\\Omega )\\le \\int _\\Omega |\\nabla w_\\Omega |^2\\,dx\\bigg /\\int _\\Omega w_\\Omega ^2\\,dx.$ Since $-\\Delta w_\\Omega =1$ , an integration by parts gives $\\int _\\Omega |\\nabla w_\\Omega |^2\\,dx=\\int _\\Omega w_\\Omega \\,dx=T(\\Omega ),$ while the Hölder inequality gives $\\int _\\Omega w^2_\\Omega \\,dx\\ge \\frac{1}{|\\Omega |}\\left(\\int _\\Omega w_\\Omega \\,dx\\right)^2=\\frac{1}{|\\Omega |}\\big (T(\\Omega )\\big )^2.$ Summarizing, we have $\\lambda _1(\\Omega )\\le \\frac{|\\Omega |}{T(\\Omega )}$ as required.",
"Remark 2.4 The infimum of $\\lambda _1(\\Omega )T(\\Omega )$ over open sets $\\Omega $ of prescribed measure is zero.",
"To see this, let $\\Omega _n$ be the disjoint union of one ball of volume $1/n$ and $n(n-1)$ balls of volume $1/n^2$ .",
"Then the radius $R_n$ of the ball of volume $1/n$ is $(n\\omega _d)^{-1/d}$ while the radius $r_n$ of the balls of volume $1/n^2$ is $(n^2\\omega _d)^{-1/d}$ , so that $|\\Omega _n|=1$ , $\\lambda _1(\\Omega _n)=\\lambda _1(B_{R_n})=\\frac{1}{R_n^2}\\lambda _1(B_1)=(n\\omega _d)^{2/d}\\lambda _1(B_1),$ and $\\begin{split}T(\\Omega _n)&=T(B_{R_n})+n(n-1)T(B_{r_n})=T(B_1)\\big (R_n^{d+2}+n(n-1)r_n^{d+2}\\big )\\\\&=T(B_1)\\omega _d^{-1-2/d}\\big (n^{-1-2/d}+(n-1)n^{-1-4/d}\\big ).\\end{split}$ Therefore $\\lambda _1(\\Omega _n)T(\\Omega _n)=\\frac{\\lambda _1(B_1)T(B_1)}{\\omega _d}\\frac{n^{2/d}+n-1}{n^{1+2/d}},$ which vanishes as $n\\rightarrow \\infty $ .",
"In the next section we investigate the inequality of Proposition REF ."
],
[
"A sharp inequality between torsion and first eigenvalue",
"We define the constant $\\mathcal {K}_d=\\sup \\left\\lbrace \\frac{\\lambda _1(\\Omega )T(\\Omega )}{|\\Omega |}\\ :\\ \\Omega \\hbox{ open in }\\mathbb {R}^d,\\ |\\Omega |<\\infty \\right\\rbrace .$ We have seen in Proposition REF that $\\mathcal {K}_d\\le 1$ .",
"The question is if the constant 1 can be improved.",
"Consider a ball $B$ ; performing the shape derivative as in [28], keeping the volume of the perturbed shapes constant, we obtain that for every field $V(x)$ $\\partial [\\lambda _1(B)T(B)](V)=T(B)\\partial [\\lambda _1(B)](V)+\\lambda _1(B)\\partial [T(B)](V)=C_B\\int _{\\partial B}V\\cdot n\\,d\\mathcal {H}^{d-1}$ for a suitable constant $C_B$ .",
"Since the volume of the perturbed shapes is constant, we have $\\int _{\\partial B}V\\cdot n\\,d\\mathcal {H}^{d-1}=0,$ where $\\mathcal {H}^{d-1}$ denotes $(d-1)$ -dimensional Hausdorff measure.",
"This shows that balls are stationary for the functional $F(\\Omega )=\\frac{\\lambda _1(\\Omega )T(\\Omega )}{|\\Omega |}.$ Below we will show, by considering rectangles, that balls are not optimal.",
"To do so we shall obtain a lower bound for the torsional rigidity of a rectangle.",
"Proposition 3.1 In a rectangle $R_{a,b}=(-b/2,b/2)\\times (-a/2,a/2)$ with $a\\le b$ we have $T(R_{a,b})\\ge \\frac{a^3b}{12}-\\frac{11a^4}{180}.$ Let us estimate the energy $\\mathcal {E}_1(R_{a,b})=\\inf \\left\\lbrace \\int _{R_{a,b}}\\left(\\frac{1}{2}|\\nabla u|^2-u\\right)\\,dx\\,dy\\ :\\ u\\in H^1_0(R_{a,b})\\right\\rbrace $ by taking the function $u(x,y)=\\frac{a^2-4y^2}{8}\\theta (x),$ where $\\theta (x)$ is defined by $\\theta (x)={\\left\\lbrace \\begin{array}{ll}1&\\hbox{,if }|x|\\le (b-a)/2\\\\(b-2|x|)/a&\\hbox{,otherwise.}\\end{array}\\right.",
"}$ We have $|\\nabla u|^2=\\left(\\frac{a^2-4y^2}{8}\\right)^2|\\theta ^{\\prime }(x)|^2+y^2|\\theta (x)|^2,$ so that $\\begin{split}\\mathcal {E}_1(R_{a,b})&\\le 2\\int _0^{a/2}\\left(\\frac{a^2-4y^2}{8}\\right)^2\\,dy\\int _0^{b/2}|\\theta ^{\\prime }(x)|^2\\,dx+2\\int _0^{a/2}y^2\\,dy\\int _0^{b/2}|\\theta (x)|^2\\,dx\\\\&\\qquad -4\\int _0^{a/2}\\frac{a^2-4y^2}{8}\\,dy\\int _0^{b/2}\\theta (x)\\,dx\\\\&=\\frac{a^4}{60}+\\frac{a^3}{12}\\left(\\frac{b-a}{2}+\\frac{a}{6}\\right)-\\frac{a^3}{6}\\left(\\frac{b-a}{2}+\\frac{a}{4}\\right)\\\\&=-\\frac{a^3b}{24}+\\frac{11a^4}{360}.\\end{split}$ The desired inequality follows since $T(R_{a,b})=-2\\mathcal {E}_1(R_{a,b})$ .",
"In $d$ -dimensions we have the following.",
"Proposition 3.2 If $\\Omega _{\\varepsilon }=\\omega \\times (-{\\varepsilon }/2,{\\varepsilon },2)$ , where $\\omega $ is a convex set in $\\mathbb {R}^{d-1}$ with $|\\omega |<\\infty $ , then $T(\\Omega _{\\varepsilon })=\\frac{{\\varepsilon }^3}{12}|\\omega |+O({\\varepsilon }^4),\\qquad \\epsilon \\downarrow 0.$ We defer the proof to Section .",
"For a ball of radius $R$ we have $\\lambda _1(B)=\\frac{j^2_{d/2-1,1}}{R^2},\\qquad T(B)=\\frac{\\omega _d R^{d+2}}{d(d+2)},\\qquad |B|=\\omega _dR^d,$ so that $F(B)=\\frac{\\lambda _1(B)T(B)}{|B|}=\\frac{j^2_{d/2-1,1}}{d(d+2)}:=\\alpha _d$ For instance, we have $\\alpha _2\\approx 0.723,\\qquad \\alpha _3\\approx 0.658,\\qquad \\alpha _4\\approx 0.612.$ Moreover, since $j_{\\nu ,1}=\\nu +O(\\nu ^{1/3}),\\ \\nu \\rightarrow \\infty $ , we have that $\\lim _{d\\rightarrow \\infty }\\alpha _d=\\frac{1}{4}$ .",
"A plot of $\\alpha _d$ is given in Figure REF .",
"Figure: The plotof α d \\alpha _d for 2≤d≤302\\le d\\le 30.We now consider a slab $\\Omega _{\\varepsilon }=\\omega \\times (0,{\\varepsilon })$ of thickness ${\\varepsilon }\\rightarrow 0$ .",
"We have by separation of variables and Proposition REF that $\\lambda _1(\\Omega _{\\varepsilon })=\\frac{\\pi ^2}{{\\varepsilon }^2}+\\lambda _1(\\omega )\\approx \\frac{\\pi ^2}{{\\varepsilon }^2},\\qquad T(\\Omega _{\\varepsilon })\\approx \\frac{{\\varepsilon }^3|\\omega |}{12},\\qquad |\\Omega _{\\varepsilon }|={\\varepsilon }|\\omega |,$ so that $F(\\Omega _{\\varepsilon })\\approx \\frac{\\pi ^2}{12}\\approx 0.822.$ This shows that in any dimension the slab is better than the ball.",
"Using domains in $\\mathbb {R}^d$ with $k$ small dimensions and $d-k$ large dimensions does not improve the value of the cost functional $F$ .",
"In fact, if $\\omega $ is a convex domain in $\\mathbb {R}^{d-k}$ and $B_k({\\varepsilon })$ a ball in $\\mathbb {R}^k$ , then by Theorem REF with $\\Omega _{\\varepsilon }=\\omega \\times B_k({\\varepsilon })$ we have that $\\lambda _1(\\Omega _{\\varepsilon })\\approx \\frac{1}{{\\varepsilon }^2}\\lambda _1\\big (B_k(1)\\big ),\\qquad T(\\Omega _{\\varepsilon })\\approx {\\varepsilon }^{k+2}|\\omega |T(B_k(1)),\\qquad |\\Omega _{\\varepsilon }|={\\varepsilon }^k|\\omega ||B_k(1)|,$ so that $F(\\Omega _{\\varepsilon })\\approx \\frac{j^2_{k/2-1,1}}{k(k+2)}\\le \\frac{\\pi ^2}{12}.$ This supports the following.",
"Conjecture 3.3 For every dimension $d$ we have $\\mathcal {K}_d=\\pi ^2/12,$ and no domain in $\\mathbb {R}^d$ maximizes the functional $F$ for $d>1$ .",
"The maximal value $\\mathcal {K}_d$ is asymptotically reached by a thin slab $\\Omega _{\\varepsilon }=\\omega \\times (0,{\\varepsilon })$ , with $\\omega \\subset \\mathbb {R}^{d-1}$ , as ${\\varepsilon }\\rightarrow 0$ ."
],
[
"The attainable set",
"In this section we bound the measure by $|\\Omega |\\le 1$ .",
"Our goal is to plot the subset of $\\mathbb {R}^2$ whose coordinates are the eigenvalue $\\lambda _1(\\Omega )$ and the torsion $T(\\Omega )$ .",
"It is convenient to change coordinates and to set for a given admissible domain $\\Omega ,$ $x=\\lambda _1(\\Omega ),\\qquad y=\\big (\\lambda _1(\\Omega )T(\\Omega )\\big )^{-1}.$ In addition, define $E=\\left\\lbrace (x,y)\\in \\mathbb {R}^2\\ :\\ x=\\lambda _1(\\Omega ),\\ y=\\big (\\lambda _1(\\Omega )T(\\Omega )\\big )^{-1}\\hbox{ for some $\\Omega $ with }|\\Omega |\\le 1\\right\\rbrace .$ Therefore, the optimization problem (REF ) can be rewritten as $\\min \\left\\lbrace \\Phi \\big (x,1/(xy)\\big )\\ :\\ (x,y)\\in E\\right\\rbrace .$ Conjecture 4.1 The set $E$ is closed.",
"We remark that the conjecture above, if true, would imply the existence of a solution of the optimization problem (REF ) for many functions $\\Phi $ .",
"Below we will analyze the variational problem in case $\\Phi (x,y)=kx+\\frac{1}{xy},$ where $k>0$ .",
"Theorem 4.2 Let $d=2,3,\\cdots $ , and let $k^*_d=\\frac{1}{2d\\omega _d^{4/d}j^2_{d/2-1,1}}.$ Consider the optimization problem $\\min \\left\\lbrace k\\lambda _1(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\right\\rbrace .$ If $0<k\\le k^*_d$ then the ball with radius $R_k=\\left(\\frac{2kdj^2_{d/2-1,1}}{\\omega _d}\\right)^{1/(d+4)}$ is the unique minimizer (modulo translations and sets of capacity 0).",
"If $k> k^*_d$ then the ball $B$ with measure 1 is the unique minimizer.",
"Consider the problem (REF ) without the measure constraint $\\min \\left\\lbrace k\\lambda _1(\\Omega )+T(\\Omega )\\ :\\ \\Omega \\subset \\mathbb {R}^d\\right\\rbrace .",
"$ Taking $t\\Omega $ instead of $\\Omega $ gives that $k\\lambda _1(t\\Omega )+T(t\\Omega )=kt^{-2}\\lambda _1(\\Omega )+t^{d+2}T(\\Omega ).$ The optimal $t$ which minimizes this expression is given by $t=\\left(\\frac{2k\\lambda _1(\\Omega )}{(d+2)T(\\Omega )}\\right)^{1/(d+4)}.$ Hence (REF ) equals $\\min \\left\\lbrace (d+4)\\left(\\frac{k^{d+2}}{4(d+2)^{d+2}}T^2(\\Omega )\\lambda _1^{d+2}(\\Omega )\\right)^{1/(d+4)}:\\ \\Omega \\subset \\mathbb {R}^d\\ \\right\\rbrace .$ By the Kohler-Jobin inequality in $\\mathbb {R}^d$ , the minimum in (REF ) is attained by any ball.",
"Therefore the minimum in (REF ) is given by a ball $B_R$ such that $\\left(\\frac{2k\\lambda _1(B_R)}{(d+2)T(B_R)}\\right)^{1/(d+4)}=1.$ This gives (REF ).",
"We conclude that the measure constrained problem (REF ) admits the ball $B_{R_k}$ as a solution whenever $\\omega _dR_k^d\\le 1$ .",
"That is $k\\le k^*_d.$ Next consider the case $k>k^*_d$ .",
"Let $B$ be the open ball with measure 1.",
"It is clear that $\\min \\lbrace k\\lambda _1(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\rbrace \\le k\\lambda _1(B)+T(B).$ To prove the converse we note that for $k>k^*_d$ , $\\min \\lbrace k\\lambda _1&(\\Omega )+T(\\Omega ):|\\Omega |\\le 1\\rbrace \\nonumber \\\\&\\ge \\min \\lbrace (k-k^*_d)\\lambda _1(\\Omega ):|\\Omega |\\le 1\\rbrace +\\min \\lbrace k^*_d\\lambda _1(\\Omega )+T(\\Omega ):|\\Omega |\\le 1\\rbrace .$ The minimum in the first term in the right hand side of (REF ) is attained for $B$ by Faber-Krahn, whereas the minimum in second term is attained for $B_{R_{k^*_d}}$ by our previous unconstrained calculation.",
"Since $|B_{R_{k^*_d}}|=|B|=1$ we have by (REF ) that $\\min \\lbrace k\\lambda _1&(\\Omega )+T(\\Omega ):|\\Omega |\\le 1\\rbrace \\nonumber \\\\&\\ge (k-k^*_d)\\lambda _1(B)+ k^*_d\\lambda _1(B)+T(B)\\nonumber \\\\&=k\\lambda _1(B)+T(B).$ Uniqueness of the above minimizers follows by uniqueness of Faber-Krahn and Kohler-Jobin.",
"It is interesting to replace the first eigenvalue in (REF ) be a higher eigenvalue.",
"We have the following for the second eigenvalue.",
"Theorem 4.3 Let $d=2,3,\\cdots $ , and let $l^*_d=\\frac{1}{2d(2\\omega _d)^{4/d}j^2_{d/2-1,1}}.$ Consider the optimization problem $\\min \\left\\lbrace l\\lambda _2(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\right\\rbrace .$ If $0<l\\le l^*_d$ then the union of two disjoint balls with radii $R_l=\\left(\\frac{ldj^2_{d/2-1,1}}{\\omega _d}\\right)^{1/(d+4)}$ is the unique minimizer (modulo translations and sets of capacity 0).",
"If $l> l^*_d$ then union of two disjoint balls with measure $1/2$ each is the unique minimizer.",
"First consider the unconstrained problem $\\min \\left\\lbrace l\\lambda _1(\\Omega )+T(\\Omega )\\ :\\ \\Omega \\subset \\mathbb {R}^d\\right\\rbrace .",
"$ Taking $t\\Omega $ instead of $\\Omega $ gives that $l\\lambda _2(t\\Omega )+T(t\\Omega )=lt^{-2}\\lambda _2(\\Omega )+t^{d+2}T(\\Omega ).$ The optimal $t$ which minimizes this expression is given by $t=\\left(\\frac{2l\\lambda _2(\\Omega )}{(d+2)T(\\Omega )}\\right)^{1/(d+4)}.$ Hence (REF ) equals $\\min \\left\\lbrace (d+4)\\left(\\frac{l^{d+2}}{4(d+2)^{d+2}}T^2(\\Omega )\\lambda _2^{d+2}(\\Omega )\\right)^{1/(d+4)}:\\ \\Omega \\subset \\mathbb {R}^d\\ \\right\\rbrace .$ It follows by the Kohler-Jobin inequality, see for example Lemma 6 in [9], that the minimizer of (REF ) is attained by the union of two disjoint balls $B_R$ and $B^{\\prime }_R$ with the same radius.",
"Since $\\lambda _2(B_R\\cup B^{\\prime }_R)=\\lambda _1(B_R)$ and $T(B_R\\cup B^{\\prime }_R)=2T(B_R)$ we have, using (REF ), that the radii of these balls are given by (REF ).",
"We conclude that the measure constrained problem (REF ) admits the union of two disjoint balls with equal radius $R_l$ as a solution whenever $2\\omega _dR_l^d\\le 1$ .",
"That is $l\\le l^*_d.$ Next consider the case $l>l^*_d$ .",
"Let $\\Omega $ be the union of two disjoint balls $B$ and $B^{\\prime }$ with measure $1/2$ each.",
"Then $\\min \\lbrace l\\lambda _2(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\rbrace \\le l\\lambda _1(B)+2T(B).$ To prove the converse we note that for $l>l^*_d$ , $\\min \\lbrace l\\lambda _2&(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\rbrace \\nonumber \\\\&\\ge \\min \\lbrace (l-l^*_d)\\lambda _2(\\Omega )\\ :\\ |\\Omega |\\le 1\\rbrace +\\min \\lbrace l^*_d\\lambda _2(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\rbrace .$ The minimum in the first term in the right hand side of (REF ) is attained for $B\\cup B^{\\prime }$ by the Krahn-Szegö inequality, whereas the minimum in second term is attained for the union of two disjoint balls with radius $R_{l^*_d}$ by our previous unconstrained calculation.",
"Since $|B_{R_{l^*_d}}|=1/2=|B|=|B^{\\prime }|$ we have by (REF ) that $\\min \\lbrace l\\lambda _2(\\Omega )+T(\\Omega )\\ :\\ |\\Omega |\\le 1\\rbrace &\\ge (l-l^*_d)\\lambda _1(B)+ l^*_d\\lambda _1(B)+2T(B)\\nonumber \\\\&=l\\lambda _1(B)+2T(B).$ Uniqueness of the above minimizers follows by uniqueness of Krahn-Szegö and Kohler-Jobin for the second eigenvalue.",
"To replace the first eigenvalue in (REF ) be the $j$ 'th eigenvalue ($j>2$ ) is a very difficult problem since we do not know the minimizers of the $j$ 'th Dirichlet eigenvalue with a measure constraint nor the minimizer of the $j$ 'th Dirichlet eigenvalue a torsional rigidity constraint.",
"However, if these two problems have a common minimizer then information similar to the above can be obtained.",
"Putting together the facts listed in Remark REF we obtain the following inequalities.",
"(i) By Faber-Krahn inequality we have $x\\ge \\pi j^2_{0,1}\\approx 18.168$ .",
"(ii) By Conjecture REF (if true) we have $y\\ge 12/\\pi ^2\\approx 1.216$ .",
"(iii) By the bound on the torsion of Remark REF v) we have $xy\\ge 8\\pi \\approx 25.133$ .",
"(iv) By the Kohler-Jobin inequality we have $y/x\\le 8/(\\pi j^4_{0,1})\\approx 0.076$ .",
"(v) The set $E$ is conical, that is if a point $(x_0,y_0)$ belongs to $E$ , then all the half-line $\\big \\lbrace (tx_0,ty_0)\\ :\\ t\\ge 1\\big \\rbrace $ in contained in $E$ .",
"This follows by taking $\\Omega _t=\\Omega /t$ and by the scaling properties iii) and iv) of Remark REF .",
"(vi) The set $E$ is vertically convex, that is if a point $(x_0,y_0)$ belongs to $E$ , then all points $(x_0,ty_0)$ with $1\\le t\\le 8/(\\pi j^4_{0,1})$ belong to $E$ .",
"To see this fact, let $\\Omega $ be a domain corresponding to the point $(x_0,y_0)\\in E$ .",
"The continuous Steiner symmetrization path $\\Omega _t$ (with $t\\in [0,1]$ ) then continuously deforms the domain $\\Omega =\\Omega _0$ into a ball $B=\\Omega _1$ , preserving the Lebesgue measure and decreasing $\\lambda _1(\\Omega _t)$ (see [13] where this tool has been developed, and Section 6.3 of [16] for a short survey).",
"The curve $x(t)=\\lambda _1(\\Omega _t),\\qquad y(t)=\\big (\\lambda _1(\\Omega _t)T(\\Omega _t)\\big )^{-1}$ then connects the point $(x_0,y_0)$ to the Kohler-Jobin line $\\left\\lbrace y=8x/(\\pi j^4_{0,1})\\right\\rbrace $ , having $x(t)$ decreasing.",
"Since $\\big (x(t),y(t)\\big )\\in E$ , the conicity of $E$ then implies vertical convexity.",
"A plot of the constraints above is presented in Figure REF .",
"Figure: The admissible region EE is contained in the dark area.Some particular cases can be computed explicitly.",
"Consider $d=2$ , and let $\\Omega =B_R\\cup B_r, \\hbox{with $B_R\\cap B_r=\\emptyset $, $r\\le R$, and }\\pi (R^2+r^2)=1.$ An easy computation gives that $\\lambda _1(\\Omega )=\\frac{j^2_{0,1}}{R^2},\\qquad T(\\Omega )=\\frac{2\\pi ^2R^4-2\\pi R^2+1}{8\\pi },$ so that the curve $y=\\frac{8\\pi x}{x^2-2\\pi j^2_{0,1}x+2\\pi ^2j^4_{0,1}},\\qquad \\pi j^2_{0,1}\\le x\\le 2\\pi j^2_{0,1}$ is contained in $E$ (see Figure REF ).",
"Figure: The dashed line corresponds to two disks of variable radii.If we consider the rectangle $\\Omega =(0,b)\\times (0,a)\\hbox{,\\ with $a\\le b$, and }ab=1,$ we have by Proposition REF $\\lambda _1(\\Omega )=\\pi ^2\\left(\\frac{1}{a^2}+\\frac{1}{b^2}\\right)=\\pi ^2\\left(\\frac{1}{a^2}+a^2\\right),\\qquad T(\\Omega )\\ge \\frac{a^3b}{12}-\\frac{11a^4}{180}=\\frac{a^2}{12}-\\frac{11a^4}{180}.$ Therefore $y\\le h\\big (x/(2\\pi ^2)\\big ),\\qquad \\hbox{where }h(t)=\\frac{90}{\\pi ^2 t\\left(11+15t-22t^2-(15+2t)\\sqrt{t^2-1}\\right)},\\quad t\\ge 1.$ By $E$ being conical the curve $y=h\\big (x/(2\\pi ^2)\\big )\\qquad \\pi ^2\\le x<+\\infty $ is contained in $E$ (see Figure REF ).",
"Figure: The dashed line is an upper bound to the line corresponding to rectangles.Besides the existence of optimal domains for the problem (REF ), the regularity of optimal shapes is another very delicate and important issue.",
"Very little is known about the regularity of optimal domains for spectral optimization problems (see for instance [12], [15], [25], [32]); the cases where only the first eigenvalue $\\lambda _1(\\Omega )$ and the torsion $T(\\Omega )$ are involved could be simpler and perhaps allow to use the free boundary methods developed in [1]."
],
[
"Torsional rigidity and the heat equation",
"It is well known that the rich interplay between elliptic and parabolic partial differential equations provide tools for obtaining results in one field using tools from the other.",
"See for example the monograph by E. B. Davies [24], and [3], [5], [6], [7], [8], [10] for some more recent results.",
"In this section we use some heat equation tools to obtain new estimates for the torsional rigidity.",
"Before we do so we recall some basic facts relating the torsional rigidity to the heat equation.",
"For an open set $\\Omega $ in $\\mathbb {R}^d$ with boundary $\\partial \\Omega $ we denote the Dirichlet heat kernel by $p_{\\Omega }(x,y;t),\\ x\\in \\Omega ,\\ y\\in \\Omega ,\\ t>0$ .",
"So $u_\\Omega (x;t):=\\int _\\Omega p_\\Omega (x,y;t)\\,dy,$ is the unique weak solution of ${\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\partial u}{\\partial t}=\\Delta u&x\\in \\Omega ,\\ t>0,\\\\\\lim _{t\\downarrow 0}u(x;t)=1&\\hbox{in }L^2(\\Omega ),\\\\u(x;t)=0&x\\in \\partial \\Omega ,\\ t>0.\\end{array}\\right.",
"}$ The latter boundary condition holds at all regular points of $\\partial \\Omega $ .",
"We denote the heat content of $\\Omega $ at time $t$ by $Q_{\\Omega }(t)=\\int _{\\Omega }u_{\\Omega }(x;t)\\,dx.$ Physically the heat content represents the amount of heat in $\\Omega $ at time $t$ if $\\Omega $ has initial temperature 1, while $\\partial \\Omega $ is kept at temperature 0 for all $t>0$ .",
"Since the Dirichlet heat kernel is non-negative, and monotone in $\\Omega $ we have that $0\\le p_\\Omega (x,y;t)\\le p_{\\mathbb {R}^d}(x,y;t)=(4\\pi t)^{-d/2}e^{-|x-y|^2/(4t)}.$ It follows by either (REF ) or by the maximum principle that $0\\le u_\\Omega (x;t)\\le 1,$ and that if $|\\Omega |<\\infty $ then $0\\le Q_\\Omega (t)\\le |\\Omega |.$ In the latter situation we also have an eigenfunction expansion for the Dirichlet heat kernel in terms of the Dirichlet eigenvalues $\\lambda _1(\\Omega )\\le \\lambda _2(\\Omega )\\le \\cdots $ , and a corresponding orthonormal set of eigenfunctions $\\lbrace \\varphi _1,\\varphi _2,\\cdots \\rbrace $ , $p_\\Omega (x,y;t)=\\sum _{j=1}^{\\infty }e^{-t\\lambda _j(\\Omega )}\\varphi _j(x)\\varphi _j(y).$ We note that the eigenfunctions are in $L^p(\\Omega )$ for all $1\\le p\\le \\infty $ .",
"It follows by Parseval's formula that $Q_\\Omega (t)=\\sum _{j=1}^{\\infty }e^{-t\\lambda _j(\\Omega )}\\left(\\int _\\Omega \\varphi _j\\,dx\\right)^2\\le e^{-t\\lambda _1(\\Omega )}\\sum _{j=1}^{\\infty }\\left(\\int _\\Omega \\varphi _j\\,dx\\right)^2=e^{-t\\lambda _1(\\Omega )}|\\Omega |.$ Since the torsion function is given by $w_\\Omega (x)=\\int _0^\\infty u_\\Omega (x;t)\\,dt,$ we have that $T(\\Omega )=\\sum _{j=1}^{\\infty }\\lambda _j(\\Omega )^{-1}\\left(\\int _\\Omega \\varphi _j\\,dx\\right)^2.$ We recover Proposition 2.3. by integrating (REF ) with respect to $t$ over $[0,\\infty )$ : $T(\\Omega )\\le \\lambda _1(\\Omega )^{-1}\\sum _{j=1}^{\\infty }\\left(\\int _\\Omega \\varphi _j\\,dx\\right)^2=\\lambda _1(\\Omega )^{-1}|\\Omega |.$ Let $M_1$ and $M_2$ be two open sets in Euclidean space with finite Lebesgue measures $|M_1|$ and $|M_2|$ respectively.",
"Let $M=M_1\\times M_2$ .",
"We have that $p_{M_1\\times M_2}(x,y;t)=p_{M_1}(x_1,y_1;t)p_{M_2}(x_2,y_2;t),$ where $x=(x_1,x_2), y=(y_1,y_2)$ .",
"It follows that $Q_M(t)=Q_{M_1}(t)Q_{M_2}(t),$ and $T(M)=\\int _0^{\\infty }Q_{M_1}(t)Q_{M_2}(t)\\,dt.$ Integrating (REF ) with respect to $t$ , and using (REF ) for $M_2$ we obtain that $T(M)\\le T(M_1)|M_2|.$ This upper bound should be “sharp” if the decay of $Q_{M_2}(t)$ with respect to $t$ is much slower than the decay of $Q_{M_1}(t)$ .",
"The result below makes this assertion precise in the case where $M_2$ is a convex set with ${\\mathcal {H}^{d_2-1}}(\\partial M_2)<\\infty $ .",
"The latter condition is for convex sets equivalent to requiring that $M_2$ is bounded.",
"Here ${\\mathcal {H}^{d_2-1}}$ denotes the $(d_2-1)$ -dimensional Hausdorff measure.",
"Theorem 5.1 Let $M=M_1\\times M_2$ , where $M_1$ is an arbitrary open set in $\\mathbb {R}^{d_1}$ with finite $d_1$ -measure and $M_2$ is a bounded convex open set in $\\mathbb {R}^{d_2}$ .",
"Then there exists a constant ${\\mathcal {C}}_{d_2}$ depending on $d_2$ only such that $T(M)\\ge T(M_1)|M_2|-{\\mathcal {C}}_{d_2}\\lambda _1(M_1)^{-3/2}|M_1|{\\mathcal {H}^{d_2-1}}(\\partial M_2).$ For the proof of Theorem REF we need the following lemma (proved as Lemma 6.3 in [4]).",
"Lemma 5.2 For any open set $\\Omega $ in $\\mathbb {R}^d$ , $u_\\Omega (x;t)\\ge 1-2\\int _{\\lbrace y\\in \\mathbb {R}^d:|y-x|>d(x)\\rbrace }p_{\\mathbb {R}^d}(x,y;t)\\,dy,$ where $d(x)=\\min \\lbrace |x-z|\\ :\\ z\\in \\partial \\Omega \\rbrace .$ [Proof of Theorem REF ] With the notation above we have that $T(M)&=T(M_1)|M_2|-\\int _0^\\infty Q_{M_1}(t)(|M_2|-Q_{M_2}(t))\\,dt\\nonumber \\\\&=T(M_1)|M_2|-\\int _0^\\infty Q_{M_1}(t)\\int _{M_2}(1-u_{M_2}(x_2;t))\\,dx_2\\,dt.$ Define for $r>0$ , $\\partial M_2(r)=\\lbrace x\\in M_2:d(x)=r\\rbrace .$ It is well known that (Proposition 2.4.3 in [16]) if $M_2$ is convex then ${\\mathcal {H}^{d_2-1}}(\\partial M_2(r))\\le {\\mathcal {H}^{d_2-1}}(\\partial M_2).$ By (REF ), (REF ) and (REF ) we obtain that $\\int _0^{\\infty }&Q_{M_1}(t)\\int _{M_2}(1-u_{M_2}(x_2;t))\\,dx_2\\,dt\\nonumber \\\\&\\le 2|M_1|{\\mathcal {H}^{d_2-1}}(\\partial M_2)\\int _0^{\\infty }dt\\,e^{-t\\lambda _1(M_1)}\\int _0^{\\infty }dr\\int _{\\lbrace z\\in \\mathbb {R}^{d_2}:|z-x|>r\\rbrace }p_{\\mathbb {R}^{d_2}}(x,z;t)\\,dz\\nonumber \\\\&=2d_2\\omega _{d_2}|M_1|{\\mathcal {H}^{d_2-1}}(\\partial M_2)\\int _0^{\\infty }dt\\,e^{-t\\lambda _1(M_1)}(4\\pi t)^{-d_2/2}\\int _0^{\\infty }dr\\,r^{d_2}e^{-r^2/(4t)}\\nonumber \\\\&={\\mathcal {C}}_{d_2}\\lambda _1(M_1)^{-3/2}|M_1|{\\mathcal {H}^{d_2-1}}(\\partial M_2),$ where ${\\mathcal {C}}_{d_2}=\\frac{\\pi ^{1/2}d_2\\Gamma ((d_2+1)/2)}{\\Gamma ((d_2+2)/2)}.$ This concludes the proof.",
"[Proof of Proposition REF ] Let $M_1=(0,\\epsilon )\\subset \\mathbb {R}$ , $M_2=\\omega \\subset \\mathbb {R}^{d-1}$ .",
"Since the torsion function for $M_1$ is given by $x(\\epsilon -x)/2,\\ 0\\le x\\le \\epsilon $ we have that $T(M_1)=\\epsilon ^3/{12}$ .",
"Then (REF ) proves the upper bound.",
"The lower bound follows from (REF ) since $\\lambda _1(M_1)=\\pi ^2/{\\epsilon }^2$ , $|M_1|=\\epsilon $ .",
"It is of course possible, using the Faber-Krahn inequality for $\\lambda _1(M_1)$ , to obtain a bound for the right-hand side of (REF ) in terms of $|M_1|^{(d_1+3)/{d_1}}{\\mathcal {H}^{d_2-1}}(\\partial M_2)$ .",
"Our next result is an improvement of Proposition REF .",
"The torsional rigidity for a rectangle follows by substituting the formulae for $Q_{(0,a)}(t)$ and $Q_{(0,b)}(t)$ given in (REF ) below into (REF ).",
"We recover the expression given on p.108 in [31]: $T(R_{a,b})=\\frac{64ab}{\\pi ^6}\\sum _{k=1,3,\\cdots }\\sum _{l=1,3,\\cdots }k^{-2}l^{-2}\\left(\\frac{k^2}{a^2}+\\frac{l^2}{b^2}\\right)^{-1}.$ Nevertheless the following result is not immediately obvious.",
"Theorem 5.3 $\\left|T(R_{a,b})-\\frac{a^3b}{12}+\\frac{31\\zeta (5)a^4}{2\\pi ^5}\\right|\\le \\frac{a^5}{15b},$ where $\\zeta (5)=\\sum _{k=1}^{\\infty }\\frac{1}{k^5}.$ A straightforward computation using the eigenvalues and eigenfunctions of the Dirichlet Laplacian on the interval together with the first identity in (REF ) shows that $Q_{(0,a)}(t)=\\frac{8a}{\\pi ^2}\\sum _{k=1,3,\\dots }k^{-2}e^{-t\\pi ^2k^2/a^2}.$ We write $Q_{(0,b)}(t)=b-\\frac{4t^{1/2}}{\\pi ^{1/2}}+\\left(Q_{(0,b)}(t)+\\frac{4t^{1/2}}{\\pi ^{1/2}}-b\\right).$ The constant term $b$ in the right-hand side of (REF ) gives, using (REF ), a contribution $\\frac{8ab}{\\pi ^2}&\\int _{[0,\\infty )}dt\\sum _{k=1,3,\\dots }k^{-2}e^{-t\\pi ^2k^2/a^2}=\\frac{8a^3b}{\\pi ^4}\\sum _{k=1,3,\\dots }k^{-4}\\nonumber \\\\&=\\frac{8a^3b}{\\pi ^4}\\left(\\sum _{k=1}^{\\infty }k^{-4}-\\sum _{k=2,4,\\dots }k^{-4}\\right)=\\frac{15a^3b}{2\\pi ^4}\\zeta (4)\\nonumber \\\\ &= \\frac{a^3b}{12},$ which jibes with the corresponding term in (REF ).",
"In a very similar calculation we have that the $-\\frac{4t^{1/2}}{\\pi ^{1/2}}$ term in the right-hand side of (REF ) contributes $-\\frac{32a}{\\pi ^{5/2}}\\int _{[0,\\infty )}dt\\,t^{1/2}\\sum _{k=1,3,\\dots }k^{-2}e^{-t\\pi ^2k^2/a^2}=-\\frac{31\\zeta (5)a^4}{2\\pi ^5},$ which jibes with the corresponding term in (REF ).",
"It remains to bound the contribution from the expression in the large round brackets in (REF ).",
"Applying formula (REF ) to the interval $(0,b)$ instead and using the fact that $\\sum _{k=1,3,\\cdots }k^{-2}=\\pi ^2/8$ gives that $Q_{(0,b)}(t)-b+\\frac{4t^{1/2}}{\\pi ^{1/2}}&=\\frac{8b}{\\pi ^2}\\sum _{k=1,3,\\dots }k^{-2}\\left(e^{-t\\pi ^2k^2/b^2}-1\\right)+\\frac{4t^{1/2}}{\\pi ^{1/2}}\\nonumber \\\\&=-\\frac{8}{b}\\sum _{k=1,3,\\dots }\\int _{[0,t]}d\\tau e^{-\\tau \\pi ^2k^2/b^2}+\\frac{4t^{1/2}}{\\pi ^{1/2}}\\nonumber \\\\ &=-\\frac{8}{b}\\int _{[0,t]}d\\tau \\left(\\sum _{k=1}^{\\infty }e^{-\\tau \\pi ^2k^2/b^2}-\\sum _{k=1}^{\\infty }e^{-4\\tau \\pi ^2k^2/b^2}\\right)+\\frac{4t^{1/2}}{\\pi ^{1/2}}.$ In order to bound the right-hand side of (REF ) we use the following instance of the Poisson summation formula.",
"$\\sum _{k\\in \\mathbb {Z}}e^{-t\\pi k^2}=t^{-1/2}\\sum _{k\\in \\mathbb {Z}}e^{-\\pi k^2/t}, \\ t>0.$ We obtain that $\\sum _{k=1}^{\\infty }e^{-t\\pi k^2}=\\frac{1}{(4t)^{1/2}}-\\frac{1}{2}+t^{-1/2}\\sum _{k=1}^{\\infty }e^{-\\pi k^2/t},\\ t>0.$ Applying this identity twice (with $t=\\pi \\tau /b^2$ and $t=4\\pi \\tau /b^2$ respectively) gives that the right-hand side of (REF ) equals $-\\frac{8}{\\pi ^{1/2}}\\int _{[0,t]}d\\tau \\left(\\tau ^{-1/2}\\sum _{k=1}^{\\infty }e^{-k^2b^2/{\\tau }}-(4\\tau )^{-1/2}\\sum _{k=1}^{\\infty }e^{-k^2b^2/{(4\\tau )}}\\right).$ Since $k\\mapsto e^{-k^2b^2/{\\tau }}$ is non-negative and decreasing, $\\sum _{k=1}^{\\infty }\\tau ^{-1/2}e^{-k^2b^2/{\\tau }}\\le \\tau ^{-1/2}\\int _{[0,\\infty )}dke^{-k^2b^2/{\\tau }}=\\pi ^{1/2}(2b)^{-1}.$ It follows that $\\left|Q_{(0,b)}(t)-b+\\frac{4t^{1/2}}{\\pi ^{1/2}}\\right|\\le \\frac{8t}{b},\\ t>0.$ So the contribution of the third term in (REF ) to $T(R_{a,b})$ is bounded in absolute value by $\\frac{64a}{\\pi ^2b}\\int _{[0,\\infty )}dt\\,t\\sum _{k=1,3,\\dots }k^{-2}e^{-t\\pi ^2k^2/a^2}&=\\frac{64a^5}{\\pi ^6b}\\sum _{k=1,3,\\dots }k^{-6}\\nonumber \\\\&=\\frac{63a^5}{\\pi ^6b}\\zeta (6)\\nonumber \\\\&=\\frac{a^5}{15b}.$ This completes the proof of Theorem REF .",
"The Kohler-Jobin theorem mentioned in Section generalizes to $d$ -dimensions: for any open set $\\Omega $ with finite measure the ball minimizes $T(\\Omega )\\lambda _1(\\Omega )^{(d+2)/2}$ .",
"Moreover, in the spirit of Theorem REF , the following inequality is proved in [9] through an elementary heat equation proof.",
"Theorem 5.4 If $T(\\Omega )<\\infty $ then the spectrum of the Dirichlet Laplacian acting in $L^2(\\Omega )$ is discrete, and $T(\\Omega )\\ge \\left(\\frac{2}{d+2}\\right)\\left(\\frac{4\\pi d}{d+2}\\right)^{d/2}\\sum _{k=1}^{\\infty }\\lambda _k(\\Omega )^{-(d+2)/2}.$ We obtain, using the Ashbaugh-Benguria theorem (p.86 in [27]) for $\\lambda _1(\\Omega )/{\\lambda _2(\\Omega )}$ , that $T(\\Omega )\\lambda _1(\\Omega )^{(d+2)/2}\\ge \\left(\\frac{2}{d+2}\\right)\\left(\\frac{4\\pi d}{d+2}\\right)^{d/2}\\Gamma \\left(1+\\frac{d}{2}\\right)\\left(1+\\left(\\frac{\\lambda _1(B)}{\\lambda _2(B)}\\right)^{(d+2)/2}\\right).$ The constant in the right-hand side of (REF ) is for $d=2$ off by a factor $\\frac{j_{0,1}^4j_{1,1}^4}{8(j_{0,1}^4+j_{1,1}^4)}\\approx 3.62$ if compared with the sharp Kohler-Jobin constant.",
"We also note the missing factor $m^{m/(m+2)}$ in the right-hand side of (57) in [9].",
"Acknowledgements.A large part of this paper was written during a visit of the first two authors at the Isaac Newton Institute for Mathematical Sciences of Cambridge (UK).",
"GB and MvdB gratefully acknowledge the Institute for the excellent working atmosphere provided.",
"The authors also wish to thank Pedro Antunes helpful discussions.",
"The work of GB is part of the project 2010A2TFX2 “Calcolo delle Variazioni” funded by the Italian Ministry of Research and University.",
"Michiel van den Berg: School of Mathematics, University of Bristol University Walk, Bristol BS8 1TW - UK [email protected] http://www.maths.bris.ac.uk/ mamvdb/ Giuseppe Buttazzo: Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo 5, 56127 Pisa - ITALY [email protected] http://www.dm.unipi.it/pages/buttazzo/ Bozhidar Velichkov: Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo 5, 56127 Pisa - ITALY [email protected]"
]
] | 1403.0114 |
[
[
"Superspecial rank of supersingular abelian varieties and Jacobians"
],
[
"Abstract An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves.",
"In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion.",
"The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves.",
"For example, it turns out that the superspecial rank determines information about the decomposition of a supersingular abelian variety up to isomorphism; namely it is a bound for the maximal number of supersingular elliptic curves appearing in such a decomposition."
],
[
"Introduction",
"If $A$ is a principally polarized abelian variety of dimension $g$ defined over an algebraically closed field $k$ of positive characteristic $p$ , then the multiplication-by-$p$ morphism $[p]=\\operatorname{Ver}\\circ \\operatorname{Fr}$ is inseparable.",
"Typically, $A$ is ordinary in that the Verschiebung morphism $\\operatorname{Ver}$ is separable, a condition equivalent to the number of $p$ -torsion points of $A$ being $p^g$ , or the only slopes of the $p$ -divisible group of $A$ being 0 and 1, or the $p$ -torsion group scheme of $A$ being isomorphic to $({\\mathbb {Z}}/p \\oplus \\mu _p)^g$ .",
"Yet the abelian varieties which capture great interest are those which are as far from being ordinary as possible.",
"In dimension $g=1$ , an elliptic curve is supersingular if it has no points of order $p$ ; if the only slope of its $p$ -divisible group is $1/2$ ; or if its $p$ -torsion group scheme is isomorphic to the unique local-local ${\\rm BT}_1$ group scheme of rank $p^2$ , which we denote by $I_{1,1}$ .",
"These characterizations are different for a principally polarized abelian variety $A$ of higher dimension $g$ .",
"One says that $A$ has $p$ -rank 0 when $A$ has no points of order $p$ ; that $A$ is supersingular when the only slope of its $p$ -divisible group is $1/2$ ; and that $A$ is superspecial when its $p$ -torsion group scheme is isomorphic to $I_{1,1}^g$ .",
"If $A$ is supersingular, then it has $p$ -rank 0, but the converse is false for $g \\ge 3$ .",
"If $A$ is superspecial, then it is supersingular, but the converse is false for $g \\ge 2$ .",
"The Newton polygon and Ekedahl-Oort type of an abelian variety usually do not determine the decomposition of the abelian variety.",
"In fact, for any prime $p$ and formal isogeny type $\\eta $ other than the supersingular one, there exists an absolutely simple abelian variety over $k$ having Newton polygon $\\eta $ [13].",
"On the other hand, consider the following results about supersingular and superspecial abelian varieties.",
"Theorem 1.1 (Oort) Let $A/k$ be a principally polarized abelian variety.",
"Then $A$ is supersingular if and only if it is isogenous to a product of supersingular elliptic curves by [22] (which uses [33]).",
"Then $A$ is superspecial if and only if it is isomorphic to a product of supersingular elliptic curves [23], see also [18].",
"The motivation for this paper was to find ways to measure the extent to which supersingular non-superspecial abelian varieties decompose up to isomorphism.",
"The $a$ -number $a:={\\rm dim}_k {\\rm Hom}(\\alpha _p, A[p])$ gives some information about this; if $A$ has $p$ -rank 0, then the number of factors in the decomposition of $A$ up to isomorphism is bounded above by the $a$ -number, see [7].",
"However, a supersingular abelian variety with large $a$ -number could still be indecomposable up to isomorphism.",
"This paper is about another invariant of $A$ , the superspecial rank, which we define in Section REF as the number of (polarized) factors of $I_{1,1}$ appearing in the $p$ -torsion group scheme of $A$ .",
"In Proposition REF , we determine which superspecial ranks occur for supersingular abelian varieties.",
"The superspecial rank of Jacobians also has an application involving Selmer groups, see Section REF .",
"In Section , we define another invariant of $A$ , the elliptic rank, which is the maximum number of elliptic curves appearing in a decomposition of $A$ up to isomorphism.",
"In Proposition REF , we prove an observation of Oort which states that, for a supersingular abelian variety $A$ , the elliptic rank equals the number of rank 2 factors in the $p$ -divisible group $A[p^\\infty ]$ .",
"Proposition REF states that the elliptic rank is bounded by the superspecial rank for an abelian variety of $p$ -rank 0.",
"As a result, for an abelian variety $A$ of $p$ -rank zero, the superspecial rank gives an upper bound for the maximal number of dimension one factors in a decomposition of $A$ up to isomorphism; this upper bound is most interesting for supersingular abelian varieties, which decompose completely up to isogeny.",
"In Section , we apply this observation to prove some results about the superspecial rank and elliptic rank of Jacobians of curves.",
"For example, in characteristic 2, Application REF states that the superspecial rank of the Jacobian of any hyperelliptic curve of 2-rank $r$ is bounded by $1 + r$ , while its elliptic rank is bounded by $1+2r$ .",
"The superspecial ranks of all the Hermitian curves are computed in Section REF ; in particular, when $n$ is even the elliptic rank of the Hermitian curve $X_{p^n}$ is zero.",
"The authors thank the organizers of the 2013 Journées Arithmétiques, the referee for valuable comments, Ritzenthaler for help with the French abstract, and Oort for sharing the idea for Proposition REF and more generally for being a source of inspiration for this work.",
"The first-named author was partially supported by grants from the Simons Foundation (204164) and the NSA (H98230-14-1-0161 and H98230-15-1-0247).",
"The second-named author was partially supported by NSF grants DMS-11-01712 and DMS-15-02227."
],
[
"Notation",
"All geometric objects in this paper are defined over an algebraically closed field $k$ of characteristic $p>0$ .",
"Some objects are defined over the ring $W(k)$ of Witt vectors over $k$ .",
"Let $\\sigma $ denote the Frobenius automorphism of $k$ and its lift to $W(k)$ .",
"Let $A$ be a principally polarized abelian variety of dimension $g$ over $k$ .",
"Here are some relevant facts about $p$ -divisible groups and $p$ -torsion group schemes."
],
[
"The $p$ -divisible group",
"By the Dieudonné-Manin classification [16], there is an isogeny of $p$ -divisible groups $A[p^\\infty ] \\sim \\oplus _{\\lambda =\\frac{d}{c+d}} {\\widetilde{G}}_{c,d}^{m_\\lambda },$ where $(c,d)$ ranges over pairs of relatively prime nonnegative integers, and ${\\widetilde{G}}_{c,d}$ denotes a $p$ -divisible group of codimension $c$ , dimension $d$ , and thus height $c+d$ .",
"The Dieudonné module ${\\widetilde{D}}_\\lambda := {\\mathbb {D}}_*({\\widetilde{G}}_{c,d})$ (see REF below) is a free $W(k)$ -module of rank $c+d$ .",
"Over $\\operatorname{Frac}W(k)$ , there is a basis $x_1, \\ldots , x_{c+d}$ for ${\\widetilde{D}}_\\lambda $ such that $F^{d}x_i=p^c x_i$ .",
"The Newton polygon of $A$ is the data of the numbers $m_\\lambda $ ; it admits an intepretation as the $p$ -adic Newton polygon of the operator $F$ on ${\\mathbb {D}}_*(A[p^\\infty ])$ .",
"The abelian variety $A$ is supersingular if and only if $\\lambda =\\frac{1}{2}$ is the only slope of its $p$ -divisible group $A[p^\\infty ]$ .",
"Letting ${\\widetilde{I}}_{1,1}={\\widetilde{G}}_{1,1}$ denote the $p$ -divisible group of dimension 1 and height 2, one sees that $A$ is supersingular if and only $A[p^\\infty ] \\sim {\\widetilde{I}}_{1,1}^g$ ."
],
[
"The $p$ -torsion group scheme",
"The multiplication-by-$p$ morphism $[p]:A \\rightarrow A$ is a finite flat morphism of degree $p^{2g}$ .",
"The $p$ -torsion group scheme of $A$ is $A[p]=\\operatorname{Ker}[p] = \\operatorname{Ker}(\\operatorname{Ver}\\circ \\operatorname{Fr}),$ where $\\operatorname{Fr}:A \\rightarrow A^{(p)}$ denotes the relative Frobenius morphism and $\\operatorname{Ver}: A^{(p)} \\rightarrow A$ is the Verschiebung morphism.",
"In fact, $A[p]$ is a $\\operatorname{BT}_1$ group scheme as defined in [24]; it is killed by $[p]$ , with $\\operatorname{Ker}(\\operatorname{Fr}) = \\operatorname{Im}(\\operatorname{Ver})$ and $\\operatorname{Ker}(\\operatorname{Ver}) = \\operatorname{Im}(\\operatorname{Fr})$ .",
"The principal polarization on $A$ induces a principal quasipolarization on $A[p]$ , i.e., an anti-symmetric isomorphism $\\psi :A[p] \\rightarrow A[p]^D$ .",
"(This definition must be modified slightly if $p=2$ .)",
"Summarizing, $A[p]$ is a principally quasipolarized (pqp) $\\operatorname{BT}_1$ group scheme of rank $p^{2g}$ .",
"Isomorphisms classes of pqp $\\operatorname{BT}_1$ group schemes over $k$ (also known as Ekedahl-Oort types) have been completely classified [24], building on unpublished work of Kraft [12] (which did not include polarizations) and of Moonen [17] (for $p \\ge 3$ ).",
"(When $p=2$ , there are complications with the polarization which are resolved in [24].)"
],
[
"Covariant Dieudonné modules",
"The $p$ -divisible group $A[p^\\infty ]$ and the $p$ -torsion group scheme $A[p]$ can be described using covariant Dieudonné theory; see e.g., [24].",
"Briefly, let ${\\widetilde{{\\mathbb {E}}}}= {\\widetilde{{\\mathbb {E}}}}(k) = W(k)[F,V]$ denote the non-commutative ring generated by semilinear operators $F$ and $V$ with relations $ FV=VF=p, \\ F \\lambda = \\lambda ^\\sigma F, \\ \\lambda V=V \\lambda ^\\sigma ,$ for all $\\lambda \\in W(k)$ .",
"There is an equivalence of categories ${\\mathbb {D}}_*$ between $p$ -divisible groups over $k$ and ${\\widetilde{{\\mathbb {E}}}}$ -modules which are free of finite rank over $W(k)$ .",
"Similarly, let ${\\mathbb {E}}= {\\widetilde{{\\mathbb {E}}}}\\otimes _{W(k)} k$ be the reduction of the Cartier ring mod $p$ ; it is a non-commutative ring $k[F,V]$ subject to the same constraints as (REF ), except that $FV = VF = 0$ in ${\\mathbb {E}}$ .",
"Again, there is an equivalence of categories ${\\mathbb {D}}_*$ between finite commutative group schemes (of rank $2g$ ) annihilated by $p$ and ${\\mathbb {E}}$ -modules of finite dimension ($2g$ ) over $k$ .",
"If $M = {\\mathbb {D}}_*(G)$ is the Dieudonné module over $k$ of $G$ , then a principal quasipolarization $\\psi :G \\rightarrow G^D$ induces a a nondegenerate symplectic form ${\\langle \\cdot ,\\cdot \\rangle :M \\times M [r]& k}$ on the underlying $k$ -vector space of $M$ , subject to the additional constraint that, for all $x$ and $y$ in $M$ , $\\langle Fx,y \\rangle = \\langle x,Vy \\rangle ^\\sigma .$ If $A$ is the Jacobian of a curve $X$ , then there is an isomorphism of ${\\mathbb {E}}$ -modules between the contravariant Dieudonné module over $k$ of ${\\rm Jac}(X)[p]$ and the de Rham cohomology group $H^1_{\\rm dR}(X)$ by [19].",
"The canonical principal polarization on $\\operatorname{Jac}(X)$ then induces a canonical isomorphism ${\\mathbb {D}}_*(\\operatorname{Jac}(X)[p]) \\simeq H^1_{\\rm dR}(X)$ ; we will use this identification without further comment.",
"For elements $A_1, \\ldots , A_r \\in {\\mathbb {E}}$ , let ${\\mathbb {E}}(A_1, \\ldots , A_r)$ denote the left ideal $\\sum _{i=1}^r {\\mathbb {E}}A_i$ of ${\\mathbb {E}}$ generated by $\\lbrace A_i \\mid 1 \\le i \\le r\\rbrace $ ."
],
[
"The $p$ -rank and {{formula:e2799d2b-3365-42d6-991d-1d25c2935752}} -number",
"For a $\\operatorname{BT}_1$ group scheme $G/k$ , the $p$ -rank of $G$ is $f={\\rm dim}_{{\\mathbb {F}}_p} {\\rm Hom}(\\mu _p, G)$ where $\\mu _p$ is the kernel of Frobenius on ${\\mathbb {G}}_m$ .",
"Then $p^f$ is the cardinality of $G(k)$ .",
"The $a$ -number of $G$ is $a={\\rm dim}_k {\\rm Hom}(\\alpha _p, G),$ where $\\alpha _p$ is the kernel of Frobenius on ${\\mathbb {G}}_a$ .",
"It is well-known that $0 \\le f \\le g$ and $1 \\le a +f \\le g$ .",
"Moreover, since $\\mu _p$ and $\\alpha _p$ are both simple group schemes, the $p$ -rank and $a$ -number are additive; $f(G\\oplus H) = f(G)+f(H)\\text{ and }a(G\\oplus H) = a(G)+a(H).$ If ${\\widetilde{G}}$ is a $p$ -divisible group, its $p$ -rank and $a$ -number are those of its $p$ -torsion; $f({\\widetilde{G}}) = f({\\widetilde{G}}[p])$ and $a({\\widetilde{G}}) = a({\\widetilde{G}}[p])$ .",
"Similarly, if $A$ is an abelian variety, then $f(A) = f(A[p])$ and $a(A) = a(A[p])$ ."
],
[
"The Ekedahl-Oort type",
"As in [24], the isomorphism type of a pqp ${\\rm BT}_1$ group scheme $G$ over $k$ can be encapsulated into combinatorial data.",
"If $G$ is symmetric with rank $p^{2g}$ , then there is a final filtration $N_1 \\subset N_2 \\subset \\cdots \\subset N_{2g}$ of ${\\mathbb {D}}_*(G)$ as a $k$ -vector space which is stable under the action of $V$ and $F^{-1}$ such that $i={\\rm dim}(N_i)$ [24].",
"The Ekedahl-Oort type of $G$ is $\\nu =[\\nu _1, \\ldots , \\nu _g], \\ {\\rm where} \\ {\\nu _i}={\\rm dim}(V(N_i)).$ The $p$ -rank is ${\\rm max}\\lbrace i \\mid \\nu _i=i\\rbrace $ and the $a$ -number equals $g-\\nu _g$ .",
"There is a restriction $\\nu _i \\le \\nu _{i+1} \\le \\nu _i +1$ on the Ekedahl-Oort type.",
"There are $2^g$ Ekedahl-Oort types of length $g$ since all sequences satisfying this restriction occur.",
"By [24], there are bijections between (i) Ekedahl-Oort types of length $g$ ; (ii) pqp ${\\rm BT}_1$ group schemes over $k$ of rank $p^{2g}$ ; and (iii) pqp Dieudonné modules of dimension $2g$ over $k$ .",
"Example 2.1 The group scheme $I_{1,1}$ .",
"There is a unique ${\\rm BT}_1$ group scheme of rank $p^2$ which has $p$ -rank 0, which we denote $I_{1,1}$ .",
"It fits in a non-split exact sequence $0 \\rightarrow \\alpha _p \\rightarrow I_{1,1} \\rightarrow \\alpha _p \\rightarrow 0.$ The structure of $I_{1,1}$ is uniquely determined over $\\overline{{\\mathbb {F}}}_p$ by this exact sequence.",
"The image of $\\alpha _p$ is the kernel of $\\operatorname{Fr}$ and $\\operatorname{Ver}$ .",
"The Dieudonné module of $I_{1,1}$ is $M_{1,1} := {\\mathbb {D}}_*(I_{1,1}) \\simeq {\\mathbb {E}}/{\\mathbb {E}}(F+V).$ If $E$ is a supersingular elliptic curve, then the $p$ -torsion group scheme $E[p]$ is isomorphic to $I_{1,1}$ ."
],
[
"Superspecial rank",
"Let $A$ be a principally polarized abelian variety defined over an algebraically closed field $k$ of characteristic $p >0$ ."
],
[
"Superspecial",
"First, recall the definition of the superspecial property.",
"Definition 3.1 One says that $A/k$ is superspecial if it satisfies the following equivalent conditions: The $a$ -number of $A$ equals $g$ .",
"The group scheme $A[p]$ is isomorphic to $I_{1,1}^g$ .",
"The Dieudonné module over $k$ of $A[p]$ is isomorphic to $M_{1,1}^g$ .",
"$A$ is isomorphic (as an abelian variety without polarization) to the product of $g$ supersingular elliptic curves.",
"A superspecial abelian variety is defined over $\\overline{{\\mathbb {F}}}_p$ , and thus over a finite field.",
"For every $g \\in {\\mathbb {N}}$ and prime $p$ , the number of superspecial principally polarized abelian varieties of dimension $g$ defined over $\\overline{{\\mathbb {F}}}_p$ is finite and non-zero."
],
[
"Definition of superspecial rank",
"Recall (Example REF ) that the $p$ -torsion group scheme of a supersingular elliptic curve is isomorphic to $I_{1,1}$ , the unique local-local pqp ${\\rm BT}_1$ group scheme of rank $p^2$ .",
"From (REF ), it follows that $I_{1,1}$ is not simple as a group scheme.",
"However, $I_{1,1}$ is simple in the category of $\\operatorname{BT}_1$ group schemes since $\\alpha _p$ is not a $\\operatorname{BT}_1$ group scheme.",
"Definition 3.2 Let $G/k$ be a $\\operatorname{BT}_1$ group scheme.",
"A superspecial factor of $G$ is a group scheme $H \\subset G$ with $H \\simeq I_{1,1}^s$ .",
"By the equivalence of categories ${\\mathbb {D}}_*$ , superspecial factors of $G$ of rank $2s$ are in bijection with ${\\mathbb {E}}$ -submodules $N \\subset {\\mathbb {D}}_*(G)$ with $N \\simeq ({\\mathbb {E}}/{\\mathbb {E}}(F+V))^s$ ; we call such an $N$ a superspecial factor of $M={\\mathbb {D}}_*(G)$ .",
"Now suppose $(G, \\psi )/k$ is a pqp $\\operatorname{BT}_1$ group scheme.",
"A superspecial factor $H$ of $G$ is polarized if the isomorphism $\\psi : G \\rightarrow G^D$ restricts to an isomorphism $\\psi _H: H \\rightarrow G^D \\twoheadrightarrow H^D$ .",
"Equivalently, a superspecial factor $N$ of $({\\mathbb {D}}_*(G), \\langle \\cdot ,\\cdot \\rangle )$ is polarized if the nondegenerate symplectic form $\\langle \\cdot ,\\cdot \\rangle :M \\times M \\rightarrow k$ restricts to a non-degenerate symplectic form $\\langle \\cdot ,\\cdot \\rangle :N\\times N \\rightarrow k$ .",
"Definition 3.3 Let $G=(G,\\psi )/k$ be a pqp $\\operatorname{BT}_1$ group scheme.",
"The superspecial rank $s(G)$ of $G$ is the largest integer $s$ for which $G$ has a polarized superspecial factor of rank $2s$ .",
"Since $I_{1,1}$ is simple in the category of $\\operatorname{BT}_1$ group schemes, the superspecial rank $s$ has an additive property similar to that for the $p$ -rank and $a$ -number (REF ); if $G$ and $H$ are pqp $\\operatorname{BT}_1$ group schemes, then $s(G\\oplus H) = s(G)+s(H).$ A $\\operatorname{BT}_1$ group scheme $G$ may fail to be simple (i.e., admit a nontrivial $\\operatorname{BT}_1$ subgroup scheme $0\\subsetneq H \\subsetneq G$ ) and yet still be indecomposable (i.e., admit no isomorphism $G \\simeq H\\oplus K$ with $H$ and $K$ nonzero).",
"This distinction vanishes in the category of pqp $\\operatorname{BT}_1$ group schemes: Lemma 3.4 Let $G/k$ be a pqp $\\operatorname{BT}_1$ group scheme, and let $H\\subset G$ be a pqp $\\operatorname{BT}_1$ sub-group scheme.",
"Let $N = {\\mathbb {D}}_*(H) \\subseteq M = {\\mathbb {D}}_*(G)$ , and let $P$ be the orthogonal complement of $N$ in $M$ .",
"Then $P$ is a pqp sub-Dieudonné module of $M$ , and $G$ admits a decomposition $G\\simeq H\\oplus K$ as pqp $\\operatorname{BT}_1$ group schemes, where $K\\subseteq G$ is the sub-group scheme with ${\\mathbb {D}}_*(K) = P$ .",
"Lemma REF is essentially present in [12]; see, e.g., [24].",
"The $k$ -vector space $P$ is an ${\\mathbb {E}}$ -module if it is stable under $F$ and $V$ .",
"It suffices to check that, for $\\beta \\in P$ , $F \\beta \\in P$ and $V \\beta \\in P$ .",
"If $\\alpha \\in N$ , the relation (REF ) implies that $\\langle F\\beta , \\alpha \\rangle = \\langle \\beta , V \\alpha \\rangle ^\\sigma = 0^\\sigma = 0$ and $\\langle V\\beta , \\alpha \\rangle = \\langle \\beta ,F \\alpha \\rangle ^{\\sigma ^{-1}} = 0^{\\sigma ^{-1}}=0.$ Thus $F \\beta $ and $V \\beta $ are in the orthogonal complement $P$ of $N$ .",
"Since $H$ is polarized, the restriction of $\\langle \\cdot ,\\cdot \\rangle $ to $N$ is perfect and so the restriction of $\\langle \\cdot ,\\cdot \\rangle $ to $P$ is perfect as well.",
"Since ${\\mathbb {D}}_*$ is an equivalence of categories, there is a decomposition $G\\simeq H \\oplus K$ as pqp group schemes.",
"It remains to verify that $K$ is a $\\operatorname{BT}_1$ group scheme, i.e., that $\\operatorname{Ker}(\\operatorname{Fr}) = \\operatorname{Im}(\\operatorname{Ver})$ and $\\operatorname{Ker}(\\operatorname{Ver})= \\operatorname{Im}(\\operatorname{Fr})$ .",
"In terms of Dieudonné modules, this is equivalent to the property that $\\operatorname{Ker}F|_{P} = V(P)$ and $\\operatorname{Ker}V|_{P} = F(P)$ .",
"This, in turn, follows from the analogous statement for $M$ and $N$ and from the fact that the decomposition $M = N \\oplus P$ is stable under $F$ and $V$ .",
"Lemma 3.5 Let $G/k$ be a pqp $\\operatorname{BT}_1$ group scheme of $p$ -rank $f$ and $a$ -number $a$ , and let $H\\subset G$ be a maximal polarized superspecial factor.",
"Then $G \\simeq H \\oplus K$ for a pqp $\\operatorname{BT}_1$ group scheme $K$ with respective $p$ -rank, superspecial rank and $a$ -number $f(K)=f$ , $s(K) =0$ , and $a(K) = a-s$ .",
"The existence of the decomposition $G\\simeq H\\oplus K$ follows from Lemma REF ; the assertions about the $p$ -rank, superspecial rank and $a$ -number of $K$ follow from the additivity of these quantities, (REF ) and (REF ).",
"Since one can always canonically pull off the étale and toric components of a finite group scheme over a perfect field, Lemma REF admits a further refinement: Lemma 3.6 Let $G/k$ be a pqp $\\operatorname{BT}_1$ group scheme with $f(G)=f$ , $s(G) = s$ , and $a(G) = a$ .",
"Then there is a local-local pqp $\\operatorname{BT}_1$ group scheme $B$ such that $G \\simeq ({\\mathbb {Z}}/p\\oplus \\mu _p)^f \\oplus I_{1,1}^s \\oplus B$ where $f(B) = s(B) = 0$ and $a(B) = a-s$ .",
"Since $k$ is perfect and $G$ is self-dual, there is a canonical decomposition of pqp group schemes $G \\simeq ({\\mathbb {Z}}/p\\oplus \\mu _p)^f \\oplus H$ .",
"Then $f(H) = 0$ , $s(H) = s(G)$ , and $a(H) = a(G)$ .",
"Now invoke Lemma REF .",
"Let $A$ be a principally polarized abelian variety of dimension $g$ .",
"On one hand, $A$ is superspecial if and only if $s(A[p]) = g$ .",
"On the other hand, if $A$ is ordinary, then $s(A[p]) =0$ .",
"More generally: Lemma 3.7 Let $G/k$ be a pqp $\\operatorname{BT}_1$ group scheme of rank $p^{2g}$ ; let $f = f(G)$ , $a=a(G)$ , and $f = f(G)$ .",
"Then $0 \\le s \\le a \\le g-f$ .",
"If $a = g-f$ , then $G \\simeq ({\\mathbb {Z}}/p\\oplus \\mu _p)^f \\oplus I_{1,1}^a$ and $s=a$ .",
"If $a\\ne g-f$ , then $s<a$ .",
"Write $G \\simeq ({\\mathbb {Z}}/p \\oplus \\mu _p)^f \\oplus B_1$ with $B_1 \\simeq I_{1,1}^s \\oplus B$ as in Lemma REF .",
"Then $a \\le g-f$ , since (using additivity) $a(G) = a(B_1)$ , and $B_1$ has rank $p^{2(g-f)}$ .",
"Moreover, $s \\le a$ since $a(I_{1,1}^s) = s$ .",
"This is true since the only pqp ${\\rm BT}_1$ group scheme of rank $p^{2(g-f)}$ with $p$ -rank 0 and $a$ -number $g-f$ is $I_{1,1}^{g-f}$ , which has superspecial rank $g-f$ by definition.",
"The hypothesis $a \\ne g-f$ implies that $B$ is non-trivial.",
"Then $a > s$ since the $a$ -number of the local-local group scheme $B$ is at least 1."
],
[
"Unpolarized superspecial rank",
"If $G/k$ is a $\\operatorname{BT}_1$ group scheme, or indeed any $p$ -torsion finite commutative group scheme, then there is also an obvious notion of an unpolarized superspecial rank, namely, the largest $u$ such that there is an inclusion $I_{1,1}^{u}\\hookrightarrow G$ .",
"In this section, we briefly explore some of the limitations of this notion.",
"For integers $r,s \\ge 1$ , let $J_{r,s}$ be the $\\operatorname{BT}_1$ group scheme with Dieudonné module $M_{r,s} := {\\mathbb {D}}_*(J_{r,s}) = {\\mathbb {E}}/{\\mathbb {E}}(F^r+V^s).$ Lemma 3.8 Suppose $r,s \\ge 2$ .",
"Then $J_{r,s}$ is an indecomposable local-local $\\operatorname{BT}_1$ group scheme.",
"There exists an inclusion $\\iota :I_{1,1} \\hookrightarrow J_{r,s}$ .",
"Part (a) is standard.",
"Indeed, using the relations $F^r = -V^s$ and $FV = VF = p$ , one sees that $F$ and $V$ act nilpotently, and thus $J_{r,s}$ is local-local; in particular, it has $p$ -rank zero.",
"Note that $M_{r,s}$ is generated over ${\\mathbb {E}}$ by a single element $x$ such that $F^r x = -V^sx$ .",
"It follows that $a(J_{r,s}) = 1$ .",
"The additivity relation (REF ) now implies that $M_{r,s}$ is indecomposable.",
"For (b), let $y \\in M_{r,s}$ be an element such that $Fy = - Vy \\ne 0$ .",
"Since $r,s \\ge 2$ , the element $y= F^{r-1}x+V^{s-1}x$ is suitable.",
"Then there is an inclusion $\\iota _*:M_{1,1} \\rightarrow M_{r,s}$ which sends a generator of $M_{1,1}$ to $y$ .",
"If $r = s$ , then $J_{r,s}$ is self-dual and admits a principal quasipolarization; in this case, let $H_{r,s} = J_{r,s}$ .",
"If $r\\ne s$ , then the Cartier dual of $J_{r,s}$ is $J_{s,r}$ ; in this case, $H_{r,s}:= J_{r,s} \\oplus J_{s,r}$ admits a principal quasipolarization.",
"In spite of Lemma REF , we find: Lemma 3.9 Suppose $r, s \\ge 2$ .",
"For any principal quasipolarization on $H_{r,s}$ , the superspecial rank of $H_{r,s}$ is zero.",
"If $r=s$ , this is immediate, since $H_{r,r}$ is indecomposable by Lemma REF and yet a polarized superspecial factor of positive rank would induce a factorization by Lemma REF .",
"Now suppose $r\\ne s$ .",
"The argument used in the classification of polarizations on superspecial $p$ -divisible groups in [15] shows that for some $u \\in \\left\\lbrace 1,2\\right\\rbrace $ , there exists an inclusion $\\iota :I_{1,1}^u \\hookrightarrow H_{r,s}$ with $G := \\iota (I_{1,1}^u)$ polarized.",
"If $G$ is contained in either $J_{r,s}$ or $J_{s,r}$ (and in particular if $u=1$ ), then we may argue as before.",
"Otherwise, consider the sum of $G$ and $J_{r,s}$ inside $H_{r,s}$ , which is not direct since $G\\cap J_{r,s} \\simeq I_{1,1}$ is nonempty.",
"By Lemma REF , $G$ has a complement $K$ in $G+J_{r,s}$ .",
"Then $J_{r,s} \\simeq I_{1,1}\\oplus K$ , contradicting the indecomposability of $J_{r,s}$ ."
],
[
"Superspecial ranks of abelian varieties",
"If $A/k$ is a principally polarized abelian variety, we define its superspecial rank to be that of its $p$ -torsion group scheme; $s(A) = s(A[p])$ .",
"Lemma REF gives constraints between the $p$ -rank, $a$ -number, and superspecial rank of $A$ .",
"It turns out that these are the only constraints on $f$ , $a$ and $s$ : Proposition 3.10 Given integers $g,f,a,s$ such that $0 \\le s < a < g-f$ , there exists a principally polarized abelian variety $A/k$ of dimension $g$ with $p$ -rank $f$ , $a$ -number $a$ and superspecial rank $s$ .",
"By [24], it suffices to show that there exists a pqp ${\\rm BT}_1$ group scheme $G$ of rank $p^{2g}$ with $p$ -rank $f$ , $a$ -number $a$ and superspecial rank $s$ .",
"Set $g_1=g-f-s, \\ {\\rm and} \\ a_1=a-s,$ and note that $a_1 \\ge 1$ and $g_1-a_1 \\ge 1$ by hypothesis.",
"Considering $G = ({\\mathbb {Z}}/p \\oplus \\mu _p)^f \\oplus I_{1,1}^s \\oplus B,$ together with the product polarization, allows one to reduce to the case of finding a pqp ${\\rm BT}_1$ group scheme $B$ of rank $p^{2g_1}$ with $p$ -rank 0, $a$ -number $a_1$ and superspecial rank 0.",
"This is possible as follows.",
"Consider the word $w$ in $F$ and $V$ given by $w=F^{g_1-a_1+1}(VF)^{a_1-1}V^{g_1-a_1+1}.$ Then $w$ is simple and symmetric with length $2g_1$ , and thus the corresponding $\\operatorname{BT}_1$ group scheme admits a canonical principal quasipolarization [24].",
"Let $L_1, \\ldots , L_{2g_1} \\in \\lbrace F, V\\rbrace $ be such that $w=L_1 \\cdots L_{2g_1}$ .",
"Consider variables $z_1, \\ldots , z_{2g_1}$ with $z_{2g_1+1}=z_1$ .",
"As in [24], the word $w$ defines the structure of a Dieudonné module on $N_w=\\oplus _{i} k \\cdot z_i$ as follows: if $L_i=F$ , let $F(z_i)=z_{i+1}$ and $V(z_{i+1})=0$ ; if $L_i=V$ , let $V(z_{i+1})=z_i$ and $F(z_i)=0$ .",
"The $a$ -number is the number of generators for $N_w$ as an ${\\mathbb {E}}$ -module.",
"By construction, $N_w$ has $a$ -number $a_1$ .",
"Since $g_1-a_1+1 \\ge 2$ , then $N_w$ has superspecial rank 0.",
"We now focus on supersingular abelian varieties Lemma 3.11 For every $g \\ge 2$ and prime $p$ , a generic supersingular principally polarized abelian variety of dimension $g$ over $k$ has superspecial rank 0.",
"A generic supersingular principally polarized abelian variety has $p$ -rank 0 and $a$ -number 1 [15].",
"This forces its Ekedahl-Oort type to be $[0,1, \\ldots , g-1]$ , its Dieudonné module to be $M_{g,g}$ , and its superspecial rank to be zero (Lemma REF ) since $g \\ge 2$ .",
"It is not difficult to classify the values of the supersingular rank which occur for supersingular abelian varieties.",
"Proposition 3.12 For every $g \\ge 2$ and prime $p$ , there exists a supersingular principally polarized abelian variety of dimension $g$ over $k$ with superspecial rank $s$ if and only if $0 \\le s \\le g-2$ or $s=g$ .",
"It is impossible for the superspecial rank to be $g-1$ since there are no local-local pqp ${\\rm BT}_1$ group schemes of rank $p^2$ other than $I_{1,1}$ .",
"For the reverse implication, recall that there exists a supersingular principally polarized abelian variety $A_1/k$ of dimension $g-s$ with $a=1$ .",
"Its Dieudonné module is $M_{g-s,g-s}$ .",
"In particular, $s(A_1)=0$ as long as $s \\le g-2$ (Lemma REF ).",
"Let $E$ be a supersingular elliptic curve.",
"Then $A=E^s \\times A_1$ , together with the product polarization, is a supersingular principally polarized abelian variety over $k$ with dimension $g$ and $s(A)=s$ .",
"Example 3.13 Let $A/k$ be a supersingular principally polarized abelian variety of dimension 3.",
"Then the $a$ -number $a=a(A)$ satisfies $1 \\le a \\le 3$ .",
"If $a=1$ , then $A[p] \\simeq J_{3,3}$ , which has superspecial rank $s=0$ .",
"If $a=2$ , then $A[p^\\infty ] \\simeq {\\widetilde{G}}_{1,1} \\times {\\widetilde{Z}}$ where ${\\widetilde{Z}}$ is supergeneral of height 4 and $a({\\widetilde{Z}})=1$ [20].",
"Then $s({\\widetilde{Z}}[p]) = 0$ (Lemma REF (c)) and thus $s(A)=1$ .",
"If $a=3$ , then $A$ has superspecial rank $s=3$ ."
],
[
"Application of superspecial rank to Selmer groups",
"Here is another motivation for studying the superspecial rank of Jacobians.",
"The superspecial rank equals the rank of the Selmer group associated with a particular isogeny of function fields in positive characteristic.",
"Let $K$ be the function field of a smooth projective connected curve $X$ over $k$ .",
"Let ${\\mathcal {E}}$ be a constant supersingular elliptic curve over $K$ .",
"Consider the multiplication-by-$p$ isogeny $f=[p]: {\\mathcal {E}} \\rightarrow {\\mathcal {E}}$ of abelian varieties over $K$ .",
"Recall the Tate-Shafarevich group ${\\mbox{{\\fontencoding {OT2}\\fontfamily {wncyr}\\fontseries {m}\\fontshape {n}\\selectfont Sh}}}(K, {\\mathcal {E}})_f={\\rm Ker}({\\mbox{{\\fontencoding {OT2}\\fontfamily {wncyr}\\fontseries {m}\\fontshape {n}\\selectfont Sh}}}(K, {\\mathcal {E}}) \\stackrel{f}{\\rightarrow } {\\mbox{{\\fontencoding {OT2}\\fontfamily {wncyr}\\fontseries {m}\\fontshape {n}\\selectfont Sh}}}(K, {\\mathcal {E}})),$ where ${\\mbox{{\\fontencoding {OT2}\\fontfamily {wncyr}\\fontseries {m}\\fontshape {n}\\selectfont Sh}}}(K,{\\mathcal {E}})={\\rm Ker}(H^1(K, {\\mathcal {E}}) \\rightarrow \\prod _{v} H^1(K_v, {\\mathcal {E}}))$ and $v$ runs over all places of $K$ .",
"The Selmer group ${\\rm Sel}(K, f)$ is the subset of elements of $H^1(K, {\\rm Ker}(f))$ whose restriction is in the image of ${\\rm Sel}(K_v, f) = {\\rm Im}({\\mathcal {E}}(K_v) \\rightarrow H^1(K_v, {\\rm Ker}(f))),$ for all $v$ .",
"There is an exact sequence $0 \\rightarrow {\\mathcal {E}}(K)/f({\\mathcal {E}}(K)) \\rightarrow {\\rm Sel}(K,f) \\rightarrow {\\mbox{{\\fontencoding {OT2}\\fontfamily {wncyr}\\fontseries {m}\\fontshape {n}\\selectfont Sh}}}(K, {\\mathcal {E}})_f \\rightarrow 0.$ Here is an earlier result, rephrased using the terminology of this paper, which provides motivation for studying the superspecial rank.",
"Theorem 3.14 (Ulmer) The rank of ${\\rm Sel}(K, [p])$ is the superspecial rank of ${\\rm Jac}(X)$ [34]."
],
[
"Elliptic curve summands of abelian varieties",
"Let $A/k$ be a principally polarized abelian variety of dimension $g$ .",
"In this section, we define the elliptic rank of $A$ to be the maximum number of elliptic curves appearing in a decomposition of $A$ up to isomorphism.",
"When $A$ has $p$ -rank 0, the elliptic rank is bounded by the superspecial rank, Proposition REF .",
"Proposition REF states that the elliptic rank is the number of rank 2 factors in the $p$ -divisible group $A[p^\\infty ]$ when $A$ is supersingular."
],
[
"Elliptic rank",
"Definition 4.1 The elliptic rank $e(A)$ of $A$ is $e(A):={\\rm max} \\lbrace e \\mid \\iota : A \\stackrel{\\simeq }{\\rightarrow } A_1 \\times (\\times _{i=1}^e E_i)\\rbrace ,$ where $E_1, \\ldots , E_e$ are elliptic curves, $A_1$ is an abelian variety of dimension $g-e$ , and $\\iota $ is an isomorphism of abelian varieties over $k$ .",
"(We remind the reader that many “cancellation problems” for abelian varieties have negative answers [31], and that the abelian variety $A_1$ in (REF ) is not necessarily unique.)",
"Here are some properties of the elliptic rank.",
"Proposition 4.2 If $A$ has $p$ -rank 0, then the elliptic rank is bounded by the superspecial rank: $e(A) \\le s(A)$ .",
"If $A$ has $p$ -rank 0, then the elliptic curves $E_1, \\ldots , E_e$ in a maximal decomposition of $A$ are supersingular.",
"Each supersingular curve in the decomposition contributes a factor of ${\\mathbb {E}}/{\\mathbb {E}}(F+V)$ to the Dieudonné module ${\\mathbb {D}}_*(A[p])$ .",
"The proof of Proposition REF shows that, for every $g \\ge 2$ and prime $p$ , there exists a supersingular principally polarized abelian variety of dimension $g$ over $k$ with elliptic rank $e$ if and only if $0 \\le e \\le g-2$ or $e=g$ .",
"Remark 4.3 It is clear that $e(A) =0$ if $A$ is simple and ${\\rm dim}(A)>1$ .",
"Recall from [13] that there exists a simple abelian variety $A$ with formal isogeny type $\\eta $ , for each non-supersingular Newton polygon $\\eta $ .",
"It follows from Proposition REF that there exist abelian varieties $A$ with $s(A) > 0$ and $e(A) =0$ for all dimensions $g \\ge 4$ ."
],
[
"Superspecial rank for $p$ -divisible groups",
"We briefly sketch a parallel version of superspecial rank in the category of $p$ -divisible groups, rather than $p$ -torsion group schemes.",
"Many of the notions and results in Section REF generalize to truncated Barsotti-Tate groups of arbitrary level, and indeed to Barsotti-Tate, or $p$ -divisible, groups.",
"Let ${\\widetilde{G}}$ be a pqp $p$ -divisible group, and let ${\\widetilde{H}} \\subseteq {\\widetilde{G}}$ be a sub-$p$ -divisible group.",
"We say that ${\\widetilde{H}}$ is polarized if the principal quasipolarization on ${\\widetilde{G}}$ restricts to one on ${\\widetilde{H}}$ .",
"Lemma REF admits an analogue for $p$ -divisible groups; for such an ${\\widetilde{H}}$ , there exists a pqp complement ${\\widetilde{K}}$ such that ${\\widetilde{G}} \\simeq {\\widetilde{H}} \\oplus {\\widetilde{K}}$ .",
"Let ${\\widetilde{I}}_{1,1}$ be the $p$ -divisible group whose Dieudonné module is ${\\widetilde{M}}_{1,1} = {\\mathbb {D}}_*({\\widetilde{I}}_{1,1}) \\simeq {\\widetilde{{\\mathbb {E}}}}/{\\widetilde{{\\mathbb {E}}}}(F+V);$ then ${\\widetilde{I}}_{1,1}[p] \\simeq I_{1,1}$ .",
"With this preparation, we define the superspecial rank ${\\widetilde{s}}({\\widetilde{G}})$ of a pqp $p$ -divisible group ${\\widetilde{G}}$ as the largest value of $s$ for which there exists a sub-pqp $p$ -divisible group of ${\\widetilde{G}}$ isomorphic to ${\\widetilde{I}}_{1,1}^{s}$ .",
"Since a decomposition of a $p$ -divisible group induces a decomposition on its finite levels, it follows that ${\\widetilde{s}}({\\widetilde{G}}) \\le s({\\widetilde{G}}[p]).$ Similarly, if $A/k$ is a principally polarized abelian variety, then any decomposition of $A$ induces a decomposition of its $p$ -divisible group.",
"So if $A$ has $p$ -rank 0, then $e(A) \\le {\\widetilde{s}}(A[p^\\infty ]).$ We thank Oort for suggesting the following result: Proposition 4.4 Let $A/k$ be a supersingular principally polarized abelian variety.",
"Then $e(A) = {\\widetilde{s}}(A[p^\\infty ]).$ Let ${\\widetilde{M}}$ be the Dieudonné module ${\\widetilde{M}}= {\\mathbb {D}}_*(A[p^\\infty ])$ , and let $E/k$ be a supersingular elliptic curve.",
"Since $A$ is principally polarized, ${\\widetilde{M}}$ is principally quasipolarized.",
"Let ${\\widetilde{s}} = {\\widetilde{s}}(A[p^\\infty ])$ .",
"By the same proof as for Lemma REF , there is a decomposition of pqp Dieudonné modules ${\\widetilde{M}} \\simeq {\\widetilde{M}}_{1,1}^{{\\widetilde{s}}} \\oplus {\\widetilde{N}},$ where ${\\widetilde{N}}$ has superspecial rank zero.",
"By [21], since ${\\widetilde{M}}$ is supersingular, (REF ) induces a corresponding decomposition $A \\simeq E^{{\\widetilde{s}}} \\oplus A_1.$ where $A_1$ is a principally polarized abelian variety of dimension $g-{\\widetilde{s}}$ with ${\\widetilde{s}}(A_1[p^\\infty ]) = 0$ .",
"Thus $e(A) \\ge {\\widetilde{s}}$ and the result follows.",
"Remark 4.5 In fact, it is not hard to give a direct proof that the existence of decomposition (REF ) implies the existence of (REF ).",
"Indeed, since $A$ is supersingular, there exists an isogeny $\\psi :E^g \\rightarrow A$ , which induces an isogeny of $p$ -divisible groups $\\psi [p^\\infty ]: {\\widetilde{I}}_{1,1}^g \\rightarrow A[p^\\infty ]$ .",
"Let $H = \\operatorname{Ker}(\\psi [p^\\infty ])$ ; it is a finite group scheme, and is thus also a sub-group scheme of $E^g$ .",
"Since $\\operatorname{End}({\\widetilde{I}}_{1,1})$ is a maximal order in a division ring over ${\\mathbb {Z}}_p$ , it is a (noncommutative) principal ideal domain (see also [14]).",
"By the theory of elementary divisors for such rings (e.g., [10]), there is an isomorphism ${\\widetilde{I}}_{1,1}^g \\simeq {\\widetilde{I}}_{1,1}^{{\\widetilde{s}}} \\times {\\widetilde{I}}_{1,1}^{g-{\\widetilde{s}}}$ under which $H$ is contained in $0\\times {\\widetilde{I}}_{1,1}^{g-{\\widetilde{s}}}$ .",
"Since $\\operatorname{End}(E^g[p^n]) \\simeq \\operatorname{End}({\\widetilde{I}}_{1,1}^g [p^n])$ for each $n \\in {\\mathbb {N}}$ , there is an analogous decomposition $E^g \\simeq E^{{\\widetilde{s}}} \\times E^{g-{\\widetilde{s}}}$ under which $H$ is contained in $0\\times E^{g-{\\widetilde{s}}}$ .",
"Then $A =E^g/N \\simeq E^{{\\widetilde{s}}} \\oplus A_1$ , where $A_1$ is supersingular but has superspecial rank zero."
],
[
"An open question",
"Consider a principally polarized abelian variety $A/k$ .",
"By Remark REF , if $A$ is not supersingular, then it can be absolutely simple ($e(A)=0$ ) and yet have positive superspecial rank ($s(A)>0$ ).",
"(Similarly, if $A$ admits ordinary elliptic curves as factors, then it is possible to have $e(A)>0$ while $s(A)=0$ .)",
"However, if $A$ has $p$ -rank 0, there are a priori inequalities $e(A) \\le {\\widetilde{s}}(A[p^\\infty ]) \\le s(A[p]).$ Proposition REF shows the first inequality is actually an equality when $A$ is supersingular.",
"This leads one to ask the following: Question 4.6 If $A/k$ is supersingular, is $e(A)=s(A)$ ?",
"If ${\\widetilde{G}}$ is a supersingular pqp $p$ -divisible group, is ${\\widetilde{s}}({\\widetilde{G}})=s({\\widetilde{G}}[p])$ ?",
"The two parts of Question REF have the same answer by Proposition REF .",
"Here is one difficulty in answering this question.",
"Remark 4.7 The $p$ -divisible group ${\\widetilde{I}}_{1,1}$ is isomorphic (over $k$ ) to the $p$ -divisible group $H_{1,1}$ introduced in [11].",
"Consequently, it is minimal in the sense of [26]; if ${\\widetilde{M}}$ is any Dieudonné module such that $({\\widetilde{M}} \\otimes _W k) \\simeq M_{1,1}^{\\oplus s}$ , then there is an isomorphism ${\\widetilde{M}} \\simeq {\\widetilde{M}}_{1,1}^{\\oplus s}$ .",
"In spite of this, because of difficulties with extensions (see, e.g., [26]), one cannot immediately conclude that ${\\widetilde{M}}$ admits ${\\widetilde{M}}_{1,1}^{\\oplus s}$ as a summand if ${\\widetilde{M}}/p {\\widetilde{M}}$ has superspecial rank $s$ .",
"Indeed, Lemma REF indicates that an appeal to minimality alone is insufficient; any argument must make use of the principal quasipolarization."
],
[
"Superspecial rank of supersingular Jacobians",
"If $X/k$ is a (smooth, projective, connected) curve, its superspecial and elliptic ranks are those of its Jacobian: $s(X)=s({\\rm Jac}(X))$ and $e(X)=e({\\rm Jac}(X))$ .",
"In this section, we address the question of which superspecial ranks occur for Jacobians of (supersingular) curves.",
"First, recall that there is a severe restriction on the genus of a superspecial curve.",
"Theorem 5.1 (Ekedahl) If $X/k$ is a superspecial curve of genus $g$ , then $g \\le p(p-1)/2$ [5], see also [1].",
"For example, if $p=2$ , then the genus of a superspecial curve is at most 1.",
"The Hermitian curve $X_p: y^p+y=x^{p+1}$ is a superspecial curve realizing the upper bound of Theorem REF .",
"In Section REF , we determine the superspecial ranks of all hyperelliptic curves in characteristic 2.",
"We determine the superspecial rank of the Jacobians of Hermitian curves in Section REF .",
"In both cases, this gives an upper bound for the elliptic rank."
],
[
"Supersingular Jacobians",
"Recall that a curve $X/{\\mathbb {F}}_q$ is supersingular if the Newton polygon of $L(X/{\\mathbb {F}}_q, t)$ is a line segment of slope $1/2$ or, equivalently, if the Jacobian of $X$ is supersingular.",
"One thing to note is that a curve $X/{\\mathbb {F}}_q$ is supersingular if and only if $X$ is minimal over ${\\mathbb {F}}_{q^c}$ for some $c \\in {\\mathbb {N}}$ .",
"Van der Geer and Van der Vlugt proved that there exists a supersingular curve of every genus in characteristic $p=2$ [36].",
"For $p \\ge 3$ , it is unknown if there exists a supersingular curve of every genus.",
"An affirmative answer would follow from a conjecture about deformations of reducible supersingular curves [25].",
"There are many constructions of supersingular curves having arbitrarily large genus.",
"Recall (from proof of Proposition REF and remarks after Proposition REF ) that there exists a (non-simple) supersingular principally polarized abelian variety of dimension $g$ over $k$ with elliptic rank $e$ if and only if $0 \\le e \\le g-2$ or $e=g$ .",
"In light of this, one can ask the following question.",
"Question 5.2 Given $p$ prime and $g \\ge 2$ and $0 \\le s \\le g-2$ , does there exist a smooth curve $X$ over $\\overline{{\\mathbb {F}}}_p$ of genus $g$ whose Jacobian is supersingular and has elliptic rank $e$ ?",
"The answer to Question REF is yes when $g=2,3$ and $e=0$ .",
"To see this, recall from the proof of Lemma REF that a generic supersingular principally polarized abelian variety of dimension $g$ has Dieudonné module ${\\mathbb {E}}/{\\mathbb {E}}(F^g+V^g)$ , which has superspecial rank $s=0$ .",
"When $g=2,3$ , such an abelian variety is the Jacobian of a smooth curve with $e=0$ .",
"One expects the answer to Question REF is yes when $g=3$ and $e=1$ also.",
"To see this, let $E$ be a supersingular elliptic curve.",
"Let $A$ be a supersingular, non-superspecial abelian surface.",
"The 3-dimensional abelian variety $B= A \\times E$ is supersingular and has superspecial rank 1.",
"If there is a principal polarization on $B$ which is not the product polarization, then $B$ is the Jacobian of a smooth curve.",
"Question REF is open for $g\\ge 4$ ."
],
[
"Superspecial rank of hyperelliptic curves when $p=2$",
"In this section, suppose $k$ is an algebraically closed field of characteristic $p=2$ .",
"Application REF states that the superspecial rank of a hyperelliptic curve over $k$ with 2-rank 0 is either 0 or 1.",
"More generally, Application REF states that the superspecial rank of a hyperelliptic curve over $k$ with 2-rank $r$ is bounded by $1+r$ .",
"A hyperelliptic curve $Y$ over $k$ is defined by an Artin-Schreier equation $y^2+y=h(x),$ for some non-constant rational function $h(x) \\in k(x)$ .",
"In [6], the authors determine the structure of the Dieudonné module $M$ of ${\\rm Jac}(Y)$ for all hyperelliptic curves $Y$ in characteristic 2.",
"A surprising feature is that the isomorphism class of $M$ depends only on the orders of the poles of $h(x)$ , and not on the location of the poles or otherwise on the coefficients of $h(x)$ .",
"In particular, consider the case that the 2-rank of $Y$ is 0, or equivalently, that $h(x)$ has only one pole.",
"In this case, the Ekedahl-Oort type is $[0,1,1,2,2, \\ldots , \\lfloor \\frac{g}{2} \\rfloor ]$ [6].",
"The $a$ -number is $\\lceil \\frac{g}{2} \\rceil $ .",
"Application 5.3 Let $Y$ be a hyperelliptic curve of genus $g$ with 2-rank 0 defined over an algebraically closed field of characteristic 2.",
"Then the superspecial rank of ${\\rm Jac}(Y)$ is $s=1$ if $g \\equiv 1 \\pmod {3}$ and is $s=0$ otherwise.",
"The elliptic rank of ${\\rm Jac}(Y)$ is $e \\le 1$ if $g \\equiv 1 \\pmod {3}$ and $e=0$ otherwise.",
"This follows by applying the algorithm in [6].",
"Specifically, by [6] (where $c=g$ ), the Dieudonné module of the group scheme with Ekedahl-Oort type $[0,1,1,2,2, \\ldots , \\lfloor \\frac{g}{2} \\rfloor ]$ is generated by variables $X_j$ for $\\lceil (g+1)/2 \\rceil \\le j \\le g$ subject to the relations $F^{e(j)+1}(X_j)+V^{\\epsilon (\\iota (j))+1}(X_{\\iota (j)})$ , where the notation is defined in [6].",
"Then $M_{1,1}$ occurs as a summand if and only if there is some $j$ such that $e(j)=\\epsilon (j)=0$ and $j=\\iota (j)$ .",
"The condition $e(j)=0$ is equivalent to $j$ being odd.",
"The conditions $\\epsilon (j)=0$ and $j=\\iota (j)$ imply that $2g-2j+1=g-(j-1)/2$ which is possible only if $g \\equiv 1 \\bmod 3$ .",
"If $g \\equiv 1 \\bmod 3$ , then $j=(2g+1)/3$ so the maximal rank of a summand isomorphic to $I_{1,1}^s$ is $s=1$ .",
"Remark 5.4 It is not known exactly which natural numbers $g$ can occur as the genus of a supersingular hyperelliptic curve over $\\overline{{\\mathbb {F}}}_2$ .",
"On one hand, if $g=2^s-1$ , then there does not exist a supersingular hyperelliptic curve of genus $g$ over $\\overline{{\\mathbb {F}}}_2$ [29].",
"On the other hand, if $h(x)=xR(x)$ for an additive polynomial $R(x)$ of degree $2^{s}$ , then $Y$ is supersingular of genus $2^{s-1}$ [35].",
"If $s$ is even, then Application REF shows that ${\\rm Jac}(Y)$ has no elliptic curve factors in a decomposition up to isomorphism, even though it decomposes completely into elliptic curves up to isogeny.",
"More generally, we now determine the superspecial ranks of hyperelliptic curves in characteristic 2 having arbitrary 2-rank.",
"Consider the divisor of poles ${\\rm div}_\\infty (h(x)) = \\sum _{j=0}^{r} d_j P_j.$ By Artin-Schreier theory, one can suppose that $d_j$ is odd for all $j$ .",
"Then ${\\rm Jac}(Y)$ has genus $g$ satisfying $2g+2=\\sum _{j=0}^r (d_j+1)$ by the Riemann-Hurwitz formula [30] and has 2-rank $f=r$ by the Deuring-Shafarevich formula [32] or [2].",
"These formulae imply that, for a given genus $g$ (and 2-rank $r$ ), there is another discrete invariant of a hyperelliptic curve $Y/k$ , namely a partition of $2g+2$ into $r+1$ positive even integers $d_j + 1$ .",
"In [6], the authors prove that the Ekedahl-Oort type of $Y$ depends only on this discrete invariant.",
"Specifically, consider the variable $x_j:=(x-P_j)^{-1}$ , which is the inverse of a uniformizer at the branch point $P_j$ in ${\\mathbb {P}}^1$ (with $x_j=x$ if $P_j = \\infty $ ).",
"Then $h(x)$ has a partial fraction decomposition of the form $h(x)=\\sum _{j=0}^r h_{j} \\big (x_j\\big ),$ where $h_j(x) \\in k[x]$ is a polynomial of degree $d_j$ .",
"Let $c_j=(d_j-1)/2$ and note that $g=r+\\sum _{j=0}^r c_j$ .",
"For $0 \\le j \\le r$ , consider the Artin-Schreier $k$ -curve $Y_j$ with affine equation $y^2 - y = h_j(x)$ .",
"Let $E_0$ be an ordinary elliptic curve over $k$ .",
"Then [6] states that the de Rham cohomology of $Y$ decomposes, as a module under the actions of Frobenius $F$ and Verschiebung $V$ , as: $H^1_{\\rm dR}(Y) \\simeq H^1_{\\rm dR}(E_0)^{r} \\oplus \\bigoplus _{j=0}^r H^1_{\\rm dR}(Y_j).$ Since $E_0$ is ordinary, it has superspecial rank 0.",
"The superspecial rank of ${\\rm Jac}(Y)$ is thus the sum of the superspecial ranks of ${\\rm Jac}(Y_j)$ .",
"Applying Application REF to $\\lbrace Y_j\\rbrace _{j=0}^r$ proves the following.",
"Application 5.5 Consider a hyperelliptic curve $Y$ defined over an algebraically closed field of characteristic 2.",
"Then $Y$ is defined by an equation of the form $y^2+y=h(x)$ with ${\\rm div}_\\infty (h(x)) = \\sum _{j=0}^{r} d_j P_j$ and $d_j$ odd.",
"Recall that $Y$ has genus $g=r+\\sum _{j=0}^r c_j$ where $c_j=(d_j-1)/2$ and $p$ -rank $r$ .",
"The superspecial rank of ${\\rm Jac}(Y)$ equals the number of $j$ such that $c_j \\equiv 1 \\bmod 3$ .",
"In particular, $s({\\rm Jac}(Y)) \\le 1 + r$ and $e(({\\rm Jac}(Y))) \\le 1 + 2r$ ."
],
[
"Hermitian curves",
"The last examples of the paper are about the superspecial rank for one of the three classes of (supersingular) Deligne-Lusztig curves: the Hermitian curves $X_q$ for $q=p^n$ for an arbitrary prime $p$ .",
"In most cases, the superspecial (and elliptic) ranks are quite small, which is somewhat surprising since these curves are exceptional from many perspectives.",
"Let $q=p^n$ .",
"The Hermitian curve $X_q$ has affine equation $y^q + y = x^{q+1}.$ It is supersingular with genus $g=q(q-1)/2$ .",
"It is maximal over ${\\mathbb {F}}_{q^2}$ because $\\#X_q\\left({\\mathbb {F}}_{q^2}\\right)=q^3+1$ .",
"The zeta function of $X_q$ is $Z(X_q/{\\mathbb {F}}_q, t)=\\frac{(1+qt^2)^g}{(1-t)(1-qt)}.$ In fact, $X_q$ is the unique curve of this genus which is maximal over ${\\mathbb {F}}_{q^2}$ [28].",
"This was used to prove that $X_q$ is the Deligne-Lusztig variety for ${\\rm Aut}(X_q)={\\rm PGU}(3,q)$ [9].",
"By [8], the $a$ -number of $X_q$ is $a=p^n(p^{n-1}+1)(p-1)/4,$ which equals $g$ when $n=1$ , equals $g/2$ when $n=2$ , and is approximately $g/2$ for $n \\ge 3$ .",
"In particular, $X_{p^n}$ is superspecial if and only if $n=1$ .",
"In [27], for all $q=p^n$ , the authors determine the Dieudonné module ${\\mathbb {D}}_*(X_q) = {\\mathbb {D}}_*(\\operatorname{Jac}(X_q)[p])$ , complementing earlier work in [3], [4].",
"In particular, [27] states that the distinct indecomposable factors of Dieudonné module ${\\mathbb {D}}_*(X_q)$ are in bijection with orbits of ${\\mathbb {Z}}/(2^n+1) -\\lbrace 0\\rbrace $ under $\\times 2$ .",
"Each factor's structure is determined by the combinatorics of the orbit, which depends only on $n$ and not on $p$ .",
"The multiplicities of the factors do depend on $p$ .",
"For example, when $n=2$ , the Dieudonné module of $X_{p^2}$ is $M_{2,2}^{g/2}$ , which has superspecial rank 0 (Lemma REF ).",
"Here is an application of these results.",
"Application 5.6 The elliptic rank of the Jacobian of the Hermitian curve $X_{p^n}$ equals 0 if $n$ is even and is at most $(\\frac{p(p-1)}{2})^n$ if $n$ is odd.",
"By Proposition REF , $e({\\rm Jac}(X_{p^n})) \\le s({\\rm Jac}(X_{p^n}))$ .",
"Applying [27], the factor ${\\mathbb {E}}/{\\mathbb {E}}(F+V)$ occurs in the Dieudonné module if and only if there is an orbit of length 2 in ${\\mathbb {Z}}/(2^n+1)$ under $\\times 2$ .",
"This happens if and only if there is an element of order three in ${\\mathbb {Z}}/(2^n+1)$ , which is true if and only if $n$ is odd.",
"If $n$ is odd, this shows that ${\\mathbb {E}}/{\\mathbb {E}}(F+V)$ is not a factor of the Dieudonné module and $s({\\rm Jac}(X_{p^n}))=0$ .",
"If $n$ is even, the multiplicity of this factor is $s({\\rm Jac}(X_{p^n}))=(\\frac{p(p-1)}{2})^n$ ."
]
] | 1403.0023 |
[
[
"Spatiotemporal correlations between plastic events in the shear flow of\n athermal amorphous solids"
],
[
"Abstract The slow flow of amorphous solids exhibits striking heterogeneities: swift localised particle rearrangements take place in the midst of a more or less homogeneously deforming medium.",
"Recently, experimental as well as numerical work has revealed spatial correlations between these flow heterogeneities.",
"Here, we use molecular dynamics (MD) simulations to characterise the rearrangements and systematically probe their correlations both in time and in space.",
"In particular, these correlations display a four-fold azimuthal symmetry characteristic of shear stress redistribution in an elastic medium and we unambiguously detect their increase in range with time.",
"With increasing shear rate, correlations become shorter-ranged and more isotropic.",
"In addition, we study a coarse-grained model motivated by the observed flow characteristics and challenge its predictions directly with the MD simulations.",
"While the model captures both macroscopic and local properties rather satisfactorily, the agreement with respect to the spatiotemporal correlations is at most qualitative.",
"The discrepancies provide important insight into relevant physics that is missing in all related coarse-grained models that have been developed for the flow of amorphous materials so far, namely the finite shear wave velocity and the impact of elastic heterogeneities on stress redistribution."
],
[
"Introduction",
"When a simple liquid is sheared, it flows homogeneously, and its flow is traditionally viewed as a uniform slide of vanishingly thin layers of fluids past each other.",
"On the other hand, if shear is applied to an amorphous solid, the response of the material is highly heterogeneous, in that small regions rearrange rapidly while the rest of the material responds elastically, in a more or less affine way [1], [2].",
"In extreme cases, the material may fracture [3], [4], [5], [6]; the shear strain is then entirely borne by a thin layer of matter which has lost its internal cohesion.",
"Material fracture is the most acute case of shear localisation, whereby the deformation is localised in one region of the system.",
"The occurrence of this phenomenon rules out the study of the flow from a homogeneous perspective.",
"But, even when sheared amorphous solids do not exhibit permanent shear localisation, there is growing evidence of the existence of correlations between the localised rearranging regions (referred to as plastic events in the following), that is, of a spatial organisation of the flow at intermediate time scales: In Ref.",
"[7], Chikkadi and co-workers observed a colloidal glass with confocal microscopy and demonstrated that the non-affine displacements in the material were spatially correlated, prior to any shear-banding instability, while Mandal, Varnik, and colleagues [8], [9] supported such experimental findings with numerical simulations.",
"The observed correlations are interpreted as the effect of the long-range elastic deformation field induced by a plastic event in the material, at the origin of avalanches of plastic events.",
"In a uniform linear elastic medium, this field is quadrupolar[10], [11], viz., $\\mathcal {G}\\left(r\\right)\\sim \\cos \\left(4\\theta \\right)/r^2$ for a plastic event occurring at the origin, in two dimensions.",
"Predicting exactly where the next plastic event will occur in the material is an intricate task that is bound to depend sensitively on detailed knowledge of the current, static configuration of the system[12], [13], [14], [15], [16].",
"Alternatively, one may choose to investigate to what extent the position of the next plastic event is influenced by that of its predecessors, in the hope that extensive information about the dynamical organization of the flow will thus be revealed.",
"The characterisation of such correlations between plastic events is the objective of this work.",
"Although our tools will be slightly different, we note that similar studies have appeared in two recent publications.",
"In [17], long-lived correlations of the local strain field observed in molecular dynamics (MD) simulations were taken as evidence of the importance of localised plastic events in a flowing liquid.",
"In [18], similar correlations were observed in a numerical model of a dense emulsion undergoing shear flow between solid plates.",
"In this work, we will concentrate on a numerical study of the flow of a very simple amorphous solid in the athermal limit.",
"We will propose a detailed description of the plastic events and their dynamical correlations, resolved both in space and time.",
"The influence of the applied shear rate is studied.",
"In order to ascertain the origin of the prominent features of the correlations, we investigate a coarse-grained model closely connected to the observed flow phenomenology in complement to the atomistic simulations.",
"The article is structured as follows: In Section , we provide the reader with the technical details pertaining to the MD simulations, and present the observable that will be used to measure the local rearrangements.",
"In Section , we report the general properties of the simulated flow, with a particular focus on the statistics of individual plastic events.",
"On the basis of these observations, a coarse-grained model is presented in Section and the general agreement of the model with the atomistic simulations is immediately assessed.",
"Finally, Section is dedicated to our main contribution, namely, a detailed study of the spatiotemporal correlations between successive plastic events and the interpretation of their salient features."
],
[
"Atomistic simulations at zero temperature",
"To probe the flow properties of amorphous solids, we resort to MD simulations of an amorphous system under shear.",
"More precisely, we simulate a binary mixture of A and B particles, with $N_{A}=32500$ and $N_{B}=17500$ , of respective diameters $\\sigma _{AA}=1.0$ and $\\sigma _{BB}=0.88$ , confined in a square box of dimensions $205\\sigma _{AA}\\times 205\\sigma _{AA}$ , with periodic boundary conditions.",
"The system is at reduced density 1.2.",
"The particles, of mass $m=1$ , interact via a pairwise Lennard-Jones potential, $V_{\\alpha \\beta }\\left(r\\right)=4\\epsilon _{\\alpha \\beta }\\left[\\left(\\frac{\\sigma _{\\alpha \\beta }}{r}\\right)^{12}-\\left(\\frac{\\sigma _{\\alpha \\beta }}{r}\\right)^{6}\\right],$ where $\\alpha ,\\beta =A,\\, B$ , $\\sigma _{AB}=0.8$ ,$\\epsilon _{AA}=1.0$ , $\\epsilon _{AB}=1.5$ , and $\\epsilon _{BB}=0.5$ .",
"The potential is truncated at $r=2.5\\sigma _{AA}$ and shifted for continuity.",
"Simple shear $\\gamma $ is imposed at rate $\\dot{\\gamma }$ by deforming the box dimensions and remapping the particle positions.",
"We conduct our study in the athermal limit, by thermostating the system to a zero-temperature, so that no fluctuating force appears in the equations of motion, viz., $\\frac{d\\mathbf {r_{i}}}{dt} & =\\mathbf {p_{i}}/m\\cr \\frac{d\\mathbf {p_{i}}}{dt} & =-\\sum _{i\\ne j}\\frac{\\partial V\\left(\\mathbf {r_{ij}}\\right)}{\\partial \\mathbf {r_{ij}}}-\\mathbf {p_{i}}/\\tau _d$ where $\\left(\\mathbf {p_{i}},\\mathbf {r_{i}}\\right)$ are the momentum and position of particle $i$ in the deforming frame.",
"Besides the interparticle forces, the motion of particle i is subject to a damping force $-\\mathbf {p_{i}}/\\tau _d$ , that models friction against solvent molecules in a mean-field way.",
"Here, $\\tau _d=1$ is the Langevin damping time.",
"The relevance of this specific implementation of friction shall be discussed in Section REF .",
"Equations REF are integrated with the velocity Verlet algorithm with $\\delta t=0.005$ .",
"In all the following, we use $\\tau _{LJ}\\equiv \\sqrt{m\\sigma _{AA}^2 / \\epsilon }$ as the unit of time and $\\sigma _{AA}$ as the unit of length.",
"To obtain the initial glassy states, we quenched the system at constant volume from the liquid state down to zero temperature at a fast rate.",
"Note that, before any data were collected, the system was always pre-sheared for $\\gamma =0.2$ to ensure that the steady state had been reached."
],
[
"Macroscopic rheology & Statistics of plastic events ",
"In this section, we analyse the global rheology of the system and collect evidence in support of the general scenario of plastic events embedded in an elastic medium outlined in the introduction.",
"We will also characterize plastic events at a statistical level."
],
[
"Flow curve",
"The dependence of the macroscopic shear stress $\\Sigma $ on the applied shear rate is shown in Fig.REF ; it is well described by the Herschel-Bulkley law $\\Sigma =0.73+2.9\\dot{\\gamma }^{0.48}$ .",
"Regarding the bulk mechanical properties of the system, plotting the stress as a function of strain at a given shear rate yields a shear modulus $\\mu \\simeq 17$ for the system (prior to deformation) and a macroscopic yield strain $\\gamma _{y}$ of order 5-10%.",
"Figure: Dependence of the macroscopic shearstress Σ\\Sigma on the applied shear rate γ ˙\\dot{\\gamma }.",
"(Blackstars) MD simulation; (blue triangles) coarse-grained model, with N=64×64N=64\\times 64 blocks.The dashed black line is a fit to the Herschel-Bulkley equation, Σ=0.73+2.9γ ˙ 0.48 \\Sigma =0.73+2.9\\dot{\\gamma }^{0.48}."
],
[
"Stress autocorrelation function",
"Turning to more local quantities, in Fig.REF (top panel) we plot the autocorrelation function $C_{\\sigma }\\left(\\Delta \\gamma \\right)\\equiv \\frac{\\left\\langle \\delta \\sigma _{xy}\\left(\\gamma \\right)\\delta \\sigma _{xy}\\left(\\gamma +\\Delta \\gamma \\right)\\right\\rangle }{\\left\\langle \\delta \\sigma _{xy}^{2}\\right\\rangle }$ of the local shear stress fluctuations $\\delta \\sigma _{xy}\\equiv \\sigma _{xy}-\\left\\langle \\sigma _{xy}\\right\\rangle $ experienced by each particle.",
"The averages are performed over time.",
"We observe a nice collapse of the data for the different shear rates.",
"This confirms that the applied strain $\\Delta \\gamma $ , and not the absolute time $t$ , causes the decorrelation in this driven athermal system, in line with the idea of periods of elastic accumulation of stress interspersed with shear-induced plastic events.",
"The master curve is fairly well fit by a stretched exponential $\\exp \\left[-\\left(\\frac{\\Delta \\gamma }{\\Delta \\gamma ^{\\star }}\\right)^{\\beta }\\right]$ , with an exponent $\\beta =0.68$ and a characteristic strain $\\Delta \\gamma ^{\\star }=0.11$ close to the macroscopic yield strain.",
"Figure: Autocorrelation function C σ ΔγC_{\\sigma }\\left(\\Delta \\gamma \\right) ofthe local shear stress fluctuations at applied shear rates (reddots) γ ˙=10 -5 \\dot{\\gamma }=10^{-5}, (blue triangles) γ ˙=10 -4 \\dot{\\gamma }=10^{-4},and (green stars) γ ˙=10 -3 \\dot{\\gamma }=10^{-3}.",
"Top panel:results from MD simulations.",
"Bottom panel: results from the coarse-grained model.",
"Thedashed lines represent a fitting to a stretched exponential C σ Δγ=exp-Δγ Δγ ☆ β C_{\\sigma }\\left(\\Delta \\gamma \\right)=\\exp \\left[\\left(\\frac{-\\Delta \\gamma }{\\Delta \\gamma ^{\\star }}\\right)^{\\beta }\\right],with β=0.68,Δγ ☆ =0.11\\left(\\beta =0.68,\\,\\Delta \\gamma ^{\\star }=0.11\\right) for theMD data and β=0.65,Δγ ☆ =0.07\\left(\\beta =0.65,\\,\\Delta \\gamma ^{\\star }=0.07\\right)for the coarse-grained results."
],
[
"Indicator of plastic activity",
"Let us now focus on plastic events.",
"In order to detect them, we make use of the $D_{min}^{2}$ quantity presented by Falk and Langer in Ref.",
"[19], which evaluates deviations from an affine deformation on a local scale.",
"This quantity has been used with noted success to characterize plasticity [19], [20], [7], [21], [8], [9], [22]; in particular, it was shown to yield results consistent with other measures of nonaffinity in Ref.",
"[21].",
"$D_{min}^{2}$ is defined locally, around a particle labelled i, as the minimum over all possible linear deformation tensors $\\epsilon $ of $D^{2}\\left(i;t,\\delta t\\right) & = & \\sum _{j}\\left[r_{ij}\\left(t+\\delta t\\right)-\\left(\\mathbb {I}+\\epsilon \\right)\\cdot r_{ij}\\left(t\\right)\\right]^{2},$ where the sum runs over all neighbours $j$ of $i$ , and $\\mathbb {I}$ denotes the identity matrix.",
"The value of the time lag $\\delta t$ was fine-tuned to provide a good signal over noise ratio while still being short enough to allow a temporal resolution of the plastic events.",
"Figure REF presents a snapshot of $D_{min}^{2}$ values in the system: one clearly sees localised plastic regions embedded in an affinely-deforming medium.",
"To provide a more dynamical view of the flow, short movies are available as Supplementary Material [23], along with their counterparts for the coarse-grained model presented in the next section.",
"Figure: Snapshot of the D min 2 D_{min}^{2} field at an applied shear rate γ ˙=10 -4 \\dot{\\gamma }=10^{-4}.Interestingly, the regions with large $D_{min}^{2}$ systematically coincide with the regions exhibiting large velocities relative to the average solvent flow.",
"This coincidence between the non-locally-affine displacement field and the singular velocity confirms that large local energy dissipation is the hallmark of a plastic event."
],
[
"Distribution of durations, magnitudes, and sizes of individual plastic\nevents",
"We now study the properties of individual plastic events in more detail.",
"First, by scrutinising a number of $D_{min}^{2}$ snapshots such as the image presented in Fig.REF , we observe that the size of plastic regions is typically a few particle diameters; this size does not depend dramatically on the shear rate.",
"This point will be confirmed in Section by a detailed analysis of the spatial correlations of the $D_{min}^{2}$ field.",
"Further insight is gained by computing the overall distribution of the measured $D_{min}^{2}$ values in Fig.REF .",
"All distributions exhibit an exponential tail, and they furthermore collapse upon rescaling with the inverse shear rate.",
"Finally, the typical lifetime of a plastic event can be extracted from the temporal decay of the $D_{min}^{2}$ autocorrelation function plotted in Fig.REF .",
"For the value of the damping time $\\tau _d$ used in this study, it is of the order of 3 time units regardless of the shear rate."
],
[
"Description of the model",
"The numerical observations reported above all support the flow scenario based on short-lived, localised, and strongly dissipative plastic events embedded in an elastic matrix.",
"Accordingly, we shall now introduce a simple, 2D coarse-grained model for the rheology of athermal amorphous solids that is motivated by these observations.",
"To start with, we discretise space into a lattice $\\left\\lbrace \\left(i,j\\right)\\right\\rbrace $ of $N=128\\times 128$ square-shaped elastoplastic blocks, each of the size of a rearranging region.",
"By default, blocks are elastic, in which case the (tensorial) deviatoric stress $\\sigma \\left(i,j\\right)$ and strain $\\epsilon \\left(i,j\\right)$ tensors carried by each block obey Hooke's law, viz., $\\left(\\begin{array}{c}\\sigma _{xx}\\left(i,j\\right)\\\\\\sigma _{xy}\\left(i,j\\right)\\end{array}\\right)=2\\mu \\left(\\begin{array}{c}\\epsilon _{xx}\\left(i,j\\right)\\\\\\epsilon _{xy}\\left(i,j\\right)\\end{array}\\right),$ where $\\mu $ is the shear modulus.",
"Here, we have postulated incompressibility, i.e., $\\epsilon _{yy}\\left(i,j\\right)=-\\epsilon _{xx}\\left(i,j\\right)$ .",
"Plasticity is incorporated into the model by allowing blocks to yield (i.e., switch to the plastic state) as soon as the following yield criterion is fulfilled, $\\left\\Vert \\sigma \\left(i,j\\right)\\right\\Vert \\equiv \\sqrt{\\sigma _{xx}^{2}\\left(i,j\\right)+\\sigma _{xy}^{2}\\left(i,j\\right)}\\geqslant \\sigma _{y}\\left(i,j\\right),$ where $\\sigma _{y}\\left(i,j\\right)$ is a fixed local yield stress, associated to an energy barrier $E_{y}\\left(i,j\\right)={\\sigma _{y}^{2}\\left(i,j\\right)}{4\\mu }$ .",
"Equation REF is simply the well-known von Mises yield criterionNote that the von Mises and the Tresca yield criteria are equivalent in two dimensions.. Every time a block yields, the value of $E_{y}\\left(i,j\\right)$ is renewed; it is randomly selected from a truncated exponential distribution, $P\\left(E_{y}\\right)=\\Theta \\left(E_{y}-E_{y}^{min}\\right)\\exp \\left(\\lambda \\left(E_{y}^{min}-E_{y}\\right)\\right),$ where $\\Theta $ is the Heaviside function.",
"The coefficient $\\lambda $ is chosen such that the average of the yield strain $\\gamma _{y}=2\\sqrt{{E_{y}}{\\mu }}$ over $P$ coincides with the MD macroscopic yield strain, $\\left\\langle \\gamma _{y}\\right\\rangle \\approx 0.1$ .",
"The introduction of a lower threshold $E_{y}^{min}$ in Eq.REF comes down to discarding too shallow metabasins in the potential energy landscape (PEL) of the subsystem modelled as an elastoplastic block.",
"A plastic block is a fluid-like inclusion embedded in an elastic region.",
"The implications of this fact are twofold.",
"First, the plastic rearrangement occurs, not instantaneously, but over a finite time scale $\\tau $ , because viscous forces oppose it[24].",
"Second, the associated distortion of the plastic region induces an additional stress in the surrounding elastic medium[10].",
"The combination of these two effects occurring in plastic blocks, along with Eq.REF for the elastic regions, leads to, $\\partial _{t}\\sigma \\left(i,j\\right)=\\mu \\dot{\\gamma }+2\\mu \\sum _{i^{\\prime },j^{\\prime }}\\mathcal {G}\\left(i-i^{\\prime },j-j^{\\prime }\\right)\\dot{\\epsilon }^{pl}\\left(i^{\\prime },j^{\\prime }\\right).$ Here, $\\dot{\\epsilon }^{pl}\\left(i^{\\prime },j^{\\prime }\\right)=\\sigma \\left(i^{\\prime },j^{\\prime }\\right)/2\\mu \\tau $ if the block $\\left(i^{\\prime },j^{\\prime }\\right)$ is plastic, ${\\bf 0}$ otherwise.",
"$\\tau $ is the local viscous time, and the propagator $\\mathcal {G}$ is such that $2\\mu \\mathcal {G}\\left(i-i^{\\prime },j-j^{\\prime }\\right)\\epsilon ^{pl}$ is the stress increment received by block $\\left(i,j\\right)$ if the block $\\left(i^{\\prime },j^{\\prime }\\right)$ endures a plastic strain $\\epsilon ^{pl}$ .",
"Note in particular that $\\mathcal {G}\\left(0,0\\right)$ has negative eigenvalues, because plastic blocks relax the stress they bear.",
"More generally, the propagator $\\mathcal {G}$ was calculated rigorously in the limit of a pointwise inclusion in a perfectly homogeneous elastic medium.",
"Its expression and the way it must be altered when simulation cell is deformed are specified elsewhere[25].",
"Obviously, the first term on the right hand side of Eq.REF contains the elastic response to the macroscopic driving and follows directly from Eq.REF , while the second term deals with the effects of plasticity.",
"The swift rearrangement of particles that characterises a plastic event corresponds to a jump between metabasins in the PEL[26].",
"In our model, it shall come to an end when a given strain $\\gamma _{c}$ has been cumulated locally in the plastic phase, i.e., when $\\int \\left\\Vert \\dot{\\epsilon }\\left(t^{\\prime }\\right)\\right\\Vert dt^{\\prime }\\geqslant \\gamma _{c},$ where the total local deformation reads, $\\dot{\\epsilon }=\\frac{\\partial _{t}\\sigma }{2\\mu }+\\dot{\\epsilon }^{pl}$ .",
"$\\gamma _{c}$ represents the typical distance between metabasins, which clearly depends on how much the PEL has been coarse-grained, that is, on $E_{y}^{min}$ .",
"For simplicity, we arbitrarily take $\\gamma _{c}=2\\sqrt{{E_{y}^{min}}{\\mu }}$ .",
"Following this simplification, $E_{y}^{min}$ is the only free parameter in the model, if one excepts the time and stress units $\\tau $ and $\\mu $ .",
"To sum up, the transitions between the elastic and plastic regimes obey, $\\text{elastic}\\underset{\\int _{pl}dt\\left\\Vert \\dot{\\epsilon }\\right\\Vert \\geqslant \\gamma _{c}}{\\overset{\\left\\Vert \\sigma \\right\\Vert \\geqslant \\sigma _{y}}{\\rightleftharpoons }}\\text{plastic}$ Finally, a coarse-grained version of convection is introduced in the system by incrementally shifting lines of blocks in the flow direction.",
"For this purpose, we keep track of the exact average displacement of each of these 'streamlines' along the flow direction, and shift the whole line once the displacement gets larger than the size of one block.",
"The implementation of convection also requires to compute the elastic propagator in a deformed frame[25]."
],
[
"Comparison of general features with the atomistic simulations",
"Figure REF presents a comparison of the flow curves obtained with the coarse-grained model and with the atomistic simulations.",
"Note that, to allow direct comparison, the time and stress units in the model must be specified.",
"A reasonably good agreement between the flow curves is obtained by setting the shear modulus to 12.5, a value comparable to the shear modulus of the atomistic system prior to deformation ($\\mu _{MD}=17$ ), and $\\tau =1.5$ , which will lead to similar plastic event life times in the MD and coarse-grained simulations.",
"The best fits of the flow curves with Herschel-Bulkley equations have very similar exponents $n\\simeq 0.5$ .",
"To quantify the global plasticity of the system, we compute the instantaneous surface density of plastic events, i.e., the fraction of blocks which are plastic at a given time.",
"In the absence of thermally-activated plastic events, this quantity increases linearly with the shear rate, from 0.05% at $\\dot{\\gamma }=10^{-5}$ to 0.36% at $\\dot{\\gamma }=10^{-4}$ and 2.8% at $\\dot{\\gamma }=10^{-3}$ .",
"These values are similar to those obtained from the atomistic simulations by integrating the tails of the $D_{min}^{2}$ distributions, in Fig.REF , down to a reasonable (but arbitrary) lower threshold: 0.07%, 0.3%, and 0.8%, respectively.",
"Turning to a more local viewpoint, the autocorrelation function of the stress fluctuations on a given block are shown in panel (b) of Fig.REF .",
"As in the MD simulations, the autocorrelations at different strain rates collapse onto a master curve.",
"Interestingly, this master curve is fitted by a stretched exponential, $\\exp \\left[-\\left(\\frac{\\Delta \\gamma }{\\Delta \\gamma ^{\\star }}\\right)^{\\beta }\\right]$ , with a stretching exponent $\\beta =0.65$ very close to the one used to fit the MD data ($\\beta =0.68$ ), although the precise value of the characteristic strain obtained here, $\\Delta \\gamma ^{\\star }=0.07$ , differs by 50%.",
"The average life time of a single plastic event in the model is of order a few $\\tau $ (remember that we set $\\tau $ to 1.5) at all shear rates.",
"More precisely, a noticeable decrease of the average life time is observed as the shear rate is increased, from $8.4$ at $\\dot{\\gamma }=10^{-5}$ to $4.2$ at $\\dot{\\gamma }=10^{-3}$ .",
"This is not unexpected, because the criterion determining the duration of a plastic event, Eq.REF , involves the total local deformation rate.",
"Indeed, the distributions of plastic event life times, shown in Fig.REF , undergo a small, but noticeable shift to shorter times at higher shear rates.",
"Figure: Histogram of the durations Δt pl \\Delta t^{pl} of plastic events, at(blue)γ ˙=10 -5 \\dot{\\gamma }=10^{-5}, (green)γ ˙=10 -4 \\dot{\\gamma }=10^{-4},and (red)γ ˙=10 -3 \\dot{\\gamma }=10^{-3} .Since we introduced a cut-off in the yield stress distribution (see Eq.REF ), the distribution of plastic event magnitudes will naturally differ from that observed in the atomistic simulations.",
"Nevertheless, this distribution is still roughly independent of the applied shear rate (data not shown)."
],
[
"Correlations between plastic events",
"Having verified the agreement of the coarse-grained model with the atomistic simulations with regard to the general flow properties, we can now move on to the study of the correlations in the flow."
],
[
"Plastic correlator",
"The individual localised rearrangements identified in Section REF are not random isolated events: in the athermal, quasi-static limit, Maloney and Lemaître [27] showed numerically that they are essentially organised in strongly correlated avalanches.",
"By investigating the transverse particle diffusivity, Lemaître and colleagues then showed that these correlations persist at finite shear rates [28] and at finite temperatures [29], [30].",
"The spatial structure of these correlations was revealed by Chikkadi, Mandal, Varnik, et al.",
"[7], [8]; these researchers provided convincing experimental and numerical evidence that the correlations between flow heterogeneities, quantified by $D^2_{min}$ , are long-ranged and all the more anisotropic as shear prevails over thermal effects, i.e., at larger Peclet numbers.",
"To do so, they monitored particle displacements in a driven “hard sphere” colloidal glass with confocal microscopy and were able to reproduce their experimental observations qualitatively with MD simulations.",
"Quantitatively, some discrepancies were found between simulations and experiments, the latter displaying longer correlations, with a power law decay in space.",
"Here, we purport to extend these studies and unveil the full dynamical picture by resolving correlations between the plastic events both in time and in space, for different shear rates, in the athermal regime.",
"The emphasis shall then be put on the causal links that exist between successive plastic events.",
"To this end, we use the following two-time, two-point plastic correlator, $\\mathcal {C}_{2}\\left(\\Delta r,\\Delta t\\right) & \\equiv & \\alpha \\Big (\\left\\langle \\overline{D_{min}^{2}\\left(r,t\\right)D_{min}^{2}\\left(r+\\Delta r,t+\\Delta t\\right)}\\right\\rangle \\\\ \\nonumber & & -\\left\\langle \\overline{D_{min}^{2}\\left(r,t\\right)}\\cdot \\overline{D_{min}^{2}\\left(r,t+\\Delta t\\right)}\\right\\rangle \\Big ),$ where the brackets denote an average over time $t$ , the bars represent an average over spatial coordinate $r$ , and the prefactor $\\alpha \\equiv \\Big [ \\left\\langle \\overline{ (D_{min}^{2}(r,t))^2 }\\right\\rangle - \\left\\langle \\overline{ (D_{min}^{2}(r,t)) }^2\\right\\rangle \\Big ]^{-1}$ is chosen such that $\\mathcal {C}_{2}\\left(\\Delta r=0,\\Delta t=0\\right)=1$ .",
"Clearly, $\\mathcal {C}_{2}$ measures the (enhanced or reduced) likelihood that a plastic event occurs at $r+\\Delta r$ if a plastic event was active at position $r$ some prescribed time $\\Delta t$ ago.",
"In the coarse-grained simulations, a sensible equivalent is, $\\mathcal {C}_{2}\\left(\\Delta r,\\Delta t\\right) & \\equiv & \\alpha ^\\prime \\Big (\\left\\langle \\overline{n\\left(r,t\\right)n\\left(r+\\Delta r,t+\\Delta t\\right)}\\right\\rangle \\nonumber \\\\& & -\\left\\langle \\overline{n\\left(r,t\\right)}\\cdot \\overline{n\\left(r,t+\\Delta t\\right)}\\right\\rangle \\Big ),$ where $n\\left(r,t\\right)=1$ if the block at position $r$ is plastic at time $t$ , $n\\left(r,t\\right)=0$ otherwise, and $\\alpha ^\\prime \\equiv \\Big [ \\left\\langle \\overline{ n(r,t)^2 }\\right\\rangle - \\left\\langle \\overline{ n(r,t) }^2\\right\\rangle \\Big ]^{-1}$ is, again, a normalisation prefactor."
],
[
"Decay of the intensity of the correlation with time",
"Plastic correlations naturally fade away with time, but one may wonder whether their decay is more appropriately described in terms of the absolute time $t$ or the strain $\\gamma $ .",
"Quite interestingly, in the atomistic simulations as well as in the coarse-grained model, absolute time turns out to be the adequate unit of measurement, as evidenced by comparing the evolution of the correlations at different shear rates.",
"It should be pointed out that this does not conflict with the decay of stress correlations as a function of the strain.",
"Stress correlations exist during the loading phase preceding the shear transformation, whose duration is typically determined by the yield strain.",
"On the other hand, the duration of the plastic activity phase is mostly determined by the local damping time, and is only weakly dependent on strain rate.",
"Correlations in plastic activity are therefore expected on the time scale on a single event, or on somewhat longer time scales in the event of correlated avalanches, but they will remain limited to finite times even for vanishingly small applied strain rates."
],
[
"Maps of plastic correlations at various shear rates",
"The plastic correlations obtained in the atomistic simulations are shown in Fig.REF , REF , and REF at different time lags for three distinct shear rates: $\\dot{\\gamma }=10^{-5}$ , $\\dot{\\gamma }=10^{-4}$ , and $\\dot{\\gamma }=10^{-3}$ .",
"The counterparts for the coarse-grained simulations are presented directly opposite to them so as to allow an easy comparison, but they will only be discussed below in section REF .",
"The presence of a spatial structure in the correlations is manifest, which is strong evidence that plastic rearrangements are indeed interdependent, and not fully isolated events.",
"The positive correlations in the streamwise and crosswise directions are strongly reminiscent of the positive lobes of the elastic propagator $\\mathcal {G}$ , which supports the idea of interactions via an elastic coupling.",
"In diagonal directions, there tend to be anticorrelations.",
"The (anti)correlations decay gradually, over approximately the same (absolute) time scale as the autocorrelation function, i.e., their value at the origin.",
"These features are common to the various shear rates studied here.",
"A closer investigation of the plots shows that the decay time tends to decrease with increasing shear rate, thereby reflecting the shear-induced decorrelation of the system, with sequences of correlated events being cutoff by the deformation.",
"Moreover, while the streamwise and crosswise lobes are hardly distinguishable at high shear rates, at lower shear rates there is clearly an asymmetry between them.",
"The propensity to shear localisation of the plastic activity is therefore enhanced at lower shear rates.",
"This is more visible in Fig.REF (top panel), where the correlations are integrated along the radial direction in different directions.",
"An enhanced propensity to shear localisation, or, more generally, flow heterogeneities, with decreasing shear rates has already been reported in the literature [31], [32], [33], although, here, some artifact associated with the use of periodic boundary conditions and finite size effects cannot be excluded[17].",
"An additional effect of the shear rate is that the anticorrelated lobes in the diagonal directions appear stronger at higher shear rates.",
"To assess the strength of the correlations, that is, to what extent they deviate from a random distribution of plastic events, we compare the probabilities that two plastic events, separated by a distance $\\Delta r$ and a time lag $\\Delta t$ , are aligned, on the one hand, along the velocity gradient direction $e_\\perp $ and, on the other hand, along the diagonal direction $e_{diag}$ with respect to the macroscopic shear.",
"We observe an enhancement of the probabilities of streamwise alignment (versus diagonal alignement) by about 10% to 20%.",
"Details are provided in Fig.REF in the Appendix.",
"We now turn to the spatial extent of the correlations.",
"In the top panel of Fig.REF , we show how they decay along the flow direction, for distinct time lags.",
"The decay, which is not purely exponential, depends only weakly on the shear rate, except at long time lags.",
"Besides, it spreads over larger and larger distances as the time lag is increased; it should however be noted that the correlations have been rescaled so as to be equal to unity close to the origin at all time lags, so that a slower spatial decay does not necessarily imply a larger absolute value far from the origin.",
"This rescaling also entails that small fluctuations will be magnified when the correlations near the origin are small, e.g., for the long time lag $\\Delta t=20$ , notably in the moderately high shear rate case.",
"At this stage, we should mention a very recent study by Varnik and co-workers [9], who reported that the spatial decay of the $D_{min}^{2}$ correlations was highly contingent on the specific implementation of the friction force in the equations of motion Note that, although these researchers have computed nominally “static” correlations, that is to say, at $\\Delta t=0$ , the time $\\delta t$ which they used to compute $D_{min}^{2}$ is very large, so that their data actually correspond to an integral of our dynamical correlations $\\mathcal {C}_{2}\\left(\\Delta r,\\Delta t\\right)$ over a wide range of time lags $\\Delta t$ .",
").",
"More precisely, only a friction force based on the relative velocity of a particle with respect to its neighbours (“contact dynamics”) could reproduce the power law decay observed in experiments on colloidal suspensions and immersed granular matter, whereas a mean-field dissipation scheme predicted a faster, exponential decay.",
"The effect of the specific implementation of the frictional force has been the subject of a wider debate: Tighe et al.",
"[34], for instance, reported that using a friction term based on relative particle displacements is key to finding suitable correlation functions in the vicinity of the jamming point, while Vagberg et al.",
"[35] claimed that a critical behaviour is found with both schemes.",
"Here, we have used a mean-field friction force; accordingly, some quantitative discrepancies may be expected between the extent of the correlations that we have found and those measured in the experimental setups of Ref.",
"[7], [9].",
"However, our choice of friction force is, arguably, the more adequate one for confined two-dimensional geometries in which particles slide along a fixed plate, for instance, bubble rafts confined in between parallel glass plates [36]."
],
[
"Successes and limitations of the coarse-grained model",
"In Section REF , we have seen that the coarse-grained model gives a rather satisfactory description of the macroscopic properties, as well as the local ones.",
"Here, we enquire how well it fares with respect to the full set of spatiotemporal correlations.",
"As shown in Figs.REF through REF , the correlation maps for the two models do bear some resemblance, but closer inspection reveals quantitative differences.",
"Among the satisfactory aspects, the coarse-grained model also indicates a decay of the correlations with absolute time and correlations display a four-fold angular symmetry.",
"By comparing the top and bottom panels of Fig.REF , we find reasonable agreement for the angular dependence of the correlations.",
"One should however admit that excessive correlations are predicted along the flow direction, especially in the near field.",
"This is an artifact associated with the use of a regular lattice: as the frame is deformed, the positive lobe of the elastic propagator in the flow direction remains aligned with one axis of the lattice, while the alignment of the perpendicular lobe with the other axis is lost.",
"Moreover, the coarse-grained simulations are able to describe the anti-correlated lobes in the diagonal directions and their enhancement at higher shear rates.",
"On the downside, it is obvious that salient features of the plastic correlations are amiss.",
"This discrepancy is interesting, because it is a hint that plastic correlations reveal some physical processes that may otherwise be left unnoticed, and that these processes have been omitted in the model.",
"First, the gradual growth with time of the correlations observed in MD is in stark contrast with the maximal extent of the correlations at vanishing time lag in the coarse-grained model, as can be seen in Fig.REF (bottom panel).",
"Indeed, within a short time lag $\\Delta t<1$ , the mesoscopic model builds up correlations over a characteristic distance of order 20, while these quasi-instantaneous correlations do not extend beyond a few unit lengths in the MD simulation.",
"This indicates that the MD correlations do not grow because more and more shear stress is redistributed as the rearrangement proceeds, but because shear waves need a finite time to propagate (whereas instantaneous equilibration was assumed in the model).",
"In other words, the acoustic delay for the propagation of strain-waves within an avalanche slows down the emergence of spatial correlations.",
"Indeed, the initial growth of the correlations is consistent with the propagation of shear waves at the transverse sound velocity (of the undamped system), viz., $c_{t}=\\sqrt{{\\mu }{\\rho }}\\simeq 4$ .",
"The gradual expansion of the strain field created by a plastic event is studied in more details in the companion paper, Ref.",
"[37], and in Ref.",
"[38], [17].",
"Note that, in this last reference, the authors also observed some advanced frontline moving at the longitudinal sound velocity $c_l > c_t$ .",
"The second major difference lies in the spatial extent of the correlations, which is much larger in the coarse-grained approach.",
"Had a frictional force based on relative velocities been used, MD may have yielded larger correlations, as suggested by ref.[9].",
"However, the large deviation between the predictions of the atomistic and coarse-grained models does point to an additional source of discrepancy.",
"We believe that the underestimation of structural disorder in the coarse-grained model is at the core of the divergence.",
"Indeed, broadening the distribution of energy barriers in the model results in somewhat shorter correlations, at the expense of a poorer fitting of the macroscopic flow properties by our essentially one-parameter model.",
"Of probably equal relevance is the use of an 'ideal' elastic propagator.",
"This propagator describes stress redistribution in a perfectly uniform elastic medium.",
"Such a description is justified on average, but is inaccurate for a specific plastic event[37], because elastic heterogeneities in the surrounding medium, i.e., the spatial variations of the local elastic constants, induce deviations from the ideal case.",
"The insufficient account of structural disorder in the model is also reflected in the vastly overestimated anisotropy of the correlations that it predicts, as measured by the directional probability enhancement (see the bottom panel of Fig.REF in the Appendix)."
],
[
"Summary and outlook",
"In conclusion, we have reported numerical simulations that confirm the basic flow scenario for amorphous solids, based on swift localised rearrangements embedded in an elastic matrix and interacting via an elastic deformation field.",
"A coarse-grained model based on this simple scenario satisfactorily reproduces the measured flow curve, the surface density of simultaneous plastic events, and the decay of the stress autocorrelation function.",
"To obtain full insight into the dynamical organisation of flow heterogeneities, we have probed the spatio-temporal correlations between plastic events, and their dependence on the shear rate.",
"As already reported in the literature, these correlations are perceivably anisotropic and exhibit the four-fold angular symmetry characteristic of the elastic propagator.",
"These correlations spread approximately at the transverse sound velocity before fading away.",
"Varying the shear rate only brings on small changes to the general picture: at higher shear rates, the near-field anticorrelations along the diagonal directions seem to be slightly more pronounced, and the velocity and velocity gradient directions are more symmetric.",
"Besides, the spatial extent of the correlations tends to decrease with increasing shear rate.",
"A coarse-grained model is able to describe the observed symmetries of the correlations, but fails to reproduce their emergence in time, owing to the neglect of the finite shear wave velocity in the model.",
"In addition, the model vastly overestimates the anisotropy in the correlations, thereby pointing to the underestimation of structural disorder in the system, at least partly because of the use of an ideal elastic propagator.",
"These two flaws are not specific to the model used here, but a general deficiency of all approaches of this type[39].",
"Consequently, should one aim for a proper description of these correlations, these missing physical aspects will need to be incorporated into the models.",
"To what extent they will alter the macroscopic flow properties predicted by the models, for instance the variable propensity to shear localisation, remains an open question.",
"More generally, it seems likely that in the future the study of plastic correlations, as described in Ref.",
"[17], [18] or in the present work, will become a powerful tool for comparison between models of various types and between models and experiments in systems in which the corresponding observables are experimentally accessible.",
"Acknowledgments The simulations were carried out using the LAMMPS molecular dynamics softwarehttp://lammps.sandia.gov.",
"JLB is supported by Institut Universitaire de France and by grant ERC-2011-ADG20110209.",
"JR acknowledges support from Université Joseph Fourier and from Institut Laue Langevin during a stay in Grenoble."
],
[
"Quantification of the anisotropy of the correlations",
"To assess the strength of the anisotropy in the plastic correlations, we compute the directional probability enhancement factor $\\alpha _\\perp $ , viz., $\\alpha _{\\perp }(\\Delta r,\\Delta t,\\dot{\\gamma }) & \\equiv & \\frac{\\langle D_{min}^{2}({\\bf r},t)\\cdot D_{min}^{2}({\\bf r}+\\Delta re_{\\perp },t+\\Delta t)\\rangle }{\\langle D_{min}^{2}({\\bf r},t)\\cdot D_{min}^{2}({\\bf r}+\\Delta re_{diag},t+\\Delta t)\\rangle },$ where $e_{\\perp }$ and $e_{diag}$ are the velocity gradient and diagonal directions, respectively.",
"This factor measures the ratio of the probabilities that two plastic events, separated by $\\Delta r$ in distance and $\\Delta t$ in time, will be aligned along the velocity gradient direction versus diagonally.",
"Figure REF compares this enhancement ratio for the MD model and the coarse-grained model.",
"Figure: Directional probability enhancement factorα ⊥ (Δr,Δt,γ ˙)\\alpha _\\perp (\\Delta r,\\Delta t,\\dot{\\gamma })for the different shear rates: (solid line) γ ˙=10 -5 \\dot{\\gamma }=10^{-5},(dash-dotted line) γ ˙=10 -4 \\dot{\\gamma }=10^{-4}, and (dottedline) γ ˙=10 -3 \\dot{\\gamma }=10^{-3}, and for various lag times.",
"Toppanel: (blue) Δt=4\\Delta t=4 (1 for panel (b)), (green)Δt=12\\Delta t=12, (red) Δt=20\\Delta t=20.",
"Bottom panel:(blue) Δt=0\\Delta t=0 (1 for panel (b)), (green) Δt=8\\Delta t=8,(red) Δt=16\\Delta t=16.",
"To allow direct comparison, we have setthe size of one coarse-grained block to r ˜=5\\tilde{r}=5."
]
] | 1403.0421 |
[
[
"The phase diagram of the antiferromagnetic XXZ model on the triangular\n lattice"
],
[
"Abstract We determine the quantum phase diagram of the antiferromagnetic spin-1/2 XXZ model on the triangular lattice as a function of magnetic field and anisotropic coupling $J_z$.",
"Using the density matrix renormalization group (DMRG) algorithm in two dimensions we establish the locations of the phase boundaries between a plateau phase with 1/3 N\\'eel order and two distinct coplanar phases.",
"The two coplanar phases are characterized by a simultaneous breaking of both translational and U(1) symmetries, which is reminiscent of supersolidity.",
"A translationally invariant umbrella phase is entered via a first order phase transition at relatively small values of $J_z$ compared to the corresponding case of ferromagnetic hopping and the classical model.",
"The phase transition lines meet at two tricritical points on the tip of the lobe of the plateau state, so that the two coplanar states are completely disconnected.",
"Interestingly, the phase transition between the plateau state and the upper coplanar state changes from second order to first order for large values of $J_z > 2.5J$."
],
[
"Appendix: Finite size scaling of the phase diagram",
"The first order phase transition to the umbrella phase for a given system size can be determined rather accurately as shown in Fig.",
"4 in the main manuscript.",
"In those simulations the magnetization is fixed and the corresponding magnetic field is determined by the derivative of the ground state energy $B(M) = E(M+1/N)-E(M)$ at the transition point.",
"This yields the phase transition lines for system sizes $6\\times 6$ , $6\\times 9$ , and $9\\times 12$ shown in Fig.",
"REF below.",
"Since the data is well behaved, it is possible to determine the corresponding continuous curves $B(J)$ for all values of $B$ by spline interpolation.",
"For each field it is then possible to use a linear fit in reciprocal system size $1/N$ as shown in Fig.",
"REF (top), which determines the estimate in the thermodynamic limit in Fig.",
"REF .",
"A reliable error estimate is difficult in this case, since the finite size data is only available for three data points and a square root behavior $1/\\sqrt{N}$ also yielded reasonable fits, which would push out the estimate of the phase transition line to higher values of $J$ by up to $J/J_z \\sim 0.3$ .",
"Therefore, the phase transition line shown should be taken as a lower estimate.",
"Figure: (Color online) Top: The finite size scaling of the first order phase transition tothe umbrella phase at B/Jz=1.4B/Jz=1.4 and B/Jz=3B/Jz=3.Lower two plots: The finite size scaling of the locations JJ (middle plot) and BB (bottom plot)of the upper and lower tri-critical point.For each finite size we find two tri-critical points where one coplanar, the 1/3 Néel, and the umbrella phase meet.",
"The location of the tri-critical points change in both $B$ and $J$ with finite size, but a reasonable estimate of the corresponding values thermodynamic limit can be made as shown in Fig.",
"REF (lower two plots).",
"Since the two points come quite close with finite size scaling it cannot be ruled out from our data that they merge into one multi-critical point in the thermodynamic limit.",
"The situation is even more complicated for the second order phase transition lines from the 1/3 Néel phase to the coplanar phase.",
"In this case we could not find any systematic finite size scaling, since the phase transition lines for $6\\times 6$ and $6\\times 9$ are very close but there is a larger change when going to $9\\times 12$ .",
"We cannot explain this behavior, but it is maybe not surprising, that it is more difficult to pinpoint the transition lines for second order phase transitions than for first order transition lines in the thermodynamic limit.",
"Figure: The energy estimate as a function of the number of sweeps forJ/J z =1.3J/J_z=1.3, M=1/6+2/NM=1/6+2/N and N=9×12N=9\\times 12 with m=1600m=1600 kept states."
],
[
"Appendix: Convergence of the DMRG data in 2D",
"In order to study two-dimensional systems with the DMRG algorithm, the sites need to be ordered along a one-dimensional chain with effectively long range interactions.",
"Therefore, neighboring sites may be rather far apart in the DMRG algorithm, so that more effort is required to capture the quantum correlations.",
"This problem is even more severe with periodic boundary conditions in both direction, which were necessary for the xxz model on the triangular lattice.",
"As a result we find that the variational state of the system is rather poorly described after the initial DMRG buildup.",
"Typically, the truncation error is not very small ($\\sim 10^{-5}$ ) and is not a reliable measure of the quality of the simulations (in fact it does not depend much on the number of states kept).",
"The finite size algorithm quickly improves this state by “sweeping”.",
"While the correction in energy maybe as large as 20% in the first sweep, the changes become several orders of magnitude smaller after just a few iterations as shown in Fig.",
"REF .",
"Figure: The energy estimate as a function of thenumber of kept states mm forJ/J z =1J/J_z=1, M=1/6M=1/6 and N=6×6N=6\\times 6.However, the convergence with the number of sweeps is also not a guarantee that the system is approaching the correct ground state, since metastable states are possible.",
"It is therefore essential to vary the number of states kept.",
"It is also possible to change the number of kept states during the sweeping procedure or change the initial buidup geometry.",
"Typically, we find that if the data has a smooth behavior with the number of states kept it also produces sensible and accurate data which fits well into the phase diagram (e.g.",
"relative to neighboring points in parameter space).",
"Hence each data point in the phase diagram has been carefully checked for consistency.",
"In principle it would also be possible to try an extrapolation fit with the number of states, but in practice this only produces tiny corrections to the phase diagram but may in turn produce artifacts.",
"Figure: The structure factors S z (Q)/NS^z(Q)/N (upper panel) and S ± (Q)/NS^{\\pm }(Q)/N (lower panel)as a function of the number of kept states mm for J/J z =1J/J_z=1, M=1/6M=1/6 and N=6×6N=6\\times 6.The following Figs.",
"REF -REF illustrate the typical behavior of the DMRG data for the energy, the structure factors, and quantum information measures at examplary values of the parameters and different system sizes as a function of number of states kept.",
"Note, that the energies alone determine much of the phase diagram since they define the magnetization plateau of the 1/3 Néel phase.",
"Figure: The energy EE dependence of the kept states at J/J z =1J/J_z=1, M=1/6M=1/6 and N=6×9N=6\\times 9.Figure: The structure factors S z (Q)/NS^z(Q)/N (left panel) and S ± (Q)/NS^{\\pm }(Q)/N (right panel) dependence of the kept states at J/J z =1J/J_z=1, M=1/6M=1/6 and N=6×9N=6\\times 9.The behavior of the energy and structure factor is shown in Fig.",
"REF -REF for system sizes 6x6 and 6x9.",
"As expected the accuracy improves with number of states kept and is fully sufficient already for ca.",
"1200 kept states.",
"In particular, the difference between the data for 800 or 1200 kept states would not be noticable in any of the plots.",
"More importantly, there are no big jumps which would be an indicator for metastable states.",
"Larger system sizes of $9\\times 12$ are more difficult.",
"As shown in Fig.",
"REF there may be large jumps when going from 600 states to 900 states, which indicates a metastable situtation.",
"However, all paramaters converge to stable values for larger values of the number of states kept.",
"We have checked convergence for up to $m=2400-3000$ at selected points.",
"Figure: Energy and quantum discord as a function of number of states kept in a 9×129\\times 12 lattice for different phases in the phase diagram and different parameters as indicated in the legends."
]
] | 1403.0008 |
[
[
"Weak Convergence to Brownian Motion on Sub-Riemannian Manifolds"
],
[
"Abstract This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold $M$.",
"To construct such a random walk we first address several issues related to the degeneracy of such a manifold.",
"In particular, we define a family of sub-Laplacian operators naturally connected to the geometry of the underlining manifold.",
"In the case when $M$ is a Riemannian (non-degenerate) manifold, we recover the Laplace-Beltrami operator.",
"We then construct the corresponding random walk, and under standard assumptions on the sub-Laplacian and $M$ we show that this random walk weakly converges to a process, horizontal Brownian motion, whose infinitesimal generator is the sub-Laplacian.",
"An example of the Heisenberg group equipped with a standard sub-Riemannian metric is considered in detail, in which case the sub-Laplacian we introduced is shown to be the sum of squares (H\\\"{o}rmander's) operator."
],
[
"Introduction",
"This paper describes a geometrically natural piecewise Hamiltonian-flow random walk in a sub-Riemannian manifold, which converges weakly to a horizontal Brownian motion on the manifold.",
"In this setting we define a sub-Laplacian by averaging over the second-order directional derivatives in the directions of the Hamiltonian flow.",
"In particular, in the case $\\mathcal {H} = TM$ , we recover the Laplace-Beltrami operator; in the case of the Heisenberg group equipped with a standard sub-Riemannian metric, we recover the sum of squares (Hörmander's) operator.",
"While the current paper presents the probabilistic aspects of this construction, the geometric exploration of this sub-Laplacian can be found in [10].",
"As we will see in Section , the sub-Laplacian we study is the one that generates the horizontal Brownian motion.",
"Over the last half century, Brownian motion on Riemannian manifolds has developed into a well-understood and rich theory.",
"Much of this development relies heavily on the Riemannian structure as one can see from the monographs [9], [11].",
"There are two major ingredients which are canonical in the Riemannian case: the Riemannian volume $\\mu $ and the corresponding Laplace-Beltrami operator $\\Delta _{LB}$ .",
"Recall that the Laplace-Beltrami operator is usually defined as $\\operatorname{div} \\operatorname{grad}$ , where $\\operatorname{div}$ is defined with respect to the Riemannian volume $\\mu $ .",
"From here, a Brownian motion on a Riemannian manifold can be described as a stochastic process with the infinitesimal generator $\\Delta _{LB}$ .",
"This approach is not easily available in the sub-Riemannian case.",
"There are several measures which might be used in lieu of the Riemannian volume such as the Hausdorff measure, Popp's measure (see [16], [1]), left or right Haar measure in the case of Lie groups.",
"Each choice of the measure will lead to a possibly different sub-Laplacian, and therefore to a different Brownian motion.",
"A more detailed analysis of sub-Laplacians and natural choices of measures is presented in [10].",
"Instead of making this choice, we develop a more classical approach of constructing a Brownian motion as the limit of an appropriately-scaled random walk.",
"Any complete list of references working in this direction on Riemannian manifolds would undoubtedly include the now-classic works [12], [15], and most relevant to our work, the isotropic transport process by M. Pinsky [17].",
"Motivated by Pinsky's approach, the sub-Laplacian we construct is canonical with respect to the limiting process of the random walk.",
"This sub-Laplacian $\\mathcal {L}$ introduced in (REF ) is elemental in the sub-Riemannian setting without some a priori canonical choice.",
"There are several fundamental issues in our construction which are not apparent in the Riemannian setting.",
"Such issues would prevent us from adopting a Pinsky-type process to a sub-Riemannian manifold without a reinterpretation of some standard objects and their relations which are taken for granted in the Riemannian setting.",
"One of these basic relations which has been exploited is the duality between the tangent and cotangent spaces.",
"This duality is not available in the sub-Riemannian setting, which led us to the realization that it seems to be more appropriate to construct the random walk in the cotangent space, rather than in the tangent space.",
"This also manifests itself in the use of a compatible Riemannian metric in the definition of the sub-Laplacian $\\mathcal {L}$ , which allows us to overcome the problem of the non-uniqueness of solutions to the Hamilton-Jacobi equations with given initial position and velocity (tangent) vector.",
"However, we show that the need for a compatible metric is illusory as $\\mathcal {L}$ actually only depends on the corresponding “vertical” bundle.",
"We further mention that there is interest in seeing how recent work by Bakry, Baudoin, Garafalo et al [2], [4], [5], [6] on generalized curvatures of such manifolds is related to dissipation of horizontal diffusions.",
"We expect further study of connections between diffusions on sub-Riemannian manifolds and corresponding generators, as well as of behavior of hypoelliptic heat kernels and corresponding functional inequalities such as in [7], [8], [3], [14]."
],
[
"Sub-Riemannian basics",
"We start by reviewing standard definitions of sub-Riemannian geometry that can be found e.g.",
"in [16] and originally were introduced by R. Strichartz in [18], [19].",
"Let $M$ be a $d$ -dimensional connected smooth manifold, with tangent and cotangent bundles $TM$ and $T^*M$ respectively.",
"Definition 2.1 For $m \\leqslant d$ , let $\\mathcal {H}$ be a smooth sub-bundle of $TM$ where each fiber $\\mathcal {H}_q$ has dimension $m$ and is equipped with an inner product which smoothly varies between fibers.",
"Then the triple $\\left( M, \\mathcal {H}, \\langle \\cdot , \\cdot \\rangle \\right)$ is called a sub-Riemannian manifold of rank $m$ ; $\\mathcal {H}$ is called a horizontal distribution on $M$ , and $\\langle \\cdot , \\cdot \\rangle $ a sub-Riemannian metric; sections of $\\mathcal {H}$ are called horizontal vector fields and curves on $M$ whose velocity vectors are always horizontal are called horizontal curves.",
"Assumption 2.2 (Hörmander's condition) Throughout this paper we assume that the distribution $\\mathcal {H}$ satisfies Hörmander's (bracket generating) condition; that is, horizontal vector fields with their Lie brackets span the tangent space $T_{q}M$ at every point $q \\in M$ .",
"Under Hörmander's condition any two points on $M$ can be connected by a horizontal curve by the Chow-Rachevski theorem.",
"Thus there is a natural sub-Riemannian distance (Carnot-Carathéodory distance) on $M$ defined as the infimum over the lengths of horizontal curves connecting two points.",
"In turn, this affords us the notion of a horizontal geodesic, a horizontal curve whose length (locally) realizes the Carnot-Carathéodory distance.",
"Due to degeneracy of the sub-Riemannian metric on the tangent bundle, it is convenient to introduce the cometric on $T^{\\ast }M$ corresponding to the sub-Riemannian structure.",
"This is a particular section of the bundle of symmetric bilinear forms on the cotangent bundle, $\\cdot , \\cdot _{q}: T^{\\ast }_{q}M \\times T^{\\ast }_{q}M \\rightarrow \\mathbb {R}, \\ q \\in M.$ We relate the cometric to the sub-Riemanian metric via the sub-Riemannian bundle map $\\beta : T^{\\ast }M \\rightarrow TM$ with image $\\mathcal {H}$ defined in the spirit of Riesz's theorem by $\\langle \\beta _q(p), v \\rangle _q = p(v)$ for all $q\\in M, p \\in T^{\\ast }_q M, \\text{ and } v \\in \\mathcal {H}_q M.$ Hence the correspondence between the sub-Riemannian metric and cometric can be summarized as $\\varphi , \\psi _{q} = \\langle \\beta _{q}(\\varphi ), \\beta _{q}(\\psi ) \\rangle _{q} = \\varphi \\left( \\beta _{q}(\\psi ) \\right) = \\psi \\left( \\beta _{q}(\\varphi ) \\right),$ for all $q \\in M,$ and $\\varphi , \\psi \\in T^{\\ast }_{q}M$ .",
"Armed with the cometric, we conclude this section by defining the corresponding sub-Riemannian Hamiltonian $H: T^{\\ast } M \\rightarrow \\mathbb {R}$ by $H\\left( q, p \\right) := \\frac{1}{2} p, p _{q}, \\ q \\in M, p \\in T^{\\ast }_{q} M$ from which we can recover the cometric via polarization.",
"Again we note the following equivalent descriptions of the map $H\\left( q, p \\right) = \\frac{1}{2}p, p _{q} = \\frac{1}{2} \\beta _{q}\\left( p \\right), \\ \\beta _{q}\\left( p \\right) _{q} = \\frac{1}{2} p \\left( \\beta _{q}\\left( p \\right) \\right).$ The Hamiltonian is used to generate the dynamics of the system, where $H\\left( q, p \\right)$ gives the (kinetic) energy of a body located at $q$ with momentum $p$ ."
],
[
"Canonical coordinates and compatible metrics",
"In the non-degenerate (Riemannian) case, the metric and cometric are matrix inverses of each other when written in any given local frame.",
"Indeed, these matrices are represented componentwise by the lowered and raised indices $g_{ij}$ and $g^{ij}$ respectively.",
"The degeneracy in the sub-Riemannian case disallows for such a relationship, leaving us with a choice of Riemannian metrics which will be compatible with a given sub-Riemannian structure.",
"The general non-canonical choice of compatible metrics will eventually lead us to defining a family of sub-Laplacians corresponding to the choice of compatible metric.",
"Definition 2.3 Let $g$ be a Riemannian metric on $M$ extending the sub-Riemannian metric; i.e., $g|_{\\mathcal {H}_q \\times \\mathcal {H}_q} = \\langle \\cdot , \\cdot \\rangle _q$ for all $q \\in M$ .",
"Then we say that $g$ is compatible with the sub-Riemannian structure, or simply that $g$ is a compatible metric.",
"Within this paper, the purpose of introducing a compatible metric $(\\cdot , \\cdot )$ is to take advantage of the induced bundle map $g : TM \\rightarrow T^*M$ defined by $g(v) = ( \\cdot , v)$ , the standard duality $TM \\leftrightarrow T^{\\ast }M$ described generally through Riesz's theorem.",
"This is a tool that we lose in the sub-Riemannian setting as we can associate to each cotangent (momentum) vector a corresponding horizontal (velocity) vector via $T^{\\ast }M \\stackrel{\\beta }{\\longrightarrow } \\mathcal {H}$ , but are unable to canonically map back $\\mathcal {H} \\stackrel{?",
"}{\\longrightarrow } T^{\\ast }M$ .",
"With a compatible metric $g$ on hand, we then recover our return $\\mathcal {H} \\stackrel{g}{\\longrightarrow } T^{\\ast }M$ .",
"However, as already mentioned, with the full strength of the Riemannian metric, we have a full bundle isomorphism $TM \\rightarrow T^{\\ast }M$ , but this is more machinery than we need since we will only be considering the mapping on the horizontal distribution; something we explore presently through an observation from [10].",
"Proposition 2.4 Let $( \\cdot , \\cdot )$ be a Riemannian metric on $M$ and let $g : TM \\rightarrow T^{\\ast }M$ be the corresponding bundle map.",
"Then $( \\cdot , \\cdot )$ is a compatible metric if and only if $\\beta \\circ g |_{\\mathcal {H}} = \\operatorname{Id}_{\\mathcal {H}}$ .",
"Further, suppose $(\\cdot , \\cdot )_1$ and $(\\cdot , \\cdot )_2$ are compatible metrics with corresponding bundle maps $g_1, g_2 : TM \\rightarrow T^{\\ast }M$ .",
"For $i = 1,2$ , let $\\mathcal {V}_i$ be the orthogonal compliment of $\\mathcal {H}$ in $TM$ with respect to $( \\cdot , \\cdot )_i$ .",
"Then $g_1(v) = g_2(v)$ for every $v \\in \\mathcal {H}$ if and only if $\\mathcal {V}_1 = \\mathcal {V}_2$ .",
"For a Riemannian metric $( \\cdot , \\cdot )$ , the corresponding bundle map $g : TM \\rightarrow T^{\\ast }M$ can be written as $g_{\\mathcal {H}} \\oplus g_{\\mathcal {V}} : \\mathcal {H} \\oplus \\mathcal {V} \\rightarrow T^{\\ast }M$ , where $\\mathcal {V}$ is the $(\\cdot ,\\cdot )$ -orthogonal compliment of $\\mathcal {H}$ in $TM$ .",
"From here, noticing that $g(\\mathcal {V})=\\operatorname{Null}(\\beta )$ , and thus $T^*M = g_{\\mathcal {H}}(\\mathcal {H}) \\oplus \\operatorname{Null}(\\beta )$ , we have $\\beta = \\beta _{\\mathcal {H}} \\oplus 0 : g_{\\mathcal {H}}(\\mathcal {H}) \\oplus \\operatorname{Null}(\\beta ) \\rightarrow TM$ .",
"Moreover, $g$ is compatible if and only if $g_{\\mathcal {H}} = \\beta _{\\mathcal {H}}^{-1}$ which in turn happens if and only if $\\beta \\circ g = \\operatorname{Id}_{\\mathcal {H}} \\oplus 0$ .",
"From here, it is easy enough to deduce that the mapping $\\mathcal {H} \\ni v \\mapsto g(v)$ depends only on $g_{\\mathcal {H}}$ and $\\mathcal {V}$ , but not on the behavior of $g_{\\mathcal {V}}$ .",
"Since, if $g$ is compatible, then $g_{\\mathcal {H}} = \\beta _{\\mathcal {H}}^{-1}$ is completely determined by $\\mathcal {V}$ and the sub-Riemannian structure, the assertions of this proposition follow.",
"With Proposition REF understood, instead of introducing a compatible metric, we could build up the remaining work by selecting a smooth vertical sub-bundle $\\mathcal {V} \\subset TM$ such that $TM = \\mathcal {H} \\oplus \\mathcal {V}$ , use this to distinguish a compliment of $\\operatorname{Null}(\\beta )$ , say $H$ , in $T^*M$ such that we have $\\beta = \\beta _{\\mathcal {H}} \\oplus 0 : H \\oplus \\operatorname{Null}(\\beta ) \\rightarrow TM$ and hence recover a “return map” with $\\mathcal {H} \\stackrel{\\beta _{\\mathcal {H}}^{-1}}{\\longrightarrow } T^*M$ .",
"As for the theory that follows, the only role that a compatible metric $g$ serves is to distinguish the vertical bundle.",
"However, for some calculational purposes, it seems advantageous to keep working in terms of a compatible metric $g$ .",
"Notation 2.5 Let $g$ be a compatible metric.",
"For local coordinates ${\\bf x} =(x^1, ..., x^d)$ on $M$ , we define the local maps $\\beta ^{ij}: M \\rightarrow \\mathbb {R}$ and $g_{ij} : M \\rightarrow \\mathbb {R}$ by $\\beta ^{ij}\\left( q \\right) := d x^i, d x^j _{q} \\text{ and } g_{ij}(q) = \\Big \\langle \\frac{\\partial }{\\partial x^i}, \\frac{\\partial }{\\partial x^j} \\Big \\rangle _q$ for all $q$ in the domain of ${\\bf x}$ .",
"The $d \\times d$ matrices with entries $\\beta ^{ij}$ and $g_{ij}$ will be denoted by $B$ and $G$ respectively.",
"As $B$ is the matrix representation of the bundle map $\\beta : T^{\\ast }M \\rightarrow TM$ in local coordinates, $G$ is the local coordinate matrix representation of the bundle map $TM \\rightarrow T^{\\ast }M$ defined by $v \\mapsto g(\\cdot ,v)$ .",
"Example 2.1 (Contact manifolds) Let $M$ be a $2n+1$ -dimensional manifold and $\\omega $ a contact 1-form on $M$ , that is, a 1-form such that $d \\omega $ is non-degenerate on $\\operatorname{Ker}(\\omega )$ .",
"Let $\\mathcal {H}:=\\operatorname{Ker}(\\omega )$ , which defines a $n$ -dimensional horizontal distribution on $M$ , called a contact distribution, and we assume that $\\mathcal {H}$ is equipped with inner product $\\langle \\cdot , \\cdot \\rangle $ .",
"The sub-Riemannian manifold $\\left( M, \\mathcal {H}, \\langle \\cdot , \\cdot \\rangle \\right)$ is called a contact sub-Riemannian manifold.",
"With any contact form $\\omega $ we can associate its Reeb vector field, which is the unique vector field $X_{0}$ satisfying the conditions $\\omega \\left( X_{0} \\right) = 1$ and $d \\omega (X_{0}, \\cdot ) = 0$ .",
"Hence for any local orthonormal frame $X_{1}, ..., X_{2n}$ for the distribution $\\mathcal {H}$ we have that $X_{0}, X_{1}, ..., X_{2n}$ is a local frame, since $X_{0}$ is transversal to $\\mathcal {H}$ .",
"Finally, if $\\langle \\cdot , \\cdot \\rangle $ is an inner product on $\\mathcal {H}$ , we can extend it to $X_{0}$ by $g\\left( X_{0}, X_{0} \\right)=1$ and setting $\\mathcal {H} \\perp X_{0}$ .",
"This $g$ is then naturally compatible with the sub-Riemannian structure.",
"Moreover, for contact sub-Riemannian manifolds there are no abnormal geodesics, that is, all geodesics are smooth and are projections of the trajectories of the Hamiltonian vector field in $T^{\\ast }M$ given by the Legendre transform of the inner product on $\\mathcal {H}$ .",
"The Heisenberg group is an example of a contact manifold where $\\omega $ is a standard symplectic form."
],
[
"Hamilton-Jacobi Equations",
"We can now re-write the Hamiltonian $H : T^{\\ast }M \\rightarrow \\mathbb {R}$ defined by (REF ) using canonical coordinates.",
"By identifying the vector $(q^1, ..., q^d, p_1, ..., p_d)$ in $\\mathbb {R}^{d \\times d}$ with the point $(q, p) \\in T^{\\ast }M$ using local coordinates for the standard identification $q^i = x^i(q)$ and $p = \\sum \\limits _{i=1}^{d} p_i d x^i$ , then $H(q,p) = \\frac{1}{2} \\sum _{i,j = 1}^{d} p_i p_j \\beta ^{ij}(q).$ A curve $(q(t), p(t)) \\in T^{\\ast }M$ satisfies the Hamilton-Jacobi equations when $&\\dot{q}^i(t) = \\frac{\\partial H}{\\partial p_i}\\left( q(t), p(t)\\right) = \\frac{1}{2} \\sum _{j =1}^{d} p_j(t) \\beta ^{ij}(q(t))\\\\&\\dot{p}_i(t) = - \\frac{\\partial H}{\\partial q^i}\\left(q(t), p(t) \\right) = \\sum _{k, j=1}^{d} p_k(t) p_j(t) \\frac{\\partial \\beta ^{kj}}{ \\partial q^i} \\left|_{q(t)}\\right.$ where we have slightly abused notation in the common way, conflating $\\frac{\\partial }{\\partial p_i}$ with the partial derivative of (REF ) in terms of $p_i$ , and $\\frac{\\partial }{\\partial q^i}$ with $\\frac{\\partial }{\\partial x^i}$ in ().",
"Equations (REF ) and () are collectively known as the Hamilton-Jacobi equations.",
"Taking a time derivative in (REF ) we get ${\\ddot{q}}^k(t) = \\sum _{i, j, l = 1}^{d} \\left\\lbrace \\beta ^{il}(q(t)) \\frac{\\partial \\beta ^{kj}}{\\partial q^l}\\left|_{q(t)}\\right.",
"- \\frac{1}{2} \\beta ^{kl}(q(t)) \\frac{\\partial \\beta ^{ij}}{\\partial q^l} \\left|_{q(t)}\\right.\\right\\rbrace p_i(t) p_j(t).$ Define the raised Christoffel symbols locally by $\\Gamma ^{ijk}(q) := - \\frac{1}{2} \\sum _{l = 1}^{d} \\left\\lbrace \\beta ^{il}(q) \\frac{\\partial \\beta ^{jk}}{\\partial x^l}\\Big |_{q} + \\beta ^{jl}(q) \\frac{\\partial \\beta ^{ik}}{\\partial x^l} \\Big |_q- \\beta ^{lk}(q) \\frac{\\partial \\beta ^{ij}}{\\partial x^l} \\Big |_{q}\\right\\rbrace .$ Rewriting (REF ) with (REF ) while suppressing the time dependence, ${\\ddot{q}}^k = - \\sum _{i, j = 1}^{d} \\Gamma ^{i j k}(q) p_i p_j.$ The negative signs in (REF ) and (REF ) are just by convention so that the acceleration term is consistent with standard Riemannian definitions.",
"Notation 2.6 We let $\\Phi $ be the flow of the Hamilton-Jacobi equations (REF ) and ().",
"That is, $\\Phi $ is a map $\\Phi : [0, \\tau ) \\times T^{\\ast }M \\longrightarrow T^{\\ast }M,$ such that if $(x,p) \\in T^{\\ast }_xM$ then $t \\mapsto \\Phi _t(x,p)$ is the curve $(q(t), p(t))$ in $T^{\\ast }M$ satisfying the Hamilton-Jacobi equations with initial conditions $q(0) = x$ and $p(0) = p$ for $t$ in some maximal interval $[0,\\tau )$ .",
"Remark 2.7 If $(q(t),p(t)) = \\Phi _t(x,p)$ , then $q(t)$ is a horizontal curve.",
"Indeed, (REF ) gaurantees that $\\dot{q}(t) = \\beta (p(t)) \\in \\mathcal {H}_{q(t)}$ ."
],
[
"Horizontal sub-Laplacians and the Heisenberg group",
"In this section we introduce a family of second order differential operators on $M$ indexed by Riemannian metrics compatible with the sub-Riemannian structure $\\left( M, \\mathcal {H}, \\langle \\cdot , \\cdot \\rangle \\right)$ .",
"In the Riemannian case when $\\mathcal {H} = TM$ , we recover the Laplace-Beltrami up to a constant scaling factor; in the Heisenberg case using the standard compatible metric introduced in Example REF , we get the familiar sums of squares Laplacian up to a constant scaling factor."
],
[
"Horizontal sub-Laplacians",
"Definition REF below introduces horizontal sub-Laplacian operators, but before we can give the definition, some notation is in order.",
"Notation 3.1 We denote the unit sphere in $\\mathcal {H}_x$ by $\\mathcal {S}^{\\mathcal {H}}_x := \\lbrace v \\in \\mathcal {H}_x : \\langle v, v \\rangle _x = 1\\rbrace $ .",
"The (unique) rotationally invariant measure on $\\mathcal {S}_x$ will be denoted $\\mathbb {U}_x$ .",
"Definition 3.2 Let $(\\cdot , \\cdot )$ be a compatible metric, and let $g$ be the corresponding bundle map $TM \\rightarrow T^{\\ast }M$ .",
"We define $\\mathcal {L} : C^{\\infty }_c(M) \\rightarrow \\mathbb {R}$ as $\\mathcal {L} f (x) := \\int _{\\mathcal {S}^{\\mathcal {H}}_x} \\left\\lbrace \\frac{d}{dt} \\Big |_0 \\frac{d}{ds} \\Big |_0 f \\big (\\Phi _{t+s}(x,g(v))\\big ) \\right\\rbrace \\mathbb {U}_x(dv).$ We will call $\\mathcal {L}$ the horizontal sub-Laplacian corresponding to $g$ .",
"As is now obvious from Proposition REF and the remarks that followed, we have the following statement.",
"Proposition 3.3 Suppose $(\\cdot , \\cdot )_1$ and $(\\cdot , \\cdot )_2$ are compatible metrics giving rise to orthogonal compliments $\\mathcal {V}_1$ and $\\mathcal {V}_2$ of $\\mathcal {H}$ , respectively.",
"For $i = 1,2$ , if $\\mathcal {L}_i$ is defined by (REF ) with respect to $(\\cdot , \\cdot )_i$ , then $\\mathcal {L}_1 = \\mathcal {L}_2$ whenever $\\mathcal {V}_1 = \\mathcal {V}_2$ ."
],
[
"A formula for $\\mathcal {L}$ in local coordinates",
"Working in local coordinates, we set $q(t) := \\pi \\left( \\Phi _t(x, p) \\right)$ , where $\\pi $ is the projection onto $M$ .",
"Defining $v = \\beta (p)$ , we get $\\begin{aligned}&\\frac{d}{dt} \\Big |_0 \\frac{d}{ds}\\Big |_0 f \\left(q(t+s)\\right) = \\frac{d}{dt}\\Big |_0 \\left\\lbrace \\sum _{i=1}^{d} \\dot{q}^i(t) \\frac{\\partial f}{\\partial x^i} \\Big |_{q(t)} \\right\\rbrace \\\\&= \\sum _{i=1}^{d}\\left\\lbrace {\\ddot{q}}^i(0) \\frac{\\partial f}{\\partial x^i}\\Big |_{x} + \\sum _{j=1}^{d} \\dot{q}^i(0) \\dot{q}^j(0) \\frac{\\partial ^2 f}{\\partial x^i \\partial x^j}\\Big |_x\\right\\rbrace \\\\&= \\sum _{i=1}^{d} \\left\\lbrace -\\sum _{k,l=1}^{d} \\Gamma ^{k l i}(x)\\, p_k p_l \\frac{\\partial f}{\\partial x^i} \\Big |_x + \\sum _{j=1}^{d} v^i v^j \\frac{\\partial ^2 f}{\\partial x^i \\partial x^j} \\Big |_x \\right\\rbrace \\\\&= \\sum _{i, j =1}^{d} \\left\\lbrace v^i v^j \\, \\frac{\\partial ^2 f}{\\partial x^i \\partial x^j} \\Big |_x - \\sum _{k=1}^{d} \\Gamma ^{i j k}(x) p_i \\, p_j \\frac{\\partial f}{\\partial x^k} \\Big |_x \\right\\rbrace \\end{aligned}$ Proposition 3.4 Let $(\\cdot , \\cdot )$ be a compatible metric with corresponding bundle map $g : TM \\rightarrow T^{\\ast }M$ .",
"For $1 \\leqslant i, j \\leqslant d$ , $\\int _{\\mathcal {S}^{\\mathcal {H}}_x} v^i v^j \\mathbb {U}_x(dv) = \\frac{1}{m} \\beta ^{ij}(x)$ and $\\int _{\\mathcal {S}^{\\mathcal {H}}_x} p_i p_j \\mathbb {U}_x(dv) = \\frac{1}{m} \\sum _{a, b=1}^{d} g_{i a} \\beta ^{ab} g_{bj}(x).$ Here $p = g(v)$ .",
"Rewrite (REF ) as $\\int _{\\mathcal {S}^{\\mathcal {H}}_x} dx^i (v) dx^j(v) \\mathbb {U}_x(dv) = \\int _{\\mathcal {S}^{\\mathcal {H}}_x} \\langle \\beta _x(dx^i), v \\rangle \\langle \\beta _x(dx^j), v \\rangle d\\mathbb {U}_x(v)\\\\= \\frac{1}{m} \\langle \\beta _x(dx^i), \\beta _x(dx^j) \\rangle = \\frac{1}{m}\\beta ^{ij}(x)$ The second equality follows from Corollary REF below.",
"From here (REF ) follows by a similar argument after realization that $p_i = \\sum \\limits _{a =1}^{d} g_{ia} v^a$ and $p_j = \\sum \\limits _{b=1}^{d} g_{jb} v^b$ .",
"Combining Proposition REF with (REF ) leads immediately to Theorem 3.5 The horizontal sub-Laplacian indexed by $g$ can be locally written as $\\begin{aligned}\\mathcal {L} = \\frac{1}{m} \\sum _{i,j = 1}^{d} \\left\\lbrace \\beta ^{ij} \\frac{\\partial ^2 }{\\partial x^i \\partial x^j} - \\sum _{a,b,k=1}^{d} \\Gamma ^{i j k} g_{ia}\\beta ^{ab} g_{bj} \\frac{\\partial }{\\partial x^k} \\right\\rbrace \\\\= \\frac{1}{m} \\sum _{i,j=1}^{d} \\left\\lbrace \\beta ^{ij} \\frac{\\partial ^2 }{\\partial x^i \\partial x^j} - \\sum _{k=1}^{d} \\Gamma ^{i j k} \\left[ GBG\\right]_{ij} \\frac{\\partial }{\\partial x^k} \\right\\rbrace \\end{aligned}$ where $\\left[ GBG \\right]_{ij}$ is the $ij$ th entry of the matrix $GBG$ and $G$ and $B$ are defined in Notation REF .",
"Remark 3.6 In the case that $\\mathcal {H} = TM$ , $B = G^{-1}$ and hence $\\mathcal {L} = \\frac{1}{m} \\sum _{i,j = 1}^{d} \\left\\lbrace \\beta ^{ij} \\frac{\\partial ^2 }{\\partial x^i \\partial x^j} - \\sum _{k=1}^{d} \\Gamma ^{i j k} g_{ij} \\frac{\\partial }{\\partial x^k} \\right\\rbrace ,$ which is the ($\\frac{1}{m}$ scaled) local formula for the Laplace-Beltrami operator on the Riemannian manifold $(M, g)$ .",
"With Proposition REF in mind, (REF ) appears deceivingly dependent on the structure of the compatible metric with the repeat appearance of its corresponding matrix $G$ .",
"However, using the notation in the proof of Proposition REF , we have $g \\circ \\beta \\circ g = g_{\\mathcal {H}} \\circ 0$ , which as the proof of and remarks following Proposition REF indicate, $g_\\mathcal {H}$ is determined by the sub-Riemannian structure once the vertical bundle $\\mathcal {V}$ is fixed.",
"The following example in the Heisenberg case illustrates this."
],
[
"An example: the Heisenberg group",
"Let $\\mathbb {H}$ be the Heisenberg group; that is, $\\mathbb {H}\\cong \\mathbb {R}^3$ with the multiplication defined by $\\left( x_{1}, y_{1}, z_{1} \\right) \\star \\left( x_{2}, y_{2}, z_{2} \\right):=\\left( x_{1}+x_{2}, y_{1}+y_{2}, z_{1}+z_{2} + \\frac{1}{2} \\omega \\left( x_{1}, y_{1}; x_{2}, y_{2}\\right)\\right),$ where $\\omega $ is the standard symplectic form $\\omega \\left( x_{1}, y_{1}; x_{2}, y_{2}\\right):=x_{1}y_{2} - y_{1}x_{2}.$ Left multiplication by $\\left( x, 0, 0 \\right)$ and $\\left( 0, y, 0 \\right)$ induce two left-invariant vector fields $X\\left( q \\right) := \\frac{\\partial }{\\partial x}\\Big |_{q} - \\frac{1}{2} y \\frac{\\partial }{\\partial z}\\Big |_{q}\\\\Y\\left( q \\right) := \\frac{\\partial }{\\partial y}\\Big |_{q} + \\frac{1}{2} x \\frac{\\partial }{\\partial z}\\Big |_{q}$ for any $q \\in \\mathbb {H}$ .",
"At each point $q \\in \\mathbb {H}$ the globally defined vector fields $X\\left( q \\right)$ and $Y\\left( q \\right)$ span a two-dimensional subspace of $T_{q}\\mathbb {H}$ ; set $\\mathcal {H}_{q}:=\\operatorname{Span} \\left\\lbrace X\\left( q \\right), Y\\left( q \\right) \\right\\rbrace $ and then $\\mathcal {H}:=\\bigcup _{q \\in \\mathbb {H}} \\mathcal {H}_{q}$ can be taken as the horizontal distribution.",
"Moreover, at each $q \\in \\mathbb {H}$ we have $[ X\\left( q \\right), Y\\left( q \\right)]=\\frac{\\partial }{\\partial z}\\Big |_{q}=: Z\\left( q \\right),$ and so Hörmander's condition is satisfied.",
"Consider $M=\\mathbb {H}$ , the horizontal distribution $\\mathcal {H}$ defined as above, and the inner product $\\langle \\cdot , \\cdot \\rangle $ on $\\mathcal {H}_{q}$ defined so that $\\lbrace X\\left( q \\right), Y\\left( q \\right) \\rbrace $ is an orthonormal basis for $\\mathcal {H}_{q}$ .",
"Recall also that in Example REF we described $\\left( \\mathbb {H}, \\mathcal {H}, \\langle \\cdot , \\cdot \\rangle \\right)$ as a contact manifold with $Z$ as a Reeb vector field.",
"A covector $\\varphi \\in T_p^{\\ast }M$ will be identified with the triple $(\\varphi _1, \\varphi _2, \\varphi _3) \\in \\mathbb {R}^3$ via $\\varphi = \\varphi _1 dx + \\varphi _2 dy + \\varphi _3 dz$ .",
"We have that for each $q = (x,y,z) \\in \\mathbb {H}$ , the sub-Riemannian bundle map $\\beta : T^{\\ast }M \\rightarrow TM$ is defined by $(\\varphi _1, \\varphi _2, \\varphi _3) \\xrightarrow{}\\left( \\varphi _1 - \\frac{1}{2}y \\varphi _3,\\varphi _2 + \\frac{1}{2}x \\varphi _3,\\frac{1}{2}(x\\varphi _2 - y \\varphi _1) + \\frac{1}{4}(y^2 + x^2)\\varphi _3\\right).$ The matrix representation of $\\beta $ with entries $\\beta ^{ij} = dx^i(\\beta (dx^j))$ is $ B(x,y,z) =\\begin{pmatrix}1 & 0 & -\\frac{y}{2} \\\\0 & 1 & \\frac{x}{2} \\\\-\\frac{y}{2} & \\frac{x}{2} & \\frac{x^2 + y^2}{4}\\end{pmatrix}$ Using the fact that $\\left( \\mathbb {H}, \\mathcal {H}, \\langle \\cdot , \\cdot \\rangle \\right)$ is a contact sub-Riemannian manifold, we can extend the sub-Riemannian metric $\\langle \\cdot , \\cdot \\rangle $ to the Riemannian metric $g$ which makes $\\lbrace X,Y,Z\\rbrace $ a global orthogonal frame with $g(Z,Z) = \\lambda > 0$ .",
"The matrix representation of $g$ with entries $g_{ij} = g( \\frac{\\partial }{\\partial x^j}, \\frac{\\partial }{\\partial x^i})$ is $ G(x,y,z) =\\begin{pmatrix}1+ \\frac{\\lambda y^2}{4} & -\\frac{\\lambda xy}{4} & \\frac{\\lambda y}{2} \\\\-\\frac{\\lambda xy}{4} & 1 + \\frac{\\lambda x^2}{4} & -\\frac{\\lambda x}{2} \\\\\\frac{\\lambda y}{2} & -\\frac{\\lambda x}{2} & \\lambda \\end{pmatrix},$ and therefore, $GBG = \\begin{pmatrix}1+ \\frac{\\lambda y^2}{4} & -\\frac{\\lambda xy}{4} & \\frac{\\lambda y}{2} \\\\-\\frac{\\lambda xy}{4} & 1 + \\frac{\\lambda x^2 }{4} & -\\frac{\\lambda x}{2} \\\\\\frac{\\lambda y}{2} & -\\frac{\\lambda x}{2} & \\lambda \\end{pmatrix}\\begin{pmatrix}1 & 0 & -\\frac{y}{2}\\\\0 & 1 & \\frac{x}{2}\\\\-\\frac{y}{2} & \\frac{x}{2} & \\frac{x^2 + y^2}{4}\\end{pmatrix}\\begin{pmatrix}1+ \\frac{\\lambda y^2}{4} & -\\frac{\\lambda xy}{4} & \\frac{\\lambda y}{2} \\\\-\\frac{\\lambda xy}{4} & 1 + \\frac{\\lambda x^2}{4} & -\\frac{\\lambda x}{2} \\\\\\frac{\\lambda y}{2} & -\\frac{\\lambda x}{2} & \\lambda \\end{pmatrix} \\\\=\\begin{pmatrix}1 & 0 & 0 \\\\0 & 1 & 0\\\\0 & 0 & 0\\end{pmatrix}.$ Here you can see the manifestation of Proposition REF through the independence of $GBG$ on any choice of $\\lambda $ .",
"Using (REF ) and (REF ), for any $k=1,2,3$ , $\\Gamma ^{11k} = \\Gamma ^{22k} = 0$ , which gives us all values needed to explicitly find (REF ) in this context.",
"$\\mathcal {L} &= \\frac{1}{2} \\sum _{i, j =1}^{3} \\left\\lbrace \\beta ^{ij} \\frac{\\partial ^2}{\\partial x^i \\partial x^j} \\right\\rbrace -\\frac{1}{2} \\sum _{i, j, k=1}^{3} \\left\\lbrace \\Gamma ^{i j k} \\left[ GBG \\right]_{ij} \\frac{\\partial }{\\partial x^k} \\right\\rbrace \\\\&= \\frac{1}{2} \\sum _{i, j =1}^{3} \\Bigg \\lbrace \\beta ^{ij} \\frac{\\partial ^2}{\\partial x^i \\partial x^j} \\Bigg \\rbrace - 0\\\\&= \\frac{1}{2}\\Bigg \\lbrace \\frac{\\partial ^2}{\\partial x^2} + \\frac{\\partial ^2}{\\partial y^2} + \\frac{1}{4} (x^2 + y^2) \\frac{\\partial ^2}{\\partial z^2} - y \\frac{\\partial ^2}{\\partial x \\partial z} + x \\frac{\\partial ^2}{\\partial y \\partial z} \\Bigg \\rbrace $ Thus we can rewrite $\\mathcal {L}$ as $\\mathcal {L} = \\frac{1}{2}\\left(X^2 + Y^2 \\right).$"
],
[
"Weak Convergence and Random Walks",
"The first part of this section discusses the weak convergence results necessary to prove the convergence of the random walk developed in Section REF to a horizontal Brownian motion.",
"The main result is Theorem REF ."
],
[
"Convergence of semigroups",
"Let $C_c^{\\infty }(M)$ and $C_c^{\\infty }(T^{\\ast }M)$ be the spaces of the smooth, compactly supported real-valued functions on $M$ and $T^{\\ast }M$ equipped with the $\\sup $ norm.",
"We identify $C_c^{\\infty }(M)$ with a closed subspace of $C_c^{\\infty }(T^{\\ast }M)$ : if $f \\in C_c^{\\infty }(M)$ then the element $\\tilde{f} \\in C_c^{\\infty }(T^{\\ast }M)$ identified with $f$ is given by $\\tilde{f}(x,p) := f(x)$ .",
"Definition 4.1 For $f \\in C_c^{\\infty }(T^{\\ast }M)$ , the Hamilton-Jacobi flow field, ${D}_{HJ} : C_c^{\\infty }(T^{\\ast }M) \\rightarrow C_c^{\\infty }(T^{\\ast }M)$ , is defined by ${D}_{HJ} f(x, p) = \\frac{d}{dt} \\Big |_{t=0} f \\left( \\Phi _t(x, p)\\right).",
"$ Remark 4.2 If $f \\in C_c^{\\infty }(M)$ , then ${D}_{HJ} f(x,p) = v(f)$ where $v = \\beta (p)$ .",
"Remark 4.3 The semigroup property of flows implies that if $f \\in C_c^{\\infty }(T^{\\ast }M)$ , then ${D}_{HJ} \\left( {D}_{HJ} f \\right)(x, p) = \\frac{d}{ds}\\Big |_{s=0} \\frac{d}{dt} \\Big |_{t=0} f\\left( \\Phi _{t+s}(x,p) \\right).$ Definition 4.4 For $f \\in C_c^{\\infty }(TM)$ , the horizontally averaged projection $\\mathcal {P} : C_c^{\\infty }(T^{\\ast } M) \\rightarrow C_c^{\\infty }(M),$ is defined by $\\mathcal {P} f(x) = \\int _{\\mathcal {S}^{\\mathcal {H}}_x} f (x, g(v)) \\mathbb {U}_x(dv).$ Here, as before, $\\mathbb {U}_x$ is the rotationally invariant (uniform) probability measure on the unit sphere $\\mathcal {S}^{\\mathcal {H}}_x$ .",
"Let us now make the following observation.",
"Proposition 4.5 For every $f \\in C_c^{\\infty }(M)$ , $\\mathcal {L}f = \\mathcal {P} {D}_{HJ} {D}_{HJ} f$ .",
"Set $\\mathcal {I}$ as the identity operator on $C_c^{\\infty }(T^{\\ast }M)$ .",
"We denote by $e^{t(\\mathcal {P} - \\mathcal {I})}$ the strongly continuous contraction semigroup on $C_c^{\\infty }(T^{\\ast }M)$ whose bounded generator is $\\mathcal {P} - \\mathcal {I}$ .",
"We denote by $e^{t {D}_{HJ}}$ the strongly continuous contraction semigroup on $C_c^{\\infty }(T^{\\ast }M)$ whose generator is ${D}_{HJ} $ .",
"Using Notation REF we have $ e^{t {D}_{HJ}} f(x, p) = f\\left(\\Phi _t(x, p)\\right).",
"$ Finally, for any $\\alpha > 0$ we denote by $T_{\\alpha }(t)$ the strongly continuous contraction semigroup on $C_c^{\\infty }(T^{\\ast }M)$ whose generator is ${D}_{HJ} + \\alpha (\\mathcal {P}-\\mathcal {I})$ .",
"This is possible since $\\mathcal {P}-\\mathcal {I}$ is bounded.",
"For more generalized notions of summing together generators we refer to [20].",
"Our aim is to prove a limit theorem of $T_{\\alpha }(\\alpha t)$ as $\\alpha \\rightarrow \\infty $ using [13].",
"To this end, we first state some prerequisites which follow easily from the definitions.",
"Lemma 4.6 The following hold.",
"1) $\\operatorname{Ran}(\\mathcal {P}) = C_c^{\\infty }(M)$ .",
"2) $\\mathcal {P} {D}_{HJ} f = 0$ for $f \\in C_c^{\\infty }(M)$ .",
"Following the notation of T. Kurtz in [13], define $ \\begin{aligned}D_0 := & \\left\\lbrace f \\in \\operatorname{Dom}({D}_{HJ}) \\cap \\operatorname{Ran}(\\mathcal {P}):\\right.\\\\& \\left.",
"\\text{ there exists } h \\in \\operatorname{Dom}({D}_{HJ}) \\text{ such that } (\\mathcal {P}-\\mathcal {I})h = - {D}_{HJ} f \\right\\rbrace .\\end{aligned}$ Using the first claim of Lemma REF , we see that $\\operatorname{Dom}({D}_{HJ}) \\cap \\operatorname{Ran}(\\mathcal {P}) = C_c^{\\infty }(M)$ .",
"Moreover, for $f \\in C_c^{\\infty }(M)$ , define $h := {D}_{HJ} f$ .",
"By the second claim of Lemma REF , $(\\mathcal {P}-\\mathcal {I}) h = -{D}_{HJ} f$ .",
"We conclude that $D_0 = C_c^{\\infty }(M)$ .",
"Before getting to Theorem REF , the main result regarding weak convergence to a sub-Riemannian Brownian motion, we first make an assumption necessary to apply the result of Kurtz we wish to use.",
"Assumption 1 We henceforth assume the semigroup $e^{t \\mathcal {L} }$ is Feller in the following sense: for every $t \\geqslant 0$ , $\\lambda > 0$ , and $h \\in C^{\\infty }_c(M)$ $x \\longmapsto \\int _0^{\\infty } e^{-\\lambda t} e^{t \\mathcal {L} }h(x) dt \\in C^{\\infty }_c(M).$ We can now formulate the main result of this section.",
"Theorem 4.7 For every $f \\in C^{\\infty }_c(M)$ , $\\lim _{\\alpha \\rightarrow \\infty } T_{\\alpha }(\\alpha t)f = e^{t \\mathcal {L}} f,$ where the limit is taken in the $\\sup $ norm.",
"By Assumption REF , for any $h \\in C^{\\infty }_c(M) = D_0$ and $\\lambda > 0$ , the function $k(x) = \\int _0^{\\infty } e^{-\\lambda t} e^{t \\mathcal {L}}h(x) dt$ is in $C_c^{\\infty }(M)$ ; moreover, $(\\lambda - \\mathcal {L}) k = h$ .",
"This shows that $C^{\\infty }_c(M) \\subset \\operatorname{Ran}(\\lambda - \\Delta _{\\mathcal {H}})$ .",
"Hence by [13], the closure of $\\mathcal {P} {D}_{HJ} {D}_{HJ}$ is the generator of a strongly continuous contraction semigroup $e^{t \\mathcal {P} {D}_{HJ} {D}_{HJ}}$ such that $\\lim _{\\alpha \\rightarrow \\infty } T_{\\alpha }(\\alpha t) f = e^{t \\mathcal {P} {D}_{HJ} {D}_{HJ}} f$ for every $f \\in C_c^{\\infty }(M)$ , where the limit is in the $\\sup $ norm.",
"As noted in Proposition REF , $\\mathcal {P} {D}_{HJ} {D}_{HJ} = \\mathcal {L}$ on $C^{\\infty }_c(M)$ .",
"This concludes the proof."
],
[
"A sub-Riemannian random walk",
"Assumption 2 We assume that the sub-Riemannian manifold $\\left( M, \\mathcal {H}, \\langle \\cdot , \\cdot , \\rangle \\right)$ is complete with respect to the Carnot-Carathéodory metric.",
"Note that in this case this sub-Riemannian manifold is also geodesically complete, that is, all geodesics are defined for all $t \\geqslant 0$ by a sub-Riemannian Hopf-Rinow theorem (e.g.",
"[18]).",
"Let $\\varepsilon > 0$ be a parameter that we eventually take to zero.",
"Let $\\lbrace e_i\\rbrace _{i=1}^{\\infty }$ be i.i.d.",
"exponential random variables with parameter 1 and define $e_0 := 0$ .",
"Let us fix $(x,p) \\in T^*M$ as our initial position and momentum and let $v = \\beta (p)$ .",
"Define $(\\xi ^{\\varepsilon }_t, p^{\\varepsilon }_t) = \\Phi _{\\varepsilon t}(x,g(v))$ for $0 \\le t < e_1$ .",
"Given $e_1$ , let $x_1^{\\varepsilon } = \\pi \\circ \\Phi _{\\varepsilon e_1}(x,g(v)) \\in T^{\\ast }M$ where $\\pi :T^{\\ast } M \\rightarrow M$ is the canonical projection, and take $v^{\\varepsilon }_1$ randomly from $\\mathcal {S}_{x^{\\varepsilon }_1}^{\\mathcal {H}}$ such that the law of $v^{\\varepsilon }_1$ is $\\mathbb {U}_{x^{\\varepsilon }_1}$ .",
"From here, for $e_1 \\le t < e_2$ , define $(\\xi ^{\\varepsilon }_t, p^{\\varepsilon }_t) = \\Phi _{\\varepsilon (t - e_1)} (x_1^{\\varepsilon }, g(v_1^{\\varepsilon }))$ .",
"Continuing recursively, for each $k\\ge 0$ , once given $\\lbrace (x_0,v_0), (x_1^{\\varepsilon }, v_1^{\\varepsilon }), ..., (x_k^{\\varepsilon }, v_k^{\\varepsilon })\\rbrace $ and $\\lbrace e_i\\rbrace _{i=1}^{k+1}$ , define $x_{k+1}^{\\varepsilon } = \\pi \\big (\\Phi _{\\varepsilon e_{k+1}}(x_k^{\\varepsilon }, g(v_k^{\\varepsilon }))\\big )$ and take $v^{\\varepsilon }_{k+1}$ randomly from $\\mathcal {S}_{x^{\\varepsilon }_{k+1}}^{\\mathcal {H}}$ such that the law of $v^{\\varepsilon }_{k+1}$ is $\\mathbb {U}_{x^{\\varepsilon }_{k+1}}$ .",
"From here, for $e_{k+1} \\le t < e_{k+2}$ define $(\\xi ^{\\varepsilon }_t, p^{\\varepsilon }_t) = \\Phi _{\\varepsilon (t - e_{k+1})} (x_{k+1}^{\\varepsilon },g( v_{k+1}^{\\varepsilon }))$ .",
"We now have a ($\\varepsilon $ -scaled) random walk $B_t^{\\varepsilon }(x,p) := (\\xi _t^{\\varepsilon }, p_t^{\\varepsilon } )$ in $T^* M$ .",
"Here, the notation $B_t^{\\varepsilon }(x,p)$ emphasizes that $(x,p)$ are the initial conditions (and $\\beta (p) = v$ is the initial horizontal velocity).",
"Define $T_t^{\\varepsilon } : C^{\\infty }_c(T^*M) \\rightarrow C^{\\infty }_c(T^*M)$ by $T_t^{\\varepsilon }f(x,p) = \\mathbb {E}\\left[ f(B_t^{\\varepsilon }(x,p))\\right].$ With this we are ready to present the final piece needed, Theorem REF , before the statement of convergence, Theorem REF .",
"Our setup to this point is such that we can use a weaker version of the argument in [17] to prove Theorem REF .",
"The inability to reproduce the stronger statement arises from the fact that $B_t(x,p_1) = B_t(x,p_2)$ when $\\beta (p_1) = \\beta (p_2)$ , even though $\\Phi _t(x,p_1)$ need not be equal to $\\Phi _t(x,p_2)$ .",
"Theorem 4.8 For every $f \\in C^{\\infty }_c(M)$ , $T^{\\varepsilon }_t f = e^{t(\\varepsilon {D}_{HJ} + \\mathcal {P}-\\mathcal {I})}f.$ Before exposing the proof of Theorem REF (which is given below in Section REF ), let us note that as a corollary, we arrive at the convergence result which is our main theorem.",
"Theorem 4.9 For every $f \\in C_c^{\\infty }(M)$ , $\\lim _{\\varepsilon \\rightarrow 0} T^{\\varepsilon }_{t/\\varepsilon ^2} f = e^{t \\mathcal {L}} f.$ From Theorem REF , it follows that if $f \\in C^{\\infty }_c(M)$ that $\\lim \\limits _{\\varepsilon \\rightarrow 0} e^{(t/\\varepsilon ^2)(\\varepsilon {D}_{HJ} + \\mathcal {P}-\\mathcal {I})}f =e^{t \\mathcal {L}} f$ .",
"Since Theorem REF shows that $T^{\\varepsilon }_t$ and $e^{t(\\varepsilon {D}_{HJ} + \\mathcal {P}-\\mathcal {I})}$ agree on $C^{\\infty }_c(M)$ , the result follows."
],
[
"The Proof of Theorem ",
"We continue with the notation introduced in Section REF .",
"For the i.i.d.",
"exponential random variables $\\lbrace e_i\\rbrace _{i=1}^{\\infty }$ and for $k \\ge 0$ , let $\\tau _k = e_0 + e_1 + \\cdots + e_k$ ; recall that $e_0 := 0$ .",
"We denote by $R^{\\varepsilon }_{\\lambda }$ the resolvent of $e^{\\varepsilon t {D}_{HJ}}$ ; that is, $R^{\\varepsilon }_{\\lambda }f(x,p) =e^{\\varepsilon t {D}_{HJ}}f(x,p) = f( \\Phi _{\\varepsilon t}(x,p)).$ We denote by $S^{\\varepsilon }_{\\lambda }$ the resolvent of $T^{\\varepsilon }_t$ ; that is, $S^{\\varepsilon }_{\\lambda } f(x, p) = \\int _0^{\\infty } e^{-\\lambda t} \\mathbb {E} \\left[ f(B^{\\varepsilon }_t(x,p))\\right] dt.$ Lemma 5.1 For any $f \\in C_c^{\\infty }(T^*M)$ , $\\mathbb {E}\\bigg [ \\int _0^{ \\tau _1} e^{-\\lambda t} f(B_t^{\\varepsilon }(x,p)) \\,dt \\bigg ] = R^{\\varepsilon }_{1+\\lambda } f(x,g \\circ \\beta (p)).$ If the initial conditions of $B_t^{\\varepsilon }$ are $(x,p)$ , then or $0 \\leqslant t < \\tau _1$ , $B_t^{\\varepsilon } =\\Phi _{\\varepsilon t}(x,g \\circ \\beta (p))$ .",
"Thusly $&\\mathbb {E}_{(x,p)}\\bigg [ \\int _0^{\\tau _1} e^{-\\lambda t} f(B^{\\varepsilon }_t) dt \\bigg ] = \\mathbb {E}_{(x,p)}\\bigg [ \\int _0^{\\tau _1} e^{-\\lambda t} f(\\Phi _{\\varepsilon t}(x,g\\circ \\beta (p))) dt \\bigg ] \\\\&= \\int _0^{\\infty }\\int _0^t e^{- s}e^{-\\lambda t} f(\\Phi _{\\varepsilon t}(x,g\\circ \\beta (p)))\\, dt\\, ds = \\int _0^{\\infty }e^{-(\\lambda + 1)t} f(\\Phi _{\\varepsilon t}(x,g \\circ \\beta (p)))\\, dt\\\\& = R^{\\varepsilon }_{1+\\lambda } f(x,g \\circ \\beta (p)).$ This concludes the proof.",
"Lemma 5.2 For any $f \\in C_c^{\\infty }(T^*M)$ , $\\mathbb {E}\\bigg [ \\int _{\\tau _1}^{\\infty } e^{-\\lambda t} f(B^{\\varepsilon }_t(x,p)) dt \\bigg ] = R^{\\varepsilon }_{1+\\lambda } \\mathcal {P} S^{\\varepsilon }_{\\lambda } f(x, g \\circ \\beta (p)).$ Notice that $\\mathbb {E}\\bigg [ \\int _{\\tau _1}^{\\infty } e^{-\\lambda t} f(B^{\\varepsilon }_t(x,p)) dt \\bigg ] = \\mathbb {E}\\bigg [ e^{- \\lambda \\tau _1} \\int _{0}^{\\infty } e^{-\\lambda t} f(B_t^{\\varepsilon }(x_1^{\\varepsilon },g(v_1^{\\varepsilon }))) dt \\bigg ],$ $\\mathbb {E}\\bigg [ \\int _{0}^{\\infty } e^{-\\lambda t} f(B_t^{\\varepsilon }(x_1^{\\varepsilon },g(v_1^{\\varepsilon }))) dt \\ \\Big | \\ (x_1^{\\varepsilon }, v_1^{\\varepsilon } ) \\bigg ] = S^{\\varepsilon }_{\\lambda } f(x_1^{\\varepsilon }, g(v^{\\varepsilon }_1)),$ and $&\\mathbb {E}\\left[ S_{\\lambda }^{\\varepsilon } f(x_1^{\\varepsilon }, g(v_1^{\\varepsilon })) \\ \\big | \\ \\tau _1= t \\right] = \\mathbb {E}\\left[ S_{\\lambda }^{\\varepsilon } f(x_t, g(U)) \\right] \\\\&\\qquad = \\int _{\\mathcal {S}^{\\mathcal {H}}_{x_t}} S_{\\lambda }^{\\varepsilon } f(x_t, g(v)) \\,\\mathbb {U}_{x_t}(dv) = \\mathcal {P} S_{\\lambda }^{\\varepsilon }f(x_t).$ where $x_t = \\pi \\circ \\Phi _{\\varepsilon t}(x, g\\circ \\beta (p))$ (as before, $\\pi : T^*M \\rightarrow M$ is the canonical projection) and $U$ is a uniform random variable on $\\mathcal {S}^{\\mathcal {H}}_{x_t}$ .",
"Putting these pieces together, $&\\mathbb {E}\\bigg [ \\int _{\\tau _1}^{\\infty } e^{-\\lambda t} f(B^{\\varepsilon }_t(x,p)) dt \\bigg ] = \\mathbb {E}\\left[ e^{-\\lambda \\tau _1} S^{\\varepsilon }_{\\lambda } f(x_1^{\\varepsilon }, v_1^{\\varepsilon }) \\right] = \\mathbb {E} \\left[ e^{-\\lambda \\tau _1} \\mathcal {P} S_{\\lambda }^{\\varepsilon }f(x_{\\tau _1})\\right] \\\\&= \\int _0^{\\infty } e^{-\\lambda t} e^{-t} \\mathcal {P} S_{\\lambda }^{\\varepsilon } f(\\Phi _{\\varepsilon t}(x,g \\circ \\beta (p))) dt = R^{\\varepsilon }_{\\lambda +1} \\mathcal {P} S_{\\lambda }^{\\varepsilon } f (x,g \\circ \\beta (p)).$ Note that the third equality used $\\mathcal {P} S_{\\lambda }^{\\varepsilon }f(x_{\\tau _1}) = \\mathcal {P} S_{\\lambda }^{\\varepsilon } f(\\Phi _{\\varepsilon t}(x,g \\circ \\beta (p)))$ by the identification of $C^{\\infty }_c(M)$ as a subset of $C^{\\infty }_c(T^*M)$ .",
"Using Lemmas REF and REF , we have $S^{\\varepsilon }_{\\lambda } f(x, p) = \\mathbb {E}\\bigg [ \\int _0^{\\tau _1} e^{-\\lambda t} f(B^{\\varepsilon }_t (x,p)) dt \\bigg ] + \\mathbb {E}\\bigg [ \\int _{\\tau _1}^{\\infty } e^{-\\lambda t} f(B^{\\varepsilon }_t (x,p)) dt \\bigg ] \\\\= R^{\\varepsilon }_{1+\\lambda } f(x,g \\circ \\beta (p)) + R^{\\varepsilon }_{1+\\lambda } \\mathcal {P}^g S^{\\varepsilon }_{\\lambda } f(x, g \\circ \\beta (p)).$ Multiplying on the left by $1 + \\lambda - \\varepsilon {D}_{HJ}$ yields $(1 + \\lambda - \\varepsilon {D}_{HJ}) S^{\\varepsilon }_{\\lambda } f(x, g \\circ \\beta (p)) = f(x, g \\circ \\beta (p)) + \\mathcal {P}^g S^{\\varepsilon }_{\\lambda } f(x, g \\circ \\beta (p)).$ That is, $(\\lambda - [\\varepsilon {D}_{HJ} + \\mathcal {P}^g - \\mathcal {I}]) S^{\\varepsilon }_{\\lambda }f(x, g \\circ \\beta (p)) = f(x, g \\circ \\beta (p)).$ In particular, for any $f \\in C^{\\infty }_c(M)$ , $(\\lambda - [\\varepsilon {D}_{HJ} + \\mathcal {P}^g - \\mathcal {I}]) S^{\\varepsilon }_{\\lambda }f = f.$ From here we can now conclude the result."
],
[
"Averaging over the unit sphere in an inner product space",
"Here we provide details of the proof of Proposition REF which are solely properties of finite-dimensional inner product spaces.",
"Proposition 5.3 Let $\\mathcal {X}$ be an $n$ -dimensional real inner product space with inner product $\\langle \\cdot , \\cdot \\rangle $ .",
"Let $S$ be the unit sphere in $\\mathcal {X}$ with respect to this inner product and set $\\mu $ as the rotationally invariant probability measure on $S$ .",
"Given any $X \\in \\mathcal {X}$ , $ \\int _S (X,\\xi )^2 \\mu (d\\xi ) = \\frac{|X|^2}{n}.",
"$ It suffices to show that if $X \\in S$ , then $\\int _S (X,\\xi )^2 \\mu (d\\xi ) = 1/n.$ To this end, suppose $X, Y \\in S$ and $l : S \\rightarrow S$ is any rotation such that $l(Y)=X$ .",
"Since the adjoint of a rotation is again a rotation, we have, $ \\int _S (X,\\xi )^2\\mu (d\\xi ) = \\int _S (l(Y),\\xi )^2 \\mu (d\\xi ) = \\int _S (Y, l^{\\ast }(\\xi ))^2 \\mu (d\\xi ) = \\int _S (Y,\\xi )^2 \\mu (d\\xi )$ where the final identity follows from the rotational invariance of $\\mu $ .",
"This shows that the value of the integral is constant for any choice of $X \\in S$ .",
"Set $a := \\int _S (X,\\xi )^2\\mu (d\\xi ).$ Take $\\lbrace X_i: 1\\leqslant i \\leqslant n \\rbrace \\subset S$ to be an orthonormal basis for $V$ , then for any $\\xi \\in S$ $1 = \\Vert \\xi \\Vert ^2 = \\sum _{i=1}^n (X_i,\\xi )^2.$ Therefore, $ 1 = \\int _S \\Vert \\xi \\Vert ^2 \\mu (d\\xi ) = \\sum _{i=1}^n \\int _S (X_i, \\xi )^2 \\mu (d\\xi ) = na $ which then implies $a = 1/n$ .",
"Corollary 5.4 Let $\\mathcal {X}$ , $S$ , and $\\mu $ be as in the previous proposition.",
"Take $X,Y \\in \\mathcal {X}$ .",
"Then $\\int _S (X,\\xi )(Y,\\xi ) \\mu (d\\xi ) = \\frac{(X,Y)}{n}.$ By the previous proposition, $\\int _S (X+Y,\\xi )^2 \\,\\mu (d\\xi ) = \\frac{|X+Y|^2}{n} = \\frac{|X|^2}{n} + \\frac{|Y|^2}{n} + 2\\frac{(X,Y)}{n}.$ On the other hand, $(X+Y,\\xi )^2 = (X,\\xi )^2 + (Y,\\xi )^2 + 2(X,\\xi )(Y,\\xi )$ .",
"Hence another application of the previous proposition yields, $\\int _S (X+Y,\\xi )^2 \\,\\mu (d\\xi ) = \\int _S \\big \\lbrace (X,\\xi )^2 + (Y,\\xi )^2 + 2(X,\\xi )(Y,\\xi ) \\big \\rbrace \\mu (d\\xi ) \\\\= \\frac{|X|^2}{n} + \\frac{|Y|^2}{n} + 2\\int _S(X,\\xi )(Y,\\xi )\\,\\mu (d\\xi ).$ Comparing terms, the result now follows.",
"Acknowledgement The authors are grateful for many helpful and motivating conversations with Alexander Teplyaev, Michael Hinz, and Dan Kelleher.",
"In large part, this paper is the result of our attempt to address several questions raised during those discussions."
]
] | 1403.0142 |
[
[
"Realizing Exterior Cromwell moves on rectangular diagrams by\n Reidemeister moves"
],
[
"Abstract If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by I.",
"A. Dynnikov.",
"Using this, Henrich and Kauffman gave an upper bound for the number of Reidemeister moves needed for unknotting a knot diagram of the trivial knot.",
"However, exchange or merge moves on the top and bottom pairs of edges of rectangular diagrams are not considered in the proof.",
"In this paper, we show that there is a rectangular diagram of the trivial knot which needs such an exchange move for being unknotted, and study upper bound of the number of Reidemeister moves needed for realizing such an exchange or merge move."
],
[
"Introduction",
"Birman and Menasco introduced arc-presentation of links in [2], and Cromwell formulated it in [3].",
"Dynnikov pointed out in [4] and [5] that Cromwell's argument in [3] almost shows that any arc-presentation of a split link can be deformed into one which is $``$ visibly split\" by a finite sequence of exchange moves.",
"He also showed that any arc-presentation of the trivial knot can be deformed into trivial one with only two arcs by a finite sequence of merge moves and exchange moves, without using divide moves which increase the number of arcs.",
"As is shown in page 41 in [3], an arc-presentation is almost equivalent to a rectangular diagram.",
"Figure: A rectangular diagram of the trivial knot with 8 vertical edgesA rectangular diagram of a link is a link diagram in the plane ${\\mathbb {R}}^2$ which is composed of vertical lines and horizontal lines such that no pair of vertical lines are colinear, no pair of horizontal lines are colinear, and the vertical line passes over the horizontal line at each crossing.",
"See Figure REF .",
"These vertical lines and horizontal lines are called edges of the rectangular diagram.",
"Every rectangular diagram has the same number of vertical edges and horizontal edges.",
"It is known that every link has a rectangular diagram (Proposition in page 42 in [3]).",
"Figure: Interior horizontal mergesCromwell moves, which are described in the next three paragraphs, are elementary moves for rectangular diagrams of links.",
"They do not change type of links.",
"Moreover, Theorem in page 45 in [3] and Proposition 4 in [4] state that, if two rectangular diagrams represent the same link, then one is obtained from the other by a finite sequence of these elementary moves and rotation moves, which is also introduced below.",
"Figure: Exterior horizontal mergesFirst, we recall merge moves.",
"If two horizontal (resp.",
"vertical) edges connected by a single vertical (resp.",
"horizontal) edge have no other horizontal (resp.",
"vertical) edges between their ordinates (resp.",
"abscissae), then we can amalgamate the three edges into a single horizontal (resp.",
"vertical) edge.",
"This move is called an interior horizontal (resp.",
"vertical) merge.",
"See Figure REF for examples of interior horizontal merge moves.",
"If the top and bottom (resp.",
"the leftmost and rightmost) horizontal (resp.",
"vertical) edges are connected by a single vertical (resp.",
"horizontal) edge, then we can amalgamate the three edges into a single horizontal (resp.",
"vertical) edge.",
"We may place the new horizontal (resp.",
"vertical) edge either at the top height or at the bottom height (resp.",
"either in the leftmost position or in the rightmost position).",
"See Figure REF .",
"We call this move an exterior horizontal (resp.",
"vertical) merge.",
"(Even when we consider rectangular link diagrams in the 2-sphere $(\\cong {\\mathbb {R}}^2 \\cup \\lbrace \\infty \\rbrace )$ , exterior merge moves are distinct from interior merge moves as moves on general link diagrams.)",
"Note that a merge move decreases the number of vertical edges and that of horizontal edges by one.",
"The inverse moves of merge moves are called divide moves.",
"Figure: Interleaved pair and non-interleaved pairsTo describe exchange moves, we need a terminology.",
"Two vertical edges are said to be interleaved, if the heights of their endpoints alternate.",
"See Figure REF .",
"Similarly, we define interleaved two horizontal edges.",
"Figure: Interior horizontal exchange movesIf two horizontal edges at mutually adjacent heights are not interleaved, then we can exchange their heights.",
"See Figure REF .",
"This move is called an interior horizontal exchange.",
"If the top horizontal edge and the bottom one are not interleaved, then we can exchange their heights.",
"We call this move an exterior horizontal exchange.",
"See Figure REF , which depicts an exterior horizontal exchange move on the rectangular diagram in Figure REF .",
"(Even when we consider rectangular link diagrams in the 2-sphere $(\\cong {\\mathbb {R}}^2 \\cup \\lbrace \\infty \\rbrace )$ , exterior exchange moves are distinct from interior exchange moves as moves on general link diagrams.)",
"Similarly, we define vertical exchange moves.",
"Figure: An exterior horizontal exchange moveThe next result of Dynnikov gives a finite algorithm to decide whether a given rectangular diagram represents the trivial knot or not.",
"The original statement is in languages on arc-presentations.",
"Theorem 1.1 [Dynnikov [4], [5]] Any rectangular diagram of the trivial knot can be deformed into trivial one with only two vertical edges and two horizontal edges by a finite sequence of merge moves and exchange moves.",
"Note that the sequence in the above theorem contains no divide moves.",
"Hence the sequence gives a monotone simplification, that is, no move in the sequence increases the number of edges.",
"There are only finitely many rectangle diagrams with a fixed number of edges.",
"Thus the above theorem gives a finite algorithm for the decision problem.",
"A Reidemeister move is a local move of a link diagram as in Figure REF .",
"An RI (resp.",
"II) move creates or deletes a monogon face (resp.",
"a bigon face).",
"An RIII move is performed on a 3-gon face, deleting it and creating a new one.",
"Any such move does not change the link type.",
"As Alexander and Briggs [1] and Reidemeister [8] showed, for any pair of diagrams $D_1$ , $D_2$ which represent the same link type, there is a finite sequence of Reidemeister moves which deforms $D_1$ to $D_2$ .",
"Figure: NO_CAPTIONIn [7], A. Henrich and L. Kauffman announced an upper bound of the number of Reidemeister moves needed for unknotting by applying Dynnikov's theorem to rectangular diagrams.",
"Lemma 7 in [7] states that no more than $n-2$ Reidemeister moves are required to perform an exchange move on a rectangular diagram with $n$ vertical edges.",
"However, the proof of Lemma 7 in [7] does not consider the exterior exchange moves.",
"In this paper, we show the next two theorems.",
"Theorem 1.2 There is a rectangular diagram of the trivial knot which needs an exterior exchange move for being deformed into the trivial rectangular diagram with two vertical edges and two horizontal edges by a sequence of exchange moves and merge moves.",
"In fact, Figure REF is one of such a rectangular diagram with the smallest number of edges.",
"Theorem REF is shown in section .",
"Theorem 1.3 Let $n$ be an integer with $n \\ge 2$ , and $\\epsilon $ the integer with $\\epsilon \\in \\lbrace 0,1 \\rbrace $ and $n \\equiv \\epsilon $ (mod 2).",
"If a rectangular diagram $D$ with $n$ vertical edges admits an exterior exchange move (resp.",
"an exterior merge move), then a sequence of $3n^2-4n-4-3\\epsilon $ (resp.",
"$(3n^2-4n-4-3\\epsilon )/2$ ) or less number of Reidemeister moves either (1) deforms $D$ into a knot diagram with no crossings, (2) deforms $D$ into a disconnected link diagram, or (3) realizes the exterior exchange move (resp.",
"the exterior merge move).",
"In addition, a sequence of $(3n^2-4n-2-3\\epsilon )/2$ or less number of Reidemeister moves either does (1) or (2) as above, or (3)$^{\\prime }$ realizes arbitrary one of the two rotation moves.",
"Figure: Rectangular diagrams with the maximal number of crossingsFigure: Rectangular diagrams with the maximal number of crossingsThis theorem is proved in section .",
"In the proof of the above theorem, we use two propositions below.",
"We say that a horizontal (resp.",
"vertical) edge is of length $|j-i|$ if it connects the $i$ th and the $j$ th vertical (resp.",
"horizontal) edges from the left (resp.",
"the bottom).",
"Proposition 1.4 Let $n$ be an integer larger than 1.",
"Let $R$ be a rectangular diagram of a link with $n$ vertical edges.",
"Then $R$ has at most $(n^2-2n-1)/2$ crossings when $n$ is odd, and at most $(n^2-2n)/2$ crossings when $n$ is even.",
"The sum of lengths of the edges of $R$ is at most $n^2-1$ when $n$ is odd, and at most $n^2$ when $n$ is even.",
"This estimation is keen.",
"The rectangular diagrams with even number of vertical edges in Figure REF and those with odd number of vertical edges in Figure REF give concrete examples which realize the maximal numbers.",
"This proposition is shown in section .",
"Figure: Jump movesAs will be shown in section , an exterior exchange (resp.",
"merge) move can be realized by a sequence of two jump moves (resp.",
"by a single jump move).",
"We recall the definition of a jump move.",
"Let $D$ be a link diagram on the plane ${\\mathbb {R}}^2$ .",
"Let $s$ be an overstrand of $D$ , that is, $s$ is a subarc of $D$ such that $s$ does not go under any crossing of $D$ and the endpoints $\\partial s$ is free from the crossings of $D$ .",
"Let $u$ be an arc with $u \\cap s = \\partial u = \\partial s$ such that $u$ is transverse to $D$ .",
"A jump move bringing $s$ to $u$ is an operation on $D$ which deletes $s$ and then adds $u$ as an overstrand.",
"Note that the resulting link diagram $D^{\\prime }$ represents the same link as $D$ .",
"See Figure REF , where the two jump moves are described, and they realize the exterior merge moves in Figure REF .",
"We define a jump move for an understrand similarly.",
"Proposition 1.5 Let $D$ be a link diagram on the plane ${\\mathbb {R}}^2$ .",
"Suppose that $D$ admits a jump move which brings an overstrand $s$ of $D$ to another arc $u$ .",
"The circle $s \\cup u$ bounds a disk, say $Q$ , in ${\\mathbb {R}}^2$ .",
"Let $\\bar{D}$ be the underlying planar graph of $D$ which is obtained by deleting over-under information of crossings of $D$ .",
"Set $D_Q = {\\rm cl}\\,(\\bar{D} \\cap {\\rm int}\\,Q)$ , where cl and int denote the closure and the interior respectively.",
"We regard the points $D_Q \\cap \\partial Q$ as vertices, where $\\partial Q$ denotes the boundary circle of $Q$ .",
"Then $D_Q$ forms a graph.",
"Let $V$ be the number of vertices of $D_Q$ in int $Q$ , and $E$ the number of edges of $D_Q$ .",
"Then a sequence of $V+E$ or less number of Reidemeister moves (1) deforms $D$ into a knot diagram with no crossings, (2) deforms $D$ into a disconnected link diagram, or (3) realizes the jump move.",
"Note that the edges in $\\bar{D} \\cap s$ are not contained in $D_Q$ .",
"A similar thing holds for a jump move for an understrand.",
"The above proposition is a correction of Remark 2 in [6], and a stronger proposition is proved in section ."
],
[
"Proof of Theorem ",
"In this section, we show Theorem REF .",
"The sequence as in Dynnikov's theorem (Theorem REF ) sometimes needs to contain exterior exchange moves.",
"In fact, the rectangle diagram shown in Figure REF represents the trivial knot.",
"It admits no merge moves since it does not have an edge of length 1 or $8-1$ .",
"We cannot apply any interior horizontal exchange move to the diagram because every pair of horizontal edges in adjacent levels are interleaved.",
"Similarly, no interior vertical exchange move can be performed on this diagram.",
"Hence every sequence as in Dynnikov's theorem on this diagram must begin with the exterior exchange move.",
"A similar argument shows that the rectangle diagram of the trviail knot shown in Figure REF admits no merge moves, no vertical exchange moves and no interior horizontal exchange moves.",
"It only admits the exterior horizontal exchange move.",
"Figure: A rectangular diagram of the trivial knot with 9 vertical edgesIt can easily be confirmed by a computer that every rectangular diagram of the trivial knot with 7 or less number of vertical edges admits a merge move or an interior exchange move, and that every rectangular diagram of the trivial knot with 8 vertical edges admits both the exterior vertical exchange move and the extrior horizontal exchange move if it admits no merge moves and no interior exchange moves."
],
[
"Proof of Proposition ",
"In this section, we prove Proposition REF .",
"Let $D$ be a rectangular diagram of a knot or a link.",
"We place $D$ in the $x$ -$y$ plane so that the $i$ th vertical line from the left is in the line $x=i$ for $i \\in \\lbrace 1,2,\\cdots , n \\rbrace $ and so that the $j$ th horizontal line from the bottom in the line $y=j$ for $j \\in \\lbrace 1,2,\\cdots , n \\rbrace $ .",
"The length of a vertical (horizontal) edge $e$ is the difference of the ordinates (resp.",
"abscissae) of the endpoints of $e$ .",
"Let $\\ell (e)$ denote it.",
"Then $e$ has at most $\\ell (e)-1$ crossing points on it.",
"We consider the sum $\\Sigma $ of the length of all the horizontal edges of $D$ .",
"Let $e_i$ be the $i$ th horizontal edge from the bottom, and $r_i$ and $l_i$ the abscissae of the right and left endpoints respectively.",
"Then we have $\\ell (e_i) = r_i - l_i$ and $\\Sigma = \\sum _{i=1}^n \\ell (e_i) = \\sum _{i=1}^n (r_i - l_i)$ .",
"We consider the multi-set $E=\\lbrace r_1, r_2, \\cdots , r_n, l_1, l_2, \\cdots , l_n \\rbrace $ , where a multi-set may contain the same element multiple times.",
"Then $E$ contains each of the natural numbers $1,2,\\cdots , n$ twice.",
"In the case where $n$ is even, $\\Sigma $ is the largest when $\\lbrace r_1, r_2, \\cdots , r_n \\rbrace = \\lbrace n,n,n-1,n-1,\\cdots , (n/2)+1, (n/2)+1 \\rbrace $ and $\\lbrace l_1, l_2, \\cdots , l_n \\rbrace = \\lbrace 1,1, 2,2, \\cdots , n/2, n/2 \\rbrace $ as multi-sets.",
"Hence $\\Sigma $ is at most $2 \\times n(n+1)/2 - 4 \\times (n/2)((n/2)+1)/2= n(n+1) - n((n/2)+1)= n^2/2$ .",
"Thus the number of crossing of $D$ is at most $\\Sigma -n \\le n(n-2)/2$ when $n$ is even.",
"This maximal number is realized by the rectangular diagrams in Figure REF .",
"In the case where $n$ is odd, $\\Sigma $ is the largest when $\\lbrace r_1, r_2, \\cdots , r_n \\rbrace = \\lbrace n,n,n-1,n-1,\\cdots , (n+3)/2, (n+3)/2, (n+1)/2 \\rbrace $ and $\\lbrace l_1, l_2, \\cdots , l_n \\rbrace = \\lbrace 1,1, 2,2, \\cdots , (n-1)/2, (n-1)/2, (n+1)/2 \\rbrace $ as multi-sets.",
"Hence $\\Sigma $ is at most $2 \\times n(n+1)/2 - 2 \\times ((n+1)/2)(((n+1)/2)+1)/2 -2 \\times ((n-1)/2)(((n-1)/2)+1)/2$ $= n(n+1) - (n+1)(n+3)/4 - (n-1)(n+1)/4= n(n+1) - (n+1)(2n+2)/4$ $= (n+1)(2n-(n+1))/2= (n+1)(n-1)/2$ .",
"Thus the number of crossing of $D$ is at most $\\Sigma -n \\le (n^2-2n-1)/2$ when $n$ is odd.",
"This maximal number is realized by the rectangular diagrams in Figure REF ."
],
[
"Proof of Theorem ",
"In this section, we show Theorem REF using Proposition REF .",
"The proof of Proposition REF is given in the next section.",
"Figure: Realizing an exterior exchange move by jump movesWe first consider an exterior exchange move on a rectangular diagram $D$ .",
"Without loss of generality, we assume that it is horizontal.",
"It can be realized by a sequence of two jump moves as in Figure REF .",
"(See section for the definition of a jump move.)",
"We can assume, without loss of generality, that the top edge is not shorter than the bottom one.",
"The first jump move brings the top edge to the bottom, and the second jump move brings the edge second to the bottom, which was the bottom one before the first jump move, to the top.",
"For the $i$ th jump move with $i=1$ or 2, the original arc $s_i$ of the rectangular diagram jumps to the arc $u_i$ , and $s_i \\cup u_i$ bounds a disk $Q_i$ in ${\\mathbb {R}}^2$ .",
"Let $D^{\\prime }$ be the rectangular diagram obtained from $D$ by the first jump move.",
"We define the graph $D_{Q_1}$ and $D^{\\prime }_{Q_2}$ as in Proposition REF .",
"Let $D_{Q_2}$ stand for $D^{\\prime }_{Q_2}$ for simplicity of notation.",
"Then int $Q_i$ contains at most $\\lbrace n(n-2) -\\epsilon \\rbrace /2$ crossings of the rectangular diagram by Proposition REF , where $\\epsilon = 1$ when $n$ is odd, and $\\epsilon =0$ when $n$ is even.",
"Each of the two vertical edges in $\\partial Q_i$ intersects at most $n-2$ horizontal edges, and such intersection points are endpoints of edges of $D_{Q_i}$ .",
"(Note that $u_1$ does not intersect the bottom edge because the top edge is not shorter than the bottom one.)",
"Since four endpoints gather at every vertex of ${\\bar{D}}_{Q_i}$ in int $Q_i$ , the disk $Q_i$ contains at most $(4 (\\lbrace n(n-2)-\\epsilon \\rbrace /2)+2(n-2))/2= n^2-n-2-\\epsilon $ edges.",
"Hence, by Proposition REF , a sequence of at most $(\\lbrace n(n-2)-\\epsilon \\rbrace /2)+(n^2-n-2-\\epsilon )$ Reidemeister moves either deforms $D$ or $D^{\\prime }$ into a knot diagram with no crossings, deforms $D$ or $D^{\\prime }$ into a disconnected link diagram, or realizes the $i$ th jump move.",
"Thus a sequence of at most $2((\\lbrace n(n-2)-\\epsilon \\rbrace /2)+(n^2-n-2-\\epsilon ))=3n^2-4n-4-3\\epsilon $ Reidemeister moves either deforms $D$ into a knot diagram with no crossings, deforms $D$ into a disconnected link diagram, or realizes the exterior exchange move.",
"A rotation move can be realized by a single jump move as shown in the first jump move in Figure REF .",
"In this case, the two vertical edges in $\\partial Q$ intersects at most $n-1$ horizontal edges, where $Q$ is the rectangle bounded by the arcs before and after the jump.",
"Hence $(3n^2-4n-2-3\\epsilon )/2$ Reidemeister moves will do.",
"Figure: Realizing an exterior merge move by a jump moveAn exterior merge move on a rectangular diagram $D$ can be realized by a single jump move as in Figures REF and REF .",
"In each example in Figure REF , the top edge and the bottom edge are in the same side of the vertical edge connecting them.",
"In Figure REF , they are in the opposite sides.",
"A similar argument as above shows the theorem for exterior merge moves."
],
[
"Proof of Proposition ",
"In this section, we prove Proposition REF , which is used in the previous section.",
"We show a little stronger proposition below.",
"Figure: the pattern of E svs E_{svs}Proposition 5.1 Let $D$ be a link diagram on the plane ${\\mathbb {R}}^2$ which admits a jump move replacing an overstrand (resp.",
"understrand) $s$ with another overstrand (resp.",
"understrand) $u$ .",
"Then, for an integer $\\Sigma $ defined below, a sequence of at most $\\Sigma $ Reidemeister moves either (1) deforms $D$ into a disconnected link diagram, (2) deforms $D$ into a knot diagram with no crossings, or (3) realizes the jump move.",
"The circle $s \\cup u$ bounds a disk $Q$ in ${\\mathbb {R}}^2$ .",
"Let $\\bar{D}$ be a graph obtained from the link diagram $D$ by ignoring the over-under informations of crossings of $D$ .",
"The crossings of $D$ become the vertices of $\\bar{D}$ .",
"Set $D_Q = {\\rm cl}(\\bar{D} \\cap {\\rm int}\\,Q)$ .",
"We regard the points $D_Q \\cap (\\partial Q)$ as vertices of the graph $D_Q$ .",
"We set $\\Sigma = V + E_i + E_{ss} + E_{\\partial } + E_s + E_{svs}$ , the sum of numbers defined as below.",
"Let $V$ be the number of vertices of $D_Q$ in int $Q$ .",
"Let $E_i$ be the number of edges of $D_Q$ which do not have an endpoint in the arc $s$ , $E_{ss}$ the number of edges of $D_Q$ which have both endpoints in int $s$ , $E_{\\partial }$ the number of edges of $D_Q$ which has a single endpoint in $\\partial s$ .",
"For a vertex $v$ of $D_Q$ in int $Q$ , let $E_{sv}$ denote the number of edges of $D_Q$ which have an endpoint at $v$ and the other one in int $s$ .",
"Then, let $E_s$ be the sum of max $(0, E_{sv}-2)$ over all vertices of $D_Q$ in int $Q$ .",
"Let $E_{svs}$ be the number of connected components $C$ of $D_Q$ as below.",
"$C$ has a vertex, say $v$ , with $E_{sv}=2$ in int $Q$ .",
"There are precisely two edges, say $e$ and $f$ , which connect $v$ and vertices, say $v_e$ and $v_f$ , in int $s$ respectively.",
"Let $t$ be the subarc of $s$ with $\\partial t = v_e \\cup v_f$ , and $R$ the disk bounded by the circle $e \\cup f \\cup t$ .",
"The other edges incident to $v$ than $e$ and $f$ are in $R$ , and $t$ contains no vertices of $C$ other than $v_e$ and $v_f$ .",
"(The arc $t$ may contain vertices of $D_Q - C$ .)",
"See Figure REF .",
"Moreover, when $E_{\\partial } = 0$ , there is a sequence of at most $\\Sigma ^{\\prime } = 2V + E_i + E_{ss} + E_{\\partial } + E_s + E_{svs}$ Reidemeister moves containing no RI moves which does (1), (2) or (3) above.",
"Note that edges with both endpoints in int $u$ and that with one endpoint in int $u$ and the other in int $Q$ are counted in $E_i$ .",
"This proposition is a correction of Lemma 4 in [6], where the term $E_{svs}$ is not considered.",
"The diagram in Figure REF (a)-1 gives a counter example to Lemma 4 in [6], where $V + E_i + E_{ss} + E_{\\partial } + E_s = 1 + 1 + 0 + 0 + 0 = 2$ , and wee need at least three Reidemeister moves to realize the jump move.",
"Moreover, the argument in the proof of Lemma 4 in [6] contains several overlooks.",
"So, we give a precise proof of the above proposition here.",
"Before that, we prove Proposition REF using the above proposition.",
"We prove Proposition REF .",
"It's enough to show that $E \\ge E_i + E_{ss} + E_{\\partial } + E_s + E_{svs}$ .",
"This can be easily seen because $E_s$ is covered by the edges connecting a vertex in int $s$ and another vertex $v$ in int $Q$ with $E_{sv} \\ge 3$ , $E_{svs}$ is covered by the edges connecting a vertex in int $s$ and another vertex $v$ in int $Q$ with $E_{sv} = 2$ , and the other terms $E_i, E_{ss}, E_{\\partial }$ are covered by the edges which do not connect a vertex in int $s$ and that in int $Q$ .",
"Figure: These moves take first priority.We prove Proposition REF .",
"When $\\Sigma =0$ , we have $V=E_i=E_{ss}=E_{\\partial }=E_s=E_{svs}=0$ , and hence all the edges of $D_Q$ connect int $s$ and int $u$ .",
"This means that $s$ and $u$ are parallel, and the diagram obtained by the jump move is the same as the original one.",
"Thus we need no Reidemeister moves, and the proposition follows in this case.",
"We consider the case where $\\Sigma > 0$ .",
"We distinguish several cases, present a sequence of Reidemeister moves in each case, and show that the number of Reidemeister moves is less than or equal to the decrease in $\\Sigma $ .",
"Then the proposition is proved by induction on $\\Sigma $ .",
"We can assume that $D$ is connected and has a crossing.",
"Otherwise, we have conclusion (1) or (2).",
"Figure: Moves (1) through (4)First, when the graph $D_Q$ has the pattern described in the left of Figure REF (a)-1 or (b), we perform the sequence of Reidemeister moves shown in those figures.",
"If there is no subgraph of $D_Q$ in the pattern of Figure REF , then we perform a Reidemeister move shown in Figures REF and REF .",
"For every integer $i$ with $1 \\le i \\le 6$ , Move (i) in Figures REF and REF is applied when Moves (1) through (i-1) cannot be applied and Move (i) can.",
"However, the move (4) must be applied to an adequate part of the link diagram, which will be described in detail later.",
"In every move in these figures, $s$ is moved keeping that it is an overstrand.",
"So, over-under informations at the crossings are not specified in the figures.",
"In the patterns described on the left side hand of Figure REF , there are two edges, say $e$ and $f$ , connecting a vertex $v$ of $D_Q$ in int $Q$ and the arc $s$ .",
"Let $R$ be the subdisk cut off from $Q$ by the arc $e \\cup f$ .",
"Then $D_Q \\cap R$ consists of $e$ and $f$ and a single loop edge having its both endpoints at $v$ .",
"In Case (a), both $e$ and $f$ have an endpoint in int $s$ .",
"Move (a)-1 in Figure REF is due to Kanako Oshiro, and consists of three Reidemeister moves.",
"This seqence of Reidemeister moves decreases $\\Sigma $ by three ($V$ by one, $E_i$ by one and $E_{svs}$ by one).",
"When $E_{\\partial } =0$ , the sequence (a)-2 of four Reidemeister moves does not contain an RI move, and decreases $2V+E_i+E_s+E_{ss}+E_{\\partial }+E_{svs}$ by four.",
"In Case (b), precisely one of $e$ and $f$ , say $e$ , has an endpoint at $\\partial s$ .",
"The sequence in Figure REF (c) is composed of three Reidemeister moves, and decreases $\\Sigma $ by three ($V$ by one, $E_i$ by one and $E_{\\partial }$ by one).",
"Figure: Moves (5) and (6)Suppose that $D_Q$ does not contain a pattern as in Figure REF .",
"We consider first Move (1) in Figure REF .",
"In this figure, an edge, say $e$ , connecting one of the two points $\\partial s$ and a vertex in int $s$ cuts off a subdisk, say $R$ , from $Q$ such that $D_Q \\cap R = e$ .",
"The RI move along $R$ decreases $\\Sigma $ by one since it decreases $E_{\\partial }$ by one.",
"In Figure REF (2), an edge, say $e$ , having both endpoints in int $s$ cuts off a subdisk, say $R$ , from $Q$ such that $D_Q \\cap R = e$ .",
"The RII move of Move (2) along $R$ decreases $\\Sigma $ by one since it decreases $E_{ss}$ by one.",
"Figure: Special cases of Move (3)We consider Move (3) in Figure REF , where two edges, say $e$ and $f$ , have an endpoint at a vertex, say $v$ , in int $Q$ and reach int $s$ .",
"The arc $e \\cup f$ cuts off a subdisk, say $R$ , from $Q$ with $D_Q \\cap R = e \\cup f$ .",
"In this case, we perform an RIII move along $R$ .",
"We will show that this decreases $\\Sigma $ by one or more.",
"We must distinguish many cases.",
"First, we consider the cases described in Figure REF (I), (II)-1, (II)-2, (III).",
"In Case (I), a loop edge has its both endpoints at $v$ .",
"Then the RIII move decreases $\\Sigma $ by one because it decreases both $V$ and $E_i$ by one and increases $E_{ss}$ by one.",
"In Case (II), two edges incident to $v$ and other than $e$ and $f$ have the other endpoints at the same vertex.",
"The RIII move decreases $V$ by one and $E_i$ by two.",
"In Case (II)-1, this may increase $E_s$ by one or two.",
"In Case (II)-2, this increases $E_{svs}$ by one.",
"Hence, in both cases of (II)-1 and (II)-2, $\\Sigma $ decreases by one or more.",
"In Case (III), all the edges incident to $v$ reach int $s$ .",
"The RIII move decreases $\\Sigma $ by one because it decreases $V$ by one, $E_s$ by two, and increases $E_{ss}$ by two.",
"Figure: contribution of gg and ww to Σ\\Sigma Suppose that the edges incident to $v$ are not in the patterns in Figure REF .",
"Then the two edges, say $g$ and $h$ , incident to $v$ and other than $e$ and $f$ are distinct, and do not share the other endpoints, say $w$ and $x$ respectively.",
"Moreover, at most one of $w$ and $x$ is in int $s$ .",
"We consider arbitrary one of $g$ and $h$ , say $g$ .",
"Either $w$ $(\\in \\partial g)$ is (i) in int $Q$ , (ii) in int $u$ , (iii) in $\\partial s$ or (iv) in int $s$ .",
"In each case, we observe the change of the contribution of $g$ and $w$ to $\\Sigma $ .",
"Precisely, we examine $E_i, E_{ss}, E_{sw}$ and $E_{svs}$ .",
"We first consider Case (i).",
"The RIII move decreases the contribution of $g$ to $E_i$ by one.",
"When $E_{sw} \\ge 2$ , the RIII move increases the contribution of $w$ to $E_{sw}$ and hence to $E_s$ by one, as shown in Figure REF (A).",
"Hence the RIII move does not change the contribution of $g$ and $w$ to $\\Sigma $ .",
"If $E_{sw} \\le 1$ , then the RIII move does not increase the contribution of $w$ to $E_s$ .",
"However, in case of $E_{sw} = 1$ , it may increase the contribution of $g$ and $w$ to $E_{svs}$ as in Figure REF (B).",
"Hence the RIII move does not change the contribution of $g$ and $w$ to $\\Sigma $ in Case (B) in Figure REF , and decreases it by one in the other cases.",
"Next, we consider Case (ii).",
"Before the RIII move, $g$ contributes $E_i$ by one.",
"Hence the contribution of $g$ and $w$ to $\\Sigma $ decreases by one after the RIII move.",
"In Case (iii), the RIII move does not change the contributions of $g$ and $w$ to $\\Sigma $ .",
"In Case (iv), the RIII move increases the contribution of $g$ to $E_{ss}$ by one.",
"Hence the contribution of $g$ and $w$ to $\\Sigma $ increases by one after the RIII move.",
"Similarly, we have four cases (i) through (iv) for the vertex $x$ , which is an endpoint of the edge $h$ .",
"We consider change of $\\Sigma $ under the RIII move.",
"It decreases $V$ by one since $s$ goes over the vertex $v$ .",
"Hence, if $\\Sigma $ does not decrease by the RIII move, then precisely one of $w$ and $x$ must be in the pattern (iv), i.e., in int $s$ .",
"(Note that we have already considered the case where both $w$ and $x$ is in the pattern (iv) in Figure REF (III).)",
"In this case, $E_{sv} = 3$ before the RIII move, and this leads to decrease of $E_s$ by one.",
"Thus, in any case, $\\Sigma $ eventually decreases by one or more.",
"Figure: Move (4) which shoud be first performedWe consider Move (4) in Figure REF (4)-1 and (4)-2, where the edge $e$ has an endpoint in $s$ $(=({\\rm int}\\,s) \\cup \\partial s)$ , the other endpoint $v$ of $e$ is in int $Q$ , an edge $f$ is incident to $v$ , the two edges $e$ and $f$ are in the boundary of the same face, and $f$ has another endpoint $w$ in int $u$ ((4)-1) or int $Q$ ((4)-2) before Move (4).",
"We must perform Move (4) at an adequate place as below.",
"First of all, if there is a pattern in Figure REF , then we immediately perform Move (4) there along an arc parallel to $g$ .",
"In Figure REF , the vertex $w$ is in int $Q$ , another edge $g$ is incident to $w$ , the edge $g$ reaches int $s$ , the arc $e \\cup f \\cup g$ cuts off a disk, say $R$ , from $Q$ , the disk $R$ contains all the edges incident to $w$ and contains none of the edges incident to $v$ other than $e$ and $f$ , and the edges incident to $v$ or $w$ do not reach int $s$ except $e$ and $g$ .",
"The endpoint of $e$ other than $v$ may be at $\\partial s$ .",
"In this case, we perform Move (4) not along an arc parallel to $e$ but along an arc parallel to $g$ and outside of $R$ .",
"This move decreases $E_i$ by one, and hence $\\Sigma $ by one.",
"(Note that $E_{svs}$ may increase if we perform Move (4) along an arc parallel to $e$ and inside of $R$ .)",
"Figure: The graph G D G_DWe consider the case where $D_Q$ does not contain the pattern in Figure REF .",
"We observe the subgraph $G_D$ of $D_Q$ as shown in Figure REF (1).",
"Precisely, let $V_1$ be the set of vertices $v$ of $D_Q$ in int $Q$ such that there are two or more edges connecting $v$ and $s \\,(= ({\\rm int}\\,s) \\cup \\partial s)$ .",
"Let $E_1$ be the set of edges of $D_Q$ which are incident to a vertex of $V_1$ and reach $s$ , $E_2$ the set of edges of $D_Q$ which have both endpoints in $s$ , and $V_2$ the set of vertices of $D_Q$ with $V_2 = (\\cup (E_1 \\cup E_2)) \\cap s$ .",
"Then we define $G_D$ to be the subgraph of $D_Q$ with $V_1 \\cup V_2$ being the set of vertices of $G_D$ and $E_1 \\cup E_2$ the set of edges of $G_D$ .",
"Note that $G_D$ does not consist of a single edge, say $e_{\\partial }$ , connecting the two points $\\partial s$ when int $s$ contains no vertex of $D_Q$ .",
"See Figure REF (2).",
"If it did, then the link diagram $D$ would have a component $e_{\\partial } \\cup s$ with no crossing, and hence, $D$ would be either a disconnected link diagram or a knot diagram with no crossing, which contradicts our assumption.",
"Figure: We cannot perform Move (4).When $G_D = \\emptyset $ , we perform Move (4) anywhere if it is applicable.",
"Let $v, e, f$ be as in Figure REF (4)-1 or 2.",
"We first consider Move (4)-2 where the edge $f$ has an endpoint in a vertex, say $w$ , in int $Q$ .",
"Note that $E_{sv} \\le 1$ and $E_{sw} \\le 1$ because of the condition $G_D = \\emptyset $ before the move.",
"Hence $E_{sv} \\le 2$ and $E_{sw} \\le 2$ after the move.",
"Thus the move decreases $E_i$ by one and does not change $E_s$ , and hence decreases $\\Sigma $ by one.",
"(It may increase $E_{svs}$ by one if there is a single edge, say $g$ , connecting $w$ and int $s$ and all the edges incident to $w$ is contained in the subdisk of $Q$ bounded by the arc $e \\cup f \\cup g$ and a subarc of $s$ .",
"However, we have already considered such a pattern in Figure REF .)",
"Next, we consider Move (4)-1 where the edge $f$ has an endpoint in int $u$ .",
"The move decreases $E_i$ by one, and hence $\\Sigma $ by one, again.",
"Note that $f$ contributes to $E_i$ before the move.",
"When $G_D \\ne \\emptyset $ , we will prove that Move (4) can be applicable if $G_D$ does not contain edges $e$ and $f$ as in one of the patterns in Figure REF , where $e$ has an endpoint at $\\partial s$ , $f$ has an endpoint in $s$ , and the arc $e \\cup f$ cuts off a disk $R$ from $Q$ with $G_D \\cap R = e \\cup f$ .",
"In addition, $E_{sv} \\le 1$ in Case (B).",
"We consider the outermost component of $G_D$ as below.",
"Among the disks obtained from $Q$ by cutting along $G_D$ , one which contains a single subarc of int $s$ or the whole of $s$ is called an outermost disk.",
"Let $R$ be an outermost disk.",
"If $R \\cap D_Q$ consists of exactly two edges one of which connects a vertex, say $v$ , in int $Q$ and a point of $\\partial s$ and the other does $v$ and a point in $s$ as in Figure REF , then we cannot perform Move (4), and go forth to Move (5) in Figure REF .",
"We take $R$ so that it does not contain any point of $\\partial s$ if there is such an outermost disk.",
"We call this Condition (*).",
"Figure: the case where int tt intersects D Q D_QFigure: y 1 =y 2 =vy_1 = y_2 = vFirst, we consider the case where the arc $t=R \\cap s$ contains a vertex of $D_Q$ other than $\\partial t$ .",
"Let $w$ be an arbitrary one of it.",
"There is an edge, say $g$ , of $D_Q$ incident to $w$ .",
"Let $x$ be the other endpoint of $g$ , and $h_1, h_2$ the edges of $D_Q$ incident to $x$ and in the boundary of the same face of $D_Q (\\subset Q)$ with $g$ .",
"See Figure REF .",
"Since we have taken the disk $R$ to be outermost, the endpoint of $h_i$ other than $x$ , say $y_i$ is in int $Q$ , and $E_{sy_i} \\le 1$ for $i=1$ or 2.",
"(If this were not the case, then for $i=1$ and 2, $E_{sy_i} \\ge 2$ , and the outermost disk $R$ would be a triangle cut from $Q$ by two edges, say $e$ and $f$ , sharing the same vertex, say $v$ , in int $Q$ , and $y_i = v$ .",
"See Figure REF .",
"Then a circle obtained by slightly shrinking the circle $h_1 \\cup h_2$ would intersect the link digaram $D$ in a single point in the edge incident to $x$ other than $g, h_1, h_2$ , a contradiction.)",
"Hence we can assume that $E_{sy_1} \\le 1$ .",
"We perform Move (4) along an arc parallel to the edge $g$ and connecting int $s$ and the edge $h_1$ .",
"See Figure REF .",
"Then we can confirm that the move decreases $\\Sigma $ by one in a similar way as in the case of $G_D = \\emptyset $ .",
"Figure: the case where int tt does not intersect D Q D_QThus we can assume that int $t = R \\cap s$ does not contain a vertex of $D_Q$ .",
"Then, $R \\cap G_D$ cannot be an arc with its both endpoints in $s$ , since we are under the assumption that $D$ is connected and that Moves (1)-(3) cannot be applied and that Move (4) can be applied.",
"Hence the outermost disk is cut from $Q$ by the union of two edges, say $e$ and $f$ , sharing a vertex, say $v$ , in int $Q$ .",
"Then all the edges incident to $v$ other than $e$ and $f$ are contained in the outermost disk $R$ .",
"(If precisely one of them were in $R$ , then a circle obtained by shrinking the circle $e \\cup f \\cup t$ would intersect $D$ in a single point.",
"If both of them were in cl $(Q-R)$ , the Move (3) would be applicable when $\\partial t \\in $ int $s$ , there would be a pattern as in Figure REF (B) when precisely one of the point of $\\partial t$ is in $\\partial s$ (we can see $E_{sv}=1$ because of Condition (*)), and $G_D$ would be of type in Figure REF (A) when $\\partial t = \\partial s$ .)",
"We perform Move (4) along an arc parallel to $e$ and contained in $R$ .",
"See Figure REF .",
"This move does not increase $E_{svs}$ (we have already considered Figure REF ) and decreases $E_i$ by one since int $t$ is free from a vertex of $D_Q$ .",
"This move may increase $E_{sv}$ .",
"If it did by two, then we would have the pattern in Figure REF (a) before the move, which we have considered.",
"If the move increases $E_{sv}$ by one, then $\\partial t \\subset $ int $s$ , and the move decreases $E_{svs}$ by one, since $e \\cup f$ forms a subgraph of the pattern in Figure REF before the move.",
"(Otherwise, $\\partial t \\cap \\partial s$ would consist of a single point, and we would have the pattern in Figure REF (b) before the move.)",
"In any way, $\\Sigma $ decreases by one.",
"We consider Move (5) in Figure REF .",
"We are under the assumption that the moves in Figures REF and REF cannot be applicable.",
"Note that either $G_D = \\emptyset $ or $G_D$ is one of the patterns in Figure REF .",
"Otherwise, we could perform Move (1), (2), (3) or (4).",
"If there is an edge, say $e$ , of $D_Q$ with one of its endpoints in $\\partial s$ , then we can apply Move (5).",
"The other endpoint, say $v$ , of $e$ is not contained in $s$ , and hence this move decreases $E_{\\partial }$ by one and does not increase $E_{ss}$ .",
"Because of the above condition on $G_D$ , we have either $E_{sv}=0$ or the pattern in Figure REF (B) before the move, which implies that the move increases none of $E_s$ and $E_{svs}$ .",
"Thus Move (5) decreases $\\Sigma $ by one.",
"Finally, we consider Move (6) in Figure REF (6)-1, (6)-2 and (6)-3.",
"We are under the assumption that Moves (1) through (5) cannot be applicable and $\\Sigma > 0$ .",
"We will show that Move (6) is applicable, and the move decreases $\\Sigma $ .",
"There are no edge with its endpoint in $\\partial s$ because Move (5) is not applicable.",
"We show that every edge, say $e$ , having an endpoint in int $s$ has the other endpoint, say $v$ , in int $u$ .",
"By the condition on $G_D$ , the vertex $v$ is not in $s$ .",
"If $v$ is contained in int $Q$ , then there is an edge, say $f$ incident to $v$ such that $e$ and $f$ are in the boundary of the same face of $D_Q$ in $Q$ .",
"By the condition on $G_D$ , the edge $f$ does not reach $s$ .",
"Hence we can apply Move (4) along an arc parallel to $e$ and connecting $s$ and $f$ , which is a contradiction.",
"Therefore, the endpoint $v$ is in int $u$ .",
"This means that every edge of $D_Q$ which reaches int $s$ also does int $u$ .",
"Since we are assuming that $\\Sigma >0$ and that $D$ is connected, there is an edge which is incident to a vertex in int $u$ and does not reach $s$ .",
"Hence we can perform Move (6) as below.",
"Among such edges, let $e$ be the one having an endpoint, say $w$ , in int $u$ such that $w$ is the nearest to a point, say $p$ , of $\\partial s$ .",
"Let $\\gamma $ be the subarc of $u$ between $p$ and $w$ .",
"We perform Move (6) along an arc parallel to $\\gamma $ when int $\\gamma $ does not contain a vertex of $D_Q$ .",
"See Figure REF (6)-3.",
"When it does, let $v$ be the vertex of $D_Q$ lying in the subarc of $u$ between $p$ and $w$ , and the nearest to $w$ .",
"The edge, say $f$ , incident to $v$ reaches int $s$ .",
"Let $\\delta $ be the subarc of $\\gamma $ between $v$ and $w$ .",
"We perform Move (6) along an arc parallel to $f \\cup \\delta $ .",
"See Figure REF (6)-1 and (6)-2.",
"These moves decrease $E_i$ , and hence $\\Sigma $ by one.",
"This completes the proof."
],
[
"Acknowledgments",
"The authors thank Kanako Oshiro for helpful comments.",
"Tatsuo Ando: Department of Mathematics, Graduate School of Science, Rikkyo University, 3-34-1 Nishi-ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan.",
"[email protected] (T. Ando), Chuichiro Hayashi and Yuki Nishikawa: Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women's University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan.",
"[email protected] (C. Hayashi) and [email protected] (Y. Nishikawa)"
]
] | 1403.0209 |
[
[
"Decentralized Hybrid Formation Control of Unmanned Aerial Vehicles"
],
[
"Abstract This paper presents a decentralized hybrid supervisory control approach for a team of unmanned helicopters that are involved in a leader-follower formation mission.",
"Using a polar partitioning technique, the motion dynamics of the follower helicopters are abstracted to finite state machines.",
"Then, a discrete supervisor is designed in a modular way for different components of the formation mission including reaching the formation, keeping the formation, and collision avoidance.",
"Furthermore, a formal technique is developed to design the local supervisors decentralizedly, so that the team of helicopters as whole, can cooperatively accomplish a collision-free formation task."
],
[
"INTRODUCTION",
"Nowadays, developing Unmanned Aerial Vehicles (UAVs) in different sizes and shapes for various applications has emerged as an attractive research area [1], [2], [3], [4].",
"A challenging problem in the aerial robotics area and cooperative control of UAVS is formation control, in which it is desired to instruct a group of agents to jointly move with a relatively fixed distance.",
"This capability improves the performance of UAVs to accomplish different tasks such as search and coverage more efficiently.",
"In the literature, there are several methods that can partly handle subcomponents of a formation mission.",
"For instance, for reaching the formation, methods such as MILP programming, navigation function, and potential field have been developed [5], [6], [7], [8].",
"Keeping the formation can be seen as a standard control problem in which the system's actual position has slightly deviated from the desired position [9], [10], [11].",
"Finally, in [12], [13], [14], [15], different scenarios for collision avoidance have been introduced using geometry approaches, predictive control, probabilistic methods, and invariant sets.",
"Nevertheless, putting all together to address the whole components of the formation mission, requires an in-depth understanding of the interplay between the components based on which a decision making unit can be embedded in the control structure of the UAVs.",
"To make this control structure reliable enough, two main problems should be addressed.",
"Firstly, this control structure has a hybrid nature, which includes both the continuous dynamics of the UAVs and the discrete dynamics of the decision making unit that interactively coexist in the system [16].",
"Although a common practice is to treat the continuous and the discrete structure of the system in a decoupled way, the ignorance of the interactions between the continuous and discrete dynamics of the system degrades the reliability of the overall system.",
"Secondly, to take the advantage of decentralized control schemes, e.g.",
"distributing the computation costs among the agents and increasing the reliability of the system against the possible failures, a decentralized controller is required.",
"To address the first problem, in [17], a hybrid supervisory control framework was introduced for the formation control of UAVs.",
"This paper addresses the second problem and presents a decentralized hybrid supervisory control of UAVs that are involved in a leader-follower formation scenario.",
"First, using the abstraction techniques, a DES model is obtained for the motion dynamics of each agent.",
"Then, the formation task is formulated by logical requirements for which we have modularly designed the discrete supervisors for different components of the formation including reaching the formation, keeping the formation, and collision avoidance.",
"In the reaching and keeping the formation, the follower UAVs can satisfy the desired performance independently.",
"However, for the collision avoidance, a tight cooperation of the UAVs is required.",
"For this purpose, a collision avoidance supervisor is designed, so that the team of UAVs as whole, can cooperatively satisfy the collision avoidance specification as a global goal.",
"Then, to render the decentralized implementation, the designed global supervisor is decomposed into local supervisors through the natural projections into local event sets.",
"The rest of this paper is organized as follows.",
"Section describes the problem formulation.",
"Section obtains an abstract model for the motion dynamics of the follower UAVs using the polar partitioning of the motion space.",
"A discrete supervisor is modularly designed in Section , and then, it is decomposed into local supervisors.",
"The paper is concluded in Section ."
],
[
"Problem formulation",
"In [18] and [19] it is shown that subject to the proper implementation of the inner-loop for an unmanned helicopter to be fast enough to track the given references, the outer loop dynamics can be approximately described as follows: $\\dot{x}=u, \\,\\,\\,\\,\\,x\\in \\mathbb {R}^2,\\,\\,\\,\\,\\,u\\in U\\subseteq \\mathbb {R}^2,$ where $x$ is the position of the UAV; $u$ is the UAV velocity reference generated by the formation algorithm, and $U$ is the convex set of velocity constraints.",
"Also, assume that the UAVs are flying at the same altitude, and the velocity of the k'th follower, $UAV_k$ , $k=1,2$ is in the following form: ${V_{follower}}_k=V_{leader}+{V_{rel}}_k.$ Now, we can consider a relatively fixed frame for each follower UAV, in which each follower moves with the relative velocity $V_{rel}$ .",
"Problem 1 Given the dynamics of the follower UAVs as (REF ) and their velocity in the form of (REF ), design the formation controller to generate the relative velocity of the followers, ${V_{rel}}_k$ , such that starting from any initial state inside the control horizon, the follower UAVs eventually reach their desired positions, while avoiding the collision with other follower UAVs.",
"Moreover, after reaching the formation, the follower UAVs should remain at the desired positions."
],
[
"Discrete model of the UAV motion dynamics over the polar partitioned space",
"To address this problem, for each UAV consider a circle with the radius of $R_m$ that is centered at its desired position.",
"With the aid of the partitioning curves $\\lbrace r_i = \\frac{{R_m }}{{n_r }-1}(i-1),\\,\\,i = 1,...,n_r \\rbrace $ and $\\lbrace \\theta _j = \\frac{2\\pi }{{n_\\theta }-1}(j-1),\\,\\,j =1,...,n_\\theta \\rbrace $ , this circle can be partitioned into $(n_r-1)(n_{\\theta }-1)$ partitioning elements.",
"In this partitioned space, an element $R_{i,j}=\\lbrace p=(r,\\theta )|\\, r_{i}\\le r\\le r_{i+1},\\,\\theta _{j}\\le \\theta \\le \\theta _{j+1}\\rbrace $ , has four vertices, $v_0,v_1,v_2,v_3$ (Fig.",
"REF ), four edges, $E_r^+$ , $E_r^-$ , $E_{\\theta }^+$ , $E_{\\theta }^-$ (Fig.",
"REF ).",
"The set $V(\\ast )$ stands for the vertices that belong to $\\ast $ ($\\ast $ can be an edge, or a region ${R}_{i,j}$ ).",
"Figure: (a) Vertices of the element R i,j R_{i,j}.",
"(b) Edges of the element R i,j R_{i,j}.As shown in [17], for a system with a multi-affine dynamics $\\dot{x} =h(x,u(x))$ defined over this polar partitioned space, two control features can be designed.",
"First, the region $R_{i,j}$ can be invariant, i.e., the trajectories of the system remain inside the region forever.",
"The other control feature is the exit edge.",
"It is possible to design a controller to drive the system's trajectory to exit from the edge $E_q^s$ , $q \\in \\lbrace r,\\theta \\rbrace $ and $s\\in \\lbrace + , - \\rbrace $ , by choosing the control values $u(v_m)$ at the vertices.",
"According to the properties of multi-affine systems, the control value at any point inside the region can be achieved based on the control values at the vertices as $u(x)= \\Sigma _{m=0}^3{\\lambda _m(x) u(v_m )}$ , which $\\lambda _m(x)$ is a coefficient that determines the weight of $u(v_m )$ in the control value $u(x)$ .",
"We denote the controller for having a region invariant by ${C_0}_k$ .",
"Also, ${C_r}_k^ +$ , ${C_r}_k^ -$ , ${C_{\\theta }}_k^ +$ , and ${C_{\\theta }}_k^-$ are respectively the controllers for having the edges ${F_r}^ +$ , ${F_r}^ -$ , ${F_{\\theta }}^ +$ , and ${F_{\\theta }}^-$ , as exit edges.",
"Further details on how to design these controllers are provided in [17].",
"Now, this model of the UAV motion dynamics over the partitioned space can be abstracted to a finite state machine and can be presented by a discrete automaton.",
"An automaton can be formally defined as follows: Definition 1 (Automaton)[20].",
"A deterministic automaton is a tuple $A := \\left(Q, q_0, E, \\delta ,Q_m\\right)$ consisting of a set of states $Q$ ; an initial state $q_0\\in Q$ ; a set of events $E$ that causes transitions between the states, and a transition relation $\\delta \\subseteq Q \\times E\\times Q$ (with a partial map $\\delta : Q \\times E \\rightarrow Q$ ), such that $(q, e,q^{\\prime })\\in \\delta $ if and only if state $q$ is transited to state $q^{\\prime }$ by event $e$ , denoted by $q\\overset{e}{\\underset{}{\\rightarrow }} q^{\\prime }$ (or $\\delta (q, e)= q^{\\prime }$ ).",
"$Q_m\\subseteq Q$ represents the marked states to assign a meaning of accomplishment to some states.",
"For supervisor automaton whose all states are marked, $Q_m$ is omitted from the tuple.",
"For this automaton, the sequence of these events forms a string.",
"We use $\\varepsilon $ to denote an empty string, and $\\Sigma ^{*}$ to denote the set of all possible strings over the set $\\Sigma $ including $\\varepsilon $ .",
"The language of the automaton $G$ , denoted by $L(G)$ , is the set of all strings that can be generated by $G$ , starting from the initial states.",
"The marked language, $L_m(G)$ , is the set of strings that belong to $L(G)$ and end with the marked states.",
"For $UAV_1$ , the discrete model of the system over the partitioned space can be described by the automaton $A_1=\\left(Q_1, {q_0}_1, E_1, \\delta _1,{Q_m}_1\\right)$ whose set of discrete states is $Q_1=\\lbrace R_1,O_1\\rbrace $ , and its event set is ${E _1} =C_1\\cup \\lbrace {C_0}_1\\rbrace \\cup D_1 \\cup Ex$ , where $C_1=\\lbrace {C_r}_1^ + ,{C_r}_1^ - ,{C_{\\theta }}_1^ + ,{C_{\\theta }}_1^ -\\rbrace $ and $D_1= \\lbrace {d_{i,j}}_1|\\,\\,1 \\le i \\le {n_r} - 1,1 \\le j \\le {n_\\theta } - 1\\rbrace $ .",
"When $UAV_1$ is in one of the regions $R_{i,j}$ , in the abstract model it is considered to be in the discrete state $R_1$ .",
"Then, one of the actuation commands belong to $C_1$ drives the UAV to one of its adjacent regions.",
"In this case, right after issuing the actuation commands, the system transits to the detection state $O_1$ and waits until the UAV enters a new region.",
"Crossing boundaries of the new region, a detection event belonging to $D_1$ will be generated which shows the UAV has entered the new region $R_{i^{\\prime },j^{\\prime }}$ .",
"The command ${C_0}_1$ , keeps the UAV in the current region and does not change the discrete state of the system.",
"We use the notation ${D_{M}}_1=\\lbrace {d_{i,j}}_1|\\,\\,1 \\le i =1,1 \\le j \\le {n_\\theta } - 1\\rbrace $ to denote the detection events, which show entering a region in the first circle, and $d_1=D_1- {D_{M}}_1=\\lbrace {d_{i,j}}_1|\\, 1 < i \\le {n_r} - 1,1 \\le j \\le {n_\\theta } - 1\\rbrace \\subseteq D_1$ for the rest of detection events.",
"Here, $Ex=\\lbrace Ca_{12F},\\,Ca_{12N},Ca_{21F},Ca_{21N},\\,\\,Stop_1,\\,Stop_2,{R_{21}},\\,{R_{12}}\\rbrace $ is the set of external events which are required for the collision avoidance and do not change the state of the system.",
"The events belong to $CA=\\lbrace Ca_{12F},\\,Ca_{12N},Ca_{21F},Ca_{21N}\\rbrace $ show the collision alarms, in which the events in $CA_1=\\lbrace C{a_{12F}}, C{a_{12N}}\\rbrace $ show that $UAV_2$ enters the alarm zone of $UAV_1$ and accordingly, the events in $CA_2=\\lbrace C{a_{21F}},C{a_{21N}}\\rbrace $ show that $UAV_1$ enters the alarm zone of $UAV_2$ .",
"The details will be discussed in Section REF .",
"The events $Stop_1$ and $Stop_2$ are the commands that request $UAV_1$ and $UAV_2$ to stop at their current position in the relative frame and the command $R_{12}$ and $R_{21}$ release them, respectively.",
"Similar definitions can be given for the DES model of $UAV_2$ .",
"The graph representation of the discrete models of $UAV_1$ and $UAV_2$ are shown in Fig.",
"REF .",
"In these graphs, the arrows starting from one state and ending to another state represent the transitions, labeled by the events belong to $E_i$ .",
"The entering arrows stand for the initial states.",
"Marked states are shown by double circles.",
"Figure: (a) DES model of UAV 1 UAV_1.",
"(b) DES model of UAV 2 UAV_2.In the DES model of $UAV_k$ , $k=1,2$ , the event set $E_k$ consists of the controllable event set ${E_c}_k=\\lbrace {C_0}_k$ , ${C_r}_k^ +$ , ${C_r}_k^ -$ , ${C_{\\theta }}_k^ +$ , ${C_{\\theta }}_k^ -$ , $Stop_1$ , $Stop_2$ , ${R_{21}}$ , ${R_{12}}\\rbrace $ and the uncontrollable event set ${E_{uc}}_{k}=\\lbrace Ca_{12F},\\,Ca_{12N},Ca_{21F},Ca_{21N}\\rbrace \\cup D_1$ .",
"The uncontrollable events are those that cannot be affected by the supervisor.",
"A language $K$ is controllable with respect to the language $L(A)$ and the event set $E_{uc}$ if and only if $\\forall s \\in K$ and $\\sigma \\in E_{uc}$ , if $s\\sigma \\in L(A)$ , then $s\\sigma \\in K $ .",
"Indeed, the controllability is the existence condition of a supervisor for the control goal described by the specification $K$ [20]."
],
[
"Designing a decentralized modular supervisor for the formation control of the UAVs",
"Given the discrete model of follower UAVs over the partitioned space, it is possible to design the supervisor to achieve a desired order of events to accomplish the formation.",
"Indeed, the supervisor, $S$ , observes the executed strings of the plant $A$ and disables the undesirable controllable events.",
"Here, we assume that all of the events are observable.",
"The generated language and marked language of the closed-loop system, $L(S/A)$ and $L_m(S/A)$ , can be constructed as follows: (1) $\\varepsilon \\in L(S/A)$ (2) $\\left[(s \\in L(S/A))\\,\\,and\\,\\,(s\\sigma \\in L(A))\\,\\,and\\,\\,(\\sigma \\in L( S))\\right] \\Leftrightarrow (s\\sigma \\in L(S/A)) $ (3) $L_m(S/A)=L(S/A)\\bigcap L_m(A)$ where $s$ is the string that has been generated so far by the plant $A$ , and $\\sigma $ is an event, which the supervisor $S$ should decide whether keep it active or not in the supervised system $S/A$ .",
"Within this framework one can use parallel composition to facilitate the control synthesis.",
"Parallel composition is a binary operation between two automata which can be defined as follows: Definition 2 (Parallel Composition [20]) Let $A_i=\\left( Q_i,q_i^0,E_i,\\delta _i,{Q_m}_i\\right)$ , $i=1,2$ , be automata.",
"The parallel composition (synchronous composition) of $A_1$ and $A_2$ is the automaton $A_1||A_2=(Q =Q_1 \\times Q_2, q_0 = (q_1^0, q_2^0), E = E_1 \\cup E_2,\\delta , Q_m={Q_m}_1$ $\\times {Q_m}_2)$ , with $\\delta $ defined as $\\forall (q_1, q_2)\\in Q, e\\in E: \\delta (\\left(q_1, q_2),e\\right)=\\\\\\left\\lbrace \\begin{array}{ll}\\left(\\delta _1(q_1, e), \\delta _2(q_2, e)\\right), & \\hbox{if $\\delta _1(q_1,e)!, \\delta _2(q_2,e)!,$}\\\\& \\hbox{$e\\in E_1 \\cap E_2$;}\\\\\\left(\\delta _1(q_1, e), q_2\\right), & \\hbox{if $\\delta _1(q_1,e)!, e\\in E_1 \\backslash E_2$;} \\\\\\left(q_1, \\delta _2(q_2, e)\\right), & \\hbox{if $\\delta _2(q_2,e)!, e\\in E_2 \\backslash E_1$;} \\\\\\hbox{undefined}, & \\hbox{otherwise.",
"}\\end{array}\\right.$ Here, the parallel composition is used to combine the plant's discrete model and the supervisor as follows: Lemma 1 [21] Let $A=(Q,\\,q_0,\\,E,\\,\\alpha ,\\,Q_m)$ , be the plant automaton and $K\\subseteq E ^*$ be the desired marked language.",
"There exists a nonblocking supervisor $S$ such that $L_m(S/A)=L_m(S\\Vert A)=K$ if $\\emptyset \\ne K=\\bar{K}\\bigcap L_m(A)$ and $K$ is controllable.",
"In this case, $S$ could be any automaton with $L(S)=L_m(S)=\\bar{K}$ .",
"Now, using the above lemma, it is possible to design the supervisor for the formation problem described in Problem REF , which includes two modules: 1- Reaching and keeping the formation and 2- Avoiding collision.",
"Next lemma describes how to design the supervisors in a modular way.",
"Lemma 2 [21] Let $A=(Q,\\,q_0,\\,E,\\,\\alpha ,\\,Q_m)$ be the plant automaton and the prefix-closed controllable languages $K_1, K_2\\subseteq E ^*$ be the desired marked specifications.",
"Suppose there exist nonblocking supervisors ${S}_1$ and ${S}_2$ such that $L_m({S}_1/A)=L_m({S}_1\\Vert A)=K_1$ and $L_m({S}_2/A)=L_m({S}_2\\Vert A)=K_2$ , then $S={S}_1\\Vert {S}_2$ is a nonblocking supervisor with $L_m(S\\Vert A)=K_1\\bigcap K_2$ .",
"$\\blacksquare $"
],
[
"Designing the supervisor for reaching and keeping the formation",
"For reaching the formation, it is sufficient to directly drive each of the follower UAVs towards one of the regions $R_{1,j}$ , $1\\le j \\le n_\\theta -1$ , located in the first circle in their corresponding partitioned motion space.",
"After reaching $R_{1,j}$ , the UAVs should remain inside it, to keep the formation.",
"The specifications ${K_F}_1$ and ${K_F}_2$ for reaching and keeping the specification for $UAV_1$ and $UAV_2$ are realized in Fig.",
"REF .",
"When the k'th follower UAV is not in the first circle, the command ${C_r}_k^-$ will be generated to push the UAV towards the origin.",
"Entering a new region, one of the events from $d_k=\\lbrace {d_{i,j}}_k|\\,\\,1 < i \\le {n_r} - 1,1 \\le j \\le {n_\\theta } - 1\\rbrace $ will appear.",
"This will continue until one of the events from ${D_{M}}_k=\\lbrace {d_{i,j}}_k|\\,\\,i=1,1 \\le j \\le {n_\\theta } - 1\\rbrace $ be generated, which shows that the formation is reached.",
"In this case, the event ${C_0}_k$ is activated, which keeps the system trajectory inside the first region.",
"If a collision alarm happens to $UAV_k$ , the formation supervisor does not change the generable language after the events belonging to $CA$ , and lets the collision avoidance supervisor handle it until the collision be avoided and the UAV be released to resume the formation task.",
"It can be seen that ${K_F}_k$ , $k=1,2$ are controllable with respect to the plant language ${L(A_k)}$ and the event set ${E_{uc}}_k$ , as they do not disable any uncontrollable event.",
"Therefore, based on Lemma REF , there exist supervisors that can control the plants $A_1$ and $A_2$ to achieve these specifications.",
"The supervisors are the realization of the above specifications in which all states are marked.",
"Marking all states of the supervisors allows the closed-loop marked states to be solely determined by the plants' marked states.",
"The supervisor for reaching the formation and keeping the formation of $UAV_k$ is denoted by ${A_F}_k$ .",
"Figure: (a) The specification for reaching and keeping the formation for UAV1.",
"(b) The specification for reaching and keeping the formation for UAV2."
],
[
"Designing the supervisor for collision avoidance",
"When $UAV_1$ is going to reach its desired position, in some situations, the other follower, $UAV_2$ , may enter the alarm zone of $UAV_1$ (Fig.",
"REF ), which requires these UAVs to cooperatively avoid the collision.",
"For this purpose, first, $UAV_1$ asks $UAV_2$ to stop in the relative frame and then, $UAV_1$ finds a path to safely get away from $UAV_2$ .",
"After avoiding the collision, $UAV_1$ releases $UAV_2$ and both UAVs resume their normal operation for reaching the formation.",
"Similar strategy is taken when $UAV_1$ enters the alarm zone of $UAV_2$ .",
"This specification, $K_C$ , is shown in Fig.",
"REF whose left side shows that after appearing one of the events ${ca_{12}}_F$ or ${ca_{12}}_N$ , $UAV_1$ realizes that $UAV_2$ has entered its alarm zone.",
"Therefore, by event $Stop_2$ , $UAV_1$ requests $UAV_2$ to stop for a while to safely manage the situation.",
"The event ${ca_{12}}_F$ shows that $UAV_2$ is in front of the path of $UAV_1$ towards its destination and hence, to avoid the collision it is sufficient that $UAV_1$ turns anticlockwise to change its azimuth angle, $\\theta $ , by activating the command $C_{\\theta } ^+$ .",
"This will continue until removing the collision alarm.",
"Then, $UAV_1$ releases $UAV_2$ , and reaching the formation can be resumed by the reaching formation supervisor which was explained in the previous section.",
"Meanwhile, if $UAV_1$ enters one of the regions in the first circle, one of the events belong to ${D_{M}}_1$ appears which means that $UAV_1$ has reached its desired formation and should remain there for the rest of mission.",
"Similarly, the right side of Fig.",
"REF , shows the collision avoidance mechanism when $UAV_1$ enters the alarm zone of $UAV_2$ .",
"If neither of collision avoidance alarms from the set $CA$ happens, then $UAV_1$ and $UAV_2$ can do their normal operations by independent enabling of events $C_1$ and $C_2$ followed by the detection signals $D_1$ and $D_2$ in any order as shown on the top of Fig.",
"REF .",
"The other module, the reaching formation supervisor, will manage this situation.",
"It can be verified that ${K_C}$ is controllable with respect to the language ${L(A_1|| A_2)}$ and the event set ${E_{uc}}_1 \\cup {E_{uc}}_2$ .",
"Therefore, based on Lemma REF , there exists a supervisor $A_c$ that can control the plants $A_1$ and $A_2$ to achieve this joint specification.",
"The supervisor is the realization of the specification $K_C$ in which all states are marked.",
"Figure: The specification for cooperative collision avoidance.The collision avoidance supervisor, $A_C$ , is a centralized supervisor which manages both $UAV_1$ and $UAV_2$ .",
"To make this supervisor decentralized and to achieve local supervisors, we will utilize our proposed decomposition scheme introduced in [22].",
"Here, local supervisors can be achieved by the projection of the global supervisor to each agent's local event set.",
"The projection of the global supervisor $A_C$ to the event set of $UAV_i$ , $E_i$ , is denoted by $P_{E_i}(A_C)$ , and can be obtained by replacing the events that belong to $E\\backslash E_i$ by $\\varepsilon $ -moves, and then, merging the $\\varepsilon $ -related states.",
"Once the local supervisor automata are derived through the natural projection, the decentralized supervisor is then obtained using the parallel composition of local supervisor automata.",
"Parallel composition captures the logical behavior of concurrent distributed systems by allowing each subsystem to evolve individually on its private events, while synchronize with its neighbors on shared events for cooperative tasks.",
"The obtained decentralized supervisor is then compared with the original global supervisor automaton using the bisimulation relation.",
"Definition 3 Consider two automata $A_i=( Q_i, q_i^0$ , $E,\\delta _i)$ , $i=1, 2$ .",
"The automaton $A_1$ is said to be similar to $A_2$ (or $A_2$ simulates $A_1$ ), denoted by $A_1\\prec A_2$ , if there exists a relation $R$ from $A_1$ to $A_2$ over $Q_1$ , $Q_2$ and with respect to $E$ , such that (1) $(q_1^0, q_2^0) \\in R$ , and (2) $\\forall \\left( {q_1 ,q_2 } \\right) \\in R,q^{\\prime }_1 \\in \\delta _1(q_1, e)$ , then $\\exists q_2^{\\prime }\\in Q_2$ such that $q^{\\prime }_2 \\in \\delta _2(q_2, e)$ , $\\left( {q^{\\prime }_1 ,q^{\\prime }_2 } \\right) \\in R$ .",
"Automata $A_1$ and $A_2$ are said to be bisimilar (bisimulate each other), denoted by $A_1\\cong A_2$ if $A_1\\prec A_2$ with a simulation relation $R_1$ , $A_2\\prec A_1$ with a simulation relation $R_2$ and $R_1^{-1} = R_2$ , where $R_1^{-1}= \\lbrace (y,x)\\in Q_2\\times Q_1|(x,y)\\in R_1\\rbrace $ .",
"Based on these definitions we can formally describe the decomposability conditions with respect to two local event sets.",
"Lemma 3 (Theorem 4 in [22]) A deterministic automaton $A = (Q, q_0, E=E_1\\cup E_2, \\delta )$ is decomposable with respect to parallel composition and natural projections $P_i$ , $i=1,2$ , such that $A\\cong P_1(A)||P_2(A)$ if and only if $A$ satisfies the following decomposability conditions (DC): $\\forall e_1 \\in E_1\\backslash E_2, e_2 \\in E_2\\backslash E_1, q\\in Q$ , $s\\in E^*$ , $DC1$ : $[\\delta (q,e_1)!\\wedge \\delta (q,e_2)!",
"]\\Rightarrow [\\delta (q, e_1e_2)!",
"\\wedge \\delta (q,e_2e_1)!",
"]$ ; $DC2$ : $\\delta (q, e_1e_2s)!\\Leftrightarrow \\delta (q, e_2e_1s)!$ ; $DC3$ : $\\forall s, s^{\\prime } \\in E^*$ , $s\\ne s^{\\prime }$ , $p_{E_1\\cap E_2}(s)$ , $p_{E_1\\cap E_2}(s^{\\prime })$ start with the same common event $a\\in E_1 \\cap E_2$ , $q\\in Q$ : $\\delta (q, s)!",
"\\wedge \\delta (q,s^{\\prime })!",
"\\Rightarrow \\delta (q,\\overline{p_1(s)}|\\overline{p_2(s^{\\prime })})!",
"\\wedge \\delta (q,\\overline{p_1(s^{\\prime })}|\\overline{p_2(s)})!$ ; $DC4$ : $\\forall i\\in \\lbrace 1, 2\\rbrace $ , $x, x_1, x_2 \\in Q_i$ , $x_1\\ne x_2$ , $e\\in E_i$ , $t\\in E_i^*$ , $x_1\\in \\delta _i (x, e)$ , $x_2\\in \\delta _i(x, e)$ : $\\delta _i (x_1, t)!",
"\\Leftrightarrow \\delta _i(x_2, t)!$ .",
"where, $\\bar{K}=\\lbrace s\\in \\Sigma ^*|(\\exists t \\in \\Sigma ^* ) st \\in K \\rbrace $ is the prefix closure of the language $K$ .",
"The decomposability conditions $DC1$ and $DC2$ respectively guarantee that any decision on the selection or order of two transitions can be done by the team of agent, while conditions $DC3$ and $DC4$ respectively ensure that the interaction of local automata $P_1(A)$ and $P_2(A)$ neither allows an illegal string that is not in $A$ , nor stops a legal string of $A$ .",
"Now assume that given the global task and local plants, a global supervisor is designed and decomposed into local supervisors such that each closed loop system (the supervised local plant with the corresponding local controller) satisfies the global task.",
"In this decentralized cooperative control architecture we are then interested to check whether the entire system satisfied the global task.",
"Problem 2 (Decentralized cooperative control problem) Consider a plant, represented by a parallel distributed system $A_P:=\\overset{2}{\\underset{i=1}{\\parallel } }A_{P_i}$ , with local event sets $E_i$ , $i=1,2$ , and let the global specification is given by a deterministic task automaton $A_S$ over $E=\\overset{2}{\\underset{i=1}{\\cup } }E_i$ .",
"Furthermore, suppose that there exist a decomposable deterministic global controller automaton $A_C \\cong \\overset{2}{\\underset{i=1}{\\parallel } }P_i(A_C)$ , so that $A_P\\parallel A_C \\cong A_S$ .",
"Then, whether the local controllers can lead the team to satisfy the global specification in a decentralized architecture, $\\overset{2}{\\underset{i=1}{\\parallel }}(A_{P_i}\\parallel P_i(A_C))\\cong A_S$ .",
"Following result considers a team of two local plants and introduces the supervisor decomposability and satisfaction of the global task by each local supervised plant as a sufficient condition for the satisfaction of global task by the team.",
"Theorem 1 (Decentralized cooperative control using supervisor decomposition) Consider a plant, represented by a parallel distributed system $A_{P_1}\\parallel A_{P_2}$ , with local event sets $E_i$ , $i=1, 2$ , and let the global specification is given by a task automaton $A_S$ over $E= E_1 \\cup E_2$ .",
"Furthermore, suppose that there exist a deterministic global controller automaton $A_C \\cong P_1(A_C) \\parallel P_2(A_C)$ , so that $A_C \\parallel A_p \\cong A_S$ .",
"Then, the entire closed loop system satisfies the global specification, in the sense of bisimilarity, i.e., $\\overset{2}{\\underset{i=1}{\\parallel }}(A_{P_i}\\parallel P_i(A_C))\\cong A_S$ , provided the decomposability conditions $DC1$ , $DC2$ , $DC3$ and $DC4$ for $A_C$ .",
"The significance of this result is the decentralized implementation of the global supervisor, $A_C$ , given in Fig.",
"REF , by decomposing $A_C$ , into local supervisors.",
"As it can be seen in $A_C$ , the successive and adjacent events from pairs of private event sets (from different local event sets) $({C_0}_1, C_2)$ , $({C_0}_1,D_2)$ , $({C_0}_2,C_1)$ , $({C_0}_2,D_1)$ , $(C_1,D_2)$ , $(C_2,D_2)$ , appear in both orders in the global supervisor automaton therefore $DC1$ and $DC2$ are satisfied.",
"Moreover, among common events $R_{12}$ , $R_{21}$ , $CA_1=\\lbrace ca_{12F},ca_{12N}\\rbrace $ , $CA_2=\\lbrace ca_{21F},ca_{21N}\\rbrace $ , $Stop_1$ , and $Stop_2$ , the events $R_{12}$ , $R_{21}$ , $Stop_1$ , and $Stop_2$ are not shared between different strings.",
"Strings just share the events $CA_1$ , $CA_2$ , where the corresponding local strings do not interleave on these events because of predecessor common events before $CA_1$ , $CA_2$ .",
"Therefore $DC3$ also is fulfilled.",
"Finally, $DC4$ is satisfied because of the determinism of local automata $P_1(A_C)$ and $P_2(A_C)$ , and hence, the supervisor automaton $A_C$ is decomposable into ${A_C}_1=P_1(A_C)$ and ${A_C}_2=P_2(A_C)$ , shown in Fig.",
"REF , so that ${A_C}_1\\parallel {A_C}_2 \\cong A_C$ .",
"Figure: (a) The local supervisor for collision avoidance for UAV 1 UAV_1.",
"(b) The local supervisor for collision avoidance for UAV 2 UAV_2."
],
[
"Verifying the algorithm through a hardware-in-the-loop simulation platform",
"To verify the proposed algorithm, we have used a hardware-in-the-loop simulation platform [23] developed for NUS UAV helicopters [24].",
"In this platform, the nonlinear dynamics of the UAVs have been replaced with their nonlinear model, and all software and hardware components that are involved in a real flight test remain active during the simulation so that the simulation results achieved from this simulator are very close to the actual flight tests.",
"This multi-UAV simulator test bed is used to verify the proposed algorithm.",
"For this purpose, consider two followers that should track a leader UAV with a desired distance, as shown in Fig.",
"REF .",
"The distance between the desired position of the $Follower_1$ and $Follower_2$ and the leader UAV are ${\\triangle _d}_1=(12,10)$ and ${\\triangle _d}_2=(-12,-10)$ , respectively.",
"The follower UAVs initially are not at the desired position.",
"The initial distance between $Follower_1$ and its desired position is ${\\triangle _0}_1=(-41.9, -0.9)$ , and the initial distance between $Follower_2$ and its desired position is ${\\triangle _0}_2=(-17.5,0.5)$ .",
"$Follower_1$ after 34.8 sec and $Follower_2$ after 14.3 sec reach the formation and then, they will keep the formation.",
"Figure: The schematic of a formation scenario with two followers and one leaderFigure: The position of the UAVs in the x-y plane.Figure: The indices of θ\\theta and rr for the traversed regions by Follower 1 Follower_1 and Follower 2 Follower_2.After 50 sec, the formation switches.",
"For the new formation, the desired distance of the followers from the leader are ${\\triangle _d}_1=(-30,-10)$ and ${\\triangle _d}_2=(0,10)$ , while their initial distances from the desired position are ${\\triangle _0}_1=(40.5,23.3)$ and ${\\triangle _0}_2=(-14.5,-23)$ .",
"When the followers are trying to reach the desired formation, at $t=55.8$ $sec$ , $Follower_2$ enters the alarm zone of $Follower_1$ .",
"As described in Section REF , to avoid collision, $Follower_1$ asks $Follower_2$ to stop in the relative frame, and then it turns to handle the situation.",
"After removing the collision alarm, both followers have resumed their normal operation to reach and keep the formation.",
"The indices of the traversed regions for $\\theta $ and $r$ are shown in Fig.",
"REF .",
"The position of the UAVs in x-y plane is shown in Fig.",
"REF ."
],
[
"CONCLUSION",
"In this paper, a collision free formation control algorithm was proposed using hybrid supervisory control techniques.",
"The proposed supervisor has a modular structure and can accomplish three main tasks: reaching the formation, keeping the formation, and collision avoidance.",
"This control structure was implemented decentralizedly so that local (decomposed) supervisors can treat the distributed agents to achieve a globally safe and collision free environment.The efficiency of the proposed approach was verified through hardware-in-loop simulation results."
],
[
"ACKNOWLEDGMENT",
"The financial supports from NSF-CNS-1239222 and NSF- EECS-1253488 for this work are greatly acknowledged."
]
] | 1403.0258 |
[
[
"Roadmap to the morphological instabilities of a stretched twisted ribbon"
],
[
"Abstract We address the mechanics of an elastic ribbon subjected to twist and tensile load.",
"Motivated by the classical work of Green and a recent experiment that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon.",
"Different types of instabilities appear in various \"corners\" of this 4D parameter space, and are addressed through distinct types of asymptotic methods.",
"Our theory employs three instruments, whose concerted implementation is necessary to provide an exhaustive study of the various parameter regimes: (i) a covariant form of the F\\\"oppl-von K\\'arm\\'an (cFvK) equations to the helicoidal state - necessary to account for the large deflection of the highly-symmetric helicoidal shape from planarity, and the buckling instability of the ribbon in the transverse direction; (ii) a far from threshold (FT) analysis - which describes a state in which a longitudinally-wrinkled zone expands throughout the ribbon and allows it to retain a helicoidal shape with negligible compression; (iii) finally, we introduce an asymptotic isometry equation that characterizes the energetic competition between various types of states through which a twisted ribbon becomes strainless in the singular limit of zero thickness and no tension."
],
[
"Overview",
"A ribbon is a thin, long solid sheet, whose thickness and length, normalized by the width, satisfy: $\\begin{aligned}{\\rm thickness\\!",
":} \\ \\ t \\ll 1 \\ \\ \\ \\ ; \\ \\ \\ \\ {\\rm length\\!",
":} \\ \\ L \\gg 1 \\ .\\end{aligned}$ The large contrast between thickness, width, and length, distinguishes ribbons from other types of thin objects, such as rods ($t \\sim 1, L \\gg 1$ ) and plates ($t \\ll 1, L\\sim 1$ ), and underlies their complex response to simple mechanical loads.",
"The unique nature of the mechanics of elastic ribbons is demonstrated by subjecting them to elementary loads – twisting and stretching – as shown in Fig.",
"REF .",
"This basic loading, which leads to surprisingly rich plethora of patterns, a few of which are shown in Fig.",
"REF , is characterized by two small dimensionless parameters: ${\\rm twist\\!",
":} \\ \\ \\eta \\ll 1 \\ \\ \\ \\ ; \\ \\ \\ \\ {\\rm tension\\!",
":} \\ \\ T \\ll 1 \\ ,$ where $\\eta $ is the average twist (per length), and $T$ is the tension, normalized by the stretching modulus Our convention in this paper is to normalize lengths by the ribbon width W, and stresses by the stretching modulus Y, which is the product of the Young modulus and the ribbon thickness (non-italicized fonts are used for dimensional parameters and italicized fonts for dimensionless parameters).",
"Thus, the actual thickness and length of the ribbon are, respectively, ${\\rm t}=t \\cdot {\\rm W}$ and ${\\rm L} =L \\cdot {\\rm W}$ , the actual force that pulls on the short edges is $T \\cdot {\\rm Y W} $ , and the actual tension due to this pulling force is ${\\rm T}= T \\cdot {\\rm Y}$ ..",
"Figure: A ribbon of length LL and width WW (and thickness tt, not shown) is submitted to a tension TT and a twist angle θ\\theta ; the twist parameter is defined as η=θ/L=θW/L\\eta =\\theta /L=\\theta {\\rm W}/{\\rm L}.",
"The longitudinal and transverse material coordinates are ss and rr, respectively.",
"n ^\\hat{n} is the unit normal to the surface, (x ^,y ^,z ^)(\\hat{x},\\hat{y},\\hat{z}) is the standard basis, (x ^,y ^)(\\hat{x},\\hat{y}) being the plane of the untwisted ribbon.Most theoretical approaches to this problem consider the behavior of a real ribbon through the asymptotic “ribbon limit\", of an ideal ribbon with infinitesimal thickness and infinite length: $t \\rightarrow 0, L \\rightarrow \\infty $ .",
"A first approach, introduced by Green [1], [2], assumes that the ribbon shape is close to a helicoid (Fig.",
"REF a), such that the ribbon is strained, and may therefore become wrinkled or buckled at certain values of $\\eta $ and $T$ (Fig.",
"REF b,c,g,h) [4], [5].",
"A second approach to the ribbon limit, initiated by Sadowsky [6] and revived recently by Korte et al.",
"[7], considers the ribbon as an “inextensible\" strip, whose shape is close to a creased helicoid – an isometric (i.e.",
"strainless) map of the unstretched, untwisted ribbon (Fig.",
"REF d).",
"A third approach, which may be valid for sufficiently small twist, assumes that the stretched-twisted ribbon is similar to the wrinkled shape of a planar, purely stretched rectangular sheet, with a wrinkle's wavelength that vanishes as $t \\rightarrow 0$ and increases with $L$ [8].",
"Finally, considering the ribbon as a rod with highly anisotropic cross section, one may approach the problem by solving the Kirchoff's rod equations and carrying out stability analysis of the solution, obtaining unstable modes that resemble the looped shape (Fig.",
"REF e) [9].",
"A recent experiment [3], which we briefly describe in Subsec.",
"REF , revealed some of the predicted patterns and indicated the validity of the corresponding theoretical approaches at certain regimes of the parameter plane $(T,\\eta )$ (Fig.",
"REF ).",
"Motivated by this development, we introduce in this paper a unifying framework that clarifies the hidden assumptions underlying each theoretical approach, and identifies its validity range in the $(T,\\eta )$ plane for given values of $t$ and $L$ .",
"Specifically, we show that a single theory, based on a covariant form of the Föppl–von Kármán (FvK) equations of elastic sheets, describes the parameter space ($T,\\eta ,t,L^{-1})$ of a stretched twisted ribbon where all parameters in Eqs.",
"(REF and REF ) are assumed small.",
"Various “corners\" of this 4D parameter space are described by distinct singular limits of the governing equations of this theory, which yield qualitatively different types of patterns.",
"This realization is illustrated in Fig.",
"REF , which depicts the projection of the 4D parameter space on the ($T,\\eta $ ) plane, and indicates several regimes that are governed by different types of asymptotic expansions."
],
[
"Experimental observations",
"The authors of [3] used Mylar ribbons, subjected them to various levels of tensile load and twist, and recorded the observed patterns in the parameter plane ($T,\\eta $ ), which we reproduce in Fig.",
"REF .",
"The experimental results indicate the existence of three major regimes that meet at a “$\\lambda $ -point\" ($T_\\lambda ,\\eta _\\lambda )$ .",
"We describe below the morphology in each of the three regimes and the behavior of the curves that separate them: $\\bullet $ The helicoidal shape (Fig.",
"REF a) is observed if the twist $\\eta $ is sufficiently small.",
"For $T < T_{\\lambda }$ , the helicoid is observed for $\\eta <\\eta _\\mathrm {lon}$ , where $\\eta _\\mathrm {lon} \\approx \\sqrt{24 T}$ is nearly independent on the ribbon thickness $t$ .",
"For $T > T_{\\lambda }$ , the helicoid is observed for $\\eta <\\eta _\\mathrm {tr}$ , where $\\eta _\\mathrm {tr}$ exhibits a strong dependence on the thickness ($\\eta _\\mathrm {tr} \\sim \\sqrt{t}$ ) and a weak (or none) dependence on the tension $T$ .",
"The qualitative change at the $\\lambda $ -point reflects two sharply different mechanisms by which the helicoidal shape becomes unstable.",
"$\\bullet $ As the twist exceeds $\\eta _\\mathrm {lon}$ (for $T<T_{\\lambda }$ ), the ribbon develops longitudinal wrinkles in a narrow zone around its centerline (Fig.",
"REF b,c).",
"Observations that are made close to the emergence of this wrinkle pattern revealed that both the wrinkle's wavelength and the width of the wrinkled zone scale as $\\sim ({t}/\\sqrt{T})^{1/2}$ .",
"This observation is in excellent agreement with Green's characterization of the helicoidal state, based on the familiar FvK equations of elastic sheets [2].",
"Green's solution shows that the longitudinal stress at the helicoidal state becomes compressive around the ribbon centerline if $\\eta > \\sqrt{24 T}$ , and the linear stability analysis of Coman and Bassom [5] yields the unstable wrinkling mode that relaxes the longitudinal compression.",
"$\\bullet $ As the twist exceeds $\\eta _\\mathrm {tr}$ (for $T>T_{\\lambda }$ ), the ribbon becomes buckled in the transverse direction (Fig.",
"REF g), indicating the existence of transverse compression at the helicoidal state that increases with $\\eta $ .",
"A transverse instability cannot be explained by Green's calculation, which yields no transverse stress [2], but has been predicted by Mockensturm [4], who studied the stability of the helicoidal state using the full nonlinear elasticity equations.",
"Alas, Mockensturm's results were only numerical and did not reveal the scaling behavior $\\eta _\\mathrm {tr} \\sim \\sqrt{t}$ observed in [3].",
"Furthermore, the nonlinear elasticity equations in [4] account for the inevitable geometric effect (large deflection of the twisted ribbon from its flat state), as well as a mechanical effect (non-Hookean stress-strain relation), whereas only the geometric effect seems to be relevant for the experimental conditions of [3].",
"$\\bullet $ Turning back to $T <T_{\\lambda }$ , the ribbon exhibits two striking features as the twist $\\eta $ is increased above the threshold value $\\eta _\\mathrm {lon}$ .",
"First, the longitudinally-wrinkled ribbon transforms to a shape that resembles the creased helicoid state predicted by [7] (Fig.",
"REF d); this transformation becomes more prominent at small tension (i.e.",
"decreasing $T$ at a fixed value of $\\eta $ ).",
"Second, the ribbon undergoes a sharp, secondary transition, described in [3] as similar to the “looping\" transition of rods [9], [10], [11], [12] (Fig REF e).",
"At a given tension $T<T_{\\lambda }$ , this secondary instability occurs at a critical twist value that decreases with $T$ , but is nevertheless significantly larger than $\\eta _\\mathrm {lon} \\approx \\sqrt{24 T}$ .",
"$\\bullet $ Finally, the parameter regime in the $(T,\\eta )$ plane bounded from below by this secondary instability (for $T<T_{\\lambda }$ ) and by the transverse buckling instability (for $T > T_{\\lambda }$ ), is characterized by self-contact zones along the ribbon (Fig.",
"REF e).",
"The formation of loops (for $T<T_{\\lambda }$ ) is found to be hysteretic unlike the transverse buckling instability (for $T>T_{\\lambda }$ ).",
"In a recent commentary , Santangelo recognized the challenge and the opportunity introduced to us by this experiment: “Above all, this paper is a challenge to theorists.",
"Here, we have an experimental system that exhibits a wealth of morphological behavior as a function of a few parameters.",
"Is there anything that can be said beyond the linear stability analysis of a uniform state?",
"How does a smooth, wrinkled state become sharply creased?",
"These are questions that have been asked before, but maybe now there is a possibility to answer them – at least in one system\".",
"The current paper is motivated by four specific puzzles: (A) What is the minimal generalization of the standard FvK equations (i.e.",
"beyond Green's calculation) that accounts for the transverse compression of the helicoidal state, and allows a quantitative description of the transverse instability of a ribbon with Hookean stress-strain relationship (i.e.",
"linear material response)?",
"(B) How does the longitudinally-wrinkled pattern evolve upon exerting a twist $\\eta $ larger than the threshold $\\eta _\\mathrm {lon}$ , where the state cannot be described any longer as a small perturbation to the compressed helicoidal shape?",
"(C) Why do the three curves, that mark the thresholds for the secondary, “looping\" instability of the helicoidal state, and the two primary instabilities (longitudinal wrinkling and transverse buckling), meet at a single triple point $(T_{\\lambda },\\eta _{\\lambda })$ ?",
"If the three thresholds are associated with distinct physical mechanisms, as was conjectured in [3], it would have been natural for them to cross at two points (at least), rather than to meet at a single point.",
"(D) What is the physical mechanism underlying the transformation of the ribbon from the longitudinally-wrinkled pattern to the creased helicoid shape upon reducing the tension $T$ ?",
"Is this a smooth crossover, or a sharp “phase transition\" that occurs at some threshold curve in the ($T,\\eta $ ) plane?"
],
[
"Main results and outline",
"Motivated by the above questions, we develop a unified theoretical framework that addresses the rich phenomenology exhibited by the stretched-twisted ribbon in the 4D parameter space spanned by the ribbon length $L$ , its thickness $t$ , the twist $\\eta $ , and the tension $T$ , where we focus on the asymptotic regime defined by Eqs.",
"(REF ,REF ).",
"Our theory leads to a phase diagram whose projection on the tension-twist plane is plotted schematically in Fig.",
"REF , and reveals three major morphological phases: the helicoid, the longitudinally wrinkled state, and a region delimited by the transverse instability.",
"This development is based on three fundamental elements: (i) A covariant version of FvK equations of elastic sheets, dubbed here “cFvK\", which is needed to describe the large deflection (from planarity) of the twisted state of the ribbon.",
"(ii) A far-from-threshold (FT) expansion of the cFvK equations that describes the state of the ribbon when the twist exceeds the threshold value $\\eta _\\mathrm {lon}$ for the longitudinal wrinkling instability.",
"(iii) A new, asymptotic isometry equation (Eq.",
"REF ), that describes the elastic energies of admissible states of the ribbon in the vicinity of the vertical axis in the parameter plane $(T,\\eta )$ .",
"We use the notion of “asymptotic isometry\" to indicate the unique nature by which the ribbon shape approaches the singular limit of vanishing thickness and tension ($t \\rightarrow 0, T \\rightarrow 0$ and fixed $\\eta $ and $L$ ).",
"Figure: The phase diagram in the tension-twist plane consists of three main regions: the helicoid, the longitudinally-wrinkled helicoid and a region delimited from below by a transverse instability.",
"These regions meet at the λ\\lambda -point.",
"The complete phase diagram is more subtle and the following parts are magnified:(a) At vanishing tension, the ribbon shape becomes closer and closer to an (asymptotic) isometry; this is investigated in Subsec. .",
"(b) The transverse buckling instability is the focus of Sec.",
", where a transition from buckling to wrinkling is predicted.",
"(c) At very low tension and twist, the longitudinal instability is described by Green's theory (see Subsec ).",
"(d) The transition from the helicoid to the far from threshold longitudinally-wrinkled helicoid is detailed in Sec. .",
"(e) At very low twist, the transverse compression due to the clamped edges overcomes the one due to the twist (see Subsec.",
").Solid lines are for quantitative predictions, dashed lines indicate scaling laws or unknown thresholds.We commence our study in Sec.",
"with the helicoidal state of the ribbon (Fig.",
"REF a) – a highly symmetric state whose mechanics was addressed by Green through the standard FvK equations [2], which is valid for describing small deviations of an elastic sheet from its planar state.",
"We employ a covariant form of the FvK theory for Hookean sheets (cFvK equations), which takes into full consideration the large deflection of the helicoidal shape from planarity.",
"Our analysis of the cFvK equations provides an answer to question (A) above, curing a central shortcoming of Green's approach, which provides the longitudinal stress but predicts a vanishing transverse stress.",
"The cFvK equations of the helicoidal state yield both components of the stress tensor, and show that the magnitude of the transverse stress is nonzero, albeit much smaller than the longitudinal one.",
"Another crucial difference between the two stress components of the helicoidal state pertains to their sign: the transverse stress is compressive throughout the whole ribbon, everywhere in the parameter plane ($T,\\eta $ ); in contrast, the longitudinal stress is compressive in a zone around the ribbon centerline only for $\\eta >\\eta _\\mathrm {lon}\\approx \\sqrt{24T}$ .",
"The compressive nature of the stress components gives rise to buckling and wrinkling instabilities that we address in Secs.",
"and .",
"In Sec.",
"we address the wrinkling instability that relaxes the longitudinal compression for $\\eta >\\sqrt{24T}$ .",
"Noticing that the longitudinally-compressed zone of the helicoidal state broadens upon increasing the ratio $\\alpha = \\eta ^2/T$ , we recognize a close analogy between the longitudinally-wrinkled state of the ribbon and wrinkling phenomena in radially-stretched sheets , , , where the size of the wrinkled zone depends on a confinement parameter, defined by a ratio between the loads exerted on the sheet.",
"Exploiting this analogy further, we find that the longitudinally-wrinkled ribbon at $\\eta >\\eta _\\mathrm {lon}$ is described by a far-from-threshold (FT) expansion of the cFvK equations, where the longitudinal stress (at any given $\\alpha >24 $ ) becomes compression-free in the singular limit of an infinitely thin ribbon, $t \\rightarrow 0$ .",
"The FT theory predicts that the broadening of the wrinkled zone with the confinement $\\alpha $ is dramatically larger than the prediction of a near-threshold (NT) approach, which is based on a perturbative (amplitude) expansion around the compressive helicoidal state.",
"Our FT theory of the longitudinally wrinkled state provides an answer to question (B) in the above list.",
"Analyzing the FT expansion in the two limits $\\alpha \\rightarrow 24$ (i.e.",
"$\\eta \\rightarrow \\sqrt{24T}$ ), and $\\alpha \\rightarrow \\infty $ (i.e.",
"fixed $\\eta $ and $T \\rightarrow 0$ ), elucidates further the nature of the longitudinally wrinkled state.",
"In the limit $\\alpha \\rightarrow 24$ , plotted schematically in Fig.",
"REF d, we find that the FT regime prevails in the domain $\\eta > \\sqrt{24 T}$ in the $(T,\\eta )$ plane, whereas the NT parameter regime, at which the state is described as a perturbation to the unwrinkled helicoidal state, shrinks to a narrow sliver close to the threshold curve as the thickness vanishes, $t\\rightarrow 0$ .",
"Analyzing the other limit, $\\alpha \\rightarrow \\infty $ , we show that the longitudinally-wrinkled state becomes an asymptotic isometry, where the strain vanishes throughout the twisted ribbon.",
"In Sec.",
"we expand more on the meaning and implications of asymptotic isometries for a stretched-twisted ribbon.",
"The FT analysis of the two limits, $\\alpha \\rightarrow 24$ and $\\alpha \\rightarrow \\infty $ , reveals the intricate mechanics of a ribbon subjected to twist $\\eta $ , whereby the longitudinally wrinkled state entails a continuous trajectory in the $(T,\\eta )$ plane, from a strainless deformation (at $T \\rightarrow 0$ ) to a fully strained helicoidal shape (at $T \\ge \\eta ^2/24$ ).",
"In Sec.",
"we turn to the transverse instability, capitalizing on our results from Secs.",
"and .",
"First, we note that the transverse stress is compressive everywhere in the $(T,\\eta )$ plane; second, we note that it is obscured by the longitudinal stress.",
"These two features imply that the threshold for the transverse instability occurs at a curve $\\eta _\\mathrm {tr}(T)$ in the $(T,\\eta )$ plane that divides it into two parts: In the first part, defined by the inequality $\\eta _\\mathrm {tr}(T) < \\sqrt{24 T}$ , the longitudinal stress is purely tensile, and the transverse instability appears as a primary instability of the helicoidal state; in the second part, defined by $\\eta _\\mathrm {tr}(T) > \\sqrt{24 T}$ , the transverse instability is preceded by the longitudinal instability, and thus materializes as a secondary instability of the helicoidal state.",
"We conclude that the “looping\" instability observed in [3] does not stem from a new physical mechanism, but simply reflects the change in nature of the transverse instability when the threshold line $\\eta _\\mathrm {tr}(T)$ crosses the curve $\\eta _\\mathrm {lon} = \\sqrt{24 T}$ that separates the longitudinally-compressed and longitudinally-tensed domains of the ($T,\\eta $ ) plane.",
"Thus, the emergence of a single “triple\" point ($T_{\\lambda },\\eta _{\\lambda }$ ) is not mysterious, but comes naturally as the intersection of these two curves in the ($T,\\eta $ ) plane.",
"This result answers question (C) in our list.",
"The cFvK equations, together with the FT analysis of the longitudinally-wrinkled state in Sec.",
", allow us to compute the deformation modes that relax the transverse compression.",
"Two results from this stability analysis are noteworthy.",
"First, assuming an infinitely long ribbon, we find that the threshold curve satisfies $\\eta _\\mathrm {tr}(T) \\sim t/\\sqrt{T}$ in both the \"low\"-tension regime ($T < T_{\\lambda }$ ) and \"large\"-tension regime ($T > T_{\\lambda }$ ), albeit with different numerical pre-factors.",
"This theoretical prediction is in strong accord with the experimental data for the transverse buckling instability and the “looping\" instability in [3].",
"Second, we find that the length of the ribbon has a dramatic effect on the dependence of the $\\lambda $ -point on the ribbon thickness $t$ , and – more importantly – on the spatial structure of the transverse instability.",
"Specifically, we predict that if $L^{-2} \\ll t \\ll 1$ , the transverse instability is buckling, and if $t \\ll L^{-2} \\ll 1$ , it may give rise to a wrinkling pattern, similarly to a stretched, untwisted ribbon [8], with a characteristic wavelength $\\lambda _\\mathrm {tr} <1 $ that becomes smaller as $T$ increases.",
"This “buckling to wrinkling\" transition is depicted in Fig.",
"REF b.",
"In Sec.",
"we turn to the edges of the $(T,\\eta )$ plane, namely, the vicinity of the vertical and horizontal axes: $(T \\!=\\!0,\\eta )$ and $(T,\\eta \\!=\\!0)$ , respectively.",
"In order to address the first limit, we briefly review the work of Korte et al.",
"[7] that predicted and analyzed the creased helicoid state.",
"We discuss the asymptotic isometry exhibited by the creased helicoid state in the singular limit $t\\rightarrow 0,T \\rightarrow 0$ , and contrast it with the asymptotic isometry of the longitudinally wrinkled state, which was noted first in Sec. .",
"We elucidate a general framework for analyzing morphological transitions between various types of asymptotic isometries in the neighborhood of the singular hyper-plane $t=0,T= 0$ in the 4D parameter space $(T,\\eta ,t,L)$ .",
"As a consequence of this discussion, we propose the scenario illustrated in Fig.",
"REF a, where the longitudinally wrinkled state undergoes a sharp transition to the creased helicoid state in the vicinity of the $(T=0,\\eta )$ line.",
"Thus, while our discussion here is less rigorous than in the previous sections (due to the complexity of the creased helicoid state [7]), we nevertheless provide a heuristic answer to question (D) in our list.",
"Since the characterization of the creased helicoid state in [7] is based on the Sadowsky's formalism of inextensible strips rather than on the FvK theory of elastic sheets, we use this opportunity to elaborate on the basic difference between the “rod-like\" and “plate-like\" approaches to the mechanics of ribbons.",
"We also recall another rod-like approach, based on implementation of the classical Kirchoff equations for a rod with anisotropic cross section [9], [10], [11], [12], and explain why it is not suitable to study the ribbon limit (Eq.",
"REF ) that corresponds to a rod with highly anisotropic cross section.",
"Finally, we turn to the vicinity of the pure-stretching line, $(\\eta =0,T)$ , and address the parameter regime where the twist $\\eta $ is so small that the ribbon does not accommodate a helicoidal shape.",
"We provide a heuristic, energy-based argument, which indicates that the helicoidal state is established if the twist $\\eta $ is larger than a minimal value that is proportional to the Poisson ratio, and scales as $1/\\sqrt{L}$ .",
"Each section (-) starts with an overview that provides a detailed description of the main results in that section.",
"Given the considerable length of this manuscript, a first reading may be focused on these overview subsections only (REF ,REF ,REF ,REF ), followed by Sec.",
", where we describe experimental challenges and propose a list of theoretical questions inspired by our work."
],
[
"Overview",
"The helicoidal state has been studied by Green [2], who computed its stress field using the standard version of the FvK equations (REF ,).",
"This familiar form, to which we refer here as the ss-FvK equations (“ss\" stands for “small slope\") is valid for small deflections of elastic sheets from their planar state .",
"The Green's stress, Eqs.",
"(REF ,), has a longitudinal component that contains terms proportional to $T$ and to $\\eta ^2$ , and no transverse component.",
"However, the experiments of [3], as well as numerical simulations , , have exhibited a buckling instability of the helicoidal state in the transverse direction, indicating the presence of transverse compression.",
"One may suspect that the absence of transverse component in Green's stress indicates that the magnitude of this component is small, being proportional to a high power of the twist $\\eta $ , which cannot be captured by the ss-FvK equations.",
"Here we resort to a covariant form of the FvK equations, which we call “cFvK\" , , , that does not assume a planar reference state, and is thus capable of describing large deviations from a planar state.",
"Notably, the large deflection of the helicoidal state from planarity does not involve large strains.",
"Hence, as long as $T,\\eta \\ll 1$ , we consider a ribbon with Hookean response, namely – linear stress-strain relationship.",
"This approach is simpler than Mockensturm's [4] (which assumes a non-Hookean, material-dependent response), and enables the analytical progress in this section and the following ones.",
"Solving the cFvK equations for the helicoidal state, we get the following expressions for the stress field in the longitudinal ($\\hat{s}$ ) and transverse ($\\hat{r}$ ) directions: $\\sigma ^{ss}_\\mathrm {hel}(r) & = T + \\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{12} \\right),\\\\\\sigma ^{rr}_\\mathrm {hel}(r) & = \\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{4} \\right) \\left[T+\\frac{\\eta ^2}{4}\\left(r^2+\\frac{1}{12} \\right) \\right],$ where $r\\in [-1/2,1/2]$ is the dimensionless transverse coordinate.",
"The longitudinal component is exactly the one found by Green [2], whereas the transverse component is nonzero, albeit of small magnitude: $\\sigma ^{rr}_\\mathrm {hel}\\sim \\eta ^2 \\sigma ^{ss}_\\mathrm {hel}$ , which explains why it is missed by the ss-FvK equations.",
"The transverse stress arises from a subtle coupling between the longitudinal stress and the geometry of the ribbon.",
"As Eqs.",
"(REF ,) show, the longitudinal stress $\\sigma ^{ss}_\\mathrm {hel}(r)$ is compressive close to the centerline $r=0$ if $\\eta ^2>24 T$ , whereas the transverse stress $\\sigma ^{rr}_\\mathrm {hel}(r)$ is compressive everywhere in the ribbon for any $(T,\\eta )$ .",
"The compressive nature of $\\sigma ^{ss}_\\mathrm {hel}(r)$ and $\\sigma ^{rr}_\\mathrm {hel}(r)$ leads to buckling and wrinkling instabilities that we address in the next sections.",
"In Sec.",
"REF we review the (standard) ss-FvK equations, and their helicoidal solution found by Green [2].",
"In Sec.",
"REF we proceed to derive the cFvK equations, following , and use this covariant formalism to determine the stress in the helicoidal state.",
"We review briefly the standard ss-FvK equations, using some basic concepts of differential geometry that will allow us to introduce their covariant version in the next subsection.",
"Assuming a small deviation from a plane, a sheet is defined by its out-of-plane displacement $z(s,r)$ and its in-plane displacements $u_s(s,r)$ and $u_r(s,r)$ ; where $s$ and $r$ are the material coordinates.",
"In this configuration, the strain is given by $\\varepsilon _{\\alpha \\beta }=\\frac{1}{2}\\left[\\partial _\\alpha u_\\beta + \\partial _\\beta u_\\alpha + (\\partial _\\alpha z)(\\partial _\\beta z)\\right].$ The greek indices $\\alpha $ and $\\beta $ take the values $s$ or $r$ .",
"We define the curvature tensor and the mean curvature as: $c_{\\alpha \\beta } & = \\partial _\\alpha \\partial _\\beta z, \\\\H & = \\frac{1}{2}{c_\\alpha ^\\alpha } = \\frac{1}{2}{\\Delta z},$ where we use the Einstein summation convention, such that $c_\\alpha ^\\alpha $ is the trace of the curvature tensor.",
"The use of upper or lower indices corresponds to the nature of the tensor (contravariant or covariant, respectively), which will become relevant in the next subsection.",
"The ss-FvK equations express the force balance in the normal direction ($\\hat{z}$ ) and the in-plane directions ($\\hat{s},\\hat{r}$ ), and involve the curvature tensor and the stress tensor $\\sigma ^{\\alpha \\beta }(s,r)$ : $c_{\\alpha \\beta }\\sigma ^{\\alpha \\beta } & = 2B\\Delta H, \\\\\\partial _\\alpha \\sigma ^{\\alpha \\beta } & = 0, $ where $B=t^2/[12(1-\\nu ^2)]$ is the bending modulus of the sheet Recall that we normalize stresses by the stretching modulus Y and lengths by the ribbon width W. The dimensional bending modulus is thus: ${\\rm Y}({\\rm W}t)^2/(12(1-\\nu ^2))$ ..",
"The stress-strain relationship is given by Hooke's law (linear material response): $\\sigma ^{\\alpha \\beta } = \\frac{1}{1+\\nu }\\varepsilon ^{\\alpha \\beta } + \\frac{\\nu }{1-\\nu ^2}\\varepsilon ^\\gamma _\\gamma \\delta ^{\\alpha \\beta }.$ where we used the Kronecker symbol $\\delta ^{\\alpha \\beta }$ ."
],
[
"Green's solution for the helicoid",
"We now apply the ss-FvK equations (REF ,) to find the stress in the helicoidal state.",
"Since this formalism assumes a small deviation of the ribbon from the plane, we approximate the helicoidal shape through: $z(s,r)=\\eta sr \\ ,$ obtained by Taylor expansion of the $z$ -coordinate of the full helicoidal shape, given below in Eq.",
"(REF ), for $|s| \\ll \\eta ^{-1}$ .",
"Its corresponding curvature tensor is: $c_{\\alpha \\beta }=\\eta \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix},$ leading to the mean curvature $H=0$ .",
"The force balance equations (REF -) now read: $\\sigma ^{sr} & = 0,\\\\\\partial _s\\sigma ^{ss} & = 0,\\\\\\partial _r\\sigma ^{rr} & = 0.$ The ss-FvK equations are supplemented by two boundary conditions: The longitudinal stress must match the tensile load exerted on the short edges, whereas the long edges are free, namely: $\\int _{-1/2}^{1/2}\\sigma ^{ss}(r)dr & = T,\\\\\\sigma ^{rr}(r=\\pm 1/2) & = 0.$ Since Eq.",
"() implies that the transverse stress is uniform across the ribbon, the boundary condition () implies that it is identically zero: $\\sigma ^{rr}(r)=0.$ With Hooke's law (REF ), this shows that $\\sigma ^{ss}=\\varepsilon ^{ss}$ .",
"Using the small slope expressions for the strain-displacement relationship (Eq.",
"REF ) and for helicoidal shape (Eq.",
"REF ), we obtain the longitudinal stress $\\sigma ^{ss}(r) = \\frac{\\eta ^2 r^2}{2}-\\chi ,$ where $\\chi =-\\partial _s u_s$ is the longitudinal contraction of the ribbon.",
"Since $\\chi $ does not depend on $s$ (due to Eq.",
"REF ) or on $r$ (due to the translational symmetry of the helicoidal shape along $\\hat{s}$ ), its value is determined by the condition (REF ): $\\chi =\\frac{\\eta ^2}{24} -T \\ .$ We thus obtain the Green's stress [2]: $\\sigma ^{ss}(r) & = T + \\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{12} \\right),\\\\\\sigma ^{rr}(r) & = 0 \\ \\ ; \\ \\ \\sigma ^{rs} = 0 \\ .",
"$"
],
[
"Covariant FvK equations",
"In order to address sheet's configurations that are far from planarity, we must avoid any reference to a planar state.",
"The shape of the sheet is now described by a surface $X(s,r)$ , and the covariant form of the force balance equations, which we call here the cFvK equations, requires us to revisit the definitions of the quantities invoked in our description of the ss-FvK equations: the strain, the curvature, and the derivative.",
"We do this by following the general approach of .",
"First, we define the surface metric as a covariant tensor: $g_{\\alpha \\beta }=\\partial _\\alpha X\\cdot \\partial _\\beta X\\ ,$ where the inverse metric is a contravariant tensor, denoted with upper indices, that satisfies $g^{\\alpha \\beta } g_{\\beta \\gamma } = \\delta ^{\\alpha }_{\\gamma }$ ($\\delta ^{\\alpha }_{\\gamma }$ is the Kronecker symbol).",
"The strain is defined as the difference between the metric and the rest metric $\\bar{g}_{\\alpha \\beta }$ : $\\varepsilon _{\\alpha \\beta }=\\frac{1}{2}\\left(g_{\\alpha \\beta }- \\bar{g}_{\\alpha \\beta } \\right).$ The curvature tensor (REF ) is now defined by $c_{\\alpha \\beta }=\\hat{n}\\cdot \\partial _\\alpha \\partial _\\beta X,$ where $\\hat{n}$ is the unit normal vector to the surface (the ss-FvK equations are based on the approximation: $\\hat{n}\\approx \\hat{z}$ ).",
"In this formulation, the covariant/contravariant nature of tensors does matter, for instance: $c_{\\alpha \\beta }\\ne c^{\\alpha \\beta }$ .",
"To lower or raise the indices, one must use the metric or its inverse, respectively: $c^\\alpha _\\beta =g_{\\beta \\gamma }c^{\\alpha \\gamma }=g^{\\alpha \\gamma }c_{\\gamma \\beta }$ .",
"The mean curvature now invokes the inverse metric, $H=c^\\alpha _\\alpha /2=g^{\\alpha \\beta }c_{\\alpha \\beta }/2$ , and the Gaussian curvature of the surface is: $K=\\frac{1}{2}\\left(c^\\alpha _\\alpha c^\\beta _\\beta - c^\\alpha _\\beta c^\\beta _\\alpha \\right)$ .",
"Hooke's law (REF ) is only slightly changed: Other terms, proportional to $t^2$ , may appear on the right hand side of Eq.",
"(REF ) ; however, they are negligible here.",
"$\\sigma ^{\\alpha \\beta } = \\frac{1}{1+\\nu }\\varepsilon ^{\\alpha \\beta } + \\frac{\\nu }{1-\\nu ^2}\\varepsilon ^\\gamma _\\gamma g^{\\alpha \\beta } \\ ,$ and the force balance equations (REF -) now read $c_{\\alpha \\beta }\\sigma ^{\\alpha \\beta } & = 2B [D_{\\alpha }D^{\\alpha }H+2H(H^2-K)], \\\\D_{\\alpha } \\sigma ^{\\alpha \\beta } & = 0.$ There are two major differences between the ss-FvK equations (REF -) and the cFvK equations (REF -).",
"First, there is a new term in the normal force balance (REF ); this term may be relevant when the equilibrium shape is characterized by a uniform, nonvanishing mean curvature (such that $|H^3|$ or $|HK|$ are comparable to or larger than $|D_{\\alpha }D^{\\alpha }H|$ ), but is negligible for a surface that can be described by small deviations from a plane or a helicoid, for which $H \\approx 0$ .",
"Second – and central to our analysis – the usual derivative $\\partial _\\alpha $ is replaced by the covariant derivative $D_\\alpha $ that takes into account the variation of the metric along the surface.",
"The covariant derivative $D_\\alpha $ is defined through the Christoffel symbols of the surface, and is given in Appendix ."
],
[
"Application to the helicoid",
"Here, we show that the helicoid is a solution of the cFvK equations and determine its stress and strain.",
"The helicoidal shape is described by $X(s,r)& = (1-\\chi )s\\hat{x}+ [r+u_r(r)]\\cos (\\eta s)\\hat{y}+ [r+u_r(r)]\\sin (\\eta s)\\hat{z}\\nonumber \\\\& = \\begin{pmatrix} (1-\\chi )s \\\\ [r+u_r(r)]\\cos (\\eta s) \\\\ [r+u_r(r)]\\sin (\\eta s) \\end{pmatrix}.",
"$ where $(\\hat{x},\\hat{y},\\hat{z})$ is the standard basis of the three-dimensional space.",
"The longitudinal contraction $\\chi $ and transverse displacement $u_r(r)$ are small (i.e.",
"both vanish when $T=\\eta =0$ ), and must be determined by our solution.",
"Expanding Eq.",
"(REF ) to leading order in $\\chi $ and $u_r(r)$ we obtain the metric: $g_{\\alpha \\beta }=\\begin{pmatrix} 1+\\eta ^2 r^2 - 2\\chi + 2\\eta ^2 ru_r(r) & 0 \\\\ 0 & 1+2u_r^{\\prime }(r) \\end{pmatrix}.$ The curvature tensor is still given by (REF ), to leading order in $\\chi $ and $u_r(r)$ , and the mean curvature in this approximation is $H=0$ .",
"It must be understood that in deriving the metric tensor, Eq.",
"(REF ), we assumed that both the twist and the exerted tension are small ($\\eta \\ll 1,T \\ll 1$ ), such that $\\chi $ and $u_r(r)$ (which appear explicitly in $g_{\\alpha \\beta }$ ) can be expressed as expansions in $\\eta $ and $T$ that vanish for $\\eta ,T \\rightarrow 0$ .",
"This natural assumption, which simplifies considerably the forthcoming analysis, implies that a consistent calculation of the stress components $\\sigma ^{ss},\\sigma ^{sr}$ , and $\\sigma ^{rr}$ , must treat them as expansions in $\\eta $ and $T$ (in Appendix we discuss this issue further).",
"With this in mind, we note that the force balance equations (REF -) become, to leading order in $\\eta $ : $\\sigma ^{sr} & = 0,\\\\\\partial _s\\sigma ^{ss} & = 0,\\\\\\partial _r\\sigma ^{rr}-\\eta ^2r\\sigma ^{ss} & = 0.",
"$ The second term in the left hand side of Eq.",
"(), which has no analog in the ss-FvK equations (REF -), encapsulates the coupling of the transverse and longitudinal stress components imposed by the non-planar helicoidal structure.",
"Its derivation, which reflects the profound role of the covariant derivative in our study, is detailed in Appendix .",
"Now, comparing the two terms in Eq.",
"() shows that for $\\eta \\ll 1$ : $\\sigma ^{rr}\\sim \\eta ^2\\sigma ^{ss}\\ll \\sigma ^{ss} \\ .$ Recalling that our computation of the stress components assumes an expansion in $\\eta $ and $T$ , the inequality (REF ) implies that the expansion of $\\sigma ^{rr}$ starts with a higher order term than the expansion of $\\sigma ^{ss}$ .",
"An immediate consequence of this observation is obtained by expressing $\\sigma ^{ss}$ and $\\sigma ^{rr}$ through Hooke's law.",
"From the metric (REF ), we deduce the strain (REF ): $\\varepsilon _{\\alpha \\beta }=\\begin{pmatrix} \\frac{\\eta ^2r^2}{2} - \\chi + \\eta ^2 ru_r(r) & 0 \\\\ 0 & u_r^{\\prime }(r) \\end{pmatrix} \\ ,$ where we substituted $\\bar{g}_{\\alpha \\beta } = \\delta _{\\alpha \\beta }$ .",
"Using Hooke's law to compute the stress components to leading order in $\\eta $ (where we anticipate that both $\\chi $ and $u_r(r)$ vanish as $\\eta \\rightarrow 0$ ), we obtain: $\\sigma ^{ss} & = \\frac{1}{1-\\nu ^2} \\left(\\frac{\\eta ^2r^2}{2}-\\chi \\right)+\\frac{\\nu }{1-\\nu ^2}u_r^{\\prime }(r),\\\\\\sigma ^{rr} & =\\frac{1}{1-\\nu ^2}u_r^{\\prime }(r)+\\frac{\\nu }{1-\\nu ^2}\\left(\\frac{\\eta ^2r^2}{2} - \\chi \\right).$ Since the force balance Eq.",
"() implies that an expansion in $\\eta $ and $T$ is valid only if $\\sigma ^{rr}$ starts with a higher order than $\\sigma ^{ss}$ , Eqs.",
"(REF ,) yield the solvability condition: $u_r^{\\prime }(r)=-\\nu \\left(\\frac{\\eta ^2r^2}{2} - \\chi \\right) \\ ,$ which guarantees that $\\sigma ^{ss} \\sim O(T,\\eta ^2)$ , whereas $\\sigma ^{rr}$ has no terms of that order (such that $\\sigma ^{rr} \\sim O(T \\eta ^2, \\eta ^4)$ ), consistently with Eq. ().",
"Inserting this result into Eq.",
"(REF ) gives the same longitudinal stress (REF ) as the small-slope approximation; the longitudinal contraction (REF ) does not change either.",
"Now that the longitudinal stress is known, the transverse component is obtained by integrating Eq.",
"() with the boundary condition (), so that finally: $\\sigma ^{ss}(r) & = T + \\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{12} \\right),\\\\\\sigma ^{rr}(r) & = \\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{4} \\right) \\left[T+\\frac{\\eta ^2}{4}\\left(r^2+\\frac{1}{12} \\right) \\right].",
"$ Comparing these equations to the Green's stress (REF -), which was obtained through the ss-FvK equations, we note two facts: First, the longitudinal component is unchanged.",
"Second, we find a compressive transverse component that originates from the coupling of the transverse and longitudinal stress components by the helicoidal geometry of the ribbon.",
"Since the transverse component is much smaller than the longitudinal one, the Green's stress is useful for studying certain phenomena, most importantly – the longitudinal instability of the helicoidal state [5].",
"However, the instability of the ribbon that stems from the compressive transverse stress is totally overlooked in Green's approach.",
"Furthermore, the covariant formalism provides a considerable conceptual improvement to our understanding since it allows to think of the helicoid (or any other shape) without assuming a planar reference shape.",
"Finally, let us re-emphasize that, although the transverse stress $\\sigma ^{rr}(r)$ is proportional to products of the small exerted strains ($T \\eta ^2$ , $\\eta ^4$ ), it originates from Hookean response of the material; its small magnitude simply reflects the small transverse strain in the helicoidal shape."
],
[
"Overview",
"If the twist is sufficiently large with respect to the exerted tension, the stress in the helicoidal state becomes compressive in the longitudinal direction in a zone around the ribbon centerline.",
"This can be easily seen from the expression (REF ): if $\\eta > \\eta _\\mathrm {lon}(T) = \\sqrt{24 T}$ , then $\\sigma _\\mathrm {hel}^{ss}(r)<0$ for $|r| < r_\\mathrm {wr}$ , where the width $r_\\mathrm {wr}$ increases with the ratio $\\eta ^2/T $ (see Fig.",
"REF ).",
"This effect reflects the helicoidal geometry, where the long edges are extended with respect to the centerline, such that the longitudinally compressive zone expands outward upon reducing the exerted tension.",
"The ratio $\\alpha = \\eta ^2/T $ , whose critical value $\\alpha =24$ signifies the emergence of longitudinal compression, plays a central role in this section and we call it the confinement parameter: ${\\rm Confinement:} \\ \\alpha \\equiv \\frac{\\eta ^2}{T}.$ Figure: Left: longitudinal stress of the helicoidal state (that approximates the stress in the NT regime), and of the far from threshold (FT) longitudinally wrinkled state, where r wr r_\\mathrm {wr} is the extent of the respective wrinkled zone.",
"The confinement is α=125\\alpha =125.",
"Right: extent of the wrinkled zone r wr r_\\mathrm {wr} in the NT regime, where it is approximated through the helicoidal state (where r wr r_\\mathrm {wr} is defined as the width of the zone under longitudinal compression), and in the FT regime.",
"Inset: the ribbon supports compression without wrinkling for 24<α<α lon 24<\\alpha <\\alpha _\\mathrm {lon}, and then the extent of the wrinkled zone interpolates between the NT and FT predictions for α lon <α<α NT - FT \\alpha _\\mathrm {lon}<\\alpha <\\alpha _\\mathrm {NT-FT}.Above α NT - FT \\alpha _\\mathrm {NT-FT}, the state is described by the FT approach.The longitudinal compression may induce a wrinkling instability, where periodic undulations of the helicoidal shape relax the compression in the zone $|r| <r_\\mathrm {wr}$ .",
"A natural way to study this instability is through linear stability analysis, which assumes that the longitudinally-wrinkled state of the ribbon can be described as a small perturbation to the compressed helicoidal state [5].",
"While this perturbative approach is useful to address the wavelength $\\lambda _\\mathrm {lon}$ of the wrinkle pattern at threshold [3], we argue that it describes the ribbon state only at a narrow, near threshold (NT) regime in the $(\\eta ,T )$ plane, above which we must invoke a qualitatively different, far from threshold (FT) approach (see Fig.",
"REF d).",
"The fundamental difference between the NT and FT theories is elucidated in Fig.",
"REF , which plots the approximated profiles of the longitudinal stress, $\\sigma _\\mathrm {hel}^{ss}(r)$ and $\\sigma _\\mathrm {FT}^{ss}(r)$ , respectively, for a given confinement $\\alpha > 24$ .",
"The NT theory assumes that the wrinkles relax slightly the compression in $\\sigma _\\mathrm {hel}^{ss}(r)$ , whereas the FT theory assumes that at a given $\\alpha >24$ the stress in the longitudinally-wrinkled ribbon approaches a compression-free profile as $t \\rightarrow 0$More precisely, the NT method is an amplitude expansion of FvK equations around the compressed helicoidal state, whereas the FT theory is an asymptotic expansion of the FvK equations around the singular limit $t \\rightarrow 0$ , carried at a fixed confinement $\\alpha $ .",
"In this limit, the longitudinally wrinkled state of the ribbon approaches the compression-free stress $\\sigma _\\mathrm {FT}^{ss}(r)$ .. For a very thin ribbon, which can support only negligible level of compression, the transition between the NT and FT regimes converges to the threshold curve $\\eta _\\mathrm {lon}(T )$ (see Fig.",
"REF d).",
"Figure: Left: Longitudinal contraction (defined with respect to the untwisted ribbon without any tension) of the helicoidal (unwrinkled) state, the FT-longitudinally-wrinkled state, and the cylindrical wrapping state as T→0T\\rightarrow 0.",
"Right: Longitudinal contractions of the helicoidal state and the FT-longitudinally-wrinkled state as a function of 1/α1/\\alpha .The sharp contrast between the NT and FT theories is further elucidated in Figs.",
"REF , REF , and REF , where the respective predictions for the spatial width of the longitudinally-wrinkled zone, the longitudinal contraction, and the energy stored in the ribbon are compared.",
"Fig.",
"REF shows that the wrinkled zone predicted by the FT theory expands beyond the compressed zone of the helicoidal state.",
"Furthermore, as the confinement $\\alpha $ increases, the FT theory predicts that the wrinkled zone invades the whole ribbon (except narrow strips that accommodate the exerted tension), whereas the compressed zone of the helicoidal state covers only a finite fraction ($1/\\sqrt{3}$ ) of the ribbon width.",
"Fig.",
"REF shows that the longidudinal contraction predicted by the FT theory is larger than the contraction of the unwrinkled helicoidal state, and the ratio between the respective contractions $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}/\\chi _\\mathrm {hel} \\rightarrow 3$ as $\\alpha \\rightarrow \\infty $ .",
"Fig.",
"REF plots the energies stored in the compressive helicoidal state ($U_\\mathrm {hel}$ ) and in the compression-free state ($U_\\mathrm {dom}$ ), demonstrating the significant gain of elastic energy enabled by the collapse of compression.",
"Focusing on the vicinity of $\\alpha =24$ , we illustrate in Fig.",
"REF a how the vanishing size of the NT parameter regime for $t \\ll 1$ results from the small (amplitude-dependent) reduction of the energy $U_\\mathrm {hel}$ versus the sub-dominant ($t$ -dependent) addition to the energy $U_\\mathrm {dom}$ .",
"The subdominant energy stems from the small bending resistance of the ribbon in the limit $t\\rightarrow 0$ .",
"Figure: Left: Dominant energy stored in the stress field of the ribbon as a function of the inverse confinement 1/α=T/η 2 1/\\alpha =T/\\eta ^2 at the helicoidal state and at the far from threshold (FT) longitudinally wrinkled state.Right: (a) Energy difference U hel -U dom U_\\mathrm {hel}-U_\\mathrm {dom} and the subdominant energy U sub U_\\mathrm {sub} due to the wrinkles close to the threshold.",
"(b) Energies of the FT longitudinally wrinkled helicoid and the cylinder wrapping at vanishing tension (1/α→01/\\alpha \\rightarrow 0); Inset: energy of the creased helicoid (CH) is added (see Subsec.",
")."
],
[
"Asymptotically isometric states:",
"Focusing on the limit $\\alpha ^{-1} \\rightarrow 0$ in Fig.",
"REF , which describes the ribbon under twist $\\eta $ and infinitely small $T $ , one observes that the dominant energy $U_\\mathrm {dom}$ becomes proportional to $T $ and vanishes as $T \\rightarrow 0$ .",
"This result reflects the remarkable geometrical nature of the FT-longitudinally-wrinkled state, which becomes infinitely close to an isometric (i.e.",
"strainless) map of a ribbon under finite twist $\\eta $ , in the singular limit $t ,T \\rightarrow 0$ .",
"At the singular hyper-plane $(t=0, T=0)$ , which corresponds to an ideal ribbon with no bending resistance and no exerted tension, the FT-longitudinally-wrinkled state is energetically equivalent to simpler, twist-accommodating isometries of the ribbon: the cylindrical shape (Fig.",
"REF ) and the creased helicoid shape (Fig.",
"REF d, [7]).",
"We argue that this degeneracy is removed in an infinitesimal neighborhood of the singular hyper-plane (i.e.",
"$t >0,T >0$ ), where the energy of each asymptotically isometric state is described by a linear function of $T $ with a $t $ -independent slope and a $t $ -dependent intercept.",
"Specifically: $U_j (t ,T ) = A_j T + B_j t ^{2\\beta _j} \\ ,$ where $j$ labels the asymptotic isometry type (cylindrical, creased helicoid, longitudinal wrinkles), and $0<\\beta _j<1$ .",
"For a fixed twist $\\eta \\ll 1$ , we argue that the intercept ($B t ^{2\\beta }$ ) is smallest for the cylindrical state, whereas the slope ($A$ ) is smallest for the FT-longitudinally-wrinkled state.",
"This scenario, which is depicted in Fig.",
"REF a, underlies the instability of the longitudinally wrinkled state in the vicinity of the axis $T = 0$ in the $(T,\\eta )$ plane.",
"The concept of asymptotic isometries has been inspired by a recent study of an elastic sheet attached to a curved substrate .",
"We conjecture that the form of Eq.",
"(REF ) is rather generic, and underlies morphological transitions also in other problems, where thin elastic sheets under geometric confinement (e.g.",
"twist or imposed curvature) are subjected to small tensile loads.",
"We start in Subsec.",
"REF with a brief review of the linear stability analysis.",
"In Subsec.",
"REF we introduce the FT theory, and discuss in detail the compression-free stress $\\sigma _\\mathrm {FT}$ and its energy $U_\\mathrm {dom}$ .",
"In Subsec.",
"REF we address the transition from the NT to the FT regime.",
"In Subsec.",
"REF we introduce the asymptotic isometries, where we explain the origin of Eq.",
"(REF ) and compare the energetic costs of the cylindrical and the longitudinally-wrinkled states.",
"In this subsection we develop a linear stability analysis of the longitudinal wrinkling, following [2], [5], [3] and focusing on scaling-type arguments rather than on exact solutions.",
"We use the small slope approximation of the FvK equations (see Subsec.",
"REF ) and its Green's solution (REF , REF -).",
"This approximation is justified here since the transverse stress $\\sigma _\\mathrm {hel}^{rr}$ is smaller by a factor $\\eta ^2$ than the longitudinal stress $\\sigma _\\mathrm {hel}^{ss}$ which is responsible for the instability.",
"Dividing $\\sigma _\\mathrm {hel}^{ss}$ by the tension $T$ we obtain a function of the transverse coordinate $r$ that depends only on the confinement parameter $\\alpha $ (REF ) and is plotted in Fig.",
"REF for three representative values of $\\alpha $ .",
"For $\\alpha > 24$ a zone $|r| < r_\\mathrm {wr}(\\alpha )$ around the ribbon centerline is under compression, and we thus expect that for a thin ribbon the threshold value for the longitudinal instability follows $\\alpha _\\mathrm {lon}(t) \\rightarrow 24$ when $t\\rightarrow 0$ .",
"A simple analysis of the function $\\sigma _\\mathrm {hel}^{ss}(r)$ leads to the following scalings for the magnitude of the compression $\\sigma _\\mathrm {hel}^{ss}(r=0)$ and the width $r_\\mathrm {wr}$ of the compressed zone near the threshold: $\\begin{aligned}\\sigma ^{ss}_\\mathrm {hel}(r=0) &\\sim T \\left( \\alpha - 24 \\right),\\\\r_\\mathrm {wr} & \\sim \\sqrt{\\alpha -24}.\\end{aligned}$ Figure: Longitudinal stress along the width of the ribbon in the helicoidal state (left) and in the far from threshold longitudinally wrinkled state (right) for different values of the confinement parameter α\\alpha .Consider now a small perturbation of the planar approximation (REF ) of the helicoidal state such that $z(s,r) \\simeq \\eta sr + \\zeta z_1(s,r)$ , where $\\zeta $ is a small parameter.",
"Substituting this expression in the normal force balance (REF ), we obtain a linear equation for $z_1(s,r)$ $\\sigma ^{ss}_\\mathrm {hel} \\partial _s^2 z_1 = B\\Delta ^2 z_1 \\ .$ Eq.",
"(REF ) should be understood as the leading order equation in an amplitude expansion of the ss-FvK equations (REF -) around the helicoidal state, where the small parameter is the amplitude $\\zeta $ of the wrinkle pattern.The absence of $s$ -dependent terms in Eq.",
"(REF ) stems from the translational symmetry in the longitudinal direction of the helicoidal state that is broken by the wrinkling instability.",
"The natural modes are thus: $z_1(s,r) = \\cos (2\\pi s/\\lambda _\\mathrm {lon}) f(r)$ , where $\\lambda _\\mathrm {lon}$ is the wrinkles wavelength and $f(r)$ is a function that vanishes outside the compressive zone of $\\sigma _{ss}^\\mathrm {hel}$ .",
"An exact calculation of $\\lambda _\\mathrm {lon}$ , $f(r)$ and the threshold $\\alpha _\\mathrm {lon}(t )$ can be found in [5], but the scaling behavior with $t $ can be obtained (as was done in [3]) by noticing that the most unstable mode is characterized by a “dominant balance\" of all forces in Eq.",
"(REF ): The restoring forces, which are associated here with the bending resistance to deflection in the two directions, $B\\partial _s^4 z_1$ and $B\\partial _r^4z_1$ , as well as the destabilizing force $\\sigma ^{ss}_\\mathrm {hel} \\partial _s^2 z_1$ .",
"Equating these forces yields the two scaling relations: $\\lambda _\\mathrm {lon} \\sim r_\\mathrm {wr}$ , and $B/\\lambda _\\mathrm {lon}^2 \\sim \\sigma ^{ss}_\\mathrm {hel}(r=0)$ .",
"With the aid of Eq.",
"(REF ) we obtain the NT scaling laws: $\\begin{aligned}\\Delta \\alpha _\\mathrm {lon} & = \\alpha _\\mathrm {lon} - 24 \\sim \\frac{t}{\\sqrt{T}},\\\\\\lambda _\\mathrm {lon} & \\sim r_\\mathrm {wr} \\sim \\frac{\\sqrt{t}}{T^{1/4}}\\end{aligned}$ These scaling laws which are based upon Eq.",
"(REF ) are only valid for $\\Delta \\alpha _\\mathrm {lon} \\ll 1$ or, equivalently for $t^2 \\ll T$ .",
"In this regime, the ribbon is so thin that the thresholds for developing a compressive zone and for wrinkling become infinitely close to each other as $t\\rightarrow 0$ .",
"In contrast, in the regime where $t^2 \\gg T$ , the ribbon is too thick compared to the exerted tension and the threshold for wrinkling is much larger (in terms of $\\alpha $ ) than the threshold for developing compression.",
"In this regime of very small tension, the linear stability analysis of the helicoid is different from the one presented above and has been performed by Green [2].",
"It resulted in a plateau in the threshold $\\eta _\\mathrm {lon}(T)$ , that we refer to as the “Green's plateau\": $\\eta _\\mathrm {lon}(T) \\longrightarrow 0.2\\, t\\quad \\textrm {for}\\quad T\\ll T_\\mathrm {sm},$ where $T_\\mathrm {sm}\\sim t^2.$ This plateau is pictured in Fig.",
"REF c. It can be obtained by a simple scaling argument, which balances, as before, the longitudinal compression and bending in the longitudinal and transverse directions: $\\eta ^2/\\lambda ^2\\sim t^2\\sim t^2/\\lambda _\\mathrm {lon}^4$ , giving $\\lambda _\\mathrm {lon}\\sim 1$ and $\\eta _\\mathrm {lon}\\sim t$ ."
],
[
"Far-from-threshold analysis",
"As the confinement gets farther from its threshold value, the wrinkle pattern starts to affect considerably the longitudinal stress and can eventually relax completely the compression.",
"The emergence of a compression-free stress field underlying wrinkle patterns has been recognized long ago in the solid mechanics and applied mathematics literature , , .",
"More recently, it has been shown that such a compression-free stress field reflects the leading order of an expansion of the FvK equations under given tensile load conditions , , .",
"In contrast to the NT analysis, which is based on amplitude expansion of FvK equations around a compressed (helicoidal) state, and whose validity is therefore limited to values of $(T,\\eta )$ at the vicinity of the threshold curve, the FT analysis is an expansion of the FvK equations around the compression-free stress, which is approached in the singular limit $t = 0$ .",
"For a sufficiently small thickness $t$ , the FT expansion is thus valid for any point $(T,\\eta )$ with confinement $\\alpha = \\eta ^2/T >24$ (see footnote REF ).",
"The leading order in the FT expansion captures the compression-free stress field $\\sigma _\\mathrm {FT}^{\\alpha \\beta }$ , which is independent on $t$ , in the asymptotic limit $t \\rightarrow 0$ .",
"The wrinkled part of the sheet (here $r <|r_\\mathrm {wr}|$ ) is identified as the zone where a principal component of the stress (here $\\sigma _\\mathrm {FT}^{ss}(r)$ ) vanishes.",
"Underlying the FT expansion there is a hierarchical energetic structure: $U_\\mathrm {FT} (\\alpha , t) = U_\\mathrm {dom}(\\alpha ) + t^{2\\beta } F(\\alpha ) \\ ,$ where $0<\\beta <1$ .",
"The dominant term $U_\\mathrm {dom}(\\alpha )$ is the elastic energy stored in the compression-free stress field, which depends on the loading conditions (through $\\alpha $ ) but not on $t$ , and on the sub-dominant term $t^{2\\beta } F(\\alpha )$ , which stems from the small bending resistance of the sheet, vanishes as $t \\rightarrow 0$ .",
"A nontrivial feature of the FT expansion, which is implicit in Eq.",
"(REF ), is the singular, degenerate nature of the limit $t \\rightarrow 0$ .",
"There may be multiple wrinkled states, all of which give rise to the same $U_\\mathrm {dom}(\\alpha )$ and $\\sigma _\\mathrm {FT}^{\\alpha \\beta }(\\alpha )$ and therefore share the same width $r_\\mathrm {wr}(\\alpha )$ of the wrinkled zone.",
"The sub-dominant term $t^{2\\beta } F(\\alpha )$ lifts this degeneracy by selecting the energetically-favorable state, and therefore determines the fine-scale features of the wrinkle pattern, namely: the wavelength $\\lambda _\\mathrm {lon}$ , the possible emergence of wrinkle cascades , , , , and so on.",
"In this paper, we focus on the dominant energy $U_\\mathrm {dom}$ , and will make only a brief, heuristic comment on the sub-dominant energy and the fine-scale features of the wrinkle pattern.",
"In the first part of this subsection we find the compression-free stress, and in the second part we study the energy $U_\\mathrm {dom}$ associated with it."
],
[
"The compression-free stress field",
"One may think of the compression-free stress field by imagining a hypothetic ribbon with finite stretching modulus but zero bending resistance.",
"When such a ribbon is twisted (with $\\alpha >24$ ), the helicoidal shape can be retained up to wrinkly undulations of infinitesimal amplitude and wavelength, that fully relax any compression.",
"This hypothetic ribbon is exactly the singular point, $t=0$ , around which we carry out the FT expansion.",
"Considering the FvK equations (REF ,), this means that the compression-free stress could be found by assuming the helicoidal shape (REF ) and searching for a stress whose longitudinal component is non-negative.",
"(Since the magnitude of the compressive transverse component $\\sigma _\\mathrm {FT}^{rr}$ is smaller by a factor of $\\eta ^2$ than the longitudinal stress, it has a negligible effect on the longitudinal instability; its effect on the transverse instability will be the subject of the next section.)",
"It must be understood though, that the longitudinal wrinkles, no matter how small their amplitude is, contain a finite fraction of the ribbon's length, which is required to eliminate compression.",
"This effect must be taken into consideration when analyzing the stress-strain relations, Eq.",
"(REF ), and leads to a “slaving\" condition on the amplitude and wavelength of the wrinkles .",
"The above paragraph translates into a straightforward computation of the compression-free stress.",
"We assume a continuous $\\sigma _\\mathrm {FT}^{ss}(r)$ , which is zero for $|r| < r_\\mathrm {wr}$ and positive for $|r| > r_\\mathrm {wr}$ (see , , , for analogous derivations of FT wrinkle patterns in radial stretching set-ups).",
"In the tensile zone there are no wrinkles that modify the helicoidal shape, and inspection of the strain (REF ) shows that the longitudinal stress must be of the form $\\eta ^2 r^2/2 + \\mathrm {cst}$ .",
"This leads to: $\\sigma _\\mathrm {FT}^{ss}(r)=\\left\\lbrace \\begin{array}{ll}0 & \\quad \\textrm {for}\\quad |r|<r_\\mathrm {wr},\\\\\\displaystyle {\\frac{\\eta ^2}{2}\\left(r^2-r_\\mathrm {wr}^2 \\right)} & \\quad \\textrm {for}\\quad |r|>r_\\mathrm {wr}.\\end{array} \\right.$ Recalling that the integral of $\\sigma _\\mathrm {FT}^{ss}(r)$ over $r$ must equal the exerted force, we obtain an implicit equation for the width $r_\\mathrm {wr}(\\alpha )$ : $(1-2r_\\mathrm {wr})^2(1+4r_\\mathrm {wr})=\\frac{24}{\\alpha }.$ Fig.",
"REF shows the longitudinal stress profile (REF ) for different values of the confinement $\\alpha $ .",
"The wrinkle's width $r_\\mathrm {wr}(\\alpha )$ , derived from Eq.",
"(REF ), is shown in Fig.",
"REF and compared to the width of the compressive zone in the helicoidal state for the corresponding values of $\\alpha $ .",
"We obtained the stress field (REF ,REF ) by requiring that, in the tensile zone $|r| >r_\\mathrm {wr}$ , the helicoidal shape with the stress $\\sigma _\\mathrm {FT}^{ss}(r)$ form a solution of the cFvK equations (REF ,REF ,), subjected to the constraint that $\\sigma _\\mathrm {FT}^{ss}(r)=0$ at $|r|<r_\\mathrm {wr}$ .",
"In order to understand how the FvK equations are satisfied also in the wrinkled zone $|r|<r_\\mathrm {wr}$ it is useful to assume the simplest type of wrinkles where the helicoidal shape is decorated with periodic undulations of wavelength $2\\pi /k$ and amplitude $f(r)$ : $X^\\mathrm {(wr)}(s,r)=\\begin{pmatrix}\\left(1-\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}\\right)s \\\\r\\cos (\\eta s) - f(r)\\cos (ks)\\sin (\\eta s)\\\\r\\sin (\\eta s) + f(r)\\cos (ks)\\cos (\\eta s)\\end{pmatrix},$ where the longitudinal contraction is given by $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}= \\frac{1}{2} \\eta ^2 r_\\mathrm {wr}^2 \\ ,$ which follows from Eq.",
"(REF ) and the continuity of $\\sigma _\\mathrm {FT}^{rr}(r)$ at $r=r_\\mathrm {wr}$ .",
"In the limit of small wrinkles amplitude and wavelength, the translationally invariant (i.e.",
"$s$ -independent) longitudinal strain is $\\varepsilon _{ss}(r)=\\frac{\\eta ^2}{2}\\left(r^2-r_\\mathrm {wr}^2 \\right)+\\frac{1}{4}k^2f(r)^2.$ Using Hookean stress-strain relation (REF ) together with the requirement $\\sigma _\\mathrm {FT}^{ss}(r) = 0$ for $|r|<r_\\mathrm {wr}$ yields $k^2 f(r)^2=2\\eta ^2 \\left(r_\\mathrm {wr}^2-r^2 \\right).$ Equation (REF ) is a “slaving\" condition (in the terminology of ) imposed on the wrinkle pattern by the necessity to collapse compression, which reflects the singular nature of the FT expansion.",
"Although $k \\rightarrow \\infty $ and $f(r) \\rightarrow 0$ as $t \\rightarrow 0$ , and $k$ and $f$ cannot be extracted from our leading order analysis, their product remains constant and is determined solely by the confinement $\\alpha $ .",
"In Appendix we show that the oscillatory ($s$ -dependent) part of the strain $\\varepsilon _{ss}(r)$ , as well as other components of the strain tensor, can also be eliminated in the limit $t \\rightarrow 0$ by modifying the deformed shape, Eq.",
"(REF ), with a wrinkle-induced longitudinal displacement $u_s(s,r)$ .",
"Finally, we use the in-plane force balance () to deduce the transverse component of the stress: $\\sigma _\\mathrm {FT}^{rr}(r) = \\left\\lbrace \\begin{array}{ll}\\displaystyle {-\\frac{\\eta ^4}{8}\\left(\\frac{1}{4}-r_\\mathrm {wr}^2 \\right)^2} & \\quad \\textrm {for}\\quad |r|<r_\\mathrm {wr},\\\\\\displaystyle {-\\frac{\\eta ^4}{8}\\left(\\frac{1}{4}-r^2\\right)\\left(\\frac{1}{4}+r^2-2r_\\mathrm {wr}^2 \\right)} & \\quad \\textrm {for}\\quad |r|>r_\\mathrm {wr}.\\end{array} \\right.$ In Sec.",
"we will employ both longitudinal and transverse components of the stress to study the transverse instability of the longitudinally wrinkled helicoid.",
"The dominant energy $U_\\mathrm {dom}$ of the FT longitudinally wrinkled state is simply the energy associated with the compression-free stress and is given by $U_\\mathrm {dom} = \\frac{1}{2}\\int _{-1/2}^{1/2} \\sigma _\\mathrm {FT}^{ss}(r)^2 dr +T\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}$ where the first term results from the strain in the ribbon and the second one is the work done by the exerted tension upon pulling apart the short edgesFor simplicity, we assume that the Poisson ratio $\\nu =0$ .",
"This does not affect any of the basic results.",
"Also, note that we neglected the contribution of the transverse stress ($\\sim \\sigma _\\mathrm {FT}^{rr}(r)^2$ ) since it comes with a factor $O(\\eta ^4)$ with respect to the terms in Eq.",
"(REF )..",
"The right hand side of Eq.",
"(REF ) is easily evaluated using Eqs.",
"(REF ,REF ,REF ), yielding $\\frac{U_\\mathrm {dom}}{T^2} =\\frac{\\alpha ^2}{1920}(1-2r_\\mathrm {wr})^3\\left(3+18r_\\mathrm {wr}+32r_\\mathrm {wr}^2\\right)+\\frac{\\alpha r_\\mathrm {wr}^2 }{2} \\ ,$ where the extent of the wrinkled zone is given by Eq.",
"(REF ).",
"The energy $U_\\mathrm {hel}$ of the compressed helicoidal state is evaluated by an equation analogous to (REF ), where $\\sigma _\\mathrm {FT}^{ss}$ and $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}$ are replaced, respectively, by Eqs.",
"(REF ,REF ), yielding: $\\frac{U_\\mathrm {hel}}{T^2}=\\frac{\\alpha ^2}{1440}+\\frac{\\alpha }{24}-\\frac{1}{2}.$ The two energies $U_\\mathrm {dom}$ and $U_\\mathrm {hel}$ are plotted in Fig.",
"REF , demonstrating the dramatic effect associated with the formation of wrinkles and the consequent collapse of compression on the elastic energy of a stretched-twisted ribbon.",
"A notable feature, clearly visible in Fig.",
"REF , is the vanishing of $U_\\mathrm {dom}$ as $T\\rightarrow 0$ for a fixed twist $\\eta $ .",
"This is elucidated by an inspection of the terms in Eq.",
"(REF ): assuming a fixed twist $\\eta $ (such that $T \\sim \\alpha ^{-1}$ ), the stress integral vanishes as $\\sim T^2$ , whereas the longitudinal compression $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}\\sim \\eta ^2$ is independent on $T$ and hence the work term scales as $\\sim T$ .",
"This low-$T$ scaling of $U_\\mathrm {dom}$ , together with the behavior of the sub-dominant energy that we describe below, underlies the asymptotic isometry equation (REF ).",
"In Subsec.",
"REF , we will argue that the linear dependence of the energy on the tension $T$ is a general feature, shared also by other types of asymptotic isometries."
],
[
"The sub-dominant energy:",
"As we noted already, computation of the sub-dominant energy requires one to consider all the wrinkled states whose energy approaches the dominant energy $U_\\mathrm {dom}(\\alpha )$ (REF ) in the limit $t \\rightarrow 0$ .",
"A complete analysis of the sub-dominant energy is beyond the scope of this paper.",
"However, we can obtain a good idea on the scaling behavior by considering a fixed confinement $\\alpha >24$ and assuming that the energetically favorable pattern consists of simply-periodic wrinkles (Eqs.",
"REF ,REF ) with $1 \\ll k \\ll t^{-1}$ .",
"We will use the bending energy of such a pattern to estimate the subdominant energy at the two limits of the confinement parameter: (a) $\\alpha $ is slightly larger than 24, which we denote as $\\Delta \\alpha = \\alpha - 24 \\ll 1$ , (b) large confinement, $\\alpha \\gg 1$ .",
"(a) Here the wrinkles are confined to a narrow zone of width $r_\\mathrm {wr} \\sim \\sqrt{ \\Delta \\alpha }$ (which follows from the Taylor expansion of Eq.",
"(REF ) around $\\alpha =24$ ).",
"Hence, the curvature of the wrinkles in both transverse and longitudinal directions is significant, and a similar argument to Subsec.",
"REF , which relies on balancing the normal forces proportional to the wrinkle amplitude $f(r)$ , implies: $k \\sim 1/r_\\mathrm {wr}$ .",
"The excess bending energy (per unit of length in the longitudinal direction) is: $U_\\mathrm {B} \\sim (B/2) \\int _{-r_\\mathrm {wr}}^{r_\\mathrm {wr}} [k^2 f(r)]^2dr$ .",
"Using the slaving condition (REF ) we obtain: $U_\\mathrm {B} \\sim \\eta ^2 t^2 (\\Delta \\alpha )^{1/2}$ .",
"(b) As $\\alpha \\gg 1$ (corresponding to the limit $T \\rightarrow 0$ for fixed twist $\\eta $ ), the exerted tension is felt only at infinitesimal strips near the long edges, and we may therefore assume that $k \\sim t^{\\beta -1}$ , where $0<\\beta <1$ is independent on $T$ .",
"A similar calculation to the above paragraph, where now $r_\\mathrm {wr} \\approx 1/2$ , yields: $U_\\mathrm {B} \\sim t^{2\\beta } \\alpha ^2 $ .",
"We thus obtain the scaling estimates for the sub-dominant energy: $U_\\mathrm {sub} \\sim \\left\\lbrace \\begin{array}{ll}\\displaystyle {\\eta ^2 t^2 \\Delta \\alpha ^{1/2}} & \\quad \\textrm {for} \\quad \\Delta \\alpha \\ll 1,\\\\\\displaystyle {t^{2\\beta } \\alpha ^2} & \\quad \\textrm {for}\\quad \\alpha \\gg 1.\\end{array} \\right.$"
],
[
"Transition from the near-threshold to the far-from-threshold regime",
"As the confinement $\\alpha $ is increased above the threshold value $\\alpha _\\mathrm {lon}$ given in Eq.",
"(REF ), we expect a transition of the width $r_\\mathrm {wr}(\\alpha )$ of the wrinkled zone from the extent of the compressive zone of the helicoidal state (REF ) to the FT result (REF ).",
"This transition is depicted in the inset to Fig.",
"REF (right).",
"The energetic mechanism underlying the NT-FT transition is described schematically in Fig.",
"REF a: In the NT regime, the energy of the wrinkled state is reduced from $U_\\mathrm {hel}(\\alpha )$ (Eq.",
"REF ) by a small amount, proportional to the wrinkle's amplitude.",
"In the FT regime, the energy $U_\\mathrm {FT}$ is expressed by Eq.",
"(REF ) where the $t$ -independent part $U_\\mathrm {dom}$ is given by Eq.",
"(REF ) and the $t$ -dependent part $U_\\mathrm {sub}$ is given by the first line of Eq.",
"(REF ).",
"Expanding the various energies for $\\Delta \\alpha \\ll 1$ , we find that the energy gain due to the collapsed compression scales as: $U_\\mathrm {hel}-U_\\mathrm {dom} \\sim T^2 \\Delta \\alpha ^{5/2}$ (solid brown curve in Fig.",
"REF a), whereas the energetic cost due to the finite-amplitude wrinkles scales as $\\sim t^2\\eta ^2 \\Delta \\alpha ^{1/2}$ (dashed purple curve).",
"Plotting these curves as a function of $\\Delta \\alpha $ we find that the FT behavior becomes energetically favorable for $\\Delta \\alpha $ above a characteristic confinement $\\Delta \\alpha _\\mathrm {NT-FT} \\sim \\frac{t}{\\sqrt{T}} \\ ,$ where we used the fact that $\\eta \\approx \\sqrt{24 T}$ for $\\Delta \\alpha \\ll 1$ .",
"We note that $\\Delta \\alpha _\\mathrm {NT-FT}$ exhibits a scaling behavior that is similar to the wrinkling threshold $\\Delta \\alpha _\\mathrm {lon}$ , Eq.",
"(REF ).",
"This scenario, which is similar to tensional wrinkling phenomena , , is depicted in Fig.",
"REF .",
"The dashed curve describes the expected behavior of the width of the wrinkled zone as $\\alpha $ increases above 24.",
"For $\\Delta \\alpha < \\Delta \\alpha _\\mathrm {lon}$ the ribbon remains in the helicoidal (unwrinkled) state; at onset, the width matches the compressed zone of the helicoidal state; as the confinement is increased further, the width overshoots the compressed zone of $\\sigma _\\mathrm {hel}^{ss}(r)$ , signifying the transformation, over a confinement interval that it comparable to $\\Delta \\alpha _\\mathrm {lon}$ , to the compression-free stress $\\sigma _\\mathrm {FT}^{ss}(r)$ ."
],
[
"Asymptotic isometries at $T \\rightarrow 0$",
"We now turn to study the vicinity of the singular hyper-plane $(T \\!=\\!",
"0,t\\!",
"=\\!",
"0)$ in the 4D parameter space, assuming fixed, small values of $\\eta $ and $L^{-1}$ .",
"Obviously, for a fixed twist $\\eta $ , the helicoidal shape contains a finite amount of strain that does not go away even if the exerted tensile load $T \\rightarrow 0$ .",
"This is seen in the behavior of $U_\\mathrm {hel}$ , which approaches in this limit (i.e.",
"$\\alpha ^{-1} \\rightarrow 0$ in Fig.",
"REF ) $\\frac{2}{7}$ of its value at the onset of the longitudinal instability ($\\alpha = 24$ ).",
"This result is consistent with our intuitive picture of the helicoid, as well as from Green's stress, Eqs.",
"(REF ,REF ), which shows that longitudes (i.e.",
"material lines $X(s,r=\\mathrm {cst})$ ) are strained in the limit $T \\rightarrow 0$ by $\\eta ^2(\\frac{1}{2}r^2-\\frac{1}{24})$ .",
"This strain stems from the helicoidal structure rather than from a tensile load, and we thus call it “geometric strain\".",
"At first, one may expect that such a $T$ -independent geometric strain is inherent to the helicoidal structure and cannot be removed by wrinkly decorations of the helicoid.",
"However, the energy $U_\\mathrm {dom}$ of the FT-longitudinally-wrinkled state, Eq.",
"(REF ), invalidates this intuitive expectation.",
"As Fig.",
"REF shows, $U_\\mathrm {dom}/U_\\mathrm {hel}$ vanishes as $T \\rightarrow 0$ , indicating that the wrinkled state becomes an asymptotic isometry of the ribbon, which can accommodate an imposed twist $\\eta $ with no strain.",
"Importantly, the subdominant energy (REF ) shows that, although the asymptotic isometry requires a diverging curvature of wrinkles, its bending cost eventually vanishes as $t \\rightarrow 0$ .",
"Hence, the longitudinal wrinkling leads to a physically admissible, nearly strainless state for the stretched-twisted ribbon, at an infinitely small neighborhood of the hyper-plane $(T = 0,t = 0)$ .",
"Equation (REF ) shows that the actual energetic cost of $U_\\mathrm {dom}$ as $T \\rightarrow 0$ is proportional to $T$ , and stems from the work done on the ribbon by the (small) tensile load, where the prefactor is the longitudinal contraction $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}$ that approaches the value $\\eta ^2/8$ in this limit.",
"Notably, the contraction $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}$ is larger than the analogous contraction $\\chi _\\mathrm {hel}$ of the unwrinkled helicoidal state (see Fig.",
"REF ).",
"This observation shows that the formation of wrinkles necessitates a slight increase in the contraction of the helicoidal shape, which implies a corresponding increase of the work done by the tensile load, but gives much more in return: an elimination of the geometric strain from the helicoidal shape.",
"The asymptotic behavior of $U_\\mathrm {FT}$ in the limit $(T \\rightarrow 0,t \\rightarrow 0)$ leads us to propose the general form of the asymptotic isometry equation (REF ), which applies to all physically admissible states of the stretched-twisted ribbon in this limit.",
"Since such states become strainless in this limit, we expect that the strain at a small finite $T$ is proportional to $T$ , such that the integral in the energetic term analogous to Eq.",
"(REF ) is proportional to $T^2$ , and is negligible in comparison to the work term that is linear in $T$ .",
"The prefactor ($A_j$ ) is nothing but the corresponding longitudinal contraction in the limit $T \\rightarrow 0$ .",
"The second term in Eq.",
"(REF ) reflects the bending cost, and the physical admissibility of the state implies the scaling $t^{2\\beta _j}$ with $\\beta _j >0$ and a prefactor $B_j$ that approaches a finite value as $T \\rightarrow 0$ The upper bound $\\beta _j \\le 1$ stems from the bending modulus, and assuming that the minimal curvature of any nontrivial state is $O(1)$ .. We demonstrate this idea by considering the simple deformation of a long, twisted ribbon: a cylindrical wrapping (Fig.",
"REF ), where the centerline, along with all other longitudes, are mapped into parallel helices.",
"Considering first the case $T=0$ , we see that the bending energy of this state is minimized by the smallest possible curvature that allows conversion of the imposed twist into a writhe.",
"This minimal curvature is $\\eta ^2$ , and is obtained when the twisted, unstretched ribbon, “collapses\" onto a plane perpendicular to its long axis, such that the longitudinal contraction is the maximal possible: $\\chi _\\mathrm {cyl} =1$ (see Fig.",
"REF ).",
"For small $T$ and $t$ , we obtain the energy: $U_\\mathrm {cyl} \\simeq T + \\eta ^4 t^2 \\ .$ Comparing $U_\\mathrm {cyl}$ to the energy $U_\\mathrm {FT}$ of the longitudinally-wrinkled helicoidal shape, we note the basic difference between these states, which is depicted in Fig.",
"REF b.",
"The formation of longitudinal wrinkles is associated with a larger cost of bending energy (i.e.",
"$\\beta <1$ in Eq.",
"REF ), and is thus less favorable at very small $T$ .",
"However, the small longitudinal contraction of the longitudinally-wrinkled state allows an energetically efficient mechanism to accommodate the exerted tensile load, and makes it favorable if $T > \\eta ^4 t^{2\\beta }$ .",
"Notably, the transition between the two states occurs at $T \\sim \\eta ^4 t^{2\\beta }$ , approaching the vertical axis in the $(T,\\eta )$ plane when $t \\rightarrow 0$ .",
"This scenario, on which we will elaborate more in Subsec.",
"REF , underlies the secondary instabilities of the helicoidal state depicted in Fig.",
"REF a.",
"The relevance of isometric maps (of 2D sheets embedded in 3D space) to the behavior of thin sheets with small but finite thickness, has been recognized and exploited in numerous studies , , , , , , , , .",
"Most studies, however, consider confining conditions that do not involve an exerted tension (i.e.",
"$T=0$ ), such that the only limit being considered is $t \\rightarrow 0$ .",
"The asymptotic isometry equation (REF ) reveals the relevance of this concept even when a small tensile load is exerted on the sheet, and provides a quantitative tool to study the energetic competition between various types of asymptotic isometries at the presence of small tension."
],
[
"Overview",
"The longitudinal wrinkling instability addressed in Sec.",
"occurs when $\\sigma ^{ss}(r)$ has a compressive zone.",
"In this section we address a different instability, whereby the ribbon buckles or wrinkles due to the compression of the transverse stress component $\\sigma ^{rr}(r)$ .",
"The transverse instability emerges when the exerted twist exceeds a threshold $\\eta _\\mathrm {tr}(T)$ , whereby the ribbon develops periodic undulations in the transverse direction (with wavelength $\\lambda _\\mathrm {tr} \\ll W$ ) or a single buckle ($\\lambda _\\mathrm {tr} \\sim W$ ).",
"Our analysis highlights two principal differences between the longitudinal and transverses instabilities, which are intimately related to the experimental observation in [3].",
"First, in contrast to the longitudinal threshold, which occurs near a curve, $\\eta _\\mathrm {lon}(T) \\approx \\sqrt{24 T}$ , that is independent on the thickness and length of the ribbon, the threshold $\\eta _\\mathrm {tr}(T)$ and the nature of the transverse instability exhibit a strong, nontrivial dependence on $t$ and $L$ .",
"Second, in contrast to the longitudinal instability, which emerges as a primary instability of the helicoidal state, the transverse instability underlies two qualitatively distinct phenomena: a primary instability of the helicoid in a “large\" tension regime ($T > T_\\lambda $ ), where the longitudinal stress is purely tensile, and a secondary instability of the helicoid preceded by the longitudinal instability at a low tension regime ($T < T_\\lambda $ ).",
"We placed the word “large\" in quotation marks since $ T_\\lambda (t,L) \\ll 1$ (see Fig.",
"REF ), hence it is fully justified to assume a Hookean response for $T_\\lambda (t,L) \\ll T \\ll 1$ .",
"This scenario implies that the tension-twist parameter space $(T,\\eta )$ consists of three major phases: A helicoidal state, a FT-longitudinally-wrinkled state, and a state delimited from below by the transverse instability.",
"This division is shown in Fig.",
"REF and strongly resembles the experimental phase diagram reported in [3].",
"In [3], the instability of the longitudinally-wrinkled state upon increasing twist was attributed to a “looping\" mechanism and was described as a new, third type of instability, separate from the longitudinal and transverse instabilities.",
"In our picture, this instability emerges simply as the transverse instability in the low tension regime, where it is superimposed on the FT longitudinally wrinkled state.",
"This insight provides a natural explanation to the appearance of a single “triple\" $\\lambda $ -point $\\left(T_\\lambda ,\\eta _\\lambda =\\sqrt{24T_\\lambda }\\right)$ in the tension-twist plane, where the threshold curve $\\eta _\\mathrm {lon}(T)$ divides $\\eta _\\mathrm {tr}(T)$ into a low-tension branch and a large-tension branch.",
"Figure: The parameter plane (T,η)(T,\\eta ) exhibits the helicoid, the far from threshold longitudinal wrinkling and the transverse instability (that is buckling here) when Lt≫1Lt\\gg 1, plotted here for t=0.005t=0.005.",
"The coordinates of the triple λ\\lambda -point are denoted (T λ ,η λ )(T_\\lambda ,\\eta _\\lambda ).Figure: Schematic phase diagram representing the two regimes L 2 t≫1L^2t\\gg 1 (Left) and L 2 t≪1L^2t\\ll 1 (Right) with the corresponding scaling laws for the coordinates T λ T_\\lambda and η λ \\eta _{\\lambda } of the λ\\lambda -point.Beyond this central result, we predict that the threshold curve $\\eta _\\mathrm {tr}(T)$ , the $\\lambda $ -point $(T_\\lambda ,\\eta _\\lambda )$ , and the wavelength $\\lambda _\\mathrm {tr}$ , exhibit a remarkable dependence on the mutual ratios of the thickness, width, and length of the ribbon.",
"This complex phenomenology is depicted in Fig.",
"REF and is summarized in the following paragraph: $\\bullet $ The threshold twist $\\eta _\\mathrm {tr}(T)$ vanishes as the ribbon thickness vanishes, $t \\rightarrow 0$ .",
"$\\bullet $ The threshold twist $\\eta _\\mathrm {tr}(T)$ diverges as $T \\rightarrow 0$ .",
"$\\bullet $ The tension $T_\\lambda (t,L)$ , which separates the regimes of low and “large\" tension, vanishes in the ribbon limit at a rate that depends in a nontrivial manner on the mutual ratios of the length, width, and thickness of the ribbon: If $t \\ll L^{-2}$ we find that $T_\\lambda \\sim (t/L)^{2/3}$ , whereas if $ L^{-2} \\ll t$ we find that $T_\\lambda \\sim t$ .",
"$\\bullet $ The mutual ratios between the length, width, and thickness in the ribbon limit (Eq.",
"REF ) affect also the type of the transverse instability.",
"Specializing for the “large\" tension regime, we find that the transverse instability may appear as a single buckle or as a periodic array of wrinkles with wavelength $\\lambda _\\mathrm {tr}$ that decreases as $T^{-1/4}$ upon increasing the tension: (a) If $L^{-1} \\ll t$ , the transverse instability appears as a single buckle of the helicoidal state.",
"(b) If $L^{-2} \\ll t \\ll L^{-1}$ the transverse instability appears as a single buckle for $T \\ll (Lt)^2$ and as a wrinkle pattern for $(Lt)^2 \\ll T \\ll 1$ .",
"(c) If $t \\ll L^{-2}$ , the transverse instability appears as a wrinkle pattern throughout the whole “large\" tension regime.",
"We start by a scaling analysis of the parameter regime that explains the above scenario.",
"Then we turn to a quantitative linear stability analysis that yields the transverse buckling threshold, as well as the shape of the buckled state for an infinitely long ribbon (or, more precisely, $L^{-2} \\ll t$ ).",
"Finally, we address at some detail the transverse instability of a ribbon with a finite length ($t \\ll L^{-2}\\ll 1$ )."
],
[
"Scaling analysis",
"Similarly to the longitudinal wrinkling, the basic mechanism of the transverse instability is simply the relaxation of compression (which is now $\\sigma ^{rr}$ ), by appropriate deformation of the helicoidal shape.",
"Taking a similar approach to Sec.",
", we can find the scaling relations for the threshold $\\eta _\\mathrm {tr}$ and the wavelength $\\lambda _\\mathrm {tr}$ , by identifying the dominant destabilizing and stabilizing normal forces associated with such shape deformation.",
"The transverse compression gives rise to a destabilizing force $\\sim \\sigma ^{rr}/\\lambda _\\mathrm {tr}^2$ .",
"The normal restoring forces are similar to the respective forces that underlie the wrinkling of a stretched (untwisted) ribbon [8]: bending resistance to deformation in the transverse direction ($\\sim B/\\lambda _\\mathrm {tr}^4$ ), and tension-induced stiffness due to the spatial variation of the deformation in the longitudinal direction ($\\sim T/L^2$ ).",
"All other normal restoring forces, in particular the bending resistance to deformation in the longitudinal direction (that scales as $\\sim B/L^4$ ) are negligible with respect to these two forces.",
"The balance between these two dominant restoring forces and the destabilizing normal force due to the compression $\\sigma ^{rr}$ may lead to buckling, namely $\\lambda _\\mathrm {tr} \\sim W=1$ , if the ribbon is extremely long ($t \\ll L^{-1}$ ), in which case the tension-induced stiffness is negligible, or to wrinkling ($\\lambda _\\mathrm {tr} \\ll W$ ), where the bending and tension-induced forces are comparable.",
"In the following paragraphs we address first the case of an extremely long ribbon, where the only dominant restoring force is associated with bending, and then show how a finite value of $L$ affects a transition from buckling to wrinkling.",
"In each case we will discuss separately the regimes of low and “large\" tension, and derive the scaling relation for the $\\lambda $ -point $(T_\\lambda ,\\eta _\\lambda )$ that separates these regimes."
],
[
"Extremely long ribbon:",
"In this case, the tension-induced stiffness is negligible, and the only significant restoring force to shape deformations is the bending resistance.",
"The transverse instability is then similar to the Euler buckling of a beam of width $W$ and thickness $t$ , and the instability mode is consequently buckling, i.e.",
"$\\lambda _\\mathrm {tr} \\sim W$ .",
"The instability threshold $\\eta _\\mathrm {tr} (T)$ is obtained when the destabilizing force becomes comparable to the stabilizing bending force, namely: $\\frac{\\sigma ^{rr}}{W^2}\\sim \\frac{B}{W^4}.$ The transverse compression $\\sigma ^{rr}(r)$ is given by Eq.",
"() in the helicoidal state (for $\\eta ^2 < 24 T$ ), and by Eq.",
"(REF ) in the FT longitudinally wrinkled state (for $\\eta ^2 > 24 T$ ).",
"Considering the asymptotic regimes $T \\gg \\eta ^2$ in Eq.",
"() and $T \\ll \\eta ^2$ in Eq.",
"(REF ), we see that in both “large\" and low tension regimes, the transverse stress scales similarly with $\\eta $ and $T$ : $\\sigma ^{rr} \\sim \\eta ^2 T.$ Substituting Eq.",
"(REF ) in Eq.",
"(REF ), we obtain the scaling of the instability threshold $\\eta _\\mathrm {tr}(T)$ for an extremely long ribbon: $\\eta _\\mathrm {tr}(T) \\sim \\frac{t}{\\sqrt{T}}.$ This scaling relation (with different numerical prefactors in the limits $T \\ll T_\\lambda $ and $T_\\lambda \\ll T \\ll 1$ ), is confirmed by our detailed calculations in Subsec.",
"REF , that are shown in Fig.",
"REF .",
"The relation (REF ) demonstrates the singular nature of the transverse instability: the threshold decreases with the ribbon thickness ($t \\rightarrow 0$ ) and diverges as the exerted tension vanishes ($T \\rightarrow 0$ ).",
"The tension $T_\\lambda $ is the horizontal coordinate of the “triple\" $\\lambda $ -point in the tension-twist parameter plane (Fig.",
"REF ) at which the transverse buckling changes its character from a primary instability at “large\" tension to a secondary instability of the helicoidal state at low tension, which is preceded by the longitudinal wrinkling instability.",
"We find $T_\\lambda $ from Eq.",
"(REF ) and the relation $\\eta _\\mathrm {lon}(T) \\sim \\sqrt{T}$ : $T_\\lambda \\sim t \\ .$"
],
[
"Ribbon of finite length:",
"Let us assume now that both tension-induced stiffness and bending resistance are significant restoring forces, which balance the destabilizing force due to transverse compression.",
"The instability onset condition (REF ) is then replaced by $\\frac{\\sigma ^{rr}}{\\lambda _\\mathrm {tr}^2}\\sim \\frac{B}{\\lambda _\\mathrm {tr}^4}\\sim \\frac{\\sigma ^{ss}}{L^2}.$ Using the scaling law (REF ) for $\\sigma ^{rr}$ and estimating $\\sigma ^{ss} \\sim T$ , we obtain the following scaling relations for the threshold and wavelength: $\\eta _\\mathrm {tr} & \\sim \\sqrt{\\frac{t}{L}} T^{-1/4}, \\\\\\lambda _\\mathrm {tr} & \\sim \\sqrt{Lt} T^{-1/4}.",
"$ In a similar way to the above paragraph, we find the coordinate $T_\\lambda $ of the $\\lambda $ -point by equating Eq.",
"(REF ) with the relation $\\eta _\\mathrm {lon}(T) \\sim \\sqrt{T}$ , yielding: $T_\\lambda \\sim \\left(\\frac{t}{L}\\right)^{2/3} \\ .$"
],
[
"From buckling to wrinkling:",
"Realizing the important effect of the ribbon length on the nature of the transverse instability, a natural question is: How long must a ribbon be such that the tension-induced stiffness becomes negligible and the scaling relations (REF ,REF ) are valid?",
"A key to address this question is the obvious inequality $\\lambda _\\mathrm {tr} \\le W$ .",
"Substituting the scaling relation (REF ) for $T_\\lambda $ in Eq.",
"(REF ), and requiring $\\lambda _\\mathrm {tr} \\ll W$ , we find that $T_\\lambda $ is characterized by the scaling relation (REF ) if $t \\ll L^{-2}$ , and by the relation (REF ) if $L^{-2} \\ll t$ .",
"This nontrivial dependence of $T_\\lambda $ on the thickness and length of the ribbon is depicted in Fig.",
"REF .",
"The behavior of $T_\\lambda $ indicates the complex nature of the ribbon limit, but a closer inspection of Eq.",
"(REF ), subjected to the condition $\\lambda _\\mathrm {tr} \\le W$ , reveals an even higher level of complexity.",
"Focusing on the large tension regime $T_\\lambda < T <1 $ , and recalling that $\\lambda _\\mathrm {tr} \\le W$ , we find that the ribbon limit is divided into three sub-regimes that exhibit qualitatively distinct types of transverse instabilities.",
"This behavior is depicted in Fig.",
"REF , and summarized below: (a) If $t \\ll L^{-2} \\ll 1$ , then $T_\\lambda $ satisfies the scaling relation (REF ) and the transverse instability appears as wrinkling, where the threshold $\\eta _\\mathrm {tr}$ and the wavelength $\\lambda _\\mathrm {tr}$ satisfy the scaling relations (REF ).",
"(b) If $ L^{-2} \\ll t \\ll L^{-1} \\ll 1$ , then $T_\\lambda $ satisfies the scaling relation (REF ), but the large tension regime splits into two parts.",
"For sufficiently small $T$ , the transverse instability appears as a buckling mode ($\\lambda _\\mathrm {tr} \\sim W$ ), and the threshold $\\eta _\\mathrm {tr}$ satisfies the scaling (REF ); for larger values of $T$ (which are nevertheless $\\ll 1$ ), the instability appears as a wrinkling mode, described by the scaling relations (REF ).",
"(c) Finally, if $ L^{-1} \\ll t \\ll 1$ , then $T_\\lambda $ satisfies the scaling relation (REF ), and the transverse instability appears as a buckling mode, with the scaling (REF ), throughout the whole regime of large tension."
],
[
"Linear stability analysis",
"In this subsection we present in detail the linear stability analysis for the case of an extremely long ribbon, assuming that the ribbon shape close to the transverse instability is well approximated by the form: $X(s,r)=\\begin{pmatrix} (1-\\chi )s + \\zeta \\eta u_{s1}(r) \\\\ [r+u_r(r)]\\cos (\\eta s) - \\zeta z_1(r)\\sin (\\eta s) \\\\ [r+u_r(r)]\\sin (\\eta s) + \\zeta z_1(r)\\cos (\\eta s)\\end{pmatrix} \\ ,$ where the perturbation's amplitude $\\zeta $ is assumed to be infinitesimal, and the functions $u_{s1}(r)$ and $z_1(r)$ correspond to the two degrees of freedom that characterize the perturbed shape and strain.",
"More precisely, it is reasonable to assume that the boundary conditions at the short edges ($s = \\pm L/2$ ) have a prominent effect only at a zone of size $W=1$ near those edges, and barely disturb the translational symmetry of the reference state in the longitudinal direction.",
"(We will comment on this assumption later in Sec. ).",
"Therefore, the eigenmodes of the system are approximated by: $\\left[ u_{sj}(r) \\cos \\left(\\frac{\\pi js + \\gamma _j}{ L}\\right), z_j (r) \\cos \\left(\\frac{\\pi js}{L}\\right) \\right]$ where $1 \\le j \\ll L$ .",
"Since we address here an instability that relaxes the transverse compression, the variation in the longitudinal direction should be minimal to avoid any energetic costs, and hence we assume that the first eigenmode to become unstable is $j=1$ (see also [8]).",
"In the next subsection we present an approximate analysis of the $j=1$ mode, but here we simplify further by neglecting the longitudinal variation altogether (i.e.",
"replacing $\\cos (\\pi js /L) \\rightarrow 1$ ).",
"Since all derivatives with respect to the variable $s$ come with negative powers of $L$ , we anticipate that this approximation is valid for sufficiently large $L$ (such that $L^{-2} \\ll t$ , as found in Subsec.",
"REF ).",
"Our linear stability analysis follows a classical approach, whose first use in elasticity theory has been attributed to Michell , The introduction of contains a useful summary of the various approaches for linear stability analysis of elastic systems.. First, we assume a small perturbation of the form (REF ) to the reference state, and expand the generalized FvK equations (REF -) to linear order in the amplitude $\\zeta $ , obtaining a set of linear homogeneous equations.",
"In general, these equations have no solution but the trivial one, $u_{s1}(r)=z_1(r)=0$ .",
"For a given tension $T$ , the transverse instability occurs at the lowest value $\\eta _\\mathrm {tr}(T)$ of the twist for which the buckling equations admit a nontrivial solution.",
"This solution is identified as the unstable transverse mode.",
"We must compute the perturbed curvature and stress tensors that enter the generalized FvK equations (REF -) upon substituting the shape (REF ), and retaining only the terms that are linear in $\\zeta $ .",
"We will limit our analysis to the simple case $\\nu =0$ .",
"Since the stress field (REF -) has been shown to be independent of the Poisson ratio, we expect that the same is true for the transverse instability.We just note that a non zero Poisson ratio would require another degree of freedom in the perturbative analysis, namely a transverse in-plane displacement $\\zeta u_{r1}(r)$ .",
"The perturbed curvature tensor is given by $c_{\\alpha \\beta }=c_{\\alpha \\beta }^{(0)}+\\zeta c_{\\alpha \\beta }^{(1)}+O(\\zeta ^2)$ , where $c^{(0)}_{\\alpha \\beta } & = \\begin{pmatrix} 0 & \\eta \\\\ \\eta & 0 \\end{pmatrix},\\\\c^{(1)}_{\\alpha \\beta } & = \\begin{pmatrix} -\\eta ^2\\left[z_1(r)-r z_1^{\\prime }(r)\\right] & 0 \\\\ 0 & z_1^{\\prime \\prime }(r) \\end{pmatrix} \\ ,$ and the perturbed stress tensor is given by $\\sigma ^{\\alpha \\beta }=\\sigma ^{\\alpha \\beta }_{(0)}+\\zeta \\sigma ^{\\alpha \\beta }_{(1)}+\\mathcal {O}( \\zeta ^2)$ , where $\\sigma _{(0)}^{\\alpha \\beta } & = \\begin{pmatrix} T+\\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{12} \\right) & 0 \\\\ 0 & \\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{4} \\right) \\left[T +\\frac{\\eta ^2}{4}\\left(r^2+\\frac{1}{12} \\right) \\right] \\end{pmatrix}, \\\\\\sigma _{(1)}^{\\alpha \\beta } & = \\frac{\\eta }{2}\\left[u_{s1}^{\\prime }(r)-z_1(r)+rz_1^{\\prime }(r) \\right] \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}.$ Notice that, to $O(\\zeta )$ , the diagonal stress components are not perturbed.",
"Furthermore, the force balance in the transverse direction (), which we evaluate as usual to $O(\\eta ^4)$ , yields the equations: $0 & = \\frac{\\eta }{2}\\left[u_{s1}^{\\prime \\prime }(r)+rz_1^{\\prime \\prime }(r) \\right],\\\\0 & = \\frac{\\eta ^3}{2}\\left(u_{s1}^{\\prime }(r)-z_1(r)+rz_1^{\\prime }(r) + 2r \\left[u_{s1}^{\\prime \\prime }(r)+rz_1^{\\prime \\prime }(r) \\right] \\right).$ These equations are solved by: $u_{s1}^{\\prime }(r)=z_1(r)-rz_1^{\\prime }(r)$ implying that the stress tensor does not deviate from its value at the reference state.",
"Turning now to the normal force balance, we substitute Eqs.",
"(REF ,,REF ) in Eq.",
"(REF ), retain the linear order in the amplitude $\\zeta $ , and obtain a $4^\\textrm {th}$ order differential equation for $z_1(r)$ : $ \\frac{t^2}{12}z_1^{(4)}(r)= -\\eta ^2\\sigma _{(0)}^{ss}\\left[z_1(r)- r z_1^{\\prime }( r) \\right] + \\sigma _{(0)}^{rr}z_1^{\\prime \\prime }(r).$ We recognize this equation as similar to the Euler buckling equation, whereby a destabilizing force $\\sigma _{(0)}^{rr}z_1^{\\prime \\prime }(r)$ that originates from the relaxation of compression by deflection is balanced by the stabilizing bending force $\\frac{t^2}{12}z_1^{(4)}(r)$ that opposes any deflection.",
"However, Eq.",
"(REF ) carries some differences from the simple Euler buckling instability.",
"First, the compression $\\sigma _{(0)}^{rr}(r)$ is not uniform across the ribbon width.",
"Second, we note the existence of another normal force, that is proportional to the longitudinal stress $\\sigma _{(0)}^{ss}$ , and originates from the fact that the reference state is non-planar.",
"Since we found that $\\sigma _{(0)}^{rr} \\sim -\\eta ^2 \\sigma _{(0)}^{ss}$ (see Eq.",
"()) we can view the right hand side of Eq.",
"(REF ) as a renormalized version of the normal force $\\sigma _{(0)}^{rr}z_1^{\\prime \\prime }(r)$ that couples the compression to the curvature.",
"We are poised to solve this equation subjected to the homogenous boundary conditions: $z_1^{\\prime \\prime }(\\pm 1/2) & = 0, \\\\z_1^{(3)}(\\pm 1/2) & = 0.",
"$ It is noteworthy that the buckling equation (REF ) is general and does not depend on the particular form (REF ) of the stress of the reference state.",
"As a consequence, the linear stability analysis can be performed over the helicoidal state, where the stress is given by Eqs.",
"(REF -), as well as over the FT-longitudinally-wrinkled state, where it is given by Eqs.",
"(REF ,REF )Recall that for a given $(\\eta ,T)$ in the regime $\\eta ^2/T >24$ , the FT approach in Subsec.",
"REF provides a longitudinally-wrinkled state whose shape is close to the helicoid, up to deviations whose amplitude vanishes as $t \\rightarrow 0$ , and whose stress is given by (REF ,REF ), up to correction that also vanish as $t \\rightarrow 0$ .",
"Therefore, the transverse linear stability analysis in this regime provides expressions for the threshold $\\eta _\\mathrm {tr}(T)$ and the unstable mode $[u_{s1}(r),z_1[(r)]$ that become accurate as $t \\rightarrow 0$ ..",
"The stress field that we have to use depends on our position in the tension-twist plane $(T,\\eta )$ with respect to the longitudinal instability line $\\eta ^2=24T$ .",
"Before turning to numerical analysis, it is useful to consider the limit $T \\ll \\eta ^2$ ($\\alpha \\rightarrow \\infty $ in the terminology of Sec.",
"), where an analytic solution of the buckling equations (REF -) is available.",
"In this limit, the exerted tension is supported by two narrow strips near the long edges of the ribbon, and the stress field (REF ,REF ) becomes: $\\sigma ^{ss}(r) & = \\frac{T}{2}\\left[\\delta \\left(r+\\frac{1}{2} \\right) + \\delta \\left(r-\\frac{1}{2} \\right) \\right],\\\\\\sigma ^{rr}(r) & = -\\frac{\\eta ^2 T}{4} \\ ,$ where $\\delta (r)$ is the Dirac-delta function.",
"Since the term $-\\eta ^2\\sigma _{(0)}^{ss}\\left[z_1(r)- r z_1^{\\prime }( r)\\right]$ is non-zero only at an infinitesimal strip near $r = \\pm 1/2$ , we can eliminate it from (REF ) by modifying the boundary condition () that becomes: $\\frac{t^2}{12}z_1^{(3)}(-1/2)=-\\frac{\\eta ^2 T}{2}\\left[z_1(-1/2)+\\frac{1}{2}z_1^{\\prime }(-1/2) \\right],$ with an analogous condition at the other edge, $r=1/2$ .",
"The buckling equation (REF ) now simply reads $z_1^{(4)}(r)=-\\frac{3\\eta ^2 T}{t^2}z_1^{\\prime \\prime }(r) \\ ,$ which is the familiar Euler buckling equation under uniform compression.",
"It admits a non-zero solution $z_1(r)=\\cos (\\pi r)$ when $\\eta $ reaches its threshold value ${\\rm small} \\ T \\ : \\ \\ \\eta _\\mathrm {tr}(T) = \\frac{\\pi }{\\sqrt{3}} \\frac{t}{\\sqrt{T}}.$ In the opposite limit, $T \\gg \\eta ^2$ , an analytic solution of Eq.",
"(REF ) is not available and a numerical solution of Eqs.",
"(REF -) yields the threshold in this limit: ${\\rm large} \\ T \\ : \\ \\ \\eta _\\mathrm {tr}(T) = 4.4 \\frac{t}{\\sqrt{T}}.$ Interestingly, the two asymptotic expressions (REF ,REF ) exhibit the scaling law (REF ), not only with the ribbon thickness $t$ , but also with the tension $T$ .",
"Our numerical results and the subsequent division of the tension-twist plane into three major phases (the helicoid, the longitudinal wrinkling and the region above the transverse instability) are shown in Fig.",
"REF for the thickness $t=0.005$ .",
"This phase diagram exhibits a striking similarity, at a quantitative level, with the phase diagram found experimentally in [3] for the same thickness.The experimental value of the length in [3] is $L=20$ .",
"The maximal tension in the experiment is $T_\\mathrm {max}=0.01$ ; from Eq.",
"(), we deduce that the minimal transverse wavelength is $\\lambda _\\mathrm {min}\\sim \\sqrt{Lt}T_\\mathrm {max}^{-1/4}=1$ .",
"This explains why buckling is observed and why the infinite length approximation is in good agreement with the experiment.",
"The numerical analysis of the buckling equation gives also the shape of the buckling mode, which we show in Fig.",
"REF for a few representative values of $T$ .",
"Choosing some (arbitrary) small amplitude, we draw the shape of the buckled ribbon in Fig.",
"REF .",
"Figure: Shape of the transverse unstable mode in the limit of an infinitely long ribbon (L→∞L\\rightarrow \\infty ) of thickness t=0.005t=0.005, as a function of the exerted tension.",
"For the range of tension applied here, the limit L→∞L\\rightarrow \\infty is relevant for lengths L>20L>20 (see Eq.",
").Figure: Shape of the ribbon undergoing a transverse instability: (a) single mode buckling of a very long ribbon, from a numerical solution of Eq.",
"() with an arbitrary amplitude (b) wrinkling of a ribbon with Lt≪1Lt\\ll 1, from Eq.",
"()."
],
[
"Effect of a finite length",
"Our analysis in this paper assumes that the ribbon is long, such that the effect of boundary conditions at $s = \\pm L/2$ is limited to the vicinity of the short edges, and the linear eigenmodes can be expressed through a Fourier series, Eq.",
"(REF ), where the most unstable one is $j=1$ .",
"In Subsec.",
"REF we went beyond this assumption and neglected the spatial variation of this unstable mode in the longitudinal direction, expecting it to make a negligible contribution to the force balance if $L$ is sufficiently large.",
"In this subsection we relax this last assumption, by taking into consideration the longitudinal variation of the perturbation.",
"This mean that the ribbon is not sufficiently long to justify a complete neglecting of the spatial variation, but it is long enough such that the mode structure is given by Eq.",
"(REF , $j=1$ ).",
"If the ribbon becomes even shorter, it is possible that the boundary effect is sufficiently strong and the assumed mode structure Eq.",
"(REF ) is not valid.",
"In Sec.",
"6 we discuss the possibility that this might happen even in the ribbon limit (i.e.",
"$L \\gg 1$ ) provided $t$ is small enough.",
"Despite its simple form, a complete analysis of the mode $j=1$ in Eq.",
"(REF ) is rather cumbersome.",
"In order to simplify our calculation we will retain only the term in the normal force balance that couples the longitudinal stress and the longitudinal curvature.",
"Such a term was found to be crucial for the wrinkling of a stretched, untwisted sheet [8].",
"We thus obtain the buckling equation $ \\frac{t^2}{12}z_1^{(4)}(r)= -\\eta ^2\\sigma _{(0)}^{ss}\\left[z_1(r)- r z_1^{\\prime }( r) \\right]+\\kappa ^2\\sigma _{(0)}^{ss}z_1(r) + \\sigma _{(0)}^{rr}z_1^{\\prime \\prime }(r),$ where $\\kappa =\\pi /L$ , subjected to the same boundary conditions that we described in the previous subsection.",
"The buckling equation can be solved numerically, and the effect of the finite ribbon length on the phase diagram is shown on Fig.",
"REF .",
"The main effect of the finite length is to increase the transverse buckling threshold; this is expected, since the new term is a stabilizing term.",
"We also note that this effect is not important at small tension.",
"Another effect is that the threshold $\\eta _\\mathrm {tr}(T)$ may become a non-monotonic function, due to the fact that the tension enhances both the compressive (destabilizing) term and the stretching (stabilizing) term, which are represented, respectively, by the third and second terms of Eq.",
"(REF ).",
"Figure: Effect of a finite ribbon length on the phase diagram: for a thickness t=5×10 -4 t=5\\times 10^{-4}, the transverse buckling threshold is plotted for L=∞L=\\infty (dashed line) and L=10L=10 (solid line).For this thickness and this tension range, the infinite length approximation is relevant for lengths such that L>t -1 T max 1/2 ≃100L > t^{-1}T_\\mathrm {max}^{1/2} \\simeq 100 (from the requirement that λ>1\\lambda >1 in Eq.",
"()).Finally, as was noted already in our scaling analysis, the unstable transverse mode transforms from buckling to wrinkling as the tension increases.",
"This transformation is shown in Fig.",
"REF .",
"Figure: Transition from transverse buckling to transverse wrinkling as a function of the tension and length.Shape of the transverse unstable modes: Left: t=5·10 -4 t=5\\cdot 10^{-4}, L=10L=10 and tensions T=10 -4 T=10^{-4}, 10 -3 10^{-3} and 10 -2 10^{-2}.",
"Right: t=0.0005t=0.0005, T=0.01T=0.01 and ribbon lengths L=5L=5, 35 and 80."
],
[
"Overview",
"In Secs.",
"- we assumed that the shape of the stretched-twisted ribbon is close to a helicoid, and employed asymptotic methods to characterize the deviations from this shape.",
"This approach allowed us to compute the curves $\\eta _\\mathrm {lon}(T)$ and $\\eta _\\mathrm {tr}(T)$ that underlie the division of the $(T,\\eta )$ plane into three major regimes (Fig.",
"REF ), and to characterize the helicoidal state (blue), the longitudinally wrinkled state (orange), and the margins of the third regime (pink), close to the transverse instability threshold.",
"The proximity to a helicoidal shape is violated at the bulk of the pink regime, which we do not address in this paper, where the self-contact zones emerge and the helicoidal shape is greatly mutilated [3].",
"Other two parameter regimes where the ribbon shape may become very different from a helicoid are the edge of the blue regime (i.e.",
"close to the horizontal line $\\eta =0$ ), where the ribbon is stretched with little twisting, and the edge of the orange regime (i.e.",
"close to the vertical line $T=0$ ), where the ribbon is twisted with little tensile load.",
"In this section we discuss the expected transformations to non-helicoidal morphologies in these parameter regimes: a nearly planar shape as $\\eta \\rightarrow 0$ , where the twist is absorbed in the vicinity of the short edges; and the formation of a creased helicoidal shape and cylindrical wrapping as $T \\rightarrow 0$ .",
"In contrast to previous sections (-), where we carried out a rigorous study based on the helicoidal solution to the cFvK equations, its linear stability analysis, and an FT analysis, our discussion in this section is more heuristic, and is based on energetic estimates and scaling arguments.",
"We start in Subsec.",
"REF with a general discussion of the difference between plate-like and rod-like approaches to the mechanics of ribbons.",
"The first approach, which we employed in previous sections, is based on the cFvK equations; the second approach consists of Kirchoff's rod equations or Sadowsky equation, the last one provides the mathematical basis for a theoretical description of the creased helicoidal state [7].",
"We take this opportunity to explain why the Kirchoff's rod equations cannot be used to study the ribbon limit (Eq.",
"REF ).",
"In Subsec.",
"REF we briefly describe the work of [7] on the creased helicoidal state, and explain how it gives rise to another type of asymptotic isometry, different from the longitudinally-wrinkled helicoidal shape and the cylindrical wrapping state.",
"In Subsec.",
"REF we turn to the vicinity of the horizontal line $\\eta =0$ , and introduce an energetic comparison that allows us to estimate the minimal twist necessary to developing a helicoidal shape for a long ribbon ($L \\gg 1$ ) subjected to tension $T$ and clamping of its short edges."
],
[
"Theory of elastic ribbons: plate-like or rod-like ?",
"In the plate-like approach, one employs the cFvK equations for elastic plates with Hookean material response to find the shape of the ribbon midplane $X(s,r)$ .",
"Except the restriction to small strains, no further assumptions are made on the deformation of the cross section or on the stress profile in the transverse direction $\\hat{r}$ .",
"This “transversal freedom\" was reflected in our analysis of the cFvK equations in sections - through the $r$ -dependence of the stresses and the consequent shape deformations, which underlie both longitudinal and transverse instabilities of the helicoidal state.",
"The transversal freedom encapsulates the conceptual difference between the plate-like approach and the rod-like approach, wherein the ribbon shape is derived from a curve $X_\\mathrm {cl}(s)$ that characterizes the shape of the centerline.",
"In the Kirchoff's method, which addresses the ribbon as a rod with highly anisotropic cross section, the ribbon is allowed to have a tensile strain and the cross section (i.e.",
"the ribbon shape in the plane perpendicular to the centerline) is assumed to retain its shape .",
"The relation between the midplane shape and the centerline is simply: ${\\rm Kirchoff \\ rod:} \\ \\ X(s,r) = X_\\mathrm {cl}(s) + r \\hat{r}(s) \\ ,$ where $\\hat{r}(s)$ is the normal to the tangent vector $\\hat{t} = dX_\\mathrm {cl}(s)/ds$ in the ribbon midplane.",
"In the Sadowsky's method, the ribbon is assumed to be strainless, and the shape of the midplane is related to the centerline by the following relation: ${\\rm Sadowsky \\ strip:} \\ \\ X(s,r) = X_\\mathrm {cl}(s) + r \\left[\\hat{b}(s) + \\frac{\\tau (s)}{\\kappa (s)} \\hat{t}(s)\\right] \\ ,$ where $\\hat{b}(s)$ is the Frenet binormal to the curve $X_\\mathrm {cl}(s)$ and $\\tau (s), \\kappa (s)$ , are its torsion and curvature [7].",
"Assuming a ribbon at mechanical equilibrium, the two methods yield strictly different sets of force balance equations that yield the centerline $X_\\mathrm {cl}(s)$ .",
"In the rest of this subsection, we briefly recall recent studies of Kirchoff equations of stretched-twisted rods with anisotropic cross section, and explain why these analyses do not pertain to the ribbon limit, Eq.",
"(REF ).",
"In the next subsection we review a recent work that employed the Sadowsky strip to describe the creased helicoidal state of a stretched twisted ribbon, and discuss the regime in the $(T,\\eta )$ plane describable by this method."
],
[
"Anisotropic Kirchoff's rod:",
"The instability of a rod with circular cross section that is subjected to tension and twist, and the consequent formation of loops, has been studied already by Love , using the Kirchoff's rod equations.",
"The theoretical works of Champneys, Thompson and van der Heijden [10], [11], [12] and of Goriely et al.",
"[9], employed the Kirchoff's rod equations to study the response to tension and twist of a rod with asymmetric (i.e.",
"non-circular) cross section.",
"An important finding of these studies was the existence of instabilities (termed “thick\" and“tapelike\" [9]), through which the straight centerline that defines a helicoidal state of the ribbon becomes unstable (see Fig.",
"3 of [9]).",
"The visual similarity of the “thick\" mode to the secondary instability of a stretched-twisted ribbon at low tension (which we described in Sec.",
"as a transverse instability superimposed on the longitudinally wrinkled ribbon), motivated the original portraying of that instability as “looping\" [3].",
"However, a close inspection of the phase diagram of [3] (Fig.",
"REF ), shows no signs of the instability predicted by [10], [11], [12].",
"Translating the results of [9] to our notations (see Appendix ) and considering the ribbon limit ($a \\ll 1$ in the terminology of [9]), we find that the theoretical prediction suggests an instability of the helicoidal state around the curve $\\eta \\approx \\sqrt{c T}$ with $1/2<c<2.8$ , at which range of parameters the experiments of [3] show a stable helicoidal state.",
"This observation indicates that using the Kirchoff's rod equation may not be suitable at the ribbon limit (Eq.",
"REF ), where the cross section is highly anisotropic.",
"Indeed, the Kirchoff's equations assume fixed values of the torsion and bending moduli (i.e.",
"independent on the exerted loads), which characterize the response of unstretched, untwisted ribbon to infinitesimal loads.",
"As was noted by Green, who considered the twisted ribbon as a 3D solid body, this assumption becomes invalid if the exerted twist $\\eta \\gg t$ (see Eq.",
"(21) of [1]).",
"As a consequence, the Kirchoff's rod equation cannot be used to describe the helicoidal state of a twisted ribbon without appropriate renormalization of the rod's moduli (that reflect the exerted twist and possibly also the tensile load)."
],
[
"The creased helicoidal state: a second look at asymptotic isometries",
"The shape of a perfectly inextensible ribbon (Eq.",
"REF ), has been addressed by Korte et al.",
"[7], who built upon earlier studies [6], , .",
"These authors found that under given twist $\\eta $ and a range of tensile loads, the ribbon admits a strainless state, whose morphology is similar to the creased helicoid state found in the experiments of [3].",
"In fact, the theory of [7] yields a family of such states, parameterized by the angle between triangular facets.",
"Importantly, the construction of [7] consists of “true\" creases, with infinitely large curvature, whose bending energy would have been infinite if the ribbon had any thickness.",
"The underlying assumption in [7] is that at a small, finite thickness, these creases are slightly smoothed (i.e.",
"the curvature diverges as $t \\rightarrow 0$ at a narrow zone whose size vanishes at the same limit), such that the overall bending energy of the crease vanishes as $t\\rightarrow 0$ .",
"Such a “stress focusing\" mechanism has been instrumental in studies of crumpled sheets .",
"In order to identify the regime in the $(T,\\eta )$ plane in which the ribbon morphology is describable by this approach we must clarify the meaning of “tensile load\" on a purely inextensible (i.e.",
"strainless) ribbon.",
"For this purpose, we will use in this paragraph dimensional parameters (denoted by non-italicized fonts), introducing explicitly the ribbon width W and stretching modulus Y, which are taken to define, respectively, the units of length and stress throughout this paper.",
"The dimensional bending modulus is ${\\rm B} \\sim {\\rm Y W}^2 t^2$ .",
"(Recall that we defined $t$ , Eq.",
"(REF ) as the ratio between the ribbon thickness and its width).",
"A thin elastic ribbon of width W has two characteristic scales for stress exerted on the midplane (i.e.",
"force/length): The first one is just the stretching modulus Y, which is the product of Young's modulus of the material (E) and the ribbon thickness ($t{\\rm W}$ ); the second scale for stress is related to the bending modulus ${\\rm B}/{\\rm W}^2 \\sim {\\rm Y} t^2 $ .",
"Since both scales are proportional to Young's modulus E, it is impossible to assume a “perfectly inextensible\" ribbon (i.e.",
"${\\rm Y}=\\infty $ ) that is nevertheless bendable (i.e.",
"${\\rm B} <\\infty $ ).",
"Hence, attributing an “inextensibility\" feature to an elastic ribbon must be understood as assuming the asymptotic limit $t \\rightarrow 0$ , such that the exerted tensile load vanishes in comparison to ${\\rm Y}$ but not in comparison to ${\\rm B}/{\\rm W}^2 = {\\rm Y} t^2$ .",
"Returning to the parameter plane $(T,\\eta )$ , we may expect that for a given twist $\\eta $ , the family of creased helicoid states predicted by [7] exists at a parameter regime $ C_1 t^2 < T < C_2 t^2$ (where $C_{1,2}$ depend on $\\eta $ ).",
"In this regime, the exerted tension is sufficiently small with respect to the stress scale set by the stretching modulus Y, such that the state of the ribbon is nearly strainless and is thus close to an isometry of the untwisted ribbon; at the same time, the tension is sufficiently large in comparison to the other stress scale set by the bending modulus and the ribbon width, ${\\rm Y}t^2$ , such that the necessary conditions for constructing a creased helicoid state by the method of [7] are satisfied."
],
[
"The creased helicoid state as an asymptotic isometry:",
"The above paragraph indicates that the analysis of [7] addresses the stretched, twisted ribbon, in the vicinity of the hyper-plane ($T\\!=\\!0 , t\\!=\\!0$ ) in the 4D parameter space spanned by the dimensionless parameters $t,L,T$ and $\\eta $ (Eq.",
"REF ).",
"As we argued in Subsec.",
"REF , the ribbon mechanics in this regime reflects a competition between distinct types of asymptotic isometries – namely, between states whose elastic energy at a small neighborhood of that singular hyper-plane is described by Eq.",
"(REF ).",
"The creased helicoid of [7] is yet another example of an asymptotic isometry, and its energy in that limit can also be expressed, as a linear function of $T$ : the intercept, $T$ -independent term of the energy (Eq.",
"REF ), is governed by the bending energy of the creases at small finite $t$ ; the term that is linear in $T$ originates from the work done by the tensile load, where the prefactor is the $t$ -independent longitudinal contraction $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {CH}}$ of the ribbon in the creased helicoid state.",
"In order to carry out a quantitative comparison between the energies of the cylindrical wrapping, longitudinal wrinkling, and creased helicoid states, we must know the longitudinal contractions ($A_j$ ), as well as the exponents ($\\beta _j$ ) that characterize the bending energy of the states at the vicinity of the hyper-plane ($T\\!=\\!0 , t\\!=\\!0$ ).",
"While we do not have yet the complete set of those values for all three types of asymptotic isometries, the schematic plot in the inset of Fig.",
"REF b seems as a plausible scenario to us: At a given $\\eta $ , the bending energy of the cylindrical wrapping (i.e.",
"the intercept of the linear function in Fig.",
"REF b) is minimal, and therefore this state should be observed if the exerted tension $T$ is very small; upon increasing $T$ , the experimental observations of [3] indicate that a creased helicoid state is formed and then gives way to a longitudinally wrinkled state, suggesting that creased helicoids are characterized by lower bending energy and larger longitudinal contraction in comparison to the longitudinal wrinkles."
],
[
"Helicoid versus planar state: from boundary-dominated to twist-dominated",
"So far, we assumed that the highly-symmetric state of a stretched-twisted ribbon, characterized by translational symmetry along the longitudinal direction $\\hat{s}$ , is the helicoid.",
"We carried out stability analysis of the helicoidal state and studied the transitions to states that break its translational symmetry.",
"However, if the ribbon is sufficiently long ($L \\gg 1$ ) and the exerted twist $\\eta $ is sufficiently small with respect to $T$ , one may envision that the unbuckled state of the ribbon is not a helicoid but rather a stretched, planar ribbon, where the exerted twist remains confined to the vicinity of the short edges (see Fig.",
"REF ).",
"Such a localized-twist state can be described as a perturbation to the well-known problem of purely stretching an elongated ribbon, where $0<T \\ll 1$ and $\\eta = 0$ [8].",
"For that problem, it was found that the clamping of the short edges together with the Poisson ratio effect gives rise to contraction of the ribbon in the transverse direction not only in the vicinity of the clamped edges, but rather throughout most of the length of the ribbon (see [8]).",
"Recalling our discussion in previous sections, one may conclude that there exist two distinct causes for transverse buckling and wrinkling in a ribbon: The first one, which we described above (Sec.",
"), is a “bulk\" mechanism, where $\\sigma ^{rr}$ becomes compressive due to the simultaneous effect of uniaxial stretching along $\\hat{s}$ and uniform twist $\\eta $ , experienced by any piece of the ribbon.",
"The second mechanism is essentially a boundary-generated effect, whereby the tendency of the ribbon to transverse contraction ($u_r(1/2)=-\\nu T/2$ in Eq.",
"(REF )) away from the clamped edges gives rise to a small transverse compression $\\sigma ^{rr}<0$ , which is also relieved by wrinkles.",
"A natural question is whether, for a given set of parameters the emergence of a compressive transverse stress $\\sigma ^{rr}$ and the consequent buckling/wrinkling instability, are governed by the bulk effect (twist) or rather by the boundary effect (clamping).",
"Figure: Picture of the ribbon shape at very small twist, (a) Helicoidal state, given for comparison with (b) Boundary dominated state where the twist is confined in the vicinity of the short edges, the central part being flat.In order to address this question, we have to compare the compressive stresses ($\\sigma ^{rr} <0$ ) associated with twist and with clamping of the short edges.",
"However, while the first one was derived above (Eq.",
"), we are not aware of a similar analytic expression for the transverse stress due to clamped edges The well-known work of Cerda and Mahadevan [8] addressed this problem in the far-from-threshold regime, where wrinkles are fully developed and the transverse compression cannot be approximated by its value at threshold.",
"The planar (unbuckled) state that underlies the wrinkling instability due to clamped boundaries was studied numerically in some recent works , .",
"However, these works did not address the ribbon limit $L \\gg 1$ that we study here.",
"A couple of papers attempted to extend the far-from-threshold approach of [8] to the near-threshold regime (by invoking effective \"inextensibility\" constraints), but the justification of this approach has yet to be established..",
"Hence, we will proceed by estimating the relevant energies.",
"We will do this by denoting $U_0 = T^2$ the energy per length of a stretched ribbon that is not clamped and not twisted, and estimating the excess energies associated with twist and clamping, which we denote, respectively, by $\\Delta U_\\mathrm {twist}$ and $\\Delta U_\\mathrm {clamp}$ .",
"Our purpose is to find the curve $\\eta ^*(T)$ in the $(T,\\eta )$ plane, below which the clamped-edge effect is significant."
],
[
"The excess energy $\\Delta U_\\mathrm {twist}$ :",
"Expecting the transition from twist-dominated to clamping-dominated instability to occur at a small value of $\\eta $ , we neglect terms of order $O(\\eta ^4)$ in comparison to terms of order $O(\\eta ^2T$ ), and thus estimate $\\Delta U_\\mathrm {twist}$ by considering a stretched, twisted, unclamped ribbon: $\\Delta U_\\mathrm {twist} \\approx \\int _{-1/2}^{1/2} \\varepsilon _{ss}(r)^2dr -U_0 \\sim T\\eta ^2 \\ ,$ where we considered only the leading order in $\\eta ^2$ , and therefore neglected the energy due to the strain $\\varepsilon _{rr}^2$ ."
],
[
"The excess energy $\\Delta U_\\mathrm {clamp}$ :",
"Consider now a stretched, clamped, untwisted ribbon.",
"In Appendix we show that the deviation of the longitudinal strain $\\varepsilon _{ss}$ from the “base\" value $T$ is proportional to the Poisson ratio $\\nu $ and is restricted to distances $\\sim 1$ from the clamped edges, at which zone the transverse and shear strain components also have nonvanishing values that are proportional to $\\nu $ .",
"This allows us to estimate: $\\Delta U_\\mathrm {clamp} \\sim \\frac{\\nu F(\\nu ) T^2}{L} \\ ,$ where $F(\\nu )$ is some smooth function of $\\nu $ that satisfies $F(\\nu ) \\rightarrow \\mathrm {cst}$ for $\\nu \\rightarrow 0$ .",
"Comparing now our estimates for the excess energies $\\Delta U_\\mathrm {twist}$ and $\\Delta U_\\mathrm {clamp}$ , we find that the transition from the clamping-dominated zone to the twist-dominated zone is expected to occur around: $\\eta ^* \\sim \\sqrt{\\frac{\\nu T}{L}} \\ ,$ confirming our expectation that $\\eta ^*(T)$ approaches the $T$ axis for small Poisson ratio and large $L$ .",
"For $\\eta <\\eta ^*$ , we expect that a transverse buckling instability is triggered by the clamped boundaries, whereas for $\\eta > \\eta ^*$ we expect the instability mechanism described in the previous sections.",
"We have to compare the above expression with the threshold for the transverse instability found in Sec.",
", $\\eta _\\mathrm {tr} \\sim \\sqrt{t}T^{-1/4}$ .",
"We find that both values are comparable when $T=T_\\mathrm {clamp}\\sim \\left(\\frac{Lt}{\\nu }\\right)^{2/3} \\ .$ If $T>T_\\mathrm {clamp}$ , the tension is sufficiently large and the effect of clamping on the transverse instability cannot be neglected.",
"Above this critical tension, the transverse instability is governed by the clamped boundaries rather than the helicoid geometry Note that if $L\\sim 1$ , as was the case in Fig.",
"3 of [3], this equation means that the transverse wrinkling reflects the clamping-induced instability mechanism of a stretched sheet [8] rather than the helicoidal mechanism described in Sec.",
".."
],
[
"Discussion",
"Our theoretical study identified distinct types of morphologies in different regimes of the 4D parameter space spanned by $T,\\eta ,t$ and $L^{-1}$ .",
"These dimensionless parameters are assumed small (Eqs.",
"REF , REF ), and one may be tempted to describe the ribbon as “thin\" ($t \\ll 1$ ), “long\" ($L^{-1} \\ll 1$ ), subjected to “small\" tensile load ($T \\ll 1$ ), and “slightly\" twisted ($\\eta \\ll 1$ ).",
"However, our analysis highlights the deceptive nature of such a colloquial description, since the ribbon exhibits markedly different behaviors in different “corners\" of the 4D parameter space.",
"In other words, the relevant parameters that govern the ribbon morphology are various ratios between the four control parameters $T,\\eta , t$ and $L$ rather than the “bare\" value of each control parameter.",
"Table: Central predictions and open questions raised in our paper.",
"The bold letter P stands for “prediction\", whereas OQ stands for “open question\".In Table REF we summarize the central phenomena predicted in our paper, as well as a few open questions raised by our analysis at some of those corners of the parameter space.",
"The challenge for an experimenter, who may be motivated by the predictions in this paper, is to construct a set-up that allows access to those distinct regimes and precise measurements of observables which characterize the transitions between them.",
"In this section we will focus on this experimental perspective, propose specific measurements, and describe a few open questions that await further theoretical and experimental study."
],
[
"Objectives",
"Let us assume a ribbon with a fixed thickness and width, such that the parameter $t$ is fixed at a very small value (say, $t \\approx 10^{-5}$ ), and address the desired range of the other control parameters.",
"$\\bullet $ Controlling $T$: As Fig.",
"REF a depicts, the tension $T$ may vary between $T_\\mathrm {min}$ , which is determined by the quality of the set-up, and $T_\\mathrm {Hook} \\ll 1$ , above which the material response can no longer be approximated through Hookean elasticity.",
"Ideally, one would like $T_\\mathrm {min} \\ll T_\\mathrm {sm}(t)$ and $T_\\mathrm {Hook} \\gg T_{\\lambda }$ , where $T_\\mathrm {sm} \\sim t^2$ (Eq.",
"REF ) and $T_{\\lambda } \\sim \\max \\left\\lbrace {t,(t/L)^{2/3}}\\right\\rbrace $ (Eqs.",
"REF ,REF ).",
"As we describe in the next subsection, varying the tensile load in the range $(T_\\mathrm {sm}, T_{\\lambda })$ will allow probing most of the phenomena associated with the nature of the longitudinally-wrinkled pattern (Subsec.",
"REF -REF ), Fig.",
"REF d), and the mechanism by which it becomes unstable at sufficiently low $T$ and given $\\eta $ (Subsec.",
"REF , Fig.",
"REF a); varying the tensile load in the range $(T_{\\lambda },T_\\mathrm {max})$ is necessary to understand the effect of $T$ on the transverse instability (Sec. ).",
"Figure: Ranges of tension and length probed by the experiments , .",
"(a) Tension: the Green's plateau for η lon (T)\\eta _\\mathrm {lon}(T) pictured in Fig.",
"c is expected for T≪T sm T\\ll T_\\mathrm {sm}, longitudinal wrinkling (near threshold and far from threshold) is expected for T sm <T<T λ T_\\mathrm {sm}<T<T_\\lambda , and transverse buckling/wrinkling becomes a primary instability of the helicoid for T>T λ T>T_\\lambda .The upper limit T Hook T_\\mathrm {Hook} is the limit of linear Hookean response of the material.",
"(b) Length: (in depicting this figure we assume L clamp =νt -1 T 3/2 ≪t -1/2 L_\\mathrm {clamp} = \\nu t^{-1} T^{3/2} \\ll t^{-1/2}, see Eq. ()).",
"For lengths L>t -1 L>t^{-1} the primary transverse instability is buckling; For L<L clamp L < L_\\mathrm {clamp} the transverse instability is governed by the clamping of the short edges, similarly to ; For L clamp ≪L≪t -1/2 L_\\mathrm {clamp} \\ll L \\ll t^{-1/2}, the transverse instability is wrinkling; finally for lengths in the range t -1/2 ≪L≪t -1 t^{-1/2} \\ll L \\ll t^{-1} we predict a crossover from buckling to wrinkling as the exerted tension TT varies from T λ T_\\lambda to T Hook T_\\mathrm {Hook} (see Fig.",
"(Left)).$\\bullet $ Controlling $L$: As Fig.",
"REF b depicts, a desired set-up should allow variation of the ribbon length $L$ from $L_\\mathrm {min}$ to $L_\\mathrm {max}$ , where $1 < L_\\mathrm {min} \\ll t^{-1/2}$ and $L_\\mathrm {max} \\gg t^{-1}$ .",
"Varying $L$ in such a range will provide an access to most of the predictions associated with the transverse instability: the wrinkling-buckling transition (Sec.",
"REF , Fig.",
"REF b), the scaling law of the triple point (Eqs.",
"REF ,REF ), and the possible existence of a localized transverse buckling mode (even for a very long ribbon), which we discuss in the next subsection.",
"$\\bullet $ Controlling $\\eta $: A good experimental set-up may allow a nearly-continuous variation of the imposed twist angle $\\theta = \\eta L$ .",
"For instance, if $\\theta $ is varied by increments of $1^\\mathrm {o}$ then the minimal twist that could be imposed is $\\eta _\\mathrm {min} \\approx 2\\pi / (360 \\ L)$ .",
"A reliable control on $\\eta _\\mathrm {min}$ is required for two purposes: In the very low tension regime ($T <T_\\mathrm {sm}(t)$ ), it may allow to address Green's “plateau\" of the longitudinal wrinkling instability $\\eta _\\mathrm {lon}(T) \\rightarrow 0.2\\, t$ (Sec.",
"REF , Fig.",
"REF c); For larger values of exerted tension, it may be necessary to probe the predicted transition from a planar state with twist confined to the short edges to a helicoidal shape (Subsec.",
"REF , Fig.",
"REF (e)).",
"Table: Typical experimental parameters, used by Green and Chopin and Kudrolli , and the corresponding ratios that are relevant for our analysis."
],
[
"Challenges",
"We are aware of two documented experiments that addressed the behavior of a stretched-twisted elastic ribbon: Green's experiment from 1937 [2], where ribbons were made of steel; and [3], which used Mylar.",
"In Tab.",
"REF , we compare the control parameters and their relevant mutual ratios in both experiments.",
"Green, who used a material with very large Young's modulus, could address the “ultra-low\" tension regime, $T \\sim T_\\mathrm {sm}(t)$ (Fig.",
"REF c), but a simple steel may exhibit a non-Hookean (or even inelastic) response at rather small $T$ , which limits its usage for addressing the regime around and above the triple point (i.e.",
"$T > T_{\\lambda }$ ).",
"In contrast, the experiment of [3] used a material with much lower Young's modulus, which allows investigation of the ribbon patterns in Fig.",
"REF (g), but the minimal exerted tension $T_\\mathrm {min}$ (associated with the experimental set-up) was not sufficiently small to probe Green's threshold plateau $\\eta _\\mathrm {lon}(T)\\rightarrow 0.2 t$ for $T \\lesssim T_\\mathrm {sm}(t)$ .",
"This comparison reveals the basic difficulty in building a single set-up that exhibits clearly the whole plethora of shapes shown in Fig.",
"REF .",
"In addition to the effect of $T_\\mathrm {min}$ and $T_\\mathrm {Hook}$ , there is an obvious restriction on $L_\\mathrm {max}$ (at most few meters in a typical laboratory).",
"Below we propose a couple of other materials, whose study – through experiment and numerical simulations – may enable a broader range of the ratios $T_\\mathrm {min}/T_\\mathrm {sm}$ , $T_\\mathrm {Hook}/T_\\lambda $ and $L_\\mathrm {max}t$ .",
"Graphene: This novel 2D material is characterized by ${\\rm t} \\sim 0.3$ nm (which we assume to approximate the “mechanical thickness\", i.e.",
"the ratio $\\sqrt{{\\rm B}/{\\rm Y}}$ ), Young's modulus ${\\rm E} = 10^3$ GPa and a yield stress of $\\sim 100$ GPa , whose ratio ($\\approx 0.1$ ) we use as an approximation of $T_\\mathrm {Hook}$ .",
"Graphene sheets can be produced with lateral scales of up to 1 mm .",
"Assuming a graphene ribbon with length 1 mm and width $0.03$ mm, we obtain $t \\approx 10^{-5}$ and $L \\approx 30$ .",
"A narrower ribbon may allow exploring a larger range of $L$ , at the expense of smaller $t$ .",
"Tensile load may be exerted on graphene by optical tweezers, which allow a force (on the short edge) in the picoNewton range, such that: $T_\\mathrm {min}/T_\\mathrm {sm} \\sim 1$ , and $T_\\mathrm {Hook}/T_{\\lambda } \\sim 10^{3}$ .",
"Ultrathin PS films: Polymer films with thickness of $30-300$ nm can be fabricated by spin coating and have been used extensively in studies of wrinkling and other elastic phenomena , .",
"Such sheets are characterized by ${\\rm E} = 3.4$ GPa, and their lateral scales may be few cm's.",
"It may thus be possible, for instance, to create ultrathin PS ribbons with $L \\approx 10^2$ and $t \\approx 10^{-4}$ .",
"Capillary forces have been used to exert tension on floating ultrathin PS sheets, where the surface tension varies from a maximum of $\\approx 70$ mN/m to 3 times lesser than this value (by using surfactants).",
"This corresponds to $T_\\mathrm {min}/T_\\mathrm {sm} \\sim 10^{4}$ , $T_\\mathrm {Hook}/T_{\\lambda } \\sim 20$ (for $t\\approx 300$ nm).",
"Thus, making ribbons from graphene or ultrathin PS films may allow a broad range of the three most relevant ratios (right columns of Tab.",
"REF ) that are necessary to explore the various asymptotic regimes of the system.",
"We recall though, that both graphene and ultrathin polymer sheets are rarely used in a free-standing form, which is the one needed for the stretch-twist experiment that we address here.",
"Let us now discuss the specific parameter regimes and the corresponding morphological instabilities addressed by our theory.",
"In each of the following paragraphs we will propose measurements and mention open theoretical questions."
],
[
"Longitudinal wrinkling at $T_\\mathrm {sm}<T<T_{\\lambda }$",
"The parameter regime $T_\\mathrm {sm} < T < T_{\\lambda }$ ($T_\\mathrm {sm}$ is defined in Eq.",
"REF ) was addressed in the experiment of Chopin and Kudrolli [3], who found that the threshold $\\eta _\\mathrm {lon}(T)$ becomes close to the curve $\\sqrt{24 T}$ at which the Green's stress predicts longitudinal compression.",
"In this regime, our discussion in Sec.",
"predicts the emergence of the FT regime rather close to the threshold curve.",
"Chopin and Kudrolli [3] used ribbons with $t = 5 \\cdot 10^{-3}$ and confirmed that the threshold $\\eta _\\mathrm {lon}(T)$ is close to the curve $\\sqrt{24T}$ , at which the Green's longitudinal stress becomes compressive.",
"Additionally, the dependence of $\\Delta \\alpha _\\mathrm {lon}$ and $\\lambda _\\mathrm {lon}$ on $t$ and $T$ (Eq.",
"REF ) was in excellent agreement with the prediction of the NT approach [5].",
"Such a value of $t$ , however, may be too large to probe the transition from the NT to the FT regime predicted in Sec.",
", since the initial width of the wrinkled zone ($r_\\mathrm {wr} \\sim \\lambda _\\mathrm {lon}$ ) is already a relatively large fraction of the ribbon width.",
"Using thinner ribbons (e.g.",
"$t \\sim 10^{-5}$ ) may lead to substantially smaller value of the initial width of the wrinkled zone, such that the different predictions of the NT and FT methods for $r_\\mathrm {wr}(\\alpha )$ (Fig.",
"REF ) may be more pronounced.",
"A useful probe for testing the predicted NT-FT transition may involve the longitudinal contraction $\\chi (\\alpha )$ (Eqs.",
"REF , REF , Fig.",
"REF ).",
"The function $\\chi (\\alpha )$ may be easier to measure in comparison to the width $r_\\mathrm {wr}(\\alpha )$ of the wrinkled zone, which requires the usage of optical tools.",
"Notably, the NT and FT predictions for the longitudinal contraction are significantly different, wherein the last one becomes 3 times smaller than the first for sufficiently large $\\alpha $ ."
],
[
"Theory:",
"Our FT analysis in Subsec.",
"REF has been focused on the dominant part of the elastic energy of the longitudinally wrinkled state, stored in the asymptotic, compression-free stress field.",
"The dominant energy is associated with macroscale features, which do not depend explicitly on $t$ , and underlies the predictions for $r_\\mathrm {wr}$ and $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}$ .",
"However, as we emphasized in Subsec.",
"REF , a complete characterization of the wrinkle pattern requires evaluation of the subdominant energy, which includes the bending cost due to wrinkles, as well as the comparable cost due to the formation of a wrinkled structure on the stretched helicoidal shape.",
"The calculation of the subdominant energy may involve some subtleties, such as internal boundary layers and the possible formation of wrinkle cascades rather than a simply periodic structure .",
"Importantly, the actual subdominant mechanics that govern the wrinkle wavelength may depend on the confinement parameter $\\alpha $ and therefore the exponent $\\beta $ that characterizes the subdominant energy (Eq.",
"REF ) may take different values in the limit $\\alpha \\rightarrow 24$ (where the helicoidal shape is highly strained) and the limit $\\alpha \\rightarrow \\infty $ , at which the longitudinally-wrinkled helicoid becomes an asymptotic isometry of the ribbon.",
"Therefore, evaluation of the subdominant energy is essential not only for finding the fine structure of the wrinkle pattern, but also to understand how it becomes unstable with respect to the creased helicoid shape as $\\alpha $ becomes large (i.e.",
"the limit $T\\rightarrow 0$ for fixed $\\eta $ ).",
"Green's theory [2] consists of a linear stability analysis of the helicoidal state (Eqs.",
"REF ,REF -) in the limit $T \\rightarrow 0$ , where the longitudinally buckled/wrinkled zone is not confined to the vicinity of the centerline, but rather expands throughout the whole width of the ribbon.",
"This (NT) analysis yields the threshold plateau $\\eta _\\mathrm {lon}(T) \\rightarrow 0.2\\,t$ as $T \\rightarrow 0$ , which was obtained (with deviation of $10\\%$ ) in Green's experiment [2].",
"Experiment: The experimental data in Green's paper [2] does show a good agreement with the theoretical prediction based on his linear stability analysis.",
"However, the available data (figures 4 and 5 of [2]) may not be sufficient to determine whether the actual instability observed by Green was a longitudinal wrinkling or a creased helicoid state.",
"This confusion is illustrated in Fig.",
"REF c, which reflects our expectation that a creased helicoid pattern (or even a cylindrical wrapping state) may be observed sufficiently close to the vertical line $T=0$ .",
"The possible emergence of a creased helicoid state directly from the helicoidal state (i.e.",
"without an intervening wrinkle pattern) may indicate that the longitudinal instability changes its supercritical (continuous) character, becoming a subcritical bifurcation at sufficiently small $T$ .",
"A careful experiment may provide a conclusive answer to this question.",
"Theory: The confusing nature of the longitudinal instability at the regime $T<T_\\mathrm {sm}$ , depicted in Fig.",
"REF c, is reflected also in the fuzziness of the NT-FT transition in this regime.",
"The FT approach (Subsec.",
"REF ) assumes that the wrinkle pattern near threshold is confined to a strip of width $r_\\mathrm {wr} <1/2$ around the centerline, and describes how $r_\\mathrm {wr}$ varies upon increasing the confinement $\\alpha $ .",
"However, for $T<T_\\mathrm {sm}$ , Green's analysis [2] shows that the ribbon is deformed across its whole width as soon as the longitudinal instability sets in.",
"A natural question is whether the FT regime of the longitudinally wrinkled helicoid terminates at a small $T \\sim T_\\mathrm {sm}$ .",
"A related question is whether a disappearance of the NT-FT transition below a certain tensile load indicates a qualitative change of the longitudinal instability from a supercritical to a subcritical type."
],
[
"From longitudinal wrinkling to the creased helicoid state",
"For a given twist $\\eta $ and sufficiently small tensile load $T$ , we expect the formation of a creased helicoid state (Fig.",
"REF ), predicted in [7] and observed in [3].",
"Our energetic argument, based on the asymptotic isometry equation, Eq.",
"(REF ), suggests the existence of a transition between the creased helicoid state and the FT-longitudinally-wrinkled state, which becomes sharply localized along a curve in the parameter plane ($T,\\eta $ ) in the limit $t \\rightarrow 0$ , as is illustrated in Fig.",
"REF a.",
"Experiment: Chopin and Kudrolli [3] did observe such a transition, but noted that “ ... the longitudinally buckled ribbon evolves continuously into a self-creased helicoid ... \".",
"The appearance of a smooth crossover, rather than a sharp transition between these states, could be attributed to the thickness parameter used in their experiment ($t \\approx 5 \\cdot 10^{-3}$ ), which may be sufficiently small to notice the “wake\" of a morphological phase transition that becomes asymptotically sharp as $t \\rightarrow 0$ , but not the acute, critical nature of the transition.",
"Future experiments may have to use significantly smaller values of $t$ in order to study this transition.",
"A useful indirect probe of the transition from a longitudinal wrinkling to a creased helicoid state may again be the longitudinal contraction of the ribbon $\\chi (T,\\eta )$ .",
"The FT approach (Subsec.",
"REF ) predicts the $T$ -dependence through the function $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}(\\alpha )$ (Fig.",
"REF ) and we expect that the method of [7] yields another function $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {CH}}(T,\\eta )$ .",
"A signature of a sharp morphological transition between the two states may be a discontinuity of the measured derivative $(\\partial \\chi / \\partial T)_{\\eta }$ at a curve $T_\\mathrm {c}(\\eta )$ in the $(T,\\eta )$ plane.",
"Theory: As we pointed out in Subsec.",
"REF , our schematic plot of the energies of the longitudinal wrinkling and the creased helicoid states (Fig.",
"REF b) reflects their asymptotic isometry in the vicinity of the hyper-plane $(T\\!=\\!0,t\\!=\\!0)$ , but has a heuristic content, since the coefficients of those linear functions are yet unknown.",
"The FT analysis of the longitudinally wrinkled state still lacks an exact evaluation of the subdominant energy, which underlies the intercept value in the corresponding linear function (red line in Fig.",
"REF b); The corresponding plot for the creased helicoid state (dashed brown in inset) lacks the values of both the slope and the intercept.",
"The slope of that line is simply the longitudinal contraction of the state, and can be computed from the construction of Korte et al.",
"[7], which describes the ribbon through the Sadowsky equation of a strainless strip.",
"The intercept of that linear function, however, cannot be computed through this framework, since the subdominant energy of the creased helicoid state stems from bending energy associated with broadening the creases into narrow zones in which the ribbon cannot be considered as strainless.",
"Thus, a reliable evaluation of the subdominant energy of the creased helicoid state may require one to use the FvK framework, where the ribbon – at least in the vicinity of the creases – is allowed to have strain.",
"It is possible that familiar concepts, such as the “minimal ridge\" can be invoked in order to approximate the subdominant energy of the creased helicoid state.",
"As we briefly described in Subsec.",
"REF , the creased helicoid shape characterizes in fact a family of states, parameterized by the angle between triangular facets.",
"Therefore, a more realistic picture of the inset to Fig.",
"REF b may consist of a series of lines that represent this family, rather than the single brown dashed line.",
"The slope and intercept of each of those lines would result, respectively, from the longitudinal contraction and the subdominant energy of each creased helicoid state.",
"Therefore, upon increasing the tensile load, a ribbon subjected to twist $\\eta $ may undergo a series of transitions between creased helicoid states before the transition to the wrinkled helicoid."
],
[
"The transverse instability: threshold",
"Our theoretical analysis of the transverse instability (Sec. )",
"predicts that the threshold curve $\\eta _\\mathrm {tr}(T)$ in the $(T,\\eta )$ plane depends on the ribbon thickness $t$ and the tension $T$ (Eq.",
"REF , Figs.",
"REF , REF ).",
"Furthermore, if the ribbon is not “infinitely\" long, such that $L \\ll t^{-1}$ , the threshold $\\eta _\\mathrm {tr}(T)$ depends also on $L$ (Fig.",
"REF ) and the nature of the transverse instability changes from buckling to wrinkling (i.e.",
"wavelength $\\lambda _\\mathrm {tr} <W$ ).",
"Experiment: For a ribbon (where $L \\gg 1$ ), Chopin and Kudrolli [3] observed only a buckling instability ($\\lambda _\\mathrm {tr}>W$ ), and reported that $\\eta _\\mathrm {tr}(T)$ (for $T > T_{\\lambda }$ ) has a plateau value, which scales as $\\sim \\sqrt{t}$ but does not depend on the tension $T$ , nor on the ribbon length $L$ .",
"The appearance of a buckling mode is consistent with the relatively long ribbon used in comparison to $t^{-1}$ [3] (see Fig.",
"REF ), but the independence of $\\eta _\\mathrm {tr}$ on $T$ disagrees with our result.",
"It is quite possible though, that the dependence on tension, $\\eta _\\mathrm {tr} \\sim T^{-1/2}$ , has simply been overlooked in those measurements.",
"Future experiments that will examine our predictions will have to use an appropriate range of ribbon lengths, depicted in Fig.",
"REF .",
"Theory: Our predictions concerning the effect of the ribbon length on the transverse instability (Subsec.",
"REF ,REF ) assume that $L$ enters through a single term in the normal force balance ($\\sigma ^{ss}/L^2)$ , which couples the longitudinal stress to the unavoidable, wrinkle-induced curvature in that direction.",
"A complete analysis should include other terms, accounting for example for the strain induced by the longitudinal variation of the longitudinal in-plane displacement.",
"These terms may affect the exact numerical values of our predictions (e.g.",
"the location of the threshold $\\eta _\\mathrm {tr}(T)$ in Fig.",
"REF ), but are unlikely to affect any of the scaling laws.",
"A nontrivial assumption in our linear stability analysis of the transverse instability is encapsulated in the form of the unstable mode (Eq.",
"REF in Sec.",
"REF ).",
"If the ribbon was truly infinitely long, then the translational symmetry of the stress in the longitudinal direction is naturally broken through such Fourier modes, and the one which requires the least curvature (i.e.",
"$j=1$ in Eq.",
"REF ) is the first to become unstable under given twist and tensile loads.",
"The translational symmetry of the helicoidal shape is not perfect, however, due to the boundary conditions at the short edges The exact boundary conditions at $s=\\pm L/2$ may depend on the specific set-up used to apply simultaneously tension and twist.",
"These may include, for instance, complete clamping (i.e.",
"$u=z= 0$ ) or partial clamping (i.e.",
"only $z = 0$ )..",
"It is natural to expect that the unstable mode “feels\" these boundary conditions at a small region near the short edges whose size is comparable to the ribbon width.",
"However, recent studies have shown that the boundary shape may have a long-range effect on the deformation of a thin sheet, with penetration length that diverges as $t \\rightarrow 0$ , .",
"If the boundary conditions at $s = \\pm L/2$ have such a long-range effect, the longitudinal structure of the unstable mode may exhibit strong deviation from the sinusoidal shape ($z_1 \\sim \\cos (\\pi s/L)$ ) even if the ribbon is very long, as long as $L < t^{-x}$ with some $x>0$ .",
"In order to address this question, one may have to carry out the transverse stability analysis, taking into full consideration the boundary conditions at $s = \\pm L/2$ , and finding the unstable mode through numerical analysis of the linear partial differential equation (i.e.",
"for $z_1(s,r)$ and $u_{s1}(s,r)$ )."
],
[
"The transverse instability: beyond threshold",
"Our analysis in Sec.",
"identified the threshold curve $\\eta _\\mathrm {tr}(T)$ and characterized the nature of the unstable modes through linear stability analysis, but this approach cannot clarify the spatial structure of the ribbon above that threshold curve.",
"The experiment of [3] found that the ribbon forms loops and self-contact zones very close to the threshold curve $\\eta _\\mathrm {tr}(T)$ , and furthermore – a strong hysteretic behavior has been observed, especially in the low tension regime ($T <T_{\\lambda }$ ) in which the transverse instability emerges as a secondary instability of the helicoidal state.",
"The emergence of a transverse instability of twisted ribbons as a precursor to the formation of loops and coils has been recognized in recent numerical studies , .",
"While it may be possible to address the formation of loops and hysteresis phenomena through the cFvK equations, an effective theory that describes the ribbon through its centerline $X_\\mathrm {cl}(s)$ may provide deeper insight into this mechanics.",
"Such an approach may be similar in spirit to the Sadowsky strip or Kirchoff rod equations (Subsec.",
"REF ), but the effective equations that govern the mechanics of a stretched-twisted ribbon above the threshold curve $\\eta _\\mathrm {tr}(T)$ are likely to be markedly different from each of these approaches."
],
[
"Summary",
"The central result of our paper is illustrated in Fig.",
"REF : A phase diagram that describes the distinct morphologies of a ribbon in the parameter plane spanned by the exerted tension $T$ and twist $\\eta $ .",
"The separation of the ($T,\\eta $ ) plane into three major parts (blue, orange, pink) that meet at a single triple point ($T_{\\lambda },\\eta _{\\lambda }$ ) has been recognized by Chopin and Kudrolli [3], who attributed this peculiar property to the presence of three operative instability mechanisms.",
"Our study reveals that this phenomenology is rooted at two basic instabilities only, whereby the ribbon responds by wrinkling/buckling to the compressive stresses in the longitudinal and transverse directions.",
"The three major morphological phases correspond to a highly symmetric helicoidal state (blue), and two states that break this symmetry through instabilities that deform the shape in the longitudinal direction (orange), in the transverse direction (pink, at $T> T_{\\lambda }$ ), or in both principal directions (pink, at $T <T_{\\lambda }$ ).",
"This insight is borne out by bringing together two theoretical elements: The cFvK equations that capture the transverse stress due to the non-planar, helicoidal shape of the twisted ribbon; and a far-from-threshold (FT) analysis of these equations, that describes the collapse of longitudinal compression enabled by the formation of wrinkles.",
"The far-from-threshold analysis of the cFvK equations revealed a profound feature of the wrinkling instability: assuming a fixed twist $\\eta $ , and reducing the exerted tension (along a horizontal line in Fig.",
"REF ), the formation of longitudinal wrinkles that decorate the helicoidal shape enables a continuous, gradual relaxation of the elastic stress from the strained helicoidal shape at $T\\!",
">\\!",
"\\eta ^2/24$ , to an asymptotically strainless state at $T \\!\\rightarrow \\!",
"0$ .",
"This remarkable feature led us to propose a general form of the asymptotic isometry equation (REF ), which characterizes the wrinkled state of the ribbon (Fig.",
"REF b,c), as well as other admissible states at the limit $T \\rightarrow 0$ , such as the cylindrical wrapping (Fig.",
"REF e) and the creased helicoid state (Fig.",
"REF d).",
"The asymptotic isometry equation provides a simple framework, in which the transitions between those morphologies in the vicinity of the vertical line ($T=0$ in Fig.",
"REF ) correspond to the intersection points between linear functions of $T$ (Fig.",
"REF b), whose intercepts and slopes are determined solely by the geometry of each state.",
"Beyond its role for the mechanics and morphological instabilities of ribbons, the asymptotic isometry equation may provide a valuable tool for studying the energetically favorable configurations of elastic sheets.",
"Notably, Eq.",
"(REF ) takes into consideration the deviations of the sheet's midplane from a perfectly strainless shape, not only due to the small thickness of the sheet but also due to a small tensile load.",
"In the context of conventional elastic sheets, whose stress-free state is planar, such as the twisted ribbon or an adhesive sheet attached to a curved substrate , the tension $T$ in Eq.",
"(REF ) can easily be recognized as the tensile load exerted on the boundary of the sheet.",
"The asymptotic isometry equation may be useful, however, also in studies of “non-Euclidean\" sheets, whose stress-free state is determined by a “target metric\", programmed by differential swelling techniques or other means , .",
"For such non-Euclidean sheets, the tension $T$ in Eq.",
"(REF ) may originate from a less direct source, such as imperfections in the prescribed metric.",
"A puzzling experimental result in this emerging field has been the surprising wrinkled shape adopted by a sheet whose target metric was prescribed to be compatible with a hyperbolic shape (constant negative $G$ ) .",
"It has been noted in that such a wrinkling pattern is consistent with an asymptotic isometry, whose bending energy, however, is higher than the simple hyperbolic shape.",
"These two isometries, may be analogous, respectively, to the cylindrical wrapping state and the longitudinally wrinkled state of the twisted ribbon, whose energetic degeneracy is lifted not only by the thickness $t$ but also by a tensile load $T$ (Fig.",
"REF b).",
"The presence of a tension-like term in the corresponding asymptotic isometry equation that describe the energy of such a non-Euclidean sheet, may clarify the experimental conditions under which the hyperbolic shape may be observed."
],
[
"Acknowledgements",
"The authors would like to thank C. Santangelo for many enlightening discussions and particularly for educating us on the covariant FvK equations; and to A. Romaguera, F. Brau, B. Audoly, and two anonymous referees, for their critical reading and useful comments on the manuscript.",
"B.D.",
"would like to thank E. Hohlfeld for many inspiring discussions on asymptotic isometries and their use for elastic sheets subjected to geometric constraints and tensile loads.",
"The authors acknowledge financial support by CNPq-Ciência sem fronteiras program, Brazil (J.C.), the KECK foundation Award 37086 (V.D.",
"), and NSF CAREER Award DMR-11-51780 (B.D.",
")."
],
[
"Hookean elasticity and leading order stresses and strains",
"Our theory addresses the “corner\" in the 4d parameter space, defined by Eqs.",
"(REF ,REF ), and therefore most of the analysis in this paper employs expansions in these parameters.",
"Why do we assume these parameters are small?",
"First, $t \\ll 1$ and $L^{-1} \\ll 1$ stem from the definition of a ribbon.",
"Second, we focus our discussion on the universal, material-independent behavior of elastic ribbons and therefore we consider a Hookean response, whereby the stress-strain relationship is linear.",
"Since Hookean response is valid only for small strains, and since the exerted tension $T$ necessarily induces strain, we must require $T \\ll 1$ .",
"Finally, the assumption $\\eta \\ll 1$ is more subtle.",
"For the unwrinkled helicoidal state, we showed in Sec.",
"REF that the components of the strain and stress tensors are proportional to positive powers of $\\eta $ , and therefore Hookean response is valid only for sufficiently small values of $\\eta $ .",
"In contrast, for the longitudinally-wrinkled helicoidal state (i.e.",
"$\\eta > \\sqrt{24 T}$ ), we showed in Sec.",
"REF that the ribbon may become nearly strainless (i.e.",
"asymptotically isometric to the undeformed ribbon) even under finite $\\eta $ , therefore the Hookean response for the wrinkled state is not limited to small values of $\\eta $ .",
"However, even for the wrinkled state the assumption $\\eta \\ll 1$ is very useful, since it enables an easy way to compute the various components of the stress tensor and allows us to characterize the wrinkled state as a sinusoidal undulation (Eq.",
"(REF ) and Appendix ).",
"Importantly, our finding that the threshold values $\\eta _\\mathrm {lon},\\eta _\\mathrm {tr}$ vanish in the asymptotic limit $t \\rightarrow 0$ (see Sec.",
"REF and Fig.",
"REF ), proves in a self-consistent manner that the basic morhoplogical instabilities of a stretched-twisted ribbon are well described by assuming $\\eta \\ll 1$ .",
"Our theory is thus valid at the leading order in $t,L^{-1}$ , $\\eta $ and $T$ , and any higher order terms are ruled out from the derivations.",
"Leading terms should be understood with respect to these expansion parameters.",
"Namely, denoting by $A$ a scalar, or a component of a vector or tensor (e.g.",
"longitudinal contraction, transverse or longitudinal stress or strain), then $A$ is expanded as $ A=\\sum _{a_1,a_2,a_3,a_4\\ge 0} T^{a_1}\\eta ^{a_2} t^{a_3} L^{-a_4}A_{(a_1,a_2,a_3,a_4)},$ and the order of $A$ is given by the four positive integers $(a_i)$ .",
"Saying that the order $(a_i^{\\prime })$ is higher than the order $(a_i)$ means that $(a_i)<(a_i^{\\prime }) \\Longleftrightarrow \\left\\lbrace \\begin{array}{l} \\forall i, a_i\\le a_i^{\\prime }\\\\\\text{and}\\\\\\exists i \\text{ such that } a_i<a_i^{\\prime }.\\end{array}\\right.$ The leading terms of $A$ are the minimal orders for the relation (REF ) with non-zero coefficient.",
"For example, the leading terms of the longitudinal stress in the helicoidal state are given in Eq.",
"(REF ), $\\sigma ^{ss}_\\mathrm {hel}=T+\\frac{\\eta ^2}{2}\\left(r^2-\\frac{1}{12} \\right)$ , they correspond to the orders $(1,0,0,0)$ and $(0,2,0,0)$ .",
"These two orders are minimal and cannot be compared.",
"The transverse stress given in Eq.",
"() has vanishing coefficients for these orders, and the minimal orders with non zero coefficients are $(1,2,0,0)$ and $(0,4,0,0)$ (i.e.",
"$T\\eta ^2$ and $\\eta ^4$ ).",
"The $s$ -independent transverse buckling equation (REF ) contains terms of order $(1,2,0,0)$ , $(0,4,0,0)$ and $(0,0,2,0)$ (respectively, $T\\eta ^2$ , $\\eta ^4$ and $t^2$ ).",
"In a given equation, several orders may appear; in this case only the minimal ones should be considered.",
"This happens in the computation of the stress in the helicoidal state.",
"Eq.",
"() shows that $\\sigma ^{rr}$ is of higher order than $\\sigma ^{ss}$ ; besides, one of the two terms in the r.h.s of Eq.",
"() has obviously the same order $(0,2,0,0)$ as $\\sigma ^{ss}$ in Eq.",
"(REF ).",
"Thus, consistency of Eqs.",
"(,REF ,) implies that the leading terms in the r.h.s mutually cancel each other (which we call a “solvability condition\"), leading to Eq.",
"(REF ) for $u_r^{\\prime }(r)$ .",
"This last equation allows to compute the longitudinal stress, Eq.",
"(REF ), from which we deduce the transverse stress, Eq.",
"(), using again Eq.",
"()."
],
[
"Covariant derivative: definition and application to the helicoid",
"For an arbitrary surface with metric $g_{\\alpha \\beta }$ , the covariant derivative is defined with the Christoffel symbols, that are given by $\\Gamma ^\\alpha _{\\beta \\gamma }=\\frac{1}{2}g^{\\alpha \\delta } \\left(\\partial _\\beta g_{\\gamma \\delta }+\\partial _\\gamma g_{\\beta \\delta }-\\partial _\\delta g_{\\beta \\gamma } \\right).$ The covariant derivative of a vector $u^\\alpha $ is then defined as $D_\\alpha u^\\beta =\\partial _\\alpha u^\\beta +\\Gamma ^\\beta _{\\alpha \\gamma }u^{\\gamma }.$ The strain tensor has two indices, so that its covariant derivative is $D_\\alpha \\sigma ^{\\beta \\gamma }=\\partial _\\alpha \\sigma ^{\\beta \\gamma }+\\Gamma ^\\beta _{\\alpha \\delta }\\sigma ^{\\delta \\gamma }+\\Gamma ^\\gamma _{\\alpha \\delta }\\sigma ^{\\beta \\delta }.$ For the helicoid with metric (REF ) $g_{\\alpha \\beta }=\\begin{pmatrix} 1+\\eta ^2 r^2 + 2\\chi + 2\\eta ^2 r u_r(r) & 0 \\\\ 0 & 1+2 u_r^{\\prime }(r)\\end{pmatrix},$ the non-zero Christoffel symbols are (to the leading order): $\\Gamma ^r_{ss} & = -\\eta ^2 r,\\\\\\Gamma ^s_{sr} & = \\eta ^2 r,\\\\\\Gamma ^r_{rr} & = u_r^{\\prime \\prime }(r).$"
],
[
"Shape of the longitudinally wrinkled helicoid far from threshold",
"Far from threshold, longitudinal wrinkles relax the longitudinal compression.",
"In the main text, we propose the following form for the wrinkles: $ X^\\mathrm {(wr)}(s,r)=\\begin{pmatrix}\\left(1-\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}\\right)s \\\\r\\cos (\\eta s) - f(r)\\cos (ks)\\sin (\\eta s)\\\\r\\sin (\\eta s) + f(r)\\cos (ks)\\cos (\\eta s)\\end{pmatrix},$ where the longitudinal contraction is given by $\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}= \\frac{1}{2} \\eta ^2 r_\\mathrm {wr}^2$ .",
"The longitudinal strain in this configuration is $\\varepsilon _{ss}(s,r)=\\frac{\\eta ^2}{2}\\left(r^2-r_\\mathrm {wr}^2 \\right)+\\frac{1}{4}k^2f(r)^2-\\eta k rf(r)\\sin (ks)-\\frac{1}{4}k^2f(r)^2\\cos (2ks).$ Setting $k^2f(r)^2=2\\eta ^2 \\left(r^2-r_\\mathrm {wr}^2 \\right)$ (Eq.",
"REF ) allows to cancel the $s$ -independent part.",
"However, since this equation implies that the product $kf$ does not vanish in the limit $t \\rightarrow 0$ , we find that the $s$ -dependent terms in the above expression for $\\varepsilon _{ss}$ remain finite as $t \\rightarrow 0$ , in apparent contradiction to our assumption that the wrinkled state becomes asymptotically strainless in the limit $t,T \\rightarrow 0$ .",
"This shortcoming can be fixed, however, by adding to the deformation (REF ) a longitudinal displacement term, $u_s(s,r)$ , $X^\\mathrm {(wr)}(s,r)=\\begin{pmatrix}\\left(1-\\raisebox {-2pt}{\\raisebox {2pt}{\\chi }_\\mathrm {FT}}\\right)s + u_s(s,r) \\\\r\\cos (\\eta s) - f(r)\\cos (ks)\\sin (\\eta s)\\\\r\\sin (\\eta s) + f(r)\\cos (ks)\\cos (\\eta s)\\end{pmatrix},$ leading to the longitudinal strain $\\varepsilon _{ss}(s,r)=-\\eta k r f(r)\\sin (ks)-\\frac{1}{4}k^2f(r)^2\\cos (2ks)+\\partial _s u_s(s,r).$ Setting $u_s(s,r)=-\\eta rf(r)\\cos (ks)+\\frac{1}{8}kf(r)^2\\sin (2 ks),$ we find that both $s$ -dependent and $s$ -independent terms of the longitudinal strain $\\varepsilon _{ss}$ in the wrinkled zone vanish for $t \\rightarrow 0$ (up to higher order terms in $\\eta $ ).",
"The configuration given by Eqs.",
"REF , REF has also transverse and shear strains, given by $\\varepsilon _{rr}(s,r) &= \\frac{1}{2}f^{\\prime }(r)^2\\cos (ks)^2, \\\\\\varepsilon _{sr}(s,r) &= -\\eta f(r)\\cos (ks) - \\frac{1}{8}kf(r)f^{\\prime }(r)\\sin (2ks).",
"$ However, in contrast to the individual terms in Eq.",
"(REF ), which are proportional to the product $kf$ (that remains finite as $t \\rightarrow 0$ ), all terms on the r.h.s.",
"of Eqs.",
"(REF ,) vanish in the limit of small thickness (since, while $kf(r)$ is finite, the amplitude $f(r) \\rightarrow 0$ in this limit)."
],
[
"The stability analysis of Kirchoff rod equations",
"Here we translate the relevant results of [9], which addressed the stability analysis of a Kirchoff rod with non-symmetric cross section to the terminology of our paper.",
"The relevant results for us pertain to the linear stability analysis of the helicoidal (“straight\") state of the ribbon.",
"This is summarized in Eqs.",
"(58) and (59) that provide the threshold for the two types of instabilities of the centerline (“tapelike\" = TL, and “thick\"=th).",
"We explain the meaning of the parameters $a,b$ and $\\rho $ .",
"The parameter $a$ (Eq.",
"(9) of [9]) is the ratio between the two principal moments of inertia of the rod ($I_1 < I_2$ ).",
"The parameter $b$ (also Eq.",
"(9) of [9]) involves also the Poisson ratio (denoted $\\sigma $ in [9], and $\\nu $ in our manuscript), and the “mixed\" moment of inertia $J$ .",
"In the limit $t \\ll 1$ : $I_1 \\sim \\mathrm {W}t^3, I_2 \\sim t\\mathrm {W}^3$ and $J \\sim t\\mathrm {W}^3$ , with some numerical coefficients that depends on the exact shape of the cross section.",
"In Eq.",
"(12), both $a$ and $b$ are evaluated for an ellipsoidal cross section, but we assume that the same expressions (i.e.",
"the exact respective ratios between $J, I_1, I_2$ ) hold also for a rectangular cross section, from which we can translate to our terminology: $a \\rightarrow t^2,\\quad b \\rightarrow \\frac{2 t^2}{1+\\nu } \\ ,$ where we assume already the limit $t \\ll 1$ and expanded $b$ to lowest order in $t$ .",
"Now, let us consider the parameter $\\rho =F_3^{(0)}/{\\kappa _3^{(0)}}^2$ (Eq.",
"(35) of [9]), where $F_3^{(0)}$ is the normalized force exerted along the centerline and $\\kappa _3^{(0)}$ is the exerted “torsion\" of the centerline.",
"We will show that the translation to our terminology is: $\\rho \\rightarrow \\frac{t^2 T}{\\eta ^2} \\ .$ To see this, first note that $F_3^{(0)}$ and $\\kappa _3^{(0)}$ are defined as the tension and the twist density in the sentence after Eq.",
"(14) of [9].",
"In order to understand the normalization, we need the normalization of lengths and forces, given, respectively in Eqs.",
"(6) and (7b).",
"Note that lengths are measured in units of $t$ (since $I_1 \\sim t^3W$ and $A \\sim tW$ ).",
"The expressions of $\\kappa _3$ and $F_3$ in our parameters is therefore: $\\kappa _3^{(0)} = (\\theta /L)/t = \\eta / t$ , and $F_3^{(0)} = \\mathrm {force}/(E tW) = \\mathrm {force}/(Y W) = T$ .",
"Substituting this expression for $F_3^{(0)}$ and $\\kappa _3^{(0)}$ in Eq.",
"(35) of [9], we find the above transformation of the parameter $\\rho $ to our parameters.",
"Importantly, $\\rho $ of [9] is inversely proportional to the ratio $\\eta ^2/T$ , and hence the unstable range of the helicoidal state (gray zones in Figure 4 of [9]) corresponds to large value of twist/tension (namely $\\eta ^2/T$ above some threshold).",
"Let us turn now to Eqs.",
"(58,59), and express them in our terminology.",
"From Eq.",
"(58) we obtain the threshold for the “tapelike\" mode to be: $\\left(\\frac{\\eta ^2}{T}\\right)_\\mathrm {TL} \\approx \\frac{1+\\nu }{1-\\nu } \\ ,$ in the limit $t \\ll 1$ , which ranges from 1 to 3 as $\\nu $ ranges from 0 to $1/2$ .",
"Eq.",
"(59) leads the threshold for the “thick\" mode, $\\left(\\frac{\\eta ^2}{T}\\right)_\\mathrm {th} \\approx \\frac{1+\\nu }{2\\left(1+2\\nu -2\\sqrt{\\nu (1+\\nu )}\\right)}\\ ,$ in the limit $t \\ll 1$ .",
"This expression ranges from $1/2$ to $2.8$ as $\\nu $ ranges from 0 to $1/2$ and it is smaller than the first threshold for any value of $\\nu $ ."
],
[
"Estimating the clamping-induced energy",
"The transverse displacement $u_r$ must vanish near the clamped edges ($s=\\pm L/2$ ), and is expected to approach $u_r \\approx -\\nu T r/2$ beyond a characteristic length $\\ell $ from the clamped edges.",
"In the region $s \\in (-L/2 +\\ell ,L/2 -\\ell )$ the Poisson contraction applies, such that the strain can be approximated as: $\\varepsilon _{ss} = T \\ , \\ \\varepsilon _{rr} = -\\nu T \\ , \\ \\varepsilon _{xy} = 0 \\ ,$ and the corresponding energy per length is: $\\left(1-\\frac{2\\ell }{L}\\right) T^2$ In the near-boundary zones $s \\in \\pm (L/2-\\ell ,L/2)$ , where $u_r$ is not determined by the Poisson effect, we may express the strain field as: $\\varepsilon _{ss} &=& \\frac{1}{1-\\nu ^2}T + \\frac{\\nu T}{\\ell } f_1 \\ , \\nonumber \\\\\\varepsilon _{rr} &=& \\frac{\\nu T}{\\ell } f_2\\ , \\varepsilon _{sr} = \\frac{\\nu T}{\\ell } f_3$ where $f_i(s/\\ell ,r)$ are $O(1)$ functions that characterize the variation of the displacement field from the clamped edge to its bulk value.",
"Note that the $\\ell $ -independent component of $\\varepsilon _{ss}$ is derived from the Hookean stress-strain relationship by assuming $\\sigma ^{ss} \\approx T$ and $\\varepsilon _{rr} \\approx 0$ .",
"Integrating over the boundary zones $s \\in \\pm (L/2-\\ell ,L/2)$ , the energy per length associated with the strain field is estimated as: $\\frac{1}{L} \\left(\\frac{T^2\\ell }{1 -\\nu ^2} + \\frac{\\nu ^2 T^2}{\\ell } \\right) \\ .$ (where some unknown numerical constants, which are independent on $\\ell $ and $\\nu $ , multiply each of the two terms in the above expression).",
"Combining the two energies, Eqs.",
"(REF ,REF ), and minimizing over $\\ell $ , we obtain: $\\Delta U_\\mathrm {clamp} \\sim \\frac{\\nu F(\\nu ) T^2}{L} \\ , \\ \\ell \\sim \\nu $ where $F(\\nu )$ is some smooth function of $\\nu $ that satisfies $F(\\nu ) \\rightarrow \\mathrm {cst}$ for $\\nu \\rightarrow 0$ ."
]
] | 1403.0267 |
[
[
"Quasiparticle dynamics and spin-orbital texture of the SrTiO3\n two-dimensional electron gas"
],
[
"Abstract Two-dimensional electron gases (2DEGs) in SrTiO$_3$ have become model systems for engineering emergent behaviour in complex transition metal oxides.",
"Understanding the collective interactions that enable this, however, has thus far proved elusive.",
"Here we demonstrate that angle-resolved photoemission can directly image the quasiparticle dynamics of the $d$-electron subband ladder of this complex-oxide 2DEG.",
"Combined with realistic tight-binding supercell calculations, we uncover how quantum confinement and inversion symmetry breaking collectively tune the delicate interplay of charge, spin, orbital, and lattice degrees of freedom in this system.",
"We reveal how they lead to pronounced orbital ordering, mediate an orbitally-enhanced Rashba splitting with complex subband-dependent spin-orbital textures and markedly change the character of electron-phonon coupling, co-operatively shaping the low-energy electronic structure of the 2DEG.",
"Our results allow for a unified understanding of spectroscopic and transport measurements across different classes of SrTiO$_3$-based 2DEGs, and yield new microscopic insights on their functional properties."
],
[
"super Quasiparticle dynamics and spin-orbital texture of the SrTiO$_3$ two-dimensional electron gas P. D. C. King [email protected] Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA Laboratory of Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca, New York 14853, USA SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY16 9SS, United Kingdom S. McKeown Walker Département de Physique de la Matière Condensée, Université de Genève, 24 Quai Ernest-Ansermet, 1211 Genève 4, Switzerland A. Tamai Département de Physique de la Matière Condensée, Université de Genève, 24 Quai Ernest-Ansermet, 1211 Genève 4, Switzerland A. de la Torre Département de Physique de la Matière Condensée, Université de Genève, 24 Quai Ernest-Ansermet, 1211 Genève 4, Switzerland T. Eknapakul P. Buaphet School of Physics and NANOTEC-SUT Center of Excellence on Advanced Functional Nanomaterials, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand S.-K. Mo Advanced Light Source, Lawrence Berkeley National Lab, Berkeley, CA 94720, USA W. Meevasana School of Physics and NANOTEC-SUT Center of Excellence on Advanced Functional Nanomaterials, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand M. S. Bahramy Quantum-Phase Electronics Center and Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan RIKEN center for Emergent Matter Science (CEMS), Wako 351-0198, Japan F. Baumberger [email protected] Département de Physique de la Matière Condensée, Université de Genève, 24 Quai Ernest-Ansermet, 1211 Genève 4, Switzerland Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY16 9SS, United Kingdom Two-dimensional electron gases (2DEGs) in SrTiO$_3$ have become model systems for engineering emergent behaviour in complex transition metal oxides.",
"Understanding the collective interactions that enable this, however, has thus far proved elusive.",
"Here we demonstrate that angle-resolved photoemission can directly image the quasiparticle dynamics of the $d$ -electron subband ladder of this complex-oxide 2DEG.",
"Combined with realistic tight-binding supercell calculations, we uncover how quantum confinement and inversion symmetry breaking collectively tune the delicate interplay of charge, spin, orbital, and lattice degrees of freedom in this system.",
"We reveal how they lead to pronounced orbital ordering, mediate an orbitally-enhanced Rashba splitting with complex subband-dependent spin-orbital textures and markedly change the character of electron-phonon coupling, co-operatively shaping the low-energy electronic structure of the 2DEG.",
"Our results allow for a unified understanding of spectroscopic and transport measurements across different classes of SrTiO$_3$ -based 2DEGs, and yield new microscopic insights on their functional properties.",
"Figure: Orbital ordering of a two-dimensional electron gas in SrTiO 3 _3.",
"(a) EE vs. kk dispersion from ARPES (hν=50h\\nu =50 eV, measured along the [10][10] direction), revealing a multi-orbital subband structure comprising co-existing ladders of light and massive dd-electron subband states.",
"The respective circular and faint elliptical Fermi surface pockets (measured with hν=51h\\nu =51 eV and the polarization along [11][11]) are shown in the inset.",
"For the dispersion plot, a normalisation (division by the average MDC) has been applied to better reveal the massive band, as shown in Supplementary Fig. 1.",
"(b) This electronic structure is well described by a self-consistent tight-binding supercell calculation.",
"The spatial dependence of the subband wavefunctions along the confinement direction, Ψ(z)\\Psi (z), reveal a pronounced real-space orbital ordering, a direct consequence of near-surface band bending (c).The ubiquitous perovskite oxide SrTiO$_3$ , a wide-gap band insulator, hosts varied bulk properties including quantum paraelectricity, dilute doping-induced superconductivity and high thermoelectric coefficients.",
"These reflect a subtle competition between interactions of the underlying quantum many-body system.",
"Intriguingly, thermodynamic and transport measurements [1], [2], [3], [4], [5], [6] indicate that the balance of these interactions can be tuned to engineer striking emergent properties when quantum confinement and doping are combined to create a two-dimensional electron gas (2DEG).",
"[1], [2], [7] A diverse and attractive array of properties have been uncovered to date in this system, including gate-tuned superconductivity, [3], [4], [9] its coexistence with ferromagnetism, [5], [6] and enhanced Seebeck coefficients, [10] establishing SrTiO$_3$ based 2DEGs as a model platform for use in future multifunctional electronic devices.",
"[1] They are most commonly realised at a polar interface to another band insulator LaAlO$_3$ , creating a narrow conducting channel that resides solely within the SrTiO$_3$ .",
"[8] Similar 2DEGs can also be created by interfacing SrTiO$_3$ to a wide array of other band or Mott insulators including NdAlO$_3$ , [11] LaTiO$_3$ , [12] and GdTiO$_3$ , [13] by chemical doping of electrons into narrow SrTiO$_3$ channels, [14], [15] analogous to $\\delta $ -doping of semiconductors such as Si, and by field-effect doping in a transistor-style configuration.",
"[9] Moreover, the recent discovery of a 2DEG formed at the free surface of a bulk SrTiO$_3$ crystal opens new avenues for its advanced spectroscopic investigation.",
"[7], [16] Exploiting this, here we present unified angle-resolved photoemission (ARPES) measurements and tight-binding supercell calculations revealing new richness of the electronic structure of this model oxide 2DEG.",
"We show how a pronounced orbital ordering mediates an unconventional spin splitting, giving rise to strongly anisotropic and subband-dependent canted spin-orbital textures.",
"The orbitally enhanced Rashba effect explains the pronounced spin splittings previously inferred from magnetotransport in this system, while simultaneously revealing a breakdown of the conventional picture used to describe these.",
"We uncover how this complex ladder of subband states are further renormalized by many-body interactions.",
"This reconciles previous discrepancies between effective masses estimated from ARPES and quantum oscillations, unifying the properties of surface and interface SrTiO$_3$ 2DEGs, and reveals a strikingly different nature of electron-phonon coupling compared to bulk SrTiO$_3$ .",
"Results Orbital ordering.",
"Figure REF summarises the generic electronic structure of SrTiO$_3$ 2DEGs, as revealed by ARPES from a SrTiO$_3$ (100) surface with saturated band bending.",
"[7] Consistent with previous reports from both surface and interface 2DEGs, [7], [16], [17], [18], [19] we find a broad bandwidth that extends up to $\\approx 250$ meV below the Fermi level.",
"Here, we can resolve a ladder of at least three light subband states that contribute concentric circular Fermi surface sheets, co-existing with just a single heavy electron band ($m^*=14\\pm 3m_\\mathrm {e}$ ) that has a much shallower binding energy of less than 50 meV and gives rise to elliptical Fermi surfaces oriented along $\\langle 10\\rangle $ .",
"From this Fermi surface topology together with the polarization dependence of our measured intensities (Supplementary Fig.",
"2), we assign not only the lowest, [16] but rather the whole ladder of observed light states as having dominantly $d_{xy}$ orbital character, while the heavy states derive from $d_{xz/yz}$ orbitals.",
"This immediately indicates a strong breaking of the $t_{2g}$ orbital degeneracy that is present in the bulk electronic structure of SrTiO$_3$ , [20] driving a pronounced orbital ordering with a polarization $P=\\frac{n(d_{xy})-n(d_{xz/yz})}{n(d_{xy})+n(d_{xz/yz})}$ , which exceeds 30%, a lower limit derived from our experimentally-resolved Fermi surface areas.",
"This is a direct consequence of the real-space anisotropy of the orbital wavefunctions combined with inversion symmetry breaking by the electrostatic potential that defines the 2DEG by creating a steep asymmetric quantum well along the $z$ -direction (Fig.",
"REF (c)).",
"As shown by our self-consistent tight-binding supercell calculations (Fig.",
"REF (b), see Methods), the resulting quantized subbands that derive from planar $d_{xy}$ orbitals have wavefunctions reminiscent of the envelope functions of a semiconductor quantum well, except that in SrTiO$_3$ they are much more localized in real space, almost to within a single unit cell for the lowest subband state.",
"In contrast, the potential variation acts as a much weaker perturbation on the out-of-plane $d_{xz/yz}$ orbitals, which have much larger hopping amplitudes along the $z$ -direction.",
"The resulting subbands sit close to the top of the potential well, leading to wavefunctions that penetrate much deeper into the bulk.",
"This disparate spatial extent of the subband states is consistent with their relative spectral weight in our surface-sensitive ARPES measurements.",
"Figure: Orbitally-enhanced spin splitting.",
"(a) Orbitally-resolved electronic structure of the 2DEG along the [10][10] direction.",
"Magnified views reveal a weak Rashba-type spin splitting around the band bottom, which becomes enhanced by approximately an order of magnitude at the crossings of the light and heavy subband states where the orbital character becomes strongly mixed, as quantified in (b) for the lowest subbands.",
"(c) The calculated Fermi surface shows how similar orbital mixing and pronounced spin splittings occur at the crossings of circular d xy d_{xy}- and elliptical d xz/yz d_{xz/yz}-derived Fermi surface sheets.",
"This gives rise to an exotic anisotropic and subband-specific coupled spin-orbital texture of the 2DEG, as evident from the magnitude (left) and direction (right) of the (d) spin, SAM, and (e) orbital, OAM, angular momenta of the four largest Fermi surface sheets.",
"The magnitude of the SAM (OAM) is represented on a false colour scale in units of ℏ/2\\hbar /2 (ℏ\\hbar ), respectively, and is shown for additional Fermi surfaces in Supplementary Fig.",
"4.Unconventional Rashba spin splitting.",
"The same breaking of inversion symmetry that drives this orbital ordering can additionally lift the spin degeneracy through a Bychkov-Rashba-type spin-orbit interaction.",
"[21], [22], [23] Focussing near the band bottom of the lowest $d_{xy}$ band (Fig.",
"REF (a,b)), we indeed find a small characteristic splitting between the calculated energy of spin-up and spin-down states, $\\delta {E}=2\\alpha {}k_{||}$ , with a Rashba parameter, $\\alpha =0.003$ eVÅ.",
"The non-negligible splitting found here, despite the modest spin-orbit interaction in 3$d$ transition metals, is indicative of the very strong electric field gradient of the confining potential.",
"From our self-consistent band bending potential calculation (Fig.",
"REF (c)), we estimate that this exceeds $2\\times 10^8$ Vm$^{-1}$ within the first 2 unit cells where the lowest subband state is confined.",
"For the more delocalized second $d_{xy}$ subband, whose wavefunction extends into regions of shallower band bending, we find a slightly smaller Rashba parameter $\\alpha =0.0014$ eVÅ, confirming that the strength of this spin splitting is controlled by the confining electric field.",
"This should therefore be directly tuneable by electrical gating, suggesting a potential route towards spintronic control in oxides.",
"Unlike typical Rashba systems such as the Au$(111)$ surface, however, here the interplay between orbital ordering and spin-orbit coupling leads to a significantly richer spin structure of the 2DEG states.",
"Close to the crossings of the light $d_{xy}$ and heavy $d_{xz/yz}$ subbands, the spin splitting increases by approximately an order of magnitude, concomitant with a strong mixing of their orbital character (Fig.",
"REF (a-c)).",
"This rationalises an increased spin splitting reported from transport when the $d_{xz/yz}$ subbands become populated in electrically-gated SrTiO$_3$ /LaAlO$_3$ interface 2DEGs.",
"[24] Moreover, the crossover from $k$ -linear to strongly enhanced spin splitting that we find here readily explains the approximately $k^3$ dependence of the splitting that has been reported.",
"[25] Our layer-projected calculations indicate that the subband wavefunctions become more delocalized in the $z$ -direction close to these band crossings, a natural consequence of the stronger overlap of neighbouring $d_{xz/yz}$ orbitals along $z$ (Supplementary Fig. 3).",
"This delocalization would naively be expected to reduce the strength of the Rashba effect.",
"In contrast, its significant enhancement here points to a dominant role of inter-orbital hopping in driving such surprisingly-large spin splittings.",
"Similar enhancements have recently also been observed in other calculations, mainly based on model 3-band Hamiltonians, [26], [27], [28] which are qualitatively entirely consistent with our results.",
"Our calculations demonstrate how this is a direct consequence of orbital ordering in the real experimentally-confirmed multi-subband structure of the SrTiO$_3$ 2DEG.",
"Moreover, as shown in Fig.",
"REF (d,e), they reveal an exotic coupled spin-orbital texture of the resulting 2DEG Fermi surfaces.",
"While an approximately perpendicular spin-momentum locking ensures tangential spin winding around the circular $d_{xy}$ sections of Fermi surface, it leads to spins aligned almost perpendicular to the Fermi surface for large sections of the extremely anisotropic $d_{xz}$ ($d_{yz}$ ) sheets.",
"Around the crossings of $d_{xy}$ and $d_{xz/yz}$ states, the spins of neighbouring Fermi surfaces align (anti-)parallel with a $\\left|\\downarrow \\right\\rangle \\!\\left|\\uparrow \\right\\rangle \\!\\left|\\uparrow \\right\\rangle \\!\\left|\\downarrow \\right\\rangle $ ordering, ensuring maximal hybridization gaps are opened.",
"At the same time, rather than being quenched to zero as might have naively been expected, we find a finite orbital angular momentum (OAM, $\\bf {L}$ ) emerges.",
"This is relatively small ($\\lesssim 0.05\\hbar $ ) for the isolated $d_{xy}$ and $d_{xz/yz}$ sections of Fermi surface (Fig.",
"REF (e) and Supplementary Fig.",
"3), but grows as large as $0.7\\hbar $ around the band crossings.",
"Moreover, we find that the OAM is oriented either parallel or antiparallel to the corresponding spin angular momentum (SAM, $\\bf {S}$ ), and so this increase in OAM maximally enhances $\\bf {L}\\cdot \\bf {S}$ , by a factor of $\\sim \\!14$ , at and in the vicinity of the hybridization gaps, comparable to the corresponding increase in spin splitting (Fig.",
"REF (b)).",
"This therefore provides a natural basis to understand the large spin splittings inferred from magnetotransport, [29], [30], [31] despite the small atomic spin-orbit interaction of SrTiO$_3$ , in terms of an orbital Rashba effect.",
"[32], [33] For isolated $d_{xy}$ sections of Fermi surface, we find that the OAM of the inner and outer spin-split branches have the same helicity, consistent with a weak spin-orbit coupling limit.",
"[33] For the $d_{xz/yz}$ sections of Fermi surface, however, the inner and outer branches have opposite OAM, reflecting additional richness as compared to model systems such as noble metal surface states.",
"[33] This causes mixed helicities of the OAM around the inner branch of each Fermi surface sheet, as compared to a complete $2\\pi $ winding for the outer branches (see arrows in Supplementary Fig. 4).",
"Importantly, we find that the dominant inter-band interactions cause the winding direction of both the OAM and SAM of the outer $d_{xy}$ -derived Fermi surfaces to abruptly switch sign across the hybridization gaps.",
"For several inner Fermi surfaces whose orbital character is strongly mixed, continuously evolving between $d_{xy}$ - and $d_{xz/yz}$ -like around the Fermi surface (Fig.",
"REF (c)), this leads to strongly frustrated spin and orbital textures, rapidly canting from tangential to radial alignment as a function of Fermi surface angle (Fig.",
"REF (d,e)).",
"This is quite distinct from the functional form of conventional Rashba splitting [21] and provides strong constraints for the influence of spin-splitting on magnetism [34] and superconductivity.",
"[35] Figure: Quasiparticle dynamics of the subband states.",
"(a) ARPES measurements along the [11][11] direction (hν=51h\\nu =51 eV), together with the peak positions of fits to MDCs and EDCs (black dots) and a cosine `bare band' dispersion (blue line).",
"The data reveal pronounced signatures of electron-phonon coupling.",
"Particularly apparent is a low-energy kink in the dispersion at ω≈30\\omega \\approx 30 meV, shown magnified inset.",
"(b) Real, Σ ' \\Sigma ^{\\prime }, and imaginary, Σ '' \\Sigma ^{\\prime \\prime }, parts of the extracted self-energy.",
"The open blue symbols show Σ '' \\Sigma ^{\\prime \\prime } before subtraction of an electron-electron scattering contribution to the total measured imaginary part of the self-energy, approximated here by the expression Σ e-e '' (ω)=βω 2 [1+0.53ln(ω/E F )]\\Sigma _{\\mathrm {e-e}}^{\\prime \\prime }(\\omega )=\\beta \\omega ^2[1+0.53 \\ln (\\omega /E_{\\mathrm {F}})] for a 2D Fermi liquid (dashed line, β=1.5\\beta =1.5 eV -1 ^{-1}, E F =0.25E_{\\mathrm {F}}=0.25 eV).",
"The solid lines are calculated electron-phonon self energies with the Eliashberg function proportional to the full phonon density of states (red/blue) or for coupling to three longitudinal optical phonons (black), which dominate the interaction in lightly doped bulk SrTiO 3 _3.",
"(c, left) Calculated electron-phonon spectral function, A(k,ω)A(k,\\omega ), along the [10][10] direction (see Methods).",
"This reveals complete bandwidth renormalization of the heavy states (particularly apparent when A(k,ω)A(k,\\omega ) is projected only onto d xz d_{xz} orbitals as shown in the right inset), and kinks in the light d xy d_{xy} subband dispersions at their crossings with this renormalized heavy state (left inset), both in good agreement with our experimental measurements, shown in the right half using curvature analysis of the raw data.Many-body interactions.",
"In Figure REF , we further uncover a pronounced role of electron-phonon interactions on this complex hierarchy of electronic states.",
"Unlike in bulk-doped SrTiO$_3$ , where the Fermi energy is typically only a few meV and the electron-phonon interaction is thus non-retarded, the occupied widths of different subbands of the 2DEG range from almost zero up to values greater than the highest phonon frequency of $\\approx 100$ meV.",
"This is an unusual situation, neither described by the adiabatic $(\\hbar \\omega _\\mathrm {D}<<E_\\mathrm {F})$ nor the anti-adiabatic $(\\hbar \\omega _\\mathrm {D}>>E_\\mathrm {F})$ approximation, and points to a complex influence of electron-phonon coupling in this system.",
"We extract the corresponding self-energy, $\\Sigma _{\\mathrm {e-ph}}(\\omega )=\\Sigma ^{\\prime }(\\omega )+i\\Sigma ^{\\prime \\prime }(\\omega )$ , from our ARPES measurements of the lowest subband along the $[11]$ direction (see methods), where we resolve an isolated band all the way up to $E_\\mathrm {F}$ .",
"The slope of $\\Sigma ^{\\prime }$ at the Fermi level yields an electron-phonon coupling strength of $\\lambda =0.7(1)$ , while its broad maximum between $\\approx 20 - 60$ meV is indicative of coupling to multiple modes.",
"Indeed, the experimentally-determined self-energy is in excellent agreement with a calculation within Eliashberg theory that assumes a coupling function $\\alpha ^2 F(\\omega )$ proportional to the entire phonon density of states associated with the motion of oxygen and Ti ions [37] and includes the realistic 2DEG electron density of states from our tight-binding calculation.",
"Together with a moderate correlation-induced mass enhancement of $\\approx 1.4$ that we estimate from a Kramers-Kronig transform of a Fermi-liquid contribution to the imaginary part of the self-energy, our analysis suggests an overall mass enhancement arising from many-body interactions of $m^{*}/m_{\\rm {band}}\\approx 2.1$ , close to the values deduced for lightly-doped bulk SrTiO$_3$ from measurements of the electronic specific heat [38] and optical spectroscopy [39].",
"The nature of the electron-phonon coupling, however, is very different.",
"In lightly-doped bulk SrTiO$_3$ , it is dominated by the long-range coupling to longitudinal optical (LO) phonons as described by the Fröhlich model.",
"[40], [39] This model predicts much weaker coupling to low-energy modes than observed here, but a significantly stronger coupling to the highest LO phonon at 100 meV, as evident from a calculation employing coupling strengths from bulk SrTiO$_3$ , [40], [41] which yield an electron-phonon self-energy in clear contrast to our experimental findings (Fig.",
"REF (b)).",
"The electric field that confines the 2DEG is known to dramatically reduce its dielectric constant.",
"[42], [7] This, together with higher carrier densities as compared to bulk SrTiO$_3$ , will lead to shorter electronic screening lengths for the 2DEG, explaining the observed suppression of long-range coupling to LO modes.",
"The enhanced coupling to low-energy phonons for the 2DEG instead leads to a pronounced kink in the dispersion of the $d_{xy}$ subbands at an energy around 30 meV.",
"We resolve these along both the $[10]$ and $[11]$ directions for the first two $d_{xy}$ subbands.",
"Crucially, the resulting enhanced quasiparticle mass, which we estimate as $1.1(2)$ m$_e$ from our measured Fermi velocities, rectifies the discrepancy between the light masses around 0.6 m$_e$ reported in earlier ARPES studies of SrTiO$_3$ surface 2DEGs [7], [16] and recent quantum oscillation experiments that revealed effective masses typically around $1m_\\mathrm {e}$ .",
"[43], [44] Intriguingly, along $[10]$ the kink energy coincides almost exactly with the crossing of the light $d_{xy}$ and heavy $d_{xz}$ subband states.",
"This behaviour is well captured by our spectral function simulations calculated from our tight-binding bare dispersions and electron-phonon self-energy.",
"These illustrate a very different effect of electron-phonon interactions on the heavy compared to the light subbands of the 2DEG, the former coupling to phonons with frequencies ranging from below to above the bare band width.",
"Our calculations reveal that electron-phonon coupling essentially results in an overall bandwidth renormalisation of these states, in agreement with our experimental data where we find the band bottom of the heavy state substantially above the value predicted by our model tight binding calculations.",
"Figure: Hierarchy and interplay of underlying degrees of freedom.",
"In the formation of the 2DEG, quantum confinement reconstructs the bulk electronic structure into a rich array of intersecting orbitally-ordered subband states.",
"The interplay of this with spin-orbit coupling lifts the spin degeneracy of these bands, particularly strong around their crossings, through an orbitally-enhanced Rashba-like interaction.",
"Electron-phonon and electron-electron interactions further renormalise these spin-split dispersions, increasing the quasiparticle masses close to the Fermi level, and causing a complete bandwidth narrowing of the shallow d xz/yz d_{xz/yz} states that pins the orbitally-ordered band crossings close to the energy of the lowest phonon mode.",
"This, in turn, sets the energy scale for enhanced spin-orbit splittings around these crossings, strongly entangling the charge, spin, orbital, and lattice degrees of freedom of the 2DEG on a low energy scale (orange shading).Discussion The combination of electron-phonon coupling with orbital ordering therefore effectively pins the crossings of the $d_{xy}$ and $d_{xz/yz}$ subbands to the low-energy peak in the phonon density of states.",
"As demonstrated above, however, we additionally find orbitally-enhanced spin-orbit splittings which become maximal around these band crossings.",
"Our direct spectroscopic measurements, together with our theoretical calculations, therefore demonstrate how a co-operative effect of orbital ordering and electron-phonon coupling sets the relevant energy scale for dominant spin splitting in this system.",
"Together, this reveals an intricate hierarchy of interactions and orderings governing the low-energy electronic structure of the SrTiO$_3$ 2DEG (Figure REF ).",
"Electrostatic screening in response to a surface or interface charge generates an electron accumulation layer confined to a narrow potential well.",
"The resulting quantum size effects drive pronounced orbital ordering, creating multiple intersections of light $d_{xy}$ - and heavy $d_{xz/yz}$ -derived subband states.",
"This leads to orbital mixing and induces a significant local orbital angular momentum, which in turn permits a pronounced spin splitting to emerge, despite modest atomic spin-orbit coupling.",
"Many-body interactions, of strikingly different form to the bulk, enhance the quasiparticle masses of these spin-split subbands, reduce their bandwidths, and renormalize the energetic locations of their intersections, thus modulating their unconventional spin splitting.",
"Together, this interplay strongly entangles the charge, spin, orbital, and lattice degrees of freedom of the SrTiO$_3$ 2DEG on an energy scale $\\le \\!30$ meV.",
"This will dominate both its transport and thermodynamic properties.",
"Already we have shown how this explains enhanced quasiparticle masses observed from quantum oscillations as well as signatures of spin splitting in magnetotransport, unifying electrical and spectroscopic measurements from surface- and interface-based SrTiO$_3$ 2DEGs.",
"More generally, it establishes how quantum size effects can dramatically manipulate the underlying electronic landscape of interacting electron liquids, setting the stage for engineering new emergent properties by dimensional confinement in transition-metal oxides.",
"Methods Angle-resolved photoemission: ARPES measurements were performed at the CASIOPEE beamline of SOLEIL synchrotron, the SIS beamline of the Swiss Light Source, and beamline 10.0.1 of the Advanced light source using Scienta R4000 hemispherical electron analysers, and with base pressures below $5\\times 10^{-11}$ mbar.",
"Single crystal SrTiO$_3$ commercial wafers were cleaved in situ at the measurement temperature of $T=20-30$ K along notches defining a $(100)$ plane.",
"Measurements were performed on stoichiometric transparent insulating samples as well as very lightly La-doped samples (Sr$_{1-x}$ La$_x$ TiO$_3$ with $x=0.001$ ) to help avoid charging.",
"2DEGs were induced at the bare surface by exposure to intense UV synchrotron light.",
"[7] The samples were exposed to irradiation doses $\\gtrsim 1000$ Jcm$^{-2}$ to saturate the formation of the 2DEG, and we experimentally confirmed that saturation was reached before starting any of the measurements presented here.",
"The data shown here was measured using $s$ -polarised photons of 51 or 55 eV, except for the Fermi surface maps shown in Supplementary Fig.",
"2 that used 43 eV $s$ - and $p$ -polarised light.",
"All measurements were performed in the second Brillouin zone.",
"Self-energy determination: To determine the electron-phonon self-energy experimentally, we fit momentum distribution curves (MDCs) and energy distribution curves (EDCs) of the lowest $d_{xy}$ subband measured along the $[11]$ direction.",
"We chose this band as its dispersion does not intersect that of the heavy $d_{xz/yz}$ subbands up to the chemical potential along this direction (Fig.",
"REF (b)), allowing us to perform a quantitative analysis over an extended energy range, free from complications associated with the hybridization of different subbands.",
"The real part of the self-energy, $\\Sigma ^{\\prime }(\\omega )$ , is given by the difference between our extracted dispersion and that of a `bare' band.",
"In order to derive a Kramers-Kronig consistent self-energy, we take the cosine bare band shown in Fig.",
"REF (a), which includes a moderate band width renormalization due to electron correlations.",
"We extract the imaginary part of the self energy, $\\Sigma ^{\\prime \\prime }(\\omega )=\\Delta {k(\\omega )}/2\\cdot \\partial \\epsilon /\\partial k$ , where $\\Delta {k}$ is the full width at half maximum of Lorentzian fits to MDCs, and $\\partial \\epsilon /\\partial k$ is the bare band dispersion.",
"This results in an imaginary part that includes a contribution from electron-electron interactions, which we approximate by the expression for a two-dimensional Fermi liquid $\\Sigma _{\\mathrm {e-e}}^{\\prime \\prime }(\\omega )=\\beta \\omega ^2[1+0.53 \\ln (\\omega /E_{\\mathrm {F}})]$ .",
"Subtracting this contribution with realistic parameter values of $\\beta =1.5$ eV$^{-1}$ and $E_{\\mathrm {F}}=0.25$ eV from our total extracted $\\Sigma ^{\\prime \\prime }$ (see Fig.",
"REF (b)) yields the imaginary part of the electron-phonon self-energy.",
"Electronic structure and self-energy calculations: To calculate the subband structure, we start from a relativistic density functional theory (DFT) calculation of bulk SrTiO$_3$ using the modified Becke-Johnson exchange potential and Perdew-Burke-Ernzerhof correlation functional as implemented in the WIEN2K program [45].",
"The muffin-tin radius of each atom $R_{\\mathrm {MT}}$ was chosen such that its product with the maximum modulus of reciprocal vectors $K_{\\mathrm {max}}$ become $R_{\\mathrm {MT}}K_{\\mathrm {max}}=7.0$ .",
"The Brillouin zone was sampled by a $15\\times 15\\times 15$ $k$ -mesh.",
"We downfold this using maximally-localized Wannier functions to generate a set of bulk tight-binding transfer integrals, and then incorporate these into a 30 unit cell supercell with additional on-site potential terms to account for band bending via an electrostatic potential variation.",
"We solve this self-consistently with Poisson's equation, incorporating an electric field-dependent dielectric constant, [42] to yield the bare band dispersions including Rashba-type spin splitting [46] of the 2DEG.",
"We stress that the total magnitude of the band bending is the only adjustable parameter, and yields a realistic electronic structure in good agreement with our spectroscopic measurements apart from our observed signatures of electron-phonon interactions which are not included at the level of DFT.",
"To incorporate these, we calculate the self-energy $\\Sigma _{\\mathrm {e-ph}}(\\omega )$ in an Eliashberg model, $\\Sigma _{\\mathrm {e-ph}}(\\omega ) = \\int _{-E_\\mathrm {F}}^\\infty N(\\epsilon )d\\epsilon \\int _{0}^{\\tilde{\\omega }_{max}}d\\tilde{\\omega } \\alpha ^2 F(\\tilde{\\omega })\\\\\\times \\bigg \\lbrace \\frac{f(-\\epsilon ,T)+n(\\tilde{\\omega },T)}{\\omega -\\epsilon -\\tilde{\\omega }+i\\delta ^{\\pm }}+\\frac{f(\\epsilon ,T)+n(\\tilde{\\omega },T)}{\\omega -\\epsilon +\\tilde{\\omega }+i\\delta ^{\\pm }}\\bigg \\rbrace $ where $N(\\epsilon )$ is the bare electronic density of states determined from our tight binding calculations and $f(\\epsilon ,T)$ and $n(\\tilde{\\omega },T)$ are the Fermi and Bose occupation factors.",
"For the coupling function $\\alpha ^2 F(\\tilde{\\omega })$ we use two different models.",
"The blue line in Fig.",
"REF (b) assumes $\\alpha ^2 F(\\tilde{\\omega })$ proportional to the entire O- and Ti-derived phonon density of states from Ref.",
"[37], while the black line is a calculation for the coupling strengths given in Refs.",
"[40], [41] for the three longitudinal optical phonons that were found to dominate the electron-phonon interaction in lightly doped bulk samples.",
"We then calculate the spectral function $A(k,\\omega )=-\\frac{1}{\\pi }\\frac{\\Sigma ^{\\prime \\prime }(\\omega )}{\\left[\\omega -\\epsilon (k)-\\Sigma ^{\\prime }(\\omega )\\right]^2+\\left[\\Sigma ^{\\prime \\prime }(\\omega )\\right]^2}$ where the bare band dispersion $\\epsilon (k)$ is taken from our tight-binding calculation.",
"To better compare with our experimental data in Fig.",
"REF (c), we project this onto different atomic orbitals, and include contributions from $d_{xy}$ and $d_{xz}$ but not $d_{yz}$ orbitals to account for transition matrix elements in our experimental geometry.",
"We additionally project the calculation onto different layers of our supercell, and incorporate an exponential attenuation of signal with depth below the surface in photoemission, assuming an inelastic mean free path of 5 Å, into our simulation.",
"Finally, we convolve the simulated spectral function with a 2D Gaussian to account for an experimental energy and momentum resolution of 0.01 eV and 0.015 Å$^{-1}$ , respectively.",
"Acknowledgements: This work was supported by the U.K. EPSRC (EP/I031014/1), the ERC (207901), the SNSF (200021-146995), the Scottish Funding Council, The Thailand Research Fund (RSA5680052), Office of the Higher Education Commission, Suranaree Univerisity of Technology, and the Japan Society for the promotion of Science (JSPS), through the `Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)', initiated by the council for Science and Technology policy (CSTP).",
"PDCK acknowledges support from the Royal Society through a University Research Fellowship (UF120096).",
"We acknowledge SOLEIL (beamline CASSIOPEE), the ALS (beamline 10.0.1), and SLS (SIS beamline) for provision of synchrotron radiation facilities, and in particular N.C. Plumb, M. Radović, and M. Shi (SLS) and P. Le Fèvre, F. Bertran and A. Taleb-Ibrahimi (SOLEIL) for technical assistance.",
"The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH11231.",
"We gratefully acknowledge C. Bell.",
"C. Berthod, V. Cooper, A. Fête, H.Y.",
"Hwang, M. Kim, J. Mannhart, D. van der Marel, and J.-M. Triscone for useful discussions.",
"Author contributions: The experimental data was measured by PDCK, SMW, AT, ADLT, TE, PB, WM and FB, and analysed by PDCK, SMW, AT and FB.",
"PDCK and MSB performed the electronic structure calculations and AT performed the electron-phonon self-energy calculations.",
"SKM maintained the ARPES endstation at the Advanced Light Source and provided experimental support.",
"PDCK and FB were responsible for overall project planning and direction, and wrote the manuscript with input and discussion from all co-authors.",
"Competing financial interests: The authors declare no competing financial interests.",
"Figure: Subband structure of the SrTiO 3 _3 2DEG.",
"(a) Raw data of the spectrum shown in Fig.",
"1a of the main manuscript, clearly revealing three light subband states as well as weak spectral weight from a single heavy band at lower binding energy.",
"(b) This latter feature is strongly enhanced by dividing by the average MDC intensity (shown inset).Figure: Orbital character from polarisation-dependent ARPES.",
"Fermi surface of the SrTiO 3 _3 2DEG measured using (a) ss-, (b) pp-polarised, and (c) the sum of ss- and pp-polarised 43 eV photons, and normalised to the same total intensity scaling.",
"A schematic of the electronic orbitals that should be dominant for the respective experimental geometries and the resulting Fermi surfaces are also shown, in good agreement with our assignment of dominant orbital characters of the subbands in the main text.Figure: Spatial extent of the 2DEG.",
"Layer-projected calculations of the electronic structure of the SrTiO 3 _3 2DEG, integrated over (a) the full 30 unit cell (u.c.)",
"supercell, and (b)–(g) individual or few unit cell regions, as labelled in the figure.Figure: Subband-dependent canted spin and orbital textures.",
"(a) In-plane spin texture (spin angular momentum, SAM) of the outer (top) and inner (bottom) branches of successive Fermi surface sheets, showing how the alignment of the spin direction evolves from radial- to tangential- as the orbital character changes from d xy d_{xy}- to d xz/yz d_{xz/yz}-like (see Fig.",
"2(c) of the main text for the orbital character), leading to frustrated spin textures for orbitally-mixed Fermi surfaces.",
"The net winding direction is shown in the centre of each plot, and the magnitude of the SAM is shown on a false colour scale in units of ℏ/2\\hbar /2.",
"(b) Equivalent plots for the orbital angular momentum (OAM), showing similar canting and mixed windings as represented by the large and small arrows.",
"The magnitude of the OAM is represented by a false colour scale in units of ℏ\\hbar , revealing how significant OAM emerges only around the band intersections where the orbital character becomes strongly mixed, and the spin splitting is thus maximised."
]
] | 1403.0520 |
[
[
"A note on the joint measurability of POVMs and its implications for\n contextuality"
],
[
"Abstract The purpose of this note is to clarify the logical relationship between joint measurability and contextuality for quantum observables in view of recent developments [1-4]."
],
[
"Introduction",
"In a recent work [2], a new proof of contextuality—in the generalized sense of Spekkens [5], [1]—was provided using positive operator-valued measures (POVMs) and the connection between joint measurability of POVMs and contextuality was explicated.",
"It was later shown in [3] that any joint measurability structure can be realized in quantum theory, leaving open the question of whether contextuality can always be demonstrated in these joint measurability structures.",
"Subsequent to these two developments, in Ref.",
"[4] a peculiar feature of POVMs with respect to joint measurability was pointed out: that there exist three measurements which are pairwise jointly measurable and triplewise jointly measurable but for which there exist pairwise joint measurements which do not admit a triplewise joint measurement.",
"In this note, I will briefly put these results in context and point out the logical relationship between joint measurability and the possibility of contextuality.",
"Also, throughout this note, `sharp measurement' will be synonymous with projection-valued measures (PVMs) and `unsharp measurement' will be synonymous with POVMs that are not PVMs.",
"Since the peculiarity of positive-operator valued measures (POVMs) in cases of interest here arises from the nonuniqueness of joint measurements, I will first prove the uniqueness of joint measurements for projection-valued measures (PVMs).",
"This will help clarify how the distinction between sharp and unsharp measurements comes to play a role in Specker's scenario [2].",
"Consider a nonempty set $\\Omega _i$ and a $\\sigma $ -algebra $\\mathcal {F}_i$ of subsets of $\\Omega _i$ , for $i\\in \\lbrace 1,\\dots ,N\\rbrace $ .",
"The POVM $M_i$ is defined as the map $M_i: \\mathcal {F}_i\\rightarrow \\mathcal {B}_+(\\mathcal {H})$ , where $\\sum _{X_i\\in \\mathcal {F}_i}M_i(X_i)=I$ and $\\mathcal {B}_+(\\mathcal {H})$ denotes the set of positive semidefinite operators on a Hilbert space $\\mathcal {H}$ .",
"$I$ is the identity operator on $\\mathcal {H}$ .",
"Therefore: $M_i\\equiv \\lbrace M_i(X_i)|X_i\\in \\mathcal {F}_i\\rbrace $ , where $X_i$ labels the elements of POVM $M_i$ .",
"$M_i$ becomes a projection-valued measure (PVM) under the additional constraint $M_i(X_i)^2=M_i(X_i)$ for all $X_i\\in \\mathcal {F}_i$ .",
"Theorem 1 Given a set of POVMs, $\\lbrace M_1,\\dots ,M_N\\rbrace $ , all of which except at most one—say $M_N$ —are PVMs, so that for $i\\in \\lbrace 1,\\dots ,N-1\\rbrace $ $M_i\\equiv \\lbrace M_i(X_i)|X_i\\in \\mathcal {F}_i, M_i(X_i)^2=M_i(X_i)\\rbrace $ and $M_N\\equiv \\lbrace M_N(X_N)|X_N\\in \\mathcal {F}_N\\rbrace ,$ the set of POVMs, $\\lbrace M_1,\\dots ,M_N\\rbrace $ , is jointly measurable if and only if they commute pairwise, i.e., $M_j(X_j)M_k(X_k)=M_k(X_k)M_j(X_j),$ for all $j,k\\in \\lbrace 1,\\dots ,N\\rbrace $ and $X_j\\in \\mathcal {F}_j, X_k\\in \\mathcal {F}_k$ .",
"In this case, there exists a unique joint POVM $M$ , defined as a map $M:\\mathcal {F}_1\\times \\mathcal {F}_2\\times \\dots \\times \\mathcal {F}_N \\rightarrow \\mathcal {B}_+(\\mathcal {H}),$ such that $M(X_1\\times \\dots \\times X_N)=M_1(X_1)M_2(X_2)\\dots M_N(X_N),$ for all $X_1\\times \\dots \\times X_N \\in \\mathcal {F}_1\\times \\dots \\times \\mathcal {F}_N.$ Proof.—This proof is adapted from, and is a generalization of, the proof of Proposition 8 in the Appendix of Ref.",
"[6].",
"The first part of the proof is for the implication: joint measurability $\\Rightarrow $ pairwise commutativity—A joint POVM for $\\lbrace M_1,\\dots ,M_N\\rbrace $ is defined as a map $M:\\mathcal {F}_1\\times \\mathcal {F}_2\\times \\dots \\times \\mathcal {F}_N \\rightarrow \\mathcal {B}_+(\\mathcal {H})$ , such that $M_i(X_i)=\\sum _{\\lbrace X_j\\in \\mathcal {F}_j|j\\ne i\\rbrace }M(X_1\\times \\dots \\times X_N)$ for all $X_i\\in \\mathcal {F}_i$ , $i\\in \\lbrace 1\\dots N\\rbrace $ .",
"Also, $M(X_1\\times \\dots \\times X_N)\\le M_1(X_1)$ , so the range of $M(X_1\\times \\dots \\times X_N)$ is contained in the range of $M_1(X_1)$ , and therefore: $M_1(X_1)M(X_1\\times \\dots \\times X_N)=M(X_1\\times \\dots \\times X_N).$ Using this relation for the complement $\\Omega _1\\backslash X_1 \\in \\mathcal {F}_1$ : $&&M_1(X_1)M(\\Omega _1\\backslash X_1\\times \\dots \\times X_N)\\nonumber \\\\&&=(I-M_1(\\Omega _1\\backslash X_1))M(\\Omega _1\\backslash X_1\\times \\dots \\times X_N)\\nonumber \\\\&&=0.$ Taking the adjoints, it follows that $M(X_1\\times \\dots \\times X_N)M_1(X_1)=M(X_1\\times \\dots \\times X_N),$ and $M(\\Omega _1\\backslash X_1\\times \\dots \\times X_N)M_1(X_1)=0.$ Denoting $M^{(i)}(X_{i+1}\\times \\dots \\times X_N)\\equiv \\sum _{\\lbrace X_j\\in \\mathcal {F}_j|j\\le i\\rbrace }M(X_1\\times \\dots \\times X_N),$ this implies: $&&M_1(X_1)M^{(1)}(X_2\\times \\dots \\times X_N)\\nonumber \\\\&=&M_1(X_1)M(X_1\\times \\dots \\times X_N)\\nonumber \\\\&&+M_1(X_1)M(\\Omega _1\\backslash X_1\\times \\dots \\times X_N)\\nonumber \\\\&=&M_1(X_1)M(X_1\\times \\dots \\times X_N)\\nonumber \\\\&=&M(X_1\\times \\dots \\times X_N).$ Taking the adjoint, $M^{(1)}(X_2\\times \\dots \\times X_N)M_1(X_1)=M(X_1\\times \\dots \\times X_N).$ Therefore: $&&M_1(X_1)M^{(1)}(X_2\\times \\dots \\times X_N)\\nonumber \\\\&=&M^{(1)}(X_2\\times \\dots \\times X_N)M_1(X_1)\\nonumber \\\\&=&M(X_1\\times \\dots \\times X_N).$ Noting that $M^{(i-1)}(X_i\\times \\dots \\times X_N)\\le M_i(X_i)$ , one can repeat the above procedure for $M_i$ , $i\\in \\lbrace 2,\\dots ,N-1\\rbrace ,$ to obtain: $&&M^{(i-1)}(X_i\\times \\dots \\times X_N)\\nonumber \\\\&=&M_i(X_i)M^{(i)}(X_{i+1}\\times \\dots \\times X_N)\\nonumber \\\\&=&M^{(i)}(X_{i+1}\\times \\dots \\times X_N)M_i(X_i).$ Doing this recursively until $i=N-1$ and noting that $M^{(N-1)}(X_N)=M_N(X_N)$ , it follows: $&&M(X_1\\times \\dots \\times X_N)\\nonumber \\\\&=&M_1(X_1)M^{(1)}(X_2\\times \\dots \\times X_N)\\nonumber \\\\&=&M^{(1)}(X_2\\times \\dots \\times X_N)M_1(X_1)\\nonumber \\\\&&\\vdots \\nonumber \\\\&=&M_1(X_1)M_2(X_2)\\dots M_{N-1}(X_{N-1})M_N(X_N)\\nonumber \\\\&=&M_N(X_N)M_{N-1}(X_{N-1})\\dots M_2(X_2)M_1(X_1).\\nonumber \\\\$ For the last equality to hold, the POVM elements must commute pairwise, so that $M(X_1\\times \\dots \\times X_N)=\\prod _{i=1}^N M_i(X_i).$ This concludes the proof of the implication, joint measurability $\\Rightarrow $ pairwise commutativity.",
"The converse is easy to see since the joint POVM defined by taking the product of commuting POVM elements, $\\lbrace M(X_1\\times \\dots \\times X_N)=\\prod _{i=1}^N M_i(X_i)|X_i\\in \\mathcal {F}_i\\rbrace ,$ is indeed a valid POVM which coarse-grains to the given POVMs, $\\lbrace M_1,\\dots ,M_N\\rbrace $ .",
"Indeed, pairwise commutativity $\\Rightarrow $ joint measurability for any arbitrary set of POVMs, $\\lbrace M_1,\\dots ,M_N\\rbrace $ , and it is only when all but (at most) one of these POVMs are PVMs that the converse—and the uniqueness of the joint POVM—holds.",
"Specker's scenario requires a set of three POVMs, $\\lbrace M_1,M_2,M_3\\rbrace $ , that are pairwise jointly measurable, i.e., $\\exists $ POVMs $M_{12}$ , $M_{23}$ , and $M_{31}$ which measure the respective pairs jointly.",
"An immediate consequence of the requirement of pairwise joint measurability of $\\lbrace M_1,M_2,M_3\\rbrace $ is that in quantum theory these three measurements cannot be realized as projective measurements (PVMs) and still be expected to show any contextuality.",
"This is because for projective measurements or projection-valued measures (PVMs), a set of three measurements that are pairwise jointly measurable—and therefore admit unique pairwise joint measurements—are also triplewise jointly measurable in the sense that there exists a unique triplewise joint measurement which coarse-grains to each pairwise implementation of the three measurements and therefore also to the single measurements.",
"From Theorem REF , it follows that if $M_i$ , $i\\in \\lbrace 1,2,3\\rbrace $ , are PVMs then they admit unique pairwise and triplewise joint PVMs: $M_{ij}(X_i\\times X_j)&=&M_i(X_i)M_j(X_j),\\\\M(X_1\\times X_2\\times X_3)&=&M_1(X_1)M_2(X_2)M_3(X_3),$ corresponding to the maps $M_{ij}:\\mathcal {F}_i\\times \\mathcal {F}_j\\rightarrow \\mathcal {B}_+(\\mathcal {H})$ and $M:\\mathcal {F}_1\\times \\mathcal {F}_2\\times \\mathcal {F}_3\\rightarrow \\mathcal {B}_+(\\mathcal {H})$ , respectively.",
"Intuitively, this is easy to see since joint measurability is equivalent to pairwise commutativity for a set of projective measurements and the joint measurement for each pair is unique [6].",
"The existence of a unique joint measurement implies that there exists a joint probability distribution realizable via this joint measurement, thus explaining the pairwise statistics of the triple of measurements noncontextually in the traditional Kochen-Specker sense.KS-noncontextuality just means that there exists a joint probability distribution over the three measurement outcomes which marginalizes to the pairwise measurement statistics.",
"Violation of a KS inequality—obtained under the assumption that a global joint distribution exists—rules out KS-noncontextuality.",
"Clearly, then, the three measurements $\\lbrace M_1, M_2, M_3\\rbrace $ must necessarily be unsharp for Specker's scenario to exhibit KS-contextuality.",
"The uniqueness of joint measurements (pairwise or triplewise) need not hold in this case.",
"I will refer to pairwise joint measurements as “2-joints” and triplewise joint measurements as “3-joints”.",
"Also, I will use the phrases `joint measurability' and `compatibility' interchangeably since they will refer to the same notion.",
"Consider the four propositions regarding the three measurements: $\\exists $ 2-joint: $\\lbrace M_1,M_2,M_3\\rbrace $ admit 2-joints, $\\nexists $ 2-joint: $\\lbrace M_1,M_2,M_3\\rbrace $ do not admit 2-joints, $\\exists $ 3-joint: $\\lbrace M_1,M_2,M_3\\rbrace $ admit a 3-joint, $\\nexists $ 3-joint: $\\lbrace M_1,M_2,M_3\\rbrace $ do not admit a 3-joint, The possible pairwise-triplewise propositions for the three measurements are: $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ , $(\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ , $(\\nexists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ .",
"Note that the proposition $(\\nexists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ is trivially excluded because triplewise compatibility implies pairwise compatibility.",
"Of the three remaining propositions, the ones of interest for contextuality are $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ and $(\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ , since the remaining one is simply about observables that do not admit any joint measurement at all and hence no nontrivial measurement contexts exist for this proposition.It is worth noting that, if $\\lbrace M_1,M_2,M_3\\rbrace $ were PVMs, then there are only two possibilities: $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ and $(\\nexists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ , since for three PVMs, $\\exists \\text{ 2-joint} \\Leftrightarrow \\exists \\text{ 3-joint}$ , because pairwise commutativity is equivalent to joint measurability and the joint measurements are unique on account of Theorem REF .",
"This is why KS-contextuality is impossible with PVMs in this scenario.",
"It may seem that for purposes of contextuality even the proposition $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ is of no interest, but there is a subtlety involved here: one is only considering whether 2-joints or a 3-joint exist for the set $\\lbrace M_1, M_2, M_3\\rbrace $ .",
"Since the statistics that is of relevance for Specker's scenario is the pairwise statistics [1], [2], one also needs to consider whether a given choice of 2-joints, $\\lbrace M_{12}, M_{23}, M_{31}\\rbrace $ , admits a 3-joint, i.e., the proposition $(\\exists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ or its negation $(\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ .",
"The four possible conjunctions are: $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})\\bigwedge (\\exists \\text{ 3-joint}|\\text{ a choice of 2-joints}),$ $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints}),$ $(\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})\\bigwedge (\\exists \\text{ 3-joint}|\\text{ a choice of 2-joints}),$ $(\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints}).$ Of these, the first conjunction rules out the possibility of KS-contextuality, so it is not of interest for the present purpose.",
"The third conjunction is false since the existence of a 3-joint for a given choice of 2-joints would also imply the existence of a 3-joint for the three measurements, hence contradicting the fact that these admit no 3-joints.",
"Thus the two remaining conjunctions of interest are: Proposition 1: $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ , Proposition 2: $(\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ $\\Leftrightarrow (\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ .",
"These two possibilities lead to the following propositions: Weak: $(\\exists \\text{ 2-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ , Strong: $(\\exists \\text{ 2-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ for all choices of 2-joints})$ $\\Leftrightarrow (\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ .",
"where Weak $\\Leftrightarrow $ Proposition 1 $\\bigvee $ Proposition 2, and Strong $\\Leftrightarrow $ Proposition 2.",
"The proposition Weak relaxes the requirement of proposition Strong that the three measurements should themselves be incompatible to only the requirement that there exists a choice of 2-joints that do not admit a 3-joint.",
"Obviously, under Strong, there exists no 3-joint for all possible choices of 2-joints: Strong $\\Rightarrow $ Weak.Note that for the case of PVMs, only the conjunction $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})\\bigwedge (\\exists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ makes sense and that it is, in fact, equivalent to the proposition $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ since there is no “choice of 2-joints” available: the 2-joints, if they exist, are unique and admit a unique 3-joint (cf.",
"Theorem REF ).",
"Consequently, the propositions Weak and Strong are not admissible for PVMs."
],
[
"Comment on Ref. {{cite:c0832b8daad9c8ae7ed9a6310e8e3ea68b2163ee}} vis-à-vis Ref. {{cite:99395bb57ef04b1d7179ebb33e3c32e27d9e2bd7}}",
"In Ref.",
"[2], contextuality—in the generalized sense of Spekkens [5] and by implication in the Kochen-Specker sense—was shown keeping in mind the proposition Strong, i.e., requiring that the three measurements $\\lbrace M_1,M_2,M_3\\rbrace $ are pairwise jointly measurable but not triplewise jointly measurable.",
"This was in keeping with the approach adopted in Ref.",
"[1], where the construction used did not violate the LSW inequality [1], [2].",
"Indeed, as shown in Theorem 1 of Ref.",
"[2], the construction used in Ref.",
"[1] could not have produced a violation because it sought a state-independent violation.",
"In Ref.",
"[4], the authors—under Proposition 1—use the construction first obtained in [2] to show a higher violation of the LSW inequality than reported in [2].",
"It is easy to check that the construction in Ref.",
"[2] recovers the violation reported in Ref.",
"[4] when the proposition Strong is relaxed to the proposition Weak: the expression for the quantum probability of anticorrelation in Ref.",
"[2] is given by $R_3^Q=\\frac{C}{6}+(1-\\frac{\\eta }{3})$ where $C>0$ for a state-dependent violation of the LSW inequality [1], [2].",
"Given a coplanar choice of measurement directions $\\lbrace \\hat{n}_1,\\hat{n}_2,\\hat{n}_3\\rbrace $ , and $\\eta $ satisfying $\\eta _l<\\eta \\le \\eta _u$ , the optimal value of $C$ —denoted as $C^{\\lbrace \\hat{n}_i\\rbrace ,\\eta }_{\\max }$ —is given by $\\nonumber &&C^{\\lbrace \\hat{n}_i\\rbrace ,\\eta }_{\\max }=2\\eta \\\\&+&\\sum _{(ij)}\\left(\\sqrt{1+\\eta ^4(\\hat{n}_i.\\hat{n}_j)^2-2\\eta ^2}-(1+\\eta ^2 \\hat{n}_i.\\hat{n}_j)\\right).$ For trine measurements, $\\hat{n}_i.\\hat{n}_j=-\\frac{1}{2}$ for each pair of measurement directions, $\\lbrace \\hat{n}_i,\\hat{n}_j\\rbrace $ .",
"Also, $\\eta _l=\\frac{2}{3}$ and $\\eta _u=\\sqrt{3}-1$ .",
"$\\eta >\\eta _l$ ensures that the three measurements corresponding to $\\lbrace \\hat{n}_1,\\hat{n}_2,\\hat{n}_3\\rbrace $ do not admit a 3-joint while $\\eta \\le \\eta _u$ is necessary and sufficient for 2-joints to exist: that is, $\\eta _l<\\eta \\le \\eta _u$ corresponds to the proposition Strong, $(\\exists \\text{ 2-joint}, \\nexists \\text{ 3-joint})$ .",
"On relaxing the requirement $\\eta _l<\\eta $ , we have $0\\le \\eta \\le \\eta _u$ .",
"This allows room for the proposition $(\\exists \\text{ 2-joint}, \\exists \\text{ 3-joint})$ when $0\\le \\eta \\le \\eta _l$ .",
"The quantity to be maximized is the quantum violation: $R_3^Q-(1-\\frac{\\eta }{3})=\\frac{C}{6}$ .",
"Substituting the value $\\hat{n}_i.\\hat{n}_j=-\\frac{1}{2}$ in Eq.",
"(REF ), the quantum probability of anticorrelation from Eq.",
"(REF ) for trine measurements is given by: $R_3^Q=\\frac{1}{2}+\\frac{\\eta ^2}{4}+\\frac{1}{2}\\sqrt{1-2\\eta ^2+\\frac{\\eta ^4}{4}},$ which is the same as the bound in Eq.",
"(11) in Theorem 3 of Ref.",
"[4].",
"The quantum violation is given by: $R_3^Q-(1-\\frac{\\eta }{3})=-\\frac{1}{2}+\\frac{\\eta }{3}+\\frac{\\eta ^2}{4}+\\frac{1}{2}\\sqrt{1-2\\eta ^2+\\frac{\\eta ^4}{4}}.$ In Ref.",
"[2], this expression was maximized under the proposition Strong ($\\eta _l<\\eta \\le \\eta _u$ ) and the quantum violation was seen to approach a maximum of $0.0336$ for $R_3^Q\\rightarrow 0.8114$ as $\\eta \\rightarrow \\eta _l=\\frac{2}{3}$ .",
"In Ref.",
"[4], the same expression was maximized while relaxing proposition Strong to proposition Weak (allowing $\\eta \\le \\eta _l$ ) and the maximum quantum violation was seen to be $0.0896$ for $R_3^Q=0.9374$ and $\\eta \\approx 0.4566$ .",
"Another comment in Ref.",
"[4] is the following: “Interestingly, there are three observables that are not triplewise jointly measurable but cannot violate LSW's inequality no matter how each two observables are jointly measured.” That is, Strong $\\nRightarrow $ Violation of LSW inequality.",
"Equally, it is also the case that Weak $\\nRightarrow $ Violation of LSW inequality.",
"Neither of these is surprising given the discussion in this note.",
"In particular, note the following implications ($0\\le \\eta \\le 1$ ): Violation of LSW inequality, i.e., $R_3^Q>1-\\frac{\\eta }{3}$ $\\Rightarrow $ Violation of KS inequality, i.e., $R_3^Q>\\frac{2}{3}$ , Violation of KS inequality, i.e., $R_3^Q>\\frac{2}{3}$ $\\Rightarrow $ Weak: $(\\exists \\text{ 2-joint})\\bigwedge (\\nexists \\text{ 3-joint}|\\text{ a choice of 2-joints})$ , Strong $\\Rightarrow $ Weak.",
"Therefore, Weak is a necessary condition for a violation of the LSW inequality.",
"It can be satisfied either under Proposition 1 (as done in [4]) or under Proposition 2 (or Strong, as done in [2])."
],
[
"Joint measurability structures",
"I end this note with a comment on the result proven in Ref.",
"[3], where it was shown constructively that any conceivable joint measurability structure for a set of $N$ observables is realizable via binary POVMs.",
"With regard to contextuality, this result proves the admissibility in quantum theory of contextuality scenarios that are not realizable with PVMs alone.",
"This should be easy to see, specifically, from the example of Specker's scenario, where PVMs do not suffice to demonstrate contextuality, primarily because they possess a very rigid joint measurability structure dictated by pairwise commutativity and their joint measurements are unique (Theorem REF ).",
"If one can demonstrate contextuality given the scenarios obtained from more general joint measurability structures then a relaxation of a sort similar to the case of Specker's scenario (from Strong to Weak) will also lead to contextuality.",
"In this sense, an implication of the result of Ref.",
"[3] is that it allows one to consider the question of contextuality for joint measurability structures which admit no PVM realization in quantum theory on account of Theorem REF .",
"In particular, for PVMs, pairwise compatibility $\\Leftrightarrow $ global compatibility because commutativity is a necessary and sufficient criterion for compatibility.",
"On the other hand, POVMs allow for a failure of the implication pairwise compatibility $\\Rightarrow $ global compatibility because pairwise compatibility is not equivalent to pairwise commutativity for POVMs: pairwise commutativity $\\Rightarrow $ pairwise compatibility, but not conversely."
],
[
"Conclusion",
"I hope this note clarifies issues that may have escaped analysis in Refs.",
"[1], [2], [3], [4].",
"In particular, the logical relationship between admissible joint measurability structures and the possibility of contextuality should be clear from the discussion here."
],
[
"Acknowledgment",
"I would like to thank Sibasish Ghosh and Prabha Mandayam for comments on earlier drafts of this article."
]
] | 1403.0470 |
[
[
"Proper cocycles and weak forms of amenability"
],
[
"Abstract Let $G$ and $H$ be locally compact, second countable groups.",
"Assume that $G$ acts in a measure class preserving way on a standard probability space $(X,\\mu)$ such that $L^\\infty(X,\\mu)$ has an invariant mean and that there is a Borel cocycle $\\alpha:G\\times X\\rightarrow H$ which is proper in a suitable, natural sense.",
"We show that if $H$ has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does $G$.",
"We observe that it is the case for a weak form of measure equivalence for pairs of discrete groups."
],
[
"Introduction",
"Let $G$ and $H$ be locally compact, second countable groups.",
"If $H$ is a closed subgroup of $G$ , it inherits weak amenability properties of $G$ .",
"Conversely, if in addition the homogeneous space $G/H$ is amenable in the sense that $L^\\infty (G/H)$ has a $G$ -invariant state, then $G$ in turn inherits weak amenability properties of $H$ : see for instance [1], [2], [7] and the recent article [9].",
"One way of proving such results is to use a natural cocycle associated to some regular Borel cross-section $\\gamma :G/H\\rightarrow G$ of the canonical projection $p:G\\rightarrow G/H$ .",
"More precisely, we define $\\alpha :G\\times G/H\\rightarrow H$ by $\\alpha (g,x)=\\gamma (gx)^{-1}g\\gamma (x)\\quad ((g,x)\\in G\\times G/H).$ Since $g\\gamma (x)H=\\gamma (gx)H$ for all $g\\in G$ and $x\\in G/H$ , we see that $\\alpha (g,x)\\in H$ as well.",
"More generally, assume that $G$ acts in a measure class preserving way on some standard probability space $(X,\\mu )$ .",
"Then a Borel cocycle is a Borel map $\\alpha :G\\times X\\rightarrow H$ such that for all $g,h\\in G$ , one has $\\alpha (gh,x)=\\alpha (g,hx)\\alpha (h,x)$ for $\\mu $ -almost every $x\\in X$ .",
"Notice that $H$ is not necessarily a closed subgroup of $G$ .",
"The aim of the present note is to prove that weak amenability properties of $H$ are inherited by $G$ when $\\alpha $ is a cocycle that satisfies a properness condition in the sense of Definition REF below, and when $L^\\infty (X)$ has a $G$ -invariant state.",
"In order to state precisely our main result, we need to recall some definitions from [2], [3], [9], [7] and [8].",
"We assume throughout the article that our groups are locally compact and second countable.",
"The definitions of the algebras $A(G)$ and $B_2(G)$ are reminded in the next section.",
"Definition 1.1 Let $G$ be a locally compact, second countable group.",
"([2]) We say that $G$ has the Haagerup property if there exists a sequence $(u_n)_{n\\ge 1}$ of normalized, positive definite functions on $G$ , such that $u_n\\in C_0(G)$ for every $n$ , and that $u_n\\rightarrow 1$ uniformly on compact subsets of $G$ .",
"($C_0(G)$ denotes the vector space of all continuous functions on $G$ that tend to 0 at infinity.)",
"([3]) The Cowling-Haagerup constant $\\Lambda _{\\mathrm {WA}}(G)$ is the infimum of all numbers $C\\ge 1$ for which there exists a sequence $(u_n)_{n\\ge 1}$ in the Fourier algebra $A(G)$ satisfying: $\\Vert u_n\\Vert _{B_2}\\le C$ for every $n$ and $u_n\\rightarrow 1$ uniformly on compact sets, where $B_2(G)$ denotes the Herz-Schur multiplier algebra of $G$ (see [9]).",
"Moreover, $G$ is said to be weakly amenable if $\\Lambda _{\\mathrm {WA}}(G)<\\infty $ .",
"([9]) The weak Haagerup constant $\\Lambda _{\\mathrm {WH}}(G)$ is the infimum of all numbers $C\\ge 1$ for which there exists a sequence $(u_n)_{n\\ge 1}\\subset B_2(G)\\cap C_0(G)$ such that $\\Vert u_n\\Vert _{B_2}\\le C$ for every $n$ and $u_n\\rightarrow 1$ uniformly on compact sets.",
"Moreover, $G$ is said to have the weak Haagerup property if $\\Lambda _{\\mathrm {WH}}(G)<\\infty $ .",
"As $A(G)\\subset B_2(G)\\cap C_0(G)$ , one always has $\\Lambda _{\\mathrm {WH}}(G)\\le \\Lambda _{\\mathrm {WA}}(G)$ , and a weakly amenable group has the weak Haagerup property.",
"Similarly, as normalized, positive definite functions are Herz-Schur multipliers of norm one, if $G$ has the Haagerup property then it has the weak Haagerup property and $\\Lambda _{\\mathrm {WH}}(G)=1$ .",
"See [9] for a discussion of these properties.",
"The following definition generalizes the notion of co-Følner groups as in [2].",
"Definition 1.2 ([6], [8], [11]) Let $G$ be a locally compact, second countable group that acts in a measure class preserving way on the standard space $(X,\\mu )$ .",
"Then we say that $(G,X)$ is an amenable pair if $L^\\infty (X,\\mu )$ has a $G$ -invariant state.",
"Finally, let $\\alpha :G\\times X\\rightarrow H$ be a Borel cocycle.",
"The following is essentially taken from [7]; see also [9].",
"We need first to fix some notation: let $A\\subset X$ be a Borel set and let $L$ be a compact subset of $H$ ; we denote by $K(A,L)$ the set of all elements $g\\in G$ for which $\\mu (X_A(g))>0$ , where $X_A(g)=\\lbrace x\\in A\\cap g^{-1}A: \\alpha (g,x)\\in L\\rbrace .$ As mentioned above, it is motivated by the case where $H$ is a closed subgroup of $G$ that we present in detail now.",
"Choose a regular Borel cross-section $\\gamma :G/H\\rightarrow G$ for the canonical projection $p:G\\rightarrow G/H$ , i.e.",
"$\\gamma $ is a Borel map such that $p(\\gamma (x))=x$ for every $x\\in G/H$ and such that, for every compact set $K\\subset G$ , the set $\\gamma (G/H)\\cap p^{-1}(p(K))$ is precompact in $G$ .",
"Moreover, for every compact set $C\\subset G/H$ , there is a compact set $C_1\\subset G$ such that $C=p(C_1)$ .",
"Then it is straightforward to see that $\\gamma (C)$ is precompact for every compact set $C\\subset G/H$ .",
"Recall that the associated cocycle $\\alpha $ is defined by $\\alpha (g,x)=\\gamma (gx)^{-1}g\\gamma (x)\\quad (g\\in G, x\\in G/H).$ Then one has: If $K\\subset G$ and $A\\subset G/H$ are compact sets, then $\\alpha (K\\times A)$ is precompact since it is contained in $\\gamma (KA)^{-1}K\\gamma (A)$ .",
"If $A\\subset G/H$ and $L\\subset H$ are compact, it is easy to see that, if $g\\in G$ is such that $X_A(g)\\ne \\emptyset $ , then $g\\in \\gamma (A)L\\gamma (A)^{-1}$ .",
"In particular $K(A,L)$ is precompact.",
"As $G$ and $G/H$ are in particular $\\sigma $ -compact, we see that the cocycle $\\alpha $ satisfies the next definition.",
"Definition 1.3 The cocycle $\\alpha :G\\times X\\rightarrow H$ is proper if it satisfies the following two conditions: for every Borel set $A\\subset X$ , for every $\\varepsilon >0$ and for every compact set $K\\subset G$ , there exists a Borel set $A_{\\varepsilon ,K}\\subset A$ such that $\\mu (A\\setminus A_{\\varepsilon ,K})<\\varepsilon $ and $\\alpha (K\\times A_{\\varepsilon ,K})$ is precompact.",
"For every compact set $L\\subset H$ and every $\\varepsilon >0$ , there exists a Borel set $A\\subset X$ such that $\\mu (X\\setminus A)\\le \\varepsilon $ and $K(A,L)$ is precompact.",
"We observe first that properness of $\\alpha $ is independent of the chosen probability measure.",
"Lemma 1.4 Let $\\alpha :G\\times X\\rightarrow H$ be a proper cocycle with respect to the probability measure $\\mu $ on $X$ , and let $\\nu $ be an equivalent probability measure on $X$ .",
"Then $\\alpha $ is proper with respect to $\\nu $ .",
"Proof.",
"This follows immediately from Theorem 6.11 of [10]: as $\\nu $ is equivalent to $\\mu $ , it follows in particular that for every $\\varepsilon >0$ , there exists $\\theta >0$ such that, if $B\\subset X$ is Borel and if $\\mu (B)\\le \\theta $ , then $\\nu (B)\\le \\varepsilon $ .",
"$\\square $ Before giving examples of such cocycles, let us state our first main result: Theorem 1.5 Let $G$ and $H$ be locally compact, second countable groups, let $G$ act on some probability space $(X,\\mu )$ so that $(G,X)$ is an amenable pair, and let $\\alpha :G\\times X\\rightarrow H$ be a proper cocycle.",
"If $H$ has the Haagerup property, then so does $G$ .",
"If $H$ is weakly amenable group then so is $G$ , and $\\Lambda _{\\mathrm {WA}}(G)\\le \\Lambda _{\\mathrm {WA}}(H).$ If $H$ has the weak Haagerup property, then so does $G$ , and $\\Lambda _{\\mathrm {WH}}(G)\\le \\Lambda _{\\mathrm {WH}}(H).$ As we will see, the proofs of the three statements rely on the same techniques, and they will be given in the next section.",
"As promised, here are examples of proper cocycles.",
"Example 1.6 Every cocycle $\\alpha $ described below is proper.",
"Let $G$ and $H$ be locally compact, second countable groups and assume that $\\sigma : G\\rightarrow H$ is a continuous homomorphism with compact kernel, and let $(X,\\mu )$ be an arbitrary standard probability $G$ -space.",
"Define $\\alpha :G\\times X\\rightarrow H$ by $\\alpha (g,x)=\\sigma (g)\\quad ((g,x)\\in G\\times X).$ More generally, let $G$ and $H$ be as in Example (1), let $G_0$ be a closed subgroup of $G$ and assume that $\\sigma :G_0\\rightarrow H$ is a continuous homomorphism with compact kernel.",
"Choose a regular Borel cross-section $\\gamma :G/G_0\\rightarrow G$ for the canonical projection and a quasi-invariant probability measure $\\nu $ on $G/G_0$ .",
"If $(Y,\\mu )$ is an arbitrary standard probability $G$ -space, equip $(G/G_0\\times Y,\\nu \\times \\mu )$ with the product action and define $\\alpha :G\\times (G/G_0\\times Y)\\rightarrow H$ by $\\alpha (g,(x,y))=\\sigma (\\gamma (gx)^{-1}g\\gamma (x)).$ This case generalizes the situation where $H=G_0$ is a closed subgroup of $G$ .",
"Let $\\pi :P\\rightarrow B$ be a (metrizable, locally compact) topological principal fiber bundle with countable structure group $H$ , and assume that a locally compact, second countable group $G$ acts continuously on $P$ so that the $G$ -action commutes with the $H$ -action.",
"Using a bounded measurable cross-section for $\\pi $ , we get a trivialization of $P\\cong H\\times B$ which preserves precompact subsets.",
"Then the corresponding action of $G$ on $H\\times B$ is given by $g\\cdot (h,x)=(\\alpha (g,x)h,gx)$ where $\\alpha $ is a Borel cocycle.",
"If the action of $G$ on $P$ is proper, then $\\alpha $ is a proper cocycle.",
"As will be proved in the last section, an important fourth family of pairs of groups that give rise to proper cocycles is the family of pairs of countable, discrete groups $\\Gamma $ and $\\Delta $ that satisfy Gromov's notion of measure equivalence.",
"We recall the latter from [4]: Definition 1.7 We say that $\\Gamma $ and $\\Delta $ are measure equivalent if there exist commuting, measure-preserving, free actions of $\\Gamma $ and $\\Delta $ on some infinite-measure standard space $(\\Sigma ,\\sigma )$ , such that $\\Gamma $ and $\\Delta $ both admit fundamental domains with finite measure.",
"(For convenience, we denote the action of $\\Gamma $ on the left and the action of $\\Delta $ on the right.)",
"Measure equivalence is a weak form of orbit equivalence: see [5], Section 3.",
"Let us generalize it slightly as follows.",
"Definition 1.8 We say that $\\Gamma $ and $\\Delta $ are amenably measure equivalent if there exist commuting, measure-preserving, free actions of $\\Gamma $ and $\\Delta $ on some infinite-measure standard space $(\\Sigma ,\\sigma )$ , such that the pairs $(\\Gamma ,\\Sigma /\\Delta )$ and $(\\Delta ,\\Gamma \\backslash \\Sigma )$ are both amenable pairs.",
"Example 1.9 Let $\\Gamma $ and $\\Delta $ be discrete subgroups of the same locally compact, second countable, unimodular group $G$ such that the homogeneous spaces $G/\\Gamma $ and $G/\\Delta $ both have $G$ -invariant means, i.e.",
"the pairs $(G,G/\\Gamma )$ and $(G,G/\\Delta )$ are amenable.",
"Then $\\Gamma $ and $\\Delta $ are amenably measure equivalent groups.",
"As claimed above, amenably measure equivalent groups give rise to proper cocycles, so that Theorem REF will be used to prove our last result.",
"Theorem 1.10 Let $\\Gamma $ and $\\Delta $ be amenably measure equivalent groups.",
"If one of them has the Haagerup property, then the other one has the same property.",
"The following equalities hold: $\\Lambda _{\\mathrm {WA}}(\\Gamma )=\\Lambda _{\\mathrm {WA}}(\\Delta )\\quad \\textrm {and}\\quad \\Lambda _{\\mathrm {WH}}(\\Gamma )=\\Lambda _{\\mathrm {WH}}(\\Delta ).$ Let us discuss instances where Theorem REF applies in the context of semidirect products.",
"Let $A$ be an amenable, countable group.",
"If it acts on some group $\\Gamma $ , then it is easy to see that $\\Gamma $ and $\\Gamma \\rtimes A$ are amenably measure equivalent.",
"The fact that $\\Gamma $ and $\\Gamma \\rtimes A$ have simultaneously the Haagerup property and that their constants $\\Lambda _{\\mathrm {WA}}$ and $\\Lambda _{\\mathrm {WH}}$ coincide is already known: it follows respectively from [2], [7] and [9].",
"Next, let us assume that the amenable group $A$ is on the other side: let $\\theta :\\Delta \\rightarrow \\mathrm {Aut}(A)$ be an action of the countable group $\\Delta $ on $A$ , and consider the following action of the semidirect product group $\\Gamma :=A\\rtimes _{\\theta }\\Delta $ on $A$ : $(a,\\delta )\\cdot b=a\\theta _{\\delta }(b)$ for all $a,b\\in A$ and all $\\delta \\in \\Delta $ .",
"Then we have: Proposition 1.11 Let $A$ , $\\Delta $ and $\\theta $ be as above and assume that there exists a sequence $(F_k)_{k\\ge 1}$ of finite subsets of $A$ such that, for all $(a,\\delta )\\in \\Gamma $ $\\frac{|a\\theta _{\\delta }(F_k)\\bigtriangleup F_k|}{|F_k|}\\rightarrow 0$ as $k\\rightarrow \\infty $ .",
"In other words, the pair $(\\Gamma ,A)$ is amenable in the sense of Definition REF .",
"Then $\\Gamma $ and $\\Delta $ are amenably measure equivalent.",
"Proof.",
"Take $\\Sigma =A\\times \\Delta $ with counting measure.",
"The action (on the left) of $\\Gamma =A\\rtimes \\Delta $ is defined by multiplication in the semidirect product, namely $(a,\\delta )\\cdot (b,\\delta ^{\\prime })=(a\\theta _{\\delta }(b),\\delta \\delta ^{\\prime })$ and the action of $\\Delta $ (on the right) is given by $(a,\\delta )\\cdot \\delta ^{\\prime }=(a,\\delta \\delta ^{\\prime })$ .",
"Then $\\Gamma \\backslash \\Sigma $ is the one-point space and $\\Sigma /\\Delta $ is isomorphic to $A$ as a $\\Gamma $ -space.",
"The amenability of the pair $(\\Gamma ,A)$ implies that $\\Gamma $ and $\\Delta $ are amenably measure equivalent.",
"$\\square $ Let us illustrate Proposition REF by an example.",
"Example 1.12 Let $1\\rightarrow Z\\rightarrow A\\rightarrow Q\\rightarrow 1$ be a central extension of some amenable group $Q$ .",
"Denote by $s:Q\\rightarrow A$ a section of the canonical projection with $s(1)=1$ .",
"Let also $h:\\Delta \\rightarrow Q$ be a homomorphism of some group $\\Delta $ .",
"With these data, define an action $\\theta $ of $\\Delta $ on $A$ by $\\theta _{\\delta }=\\mathrm {Ad}(s\\circ h(\\delta ))\\quad \\forall \\delta \\in \\Delta .$ As each automorphism $\\theta _{\\delta }$ is inner, $(A,\\Delta ,\\theta )$ fulfills the conditions of Proposition REF : indeed, since $A$ is amenable, the direct product group $A\\times A$ is as well, and the action $(a,b)\\cdot x=axb^{-1}$ of $A\\times A$ on $A$ admits a sequence $(F_k)_{k\\ge 1}$ of finite subsets of $A$ such that $\\frac{|(a,b)\\cdot F_k\\bigtriangleup F_k|}{|F_k|}\\rightarrow 0$ as $k\\rightarrow \\infty $ for all $a,b\\in A$ .",
"Remark also that $s\\circ h$ is not a homomorphism in general, thus $\\Gamma :=A\\rtimes _{\\theta }\\Delta $ is not isomorphic to the direct product $A\\times \\Delta $ .",
"$\\square $ Acknowledgements.",
"I am very grateful to Tadeusz Januszkiewicz and the referee for their careful reading of the manuscript and their valuable comments."
],
[
"Proof of Theorem ",
"Let $G$ and $H$ be locally compact groups; we assume that they are second countable even if definitions below make sense for arbitrary locally compact groups.",
"The Fourier-Stieltjes algebra of $G$ is the set of all coefficient functions of unitary representations of $G$ , thus, for every $u\\in B(G)$ there exists a unitary representation $(\\pi ,\\mathcal {H})$ of $G$ and two vectors $\\xi ,\\eta \\in \\mathcal {H}$ such that $u(g)=\\langle \\pi (g)\\xi |\\eta \\rangle $ for every $g\\in G$ .",
"It is a Banach algebra with respect to the norm $\\Vert u\\Vert _B=\\inf \\Vert \\xi \\Vert \\Vert \\eta \\Vert $ where the infimum is taken over all representations of $u$ as above.",
"The Fourier algebra of $G$ is the set of all coefficient functions associated to the left regular representation $\\lambda $ of $G$ (which acts on $L^2(G)$ ).",
"It is the norm closure of the algebra of compactly supported continuous functions $C_c(G)\\cap B(G)$ in the algebra $B(G)$ .",
"A Herz-Schur multiplier of $G$ is a continous function $u:G\\rightarrow \\mathbb {C}$ for which there exists a separable Hilbert space $\\mathcal {H}$ and two bounded, continuous functions $\\xi ,\\eta :G\\rightarrow \\mathcal {H}$ such that $u(h^{-1}g)=\\langle \\xi (g)|\\eta (h)\\rangle \\quad (g,h\\in G).$ It turns out that the set $B_2(G)$ of all Herz-Schur multipliers on $G$ is a Banach algebra with respect to the pointwise product and to the norm $\\Vert u\\Vert _{B_2}=\\inf \\Vert \\xi \\Vert _\\infty \\Vert \\eta \\Vert _\\infty $ where the infimum is taken over all representations of $u$ as above.",
"Assume from now on that $G$ acts in a measure class preserving way on a standard probability space $(X,\\mu )$ and that $\\alpha :G\\times X\\rightarrow H$ is a (not necessarily proper) Borel cocycle.",
"We denote by $(g,x)\\mapsto \\chi (g,x)$ the Radon-Nikodym derivative related to the action of $G$ on $X$ and characterized by $\\int \\limits _X f(gx)\\chi (g,x)d\\mu (x)=\\int \\limits _X f(x)d\\mu (x)\\quad \\forall f\\in L^1(X,\\mu ).$ It satisfies the cocycle relation: $\\chi (gh,x)=\\chi (g,hx)\\chi (h,x)$ for all $g,h\\in G$ and $\\mu $ -a.e.",
"$x\\in X$ .",
"Taking $f=1_X$ , we have: $\\int \\limits _X\\chi (g,x)d\\mu (x)=1$ for every $g\\in G$ .",
"For future use, let us observe that for every $g\\in G$ and every Borel set $B\\subset X$ , one has, by Cauchy-Schwarz Inequality: $\\int \\limits _B\\sqrt{\\chi (g,x)}d\\mu (x)\\le \\mu (B)^{1/2}\\left(\\int \\limits _X\\chi (g,x)d\\mu (x)\\right)^{1/2}=\\mu (B)^{1/2}$ and $\\int \\limits _{g^{-1}B}\\sqrt{\\chi (g,x)}d\\mu (x)\\le \\mu (X)^{1/2}\\left(\\int \\limits _X 1_B(gx)\\chi (g,x)d\\mu (x)\\right)^{1/2}=\\mu (B)^{1/2}.$ The proof of Theorem REF relies on two auxiliary results.",
"Lemma 2.1 Let $u\\in L^\\infty (H)$ .",
"Define $\\hat{u}:G\\rightarrow \\mathbb {C}$ by $\\hat{u}(g)=\\int \\limits _X u(\\alpha (g,x))\\sqrt{\\chi (g,x)}d\\mu (x)\\quad \\forall g\\in G.$ If $u\\in B_2(H)$ is a Herz-Schur multiplier on $H$ , then $\\hat{u}\\in B_2(G)$ and $\\Vert \\hat{u}\\Vert _{B_2}\\le \\Vert u\\Vert _{B_2}$ .",
"If furthermore $u$ is positive definite, so is $\\hat{u}$ .",
"If $\\alpha $ is proper and if $u\\in C_0(H)$ then $\\hat{u}\\in C_0(G)$ .",
"In particular, if $u\\in B_2(H)\\cap C_0(H)$ , then $\\hat{u}\\in B_2(G)\\cap C_0(G)$ .",
"If $\\alpha $ is proper and if $u\\in A(H)$ , then $\\hat{u}\\in A(G)$ .",
"Proof.",
"(a) There exist a separable Hilbert space $\\mathcal {H}$ and bounded, continuous functions $\\xi ,\\eta :H\\rightarrow \\mathcal {H}$ such that $u(t^{-1}s)=\\langle \\xi (s)|\\eta (t)\\rangle $ for all $s,t\\in H$ ; $\\Vert u\\Vert _{B_2}\\le \\Vert \\xi \\Vert _\\infty \\Vert \\eta \\Vert _\\infty $ .",
"Define $\\hat{\\xi },\\hat{\\eta }:G\\rightarrow L^2(X,\\mu ,\\mathcal {H})$ by $\\hat{\\xi }(g)(x)=\\xi (\\alpha (g^{-1},x)^{-1})\\sqrt{\\chi (g^{-1},x)}$ and $\\hat{\\eta }(g)(x)=\\eta (\\alpha (g^{-1},x)^{-1})\\sqrt{\\chi (g^{-1},x)}.$ One has for every $g\\in G$ : $\\Vert \\hat{\\xi }(g)\\Vert ^2 &=& \\int \\limits _X\\Vert \\xi (\\alpha (g^{-1},x))\\Vert ^2\\chi (g^{-1},x)d\\mu (x)\\\\&\\le &\\Vert \\xi \\Vert _\\infty ^2\\int \\limits _X\\chi (g^{-1},x)d\\mu (x)=\\Vert \\xi \\Vert _\\infty ^2.$ Similarly, $\\Vert \\hat{\\eta }\\Vert _\\infty \\le \\Vert \\eta \\Vert _\\infty $ .",
"We are going to prove that, for all $g,h\\in G$ , one has: $\\hat{u}(h^{-1}g)=\\langle \\hat{\\xi }(g)|\\hat{\\eta }(h)\\rangle .$ It turns out that $\\hat{u}$ is a continuous function on $G$ by Appendix C of [9] (even though $\\hat{\\xi }$ and $\\hat{\\eta }$ are not necessarily continuous).",
"Observe that the cocycle relation $\\alpha (gh,x)=\\alpha (g,hx)\\alpha (h,x)$ for all $g,h\\in G$ and for $\\mu $ -a.e $x\\in X$ implies that $\\alpha (h^{-1}g,g^{-1}x)\\alpha (g^{-1},x)=\\alpha (h^{-1},x)$ and similarly $\\chi (h^{-1}g,g^{-1}x)\\chi (g^{-1},x)=\\chi (h^{-1},x)$ for all $g,h\\in G$ and $\\mu $ -a.e.",
"$x\\in X$ .",
"Fix $g,h\\in G$ .",
"One has: $\\langle \\hat{\\xi }(g)|\\hat{\\eta }(h)\\rangle &=&\\int \\limits _X\\langle \\xi (\\alpha (g^{-1},x)^{-1})|\\eta (\\alpha (h^{-1},x)^{-1})\\rangle \\sqrt{\\chi (g^{-1},x)\\chi (h^{-1},x)}d\\mu (x)\\\\&=&\\int \\limits _X u(\\alpha (h^{-1},x)\\alpha (g^{-1},x)^{-1})\\sqrt{\\chi (g^{-1},x)\\chi (h^{-1},x)}d\\mu (x)\\\\&=&\\int \\limits _X u(\\alpha (h^{-1}g,g^{-1}x))\\sqrt{\\chi (g^{-1},x)\\chi (h^{-1},x)}d\\mu (x)\\\\&=&\\int \\limits _X u(\\alpha (h^{-1}g,g^{-1}x))\\chi (g^{-1},x)\\sqrt{\\chi (h^{-1}g,g^{-1}x)}d\\mu (x)\\\\&=&\\int \\limits _X u(\\alpha (h^{-1}g,x))\\sqrt{\\chi (h^{-1}g,x)}d\\mu (x)=\\hat{u}(h^{-1}g).$ If furthermore $u$ is positive definite, then one can take $\\eta =\\xi $ , and it is straightforward to see that $\\hat{u}$ is positive definite on $G$ as well.",
"(In fact, let $(\\pi _u,\\mathcal {H}_u,\\xi _u)$ be the Gel'fand-Naimark-Segal triple associated to $u$ .",
"Then, as $u(h)=\\langle \\pi _u(h)\\xi _u|\\xi _u\\rangle $ for every $h\\in H$ , we see that the function $h\\mapsto \\xi (h)=\\pi _u(h)\\xi _u$ works.)",
"(b) Let now $u\\in C_0(H)$ .",
"We assume without loss of generality that $\\Vert u\\Vert _\\infty \\le 1$ .",
"Fix $\\varepsilon >0$ .",
"There exists a compact set $L\\subset H$ such that $|u(h)|\\le \\frac{\\varepsilon }{2}$ for all $h\\notin L$ .",
"Choose next a Borel set $A\\subset X$ which satisfies: $\\mu (X\\setminus A)\\le \\frac{\\varepsilon ^2}{16}$ and for which $K=\\overline{K(A,L)}$ is compact.",
"Fix $g\\in G\\setminus K$ .",
"One has: $|\\hat{u}(g)| &\\le &\\int \\limits _{\\lbrace \\alpha (g,x)\\in L\\rbrace } |u(\\alpha (g,x))|\\sqrt{\\chi (g,x)}d\\mu (x)\\\\& &+\\int \\limits _{\\lbrace \\alpha (g,x)\\notin L\\rbrace }|u(\\alpha (g,x))|\\sqrt{\\chi (g,x)}d\\mu (x)\\\\&\\le &\\int \\limits _{X\\setminus (A\\cap g^{-1}A)}|u(\\alpha (g,x))|\\sqrt{\\chi (g,x)}d\\mu (x)+\\frac{\\varepsilon }{2}\\\\&\\le &\\int \\limits _{X\\setminus A}\\sqrt{\\chi (g,x)}d\\mu (x)+\\int \\limits _{g^{-1}(X\\setminus A)}\\sqrt{\\chi (g,x)}d\\mu (x)+\\frac{\\varepsilon }{2}\\\\&\\le &2\\mu (X\\setminus A)^{1/2}+\\frac{\\varepsilon }{2}\\le \\varepsilon .$ (c) The assertion is similar to that contained in Proposition 2.8 of [7], but we give a proof for the sake of completeness.",
"We have to prove that $\\hat{u}$ is in the norm closure of $C_c(G)\\cap B(G)$ in the Fourier-Stieltjes algebra $B(G)$ .",
"Thus, let us assume that $u(h)=\\langle \\lambda (h)\\xi |\\eta \\rangle $ with $\\xi ,\\eta \\in C_c(H)$ .",
"Then it is straighforward to check that $\\hat{u}(g)=\\langle \\lambda _\\alpha (g)1_X\\otimes \\xi |1_X\\otimes \\eta \\rangle \\quad (g\\in G)$ where $\\lambda _\\alpha $ is the unitary representation of $G$ on $L^2(X,\\mu ,L^2(H))$ defined by $(\\lambda _\\alpha (g)\\zeta )(x)=\\lambda (\\alpha (g^{-1},x)^{-1})\\zeta (g^{-1}x)\\sqrt{\\chi (g^{-1},x)}$ for $\\zeta \\in L^2(X,\\mu ,L^2(H))$ .",
"Hence $\\hat{u}$ is a norm limit in $B(G)$ of functions of the form $\\hat{u}_A(g)=\\langle \\lambda _\\alpha (g)1_A\\otimes \\xi |1_A\\otimes \\eta \\rangle $ with $A\\subset X$ Borel.",
"Denote by $L$ the support of $u$ .",
"For every Borel set $A\\subset X$ such that $K(A,L)$ is precompact, we have $\\hat{u}_A(g)=\\int \\limits _{A\\cap g^{-1}A}u(\\alpha (g,x))\\sqrt{\\chi (g,x)}d\\mu (x) \\quad (g\\in G).$ If $g\\notin \\overline{K(A,L)}$ , the latter being compact in $G$ , then $\\mu (\\lbrace x\\in A\\cap g^{-1}A:\\alpha (g,x)\\in L\\rbrace )=0$ by Definition REF and this implies that $\\hat{u}_A(g)=0$ .",
"$\\square $ Assume now that $(G,X)$ is an amenable pair.",
"Denote by $\\beta $ the action of $G$ on $L^1(X,\\mu )$ given by $\\beta _g(f)(x)=f(g^{-1}x)\\chi (g^{-1},x)$ Since the set of normal states is weak* dense in the set of all states of $L^\\infty (X)$ , the amenability of $(G,X)$ is equivalent to the existence of a sequence $(f_n)\\subset L^1(X,\\mu )$ such that $f_n\\ge 0$ and $\\Vert f_n\\Vert _1=\\int \\limits _X f_n(x)d\\mu (x)=1$ for every $n$ ; for every compact set $K\\subset G$ , $\\sup _{g\\in K}\\Vert \\beta _g(f_n)-f_n\\Vert _1\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Lemma 2.2 If the pair $(G,X)$ is amenable, then there exists a sequence of probability measures $(\\mu _n)$ on $X$ such that: $\\mu _n$ is equivalent to $\\mu $ for every $n$ ; for every compact set $K\\subset G$ , $\\sup _{g\\in K}\\int \\limits _X |\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)\\rightarrow 0$ as $n\\rightarrow \\infty $ , where $\\chi _n(g,\\cdot )=dg_*^{-1}\\mu _n/d\\mu _n$ denotes the corresponding Radon-Nikodym derivative.",
"Proof.",
"Let $(f_n)$ be a sequence in $L^1(X,\\mu )$ as above.",
"Adding the constant function $\\frac{1}{n}1_X$ to $f_n$ and renormalizing if necessary, we assume that $(f_n)$ satisfies conditions (1) and (2) above, and that there exists a constant $c_n>0$ for every $n$ such that $f_n\\ge c_n$ for every $n$ .",
"Define $\\mu _n$ by $\\int \\limits _Xf(x)d\\mu _n(x)=\\int \\limits _X f(x)f_n(x)d\\mu (x).$ Then $L^\\infty (X,\\mu _n)=L^\\infty (X,\\mu )$ isometrically, and we have for $f\\in L^\\infty (X,\\mu )$ , $g\\in G$ and $n\\ge 1$ : $\\left|\\int \\limits _X f(x)\\lbrace \\chi _n(g,x)-1\\rbrace d\\mu _n(x)\\right| &=&\\left|\\int \\limits _X f(g^{-1}x)f_n(x)d\\mu (x)-\\int \\limits _X f(x)f_n(x)d\\mu (x)\\right|\\\\&=&\\left|\\int \\limits _X f(x)\\lbrace f_n(gx)\\chi (g,x)-f_n(x)\\rbrace d\\mu (x)\\right|\\\\&=&\\left|\\int \\limits _X f(x)\\lbrace \\beta _{g^{-1}}(f_n)(x)-f_n(x)\\rbrace d\\mu (x)\\right|\\\\&\\le &\\Vert f\\Vert _\\infty \\Vert \\beta _{g^{-1}}(f_n)-f_n\\Vert _1.$ This implies that $\\Vert \\chi _n(g,\\cdot )-1\\Vert _{L^{1}(X,\\mu _n)}=\\int \\limits _X|\\chi _n(g,x)-1|d\\mu _n(x)\\le \\Vert \\beta _{g^{-1}}(f_n)-f_n\\Vert _{L^{1}(X,\\mu )}$ for every $g\\in G$ and every $n$ .",
"Let $K$ be a compact subset of $G$ and $\\varepsilon >0$ .",
"There exists $N>0$ such that $\\sup _{g\\in K}\\Vert \\beta _{g^{-1}}(f_n)-f_n\\Vert _{L^{1}(X,\\mu )}\\le \\varepsilon $ for every $n\\ge N$ , so that we get for $g\\in K$ and $n\\ge N$ : $\\int \\limits _X|\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x) &\\le &\\int \\limits _X |\\sqrt{\\chi _n(g,x)}-1||\\sqrt{\\chi _n(g,x)}+1|d\\mu _n(x)\\\\&=&\\int \\limits _X|\\chi _n(g,x)-1|d\\mu _n(x)\\le \\varepsilon .$ $\\square $ Proof of Theorem REF .",
"We are going to prove statement (c) of Theorem 1.4.",
"The proofs of (a) and (b) are special cases that will be discussed briefly afterwards.",
"Thus let us assume that $H$ has the weak Haagerup property.",
"Fix $C>\\Lambda _{\\mathrm {WH}}(H)$ , $K\\subset G$ compact and $\\varepsilon >0$ .",
"We are going to prove that there exists $\\hat{u}\\in B_2(G)\\cap C_0(G)$ such that: $\\Vert \\hat{u}\\Vert _{B_2}\\le C$ ; $\\sup _{g\\in K}|\\hat{u}(g)-1|\\le \\varepsilon $ .",
"Let $(\\mu _n)$ be as in Lemma 2.2, and let $n$ be large enough in order that $\\sup _{g\\in K}\\int \\limits _X|\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)\\le \\frac{\\varepsilon }{4(C+1)}=:\\varepsilon ^{\\prime }.$ By condition (i) of Definition REF and Lemma 1.4, there exists a Borel set $A=A_{\\varepsilon ^{\\prime },K}$ such that $\\alpha (K\\times A)$ is precompact and $\\mu _n(X\\setminus A)\\le \\frac{\\varepsilon }{4(C+1)}.$ Let $L$ denote the closure of $\\alpha (K\\times A)$ in $H$ and choose $u\\in B_2(H)\\cap C_0(H)$ such that $\\Vert u\\Vert _{B_2}\\le C\\quad \\textrm {and}\\quad \\sup _{h\\in L}|u(h)-1|\\le \\frac{\\varepsilon }{4}$ and put $\\hat{u}(g)=\\int \\limits _X u(\\alpha (g,x))\\sqrt{\\chi _n(g,x)}d\\mu _n(x)$ for $g\\in G$ .",
"Lemma REF implies that $\\hat{u}\\in B_2(G)\\cap C_0(G)$ and that $\\Vert \\hat{u}\\Vert _{B_2}\\le C$ .",
"We have for every $g\\in K$ : $|\\hat{u}(g)-1| &\\le &\\int \\limits _{A}|u(\\alpha (g,x))\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)\\\\& &+\\int \\limits _{A^c}|u(\\alpha (g,x))\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x).$ Then $\\int \\limits _{A}|u(\\alpha (g,x))\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)& \\le &\\int \\limits _{A}|u(\\alpha (g,x))||\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)\\\\& &+\\int \\limits _{A}|u(\\alpha (g,x))-1|d\\mu _n(x)\\\\&\\le &C\\cdot \\int \\limits _X|\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)+\\frac{\\varepsilon }{4}\\\\& \\le &\\frac{\\varepsilon }{2}.$ Next, $\\int \\limits _{A^c}|u(\\alpha (g,x))\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)$ $& \\le &\\int \\limits _{A^c} |u(\\alpha (g,x))||\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)\\\\& &+\\int \\limits _{A^c} |u(\\alpha (g,x))-1|d\\mu _n(x)\\\\&\\le &C\\cdot \\int \\limits _X |\\sqrt{\\chi _n(g,x)}-1|d\\mu _n(x)\\\\& &+(C+1)\\mu _n(X\\setminus X(K,L))\\\\&\\le & \\frac{\\varepsilon }{2}.$ This ends the proof of statement (c).",
"If $H$ satisfies condition (a), given $K\\subset G$ compact and $\\varepsilon >0$ , the same construction as above from a positive definite, normalized function $u\\in C_0(H)$ gives a positive definite function $\\hat{u}\\in C_0(G)$ that satisfies $\\sup _{g\\in K}|\\hat{u}(g)-1|\\le \\varepsilon .$ This proves that $G$ has the Haagerup property.",
"Finally, if $H$ satisfies condition (b), if $C>\\Lambda _{\\mathrm {WA}}(H)$ , $K\\subset G$ compact and $\\varepsilon >0$ are given, choosing $u\\in A(H)$ with $\\Vert u\\Vert _{B_2}\\le C$ as in the first part of the proof, we get $\\hat{u}\\in A(G)$ satisfying (1) and (2) above.",
"This proves that $G$ is weakly amenable and that $\\Lambda _{\\mathrm {WA}}(G)\\le \\Lambda _{\\mathrm {WA}}(H)$ .",
"$\\square $"
],
[
"Proof of theorem ",
"Let $\\Gamma $ and $\\Delta $ be amenably measure equivalent groups as in Theorem REF and let $(\\Sigma ,\\sigma )$ be a standard infinite measure space on which $\\Gamma $ and $\\Delta $ act, the first one acting freely on the left and the second one freely on the right, both actions preserving the infinite measure $\\sigma $ , and such that the pairs $(\\Gamma ,\\Sigma /\\Delta )$ and $(\\Delta ,\\Gamma \\backslash \\Sigma )$ are amenable.",
"We fix our notation and recall some needed facts from [4]: we choose Borel cross-sections from $\\Gamma \\backslash \\Sigma $ and $\\Sigma /\\Delta $ to $\\Sigma $ , and we denote by $Y$ and $X$ their respective ranges so that $\\Sigma =\\bigsqcup _{\\delta \\in \\Delta }X\\delta =\\bigsqcup _{\\gamma \\in \\Gamma }\\gamma Y,$ and that they are standard measure spaces endowed with the corresponding restrictions of $\\sigma $ , say $\\mu =\\sigma |_X$ and $\\nu =\\sigma |_Y$ .",
"The action of $\\Gamma $ on $\\Sigma /\\Delta $ is isomorphic to the following action of $\\Gamma $ on $X$ : for each pair $(\\gamma ,x)\\in \\Gamma \\times X$ , there exists a unique element $\\alpha (\\gamma ,x)\\in \\Delta $ such that $\\gamma x\\in X\\alpha (\\gamma ,x)$ .",
"Hence the element $\\gamma \\cdot x:=\\gamma x\\alpha (\\gamma ,x)^{-1}$ belongs to $X$ , and the mapping $(\\gamma ,x)\\mapsto \\gamma \\cdot x$ defines an action of $\\Gamma $ on $X$ , and $\\alpha $ is a Borel cocycle with values in $\\Delta $ : $\\alpha (\\gamma _1\\gamma _2,x)=\\alpha (\\gamma _1,\\gamma _2\\cdot x)\\alpha (\\gamma _2,x)\\quad \\forall x\\in X,\\ \\forall \\gamma _1,\\gamma _2\\in \\Gamma .$ Similarly, the action of $\\Delta $ on the orbit space $\\Gamma \\backslash \\Sigma $ is isomorphic to the following action of $\\Delta $ on $Y$ : for each pair $(y,\\delta )\\in Y\\times \\Delta $ there exists a unique $\\beta (y,\\delta )\\in \\Gamma $ such that $y\\delta \\in \\beta (y,\\delta )Y$ .",
"Then set $y\\cdot \\delta =\\beta (y,\\delta )^{-1}y\\delta \\in Y$ , so that this defines an action of $\\Delta $ on $Y$ on the right, and $\\beta $ is a Borel cocycle for this action, viz $\\beta (y,\\delta _{1})\\beta (y\\cdot \\delta _{1},\\delta _{2})=\\beta (y,\\delta _{1}\\delta _{2})\\quad \\forall y\\in Y,\\ \\forall \\delta _{1},\\delta _{2}\\in \\Delta .$ Observe that $\\mu $ (resp.",
"$\\nu $ ) is $\\Gamma $ -invariant (resp.",
"$\\Delta $ -invariant), but that it need not be finite.",
"However, there are invariant states on $L^{\\infty }(X)$ and on $L^{\\infty }(Y)$ .",
"Theorem REF will be a straightforward consequence of Theorem REF for two reasons: the cocycle $\\alpha $ is proper as the following lemma shows, and amenably measure equivalence is a symmetric property.",
"Lemma 3.1 Retaining notation above, $\\alpha $ is a proper cocycle from $\\Gamma \\times X$ to $\\Delta $ .",
"More precisely: Let $1\\in K\\subset \\Gamma $ be finite, let $A\\subset X$ be a Borel set with finite measure and let $\\varepsilon >0$ .",
"Then there exists a Borel set $A_{\\varepsilon ,K}\\subset A$ and a finite set $F\\subset \\Delta $ such that $\\sigma (A\\setminus A_{\\varepsilon ,K})<\\varepsilon $ and $\\bigcup _{\\gamma \\in K}\\gamma A_{\\varepsilon ,K}\\subset \\bigsqcup _{\\delta \\in F}X\\delta .$ In particular $\\alpha (K\\times A_{\\varepsilon ,K})$ is a finite set and $\\alpha $ satisfies condition (i) of Definition REF .",
"Let $1\\in F\\subset \\Delta $ be a finite set and let $\\mathcal {A}_F$ be the set of Borel subsets $A$ of $X$ with finite measure for which there exists a finite set $K=K(A,F)\\subset \\Gamma $ such that $\\bigsqcup _{\\delta \\in F}A\\delta \\subset \\bigsqcup _{\\gamma \\in K}\\gamma Y.$ Then the elements of $\\mathcal {A}_F$ have the following properties: For every Borel set $A\\subset X$ with finite measure and for every $\\varepsilon >0$ , there exists $A_\\varepsilon \\in \\mathcal {A}_F$ such that $A_\\varepsilon \\subset A$ and $\\mu (A\\setminus A_\\varepsilon )<\\varepsilon $ .",
"Let $A\\in \\mathcal {A}_F$ and let $K$ be a finite subset of $\\Gamma $ such that $\\bigsqcup _{\\delta \\in F}A\\delta \\subset \\bigsqcup _{\\gamma \\in K}\\gamma Y.$ For $\\gamma \\in \\Gamma $ , set, as in Definition REF , $X_{A}(\\gamma )=\\lbrace x\\in A\\cap \\gamma ^{-1}\\cdot A:\\alpha (\\gamma ,x)\\in F\\rbrace .$ Then $X_{A}(\\gamma )=\\emptyset $ for every $\\gamma \\notin KK^{-1}$ .",
"In particular, $\\alpha $ satisfies condition (ii) of Definition REF .",
"Proof.",
"(a) Using induction on $|K|$ , it suffices to prove the claim for singleton sets.",
"Thus, let $A\\subset X$ be a Borel set with finite measure, let $\\gamma \\in \\Gamma $ and let $\\varepsilon >0$ .",
"There exists a finite set $F\\subset \\Delta $ such that $\\sum _{\\delta \\notin F}\\sigma (A\\cap \\gamma ^{-1}X\\delta )<\\varepsilon .$ Then the Borel set $A_\\varepsilon :=\\bigsqcup _{\\delta \\in F}A\\cap \\gamma ^{-1}X\\delta $ is a subset of $A$ such that $\\sigma (A\\setminus A_\\varepsilon )<\\varepsilon $ and $\\gamma A_\\varepsilon \\subset \\bigsqcup _{\\delta \\in F}X\\delta .$ (b1) Put $AF=\\bigsqcup _{\\delta \\in F}A\\delta $ .",
"Since $\\sigma (AF)=|F|\\sigma (A)=|F|\\mu (A)<\\infty $ , there exists a finite set $K\\subset \\Gamma $ such that $\\sum _{\\gamma \\notin K}\\sigma (AF\\cap \\gamma Y)<\\varepsilon .$ Put $Z=AF\\cap \\left(\\bigsqcup _{\\gamma \\in K}\\gamma Y\\right)$ .",
"For every $\\delta \\in F$ , put $Z_\\delta =(Z\\delta ^{-1})\\cap X$ , so that $Z=\\bigsqcup _{\\delta \\in F}Z_\\delta \\delta .$ Finally, put $A_\\varepsilon =\\bigcap _{\\delta \\in F}Z_\\delta .$ Then $A_\\varepsilon \\subset Z_\\delta $ for every $\\delta \\in F$ , hence $A_\\varepsilon \\delta \\subset Z_\\delta \\delta \\subset Z$ for every $\\delta \\in F$ , so that $\\bigsqcup _{\\delta \\in F}A_\\varepsilon \\delta \\subset Z$ .",
"One has: $\\mu (A\\setminus A_\\varepsilon ) &=&\\sigma (A\\setminus A_\\varepsilon )\\\\&=&\\sigma \\left(A\\cap \\left(\\bigcup _{\\delta \\in F}Z_\\delta ^c\\right)\\right)\\\\&\\le &\\sum _{\\delta \\in F}\\sigma (A\\cap Z_\\delta ^c)=\\sum _{\\delta \\in F}\\sigma (A\\delta \\setminus Z_\\delta \\delta )\\\\&=&\\sigma (AF\\setminus Z)<\\varepsilon .$ This ends the first part of the proof of the lemma.",
"(b2) Let $A$ and $K$ be as stated, and let $\\gamma \\in \\Gamma $ .",
"If $X_{A}(\\gamma )$ contains some element $x$ , then $\\alpha (\\gamma ,x)\\in F$ and $\\gamma x\\in AF$ .",
"It follows that $x\\in \\left(\\bigsqcup _{\\gamma ^{\\prime }\\in K}\\gamma ^{\\prime }Y\\right)\\cap \\left(\\bigsqcup _{\\gamma ^{\\prime \\prime }\\in K}\\gamma ^{-1}\\gamma ^{\\prime \\prime }Y\\right).$ This implies that there are $\\gamma ^{\\prime },\\gamma ^{\\prime \\prime }\\in K$ such that $\\gamma =\\gamma ^{\\prime \\prime }\\gamma ^{\\prime -1}\\in KK^{-1}$ .",
"$\\square $"
]
] | 1403.0207 |
[
[
"Reconfiguring Independent Sets in Claw-Free Graphs"
],
[
"Abstract We present a polynomial-time algorithm that, given two independent sets in a claw-free graph $G$, decides whether one can be transformed into the other by a sequence of elementary steps.",
"Each elementary step is to remove a vertex $v$ from the current independent set $S$ and to add a new vertex $w$ (not in $S$) such that the result is again an independent set.",
"We also consider the more restricted model where $v$ and $w$ have to be adjacent."
],
[
"Introduction",
"Reconfiguration problems.",
"To obtain a reconfiguration version of an algorithmic problem, one defines a reconfiguration rule – a (symmetric) adjacency relation between solutions of the problem, describing small transformations one is allowed to make.",
"The main focus is on studying whether one given solution can be transformed into another by a sequence of such small steps.",
"We call it a reachability problem.",
"For example, in a well-studied reconfiguration version of vertex coloring [7], [9], [10], [2], [1], [3], [19], we are given two $k$ -colorings of the vertices of a graph and we should decide whether one can be transformed into the other by recoloring one vertex at a time so that all intermediate solutions are also proper $k$ -colorings.",
"A useful way to look at reconfiguration problems is through the concept of the solution graph.",
"Given a problem instance, the vertices of the solution graph are all solutions to the instance, and the reconfiguration rule defines its edges.",
"Clearly, one solution can be transformed into another if they belong to the same connected component of the solution graph.",
"Other well-studied questions in the context of reconfiguration are as follows: can one efficiently decide (for every instance) whether the solution graph is connected?",
"Can one efficiently find shortest paths between two solutions?",
"Common non-algorithmic results are giving upper and lower bounds on the possible diameter of components of the solution graph, in terms of the instance size, or studying how much the solution space needs to be increased in order to guarantee connectivity.",
"Reconfiguration is a natural setting for real-life problems in which solutions evolve over time and an interesting theoretical framework that has been gradually attracting more attention.",
"The theoretical interest is based on the fact that reconfiguration problems provide a new perspective and offer a deeper understanding of the solution space as well as a potential to develop heuristics to navigate that space.",
"Reconfiguration paradigm has been recently applied to a number of algorithmic problems: vertex coloring [7], [8], [10], [9], list-edge coloring [18], clique, set cover, integer programming, matching, spanning tree, matroid bases [16], block puzzles [15], satisfiability [14], independent set [15], [16], [21], shortest paths [4], [5], [20], and dominating set [30]; recently also in the setting of parameterized complexity [26].",
"A recent survey [31] gives a good introduction to this area of research.",
"Reconfiguration of independent sets.",
"The topic of this paper is reconfiguration of independent sets.",
"An independent set in a graph is a set of pairwise nonadjacent vertices.",
"We will view the elements of an independent set as tokens placed on vertices.",
"Three different reconfiguration rules have been studied in the literature: token sliding (TS), token jumping (TJ), and token addition/removal (TAR).",
"The reconfiguration rule in the TS model allows to slide a token along an edge.",
"The reconfiguration rule in the TJ model allows to remove a token from a vertex and place it on another unoccupied vertex.",
"In the TAR model, the reconfiguration rule allows to either add or remove a token as long as at least $k$ tokens remain on the graph at any point, for a given integer $k$ .",
"In all three cases, the reconfiguration rule may of course only be applied if it maintains an independent set.",
"A sequence of moves following these rules is called a TS-sequence, TJ-sequence, or $k$ -TAR-sequence, respectively.",
"Note that the TS model is more restricted than the TJ model, in the sense that any TS-sequence is also a TJ-sequence.",
"Kamiński et al.",
"[21] showed that the TAR model generalizes the TJ model, in the sense that there exists a TJ-sequence between two solutions $I$ and $J$ with $|I|=|J|$ if and only if there exists a $k$ -TAR-sequence between them, with $k=|I|-1$ .",
"TS seems to have been introduced by Hearn and Demaine [15], TAR was introduced by Ito et al.",
"[17] and TJ by Kamiński et al. [23].",
"In all three models, the corresponding reachability problems are PSPACE-complete in general graphs [17] and even in perfect graphs [21] or in planar graphs of maximum degree 3 [15] (see also [7]).",
"We remark that in [15], only the TS-model was explicitly considered, but since only maximum independent sets are used, this implies the result for the TJ model (see Proposition REF below) and for the TAR model (using the aforementioned result from [21]).",
"Claw-free graphs.",
"A claw is the tree with four vertices and three leaves.",
"A graph is claw-free if it does not contain a claw as an induced subgraph.",
"A claw is not a line graph of any graph and thus the class of claw-free graphs generalizes the class of line graphs.",
"The structure of claw-free graphs is not simple but has been recently described by Chudnovsky and Seymour in the form of a decomposition theorem [11].",
"There is a natural one-to-one correspondence between matchings in a graph and independent sets in its line graph.",
"In particular, a maximum matching in a graph corresponds to a maximum independent set in its line graph.",
"Hence, Edmonds' maximum matching algorithm [13] gives a polynomial-time algorithm for finding maximum independent sets in line graphs.",
"This results has been extended to claw-free graphs independently by Minty [25] and Sbihi [28].",
"Both algorithms work for the unweighted case, while the algorithm of Minty, with a correction proposed by Nakamura and Tamura in [27], applies to weighted graphs (see also [29]).",
"A fork is the graph obtained from the claw by subdividing one edge.",
"Every claw-free graph is also fork-free.",
"Milanič and Lozin gave a polynomial-time algorithm for maximum weighted independent set in fork-free graphs [24].",
"This generalizes all aforementioned results for claw-free graphs.",
"Our results.",
"In this paper, we study the reachability problem for independent set reconfiguration, using the TS and TJ model.",
"Our main result is that these problems can be solved in polynomial time for the case of claw-free graphs.",
"Along the way, we prove some results that are interesting in their own right.",
"For instance, we show that for connected claw-free graphs, the existence of a TJ-sequence implies the existence of a TS-sequence between the same pair of solutions.",
"This implies that for connected claw-free and even-hole-free graphs, the solution graph is always connected, answering an open question posed in [21].",
"Since claw-free graphs generalize line graphs, our results generalize the result by Ito et al.",
"[17] on matching reconfiguration.",
"Since a vertex set $I$ of a graph $G$ is an independent set if and only if $V(G)\\backslash I$ is a vertex cover, our results also apply to the recently studied vertex cover reconfiguration problem [26].",
"The new techniques we introduce can be seen as an extension of the techniques introduced for finding maximum independent sets in claw-free graphs, and we expect them to be useful for addressing similar reconfiguration questions, such as efficiently deciding whether the solution graph is connected.",
"Some proof details are omitted.",
"Statements for which further details can be found in the appendix are marked with a star."
],
[
"Preliminaries",
"For graph theoretic terminology not defined here, we refer to [12].",
"For a graph $G$ and vertex set $S\\subseteq V(G)$ , we denote the subgraph induced by $S$ by $G[S]$ , and denote $G-S=G[V\\backslash S]$ .",
"The set of neighbors of a vertex $v\\in V(G)$ is denoted by $N(v)$ , and the closed neighborhood of $v$ is $N[v]=N(v)\\cup \\lbrace v\\rbrace $ .",
"A walk from $v_0$ to $v_k$ of length $k$ is a sequence of vertices $v_0,v_1,\\ldots ,v_k$ such that $v_iv_{i+1}\\in E(G)$ for all $i\\in \\lbrace 0,\\ldots ,k-1\\rbrace $ .",
"It is a path if all of its vertices are distinct, and a cycle if $k\\ge 3$ , $v_0=v_k$ and $v_0,\\ldots ,v_{k-1}$ is a path.",
"We use $V(C)$ to denote the vertex set of a path or cycle, viewed as a subgraph of $G$ .",
"A path or graph is called trivial if it contains only one vertex.",
"Edges of a directed graph or digraph $D$ are called arcs, and are denoted by the ordered tuple $(u,v)$ .",
"A directed path in $D$ is a sequence of distinct vertices $v_0,\\ldots ,v_k$ such that for all $i\\in \\lbrace 0,\\ldots ,k-1\\rbrace $ , $(v_i,v_{i+1})$ is an arc of $D$ .",
"We denote the distance of two vertices $u,v\\in V(G)$ by $\\mbox{d}_G(u,v)$ .",
"By $\\mbox{diam}(G)$ we denote the diameter of a connected graph $G$ , defined as $\\max _{u,v\\in V(G)} \\mbox{d}_G(u,v)$ .",
"For a vertex set $S$ of a graph $G$ and integer $i\\in \\mathbb {N}$ , we denote $N_i(S)=\\lbrace v\\in V(G)\\backslash S : |N(v)\\cap S|=i\\rbrace $ .",
"For a graph $G$ , by $\\mbox{TS}_k(G)$ we denote the graph that has as its vertex the set of all independent sets of $G$ of size $k$ , where two independent sets $I$ and $J$ are adjacent if there is an edge $uv\\in E(G)$ with $I\\backslash J=\\lbrace u\\rbrace $ and $J\\backslash I=\\lbrace v\\rbrace $ .",
"We say that $J$ can be obtained from $I$ by sliding a token from $u$ to $v$, or by the move $u\\rightarrow v$ for short.",
"A walk in $\\mbox{TS}_k(G)$ from $I$ to $J$ is called a TS-sequence from $I$ to $J$.",
"We write $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ to indicate that there is a TS-sequence from $I$ to $J$ .",
"Analogously, by $\\mbox{TJ}_k(G)$ we denote the graph that has as its vertex set the set of all independent sets of $G$ of size $k$ , where two independent sets $I$ and $J$ are adjacent if there is a vertex pair $u,v\\in V(G)$ with $I\\backslash J=\\lbrace u\\rbrace $ and $J\\backslash I=\\lbrace v\\rbrace $ .",
"We say that $J$ can be obtained from $I$ by jumping a token from $u$ to $v$.",
"A walk in $\\mbox{TS}_k(G)$ from $I$ to $J$ is called a TJ-sequence from $I$ to $J$.",
"We write $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ to indicate that there exists a TJ-sequence from $I$ to $J$ .",
"Note that $\\mbox{TS}_k(G)$ is a spanning subgraph of $\\mbox{TJ}_k(G)$ .",
"The reachability problem for token sliding (resp.",
"token jumping) has as input a graph $G$ and two independent sets $I$ and $J$ of $G$ with $|I|=|J|$ , and asks whether $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ (resp.",
"$I\\leftrightarrow _{\\mbox{\\sc tj}}J$ ).",
"These problems are called TS-Reachability and TJ-Reachability, respectively.",
"If $H$ is a claw with vertex set $\\lbrace u,v,w,x\\rbrace $ such that $N(u)=\\lbrace v,w,x\\rbrace $ , then $H$ is called a $u$ -claw with leaves $v,w,x$ .",
"Sets $I\\backslash \\lbrace v\\rbrace $ and $I\\cup \\lbrace v\\rbrace $ are denoted by $I-v$ and $I+v$ respectively.",
"The symmetric difference of two sets $I$ and $J$ is denoted by $I\\Delta J=(I\\backslash J)\\cup (J\\backslash I)$ .",
"The following observation is used implicitly in many proofs: if $I$ and $J$ are independent sets in a claw-free graph $G$ , then every component of $G[I\\Delta J]$ is a path or an even length cycle.",
"By $\\alpha (G)$ we denote the size of the largest independent set of $G$ .",
"An independent set $I$ is called maximum if $|I|=\\alpha (G)$ .",
"A vertex set $S\\subseteq V(G)$ is a dominating set if $N[v]\\cap S\\ne \\emptyset $ for all $v\\in V(G)$ .",
"Observe that a maximum independent set is a dominating set, thus the only possible token jumps from it are between adjacent vertices, and hence all are token slides: Proposition 1 Let $I$ and $J$ be maximum independent sets in a graph $G$ .",
"Then, $TS_k(G) = TJ_k(G)$ .",
"In particular, $I \\leftrightarrow _{\\mbox{\\sc ts}}J$ if and only if $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ ."
],
[
"The Equivalence of Sliding and Jumping",
"In our main result (Theorem REF ), we will consider equal size independent sets $I$ and $J$ of a claw-free graph $G$ , and show that in polynomial time, it can be verified whether $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ and whether $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"In this section, we show that if $G$ is connected and $G[I\\Delta J]$ contains no cycles, then $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"From this, we will subsequently conclude that for connected claw-free graphs $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ holds if and only if $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ , even in the case of nonmaximum independent sets.",
"Lemma 2 (*) Let $I$ and $J$ be independent sets in a connected claw-free graph $G$ with $|I|=|J|$ .",
"If $G[I\\Delta J]$ contains no cycles, then $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"Proof sketch: We show that $I$ or $J$ can be modified using token slides such that the two resulting independent sets are closer to each other in the sense that either $|I\\setminus J|$ is smaller, or it is unchanged and the minimum distance between vertices $u,v$ with $u\\in I\\setminus J$ and $v\\in J\\setminus I$ is smaller.",
"The claim follows by induction.",
"(See the appendix for an induction statement with a bound on the length of the reconfiguration sequence.)",
"Suppose first that $G[I\\Delta J]$ contains at least one nontrivial component $C$ .",
"Since it is not a cycle by assumption, it must be a path.",
"Choose an end vertex $u$ of this path, and let $v$ be its unique neighbor on the path.",
"If $u\\in J$ then $N(u)\\cap I=\\lbrace v\\rbrace $ , so we can obtain a new independent set $I^{\\prime }=I+u-v$ from $I$ using a single token slide.",
"The new set $I^{\\prime }$ is closer to $J$ in the sense that $|I^{\\prime }\\backslash J|<|I\\backslash J|$ , so we may use induction to conclude that $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc ts}}J$ , and thus $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"On the other hand, if $u\\in I$ then we can obtain a new independent set $J^{\\prime }=J-v+u$ from $J$ , and conclude the proof similarly by applying the induction assumption to $J^{\\prime }$ and $I$ .",
"In the remaining case, we may assume that $G[I\\Delta J]$ consists only of isolated vertices.",
"Choose $u\\in I\\backslash J$ and $v\\in J\\backslash I$ , such that the distance $d:=\\mbox{d}_G(u,v)$ between these vertices is minimized.",
"Starting with $I$ , we intend to slide the token on $u$ to $v$ , to obtain an independent set $I^{\\prime }=I-u+v$ that is closer to $J$ .",
"To this end, we choose a shortest path $P=v_0,\\ldots ,v_d$ in $G$ from $v_0=u$ to $v_d=v$ .",
"If the token can be moved along this path while maintaining an independent set throughout, then $I\\leftrightarrow _{\\mbox{\\sc ts}}I^{\\prime }$ , and the proof follows by induction as before.",
"So now suppose that this cannot be done, that is, at least one of the vertices on $P$ is equal to or adjacent to a vertex in $I-u$ .",
"In that case, we choose $i$ maximum such that $N(v_i)\\cap I\\ne \\emptyset $ .",
"Using some simple observations (including the fact that $G$ is claw-free), one can now show that $N(v_i)\\cap I$ consists of a single vertex $x$ .",
"By choice of $v_i$ , starting with $I$ , the token on $x$ can be moved along the path $x,v_i,v_{i+1},\\ldots ,v_d$ while maintaining an independent set throughout.",
"This yields an independent set $I^{\\prime \\prime }=I-x+v$ , with $I\\leftrightarrow _{\\mbox{\\sc ts}}I^{\\prime \\prime }$ .",
"It can also easily be shown that $\\mbox{d}_G(u,x)<\\mbox{d}_G(u,v)$ and $\\mbox{d}_G(x,v)<\\mbox{d}_G(u,v)$ .",
"So considering the choice of $u$ and $v$ , it follows that $x\\in I\\cap J$ , and thus $|I^{\\prime \\prime }\\backslash J|=|I\\backslash J|$ .",
"Since now the pair $u\\in I^{\\prime \\prime }\\backslash J$ and $x\\in J\\backslash I^{\\prime \\prime }$ has a smaller distance $\\mbox{d}_G(u,x)<\\mbox{d}_G(u,v)=d$ , we may assume by induction that $I^{\\prime \\prime }\\leftrightarrow _{\\mbox{\\sc ts}}J$ , and thus $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .$\\Box $ Corollary 3 Let $I$ and $J$ be independent sets in a connected claw-free graph $G$ .",
"Then $I \\leftrightarrow _{\\mbox{\\sc ts}}J$ if and only if $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"Proof: Clearly, a TS-sequence from $I$ to $J$ is also a TJ-sequence.",
"For the nontrivial direction of the proof, it suffices to show that any token jump can be replaced by a sequence of token slides.",
"Let $J$ be obtained from $I$ by jumping a token from $u$ to $v$ .",
"Then $G[I\\Delta J]$ contains only two vertices and therefore no cycles.",
"Then Lemma REF shows that $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"$\\Box $ We now consider implications of the above corollary for graphs that are claw- and even-hole-free.",
"A graph is even-hole-free if it contains no even cycle as an induced subgraph.",
"Kamiński et al.",
"[21] proved the following statement.",
"Theorem 4 ([21]) Let $I$ and $J$ be two independent sets of a graph $G$ with $|I|=|J|$ .",
"If $G[I\\Delta J]$ contains no even cycles, then there exists a TJ-sequence from $I$ to $J$ of length $|I\\backslash J|$ , which can be constructed in linear time.",
"In particular, if $G$ is even-hole-free, then $\\mbox{TJ}_k(G)$ is connected (for every $k$ ).",
"However, $\\mbox{TS}_k(G)$ is not necessarily connected (consider a claw with two tokens).",
"This motivated the question asked in [21] whether for connected, claw-free and even-hole-free graph $G$ , $\\mbox{TS}_k(G)$ is connected.",
"Combining Corollary REF with Theorem REF shows that the answer to this question is affirmative.",
"Corollary 5 Let $G$ be a connected claw-free and even-hole-free graph.",
"Then $\\mbox{TS}_k(G)$ is connected."
],
[
"Nonmaximum Independent Sets",
"We now continue studying connected claw-free graphs.",
"Lemma REF shows that it remains to consider the case that $G[I\\Delta J]$ contains (even length) cycles.",
"In this section, we show that when $I$ and $J$ are not maximum independent sets of $G$ , such cycles can always be resolved.",
"This requires various techniques developed in the context of finding maximum independent sets in claw-free graphs and the following definitions.",
"A vertex $v\\in V(G)$ is free (with respect to an independent set $I$ of $G$ ) if $v \\notin I$ and $|N(v)\\cap I|\\le 1$ .",
"Let $W=v_0,\\ldots ,v_k$ be a walk in $G$ , and let $I\\subseteq V(G)$ .",
"Then $W$ is called $I$ -alternating if $|\\lbrace v_i,v_{i+1}\\rbrace \\cap I|=1$ for $i=0,\\dots ,k-1$ .",
"In the case that $W$ is a path, $W$ is called chordless if $G[\\lbrace v_0,\\ldots ,v_k\\rbrace ]$ is a path.",
"In the case that $W$ is a cycle (so $v_0=v_k$ ), $W$ is called chordless if $G[\\lbrace v_0,\\ldots ,v_{k-1}\\rbrace ]$ is a cycle.",
"A cycle $W=v_0,\\ldots ,v_k$ is called $I$ -bad if it is $I$ -alternating and chordless.",
"A path $W=v_0,\\ldots ,v_k$ with $k\\ge 2$ is called $I$ -augmenting if it is $I$ -alternating and chordless, and $v_0$ and $v_k$ are both free vertices.",
"This definition of $I$ -augmenting paths differs from the usual definition, as it is used in the setting of finding maximum independent sets, since the chordless condition is stronger than needed in such a setting.",
"However, we observe that in a claw-free graph $G$ , the two definitions are equivalent (see Proposition REF in the appendix) so we may apply well-known statements about $I$ -augmenting paths proved elsewhere.",
"In particular, we use the following two results originally proved by Minty [25] and Sbihi [28] (see also [29]).",
"Theorem 6 ([29]) Let $I$ be an independent set in a claw-free graph $G$ .",
"It can be decided in polynomial time whether an $I$ -augmenting path between two given free vertices $x$ and $y$ exists, and if so, compute one.",
"Proposition 7 ([29]) Let $I$ be a nonmaximum independent set in a claw-free graph $G$ .",
"Then $I$ is not a dominating set, or there exists an $I$ -augmenting path.",
"We use Proposition REF to handle the case of nonmaximum independent sets.",
"The next statement is formulated for token jumping, and (by Corollary REF ) implies the same result for token sliding only in the case of connected graphs.",
"Lemma 8 (*) Let $I$ be a nonmaximum independent set in a claw-free graph $G$ .",
"Then for any independent set $J$ with $|J|=|I|$ , $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ holds.",
"Proof sketch: By Theorem REF , it suffices to consider the case where $G[I\\Delta J]$ contains at least one cycle $C$ .",
"Let $C=u_1,v_1,u_2,v_2,\\ldots ,v_k,u_1$ , so that $u_i\\in I$ and $v_i\\in J$ for all $i$ .",
"Suppose first that $I$ is not a dominating set.",
"Then we can choose a vertex $w$ with $N[w]\\cap I=\\emptyset $ .",
"With a single token jump, we can obtain the independent set $I^{\\prime }=I+w-u_1$ from $I$ .",
"Next, apply the moves $u_k\\rightarrow v_k$ , $u_{k-1}\\rightarrow v_{k-1}$ ,..., $u_2\\rightarrow v_2$ , in this order.",
"(This is possible since $C$ is chordless.)",
"Finally, jump the token from $w$ to $v_1$ .",
"It can be verified that this yields a token jumping sequence from $I$ to $I^{\\prime }=I\\Delta V(C)$ .",
"This way, all cycles can be resolved one by one, until no more cycles remain and Theorem REF can be applied to prove the statement.",
"On the other hand, if $I$ is a dominating set, then Proposition REF shows that there exists an $I$ -augmenting path $P=v_0,u_1,v_1,\\ldots ,u_d,v_d$ , with $u_i\\in I$ for all $i$ .",
"Since $v_d$ is a free vertex, we can first apply the moves $u_d\\rightarrow v_d$ , $u_{d-1}\\rightarrow v_{d-1}$ ,...$u_1\\rightarrow v_1$ , in this order (which can be done since $P$ is chordless), to obtain an independent set $I^{\\prime }$ from $I$ , with $I\\leftrightarrow _{\\mbox{\\sc ts}}I^{\\prime }$ .",
"Then $v_0$ is not dominated by $I^{\\prime }$ , so the previous argument can be applied to show that $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc tj}}J$ , which implies $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"$\\Box $"
],
[
"Resolving Cycles",
"It now remains to study the case where $G[I\\Delta J]$ contains (even) cycles and both $I$ and $J$ are maximum independent sets.",
"In this case, there may not be a TS-sequence from $I$ to $J$ (even though we assume that $G$ is connected and claw-free) – consider for instance the case where $G$ itself is an even cycle.",
"In this section, we characterize the case where $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ holds, by showing that this is equivalent with every cycle being resolvable in a certain sense (Theorem REF below).",
"Subsequently, we show that resolvable cycles fall into two cases: internally or externally resolvable cycles, which are characterized next.",
"We first define the notion of resolving a cycle.",
"Cycles in $G[I\\Delta J]$ are clearly both $I$ -bad and $J$ -bad.",
"The $I$ -bipartition of an $I$ -bad cycle is the ordered tuple $[V(C)\\cap I,V(C)\\backslash I]$ .",
"We say that an $I$ -bad cycle $C$ with $I$ -bipartition $[A,B]$ is resolvable (with respect to $I$ ) if there exists an independent set $I^{\\prime }$ such that $I\\leftrightarrow _{\\mbox{\\sc ts}}I^{\\prime }$ and $G[I^{\\prime }\\cup B]$ contains no cycles.",
"A corresponding TS-sequence from $I$ to $I^{\\prime }$ is called a resolving sequence and is said to resolve $C$.",
"By combining such a resolving sequence with a sequence of moves similar to the previous proof, and then reversing the moves in the sequence from $I^{\\prime }$ to $I$ , except for moves of tokens on the cycle, one can show that every resolvable cycle can be `turned': Lemma 9 (*) Let $I$ be an independent set in a claw-free graph $G$ and let $C$ be an $I$ -bad cycle.",
"If $C$ is resolvable with respect to $I$ , then $I\\leftrightarrow _{\\mbox{\\sc ts}}I\\Delta V(C)$ .",
"We can now prove the following useful characterization: $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ if and only if every cycle in $G[I\\Delta J]$ is resolvable.",
"By symmetry, it does not matter whether one considers resolvability with respect to $I$ or to $J$ .",
"Theorem 10 Let $I$ and $J$ be independent sets in a claw-free connected graph $G$ .",
"Then $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ if and only if every cycle in $G[I\\Delta J]$ is resolvable with respect to $I$ .",
"Proof: Consider an $I$ -bad cycle $C$ in $G[I\\Delta J]$ with $I$ -bipartition $[A,B]$ , and a TS-sequence from $I$ to $J$ .",
"Since $N_2(B)$ eventually contains no tokens, this sequence must contain a move $u\\rightarrow v$ with $u\\in N_2(B)$ and $v\\notin N_2(B)$ .",
"The first such move can be shown to resolve the cycle.",
"(See Lemma REF in the appendix for details.)",
"The other direction is proved by induction on the number $k$ of cycles in $G[I\\Delta J]$ .",
"If $k=0$ , then by Lemma REF , $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"If $k\\ge 1$ , then consider an $I$ -bad cycle $C$ in $G[I\\Delta J]$ .",
"Let $I^{\\prime }=I\\Delta V(C)$ .",
"By Lemma REF , $I\\leftrightarrow _{\\mbox{\\sc ts}}I^{\\prime }$ .",
"The graph $G[I^{\\prime }\\Delta J]$ has one cycle fewer than $G[I\\Delta J]$ .",
"Every cycle in $G[I^{\\prime }\\Delta J]$ remains resolvable with respect to $I^{\\prime }$ (one can first consider a TS-sequence from $I^{\\prime }$ to $I$ , and subsequently a TS-sequence from $I$ that resolves the cycle).",
"So by induction, $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc ts}}J$ , and therefore, $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .$\\Box $ Finally, we show that if an $I$ -bad cycle $C$ can be resolved, it can be resolved in at least one of two very specific ways.",
"Let $[A,B]$ be the $I$ -bipartition of $C$ .",
"A move $u\\rightarrow v$ is called internal if $\\lbrace u,v\\rbrace \\subseteq N_2(B)$ and external if $\\lbrace u,v\\rbrace \\subseteq N_0(B)$ .",
"A resolving sequence $I_0,\\ldots ,I_m$ for $C$ is called internal (or external) if every move except the last is an internal (respectively, external) move.",
"(Obviously, to resolve the cycle, the last move can neither be internal nor external, and can in fact be shown to always be a move from $N_2(B)$ to $N_1(B)$ .)",
"The $I$ -bad cycle $C$ is called internally resolvable resp.",
"externally resolvable if such sequences exist.",
"Lemma 11 (*) Let $I$ be an independent set in a claw-free graph $G$ and let $C$ be an $I$ -bad cycle.",
"Then any shortest TS-sequence that resolves $C$ is an internal or external resolving sequence.",
"Proof sketch: Let $[A,B]$ be the $I$ -bipartition of $C$ .",
"Since $G$ is claw-free, it follows that there are no edges between vertices in $N_2(B)$ and $N_0(B)$ .",
"This can be used to show that informally, any resolving sequence for $C$ remains a resolving sequence after either omitting all noninternal moves or omitting all nonexternal moves, while keeping the last move, which subsequently resolves the cycle.",
"$\\Box $ Theorem REF and Lemma REF show that to decide whether $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ , it suffices to check whether every cycle in $G[I\\Delta J]$ is externally or internally resolvable.",
"Next we give characterizations that allow polynomial-time algorithms for deciding whether an $I$ -bad cycle is internally or externally resolvable.",
"For the external case, we use the assumption that $I$ is a maximum independent set to show that in a shortest external resolving sequence $I_0,\\ldots ,I_m$ , every token moves at most once (that is, for every move $u\\rightarrow v$ , both $u\\in I_0$ and $v\\in I_m$ hold), so these moves outline an augmenting path in a certain auxiliary graph.",
"Theorem 12 [*] Let $I$ be a maximum independent set in a claw-free graph $G$ and let $C$ be an $I$ -bad cycle with $I$ -bipartition $[A,B]$ .",
"Then $C$ is externally resolvable if and only if there exists an $(I\\backslash A)$ -augmenting path in $G-A-B$ between a pair of vertices $x\\in N_0(B)$ and $y\\in N_1(B)$ .",
"For a given $I$ -bad cycle $C$ with $I$ -bipartition $[A,B]$ , there is a quadratic number of vertex pairs $x\\in N_0(B)$ and $y\\in N_1(B)$ that need to be considered, and for every such a pair, testing whether there is an $(I\\backslash A)$ -augmenting path between these in $G-A-B$ can be done in polynomial time (Theorem REF ).",
"So from Theorem REF we conclude: Corollary 13 Let $I$ be a maximum independent set in a claw-free graph $G$ , and let $C$ be an $I$ -bad cycle.",
"In polynomial time, it can be decided whether $C$ is externally resolvable.",
"Next, we characterize internally resolvable cycles.",
"Shortest internal resolving sequences cannot be as easy to describe as external ones, since a token can move several times (see Figure REF ).",
"Nevertheless, these sequences can be shown to have a very specific structure, which can be characterized using paths in the following auxiliary digraphs.",
"To define these digraphs, consider an $I$ -bad cycle $C=c_0,c_1,\\ldots ,c_{2n-1},c_0$ in $G$ , with $c_i\\in I$ for even $i$ .",
"Let $[A,B]$ be the $I$ -bipartition of $C$ .",
"For every $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ , define the corresponding layer as follows: $L_i=\\lbrace v\\in V(G) \\mid N(v)\\cap B=N(c_{2i})\\cap B\\rbrace $ .",
"So when starting with $I$ and using only internal moves, it can be seen that the token that starts on $c_{2i}$ will stay in the layer $L_i$ .",
"For such an $I$ -bad cycle $C$ of length at least 8, define $D(G,C)$ to be a digraph with vertex set $V(G)$ , with the following arc set.",
"For every $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ and all pairs $u\\in L_i, v\\in L_{(i+1)\\bmod n}$ with $uv\\notin E(G)$ , add an arc $(u,v)$ .",
"For every $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ and $b\\in N_1(B)$ with $N(b)\\cap B=\\lbrace c_{(2i-1) \\bmod 2n}\\rbrace $ , and every $v\\in L_i$ with $bv\\notin E(G)$ , add an arc $(b,v)$ .",
"Also, we denote the reversed cycle by $C^{rev}=c_0,c_{2n-1},\\ldots ,c_1,c_0$ .",
"This defines a similar digraph $D(G,C^{rev})$ (where arcs between layers are reversed, and arcs from $N_1(B)$ go to different layers).",
"These graphs can be used to characterize whether $C$ is internally resolvable.",
"Figure: An example of a claw-free graph GG with an internally resolvable cycle, along with the corresponding auxiliary digraph D(G,C)D(G,C).Theorem 14 (*) Let $I$ be an independent set in a claw-free graph $G$ .",
"Let $C=c_0,c_1,\\dots ,c_{2n-1},c_{0}$ be an $I$ -bad cycle ($c_0\\in I$ ) with $I$ -bipartition $[A,B]$ , of length at least 8.",
"Then $C$ is internally resolvable if and only if $D(G,C)$ or $D(G,C^{rev})$ contains a directed path from a vertex $b\\in N_1(B)$ with $N(b)\\cap I\\subseteq A$ to a vertex in $A$ .",
"Corollary 15 Let $I$ be an independent set in a claw-free graph $G$ on $n$ vertices and let $C$ be an $I$ -bad cycle.",
"It can be decided in polynomial time whether $C$ is internally resolvable.",
"Proof: If $C$ has length at least 8, then Theorem REF shows that it suffices to make a polynomial number of depth-first-searches in $D(G,C)$ and $D(G,C^{rev})$ .",
"Otherwise, let $[A,B]$ be the $I$ -bipartition of $C$ .",
"$|A|\\le 3$ , so there are only $O(n^3)$ independent sets $I^{\\prime }$ with $|I^{\\prime }|=|I|$ and $I\\backslash A\\subseteq I^{\\prime }$ .",
"So in polynomial time we can generate the subgraph of $\\mbox{TS}_k(G)$ induced by these sets, and search whether it contains a path from $I$ to an independent set $I^*$ with $I\\backslash A\\subseteq I^*$ where $G[B\\cup I^*]$ contains no cycle.",
"$C$ is internally resolvable if and only if such a path exists.",
"$\\Box $"
],
[
"Summary of the Algorithm",
"We now summarize how the previous lemmas yield a polynomial time algorithm for TS-Reachability and TJ-Reachability in claw-free graphs.",
"Theorem 16 Let $I$ and $J$ be independent sets in a claw-free graph $G$ .",
"We can decide in polynomial time whether $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ and whether $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"Proof: Assume $|I|=|J|$ ; otherwise, we immediately return NO.",
"We first consider the case when $G$ is connected.",
"By Corollary REF , since $G$ is connected, $I \\leftrightarrow _{\\mbox{\\sc ts}}J$ if and only if $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ , thus we only need to consider the sliding model.",
"We test whether $I$ and $J$ are maximum independent sets of $G$ , which can be done in polynomial time (by combining Proposition REF and Theorem REF ; see also [25], [28], [29]).",
"If not, then by Lemma REF , $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ holds, and thus $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ , so we may we return YES.",
"Now consider the case that both $I$ and $J$ are maximum independent sets.",
"Theorem REF shows that $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ if and only if every cycle in $G[I\\Delta J]$ is resolvable with respect to $I$ .",
"By Lemma REF , it suffices to check for internal and external resolvability of such cycles.",
"This can be done in polynomial time by Corollary REF (since $I$ is a maximum independent set of $G$ ) and Corollary REF .",
"We return YES if and only if every cycle in $C$ was found to be internally or externally resolvable, and NO otherwise.",
"Now let us consider the case when $G$ is disconnected.",
"Clearly tokens cannot slide between different connected components, so for deciding whether $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ , we can apply the argument above to every component, and return YES if and only if the answer is YES for every component.",
"If $I$ is a not a maximum independent set then Lemma REF shows that $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ always holds.",
"If $I$ is maximum, then Proposition REF shows that $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ holds if and only if $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"$\\Box $"
],
[
"Discussion",
"The results presented here have two further implications.",
"Firstly, combined with techniques from [6], it follows that $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ can be decided for any graph $G$ that can be obtained from a collection of claw-free graphs using disjoint union and complete join operations.",
"See [6] for more details.",
"Secondly, a closer look at constructed reconfiguration sequences (in the appendix) shows that when $G$ is claw-free, components of both $\\mbox{TS}_k(G)$ and $\\mbox{TJ}_k(G)$ have diameter bounded polynomially in $|V(G)|$ .",
"This is not surprising, since the same behavior has been observed many times.",
"To our knowledge, the only known examples of polynomial time solvable reconfiguration problems that nevertheless require exponentially long reconfiguration sequences are on artificial instance classes, which are constructed particularly for this purpose (see e.g.",
"[7], [22])."
],
[
"Details for Section ",
"We now give a detailed induction proof of Lemma REF , including a bound on the length of the resulting TS-sequence.",
"Lemma 17 Let $I$ and $J$ be independent sets in a connected claw-free graph $G$ , with $|I|=|J|$ .",
"If $G[I\\Delta J]$ contains no cycles then there is a TS-sequence from $I$ to $J$ of length at most $2\\cdot |I\\backslash J|\\cdot \\mbox{diam}(G)$ .",
"Proof: For two vertex sets $I$ and $J$ with $I\\backslash J\\ne \\emptyset $ and $J\\backslash I\\ne \\emptyset $ , define the minimum distance $\\mbox{md}(I,J)$ to be the minimum of $\\mbox{d}_G(u,v)$ over all pairs $u\\in I\\backslash J$ and $v\\in J\\backslash I$ .",
"In addition, define $\\phi (I,J)=(|I\\backslash J|-1)\\cdot \\mbox{diam}(G) + \\mbox{md}(I,J)$ .",
"We will prove by induction on $\\phi (I,J)$ that if $I$ and $J$ are two independent sets with $|I|=|J|$ such that $G[I\\Delta J]$ contains no cycles, then there exists a TS-sequence from $I$ to $J$ of length at most $2\\phi (I,J)$ .",
"Since $2\\phi (I,J)\\le 2\\cdot |I\\backslash J|\\cdot \\mbox{diam}(G)$ , this proves the lemma.",
"First let us consider the case that $\\mbox{md}(I,J)=1$ .",
"This means that $G[I\\Delta J]$ contains at least one edge.",
"Since $G$ is claw-free, $G[I\\Delta J]$ has maximum degree 2.",
"But we assumed that it contains no cycles, so it is a collection of paths, with at least one path $P$ of length at least 1.",
"Choose an end vertex $v$ of $P$ .",
"Suppose first that $v\\in J$ .",
"Let $u$ be the vertex on $P$ that is adjacent to $v$ (so $u\\in I$ ).",
"Then in $I$ , the token from $u$ can be moved to $v$ , to obtain a new independent set $I^{\\prime }$ .",
"In the case that $\\phi (I,J)=1$ (the induction base), $|I\\backslash J|=1$ , so $I^{\\prime }=J$ and we exhibited a TS-sequence of length 1 between $I$ and $J$ , which proves the claim.",
"Otherwise, note that $G[I^{\\prime }\\Delta J]$ again contains no cycles, so by induction, there exists a TS-sequence from $I^{\\prime }$ to $J$ of length at most $2(|I^{\\prime }\\backslash J|-1)\\cdot \\mbox{diam}(G) + 2\\mbox{md}(I^{\\prime },J)\\le $ $2(|I\\backslash J|-2)\\cdot \\mbox{diam}(G) + 2\\mbox{diam}(G)=$ $2(|I\\backslash J|-1)\\cdot \\mbox{diam}(G)$ .",
"Since $I^{\\prime }$ was obtained from $I$ using one token slide, we conclude that there exists a TS-sequence from $I$ to $J$ of length at most $2(|I\\backslash J|-1)\\cdot \\mbox{diam}(G)+1\\le \\phi (I,J)$ , which proves the claim.",
"If the chosen end vertex $v$ of the path $P$ is in $I$ , then from $J$ we obtain $J^{\\prime }$ by sliding the adjacent token to $v$ , and the statement can be proved analogously.",
"Now suppose that $\\mbox{md}(I,J)\\ge 2$ , let $d=\\mbox{md}(I,J)$ .",
"Choose $u\\in I\\backslash J$ and $v\\in J\\backslash I$ such that $\\mbox{d}_G(u,v)=d$ , and let $P=v_0,\\ldots ,v_d$ be a shortest path between $v_0=u$ and $v_d=v$ .",
"We intend to slide the token from $u$ to $v$ along the path $P$ .",
"Define $I_i = I - u + v_i$ for $i=0,\\dots ,d$ .",
"If these are all independent sets, then they form a TS-sequence of length $d$ from $I_0=I$ to a set $I_d$ that satisfies $|I_d\\backslash J|<|I\\backslash J|$ .",
"Then we can prove the statement by applying the induction assumption to $I_d$ and $J$ , analogously to before.",
"Otherwise, let $i$ be the maximum index such that $I_i$ is not an independent set, and let $x\\in I-u$ be a token adjacent to $v_i$ .",
"(Informally: we choose a token $x$ adjacent to or on $P$ , as close as possible to $v$ .",
"If $x$ lies on $P$ , then this implies $x=v_{i-1}$ .)",
"Note that $i<d$ : Otherwise either $J$ is not an independent set (if $x\\in J$ ) or $d=1$ (if $x\\notin J$ ), both contradictions.",
"In addition, $i\\ge 2$ holds: $i\\ge 1$ is obvious, and if $i=2$ there there would be a $v_1$ -claw with leaves $u$ , $x$ and $v_2$ .",
"We first argue that it is possible to slide the token from $x$ to $v$ , along the path $x,v_i,v_{i+1},\\ldots ,v_d$ .",
"By choice of $i$ , there is no vertex in $I$ adjacent to $v_j$ for $j>i$ .",
"If there is a vertex $y\\in I-x$ that is also adjacent to $v_i$ , then $G$ contains a $v_i$ -claw with leaves $x,y,v_{i+1}$ , a contradiction.",
"This shows that $I$ can be reconfigured to $I^{\\prime }=I-x+v$ , using $d-i+1$ moves.",
"It remains to show that we may apply the induction assumption to $I^{\\prime }$ and $J$ .",
"Clearly, $G[I^{\\prime }\\Delta J]$ again contains no cycles.",
"Since $P$ is a shortest path, and $i\\ge 2$ , it holds that $\\mbox{d}_G(x,v)\\le d-i+1<d=\\mbox{d}_G(u,v)$ .",
"So by choice of $u$ and $v$ , it follows that $x\\in J\\cap I$ .",
"Therefore, $|I^{\\prime }\\backslash J|=|I\\backslash J|$ .",
"However, the minimum distance $\\mbox{md}(I^{\\prime },J)$ is now at most $d(u,x)$ .",
"If $x\\in V(P)$ , then $x=v_{i-1}$ and $\\mbox{d}_G(u,x)\\le i-1$ ; otherwise $x$ is adjacent to $v_{i-1}$ (since there is no $v_i$ -claw with leaves $v_{i-1}$ , $v_{i+1}$ and $x$ ), so $\\mbox{d}_G(u,x)\\le i$ .",
"Since $i<d=\\mbox{md}(I,J)$ , we conclude that $\\phi (I^{\\prime },J)<\\phi (I,J)$ , and thus we may apply the induction assumption to $I^{\\prime }$ and $J$ .",
"Combining this with the fact that we have a TS-sequence from $I$ to $I^{\\prime }$ of length $d-i+1$ , and that $d+i+1\\le 2d=2\\mbox{md}(I,J)$ , we conclude that there exists a TS-sequence from $I$ to $J$ of length at most $d-i+1 + 2(|I^{\\prime }\\backslash J|-1)\\cdot \\mbox{diam}(G) + 2\\mbox{md}(I^{\\prime },J)\\le $ $d-i+1 + 2(|I\\backslash J|-1)\\cdot \\mbox{diam}(G) + 2i\\le $ $2(|I\\backslash J|-1)\\cdot \\mbox{diam}(G) + 2\\mbox{md}(I,J)$ .",
"This concludes the proof of the induction step.",
"$\\Box $"
],
[
"Details for Section ",
"First we show that in claw-free graphs, our definition of $I$ -augmenting paths is equivalent with the definition used in the setting of finding maximum independent sets.",
"The usual definition, as used e.g.",
"in [29], is given in the next proposition.",
"Proposition 18 Let $I$ be an independent set in a claw-free graph $G$ .",
"An $I$ -alternating walk $W=w_0,v_1,w_1,\\ldots ,v_k,w_k$ is an $I$ -augmenting path if and only if $w_0,w_k\\notin I$ and $I^{\\prime }=I\\backslash \\lbrace v_1,\\ldots ,v_k\\rbrace \\cup \\lbrace w_0,\\ldots ,w_k\\rbrace $ is an independent set.",
"Proof: Suppose $W$ is an $I$ -augmenting path.",
"Then clearly $w_0,w_k\\notin I$ and $v_i\\in I$ for $i=0,\\dots ,k$ .",
"The vertices $w_0$ and $w_k$ have no neighbors in $I\\backslash \\lbrace v_1,\\ldots ,v_k\\rbrace $ since they are free.",
"If a vertex $w_i$ with $1\\le i\\le k-1$ has a neighbor $x\\in I\\backslash \\lbrace v_1,\\ldots ,v_k\\rbrace $ , then $G$ contains a $w_i$ -claw with leaves $v_{i},v_{i+1},x$ , a contradiction.",
"Two vertices $w_i$ and $w_j$ are not adjacent since $W$ is chordless.",
"Hence $I^{\\prime }$ is an independent set again, which proves one direction of the statement.",
"Now suppose $I^{\\prime }$ is an independent set and $w_0,w_k\\notin I$ .",
"We prove that $W$ is an $I$ -augmenting path.",
"$W$ is chordless, otherwise there would be an edge $v_iw_j$ with $j\\notin \\lbrace i-1,i\\rbrace $ – but then $G$ contains a $v_i$ -claw with leaves $w_j,w_{i-1},w_i$ , a contradiction.",
"Therefore $w_0$ and $w_k$ are free with respect to $I$ , so $W$ is an $I$ -augmenting path.",
"$\\Box $ We now give a detailed proof of Lemma REF .",
"The proof is split up into three steps.",
"Proposition 19 Let $A$ and $B$ be two independent sets in a claw-free graph $G$ , such that $G[A\\Delta B]$ is a collection of cycles.",
"Then for any vertex $v\\in V(G)$ : if $N[v]\\cap A=\\emptyset $ then $N[v]\\cap B=\\emptyset $ .",
"Proof: Choose a vertex $v$ with $N[v]\\cap A=\\emptyset $ , and suppose to the contrary that there exists a vertex $w\\in N[v]\\cap B$ .",
"Then $w\\in B\\backslash A$ , so $w$ is part of a cycle $C$ in $G[A\\Delta B]$ .",
"Let $x$ and $y$ be the neighbors of $w$ on the cycle, so $\\lbrace x,y\\rbrace \\subseteq A$ .",
"Then neither $x$ nor $y$ is adjacent to $v$ , so there exists a $w$ -claw with leaves $v,x,y$ , a contradiction.",
"$\\Box $ Proposition 20 Let $I$ and $J$ be independent sets in a claw-free graph $G$ such that $I$ is not a dominating set and $G[I\\Delta J]$ is a collection of cycles.",
"Then $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"Proof: The proof is by induction over the number of cycles in $G[I\\Delta J]$ .",
"If there are no cycles, then $I=J$ so $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ trivially holds.",
"Now consider a cycle $C$ in $G[I\\Delta J]$ .",
"Let $v$ be a vertex with $N[v]\\cap I=\\emptyset $ (which exists since $I$ is not dominating), and choose a vertex $u\\in V(C)\\cap I$ .",
"Then $I^{\\prime }=I+v-u$ is again an independent set, and clearly, $I\\leftrightarrow _{\\mbox{\\sc tj}}I^{\\prime }$ .",
"Next, let $I^{\\prime \\prime }=I\\Delta V(C)$ , so $G[I\\Delta I^{\\prime \\prime }]$ consists only of the cycle $C$ .",
"Proposition REF shows that $N[v]\\cap I^{\\prime }=\\emptyset $ , so $G[I^{\\prime }\\Delta I^{\\prime \\prime }]$ contains no cycles (it consists of one odd length path and one isolated vertex).",
"Then by Theorem REF , $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc tj}}I^{\\prime \\prime }$ , and thus $I\\leftrightarrow _{\\mbox{\\sc tj}}I^{\\prime \\prime }$ .",
"Now $G[I^{\\prime \\prime }\\Delta J]$ is again a collection of cycles, but contains exactly one cycle fewer than $G[I\\Delta J]$ (namely $C$ ), so by induction we may conclude that $I^{\\prime \\prime }\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"Together, this shows that $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"$\\Box $ Proof of Lemma REF : First consider the case that $I$ is not a dominating set.",
"Let $H$ be the subgraph of $G[I\\Delta J]$ that consists of all cycle components.",
"(So possibly $H$ is the empty graph.)",
"Let $I^{\\prime }=I\\Delta V(H)$ .",
"So $G[I^{\\prime }\\Delta J]$ contains no cycles, and $G[I\\Delta I^{\\prime }]=H$ .",
"Then by Theorem REF , $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc tj}}J$ , and by Proposition REF , $I\\leftrightarrow _{\\mbox{\\sc tj}}I^{\\prime }$ .",
"Together this shows that $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"Otherwise, Proposition REF shows that there exists an $I$ -augmenting path $P$ .",
"Write $P=u_0,v_1,u_1,v_2,\\ldots ,v_k,u_k$ , with $v_i\\in I$ for all $i$ .",
"Then $I^{\\prime }=I\\backslash \\lbrace v_1,\\ldots ,v_k\\rbrace \\cup \\lbrace u_1,\\ldots ,u_k\\rbrace $ is again an independent set with $|I^{\\prime }|=|I|$ , and $G[I\\Delta I^{\\prime }]$ consists of a single (even length) path.",
"So by Theorem REF , $I\\leftrightarrow _{\\mbox{\\sc tj}}I^{\\prime }$ .",
"Since $u_0$ is a free vertex for $I$ , it is not dominated by $I^{\\prime }$ .",
"We conclude that $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc tj}}J$ , and thus $I\\leftrightarrow _{\\mbox{\\sc tj}}J$ .",
"$\\Box $"
],
[
"Proof Details for Lemmas ",
"For our detailed proofs of the statements from Section , it is useful to first characterize the neighborhood of $I$ -bad cycles using some simple observations (Proposition REF ), and next characterize (shortest) resolving sequences (Lemma REF ).",
"For a vertex set $S\\subseteq V(G)$ , we denote $N(S)=\\bigcup _{v\\in S} N(v)$ .",
"(We will apply this only to independent sets $S$ , so then $S\\cap N(S)=\\emptyset $ .)",
"Proposition 21 Let $I$ be an independent set of a claw-free graph $G$ , and let $C$ be an $I$ -bad cycle with $I$ -bipartition $[A,B]$ .",
"Then the following properties hold: For all $i\\ge 3$ , $N_i(B)=\\emptyset $ .",
"There are no edges between vertices in $N_2(B)$ and $N_0(B)$ .",
"For any $v\\in V(G)$ : $v\\in N(B) \\backslash A$ if and only if $v\\in N(A)\\backslash B$ .",
"$N(B) \\cap I = A$ .",
"Proof: This follows since $B$ is an independent set and $G$ is claw-free.",
"Suppose to the contrary that $vw\\in E(G)$ with $v\\in N_2(B)$ and $w\\in N_0(B)$ .",
"Let $N(v)\\cap B=\\lbrace x,y\\rbrace $ .",
"Then $G$ contains a $v$ -claw with leaves $v,x,y$ , a contradiction.",
"Suppose $v \\in N(B)\\backslash A$ and $v\\notin N(A)\\backslash B$ (the symmetric case is analogous).",
"Since $v\\in N(B)$ , we have $v\\notin B$ , so $v\\notin N(A)$ .",
"Choose any vertex $x \\in B\\cap N(v)$ .",
"It has two neighbors $y,z\\in A$ , so $G$ contains an $x$ -claw with leaves $v,y,z$ , a contradiction.",
"Suppose there is a token $v\\in I\\backslash A$ in the neighborhood of $B$ .",
"By the previous claim, $v \\in N(A)$ , which contradicts that $I$ is an independent set.$\\Box $ Shortest resolving sequences have a very specific and useful structure, which is characterized in the following lemma.",
"In particular, this lemma shows that a TS-sequence (starting with $I$ ) resolves an $I$ -bad cycle with $I$ -bipartition $[A,B]$ as soon as the first token slides from $N_2(B)$ to $N_1(B)$ , but not earlier.",
"Let $C$ be an $I$ -bad cycle with $I$ -bipartition $[A,B]$ .",
"We say that a sequence $I_0,\\ldots ,I_m$ with $I_0=I$ contains a resolving sequence for $C$ if for some $i\\in \\lbrace 0,\\ldots ,m\\rbrace $ , $G[I_i\\cup B]$ contains no cycle (so $I_0,\\ldots ,I_i$ is a resolving sequence, although $I_0,\\ldots ,I_m$ may not be one).",
"Lemma 22 Let $I$ be an independent set of a claw-free graph $G$ , and let $C$ be an $I$ -bad cycle with $I$ -bipartition $[A,B]$ .",
"Let $S=I_0,\\ldots ,I_m$ be a TS-sequence with $I_0=I$ .",
"Then the following properties hold: $S$ contains a resolving sequence for $C$ if and only if it contains a move $u\\rightarrow v$ with $u\\in N_2(B)$ and $v\\in N_1(B)$ .",
"For every index $i$ such that $I_0,\\ldots ,I_i$ contains no resolving sequence for $C$ : $I_i\\subseteq N_2(B)\\cup N_0(B)$ , and $I_i$ is obtained from $I_{i-1}$ by a move $u\\rightarrow v$ with $N(u)\\cap B=N(v)\\cap B$ .",
"Proof: Call an independent set $J$ $B$ -cyclic if there is one cycle in $G[J\\cup B]$ that contains all vertices of $B$ .",
"Since $J$ is an independent set and $G[J\\cup B]$ has maximum degree 2, this implies that $G[J\\cup B]$ consists of exactly one cycle and a number of isolated vertices.",
"Furthermore, it implies that every $v\\in B$ has exactly two neighbors in $J$ , which in turn are in $N_2(B)$ .",
"Now consider an independent set $I_{i}$ in the sequence $S$ , that is obtained from a $B$ -cyclic set $I_{i-1}$ using the move $u\\rightarrow v$ .",
"Since any vertex in $B$ has at most two neighbors in any independent set of $G$ , and $I_{i-1}$ is $B$ -cyclic, we deduce that $N(v)\\cap B\\subseteq N(u)\\cap B$ .",
"Clearly, $I_i$ is again $B$ -cyclic if and only if $N(v)\\cap B=N(u)\\cap B$ .",
"So if $I_i$ is not $B$ -cyclic, then $|N(v)\\cap B|\\le 1$ and $|N(u)\\cap B|\\ge 1$ .",
"Since $I_{i-1}\\subseteq N_0(B)\\cup N_2(B)$ , it follows that $u\\in N_2(B)$ , and since vertices in $N_2(B)$ have no neighbors in $N_0(B)$ (Proposition REF (REF )), it follows that $v\\in N_1(B)$ .",
"In this case, we argue that $G[I_i\\cup B]$ contains no cycle: If to the contrary $G[I_i\\cup B]$ contains a cycle $C^{\\prime }$ , then $C^{\\prime }$ does not contain $v$ .",
"So it would also be a cycle in $G[I_{i-1}\\cup B]$ , which contains neither $u$ nor its neighbors in $B$ , contradicting that $I_{i-1}$ is $B$ -cyclic.",
"Summarizing, we have shown that if $I_i$ is obtained from $I_{i-1}$ using the move $u\\rightarrow v$ and $I_{i-1}$ is $B$ -cyclic, then: If $I_i$ is again $B$ -cyclic, then $N(u)\\cap B=N(v)\\cap B$ .",
"If $I_i$ is not $B$ -cyclic, then $u\\in N_2(B)$ and $v\\in N_1(B)$ , and $G[I_i\\cup B]$ contains no cycles.",
"We use this to prove the properties in the lemma statement.",
"Note that $I_0=I$ is $B$ -cyclic, so if $S$ contains a non-$B$ -cyclic set, then the first such set $I_i$ has $i\\ge 1$ and is preceded by a $B$ -cyclic set $I_{i-1}$ .",
"If $S$ contains a resolving sequence, then clearly it contains a non-$B$ -cyclic set, so by considering the first non $B$ -cyclic set $I_i$ and applying (REF ), we conclude that $S$ contains a move from $N_2(B)$ to $N_1(B)$ .",
"On the other hand, if $S$ contains such a move, then from (REF ) it follows that $S$ contains a non-$B$ -cyclic set, and therefore by (REF ), it contains a resolving sequence for $C$ .",
"This proves Property (REF ).",
"Property (REF ) implies that if a subsequence $I_0,\\ldots ,I_i$ contains no resolving sequence for $C$ , then all these sets are $B$ -cyclic, so Property (REF ) follows from (REF ).",
"$\\Box $ Now we can prove Lemma REF and Lemma REF in detail.",
"Proof of Lemma REF : Denote $J=I\\Delta V(C)$ , and let $[A,B]$ be the $I$ -bipartition of $C$ .",
"Consider a shortest TS-sequence $I_0,\\ldots ,I_m$ that resolves $C$ .",
"Suppose first that $m=1$ , so $I_1$ is obtained from $I$ by a move $u\\rightarrow v$ with $u\\in N_2(B)$ and $v\\in N_1(B)$ (Lemma REF (REF )).",
"Since $G$ contains no $u$ -claw, it follows that $N(v)\\cap B\\subset N(u)\\cap B$ , so we can label the vertices of $C$ $c_1,\\ldots ,c_{2n}$ in order along the cycle such that $u=c_1$ and $N(v)\\cap B=\\lbrace c_2\\rbrace $ .",
"Then the following sequence of moves yields $J$ , when starting with $I$ : $c_1\\rightarrow v$ , $c_{2n-1}\\rightarrow c_{2n}$ , $c_{2n-3}\\rightarrow c_{2n-2},\\ldots ,c_3\\rightarrow c_4,v\\rightarrow c_2$ .",
"Using the fact that $C$ is a chordless cycle and that $N(B)\\cap I=A$ (Proposition REF (REF )), it is easily verified that every vertex set in the resulting sequence is an independent set, so $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"Now suppose that $m\\ge 2$ .",
"Let $I^{\\prime }=I_{m-1}$ .",
"Then there exists an $I^{\\prime }$ -bad cycle $[A^{\\prime },B]$ , since until this point in the TS-sequence, tokens that started on $A$ (i.e.",
"tokens on $N_2(B)$ ) only moved to vertices with exactly the same neighbors in $B$ (Lemma REF (REF )).",
"From $I^{\\prime }$ we can obtain $J^{\\prime }=I^{\\prime }\\Delta V(C)$ in the same way as shown the previous paragraph.",
"It remains to show that from $J^{\\prime }$ , $J$ can be obtained, by essentially reversing all moves outside the neighborhood of the cycle, while moving no tokens on B.",
"More precisely, for every $i\\in \\lbrace 0,\\ldots ,m-1\\rbrace $ , define $I^{\\prime }_i=(I_i\\backslash N(B))\\cup B$ .",
"Note that $I^{\\prime }_0=J$ and that $I^{\\prime }_{m-1}=J^{\\prime }$ .",
"We argue that (after removing repetitions), $I^{\\prime }_{m-1},\\ldots ,I^{\\prime }_0$ yields a TS-sequence from $J^{\\prime }$ to $J$ : By Lemma REF (REF ), for any $i<m-1$ , if $I^{\\prime }_{i+1}\\ne I^{\\prime }_i$ , then $I^{\\prime }_{i+1}$ can be obtained from $I^{\\prime }_i$ by a move $u\\rightarrow v$ where both $u$ and $v$ are part of $N_0(B)$ .",
"This way, it can be verified that for every $i$ , $I^{\\prime }_i$ is an independent set, so $J^{\\prime }\\leftrightarrow _{\\mbox{\\sc ts}}J$ .",
"Combining this with $I\\leftrightarrow _{\\mbox{\\sc ts}}I^{\\prime }$ and $I^{\\prime }\\leftrightarrow _{\\mbox{\\sc ts}}J^{\\prime }$ shows that $I\\leftrightarrow _{\\mbox{\\sc ts}}J$ .$\\Box $ Proof of Lemma REF : Denote the $I$ -bipartition of $C$ by $[A,B]$ .",
"Consider a shortest TS-sequence $S=I_0,I_1,\\dots ,I_m$ that resolves $C$ .",
"Let $u\\rightarrow v$ be the last move of this sequence, so $u\\in N_2(B)$ and $v\\in N_1(B)$ (Lemma REF (REF )).",
"By Proposition REF (REF ), $|N(v)\\cap A|\\ge 1$ , and clearly, $|N(v)\\cap I|\\le 2$ .",
"So one of the following cases applies to the neighborhood of $v$ .",
"Case 1: $N(v)\\cap I=\\lbrace x\\rbrace $ for some $x\\in A$ .",
"Then the move $x\\rightarrow v$ yields an independent set again, and since $v\\in N_1(B)$ , it resolves $C$ (Lemma REF (REF )), so $C$ is both internally and externally resolvable.",
"Case 2: $N(v)\\cap I=\\lbrace x,y\\rbrace $ for some $x\\in A$ and $y\\notin A$ .",
"In this case, we omit all internal moves, to obtain an external TS-sequence that resolves $C$ .",
"More precisely, for every $i$ , define $I^{\\prime }_i=(I_i\\backslash N(B))\\cup A$ , and consider the sequence $I^{\\prime }_0,\\ldots ,I^{\\prime }_{m-1}$ .",
"For every $i$ such that $I^{\\prime }_i\\ne I^{\\prime }_{i+1}$ , it holds that $I^{\\prime }_{i+1}$ is obtained from $I^{\\prime }_i$ by a move $u_i\\rightarrow v_i$ where both $u_i$ and $v_i$ are in $N_0(B)$ (Lemma REF (REF )).",
"Since there are no edges between $N_0(B)$ and $A\\subseteq N_2(B)$ (Proposition REF (REF )), every $I^{\\prime }_i$ is an independent set, and thus this is a TS-sequence.",
"Now $x\\in I^{\\prime }_{m-1}$ because $x\\in A$ , and $I_{m-1}\\cap N(v)=\\lbrace u\\rbrace \\subseteq N_2(B)$ , so $N(v)\\cap I^{\\prime }_{m-1}=\\lbrace x\\rbrace $ .",
"Therefore, from $I^{\\prime }_{m-1}$ , the move $x\\rightarrow v$ can be made, to resolve $C$ (Lemma REF (REF )).",
"This shows that $C$ is externally resolvable.",
"Case 3: $N(v)\\cap I=\\lbrace x,y\\rbrace $ for some $x,y\\in A$ .",
"In this case, we omit all external moves, to obtain an internal TS-sequence that resolves $C$ .",
"More precisely, for every $i$ , define $I^{\\prime }_i=(I\\backslash N(B))\\cup (I_i\\cap N(B))$ , and consider the sequence $I^{\\prime }_0,\\ldots ,I^{\\prime }_{m-1}$ .",
"For every $i<m-1$ such that $I^{\\prime }_i\\ne I^{\\prime }_{i+1}$ , it holds that $I^{\\prime }_{i+1}$ is obtained from $I^{\\prime }_i$ by a move $u_i\\rightarrow v_i$ where both $u_i$ and $v_i$ are in $N_2(B)$ (Lemma REF (REF )).",
"Since there are no edges between $N_2(B)$ and $(I\\backslash N(B))\\subseteq N_0(B)$ (Proposition REF (REF )), every $I^{\\prime }_i$ is an independent set, and thus this is a TS-sequence.",
"Since $I_{m-1}\\cap N(v)=\\lbrace u\\rbrace $ and $N(v)\\cap I\\subseteq N(B)$ , it also holds that $I^{\\prime }_{m-1}\\cap N(v)=\\lbrace u\\rbrace $ , so from $I^{\\prime }_{m-1}$ , the move $u\\rightarrow v$ can be made, to resolve $C$ (Lemma REF (REF )).",
"This shows that $C$ is internally resolvable.",
"$\\Box $"
],
[
"The Proof of Theorem ",
"We prove in this section that to verify whether an $I$ -bad cycle $C$ is externally resolvable it suffices to search for a certain type of $I$ -augmenting paths in $G-V(C)$ .",
"The key observation is that in a shortest TS-sequence that externally resolves $C$ , no token moves more than once, provided that $I$ is a maximum independent set (Lemma REF below).",
"In that case, the token moves easily yield a set of $I$ -alternating paths, as shown in the next lemma.",
"Note that Lemma REF may fail if we drop the assumption that $I$ is a maximum independent set (for example, consider the graph from Figure REF with dashed edges removed, and let $I$ contain vertex $x$ together with all round white vertices, except for $y$ .",
"We note that $I$ is then maximal, but not maximum).",
"To be precise, we say that in a TS-sequence $I_0,\\ldots ,I_n$ , every token moves at most once if for all $i,j\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ with $i\\le j$ : $v\\in I_{j}\\backslash I_{j+1}$ implies that $v\\in I_i$ .",
"Proposition 23 Let $I_0,\\dots ,I_m$ be a TS-sequence in a claw-free graph $G$ in which every token moves at most once.",
"Then every component of $G[I_0\\Delta I_m]$ is a path of odd length.",
"Proof: The proof is by induction on $m$ .",
"Let $I_m$ be obtained from $I_{m-1}$ by the move $x\\rightarrow y$ .",
"By induction, $G^{\\prime }=G[I_0\\Delta I_{m-1}]$ is a set of paths of odd length.",
"Since every token moves at most once, $G^{\\prime }$ contains neither $x$ nor $y$ .",
"By claw-freeness, $G[I_0\\Delta I_m]$ has maximum degree two, and since it is obtained from $G^{\\prime }$ by adding two adjacent vertices (plus incident edges), no even length path can be introduced.",
"So it now suffices to show that in $G[I_0\\Delta I_m]$ , there is no cycle containing $x$ and $y$ .",
"The vertex $x$ is part of both $I_0$ and $I_{m-1}$ , so it has no neighbors in $I_0\\Delta I_{m-1}$ .",
"Therefore it has degree 1 in $G[I_0\\Delta I_m]$ , which shows that it is not part of a cycle, and thus $G[I_0\\Delta I_m]$ is a collection of odd length paths again.$\\Box $ We will often use the following simple proposition.",
"Proposition 24 Let $I$ and $J$ be two independent sets in a graph $G$ .",
"If every component in $G[I\\Delta J]$ is an odd length path, and $|I\\backslash J|=p$ , then there exists a TS-sequence from $I$ to $J$ of length $p$ .",
"Proof: We prove the statement by induction on $|I\\backslash J|$ .",
"The case $I=J$ is trivial, so now assume that there exists at least one odd length path component $P$ in $G[I\\Delta J]$ .",
"Then $P$ has an end vertex $v\\in J\\backslash I$ with neighbor $u\\in I\\backslash J$ , but with no other neighbors in $I$ .",
"So from $I$ , we can make the move $u\\rightarrow v$ , which yields $I^{\\prime }$ , such that every component of $G[I^{\\prime }\\Delta J]$ is again an odd length path, and $|I^{\\prime }\\backslash J|=p-1$ .",
"The statement now follows by induction.",
"$\\Box $ Lemma 25 Let $I$ be a maximum independent set in a claw-free graph $G$ , and let $C$ be an externally resolvable $I$ -bad cycle.",
"Then in any shortest external resolving sequence for $C$ , every token moves at most once.",
"Proof: The following proof is illustrated in Figure REF .",
"Let $[A,B]$ be the $I$ -bipartition of $C$ .",
"Let $S=I_0,\\ldots ,I_m$ be a shortest external resolving sequence for $C$ .",
"By definition of external resolving sequence, all moves but the last one are between vertices in $N_0(B)$ , and the last move of the sequence is $u\\rightarrow v$ for some $u\\in N_2(B)$ and $v\\in N_1(B)$ (Lemma REF (REF )).",
"We prove the statement by induction on $m$ .",
"If $m=1$ then obviously, no token moves twice.",
"Figure: An illustration of the proof of Lemma .Now suppose that $m\\ge 2$ .",
"Then $A\\subseteq I_1$ , so $C$ is also an $I_1$ -bad cycle with $I_1$ -bipartition $[A,B]$ , and $I_1,\\ldots ,I_m$ is a shortest external resolving sequence for $C$ with respect to $I_1$ .",
"By induction, no token moves twice in this sequence.",
"So every component of $G[I_1\\Delta I_m]$ is an odd length path (Proposition REF ), which is clearly both $I_1$ -alternating and $I_m$ -alternating.",
"Let $P=v_1,u_1,v_2,u_2,\\ldots ,v_k,u_k$ be the path in $G[I_1\\Delta I_m]$ that contains $u$ and $v$ , labeled such that $u_i\\in I_1$ and $v_i\\in I_m$ for every $i$ .",
"So $u=u_{k^{\\prime }}$ and $v=v_{k^{\\prime }}$ for some index ${k^{\\prime }}$ .",
"Then it is easily verified that from $I_1$ , we can make the sequence of moves $u_1\\rightarrow v_1,\\ldots ,u_{k^{\\prime }}\\rightarrow v_{k^{\\prime }}$ , maintaining an independent set throughout.",
"This TS-sequence resolves $C$ using ${k^{\\prime }}$ moves (Lemma REF (REF )).",
"Since the TS-sequence $I_1,\\ldots ,I_m$ is a shortest TS-sequence for $I_1$ that resolves $C$ , we conclude that these are all moves from the sequence.",
"So $k=k^{\\prime }$ , and the path $P$ is the only component in $G[I_1\\Delta I_m]$ , and thus $m=k+1$ .",
"If in the entire sequence $I_0,\\ldots ,I_m$ no token moves twice, then there is nothing to prove, so now assume that at least one token moves twice.",
"Since no token moves twice in the subsequence $I_1,\\ldots ,I_m$ , the first move from $I_0$ to $I_1$ is $x\\rightarrow y$ , and later in the sequence, a move $y\\rightarrow z$ occurs.",
"Then $y$ and $z$ lie on the path $P$ , so $y=u_j$ and $z=v_j$ for some index $j$ .",
"We start with a few simple observations: $j<k$ .",
"If to the contrary $j=k$ , then $y=u$ and $z=v$ , so $y\\in N_2(B)$ .",
"But then $x\\in N_2(B)$ (Lemma REF (REF )), which contradicts that $I_0,\\ldots ,I_m$ is an external resolving sequence.",
"$N(x)\\cap V(P)\\subseteq \\lbrace v_1,v_j,u_j,v_{j+1}\\rbrace $ .",
"This holds because $x$ cannot be adjacent to a vertex $u_{j^{\\prime }}$ with $j^{\\prime }\\ne j$ , since these vertices are both part of the independent set $I_0$ .",
"Furthermore, any vertex $v_{j^{\\prime }}$ with $j^{\\prime }\\notin \\lbrace 1,j,j+1\\rbrace $ has neighbors $u_{j^{\\prime }}$ and $u_{j^{\\prime }-1}$ , which are both in $I_0$ , so an edge $xv_{j^{\\prime }}$ would yield a $v_{j^{\\prime }}$ -claw.",
"$xv_{j+1}\\in E(G)$ .",
"Assume to the contrary that $xv_{j+1}\\notin E(G)$ .",
"We can then argue that $J:=(I_0\\backslash \\lbrace u_{j+1},\\ldots ,u_k\\rbrace )\\cup \\lbrace v_{j+1},\\ldots ,v_k\\rbrace $ is also an independent set: $P$ is chordless, so none of the added vertices $\\lbrace v_{j+1},\\ldots ,v_k\\rbrace $ are adjacent to vertices in $\\lbrace u_1,\\ldots ,u_{j-1}\\rbrace $ .",
"By the previous observation and the assumption $xv_{j+1}\\notin E(G)$ , $x$ is also not adjacent to any of the added vertices.",
"Finally, considering $I_m$ , which contains the added vertices and the vertices $I_0\\backslash V(P)$ , we conclude that the added vertices are not adjacent to vertices of $I_0\\backslash V(P)$ .",
"Note that $G[I_0\\Delta J]$ consists of a single odd path on $2(k-j)$ vertices, so there exists a (shorter) TS-sequence of length $k-j$ which resolves $C$ (Proposition REF ), a contradiction.",
"To complete the proof we consider four cases.",
"Case 1: $v_{j}\\notin N(x)$ and $v_1\\notin N(x)$ .",
"If $j=1$ , then $v_1=v_j$ has no neighbors in $I_0$ .",
"Otherwise, both $v_1$ and $v_j$ are free vertices with respect to $I_0$ (their only $I_0$ -neighbors are $u_1$ and $u_{j-1}$ , respectively), so $v_1,u_1,\\ldots ,u_{j-1},v_j$ is an $I_0$ -augmenting path.",
"In both cases, this contradicts that $I=I_0$ is a maximum independent set.",
"Case 2: $v_{j}\\in N(x)$ and $v_1\\notin N(x)$ .",
"Then the previous observations show that in $G[I_0\\Delta I_m]$ , we have the following odd path component: $v_1,u_1,\\ldots ,v_j,x,v_{j+1},u_{j+1},\\ldots ,v_k,u_k,$ containing $2k$ vertices.",
"Hence a TS-sequence of shorter length $k<m$ that resolves $C$ is possible (Proposition REF ), a contradiction.",
"Case 3: $v_{j}\\notin N(x)$ and $v_1\\in N(x)$ .",
"Then the previous observations show that in $G[I_0\\Delta I_m]$ , we have the following odd path component: $v_j,u_{j-1},v_{j-1},\\ldots ,u_1,v_1,x,v_{j+1},u_{j+1},\\ldots ,v_k,u_k,$ containing $2k$ vertices.",
"Hence a TS-sequence of shorter length $k<m$ that resolves $C$ is possible (Proposition REF ), a contradiction.",
"Case 4: $v_{j}\\in N(x)$ and $v_1\\in N(x)$ .",
"If $j=1$ , then $y=u_j=u_1$ and the first two moves $x\\rightarrow y, u_1\\rightarrow v_1$ can be replaced by one move $x\\rightarrow v_1$ , giving a shorter TS-sequence, a contradiction.",
"Otherwise, $x$ has three neighbors $v_1$ , $v_j$ and $v_{j+1}$ in an independent set $I_m$ , contradicting claw-freeness.",
"We have obtained a contradiction in every case, so we conclude that in a shortest external TS-sequence of length $m$ , no token moves twice.",
"This concludes the inductive step of the proof, and the statement follows by induction.",
"$\\Box $ We can now combine Proposition REF and Lemma REF to prove Theorem REF .",
"Proof of Theorem REF : Denote $G^{\\prime }=G-A-B$ and $I^{\\prime }=I\\backslash A$ , so $I^{\\prime }$ is an independent set of $G^{\\prime }$ .",
"Suppose first that $G^{\\prime }$ contains such an $I^{\\prime }$ -augmenting path $P=u_0$ ,$v_0$ ,$u_1$ ,$v_1$ ,$\\dots $ ,$v_{k-1}$ ,$u_k$ with $u_0\\in N_0(B)$ and $u_k\\in N_1(B)$ .",
"(So $k\\ge 1$ .)",
"We prove that then $C$ is externally resolvable.",
"Since $u_0\\notin N(B)$ and $u_0\\notin B$ , we observe that $u_0\\notin N(A)$ (Proposition REF (REF )).",
"Secondly, we argue that $|N(u_k)\\cap A|=1$ : by Proposition REF (REF ), $u_k$ has at least one neighbor $w$ in $A$ .",
"Since $u_k$ is also adjacent to $v_{k-1}\\in I\\backslash A$ and there is no $u_k$ -claw, $w$ is its only neighbor in $A$ .",
"All other vertices $u_j$ with $1\\le j\\le k-1$ are adjacent to $v_{j-1}$ and $v_j$ , which are both in $I\\backslash A$ , so they have no other neighbors in $I$ , in particular not in $A$ .",
"Since $u_0$ is free with respect to $I^{\\prime }$ , it has no neighbor in $I$ other than $v_0$ .",
"Since $P$ is also chordless, it follows that we can apply the moves $v_0\\rightarrow u_0$ ,...,$v_{k-1}\\rightarrow u_{k-1}$ to $I$ while maintaining an independent set, which yields $J$ .",
"We have that $N(u_k)\\cap J=\\lbrace w\\rbrace $ , so the move $w\\rightarrow u_k$ (with $w\\in A$ and $u_k\\in N_1(B)$ ) is subsequently possible, and resolves the cycle $C$ (Lemma REF (REF )), and thus $C$ is externally resolvable.",
"We now prove the other direction.",
"Suppose that $C$ is externally resolvable, and consider a shortest external resolving sequence $S=I_0,\\ldots ,I_m$ (with $I_0=I$ ).",
"By definition of external resolving sequence and Lemma REF (REF ), the last move is $u\\rightarrow v$ for some $u\\in N_2(B)$ and $v\\in N_1(B)$ , and every other move is between two vertices in $N_0(B)$ .",
"By Lemma REF , every token moves at most once in $S$ .",
"So $G[I_0\\Delta I_m]$ is a set of odd paths (Proposition REF ), in which all vertices except $u$ and $v$ are in $N_0(B)$ .",
"Clearly these paths are all both $I_0$ -alternating and $I_m$ -alternating.",
"Consider the path $P$ that contains $u$ and $v$ and denote $P=v_0,u_0,v_1,u_1,\\ldots ,v_k,u_k$ , with $v_i\\in I_m$ and $u_i\\in I_0$ for all $i$ .",
"So for every $i$ , $u_i\\rightarrow v_i$ is a move in the TS-sequence $S$ , and $u_k\\rightarrow v_k$ must be the last of these moves on $P$ , since an independent set should be maintained throughout.",
"So $u=u_k$ and $v=v_k$ .",
"We now argue that $P^{\\prime }=v_0,u_0,v_1,u_1,\\ldots ,u_{k-1},v_k$ is the desired $I^{\\prime }$ -augmenting path in $G^{\\prime }$ : clearly the path is $I^{\\prime }$ -alternating and chordless.",
"The vertex $v_0$ is not adjacent to any $I$ -vertex $x$ other than $u_0$ , because otherwise $I_m$ is not an independent set (if $x\\in I_m$ ), or $P$ is not a component of $G[I_0\\Delta I_m]$ (if a move $x\\rightarrow y$ occurs in $S$ ).",
"So $v_0$ is a free vertex with respect to both $I$ and $I^{\\prime }$ .",
"The vertex $v_k=v$ is adjacent to both $u_{k-1}$ and $u_k=u$ (in $G$ ), so it is not adjacent to any other vertex from $I$ .",
"Therefore, in $G^{\\prime }$ , it is a free vertex with respect to $I^{\\prime }$ (which does not contain $u$ ).",
"This shows that $P^{\\prime }$ is an $I^{\\prime }$ -augmenting path for $G^{\\prime }$ , between vertices in $N_0(B)$ and $N_1(B)$ .",
"$\\Box $"
],
[
"The Proof of Theorem ",
"Lemma REF proves the forward direction of Theorem REF , and subsequently, Lemma REF proves the backward direction.",
"Lemma 26 Let $I$ be an independent set in a claw-free graph $G$ , and $C=c_0,c_1,\\dots ,c_{2n-1},c_{0}$ be an $I$ -bad cycle with $n\\ge 3$ and $c_0\\in I$ , and $I$ -bipartition $[A,B]$ .",
"If $C$ is internally resolvable then $D(G,C)$ or $D(G,C^{rev})$ contains a directed path from a vertex $b\\in N_1(B)$ with $N(b)\\cap I\\subseteq A$ to a vertex in $A$ .",
"Proof: Let $I_0,\\dots ,I_m$ be a shortest internal resolving TS-sequence for $C$ , so $I_0=I$ .",
"By definition of internally resolvable and Lemma REF (REF ), the last move is is from $N_2(B)$ to a vertex $b\\in N_1(B)$ , and all other moves $u\\rightarrow v$ satisfy $u,v\\in N_2(B)$ and $N(u)\\cap B = N(v)\\cap B$ .",
"We shall prove that $A$ is reachable from $b$ by a directed path in $D(G,C)$ or $D(G,C^{rev})$ .",
"For every $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ , $I_0$ contains exactly one vertex of $L_i$ (namely $c_{2i}$ ), and this accounts for all vertices in $I_0\\cap N_2(B)$ .",
"Since $N(u)\\cap B = N(v)\\cap B$ holds for every move $u\\rightarrow v$ , this property is maintained for every $I_j$ .",
"So for all $j\\in \\lbrace 0,\\dots , m-1\\rbrace $ and $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ , we may denote by $v^i_j$ the unique vertex in $I_j \\cap L_i$ .",
"W.l.o.g.",
"assume $N(b)\\cap B=\\lbrace c_1\\rbrace $ .",
"Since a token is moved to $b$ in the last move, $b$ is free in $I_{m-1}$ , so it cannot be adjacent to both $v^{0}_{m-1}$ and $v^{1}_{m-1}$ .",
"Assume w.l.o.g.",
"that it is nonadjacent to $v^1_{m-1}$ (otherwise reverse the order of vertices of $C$ , such that the remainder of the proof applies to $D(G,C^{rev})$ instead of $D(G,C)$ ).",
"Then by definition, $D(G,C)$ contains an arc $(b,v^1_{m-1})$ .",
"For all $j$ , denote $I^{\\prime }_j=I_j\\cap N(B)$ .",
"So we conclude that at least one vertex of $I^{\\prime }_{m-1}$ is reachable from $b$ in $D(G,C)$ .",
"For every $j=0,\\dots ,m-1$ and every $i$ , $D(G,C)$ contains an arc from $v^i_j$ to $v^{(i+1)\\bmod n}_j$ , because these vertices are both in $I_j$ and are therefore nonadjacent in $G$ .",
"Therefore if for some $j=0\\dots m-1$ , at least one vertex in $I^{\\prime }_j=\\lbrace v^0_j,\\dots ,v^{n-1}_j\\rbrace $ is reachable from $b$ in $D(G,C)$ , then all vertices of $I^{\\prime }_j$ are reachable.",
"But $I^{\\prime }_{j-1}\\cap I^{\\prime }_j\\ne \\emptyset $ (they share in fact $n-1$ vertices), so by a simple induction proof it follows that every vertex of every $I^{\\prime }_j$ is reachable from $b$ in $D(G,C)$ .",
"In particular, this shows that there is a directed path from $b$ to $I^{\\prime }_0=A$ in $D(G,C)$ .",
"It remains to show that $b$ has no neighbors in $I\\backslash A$ .",
"Suppose that some vertex $x\\in I\\backslash A$ is adjacent to $b$ .",
"Since $x\\notin N_2(B)$ (Proposition REF (REF )), the token on $x$ is never moved in the TS-sequence, so $x\\in I_m$ .",
"But its neighbor $b$ is also part of the independent set $I_m$ , a contradiction.",
"$\\Box $ Lemma 27 Let $I$ be an independent set in a claw-free graph $G$ , and $C=c_0,c_1,\\dots ,c_{2n-1},c_{0}$ be an $I$ -bad cycle with $n\\ge 4$ and $c_0\\in I$ , and $I$ -bipartition $[A,B]$ .",
"If $D(G,C)$ or $D(G,C^{rev})$ contains a directed path from a vertex $b\\in N_1(B)$ with $N(b)\\cap I\\subseteq A$ to a vertex in $A$ , then $C$ is internally resolvable.",
"Proof: W.l.o.g.",
"we may assume that $D(G,C)$ contains such a path, and that $N(b)\\cap B=\\lbrace c_1\\rbrace $ .",
"Let $P=u_0,\\dots ,u_m$ be a shortest path in $D(G,C)$ from $u_0=b$ to a vertex $u_m\\in A$ .",
"Throughout this proof, we take layer indices modulo $n$ and cycle indices modulo $2n$ , so $L_i$ denotes $L_{i \\bmod n}$ and $c_i$ denotes $c_{i\\bmod 2n}$ .",
"By definition of $D(G,C)$ , any arc $(u_0,x)$ must have $x\\in L_1$ .",
"Thus $u_1\\in L_1$ and similarly, $u_j\\in L_j$ follows inductively for all $j=1,\\dots ,m$ .",
"The vertex $u_m$ is the first vertex of $P$ in $A$ , with $u_m\\in L_m$ so $u_m=c_{2m}$ .",
"For all $j$ , $D(G,C)$ contains an arc from $c_{2j}$ to $c_{2j+2}$ , so we can extend $P$ to a directed path $P^{\\prime }=u_0,\\ldots ,u_{m+n-1}$ by defining $u_{m+j}=c_{2(m+j)}$ for $j=1\\dots n-1$ .",
"The following properties hold for $P^{\\prime }$ , and will be used often in the remainder of the proof: $& N(u_0) \\cap B = \\lbrace c_1\\rbrace ,\\\\& N(u_j) \\cap B = \\lbrace c_{2j-1}, c_{2j+1}\\rbrace \\mbox{ for } j=1,\\dots ,m+n-1.$ The idea is to reconfigure $A$ to subsequent infixes of $P^{\\prime }$ .",
"More precisely, for $j=0,\\dots ,m$ , define $I_j = \\lbrace u_{m-j},u_{m-j+1},\\dots ,u_{m-j+n-1}\\rbrace \\cup (I\\backslash A),$ and consider the sequence $I_0,\\ldots ,I_m$ .",
"This sequence starts with $I_0 = I$ (since $A=\\lbrace u_m,\\ldots ,u_{m+n-1}\\rbrace $ ).",
"Since $I_j=I_{j-1}+u_{m-j}-u_{m-j+n}$ for $j=1,\\dots ,m$ , the consecutive steps correspond to replacing $u_{m-j+n}$ by $u_{m-j}$ .",
"We will now show that for every $j$ , $I_j$ is an independent set, and that $u_{m-j+n}u_{m-j}\\in E(G)$ (so $u_{m-j+n}\\rightarrow u_{m-j}$ is a valid move), which shows that $I_0,\\ldots ,I_m$ is a TS-sequence.",
"Considering the last move, it then follows that this sequence resolves $C$ (Lemma REF (REF )), and is in fact an internal resolving sequence.",
"Summarizing, to prove the lemma, it now suffices to show that: $u_{m-j}u_{m-j+n}\\in E(G)$ for all $j\\in \\lbrace 1,\\ldots ,m\\rbrace $ , and $u_ju_{j^{\\prime }}\\notin E(G)$ , for all $j,j^{\\prime }\\in \\lbrace 0,\\dots ,m+n-1\\rbrace $ with $1\\le j^{\\prime }-j \\le n-1$ .",
"Indeed, the second condition ensures that each $I_j$ is an independent set: $u_0=b$ has no neighbors in $I\\backslash A$ by assumption, and the vertices $u_j$ for $j\\ge 1$ have no neighbors in $I\\backslash A\\subseteq N_0(B)$ because there are no edges between $N_0(B)$ and $N_2(B)$ (Proposition REF (REF )).",
"To prove the above statements, we will prove a few claims, that are marked with Greek letters for later reference.",
"Claim $\\alpha $: $u_{j}u_{j+1} \\notin E(G)$ for all $j$ .",
"This claim follows directly from the definition of $D(G,C)$ and the fact that these are consecutive path vertices.",
"Claim $\\beta $: $u_{j}u_{j^{\\prime }} \\notin E(G)$ for $2\\le j^{\\prime }-j\\le n-2$ .",
"If $j>0$ then $u_j$ and $u_{j^{\\prime }}$ belong to $L_j$ and $L_{j^{\\prime }}$ , respectively.",
"So their neighbors in $B$ are exactly $c_{2j-1},c_{2j+1}$ and $c_{2j^{\\prime }-1},c_{2j^{\\prime }+1}$ , respectively.",
"By choice of $j$ and $j^{\\prime }$ , these are four different vertices.",
"Thus if $u_ju_{j^{\\prime }}\\in E(G)$ , then there would be a $u_{j^{\\prime }}$ -claw with leaves $u_j,c_{2j^{\\prime }-1},c_{2j^{\\prime }+1}$ , a contradiction.",
"If $j=0$ , then the proof is analogous, except that $u_j$ has exactly one neighbor in $B$ , namely $c_1=c_{2j+1}$ .",
"This concludes the proof of Claim $\\beta $ .",
"Together, Claims $\\alpha $ and $\\beta $ prove statement (REF ) above for all cases except $j^{\\prime }-j=n-1$ .",
"So we conclude that it now remains to show that: Claim $\\gamma $: $u_ju_{j+n}\\in E(G)$ for all $j\\in \\lbrace 0,\\dots ,m-1\\rbrace $ and Claim $\\delta $: $u_ju_{j+n-1}\\notin E(G)$ for all $j\\in \\lbrace 0,\\dots ,m\\rbrace $ .",
"We prove these claims by induction on $j$ .",
"$\\delta (0)$ : $u_0u_{n-1}\\notin E(G)$ , for otherwise there would be a $u_{n-1}$ -claw with leaves $u_0,c_{2n-3},c_{2n-1}$ (recall that $N(u_0)\\cap B=c_1$ , and that $n\\ge 3$ ).",
"$\\gamma (0)$ : We wish to prove that $u_0u_n\\in E(G)$ .",
"We first observe that $N(u_0)\\cap A\\subseteq \\lbrace c_0,c_2\\rbrace $ .",
"Indeed, if $u_0$ would be adjacent to another vertex $c_{2i}\\in A$ , then there is a $c_{2i}$ -claw with leaves $u_0,c_{2i-1},c_{2i+1}$ .",
"The vertex $u_0$ is adjacent to at least one of $c_0$ and $c_2$ ; otherwise there would be a $c_1$ -claw with leaves $u_0,c_0,c_2$ .",
"If $u_0$ is adjacent to exactly one of them, then $u_0$ is already free in $I$ , so $C$ can trivially be (internally) resolved in one move.",
"So now we may assume that $N(u_0)\\cap A=\\lbrace c_0,c_2\\rbrace $ .",
"If $m\\le n$ then $u_n\\in A$ so $u_n=c_0$ , which shows that $u_0u_n\\in E(G)$ .",
"Now suppose $m=n+1$ , so $u_{n+1}=c_2$ .",
"Claim $\\alpha $ shows that $u_nc_2=u_nu_{n+1}\\notin E(G)$ .",
"Since $u_n\\in L_0$ , it holds that $u_nc_1\\in E(G)$ .",
"We conclude that $u_nc_0\\in E(G)$ , for otherwise there would be a $c_1$ -claw with leaves $c_0,c_2,u_n$ .",
"Next, we note that $u_{n-1}c_0 \\in E(G)$ , because otherwise there would be a shorter path $u_0,\\dots ,u_{n-1},c_0$ in $D(G,C)$ .",
"Furthermore, $u_0 u_{n-1} \\notin E(G)$ holds by $\\delta (0)$ , and $u_{n-1} u_n \\notin E(G)$ by $\\alpha $ .",
"Combining these facts, we conclude that $u_0u_n\\in E(G)$ , because otherwise there would be a $c_0$ -claw with leaves $u_0,u_n,u_{n-1}$ in $G$ .",
"In the remaining case, $m\\ge n+2$ holds.",
"Then $u_nc_2\\in E(G)$ , for otherwise there would be a shorter path $u_0,\\dots ,u_n,c_2$ in $D(G,C)$ .",
"In this case $u_0u_n\\in E(G)$ follows since otherwise, there would be a $c_2$ -claw with leaves $u_0,u_n,c_3$ .",
"This concludes the proof of Claim $\\gamma $ for the case $j=0$ .",
"$\\delta (1)$ : By $\\gamma (0)$ , it holds that $u_0u_n\\in E(G)$ .",
"Recall that $u_0u_1\\notin E(G)$ , $c_{2n-1}u_1\\notin E(G)$ and $c_{2n-1}u_0\\notin E(G)$ .",
"So $u_1u_n\\notin E(G)$ , because otherwise there would be a $u_n$ -claw with leaves $c_{-1},u_0,u_1$ .",
"$\\delta (j)\\Rightarrow \\gamma (j)$ for $j=1,\\dots ,m-1$ : By $\\delta (j)$ , $u_j u_{j+n-1}\\notin E(G)$ holds, and by $\\alpha $ , $u_{j+n}u_{j+n-1}\\notin E(G)$ holds.",
"So $u_ju_{j+n}\\in E(G)$ , for otherwise there would be a $c_{2j-1}$ -claw with leaves $u_j,u_{j+n},u_{j+n-1}$ .",
"$\\gamma (j)\\Rightarrow \\delta (j+1)$ for $j=1,\\dots ,m-1$ : We observe that $u_{j+n}u_{j-1}\\in E(G)$ , for otherwise there would be a shorter path $u_0,\\dots ,u_{j-1},u_{j+n},\\dots ,u_m$ in $D(G,C)$ .",
"Next, $u_{j+n}u_{j} \\in E(G)$ by $\\gamma (j)$ , $u_{j-1}u_j\\notin E(G)$ and $u_j u_{j+1}\\notin E(G)$ by $\\alpha $ , and $u_{j-1}u_{j+1} \\notin E(G)$ by $\\beta $ , using that $n\\ge 4$ .",
"We conclude that $u_{j+1}u_{j+n}\\notin E(G)$ , for otherwise there would be a $u_{j+n}$ -claw with leaves $u_{j-1},u_j,u_{j+1}$ .",
"This concludes the induction proof of Claims $\\gamma $ and $\\delta $ , and therefore the proof of the lemma.",
"$\\Box $"
],
[
"An Example of a Nontrivial Internal Resolving Sequence",
"In Figure REF on the next page, the construction of the graph $D(G,C)$ is illustrated.",
"This figure shows an example where an elaborate TS-sequence is required to (internally) resolve the given cycle.",
"Figure: An example of a graph GG, with an II-bad cycle of length 14, which is internally resolvable in m=18m=18 steps.",
"The vertices of II are drawn as circles, and the other vertices of the II-bad cycle as squares.Half edges at the boundary of the figure continue on the other side.",
"Vertices in each column L 1 ,⋯,L 7 L_1,\\dots ,L_7 form a clique.",
"The directed path from the vertex bb to II in D(G,C)D(G,C) is also shown as a red dotted line to clarify the structure of GG."
]
] | 1403.0359 |
[
[
"A priori analysis: an application to the estimate of the uncertainty in\n course grades"
],
[
"Abstract The a priori analysis (APA) is discussed as a tool to assess the reliability of grades in standard curricular courses.",
"This unusual, but striking application is presented when teaching the section on data treatment of a Laboratory Course to illustrate the characteristics of the APA and its potential for widespread use, beyond the traditional Physics Curriculum.",
"The conditions necessary for this kind of analysis are discussed, the general framework is set out and a specific example is given to illustrate its various aspects.",
"Students are often struck by this unusual application and are more apt to remember the APA.",
"Instructors may also benefit from some of the gathered information, as discussed in the paper."
],
[
"Introduction",
"Teaching statistical data treatment techniques, generally done within the framework of a laboratory course in the Physics Curriculum, is a task that is known to many of us as an unrewarding one.",
"On the one hand, the topics to be treated need attention to detail and to the basic assumptions which ensure the validity of the whole analysis.",
"On the other hand, students often find the topic boring and unappealing.",
"Nonetheless, it is a job that needs to be accomplished, in the same way one has to learn many other basic techniques indispensable for the conduct of one's work.",
"Thus, it is always helpful to try and find ways of rendering the material more appealing to students, for example by using some unusual and unexpected applications of data treatment.",
"One such example, which infallibly attracts students' attention, is the use of the A Priori Analysis (APA) to estimate the uncertainty on the grade which they receive in the same laboratory course where I present this material.",
"Besides attracting the students' attention, this example never fails to arouse some curiosity – often disguised – since the concept of a grade not being given with perfect certainty appears to surprise a number of students (and perhaps unsettle some of them a little bit).",
"Thus, in addition to giving students an illustration of the APA which brings the message very close to home, this application carries a triple weight: 1. giving a practical implementation of a concept which may otherwise be relegated to the category of techniques to be set aside (and forgot)Not too many students in a class will be confronted with the needs of a true APA in the context of their future careers.",
"; 2. introducing the pedagogically important concept that grades, like everything else in real life, are affected by an intrinsic uncertainty; 3. showing that the tools that are learnt within the Physics Curriculum have an application to real life.",
"Instructors may also find that this application of the APA offers some useful information, as illustrated in the course of the paper and discussed in more detail in the conclusions."
],
[
"A Priori Analysis",
"Although not always included among the experimental analysis techniques taught in the basic curriculum, the APA is a powerful method which makes it possible to identify the different sources of uncertainty and quantify their influence on the outcome of an experiment.",
"Two applications of the APA are obvious: 1. estimating the size of errors which can be expected before performing an experiment – particularly interesting for long and complex or expensive experiments – and 2. evaluating the influence of the individual error sources, thus permitting the identification and/or removal of the strongest uncertainty contributions.",
"In this sense in Physics one could say that the APA is most useful in experiment design and/or performance evaluation.",
"It is also useful to check whether the errors obtained are consistent with those expected, ths enabling an independent test of the error size and, possibly, the detection of mistakes (or systematic errors).",
"This is often beyond the reach of a student lab, though.",
"However, in addition to these two more immediate applications, the APA becomes a valuable tool of error assessment whenever the a posteriori analysis – through repetition of the experiment – is not possibleA good example within the physical sciences can be given when considering working on existing datasets – e.g., old climatologic or meteorological data, where only measurements without uncertainties may have been registered..",
"Since the concept of repetition may be arguable in the example that I will discuss (tests can be repeated, and multiple tests are regularly administered in a course), it is useful to recall the constraints which need to be fulfilled for the a posteriori analysis to hold [1].",
"Indeed, since mean and standard deviations are estimated through the repetition of the measurement, each outcome must be statistically independent from the previous ones (i.e., the fluctuations from one measurement to the other be random).",
"If this condition is not fulfilled, the estimated mean and standard deviation do not hold and the results lose significance.",
"The statistical independence hypothesis certainly does not hold for at least two reasons: 1. it is virtually impossible to devise tests which are perfectly equivalent, thus the experimental conditions are not constant; 2. one cannot repeat the experiment by having students take repeated (equivalent) tests, since the (desirable, and normally observed) progressive improvement accompanying successive tests would skew the results (particularly the standard deviation).",
"Hence, the condition of statistical independence is clearly violated.",
"While one could think of finding ways of compensating for the difficulty presented in (1.)",
"by taking large ensemble averages (i.e., repeated tests for each student) which could smooth out the differences if the tests are sufficiently well constructed, the obstacle presented by point (2.)",
"is unsurmountable and even contradicts the possible solution just envisaged for point (1.).",
"Indeed, in addition to obtaining meaningless results with test repetition, one cannot even think of replacing multiple tests with ensemble averages – on the class size – taken on a single test, since their indicators (average and standard deviation) cannot give any information on the grade of each individual student!",
"The variability in level for an entire class is tipically much wider than the accuracy with which we can estimate the grade for an individual, since the former represents the excursion in achievement due to different levels of individual ability, assiduity, performance, engagement, etc.",
"Indeed, if this were not the case individual grades would be meaningless.",
"The a priori estimate of the uncertainty is therefore an interesting indicator which, far from providing the correct uncertainty, gives for it at least a reasonable estimate.",
"As always true for a priori uncertainties, the quality of the final outcome – at the end of the analysis – is strictly related to the reliability of the guesses for the uncertainty of each test component.",
"The appropriation of this concept is pedagogically very important, as it teaches the student to critically analyze the problem and shows that within the framework of an APA a critical eye and repeated tests (with different estimates for the different error components) play a major role in the process.",
"Testing various estimated initial uncertainties is all the more useful the more numerous the tests which compose the final grade (although the case I discuss in detail turns out to be very simple).",
"As such, lab courses may be the most interesting examples, but the technique is applicable to any kind of course."
],
[
"Mathematical formulation",
"One can generally formulate the problem as follows.",
"An ensemble of $N$ tests of different nature – lab report, written test, final exam, etc.",
"–, with individual grades $Gj$ , combine with weight coefficients $w_j$ to give the global grade $G$ : $G = \\sum _{j = 1}^N w_j G_j \\, .$ Each category of test may itself be subdivided into different subtests and result therefore from an average over homogeneous grades: $G_j = \\sum _{k = 1}^M \\frac{G_{j,k}}{M} \\, ,$ where $M$ is the number of homogeneous tests in each category.",
"A concrete exampleThis is the structure of grades of the laboratory course which I use as an example for the students.",
"can illustrate the structure of the grades more easily.",
"Suppose that the grade be composed of: 1.",
"Work performance during the lab session; 2.",
"Quality of the reporting in the labbook; 3.",
"Evaluation of the lab report; 4.",
"Final exam; thus of $N=4$ different kinds of tests.",
"Each category of test may contain repetitions of individual tests.",
"For instance, a student will do several, $M$ , experiments and therefore will have at least $M$ notes in category 1.",
"We first define the uncertainty in the homogeneous test category as $\\sigma _{G_j} = \\frac{1}{M} \\sqrt{\\sum _{k=1}^M \\sigma _{j,k}^2} \\, ,$ where $\\sigma _{j,k}$ represents the uncertainty estimated (in the APA) for each individual test within a category.",
"Thus, equation REF provides the general expression for the a priori estimate of the uncertainty in each grade categoryThis can be useful when one test from an ensemble proves more difficult to grade, or exhibits larger fluctuations in students' performance (e.g., due to a harder problem set)..",
"In most cases, however, the uncertainty can be estimated to be the same for all $k$ tests of a certain category $j$ (e.g., homework, lab report, etc.).",
"In such a case the uncertainty, equation REF , simplifies to become [1] $\\sigma _{G_j} & = & \\frac{1}{M} \\sqrt{ \\sum _{k=1}^M \\sigma _j^2} \\, , \\\\& = & \\frac{\\sigma _j}{\\sqrt{M}} \\, .$ In order to obtain the uncertainty on the final grade, it suffices to propagate the individual uncertainties $\\sigma _{G_j}$ through the general definition, equation REF , to obtain [1] $\\sigma _G & = & \\sqrt{\\sum _{j=1}^N \\left( \\frac{\\partial G}{\\partial G_j} \\right)^2 \\sigma _{G_j}^2} \\, , \\\\& = & \\sqrt{ \\sum _{j=1}^N \\left( w_j^2 \\frac{\\sigma _j^2}{M} \\right) } \\, ,$ where the former expression is general and the latter applies to the case of equal estimated uncertainties within a test category (cf.",
"equation REF )."
],
[
"Estimating the a priori uncertainties",
"The most interesting, and challenging, part of the work comes when one has to determine reasonable estimates for the uncertainties to be attributed to each individual type of testThe nature of the problem is the same if different uncertainties are attributed to the individual test, as previously mentioned.. For clarity, I will proceed with a concrete example: the laboratory course in which this material is presented.",
"The structure of the course is such that students are evaluated in four different categories (cf.",
"table REF ).",
"Table: The Repetitions column corresponds to the number of tests in the corresponding category.",
"Different lab sessions (4 in this example) lead to one report, thus the number of lab reports is four times smaller than the number of grades in the participation or labbook sections.Assuming that the uncertainties be homogeneous for each category of test, we apply equations and and therefore need to estimate the values of $\\sigma _j$ .",
"The estimates are given in table REF (second column) and are based on the following considerations (items labelled according to the test category, cf.",
"tables REF and REF – for a description of French grades look at table's REF caption): $p$ assuming $\\sigma _{p} = 1$ amounts to saying that an error in grading by $\\pm 3$ units has a probability of occurrence $P < 0.003 $Gaussian-distributed errors are assumed here [1]..",
"Translated in percentage $ 3 \\widetilde{\\sigma }_p = 0.15$ , which is quite a large interval.",
"Such a large error bar is introduced on the basis of the nature of the evaluation: different lab monitors give an estimate of the performance of each individual student – working in a small group (typically two or three) – by observing their work, discussing with the group and asking occasional questions.",
"Each monitor is required to follow several groups (typically between four and six) and differences in evaluation among monitors, as well as fluctuations for a same monitor due to variable working conditions, are unavoidable.",
"$l$ $\\sigma _l = 0.5$ amounts to assigning $\\pm 1.5$ points to the uncertainty with probability $P > 0.997$ of the true grade falling within the interval.",
"In percentage this amounts to $3 \\widetilde{\\sigma }_l = 0.075$ .",
"The variability in the evaluation is estimated to be lower than for p-tests due to the fact that labbooks, as a written document, can be more reliably evaluated.",
"The estimated uncertainty could be smaller if all grading were done by one and the same person (not the case in this context).",
"The size of $\\sigma _l$ is therefore chosen to reflect the added variability coming from an ensemble of graders.",
"$r$ We assign the same error estimates to this test as those chosen for $\\sigma _l$ for the reasons exposed in the preceding point.",
"$o$ This kind of test requires a closer look at its details.",
"Being an oral examination – even though conducted by a panel of (at least) three examiners – it is somewhat more susceptible to fluctuations (in the questions, their evaluation, and in the student's reactions).",
"Therefore, we assign to it a value of $\\sigma _o = 0.7$ which amounts to considering a full 95% confidence intervalIt is interesting here to use a different way of evaluating the confidence interval.",
"Indeed, contrary to common practice in Physics, in most applied sciences – e.g.",
"Risk Assessment, Geosciences, etc.",
"– the standard error bar associated with any given quantitative estimate is $\\pm 2 \\sigma $ , rather than $\\pm \\sigma $ by virtue of its larger confidence level ($\\tilde{9}5\\%$ ), more useful for practical purposes.",
"It is therefore instructive to use this example to present this aspect of uncertainty estimates to physics students.",
"($\\pm 2 \\sigma _o$ ) [1] to a spread of (approximately) three points.",
"Translated into percentages, $\\widetilde{\\sigma }_o = 0.035$ (i.e.",
"$3 \\widetilde{\\sigma }_o = 0.105$ ).",
"In other words, we expect the probability of a grade outside this interval to be below $5\\%$ .",
"Table: The French University Grading System (FUGS) attributes the grades in x/20x/20 (passing grade x=10x=10), where xx represents the grade attributed to the test.",
"The numerical estimates are given in absolute values, i.e., in FUGS units, but – in order to improve readability – are also repeated in percentage.",
"The latter are identified by the corresponding quantities marked by a tilde v ˜\\widetilde{v} (vv being the generic variable).",
"The conversion gives rise to a resuilt with an excess of digits for some grade categories (kept here to be consistent with the absolute estimates, used for the calculations).The propagation of the a priori uncertainties on each individual grade for each kind of test follows equation and produces the values of $\\sigma _{G_j}$ given in Table REF .",
"Notice that $M=1$ for the oral test (label $o$ ), therefore no uncertainty improvement ensues for this grade.",
"Computing the propagation of the a priori uncertainty on the final grade, equation , we obtain $\\sigma _G & = & 0.3 \\, {\\rm points} \\, , \\\\& = & 0.015 \\, {\\rm (in \\, percent)}$ which amounts to a $\\Delta G \\simeq 0.5$ points (or $\\widetilde{\\Delta G} \\simeq 0.025$ ) with probability $P \\simeq 0.9$From [2] we obtain that $P (x) = 0.9$ when $x \\simeq 1.645$ .",
"Thus, computing $\\Delta G = 1.645 \\times \\sigma _G$ (equation REF ) we obtain $\\Delta G \\simeq 0.4935 \\approx 0.5$ , as in the text.",
"of obtaining the actual grade within this interval.",
"We thus conclude that the a priori estimate for the grade each student receives in this course is $\\pm 0.5$ points (or $2.5\\%$ ) with a confidence level of $90\\%$ ."
],
[
"Discussion",
"Aside from the numerical result just obtained, equation REF , it is very instructive to look at the details of the contributions which compose the final value $\\sigma _G$ .",
"Table REF provides the breakup of the various contributions, where we notice that the smallest one comes from the $p$ component.",
"We immediately recognize that the influence on the final uncertainty coming from the participation grade ($p$ ) is entirely negligible (by two orders of magnitude), in spite of its intrinsic variability and of the large a priori uncertainty we have consequently assigned to it.",
"This results from the combined effect of the larger number of tests in this category ($M_p = 12$ ) and of the small weight assigned to this category ($w_p = 0.1$ , cf.",
"table REF ).",
"The relative contributions of the labbook ($\\sigma _l$ ) and report ($\\sigma _r$ ) uncertainties are different in table REF due to their different number of samples ($M_l = 12$ and $M_r = 3$ ), which reduce by $M^{-\\frac{1}{2}}$ their uncertainty (equation ).",
"Overall, however, even the weighted contribution coming from $G_r$ is negligible – by more than one order of magnitude – when compared to that of the oral exam.",
"We thus conclude that only the uncertainty on the latter matters in the determination of the uncertainty on the final grade, owind to the larger size of $\\sigma _o$ , the single event ($M_o=1$ ), and especially its dominant weight ($w_0 = 0.4$ , table REF ).",
"One should not confuse the influence of each grade category on the final outcome with the dominance of the uncertainty of the oral test on the overall uncertainty.",
"Each grade category contributes, proportionally to its weight, to the course grade, but the confidence interval is determined exclusively by the oral test, all other forms of grading providing a much more “accurate\" evaluation.",
"This result has the following implications: a. given the very sizeable difference it error contribution (table REF ) modifying the estimates of the a priori uncertainties which we have assigned to the various kind of tests (except for the oral test) will not influence the size of the uncertainty.",
"Thus, except for the oral test, we realize a posteriori that our careful estimates in the preceding section do not hold any relevance; b. given that the only dependence of the estimated a priori final uncertainty has a linear dependence onto the estimated error assigned to the oral test, we know that modyfing the latter linearly translates onto the reliability of the global grade (weighted by $w_o$ ); c. there is no need to worry about the reliability of the grades for the first three kinds of tests, i.e.",
"about the variability originating from the involvement of several monitors in the various grading steps.",
"The last point is important for instructors who may worry that, in particular for the participation grade, the instrinsic variability due to multiple evaluators, and the ensuing point spread, may distort the reliability of the course's global grade.",
"This also means that one can confidently introduce different measures of evaluation – in particular some which give the benefit of an immediate return to the students, such as the participation grade – without risking a substantial impact onto the final grade.",
"One final point: the a priori estimate of the uncertainty on the final grade also gives a measure of the precision with which the latter can be given.",
"In the specific case of the example used, a good discretization is $\\Delta G$ , i.e., using a scale in points with integer and half-integer values (or 2.5% in relative precision).",
"This can be used to explain to students what is a reasonable scale in grade spacing.",
"Of course, the actual value will depend on the structure of the course and on the number of test categories (and of test number in each category)."
],
[
"Conclusions",
"The simple, but striking, application of the APA to course grades illustrates quite effectively the intrinsic nature of this kind of analysis and its main features.",
"It clearly shows the technique's importance in all those situations where measurements cannot be repeated to obtain a posteriori error estimates, and the power of its predictions.",
"At the same time, the analysis has shown the need for a careful assessment of the individual error sources to be assigned to the primary, measured quantities, and the futility of part of the work, rendered irrelevant by the intrinsic structure of the analyzed quantities (composition of the grade and of its uncertainty).",
"Students are often taken aback both by the fact that an aspect of their curriculum can be analyzed in detail with techniques seemingly exclusively devised for lab experiment analysis, and by the information which can be gathered by this analysis.",
"This example should also serve as an encouragement for testing the application of data treatment techniques to real-life everyday's problems.",
"As a bonus, we have shown that instructors may gather precious information on the uncertainty which affects their grades and on the confidence level of each grade component, while gaining some freedom in experimenting with innovative ways of introducing partial grades.",
"The latter can be beneficial to giving students early and welcome feedback, while ensuring that the reliability of the course grade is not affected by evaluation components which are more prone to larger fluctuations.",
"This quantitative analysis, though partly subjective (in the assignment of the a priori error components) may also be helpful in arguing in favour of the introduction of complementary grade parts in the discussions with skeptical collegues or Department Directors.",
"I am grateful to all the students (in excess of five hundred) who, having taken this course over the last decade, have stimulated the development of new ideas and examples."
]
] | 1403.0419 |
[
[
"Point compression for the trace zero subgroup over a small degree\n extension field"
],
[
"Abstract Using Semaev's summation polynomials, we derive a new equation for the $\\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\\mathbb{F}_q$.",
"Using this equation, we produce an optimal-size representation for such points.",
"Our representation is compatible with scalar multiplication.",
"We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity).",
"The algorithms are efficient for trace zero varieties coming from small degree extension fields.",
"We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions."
],
[
"Introduction",
"Given a (hyper)elliptic curve defined over $$ and a field extension $|$ , consider the $$ -rational points of trace zero.",
"They form a subgroup of the group of $$ -rational points of the curve, and can be realized as the $$ -rational points of an abelian variety built by Weil restriction from the original curve, called the trace zero variety.",
"The trace zero subgroup was first proposed for use in cryptography by Frey [15], and further studied by Naumann [35], Weimerskirch [44], Blady [5], Lange [31], [32], Avanzi–Cesena [1], [8], and Diem-Scholten [12].",
"Trace zero subgroups are interesting because they allow efficient arithmetic, due to a speed-up of the standard scalar multiplication using the Frobenius endomorphism.",
"This is analogous to the use of endomorphisms to speed up scalar multiplication on Koblitz curves (see [30]) and GLV–GLS curves (see [19], [17]), which are the basis for several recent implementation speed records for elliptic curve arithmetic (see [34], [14], [6]).",
"The trace zero subgroup is of interest in the context of pairing-based cryptography.",
"Rubin and Silverberg have shown in [37], [40] that the security of pairing-based cryptosystems can be improved by using abelian varieties of dimension greater than one in place of elliptic curves.",
"Jacobians of hyperelliptic curves and trace zero varieties are therefore the canonical examples for such applications.",
"Since the trace zero subgroup is contained in the group of $$ -rational points of the (Jacobian of the) curve, the DLP in the trace zero subgroup is at most as hard as the DLP in the curve.",
"It is easy to show that in fact the DLP's in the two groups have the same complexity.",
"From a mathematical point of view therefore, trace zero variety cryptosystems may be regarded as the (hyper)elliptic curve analog of torus-based cryptosystems such as LUC [43], Gong–Harn [25], XTR [33], and CEILIDIH [38].",
"The hardness of the discrete logarithm problem in a group is closely connected with the size of the representation of the group elements.",
"Usually, the hardness of the DLP is measured as a function of the group size.",
"However, for practical purposes, the comparison with the size of the representation of group elements is a better indicator, since it quantifies the storage and transmission costs connected with using the corresponding cryptosystem.",
"Therefore, in order to make the comparison between DLP complexity and group size a fair one, we are interested in a compact representation that reflects the size of the group.",
"Such an optimal-size representation consists of $\\log _2 N$ bits, where $N$ is the size of the group.",
"See also [26] for a discussion on the significance of compact representations.",
"An optimal-size representation for elliptic curves is well-known.",
"In the cryptographic setting, it is standard procedure to represent an elliptic curve point by its $x$ -coordinate only, since the $y$ -coordinate can easily be recomputed, up to sign, from the curve equation.",
"If desired, the sign can be stored in one extra bit of information.",
"Representing a point via its $x$ -coordinate gives an optimal representation for the elements of the group of $$ -rational points of an elliptic curve: Each of the approximately $q^n$ points can be represented by one element of $$ , or $n$ elements of $$ after choosing a basis of the field extension.",
"Notice moreover that storing the sign of the $y$ -coordinate is unnecessary, since this representation is compatible with scalar multiplication of points: For any $k\\in $ , the $x$ -coordinates of the points $kP$ and $-kP$ coincide.",
"The trace zero variety of an elliptic curve with respect to a prime extension degree $n$ has dimension $n-1$ , and we are interested in the $$ -rational points.",
"Hence, an optimal representation should have $\\log _2 q^{n-1}$ bits, or consist of $n-1$ elements of $$ .",
"For practical purposes, it is important that the representation can be efficiently computed (“compression”) and that the original point can be easily recovered, possibly up to some small ambiguity, from the representation (“decompression”).",
"Naumann [35], Rubin–Silverberg [37], [39], [42], and Lange [32] propose compact representations with compression and decompression algorithms for genus 1 and genus 2 curves, respectively.",
"The work by Eagle–Galbraith–Ong [13] on point compression methods for Koblitz curves is also related.",
"In this paper, we concentrate on extension fields of degree $n=3$ or 5.",
"This is due to the fact that an index calculus attack [20] and a cover attack [9], [10], [11] apply to $T_n$ , making it vulnerable for large values of $n$ .",
"In this work we briefly discuss these attacks and come to the conclusion that there are no security issues for $n=3$ .",
"For $n=5$ the cover attacks can be avoided by imposing extra conditions, and the known index calculus attacks do not threaten the security of pairing-based cryptosystems involving trace zero subgroups of supersingular curves.",
"The main purpose of this paper is introducing a new representation for the points on the trace zero variety of an elliptic curve.",
"The compression and decompression algorithms are more efficient than that of [42], and points are recovered with smaller ambiguity.",
"In addition, our representation is (to the extent of our knowledge) the only one that is compatible with scalar multiplication of points, which is the only operation needed in Diffie–Hellman-based cryptographic protocols.",
"The paper is structured as follows: In Section , we fix the notation, give the relevant definitions, and briefly recall the standard representation for points on the trace zero variety.",
"We also discuss the simple case of the trace zero variety for a quadratic field extension.",
"Using Semaev's summation polynomials, in Section we derive a single equation whose $$ -solutions describe the $$ -points of the trace zero variety, up to a few well-described exceptions (see Lemma and Proposition ).",
"In Section , using the equation that we produced in the previous section, we propose a new representation for the points on the trace zero variety.",
"The size of the representation is optimal, and we give efficient compression and decompression algorithms.",
"In Sections and we analyze in detail what our method produces for the cases $n=3$ and 5.",
"We give explicit equations and concrete examples computed with Magma and comment on security issues for these parameters.",
"It is generally agreed that 3 and 5 are the practically relevant extension degrees in the case of elliptic curves (see e.g.",
"[32])."
],
[
"Preliminaries",
"Let $$ be a finite field with $q$ elements, and let $E$ be an elliptic curve defined over $$ by an affine Weierstraß equation.",
"We consider the group $E()$ of $$ -rational points of $E$ for field extensions of prime degree $n$ .",
"The group operation is point addition, and the neutral element is the point at infinity, denoted by $Ø$ .",
"We denote indeterminates by lower case letters and finite field elements by upper case letters.",
"The Frobenius endomorphism on $E$ is defined by $\\varphi : E \\rightarrow E,~ (X,Y) \\mapsto (X^q, Y^q), ~Ø\\mapsto Ø.$ One can define a trace map $: E() \\mapsto E(), ~P \\mapsto P + \\varphi (P) + \\varphi ^2(P) + \\ldots + \\varphi ^{n-1}(P),$ relative to the field extension $|$ .",
"The kernel of the trace map is the trace zero subgroup of $E()$ , which we denote by $T_n$ .",
"By the process of Weil restriction, the points of $T_n$ can be viewed as the $$ -rational points of an abelian variety $V$ of dimension $n-1$ defined over $$ .",
"$V$ is called the trace zero variety.",
"In trace zero subgroups, arithmetic can be made more efficient by using the Frobenius endomorphism, following a similar approach to Koblitz curves and GLV–GLS curves.",
"They turn out to be extremely interesting in the context of pairing-based cryptography, where they achieve the largest security parameters in some cases, as discussed in [37], [40], [1], [8].",
"It is easy to show that the DLP in $E()$ is as hard as the DLP in $T_n$ .",
"An explanation is given in [27] for the analogous case of algebraic tori: The trace maps a DLP in $E()$ to a DLP in $E()$ .",
"By solving it in the smaller group, the discrete logarithm is obtained modulo the order of $E()$ .",
"The remaining modular information required to compute the full discrete logarithm comes from solving a DLP in $T_n$ .",
"A formal argument, which applies to any short exact sequence of algebraic groups, is given in [18].",
"Consider the exact sequence $ 0 \\longrightarrow T_n \\longrightarrow E() \\overset{}{\\longrightarrow } E() \\longrightarrow 0.",
"$ Then solving a DLP in $E()$ has the same complexity as solving a DLP in $T_n$ and a DLP in $E()$ .",
"In the conclusions of [1], large bandwidth is mentioned as the only drawback of using trace zero subgroups in pairing-based cryptography.",
"In this paper we solve this problem by finding an optimal representation for the elements of the trace zero subgroup.",
"Let $G$ be a finite set.",
"A representation for the elements of $G$ is a bijection between $G$ and a set of binary strings of fixed length $\\ell $ .",
"Equivalently, it is an injective map from $G$ to $_2^{\\ell }$ .",
"A representation is optimal if $|G|\\sim 2^{\\ell }$ , i.e.",
"if we need approximately $\\log _2 |G|$ bits to represent an element of $G$ .",
"Abusing terminology, in this paper we call representation a map from $G$ to $_2^{\\ell }$ with the property that an element of $_2^{\\ell }$ has at most $d$ inverse images, for some small fixed $d$ .",
"In this case, we say that the representation identifies classes of at most $d$ elements, namely those that have the same representation.",
"Notice that the number of classes is about $|G|/d \\sim |G|$ , if $d$ is a small constant.",
"Since the elements of a finite field $$ can be represented via binary strings of length $\\log _2 q$ , a representation for $G$ can be given via a bijection between $G$ and a subset of $_q^m$ , for some $m$ and some prime power $q$ .",
"Such a representation is optimal if and only if $|G|\\sim q^m$ .",
"Representing points of an elliptic curve via their $x$ -coordinate is a standard example of optimal representation.",
"It is customary to represent a point $(X,Y)\\in E()$ via its $x$ -coordinate $X\\in $ .",
"The $y$ -coordinate can then be recovered, up to sign, from the curve equation.",
"If desired, the sign can be stored in one extra bit of information.",
"Such a representation is optimal, since by Hasse's Theorem $|E()|\\sim q^n$ .",
"The representation from the previous example identifies pairs of points, since $P$ and $-P$ have the same $x$ -coordinate.",
"We often say that the $x$ -coordinate is a representation for the equivalence class consisting of $P$ and $-P$ .",
"The representation that we propose in this paper identifies a small number of points as well.",
"Before we discuss our representation, we notice that representing a point $P\\in T_n$ via its $x$ -coordinate is no longer optimal.",
"Since a point $P=(X,Y)\\in T_n$ is an element of $E()$ , we can represent $P$ via $X\\in $ .",
"Choosing an $$ -basis of $$ , we can represent $X\\in $ as an $n$ -tuple $(X_0,\\ldots ,X_{n-1})\\in _q^n$ .",
"Representing $P\\in T_n$ as $X\\in $ or as $(X_0,\\ldots ,X_{n-1})\\in _q^n$ however is not optimal, since $|T_n|\\sim q^{n-1}$ .",
"In this paper we find a representation for the elements of $T_n$ , via $n-1$ coordinates in $$ .",
"Our representation is not injective, but it identifies a small number of points.",
"Our approach is the following: We start from the representation of $P\\in T_n$ as an $n$ -tuple $\\rho (P)=(X_0,\\ldots ,X_{n-1})\\in _q^n$ , and write an equation in $[x_0,\\ldots ,x_{n-1}]$ which vanishes on $\\rho (P)$ for all $P\\in T_n$ .",
"This allows us to drop one coordinate of $\\rho (P)$ and reconstruct it using the equation.",
"Therefore, we can represent elements of $T_n$ via $n-1$ coordinates in $$ , which is optimal.",
"We now fix some notation that we will use when writing explicit equations in Sections , , and .",
"Let $$ be the finite field with $q$ elements, $n$ a prime.",
"For the sake of concreteness, we assume that $n\\mid q-1$ .",
"Due to its simplicity, we always consider this case when writing explicit equations.",
"All of our arguments however work for any $n$ and $q$ , see also Remark .",
"If $n\\mid q-1$ , thanks to Kummer theory we can write the extension field as $= [\\zeta ]/(\\zeta ^n - \\mu ),$ where $\\mu $ is not an $n$ -th power in $$ .",
"Where necessary, we take $1, \\zeta , \\ldots , \\zeta ^{n-1}$ as a basis of the field extension.",
"When doing Weil restriction, we associate $n$ new variables $x_0,\\ldots ,x_{n-1}$ to the variable $x$ .",
"They are related via $x =x_0 + x_1 \\zeta + \\ldots + x_{n-1} \\zeta ^{n-1}.$ We abuse terminology and use the term Weil restriction not only for the variety, but also for the process of writing equations for the Weil restriction.",
"In particular for us, Weil restriction is a procedure that can be applied to a polynomial defined over $$ and results in $n$ polynomials with $n$ times as many variables, and coefficients in $$ .",
"If $n$ does not divide $q-1$ , we choose a normal basis $\\lbrace \\alpha ,\\alpha ^q,\\ldots ,\\alpha ^{q^{n-1}}\\rbrace $ of $$ over $$ and Weil restriction coordinates $x = x_0\\alpha + x_1 \\alpha ^q + \\ldots + x_{n-1} \\alpha ^{q^{n-1}}.$ It is easy to show that the case $n=2$ allows a trivial optimal representation for the elements of $T_n$ .",
"Hence in the next sections we concentrate on the more interesting case of odd primes $n$ .",
"The trace zero subgroup $T_2$ of $E(_{q^2})$ can be described as $T_2=\\lbrace (X,Y)\\in E(_{q^2}) \\mid X \\in , \\;Y\\notin \\rbrace \\cup E[2]().$ In particular, representing a point $(X,Y)\\in T_2$ by $X \\in $ yields a representation of optimal size.",
"We first prove that $T_2$ is contained in the union of sets on the right hand side of the equality.",
"Let $P\\in T_2$ , $P\\ne Ø$ , so $P=(X,Y)\\in E(_{q^2})$ .",
"If $P\\in E()$ , then $2P=Ø$ , hence $P\\in E[2]()$ .",
"If $P\\notin E()$ , then $(X,Y)=-(X^q,Y^q)$ .",
"In particular $X=X^q$ , so $X\\in $ , which also implies $Y\\notin $ .",
"To prove the other inclusion, observe that by definition $P\\in E[2]()$ satisfies $2P=Ø$ , so $P\\in T_2$ .",
"Let $P=(X,Y)\\in E(_{q^2})$ with $X\\in $ , $Y\\notin $ .",
"Since $X\\in $ , the points $(X,Y)$ and $\\varphi (X,Y)=(X,Y^q)$ are distinct points on $E$ which lie on the same vertical line $x-X=0$ .",
"Hence $(X,Y)+\\varphi (X,Y)=Ø$ and $(X,Y)\\in T_2$ .",
"The next proposition will be useful when writing equations for the $$ -rational points of the trace zero variety.",
"For a multivariate polynomial $h$ , we denote by $\\deg _{x_i}(h)$ the degree of $h$ in the variable $x_i$ .",
"Let $h \\in [x_0,\\ldots ,x_{n-1}]$ be a polynomial with $h(X_0,\\ldots ,X_{n-1}) = 0$ for all $(X_0,\\ldots ,X_{n-1}) \\in _q^n$ , and assume that $\\deg _{x_i}(h) < q$ for $i \\in \\lbrace 0,\\ldots ,n-1\\rbrace $ .",
"Then $h$ is the zero polynomial.",
"Write $V(h) = \\lbrace (X_0,\\ldots ,X_{n-1}) \\in ^{\\,n} \\mid h(X_0,\\ldots ,X_{n-1}) = 0\\rbrace \\subseteq ^{\\,n}$ for the zero locus of $h$ over the algebraic closure of $$ and $I(V) = \\lbrace f \\in [x_0,\\ldots ,x_{n-1}] \\mid f(X_0,\\ldots ,X_{n-1}) = 0 \\text{ for all } (X_0,\\ldots ,X_{n-1}) \\in V \\rbrace $ for the ideal of the polynomials vanishing on some $V \\subseteq ^{\\,n}$ .",
"First we show that $I(_q^n) = J_n$ where $J_n = (x_0^q-x_0,\\ldots ,x_{n-1}^q-x_{n-1})$ .",
"We proceed by induction on $n$ .",
"The claim holds for $n=1$ , since the elements of $$ are exactly those elements of $$ that satisfy the equation $x_0^q-x_0$ .",
"Assuming that the statement is true for $n-1$ , we have $I(_q^n) & = & \\bigcap _{(\\alpha _0,\\ldots ,\\alpha _{n-1}) \\in _q^n} (x_0-\\alpha _0,\\ldots ,x_{n-1}-\\alpha _{n-1})\\\\& = & \\bigcap _{\\alpha _0 \\in } \\bigcap _{(\\alpha _1,\\ldots ,\\alpha _{n-1})\\in _q^{n-1}}(x_0-\\alpha _0,\\ldots ,x_{n-1}-\\alpha _{n-1})\\\\& = & \\bigcap _{\\alpha _0 \\in } (x_0-\\alpha _0,x_1^q-x_1,\\ldots ,x_{n-1}^q-x_{n-1})\\\\& = & \\left(\\prod _{\\alpha _0 \\in } (x_0-\\alpha _0),x_1^q-x_1,\\ldots ,x_{n-1}^q-x_{n-1}\\right)\\\\& = & J_n.$ Now we show that $h=0$ .",
"Since $h$ vanishes on $_q^n$ , we have $_q^n \\subseteq V(h) \\subseteq ^{\\,n}$ , which implies $h\\in I(V(h)) \\subseteq I(_q^n) = J_n$ .",
"The leading terms of $x_0^q-x_0,\\ldots ,x_{n-1}^q-x_{n-1}$ with respect to any term order are $x_0^q,\\ldots ,x_{n-1}^q$ , in particular they are pairwise coprime.",
"Hence the polynomials $x_0^q-x_0,\\ldots ,x_{n-1}^q-x_{n-1}$ are a Gröbner basis of $J_n$ .",
"Therefore, $h\\in J_n$ implies that $h$ reduces to zero using the generators of $J_n$ , i.e.",
"if we divide $h$ by $x_i^q-x_i$ whenever the leading term of $h$ is divisible by $x_i^q$ , we must obtain remainder zero when no more division is possible.",
"But since $\\deg _{x_i}(h) < q$ for all $i$ , $h$ is equal to the remainder of the division of $h$ by $x_0^q-x_0,\\ldots ,x_{n-1}^q-x_{n-1}$ , hence $h = 0$ ."
],
[
"An equation for the trace zero subgroup",
"In this section we use Semaev's summation polynomials [41] to write an equation for the set of $$ -rational points of the trace zero variety.",
"The equation involves the $x$ -coordinates only and will help us in finding a better representation for the elements of the trace zero subgroup.",
"Semaev introduced the summation polynomials in the context of attacking the elliptic curve discrete logarithm problem.",
"They give polynomial conditions describing when a number of points on an elliptic curve sum to $Ø$ , involving only the $x$ -coordinates of the points.",
"Let $$ be a finite field of characteristic different from 2 and 3 and let $E$ be a smooth elliptic curve defined by the affine equation $E : y^2 = x^3 + Ax + B,$ with coefficients $A, B \\in $ .",
"Define the $m$ -th summation polynomial $f_m$ recursively by $\\begin{array}{rcl}f_3(z_1,z_2,z_3) & = & (z_1 - z_2)^2z_3^2 - 2((z_1+z_2)(z_1z_2 + A) + 2B)z_3 + (z_1z_2-A)^2 - 4B(z_1+z_2) \\\\f_m(z_1,\\ldots ,z_m) & = & _z(f_{m-k}(z_1,\\ldots ,z_{m-k-1},z),f_{k+2}(z_{m-k},\\ldots ,z_m,z))\\end{array} $ for $m \\ge 4$ and $m-3 \\ge k \\ge 1$ .",
"We briefly recall the properties of summation polynomials that we will need.",
"[[41], Theorem 1] For any $m \\ge 3$ , let $Z_1,\\ldots , Z_m$ be elements of the algebraic closure $$ of $$ .",
"Then $f_m(Z_1,\\ldots ,Z_m) = 0$ if and only if there exist $Y_1,\\ldots ,Y_m \\in $ such that the points $(Z_i,Y_i)$ are on $E$ and $(Z_1,Y_1) + \\ldots + (Z_m,Y_m) = Ø$ in the group $E()$ .",
"Furthermore, $f_m$ is absolutely irreducible and symmetric of degree $2^{m-2}$ in each variable.",
"The total degree is $(m-1)2^{m-2}$ .",
"Definition is the original definition that Semaev gave in [41].",
"Semaev polynomials can be defined and computed also over a finite field of characteristic 2 or 3.",
"Although the formulas look different, the properties are analogous to those stated in Theorem .",
"Hence all the results that we prove in this paper hold, with the appropriate adjustments, over a finite field of any charasteristic.",
"Since the points in $T_n$ are characterized by the condition that their Frobenius conjugates sum to zero, we can use the Semaev polynomial to give an equation only in $x$ .",
"It is clear that $(X,Y) \\in T_n$ implies $f_n(X,X^q,\\ldots ,X^{q^{n-1}}) = 0$ .",
"The opposite implication has some obvious exceptions.",
"For any prime $n$ , let $T_n$ denote the trace zero subgroup associated with the field extension $|$ .",
"We have $\\bigcup _{k=0}^{\\lfloor \\frac{n}{2}\\rfloor -1} (E[n-2k](_q)+E[2]\\cap T_n)\\subseteq \\lbrace (X,Y) \\in E(_{q^n}) \\mid f_n(X,X^q,\\ldots ,X^{q^{n-1}}) = 0 \\rbrace \\cup \\lbrace Ø\\rbrace .$ Let $k \\in \\lbrace 0,\\ldots ,\\lfloor \\frac{n}{2} \\rfloor \\rbrace $ , and let $P = Q + R$ with $Q \\in E[n-2k](), R \\in E[2] \\cap T_n$ .",
"Then we have $&& \\underbrace{P + \\varphi (P) + \\ldots + \\varphi ^{n-2k-1}(P)}_{n - 2k \\text{ summands}} + \\underbrace{\\varphi ^{n-2k}(P) - \\varphi ^{n-2k+1}(P) + \\ldots - \\varphi ^{n-1}(P)}_{2k \\text{ summands with alternating signs}}\\\\& = & \\underbrace{Q+\\ldots + Q}_{n-2k \\text{ summands}} + \\underbrace{Q - Q + \\ldots - Q}_{2k \\text{ summands with alternating signs}} + R + \\varphi (R) + \\ldots + \\varphi ^{n-1}(R)\\\\& = & (n-2k)Q + (R)\\\\& = & Ø,$ where for the first equality we use that $Q \\in E()$ and $R \\in E[2]$ , and for the third equality we use that $Q \\in E[n-2k]$ and $R \\in T_n$ .",
"Notice that the points of the form $P=Q+R$ with $Q\\in E[n-2k](_q)$ and $R\\in E[2]\\cap T_n$ are not trace zero points if $Q\\ne Ø$ and $3\\le n-2k\\le n-2$ .",
"For the interesting cases $n=3$ and 5 we prove that these are the only exceptions.",
"Let $T_n$ be the trace zero subgroup associated with the field extension $|$ .",
"We have $\\begin{array}{lcl}T_3 & = & \\lbrace (X,Y) \\in E(_{q^3}) \\mid f_3(X,X^q,X^{q^2}) = 0\\rbrace \\cup \\lbrace Ø\\rbrace \\\\T_5 \\cup (E[3](_q)+E[2]\\cap T_5) & = & \\lbrace (X,Y) \\in E(_{q^5}) \\mid f_5(X,X^q,\\ldots ,X^{q^4}) = 0 \\rbrace \\cup \\lbrace Ø\\rbrace .\\end{array}$ Let $P = (X,Y) \\in E(_{q^3})$ with $f_3(X,X^q,X^{q^2}) = 0$ .",
"Then by the properties of the Semaev polynomial, there exist $Y_0,Y_1,Y_2 \\in $ such that $(X,Y_0) + (X^q,Y_1) + (X^{q^2},Y_2) = Ø$ .",
"Obviously we have $Y_i = \\pm Y^{q^i}, i = 0,1,2$ , so $P \\pm \\varphi (P) \\pm \\varphi ^2(P) = Ø$ .",
"We have to show that all signs are “+”.",
"Suppose $P - \\varphi (P) + \\varphi ^2(P) = Ø$ .",
"By applying $\\varphi $ , we get $\\varphi (P) - \\varphi ^2(P) + P = Ø$ .",
"Adding these two equations gives $2P = Ø$ , implying that $P = -P$ , hence $P+\\varphi (P)+\\varphi ^2(P)=Ø$ .",
"In particular, $P \\in T_3$ .",
"The rest follows by symmetry.",
"Now let $P = (X,Y) \\in E(_{q^5})$ with $f_5(X,X^q,\\ldots ,X^{q^4}) = 0$ .",
"Then as before, $P \\pm \\varphi (P) \\pm \\varphi ^2(P) \\pm \\varphi ^3(P) \\pm \\varphi ^4(P) = Ø$ .",
"If all signs are “+”, then $P \\in T_5$ .",
"We treat all other cases below.",
"[one minus] Assume $P + \\varphi (P) + \\varphi ^2(P) + \\varphi ^3(P) -\\varphi ^4(P) = Ø$ .",
"Applying $\\varphi $ to the equation and adding the two equations, we get $2\\varphi (P) + 2\\varphi ^2(P) + 2\\varphi ^3(P) =Ø$ , and by substituting into twice the first equation, $2P =\\varphi ^4(2P)$ .",
"Hence $2P \\in E(_{q^4}) \\cap E(_{q^5}) =E(_q)$ , so $2P\\in E[3]()$ .",
"Now $P = Q + R\\in E[6]$ is the sum of $Q\\in E[3]$ and $R\\in E[2]$ .",
"We have $Q = -2Q = -2P \\in E[3]()$ .",
"From the original equation $P + \\varphi (P) + \\varphi ^2(P) + \\varphi ^3(P) -\\varphi ^4(P) = Ø$ , we get an analogous equation in $R$ , which together with $R \\in E[2]$ gives $R \\in T_5$ .",
"[two minuses in a row] Assume $P + \\varphi (P) + \\varphi ^2(P) -\\varphi ^3(P) - \\varphi ^4(P) = Ø$ .",
"Applying $\\varphi ^2$ and adding, we get $2\\varphi ^2(P) = Ø$ , hence $P = -P$ and therefore $P \\in T_5$ .",
"[two minuses not in a row] Finally, assume $P + \\varphi (P) -\\varphi ^2(P) + \\varphi ^3(P) - \\varphi ^4(P) = Ø$ .",
"Applying $\\varphi $ and adding, we get $2 \\varphi (P) = Ø$ , hence $P = -P$ and therefore $P \\in T_5$ .",
"The other cases follow by symmetry, thus proving the claim.",
"In the sequel, we use $f_n$ as an equation for $T_n$ .",
"In practice however, for any root $X\\in $ of $f_n(x,x^q,\\ldots ,x^{q^{n-1}})$ we need to be able to decide efficiently whether $(X,Y)\\in T_n$ .",
"For $n=3$ we only need to check that $Y\\in _{q^3}$ .",
"This guarantees that $(X,Y)\\in T_3$ , by Proposition .",
"For $n=5$ , by Proposition we have to exclude from the solutions of $f_5=0$ the points $(X,Y)\\in E$ such that $Y\\notin _{q^5}$ and the points of the form $Q+R$ where $Ø\\ne Q\\in E[3]()$ and $R\\in E[2]\\cap T_5$ .",
"Let $\\mathcal {L}$ be the set of the $x$ -coordinates of the elements $Q+R\\in E[3]()+E[2]\\cap T_5$ with $Q\\ne Ø$ .",
"Then $\\mathcal {L}$ has cardinality at most 16.",
"A root $X\\in _{q^5}$ of $f_5(x,x^q,\\ldots ,x^{q^4})$ corresponds to a point $(X,Y)\\in T_5$ if and only if $X\\notin \\mathcal {L}$ and $Y\\in _{q^5}$ .",
"The $x$ -coordinates of the points of $T_n$ correspond to zeros of the Weil restriction of the polynomial $f_n(x,\\ldots ,x^{q^{n-1}})$ .",
"Since $E$ is defined over $$ , then $f_n(x,\\ldots ,x^{q^{n-1}})\\in [x]$ .",
"Therefore, for any $\\alpha \\in $ we have $f_n(\\alpha ,\\ldots ,\\alpha ^{q^{n-1}})^q=f_n(\\alpha ^q,\\ldots ,\\alpha ^{q^{n-1}},\\alpha )=f_n(\\alpha ,\\ldots ,\\alpha ^{q^{n-1}}),$ where the second equality follows from the symmetry of the Semaev polynomial.",
"It follows that $f_n(\\alpha ,\\ldots ,\\alpha ^{q^{n-1}})\\in ~~~\\mbox{for all}~\\alpha \\in .$ We use the relations (REF ) to write equations for the Weil restriction.",
"Notice that since we are only interested in the $$ -rational points of the Weil restriction, we may reduce the equations that we obtain modulo $x_i^q-x_i$ for $i=0,\\ldots ,n-1$ .",
"Hence we obtain equations in $x_0,\\ldots ,x_{n-1}$ of degree less than $q$ in each indeterminate.",
"Now (REF ) together with Proposition implies that the last $n-1$ equations are identically zero.",
"Therefore, although Weil restriction could produce up to $n$ equations, by reducing modulo the equations $x_i^q-x_i$ we obtain only one equation at the end.",
"We denote this new equation by $\\tilde{f}_n(x_0,\\ldots ,x_{n-1}) = 0.$ We stress that its $$ -solutions correspond to the elements of $T_n$ , together with some extra points described in Lemma and Proposition .",
"In Remark we discussed how to distinguish the extra solutions.",
"Since we reduce the Weil restriction of $f_n(x,x^q,\\ldots ,x^{q^{n-1}})$ modulo $x_i^q-x_i$ , the $q$ th powers disappear, and we are left with an equation $\\tilde{f}_n$ of the same degree as the original Semaev polynomial $f_n$ .",
"Concerning the representation, we now have an equation that is compatible with dropping the $y$ -coordinate.",
"It is a natural idea to drop one $X_i$ in order to obtain a compact representation, mimicking the approach of [35], [32], [42].",
"The decompression algorithm could then use $\\tilde{f}_n$ to recompute the missing coordinate.",
"However, since $\\tilde{f}_n$ has relatively large degree, this would identify more points than desired.",
"Moreover, the computation of the Weil restriction of the Semaev polynomials requires a large amount of memory.",
"It is already very demanding for $n=5$ .",
"We present a modified approach to the problem in the next section."
],
[
"An optimal representation",
"As the Semaev polynomials are symmetric in nature, they can be written in terms of the symmetric functions.",
"We write $ f_n(z_1,\\ldots ,z_n) = g_n(e_1(z_1,\\ldots ,z_n),\\ldots ,e_n(z_1,\\ldots ,z_n)),$ where $e_i$ are the elementary symmetric polynomials $e_i(z_1,\\ldots ,z_n) = \\sum _{1 \\le j_1 < \\ldots < j_i \\le n} z_{j_1} \\cdot \\ldots \\cdot z_{j_i},$ and call $g_n$ the “symmetrized” $n$ -th Semaev polynomial.",
"The advantage over the original Semaev polynomial is that $g_n$ has lower degree (e.g.",
"2 instead of 4 for $n=3$ , and 8 instead of 32 for $n=5$ ) and fewer $$ -solutions, as it respects the inherent symmetry of the sum (i.e.",
"where $f_n$ has as solutions all permutations of possible $x$ -coordinates, $g_n$ has only one solution, the symmetric functions of these coordinates).",
"See [29] for how to efficiently compute the symmetrized Semaev polynomials.",
"In this sense, $ g_n(s_1,\\ldots ,s_n) = 0$ also describes the points of $T_n$ via the relations $s_i = e_i(x,x^q,\\ldots ,x^{q^{n-1}}), ~i = 1,\\ldots ,n.$ Notice that for $X \\in $ , we have $e_i(X,X^q,\\ldots ,X^{q^{n-1}}) \\in $ .",
"Summarizing, $g_n$ is a polynomial with $$ -coefficients by equation (REF ), as well as the polynomials $\\tilde{e_i}$ that we obtain by Weil restriction from the symmetric functions in the $q$ -powers of $x$ : $ s_i = \\tilde{e}_i(x_0,\\ldots ,x_{n-1}),~ i = 1,\\ldots ,n.$ Furthermore, we get exactly one new relation per equation (reducing modulo $x_i^q-x_i$ and applying Proposition , as before).",
"Hence we have a total of $n$ equations in the Weil restriction coordinates describing the symmetric functions.",
"The $q$ th powers in the exponents disappear thanks to the reduction, and each $\\tilde{e}_i$ is homogeneous of degree $i$ .",
"A combination of the equations (REF ) and (REF ) enables us to give a compact representation of the affine points of $T_n=V()$ .",
"It can be computed with the compression algorithm, the full point can be recovered (up to some small ambiguity) with the decompression algorithm.",
"Compression.",
"Input: $P = (X_0,\\ldots ,X_{n-1},Y_0,\\ldots ,Y_{n-1}) \\in V()$ Compute the symmetric functions of the Frobenius conjugates of $X$ : $S_i = \\tilde{e}_i(X_0,\\ldots ,X_{n-1}),~ i = 1,\\ldots ,n-1$ Output: $(S_1,\\ldots ,S_{n-1}) \\in _q^{n-1}$ Decompression.",
"Input: $(S_1,\\ldots ,S_{n-1}) \\in _q^{n-1}$ Solve $g_n(S_1,\\ldots ,S_{n-1},t) = 0$ for $t$ .",
"For each solution $\\tau $ , find a solution (if it exists) of the system $ \\begin{array}{rcl}S_1 & = & \\tilde{e}_1(x_0,\\ldots ,x_{n-1}) \\\\& \\vdots & \\\\S_{n-1} & = & \\tilde{e}_{n-1}(x_0,\\ldots ,x_{n-1}) \\\\\\tau & = & \\tilde{e}_n(x_0,\\ldots ,x_{n-1}).\\end{array}$ For the found solution $(X_0^{(j)},\\ldots ,X_{n-1}^{(j)})$ , recompute one of the $y$ -coordinates $Y^{(j)}$ belonging to $X^{(j)} = X_0^{(j)} + \\ldots + X_{n-1}^{(j)} \\zeta ^{n-1}$ using the curve equation.",
"If $(X^{(j)},Y^{(j)})\\in T_n$ , then add $\\pm P = (X^{(j)},\\pm Y^{(j)})$ and all their Frobenius conjugates to the set of output points.",
"Output: All points of $T_n=V()$ that have $(S_1,\\ldots ,S_{n-1})$ as compact representation Because of Lemma , in the last step of the decompression algorithm, for each root $X^{(j)}$ of the polynomial $f_n$ one needs to check that the point $(X^{(j)},Y^{(j)})\\in T_n$ .",
"This step can in practice be eliminated for $n=3,5$ , as discussed in Remark .",
"For a small set of points, equation (REF ) vanishes when evaluated in the given $S_1,\\ldots ,S_{n-1}$ .",
"For such points $P$ , any $t\\in $ solves the equation $g_n(S_1,\\ldots ,S_{n-1},t)=0$ , making the computational effort for decompressing $(P)$ very large.",
"Therefore, our decompression algorithm is not practical for such points.",
"However, for almost all points $P \\in V()$ the polynomial $g_n(S_1,\\ldots ,S_{n-1},t)$ has only a small number of roots in $t$ (upper bounded by the degree of $g_n$ in the variable $t$ ).",
"For our analysis, we assume that we are in the latter case.",
"Since the points of $V()$ are described by $g_n(\\tilde{e}_1(x_0,\\ldots ,x_{n-1}),\\ldots ,\\tilde{e}_n(x_0,\\ldots ,x_{n-1}))$ , we have $P \\in ((P))$ .",
"The relevant question is how many more points the output may contain.",
"First of all, by compressing a point, we lose the ability to distinguish between Frobenius conjugates of points, since for each solution of system (REF ), all Frobenius conjugates are also solutions.",
"This can be compared to the fact that when using the “standard” compression, we lose the ability to distinguish between points and their negatives.",
"If desired, a few extra bits can be used to remember that information.",
"Alternatively, we can think of working in $T_n$ modulo an equivalence relation that identifies the Frobenius conjugates of each point and its negative.",
"This reduces the size of the group $T_n$ by a factor $2n$ , which is a small price to pay considering the amount of memory saved by applying the compression, especially since $n$ is small in practice.",
"Notice also that it is enough to compute one solution of system (REF ), since the set of all solutions consists precisely of the Frobenius conjugates of one point.",
"This is because any polynomial in $n$ variables which is left invariant by any permutation of the variables can be written uniquely as a polynomial in the elementary symmetric functions $e_1,\\ldots ,e_n$ .",
"Now, how many different equivalence classes of points can be output by the decompression algorithm depends only on the degree of $g_n$ in the last indeterminate.",
"For $n=3$ the degree is one and decompression therefore outputs only a single class.",
"As $n$ grows, the degree of the Semaev polynomial also grows, thus producing more ambiguity in the recovery process.",
"This also reflects the growth in the number of extra points which satisfy the equation coming from the Semaev polynomial, as seen in Lemma .",
"Notice moreover that there may be solutions $\\tau $ of $g_n(S_1,\\ldots ,S_{n-1},t)=0$ for which system (REF ) has no solutions, and that not all the solutions of system (REF ) produce an equivalence class of points on the trace zero variety.",
"E.g., if $X\\in $ satisfies $f_n(X,X^q,\\ldots ,X^{q^{n-1}})=0$ , the corresponding point $P=(X,Y)\\in E$ may have $Y\\in _{q^{2n}}\\setminus $ .",
"In this case $P\\notin T_n$ .",
"Since our algorithms are most useful for $n=3$ and 5, an asymptotic complexity analysis for general $n$ does not make much sense.",
"In fact, it is easy to count the number of additions, multiplications, and squarings in $$ needed to compute the representation just from looking at the formulas for $s_1,\\ldots ,s_{n-1}$ .",
"We do this for the cases $n=3$ and 5 in Sections and , respectively.",
"There, we also discuss the efficiency of our decompression algorithm and how it compares to the approaches of [35], [42].",
"In order to compute with points of $T_n$ , we suggest to decompress a point, perform the operation in $E()$ , and compress again the result.",
"Since compression and decompression is very efficient, this adds only little overhead.",
"In an environment with little storage and/or bandwidth capacity, the memory savings of compressed points may well be worth this small trade-off with the efficiency of the arithmetic.",
"Also notice that scalar multiplication of trace zero points in $E()$ is more efficient than scalar multiplication of arbitrary points of $E()$ , due to a speed-up using the Frobenius endomorphism, as pointed out by Frey [15] and studied in detail by Lange [31], [32] and subsequently by Avanzi and Cesena [1].",
"Our recommendation corresponds to usual implementation practice in the setting of point compression: Even when a method to compute with compressed points is available, it is usually preferable to perform decompression, compute with the point in its original representation, and compress the result.",
"For example, Galbraith-Lin show in [16] that although it is possible to compute pairings using the $x$ -coordinates of the input points only, it is more efficient in most cases (namely, whenever the embedding degree is greater than 2) to recompute the $y$ -coordinates of the input points and perform the pairing computation on the full input points.",
"As a second example, let us consider the following two methods for scalar multiplication by $k$ of an elliptic curve point $P = (X,Y)$ when only $X$ is given: Use the Montgomery ladder, which computes the $x$ -coordinate of $kP$ from $X$ only.",
"Find $Y$ by computing a square root, apply a fast scalar multiplication algorithm to (X,Y), and return only the $x$ -coordinate of the result.",
"All recent speed records for scalar multiplication on elliptic curves have been set using algorithms that need the full point $P$ , in other words with the second approach, see e.g.",
"[4], [34], [36], [14].",
"Timings typically ignore the additional cost for point decompression, but there is strong evidence that on a large class of elliptic curves the second approach is faster.",
"This is the basis for our suggestion to follow the second approach when working with compressed points of $T_n$ ."
],
[
"Explicit equations for extension degree 3",
"We give explicit equations for $n=3$ , where we write $_{q^3} = [\\zeta ]/(\\zeta ^3-\\mu )$ and use $1,\\zeta ,\\zeta ^2$ as a basis for $_{q^3}|$ .",
"For completeness, we start with the standard equations for the trace zero variety (see [15]), although we do not make further use of them in our approach.",
"They describe an open affine part of the trace zero variety (i.e.",
"they hold when $x_1, x_2 \\ne 0$ ): $ \\begin{array}{rcl}y_0^2 + 2 \\mu y_1 y_2 & = & x_0^3 + \\mu x_1^3 + \\mu ^2 x_2^3 + 6 \\mu x_0 x_1 x_2 + A x_0 + B\\\\2 y_0 y_1 + \\mu y_2^2 & = & 3 x_0^2 x_1 + 3 \\mu x_0 x_2^2 + 3 \\mu x_1^2 x_2 + A x_1\\\\2 y_0 y_2 + y_1^2 & = & 3 x_0^2 x_2 + 3 x_0 x_1^2 + 3 \\mu x_1 x_2^2 + A x_2\\\\x_1 y_2 & = & x_2 y_1.\\end{array}$ The equation that we found in Section only involves the $x$ -coordinate and is $f_3(x,x^q,x^{q^2}) & = & x^{2q^2+2q} - 2x^{2q^2+q+1} + x^{2q^2+q} - 2x^{q^2+2q+1} -2x^{q^2+q+2}-2Ax^{q^2+q} \\\\& & - 2Ax^{q^2+1} - 4Bx^{q^2} + x^{2q+2} - 2Ax^{q+1} - 4Bx^q- 4Bx + A^2.$ For Weil restriction, we write $x = x_0 + x_1 \\zeta + x_2 \\zeta ^2$ and get $ \\begin{array}{rcl}x & = & x_0 + x_1 \\zeta + x_2 \\zeta ^2\\\\x^q & = & x_0 + \\mu ^b x_1 \\zeta + \\mu ^{2b} x_2 \\zeta ^2 \\\\x^{q^2} & = & x_0 + \\mu ^{2b} x_1 \\zeta + \\mu ^b x_2 \\zeta ^2,\\end{array} $ where $b = \\frac{q-1}{3}$ .",
"The second and third equalities follow from observing that we can substitute $x_i$ for $x_i^q$ when looking for $$ -solutions.",
"This gives $ \\begin{array}{rcl}\\tilde{f}_3(x_0,x_1,x_2) & = & -3x_0^4 - 12 \\mu ^2 x_0 x_2^3 - 12 \\mu x_0 x_1^3 + 18 \\mu x_0^2 x_1 x_2\\\\& & + 9 \\mu ^2 x_1^2 x_2^2 - 6Ax_0^2 + 6A \\mu x_1 x_2 - 12Bx_0 + A^2.\\end{array}$ The symmetrized third Semaev polynomial is $ g_3(s_1,s_2,s_3) = s_2^2 - 4s_1s_3 - 4Bs_1 - 2As_2 + A^2$ and describes the trace zero subgroup via $ \\begin{array}{rclcl}s_1 & = & x + x^q + x^{q^2} & = & 3x_0 \\\\s_2 & = & x^{1+q} + x^{1+q^2} + x^{q+q^2} & = & 3x_0^2 - 3\\mu x_1x_2\\\\s_3 & = & x^{1+q+q^2} & = & x_0^3 - 3 \\mu x_0 x_1 x_2 + \\mu x_1^3 + \\mu ^2 x_2^3.\\end{array}$ So for compression of a point $(x_0,x_1,x_2,y_0,y_1,y_2)$ , we use the coordinates $ (s_1,s_2) = (3x_0,3x_0^2 - 3\\mu x_1x_2), $ and for decompression, we have to solve $g_3(s_1,s_2,s_3) = 0$ for $s_3$ , where $g_3$ is given by equation (REF ).",
"Since the equation is linear in $s_3$ , the missing coordinate can be recovered uniquely, except when $s_1 = 0$ .",
"This is the case only for a small set of points.",
"Notice moreover that the points $(0,s_2,s_3)$ with $s_2^2-2As_2+A^2=0$ satisfy equation (REF ) for every $s_3$ .",
"The only ambiguity in decompression comes from solving system (REF ), which yields the Frobenius conjugates $x,x^q,x^{q^2}$ of the original $x$ .",
"So for $n = 3$ this gives an optimal representation in our sense.",
"The following representation is equivalent to the above, but easier to compute.",
"Set $ t_1 = x_0,~t_2 = x_1x_2,~t_3 = x_1^3 + \\mu x_2^3,$ and take $(t_1, t_2)$ as a representation.",
"The relation between the two sets of coordinates is $s_1 = 3t_1,~s_2 = 3t_1^2 - 3 \\mu t_2,~s_3 = t_1^3 - 3\\mu t_1t_2 + \\mu t_3.$ In this case, we recover $t_3$ from the equation $ -3t_1^4 + 18 \\mu t_1^2 t_2 + 9 \\mu ^2 t_2^2 - 12 \\mu t_1t_3 - 12Bt_1 -6At_1^2 + 6A\\mu t_2 + A^2.",
"$ The equation is linear in $t_3$ , thus making point recovery unique whenever $t_1 \\ne 0$ , but the total degree is higher.",
"Compared to the representation $(s_1,s_2)$ , fewer operations are needed for compression and for computing the solutions of the system during decompression.",
"Thus, compression and decompression for this variant of the representation are more efficient.",
"We give timings for 10, 20, 40, 60, and 79 bit fields in Table REF , where we see that compression is about a factor 3 to 4 faster and decompression is slightly faster for the second method.",
"Notice that decompression timings are for recomputing the $x$ -coordinate only.",
"All computations were done with Magma version 2.19.3 [7], running on one core of an Intel Xeon Processor X7550 (2.00 GHz) on a Fujitsu Primergy RX900S1.",
"Our Magma programs are straight forward implementations of the methods presented here and are only meant as an indication.",
"No particular effort has been put into optimizing them.",
"Table: Average time in milliseconds for compression/decompression of one point when n=3n=3We give a concrete example, before concluding the section with a more detailed analysis of the efficiency of our algorithms.",
"Let $E$ be the curve $y^2 = x^3 + x + 368$ over $$ , where $q = 2^{79}-67$ is a 79-bit prime, and $\\mu = 3$ .",
"The trace zero subgroup of $E(_{q^3})$ has prime order of 158 bits.",
"We choose a random point (to save some space, we write only $x$ -coordinates) ${\\small \\begin{array}{c}P = 260970034280824124824722 + 431820813779055023676698 \\zeta + 496444425404915392572065 \\zeta ^2 \\in T_3\\end{array} } $ and compute ${\\small \\begin{array}{l}(P) = (178447193035157787121145, 159414355696879147312583) \\smallskip \\\\(178447193035157787121145, 159414355696879147312583) = \\\\~~~\\lbrace 260970034280824124824722 + 431820813779055023676698 \\zeta + 496444425404915392572065 \\zeta ^2 , \\\\~~~~ 260970034280824124824722 + 318397306102476549147695 \\zeta + 124410673032925784958936 \\zeta ^2 , \\\\~~~~ 260970034280824124824722 + 458707699733097601881649 \\zeta + 88070721176787997175041 \\zeta ^2 \\rbrace \\end{array}}$ where the results of decompression are exactly the Frobenius conjugates of $P$ .",
"In our Magma implementation, we solve system (REF ) over $$ similarly to how one would do it by hand, as described below.",
"Note that the solutions could also be found by computing the roots of the polynomial $x^3 - s_1x^2 + s_2x - s_3$ over $_{q^3}$ , but since the system is so simple for $n=3$ , solving the system directly is faster in all instances.",
"When using the second variant of the representation, we compute $ {\\begin{array}{c}(t_1,t_2) = (260970034280824124824722, 492721032528256431308437)\\end{array}}$ and naturally get the same result for decompression by solving system (REF ) in a similar way.",
"Operation count for representation in the $s_i$ .",
"Where possible, we count squarings (S), multiplications (M), and divisions (D) in $$ .",
"We do not count multiplication by constants, since they can often be chosen small (see [32]), and multiplication can then be performed by repeated addition.",
"Compressing a point clearly takes 1S+1M.",
"Decompression requires the following steps.",
"Evaluating $g_3(s_1,s_2,s_3)$ in the first two indeterminates and solving for the third indeterminate means computing $s_3 = \\frac{1}{4s_1}(s_2(s_2-2A) - 4Bs_1 + A^2)$ , which takes 1M+1D.",
"Given $s_1,s_2,s_3,$ we need to solve system (REF ) for $x$ , or for $x_0,x_1,x_2$ .",
"The most obvious way would be to compute the roots of the univariate polynomial $x^3 - s_1x^2 + s_2x - s_3$ over $_{q^3}$ .",
"Finding all roots of a degree $d$ polynomial over $$ takes $O(n^{\\log _2 3} d^{\\log _2 3} \\log d \\log (dq^n))$ operations in $$ using Karatsuba's algorithm for polynomial multiplication (see [22]).",
"In our case, the degree and $n$ are constants, and hence factoring this polynomial takes $O(\\log q)$ operations in $$ .",
"However, since the system is so simple, in practice it is better to solve directly for $x_0,x_1,x_2$ over $$ .",
"We know that the system has exactly three solutions (except in very few cases, where it has a unique solution in $$ , i.e.",
"$x_1 = x_2 = 0$ ).",
"We get $x_0$ from $s_1$ for free.",
"Assuming that $x_1 \\ne 0$ (the special case when $x_0=0$ is easier than this general case), we can solve the second equation for $x_2$ , plug this into the third equation, and multiply by the common denominator $27 \\mu ^3 x_1^3$ .",
"In this way, we obtain the equation $27 \\mu ^4 x_1^6 + 27 \\mu ^3( x_0 (s_2 - 2 x_0^2) - s_3) x_1^3 + \\mu ^2 (3x_0^2-s_2)^3,$ which must be solved for $x_1$ .",
"The coefficient of $x_1^6$ is a constant, the coefficient of $x_1^3$ can be computed with 1S+1M, and the constant term can then be computed with 1S+1M.",
"Now we can solve for $x_1^3$ with the quadratic formula, which takes 1S and a square root in $$ for the first value, which will have either no or three distinct cube roots.",
"In case it has none, we compute the second value for $x_1^3$ , using only an extra addition, and the three distinct cube roots of this number.",
"This gives a total of 3 values for $x_1$ .",
"Finally, we can compute $x_2 = \\frac{3 x_0^2 - s_2}{3 \\mu x_1}$ , which takes 1D for the first, and a multiplication by the inverse of a cube root of unity for the other two values.",
"Altogether, solving system (REF ) takes a total of at most 3S+2M+1D, 1 square root, and 2 cube roots in $$ .",
"Finally, for each of the at most 3 values for $x$ , we recompute a corresponding $y$ -coordinate from the curve equation and check that it belongs to $_{q^3}$ .",
"Since these are standard procedures for elliptic curves, we do not count operations for these tasks.",
"Therefore, the decompression algorithm takes at most 3S+3M+2D, one square root, and two cube roots in $$ .",
"The cost of computing the roots depends on the specific choice of the field and on the implementation, but it clearly dominates this computation.",
"Operation count for representation in the $t_i$ .",
"In this case, compression takes only 1M.",
"For decompression, we proceed as follows.",
"Given $t_1$ and $t_2$ , we recover $t_3$ from the equation $t_3 = \\frac{1}{12 \\mu t_1}(-3t_1^4 + (18 \\mu t_1^2 + 9 \\mu ^2 t_2 + 6A \\mu )t_2 - 12Bt_1 - 6At_1^2 + A^2)$ .",
"This takes 2S+1M+1D.",
"To solve system (REF ), again assuming $x_1 \\ne 0$ , we have to find the roots of the equation $ x_1^6 - t_3x_1^3 + \\mu t_2^3 .$ The coefficients of this equation can be computed with a total of 1S+1M.",
"We proceed as above to compute 3 values for $x_1$ using 1S, 1 square root, and 2 cube roots.",
"Finally, we compute $x_2 = \\frac{t_2}{x_1}$ using 1D.",
"Thus, solving the system takes a total of at most 2S+1M+1D, 1 square root, and 2 cube roots.",
"In total, decompression takes at most 4S+2M+2D, 1 square root, and 2 cube roots.",
"The cost of this computation is comparable to the decompression using $s_i$ .",
"This corresponds to our experimental results with Magma (see Table REF ).",
"Comparison with Silverberg's method.",
"The representation of [42] consists of the last $n-1$ Weil restriction coordinates, together with three extra bits, say $0 \\le \\nu \\le 3$ to resolve ambiguity in recovering the $x$ -coordinate and $0 \\le \\lambda \\le 1$ to determine the sign of the $y$ -coordinate.",
"So in our notation, Silverberg proposes to represent a point $(x,y) \\in T_3$ is via the coordinates $(x_1,x_2,\\nu ,\\lambda )$ .",
"The compression and decompression algorithms (in characteristic not equal to 3) carry out essentially the same steps: Compute a univariate polynomial of degree 4.",
"The coefficients are polynomials over $$ in 2 indeterminates of degree at most 4.",
"Compute the (up to 4) roots of this polynomial.",
"During compression, this determines $\\nu $ .",
"During decompression, $\\nu $ determines which root is the correct one, and it is then used to compute $x_0$ via addition and multiplication with constants.",
"During decompression, compute the $y$ -coordinate from the curve equation, using $\\lambda $ to determine its sign.",
"We disregard this step when estimating the complexity.",
"Since [42] does not contain a detailed analysis of the decompression algorithm, we cannot compare the exact number of operations.",
"However, the essential difference with our approach is that Silverberg's compression and decompression algorithms both require computing the roots of a degree 4 polynomial over $$ .",
"For compression, this is clearly more expensive than our method, which consists only of evaluating some small expressions.",
"For decompression, this is also less efficient than our method, which computes only a root of a quadratic polynomial, since running a root finding algorithm, or using explicit formulas for the solutions (i.e.",
"solving the quartic by radicals), is much more complicated than computing the roots of our equation.",
"One might argue that it is possible to represent $(x,y)$ via the coordinates $(x_1,x_2)$ only.",
"In such a case, compression would consist simply of dropping $y$ and $x_0$ and would therefore have no computational cost.",
"Without remembering $\\nu $ and $\\lambda $ to resolve ambiguity, this representation would identify up to 4 $x$ -coordinates and up to 8 full points.",
"This is not much worse than our representation, which identifies up to 3 $x$ -coordinates and 6 full points.",
"However, it is not clear that this identification is compatible with scalar multiplication of points.",
"Therefore, one may want to use at least $\\nu $ to distinguish between the recovered $x$ -coordinates.",
"This is in contrast with our situation, where we know exactly which points are recovered during decompression (i.e.",
"the three Frobenius conjugates of the original point).",
"Identifying these three points is compatible with scalar multiplication, since $P =\\varphi ^i(Q)$ implies $kP = \\varphi ^i(kQ)$ for all $k \\in $ and $P,Q\\in T_3$ , and so no extra bits are necessary.",
"Comparison with Naumann's method.",
"Naumann [35] studies trace zero varieties for $n=3$ .",
"He does not give explicit compression and decompression algorithms, but he derives an equation for the trace zero subgroup that may be used for such.",
"In fact, his equation is identical to our equation (REF ), the Weil restriction of the (unsymmetrized) Semaev polynomial.",
"However, he obtains it in a different way, namely, by eliminating from system (REF ).",
"Naumann suggests a compression method analogous to the one of Silverberg: A point is represented via the coordinates $(x_1,x_2,\\nu ,\\lambda )$ .",
"For decompression, $x_0$ is recomputed from a quartic equation, $0 \\le \\nu \\le 3$ determines which root of the equation is the correct $x_0$ , and $0\\le \\lambda \\le 1$ determines the sign of the $y$ -coordinate.",
"Hence the quartic equation must be solved during both compression and decompression.",
"Naumann's equation is different from Silverberg's, yet the analysis of his method is analogous to that of Silverberg's method, and the conclusions are the same.",
"In particular, his algorithms are less efficient than ours, and it is not clear whether it is possible to drop $\\nu $ from the representation and still have a well defined scalar multiplication.",
"Security issues.",
"To the extent of our knowledge, there are no known attacks on the DLP in $T_3$ whose complexity is lower than generic (square root) attacks, provided that one chooses the parameters according to usual cryptographic practice.",
"In particular, the group should have prime or almost prime order and be sufficiently large (e.g.",
"160 or 200 bits).",
"We stress that index calculus methods, as detailed in [20] among many other works, do not yield an attack which is better than generic (square root) attacks in this setting, since the trace zero variety has dimension 2."
],
[
"Explicit equations for extension degree 5",
"The fifth Semaev polynomial is too big to be printed here, but a computer program can easily work with it.",
"It has total degree 32 and degree 8 in each indeterminate.",
"The symmetrized fifth Semaev polynomial has total degree 8 and degree 6 in the last indeterminate.",
"In fact, it has degree 6 in the first, third and fifth indeterminate, and degree 8 in the second and fourth indeterminate.",
"We can compute it efficiently with Magma.",
"It has a small number of terms compared to the original polynomial, but printing it here would still take several pages.",
"The fact that we recover the missing coordinate from a degree 6 polynomial introduces some indeterminacy in the decompression process.",
"However, extensive Magma experiments for different field sizes and curves show that for more than 90% of all points in $T_5$ , only a single class of Frobenius conjugates is recovered.",
"For another 9%, two classes (corresponding to 10 $x$ -coordinates) are recovered.",
"Thus the ambiguity is very small for a great majority of points.",
"In any case, this improves upon the approach of [42], where the missing coordinate is recovered from a degree 27 polynomial, thus possibly yielding 27 different $x$ -coordinates.",
"The Weil restriction of the symmetric functions is $s_1 & = & 5 x_0 \\\\s_2 & = & 10 x_0^2-5 \\mu x_1x_4-5 \\mu x_2x_3 \\\\s_3 & = & 10 x_0^3+5 \\mu ^2x_3^2x_4+5 \\mu ^2x_2x_4^2+5 \\mu x_1x_2^2+5 \\mu x_1^2x_3-15\\mu x_0x_1 x_4-15 \\mu x_0x_2x_3\\\\s_4 & = & 5 x_0^4-15\\mu x_0^2x_1 x_4-15\\mu x_0^2x_2x_3 -5 \\mu x_1^3x_2-5 \\mu ^2x_1x_3^3-5 \\mu ^2x_2^3x_4-5 \\mu ^3x_3x_4^3+5 \\mu ^2x_2^2x_3^2\\\\& & +5 \\mu ^2x_1^2x_4^2+10 \\mu x_0x_1^2x_3+10 \\mu x_0x_1x_2^2+10 \\mu ^2x_0x_3^2x_4+10 \\mu ^2x_0x_2x_4^2 -5 \\mu ^2x_1x_2x_3x_4\\\\s_5 & = & x_0^{5}+\\mu ^3x_3^{5}+\\mu ^4x_4^{5}+\\mu x_1^5+\\mu ^2x_2^5-5 \\mu ^2x_1x_2^3x_3-5 \\mu ^3x_1x_2x_4^3-5 \\mu ^3x_2x_3^3x_4-5 \\mu x_0x_1^3x_2\\\\& & -5 \\mu ^2x_0x_1x_3^3-5 \\mu ^2x_0x_2^3x_4-5 \\mu ^3x_0x_3x_4^3-5 \\mu ^2x_1^3x_3x_4-5 \\mu x_0^3x_1x_4-5 \\mu x_0^3x_2x_3\\\\& & +5 \\mu x_0^2x_1^2x_3+5 \\mu x_0^2x_1x_2^2+5 \\mu ^2x_0^2x_2x_4^2+5 \\mu ^2x_0^2x_3^2x_4+5 \\mu ^2x_0x_1^2x_4^2+5 \\mu ^2x_0x_2^2x_3^2\\\\& & +5 \\mu ^2x_1^2x_2^2x_4+5 \\mu ^2x_1^2x_2x_3^2+5 \\mu ^3x_1x_3^2x_4^2+5 \\mu ^3x_2^2x_3x_4^2-5 \\mu ^2x_0x_1x_2x_3x_4.$ The compression algorithm computes $s_1,\\ldots ,s_4$ according to these formulas over $$ .",
"The decompression algorithm solves a degree 6 equation for $s_5$ and then recomputes the $x$ -coordinate of the point.",
"For the last step, we test two methods: We compute $x$ by factoring the polynomial $x^5 - s_1 x^4 + s_2x^3 - s_3x^2 + s_4x - s_5$ over $_{q^5}$ , and we compute $x_0,\\ldots ,x_4$ by solving the above system over $$ with a Gröbner basis computation.",
"Our experiments show that polynomial factorization can be up to 20 times as fast as computing a lexicographic Gröbner basis in Magma for some choices of $q$ , and the entire decompression algorithm can be up to a factor 6 faster when implementing the polynomial factorization method.",
"We give some exemplary timings for both methods for fields of 10, 20, 30, 40, 50 and 60 bits in Table REF .",
"However, these experimental results can only be an indication: In Magma, the performance of the algorithms depends on the specific choice of $q$ .",
"In addition, any implementation exploiting a special shape of $q$ would most likely produce better results.",
"As for $n=3$ , we suggest an equivalent representation $(t_1,t_2,t_3,t_4)$ where $ \\begin{array}{rcl}t_1 & = & x_0\\\\t_2 & = & x_1x_4 + x_2x_3\\\\t_3 & = & x_1^2x_3 + x_1 x_2^2 + \\mu x_3^2 x_4 + \\mu x_2 x_4^2\\\\t_4 & = & \\mu x_2^2 x_3^2 + \\mu x_1^2 x_4^2 - \\mu x_1 x_3^3 - x_1^3 x_2 - \\mu x_2^3 x_4 - \\mu ^2 x_3 x_4^3 + \\mu x_1 x_2 x_3 x_4\\\\t_5 & = & x_1^5 + \\mu x_2 ^ 5 + \\mu ^2 x_3^5 + \\mu ^3 x_4^5 + 5 \\mu x_1^2 x_2 x_3^2 + 5 \\mu x_1^2 x_2^2 x_4 + 5 \\mu ^2 x_2^2 x_3 x_4^2 \\\\& &+ 5 \\mu ^2 x_1 x_3^2 x_4^2 - 5 \\mu x_1^3 x_3 x_4 - 5 \\mu ^2 x_2 x_3^3 x_4 - 5 \\mu ^2 x_1 x_2 x_4^3 - 5 \\mu x_1 x_2^3 x_3\\end{array}$ and $ \\begin{array}{rcl}s_1 & = &5 t_1 \\\\s_2 & = & 10 t_1^2 - 5 \\mu t_2 \\\\s_3 & = & 10 t_1^3 - 15 \\mu t_1t_2 + 5 \\mu t_3\\\\s_4 & = & 5 t_1^4 - 15 \\mu t_1^2t_2 + 10 \\mu t_1 t_3 + 5 \\mu t_4\\\\s_5 & = & t_1^5 - 5 \\mu t_1^3 t_2 + 5 \\mu t_1^2 t_3 + 5 \\mu t_1 t_4 + \\mu t_5.\\end{array}$ Compared to the representation in the $s_i$ , this representation gives a faster compression, but a slower decompression.",
"Therefore, this approach may be useful in a setting where compression must be particularly efficient.",
"For decompression, the missing coordinate $t_5$ can be recomputed from a degree 6 equation, which we obtain by substituting the relations (REF ) into the symmetrized fifth Semaev polynomial.",
"Afterwards we may either recompute $s_1,\\ldots ,s_5$ from $t_1,\\ldots ,t_5$ according to system (REF ) and solve $x^5 - s_1 x^4 + s_2x^3 - s_3x^2 + s_4x - s_5$ for $x$ , or else we may solve system (REF ) directly for $x_0,\\ldots ,x_4$ with Gröbner basis techniques.",
"The polynomial factorization method is equivalent to using the representation in the $s_i$ , only that some of the computations are shifted from the compression to the decompression algorithm.",
"The Gröbner basis method (use $t_i$ and compute Gröbner basis, “second method”) compares to using $s_i$ with Gröbner basis (“first method”) as given in Table REF .",
"We see that the second method is a factor 2 to 3 faster in compression, but slower in decompression.",
"The reason for this is that the polynomial used to recompute the missing coordinate is more complicated for the second method, and evaluation of polynomials is quite slow in Magma.",
"Solving for the missing coordinate takes 5 times longer for the second method.",
"The solution of system (REF ), which we achieve by computing a lexicographic Gröbner basis and solving the resulting triangular system in the obvious way, takes the same amount of time in both cases.",
"Table: Average time in milliseconds for compression/decompression of one point when n=5n=5We now give an example of our compression/decompression algorithms, including two points $P$ on the trace zero variety where $((P))$ produces the minimum and maximum possible number of outputs.",
"Let $E$ be the curve $y^2 = x^3 + x + 135$ over $$ , where $q = 2^{60}-695$ is a 60-bit prime, and $\\mu = 3$ .",
"The trace zero subgroup of $E(_{q^5})$ has prime order of 240 bits.",
"We choose a random point $P & = & 697340666673436518 + 801324486821916366 \\zeta + 191523769921581598 \\zeta ^2 \\\\&&+ 193574581008452232 \\zeta ^3 + 808272437423069772 \\zeta ^4 \\in T_5$ and compute ${\\small \\begin{array}{l}(P) = (27938819546643747, 599177118073319826, 587362643323803394, 899440023033601132) \\smallskip \\\\(27938819546643747, 599177118073319826, 587362643323803394, 899440023033601132)\\\\~~~ = \\lbrace 697340666673436518 + 801324486821916366 \\zeta + 191523769921581598 \\zeta ^2 \\\\~~~~~~~~~~ + 193574581008452232 \\zeta ^3 + 808272437423069772 \\zeta ^4, \\\\~~~~~~~~~~ 697340666673436518 + 836712212802745328 \\zeta + 506907366758395901 \\zeta ^2 \\\\~~~~~~~~~~ + 517000572714098077 \\zeta ^3 + 268866625974497959 \\zeta ^4, \\\\~~~~~~~~~~ 697340666673436518 + 960543166171367987 \\zeta + 126552294958642222 \\zeta ^2\\\\~~~~~~~~~~ + 448251978051599093 \\zeta ^3 + 74315924307841334 \\zeta ^4 , \\\\~~~~~~~~~~ 697340666673436518 + 810370833605859760 \\zeta + 539948230971075773 \\zeta ^2 \\\\~~~~~~~~~~ + 1032750511909194579 \\zeta ^3 + 944608723064092684 \\zeta ^4 , \\\\~~~~~~~~~~ 697340666673436518 + 49813814418649402 \\zeta + 940911346603997068 \\zeta ^2\\\\~~~~~~~~~~ + 114265365530348581 \\zeta ^3+ 209779298444190813 \\zeta ^4 \\rbrace .\\end{array}}$ When using the second variant of the representation, we compute ${\\small \\begin{array}{c} (t_1,t_2,t_3,t_4) = (697340666673436518, 553115374027544004, 315951679773440541, 285024754797056479).\\end{array}}$ For this point, the results of decompression are exactly the Frobenius conjugates of $P$ .",
"However, this is not always the case.",
"In rare cases, the algorithm may recover up to six classes of Frobenius conjugates.",
"We give an example of a point for which three classes of Frobenius conjugates are recovered: $P & = & 760010909342414570 + 568064535058825884 \\zeta + 244006548504894796 \\zeta ^2\\\\&&+ 446522043528586762 \\zeta ^3 + 731314735984238952 \\zeta ^4 \\in T_5.$ Operation count for representation in the $s_i$ .",
"Given $x_0,\\ldots ,x_4$ , the numbers $t_1,\\ldots ,t_4$ can be computed with a total of 5S+13M according to (REF ).",
"Then $s_1,\\ldots , s_4$ can be computed from those numbers with 2S+3M as given in (REF ).",
"This seems to be the best way to compute $s_1,\\ldots ,s_4$ , since these formulas group the terms that appear several times.",
"Hence compression takes a total of 7S+16M.",
"For decompression, the most costly part of the algorithm is factoring the polynomials.",
"First, the algorithm has to factor a degree 6 polynomial over $$ , and next, a degree 5 polynomial over $_{q^5}$ .",
"The asymptotic complexity for both of these is $O(\\log q)$ operations in $$ .",
"Operation count for representation in the $t_i$ .",
"Compression takes 5S+13M.",
"For decompression, we can either recompute $s_1,\\ldots ,s_5$ from $t_1,\\ldots ,t_5$ and factor the polynomial, in which case this approach is exactly the same as the above.",
"Or else we can solve system (REF ) by means of a Gröbner basis computation over $$ .",
"Since there are no practically meaningful bounds for Gröbner basis computations, a complexity analysis of this approach makes no sense.",
"Comparison with Silverberg's method.",
"Concrete equations are presented in [42] for the case where the ground field has characteristic 3.",
"The most costly parts of the compression and decompression algorithms are computing the resultant of two polynomials of degree 6 and 8 with coefficients in $$ , and finding the roots of a degree 27 polynomial over $$ .",
"In general, resultant computations are difficult, and the polynomial to be factored has much larger degree than those in our algorithm.",
"In Silverberg's approach, five extra bits are required to distinguish between the possible 27 roots of the polynomial.",
"Although neither Silverberg nor we give explicit equations for larger $n$ , our understanding is that our algorithm scales better with increasing $n$ , since our method is more natural and respects the structure of the group.",
"Security issues.",
"We briefly discuss the security issues connected with use of $T_5$ in DL-based and pairing-based cryptosystems.",
"Since $T_5$ is a group of size $q^4$ , generic algorithms that solve the DLP in $T_5$ have complexity $O(q^2)$ .",
"Security threats in the context of DL-based cryptosystems are posed by algorithms for solving the DLP that achieve lower complexity.",
"There are two types of algorithms that one needs to consider: First, cover attacks aim to transfer the DLP in $E(_{q^5})$ to the DLP in the Picard group of a curve of larger (but still rather low) genus, see [21], [9].",
"The DLP is then solved there using index calculus methods.",
"Combining the results of [9] and [10], it is sometimes possible to map the DLP from $T_5$ into the Picard group of a genus 5 curve (which is usually not hyperelliptic), where it can be solved with probabilistic complexity $\\tilde{O}(q^{4/3})$ following the approach of [11].",
"However, only a very small proportion of curves is affected by this attack, and such curves should be avoided in practice.",
"Moreover, in order to avoid isogeny attacks, the curve should be chosen such that 4 does not divide the order of $T_5$ , see [9].",
"Second, the index calculus attack of [20] applies to $T_5$ and has complexity $\\tilde{O}(q^{3/2})$ .",
"This makes $T_5$ not an ideal group to use in a DL-based cryptosystem.",
"Notice that in practice, however, the constant in the $O$ is very large, since the attack requires Gröbner basis computations, which are very time consuming (their worst case complexity is doubly exponential in the size of the input), and often do not terminate in practice.",
"It is our impression that more in depth study is needed in order to give a precise estimate of the feasibility of such an attack for a practical choice of the parameters.",
"We carried on preliminary experiments, which indicate that a straightforward application of the method from [20] to $T_5$ yields a system of equations which is very costly to compute (it requires computing the Weil descent of the fifth Semaev polynomial) and which Magma cannot solve in several weeks and using more than 300 GB of memory on the same machine that we used to carry out the experiments reported on in Sections and of this article.",
"Notice that solving such a system would (possibly) produce one relation, to be then used in an index calculus attack.",
"Therefore, in practice one would need to solve many such systems, in order to produce the relations needed for the linear algebra step of the index calculus attack.",
"Trace zero varieties are even more interesting in the context of pairing-based cryptography.",
"The main motivation comes from [40], where Rubin and Silverberg show that supersingular abelian varieties of dimension greater than one offer more security than supersingular elliptic curves, for the same group size.",
"Trace zero varieties are explicitly mentioned in [40] as one of the most relevant examples of abelian varieties for pairing-based cryptography.",
"In order to estimate the security of $T_5$ in pairing-based cryptosystems, one needs to compare the complexities of solving the DLP in $T_5$ and in $_{q^{5k}}$ , where $k$ is the embedding degree, i.e., the smallest integer $k$ such that $_{q^{5k}}$ contains the image of the pairing.",
"A first observation is that, since the results of [40] hold over fields of any characteristic, one should avoid fields of small characteristic, so that the recent attacks from [23], [28], [24], [2], [3] do not apply.",
"Over a field of large characteristic, the cover and index calculus attacks that we discussed in the previous paragraph do not seem to pose a serious security concern in the context of pairing-based cryptography.",
"This is due to the fact that, for most supersingular elliptic curves, the Frey-Rück or the MOV attack have lower complexity than cover and index calculus attacks in the lines of [20], [21], [9], [11].",
"In some cases however, the choice of the security parameter may need to be adjusted, according to the complexity of these index calculus attacks.",
"As an example, let us discuss the choice of parameters for a pairing with 80-bits security.",
"One needs a field of about 1024-bits as the target of the pairing (avoiding fields of small characteristic).",
"If we assume that the pairing ends up in an extension field of degree $k=2$ of the original field $_{q^5}$ (this is the case for most supersingular elliptic curves), then $q$ should be a 102-bit number.",
"A $q^{3/2}$ attack on the group $T_5$ on which the pairing is defined would result in 153-bit security, while a $q^{4/3}$ attack would result in 136-bit security.",
"However, on the side of the finite field the system has an 80-bit security, so the attacks from [20], [21], [9], [11] end up not influencing the overall security of the pairing-based cryptosystem in this case.",
"A related comment is that an interesting case for pairings is when the DLP in $T_5$ and in the finite field extension $_{q^{5k}}$ where the pairing maps have the same complexity.",
"In order to achieve this in our previous example, we would need to have a security parameter $k = 4$ , which can be achieved by supersingular trace zero varieties.",
"In this case, the complexity of solving a DLP in $T_5$ and in $_{q^{20}}$ are both about 80-bits when $q\\sim 2^{53}$ .",
"Summarizing, the complexity of the DLP in $T_5$ coming from the works [20], [21], [9], [11] influences the choice of the specific curves that we use in pairing-based applications, since it influences the security parameter $k$ that makes the hardness of solving the DLP in $T_5$ and in $_{q^{5k}}$ comparable, and the value of $k$ depends on the choice of the curve.",
"However, in general it does not influence the size $q$ of the field that we work on, since an attack can influence the value of $q$ only if it has lower complexity than the Frey-Rück or the MOV attack for supersingular elliptic curves.",
"Therefore, using trace zero varieties instead of elliptic curve groups in pairing-based cryptography has the advantages of enhancing the security and allowing for more flexibility in the setup of the system."
],
[
"Conclusion",
"The Semaev polynomials give rise to a useful equation describing the $$ -rational points of the trace zero variety.",
"Its significance is that it is one single equation in the $x$ -coordinates of the elliptic curve points, but unfortunately its degree grows quickly with $n$ .",
"Using this equation, we obtain an efficient method of point compression and decompression.",
"It computes a representation for the $$ -points of the trace zero variety that is optimal in size for $n=3$ and for $n=5$ .",
"Our polynomials have lower degree than those used in the representations of [42] (1 compared to 4 for $n=3$ , and 6 compared to 27 for $n=5$ ) and [35] (1 compared to 3 for $n=3$ ), thus allowing more efficient compression and decompression and less ambiguity in the recovery process.",
"Finally, our representation is interesting from a mathematical point of view, since it is the first representation (to our knowledge) that is compatible with scalar multiplication of points.",
"Acknowledgements We thank Pierrick Gaudry and Peter Schwabe for helpful discussions and Tanja Lange for pointing out the work of Naumann.",
"We are grateful to the mathematics department of the Univerity of Zürich for access to their computing facilities.",
"The authors were supported by the Swiss National Science Foundation under Grant No.",
"123393."
]
] | 1403.0126 |
[
[
"Clinical Facts Along With a Feedback Control Perspective Suggest That\n Increased Response Time Might be the Cause of Parkinsonian Rest Tremor"
],
[
"Abstract Parkinson's disease (PD) is a neurodegenerative disorder characterized by increased response times leading to a variety of biomechanical symptoms such as tremors, stooping and gait instability.",
"Although the deterioration in biomechanical control can intuitively be related to sluggish response times, how the delay leads to such biomechanical symptoms as tremor is not yet understood.",
"Only recently has it been explained from the perspective of feedback control theory that delay beyond a threshold can be the cause of Parkinsonian tremor [1].",
"This paper correlates several observations from this perspective to clinical facts and reinforces them with simple numerical and experimental examples.",
"This work provides a framework towards developing a deeper conceptual understanding of the mechanism behind PD symptoms.",
"Furthermore, it lays a foundation for developing tools for diagnosis and progress tracking of the disease by identifying some key trends."
],
[
"Introduction",
"Patients suffering from Parkinson's disease (PD) experience a variety of biomechanical symptoms including tremors, stooping, rigidity, and gait instability [2].",
"Since the discovery of this disease in 1817 [3], the connections between these apparently varied biomechanical symptoms has puzzled researchers and have led to a range of hypotheses and conjectures about the source of these symptoms and whether there is a single underlying explanation for these symptoms or not [4], [5].",
"PD is a neuro-degenerative disease and is also characterized by a permanent increase in response time in both voluntary and involuntary motor responses.",
"Wilson [6] first noted the impairment of response time in Parkinson's disease, stating that \"Recent measurement with special apparatus of the muscular response to a single visual stimulus have given the figures of 0.24 seconds for normal individuals and 0.36 for the the subjects of paralysis agitans\".",
"Several detailed studies [7], [8], [9], [10] since then have come up with similar conclusions.",
"This has added another dimension to the mystery surrounding the symptoms.",
"Although the deterioration in biomechanical control can intuitively be related to sluggish response times, how the increase in response time leads to such biomechanical symptoms such as tremor and stooping is not yet understood [11].",
"In fact it is argued that tremor may have a pathophysiology different from most other Parkinson's symptoms [4], [5].",
"It is widely believed, that Parkinsonian tremor occurs most likely due to oscillating neuronal activity within the central nervous system [12].",
"Neroprosthetic therapies such as deep brain stimulation [13] suppress Parkinsonian tremor, however, the fundamental mechanism behind these therapies also unresolved [14].",
"On another front, empirical mathematical models are proposed that can simulate Parkinsonian tremor, for instance, a limit-cycle-exhibiting system such as the Van-Der-Pol oscillator can be fit to experimentally measured data [15].",
"But such an approach lacks physical underpinnings and does not provide any insights into some of the key features of Parkinsonian tremor.",
"For example, why a PD patient trying to keep still would exhibit tremors (referred to as rest tremors) [2], whereas a PD patient involved in engrossing physical or mental activity may not exhibit tremors.",
"Recently, [1] presented arguments based on a control-system analogy that supports the hypothesis that Parkinsonian tremor may indeed be limit cycle oscillations, and established a direct logical connection between increased response time and limit-cycle behaviour of the Parkinsonian tremors.",
"Since then, similar connections between increased time delays and limit cycle oscillations (although not in the context of Parkinson's disease) have also been drawn [16].",
"In the current work, we analyse the hypothesis in [1] that an increased sensorimotor loop delay (that is observed as an increased response time) causes rest tremors in PD with two specific objectives.",
"First, we wish to draw qualitative observations based on this hypothesis that are supported by clinical facts, feedback control arguments, and simple numerical and experimental examples (see Figure REF ).",
"Second, based on this hypothesis, we explore possibilities for using biomechanics analysis of tremor data for progress tracking, diagnosis, and early diagnosis of PD from tremor motion data.",
"The current work therefore builds a framework towards developing a deeper conceptual understanding of the mechanism behind PD rest tremors, and uses both simple feedback control theory arguments and simple simulation and experimental examples to setup possibilities for progress tracking, diagnosis and early diagnosis before onset of tremors.",
"While the focus of the current work is neither developing detailed diagnosis methodologies and algorithms nor making connections with the other biomechanics symptoms of PD, this works lays the foundation for such development in the future.",
"For exploring these possibilities in further details in the future, one can use a combination of patient tests and simulations with models of human posture, motor control, or gait (such as in [17]) as next steps.",
"Figure: The contributions of this work."
],
[
"Explaining Clinical Facts Using Feedback Control Representation",
"In [1], with the help of a simple feedback control representation of the motor control of a body part, the authors proposed that an increase sensorimotor loop delay as compared to healthy individuals is the primary cause for tremors in Parkinson's patients.",
"Therefore, while the overall sensorimotor loop delay could be thought of as a combination of various delays in various portions of the sensorimotor loop, we lump them together in one position and later in Section discuss how the specific locations of the various delays do not affect the overall observations derived from the analysis below.",
"With this background, we proceed with the assumption that it is indeed true that the increased sensorimotor loop delay is the cause for Parkinsonian tremor, and adopt the feedback control system framework presented in [1] as depicted in Figure REF .",
"With the help of different computational and physical examples of feedback control systems with delay, we find that several qualitative features of the resulting limit-cycle oscillations match with the clinical facts about the tremor.",
"Such findings lend further credibility the feedback control system framework presented in [1], in which the increased sensorimotor loop delay causes the tremors.",
"A strength of this finding is that these qualitative features do not depend on the exact details of the plant and the controller.",
"We further note that these qualitative features may be explained readily with intuitive feedback control arguments.",
"However, the data produced by these toy examples provide common insightful trends and features of tremor that can not only be leveraged for diagnosis and monitoring of PD, but also pave the way to developing early diagnosis and treatment strategies.",
"In the feedback control framework depicted in Figure REF the mathematical model that governs the dynamics of any body part (e.g.",
"hand) in the absence of any neural control is what is referred to as a plant in control-system notation.",
"The feedback path represents all sensory feedbacks, and the controller represents the neurosystem's logic that continuously compares the actual velocity (from sensory feedback) with the desired velocity to determine muscle actions.",
"As discussed earlier, the sensorimotor loop delay is lumped in to one €œtransport delay in the closed-loop feedback system.",
"Finally, the physiological limit of the transmission of neural control actions [18] is represented as a saturation function that imposes a constraint on control input to the plant.",
"Based on the above feedback control framework, we employ three examples that are explained in detail in the Appendix A, a numerical example and two table-top motion control experiment examples.",
"While all three examples follow the same feedback control architecture and all three have saturation and delay, the two crucial features to explain the possible mechanism of parkinsonian tremor, they otherwise have very different plants and controllers.",
"The pendulum simulation example is a second-order stable system with a proportional-derivative-integral controller, the servo motor control experimental example is a first-order, linear, stable system with a proportional-derivative controller and the inverted-pendulum experimental example is a fourth-order, non-linear, unstable system with LQR controller.",
"Thus, together they serve to qualitatively explain the clinical facts by highlighting how the feedback control perspective can explain the facts regardless of variations between patients.",
"Through this logic, it is also obvious that a more realistic and fully featured mathematical model of the neural control and of the mechanics of human body are not needed to explain these clinical facts.",
"We describe results for the servo position control experiment in more detail and keep discussion relating to the rotary inverted pendulum experiment brief whenever they show similar results.",
"Figure: The closed-loop feedback system representing motor control in patients with Parkinson's disease.",
"[Note: Transport Delay can be anywhere inside the loop.]"
],
[
"Clinical Facts",
"1.",
"Parkinsonian tremor is a tremor-at-rest [6].",
"With all three examples, it is clear from the simulations and experiments that a delay beyond a certain threshold triggers oscillatory behaviour as seen in figures REF and REF .",
"It is important to note that there is zero intended velocity in all three cases thus representing rest position.",
"This can be readily explained from feedback control theory in the following way.",
"It is well-known that any stable feedback control loop [19] with loop gain more than one will become unstable if a large enough delay is introduced in the loop, and further due to the presence of the saturation in the loop, the instability is not driving the oscillations to infinity, but rather a regular bounded oscillatory behaviour (self-sustained oscillations).",
"Another way to explain this is that in the phase space, an increased delay (beyond a certain threshold) renders the origin unstable, and even with the smallest disturbance or initial condition, the response is driven to a stable limit cycle oscillations due to the saturation in the loop.",
"In fact, observing the phase plot, that is the plot of angular velocity v/s angular acceleration, in both the numerical and experimental examples for various initial conditions confirms this explanation as it yields trajectories converging to a stable limit cycle as seen in Figures REF and REF .",
"Hence, these examples not only support the rest tremors observations in PD but also is able to explain in relatively simple terms why it is a tremor at rest.",
"Figure: Angular velocity for various time delays with zero intended velocity, initial angular position of 0.1rad0.1rad and saturation limits as -100 to 100 (Pendulum numerical example).Figure: Angular velocity for various time delays with zero intended velocity and saturation limits as -1 to 1 (Servo motor experimental example).Figure: Trajectories starting from various initial conditions converging to a limit cycle in phase space (Pendulum numerical example).Figure: Trajectories starting from various initial conditions converging to a limit cycle in phase space (Servo motor experimental example).2.",
"Rest tremor often disappears when large-scale voluntary motion is attempted [2].",
"Numerical and experimental example show that when a significant intended velocity is used (large-scale voluntary motion), the tremor disappears as seen in Figure REF and Figure REF ).",
"While the trace of the tremor is still present while the amplitude is low, for higher value of amplitude, trace of tremor disappears.",
"This can be readily explained as a nonlinear effect because the saturation in the loop can lead to amplitude-dependent behavior and thus shows different behavior for very small amplitude or zero intended velocity versus large intended velocity.",
"3.",
"Rest tremor also disappears when patients sleep or engage in engrossing mental activity [2].",
"When a patient is asleep or engaged in engrossing mental activity, one can argue that their sensory feedback is cut off or compromised.",
"It is then obvious from the explanation in point 1 above that if the feedback path is disrupted, then the tremors would disappear.",
"This explains why Parkinsonian rest tremor, in contrast to “Essential tremor” [2], disappears when patients suffering from Parkinson's disease patients sleep or engage in engrossing mental activity.",
"This argument further explains why when patients undergo surgical treatments involving damage of brain's control circuitry rest tremors disappear [20], [21].",
"Figure: Velocity plots for a sinusoidal intended velocity of frequency 1 Hz, saturation limits as -100 to 100 and amplitude 0, 10 and 30 units (Pendulum numerical example).Figure: Velocity plots for a sinusoidal intended velocity of frequency 1.5 Hz, saturation limits as -1 to 1 and amplitude 0, 1 and 10 units (Servo motor experimental example).4.",
"As the disease progresses a decrease in frequency of tremor is observed and a corresponding increase in the amplitude of tremor is observed [22].",
"We first conclude that the progression of the disease is directly linked to increased response times and therefore linked to an increase in sensorimotor loop delay.",
"Numerical and experimental results with an increased transport delay clearly show a decrease in the frequency of oscillation (tremor).",
"From figures REF and REF it is clear that there exist a inverse relationship between the loop delay and frequency of tremor and that results in an increase in the amplitude (Figures REF and REF ).",
"Figure: Delay vs. frequency of oscillation with saturation limits as -4 to 4 (Pendulum numerical example).Figure: Delay vs. frequency of oscillation with saturation limits as -10 to 10 (Servo motor experimental example).The increase in the amplitude can be explained through the following argument.",
"Consider a simple oscillating signal such as $a=sin(\\omega t)$ where $\\omega $ is an angular frequency and $a$ is an acceleration.",
"Noting that its integral is $v=-cos(\\omega t)/\\omega $ , it is obvious that for periodic oscillations the amplitude of the velocity signal is inversely proportional to the frequency of the acceleration signal.",
"Here, since the amplitude of the acceleration signal is limited by the saturation levels, it is not surprising that for the same saturation level, the amplitude of velocity signal is higher for lower frequency oscillations and vice versa.",
"It should also be noted here that this explanation holds regardless of whether the plant has low-pass filter characteristics or not.",
"We do not show all the results with the inverted pendulum setup due to space limitation, but trends similar to the one observed with the servo position control setup are seen (for an example Figure REF ).",
"We also note that the results of the simulation and experimental examples are qualitatively similar.",
"In the next section, we explore if this leads to specific features that can be used of diagnosis of parkinson's disease and estimate its severity.",
"Figure: Trajectories starting from various initial conditions converging to a limit cycle in phase space (Rotary Inverted Pendulum experimental example)."
],
[
"Possibilities for Progress Tracking and Diagnosis",
"From the discussion in the previous section, it is emerging clearly that there are trends in frequency and amplitude of tremors that can possibly be leveraged for tracking the progression of PD over a period of time and observing the effect of treatment strategies.",
"For instance, one could develop a simple pocket device which has accelerometers and/or gyro sensors and record tremor data (accelerations and/or angular velocities) from PD patients.",
"One could make measurements at regular intervals, (say, every three months) and based on the frequency of the measured tremors track the progression of the disease using trends such as observed in Figures REF and REF .",
"It may also be possible to evaluate the efficacy of treatment strategies using this progress tracking.",
"In reality however, one would expect that it is very likely that PD tremors may show a combination of frequencies and may have some irregularities and thus making picking out a clearly defined frequency or amplitude from tremor data challenging.",
"To resolve this, there could be two approaches.",
"One approach is to explore other tremor parameters that may be more robust indicators of the underlying sensorimotor loop delay.",
"A second approach is that even if a more robust parameter is not available, take multiple independent measurements of parameters that indirectly relate to sensorimotor loop delay and then fuse the estimates (e.g.",
"using a Maximum-likelihood method) to obtain a robust indication of the sensorimotor loop delay.",
"This is a common practice in sensor fusion wherein several inexpensive low quality sensor measurements are fused to obtain a high-quality estimate.",
"On further exploration, two other parameters that correlate with the delay in the sensorimotor loop and therefore provide possibilities of tracking the progression of disease are the area and aspect ratio of the limit cycle/trajectory formed in the phase space.",
"Figure REF shows the variations of aspect ratio and area of the limit cycle as a function of delay and saturation and shows that both of them are directly correlated to delay.",
"We also note that aspect ratio in Figure REF is practically unaffected by the physiological saturation while area is affected by the physiological saturation.",
"In any case, it is to be noted that amplitude, frequency, area of limit cycle and aspect ratio of limit cycle serves as four independent measurements that are correlated with sensorimotor loop delay.",
"The potential of using these parameters for progress tracking have to be confirmed through patient studies.",
"A more challenging possibility that cannot be ruled out at the moment is to eventually use these parameters for diagnosis of PD.",
"These would however have to be developed after extensive studies to factor out patient-to-patient variabilities.",
"In the next section, we explore the possibility of early diagnosis of PD even before the appearance of tremors.",
"Figure: (a) Aspect ratio of limit cycle as a function of saturation and delay.",
"(b) Area of the limit cycle as a function of delay and saturation."
],
[
"Early Diagnosis of Parkinson's Disease",
"So far, we have discussed the possibility for tracking and diagnosis of PD by analyzing the features of parkinsonian tremor using hand tremor data.",
"So now the question we want to explore is if it is possible to diagnose PD in patients before the onset of rest tremors.",
"While the rest tremor is one of the more common clinical features of PD, around $31\\%$ of PD patient do not have rest tremor during the onset of the disease [2], [23].",
"So in this section, we explore the possibility of diagnosing PD before onset of tremor using a bio mechanical approach.",
"Since it has been established earlier that rest tremor arises from a delay-induced instability, the key idea explored in this section is if the propensity to develop an instability can be seen even before the instability actually develops.",
"While it is obvious that a delay beyond a certain threshold will lead to instability in a feedback system, one way to quantify it is as follows.",
"A delay in the time domain can be represented as $e^{-T_d s}$ in the frequency domain, where $T_d$ is the delay and $s$ is the Laplace variable.",
"$e^{-T_d s}$ can be approximated using a first-order Pade approximation [19] as given by $e^{-T_d s}\\simeq \\frac{1-(T_ds/2)}{1+(T_ds/2)} \\hspace{8.53581pt} ,$ Thus, the Pade approximation suggests that the delay can be viewed as approximately a non-minimum phase zero and a corresponding stable pole pair in the system.",
"If the saturation is ignored for the moment (for small-amplitude motion in the stable regime), the loop transfer function of the servo position control experiment with the Proportional-Derivative controller and the delay is given by (as per Figure REF ) $L(s)= \\frac{1-(T_ds/2)}{1+(T_ds/2)} \\times \\frac{10.38s+519}{s^2+7.89s} \\hspace{8.53581pt},$ Viewing $T_d$ as the variable parameters, and putting the characteristics equation in the root locus form, we get $1+(T_d)\\frac{s^3+18.27s^2-519s}{2s^2+36.54s+1038} = 0 \\hspace{5.69054pt}.$ Since this form shows a non minimum phase zero, it is obvious that as the delay $T_{d}$ increases at least one closed loop pole will move towards the RHP (and beyond a certain threshold goes unstable and induces oscillations).",
"We exploit this fact for early diagnosis of parkinson's disease.",
"Since the PD rest tremor appears to be arising from an instability, estimating the eigenvalues of the underlying dynamics will likely give us an idea about the presence of the disease, its severity, and propensity to develop tremors.",
"For instance, eigenvalues that are stable but close to imaginary axis may indicate the possibility of PD (without yet displaying any tremor symptoms).",
"Or if eigenvalues appear to be moving closer to the RHP in successive tests conducted over a period of time, then it may again indicate early development of parkinson's disease before manifestation of tremors.",
"Figure: Movement of the poles in the complex plane as a function of delay (Servo motor experimental example).To estimate the eigenvalues from motion of a body part, let's say an arm, standard methods such as Prony's method [24] and Matrix Pencil method [25] can be used.",
"Both these methods estimate eigenvalues of the system dynamics by extracting complex exponentials from time series data.",
"Earlier experience with these methods [26] suggest that the Matrix Pencil method is more robust than Prony's method.",
"Using data from the servo motor experimental example mentioned earlier with varying delays in the loop, and using the matrix pencil method, the estimated eigenvalues are shown in Figure REF , clearly showing the estimated eigenvalues moving to the RHP.",
"With real subjects, one could design a simple motor control task and record the motion data for the task performed.",
"From this data, once we find the eigenvalues using Matrix Pencil Method, if estimated eigenvalues appear to be moving closer to RHP over the tests conducted over a period of time than that would indicate a possibility of increase in the sensorimotor loop delay.",
"This in turn can be used for early diagnosis of PD patients even if rest tremors are not visible."
],
[
"Analysis of Location of Delay",
"Although we performed the analysis with the delay in the forward path (between the controller and plant) as indicated in Figure REF we note that the position of the delay in the sensorimotor loop doesn't affect the phase portrait and therefore our conclusions.",
"To understand this, consider the case in which we have an additional delay ($t_{d2}$ ) in the feedback path with delay ($t_{d1}$ ) in the forward path.",
"In this case, the output (ignoring the saturation block) becomes, $ Y(s)=e^{-t_{d1}s}{\\tilde{G}} R(S)$ , where $\\tilde{G}=\\frac{G(s)}{1+G(s) e^{-t_ds}}$ with $t_d=t_{d1}+t_{d2}$ .",
"Note here that the nature of the response is determined by $\\tilde{G}$ which has the total delay in its denominator, while $e^{-t_{d1}s}$ only serves to shift the output $Y(s)$ .",
"Therefore, in the phase space, the trajectories would only depend on the total combined delay in the sensorimotor loop and is unaffected by the actual positions of the delay elements."
],
[
"Conclusions",
"We started with the hypothesis that an increased sensorimotor loop delay is the cause of rest tremors in Parkinson's disease.",
"With this starting point, with the use of two simple experimental examples and a numerical example along with simple feedback control arguments, were able to explain several qualitative clinical observations related tor best tremors in PD.",
"Thus, this helps strengthen the hypothesis while laying a foundation for developing a better understanding of the mechanism behind rest tremors.",
"Further, using numerical and experimental examples, we further explored the possibility of using the tremor amplitude, frequency, area of limit cycle and aspect ratio of the limit cycle for tracking the progress of the disease and for diagnosis of the PD.",
"Further, we presented the possibility of developing a novel early diagnosis tool that exploited the fact that tremor is a instability-driven oscillation and hence use tendency to develop an instability (by tracking eigenvalues) as an early indicator for PD."
],
[
"Acknowledgment",
"The authors gratefully acknowledge the support received from IIT Gandhinagar and the Ministry of Human Resources in the form of the fellowships for first author.",
"Funding: From IIT Gandhinagar and the Ministry of Human Resources in the form of the fellowships for first author.",
"Conflicts of Interest: None declared Ethical Approval: Not required"
],
[
"Appendix A",
"Numerical Example: We use a simple pendulum as the plant and a simple PID law as the controller in the simulation example.",
"The simple pendulum has length L, mass m, and the damping coefficient c. The state-space form for the simple pendulum model linearized about the stable equilibrium is $\\begin{split}\\dot{x} \\hspace{5.69054pt} = & \\hspace{5.69054pt} Ax + Bu,\\\\y \\hspace{5.69054pt} = & \\hspace{5.69054pt} Cx + Du,\\\\\\end{split}$ where $x= [\\begin{matrix}\\theta & \\dot{\\theta }\\end{matrix}]$ $\\in R^{2}$ is the state vector with $\\theta $ being the angle of the pendulum, $u \\in R$ is the controlling torque on the pendulum as determined by the controller based on the feedback $y \\in R$ , which is the measured angular velocity of the pendulum, and $A=\\begin{bmatrix}0 & 1 \\\\-g/L & -c/mL^2\\end{bmatrix},B=\\begin{bmatrix}0 \\\\1/mL^2\\end{bmatrix},C=\\begin{bmatrix}0 & 1\\end{bmatrix}, D = 0.$ For our simulation purpose, we use $L$ = 0.65$m$ , $m$ = 3.5$Kg$ , $c$ =3.375$Kg.m/s$ , satoration limits of -100N-m and 100N-m and a Proportional-Integral-Derivative (PID) controller with the proportional gain $k_P = 15mL^2$ , integral gain $k_I = 4mL^2$ and derivative gain $k_D = 0.5mL^2$ .",
"The simulations are performed in Matlab SIMULINK.",
"Experimental Examples: We consider two motion control experimental examples, an angular position control experiment for a servo motor and a rotary inverted pendulum balancing experiment.",
"The former is a first-order, linear, stable system with a proportional-derivative controller and the latter is a fourth-order, non-linear, unstable system with LQR controller.",
"Thus, these two systems provide two very different types of systems.",
"With these motion-control experiments, we first construct and verify a stable closed-loop control system with Proportional-Derivative and LQR controllers, respectively, and then experimentally observe the effect of delay and saturation by replacing the dynamic of simple pendulum (in case of simulation example) with the dynamics of servo motor or rotary inverted pendulum as body dynamics (Figure 1).",
"Both of these experiments are based on a QUBE Rotary Servo Experiment from QUANSER as shown in Figure REF .",
"Figure: QUBE Rotary Servo Experiment setup.",
"(a) Position control experiment setup (left) and (b) Inverted pendulum setup (right).We note that since the focus of the paper is on qualitative observations relating to Parkinsonian tremor, the values of the parameters in the examples are indicative values and not related to parameters in physics human body."
]
] | 1403.0296 |
[
[
"Measurement of TeV atmospheric muon charge ratio with the full OPERA\n data"
],
[
"Abstract The OPERA detector, designed to search for $\\nu_{\\mu} \\to \\nu_{\\tau}$ oscillations in the CNGS beam, is located in the underground Gran Sasso laboratory, a privileged location to study TeV-scale cosmic rays.",
"For the analysis here presented, the detector was used to measure the atmospheric muon charge ratio in the TeV region.",
"OPERA collected charge-separated cosmic ray data between 2008 and 2012.",
"More than 3 million atmospheric muon events were detected and reconstructed, among which about 110000 multiple muon bundles.",
"The charge ratio $R_{\\mu} \\equiv N_{\\mu^+}/N_{\\mu^-}$ was measured separately for single and for multiple muon events.",
"The analysis exploited the inversion of the magnet polarity which was performed on purpose during the 2012 Run.",
"The combination of the two data sets with opposite magnet polarities allowed minimizing systematic uncertainties and reaching an accurate determination of the muon charge ratio.",
"Data were fitted to obtain relevant parameters on the composition of primary cosmic rays and the associated kaon production in the forward fragmentation region.",
"In the surface energy range 1-20 TeV investigated by OPERA, $R_{\\mu}$ is well described by a parametric model including only pion and kaon contributions to the muon flux, showing no significant contribution of the prompt component.",
"The energy independence supports the validity of Feynman scaling in the fragmentation region up to $200$ TeV/nucleon primary energy."
],
[
"Introduction",
"Underground experiments detect the penetrating remnants of primary cosmic ray interactions in the atmosphere, namely muons and neutrinos.",
"These are the decay products of charged mesons contained in the particle cascade, mainly pions and kaons.",
"At very high energies also charmed particles are expected to contribute.",
"The muon charge ratio $R_{\\mu } \\equiv N_{\\mu ^+}/N_{\\mu ^-}$ , defined as the number of positive over negative charged muons, is studied since many decades.",
"It provides an understanding of the mechanism of multiparticle production in the atmosphere in kinematic regions not accessible to accelerators, as well as information on the primary cosmic ray composition.",
"A charge ratio larger than unity reflects the abundance of protons over heavier nuclei in the primary cosmic radiation.",
"The charge asymmetry is preserved in the secondary hadron production, and consequently in the muon fluxes, due to the steepness of the primary spectrum which enhances the forward fragmentation region [1].",
"The kaon contribution to the muon flux increases with the muon energy.",
"Since the production of positive kaons is favoured by the associated production $\\Lambda K^+$ , the muon charge ratio is expected to rise with energy.",
"Assuming the hypothesis of complete scaling we expect an energy independent charge ratio above the TeV energy region at sea level [1] once the kaon contribution to the muon flux reached its asymptotic value [2].",
"At higher energies, around $O(100)$ TeV, the heavy flavor contribution, as well as changes in the primary composition, may become significant.",
"The OPERA experiment, described in detail in Ref.",
"[3], is a hybrid electronic detector/emulsion apparatus, located in the underground Gran Sasso laboratory, at an average depth of 3800 meters of water equivalent.",
"The main physics goal of the experiment is the first observation of neutrino oscillations in direct appearance mode in the $\\nu _{\\mu } \\rightarrow \\nu _{\\tau }$ channel [4], [5], [6].",
"OPERA already reported a first measurement of the atmospheric muon charge ratio at TeV surface energies using the 2008 Run data [7].",
"Here we present the final results obtained with the complete statistics.",
"OPERA continuously accumulated cosmic ray data with the electronic detectors of the target over the whole year from 2008 up to 2012.",
"However the magnetic spectrometers were active only during the CNGS Physics Runs, being switched off during the CNGS winter shutdowns.",
"As it was done in Ref.",
"[7], we used the momentum and charge reconstruction obtained via the Precision Trackers (PT) of the OPERA spectrometers [8].",
"Layers of vertical drift tubes are arranged in PT stations instrumenting the two identical dipole magnets.",
"The momentum and charge information is given by the angle $\\Delta \\phi $ in the bending plane, i.e.",
"the difference between the track directions reconstructed by the two PT stations before and after each magnet arm.",
"For nearly horizontal muons up to four bending angles can be measured in the two dipole magnets."
],
[
"Data Analysis",
"The cosmic ray data used for this analysis were collected during the five CNGS Physics Runs between 2008 and 2012.",
"In the first four years (2008-2011) the magnetic field was directed upward in the first arm of both dipoles and in the opposite direction in the second arm (standard polarity, SP).",
"In 2012 the coil currents were reversed and the spectrometer operated in inverted polarity (IP) mode.",
"A pre-selection was applied in order to select only stable conditions of detector operation.",
"Short periods with increased electronic noise or with any subdetector under test were removed, as well as periods in which the magnets were not in nominal conditions.",
"Details on atmospheric muon event selection, reconstruction and analysis can be found in Refs.",
"[7], [9].",
"The final SP data correspond to 625.0 live days, distributed among the Runs as shown in Table REF .",
"The final IP exposure is equivalent to 234.8 live days.",
"Table: Data sets with magnet configuration in standard polarity (SP) and inverted polarity (IP).",
"For each Runthe number of cosmic muon events, the number of muon bundles therein and the live time after the pre-selection of good quality running periods are reported.In the total SP + IP live time, 3044281 cosmic muon events were recorded.",
"Among them, 113662 are muon bundles, i.e.",
"events with a muon multiplicity $n_{\\mu }$ greater than 1.",
"To reconstruct the muon charge, the track has to cross at least one magnet arm yielding a measurement of the bending angle $\\Delta \\phi $ by the PT system.",
"This resulted in the reconstruction of momentum and charge for 650492 muons in SP ($28.7$ % of the total muon events) and 244626 muons in IP ($28.9$ % of the total muon events).",
"In order to improve the charge identification purity, the two selection criteria used in Ref.",
"[7] were applied to the data.",
"The first selection is a track quality cut.",
"The $\\Delta \\phi $ bending angle measurement is provided by the PT track reconstruction which is spoiled in events containing a large number of fired tubes, typically due to radiative processes.",
"When the number of PT hits is much larger than the number expected from geometrical considerations [7], [9] the event is rejected.",
"The second selection acts on the charge discrimination power.",
"Events with a bending angle smaller than 3 times the angular resolution were rejected.",
"This corresponds to a maximum detectable momentum up to 1 TeV/c [9].",
"A further cut was applied to remove a few events with very large deflections ($|\\Delta \\phi | > 100$ mrad), either due to the scattering of low momentum muons ($p_{\\mu } \\le 5$ GeV/c) or mimicked by secondary particles produced in high energy events.",
"Muons induced by atmospheric neutrinos coming from below were removed from the data set on the basis of time-of-flight measurements.",
"Contributions from muon backscattering or up-going charged particles induced by muons were computed according to Ref.",
"[10] and found to be negligible.",
"The numbers of single and multiple muons surviving all the selection cuts and used in the computation of the muon charge ratio are reported in Table REF .",
"Table: Final statistics for the muon charge ratio measurement; the number of muons surviving the cuts is quoted for bothmagnet polarity configurations.For muon bundles we provide the total number of muons and not the number of events."
],
[
"Systematic uncertainties and unbiased charge ratio",
"The comparison between the two data sets with opposite magnet polarity (SP and IP) allows checking systematic uncertainties affecting the muon charge ratio.",
"These can be cancelled out using a proper combination of the two data samples (see ).",
"The two main sources of systematic uncertainties are due to alignment and charge misidentification.",
"In principle a different acceptance for $\\mu ^{+}$ and $\\mu ^{-}$ could also contribute to the overall systematic uncertainty.",
"However the symmetry of the detector geometry allows to safely neglect this contribution.",
"An indirect confirmation is given by the compatibility of the charge ratio values computed separately in the two arms of the same magnet, where the magnetic field has opposite directions [9].",
"Using the SP and IP data sets, we checked the symmetry in the acceptance for each magnet arm.",
"According to the reference frame defined in Ref.",
"[7], where the $z$ -axis points toward the CNGS direction, muons travelling toward the positive $z$ -axis are defined as south-oriented (SO), while muons travelling toward the negative $z$ -axis are defined as north-oriented (NO).",
"A muon crosses a magnet arm in one of these two possible “orientations”.",
"South-oriented $\\mu ^{+}$ and north-oriented $\\mu ^{-}$ are deflected toward west in the first arm in SP mode.",
"The reversals of either the muon incoming orientation or the polarity mode are equivalent ways to exchange the muon bending sign.",
"We computed the ratio $A_i$ of the number $N_i$ of charge-reconstructed muons in SP mode to the number in IP mode (normalized by their polarity live time), $A_i = (N_i)_{SP}$ /$(N_i)_{IP}$ for the two orientations in each magnet arm $i$ .",
"The results are reported in Table REF .",
"The values of $A_i$ obtained in one orientation are all compatible with the values obtained in the other orientation, as expected from a charge-symmetric spectrometer.",
"The individual comparison between $A_i$ (SO) and $A_i$ (NO) for each arm disposes of possible small live time differences among PT stations.",
"The results are consistent with unity within statistical errors.",
"Table: Ratio between SP and IP numbers of charge-reconstructed muonsin each magnet arm.",
"The normalization by the relative polarity live time is globally applied.We have investigated the systematic uncertainty related to the alignment of the PT system.",
"The SP and IP bending angle distributions were compared separately for south- and north-oriented muons in each magnet arm.",
"In case of perfect alignment, the two distributions (normalized by their respective live times) would coincide.",
"In the data, a systematic bending angle shift $|\\delta \\phi _s| \\sim (0.10 \\pm 0.03)$ mrad was observed on average (in each magnet arm, for $ i = 1, \\dots ,4$ : $|\\delta \\phi _{s,i}| = \\lbrace <0.03, \\, 0.07, \\, 0.10, \\, 0.15\\rbrace $ mrad).",
"Inverting the muon orientation, $\\delta \\phi _{s,i}$ preserves the absolute value and flips the sign, as expected in case of misalignment.",
"Note that the absolute value is compatible with the alignment systematic uncertainty $\\delta \\phi _{syst} = 0.08$ mrad given in [7].",
"The observed global shift $\\delta \\phi _s$ is however an average value.",
"It is a cumulative result of local distortions, tilts and bendings which depend on the muon position, zenith and azimuth.",
"We therefore did not globally correct for $\\delta \\phi _s$ since the combination of IP and SP data allows to completely remove this systematics at a local level.",
"As detailed in the , the unbiased charge ratio $\\hat{R}_{\\mu }$ is obtained by the normalized sum of $\\mu ^{+}$ over the normalized sum of $\\mu ^{-}$ : $\\hat{R}_{\\mu } = \\frac{\\frac{N^+_{SP}}{l_{SP}} + \\frac{N^+_{IP}}{l_{IP}}}{\\frac{N^-_{SP}}{l_{SP}} + \\frac{N^-_{IP}}{l_{IP}}} = \\frac{R_{\\mu } (1 - \\eta ) + \\eta }{(1 - \\eta ) + \\eta R_{\\mu }}$ where $l_{SP,IP}$ is the respective polarity live time and $\\eta $ is the charge misidentification probability.",
"This combination provides a charge ratio in which the effects induced by misalignments cancel out.",
"Indeed, the last equation is exactly the relation between the reconstructed $\\hat{R}_{\\mu }$ and the true $R_{\\mu }$ charge ratio in case of perfect alignment [7].",
"Inverting this relation, the charge ratio $R_{\\mu }$ is obtained from the measured $\\hat{R}_{\\mu }$ corrected by the misidentification probability.",
"In principle, all the systematic contributions due to misalignment cancel with this combination of SP and IP data.",
"The residual systematic errors which do not cancel are estimated by the difference between the charge ratio values computed separately for SO and NO orientations.",
"Since the alignment bias has opposite sign in the two orientations, we take $|R_{\\mu }(NO) - R_{\\mu }(SO)|$ as the systematic uncertainty related to our combination procedure.",
"It was found $\\delta R_{\\mu } = 0.001$ for single muon events and $\\delta R_{\\mu } = 0.013$ for multiple muon events.",
"In the latter the statistical contribution is dominant.",
"The second source of systematic uncertainty considered is related to the determination of $\\eta $ .",
"The charge misidentification computed with Monte Carlo is $\\eta _{\\footnotesize {\\textrm {MC}}} = 0.030$ , nearly independent on the muon momentum in the range 5 GeV/c $< \\over {_{\\sim }}p_{\\mu } < \\over {_{\\sim }}$ 1 TeV/c [9].",
"We estimated the systematic uncertainty of $\\eta $ using a subsample of experimental data, i.e.",
"the muon tracks reconstructed in both arms of each spectrometer.",
"The probability of wrong charge assignment was evaluated counting the fraction of tracks with different charges, and the experimental $\\eta _{\\footnotesize {\\textrm {data}}}$ was derived.",
"The difference between $\\eta _{\\footnotesize {\\textrm {data}}}$ and $\\eta _{\\footnotesize {\\textrm {MC}}}$ is $\\delta \\eta = 0.007 \\pm 0.002$ [7].",
"This corresponds to a one-sided systematic uncertainty on the charge ratio $\\delta R_{\\mu } = 0.007$ .",
"The final systematic uncertainty is the quadratic sum of the misalignment and the misidentification contributions."
],
[
"Results",
"The charge ratio of single muons impinging on the apparatus was computed combining the two polarity data sets according to Eq.",
"REF .",
"After the correction for charge misidentification and detector misalignment, the final measurement with the complete 5-year statistics yields the result: $R_{\\mu } (n_{\\mu } = 1) = 1.377 \\pm 0.006 (stat.)",
"^{+0.007}_{-0.001} (syst.)",
"$ The charge ratio of multiple muon events was computed using all the muon charges reconstructed in events with $n_{\\mu } > 1$ .",
"It is not computed within the bundle itself, but summing up all the positive and the negative charges belonging to the bundle subsample.",
"The result after polarity combination and correction for misidentification is significantly lower than the single muon value: $R_{\\mu } (n_{\\mu } > 1) = 1.098 \\pm 0.023 (stat.)",
"^{+0.015}_{-0.013} (syst.)",
"$ The smaller value of the charge ratio for multiple muon events originates from two effects.",
"First, as pointed out in [7], the multiple muon sample naturally selects heavier primaries, thus a neutron enriched primary beam ($\\langle A \\rangle \\simeq 3.4$ for single muons, $\\langle A \\rangle \\simeq 8.5$ for bundles).",
"Second, the selection of muon bundles biases the Feynman-$x$ distribution towards the central region ($x_F \\simeq E_{\\footnotesize {\\textrm {secondary}}}/E_{\\footnotesize {\\textrm {primary}}} \\rightarrow 0$ ), in which the sea quark contribution to secondary particle production becomes relevant [9].",
"Both processes cause a decrease in the charge ratio.",
"The single muon charge ratio was projected at the Earth surface using a Monte Carlo based unfolding technique for the muon energy $\\mathcal {E}_{\\mu }$ [9].",
"As a first attempt, only pion and kaon contributions to the total muon flux are considered.",
"We used the analytic approximation described in [7] to infer the fractions of charged mesons decaying into a positive muon, $f_{\\pi ^+}$ and $f_{K^+}$ .",
"This approach does not yet consider any energy dependence of the proton excess in the primary composition.",
"In this case the muon flux and charge ratio depend on the vertical surface energy $\\mathcal {E}_{\\mu } \\cos \\theta ^*$ , where $\\theta ^*$ is the zenith angle at the muon production point [11].",
"$R_{\\mu }$ is computed as a function of the vertical surface muon energy, binned according to the energy resolution, which is of the order of d$(\\log _{10} \\mathcal {E}_{\\mu }/\\textrm {GeV}) \\simeq 0.15$ in a logarithmic scale [9].",
"In each bin the two polarity data sets are combined and the obtained value is corrected for the charge misidentification.",
"The two contributions to the systematic uncertainty are computed and added in quadrature.",
"The results are shown in Fig.",
"REF , together with data from other experiments (L3+C [15], MINOS Near and Far Detectors [16], [17], CMS [18] and Utah [19]).",
"The information for each of the four $\\mathcal {E}_{\\mu } \\cos \\theta ^*$ bins are presented in Table REF : the energy range, the most probable value of the energy distribution in the bin, the average zenith angle, the charge ratio $R_{\\mu }$ , the statistical and systematic uncertainties.",
"Table: The charge ratio in bins of ℰ μ cosθ * \\mathcal {E}_{\\mu } \\cos \\theta ^*.",
"Here reported are the energy bin range, the most probable value of the energy distribution in the bin (MPV, evaluated using the full Monte Carlo simulation described in ), the average zenith angle,the charge ratio and the statistical and systematic uncertainties.Following the procedure described in [7], we fitted our data and those from [15] (for the high and low energy regions) in order to infer the fractions $f_{\\pi ^+}$ and $f_{K^+}$ .",
"In this approach, the atmospheric charged kaon/pion production ratio $R_{K/ \\pi }$ had to be fixed.",
"For this, we took the weighted average of experimental values reviewed in [20], $R_{K/ \\pi } = 0.127$ .",
"The fit yields $f_{\\pi ^+} = 0.5512 \\pm 0.0014$ and $f_{K^+} = 0.705 \\pm 0.014$ , corresponding to a muon charge ratio from pion decay $R_{\\pi } = 1.2281 \\pm 0.0007$ and a muon charge ratio from kaon decay $R_{K} = 2.39 \\pm 0.07$ .",
"Figure: The muon charge ratio measured by OPERA as a function of the vertical surface energy ℰ μ cosθ * \\mathcal {E}_{\\mu } \\cos \\theta ^* (black points).Our data are fitted together with the L3+C data (open triangles).The fit result is shown by the continuous line.",
"The dashed, dotted and dash-dotlines are, respectively, the fit results with the inclusion of the RQPM , QGSM and VFGS models for prompt muon production in the atmosphere.",
"The vertical inner bars denote the statistical uncertainty, the full bars show the total uncertainty.Results from other experiments, MINOS Near and Far Detectors , , CMS and Utah , are shown for comparison.Taking into account various models for charm production, namely RQPM [21], QGSM [21] and VFGS [22], the positive pion and kaon fractions obtained from the fit are unchanged within statistical errors.",
"The results are shown in Fig.",
"REF .",
"The prompt muon component does not significantly contribute to $R_{\\mu }$ up to $\\mathcal {E}_{\\mu } \\cos \\theta ^* < \\over {_{\\sim }}10$ TeV.",
"Recently, an enlightening analytic description of the muon charge ratio considering an explicit dependence on the relative proton excess in the primary cosmic rays, $\\delta _0 = (p - n)/(p + n)$ , was presented in [2]: $R_{\\mu } = \\left[ \\frac{f_{\\pi ^+}}{1 + B_{\\pi } \\mathcal {E}_{\\mu } \\cos \\theta ^*/\\epsilon _{\\pi }} + \\frac{\\frac{1}{2} (1+ \\alpha _K \\beta \\delta _0)A_{K}/A_{\\pi }}{1 + B^+_{K} \\mathcal {E}_{\\mu } \\cos \\theta ^*/\\epsilon _{K}} \\right] \\\\ \\nonumber \\times \\left[ \\frac{1- f_{\\pi ^+}}{1 + B_{\\pi } \\mathcal {E}_{\\mu } \\cos \\theta ^*/\\epsilon _{\\pi }} + \\frac{ (Z_{NK^-}/Z_{NK}) A_{K}/A_{\\pi }}{1 + B_{K} \\mathcal {E}_{\\mu } \\cos \\theta ^*/\\epsilon _{K}} \\right]^{-1}$ Here $p$ and $n$ fluxes are defined as $p = \\sum _i Z_i \\, \\Phi _i (E_N); \\quad n = \\sum _i (A_i - Z_i) \\, \\Phi _i (E_N)$ where the index $i$ runs over the primary ions (H, He, CNO, Mg-Si, Fe) and $E_N$ is the primary nucleon energy.",
"The contributions from decays of pions and kaons are included in the kinematic factors $A_i, B_i, \\epsilon _i \\, (i = \\pi , K)$ described in [2], [11].",
"An analogous contribution from charm decay is foreseen at high energies but still not observed.",
"The spectrum weighted moments $Z_{ij}$ [2] are contained in $\\beta $ and $\\alpha _K$ : $\\beta = \\frac{1- Z_{pp} - Z_{pn}}{1- Z_{pp} + Z_{pn}}; \\quad \\alpha _{K} = \\frac{Z_{pK^+} - Z_{pK^-}}{Z_{pK^+} + Z_{pK^-}}$ Isospin symmetry allows expressing the pion contribution in terms of $f_{\\pi ^+}$ , where $f_{\\pi ^+} = \\frac{1+\\beta \\delta _0 \\alpha _{\\pi }}{2}$ Here $\\alpha _{\\pi }$ is obtained replacing the subscript $K$ with the subscript $\\pi $ in $\\alpha _K$ .",
"We extracted from the data the composition parameter $\\delta _0$ and the factor $Z_{pK^+}$ related to the associated production $\\Lambda \\, K^+$ in the forward region.",
"The $Z_{pK^+}$ moment is still poorly known and its predicted value considerably differs for different Monte Carlo codes [12], [13].",
"In Eq.",
"REF the charge ratio does not exclusively depend on the vertical surface energy.",
"Since the spectra of primary nuclei have different spectral indices, the parameter $\\delta _0$ depends on the primary nucleon energy $E_N$ .",
"In the energy range of interest the approximation $E_N \\simeq 10 \\times \\mathcal {E}_{\\mu }$ can be used [2].",
"The correct way of taking into account the different dependencies is to simultaneously fit Eq.",
"REF as a function of the two variables $(\\mathcal {E}_{\\mu }, \\cos \\theta ^*)$ .",
"In each $(\\mathcal {E}_{\\mu }, \\cos \\theta ^*)$ bin the data sets with opposite polarities are combined and $\\hat{R}_{\\mu }$ is corrected for the charge misidentification.",
"The pion moments $Z_{p \\pi ^+}$ and $Z_{p \\pi ^-}$ were set to the values reported in [2], since the fraction of positive pions in the atmosphere $f_{\\pi ^+} = 0.5512 \\pm 0.0014$ derived in this work is robust and consistent with previous measurements [16], [17] and with the $Z_{N\\pi }$ values based on fixed target data [14].",
"The moment $Z_{pK^-}$ was also set to the value given in [2], since for $K^-$ there is no counterpart of the associated production $\\Lambda \\, K^+$ .",
"On the other hand $K^-$ are equally produced in $K^+ K^-$ pairs by protons and neutrons ($Z_{pK^-} \\simeq Z_{nK^-}$ ).",
"A linear energy dependence in logarithmic scale of the parameter $\\delta _0$ was assumed, $\\delta _0 = a + b \\log _{10} (E_{N}$ /GeV/nu-cleon$)$ , as suggested by direct measurements of the primary composition and by the Polygonato model [23].",
"We fixed the slope at $b = -0.035$ which was obtained fitting the values reported in [2].",
"We made a two-dimensional fit of OPERA and L3+C data as a function of $(\\mathcal {E}_{\\mu }, \\cos \\theta ^*)$ to Eq.",
"REF with $\\delta _0$ and $Z_{pK^+}$ as free parameters.",
"The fit yields the composition parameter at the average energy measured by OPERA $\\langle \\mathcal {E}_{\\mu } \\rangle = 2$ TeV (corresponding to $\\langle E_{N} \\rangle \\approx 20$ TeV/nucleon) $\\delta _0 (\\langle \\mathcal {E}_{\\mu } \\rangle ) = 0.61 \\pm 0.02$ and the factor $Z_{p K^+} = 0.0086 \\pm 0.0004$ .",
"The result of the fit in two variables $(\\mathcal {E}_{\\mu }, \\cos \\theta ^*)$ is projected on the average OPERA zenith $\\langle \\cos \\theta ^* \\rangle \\simeq 0.7$ and is shown in Fig.",
"REF together with the measured charge ratio as a function of the surface muon energy.",
"The energy independence of the charge ratio above the TeV supports the validity of the Feynman scaling in the fragmentation region.",
"Figure: Our measurement of the muon charge ratio as a function of the surface energy ℰ μ \\mathcal {E}_{\\mu } (black points).The two-dimensional fit in (ℰ μ ,cosθ * )(\\mathcal {E}_{\\mu }, \\cos \\theta ^*) yields a measurement of the composition parameter δ 0 \\delta _0 and of the factor Z pK + Z_{p K^+}.The fit result is projected on the average OPERA zenith 〈cosθ * 〉≃0.7\\langle \\cos \\theta ^* \\rangle \\simeq 0.7 and shown by the continuous line.Results from other experiments, L3+C (only for 0.675<cosθ<0.750.675 < \\cos \\theta < 0.75) , MINOS Near and Far Detectors , , CMS and Utah ,are also shown for comparison."
],
[
"Conclusions",
"The atmospheric muon charge ratio $R_{\\mu }$ was measured with the complete statistics accumulated along the five years of data taking.",
"The combination of the two data sets collected with opposite magnet polarities allows reaching the most accurate measurement in the high energy region to date.",
"The underground charge ratio was evaluated separately for single and for multiple muon events.",
"For single muons, the integrated $R_{\\mu }$ value is $R_{\\mu } (n_{\\mu } = 1) = 1.377 \\pm 0.006 (stat.)",
"^{+0.007}_{-0.001} (syst.",
")$ while for muon bundles $R_{\\mu } (n_{\\mu } > 1) = 1.098 \\pm 0.023 (stat.)",
"^{+0.015}_{-0.013} (syst.",
")$ The integral value and the energy dependence of the charge ratio for single muons are compatible with the expectation from a simple model [2], [14] which takes into account only pion and kaon contributions to the atmospheric muon flux.",
"We extracted the fractions of charged pions and kaons decaying into positive muons, $f_{\\pi ^+} = 0.5512 \\pm 0.0014$ and $f_{K^+} = 0.705 \\pm 0.014$ .",
"Considering the composition dependence embedded in Eq.",
"REF , we inferred a proton excess in the primary cosmic rays $\\delta _0 = 0.61 \\pm 0.02$ at the energy $\\langle E_{N} \\rangle \\approx 20$ TeV/nucleon and a spectrum weighted moment $Z_{p K^+} = 0.0086 \\pm 0.0004$ .",
"The observed behaviour of $R_{\\mu }$ as a function of the surface energy from $\\sim 1$ TeV up to 20 TeV (about 200 TeV/nu-cleon for the primary particle) shows no deviations from a simple parametric model taking into account only pions and kaons as muon parents, supporting the hypothesis of limiting fragmentation up to primary energies/nucleon around 200 TeV.",
"We thank CERN for the successful operation of the CNGS facility and INFN for the continuous support given to the experiment during the construction, installation and commissioning phases through its LNGS laboratory.",
"We warmly acknowledge funding from our national agencies: Fonds de la Recherche Scientifique-FNRS and Institut InterUniversitaire des Sciences Nucléaires for Belgium, MoSES for Croatia, CNRS and IN2P3 for France, BMBF for Germany, INFN for Italy, JSPS (Japan Society for the Promotion of Science), MEXT (Ministry of Education, Culture, Sports, Science and Technology), QFPU (Global COE programme of Nagoya University, Quest for Fundamental Principles in the Universe supported by JSPS and MEXT) and Promotion and Mutual Aid Corporation for Private Schools of Japan for Japan, SNF and the University of Bern for Switzerland, the Russian Foundation for Basic Research (grant no.",
"09-02-00300 a, 12-02-12142 ofim), the Programs of the Presidium of the Russian Academy of Sciences Neutrino physics and Experimental and theoretical research-es of fundamental interactions connected with work on the accelerator of CERN, the Programs of Support of Leading Schools (grant no.",
"3517.2010.2), and the Ministry of Education and Science of the Russian Federation for Russia, the National Research Foundation of Korea Grant No.",
"2011-0029457 for Korea and TUBITAK, the Scientific and Technological Research Council of Turkey, for Turkey.",
"We are also indebted to INFN for providing fellowships and grants to non-Italian researchers.",
"We thank the IN2P3 Computing Centre (CC-IN2P3) for providing computing resources for the analysis and hosting the central database for the OPERA experiment.",
"We are indebted to our technical collaborators for the excellent quality of their work over many years of design, prototyping and construction of the detector and of its facilities."
],
[
"Combination of data sets",
"A systematic shift of the bending angle distribution biases the integral value of the muon charge ratio.",
"Moreover, since the unfolding of the surface muon energy is based on the underground muon momentum, a curvature bias has an important effect on the bin-to-bin migration matrix, i.e.",
"the probability of measuring a surface energy $\\mathcal {E}_{i}$ at a true energy $\\mathcal {E}_{j}$ .",
"Due to misalignment there are in principle two different migration matrices $U^+$ and $U^-$ for each magnet polarity.",
"Given the symmetry of the detector, the exchange of the magnet polarity is equivalent to the exchange of the charge sign (see Sect.",
"REF ), thus $U^+_{SP} = U^-_{IP}$ and coherently $U^+_{IP} = U^-_{SP}$ .",
"In general, with a curvature bias that shifts the bending angle distribution, a different charge misidentification $\\eta ^+$ and $\\eta ^-$ for positive and negative muons is expected for both standard and inverted magnet polarity.",
"Given the symmetry of the detector, the relations $\\eta ^+_{SP} = \\eta ^-_{IP}$ and $\\eta ^-_{SP} = \\eta ^+_{IP}$ are valid.",
"However we verified that after the application of the second selection criterion (the bending angle cut) the charge misidentification $\\eta $ is insensitive to the charge sign.",
"We applied a rigid curvature bias $\\delta \\phi _s$ and observed that the bin construction clearly separates positive and negative bins.",
"Therefore a symmetric misidentification $\\eta = \\eta ^+ = \\eta ^-$ is assumed.",
"Each energy bin content $N^{\\pm }$ is the integral of the true charged muon flux $\\Phi _{\\mu }$ convolved with the migration matrix $U$ and corrected for the charge misidentification.",
"For the standard polarity SP we have: $N^+_{SP} & = & \\int _{E_1}^{E_2} dE^{^{\\prime }} \\int _{-\\infty }^{+\\infty } [ U^+_{SP} (E,E^{^{\\prime }}) \\Phi _{\\mu }^+ (E) (1 - \\eta ) + \\nonumber \\\\& & + U^-_{SP}(E,E^{^{\\prime }}) \\Phi _{\\mu }^- (E) \\eta ] \\, dE$ $N^-_{SP} & = & \\int _{E_1}^{E_2} dE^{^{\\prime }} \\int _{-\\infty }^{+\\infty } [U^-_{SP} (E,E^{^{\\prime }}) \\Phi _{\\mu }^- (E) (1 - \\eta ) + \\nonumber \\\\& & + U^+_{SP}(E,E^{^{\\prime }}) \\Phi _{\\mu }^+ (E) \\eta ] \\, dE$ where $E_1, E_2$ are the lower and upper bounds of the reconstructed energy bin.",
"The positive flux contribution can be rewritten in terms of the true charge ratio $R_{\\mu }$ and the negative flux: $\\Phi _{\\mu }^+ (E) = R_{\\mu } \\, \\Phi _{\\mu }^- (E)$ Writing the same equations for the inverted polarity IP, the symmetries described above are taken into account: $N^+_{IP} & = & \\int _{E_1}^{E_2} dE^{^{\\prime }} \\int _{-\\infty }^{+\\infty } [ U^-_{SP} (E,E^{^{\\prime }}) R_{\\mu } \\, \\Phi _{\\mu }^- (E) (1 - \\eta ) + \\nonumber \\\\& & + U^+_{SP}(E,E^{^{\\prime }}) \\Phi _{\\mu }^- (E) \\eta ] \\, dE$ $N^-_{IP} & = & \\int _{E_1}^{E_2} dE^{^{\\prime }} \\int _{-\\infty }^{+\\infty } [ U^+_{SP} (E,E^{^{\\prime }}) \\Phi _{\\mu }^- (E) (1 - \\eta ) + \\nonumber \\\\& & + U^-_{SP}(E,E^{^{\\prime }}) R_{\\mu } \\, \\Phi _{\\mu }^- (E) \\eta ] \\, dE$ Thanks to the symmetric detector setup, the data combination able to cancel the misalignment systematic errors is the ratio $(N_{SP}^+ + N_{IP}^+)/(N_{SP}^- + N_{IP}^-)$ , where the numbers are normalized by the respective polarity live times.",
"Indeed, writing the integrands only, we obtain: $N^+_{SP} + N^+_{IP} = (U^+_{SP}+ U^-_{SP}) (R_{\\mu } \\, \\Phi _{\\mu }^- (1 - \\eta ) + \\Phi _{\\mu }^- \\eta )$ $N^-_{SP} + N^-_{IP} = (U^-_{SP} + U^+_{SP}) (\\Phi _{\\mu }^- (1 - \\eta ) + R_{\\mu } \\, \\Phi _{\\mu }^- \\eta )$ Thus the unbiased charge ratio is given by the normalized sum of $\\mu ^{+}$ over the normalized sum of $\\mu ^{-}$ : $\\hat{R}_{\\mu } = \\frac{\\frac{N^+_{SP}}{l_{SP}} + \\frac{N^+_{IP}}{l_{IP}}}{\\frac{N^-_{SP}}{l_{SP}} + \\frac{N^-_{IP}}{l_{IP}}} = \\frac{R_{\\mu } (1 - \\eta ) + \\eta }{(1 - \\eta ) + \\eta R_{\\mu }}$ The last equation is exactly the relation between the reconstructed $\\hat{R}_{\\mu }$ and true $R_{\\mu }$ charge ratio in case of perfect alignment [7]."
]
] | 1403.0244 |
[
[
"Quantum networks: Anti-core of spin chains"
],
[
"Abstract The purpose of this paper is to exhibit a quantum network phenomenon - the anti-core---that goes against the classical network concept of congestion core.",
"Classical networks idealized as infinite, Gromov hyperbolic spaces with least-cost path routing (and subject to a technical condition on the Gromov boundary) have a congestion core, defined as a subnetwork that routing paths have a high probability of visiting.",
"Here, we consider quantum networks, more specifically spin chains, define the so-called maximum excitation transfer probability $p_{\\max}(i,j)$ between spin $i$ and spin $j$, and show that the central spin has among all other spins the lowest probability of being excited or transmitting its excitation.",
"The anti-core is singled out by analytical formulas for $p_{\\mathrm{max}}(i,j)$, revealing the number theoretic properties of quantum chains.",
"By engineering the chain, we further show that this probability can be made vanishingly small."
],
[
"Introduction",
"Probably the most significant result of the Gromov analysis of classical networks [4], [6] is existence of a congestion core.",
"Under a network protocol that sends the packets along least cost paths, the core can be qualitatively defined as a point where most of the geodesics (least cost paths) converge, creating packet drops, high retransmission rates, and other nuisances under the TCP-IP protocol [10].",
"Existence of the core has been experimentally observed [11] and mathematically proved [8] if the network is Gromov hyperbolic, subject to some highly technical conditions related to the Gromov boundary [2].",
"A Gromov hyperbolic network can intuitively be defined as a network that “looks like” a negatively curved Riemannian manifold (e.g., a saddle) when viewed from a distance.",
"See, e.g., [3] for a precise definition.",
"Next to classical networks, one can envision quantum networks: the nodes are spins that can be up $\\left|\\uparrow \\right\\rangle $ (not excited) or down $\\left|\\downarrow \\right\\rangle $ (excited) and the links are quantum mechanical couplings of the XX or Heisenberg type.",
"Given some random source-destination pair $(i,j)$ , a valid question is whether some spin $\\omega $ could act as a “core,” that is, a spin that could be excited no matter what the source and the destination are.",
"For a linear chain, one would expect such a congestion core in the center as classically any excitation in one half of the chain would have to transit the center of the chain to reach the other half.",
"In this work, we demonstrate that quantum-mechanically the transmission of excitations does not need to occur this way, and in fact the center $\\omega $ of an odd-length spin chain can act as an “anti-core,” excitation of which is avoided.",
"This “anti-core,\" or “anti-gravity” center as it was originally called, was first observed in [7].",
"The anti-core $\\omega $ was defined as a point of high inertia, $\\sum _id^\\alpha (i,\\omega )$ , $\\alpha \\ge 1$ , as opposed to the classical congestion core that has minimum inertia owing to the negative curvature of the underlying space [9].",
"The inertia quantifies how difficult communication to and from the anti-core is.",
"As it has been done along this line of work, a pre-metric $d(\\cdot ,\\cdot )$ based on the Information Transfer Capacity (ITC) (see Sec. )",
"is employed.",
"Unlike standard quantum mechanical distances [13], [15], [12], this “distance\" measure aims to quantify not how distant two fixed quantum states are, but how close to a desired target state a quantum state can get under the evolution of a particular Hamiltonian.",
"The initial and target states are typically orthogonal.",
"In this paper, we provide an analytical justification of the numerically observed anti-core phenomenon in spin chains with XX coupling, starting with finite-length chains, extending the ITC concept to semi- and bi-infinite cases (Sec.",
"), and finally proving that $d(\\omega ,j) \\ge d(i,j)$ , $\\forall j \\ne \\omega $ in Sec. .",
"We further show that by adding a bias on the central spin its “anti-core” property can be made stronger in the sense that the probability of transmission of the excitation to and from it is infinitesimally small (Sec. ).",
"The remaining nagging question is why was it observed in [7] that spin chains appear Gromov-hyperbolic and have an anti-core, while classical networks are Gromov hyperbolic with the opposite core?",
"This will be clarified in Sec.",
"by means of a spin chain example, showing that its Gromov boundary has only one point, while classical networks need to have at least two points in their Gromov boundary for the core to emerge."
],
[
"Metrization of homogeneous spin chains",
"We consider a linear array of two-level systems (spin $\\tfrac{1}{2}$ particles) with uniform coupling between adjacent spins (homogeneous spin chain) made up of an odd number $N$ of physically equally spaced spins with coupling Hamiltonian $H = \\sum _{i=1}^{N-1}\\left(\\sigma ^x_i\\sigma ^x_{i+1} + \\sigma ^y_i\\sigma ^y_{i+1}+ \\epsilon \\sigma ^z_i\\sigma ^z_{i+1} \\right).$ Here we shall be primarily interested in the case of XX coupling, for which $\\epsilon =0$ .",
"The factor $\\sigma ^{x,y,z}_i$ is the Pauli matrix along the $x,y,$ or $z$ direction of spin $i$ in the array, i.e., $\\sigma ^{x,y,z}_i =I_{2\\times 2} \\otimes \\ldots \\otimes I_{2 \\times 2} \\otimes \\sigma ^{x,y,z}\\otimes I_{2\\times 2}\\otimes \\ldots \\otimes I_{2 \\times 2},$ where the factor $\\sigma ^{x,y,z}$ occupies the $i$ th position among the $N$ factors and $\\sigma ^{x,y,z}$ is one of the single spin Pauli operators $ \\sigma ^x= \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\quad \\sigma ^y= \\begin{pmatrix} 0 & -\\imath \\\\ \\imath & 0 \\end{pmatrix}, \\quad \\sigma ^z= \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}.$ It is easily seen that $H$ is real and symmetric."
],
[
"Single excitation subspace ",
"The $2^N \\times 2^N$ Hamiltonian commutes with the operator $S=\\sum _{i=1}^N \\sigma _i^{z}$ which counts the total number of excitations.",
"The Hilbert space can therefore be decomposed into subspaces corresponding to the number of excitations.",
"Define $|i\\rangle =|\\uparrow \\cdots \\uparrow \\downarrow \\uparrow \\cdots \\uparrow \\rangle $ to be the quantum state in which the excitation is on spin $i$ .",
"The single excitation subspace $\\mathcal {H}_1$ is spanned by $\\lbrace |i\\rangle :i=1,\\ldots ,N\\rbrace $ .",
"Restricted to this subspace, the Hamiltonian in this natural basis takes the form $H_1 = \\begin{pmatrix}\\epsilon & 1 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 0 \\\\1 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\0 & 0 & \\ldots & 0 & 1 & 0 & \\ldots & 0 & 0 \\\\0 & 0 & \\ldots & 1 & 0 & 1 & \\ldots & 0 & 0 \\\\0 & 0 & \\ldots & 0 & 1 & 0 & \\ldots & 0 & 0\\\\\\vdots & \\vdots & & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 1 \\\\0 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 1 & \\epsilon \\end{pmatrix}.$ For XX coupling ($\\epsilon =0$ ), $H_1$ becomes the $N \\times N$ Toeplitz matrix $T_N$ made up of zeros on the diagonal, ones on the super- and subdiagonal and zeros everywhere else.",
"Table REF shows the eigenvalues and eigenvectors of $H_1$ .",
"Table: Eigenvalues and eigenvectors of Single Excitation SubspaceHamiltonian H 1 H_1 under XX coupling ."
],
[
"Information Transfer Capacity (ITC) semi-metric",
"The probability for the system to transfer from state $|i\\rangle $ to state $|j\\rangle $ in an amount of time $t$ , that is, the probability of transfer of the excitation from spin $i$ to spin $j$ in an amount of time $t$ , is $p_t(i,j)=|\\langle i|e^{-\\imath H_1 t}|j\\rangle |^2.$ Observe that $\\sum _{j=1}^Np_t(i,j)=1$ .",
"In order to remove the dependency of the probability distribution on the time, we proceed as in Refs [5], [7]: $p_t(i,j)&= \\left|\\sum _{k=1}^N e^{-\\imath \\lambda _k t}\\langle i|v_k\\rangle \\langle v_k|j\\rangle \\right|^2 \\\\&\\le \\left|\\sum _{k=1}^N |\\langle i|v_k\\rangle \\langle v_k|j\\rangle |\\right|^2 =:p_{\\mathrm {max}}(i,j).$ We refer to $p_{\\mathrm {max}}(i,j)$ as maximum transfer probability from $|i\\rangle $ to $|j\\rangle $ or Information Transfer Capacity (ITC) between $|i\\rangle $ and $|j\\rangle $ .",
"Its explicit formulation for XX chains is easily obtained from (REF ) and Table REF : $\\sqrt{p_{\\mathrm {max}}(i,j)}=\\frac{2}{N+1}\\sum _{k=1}^{2n+1}\\left|\\sin \\frac{\\pi k i}{2(n+1)}\\right|\\left|\\sin \\frac{\\pi k j}{2(n+1)}\\right|.$ Lemma 1 $p_{\\mathrm {max}}(i,j)\\le 1$ and $p_{\\mathrm {max}}(i,i)=1$ for all $i,j=1,\\ldots ,N$ .",
"Proof.",
"$p_{\\max }(i,i)=1$ follows directly from (REF a), setting $i=j$ and $t=0$ and noting that the eigenvectors $|v_m\\rangle $ form an orthonormal basis.",
"$p_{\\rm max}(i,j)\\le 1$ then follows from a Cauchy-Schwartz argument.",
"$\\blacksquare $ The preceding lemma tells us that in order to define a (pre)metric from $p_{\\mathrm {max}}(i,j)$ , it is legitimate to take the $\\log $ and define $d(i,j):=-\\log p_{\\mathrm {max}}(i,j)$ on the single excitation subspace of the chain.",
"From Lemma REF , $d(i,i)=0$ and $d(i,j) \\ge 0$ , and clearly $d(i,j)=d(j,i)$ .",
"Observe, however, that $d(i,j)$ can vanish for $i\\ne j$ and the triangle inequality need not be satisfied, so that $d(i,j)$ is just a pre-metric, but this will be sufficient for our purposes.",
"The definition of $d(i,j)$ bears some commonality with sensor networks [1], where the Packet Reception Rate $\\mathrm {PRR}(i,j)$ from sensor $i$ to sensor $j$ —that is, the probability of successful transmission of packets from $i$ to $j$ —defines a premetric $d(i,j)=-\\log \\mathrm {PRR}(i,j)$ ."
],
[
"Infinite chains",
"In this section, we develop some asymptotic formulas for $\\sqrt{p_{\\mathrm {max}}(i,j)}$ for infinite-length chains in order to show that the central spin $n+1$ of a chain of odd length $N=2n+1$ has the lowest probability of being excited, hence justifying the terminology of “anti-core,” even for $N \\rightarrow \\infty $ .",
"This will further reveal a classical-quantum discrepancy: Classical dynamical systems interconnected in an homogeneous infinite chain architecture exhibit the so-called shift-invariance, that is, those dynamical interactions depending on the positions $i$ and $j$ of two systems in the chain in fact depend only on the distance $|i-j|$ .",
"As a corollary of the asymptotic formulas, this well known shift-invariance does not carry over to the quantum chains—no matter how the chain is extended to infinity, two spins in their transfer probability interaction keep properties specific to some number theoretic properties of their positions $i$ and $j$ .",
"Moreover, in a classical chain, the interaction at infinity is insensitive to the way the limit is taken: either the chain starts at a specific system, say 1, and extends to infinity as $(1,2,3,...), \\quad (\\mbox{``semi-infinite chain,'' written }\\rightarrow )$ or the chain starts at its center $\\omega $ and extends both ways as $(...,\\omega -2,\\omega -1,\\omega ,\\omega + 1, \\omega +2,...),\\quad \\mbox{(``doubly-infinite chain'' written} \\leftrightarrow ).$ It is another quantum mechanical effect that the two infinite chains do not yield the same asymptotic transfer probabilities."
],
[
"Semi-infinite chains",
"Theorem 1 For a semi-infinite XX chain, the maximum transition probabilities are given by $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}&= \\frac{4}{\\pi ^2}\\left( 2+\\sum _{m=2,4,...} \\frac{4}{(m^2{\\bf j}^2-1)(m^2{\\bf i}^2-1)} \\right)\\\\&= \\frac{8}{\\pi ^2} \\left(\\frac{\\bf {i}^2}{\\bf {i}^2-\\bf {j}^2} \\left(\\frac{\\pi }{2\\bf {i}} \\right)\\cot \\left(\\frac{\\pi }{2\\bf {i}} \\right)-\\frac{\\bf {j}^2}{\\bf {i}^2-\\bf {j}^2} \\left(\\frac{\\pi }{2\\bf {j}}\\right)\\cot \\left(\\frac{\\pi }{2\\bf {j}} \\right)\\right),$ where ${\\bf {i}}=i/\\mathrm {gcd}(i,j)$ and ${\\bf {j}}=j/\\mathrm {gcd}(i,j)$ , and $\\mathrm {gcd}(i,j)$ denotes the greatest common divisor of $i$ and $j$ .",
"Proof.",
"The proof is in Appendix .",
"$\\blacksquare $ Lemma REF provided some “probability” interpretations of $p_{\\max }(i,j)$ for finite chains.",
"We show that the same interpretation holds for infinite chains.",
"Lemma 2 $p_{\\mathrm {max}}^{\\rightarrow }(i,i)=1$ and $p_{\\mathrm {max}}^{\\rightarrow }(i,j)<1$ for $i \\ne j$ .",
"Proof.",
"For $i=j$ , $\\mathrm {gcd}(i,j)=i=j$ , so that ${\\bf i}={\\bf j}=1$ and it remains to show that $\\frac{4}{\\pi ^2}\\left( 2 + \\sum _{m=2}^{\\infty } \\frac{4}{(m^2-1)^2} \\right)=1.$ This can be derived as follows.",
"From the definition of the Riemann $\\zeta $ function, the following is easily verified: $2 \\sum _{m=2,4,...}^\\infty \\frac{1}{m^s}=2^{1-s}\\zeta (s).$ Observing that the left-hand side is $2\\left(\\zeta (s)-\\sum _{m=1,3,...}\\frac{1}{m^s}\\right)$ , it follows that $\\zeta (s)(1-2^{-s})=\\sum _{\\mu =1}^\\infty \\frac{1}{(2\\mu -1)^s}.$ Setting $s=2$ and remembering that $\\zeta (2)=\\pi ^2/6$ (Euler formula) give $\\sum _{\\mu =1}^\\infty \\frac{1}{(2\\mu -1)^2}=\\frac{\\pi ^2}{8}.$ Therefore, $\\sum _{m=2,4,...}\\frac{16}{(m^2-1)^2}=4\\left(\\frac{\\pi ^2}{8}+\\left(\\frac{\\pi ^2}{8}-1\\right)-1\\right)=\\pi ^2-8.$ The above and the infinite series representation yields $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}< \\frac{4}{\\pi ^2}\\left( 2+ \\sum _{m=2,4,...} \\frac{4}{(m^2-1)^2}\\right)=1.$ $\\blacksquare $ It is interesting to observe from the infinite series representation that $p_{\\mathrm {max}}^{\\rightarrow }(i,j)$ dips when $i$ and $j$ are relatively prime.",
"In particular, relative to the anti-core $j=n+1$ , the deepest dips happen at $i=1$ and $i=2n+1$ , since $\\mathrm {gcd}(1,n+1)=1$ and $\\mathrm {gcd}(n+1,2n+1)=1$ .",
"The opposite phenomenon happens when $i$ and $j$ share prime factors.",
"In this case, ${\\bf i}$ and ${\\bf j}$ drop, hence by the infinite series representation $p_{\\mathrm {max}}(i,j)$ shoots up.",
"This explains the “ripples” in the $\\sqrt{p_{\\mathrm {max}}(i,j)}$ plots of Fig.",
"REF .",
"Even though this figure is the case of a finite length chain, the “ripple” phenomenon is well explained by the asymptotic formula.",
"As a word of warning, the “spikes” near the anti-core of Fig.",
"REF should not be misconstrued as “cores.” Indeed, the first spike occurs at $j=87$ , so all it is depicting is the trivial fact that $p_{\\mathrm {max}}(87,87)=1$ ; this implies, by mirror symmetry relative to the middle spin, that $p_{\\mathrm {max}}(87,115)=1$ as well.",
"Corollary 1 The diameter of the semi-infinite chain is finite and is achieved along a sequence $\\lbrace i_{k\\in \\mathbb {N}}\\rbrace $ of prime numbers such that $\\lim _{k \\rightarrow \\infty }i_k=\\infty $ .",
"Proof.",
"From the infinite series representation, it is clear that $p^{\\rightarrow }_{\\mathrm {max}}(i,j)\\ge 64/\\pi ^4$ .",
"Hence $\\sup _{i\\ne j}d^{\\rightarrow }(i,j)\\le -\\log (64/\\pi ^4)$ .",
"To show that this can be achieved, it suffices to observe that the infinite series goes to 0 along an $i$ -sequence (or $j$ -sequence) of prime numbers.",
"$\\blacksquare $"
],
[
"Doubly-infinite chains",
"In the doubly infinite chain case, the position of the spins is referenced to $\\omega $ .",
"Hence, define $i^{\\prime }=i-\\omega $ and $j^{\\prime }=j-\\omega $ .",
"Furthermore, ${\\bf i^{\\prime }}=i^{\\prime }/\\mathrm {gcd}(i^{\\prime },j^{\\prime })$ and ${\\bf j^{\\prime }}=j^{\\prime }/\\mathrm {gcd}(i^{\\prime },j^{\\prime })$ .",
"Theorem 2 Consider an homogeneous XX chain of odd length $N=2n+1$ with the positions $i^{\\prime }$ , $j^{\\prime }$ of the spins referenced relative to the center $n+1$ .",
"Assume $i^{\\prime }$ and $j^{\\prime }$ are positive.",
"If both $i^{\\prime }$ and $j^{\\prime }$ are odd or both $i^{\\prime }$ and $j^{\\prime }$ are even with the same power of 2 in their prime number factorization, we have $\\sqrt{p_{\\mathrm {max}}^\\leftrightarrow (i^{\\prime },j^{\\prime })}&= \\frac{4}{\\pi ^2}\\left( 2+\\sum _{m=2,4,...}\\frac{4}{(m^2{\\bf i^{\\prime }}^2-1)(m^2 {\\bf j^{\\prime }}^2-1)} \\right)\\\\&=\\frac{8}{\\pi ^2}\\left(\\frac{1}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left({\\bf i^{\\prime }}^2 \\left(\\frac{\\pi }{2{\\bf i^{\\prime }}}\\right) \\cot \\left(\\frac{\\pi }{2 {\\bf i^{\\prime }}}\\right)- {\\bf j^{\\prime }}^2 \\left(\\frac{\\pi }{2{\\bf j^{\\prime }}}\\right) \\cot \\left(\\frac{\\pi }{2 {\\bf j^{\\prime }}}\\right)\\right)\\right).$ If $i^{\\prime }$ and $j^{\\prime }$ are even with different powers of 2 in their prime number factorization or $i^{\\prime }$ is odd and $j^{\\prime }$ is even, $\\sqrt{p_{\\mathrm {max}}^\\leftrightarrow (i^{\\prime },j^{\\prime })}&= \\frac{4}{\\pi ^2} \\left( 2+\\sum _{m=4,8,...} \\frac{4}{(m^2{\\bf i^{\\prime }}^2-1)(m^2{\\bf j^{\\prime }}^2-1)}\\right) \\\\&=\\frac{8}{\\pi ^2} \\left(\\frac{{\\bf i^{\\prime }}^2}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left(\\frac{\\pi }{4{\\bf i^{\\prime }}}\\right) \\cot \\left( \\frac{\\pi }{4{\\bf i^{\\prime }}}\\right)- \\frac{{\\bf j^{\\prime }}^2}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2} \\left( \\frac{\\pi }{4{\\bf j^{\\prime }}}\\right)\\cot \\left( \\frac{\\pi }{4{\\bf j^{\\prime }}}\\right) \\right).$ Proof.",
"See Appendix .",
"$\\blacksquare $ Theorem 3 For a homogeneous XX chain of length $N=2n+1$ with the positions 0, $j^{\\prime }$ of the spins referenced relative to the center $n+1$ and $j^{\\prime } > 0$ $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }(0,j^{\\prime })}=\\frac{2}{\\pi }\\approx 0.636619.$ Proof.",
"The result follows from the integral formulas of Sec.",
"REF of Appendix .",
"$\\blacksquare $ As a corollary of this theorem, we show that its asymptotic formula predicts the magnitude of the dip of Figure REF .",
"Observe the following: $&\\sqrt{p_{\\mathrm {max}}^{[1:201]}(87,101)} \\\\&= \\sqrt{p_{\\mathrm {max}}^{[1:201]}(101,87)} \\quad (\\mbox{by symmetry of the }p_{\\mathrm {max}}\\mbox{ function}) \\\\&=\\sqrt{p_{\\mathrm {max}}^{[1:201]}(101,115)} \\quad (\\mbox{by mirror symmetry of chain relative to center})\\\\&\\approx 0.63\\quad (\\mbox{by inspection of Fig.",
"}~\\ref {f:dramatic_anti_core}).$ Next, translating the finite chain to the doubly-infinite model, one would expect $ \\sqrt{p_{\\mathrm {max}}^{[1:201]}(101,115)} \\approx \\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }(0,14)}, $ which given the above numerical observations holds remarkably accurately.",
"Although $N<\\infty $ , the dip value of $\\sqrt{p_{\\mathrm {max}}^{[1:101]}(87,101)}$ is consistent with the asymptotic value given by Theorem REF .",
"Observe from Theorems REF and REF that the “probability” interpretation of $p_{\\mathrm {max}}^{\\leftrightarrow }$ holds the same way as it did for the semi-infinite chain.",
"The details are left out.",
"Corollary 2 The diameter of the doubly-infinite chain is finite and is achieved for $d^{\\leftrightarrow }(0,j^{\\prime })=-2\\log (2/\\pi )$ .",
"Proof.",
"It is easily seen from the infinite series representations of $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }(i^{\\prime },j^{\\prime })}$ in both cases of Theorem REF that $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }(i^{\\prime }j^{\\prime })}\\ge 8/\\pi ^2$ , $\\forall i^{\\prime }, j^{\\prime } \\ne 0$ .",
"This together with Theorem REF implies that the diameter is finite.",
"Furthermore, observe that the bound $8/\\pi ^2$ is reached along an infinite sequence $\\lbrace i^{\\prime }_{k\\in \\mathbb {N}}\\rbrace $ of prime numbers such that $\\lim _{k \\rightarrow \\infty } i^{\\prime }_k =\\infty $ , which guarantees that ${\\bf i^{\\prime }}_{k}:=i^{\\prime }_k/\\mathrm {gcd}(i^{\\prime }_k,j^{\\prime }) \\rightarrow \\infty $ at infinity.",
"This together with $2/\\pi < 8/\\pi ^2$ implies that the diameter is $-2 \\log (2/\\pi )$ .",
"$\\blacksquare $ The fact that the diameter is achieved for one spin at $i^{\\prime }=0$ reveals the “anti-core.”"
],
[
"Minimum probability",
"Inspired from congestion phenomena in classical communications [8], it was numerically observed in [7] that for chains of odd length $N=2n+1$ the inertia of the quantum network relative to the spin $j$ , $I^{(\\alpha )}(j):=\\sum _{i=1}^N d^\\alpha (i,j)$ , $\\alpha =2$ , is maximal for $j=\\omega :=n+1$ .",
"We now show that a stronger result holds: $\\arg \\max _j d(i,j) = \\omega , \\quad \\forall i \\ne \\omega ,$ In other words, for each spin other than the center, the center is the farthest away, which of course implies that $I^{(\\alpha )}(j)$ is maximum for $j=\\omega $ .",
"The preceding can be rephrased as $\\arg \\min _j p_{\\mathrm {max}}(i,j) = \\omega , \\quad \\forall i \\ne \\omega .$ Given the explicit expression for $p_{\\max }(i,j)$ in (REF ), the claim that $\\sqrt{p_{\\mathrm {max}}(i,j)}$ is achieved for $j=n+1$ amounts to proving the following: Theorem 4 For XX chains of odd length $N=2n+1$ , we have $\\frac{2}{N+1}\\sum _{k=1}^{2n+1} \\left| \\sin \\frac{\\pi k i }{2(n+1)} \\sin \\frac{\\pi k j}{2(n+1)}\\right| \\ge \\frac{2}{N+1} \\sum _{k=1}^{2n+1} \\left| \\sin \\frac{\\pi k i }{2(n+1)}\\sin \\frac{\\pi k }{2}\\right|$ as $n \\rightarrow \\infty $ .",
"Proof.",
"Firstly, we evaluate the asymptotic value of the right-hand side, that is, the maximum excitation transition probability from spin $i$ to spin $(n+1)=\\omega $ (or from spin $\\omega $ to spin $N$ ) for an infinite ($N\\rightarrow \\infty $ ) chain with XX coupling.",
"From [14] or Table REF , we have $ \\lim _{n \\rightarrow \\infty } \\sqrt{p_{\\mathrm {max}}(i,\\omega )}=\\lim _{n\\rightarrow \\infty }\\frac{1}{n+1} \\sum _{k=1}^{2n+1}\\left|\\sin \\frac{\\pi k i}{2(n+1)}\\sin \\frac{\\pi k}{2}\\right|.",
"$ The even terms are zero and letting $k=2l+1$ we have $ \\sqrt{p_{\\mathrm {max}}(i,n+1)}= \\frac{1}{n+1}\\sum _{l=0}^{n}\\left| \\sin \\left( \\frac{\\pi i(2l+1)}{2(n+1)}\\right) \\right|.",
"$ Setting $x=(2l+1)/(2(n+1))$ , we have $\\sqrt{p_{\\mathrm {max}}(i,n+1)}=\\int _0^1 | \\sin ( \\pi i x) | dx= i \\int _0^{1/i} \\sin \\pi i x dx=\\frac{2}{\\pi }.$ Next, from the above and the infinite series representation of Theorem REF , it suffices to show that $\\frac{4}{\\pi ^2}\\left( 2+\\sum _{m\\in M} \\frac{4}{(m^2{\\bf j}^2-1)(m^2{\\bf i}^2-1)} \\right)\\ge \\frac{2}{\\pi }.$ Observe that $\\frac{4}{\\pi ^2}\\Big ( 2+\\sum _{m\\in M} \\frac{4}{(m^2{\\bf j}^2-1)(m^2{\\bf i}^2-1)}\\Big )\\ge \\frac{8}{\\pi ^2} > \\frac{2}{\\pi }$ and the Theorem is proved.",
"$\\blacksquare $ Thus we have identified spin $\\omega =n+1$ as having minimal probability of any excitation being transferred to or from it.",
"To put it another way, the spin $\\omega $ is maximally distant from all other spins.",
"We shall call the corresponding probability amplitude “$\\omega $ -small” and the corresponding distance “$\\omega $ -large.” The “$\\omega $ -small” property is illustrated in Fig.",
"REF .",
"Figure: Square root of maximum transition probability between spinsi=1,87i=1,87 and all jj-spins for an XX chain with N=201N=201.",
"The sharp dropat j=101j=101 illustrates the “ω\\omega -small” property.The two spikes near the anti-core are not congestion cores, as observed in Sec.",
"."
],
[
"Transport properties",
"Here we examine the transport properties of the center $\\omega =n+1$ and justify its “anti-core” properties.",
"To this end, we consider the path integral representation.",
"Starting with $\\langle i|e^{-\\imath H_1 \\tau }|k\\rangle =\\sum _{\\ell =1}^N \\langle i|e^{-\\imath H_1 s} |\\ell \\rangle \\langle \\ell | e^{-\\imath H_1(\\tau - s)}|k\\rangle ,$ we obtain $\\langle i|e^{-\\imath H_1 t}|j\\rangle = \\sum _{k,\\ell =1}^N \\langle i|e^{-\\imath H_1 s} |\\ell \\rangle \\langle \\ell | e^{-\\imath H_1 (\\tau - s)}|k\\rangle \\langle k |e^{-\\imath H_1(t-\\tau )}|j\\rangle .$ By iterating, we get $\\langle i|e^{-\\imath H_1 t}|j\\rangle &= \\sum _{k_1,k_2,\\ldots ,k_{n-2}=1}^N \\langle i|e^{-\\imath H_1 t_1} |k_1\\rangle \\langle k_1| e^{-\\imath H_1 t_2 )}|k_2\\rangle \\ldots \\langle k_{n-2} |e^{-\\imath H_1 t_{n-1}}|j\\rangle \\\\&= \\sum _{k_1,k_2,\\ldots ,k_{n-2}=1}^N \\prod _{i=1,\\ldots ,n-1}\\langle k_{i-1}|e^{-\\imath H_1 t_i}|k_i\\rangle $ with $k_0=i$ , $k_{n-1}=j$ , $t_1+t_2+\\ldots +t_{n-1}=t$ , and $n \\le N$ .",
"It follows that $\\sqrt{p_t(i,j)}&\\le \\sum _{k_1,k_2,\\ldots ,k_{n-2}=1}^N \\prod _{i=1,\\ldots ,n-1}\\sqrt{p_{t_i}(k_{i-1},k_i)} \\\\&\\le \\sum _{k_1,k_2,\\ldots ,k_{n-2}=1}^N \\prod _{i=1,\\ldots ,n-1}\\sqrt{p_{\\mathrm {max}}(k_{i-1},k_i)}.$ Since the above is valid for all $t$ 's, we get $\\sqrt{p_{\\mathrm {max}}(i,j)}\\le \\sum _{k_1,k_2,\\ldots ,k_{n-2}=1}^N\\prod _{i=1,\\ldots ,n-1}\\sqrt{p_{\\mathrm {max}}(k_{i-1},k_i)}.$ The above means that an excitation from the source $i$ to the destination spin $j$ takes all possible length-$n$ paths from $i$ to $j$ , including those paths transiting through $\\frac{N+1}{2}=\\omega $ .",
"For those paths, any term of the form $p_{\\mathrm {max}}(\\omega ,k_i)$ or $p_{\\mathrm {max}}(k_{i-1},\\omega )$ is $\\omega $ -small, making the norm of the product in the right-hand side $\\omega $ -small.",
"Thus, for any transfer of excitation from $i$ to $j$ , the probability of exciting $\\omega $ along the way is $\\omega $ -small.",
"If we consider the probability of excitation of $\\omega $ as its “congestion,” then $\\omega $ remains clear of congestion, for transfer from any source $i\\ne \\omega $ to any destination $j\\ne \\omega $ .",
"Thus $\\omega $ appears to be the anti-thesis of the concept of core; let us agree to call it “anti-core.”"
],
[
"Anti-core in engineered chains",
"As observed earlier, the diameter of a homogeneous XX chain remains finite even as the length of the chain goes to infinity.",
"We now examine whether we can modify the chain to increase its diameter to infinity.",
"One way to achieve this is to apply a local potential $\\zeta $ to the central spin $\\omega $ .",
"This has the effect of perturbing the single excitation Hamiltonian, in turn distorting the original homogeneous distance to $d_\\zeta $ , so that in the limit $\\zeta \\rightarrow \\infty $ the ITC diameter increases ad infinitum, hence getting close to the coarse geometry paradigm of dealing with objects of infinite size.",
"The anti-core phenomenon is amplified in the sense that $\\lim _{\\zeta \\rightarrow \\infty }\\sqrt{p_{\\mathrm {max}}(1,\\omega )} =0$ .",
"This provides a tunneling barrier interpretation of the anti-core.",
"We prove that the $d_{\\zeta \\rightarrow \\infty }$ diameter of the engineered chain goes to $\\infty $ under two different scenarios: $N<\\infty $ and $N=\\infty $ .",
"The $N<\\infty $ proof is in the spirit of the main body of the paper; the $N=\\infty $ proof is operator-theoretic and relies on the assumption that $H_1$ is a doubly-infinite matrix, hence eradicating the “border effects.”"
],
[
"Finite Chains",
"With the applied potential the Hamiltonian in the single excitation subspace becomes $H_1^{(\\zeta )}= \\begin{pmatrix}0 & 1 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 0 \\\\1 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & & \\vdots &\\vdots \\\\0 & 0 & \\ldots & 0 & 1 & 0 & \\ldots & 0 & 0 \\\\0 & 0 &\\ldots & 1 & \\zeta & 1 & \\ldots & 0 & 0 \\\\0 & 0 & \\ldots & 0 & 1 & 0 & \\ldots & 0 & 0\\\\\\vdots & \\vdots & & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 1 \\\\0 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 1 & 0\\end{pmatrix}.$ The eigenstructure of this new Hamiltonian yields the new ITC distance $d_\\zeta $ .",
"Theorem 5 For an XX chain of odd length $N<\\infty $ , $\\lim _{\\zeta \\rightarrow \\infty }d_\\zeta (1,\\omega )=\\infty $ .",
"Proof.",
"To emphasize the dependency on the number $N$ of spins, observe that $H_1^{(\\zeta )}=T_N+\\zeta E,$ where $E$ is the $N \\times N$ matrix made up of 0's everywhere except for a 1 in position $\\left(\\frac{N+1}{2},\\frac{N+1}{2}\\right)$ ; and $T_N$ is the $N \\times N$ finite Toeplitz matrix made up of 0's on the diagonal, 1's on the super-diagonal, 1's on the sub-diagonal, and 0's everywhere else.",
"Recall that the determinant of the sum of two matrices equals the sums of the determinants of all matrices made up with some columns of one matrix and the complementary columns of the other matrix.",
"Applying the latter to $\\det ((\\lambda I_N -T_N)-\\zeta E))$ yields the characteristic polynomial $\\det (\\lambda I - T_N) -\\zeta \\left(\\det \\left(\\lambda I-T_{\\frac{N-1}{2}}\\right)\\right)^2.$ From classical root-locus techniques, it follows that, as $\\zeta \\rightarrow \\infty $ , exactly one eigenvalue $\\lambda _N$ goes to $\\infty $ , while the $(N-1)$ remaining ones converge to the roots of $\\left(\\det \\left(\\lambda I-T_{\\frac{N-1}{2}}\\right)\\right)^2=0$ .",
"The eigenvector equations $(T_N+\\zeta E)v_k=\\lambda _k(\\zeta ) v_k$ split, asymptotically as $\\zeta \\rightarrow \\infty $ , into two subsets: one for $\\lambda _N \\rightarrow \\infty $ and the others for $\\frac{\\lambda _k}{\\zeta }\\rightarrow 0$ ; that is, resp., $Ev_N&= \\left( \\lim _{\\zeta \\rightarrow \\infty } \\frac{\\lambda _N(\\zeta )}{\\zeta }\\right) v_N,\\\\Ev_k&= 0, \\quad k\\ne N.$ Next, again from root-locus techniques, it follows that $\\lim _{\\zeta \\rightarrow \\infty } \\frac{\\lambda _N(\\zeta )}{\\zeta }=1,$ so that $v_{N,k}=0$ , $k \\ne \\omega $ and $v_{N,\\omega }=1$ .",
"On the other hand, it is obvious that $v_{k,\\omega }=0$ , $k\\ne N$ .",
"Therefore $p_{\\mathrm {max}}(1,\\omega )=\\sum _{k=1}^N|\\langle 1|v_k\\rangle \\langle v_k|\\omega \\rangle |=0$ and $\\lim _{\\zeta \\rightarrow \\infty }d_{\\zeta }(1,\\omega )=\\infty $ .",
"$\\blacksquare $"
],
[
"Example",
"To illustrate several important points, we consider a very simple example, which has the advantage of being analytically tractable.",
"Consider the Hamiltonian (REF ) for the $N=3$ case.",
"The diagonal matrix of eigenvalues of $H_1$ can be computed symbolically as $\\begin{pmatrix}\\zeta /2 - (z^2 + 8)^{1/2}/2 & 0 & 0\\\\0& 0 & 0\\\\0& 0 & \\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2\\end{pmatrix}.$ The normalized eigenvectors are computed as $v_1=\\begin{pmatrix}1/((\\zeta /2 - (\\zeta ^2 + 8)^{1/2}/2)^2 + 2)^{1/2}\\\\(\\zeta /2 - (\\zeta ^2 + 8)^{1/2}/2)/((\\zeta /2 - (\\zeta ^2 + 8)^{1/2}/2)^2 + 2)^{1/2}\\\\1/((\\zeta /2 - (\\zeta ^2 + 8)^{1/2}/2)^2 + 2)^{1/2}\\end{pmatrix},$ $ v_2=\\begin{pmatrix}-2^{1/2}/2\\\\0\\\\2^{1/2}/2\\end{pmatrix},$ $v_3=\\begin{pmatrix}1/((\\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2)^2 + 2)^{1/2}\\\\(\\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2)/((\\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2)^2 + 2)^{1/2}\\\\1/((\\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2)^2 + 2)^{1/2}\\end{pmatrix}.$ Using those eigenvectors to symbolically compute $p_{\\mathrm {max}}(1,2)$ yields $&p_{\\mathrm {max}}(1,2)=\\\\&\\left(\\left|(\\zeta /2 - (\\zeta ^2 + 8)^{1/2}/2)/((\\zeta /2 - (\\zeta ^2 + 8)^{1/2}/2)^2 + 2\\right|\\right.\\\\+&~~\\left.\\left|(\\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2)/((\\zeta /2 + (\\zeta ^2 + 8)^{1/2}/2)^2 + 2\\right|\\right)^2\\sim 4 \\zeta .$ After symbolic computation of $p_{\\mathrm {max}}(1,3)$ and symbolically simplifying the expression, it is observed that $p_{\\mathrm {max}}(1,3)=1$ .",
"The results are translated into distances and plotted in Fig.",
"REF , left.",
"Figure: Simple 3-spin (left) and 5-spin (right) chain examples with bias ζ\\zeta at centershowing logarithmic behavior of the distance from the center to the outer spinsand vanishing (left) and bounded (right) distance between the outer spins.There are several important observations to be made from this simple example: The logarithmic behavior of the distance between the anti-core 2 and the outer spin, $d(1,2)=\\Theta (\\log (\\zeta ))$ , is confirmed analytically from the symbolic expression of the eigenvectors.",
"The same applies to $d(1,3)=0$ .",
"Because $d(1,3)=0$ and $d(2,1),d(2,3)\\rightarrow \\infty $ as $\\zeta \\rightarrow \\infty $ , the geodesic triangle 123 degenerates to the ray $[2,1]=[2,3]$ .",
"(A ray is an isometric embedding of $[0,\\infty )$ to a metric space [3], intuitively meaning that a ray starts at a finite point and extends to infinity along a length minimizing path.)",
"As such, because $\\triangle 123$ is “flat,” it is a Gromov $\\delta $ -slim triangle.",
"(A triangle is $\\delta $ -slim [3] if any edge is contained in the union of the $\\delta $ -neighborhoods of the other two edges.)",
"Thus the 123 chain is a, albeit trivial, Gromov hyperbolic space.",
"(A metric space is Gromov hyperbolic [3] if there exists a $\\delta < \\infty $ such that all of its geodesic triangles are $\\delta $ -slim.)",
"The rays $[2,1]$ and $[2,3]$ are going to infinity while keeping their Hausdorff distance finite, in fact vanishing.",
"So they converge to the same point on the Gromov boundary.",
"(The Gromov boundary [3] is the equivalence class of rays keeping their distance finite.)",
"Thus the 123 chain has its Gromov boundary reduced to a singleton.",
"The preceding fact (Gromov boundary reduced to a singleton) is the major topological discrepancy between classical and quantum networks.",
"Most classical networks have at least two points in their Gromov boundary, creating a core [2] as opposed to the anti-core of quantum communications.",
"The preceding 3-spin example has been extended to the same 5-spin case (Fig.",
"REF , right) with all results proved by symbolic manipulations, which unfortunately become prohibitively long to be included here.",
"The results are the same, except that the distance between the outer spins remains finite (rather than vanishing), but this suffices to come to the conclusion that the chain has only one point in its Gromov boundary.",
"We conjecture that this is a general feature."
],
[
"Infinite Chains",
"A proof of the infinite diameter property of the engineered chain with bias $\\zeta $ at the center $\\omega $ can be developed under infinite number of spins hypothesis, $N \\rightarrow \\infty $ .",
"Indeed, in this case, $H_1$ becomes the doubly infinite Toeplitz (also referred to as Laurent or multiplication) operator $T_s$ with symbol $s(\\exp (i\\theta ))=\\exp (i\\theta )+\\exp (-i\\theta )$ , and $H_1^{(\\zeta )}$ is a compact perturbation $\\zeta E$ of $T_s$ , where $E=|e\\rangle \\langle e|$ with $e=(\\ldots 0,0,1,0,0\\ldots )^{\\prime }$ the unit basis vector of the Hilbert space $\\ell ^2(-\\infty ,+\\infty )$ of square summable doubly infinite sequences.",
"By a well known perturbation theory result, the spectrum of $H_1^{(\\zeta )}:\\ell ^2(-\\infty ,+\\infty ) \\rightarrow \\ell ^2(-\\infty ,+\\infty )$ consists of the interval $s([0,2\\pi ))=[-2,+2]$ plus another eigenvalue of finite multiplicity that converges asymptotically to $\\zeta $ .",
"The eigen-equation $H_1^{(\\zeta )}v_\\lambda =\\lambda v_\\lambda $ becomes $ (T_s-\\lambda I)v_\\lambda =- \\zeta Ev_\\lambda =-\\zeta v_{\\lambda ,\\omega } e,$ where $v_{\\lambda ,\\omega }$ denotes the central component of the eigenvector $v_\\lambda $ .",
"In the Fourier or $z$ -domain, $T_s-\\lambda I$ is just the multiplication by $(z+z^{-1}-\\lambda )$ operator.",
"Therefore, the above can be resolved as $\\hat{v}_\\lambda = \\frac{-\\zeta v_{\\lambda ,\\omega }}{z+z^{-1}-\\lambda }, $ where $\\hat{v}_\\lambda $ denotes the Fourier or $z$ -transform of $v_\\lambda $ .",
"Taking $\\lambda \\in (-2,2)$ , it is easily verified that the Laurent expansion of the above converges on the unit circle; hence the inverse Fourier or $z$ -transform is in $\\ell ^2(-\\infty ,+\\infty )$ .",
"This provides an example of the rather unusual circumstance under which a continuous spectrum (here $[-2,+2]$ of $T_s$ ) is converted in to pure point spectrum (here $(-2,+2)$ of $T_s+\\zeta E$ ) by a compact perturbation.",
"The formula for the probability becomes $ p_{\\mathrm {max}}(i,j)=\\int _{-2}^{+2} |\\langle v_\\lambda | i \\rangle \\langle j | v_\\lambda \\rangle | d \\lambda + |\\langle v_\\zeta | i \\rangle \\langle j | v_\\zeta \\rangle |,$ where $v_\\zeta $ is the eigenvector corresponding to the asymptotic eigenvalue $\\zeta $ .",
"Here, we are specifically interested in the case $p_{\\mathrm {max}}(\\omega ,\\infty )$ of the probability of transition from the center to infinity.",
"Take $\\lambda \\in (-2,+2)$ .",
"We need to evaluate $\\langle v_\\lambda | \\omega \\rangle $ and $\\langle v_\\lambda |\\infty \\rangle $ .",
"Recall that Parseval's theorem $\\sum _{m=-\\infty }^{+\\infty }a_mb_m=\\frac{1}{2\\pi i} \\oint \\hat{a}(z)\\hat{b}(z^{-1}) \\frac{dz}{z}$ allows us to compute an inner product by residue calculation.",
"Using the recipe yields $\\langle v_\\lambda | \\omega \\rangle &= \\frac{1}{2\\pi \\imath }\\oint 1 \\frac{-\\zeta v_{\\lambda ,\\omega }}{z+z^{-1}-\\lambda } \\frac{dz}{z}= \\frac{1}{2\\pi \\imath }\\oint 1 \\frac{-\\zeta v_{\\lambda ,\\omega }}{z^2-\\lambda z +1} dz\\\\&=\\mathrm {Residue}\\left( \\frac{-\\zeta v_{\\lambda ,\\omega }}{z^2-\\lambda z +1}\\right)_{p,\\bar{p}},$ where $p,\\bar{p}$ , with $|p|<1, |\\bar{p}|<1$ , are the poles of the integrand, that is, the zeros of $z^2-\\lambda z + 1$ .",
"This yields $\\langle v_\\lambda | \\omega \\rangle = \\frac{-\\zeta v_{\\lambda ,\\omega }}{p-\\bar{p}} + \\frac{-\\zeta v_{\\lambda ,\\omega }}{\\bar{p}-p}=0.$ Next, we look at the term $\\langle v_\\lambda |\\infty \\rangle $ as the limit of $\\langle v_\\lambda |M\\rangle $ as $M \\rightarrow \\infty $ .",
"We have $\\langle v_\\lambda |M\\rangle &= \\frac{1}{2\\pi \\imath }\\oint \\frac{-\\zeta v_{\\lambda ,\\omega }}{z+z^{-1}-\\lambda }z^M \\frac{dz}{z}= \\mathrm {Residue}\\left( \\frac{-\\zeta v_{\\lambda ,\\omega } z^M}{z^2-\\lambda z +1} \\right)_{p,\\bar{p}}\\\\&= \\frac{-\\zeta v_{\\lambda ,\\omega }p^M}{p-\\bar{p}} + \\frac{-\\zeta v_{\\lambda ,\\omega }\\bar{p}^M}{\\bar{p}-p}.$ Since $|p|<1$ , the limit of the above as $M\\rightarrow \\infty $ vanishes.",
"Last, we look at the isolated eigenvalue case, $|\\langle v_k | \\omega \\rangle \\langle M | v_k \\rangle |$ .",
"Since $v_k \\approx e$ , and the excited state $|M\\rangle $ is a basis vector orthogonal to $e$ , we have $\\langle M | v_k \\rangle =0$ .",
"Hence the probability $p_{\\mathrm {max}}(\\infty ,\\omega )=0$ and the distance $d_\\zeta (\\omega ,\\infty )$ is infinity."
],
[
"Discussion and Conclusions",
"The early numerical observation [7] that homogeneous odd length $N$ -spin chains have an “anti-gravity” center has been analytically confirmed in the $N \\rightarrow \\infty $ limit by developing closed-form formulas for the asymptotic maximum excitation transfer probability, and by showing that the anti-core has the lowest probability of being excited or of transmitting its excitation.",
"As shown in Sec.",
", the phenomena exhibited in Fig.",
"REF can be accurately explained by the $N=\\infty $ asymptotic formula.",
"The existence of an anti-core at the center of a linear array of spins shows that excitations in a spin network do not propagate as they would in a classical network.",
"In a classical linear network any excitation in one half of the chain must transit through the center to reach the other half, and the center would thus be expected to be a congestion core.",
"In quantum networks, however, excitations can be transferred from one end of a chain to the other without passing through the center due to the intrinsic entanglement present in the eigenstates of the system.",
"When a single excitation is created in one location what is really created is a wavepacket, which is a superposition of many eigenstates of the system Hamiltonian that subsequently evolve and interfere.",
"High probability of transmission requires constructive interference at any particular node at some time, and the existence of an anti-core shows that, surprisingly, there is least constructive interference for the center of the chain.",
"As we have also shown, it is possible to engineer chains such that the diameter of the chain goes to infinity even if the physical number of spins is finite.",
"The simplest way to achieve this is by applying a bias to the central spin in an odd-length chain.",
"By increasing the bias we can increase the diameter even for a finite chain and achieve infinite diameter in the limit of infinite bias.",
"This can be explained in terms of the bias moving the central spin further and further away from the other spins and therefore effectively decoupling the chains.",
"However, for any finite bias, no matter how large, an excitation in one half of the chain can tunnel through the obstruction in the center given sufficient time, allowing almost perfect excitation transfer between the end spins of the engineered chain.",
"Such finite length, infinite diameter chains lend themselves to a coarse Gromov analysis, as Section shows.",
"The specific feature, demonstrated on chains of limited length but conjectured to hold for longer chains, is a Gromov boundary reduced to a singleton.",
"This strongly contrasts with the classical network paradigm of a Gromov boundary with at least two points, creating the congestion core [2].",
"Although there is early indication that the difference in cardinalities of the Gromov boundaries might be at least part of the explanation of the core versus anti-core discrepancy, more analysis is needed to prove a general fact and is left for further research."
],
[
"Proof of Theorem ",
"We proceed from $\\sqrt{p_{\\mathrm {max}}^{[1:N]}(i,j)}=\\frac{2}{N+1} \\sum _{k=1}^{2n+1}\\left| \\sin \\frac{\\pi k i}{2(n+1)} \\sin \\frac{\\pi k j}{2(n+1)} \\right|,$ where $N=2n+1$ is the (odd) number of spins and $i$ and $j$ are the positions of the two spins relative to the left-most spin (1).",
"Since the number of spins will be taken to infinity, we make the dependency on such number explicit."
],
[
"Asymptotic maximum transfer probability",
"Defining $x_k^{\\prime }=k/(2(n+1))$ for $k=0,\\ldots ,2n+1$ , the right-hand side of (REF ) becomes $\\sqrt{p_{\\mathrm {max}}^{[1:N]}(i,j)} =2\\sum _{k=1}^{2n+1} \\left| \\sin \\pi x_k^{\\prime } i\\right| \\left| \\sin \\pi x_k^{\\prime } j \\right| (x_{k}^{\\prime }-x_{k-1}^{\\prime }).$ Taking the limit $n \\rightarrow \\infty $ , the above becomes $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}&:=\\lim _{n \\rightarrow \\infty } \\sqrt{p_{\\mathrm {max}}^{[1:N]}(i,j)} \\\\&= 2 \\int _0^1 |\\sin \\pi i x^{\\prime }||\\sin \\pi j x^{\\prime }| \\, dx^{\\prime }= 4 \\int _0^{1/2} |\\sin 2 \\pi i x||\\sin 2 \\pi j x|\\, dx.$ Since $|\\sin 2 \\pi i x| = \\left|\\sin \\left(2 \\pi i\\left(x+\\frac{1}{2}\\right)\\right)\\right|$ , the above becomes $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}=& 2 \\int _0^{1/2} \\!\\!\\!",
"|\\sin 2 \\pi i x||\\sin 2 \\pi j x|dx\\\\&~~~~~+2 \\int _0^{1/2} \\!\\!\\!",
"\\left| \\sin 2 \\pi i \\left( \\tfrac{1}{2}+x \\right) \\right|\\left| \\sin 2 \\pi j \\left(\\tfrac{1}{2}+x \\right)\\right|dx\\\\=& 2 \\int _0^1 |\\sin 2 \\pi i x||\\sin 2 \\pi j x|dx.$ Next, observe that $|\\sin 2 \\pi i x|=\\sin (2 \\pi i x) s_i(x)$ where $s_i(x)$ is a periodic square wave with fundamental $\\sin 2 \\pi i x$ and Fourier decomposition $ s_i(x)=\\frac{4}{\\pi } \\sum _{p=1,3,\\ldots } \\frac{1}{p}\\sin 2 \\pi i p x.",
"$ Therefore, the absolute values in the integral representation of $\\sqrt{p_{\\mathrm {max}}(i,j)}$ can be removed as follows: $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}=\\frac{32}{\\pi ^2}\\sum _{p,q=1,3,\\ldots }\\frac{1}{pq}\\int _0^1(\\sin 2 \\pi i x \\sin 2 \\pi i p x)( \\sin 2 \\pi j x \\sin 2 \\pi j q x) dx.$ Next, utilizing several well-known trigonometric identities, we get, successively, ${\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}}\\\\=&\\frac{8}{\\pi ^2} \\sum _{p,q}\\frac{1}{pq}\\int _0^1(\\cos 2 \\pi i (p-1)x-\\cos 2 \\pi i (p+1)x)\\\\&~~~~~~~~~~~~~~~~~~~~~~~\\cdot (\\cos 2 \\pi j (q-1)x - \\cos 2 \\pi j (q+1)x) \\, dx\\\\=& \\frac{4}{\\pi ^2} \\sum _{p,q}\\frac{1}{pq}\\int _0^1 ((\\cos 2 \\pi (i(p-1)-j(q-1))x+\\cos 2 \\pi (i(p-1)+j(q-1))x) \\\\~&\\qquad \\qquad -(\\cos 2 \\pi (i(p-1)-j(q+1)x)+\\cos 2 \\pi (i(p-1)+j(q+1))x)\\\\~&\\qquad \\qquad -(\\cos 2 \\pi (i(p+1)-j(q-1))x+\\cos 2\\pi ( i (p+1)+j(q-1))x)\\\\~&\\qquad \\qquad +(\\cos 2 \\pi (i(p+1)-j(q+1))x+\\cos 2 \\pi (i(p+1)+j(q+1))x)\\, )dx\\\\=&\\frac{4}{\\pi ^2} \\sum _{p,q}\\frac{1}{pq}(\\mathbb {1}_{i(p-1)-j(q-1)=0}+\\mathbb {1}_{i(p-1)+j(q-1)=0}\\\\~& \\qquad \\qquad -\\mathbb {1}_{i(p-1)-j(q+1)=0}-\\mathbb {1}_{i(p-1)+j(q+1)=0}\\\\~& \\qquad \\qquad -\\mathbb {1}_{i(p+1)-j(q-1)=0}-\\mathbb {1}_{i(p+1)+j(q-1)=0}\\\\~& \\qquad \\qquad +\\mathbb {1}_{i(p+1)-j(q+1)=0}+\\mathbb {1}_{i(p+1)+j(q+1)=0}\\, ),$ where $\\mathbb {1}_{L}$ takes the value 1 if the logical statement $L$ is true and 0 otherwise.",
"Observe that, since $i,p,j,q \\ge 1$ , $ \\mathbb {1}_{i(p-1)+j(q+1)=0}=0, \\:\\mathbb {1}_{i(p+1)+j(q-1)=0}=0, \\:\\mathbb {1}_{i(p+1)+j(q+1)=0}=0$ and that $\\mathbb {1}_{i(p-1)+j(q-1)=0}$ takes the value 1 only for $p=q=1$ .",
"Thus, in the above, the sum over $p,q$ of the right-hand side terms amounts to 1.",
"Next, we look at the left-hand side terms of the sum over $p,q$ .",
"For such a statement as $i(p-1)-j(q-1)=0$ to be true, we need $p-1=m{\\bf j}$ and $q-1=m{\\bf i}$ for some $m \\in \\mathbb {N}$ , where $\\mathbf {j}=j/\\mathrm {gcd}(i,j)$ and ${\\bf i}=i/\\mathrm {gcd}(i,j)$ .",
"This yields $p=m{\\mathbf {j}}+1$ and $q=m{\\bf i}+1$ .",
"Observe that $m=0$ is an admissible value, since this yields $p=1$ , $q=1$ and hence $i(p-1)-j(q-1)=0$ .",
"Recapitulating and following up with the same argument on the other logical statements, we find $i(p-1)-j(q-1)=0 & \\Leftrightarrow \\left\\lbrace \\begin{array}{c}p=m{\\bf j}+1,\\\\q=m{\\bf i}+1,\\end{array}\\right.\\\\i(p-1)-j(q+1)=0 & \\Leftrightarrow \\left\\lbrace \\begin{array}{c}p=m{\\bf j}+1,\\\\q=m{\\bf i}-1,\\end{array}\\right.\\\\i(p+1)-j(q-1)=0 & \\Leftrightarrow \\left\\lbrace \\begin{array}{c}p=m{\\bf j}-1,\\\\q=m{\\bf i}+1,\\end{array}\\right.\\\\i(p+1)-j(q+1)=0 & \\Leftrightarrow \\left\\lbrace \\begin{array}{c}p=m{\\bf j}-1,\\\\q=m{\\bf i}-1.\\end{array}\\right.$ Note that, since $p,q \\ge 1$ , the solution $m=0$ is not admissible for the second, third, and forth cases, since this would entail either $p$ or $q$ or both of them to equal $-1$ .",
"In addition, $p$ and $q$ must be restricted to be odd, that is, both $m{\\bf i}$ and $m{\\bf j}$ must be even.",
"Since ${\\bf i}$ and ${\\bf j}$ are relatively prime, they cannot be both even; thus $m$ must be even.",
"The solution $m=0$ is still acceptable for the first case, since it makes both $p$ and $q$ odd.",
"To summarize, the sum over $p,q=1,3,\\ldots $ reduces to 2 plus a sum over $m \\in \\mathbb {N}^*$ , subject to the restrictions that $m{\\bf i}-1\\ne 0$ and $m {\\bf j}-1 \\ne 0$ .",
"Changing the sum over $p,q$ to a sum over $m$ , it follows that $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}=& \\frac{4}{\\pi ^2} \\left[ 2+ \\sum _{m\\in M}\\frac{1}{(m{\\bf j}+1)(m{\\bf i}+1)}-\\frac{1}{(m{\\bf j}+1)(m{\\bf i}-1)} \\right.",
"\\nonumber \\\\& \\;\\qquad \\qquad \\left.",
"-\\frac{1}{(m{\\bf j}-1)(m{\\bf i}+1)}+\\frac{1}{(m{\\bf j}-1)(m{\\bf i}-1))} \\right]\\nonumber \\\\=& \\frac{4}{\\pi ^2}\\left( 2+\\sum _{m\\in M} \\frac{4}{(m^2{\\bf j}^2-1)(m^2{\\bf i}^2-1)} \\right),$ where $M = \\lbrace m \\in \\mathbb {N}^*: m \\mbox{ is even}, m^2{\\bf i}^2-1\\ne 0, m^2{\\bf j}^2-1 \\ne 0\\rbrace $ .",
"This proves the infinite series representation of Theorem REF .",
"$\\blacksquare $"
],
[
"Closed form of asymptotic transfer probability",
"Next, we express the infinite series of Theorem REF in terms of elementary functions.",
"First, consider the partial fraction decomposition $\\frac{1}{(m^2{\\bf j}^2-1)(m^2{\\bf i}^2-1)}=\\frac{A(\\bf {i},\\bf {j})}{(m^2{\\bf j}^2-1)}+\\frac{A(\\bf {j},\\bf {i})}{(m^2{\\bf i}^2-1)},\\qquad A(\\bf {i},\\bf {j}) := \\frac{\\bf {i}^2}{\\bf {j}^2-i^2}.$ Next, observe the following lemma: Lemma 3 $\\sum _{m=2,4,6,...} \\frac{1}{(m^2{\\bf i}^2-1)} =\\frac{1}{2} \\left( 1-\\frac{\\pi }{2\\bf {i}} \\cot \\frac{\\pi }{2\\bf {i}}\\right).$ Proof.",
"In the known expression for the cotangent, $\\pi \\cot (\\pi z)=\\frac{1}{z}+2z \\sum _{n=1}^{\\infty }\\frac{1}{z^2-n^2},$ set $z=1/2\\bf {i}$ .",
"This yields $\\sum _{n=1}^\\infty \\frac{1}{(2n\\bf {i})^2-1}=\\frac{1}{2}\\left( 1- \\left( \\frac{\\pi }{2\\bf {i}} \\right) \\cot \\left(\\frac{\\pi }{2\\bf {i}} \\right)\\right).$ Setting $m=2n$ yields the result.",
"$\\blacksquare $ Putting everything together using the lemma yields $\\sqrt{p_{\\mathrm {max}}^{\\rightarrow }(i,j)}=\\frac{8}{\\pi ^2} \\left( \\frac{\\bf {i}^2}{\\bf {i}^2-\\bf {j}^2}\\left(\\frac{\\pi }{2\\bf {i}}\\right)\\cot \\left(\\frac{\\pi }{2\\bf {i}} \\right)-\\frac{\\bf {j}^2}{\\bf {i}^2-\\bf {j}^2} \\left(\\frac{\\pi }{2\\bf {j}} \\right)\\cot \\left(\\frac{\\pi }{2\\bf {j}}\\right) \\right)$ and Theorem REF is proved.",
"$\\blacksquare $"
],
[
"Proofs of Theorem ",
"We proceed from $ \\sqrt{p_{\\mathrm {max}}^{[1:N]}(i,j)}=\\frac{2}{N+1} \\sum _{k=1}^{2n+1}\\left| \\sin \\frac{\\pi k i}{2(n+1)} \\sin \\frac{\\pi k j}{2(n+1)} \\right|, $ where $N=2n+1$ is the (odd) number of spins and $i$ and $j$ are the positions of the two spins relative to the central spin (n+1).",
"Since the number of spins will be taken to infinity, we make the dependency on such number explicit."
],
[
"Referencing max. transfer probability to anti-core",
"The first operation is to do the change of variable $k^{\\prime }=k-(n+1)$ and convert the sum as $k$ goes from 1 to $2n+1$ to a sum where $k^{\\prime }$ goes from $-n$ to $+n$ .",
"After some manipulation, the following is found: $\\sqrt{p_{\\mathrm {max}}^{[1:N]}(i,j)}=\\frac{2}{N+1} \\sum _{k^{\\prime }=-n}^{+n}\\left|f\\left( \\frac{\\pi k^{\\prime } i}{2(n+1)}\\right)g\\left(\\tfrac{\\pi k^{\\prime } j}{2(n+1)}\\right)\\right|,$ where $f,g$ are given in Table REF .",
"Table: The functions ff and gg.The next step is the change of variables $i^{\\prime }=i-(n+1)$ and $j^{\\prime }=j-(n+1)$ .",
"With this change of variables, the position of the spins are relative to the anti-core, $n+1$ .",
"This change of variables leads to the following: $\\sqrt{p^{[-n:+n]}_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}&:= \\sqrt{p_{\\mathrm {max}}^{[1:N]}(i^{\\prime }+(n+1),j^{\\prime }+(n+1))}\\\\& = \\frac{2}{N+1} \\sum _{k^{\\prime }=-n,\\mathrm {even}}^{+n}\\left|f\\left( \\frac{\\pi k^{\\prime } i^{\\prime }}{2(n+1)}\\right)g\\left( \\frac{\\pi k^{\\prime } j^{\\prime }}{2(n+1)}\\right)\\right|\\\\& \\quad +\\frac{2}{N+1}\\sum _{k^{\\prime }=-n,\\mathrm {odd}}^{+n}\\left|\\bar{f}\\left( \\frac{\\pi k^{\\prime } i^{\\prime }}{2(n+1)}\\right)\\bar{g}\\left( \\frac{\\pi k^{\\prime } j^{\\prime }}{2(n+1)}\\right)\\right|,$ where $f,g$ are still given by Table REF and $\\bar{f}(\\cdot )=\\cos (\\cdot ),\\sin (\\cdot )$ whenever $f(\\cdot )=\\sin (\\cdot ),\\cos (\\cdot )$ , resp., with the same definition for $\\bar{g}$ .",
"Since the most recent formula is in terms of $i^{\\prime },j^{\\prime }$ , we rewrite Table REF in terms of $i^{\\prime },j^{\\prime }$ and $n$ in Table REF .",
"Table: The functions ff and gg in terms of i ' ,j ' i^{\\prime },j^{\\prime } and nn."
],
[
"Towards asymptotic maximum probability",
"In anticipation of letting $n \\rightarrow \\infty $ , define $x_{k^{\\prime }}:=\\frac{k^{\\prime }}{4(n+1)}$ and the preceding sums can be rewritten as $\\sqrt{p^{[-n:+n]}_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}&= 2\\sum _{k^{\\prime }=-n,\\mathrm {even}}^{+n}\\left|f\\left( 2 \\pi x_{k^{\\prime }}i^{\\prime }\\right) g\\left( 2\\pi x_{k^{\\prime }} j^{\\prime }\\right)\\right|\\left(x_{k^{\\prime }+2}-x_{k^{\\prime }} \\right)\\\\&\\quad + 2\\sum _{k^{\\prime }=-n,\\mathrm {odd}}^{+n}\\left|\\bar{f}\\left( 2 \\pi x_{k^{\\prime }}i^{\\prime } \\right)\\bar{g}\\left( 2\\pi x_{k^{\\prime }} j^{\\prime } \\right)\\right| \\left(x_{k^{\\prime }+2}-x_{k^{\\prime }} \\right).$ Letting $n \\rightarrow \\infty $ yields $\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}&:= \\lim _{n\\rightarrow \\infty }\\sqrt{p^{[-n:+n]}_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}\\\\&= 2\\int _{-1/4}^{+1/4} |f(2\\pi xi^{\\prime })g(2\\pi x j^{\\prime })| dx+ 2\\int _{-1/4}^{+1/4}|\\bar{f}(2\\pi xi^{\\prime })\\bar{g}(2\\pi x j^{\\prime })| dx.$ In order to make the integrations more straightforward and to follow a procedure that parallels the one of Appendix , it is convenient to change the integration limits by making use of the periodicity of the integrands as functions of $x$ .",
"Observe that both $fg$ and $\\bar{f}\\bar{g}$ have decompositions in terms of sines or cosines of arguments $2\\pi x (i^{\\prime } \\pm j^{\\prime })$ .",
"Write the generic term as $\\left\\lbrace \\begin{array}{c} \\sin \\\\ \\cos \\end{array}\\right\\rbrace (2\\pi x (i^{\\prime } \\pm j^{\\prime }))$ .",
"If $i^{\\prime }\\pm j^{\\prime }$ is even, observe that $\\left\\lbrace \\begin{array}{c}\\sin \\\\\\cos \\end{array}\\right\\rbrace (2\\pi (x+1/2) (i^{\\prime } \\pm j^{\\prime }))=\\left\\lbrace \\begin{array}{c}\\sin \\\\\\cos \\end{array}\\right\\rbrace (2\\pi x (i^{\\prime } \\pm j^{\\prime }))$ .",
"If $i^{\\prime }\\pm j^{\\prime }$ is odd, $\\left\\lbrace \\begin{array}{c}\\sin \\\\\\cos \\end{array}\\right\\rbrace (2\\pi (x+1/2) (i^{\\prime } \\pm j^{\\prime }))=-\\left\\lbrace \\begin{array}{c}\\sin \\\\\\cos \\end{array}\\right\\rbrace (2\\pi x (i^{\\prime } \\pm j^{\\prime }))$ .",
"In either case, $|fg|$ and $|\\bar{f}\\bar{g}|$ have period $1/2$ .",
"With this property, the previous integrals can be rewritten as $\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}= \\int _{-1/2}^{+1/2} |f(2\\pi xi^{\\prime })g(2\\pi x j^{\\prime })| dx+\\int _{-1/2}^{+1/2}|\\bar{f}(2\\pi xi^{\\prime })\\bar{g}(2\\pi x j^{\\prime })| dx.$ Observe that $p_{\\mathrm {max}}(i^{\\prime },j^{\\prime }) \\le 1$ , as easily seen from a Cauchy-Schwarz argument.",
"Also observe that $p_{\\mathrm {max}}(i^{\\prime },i^{\\prime })=1$ ; indeed, if $i^{\\prime }=j^{\\prime }$ , Table REF reveals that the integrands are of the form $|\\cos (2\\pi i^{\\prime }x)\\cos (2\\pi i^{\\prime }x)|$ or $|\\sin (2\\pi i^{\\prime }x)\\sin (2\\pi i^{\\prime }x)|$ , from which the assertion is trivial.",
"The next step is to make $|f|$ , $|g|$ , $|\\bar{f}|$ , $|\\bar{g}|$ more manageable by expressing them as $f(2\\pi xi^{\\prime })s_{i^{\\prime }}(x)$ , $g(2\\pi xj^{\\prime })s_{j^{\\prime }}(x)$ , $\\bar{f}(2\\pi x i^{\\prime })s_{i^{\\prime }}(x)$ , $\\bar{g}(2\\pi xj^{\\prime })s_{j^{\\prime }}(x)$ if $f$ , $g$ , $\\bar{f}$ , $\\bar{g}$ are sines and by $f(2\\pi xi^{\\prime })c_{i^{\\prime }}(x)$ , $g(2\\pi x j^{\\prime })c_{j^{\\prime }}(x)$ , $\\bar{f}(2\\pi x i^{\\prime })c_{i^{\\prime }}(x)$ , $\\bar{g}(2\\pi x j^{\\prime })c_{j^{\\prime }}(x)$ if they are cosines.",
"In the preceding, $s_{i^{\\prime }}(x)$ and $c_{i^{\\prime }}(x)$ are odd and even, resp., square waves of unit amplitude and of period 1, with Fourier decompositions $s_{i^{\\prime }}(x)&=\\frac{4}{\\pi } \\sum _{p=1,3,...} \\frac{1}{p} \\sin (2 \\pi i^{\\prime }px),\\\\c_{i^{\\prime }}(x)&=\\frac{4}{\\pi } \\sum _{p=1,3,...} \\frac{(-1)^{\\frac{p-1}{2}}}{p} \\cos (2 \\pi i^{\\prime }px).$ At this stage, it is necessary to be more specific as to what $f$ , $g$ , $\\bar{f}$ , $\\bar{g}$ are."
],
[
"$i^{\\prime }$ and {{formula:4f4a6c6c-cee9-4efa-a0b6-029c697fc6e2}} even",
"If $i^{\\prime }$ and $j^{\\prime }$ are even, and if we let $n \\rightarrow \\infty $ along the even number subsequence of $\\mathbb {N}$ , we need to take $f(\\cdot )=\\cos (\\cdot )$ and $g(\\cdot )=\\cos (\\cdot )$ , as seen from Table REF , together with $\\bar{f}(\\cdot )=\\sin (\\cdot )$ and $\\bar{g}(\\cdot )=\\sin (\\cdot )$ .",
"If on the other hand, we let $n \\rightarrow \\infty $ along the odd number subsequence of $\\mathbb {N}$ , we need to take $f(\\cdot )=\\sin (\\cdot )$ and $g=\\sin (\\cdot )$ , together with $\\bar{f}(\\cdot )=\\cos (\\cdot )$ and $\\bar{g}(\\cdot )=\\cos (\\cdot )$ .",
"However, because of the symmetry of formula (REF ), both subsequences yield the same result: ${\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}}\\\\&= \\int _{-1/2}^{+1/2}|\\cos (2\\pi xi^{\\prime }) \\cos (2\\pi x j^{\\prime })| dx+\\int _{-1/2}^{+1/2}|\\sin (2\\pi xi^{\\prime }) \\sin (2\\pi x j^{\\prime })| dx\\\\&= \\int _{-1/2}^{+1/2}\\cos (2\\pi xi^{\\prime }) c_{i^{\\prime }}(x)\\cos (2\\pi x j^{\\prime }) c_{j^{\\prime }}(x) dx \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\int _{-1/2}^{+1/2}\\sin (2\\pi xi^{\\prime }) s_{i^{\\prime }}(x)\\sin (2\\pi x j^{\\prime }) s_{j^{\\prime }}(x) dx.$"
],
[
"$i^{\\prime }$ and {{formula:a25e8783-fd00-4c42-bae4-87af3bd37896}} odd",
"The argument is the same as before and the preceding formula still holds."
],
[
"$i^{\\prime }$ odd and {{formula:fe110625-fe45-465a-9573-471a0af966d4}} even",
"From Table REF , we could let $n \\rightarrow \\infty $ along the even number subsequence of $\\mathbb {N}$ , in which case we need to take $f(\\cdot )=\\sin (\\cdot )$ and $g(\\cdot )=\\cos (\\cdot )$ .",
"If we let $n \\rightarrow \\infty $ along the odd number subsequence of $\\mathbb {N}$ , we need to take $f(\\cdot )=\\cos (\\cdot )$ and $g(\\cdot )=\\sin (\\cdot )$ .",
"In either case, because of the symmetry of (REF ), the result is the same and is given by ${\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}}\\\\&=\\int _{-1/2}^{+1/2} |\\sin (2\\pi xi^{\\prime }) \\cos (2\\pi x j^{\\prime })| dx+\\int _{-1/2}^{+1/2} |\\cos (2\\pi xi^{\\prime }) \\sin (2\\pi x j^{\\prime })| dx\\\\&=\\int _{-1/2}^{+1/2} \\sin (2\\pi xi^{\\prime }) s_{i^{\\prime }}(x)\\cos (2\\pi x j^{\\prime }) c_{j^{\\prime }}(x) dx\\\\&~~~~~~~~~~~~~~~~~~~~~~~~+\\int _{-1/2}^{+1/2} \\cos (2\\pi xi^{\\prime }) c_{i^{\\prime }}(x)\\sin (2\\pi x j^{\\prime }) s_{j^{\\prime }}(x) dx.$"
],
[
"$i^{\\prime }$ even and {{formula:d716eb57-e34c-443d-9758-0a428e50a7b2}} odd",
"The formula of the preceding section remains valid.",
"To prove it, just interchange the role of $i^{\\prime }$ and $j^{\\prime }$ ."
],
[
"Towards integration by quadrature",
"From the above, it follows that all cases share a few quadrature integrals: ${\\int _{-1/2}^{+1/2}\\sin (2\\pi xi^{\\prime }) s_{i^{\\prime }}(x)\\sin (2\\pi x j^{\\prime }) s_{j^{\\prime }}(x)dx}\\\\&=\\frac{16}{\\pi ^2} \\sum _{p,q=1,3,...} \\frac{1}{pq}\\int _{-1/2}^{1/2}\\sin (2\\pi i^{\\prime }x)\\sin (2\\pi i^{\\prime }px)\\sin (2\\pi j^{\\prime }x)\\sin (2\\pi j^{\\prime }qx)dx\\\\&=\\frac{4}{\\pi ^2} \\sum _{p,q}\\frac{1}{pq}\\int _{-1/2}^{1/2}(\\cos (2\\pi i^{\\prime }(p-1)x)-\\cos (2\\pi i^{\\prime } (p+1)x))\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\cdot (\\cos (2\\pi j^{\\prime }(q-1)x)-\\cos (2\\pi j^{\\prime } (q+1)x))dx\\\\&=\\frac{2}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\int _{-1/2}^{1/2}(\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q-1))x)\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q+1))x)\\\\& \\quad -\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x)+\\sum \\cos (2\\pi (i^{\\prime }(p+1) \\pm j^{\\prime }(q+1))x))dx.$ The right-hand side of the last equality involves expressions like $\\cos (2\\pi (a+b)x)+\\cos (2\\pi (a-b)x)$ .",
"To simplify the notation, we wrote such expressions as $\\sum \\cos (2\\pi (a\\pm b)x)$ .",
"${(-1)^{\\frac{p+q}{2}-1}\\int _{-1/2}^{+1/2}\\cos (2\\pi xi^{\\prime }) c_{i^{\\prime }}(x)\\cos (2\\pi x j^{\\prime }) c_{j^{\\prime }}(x)dx}\\\\&= \\frac{16}{\\pi ^2} \\sum _{p,q=1,3,...}\\frac{1}{pq} \\int _{-1/2}^{1/2}\\cos (2\\pi i^{\\prime }x)\\cos (2\\pi i^{\\prime }px)\\cos (2\\pi j^{\\prime }x)\\cos (2\\pi j^{\\prime }qx)dx\\\\&=\\frac{4}{\\pi ^2} \\sum _{p,q}\\frac{1}{pq} \\int _{-1/2}^{1/2}(\\cos (2\\pi i^{\\prime }(p+1)x)+\\cos (2\\pi i^{\\prime } (p-1)x))\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~\\cdot (\\cos (2\\pi j^{\\prime }(q+1)x)+\\cos (2\\pi j^{\\prime } (q-1)x))dx\\\\&=\\frac{2}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\int _{-1/2}^{1/2}(\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q+1))x)\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x)\\\\&\\quad +\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q+1))x)+\\sum \\cos (2\\pi (i^{\\prime }(p-1) \\pm j^{\\prime }(q-1))x))dx;$ ${(-1)^{\\frac{q-1}{2}}\\int _{-1/2}^{+1/2}\\sin (2\\pi xi^{\\prime }) s_{i^{\\prime }}(x)\\cos (2\\pi x j^{\\prime }) c_{j^{\\prime }}(x)dx}\\\\&= \\frac{16}{\\pi ^2} \\sum _{p,q=1,3,...}\\frac{1}{pq} \\int _{-1/2}^{1/2}\\sin (2\\pi i^{\\prime }x)\\sin (2\\pi i^{\\prime }px)\\cos (2\\pi j^{\\prime }x)\\cos (2\\pi j^{\\prime }qx)dx\\\\&=\\frac{4}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\int _{-1/2}^{1/2}(\\cos (2\\pi i^{\\prime }(p-1)x)-\\cos (2\\pi i^{\\prime } (p+1)x))\\\\&~~~~~~~~~~~~~~~~~~~~~~~~\\cdot (\\cos (2\\pi j^{\\prime }(q-1)x)+\\cos (2\\pi j^{\\prime } (q+1)x))dx\\\\&=\\frac{2}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\int _{-1/2}^{1/2}(\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q-1))x)\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q+1))x)\\\\& \\quad -\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x)-\\sum \\cos (2\\pi (i^{\\prime }(p+1) \\pm j^{\\prime }(q+1))x))dx;$ ${(-1)^{\\frac{p-1}{2}}\\int _{-1/2}^{+1/2}\\cos (2\\pi xi^{\\prime }) c_{i^{\\prime }}(x)\\sin (2\\pi x j^{\\prime }) s_{j^{\\prime }}(x)dx}\\\\&= \\frac{16}{\\pi ^2} \\sum _{p,q=1,3,...}\\frac{1}{pq} \\int _{-1/2}^{1/2}\\cos (2\\pi i^{\\prime }x)\\cos (2\\pi i^{\\prime }px)\\sin (2\\pi j^{\\prime }x)\\sin (2\\pi j^{\\prime }qx)dx\\\\&=\\frac{4}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\int _{-1/2}^{1/2}(\\cos (2\\pi i^{\\prime }(p+1)x)+\\cos (2\\pi i^{\\prime } (p-1)x))\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~\\cdot (\\cos (2\\pi j^{\\prime }(q-1)x)-\\cos (2\\pi j^{\\prime } (q+1)x))dx\\\\&=\\frac{2}{\\pi ^2} \\sum _{p,q}\\frac{1}{pq} \\int _{-1/2}^{1/2}(\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x)\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q+1))x)\\\\& \\quad +\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q-1))x)-\\sum \\cos (2\\pi (i^{\\prime }(p-1) \\pm j^{\\prime }(q+1))x))dx.$"
],
[
"Asymptotic max. transfer probability around anti-core",
"Here we proceed from the general formula (REF ) for $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }(i^{\\prime },j^{\\prime })}$ , utilize the quadrature integrals of the preceding section, and derive, first, an infinite series representation of the asymptotic maximum transfer probability and, finally, a representation in terms of special functions.",
"Since the general formula (REF ) is in terms of function $f$ , $g$ , $\\bar{f}$ , $\\bar{g}$ that depend on whether $i^{\\prime }$ and $j^{\\prime }$ are even or odd (see Section REF ), it is necessary to examine each case in particular.",
"From Section REF , it follows that the case where both $i^{\\prime }$ and $j^{\\prime }$ are even and the case where both $i^{\\prime }$ and $j^{\\prime }$ are odd are the same.",
"From the same Section REF , the case where $i^{\\prime }$ is even and $j^{\\prime }$ odd is the same as the case where $i^{\\prime }$ odd and $j^{\\prime }$ even, but is different from the preceding one.",
"So, there are essentially two cases to be distinguished."
],
[
"Both $i^{\\prime }$ and {{formula:02cde6d2-6add-4a38-a6b0-4ab959b35ebc}} even or both {{formula:0d33e321-b2cb-400a-b28d-abb691f1315b}} and {{formula:4bc317bb-b255-4a8f-bc0c-cede08ee89d5}} odd",
" where $c(p,q) = \\frac{1}{2} \\left(1 +(-1)^{\\frac{p+q}{2}-1}\\right), \\quad d(p,q) = \\frac{1}{2} \\left(1 +(-1)^{\\frac{p+q}{2}}\\right).$ $c(\\cdot ,\\cdot )$ and $d(\\cdot ,\\cdot )$ are functions taking value 0 or 1, and complementary in the sense that $c(p,q)+d(p,q)=1$ .",
"Next, we find that $&\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}\\\\&=\\frac{4}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\left(c(p,q)\\mathbb {I}_{2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q-1))x=0}+c(p,q)\\mathrm {I}_{2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q+1))x=0 } \\right.\\\\&\\quad -d(p,q)\\mathrm {I}_{2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q+1))x=0}\\left.-d(p,q)\\mathrm {I}_{2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x=0}\\right),$ where ${\\bf i^{\\prime }}= \\frac{i^{\\prime }}{\\mathrm {gcd}(i^{\\prime },j^{\\prime })}, \\quad {\\bf j^{\\prime }}= \\frac{j^{\\prime }}{\\mathrm {gcd}(i^{\\prime },j^{\\prime })}.$ Observe that ${\\bf i^{\\prime }}$ , ${\\bf j^{\\prime }}$ are relatively prime; hence they could not be both even.",
"The developments follow closely the semi-infinite chain case, except that, here, ${\\bf i^{\\prime }}$ and ${\\bf j^{\\prime }}$ are not restricted to be positive.",
"Hence we have to consider several cases:"
],
[
"${\\bf i^{\\prime }},{\\bf j^{\\prime }}\\ge 1$",
"The following is easily observed: $\\sum _{p,q=1,3,...}\\frac{c(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p-1)-{\\bf j^{\\prime }}(q-1)=0}(p,q)&=\\sum _{m=0,2,...}\\frac{c(m {\\bf j^{\\prime }}+1,m{\\bf i^{\\prime }}+1)}{(m{\\bf j^{\\prime }}+1)(m{\\bf i^{\\prime }}+1)},\\\\\\sum _{p,q=1,3,...}\\frac{c(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p-1)+{\\bf j^{\\prime }}(q-1)=0}(p,q)&=c(1,1),\\\\\\sum _{p,q=1,3,...}\\frac{c(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p+1)+{\\bf j^{\\prime }}(q+1)=0}(p,q)&=0,\\\\\\sum _{p,q=1,3,...}\\frac{c(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p+1)-{\\bf j^{\\prime }}(q+1)=0}(p,q)&=\\sum _{m=2,4,...}\\frac{c(m {\\bf j^{\\prime }}-1,m {\\bf i^{\\prime }}-1)}{(m{\\bf j^{\\prime }}-1)(m{\\bf i^{\\prime }}-1)}.$ The situation is pretty much the same for the terms involving $d(p,q)$ : $\\sum _{p,q=1,3,...}\\frac{d(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p-1)-{\\bf j^{\\prime }}(q+1)=0}(p,q)&=\\sum _{m=2,4,...}\\frac{d(m {\\bf j^{\\prime }}+1,m{\\bf i^{\\prime }}-1)}{(m{\\bf j^{\\prime }}+1)(m{\\bf i^{\\prime }}-1)},\\\\\\sum _{p,q=1,3,...}\\frac{d(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p-1)+{\\bf j^{\\prime }}(q+1)=0}(p,q)&=0,\\\\\\sum _{p,q=1,3,...}\\frac{d(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p+1)+{\\bf j^{\\prime }}(q-1)=0}(p,q)&=0,\\\\\\sum _{p,q=1,3,...}\\frac{d(p,q)}{pq} \\mathbb {I}_{{\\bf i^{\\prime }}(p+1)-{\\bf j^{\\prime }}(q-1)=0}(p,q)&=\\sum _{m=2,4,...}\\frac{d(m {\\bf j^{\\prime }}-1,m {\\bf i^{\\prime }}+1)}{(m{\\bf j^{\\prime }}-1)(m{\\bf i^{\\prime }}+1)}.$ Here we have to make a distinction between the two cases: both $i^{\\prime }$ and $j^{\\prime }$ odd and both $i^{\\prime }$ and $j^{\\prime }$ even.",
"We start with the easy case where both i' and j' are odd.",
"In this case indeed, both ${\\bf i^{\\prime }}\\pm {\\bf j^{\\prime }}$ is even.",
"This along with $m$ is even yields $c(m{\\bf j^{\\prime }}+1,m {\\bf i^{\\prime }}+1) &=1,\\\\c(1,1) &=1,\\\\c(m{\\bf j^{\\prime }}-1,m {\\bf i^{\\prime }}-1) &=1,\\\\d(m{\\bf j^{\\prime }}+1,m {\\bf i^{\\prime }}-1) &=1,\\\\d(m{\\bf j^{\\prime }}-1,m {\\bf i^{\\prime }}+1) &=1.$ Putting everything together, we find, using partial fraction decompositions, $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }(i^{\\prime },j^{\\prime })}&=\\frac{4}{\\pi ^2}\\left( 2+\\sum _{m=2,4,...}\\frac{4}{(m^2{\\bf i^{\\prime }}^2-1)(m^2 {\\bf j^{\\prime }}^2-1)} \\right) \\\\&=\\frac{4}{\\pi ^2}\\left( 2+\\frac{4}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2} \\sum _{m=2,4,...} \\left( \\frac{{\\bf j^{\\prime }}^2}{m^2{\\bf j^{\\prime }}^2-1}-\\frac{{\\bf i^{\\prime }}^2}{m^2 {\\bf i^{\\prime }}^2-1}\\right) \\right).$ Finally, recall (Lemma REF ) that the infinite sums can be expressed in terms of cotangents; this yields $\\sqrt{p_{\\mathrm {max}}^\\leftrightarrow (i^{\\prime },j^{\\prime })}= \\frac{8}{\\pi ^2} \\left( \\frac{1}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left({\\bf i^{\\prime }}^2 \\left(\\frac{\\pi }{2{\\bf i^{\\prime }}}\\right) \\cot \\left(\\frac{\\pi }{2 {\\bf i^{\\prime }}}\\right)- {\\bf j^{\\prime }}^2 \\left(\\frac{\\pi }{2{\\bf j^{\\prime }}}\\right) \\cot \\left(\\frac{\\pi }{2 {\\bf j^{\\prime }}}\\right)\\right) \\right).$ The case where both $i^{\\prime }$ and $j^{\\prime }$ are even is more complicated.",
"If $i^{\\prime }$ and $j^{\\prime }$ have the same power of 2 in their prime number factorization, then ${\\bf i^{\\prime }}\\pm {\\bf j^{\\prime }}$ is even and the preceding formula holds.",
"If the powers of 2 are different, then ${\\bf i^{\\prime }}\\pm {\\bf j^{\\prime }}$ is odd and it is easily verified that $\\left.\\begin{array}{c}c(m{\\bf j^{\\prime }}+1,m{\\bf i^{\\prime }}+1)\\\\c(m{\\bf j^{\\prime }}-1,m{\\bf i^{\\prime }}-1)\\\\d(m{\\bf j^{\\prime }}+1,m{\\bf i^{\\prime }}-1)\\\\d(m{\\bf j^{\\prime }}-1,m{\\bf i^{\\prime }}+1)\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{cll}1 & \\mbox{for}&m=0,4,8,12,..., \\\\0 & \\mbox{otherwise}.&\\end{array}\\right.$ With the above, we get $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }({\\bf i^{\\prime }},{\\bf j^{\\prime }})}&=\\frac{4}{\\pi ^2}\\left(2 +\\sum _{m=4,8,...}\\frac{1}{(m{\\bf j^{\\prime }}+1)(m{\\bf i^{\\prime }}-1)} + \\frac{1}{(m{\\bf j^{\\prime }}+1)(m{\\bf i^{\\prime }}+1)} \\right.\\\\& \\qquad \\qquad -\\left.\\frac{1}{(m{\\bf j^{\\prime }}^{\\prime }+1)()m{\\bf i^{\\prime }}-1}-\\frac{1}{(m{\\bf j^{\\prime }}-1)(m{\\bf i^{\\prime }}+1)}\\right)\\\\&= \\frac{4}{\\pi ^2}\\left( 2+\\sum _{m=4,8,...} \\frac{4}{(m^2{\\bf j^{\\prime }}^2-1)(m^2{\\bf i^{\\prime }}^2-1)} \\right)\\\\&= \\frac{4}{\\pi ^2}\\left( 2+ \\frac{4}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\sum _{m=4,8,...}\\left( \\frac{{\\bf j^{\\prime }}^2}{m^2{\\bf j^{\\prime }}^2-1}-\\frac{{\\bf i^{\\prime }}^2}{m^2{\\bf i^{\\prime }}^2-1}\\right)\\right).$ In order to derive a closed-form representation of the infinite series, we need the following lemma: Lemma 4 $ \\sum _{m=4,8,...}\\frac{1}{m^2{\\bf i^{\\prime }}^2-1}=\\frac{1}{2}\\left( 1-\\left(\\frac{\\pi }{4{\\bf i^{\\prime }}}\\right)\\cot \\left( \\frac{\\pi }{4{\\bf i^{\\prime }}}\\right) \\right).",
"$ Proof.",
"The proof is the same as that of Lemma REF , except that instead of setting $z=1/2{\\bf i}$ we set $z=1/4{\\bf i^{\\prime }}$ .",
"$\\blacksquare $ Using the lemma, we finally get the closed-form formula: $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }({\\bf i^{\\prime }},{\\bf j^{\\prime }})}=\\frac{8}{\\pi ^2} \\left(\\frac{{\\bf i^{\\prime }}^2}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left( \\frac{\\pi }{4{\\bf i^{\\prime }}}\\right) \\cot \\left( \\frac{\\pi }{4{\\bf i^{\\prime }}}\\right)-\\frac{{\\bf j^{\\prime }}^2}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left( \\frac{\\pi }{4{\\bf j^{\\prime }}}\\right) \\cot \\left( \\frac{\\pi }{4{\\bf j^{\\prime }}}\\right)\\right).$"
],
[
"${\\bf i^{\\prime }}< 0 < {\\bf j^{\\prime }}$",
"The preceding formula remains valid, after replacing $i^{\\prime }$ by $-i^{\\prime }$ ."
],
[
"$i^{\\prime }$ odd and {{formula:b38659d8-d792-4dbe-b453-c497ce49ac97}} even",
"From the integral representation, we get $&\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}\\\\&=\\frac{4}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\int _{-1/2}^{1/2}\\left( (-1)^{\\frac{p-1}{2}}c(p,q)\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q-1))x)\\right.\\\\& \\qquad -(-1)^{\\frac{p-1}{2}}c(p,q)\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q+1))x)\\\\& \\qquad -(-1)^{\\frac{p-1}{2}}d(p,q)\\sum \\cos (2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q+1))x)\\\\& \\qquad +(-1)^{\\frac{p-1}{2}}\\left.d(p,q)\\sum \\cos (2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x)\\right)dx.$ Next, we find that $\\sqrt{p^{\\leftrightarrow }_{\\mathrm {max}}(i^{\\prime },j^{\\prime })}&=\\frac{4}{\\pi ^2} \\sum _{p,q} \\frac{1}{pq}\\left( (-1)^{\\frac{p-1}{2}}c(p,q)\\mathbb {I}_{2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q-1))x=0}\\right.\\\\& \\qquad -(-1)^{\\frac{p-1}{2}}c(p,q)\\mathrm {I}_{2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q+1))x=0}\\\\& \\qquad -(-1)^{\\frac{p-1}{2}}d(p,q)\\mathrm {I}_{2\\pi (i^{\\prime }(p-1)\\pm j^{\\prime }(q+1))x=0}\\\\& \\qquad \\left.+(-1)^{\\frac{p-1}{2}}d(p,q)\\mathrm {I}_{2\\pi (i^{\\prime }(p+1)\\pm j^{\\prime }(q-1))x=0}\\right).$ Despite the extra difficulties created by the $c(\\cdot ,\\cdot )$ and $d(\\cdot ,\\cdot )$ functions and the various signs, the pattern remains the same as before: the indicators are nonvanishing only if $ p=m{\\bf j^{\\prime }}\\pm 1, \\quad q={\\bf i^{\\prime }}\\pm 1 $ for some even $m$ ."
],
[
"${\\bf i^{\\prime }}, {\\bf j^{\\prime }}>0$",
"Since $i^{\\prime }$ is odd and $j^{\\prime }$ is even, $\\mathrm {gcd}(i^{\\prime },j^{\\prime })$ does not contain any positive power of 2 in its prime divisors; therefore, ${\\bf i^{\\prime }}$ remains odd and ${\\bf j^{\\prime }}$ remains even; in other words, ${\\bf i^{\\prime }}+{\\bf j^{\\prime }}$ is odd.",
"From this observation, tedious but elementary manipulations lead to the following: $ \\left.\\begin{array}{c}c(m{\\bf j^{\\prime }}+1,m{\\bf i^{\\prime }}+1)\\\\c(m{\\bf j^{\\prime }}-1,m{\\bf i^{\\prime }}-1)\\\\d(m{\\bf j^{\\prime }}+1,m{\\bf i^{\\prime }}-1)\\\\d(m{\\bf j^{\\prime }}-1,m{\\bf i^{\\prime }}+1)\\end{array}\\right\\rbrace =\\left\\lbrace \\begin{array}{cll}1 & \\mbox{for}&m=0,4,8,12,..., \\\\0 & \\mbox{otherwise}.&\\end{array}\\right.",
"$ Next, tedious but elementary manipulation reveal that $ (-1)^{\\frac{p-1}{2}}=\\left\\lbrace \\begin{array}{ccc}1 & \\mbox{if} & p=m{\\bf j^{\\prime }}+ 1,\\\\0 & \\mbox{if} & p=m{\\bf j^{\\prime }}- 1.\\end{array}\\right.$ Putting everything together yields $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }({\\bf i^{\\prime }},{\\bf j^{\\prime }})}&=\\frac{4}{\\pi ^2}\\left(2 +\\sum _{m=4,8,...}\\frac{1}{(m{\\bf j^{\\prime }}+1)(m{\\bf i^{\\prime }}-1)} + \\frac{1}{(m{\\bf j^{\\prime }}+1)(m{\\bf i^{\\prime }}+1)} \\right.",
"\\\\& \\qquad \\qquad -\\left.\\frac{1}{(m{\\bf j^{\\prime }}^{\\prime }+1)()m{\\bf i^{\\prime }}-1}-\\frac{1}{(m{\\bf j^{\\prime }}-1)(m{\\bf i^{\\prime }}+1)}\\right)\\\\&=\\frac{4}{\\pi ^2}\\left( 2+\\sum _{m=4,8,...} \\frac{4}{(m^2{\\bf j^{\\prime }}^2-1)(m^2{\\bf i^{\\prime }}^2-1)} \\right)\\\\&=\\frac{4}{\\pi ^2}\\left( 2+ \\frac{4}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\sum _{m=4,8,...}\\left( \\frac{{\\bf j^{\\prime }}^2}{m^2{\\bf j^{\\prime }}^2-1}-\\frac{{\\bf i^{\\prime }}^2}{m^2{\\bf i^{\\prime }}^2-1}\\right)\\right).$ In order to derive the closed form solution, we invoke Lemma REF and find that $\\sqrt{p_{\\mathrm {max}}^{\\leftrightarrow }({\\bf i^{\\prime }},{\\bf j^{\\prime }})}=\\frac{8}{\\pi ^2} \\left(\\frac{{\\bf i^{\\prime }}^2}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left( \\frac{\\pi }{4{\\bf i^{\\prime }}}\\right) \\cot \\left( \\frac{\\pi }{4{\\bf i^{\\prime }}}\\right)-\\frac{{\\bf j^{\\prime }}^2}{{\\bf i^{\\prime }}^2-{\\bf j^{\\prime }}^2}\\left( \\frac{\\pi }{4{\\bf j^{\\prime }}}\\right) \\cot \\left( \\frac{\\pi }{4{\\bf j^{\\prime }}}\\right)\\right).$"
],
[
"${\\bf i^{\\prime }}< 0 <{\\bf j^{\\prime }}$",
"Again the preceding formula remains valid after replacing $i^{\\prime }$ by $-i^{\\prime }$ ."
]
] | 1403.0159 |
[
[
"Electron Transport in MoWSeS Monolayers in Presence of an External\n Electric Field"
],
[
"Abstract The influence of an external electric field on single-layer transition-metal dichalcogenides TX2 with T = Mo, W and X = S, Se (MoWSeS) have been investigated by means of density-functional theory within two-dimensional periodic boundary conditions under consideration of relativistic effects including the spin-orbit interactions.",
"Our results show that the external field modifies the band structure of the monolayers, in particular the conduction band.",
"This modification has, however, very little influence on the band gap and effective masses of holes and electrons at the K point, and also the spin-orbit splitting of these monolayers is almost unaffected.",
"Our results indicate a remarkable stability of the electronic properties of TX2 monolayers with respect to gate voltages.",
"A reduction of the electronic band gap is observed only starting from field strengths of 2.0 V per {\\AA} (3.5 V per {\\AA}) for selenides (sulphides), and the transition to a metallic phase would occur at fields of 4.5 V per {\\AA} (6.5 V per {\\AA})."
],
[
"Acknowledgements",
"Financial support by Deutsche Forschungsgemeinschaft (DFG, HE 3543/17-1) and the European Commission through the Initial Training Network (ITN) MoWSeS (GA FP7-PEOPLE-2012-ITN) and the Industrial Academic Partnership Pathways (IAPP) QUASINANO (GA FP7-PEOPLE-2009-IAPP) is acknowledged."
]
] | 1403.0552 |
[
[
"A distributional equality for suprema of spectrally positive L\\'evy\n processes"
],
[
"Abstract Let $Y$ be a spectrally positive L\\'evy process with $E Y_1<0$, $C$ an independent subordinator with finite expectation, and $X=Y+C$.",
"A curious distributional equality proved in Huzak et al., Ann.",
"Appl.",
"Probab.",
"14 (2004) 1278--1397, states that if $E X_1<0$, then $\\sup_{0\\le t <\\infty}Y_t$ and the supremum of $X$ just before the first time its new supremum is reached by a jump of $C$ have the same distribution.",
"In this paper we give an alternative proof of an extension of this result and offer an explanation why it is true."
],
[
"Introduction",
"Let $Y=(Y_t)_{t\\ge 0}$ be a one-dimensional spectrally positive Lévy process such that $Y_1$ is integrable and ${E}Y_1<0$ .",
"By the law of large numbers, cf.",
"[6], $\\lim _{t\\rightarrow \\infty }Y_t=-\\infty $ a.s., and consequently $\\overline{Y}_{\\infty }:=\\sup _{t\\ge 0}Y_t <\\infty $ .",
"Assume further that $C=(C_t)_{t\\ge 0}$ is a subordinator without drift independent of $Y$ with jumps denoted by $\\Delta C_t=C_t-C_{t-}$ .",
"By setting $X_t:=Y_t+C_t$ we see that $X=(X_t)_{t\\ge 0}$ is again a spectrally positive Lévy process.",
"Its supremum process is defined by $\\overline{X}_t:=\\sup _{0\\le s \\le t}X_s$ .",
"Let $\\sigma :=\\inf \\lbrace t>0:\\, \\Delta C_t>\\overline{X}_{t-}-X_{t-}\\rbrace $ be the first time the supremum of $X$ is reached by a jump of the subordinator $C$ .",
"As a consequence of spectral positivity of $Y$ it holds that $\\sigma >0$ a.s. (cf.",
"[3] and [7]).",
"Assume further that $C_1$ has finite expectation satisfying ${E}X_1={E}Y_1+{E}C_1 <0$ .",
"Then the following distributional equality was proved in [3]: $\\sup _{0\\le t <\\infty } Y_t \\stackrel{d}{=} \\sup _{0\\le t<\\sigma } X_t\\, .$ Note that the right-hand side is the supremum of the process $X=Y+C$ just before the first time the new supremum of $X$ is reached by a jump of the subordinator $C$ .",
"As such, one might expect that its distribution depends on the subordinator $C$ .",
"A curious fact about (REF ) is that the right-hand side is independent of $C$ (as long as ${E}Y_1+{E}C_1<0$ ).",
"The proof given in [3] does not reveal why this is so – the equality (REF ) was obtained by deriving the Pollaczek-Khintchine formula for the overall supremum $\\sup _{0\\le t<\\infty }X_t$ in two different ways and by equating factors in the Laplace transforms.",
"The goal of this paper is to give an alternative proof of (a slight extension) of (REF ) which hopefully sheds more light on why this equality holds true and what are the limitations of further extensions of the formula.",
"More precisely, we will prove the following result.",
"Theorem 1.1 Let $Y=(Y_t)_{t\\ge 0}$ be a spectrally positive Lévy process such that $Y_1$ is integrable and ${E}Y_1<0$ , let $C=(C_t)_{t\\ge 0}$ be a subordinator without a drift independent of $Y$ such that $C_1$ is integrable and ${E}Y_1 +{E}C_1 \\le 0$ .",
"If $X=Y+C$ and $\\sigma $ is defined by (REF ), then the distributional equality (REF ) holds true.",
"Note that we extend [3] to the case when ${E}X_1=0$ .",
"On the other hand, we also show that when ${E}X_1 >0$ (REF ) is no longer valid.",
"The proof of Theorem REF is split into two parts.",
"The first part deals with the case when both $Y$ and $C$ are compound Poisson processes.",
"Then $Y$ can be written as $Y_t= -ct+Z_t $ where $Z=(Z_t)_{t\\ge 0}$ is a subordinator with finite Lévy measure such that ${E}Z_1 <c$ .",
"Similarly, $X_t=-ct + Z_t +C_t$ .",
"Then $\\sup _{0\\le t<\\infty }Y_t$ can be written as a sum of geometrically many overshoots leading to a new supremum.",
"One minus the parameter of this geometric distribution is the probability that $Y$ goes above level zero, while the distribution of each overshoot is equal to the distribution of the overshoot of level zero (conditional on the fact that level zero is reached).",
"On the other hand, $\\sup _{0\\le t<\\sigma }X_t$ is also a sum of geometrically many overshoots leading to a new supremum reached by a jump of $Z$ .",
"One minus the parameter of the geometric distribution is equal to the probability that $X$ goes over zero by a jump of $Z$ , while each of the overshoots has the same distribution as the distribution of the overshoot of level zero (conditional on the fact that level zero is reached by a jump of $Z$ ).",
"An explanation that the corresponding quantities in these two situations are equal relies on Takács' formula [8] and a fluctuation identity for spectrally one-sided Lévy processes [1].",
"In this part we use and extend the arguments from [4].",
"We further show that if ${E}X_1>0$ , then $\\sup _{0\\le t<\\infty }Y_t$ and $\\sup _{0\\le t<\\sigma }X_t$ have different distributions.",
"This part of the proof is explained in Section of the paper.",
"In Section we give two approximation results.",
"The first one roughly says that if a Lévy process $Y$ is a distributional limit of a sequence $(Y^{(n)})_{n\\ge 1}$ of Lévy processes, $C$ is a subordinator with finite Lévy measure, and if (REF ) holds for approximating processes, then it also holds for the limiting process, see Proposition .",
"The second result is of a similar nature, only the subordinator $C$ with infinite Lévy measure is approximated by a sequence of subordinators of finite Lévy measures, see Proposition REF .",
"Both results rely on certain approximations of functions in Skorohod's space $, see Lemmas \\ref {l:approx-CPP} and \\ref {l:approx-general}.",
"Proofs of the lemmas are quite technical and the reader may want to continue with the general case before delving into proofs.$ In Section we check that conditions required in approximation results are valid and give the proof of Theorem REF for the general case.",
"We finish the paper with a discussion on how essential is the assumption on spectral positivity of $Y$ for validity of Theorem REF and show that the theorem fails in case $Y$ does not creep downwards (i.e.",
"it continuously crosses every level from above with probability zero, cf.",
"[1] for details)."
],
[
" The case of compound Poisson process",
"Let $Z=(Z_t)_{t\\ge 0}$ be a subordinator with no drift, finite Lévy measure $\\nu _Z$ and the Laplace exponent $\\phi _Z$ given by $\\phi _Z(\\lambda )=\\int _{(0,\\infty )}(1-e^{-\\lambda x})\\, \\nu _Z(dx)\\, .$ Assume further that $\\mu _Z:={E}Z_1=\\int _0^{\\infty }\\nu _Z(x,\\infty )\\, dx <\\infty $ .",
"Let $C=(C_t)_{t\\ge 0}$ be another subordinator, independent of $Z$ , with no drift, finite Lévy measure $\\nu _C$ and the Laplace exponent $\\phi _C$ .",
"Assume also that $\\mu _C:={E}C_1 < \\infty $ .",
"Let $(\\mathcal {F}_t)_{t\\ge 0}$ be the natural filtration generated by subordinators $Z$ and $C$ , augmented in the usual way.",
"By independence, $Z$ and $C$ do not jump at the same time.",
"We will tacitly use this fact throughout the paper.",
"Let $c>0$ .",
"The process $X=(X_t)_{t\\ge 0}$ defined by $X_t:=-ct+Z_t+C_t$ is a spectrally positive Lévy process such that $X_1$ is integrable and ${E}X_1=-c+\\mu _Z+\\mu _C$ .",
"Note that $Z$ and $C$ have symmetric roles in $X$ .",
"The process $-X$ is a spectrally negative Lévy process with the Laplace exponent $\\psi =\\psi _X$ defined by ${E}\\left[e^{\\lambda (-X_t)}\\right]=e^{t \\psi _X(\\lambda )}\\, ,\\qquad \\lambda >0\\, ,$ and $\\psi _X:[0,\\infty )\\rightarrow (-\\infty ,\\infty )$ is strictly convex (cf. [1]).",
"Let $\\Phi _X(0)$ be the largest solution of the equation $\\psi _X(\\lambda )=0$ .",
"Then $\\psi _X:[ \\Phi _X(0), \\infty )\\rightarrow [0,\\infty )$ is a bijection, and its inverse is denoted by $\\Phi _X$ .",
"Note that $\\Phi _X(0)=0$ if and only if ${E}X_1\\le 0$ .",
"For $y\\le 0$ let $T_y^X:=\\inf \\lbrace t\\ge 0:\\, X_t<y\\rbrace =\\inf \\lbrace t\\ge 0:\\, X_t=y\\rbrace \\, ,$ where the equality follows from the fact that $X$ is spectrally positive.",
"We deduce from [1] that $(T_y^X)_{y\\le 0}$ is a (possibly killed) subordinator with the Laplace exponent $\\Phi _X$ .",
"In particular, ${P}(T_y^X<\\infty )=e^{\\Phi _X(0)y}\\, ,\\quad y\\le 0\\, .$ Let $\\tau _0^X:=\\inf \\lbrace t>0:\\, X_t>0\\rbrace $ be the first passage time of $X$ above the level zero.",
"Note that at $\\tau _0^X$ the process $X$ makes a jump over zero, and that either $Z$ or $C$ can make this jump.",
"In the next proposition we compute the probability that the jump was made by $Z$ and the distribution of the overshoot.",
"The same result was proved in [4] in case when ${E}X_1 <0$ .",
"A related result is given in [1].",
"Proposition 2.1 Let $X_0=0$ .",
"For $y\\le 0$ and $x>0$ , ${P}\\big (\\tau _0^X<\\infty , X_{\\tau _0^X-}\\in dy, X_{\\tau _0^X}\\in dx, \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )= \\frac{1}{c} \\, e^{\\Phi _X(0)y} \\nu _Z(-y+dx)\\, dy\\, .$ Consequently, ${{P}\\big (\\tau _0^X<\\infty , X_{\\tau _0^X}\\in dx, \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )}\\nonumber \\\\&=&\\frac{1}{c} \\, \\left( \\nu _Z(x,\\infty )-\\Phi _X(0)e^{\\Phi _X(0)x}\\int _x^{\\infty }e^{-\\Phi _X(0)u}\\nu _Z(u,\\infty )\\, du\\right)\\, dx$ and ${P}\\big (\\tau _0^X<\\infty , \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )= \\frac{1}{c} \\int _0^{\\infty }e^{-\\Phi _X(0)u} \\nu _Z(u,\\infty )\\, du\\, .$ The proof relies on the following two results: The first one is (a version of) the remarkable formula due to Takács, see [8] which states that ${P}\\left(\\sup _{0\\le s\\le t} X_s >0 \\, \\Big | \\, X_t\\right)=1-\\left(-\\frac{X_t}{ct}\\right)^+\\, .$ The second result is the another remarkable identity valid for one-sided Lévy process, see [1], which in our case says that $t {P}(T_y^X\\in dt)\\, dy = (-y){P}(X_t \\in dy)\\, dt \\, \\quad \\text{as measures on } [0,\\infty )\\times (-\\infty , 0]\\, .$ Recall that $\\overline{X}_t=\\sup _{0\\le s \\le t}X_s$ .",
"By use of the compensation formula applied to the two-dimensional Poisson point process $(\\Delta Z_t, \\Delta C_t)_{t\\ge 0}$ with the characteristic measure concentrated on positive coordinate axes, we see that ${{P}\\big (\\tau _0^X<\\infty , X_{\\tau _0^X-}\\in dy, X_{\\tau _0^X}\\in dx, \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )}\\\\&=& {E}\\left(\\sum _{0<t<\\infty } I_{(X_{t-}\\in dy, \\overline{X}_{t-}\\le 0)} I_{(\\Delta Z_t\\in -y+dx)}\\right) \\\\&=& {E}\\left(\\int _0^{\\infty } dt\\, I_{(X_{t-}\\in dy, \\overline{X}_{t-}\\le 0)}\\nu _Z(-y+dx)\\right)\\\\&=& \\int _0^{\\infty }dt\\, {P}(X_t\\in dy, \\overline{X}_t\\le 0)\\nu _Z(-y+dx)\\, .$ The compensation formula was used in the second equality, while the last equality follows from the fact that $X_{t-}=X_t$ for a.e.",
"$t$ .",
"By using first (REF ), then (REF ) and finally (REF ) we see that the last line is equal to ${\\int _0^{\\infty }dt\\, {P}(X_t\\in dy) \\frac{-y}{ct} \\nu _Z(-y+dx)=\\frac{1}{c} \\int _0^{\\infty }{P}(T_y^X\\in dt)\\, dy \\, \\nu _Z(-y+dx)} \\\\&=&\\frac{1}{c}{P}(T_y^X<\\infty )\\nu _Z(-y+dx)\\, dy =\\frac{1}{c} \\, e^{\\Phi _X(0)y} \\nu _Z(-y+dx)\\, dy\\, .$ This proves (REF ).",
"By integrating over $y$ we obtain (REF ).",
"Indeed, define the measure $\\rho $ on $(0,\\infty )$ by $\\rho (A):=\\frac{1}{c}\\int _{-\\infty }^0 e^{\\Phi _X(0)y}\\nu _Z(-y+A)\\, dy=\\frac{1}{c}\\int _0^{\\infty }e^{-\\Phi _X(0)u}\\nu _Z(u+A)\\, du\\, ,\\quad A\\subset (0,\\infty )\\, .$ When $A=(x,\\infty )$ , $x>0$ , we get $\\rho (x,\\infty )&=&\\frac{1}{c} \\int _0^{\\infty } e^{-\\Phi _X(0)u} \\nu (u+x,\\infty )\\, du\\\\&=&\\frac{1}{c}\\, e^{\\Phi _X(0) x}\\int _x^{\\infty }e^{-\\Phi _X(0)u}\\nu _Z(u,\\infty )\\, du\\, .$ It follows that the measure $\\rho $ has a density $\\rho (x)=-\\frac{d}{dx}\\rho (x,\\infty )$ .",
"By differentiating we get that $\\rho (x)=\\frac{1}{c} \\nu _Z(x,\\infty )-\\Phi _X(0)\\rho (x,\\infty )$ which proves (REF ).",
"Finally, ${P}\\big (\\tau _0^X<\\infty , \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )=\\rho (0,\\infty )=\\frac{1}{c} \\int _0^{\\infty }e^{-\\Phi _X(0)u} \\nu _Z(u,\\infty )\\, du\\, ,$ proving (REF ).",
"By applying Proposition REF to the process $Y_t:=-ct+Z_t$ the following identities follow: $& {P}\\big (\\tau _0^Y<\\infty , Y_{\\tau _0^Y-}\\in dy, Y_{\\tau _0^Y}\\in dx \\big )= \\frac{1}{c} \\, e^{\\Phi _Y(0)y} \\nu _Z(-y+dx)\\, dy \\\\& {P}\\big (\\tau _0^Y<\\infty , Y_{\\tau _0^Y}\\in dx \\big )=\\frac{1}{c} \\, \\left( \\nu _Z(x,\\infty )-\\Phi _Y(0)e^{\\Phi _Y(0)x}\\int _x^{\\infty }e^{-\\Phi _Y(0)u}\\nu _Z(u,\\infty )\\, du\\right)\\, dx \\\\& {P}\\big (\\tau _0^Y<\\infty \\big )= \\frac{1}{c} \\int _0^{\\infty }e^{-\\Phi _Y(0)y} \\nu _Z(y,\\infty )\\, dy\\, .$ Corollary 2.2 Suppose that $\\mu _Z+\\mu _C\\le c$ .",
"Then ${P}\\big (\\tau _0^X<\\infty , X_{\\tau _0^X}\\in dx, \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )={P}\\big (\\tau _0^Y<\\infty , Y_{\\tau _0^Y}\\in dx \\big )=\\frac{1}{c}\\, \\nu _Z(x,\\infty )\\, dx\\, ,$ ${P}\\big (\\tau _0^X<\\infty , \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )={P}\\big (\\tau _0^Y<\\infty \\big )=\\frac{\\mu _Z}{c}\\, .$ and consequently ${P}\\big ( X_{\\tau _0^X}\\in dx \\, \\big | \\tau _0^X<\\infty ,\\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X}\\big )={P}\\big (Y_{\\tau _0^Y}\\in dx \\, \\big |\\, \\tau _0^Y<\\infty \\big )=\\frac{\\nu _Z(x,\\infty )\\, dx}{\\mu _Z}\\, .$ The assumption implies that ${E}X_1 \\le 0$ , hence $\\Phi _X(0)=\\Phi _Y(0)=0$ .",
"Now the first equality follows from (REF ) and (), the second one is the consequence of (REF ), () and the fact that $\\mu _Z=\\int _0^{\\infty }\\nu _Z(x,\\infty )\\, dx$ , while the third is immediate from the first two.",
"We are now ready to prove Theorem REF in the compound Poisson case.",
"As above, $Y_t=-ct+Z_t$ , $X_t=-ct+Z_t+C_t$ where $\\mu _Z+\\mu _C\\le c$ .",
"Since ${E}Y_1<0$ , $\\sup _{t\\ge 0}Y_t <\\infty $ a.s.",
"This supremum can be written as the geometric sum of overshoots leading to a new supremum.",
"The distribution of the overshoots and the parameter of the geometric random variable (i.e.",
"the probability of reaching the new supremum) are given by () and ().",
"On the other hand, $\\sup _{0\\le t <\\sigma }X_t$ is also a geometric sum of those overshoots leading to the new supremum of $X$ obtained by jumps of $Z$ .",
"The distribution of such overshoots and the parameter of the geometric random variable are by (REF ) and (REF ) equal as before.",
"This shows that $\\sup _{t\\ge 0}Y_t$ and $\\sup _{0\\le t <\\sigma }X_t$ have the same distribution.",
"We now make these arguments more precise.",
"Proof of Theorem REF – compound Poisson process case.",
"Let $\\tau ^{(0)}=0$ , $\\tau ^{(1)}=\\inf \\lbrace t>0:\\, Y_t>\\overline{Y}_{t-}\\rbrace $ and for $n\\ge 2$ define $\\tau ^{(n)}=\\inf \\lbrace t>\\tau ^{(n-1)}:\\, Y_t>\\overline{Y}_{t-}\\rbrace $ on $\\lbrace \\tau ^{(n-1)}<\\infty \\rbrace $ .",
"Note that $\\tau ^{(1)}=\\tau _0^Y$ .",
"On $\\lbrace \\tau ^{(n)}<\\infty \\rbrace $ define $I_n:= Y_{\\tau ^{(n)}}-\\overline{Y}_{\\tau ^{(n-1)}}$ , and let $N:=\\max \\lbrace n:\\, \\tau ^{(n)}<\\infty \\rbrace $ .",
"Then $\\overline{Y}_{\\infty }=\\sup _{t\\ge 0}Y_t=\\sum _{n=1}^N I_n$ (with the convention that $\\sum _{n=1}^0 = 0$ ).",
"By the strong Markov property $N$ has geometric distribution with parameter ${P}(\\tau ^{(1)}=\\infty )=1-\\frac{\\mu _Z}{c}$ , and for all $n\\ge 1$ , conditionally on $\\tau ^{(n)}<\\infty $ , $I_n$ has the distribution $\\frac{1}{\\mu _Z}\\nu _Z(x,\\infty )\\, dx$ , see (REF ).",
"On the other hand, let $\\varsigma ^{(0)}=0$ , and let $\\widetilde{\\varsigma } =\\widetilde{\\varsigma }^{(1)}:=\\inf \\lbrace t>0:\\, X_t>\\overline{X}_{t-},\\, \\Delta X_t=\\Delta Z_t\\rbrace =\\inf \\lbrace t>0:\\, \\Delta Z_t>\\overline{X}_{t-}-X_{t-}\\rbrace $ be the first time the new supremum of $X$ is reached by the jump of $Z$ .",
"Inductively, for $n\\ge 2$ we define $\\widetilde{\\varsigma }^{(n)}:=\\inf \\lbrace t> \\widetilde{\\varsigma }^{(n-1)}:\\, X_t>\\overline{X}_{t-},\\, \\Delta X_t=\\Delta Z_t\\rbrace $ on $\\lbrace \\widetilde{\\varsigma }^{(n-1)}<\\infty \\rbrace $ .",
"We are interested in times $\\widetilde{\\varsigma }^{(n)}$ only if they occur before $\\sigma $ .",
"Hence we define $\\varsigma ^{(n)}:=\\widetilde{\\varsigma }^{(n)}I_{(\\widetilde{\\varsigma }^{(n)}<\\sigma )}+\\infty I_{(\\widetilde{\\varsigma }^{(n)}>\\sigma )}$ , $n\\ge 1$ .",
"Let further $J_n:= X_{\\varsigma ^{(n)}}-\\overline{X}_{\\varsigma ^{(n-1)}}$ on $\\lbrace \\varsigma ^{(n)}<\\infty \\rbrace $ , and $M:=\\max \\lbrace n:\\, \\varsigma ^{(n)}<\\infty \\rbrace $ .",
"Then $\\sup _{0\\le t<\\sigma }X_t=\\sum _{n=1}^{M}J_n\\, .$ Again, by the strong Markov property at stopping times $\\varsigma ^{(n)}$ , $M$ has geometric distribution with parameter ${P}(\\varsigma ^{(1)}=\\infty )=1-{P}(\\tau _0^X<\\infty , \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X})=1-\\frac{\\mu _Z}{c}$ , see (REF ).",
"Further, by (REF ), for all $n\\ge 1$ , conditionally on $\\varsigma ^{(n)}<\\infty $ , $J_n$ has the distribution $\\frac{1}{\\mu _Z}\\nu _Z(x,\\infty )\\, dx$ .",
"This finishes the proof.",
"$\\Box $ Proposition 2.3 Assume that $\\mu _Z<c<\\mu _Z+\\mu _C$ .",
"Then $\\sup _{t\\ge 0} Y_t$ and $\\sup _{0\\le t <\\sigma } X_t$ have different distributions.",
"For the proof we need the following simple result.",
"Lemma 2.4 Let $(\\xi _n)_{n\\ge 1}$ be an i.i.d.",
"sequence of strictly positive random variables, $S_n=\\xi _1+\\cdots +\\xi _n$ , and let $N$ be an independent geometric random variable with parameter $1-\\rho \\in (0,1)$ .",
"Similarly, let $(\\eta _n)_{n\\ge 1}$ be an i.i.d.",
"sequence of strictly positive random variables, $T_n=\\eta _1+\\cdots +\\eta _n$ , and let $M$ be an independent geometric random variable with parameter $1-\\varrho \\in (0,1)$ .",
"If $S_N\\stackrel{d}{=}T_M$ , then $\\rho =\\varrho $ and $\\xi _1\\stackrel{d}{=}\\eta _1$ .",
"Let $f(\\lambda )={E}\\left[e^{-\\lambda \\xi _1}\\right]$ and $g(\\lambda )={E}\\left[e^{-\\lambda \\eta _1}\\right]$ .",
"Then the Laplace transforms of $S_N$ and $T_M$ are given by ${E}\\left[e^{-\\lambda S_N}\\right]=\\frac{1-\\rho }{1-\\rho f(\\lambda )}\\, ,\\qquad {E}\\left[e^{-\\lambda T_M}\\right]=\\frac{1-\\varrho }{1-\\varrho g(\\lambda )}\\, .$ By the assumption, these two Laplace transforms are equal.",
"By letting $\\lambda \\rightarrow \\infty $ and using that $\\lim _{\\lambda \\rightarrow \\infty }f(\\lambda )=\\lim _{\\lambda \\rightarrow \\infty }g(\\lambda )=0$ , we first get that $\\varrho =\\rho $ , and then $g=f$ .",
"Proof of Proposition REF .",
"We use the notation from the proof of Theorem REF given above.",
"The representations (REF ) and (REF ) are still valid.",
"On the other hand, by the assumption we have that $\\Phi _Y(0)=0$ while $\\Phi _X(0)>0$ .",
"Therefore, ${P}(\\varsigma ^{(1)}=\\infty )&=&1-{P}(\\tau _0^X<\\infty , \\Delta X_{\\tau _0^X}=\\Delta Z_{\\tau _0^X})\\\\&=&1-\\frac{1}{c} \\int _0^{\\infty }e^{-\\Phi _X(0)y} \\nu _Z(y,\\infty )\\, dy \\\\&> & 1-\\frac{1}{c} \\int _0^{\\infty } \\nu _Z(y,\\infty )\\, dy\\\\&=&1-\\frac{\\mu _Z}{c}={P}(\\tau ^{(1)}=\\infty )\\, .$ The claim now follows from Lemma REF .",
"$\\Box $ Remark 2.5 (a) Assume that $\\mu _C={E}C_1=\\infty $ so that $\\mu _Z<c<\\mu _Z+\\mu _C$ .",
"It is easy to see that Proposition REF is still valid, hence we conclude that Proposition REF also holds.",
"(b) Assume that $\\mu _Z>c$ .",
"Then $Y$ drifts to $+\\infty $ , hence ${P}(\\tau _0^Y<\\infty )=1$ .",
"We check that () also gives this result.",
"Indeed, since $\\Phi _Y(0)>0$ solves the equation $\\psi _Y(\\lambda )=c\\lambda -\\phi _Z(\\lambda )=0$ , we have that $c\\Phi _Y(0)= \\phi _Z(\\Phi _Y(0))$ .",
"By use of $\\phi _Z(\\lambda )=\\int _{(0,\\infty )}(1-e^{-\\lambda x})\\, \\nu _Z(dx)=\\lambda \\int _0^{\\infty }e^{-\\lambda x}\\nu _Z(x,\\infty )\\, dx\\, ,$ we see that $c=\\frac{1}{\\Phi _Y(0)}\\, \\phi _Z(\\Phi _Y(0))=\\int _0^{\\infty }e^{-\\Phi _Y(0)x}\\nu _Z(x,\\infty )\\, dx\\, .$"
],
[
"Two approximation results",
"Let $[0,\\infty ),{R})$ denote the space of all functions $x:[0,\\infty )\\rightarrow {R}$ that are right-continuous and have left limits.",
"Endowed with the Skorohod $J_1$ -topology $ becomes a Polish space (cf.~\\cite [Chapter VI]{JS}).",
"Let $x(t):=0s tx(s)$.$ Lemma 3.1 Assume that $(y_n)_{n\\ge 1}\\subset , $ y and $y_n\\rightarrow y$ in $.",
"Let $ c be of the form $c(t)=\\sum _{k\\ge 1}\\Delta c(s_k)I_{(s_k\\le t)}$ where $0<s_1<s_2<\\dots $ , $\\lim _k{s_k}=\\infty $ , and $\\Delta c(s_k)>0$ for all $k\\ge 1$ .",
"Let $x:=y+c$ , $x_n:=y_n+c$ , $n\\ge 1$ , and assume that $\\Delta y(s_k)=\\Delta y_n(s_k)=0 \\quad \\text{for all }n\\ge 1 \\text{ and }k\\ge 1\\, ,$ and $\\Delta c(s_k)\\ne \\overline{x}(s_k-)-x(s_k-)\\quad \\text{for all }k\\ge 1\\, .$ Define $\\sigma &:=&\\inf \\lbrace t>0:\\, \\Delta c(t)> \\overline{x}(t-)-x(t-)\\rbrace \\, ,\\\\\\sigma _n &:=&\\inf \\lbrace t>0:\\, \\Delta c(t)> \\overline{x}_n(t-)-x_n(t-)\\rbrace \\, .$ Then $\\sigma =\\lim _{n\\rightarrow \\infty }\\sigma _n$ and $\\overline{x}(\\sigma -)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(\\sigma _n-)$ .",
"Remark 3.2 Note that the function $c$ is a typical realization of the subordinator $C$ with finite Lévy measure.",
"Assumption (REF ) says that $y$ and $y_n$ do not jump at times when $c$ has a jump.",
"Finally, assumption (REF ) says that no jump of $c$ will hit the exact level of the current maximum of the function $x$ .",
"Proof of Lemma REF .",
"We first note that $y_n(t)\\rightarrow y(t)$ at every continuity point of $y$ .",
"Further, by [5] and the assumption (REF ) we have $x_n\\rightarrow x$ in $.$ Since $x_n=y_n$ and $x=y$ on $[0,s_1)$ , and since $y$ and $y_n$ are continuous at $s_1$ , it follows that $\\overline{x}(s_1-)=\\overline{y}(s_1-)\\, &\\qquad & x(s_1-)=y(s_1-)=y(s_1)\\, ,\\\\\\overline{x}_n(s_1-)=\\overline{y}_n(s_1-)\\, &\\qquad & x_n(s_1-)=y_n(s_1-)=y_n(s_1)\\, .$ Further, $y(s_1)=\\lim _{n\\rightarrow \\infty }y_n(s_1)$ implies that $x(s_1-)=\\lim _{n\\rightarrow \\infty }x_n(s_1-)\\, .$ By continuity of $y$ at $s_1$ it follows from [5] that $\\overline{y}(s_1)=\\lim _{n\\rightarrow \\infty }\\overline{y}_n(s_1)$ .",
"Again by continuity of $y$ at $s_1$ we see that $\\overline{y}(s_1)=\\overline{y}(s_1-)$ and similarly for $\\overline{y}_n$ .",
"We conclude that $\\overline{x}(s_1-)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(s_1-)\\, .$ Now (REF ) and (REF ) give together that $\\overline{x}(s_1-)-x(s_1-)=\\lim _{n\\rightarrow \\infty }\\big (\\overline{x}_n(s_1-)-x_n(s_1-)\\big )\\, .$ Suppose that $\\sigma =s_1$ , i.e.",
"$\\Delta c(s_1)>\\overline{x}(s_1-)-x(s_1-)$ .",
"It follows from (REF ) that there exists $n^{\\prime }_1\\in {N}$ such that for all $n\\ge n^{\\prime }_1$ it holds that $\\Delta c(s_1)>\\overline{x}_n(s_1-)-x_n(s_1-)$ .",
"Therefore, $\\sigma _n=s_1$ for all $n\\ge n^{\\prime }_1$ , and $\\overline{x}_n(\\sigma _n-)=\\overline{x}_n(s_1-)$ .",
"In particular, it holds that $\\lim _{n\\rightarrow \\infty }\\sigma _n=s_1=\\sigma $ , and $\\lim _{n\\rightarrow \\infty }\\overline{x}_n(\\sigma _n-)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(s_1-)=\\overline{x}(s_1-)=\\overline{x}(\\sigma -)$ .",
"Suppose now that $\\Delta c(s_1)<\\overline{x}(s_1-)-x(s_1-)$ which is by (REF ) same as $\\sigma \\ne s_1$ .",
"Then there exists $n_1\\in {N}$ such that for all $n\\ge n_1$ it holds that $\\Delta c(s_1)<\\overline{x}_n(s_1-)-x_n(s_1-)$ , i.e.",
"$\\sigma _n\\ne s_1$ .",
"So far we have shown that if $\\sigma =s_1$ , the claim of the lemma is true.",
"If $\\sigma \\ne s_1$ , we consider the interval $[0,s_2]$ .",
"Set $y^{(1)}=y$ , $y^{(1)}_n=y_n$ and $c^{(1)}=c$ .",
"Define $y^{(2)}$ , $y^{(2)}_n$ and $c^{(2)}$ in the following way: $y^{(2)}(t)&=&y^{(1)}(t)+\\Delta c(s_1)\\, ,\\\\y^{(2)}_n(t)&=&y^{(1)}_n(t)+\\Delta c(s_1)\\, ,\\\\c^{(2)}(t)&=&c^{(1)}(t)-\\Delta c(s_1)=\\sum _{j\\ge 2}\\Delta c(s_j)\\, ,$ (the jump $\\Delta c(s_1)$ is moved from $c$ to $y$ and $y_n$ ).",
"Then $y^{(2)}, y^{(2)}_n\\in , $ ny(2)n = y(2)$ in $ , $c^{(2)}$ has the same form as $c^{(1)}$ (piecewise constant with positive jumps in $s_2<s_3< \\dots $ ), and it holds that $x=y^{(2)}+c^{(2)} \\quad \\textrm {and }\\quad x_n=y^{(2)}_n+c^{(2)}\\, .$ The functions $y^{(2)}$ and $y^{(2)}_n$ satisfy the assumption (REF ) for all $k\\ge 2$ : $\\Delta y^{(2)}(s_k)=\\Delta y^{(2)}_n(s_k)=0$ , $n\\ge 1$ , $k\\ge 2$ .",
"The assumption (REF ) is also valid if $c=c^{(1)}$ is replaced by $c^{(2)}$ .",
"In the same way as in the first part of the proof we conclude that $& & \\overline{x}(s_2-)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(s_2-)\\, ,\\\\& & \\overline{x}(s_2-)-x(s_2-)=\\lim _{n\\rightarrow \\infty }\\big (\\overline{x}_n(s_2-)-x_n(s_2-)\\big )\\, .$ Suppose that $\\sigma =s_2$ , i.e.",
"$\\Delta c(s_2)=\\Delta c^{(2)}(s_2)>\\overline{x}(s_2-)-x(s_2-)$ .",
"It follows from () that there exists $n^{\\prime }_2\\in {N}$ such that for all $n\\ge n^{\\prime }_2$ it holds $\\Delta c(s_2)>\\overline{x}_n(s_2-)-x_n(s_2-)$ .",
"Since $\\sigma \\ne s_1$ , we have that $\\sigma _n\\ne s_1$ for $n\\ge n_1$ , hence for $n\\ge n_1\\vee n^{\\prime }_2$ it holds that $\\sigma _n=s_2$ .",
"This immediately implies $\\lim _{n\\rightarrow \\infty }\\sigma _n=s_2=\\sigma $ .",
"From (REF ) we conclude that $\\lim _{n\\rightarrow \\infty }\\overline{x}_n(\\sigma _n-)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(s_2-)=\\overline{x}(s_2-)=\\overline{x}(\\sigma -)$ .",
"If $\\Delta c(s_2)=\\Delta c^{(2)}(s_2)<\\overline{x}(s_2-)-f(s_2-)$ , then there exists $n_2\\in {N}$ such that for all $n\\ge n_2$ it holds that $\\Delta c(s_2)<\\overline{x}_n(s_2-)-x_n(s_2-)$ , i.e.",
"$\\sigma _n\\ne s_2$ .",
"The proof continuous by induction.",
"$\\Box $ If $Y$ , $Y^{(n)}$ , $n\\ge 1$ , are Lévy processes, we will write $Y^{(n)}\\Rightarrow Y$ for the weak convergence of induced probability measures on $.",
"We use the analogous notation for the weak convergence of random variables.\\begin{prop}Assume that Y is a Lévy process with infinite Lévy measure and (Y^{(n)})_{n\\ge 1} a sequence of Lévy processes such that Y^{(n)}\\Rightarrow Y.",
"Let C be an independent subordinator with finite Lévy measure.",
"Define X:=Y+C, X^{(n)}:=Y^{(n)}+C, n\\ge 1, and let\\sigma ^{(n)}:=\\inf \\lbrace t>0:\\, \\Delta C_t> \\overline{X}^{(n)}_{t-}-X^{(n)}_{t-}\\rbrace \\, .If\\begin{equation}\\sup _{0\\le t<\\infty }Y^{(n)}_t\\Rightarrow \\sup _{0\\le t <\\infty }Y_t\\end{equation}and\\begin{equation}\\sup _{0\\le t<\\infty }Y^{(n)}_t \\stackrel{d}{=}\\sup _{0\\le t <\\sigma ^{(n)}}X^{(n)}_t \\qquad \\text{for all }n\\ge 1 \\, ,\\end{equation}then also\\sup _{0\\le t<\\infty }Y_t \\stackrel{d}{=}\\sup _{0\\le t <\\sigma }X_t\\, .\\end{prop}\\begin{proof}Since is separable, it follows from Skorohod^{\\prime }s representation theorem, see \\cite [Theorem 6.7]{Bil}), that we can assume that processes Y and Y^{(n)}, n\\ge 1, are all defined on the same probability space (\\Omega , \\mathcal {F}, {P}) and that Y^{(n)}(\\omega )\\rightarrow Y(\\omega ) in for every \\omega \\in \\Omega .",
"We note that here Y^{(n)}(\\omega ) and Y(\\omega ) are regarded as functions in .",
"Without loss of generality we may assume that C is defined on the same probability space (\\Omega , \\mathcal {F}, {P}) and is independent of Y and (Y^{(n)})_{n\\ge 1}.",
"Clearly, for a.e.~\\omega \\in \\Omega , the function C(\\omega ) is of the form (\\ref {e:form-of-c}).Since C is independent of Y and Y^{(n)}, n\\ge 1,the assumption (\\ref {e:no-sim-jumps}) holds {P}-almost surely.Further, since X has infinite Lévy measure, the assumption (\\ref {e:strict-overshoot}) holds {P}-a.s.~by \\cite [Proposition VI 4]{Ber}.",
"We deduce from Lemma \\ref {l:approx-CPP} that\\sup _{0\\le t <\\sigma ^{(n)}}X^{(n)}_t \\rightarrow \\sup _{0\\le t <\\sigma }X_t\\, \\quad \\text{a.s.}The claim now follows from assumptions (\\ref {e:approx-1-a}) and (\\ref {e:approx-1-b}).\\end{proof}$ Lemma 3.3 Let $y\\in and let $ c be a non-decreasing function without continuous part such that $\\Delta y(s)\\Delta c(s)=0\\, \\quad \\text{for all }s>0\\, .$ For $n\\ge 1$ define $c_n(t):=\\sum _{0<s\\le t} \\Delta c(s) I_{\\left(\\Delta c(s)>\\frac{1}{n}\\right)}\\, .$ Let $x:=y+c$ , $x_n:=y+c_n$ , $n\\ge 1$ , and assume that $\\Delta c(t)\\ne \\overline{x}(t-)-x(t-)\\quad \\text{for all }t>0\\, .$ Define $\\sigma &:=&\\inf \\lbrace t>0:\\, \\Delta c(t)> \\overline{x}(t-)-x(t-)\\rbrace \\, ,\\\\\\sigma _n &:=&\\inf \\lbrace t>0:\\, \\Delta c_n(t)> \\overline{x}_n(t-)-x_n(t-)\\rbrace \\, .$ Then $\\sigma =\\lim _{n\\rightarrow \\infty }\\sigma _n$ and $\\overline{x}(\\sigma -)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(\\sigma _n-)$ where $\\overline{x}(0-):=0$ .",
"First note that $c(t)=\\sum _{0<s\\le t}\\Delta c(s)$ .",
"Further, $c_n\\in , the sequence $ (cn)n1$ is non-decreasing and converges to $ c$ uniformly on every finite interval $ [0,t]$.$ Case 1: $\\sigma >0$ .",
"Suppose that $\\sigma <\\infty $ and define $\\eta :=\\Delta c(\\sigma )- (\\overline{x}(\\sigma -)-x(\\sigma -))>0$ .",
"Let $0<\\epsilon <\\eta $ be arbitrary.",
"In particular, $\\Delta c(\\sigma )>\\eta >\\epsilon $ .",
"Choose $n_0\\in {N}$ such that $1/n_0 <\\epsilon $ and for all $n\\ge n_0$ it holds $c(\\sigma )-c_n(\\sigma )=\\sum _{0<s\\le \\sigma }\\Delta c(s) I_{\\left(\\Delta c(s) \\le \\frac{1}{n}\\right)}<\\frac{\\epsilon }{4}\\, .$ Note that for $n\\ge n_0$ , the function $c_n$ has a jump at $\\sigma $ : $\\Delta c_n(\\sigma )=\\Delta c(\\sigma )$ .",
"Further, on $[0,\\sigma ]$ we have $x-\\frac{\\epsilon }{4}=y+c-\\frac{\\epsilon }{4}<y+c_n=x_n\\le x\\, .$ If $ s \\in [0,\\sigma )$ is such that $x(s) > \\overline{x}(\\sigma -)-\\epsilon /4$ , then $x_n(s)>x(s )-\\epsilon /4>\\overline{x}(\\sigma -)-\\epsilon /2$ , which implies $\\overline{x}(\\sigma -)-\\frac{\\epsilon }{2} \\le \\overline{x}_n(\\sigma -) \\le \\overline{x}(\\sigma -)\\, .$ It also holds that $x(\\sigma -)-\\frac{\\epsilon }{4}\\le x_n(\\sigma -) \\le x(\\sigma -)\\, .$ From the last two displays we see that for $n\\ge n_0$ $\\overline{x}(\\sigma -)-x(\\sigma -)-\\frac{\\epsilon }{2}<\\overline{x}_n(\\sigma -)-x_n(\\sigma -)<\\overline{x}(\\sigma -)-x(\\sigma -)+\\frac{\\epsilon }{2}\\, .$ In particular, for all $n\\ge n_0$ $\\Delta c_n(\\sigma )-\\big (\\overline{x}_n(\\sigma -)-x_n(\\sigma -)\\big )&=&\\Delta c(\\sigma )-\\big (\\overline{x}_n(\\sigma -)-x_n(\\sigma -)\\big )\\\\&>&\\Delta c(\\sigma )-\\big (\\overline{x}(\\sigma -)-x(\\sigma -)+\\frac{\\epsilon }{2}\\big )\\\\&>&\\frac{\\epsilon }{2}\\, .$ This means that a new supremum of $x_n$ is reached by a jump of $c_n$ at time $\\sigma $ .",
"Therefore, $\\sigma _n\\le \\sigma $ .",
"To prove the opposite inequality, suppose that $0<t<\\sigma $ .",
"Together with the assumption (REF ) this implies that $\\Delta c(t)- (\\overline{x}(t-)-x(t-))<0$ .",
"Set $\\epsilon =-\\big (\\Delta c(t)- (\\overline{x}(t-)-x(t-))\\big )$ .",
"The same argument as above implies that for all $n$ large enough it holds $\\Delta c_n(t)-\\big (\\overline{x}_n(t-)-x_n(t-)\\big )<-\\frac{\\epsilon }{2}\\, .$ This means that $\\sigma _n>t$ .",
"We conclude that $\\sigma _n \\ge \\sigma $ .",
"Note that this argument is valid also in the case when $\\sigma =\\infty $ .",
"Together with the first part of the proof this implies $\\sigma _n=\\sigma $ for all sufficiently large $n$ .",
"In particular, $\\sigma =\\lim _{n\\rightarrow \\infty }\\sigma _n$ .",
"Suppose that $\\sigma <\\infty $ .",
"Since $\\epsilon $ is arbitrary, it follows from (REF ) that $\\overline{x}(\\sigma -)=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(\\sigma _n-)$ .",
"If $\\sigma =\\infty $ , then $\\overline{x}(\\infty )=\\lim _{n\\rightarrow \\infty }\\overline{x}_n(\\infty )$ .",
"Case 2: $\\sigma =0$ .",
"We claim that $\\limsup _{n\\rightarrow \\infty }\\sigma _n=0$ .",
"Suppose not, and let $\\delta :=\\limsup _{n\\rightarrow \\infty }\\sigma _n>0$ .",
"Since $\\sigma =0$ , there exists $t\\in (0,\\delta /2)$ such that $\\eta :=\\Delta c(t)- (\\overline{x}(t-)-x(t-))>0$ .",
"Let $0<\\epsilon <\\eta $ be arbitrary.",
"Now we follow the first part of the proof of Case 1 replacing $\\sigma $ by $t$ to conclude that there exists $n_0\\in {N}$ such that for all $n\\ge n_0$ $\\Delta c_n(t)-\\big (\\overline{x}_n(t-)-x_n(t-)\\big )>\\frac{\\epsilon }{2}\\, .$ It follows that $\\sigma _n\\le t<\\delta /2$ for all $n\\ge n_0$ which is a contradiction with $\\limsup _{n\\rightarrow \\infty }\\sigma _n=\\delta $ .",
"Since $\\overline{x}_n(\\sigma _n-)\\le \\overline{x}(\\sigma _n-)$ , see (REF ), we conclude that $\\limsup _{n\\rightarrow \\infty }\\overline{x}_n(\\sigma _n-)\\le \\limsup _{n\\rightarrow \\infty }\\overline{x}(\\sigma _n-)=0\\, .$ Let $C=(C_t)_{t\\ge 0}$ be a subordinator (without drift) with infinite Lévy measure.",
"For $n\\ge 1$ define the process $C^{(n)}=(C^{(n)}_t)_{t\\ge 0}$ by $C^{(n)}_t:=\\sum _{0<s\\le t}\\Delta C(s) I_{\\left(\\Delta C(s)>\\frac{1}{n}\\right)}\\, .$ Clearly, $C^{(n)}$ is a subordinator (without drift) with the finite Lévy measure $\\nu _n:=\\nu _{|\\left(\\frac{1}{n},\\infty \\right)}$ .",
"Proposition 3.4 Assume that $Y$ is a Lévy process and $C$ an independent subordinator with infinite Lévy measure.",
"Define $X:=Y+C$ , $X^{(n)}:=Y+C^{(n)}$ , $n\\ge 1$ , and let $\\sigma ^{(n)}:=\\inf \\lbrace t>0:\\, \\Delta C^{(n)}_t > \\overline{X}^{(n)}_{t-}-X^{(n)}_{t-}\\rbrace \\, .$ If $\\sup _{0\\le t<\\infty }Y_t \\stackrel{d}{=}\\sup _{0\\le t <\\sigma ^{(n)}}X^{(n)}_t \\qquad \\text{for all }n\\ge 1 \\, ,$ then also $\\sup _{0\\le t<\\infty }Y_t \\stackrel{d}{=}\\sup _{0\\le t <\\sigma }X_t\\, ,$ where $\\sup _{0\\le t <\\sigma }X_t:=0$ in case $\\sigma =0$ .",
"In the same way as in the proof of Proposition we see that the assumptions (REF ) and (REF ) hold for a.e.",
"$\\omega \\in \\Omega $ .",
"It follows from Lemma REF that $\\sup _{0\\le t <\\sigma ^{(n)}}X^{(n)}_t \\rightarrow \\sup _{0\\le t <\\sigma }X_t \\quad \\textrm {a.s.}$ Together with the assumption (REF ) this proves the claim.",
"Remark 3.5 Note that if $Y$ and $C$ in Proposition REF are such that $\\sigma =0$ a.s. and $\\sup _{t\\ge 0}Y_t$ is not identically zero, then (REF ) cannot hold.",
"We will come to this again at the end of Section ."
],
[
"The general case",
"Let $Y=(Y_t)_{t\\ge 0}$ be a spectrally positive Lévy process such that $\\gamma :={E}Y_1<0$ .",
"The characteristic exponent of $Y$ will be denoted by $\\Psi _Y$ , that is ${E}[\\exp \\lbrace iz Y_t\\rbrace ]=\\exp \\lbrace t\\Psi _Y(z)\\rbrace $ .",
"Then $\\Psi _Y(z)=-\\frac{1}{2} az^2 +i\\gamma z +\\int _{(0,\\infty )}\\left(e^{izx}-1-izx\\right)\\, \\nu (dx)\\, ,$ where $a\\ge 0$ is the diffusion coefficient.",
"With the centering function $c(x)\\equiv 1$ , the Lévy triplet of $Y$ is equal to $(a,\\gamma ,\\nu )$ , cf. [6].",
"Throughout this section we assume that the Lévy measure $\\nu $ of $Y$ is infinite.",
"We start by recording the well-known fact that such a process can be approximated by a sequence of spectrally positive Lévy processes $(Y^{(n)})_{n\\ge 0}$ with finite Lévy measures.",
"This approximation is in the sense of weak convergence of one-dimensional distributions as well as weak convergence in the Skorohod space $.$ Lemma 4.1 Let $Y=(Y_t)_{t\\ge 0}$ be a spectrally positive Lévy process such that $\\gamma :={E}Y_1<0$ .",
"There exists a sequence $(Y^{(n)})_{n\\ge 1}$ of spectrally positive Lévy processes with finite Lévy measures such that ${E}Y_1^{(n)}=\\gamma $ , $Y_1^{(n)}\\Rightarrow Y_1$ and $Y^{(n)}\\Rightarrow Y$ in $.$ Let $(a,\\gamma ,\\nu )$ be the Lévy triplet of $Y$ (with the centering function $c(x)\\equiv 1$ ).",
"Let $(x_n)_{n\\ge 1}$ be a sequence in $(0,1)$ such that $\\lim _{n\\rightarrow \\infty } x_n=0$ .",
"For $n\\ge 1$ we let $Y^{(n)}$ be the Lévy process with the triplet $(0,\\gamma , \\nu _n)$ (again with the centering function $c(x)\\equiv 1$ ) where $\\nu _n:=\\nu _{|(\\frac{1}{n},\\infty )}+\\frac{a}{x_n^2}\\delta _{x_n}\\, .$ Clearly, $\\nu _n$ is a finite measure concentrated on $(0,\\infty )$ and ${E}Y_1^{(n)}=\\gamma $ .",
"It is straightforward to check that the characteristic exponent $\\Psi _{Y^{(n)}}$ converges pointwise to $\\Psi _Y$ .",
"This is clearly equivalent to the weak convergence $Y_1^{(n)}\\Rightarrow Y_1$ .",
"Finally, the weak convergence of processes $Y^{(n)}\\Rightarrow Y$ follows from [5].",
"The approximating process $Y^{(n)}$ can be written in the form $Y^{(n)}_t=-c^{(n)}t+Z^{(n)}_t$ where $c^{(n)}>0$ , $Z^{(n)}$ is a subordinator with finite Lévy measure and no drift, and $-c^{(n)}+{E}Z^{(n)}_1=\\gamma <0$ .",
"Proposition 4.2 Let $Y$ and $(Y^{(n)})_{n\\ge 1}$ be as in Lemma REF .",
"Then $\\sup _{0\\le t <\\infty } Y^{(n)}_t \\Rightarrow \\sup _{0\\le t<\\infty }Y_t\\, .$ Let $\\psi $ denote the Laplace exponent of the spectrally negative dual process $\\widehat{Y}=-Y$ in the sense of [1].",
"Then $\\psi (\\lambda )=\\Psi _Y(i\\lambda )$ , $\\lambda \\ge 0$ .",
"It is a straightforward consequence of [1] that ${E}\\left[\\exp \\left\\lbrace \\lambda \\inf _{0\\le t<\\infty }\\widehat{Y}_t\\right\\rbrace \\right]=-\\gamma \\frac{\\lambda }{\\psi (\\lambda )}\\, ,\\quad \\lambda >0\\, ,$ (cf.",
"also [3]).",
"In terms of the process $Y$ this reads as ${E}\\left[\\exp \\left\\lbrace -\\lambda \\sup _{0\\le t<\\infty }Y_t\\right\\rbrace \\right]=-\\gamma \\frac{\\lambda }{\\Psi _Y(i\\lambda )}\\, ,\\quad \\lambda >0\\, .$ The same relation holds for the approximating processes: ${E}\\left[\\exp \\left\\lbrace -\\lambda \\sup _{0\\le t<\\infty }Y^{(n)}_t\\right\\rbrace \\right]=-\\gamma \\frac{\\lambda }{\\Psi _{Y^{(n)}}(i\\lambda )}\\, ,\\quad \\lambda >0\\, .$ Since $\\Psi _{Y^{(n)}}\\rightarrow \\Psi _Y$ pointwise, we see that the Laplace transforms of $\\sup _{0\\le t<\\infty }Y^{(n)}_t$ converge to the Laplace transform of $\\sup _{0\\le t<\\infty }Y_t$ .",
"This proves the claim.",
"Proof of Theorem REF – general case.",
"Let $Y$ be a spectrally positive Lévy process with infinite Lévy measure satisfying ${E}Y_1< 0$ .",
"We first consider an independent subordinator $C$ (without drift) with finite Lévy measure, and set $X:=Y+C$ .",
"By Lemma REF there exists a sequence $(Y^{(n)})_{n\\ge 1}$ of spectrally positive Lévy processes with finite Lévy measures such that ${E}Y_1^{(n)}={E}Y_1$ , $Y_1^{(n)}\\Rightarrow Y_1$ and $Y^{(n)}\\Rightarrow Y$ in $.",
"By Proposition \\ref {p:approx-sup} we have $ 0t < Y(n)t 0t<Yt$.",
"Let $ X(n):=Y(n)+C$.",
"Then (\\ref {e:approx-1-b}) is true by the proof of Theorem \\ref {t:main} for the compound Poisson case given in Section \\ref {sec-2}.",
"Now it follows from Proposition \\ref {p:approx-1} that $ 0t<Ytd= 0t<Xt$.$ In the second step we take an independent subordinator $C$ (without drift) with infinite Lévy measure and set $X:=Y+C$ .",
"For each $n\\ge 1$ define the subordinator $C^{(n)}$ by (REF ) and let $X^{(n)}:=Y+C^{(n)}$ .",
"Then by what has been just proved we have that $\\sup _{0\\le t<\\infty }Y_t\\stackrel{d}{=} \\sup _{0\\le t<\\sigma }X^{(n)}_t$ .",
"We finish the proof by invoking Proposition REF .",
"$\\Box $ At the end of the paper we discuss briefly how essential is the assumption on spectral positivity of $Y$ for validity of Theorem REF .",
"Suppose that $Y$ is a general Lévy process, not necessarily spectrally positive, which is not a sum of negative subordinator and negative drift, and such that ${E}Y_1<0$ .",
"Then $\\sup _{t\\ge 0} Y_t$ is finite a.s. and not identically zero.",
"Let $C$ be an independent subordinator without drift, define $X=Y+C$ , and assume that $\\sigma =\\inf \\lbrace t> 0:\\, \\Delta C_t>\\overline{X}_{t-}-X_{t-}\\rbrace =0$ .",
"Then clearly $\\sup _{0\\le t <\\sigma }X_t=0$ and hence cannot be equal in distribution to $\\sup _{t\\ge 0} Y_t$ .",
"Moreover, if $C^{(n)}$ is a sequence of subordinators defined as in (REF ), $X^{(n)}=Y+C^{(n)}$ and $\\sigma ^{(n)}=\\inf \\lbrace t>0:\\, \\Delta C^{(n)}_t>\\overline{X}^{(n)}_{t-}-X^{(n)}_{t-}\\rbrace $ , then $\\sigma ^{(n)}>0$ and according to Remark REF , it cannot hold that $\\sup _{0\\le t<\\infty }Y_t \\stackrel{d}{=}\\sup _{0\\le t <\\sigma ^{(n)}}X^{(n)}_t \\qquad \\text{for all }n\\ge 1 \\, .$ This means that there exists a subordinator $\\widetilde{C}$ with finite Lévy measure, independent of $Y$ such that if $\\widetilde{X}=Y+\\widetilde{C}$ and $\\widetilde{\\sigma }=\\inf \\lbrace t> 0:\\, \\Delta \\widetilde{C}_t>\\overline{\\widetilde{X}}_{t-}-\\widetilde{X}_{t-}>0\\rbrace $ , then $\\sup _{t\\ge 0}Y_t$ and $\\sup _{0\\le t<\\widetilde{\\sigma }}\\widetilde{X}_t$ have different distributions.",
"Hence Theorem REF cannot hold for $Y$ even in case of subordinators with finite Lévy measure.",
"The necessary and sufficient condition for $\\sigma =0$ was given in [7].",
"Let $Y$ be a Lévy process of unbounded variation, let $C$ be an independent subordinator with the Lévy measure $\\nu $ , and let $X=Y+C$ .",
"Denote by $V(x)$ the renewal function of the descending ladder height process of $Y$ (i.e.",
"the ascending ladder height of the dual process $-Y$ ), cf. [1].",
"Then $\\sigma >0$ a.s. if and only if $\\int _0^1 V(x)\\, \\nu (dx)<\\infty \\, .$ In case $Y$ is spectrally positive, it holds that $V(x)=x$ , hence (REF ) is automatically satisfied because of integrability property of $\\nu $ .",
"More generally, $Y$ creeps downwards (i.e.",
"with positive probability crosses every level from above continuously), if and only if there exists a constant $c>0$ such that $V(x)\\le cx$ for all $x\\ge 0$ , see [1].",
"Assume that $Y$ is of unbounded variation such that it is not true that $V(x)\\le cx$ for $x\\in [0,1]$ for any constant $c>0$ .",
"Then it follows that $\\lim _{x\\rightarrow 0+}V(x)/x=+\\infty $ .",
"Let $\\beta (x):=\\inf _{0<t\\le x}\\frac{V(t)}{t}\\, .$ Then $\\beta $ is non-increasing and $\\lim _{x\\rightarrow 0+}\\beta (x)=+\\infty $ .",
"Denote by $\\beta (dx)$ the measure on $(0,1)$ corresponding to the function $\\beta $ and define $\\widetilde{\\nu }(dx):=\\frac{\\beta (dx)}{\\beta (x)^2}\\, .$ By a change of variable it is easy to see that $\\int _0^1 \\beta (x)\\widetilde{\\nu }(dx)=\\int _0^1 \\frac{\\beta (dx)}{\\beta (x)}=\\infty \\qquad \\text{and} \\qquad \\int _0^1 \\widetilde{\\nu }(dx)=\\int _0^1 \\frac{\\beta (dx)}{\\beta (x)^2}<\\infty \\, .$ Finally, let $\\nu (dx):=\\widetilde{\\nu }(dx)/x$ .",
"Then $\\int _0^1 V(x)\\, \\nu (dx)\\ge \\int _0^1 \\beta (x)\\, \\widetilde{\\nu }(dx)=\\infty \\qquad \\text{and} \\qquad \\int _0^1 x\\, \\nu (dx) =\\int _0^1\\widetilde{\\nu }(dx)<\\infty \\, .$ Hence $\\nu $ is a Lévy measure such that (REF ) is not satisfied.",
"If $C$ is a subordinator independent of $Y$ with Lévy measure $\\nu $ , then $\\sigma =0$ a.s. We conclude that Theorem REF cannot hold for the Lévy process $Y$ .",
"To summarize, if $Y$ is of unbounded variation, ${E}Y_1<0$ , and $Y$ does not creep downwards, then there exists a subordinator $C$ with finite Lévy measure such that $\\sup _{t\\ge 0}Y_t$ and $\\sup _{0\\le t<\\sigma }X_t$ have different distributions.",
"Acknowledgement: We thank the referees for careful reading of the paper and helpful remarks.",
"This work has been supported in part by Croatian Science Foundation under the project 3526.",
"Ivana Geček Tuđen Department of Mathematics, University of Zagreb, Zagreb, Croatia Email: [email protected] Zoran Vondraček Department of Mathematics, University of Zagreb, Zagreb, Croatia Email: [email protected]"
]
] | 1403.0431 |
[
[
"Reduction of a family of ideals"
],
[
"Abstract In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions.",
"As a corollary we generalize the result of A. P\\l{}oski \\cite{ploski} on the semicontinuity of the \\L{}ojasiewicz exponent in a multiplicity-constant deformation."
],
[
"Introduction",
"Let $R$ be a ring and $I$ an ideal.",
"We say that an ideal $J$ is a reduction of $I$ if it satisfies the following condition: $J\\subset I,\\quad \\text{and for some}\\quad r>0\\quad \\text{we have}\\quad I^{r+1}=JI^{r}.$ The notion of reduction is closely related to the notions of Hilbert-Samuel multiplicity and integral closure of an ideal.",
"Recall that if $\\left(R,\\mathfrak {m}\\right)$ is a Noetherian local ring of dimension $n$ and $I$ is an $\\mathfrak {m}$ -primary ideal of $R$ , then the Hilbert-Samuel multiplicity of $I$ is given by the formula $e(I)=n!\\lim _{k\\rightarrow \\infty }\\frac{\\mathrm {length}_{R}R/I^{k}}{k^{n}}.$ For the multiplicity theory in local rings see for example [5] or [2].",
"Let $I$ be an ideal in a ring $R$ .",
"An element $x\\in R$ is said to be integral over $I$ if there exists an integer $n$ and elements $a_{k}\\in I^{k}$ , $k=1,\\ldots ,n$ , such that $x^{n}+a_{1}x^{n-1}+\\cdots +a_{n}=0.$ The set of all elements of $R$ that are integral over $I$ is called the integral closure of $I$ , and is denoted $\\overline{I}$ .",
"If $I=\\overline{I}$ then $I$ is called integrally closed.",
"It is well known that $\\overline{I}$ is an ideal.",
"The relationship between the above notions is given in the following Theorem due to D. Rees: Theorem 1 (Rees, [2]) Let $(R,\\mathfrak {m})$ be a formally equidimensional Noetherian local ring and let $J\\subset I$ be two $\\mathfrak {m}$ -primary ideals.",
"Then the following conditions are equivalent: $J$ is a reduction of $I$ ; $e\\left(I\\right)=e\\left(J\\right)$ ; $\\overline{I}=\\overline{J}$ .",
"It is an important fact that a reduction of an ideal is often generated by a system of parameters.",
"More precisely we have Theorem 2 ([5]) Let $(R,\\mathfrak {m})$ be a $d$ -dimensional Noetherian local ring, and suppose that $k=R/\\mathfrak {m}$ is an infinite field; let $I=(u_{1},\\ldots ,u_{s})$ be an $\\mathfrak {m}$ -primary ideal.",
"Then there exist a finite number of polynomials $D_{\\alpha }\\in k[Z_{ij};1\\leqslant i\\leqslant d,1\\leqslant j\\leqslant s]$ , $1\\leqslant \\alpha \\leqslant \\nu $ such that if $y_{i}=\\sum a_{ij}u_{j}$ , $i=1,\\ldots ,d$ and at least one of $D_{\\alpha }(\\overline{a}_{ij};1\\leqslant i\\leqslant d,1\\leqslant j\\leqslant s)\\ne 0$ , then the ideal $(y_{1},\\ldots ,y_{d})R$ is a reduction of $I$ and $\\lbrace y_{1},\\ldots ,y_{d}\\rbrace $ is a system of parameters of $R$ .",
"Let $\\left(\\mathcal {O}_{n},\\mathfrak {m}_{n}\\right)$ be the ring of germs of holomorphic functions $\\left(\\mathbb {C}^{n},0\\right)\\rightarrow \\mathbb {C}$ .",
"The aim of this note is to prove the following: Theorem 3 Let $F=F_{t}(x)=F(x,t)\\colon (\\mathbb {C}^{n}\\times \\mathbb {C},0)\\rightarrow (\\mathbb {C}^{m},0)$ be a holomorphic map.",
"Assume that $(F_{t})\\mathcal {O}_{n}$ is an $\\mathfrak {m}_{n}$ -primary ideal for all $t$ .",
"Then there exists a complex linear map $\\pi \\colon \\mathbb {C}^{m}\\rightarrow \\mathbb {C}^{n}$ such that for all $t$ the ideal $\\left(\\pi \\circ F_{t}\\right)\\mathcal {O}_{n}$ is a reduction of $\\left(F_{t}\\right)\\mathcal {O}_{n}$ .",
"In the next section we get as a corollary that if the above family $\\left(F_{t}\\right)\\mathcal {O}_{n}$ is of constant multiplicity then the Łojasiewicz exponent in this family is a lower semicontinuos function of $t$ .",
"A. Płoski proved this result under additional restriction $m=n$ but with space of parameters of arbitrary dimension.",
"The proof of Theorem REF is based on some geometric property of Hilbert-Samuel multiplicity, given in section 3."
],
[
"Semicontinuity of the Łojasiewicz exponent",
"Let $(R,\\mathfrak {m})$ be a local ring and let $I$ be an $\\mathfrak {m}$ -primary ideal.",
"By the Łojasiewicz exponent $\\mathcal {L}(I)$ of $I$ we define the infimum of $\\left\\lbrace \\frac{p}{q}:\\mathfrak {m}^{p}\\subset \\overline{I^{q}}\\right\\rbrace .$ It was proved in [3] that if $F\\colon \\left(\\mathbb {C}^{n},0\\right)\\rightarrow \\left(\\mathbb {C}^{m},0\\right)$ is a holomorphic map with an isolated zero at the origin and $I:=\\left(F\\right)\\mathcal {O}_{n}$ , then $\\mathcal {L}\\left(I\\right)$ is an optimal exponent $\\nu $ in the inequality $\\left|F\\left(x\\right)\\right|\\geqslant C\\left|x\\right|^{\\nu },$ where $C$ is some positive constant and $x$ runs through sufficiently small neighbourhood of $0\\in \\mathbb {C}^{n}$ .",
"Lemma 4 Let $\\left(R,\\mathfrak {m}\\right)$ be a Noetherian local ring.",
"If $I$ is an $\\mathfrak {m}$ -primary ideal of $R$ and $J$ is a reduction of $I$ then $\\mathcal {L}\\left(I\\right)=\\mathcal {L}\\left(J\\right)$ .",
"Obviously $\\mathcal {L}(I)\\leqslant \\mathcal {L}(J)$ .",
"Assume that $\\mathfrak {m}^{p}\\subset \\overline{I^{q}}$ .",
"Since $J$ is a reduction of $I$ , then also $J^{q}$ is a reduction of $I^{q}$ [2].",
"Thus $\\overline{J^{q}}=\\overline{I^{q}}$ by Theorem REF , which gives $\\mathfrak {m}^{p}\\subset \\overline{J^{q}}$ .",
"This proves the inequality $\\mathcal {L}(J)\\leqslant \\mathcal {L}(I)$ and ends the proof.",
"Corollary 5 (A. Płoski for $m=n$ , [6]) Let $F\\colon \\left(\\mathbb {C}^{n}\\times \\mathbb {C},0\\right)\\rightarrow \\left(\\mathbb {C}^{m},0\\right)$ be a holomorphic map.",
"Put $I_{t}:=\\left(F_{t}\\right)\\mathcal {O}_{n}$ .",
"If the function $t\\mapsto e\\left(I_{t}\\right)$ is constant and finite then the function $t\\mapsto \\mathcal {L}\\left(I_{t}\\right)$ is lower semicontinuos.",
"By Theorem REF there exists a linear map $\\pi \\colon \\mathbb {C}^{m}\\rightarrow \\mathbb {C}^{n}$ such that $J_{t}:=\\left(\\pi \\circ F_{t}\\right)\\mathcal {O}_{n}$ is a reduction of $I_{t}$ for all $t$ .",
"Thus $\\mathcal {L}\\left(J_{t}\\right)=\\mathcal {L}\\left(I_{t}\\right)$ and $e\\left(J_{t}\\right)=e\\left(I_{t}\\right)$ by Theorem REF and Lemma REF .",
"Consequently $t\\mapsto e\\left(J_{t}\\right)$ is constant and finite and the assertion follows from the case $m=n$ proved by A. Płoski."
],
[
"Improper intersection multiplicity",
"Let $I$ be an $\\mathfrak {m}_{n}$ -primary ideal of $\\mathcal {O}_{n}$ and let $f_{1},\\ldots ,f_{m}$ be its generators.",
"We put $f=\\left(f_{1},\\ldots ,f_{m}\\right)$ .",
"It is well known that if $m=n$ then $e(I)=\\dim _{\\mathbb {C}}\\mathcal {O}_{n}/I.$ On the other hand, if $m>n$ then we may define so-called improper intersection multiplicity $i_{0}\\left(I\\right)$ of $I$ as the improper intersection multiplicity $i(\\mathrm {graph}f\\cdot (\\mathbb {C}^{n}\\times \\lbrace 0\\rbrace );(0,0))$ of $\\mathrm {graph}f$ and $\\mathbb {C}^{n}\\times \\lbrace 0\\rbrace $ at the point $(0,0)\\in \\mathbb {C}^{n}\\times \\mathbb {C}^{m}$ (see [1]).",
"Let $C_{f}$ be the (Whitney) tangent cone of the germ of the image of $f$ at the origin.",
"The following observation is due to S. Spodzieja.",
"Theorem 6 ([7]) The number $i_{0}\\left(I\\right)$ is well defined.",
"Moreover, if $\\pi \\colon \\mathbb {C}^{m}\\rightarrow \\mathbb {C}^{l}$ is a linear map such that $\\ker \\pi \\cap C_{f}=\\lbrace 0\\rbrace $ , then the ideal $J$ generated by $\\pi \\circ f$ is $\\mathfrak {m}_{n}$ -primary and we have $i_{0}(I)=i_{0}(J)$ .",
"If additionally $l=n$ then $i_{0}(I)=e(J)$ .",
"Corollary 7 If $I$ is an $\\mathfrak {m}_{n}$ -primary ideal in $\\mathcal {O}_{n}$ , then $i_{0}(I)=e(I)$ .",
"Let $I=(f_{1},\\ldots ,f_{m})\\mathcal {O}_{n}$ .",
"By Theorems REF and REF there exists linear combinations $g_{i}=\\sum a_{ij}f_{j}$ , $i=1,\\ldots ,n$ such that $J=(g_{1},\\ldots ,g_{n})\\mathcal {O}_{n}$ is a reduction of $I$ , $\\lbrace g_{1},\\ldots ,g_{n}\\rbrace $ is a system of parameters of $\\mathcal {O}_{n}$ and $i_{0}(I)=i_{0}(J)=e(J)$ .",
"From Theorem REF we get $e(I)=e(J)$ .",
"This ends the proof.",
"Corollary 8 If $\\pi \\colon \\mathbb {C}^{m}\\rightarrow \\mathbb {C}^{l}$ is a linear map such that $\\ker \\pi \\cap C_{f}=\\lbrace 0\\rbrace $ , then the ideal $J$ generated by $\\pi \\circ f$ is a reduction of $I$ .",
"We have $J\\subset I$ and $e\\left(J\\right)=e\\left(I\\right)$ .",
"This and Theorem REF give the assertion."
],
[
"Elementary blowing-up",
"Here we recall the notion of an elementary blowing-up after [4].",
"Let $U\\subset \\mathbb {C}^{n}$ be an open and connected neighbourhood of $0\\in \\mathbb {C}^{n}$ ; let $f=(f_{0},\\ldots ,f_{m})\\ne 0$ be a sequence of holomorphic functions on $U$ .",
"Put $S=\\lbrace x\\in U:f(x)=0\\rbrace $ and $E(f)=\\lbrace (x,u)\\in U\\times \\mathbb {P}^{m}:f_{i}(x)u_{j}=f_{j}(x)u_{i},i,j=0,\\ldots ,m\\rbrace ,$ where $u=[u_{0}:\\cdots :u_{m}]\\in \\mathbb {P}^{m}$ .",
"Let $Y$ be the closure of $E(f)\\setminus S$ in $U\\times \\mathbb {P}^{m}$ .",
"The natural projection $\\pi :Y\\rightarrow U$ is called the (elementary) blowing-up of $U$ by means of $f_{0},\\ldots ,f_{m}$ .",
"The analytic subset $S$ is called a centre of the blowing-up and its inverse image $\\pi ^{-1}(S)\\subset Y$ is called the exceptional set of the blowing-up.",
"Proposition 9 Under above notations we have: $Y$ is an analytic subset of $U\\times \\mathbb {P}^{m}$ ; $\\pi $ is proper, its range is $U$ and the restriction $\\pi _{|Y\\setminus \\pi ^{-1}(S)}$ is a biholomorphism onto $U\\setminus S$ ; $Y$ is irreducible; The exceptional set $\\pi ^{-1}(S)$ is analytic in $U\\times \\mathbb {P}^{m}$ and it is of pure dimension $n-1$ .",
"Although the above proposition is well known, we think that point (REF ) is worth proving.",
"Let us consider the analytic map $F\\colon U\\times \\mathbb {P}^{m}\\ni (x,u)\\mapsto (f(x),u)\\in \\mathbb {C}^{m+1}\\times \\mathbb {P}^{m}.$ Let $y_{0},\\ldots ,y_{m}$ be coordinates in $\\mathbb {C}^{m+1}$ .",
"If we denote by $\\pi _{m+1}\\colon \\Pi _{m+1}\\rightarrow \\mathbb {C}^{m+1}$ the blowing-up of $\\mathbb {C}^{m+1}$ by means of $y_{0},\\ldots ,y_{m}$ then for the restriction $\\widetilde{f}=F_{|Y}$ we get the following commutative diagram of analytic maps: ${\\begin{matrix}Y &\\xrightarrow{}& \\Pi _{m+1} \\\\\\mathbox{mphantom}{\\scriptstyle \\pi }\\downarrow {\\scriptstyle \\pi }&& \\mathbox{mphantom}{\\scriptstyle \\pi _{m+1} }\\downarrow {\\scriptstyle \\pi _{m+1} }&&\\\\U &\\xrightarrow{}& \\mathbb {C}^{m+1}\\end{matrix}}$ Take $(x_{0},u_{0})\\in \\pi ^{-1}(0)$ .",
"Let $\\Omega \\subset \\Pi _{m+1}$ be a neighbourhood of $(0,u_{0})$ , $h\\colon \\Omega \\rightarrow \\mathbb {C}$ an analytic function such that $\\pi _{m+1}^{-1}(0)\\cap \\Omega =\\lbrace (y,u)\\in \\Omega :h(y,u)=0\\rbrace .$ Let $\\widetilde{\\Omega }\\subset Y$ be a neighbourhood of $(x_{0},u_{0})$ such that $\\widetilde{f}(\\widetilde{\\Omega })\\subset \\Omega $ .",
"Since $\\widetilde{f}^{-1}(\\pi _{m+1}^{-1}(0))=\\pi ^{-1}(S)$ we get $\\pi ^{-1}(S)\\cap \\widetilde{\\Omega }=\\lbrace (x,u)\\in \\widetilde{\\Omega }:h\\circ \\widetilde{f}(x,u)=0\\rbrace .$ Thus there exists a neighbourhood $\\Delta \\subset U\\times \\mathbb {P}^{m}$ of $(x_{0},u_{0})$ and an analytic set $V\\subset \\Delta $ of pure dimension $n+m-1$ such that $\\pi ^{-1}(S)\\cap \\Delta =V\\cap Y\\cap \\Delta .$ This gives $\\dim _{(x_{0},u_{0})}\\pi ^{-1}(S)\\geqslant \\dim _{(x_{0},u_{0})}Y-1=n-1.$ Since $Y$ is irreducible and $\\pi ^{-1}(S)Y$ we get that $\\dim _{p}\\pi ^{-1}(S)=n-1$ for any $p\\in \\pi ^{-1}(S)$ .",
"This ends the proof."
],
[
"Proof of Theorem ",
"Lemma 10 Let $F\\colon (\\mathbb {C}^{n}\\times \\mathbb {C},0)\\rightarrow (\\mathbb {C}^{m+1},0)$ , $m\\geqslant n$ be a holomorphic map.",
"Assume that 0 is an isolated point of $F_{t}^{-1}(0)$ for $|t|<\\delta $ .",
"Then there exists $\\delta >\\epsilon >0$ and a complex line $V\\subset \\mathbb {C}^{m+1}$ , such that $V\\cap C_{F_{t}}=\\lbrace 0\\rbrace $ for $|t|<\\epsilon $ .",
"Let $F\\colon U\\rightarrow \\mathbb {C}^{m+1}$ , where $U\\subset \\mathbb {C}^{n}\\times \\mathbb {C}$ is a connected neighbourhood of the origin.",
"Put $S=\\lbrace (z,t)\\in U:F(z,t)=0\\rbrace $ and let $\\pi \\colon U\\times \\mathbb {P}^{m}\\supset Y\\rightarrow U$ be the elementary blowing-up of $U$ by $F$ .",
"By Proposition REF its exceptional set $E:=\\pi ^{-1}(S)$ is an analytic set of pure dimension $n$ .",
"Let $\\mathcal {E}$ be a set of those irreducible components $W$ of $E$ for which origin in $\\mathbb {C}^{n+1}$ is an accumulation point of $\\pi (W)\\cap (\\lbrace 0\\rbrace \\times \\mathbb {C})$ .",
"Then $\\mathcal {E}$ is finite.",
"Denote by $\\widetilde{C_{F_{t}}}$ the image of the cone $C_{F_{t}}$ in $\\mathbb {P}^{m}$ .",
"Observe that $\\lbrace (0,t)\\rbrace \\times \\widetilde{C_{F_{t}}}\\subset \\bigcup \\mathcal {E},\\quad |t|<\\delta .$ On the other hand for any $W\\in \\mathcal {E}$ we have $\\dim W\\cap (\\lbrace 0\\rbrace \\times \\mathbb {P}^{m})\\leqslant n-1<m.$ Thus there exists $\\epsilon >0$ and an open set $G\\subset \\mathbb {P}^{m}$ such that $(\\lbrace (0,t)\\rbrace \\times G)\\cap \\bigcup \\mathcal {E}=\\emptyset ,\\quad 0<|t|<\\epsilon $ As a result if $V$ is a line in $\\mathbb {C}^{m+1}$ corresponding to some point in $G$ then $V\\cap C_{F_{t}}=\\lbrace 0\\rbrace $ for $0<|t|<\\epsilon $ .",
"Since $G$ is not a subset of $C_{F_{0}}$ we get the assertion.",
"[Proof of Theorem REF ] Induction on $m$ .",
"In the case $m=n$ there is nothing to prove.",
"Let us assume that the assertion is true for some $m\\geqslant n$ and let $F\\colon (\\mathbb {C}^{n}\\times \\mathbb {C},0)\\rightarrow (\\mathbb {C}^{m+1},0)$ be a holomorphic map such that the ideals $\\left(F_{t}\\right)\\mathcal {O}_{n}$ are $\\mathfrak {m}_{n}$ -primary.",
"By Lemma REF there exists $\\epsilon >0$ and a linear mapping $\\pi ^{\\prime }\\colon \\mathbb {C}^{m+1}\\rightarrow \\mathbb {C}^{m}$ such that $\\ker \\pi ^{\\prime }\\cap C_{F_{t}}=\\left\\lbrace 0\\right\\rbrace $ for $|t|<\\epsilon $ .",
"Thus, by Corollary REF the ideal $\\left(\\pi ^{\\prime }\\circ F_{t}\\right)\\mathcal {O}_{n}$ is a reduction of $\\left(F_{t}\\right)\\mathcal {O}_{n}$ .",
"On the other hand, by induction hypothesis, there exists a linear map $\\pi ^{\\prime \\prime }\\colon \\mathbb {C}^{m}\\rightarrow \\mathbb {C}^{n}$ such that $\\left(\\pi ^{\\prime \\prime }\\circ \\pi ^{\\prime }\\circ F_{t}\\right)\\mathcal {O}_{n}$ is a reduction of $\\left(\\pi ^{\\prime }\\circ F_{t}\\right)\\mathcal {O}_{n}$ for small $t$ .",
"Thus if we put $\\pi :=\\pi ^{\\prime \\prime }\\circ \\pi ^{\\prime }$ we get the assertion."
]
] | 1403.0123 |
[
[
"Confidence intervals for average success probabilities"
],
[
"Abstract We provide Buehler-optimal one-sided and some valid two-sided confidence intervals for the average success probability of a possibly inhomogeneous fixed length Bernoulli chain, based on the number of observed successes.",
"Contrary to some claims in the literature, the one-sided Clopper-Pearson intervals for the homogeneous case are not completely robust here, not even if applied to hypergeometric estimation problems."
],
[
"Introduction and results",
"The purpose of this paper is to provide optimal one-sided (Theorem REF ) and some valid two-sided (Theorems REF and REF ) confidence intervals for the average success probability of a possibly inhomogeneous fixed length Bernoulli chain, based on the number of observed successes.",
"For this situation, intervals proposed in the literature known to us are, if at all clearly specified, in the one-sided case either not optimal or erroneously claimed to be valid, see Remarks REF and REF below, and in the two-sided case either improved here, see Remark REF , or not previously proven to be valid.",
"To be more precise, let $\\mathrm {B}_p$ for $p\\in [0,1]$ , $\\mathrm {B}_{n,p}$ for $n\\in {\\mathbb {N}}_0$ and $p\\in [0,1]$ , and $\\mathrm {BC}_p\\operatornamewithlimits{\\mbox{\\LARGE $\\ast $}}\\nolimits _{j=1}^n \\mathrm {B}_{p_j}$ for $n\\in {\\mathbb {N}}_0$ and $p\\in [0,1]^n$ denote the Bernoulli, binomial, and Bernoulli convolution (or Poisson-binomial) laws with the indicated parameters.",
"For $a,b\\in {\\mathbb {R}}\\cup \\lbrace -\\infty ,\\infty \\rbrace $ let $\\mathopen ]a,b\\mathclose ]\\lbrace x\\colon a < x \\le b\\rbrace $ and let the other intervals be defined analogously.",
"Then, for $n\\in {\\mathbb {N}}$ and $\\beta \\in \\mathopen ]0,1\\mathclose [$ , and writing $\\overline{p}\\frac{1}{n}\\sum \\nolimits _{j=1}^n p_j$ for $p\\in [0,1]^n$ , we are interested in $\\beta $ -confidence regions for the estimation problem $ \\left( \\left(\\mathrm {BC}_p\\colon p\\in [0,1]^n\\right), [0,1]^n\\ni p\\mapsto \\overline{p}\\right),$ that is, in functions $\\mathrm {K}\\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ satisfying $ \\mathrm {BC}_p\\left(\\mathrm {K}\\ni \\overline{p}\\right) \\ge \\beta $ for $p\\in [0,1]^n$ .",
"Clearly, every such $\\mathrm {K}$ is also a $\\beta $ -confidence region for the binomial estimation problem $ \\left( (\\mathrm {B}_{n,p}\\colon p\\in [0,1]),{\\mbox{\\rm id}}_{[0,1]} \\right),$ that is, satisfies $\\mathrm {B}_{n,p}\\left(\\mathrm {K}\\ni p\\right) \\ge \\beta $ for $p\\in [0,1]$ , but the converse is false by Remark REF below.",
"However, a classical Chebyshev-Hoeffding result easily yields the following basic fact.",
"Theorem 1.1 Let $n\\in {\\mathbb {N}}$ and $\\beta \\in \\mathopen ]0,1\\mathclose [$ .",
"For $m\\in \\lbrace 0,\\ldots ,n\\rbrace ,$ let $\\mathrm {K}^{\\prime }_m$ be a $\\beta $ -confidence region for $\\left( (\\mathrm {B}_{m,p}\\colon p\\in [0,1]),{\\mbox{\\rm id}}_{[0,1]}\\right)$ .",
"Then a $\\beta $ -confidence region $\\mathrm {K}$ for (REF ) is given by $\\mathrm {K}(x){-10mu}\\bigcup \\limits _{\\genfrac{}{}{0.0pt}{}{l\\in \\lbrace 0,\\dots ,x\\rbrace ,}{m\\in \\lbrace x-l,\\dots ,n-l\\rbrace }} {-10mu} \\left(\\tfrac{m}{n} \\mathrm {K}^{\\prime }_m(x-l) + \\tfrac{l}{n} \\right)\\,\\ \\supseteq \\,\\ \\mathrm {K}_n^{\\prime }(x)\\quad \\text{ for }x\\in \\lbrace 0,\\ldots ,n\\rbrace .$ Proofs of the three theorems of this paper are presented in section below.",
"If the above $\\mathrm {K}_m^{\\prime }$ are taken to be one-sided intervals of Clopper and Pearson [5], then the resulting $\\mathrm {K}$ turns out to be Buehler-optimal and, if $\\beta $ is not unusually small, the formula for $\\mathrm {K}$ simplifies drastically, as stated in Theorem REF below for uprays: A set $J\\subseteq [0,1]$ is an upray in $[0,1]$ if $x\\in J, y\\in [0,1], x\\le y$ jointly imply $y\\in J$ .",
"This is equivalent to $J$ being of the form $[a,1]$ or $]a,1]$ for some $a\\in [0,1]$ .",
"A function $\\mathrm {K}\\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ is an upray if each of its values $\\mathrm {K}(x)$ is an upray in $[0,1]$ .",
"For $\\beta \\in \\mathopen ]0,1\\mathclose [$ and with $g^{}_{n}(x)&& g^{}_{n,\\beta }(x)\\,\\ \\,\\ \\text{ the } p \\in [0,1]\\text{ with } \\mathrm {B}_{n,p}(\\lbrace x,\\dots ,n\\rbrace ) = 1-\\beta $ for $n\\in {\\mathbb {N}}$ and $x\\in \\lbrace 1,\\dots ,n\\rbrace $ , which is well-defined due to the strict isotonicity of $p\\mapsto \\mathrm {B}_{n,p}(\\lbrace x,\\dots ,n\\rbrace )$ and which yields in particular the special values $ g^{}_{n}(1) = 1-\\beta ^{1/n} &\\text{ and }&g^{}_{n}(n) = \\left(1-\\beta \\right)^{1/n}$ and the fact that $ g^{}_{n,\\beta }(x) \\,\\ \\text{ is strictly }\\left\\lbrace \\begin{array}{l}\\text{increasing}\\\\\\text{decreasing}\\end{array} \\right\\rbrace \\text{ in } \\left\\lbrace \\begin{array}{l} x\\\\ \\beta \\end{array} \\right\\rbrace ,$ the Clopper-Pearson $\\beta $ -confidence uprays $\\mathrm {K}^{}_{\\mathrm {CP,}n}\\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ are given by $\\qquad \\mathrm {K}^{}_{\\mathrm {CP,}n}(x) && \\mathrm {K}^{}_{\\mathrm {CP,}n,\\beta }(x) \\,\\ \\,\\ \\left\\lbrace \\begin{array}{ll}\\left[0,1\\right] & \\text{if } x=0,\\\\\\left]g^{}_{n}(x), 1\\right] & \\text{if } x\\in \\lbrace 1,\\dots ,n\\rbrace \\end{array} \\right\\rbrace $ for $n\\in {\\mathbb {N}}_0,$ and in particular $ \\qquad \\mathrm {K}^{}_{\\mathrm {CP,}n}(1)\\,\\ = \\,\\ \\left]1-\\beta ^{1/n},1\\right] & \\text{ and } &\\mathrm {K}^{}_{\\mathrm {CP,}n}(n) \\,\\ =\\,\\ \\left] (1-\\beta )^{1/n},1\\right]$ for $n\\in {\\mathbb {N}}.$ An upray $\\mathrm {K}\\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ is isotone if it is isotone with respect to the usual order on $\\lbrace 0, \\ldots ,n\\rbrace $ and the order reverse to set inclusion on $2^{[0,1]}$ , that is, if we have the implication $x,y\\in \\lbrace 0, \\ldots ,n\\rbrace ,\\, x < y &\\Rightarrow & \\mathrm {K}(x) \\supseteq \\mathrm {K}(y),$ and strictly isotone if “$\\supseteq $ ” above can be sharpened to “$\\supsetneq $ ”.",
"For example, each of the above $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ is strictly isotone by (REF ) and (REF ).",
"An isotone $\\beta $ -confidence upray for (REF ) is (Buehler-)optimal (see Buehler [2] and the recent discussion by Lloyd and Kabaila [11], prompted by rediscoveries by Wang [16]) if every other isotone $\\beta $ -confidence upray $\\mathrm {K}^\\ast $ for (REF ) satisfies $\\mathrm {K}(x)\\subseteq \\mathrm {K}^\\ast (x) $ for every $x\\in \\lbrace 0,\\ldots ,n\\rbrace $ .",
"Finally, a not necessarily isotone $\\beta $ -confidence upray $\\mathrm {K}$ for (REF ) is admissible in the set of all confidence uprays for (REF ) if for every other $\\beta $ -confidence upray $\\mathrm {K}^\\ast $ for (REF ) with $\\mathrm {K}^\\ast (x)\\subseteq \\mathrm {K}(x)$ for each $x\\in \\lbrace 0,\\dots ,n\\rbrace $ we have $\\mathrm {K}^\\ast =\\mathrm {K}.$ Let us put $\\beta _n && \\mathrm {B}_{n,\\frac{1}{n}}(\\lbrace 0,1\\rbrace ) \\qquad \\text{ for }n\\in {\\mathbb {N}},$ so that $\\beta _1=1$ , $\\beta _2=\\frac{3}{4}$ , $\\beta _3 = \\frac{20}{27}$ , and $\\beta _n\\downarrow \\frac{2}{\\mathrm {e}} = 0.735\\ldots $ , with the strict antitonicity of $(\\beta _n)$ following from Jogdeo and Samuels [9] so that we have in particular $\\beta _n &\\le & \\tfrac{3}{4} \\qquad \\text{ for }n\\ge 2.$ Theorem 1.2 Let $n\\in {\\mathbb {N}}$ and $\\beta \\in \\mathopen ]0,1\\mathclose [$ , and let $\\mathrm {K}$ be as in Theorem REF with the $\\mathrm {K}_m^{\\prime }\\mathrm {K}^{}_{\\mathrm {CP},m}$ as defined in (REF ).",
"Then $\\mathrm {K}$ is the optimal isotone $\\beta $ -confidence upray for (REF ), is admissible in the set of all $\\beta $ -confidence uprays for (REF ), is strictly isotone, and has the effective level $\\inf _{p\\in [0,1]^n} \\mathrm {BC}_p\\left(\\mathrm {K}\\ni \\overline{p}\\right) = \\beta $ .",
"We have $\\mathrm {K}(x)&=&\\left\\lbrace \\begin{array}{ll}\\left[0,1\\right] & \\text{if } x=0,\\\\\\left]\\frac{1-\\beta }{n},1\\right] & \\text{if } x=1,\\\\\\left]g^{}_{n}(x), 1\\right] & \\text{if } x\\in \\lbrace 2,\\dots ,n\\rbrace \\text{ and } \\beta \\ge \\beta _n .\\end{array}\\right.$ Remark 1.3 Nestedness is preserved by the construction in Theorem REF : Suppose that we apply Theorem REF to several $\\beta \\in \\mathopen ]0,1\\mathclose [$ and that we accordingly write $\\mathrm {K}^{\\prime }_{m,\\beta }$ and $\\mathrm {K}^{}_\\beta $ in place of $\\mathrm {K}^{\\prime }_m$ and $\\mathrm {K}$ .",
"If now $\\beta ,\\tilde{\\beta }\\in \\mathopen ]0,1\\mathclose [$ with $\\beta <\\tilde{\\beta }$ are such that $\\mathrm {K}^{\\prime }_{m,\\beta }(x)\\subseteq \\mathrm {K}^{\\prime }_{m,\\tilde{\\beta }}(x) $ holds for $m\\in \\lbrace 0,\\ldots ,n\\rbrace $ and $x\\in \\lbrace 0,\\ldots ,m\\rbrace $ , then, obviously, $\\mathrm {K}_{\\beta }(x)\\subseteq \\mathrm {K}_{\\tilde{\\beta }}(x)$ holds for $x\\in \\lbrace 0,\\ldots ,n\\rbrace $ .",
"By the second line in (REF ) and by (REF ), the Clopper-Pearson uprays are nested, and hence so are the uprays of Theorem REF .",
"Analogous remarks apply to the confidence downrays of Remark REF and to the two-sided confidence intervals of Theorem REF .",
"Remark 1.4 Let $n\\ge 2$ and $\\beta \\in \\mathopen ]0,1\\mathclose [$ .",
"As noted by Agnew [1] but ignored by later authors, compare Remark REF below, $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ is not a $\\beta $ -confidence region for (REF ).",
"This is obvious from Theorem REF and $\\mathrm {K}^{}_{\\mathrm {CP,}n}(1) \\subsetneq \\mathrm {K}(1)$ , using either the optimality of $\\mathrm {K}$ and the isotonicity of $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ , or the admissibility of $\\mathrm {K}$ and $\\mathrm {K}^{}_{\\mathrm {CP,}n}(x)\\subseteq \\mathrm {K}(x)$ for every $x$ .",
"If $\\beta \\ge \\beta _n$ , then Theorem REF further implies that the effective level of $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ as a confidence region for (REF ) is $\\gamma _n && 1-n\\left(1-\\beta ^{1/n}\\right) \\,\\ \\in \\,\\ \\mathopen ]1+\\log (\\beta ),\\beta \\mathclose [,$ as for $p\\in [0,1]^n$ with $\\overline{p}\\notin \\mathopen ]\\frac{1-\\beta }{n},g^{}_{n}(1)\\mathclose ]$ , formula (REF ) yields $\\mathrm {BC}_p\\left(\\mathrm {K}^{}_{\\mathrm {CP,}n} \\ni \\overline{p}\\right)$ $=$ $\\mathrm {BC}_p\\left(\\mathrm {K}\\ni \\overline{p}\\right)$ $\\ge $ $\\beta ,$ and considering $p_1 = n g^{}_{n}(1)\\le 1$ and $p_2 = \\ldots = p_{n} = 0$ at the second step below yields $\\inf \\limits _{\\overline{p}\\in \\mathopen ]\\frac{1-\\beta }{n},g^{}_{n}(1)\\mathclose ]} \\mathrm {BC}_p\\left(\\mathrm {K}^{}_{\\mathrm {CP},n} \\ni \\overline{p}\\right)&=& \\inf \\limits _{\\overline{p}\\in \\mathopen ]\\frac{1-\\beta }{n},g^{}_{n}(1)\\mathclose ]} \\prod \\limits _{j=1}^n(1-p_j)\\\\&=& 1-n g^{}_{n}(1) \\,\\ =\\,\\ \\gamma _n.$ Since $\\gamma _n\\downarrow 1+\\log (\\beta )<\\beta $ for $n\\rightarrow \\infty $ , it follows for $\\beta >\\frac{2}{\\mathrm {e}}$ that the $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ are not even asymptotic $\\beta $ -confidence regions for (REF ).",
"Remark 1.5 The only previous $\\beta $ -confidence upray for (REF ) known to us was provided by Agnew [1] as $\\mathrm {K}_{\\mathrm {A}}(x)[g^{}_{\\mathrm {A}}(x),1]$ with $g^{}_{\\mathrm {A}}(0)0$ and $g^{}_{\\mathrm {A}}(x)g^{}_n(x) \\wedge \\frac{x-1}{n}$ for $x\\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"But $\\mathrm {K}_{\\mathrm {A}}$ is strictly worse than the optimal isotone $\\mathrm {K}$ from Theorem REF , since $\\mathrm {K}_{\\mathrm {A}}$ is isotone as well, with $\\mathrm {K}_{\\mathrm {A}}(1)=[0,1] \\supsetneq \\mathrm {K}(1)$ .",
"On the other hand, Lemma REF below shows that actually $g^{}_{\\mathrm {A}}(x)=g^{}_n(x)$ for $\\beta \\ge \\beta _n$ and $x\\in \\lbrace 2,\\ldots ,n\\rbrace $ , which is a precise version of an unproven claim in the cited reference.",
"Remark 1.6 The condition $\\beta \\ge \\beta _n$ in (REF ) can not be omitted: For $n\\in {\\mathbb {N}}$ , let $A_n\\lbrace \\beta \\in \\mathopen ]0,1\\mathclose [ \\colon \\text{If } \\mathrm {K}\\text{ is as in Theorem~\\ref {Thm:lower}, then }\\mathrm {K}(x)= \\mathopen ]g^{}_n(x),1\\mathclose ]\\text{ for }x\\in \\lbrace 2,\\ldots ,n\\rbrace \\rbrace $ .",
"Then $\\mathopen [\\beta _n,1\\mathclose [ \\subseteq A_n$ , by Theorem REF .",
"Numerically, we found for example also $\\beta _n-0.001 \\in A_n$ for $2\\le n\\le 123$ , but $\\mathrm {K}(2)\\supsetneq \\mathopen ]g^{}_{n}(2),1\\mathclose ]$ for $\\beta = \\beta _n-0.001$ and $124\\le n\\le 3000$ .",
"Remark 1.7 The $\\beta $ -confidence upray $\\mathrm {K}$ for (REF ) from Theorem REF considered merely as a $\\beta $ -confidence interval shares with $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ as a $\\beta $ -confidence interval for (REF ) the defect of not being admissible in the set of all $\\beta $ -confidence intervals, since with $c\\left(\\inf \\mathrm {K}(n)\\right)\\vee \\left(1-(1-\\beta )^{1/n}\\right)$ and $\\mathrm {K}^\\ast (x) &&\\left\\lbrace \\begin{array}{ll}[0,c] \\,\\subsetneq \\, \\mathrm {K}(0) & \\text{if } x=0,\\\\\\mathrm {K}(x) & \\text{if } x\\in \\lbrace 1,\\dots ,n\\rbrace ,\\end{array}\\right.$ we have $\\mathrm {BC}_p(\\mathrm {K}^\\ast \\ni \\overline{p}) = \\mathrm {BC}_p(\\mathrm {K}\\ni \\overline{p}) \\ge \\beta $ if $\\overline{p}\\le c$ , and, if $\\overline{p}>c,$ $\\mathrm {BC}_p(\\mathrm {K}^\\ast \\ni \\overline{p})$ $=$ $\\mathrm {BC}_p(\\lbrace 1,\\dots ,n\\rbrace )$ $=$ $1-\\prod \\nolimits _{j=1}^n (1-p_j)$ $\\ge $ $1-(1-\\overline{p})^n > 1-(1-c)^n\\ge \\beta .$ Remark 1.8 Since $\\mathrm {K}$ is a $\\beta $ -confidence region for (REF ) iff $\\lbrace 0,\\ldots ,n\\rbrace \\ni x\\mapsto 1-\\mathrm {K}(n-x)$ is one, Theorem REF and Remarks REF –REF yield obvious analogs for downrays, that is confidence regions with each value being $\\mathopen [0,b\\mathclose [$ or $[0,b]$ for some $b\\in [0,1]$ : A downray $\\Lambda \\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ is isotone if $\\Lambda (x)\\subseteq \\Lambda (y)$ holds for $x<y$ .",
"The Clopper-Pearson downrays $\\Lambda ^{}_{\\mathrm {CP,}n} \\Lambda ^{}_{\\mathrm {CP,}n,\\beta }$ defined by $\\Lambda ^{}_{\\mathrm {CP,}n,\\beta }(x)1- \\mathrm {K}^{}_{\\mathrm {CP,}n,\\beta }(n-x)$ are isotone, and Theorem REF remains valid if we replace $\\mathrm {K}^{}_{\\mathrm {CP,}m}$ by $\\Lambda ^{}_{\\mathrm {CP,}m}$ , upray by downray, and (REF ) by $\\qquad \\mathrm {K}(x)&=&\\left\\lbrace \\begin{array}{ll}\\left[0,1-g^{}_{n}(n-x)\\right[ & \\text{if } x\\in \\lbrace 0,\\dots ,n-2\\rbrace \\text{ and } \\beta \\ge \\beta _n, \\\\\\left[0, 1-\\frac{1-\\beta }{n}\\right[ & \\text{if } x=n-1,\\\\\\left[0,1\\right] & \\text{if } x=n.\\end{array}\\right.$ Remark 1.9 Papers erroneously claiming the Clopper-Pearson uprays or downrays to be $\\beta $ -confidence regions for (REF ) include Kappauf and Bohrer [10], Byers et al.",
"[3], and Cheng et al. [4].",
"The analogous claim of Ollero and Ramos [12] for a certain subfamily of $(\\mathrm {BC}_p\\colon p\\in [0,1]^n),$ which includes the hypergeometric laws with sample size parameter $n,$ is refuted in Remark REF below.",
"The common source of error in these papers seems to be an unclear remark of Hoeffding [8] related to the fact that, by [8] or by David [6], certain tests for $p\\mapsto p$ in the binomial model $(\\mathrm {B}_{n,p}\\colon p\\in [0,1])$ keep their level as tests for $p\\mapsto \\overline{p}$ in $(\\mathrm {BC}_p\\colon p\\in [0,1]^n)$ .",
"Let us further note that [12] should have cited Vatutin and Mikhailov [15] concerning the representability of hypergeometric laws as Bernoulli convolutions.",
"Remark 1.10 The core of the unclear remark in [8] mentioned in Remark REF is “that the usual (one-sided and two-sided) tests for the constant probability of `success' in $n$ independent (Bernoulli) trials can be used as tests for the average probability of success when the probability of success varies from trial to trial.” We specify and generalise this in the following way: Let $n \\in {\\mathbb {N}},$ $p_1 \\le p_2 \\in [0,1],$ $\\gamma _-,\\gamma _+\\in [0,1],$ $c_-\\le \\lfloor np_1\\rfloor -1,$ and $c_+\\ge \\lceil np_2 \\rceil +1.$ Then the randomised test $\\psi \\mathbf {1}_{\\lbrace 0,\\dots ,c_{-}-1\\rbrace } + \\gamma _{-}\\mathbf {1}_{\\lbrace c_{-}\\rbrace } + \\gamma _{+}\\mathbf {1}_{\\lbrace c_{+}\\rbrace } +\\mathbf {1}_{\\lbrace c_{+}+1,\\dots ,n\\rbrace }$ for the hypothesis $[p_1,p_2]$ in the binomial model $\\left(\\mathrm {B}_{n,p}\\colon p\\in [0,1]\\right)$ keeps its level as a randomised test for $\\left\\lbrace p\\in [0,1]^n\\colon \\overline{p}\\in [p_1,p_2]\\right\\rbrace $ in the model $\\left(\\mathrm {BC}_p\\colon p\\in [0,1]^n\\right),$ because for every $p$ with $\\overline{p}\\in [p_1,p_2]$ it follows from [8] that we have $\\mathrm {BC}_p\\psi \\,\\ = &\\,\\ \\gamma _- \\mathrm {BC}_p(\\lbrace 0,\\dots ,c_-\\rbrace ) + (1-\\gamma _-)\\mathrm {BC}_p(\\lbrace 0,\\dots ,c_--1\\rbrace )\\\\&\\,\\ +\\gamma _+ \\mathrm {BC}_p(\\lbrace c_+,\\dots ,n\\rbrace ) + (1-\\gamma _+)\\mathrm {BC}_p(\\lbrace c_++1,\\dots ,n\\rbrace )\\\\\\le &\\,\\ \\mathrm {B}_{n,\\overline{p}} \\psi .$ But this statement does not always apply to the one-sided tests based on the Clopper-Pearson uprays: Let $n=2$ and $\\beta \\in \\mathopen ]0,1\\mathclose [.$ Let $r\\in \\mathopen [0,1\\mathclose ],$ $H\\mathopen [0,r\\mathclose ],$ and $\\psi \\mathbf {1}_{\\lbrace \\mathrm {K}_{\\mathrm {CP},n}\\cap H =\\emptyset \\rbrace },$ so that we have $\\sup \\nolimits _{p\\in H}\\mathrm {B}_{n,p}\\psi \\le 1-\\beta .$ But, if for example $r=1-\\sqrt{\\beta },$ the test simplifies to $\\psi =\\mathbf {1}_{\\lbrace 1,2\\rbrace },$ and for $p(r-\\varepsilon ,r+\\varepsilon )$ for an $\\varepsilon >0$ small enough, we have $\\overline{p}\\in H$ and $\\mathrm {BC}_p\\psi = 1 -\\mathrm {BC}_p(\\lbrace 0\\rbrace ) = 1-\\beta +\\varepsilon ^2 >1-\\beta .$ Remark 1.11 Clopper-Pearson uprays can be invalid for hypergeometric estimation problems: For $N\\in {\\mathbb {N}}_0,$ $n\\in \\lbrace 0,\\dots ,N\\rbrace $ , and $p\\in \\left\\lbrace \\tfrac{j}{N}\\colon j\\in \\left\\lbrace 0,\\dots ,N\\right\\rbrace \\right\\rbrace ,$ let $\\mathrm {H}^{}_{n,p,N}$ denote the hypergeometric law of the number of red balls drawn in a simple random sample of size $n$ from an urn containing $Np$ red and $N(1-p)$ blue balls, so that we have $\\mathrm {H}^{}_{n,p,N}(\\lbrace k\\rbrace ) = \\binom{Np}{k} \\binom{N(1-p)}{n-k} / \\binom{N}{n}$ for $k\\in {\\mathbb {N}}_0.$ For $\\beta \\in \\mathopen ]0,1\\mathclose [$ and fixed $n$ and $N,$ in general, $\\mathrm {K}^{}_{\\mathrm {CP,}n}$ is not a $\\beta $ -confidence region for the estimation problem $\\left(\\left(\\mathrm {H}^{}_{n,p,N}\\colon p\\in \\left\\lbrace \\tfrac{j}{N}\\colon j\\in \\left\\lbrace 0,\\dots ,N\\right\\rbrace \\right\\rbrace \\right), p\\mapsto p\\right),$ because if, for example, $n \\ge 2$ and $\\beta =\\left(1-\\tfrac{1}{N} \\right)^n,$ then for $p=g^{}_{n}(1)$ we have $p = 1-\\beta ^{1/n} = \\tfrac{1}{N}$ and so $\\mathrm {H}^{}_{n,p,N}\\left(\\mathrm {K}^{}_{\\mathrm {CP,}n}\\ni p\\right) = \\mathrm {H}^{}_{n,p,N}\\left(\\lbrace 0\\rbrace \\right)= \\binom{N(1-p)}{n} / \\binom{N}{n} = \\prod \\nolimits _{j=0}^{n-1}\\tfrac{N(1-p)-j}{N-j} < (1-p)^n = \\beta .$ In contrast to Remark REF , we have the following positive result for the two-sided Clopper-Pearson $\\beta $ -confidence intervals $\\mathrm {M}^{}_{\\mathrm {CP,}n}$ for (REF ), as defined in (REF ) below.",
"Theorem 1.12 Let $n\\in {\\mathbb {N}}$ , $\\beta \\in \\mathopen ]0,1\\mathclose [$ , and $\\qquad \\mathrm {M}^{}_{\\mathrm {CP,}n}(x)&&\\mathrm {K}^{}_{\\mathrm {CP,}n,\\frac{1+\\beta }{2}}(x) \\cap \\Lambda ^{}_{\\mathrm {CP,}n,\\frac{1+\\beta }{2}}(x)\\quad \\text{ for }x\\in \\lbrace 0,\\ldots ,n\\rbrace $ with $\\mathrm {K}^{}_{\\mathrm {CP,}n,\\frac{1+\\beta }{2}}$ as in (REF ) and $\\Lambda ^{}_{\\mathrm {CP,}n,\\frac{1+\\beta }{2}}$ as in Remark REF .",
"If $\\beta \\ge 2\\beta _n - 1$ or $n=1$ , hence in particular if $\\beta \\ge \\frac{1}{2}$ , then $\\mathrm {M}^{}_{\\mathrm {CP,}n}$ is a $\\beta $ -confidence interval for (REF ).",
"Remark 1.13 The interval $\\mathrm {M}^{}_{\\mathrm {CP,}n}$ of Theorem REF improves on the two-sided interval for (REF ) obtained by Agnew [1] in the obvious way from his one-sided ones.",
"Remark 1.14 In contrast to Remark REF , we do not know whether the condition “$\\beta \\ge 2\\beta _n - 1$ or $n=1$ ” in Theorem REF might be omitted.",
"Remark 1.15 The robustness property of the two-sided Clopper-Pearson intervals given by Theorem REF does not extend to every other two-sided interval for (REF ), for example if $n=2$ not to the Sterne [13] type $\\beta $ -confidence interval $\\mathrm {K}^{}_{\\mathrm {S,}n}$ for (REF ) of Dümbgen [7]: For $\\beta \\in \\mathopen ]0,1\\mathclose [$ and $n\\in {\\mathbb {N}},$ $\\mathrm {K}^{}_{\\mathrm {S,}n}$ is given by $\\mathrm {K}^{}_{\\mathrm {S,}n}(x) && \\mathrm {K}^{}_{\\mathrm {S,}n,\\beta }(x)\\\\&& \\left\\lbrace p\\in \\mathopen [0,1\\mathclose ] \\colon \\mathrm {B}_{n,p}\\left( \\left\\lbrace k\\colon \\mathrm {B}_{n,p}(\\lbrace k\\rbrace )\\le \\mathrm {B}_{n,p}(\\lbrace x\\rbrace )\\right\\rbrace \\right)\\,\\ > \\,\\ 1-\\beta \\right\\rbrace .$ If, for example, $n=2$ and $\\beta > \\beta _2$ we have in particular $\\mathrm {K}^{}_{\\mathrm {S,}2}(0) = \\left[0, 1- g^{}_{2}(2)\\right[,$ $\\mathrm {K}^{}_{\\mathrm {S,}2}(1) = \\left]g^{}_{2}(1), 1-g^{}_{2}(1)\\right[,$ and $\\mathrm {K}^{}_{\\mathrm {S,}2}(2) = \\left]g^{}_{2}(2),1\\right],$ and indeed $\\mathrm {K}^{}_{\\mathrm {S,}2}$ is not valid for (REF ), because for $p\\in \\mathopen [0,1\\mathclose ]^2$ with $\\overline{p}= g^{}_{2}(1)$ and $p_1 \\ne p_2$ we have $\\mathrm {BC}_p\\left(\\mathrm {K}^{}_{\\mathrm {S,}2}\\ni \\overline{p}\\right)= \\mathrm {BC}_p\\left(\\left\\lbrace 0\\right\\rbrace \\right)= \\prod \\limits _{j=1}^2 (1-p_j)< \\left(1-\\overline{p}\\right)^2= \\left(1-g^{}_{2}(1)\\right)^2= \\beta .",
"$ For $n=2$ and $\\beta > \\beta _2$ we get a $\\beta $ -confidence interval for (REF ), say $\\tilde{\\mathrm {K}},$ from Theorem REF by setting $\\mathrm {K}^{\\prime }_m\\mathrm {K}^{}_{\\mathrm {S,}m}$ for $m\\in \\lbrace 0,1,2\\rbrace ,$ namely $\\tilde{\\mathrm {K}}(0) = \\left[0,1-(1-\\beta )^{1/2}\\right[ \\ ,\\ \\tilde{\\mathrm {K}}(1) = \\left]\\tfrac{1-\\beta }{2},\\tfrac{1+\\beta }{2}\\right[ \\ ,\\ \\tilde{\\mathrm {K}}(2) = \\left](1-\\beta )^{1/2},1\\right].$ One computes that $\\tilde{\\mathrm {K}}(x) \\subsetneq \\mathrm {M}^{}_{\\mathrm {CP,}2}(x)$ for $x\\in \\lbrace 0,1,2\\rbrace ,$ with $\\mathrm {M}^{}_{\\mathrm {CP,}2}$ as defined in Theorem REF .",
"We do not know whether these inclusions are true for every $n$ and usual $\\beta ,$ but in fact we do not even know whether $\\mathrm {K}^{}_{\\mathrm {S,}n} (x) \\subseteq \\mathrm {M}^{}_{\\mathrm {CP,}n} (x)$ holds universally."
],
[
"Proofs of the theorems",
"We obviously have $\\mathrm {K}(x)\\subseteq [0,1]$ and, by considering $l=0$ and $m=n$ , $\\mathrm {K}(x)\\supseteq \\mathrm {K}^{\\prime }_n(x)$ for every $x$ .",
"If $\\varphi \\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow {\\mathbb {R}}$ is any function and $\\pi \\in [0,1]$ , then, by Hoeffding's (1956, Corollary 2.1) generalization of Tchebichef [14], the minimum of the expectation $\\mathrm {BC}_{p} \\varphi $ as a function of $p\\in [0,1]^n$ subject to $\\overline{p}=\\pi $ is attained at some point $p$ whose coordinates take on at most three values and with at most one of these distinct from 0 and 1.",
"Given $p\\in [0,1]^n$ , the preceding sentence applied to $\\pi \\overline{p}$ and to $\\varphi $ being the indicator of $\\lbrace \\mathrm {K}\\ni \\pi \\rbrace $ yields the existence of $r,s\\in \\lbrace 0,\\ldots ,n\\rbrace $ with $r+s\\le n$ and of an $a\\in [0,1]$ with $r+sa=n\\pi $ and $\\mathrm {BC}_{p}\\left(\\mathrm {K}\\ni \\overline{p} \\right)&\\ge & \\left(\\delta _r\\ast \\mathrm {B}_{s,a}\\right)\\left(\\lbrace x\\in \\lbrace r,\\ldots ,r+s\\rbrace \\colon \\mathrm {K}(x) \\ni \\pi \\rbrace \\right) \\\\&\\ge & \\left(\\delta _r\\ast \\mathrm {B}_{s,a}\\right)\\left(\\lbrace x\\in \\lbrace r,\\ldots ,r+s\\rbrace \\colon \\tfrac{s}{n}\\mathrm {K}_s^{\\prime }(x-r)+\\tfrac{r}{n} \\ni \\pi \\rbrace \\right) \\\\&=& \\mathrm {B}_{s,a}\\left(\\mathrm {K}_s^{\\prime } \\ni a \\right) \\\\&\\ge & \\beta $ by bounding in the second step the union defining $\\mathrm {K}(x)$ by the set with the index $(l,m)=(r,s)$ .",
"For proving Theorem REF , we use Lemma REF prepared by Lemma REF .",
"Let $F^{}_{n,p}$ and $f^{}_{n,p}$ denote the distribution and density functions of the binomial law $\\mathrm {B}_{n,p}$ .",
"Lemma 2.1 Let $n\\in {\\mathbb {N}}$ .",
"Then $F^{}_{n,\\frac{x}{n}}(x) &<& F^{}_{n,\\frac{1}{n}}(1) \\quad \\text{ for }x\\in \\lbrace 2,\\ldots ,n-1\\rbrace .$ If $x\\in {\\mathbb {N}}$ with $x\\le \\frac{n-1}{2}$ , then for $p\\in \\mathopen ]\\frac{x}{n},\\frac{x+1}{n}\\mathclose [$ , we have $yx+1-np>0$ , hence $\\frac{f^{}_{n-1,p}\\left(x\\right)}{f^{}_{n,\\frac{x+1}{n}}\\left(x+1\\right)}& = \\frac{f^{}_{n-1,p}\\left(x\\right)}{f^{}_{n-1,\\frac{x+1}{n}}\\left(x\\right)}\\\\& = \\frac{\\left(1+\\frac{y}{n-x-1}\\right)^{n-x-1}}{\\left(1+\\frac{y}{np}\\right)^x}> \\frac{\\left(1+\\frac{y}{n-x-1}\\right)^{n-x-1}}{\\left(1+\\frac{y}{x}\\right)^x}\\ge \\,\\ 1,$ using the isotonicity of $\\mathopen ]0,\\infty \\mathclose [ \\ni t \\mapsto \\left(1+\\frac{y}{t} \\right)^t$ in the last step, and hence we get $F^{}_{n,\\frac{x}{n}}(x) - F^{}_{n,\\frac{x+1}{n}}(x+1) &=&n\\int \\limits _{\\frac{x}{n}}^{\\frac{x+1}{n}}f^{}_{n-1,p}\\left(x\\right)\\mathrm {d}p\\,-\\, f^{}_{n,\\frac{x+1}{n}}\\left(x+1\\right) \\,\\ >\\,\\ 0;$ consequently (REF ) holds under the restriction $x\\le \\frac{n+1}{2}$ .",
"If now $x\\in {\\mathbb {N}}$ with $\\frac{n+1}{2}\\le x\\le n-1$ , then $1\\le k n-x <\\frac{n}{2}$ , and hence an inequality attributed to Simmons by Jogdeo and Samuels [9] yields $F^{}_{n,\\frac{k}{n}}(k-1) > 1- F^{}_{n,\\frac{k}{n}}(k)$ , so that $F^{}_{n,\\frac{x}{n}}(x)&=& 1- F^{}_{n,\\frac{k}{n}}(k-1) \\,\\ <\\,\\ F^{}_{n,\\frac{k}{n}}(k)\\,\\ \\le \\,\\ F^{}_{n,\\frac{1}{n}}(1),$ using in the last step (REF ) in a case already proved in the previous sentence.",
"Lemma 2.2 Let $n\\in {\\mathbb {N}}$ , $\\beta \\in [\\beta _n,1[$ , and $x\\in \\lbrace 2,\\ldots ,n\\rbrace $ .",
"Then $g_n(x)\\le \\frac{x-1}{n}$ .",
"Using Lemma REF , we get $ F^{}_{n,\\frac{x-1}{n}}(x-1) \\le F^{}_{n,\\frac{1}{n}}(1)=\\beta _n \\le \\beta =F^{}_{n,g^{}_n(x)}(x-1)$ , and hence the claim.",
"To simplify the defining representation of $\\mathrm {K}$ in the present case, let us put $\\qquad g(x) &&\\min \\limits _{\\genfrac{}{}{0.0pt}{}{l\\in \\lbrace 0,\\dots ,x-1\\rbrace ,}{m\\in \\lbrace x-l,\\dots ,n-l\\rbrace }}\\left(\\tfrac{m}{n} g^{}_{m}(x-l) + \\tfrac{l}{n} \\right)\\qquad \\text{ for }x\\in \\lbrace 1,\\ldots ,n\\rbrace .$ For $x\\in \\lbrace 0,\\ldots ,n\\rbrace $ , we have, using (REF ), $\\mathrm {K}(x)&\\supseteq &\\tfrac{n-x}{n}\\mathrm {K}_{\\mathrm {CP},n-x}(x-x)+\\tfrac{x}{n} \\,\\ = \\,\\ \\left[\\tfrac{x}{n},1\\right],$ hence in particular $\\mathrm {K}(0)=[0,1]$ .",
"For $x\\in \\lbrace 1,\\ldots ,n\\rbrace $ , we have, with $(l,m)$ denoting some pair where the minimum in (REF ) is attained, $\\mathrm {K}(x)&\\supseteq &\\tfrac{m}{n}\\mathrm {K}_{\\mathrm {CP},m}(x-l) +\\tfrac{l}{n}\\,\\ =\\,\\ \\left]g(x),\\tfrac{l+m}{n}\\right]\\,\\ \\supseteq \\,\\ \\left]g(x),\\tfrac{x}{n}\\right]$ and, using $g^{}_{x}(x)<1$ at the third step below, $\\mathrm {K}(x) \\setminus \\left]g(x),1\\right]&\\subseteq & \\bigcup _{m\\in \\lbrace 0,\\ldots ,n-x\\rbrace }\\left( \\tfrac{m}{n} \\mathrm {K}_{\\mathrm {CP},m}(x-x)+\\tfrac{x}{n} \\right)\\,\\ \\subseteq \\,\\ \\left[\\tfrac{x}{n},1 \\right] \\\\&\\subseteq & \\left] \\tfrac{x}{n} g^{}_{x}(x-0)+\\tfrac{0}{n} , 1\\right]\\,\\ \\subseteq \\,\\ \\mathopen ]g(x),1\\mathclose ].$ Combining the above yields $\\mathrm {K}(x) &=& \\left\\lbrace \\begin{array}{ll} [0,1] &\\text{ if }x=0,\\\\\\mathopen ]g(x),1\\mathclose ] &\\text{ if }x\\in \\lbrace 1,\\ldots ,n\\rbrace ,\\end{array}\\right.$ so in particular $\\mathrm {K}$ is indeed an upray, and (REF ) holds in its trivial first case.",
"Using (REF ) and the isotonicity of $t\\mapsto \\left(\\beta ^t-1\\right)/t$ due to the convexity of $t\\mapsto \\beta ^t$ yields $g(1) &=& \\min _{m=1}^n \\tfrac{m}{n}g^{}_{m}(1)\\,\\ = \\,\\ \\tfrac{1}{n} \\min _{m=1}^n m\\left(1-\\beta ^{ 1/m }\\right) \\,\\ = \\,\\ \\tfrac{1-\\beta }{n}$ and hence (REF ) also in the second case.",
"The last case is treated at the end of this proof.",
"$\\mathrm {K}$ is strictly isotone, since, for $x\\in \\lbrace 2,\\dots ,n\\rbrace $ , we get, using $g^{}_{m}(x-1) < g^{}_{m}(x)$ for $2 \\le x \\le m \\le n$ due to (REF ), $g(x)& = & \\min \\limits _{m\\in \\lbrace x,\\dots ,n\\rbrace } \\tfrac{m}{n} g^{}_{m}(x) \\nonumber \\\\& & \\wedge \\min \\limits _{\\genfrac{}{}{0.0pt}{}{l\\in \\lbrace 1,\\dots ,x-1\\rbrace ,}{m\\in \\lbrace x-(l-1)-1,\\dots ,n-(l-1)-1\\rbrace }}\\left( \\tfrac{m}{n} g^{}_{m}(x-1-(l-1)) +\\tfrac{l-1}{n} +\\tfrac{1}{n}\\right) \\nonumber \\\\& > & \\min \\limits _{m\\in \\lbrace x-1,\\dots ,n\\rbrace } \\tfrac{m}{n} g^{}_{m}(x-1)\\wedge \\!\\!\\!",
"\\min \\limits _{\\genfrac{}{}{0.0pt}{}{l\\in \\lbrace 0,\\dots ,x-1-1\\rbrace ,}{m\\in \\lbrace x-1-l,\\dots ,n-1-l\\rbrace }} \\!",
"\\!\\!\\left(\\tfrac{m}{n} g^{}_{m}(x-1-l) +\\tfrac{l}{n} \\right) \\nonumber \\\\& \\ge & g(x-1).$ By considering $p=(1-\\beta ,0,\\ldots ,0)\\in [0,1]^n$ at the first step below, and using $\\mathrm {K}(1)=\\mathopen ]\\frac{1-\\beta }{n},1\\mathclose ]\\lnot \\ni \\tfrac{1-\\beta }{n}$ and the isotonicity of $\\mathrm {K}$ at the second, we get $\\inf _{p\\in [0,1]^n}\\mathrm {BC}_{p}(\\mathrm {K}\\ni \\overline{p})&\\le & \\mathrm {B}_{1-\\beta }\\left(\\mathrm {K}\\ni \\tfrac{1-\\beta }{n}\\right)\\,\\ = \\,\\ \\mathrm {B}_{1-\\beta }\\left(\\lbrace 0\\rbrace \\right) \\,\\ =\\,\\ \\beta $ and hence, by Theorem REF , $\\inf _{p\\in [0,1]^n}\\mathrm {BC}_{p}(\\mathrm {K}\\ni \\overline{p})=\\beta $ .",
"To prove the optimality of $\\mathrm {K}$ , let us assume that $\\tilde{\\mathrm {K}}\\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ is another isotone upray and that we have an $x^{\\prime }\\in \\lbrace 0,\\ldots ,n\\rbrace $ with $\\tilde{\\mathrm {K}}(x^{\\prime }) \\,\\ \\subsetneq \\,\\ \\mathrm {K}(x^{\\prime }).$ We have to show that $\\inf _{p\\in [0,1]^n} \\mathrm {BC}_p(\\tilde{\\mathrm {K}}\\ni \\overline{p}) < \\beta $ .",
"If $x^{\\prime }=0$ , then $\\mathrm {K}(x^{\\prime })=[0,1]$ and, since $\\tilde{\\mathrm {K}}(0)$ is an upray in $[0,1]$ , (REF ) yields $0\\notin \\tilde{\\mathrm {K}}(0)$ , and hence $\\inf _{p\\in [0,1]^n} \\mathrm {BC}_p\\left(\\tilde{\\mathrm {K}}\\ni \\overline{p}\\right)&\\le & \\delta _{0}\\left(\\tilde{\\mathrm {K}}\\ni 0 \\right)\\,\\ =\\,\\ 0 \\,\\ <\\,\\ \\beta .$ If $x^{\\prime }\\in \\lbrace 1,\\dots ,n\\rbrace $ , then, using (REF ) and (REF ), we get $\\mathrm {K}(x^{\\prime })=\\mathopen ]\\frac{m}{n}g^{}_{m}(x^{\\prime }-l)+\\frac{l}{n},1\\mathclose ]$ for some $l\\in \\lbrace 0,\\dots ,x^{\\prime }-1\\rbrace $ and $m\\in \\lbrace x^{\\prime }-l,\\dots ,n-l\\rbrace $ , and since $g^{}_{m}(x^{\\prime }-l)<1$ , we find an $a\\in \\mathopen ]g^{}_{m}(x^{\\prime }-l),1\\mathclose ]$ with $\\frac{m}{n}a+\\frac{l}{n}\\notin \\tilde{\\mathrm {K}}(x^{\\prime })$ , hence $\\frac{m}{n}a+\\frac{l}{n}\\notin \\tilde{\\mathrm {K}}(y)$ for $y\\in \\lbrace x^{\\prime },\\ldots , n\\rbrace $ by the isotonicity of $\\tilde{\\mathrm {K}}$ , and hence $\\inf \\limits _{p\\in [0,1]^n} \\mathrm {BC}_p(\\tilde{\\mathrm {K}}\\ni \\overline{p})&\\le & \\mathrm {B}_{m,a}\\left(\\left\\lbrace x\\in \\lbrace 0,\\dots ,n\\rbrace \\colon \\tilde{\\mathrm {K}}(x+l)\\ni \\tfrac{l+ma}{n}\\right\\rbrace \\right)\\\\&\\le & \\mathrm {B}_{m,a}(\\lbrace 0,\\dots ,x^{\\prime }-l-1\\rbrace ) \\\\& < & \\mathrm {B}_{m,g^{}_{m}(x^{\\prime }-l)}(\\lbrace 0,\\dots ,x^{\\prime }-l-1\\rbrace )\\\\& = & \\beta .$ To prove the admissibility of $\\mathrm {K},$ assume that there was a $\\beta $ -confidence upray $\\mathrm {K}^\\ast $ for (REF ) with $\\mathrm {K}^\\ast (x)\\subseteq \\mathrm {K}(x)$ for each $x\\in \\lbrace 0,\\dots ,n\\rbrace $ and $\\mathrm {K}^\\ast (x^{\\prime })\\subsetneq \\mathrm {K}(x^{\\prime })$ for some $x^{\\prime }.$ Then, since $\\mathrm {K}$ is strictly isotone, $\\mathrm {K}^{\\ast \\ast }(x) &&\\left\\lbrace \\begin{array}{ll} \\mathrm {K}(x)&\\text{ if }x\\ne x^{\\prime }\\\\\\mathrm {K}^\\ast (x^{\\prime })\\cup \\mathrm {K}(x^{\\prime }+1)&\\text{ if }x= x^{\\prime } < n,\\\\\\mathrm {K}^\\ast (x^{\\prime }) &\\text{ if }x= x^{\\prime } = n\\end{array}\\right\\rbrace \\,\\ \\supseteq \\,\\ \\mathrm {K}^{\\ast }(x)$ would define an isotone $\\beta $ -confidence upray for (REF ) with $\\mathrm {K}^{\\ast \\ast }(x^{\\prime })\\subsetneq \\mathrm {K}(x^{\\prime })$ , contradicting the optimality of $\\mathrm {K}$ .",
"To prove finally the last case of (REF ), let $n\\ge 2$ and $\\beta \\ge \\beta _n$ , and let now $\\tilde{\\mathrm {K}}\\colon \\lbrace 0,\\ldots ,n\\rbrace \\rightarrow 2^{[0,1]}$ be defined by the right hand side of (REF ).",
"If $p\\in [0,1]^n$ with $\\overline{p}\\in [0,\\frac{1-\\beta }{n}],$ then $\\mathrm {BC}_p(\\tilde{\\mathrm {K}}\\ni \\overline{p})&\\ge & \\mathrm {BC}_p(\\lbrace 0\\rbrace ) \\,\\ =\\,\\ \\prod \\limits _{j=1}^n (1-p_j)\\,\\ \\ge \\,\\ 1-\\sum \\limits _{j=1}^n p_j \\,\\ =\\,\\ 1-n\\overline{p}\\,\\ \\ge \\,\\ \\beta .$ If $p\\in [0,1]^n$ with $\\overline{p}\\in \\mathopen ]\\frac{1-\\beta }{n},1\\mathclose ]$ , then with $g^{}_n(n+1)1$ either there is a $c\\in \\lbrace 2,\\ldots ,n\\rbrace $ with $\\overline{p}\\in \\mathopen ]g^{}_n(c), g^{}_n(c+1) \\mathclose ]$ , or $\\overline{p}\\in \\mathopen ]\\frac{1-\\beta }{n},g^{}_{n}(2)\\mathclose ]$ and we put $c1$ ; in either case then $n \\overline{p}\\le n g^{}_{n}(c+1) \\le 1$ by Lemma REF , and hence an application of Hoeffding [8] at the second step below yields $\\mathrm {BC}_p\\left(\\tilde{\\mathrm {K}}\\ni \\overline{p}\\right) &=& \\mathrm {BC}_p\\left(\\lbrace 0,\\ldots ,c\\rbrace \\right)\\,\\ \\ge \\,\\ F^{}_{n,\\overline{p}} \\left(c\\right) \\,\\ \\ge \\,\\ F^{}_{n,g^{}_{n}(c+1)} \\left(c\\right)\\,\\ \\ge \\,\\ \\beta .$ Hence $\\tilde{\\mathrm {K}}$ is a $\\beta $ -confidence upray for (REF ) and satisfies $\\tilde{\\mathrm {K}}(x)\\subseteq \\mathrm {K}(x)$ for each $x$ , and so the admissibility of $\\mathrm {K}$ yields $\\tilde{\\mathrm {K}}=\\mathrm {K}$ , and hence (REF ).",
"Let $\\gamma \\frac{1+\\beta }{2}$ , let $\\mathrm {K}_\\gamma $ be the $\\gamma $ -confidence upray from Theorem REF , and let $\\Lambda _\\gamma $ be the analogous $\\gamma $ -confidence downray from Remark REF .",
"Then, by subadditivity, $\\mathrm {M}_\\beta (x)\\mathrm {K}_\\gamma (x)\\cap \\Lambda _\\gamma (x)$ for $x\\in \\lbrace 0,\\ldots ,n\\rbrace $ defines a $\\beta $ -confidence interval for (REF ).",
"If $n=1$ , then $\\mathrm {K}^{}_{\\mathrm {CP},n} = \\mathrm {M}_\\beta $ , hence the claim.",
"So let $\\beta \\ge 2\\beta _n-1$ , that is, $\\gamma \\ge \\beta _n$ .",
"Then (REF ) and (REF ), with $\\gamma $ in place of $\\beta $ , yield $\\mathrm {M}^{}_{\\mathrm {CP},n}(x) = \\mathrm {M}^{}_\\beta (x)$ for $x\\notin \\lbrace 1,n-1\\rbrace .$ So, if $\\overline{p}\\notin \\left( \\mathrm {M}^{}_{\\mathrm {CP},n}(1) \\setminus \\mathrm {M}^{}_\\beta (1)\\right)\\cup \\left( \\mathrm {M}^{}_{\\mathrm {CP},n}(n-1) \\setminus \\mathrm {M}^{}_\\beta (n-1)\\right),$ we have $\\mathrm {BC}_p\\left(\\mathrm {M}^{}_{\\mathrm {CP},n}\\ni \\overline{p}\\right) = \\mathrm {BC}_p\\left(\\mathrm {M}^{}_\\beta \\ni \\overline{p}\\right)\\ge \\beta .$ Otherwise $\\overline{p}\\in \\left]\\frac{1-\\gamma }{n},g^{}_{n,\\gamma }(1)\\right]$ or $\\overline{p}\\in \\left[g^{}_{n,\\gamma }(n-1),1-\\frac{1-\\gamma }{n}\\right[.$ In the first case, $\\overline{p}\\in \\left] \\frac{1-\\gamma }{n}, g^{}_{n,\\gamma }(1) \\right]= \\left] \\frac{1-\\gamma }{n}, 1-\\gamma ^{1/n} \\right]\\subseteq \\left[0,1-(1-\\gamma )^{1/n}\\right[ = \\mathrm {M}^{}_{\\mathrm {CP},n}(0)$ and from $\\overline{p}\\in \\mathrm {M}^{}_{\\mathrm {CP},n}(0)$ and $\\overline{p}\\le 1-\\gamma ^{1/n}$ we get $\\mathrm {BC}_p\\left(\\mathrm {M}^{}_{\\mathrm {CP},n}\\ni \\overline{p}\\right) \\ge \\mathrm {BC}_p(\\lbrace 0\\rbrace )& = \\prod \\limits _{j=1}^n (1-p_j) \\\\& \\ge 1-n\\overline{p} \\ge 1-n\\left(1-\\gamma ^{1/n}\\right)\\ge \\gamma > \\beta .$ In the second case, analogously, $\\overline{p}\\in \\left[g^{}_{n,\\gamma }(n-1),1-\\frac{1-\\gamma }{n}\\right[= \\left[\\gamma ^{1/n},1-\\frac{1-\\gamma }{n}\\right[\\subseteq \\mathrm {M}^{}_{\\mathrm {CP},n}(n)$ and from $\\overline{p}\\in \\mathrm {M}^{}_{\\mathrm {CP},n}(n)$ and $\\overline{p}\\ge \\gamma ^{1/n}$ we get $\\mathrm {BC}_p\\left(\\mathrm {M}^{}_{\\mathrm {CP},n}\\ni \\overline{p}\\right) \\ge \\mathrm {BC}_p(\\lbrace n\\rbrace )= \\prod \\nolimits _{j=1}^n p_j\\ge {\\overline{p}}^{\\, n}\\ge \\gamma > \\beta .$"
],
[
"Acknowledgement",
"We thank Jona Schulz for help with the proof of Lemma REF , and the referee for suggesting to address nestedness."
]
] | 1403.0229 |
[
[
"A physical model for the evolving UV luminosity function of high\n redshift galaxies and their contribution to the cosmic reionization"
],
[
"Abstract [Abridged] We present a physical model for the evolution of the ultraviolet (UV) luminosity function (LF) of high-z galaxies taking into account in a self-consistent way their chemical evolution and the associated evolution of dust extinction.",
"The model yields good fits of the UV and Lyman-alpha LFs at z>~2.",
"The weak evolution of both LFs between z=2 and z=6 is explained as the combined effect of the negative evolution of the halo mass function, of the increase with redshift of the star formation efficiency, and of dust extinction.",
"The slope of the faint end of the UV LF is found to steepen with increasing redshift, implying that low luminosity galaxies increasingly dominate the contribution to the UV background at higher and higher redshifts.",
"The observed range of UV luminosities at high-z implies a minimum halo mass capable of hosting active star formation M_crit <~ 10^9.8 M_odot, consistent with the constraints from hydrodynamical simulations.",
"From fits of Lyman-alpha LFs plus data on the luminosity dependence of extinction and from the measured ratios of non-ionizing UV to Lyman-continuum flux density for samples of z=~3 Lyman break galaxies and Lyman-alpha emitters, we derive a simple relationship between the escape fraction of ionizing photons and the star formation rate, impling larger escape fraction for less massive galaxies.",
"Galaxies already represented in the UV LF (M_UV <~ -18) can keep the universe fully ionized up to z=~6, consistent with (uncertain) data pointing to a rapid drop of the ionization degree above z~6.",
"On the other side, the electron scattering optical depth, tau_es, inferred from CMB experiments favor an ionization degree close to unity up to z=~9-10.",
"Consistency with CMB data can be achieved if M_crit =~ 10^8.5 M_odot, implying that the UV LFs extend to M_UV =~ -13, although the corresponding tau_es is still on the low side of CMB-based estimates."
],
[
"Introduction",
"One of the frontiers of present day astrophysical/cosmological research is the understanding of the transition from the “dark ages”, when the hydrogen was almost fully neutral, to the epoch when stars and galaxies began to shine and the intergalactic hydrogen was almost fully re-ionized.",
"Observations with the Wide Field Camera 3 (WFC-3) on the Hubble Space Telescope [42], [15], [105], [106], [36], [128] have substantially improved the observational constraints on the abundance and properties of galaxies at cosmic ages of less than 1 Gyr.",
"Determinations of the ultraviolet (UV) luminosity function (LF) of galaxies at $z=7$ –8 have been obtained by [10], [13], [147], [104], [138], [94], [93], [166], [165], [84], [83], and [16].",
"Estimates over limited luminosity ranges were provided by [10], [105], and [93] at $z=9$ , and by [13] and [105], [106] at $z=10$ .",
"Constraints on the UV luminosity density at redshifts up to 12 have been presented by [36].",
"Since galaxies at $z\\geqslant 6$ are the most likely sources of the UV photons capable of ionizing the intergalactic hydrogen, the study of the early evolution of the UV luminosity density is directly connected with the understanding of the cosmic reionization.",
"Several studies [128], [74], [3], [51] have adopted parameterized models for the evolving UV luminosity density.",
"These models are anchored to the observed high-$z$ LFs and are used to investigate plausible reionization histories, consistent with other probes of the redshift-dependent ionization degree and primarily with the electron scattering optical depth measured by the Wilkinson Microwave Anisotropy Probe [57].",
"There are also several theoretical models for the evolution of the LFs of Ly$\\alpha $ emitters (LAEs) and of Lyman break galaxies (LBGs), using different approaches.",
"These include various semi-analytic galaxy formation models [156], [82], [71], [125], [46], [96], [76], [47], smoothed particle hydrodynamics (SPH) simulations [31], [29], [101], [132], [44], [63], [64] as well as analytic models [92], [30], [133], [99], [98], [131], [155].",
"Each approach has known strengths and weaknesses.",
"Given the complexity and the large number of variable parameters in many models, an analytic physical approach is particularly useful in understanding the role and the relative importance of the ingredients that come into play.",
"We adopt such approach, building on the work by [92].",
"As mentioned above, spectacular advances in direct determinations of the UV LFs of galaxies at epochs close to the end of reionization have been recently achieved.",
"These imply much stronger constraints on models, particularly on the analytic ones, that need to be revisited.",
"Further constraints, generally not taken into account by all previous studies, have been provided by far-IR/(sub-)millimeter data, that probe other phases of the early galaxy evolution, concurring to delineate the complete picture.",
"A key novelty of the model is that it includes a self-consistent treatment of dust absorption and re-emission, anchored to the chemical evolution of the interstellar medium (ISM).",
"This allows us to simultaneously account for the demography of both UV- and far-IR/(sub-)millimeter-selected high-$z$ star-forming galaxies.",
"In addition, the model incorporates a variety of observational constraints that are only partially taken into account by most previous studies.",
"Specifically, we take into account constraints on: the escape fraction of ionizing photons coming from both continuum UV and Ly$\\alpha $ line LFs and from measurements of the ratio of non-ionizing to ionizing emission from galaxies; the luminosity/stellar mass–metallicity [89], [91] and the stellar mass–UV luminosity [151], [152] relations; the amplitude of the ionizing background at several redshifts up to $z=6$ [26], [162], [18], [74], [5], [103].",
"The successful tests of the model against a wide variety of data constitute a solid basis for extrapolations to luminosity and redshift ranges not yet directly probed by observations.",
"The plan of the paper is the following.",
"In Section , we outline the model describing its basic ingredients.",
"In Section , we exploit it to compute the cosmic-epoch dependent UV LF, allowing for the dust extinction related to the chemical evolution of the gas.",
"In Section , we compute the production rate of ionizing photons and investigate their absorption rates by both dust and neutral hydrogen (HI), as constrained by measurements of the Ly$\\alpha $ line LFs at various redshifts and of the ratios of non-ionizing UV to Lyman-continuum luminosities.",
"We then derive the fraction of ionizing photons that can escape into the intergalactic medium (IGM) and use the results to obtain the evolution with redshift of the volume filling factor of the intergalactic ionized hydrogen.",
"The main conclusions are summarized in Section .",
"Throughout this paper we adopt a flat $\\Lambda \\rm CDM$ cosmology with matter density $\\Omega _{\\rm m} = 0.32$ , $\\Omega _{\\rm b} = 0.049$ , $\\Omega _{\\Lambda } = 0.68$ , Hubble constant $h=H_0/100\\, \\rm km\\,s^{-1}\\,Mpc^{-1} = 0.67$ , spectrum of primordial perturbations with index $n = 0.96$ , and normalization $\\sigma _8 = 0.83$ [123].",
"All the quoted magnitudes are in the AB system [109]."
],
[
"Outline of the model",
"In this Section we present the basic features of the model used to compute the redshift-dependent UV and Lyman-$\\alpha $ (Ly$\\alpha $ ) LFs.",
"The model is essentially the same used by [17] in his study of the evolution of the IR LF.",
"However, the [17] study dealt with a later evolutionary phase, when star formation was dust-enshrouded.",
"Here we are interested in the phase in which dust extinction was low but growing as the gas was being enriched in metals and the dust abundance was correspondingly increasing.",
"The associated evolution with galactic age of dust extinction is discussed in Section REF .",
"In Section REF , the model, integrated with this additional ingredient, is tested against data on high-$z$ galaxies not used to constrain the model parameters."
],
[
"Basic ingredients",
"Our approach is in line with the scenario worked out by [48] and further elaborated by [80], [78], [92], and [17].",
"It exploits the outcomes of many intensive $N$ -body simulations and semi-analytic studies [168], [160], [77] according to which pre-galactic halos undergo an early phase of fast collapse, including a few major merger events, during which the central regions reach rapidly a dynamical quasi-equilibrium; a subsequent slower accretion phase follows, which mainly affects the halo outskirts.",
"We take the transition between the two phases as the galaxy formation time.",
"The model envisages that, during the fast collapse phase, the baryons falling into the newly created potential wells are shock-heated to the virial temperature.",
"This assumption may be at odds with analytic estimates and SPH simulations showing that the amount of shock-heated gas is relatively small for halo masses below $\\sim 10^{12}\\,M_\\odot $ [70], [33], i.e., in the mass range of interest here (see below).",
"However, the analytic estimates assume spherical collapse and the SPH simulations consider a smooth initial configuration.",
"In both cases, the results do not necessarily apply to the halo build up during the fast collapse phase, dominated by a few major mergers.",
"A crucial test of the star formation history implied by the model is provided by the data discussed in the following.",
"An alternative picture for the formation of galaxies is the “cold stream driven” scenario, according to which massive galaxies (halo masses above $\\sim 10^{12}\\,M_\\odot $ ) formed from steady, narrow, cold gas streams, fed by dark matter filaments from the cosmic web, that penetrate the shock-heated media of massive dark matter haloes [32].",
"However, this mechanism may require an implausibly high star formation efficiency [146], [78] and a too high bias factor of galaxies at $z\\simeq 2$ [27], [164].",
"Also observational evidence of cold flows has been elusive.",
"The recent adaptive optics–assisted SINFONI observations of a $z=2.3285$ star-forming galaxy obtained at the Very Large Telescope [7] showed evidence of cold material at about one-third of the virial size of the halo.",
"The gas appears to be co-rotating with the galaxy and may eventually inflow towards the center to feed star formation.",
"However, this gas does not necessarily come from the intergalactic space; it may well be the result of cooling of the gas inside the halo, as implied by our scenario.",
"Anyway, as we will see in the following, massive galaxies are of only marginal importance in the present context.",
"In our framework, the galaxy LF is directly linked to the halo formation rate, $d^3 N(M_{\\rm vir}, \\tau )/ dM_{\\rm vir}\\, d\\tau \\,dV$ , as a function of halo mass, $M_{\\rm vir}$ , and cosmic time, $\\tau $ (see Figure REF ).",
"The halo formation rate is approximated by the positive term of the derivative with respect to $\\tau $ of the cosmological halo mass function [135].",
"[79], using an excursion set approach, showed that this is a good approximation.",
"For the halo mass function, we used the analytic approximation by [142].",
"In converting the halo formation rate into the UV LF, we also need to take into account the survival time, $t_{\\rm destr}$ , of the halos that are subject to merging into larger halos.",
"A short halo survival time would obviously decrease the number density of halos of given mass that are on at a given time.",
"At the high redshifts of interest here, we expect that a large fraction of halos undergo major mergers, loosing their individuality; therefore the survival time could impact our estimate of the LF.",
"The survival time is difficult to define unambiguously.",
"We adopt the value obtained from the negative term of the time derivative, $(\\partial _t N)_{-}$ , of the [124] or of the [142] mass functions, i.e., $t_{\\rm destr} \\equiv N/ (\\partial _t N)_{-}= t_{\\rm Hubble}(z)$ , independent of halo mass.",
"In our calculations of the UV LF (Equation (REF )) we cut the integration over time at $t_{\\rm destr}$ .",
"As we will see in the following, in the mass and redshift ranges of interest here, the duration of the UV bright phase of galaxy evolution is generally substantially shorter than $t_{\\rm Hubble}(z)$ and therefore also shorter than $t_{\\rm destr}$ .",
"Therefore, the results are only weakly affected by the cut.",
"Hence a more refined treatment of this tricky issue is not warranted.",
"The history of star formation and of accretion into the central BH were computed by numerically solving the set of equations laid down in the Appendix of [17].",
"These equations describe the evolution of the gas along the following lines.",
"Initially, a galactic halo of mass $M_{\\rm vir}$ contains a cosmological gas mass fraction $f_{\\rm b} = M_{\\rm gas}/M_{\\rm vir}=0.17$ , distributed according to a [102] density profile, with a moderate clumping factor and primordial composition.",
"The gas is heated to the virial temperature at the virialization redshift, $z_{\\rm vir}$ , then cools and flows towards the central regions.",
"The cooling is fast especially in the denser central regions, where it triggers a burst of star formation.",
"We adopt a [22] initial mass function (IMF).",
"The gas cooling rate is computed adopting the cooling function given by [154].",
"The evolution of the metal abundance of the gas is followed using the classical equations and stellar nucleosynthesis prescriptions, as reported, for instance, in [130].",
"An appreciable amount of cooled gas loses its angular momentum via the drag by the stellar radiation field, settles down into a reservoir around the central supermassive black hole (BH), and eventually accretes onto it by viscous dissipation, powering the nuclear activity.",
"The supernova (SN) explosions and the nuclear activity feed energy back to the gaseous baryons, and regulate the ongoing star formation and the BH growth.",
"The feedback from the active nucleus has a key role in quenching the star formation in very massive halos but is only of marginal importance for the relatively low halo masses of interest here.",
"This implies that the results presented in this paper are almost insensitive to the choice for the parameters governing the BH growth and feedback.",
"The really important model parameters are only those that regulate the star formation rate (SFR), hence also the chemical evolution.",
"These are the clumping factor $C$ , that affects the cooling rate, the ratio between the gas inflow timescale, $t_{\\rm cond}$ , and the star formation timescale, $t_\\star $ , $s=t_{\\rm cond}/t_\\star $ , and the SN feedback efficiency, $\\epsilon _{\\rm SN}$ .",
"For the first two of the latter parameters, the clumping factor $C$ and ratio $s$ , as well as for all the parameters controlling the growth of and the feedback from the active nucleus that, albeit almost irrelevant, are nevertheless included in our calculations, we adopt the values determined by [80] and also used by [78] and [17], i.e., $C=7$ and $s=5$ .",
"A discussion on the plausible ranges for all the model parameters can be found in [17].",
"On the other hand, we have made a different choice for $\\epsilon _{\\rm SN}$ .",
"This is motivated by the fact that the earlier papers dealt with relatively high halo masses ($M_{\\rm vir}\\gtrsim 10^{11.2}\\ M_\\odot $ ) while, at the high redshifts of interest here, the masses of interest are substantially lower.",
"As pointed out by [140], the data indicate that the star formation efficiency in low-mass halos must be lower than implied by the SN feedback efficiency gauged for higher halo masses.",
"In other words, as extensively discussed in the literature, external (UV background) or internal (SN explosions, radiation from massive low-metallicity stars and, possibly, stellar winds) heating processes can reduce the SFR in low-mass halos [119], [52], [43], [73], [118], [163], [148] and completely suppress it below a critical halo mass, $M_{\\rm crit}$ .",
"We take this into account by increasing the SN feedback efficiency with decreasing halo mass and introducing a low-mass cutoff, i.e., considering only galaxies with $M_{\\rm vir}\\geqslant M_{\\rm crit}$ .",
"We set $\\epsilon _{\\rm SN} = 0.05 \\lbrace 3 + {\\rm erf}[-\\log (M_{\\rm vir}/10^{11}\\,M_\\odot )/0.5]\\rbrace / 2,$ that converges, for $M_{\\rm vir}> 10^{11}\\,M_\\odot $ , to the constant value, $\\epsilon _{\\rm SN} = 0.05$ , used by [17].",
"The critical halo mass can be estimated equating the gas thermal speed in virialized halos to the escape velocity to find $M_{\\rm crit}\\simeq 2\\times 10^8(T/2\\times 10^4\\,\\hbox{K})^{3/2}\\,M_\\odot $ [52], where $2\\times 10^4\\,\\hbox{K}$ corresponds to the peak of the hydrogen cooling curve.",
"If the balance between heating and cooling processes results in higher gas temperatures, $M_{\\rm crit}$ can increase even by a large factor.",
"An accurate modeling of the physical processes involved is difficult and numerical simulations yield estimates of $M_{\\rm crit}/M_\\odot $ in the range from $4\\times 10^8$ to $\\gtrsim 10^9$ [52], [73], [118], [148], with a trend toward lower values at $z >6$ when the IGM was cooler [34], [72], [108] and the intensity of the UV background is expected to decrease.",
"If, on one side, heating properties can decrease the SFR in small galactic halos, the production of UV photons per unit stellar mass is expected to substantially increase with redshift, at least at $z\\gtrsim 3$ .",
"[117] argued that the characteristic stellar mass, corresponding to the peak in the IMF per unit logarithmic mass interval, scales as the square of the temperature, $T_{\\rm mc}$ , of the giant molecular clouds within which stars form.",
"But the typical local value of $T_{\\rm mc}$ ($\\sim 10\\,$ K) is lower than the cosmic microwave background temperature at $z\\gtrsim 3$ , implying a rapid increase with $z$ of the characteristic stellar mass, hence of the fraction of massive stars producing UV photons.",
"On the other hand, a higher fraction of massive stars also implies a more efficient gas heating, hence a stronger quenching of the SFR.",
"This complex set of phenomena tend to counterbalance each other.",
"Therefore, in the following we adopt, above $M_{\\rm crit}$ , the SFR given by the model and the UV (ionizing and non-ionizing) photon production rate appropriate to the [22] IMF, allowing for the possibility of a correction factor to be determined by comparison with the data."
],
[
"Evolution of galaxy properties",
"The model outlined above allows us to compute the evolution with galactic age of the SFR, of the mass in stars, and of the metal abundance of the gas as a function of the halo mass, $M_{\\rm vir}$ , and of the halo virialization redshift, $z_{\\rm vir}$ .",
"Illustrative examples for four values of the halo mass at fixed $z_{\\rm vir}=6$ (left panels) and for four values of $z_{\\rm vir}$ and two values of the halo mass (right panels) are shown in Figure REF .",
"A few points are worth noting: i) the active star formation phase in the most massive halos ($M_{\\rm vir}\\gtrsim 10^{12}\\,M_\\odot $ ) is abruptly terminated by the AGN feedback, while in less massive halos, where the feedback is dominated by stellar processes, the SFR declines more gently; ii) at fixed halo mass, the SFR is initially higher at higher redshifts (mainly because the higher densities imply faster cooling of the gas) and declines earlier; iii) the higher star formation efficiency at higher $z$ implies that the stellar to halo mass ratio and the metallicity are higher for galaxies that formed at higher redshifts.",
"The different behavior of these quantities for different halo masses and different redshifts determines the relative contributions of the various masses to the LFs and their evolution with cosmic time, as discussed in the following.",
"For comparisons with the data we further need to take into account the dust extinction.",
"The latter is proportional to the dust column density which is proportional to the gas column density.",
"In turn, the dust column density is proportional to the gas column density, $N_{\\rm gas}$ , times the metallicity, $Z_{\\rm g}$ , to some power reflecting the fraction of metals locked into dust grains.",
"We thus expect that at our reference UV wavelength, 1350 ${\\rm \\mathring{A}}$ , the dust extinction $A_{1350}\\propto \\dot{M}_{\\mathrm {\\star }}^{\\alpha } Z_{\\rm g}^\\beta $ , where $\\dot{M}_{\\mathrm {\\star }}$ is the star formation rate.",
"[92] found that a relationship of this kind ($A_{1350}\\approx 0.35\\,(\\dot{M}_{\\star }/M_{\\odot }\\,\\mathrm {yr}^{-1})^{0.45}\\,(Z_{\\rm g}/Z_{\\odot })^{0.8}$ ) does indeed provide a good fit of the luminosity-reddening relation found by [141].",
"We have adopted a somewhat different relation $A_{1350}\\approx 0.75\\,\\left(\\frac{\\dot{M}_{\\star }}{M_{\\odot }\\,\\mathrm {yr}^{-1}}\\right)^{0.25}\\, \\left(\\frac{Z_{\\rm g}}{Z_{\\odot }}\\right)^{0.3},$ that we have found to provide a better fit of the UV LFs of LBGs [127], [9], [93] still being fully consistent with the [141] data (cf.",
"Figure REF ) as well as with observational estimates of the escape fractions ($f^{\\rm esc}_\\lambda =\\exp (-A_\\lambda /1.08)$ , see below) of UV and Ly$\\alpha $ photons (Figure REF ).",
"As illustrated by the bottom-left panel of Figure REF , the UV extinction is always low for the least massive galaxies ($M_{\\rm vir}\\lesssim 10^{11}\\,M_\\odot $ ), but increases rapidly with increasing $M_{\\rm vir}$ .",
"This implies a strong correlation between dust attenuation and halo/stellar mass, thus providing a physical explanation for the correlation reported by [55].",
"The bottom-right panel of the figure shows that, at fixed halo/stellar mass, the model implies a mild increase of the attenuation with increasing $z$ , paralleling the small increase in the gas metallicity due to the higher star formation efficiency.",
"The observed absolute magnitude in the AB system of a galaxy at our reference UV wavelength, $\\lambda = 1350\\,$ Å, is related to its SFR and to dust extinction $A_\\lambda $ as $M_{1350}^{\\rm obs}=51.59 - 2.5\\, \\log {L_{1350}^{\\rm int}\\over \\mathrm {erg\\,s}^{-1}\\,\\mathrm {Hz}^{-1}} + A_{1350},$ with ${L_{1350}^{\\rm int}} = k_{\\rm UV} {\\dot{M}_\\star },$ where $L_{1350}^{\\rm int}$ is the intrinsic monochromatic luminosity at the frequency corresponding to 1350 Å.",
"On account of the uncertainties on the IMF at high $z$ , mentioned above, we treat the normalization factor $k_{\\rm UV}\\, (\\hbox{erg}\\, \\hbox{s}^{-1}\\, \\hbox{Hz}^{-1}\\, M_\\odot ^{-1}\\, \\hbox{yr})$ as an adjustable parameter.",
"Using the evolutionary stellar models of [39], [40], we find that $k_{\\rm UV}$ varies with galactic age and gas metallicity as illustrated by Figure REF .",
"The main contributions to the UV LF come from galactic ages $\\gtrsim 10^7\\,$ yr, when $\\log (k_{\\rm UV})\\gtrsim 27.8$ and is somewhat higher for lower metallicity.",
"We adopt $\\log (k_{\\rm UV}) = 28$ as our reference value.",
"We remind that the relationship between the absolute AB magnitude at the effective frequency $\\nu $ , $M_{\\nu }$ , and the luminosity is $\\log [(\\nu \\,L_\\nu )/L_\\odot ]=2.3995-\\log (\\lambda /1350\\,{\\rm \\mathring{A}})-0.4\\,M_{\\nu }$ where $L_\\odot =3.845\\times 10^{33}\\,\\hbox{erg}\\,\\hbox{s}^{-1}$ .",
"We adopt the standard “dust screen” model for dust extinction, so that the observed luminosity is related to the intrinsic one by $L^{\\rm obs}(\\lambda ) = L^{\\rm int}(\\lambda ) 10^{-0.4A_\\lambda },$ with the [19] reddening curve, holding for $0.12\\,{\\rm \\mu m}\\, \\leqslant \\lambda \\leqslant 0.63\\,{\\rm \\mu m}$ , $A_\\lambda /E_s(B-V) \\simeq 2.659 (-2.156 + 1.509/\\lambda - 0.198/\\lambda ^2 + 0.011/\\lambda ^3) + R_V$ where $E_s(B-V)$ is the color excess of the galaxy stellar continuum and $R_V = 4.05$ .",
"This gives $A_{1350}/E_s(B-V) \\simeq 11.0$ and $A_{1216} \\simeq 1.08 A_{1350}$ .",
"[107] found that the dust reddening at $z \\simeq 4$ is better described by a Small Magellanic Cloud (SMC) extinction curve [121] rather than by the Calzetti curve.",
"However, for our purposes the only relevant quantity is the $A_{1216}/A_{1350}$ ratio.",
"Using the [121] fitting function for the SMC curve we find $(A_{1216}/A_{1350})_{\\rm SMC}=1.14$ , very close to the Calzetti ratio (1.08).",
"Thus our results would not appreciably change adopting the SMC law."
],
[
"Testing the model against high-$z$ data",
"Some average properties, weighted by the halo formation rate, of galaxies at $z_{\\rm obs}=2$ , 6, and 12 are shown, as a function of the observed luminosity, in Figures REF and REF .",
"The weighted averages have been computed as follows.",
"We sample the formation rate of halos with mass $M_{\\rm vir}$ virializing at $t_{\\rm vir}$ , $\\dot{n}(M_{\\rm vir}, t_{\\rm vir}) \\equiv d^3N(M_{\\rm vir}, t_{\\rm vir})/d\\log M_{\\rm vir}\\,dt\\,dV$ , in steps of $\\Delta \\log M_{\\rm vir} = 0.08$ and cosmic time interval $\\Delta t_{\\rm vir} = 4\\,$ Myr, and select those having some property $X$ , like the UV, $L_{1350}$ luminosity; the Ly$\\alpha $ , $L_{1216}$ , luminosity; or the stellar mass (actually we set $X=\\log (\\hbox{quantity})$ ) in a given range, $X_i \\in (X - \\Delta X/2, X + \\Delta X/2]$ , at the selected $z_{\\rm obs}$ .",
"The average value, $\\mu _Y(X, \\Delta X)$ , of some other property $Y$ (e.g., age of stellar populations, $t_{\\rm age}$ , gas metallicity, $Z_{\\rm g}$ , SFR, etc.)",
"is then computed as $\\mu _Y(X, \\Delta X) = \\frac{\\sum _i Y_i \\,\\dot{n}_i \\, \\theta _{\\rm H}(X - \\Delta X / 2 < X_i \\leqslant X + \\Delta X / 2)}{ \\sum _i \\dot{n}_i \\, \\theta _{\\rm H}(X - \\Delta X / 2 < X_i \\leqslant X + \\Delta X / 2)},$ where $\\theta _{\\rm H}$ ($\\theta _{\\rm H} (x)= 1\\ \\mathrm {if}\\ x\\ \\mathrm {is\\ true}$ , $\\theta _{\\rm H} (x)= 0\\ \\mathrm {otherwise}$ ) is the Heaviside function.",
"The comoving number density of galaxies satisfying the selection criterion is $n(X, \\Delta X) = \\frac{1}{\\Delta X} \\sum _i \\dot{n}_i \\cdot \\Delta \\log M_{\\rm vir} \\cdot \\Delta t_{\\rm vir} \\cdot \\theta _{\\rm H}(X - \\Delta X / 2 < X_i \\leqslant X + \\Delta X / 2).$ The data we compare with are specified in the panels and in the captions of the figures.",
"Note that these data were not used to constrain the model parameters.",
"As illustrated by Figure REF , the model compares favorably with observational estimates of various galaxy properties at different redshifts.",
"Stellar population ages are, on average, substantially younger for the brighter galaxies, which are associated to the most massive halos.",
"This is fully consistent with the well established “downsizing” scenario since it happens because massive galaxies have high SFRs that may reach $\\hbox{SFR}\\sim 100\\,M_\\odot \\,\\hbox{yr}^{-1}$ and therefore develop their stellar populations faster.",
"The chemical enrichment and the associated dust obscuration grow also faster in these galaxies that become soon UV-faint.",
"This causes a downward trend of $t_{\\rm age}$ with increasing $L^{\\rm obs}_{1350}$ , consistent with the data (upper left panel of Figure REF ).",
"To compare the predictions of our model with observations on the metallicity–luminosity and metallicity–stellar mass relations at different redshifts, we have converted the values of $[12 + \\log ({\\rm O/H})]$ given by [89] and [81] to $Z_{\\rm g}$ in solar units adopting a solar oxygen abundance $[12 + \\log ({\\rm O/H})]_\\odot = 8.69$ [4].",
"In Figure REF the stellar masses estimated by [89] assuming a Salpeter IMF have been divided by a factor of 1.4 to convert them to the Chabrier IMF used in the model.",
"Also we have plotted the metallicity estimates by [37] for a sample of 87 LBGs at $z \\sim 2.2$ as revised by [89] using an improved calibration.",
"[25] found an offset in the relation for $z\\sim 2$ galaxies compared to [37].",
"They argue that the difference originates from the use of different metallicity estimators with locally calibrated metallicity relations that may not be appropriate for the different physical conditions of star-forming regions at high redshifts.",
"The dependence of gas-phase metallicity on stellar mass, with lower $M_\\star $ galaxies having lower metallicities, is accounted for by the model, at least at $z_{\\rm obs}=2.2$ (Figure REF ).",
"This trend is also present in the data on $z_{\\rm obs}=3.4$ galaxies by [89] which indicate a decrease by 0.6 dex of the amplitude of the $Z_{\\rm g}$ –$M_\\star $ relation compared to that observed at $z_{\\rm obs}=2.2$ .",
"Such strong evolution is at odds with model predictions, according to which the evolution should be quite weak, as shown in the bottom-right panel of Figure REF .",
"However, as argued by [25], the strong evolution may be an artifact due to the use of locally calibrated metallicity indicators which do not account for evolution in the physical conditions of star-forming regions.",
"In the previous Section we have introduced all the ingredients needed to compute the population properties of high-$z$ galaxies.",
"We now proceed with the calculation of the cosmic epoch-dependent UV LF and discuss the role of dust extinction in shaping it (Section REF ), the constraints on its faint end (Section REF ) that, as we shall see, is important in the context of understanding the cosmic reionization, its cosmological evolution (Section REF ), and the transition from the dust-free to the dust-enshrouded star formation phases (Section REF ).",
"The comoving differential UV LF $\\Phi (\\log L^{\\rm obs}_{1350}, z)$ , i.e., the number density of galaxies per unit $\\log L^{\\rm obs}_{1350}$ interval at redshift $z$ , can be computed coupling the halo formation rate with the relationship between halo mass and SFR as a function of cosmic time, $\\tau $ , and with the relationship between SFR and UV luminosity (Equation (REF )) to obtain $\\Phi (\\log L^{\\rm obs}_{1350}, z)\\!\\!",
"=\\!\\!\\!",
"\\int ^{M^{\\rm max}_{\\rm vir}}_{M^{\\rm min}_{\\rm vir}}\\!\\!\\!\\!\\!",
"dM_{\\rm vir} \\int ^{z^{\\rm max}_{\\rm vir}}_z\\!\\!\\!\\!\\!\\!",
"dz_{\\rm vir} \\Big |\\frac{d \\tau _{\\rm vir}}{dz_{\\rm vir}}\\Big | {d^3 N\\over d M_{\\rm vir}\\, d \\tau _{\\rm vir}\\, dV } \\, \\theta _{\\rm H}\\bigl [t(z) - t(z_{\\rm vir}) < t_{\\rm destr}(z_{\\rm vir})\\bigr ],$ where $L^{\\rm obs}_{1350}$ is the observed luminosity, attenuated by dust, and $\\theta _{\\rm H}(x)$ is the Heaviside function.",
"We set $z^{\\rm max}_{\\rm vir}=16$ , $M^{\\rm max}_{\\rm vir}= 10^{13.3}\\,M_\\odot $ , and $M^{\\rm min}_{\\rm vir} = M_{\\rm crit}$ .",
"Figure REF shows that the model is nicely consistent with observational estimates of the UV LF over the full redshift range from $z \\sim 3$ to $z \\sim 10$ for $k_{\\rm UV}=1.0\\times 10^{28}\\,\\hbox{erg}\\,\\hbox{s}^{-1}\\,\\hbox{Hz}^{-1}\\,M_\\odot ^{-1}\\,\\hbox{yr}$ (Equation (REF ), see also Figure REF ).",
"Observational determinations of UV LFs were made at various rest-frame wavelengths.",
"However, since the UV continuum slope, $\\beta $ ($f_\\lambda \\propto \\lambda ^\\beta $ ), of high-$z$ galaxies is close to $\\beta =-2$ [15], [21], the color correction $M_{\\rm AB, \\lambda 1}-M_{\\rm AB,\\lambda 2}=-2.5(\\beta +2)\\log (\\lambda _1/\\lambda _2)$ is small and has been neglected."
],
[
"The role of dust extinction",
"Dust extinction is differential in luminosity as is clearly seen comparing the solid black lines (no extinction) in Figure REF with the dot-dashed blue lines (extinction included).",
"This follows from the faster metal enrichment in the more massive objects (see Figure REF ), where the SN and radiative feedbacks are less effective in depressing the SFR.",
"Figure REF also shows that the least massive galaxies ($M_{\\rm vir} < 10^{11}\\,M_\\odot $ ) have low extinction throughout their lifetime.",
"In conjunction with the fast decline of the number density of more massive halos, this results in an increasing dominance of the contribution of low mass galaxies to the UV LF at higher and higher $z$ .",
"Figure REF indeed shows that the observed portion of the UV LF at $z\\geqslant 7$ is accounted for by galaxies with $M_{\\rm vir} < 10^{11.2}\\,M_\\odot $ , and the effect of dust extinction is negligible over most of the observed luminosity range.",
"This offers the interesting possibility of reconstructing the halo formation rate from the UV LF without the complication of uncertain extinction corrections."
],
[
"The faint end of the UV luminosity function",
"The LF has, at the faint end, a break corresponding to $M_{\\rm crit}$ , the mass below which the star formation is suppressed by external and internal heating processes (see Section ).",
"In Figure REF $M_{\\rm crit}=10^{8.5}\\,M_\\odot $ but in the panels comparing the model with the data at various redshifts the thin dotted and solid blue lines show the effect of increasing $M_{\\rm crit}$ to $10^{9.8}\\,M_\\odot $ and $10^{11.2}\\,M_\\odot $ , respectively.",
"This shows that the data only constrain $M_{\\rm crit}$ to be $\\lesssim 10^{9.8}\\,M_\\odot $ , consistent with estimates from simulations, summarized in Section REF .",
"Interestingly, [99], using a substantially different model, find that the observed UV LFs at $z=6$ , 7, and 8 are best fit with a minimum halo mass per galaxy $\\log (M_{\\rm min}/M_\\odot )= 9.4 (+0.3,-0.9)$ in good agreement with our estimate.",
"These authors simply assume negligible dust extinction at these redshifts; our model provides a quantitative physical account of the validity of this assumption.",
"[125], using the Durham GALFORM semi-analytical galaxy formation model, also find that the minimum halo mass that contributes substantially to the production of UV photons at these redshifts is $\\log (M_{\\rm min}/{h}^{-1}\\,M_\\odot ) \\simeq 9$ .",
"A lower value, $\\log (M_{\\rm min}/\\,M_\\odot ) \\simeq 8.2$ , was obtained by [63] from their cosmological hydrodynamical simulations.",
"[157] derive from abundance matching that the galaxies currently detected by HST live in dark matter halos with $M_{\\rm H} \\gtrsim 10^{10}\\,M_\\odot $ , and they predict a weak decrease of the star formation efficiency with decreasing halo mass down to a minimum halo mass for star formation $M_{\\rm H} \\sim 10^8\\,M_\\odot $ , under the assumption that the luminosity function remains a steep power law at the faint end.",
"The higher feedback efficiency for less massive halos (Equation (REF )) makes the slope of the faint end of the UV LF (above the break) flatter than that of the low mass end of the halo formation rate function (illustrated by Figure REF ).",
"However the former slope reflects to some extent the steepening with increasing redshift of the latter.",
"Just above the low-luminosity break corresponding to $M_{\\rm crit}$ , the slope $-d\\log \\Phi (L^{\\rm obs}_{1350},z)/d\\log L^{\\rm obs}_{1350}$ (where $\\Phi $ is per unit $d\\log L^{\\rm obs}_{1350}$ ) steadily increases from $\\simeq 0.5$ at $z=2$ , to $\\simeq 0.7$ at $z=6$ , and to $\\simeq 1$ at $z=10$ (see the bottom-right panel of Figure REF ).",
"The steepening of the faint end of the UV LF with increasing $z$ implies an increasing contribution of low luminosity galaxies to the UV background.",
"The dust extinction, differential with halo mass, further steepens the bright end of the observed UV LF."
],
[
"Evolution of the UV luminosity function",
"As shown by the bottom-right panel of Figure REF , the model implies that the evolution of the UV LF from $z=2$ to $z=6$ is weak.",
"In particular the fast decrease with increasing redshift of the high mass halo formation rate (Figure REF ) is not mirrored by the bright end of the LF.",
"This is partly due, again, to the fast metal enrichment of massive galaxies that translates into a rapid increase of dust extinction and, consequently, in a short duration of their UV bright phase (Figure REF ).",
"Thus the contribution to the UV LF of galaxies in halos more massive than $\\sim 10^{11.2}\\, M_\\odot $ (thin solid blue line in Figure REF ) decreases rapidly with increasing redshift and galaxies with $M_{\\rm vir} \\gg 10^{11.2}\\, M_\\odot $ contribute very little.",
"The UV LF at high $z$ is therefore weakly sensitive to the rapid variation of the formation rate of the latter galaxies.",
"The milder decrease of the formation rates of less massive galaxies is largely compensated by the increase of the star formation efficiency due to the faster gas cooling (see Figure REF ).",
"At $z>6$ , however, the decrease of the formation rate even of intermediate mass galaxies prevails and determines a clear negative evolution of the UV LF."
],
[
"From dust-free to dust-enshrouded star formation",
"As illustrated by the bottom-left panel of Figure REF , the star formation in massive galaxies is dust enshrouded over most of its duration.",
"For example, the figure shows that the extinction at 1350 Å of a galaxy with $M_{\\rm vir}=10^{12}\\,M_\\odot $ steeply grows already at galactic ages of a few times $10^7\\,$ yr, reaching two magnitudes at an age of $4\\times 10^{7}\\,$ yr.",
"The total duration of the star formation phase for these galaxies is $\\simeq 7\\times 10^8[(1+z)/3.5]^{-1.5}\\,$ yr [17], which implies that they show up primarily at far-IR/(sub-)millimeter wavelengths.",
"Thus the present framework entails a continuity between the high-$z$ galaxy SFR function derived from the UV LF (dominated by low-mass galaxies) and that derived from the infrared (IR; 8–$1000\\,\\mu $ m) LF (dominated by massive galaxies), investigated by [78] and [17].",
"This continuity is borne out by observational data at $z = 2$ and 3, as shown by Figure REF .",
"Once proper allowance for the effect of dust attenuation is made, the model accurately matches the IR and the UV LFs and, as expected, UV and IR data cover complementary SFR ranges.",
"The SFR histories inferred from UV and far-IR data are compared in Figure REF [41].",
"This figure shows that the ratio of dust-obscured to unobscured SFR increases with increasing redshift until it reaches a broad maximum at $z\\sim 2$ –3 and decreases afterwards.",
"This entails a prediction for the evolution of the IR luminosity density, $\\rho _{\\rm IR}$ , beyond $z\\simeq 3.5$ , where it cannot yet be determined observationally.",
"According to the present model, at these redshifts, $\\rho _{\\rm IR}$ decreases with increasing $z$ faster than the UV luminosity density, $\\rho _{\\rm UV}$ .",
"In other words, the extinction correction needed to determine the SFR density from the observed $\\rho _{\\rm UV}$ is increasingly small at higher and higher redshifts.",
"This is because the massive halos, that can reach high values of dust extinction, become increasingly rare at high $z$ .",
"We now move from the non-ionizing to the ionizing UV.",
"Two issues need to be addressed: the production rate of ionizing photons as a function of halo mass and galactic age and the fraction of them that manages to escape into the IGM, contributing to its ionization rate.",
"Key information on both issues is provided by the Ly$\\alpha $ line LF, observationally determined up to $z\\simeq 8$ with some constraints also at $z\\simeq 9$ .",
"As discussed in Section REF , the observed Ly$\\alpha $ luminosity of a galaxy is directly proportional to the production rate of ionizing photons, to their absorption fraction by HI in the ISM, and to the fraction of them that escapes absorption by dust.",
"It is thus a sensitive probe of the rate at which such photons can escape into the IGM, ionizing it.",
"Equipped with the information obtained from the redshift-dependent Ly$\\alpha $ line LFs and taking into account recent measurements of the ratio of non-ionizing UV to Lyman-continuum emission, in Section REF we evaluate the injection rate of ionizing photons into the IGM and discuss the cosmic reionization.",
"This is done for several choices, taken from the literature, of the IGM clumping factor, an ingredient not provided by the model.",
"The results are discussed vis-a-vis with a variety of observational constraints, including those from the electron scattering optical depth measured by CMB experiments."
],
[
"Ly$\\alpha $ emitters",
"The ionizing photons ($\\lambda \\leqslant 912\\,{\\rm \\mathring{A}}$ ) can be absorbed by both dust and HI.",
"About $2/3$ of those absorbed by HI are converted into Ly$\\alpha $ photons [110], [134], so that the LFs of LAEs provide information on their production rate.",
"The Ly$\\alpha $ line luminosity before extinction is then $L^{\\mathrm {int}}_{{\\mathrm {Ly}}\\alpha }= {2\\over 3}\\,\\dot{N}^{\\mathrm {int}}_{912 }\\,f^{\\rm dust}_{912}\\,(1-f^{\\rm HI}_{912})h_{\\rm P}\\nu _{{\\mathrm {Ly}}\\alpha }$ where $\\dot{N}^{\\mathrm {int}}_{912 }$ is the production rate of ionizing photons while $f^{\\rm dust}_{912}=\\exp (-\\tau _{\\rm dust,ion})$ and $f^{\\rm HI}_{912}=\\exp (-\\tau _{\\rm HI})$ are the fractions of them that escape absorption by dust and by HI, respectively, and $h_{\\rm P}$ is the Planck constant.",
"We model $\\tau _{\\rm HI}$ as $\\tau _{\\rm HI} = \\tau ^0_{\\rm HI} \\left( \\frac{\\dot{M}_\\star }{M_\\odot \\,\\rm yr^{-1}} \\right)^{\\alpha _{\\rm HI}} + \\beta _{\\rm HI},$ where the first term on the right-hand side refers to the contribution from high density star-forming regions while the second term refers to the contribution of diffuse HI.",
"Since the [19] law does not provide the dust absorption optical depth of ionizing photons, we set $\\tau _{\\rm dust,ion}=A_{912}/1.08=\\gamma \\,A_{1350}/1.08$ where $A_{1350}$ is given by Equation (REF ) and $\\gamma $ is a parameter of the model.",
"The evolution with galactic age of $\\dot{N}^{\\mathrm {int}}_{912 }$ , for a [22] IMF, a constant SFR of $1\\,M_\\odot \\,\\hbox{yr}^{-1}$ , and three values of the gas metallicity is illustrated by Figure REF .",
"We adopt an effective ratio $k_{\\rm ion}\\equiv \\dot{N}^{\\rm int}_{912}/\\dot{M}_\\star = 4.0 \\times 10^{53}\\, \\hbox{photons}\\, \\hbox{s}^{-1}\\, M^{-1}_\\odot \\ {\\rm yr}$ , appropriate for the typical galactic ages and metallicities of our sources.",
"The figure also shows the evolution of the intrinsic ratio of Lyman-continuum ($\\lambda \\leqslant 912\\,{\\rm \\mathring{A}}$ ; $L^{\\rm int}_{912} = \\dot{N}^{\\rm int}_{912} h_{\\rm P} \\rm \\ erg\\ s^{-1}\\ Hz^{-1}$ ) to UV luminosity at 1350 Å, $R_{\\rm int}\\equiv L^{\\rm int}_{912}/L^{\\rm int}_{1350}$ .",
"For our choice of $k_{\\rm ion}$ and $k_{\\rm UV}$ (Equation (REF )) we have $R_{\\rm int} \\simeq 0.265$ .",
"The interstellar dust attenuates $L^{\\mathrm {int}}_{{\\mathrm {Ly}}\\alpha }$ by a factor $f^{\\rm dust}_{{\\rm Ly}\\alpha }= \\exp (-\\tau ^{\\mathrm {dust}}_{{\\mathrm {Ly}}\\alpha })$ , where $\\tau ^{\\mathrm {dust}}_{{\\mathrm {Ly}}\\alpha }$ is the dust optical depth at the Ly$\\alpha $ wavelength (1216 Å).",
"The physical processes governing the escape of Ly$\\alpha $ photons from galaxies are complex.",
"Dust content, neutral gas kinematics, and the geometry of the neutral gas seem to play the most important roles.",
"For objects with low dust extinction, such as those relevant here, the detailed calculations of the Ly$\\alpha $ radiation transfer by [35] show that the Ly$\\alpha $ is more attenuated than the nearby UV continuum by a factor $\\simeq 2$ , consistent with the observational result [49] that the SFRs derived from the Ly$\\alpha $ luminosity are $\\simeq 3$ times lower than those inferred from the rest-frame UV continuum.",
"With reference to the latter result, it must be noted that part of the attenuation of the Ly$\\alpha $ luminosity must be ascribed to the IGM (see below) and that the discrepancy between Ly$\\alpha $ and UV continuum SFRs may be due, at least in part, to uncertainties in their estimators.",
"A good fit of the Ly$\\alpha $ line LFs is obtained setting $f^{\\rm dust}_{{\\rm Ly}\\alpha }\\simeq f^{\\rm dust}_{1216}/1.6$ , where $f^{\\rm dust}_{1216} = \\exp (-A_{1216}/1.08)$ , consistent with the above results.",
"The fraction $f^{\\mathrm {IGM}}_{{\\mathrm {Ly}}\\alpha }=\\exp (-\\tau ^{\\mathrm {IGM}}_{{\\mathrm {Ly}}\\alpha })$ of Ly$\\alpha $ photons that survive the passage through the IGM was computed following [85], taking into account only the absorption of the blue wing of the line $f^{\\rm IGM}_{\\rm Ly\\alpha } = 0.5 \\lbrace 1 + \\exp [-0.0036(1+z)^{3.46}]\\rbrace .$ The strong attenuation by dust within the galaxy and by HI in the IGM implies that estimates of the SFR of high-$z$ LBGs and LAEs from the observed Ly$\\alpha $ luminosity require careful corrections and are correspondingly endowed with large uncertainties.",
"Vice versa, the statistics of LAEs and LBGs at high redshifts are sensitive absorption/extinction probes.",
"Then the observed Ly$\\alpha $ line luminosity writes $L_{{\\mathrm {Ly}}\\alpha }^{\\mathrm {obs}}\\simeq 4.36 \\times 10^{42}\\,\\left(\\frac{\\dot{M}_{\\star }}{M_{\\odot }\\,\\mathrm {yr}^{-1}}\\right)\\,f^{\\rm dust}_{912}\\,(1-f^{\\mathrm {HI}}_{912}) \\,f^{\\mathrm {dust}}_{{\\mathrm {Ly}}\\alpha }\\,f^{\\mathrm {IGM}}_{{\\mathrm {Ly}}\\alpha }~\\mathrm {erg\\,s}^{-1}.$ On the whole, this equation contains four parameters: the three parameters in the definition of $\\tau _{\\rm HI}$ (Equation (REF )), i.e., $\\tau ^0_{\\rm HI}$ , $\\alpha _{\\rm HI}$ , and $\\beta _{\\rm HI}$ , plus $\\gamma $ (Equation (REF )).",
"We have attempted to determine their values fitting the Ly$\\alpha $ line LFs by [6] at $z \\sim 3$ and by [113] at $z \\sim 3.8$ .",
"However, we could not find an unambiguous solution because of the strong degeneracy among the parameters.",
"To break the parameters' degeneracy, we fixed $\\alpha _{\\rm HI}= 0.25$ , in analogy to Equation (REF ), and $\\tau ^0_{\\rm HI} = 0.3$ .",
"Furthermore, we discarded the solutions that imply too high emission rates of ionizing photons and too low electron scattering optical depth (see below).",
"Based on these, somewhat loose, criteria, we have chosen $\\beta _{\\rm HI}\\simeq 1.5$ and $\\gamma \\simeq 0.85$ .",
"A check on the validity of our choices is provided by recent Lyman-continuum imaging of galaxies at $z\\simeq 3$ .",
"[103] measured the average ratios of non-ionizing UV to Lyman-continuum flux density corrected for IGM attenuation, $\\eta _{\\rm obs}$ , for a sample of LBGs and a sample of LAEs, both at $z\\sim 3$ .",
"Such ratios were found to be $\\eta _{\\rm LBG}=18.0^{+34.8}_{-7.4}$ for LBGs (rest-frame UV absolute magnitudes $-22\\leqslant M_{\\rm UV} \\leqslant -20$ ) and $\\eta _{\\rm LAE}=3.7^{+2.5}_{-1.1}$ for LAEs ($-20< M_{\\rm UV} \\leqslant -18.3$ ).",
"[97] probed the Lyman-continuum spectral region of 49 LBGs and 70 LAEs spectroscopically confirmed at $z \\sim 2.85$ , as well as of 58 LAE photometric candidates in the same redshift range.",
"After correcting for foreground galaxy contamination and HI absorption in the IGM, the average values for their samples are $\\eta _{\\rm LBG}=82\\pm 45$ , $\\eta _{\\rm LAE}=7.6\\pm 4.1$ for the spectroscopic sample and $\\eta _{\\rm LAE}=2.6\\pm 0.8$ for the full LAE sample.",
"The observed ratios are equal to the intrinsic ones $\\eta _{\\rm int} \\simeq R_{\\rm int}^{-1}$ attenuated by dust and HI absorption (the latter only for ionizing photons) $\\eta _{\\rm obs}=\\eta _{\\rm int}{f^{\\rm dust}_{1350}\\over f^{\\rm dust}_{912}\\cdot f^{\\mathrm {HI}}_{912}}.$ The intrinsic ratio we have adopted, $\\eta _{\\rm int} \\simeq R_{\\rm int}^{-1}\\simeq 3.77$ , is within the range measured for LAEs in both the [103] and the [97] samples, implying that the attenuation both by dust and by HI is small, as expected in the present framework (cf.",
"Figure REF ).",
"LBGs in both samples have SFRs in the range 5–$250\\,M_\\odot \\,{\\rm yr^{-1}}$ , with median values around $50\\,M_\\odot \\,{\\rm yr^{-1}}$ , and gas metallicities $Z_{\\rm g}\\simeq 0.7 \\pm 0.3\\,Z_\\odot $ .",
"After Equation (REF ) the optical depth for absorption of ionizing photons by HI is $\\tau _{\\rm HI,LBG} = \\ln (\\eta _{\\rm obs,LBG}/\\eta _{\\rm int}) - \\ln (f^{\\rm dust}_{1350}/f^{\\rm dust}_{912}) = \\ln (\\eta _{\\rm obs,LBG}/\\eta _{\\rm int}) - (\\gamma - 1) A_{1350} / 1.08$ , giving $\\tau _{\\rm HI,LBG} \\simeq 1.8^{+1.3}_{-0.6} $ for $\\eta _{\\rm obs,LBG} = 18.0^{+34.8}_{-7.4}$ [103] and $\\tau _{\\rm HI,LBG} \\simeq 3.3^{+0.6}_{-0.9} $ for $\\eta _{\\rm obs,LBG} = 82 \\pm 45$ [97].",
"With our choice for the parameters, Equation (REF ) gives $\\tau _{\\rm HI} = \\tau ^0_{\\rm HI} \\dot{M}_\\star ^{\\alpha _{\\rm HI}} + \\beta _{\\rm HI} \\simeq 2.3^{+0.4}_{-0.4}$ , consistent with both observational estimates.",
"As illustrated by Figure REF , the Ly$\\alpha $ line LFs yielded by the model compare quite well with observational determinations at several redshifts, from $z=3$ to $z=7.7$ .",
"The bottom-right panel shows that the intrinsic evolution of the Ly$\\alpha $ line LF is remarkably weak from $z=2$ to $z=6$ , even weaker than in the UV case (Figure REF ), and similarly to the case of the UV LF, its faint portion is predicted to steepen with increasing redshift.",
"The observed evolution at high-$z$ is more strongly negative than the intrinsic one due to the increasing attenuation by the IGM.",
"The figure also demonstrates that, at $z\\geqslant 5.7$ , observational estimates based on photometric samples (open symbols), only partially confirmed in spectroscopy, tend to be systematically higher than those based on purely spectroscopic samples (filled symbols).",
"Therefore, more extensive spectroscopic confirmation is necessary before firm conclusions on the high-$z$ evolution of the Ly$\\alpha $ line LF can be drawn.",
"The average properties, weighted by the halo formation rate, of LAEs at $z=2$ , 6, and 12 are shown, as a function of the observed Ly$\\alpha $ luminosity, in the right-hand panels of Figure REF .",
"Compared to LBGs (left panels of the same figure), LAEs have somewhat younger ages, implying somewhat lower stellar masses, SFRs, and metallicities at given $M_{\\rm vir}$ .",
"The latter two factors combine to give substantially lower dust extinction."
],
[
"Escape fraction of ionizing photons and reionization",
"The injection rate of ionizing photons into the IGM is $\\dot{N}^{\\rm esc}_{912}=\\dot{N}^{\\rm int}_{912} f^{\\rm esc}_{912} = k_{\\rm ion} {\\dot{M}_\\star } f^{\\rm esc}_{912} \\ \\hbox{photons}\\,\\hbox{s}^{-1},$ where $f^{\\rm esc}_{912} \\equiv f^{\\rm dust}_{912} \\times f^{\\rm HI}_{912}$ is the fraction of ionizing photons emerging at the galaxy boundary.",
"The dependencies on UV magnitude and redshift of $f^{\\rm dust}_{912}$ , $f^{\\rm HI}_{912}$ , and $f^{\\rm esc}_{912}$ , weighted by the halo formation rate, are illustrated in Figure REF .",
"As shown by Equations (REF ) and (REF ), the optical depths for absorption by both dust and HI (and thus the corresponding escape fractions) are determined by the intrinsic properties of the galaxies (the SFR and, in the case of dust absorption, the metallicity), mostly controlled by the halo mass.",
"They are thus weakly dependent on redshift.",
"Lower mass galaxies have lower SFRs and, more importantly, lower metallicities.",
"This translates into substantially lower optical depths ($<1$ ), i.e., substantially higher escape fractions.",
"For brighter galaxies, which have optical depths $\\geqslant 1$ , the escape fractions are exponentially sensitive to the weak redshift dependence of the metallicity at given halo mass (see Figure REF ).",
"Thus the small decrease with $z$ of the metallicity translates, for bright galaxies, into a significant increase of $f^{\\rm dust}_{912}$ .",
"These luminosity and redshift dependencies do not apply to $f^{\\rm HI}_{912}$ which, not being affected by metallicity, is almost redshift independent at all luminosities.",
"In the bottom-right panel of Figure REF the escape fractions of ionizing photons given by the model for two ranges of observed UV luminosity are compared with observational estimates at several redshifts.",
"The agreement is generally good.",
"The redshift-dependent emission rate function, $\\phi (\\log \\dot{N}^{\\rm esc}_{912}, z)$ , can be constructed in the same way as the UV LF (Section ).",
"The average injection rate of ionizing photons into the IGM per unit comoving volume at redshift $z$ is $\\langle \\dot{N}^{\\rm esc}_{912} \\rangle (z) = \\int \\dot{N}^{\\rm esc}_{912} \\phi (\\log \\dot{N}^{\\rm esc}_{912}, z) d\\log \\dot{N}^{\\rm esc}_{912}.$ Figure REF compares the average injection rate of ionizing photons into the IGM, $\\langle \\dot{N}^{\\rm esc}_{912}\\rangle (z)$ , as a function of redshift yielded by the model for two choices of the critical halo mass ($M_{\\rm crit}=10^{8.5}\\,M_\\odot $ and $M_{\\rm crit}=10^{10}\\,M_\\odot $ ) with observational estimates.",
"The original data refer to different quantities such as the proper hydrogen photoionization rate, $\\Gamma _{\\rm HI}(z)$ , the average specific intensity of UV background, $J_\\nu (z)$ , and the comoving spatially averaged emissivity, $\\epsilon _\\nu (z)$ .",
"The conversion of these quantities into $\\langle \\dot{N}^{\\rm esc}_{912}\\rangle (z)$ was done using the formalism laid down by [74].",
"The values of $\\langle \\dot{N}^{\\rm esc}_{912}\\rangle $ obtained from the model are above the estimates by [74] but consistent with (although on the high side of) the more recent results by [5] and [103].",
"The reionization of the IGM is described in terms of the evolution of the volume filling factor of HII regions, $Q_{\\rm HII}(t)$ , which is ruled by [86] $\\frac{dQ_{\\rm HII}}{dt} \\simeq \\frac{\\langle \\dot{N}^{\\rm esc}_{912}\\rangle }{\\bar{n}_{\\rm H}} - \\frac{Q_{\\rm HII}}{\\bar{t}_{\\rm rec}},$ where $\\bar{n}_{\\rm H} = X \\rho _{\\rm c} \\Omega _{\\rm b} / m_{\\rm p} \\simeq 2.5 \\times 10^{-7} X (\\Omega _{\\rm b} h^2/0.022)$ $\\hbox{cm}^{-3}$ is the mean comoving hydrogen number density in terms of the primordial mass fraction of hydrogen $X = 0.75$ , of the present-day critical density $\\rho _{\\rm c} = 1.878 h^2 \\times 10^{-29}\\ {\\rm g\\ cm^{-3}}$ , and of the proton mass $m_{\\rm p} = 1.67 \\times 10^{-24}\\ {\\rm g}$ .",
"The mean recombination time is given by [86], [74] $\\bar{t}_{\\rm rec} & = & \\frac{1}{f_e \\bar{n}_{\\rm H} (1+z)^3 \\alpha _{\\rm B}(T_0) C_{\\rm HII}} = \\nonumber \\\\&=& {0.51\\over f_e} \\Big ( \\frac{\\Omega _{\\rm b}}{0.049} \\Big )^{-1} \\Big ( \\frac{1+z}{7} \\Big )^{-3} \\Big ( \\frac{T_0}{2 \\times 10^4\\ {\\rm K}} \\Big )^{0.7} \\Big ( \\frac{C_{\\rm HII}}{6} \\Big )^{-1}\\ {\\rm Gyr},$ where $f_e$ is the number of free electrons per hydrogen nucleus in the ionized IGM, assumed to have a temperature $T_0 = 2 \\times 10^4\\ \\rm K$ , and $C_{\\rm HII} \\equiv \\langle \\rho ^2_{\\rm HII} \\rangle / \\langle \\rho _{\\rm HII} \\rangle ^2$ is the clumping factor of the ionized hydrogen.",
"$f_e$ depends on the ionization state of helium; we assumed it to be doubly ionized ($f_e = 1 + Y/2X \\simeq 1.167$ ) at $z < 4$ and singly ionized ($f_e = 1 + Y/4X \\simeq 1.083$ ) at higher redshifts [128].",
"The clumping factor has been extensively investigated using numerical simulations [45].",
"A series of drawbacks have been progressively discovered and corrected.",
"The latest studies, accounting for the photo-ionization heating, that tends to smooth the diffuse IGM, and for the IGM temperature, which could suppress the recombination rate further, generally find relatively low values of $C_{\\rm HII}$ [120], [95], [145], [45], [74].",
"Alternatively, the clumping factor can be computed as the second moment of the IGM density distribution, integrating up to a maximum overdensity [75].",
"We adopt, as our reference, the model $C_{\\rm HII,T_b, x_{\\rm HII}>0.95}$ by [45], but we will discuss the effect of different choices.",
"Additional constraints on the reionization history are set by the electron scattering optical depth, $\\tau _{\\rm es}$ , measured by Cosmic Microwave Background (CMB) anisotropy experiments.",
"The optical depth of electron scattering up to redshift $z$ is given by $\\tau _{\\rm es}(\\leqslant z) = \\int ^z_0 dz^{\\prime } \\Big | \\frac{dt}{dz^{\\prime }} \\Big | c \\sigma _{\\rm T} n_e(z^{\\prime }) = \\int ^z_0 dz^{\\prime } \\frac{c (1+z^{\\prime })^2}{H_0 \\sqrt{\\Omega _{\\Lambda } + \\Omega _{\\rm m} (1+z^{\\prime })^3}} f_e Q_{\\rm HII}(z^{\\prime }) \\sigma _{\\rm T} \\bar{n}_{\\rm H},$ where $n_e \\simeq f_e Q_{\\rm HII} \\bar{n}_{\\rm H} (1+z)^3$ is the mean electron density, $c$ is the speed of light, and $\\sigma _{\\rm T} \\simeq 6.65 \\times 10^{-25}\\ {\\rm cm^2}$ is the Thomson cross section.",
"WMAP 9-yr data alone give $\\tau _{\\rm es}=0.089\\pm 0.014$ , slightly decreasing to $\\tau _{\\rm es}=0.081\\pm 0.012$ when they are combined with external data sets [57].",
"The combination of the Planck temperature power spectrum with a WMAP polarisation low-multipole likelihood results in an estimate closely matching the WMAP 9-yr value, $\\tau _{\\rm es}=0.089^{+0.012}_{-0.014}$ [123].",
"However, replacing the WMAP polarised dust template with the far more sensitive Planck/HFI 353 GHz polarisation map lowers the best fit value to $\\tau _{\\rm es}=0.075\\pm 0.013$ [122]; this result has however to be taken as preliminary.",
"The evolution of $Q_{\\rm HII}(t)$ and of the electron scattering optical depth $\\tau _{\\rm es}(\\leqslant z)$ yielded by the model adopting the critical halo mass $M_{\\rm crit} = 10^{8.5}\\ M_\\odot $ , the effective escape fraction $f^{\\rm esc}_{912}$ laid down before, and the evolutionary law for the IGM clumping factor $C_{\\rm HII}$ by [45], are shown by the thick solid black lines in the main panel of Figure REF .",
"Although this model gives a $\\tau _{\\rm es}$ consistent with the determination by [123] and less than $2\\sigma $ below those by [57] and [123], it yields a more extended fully ionized phase than indicated by the constraints on $Q_{\\rm HII}$ at $z\\geqslant 6$ collected by [128] who, however, cautioned that they are all subject to substantial systematic or modeling uncertainties.",
"Indications pointing to a rapid drop of the ionization degree above $z\\simeq 6$ include hints of a decrease of the comoving emission rates of ionizing photons (see Figure REF ), of sizes of quasar near zones, and of the Ly$\\alpha $ transmission through the IGM [128].",
"As we have seen before, the observed UV LFs imply $M_{\\rm crit}\\lesssim 10^{10}\\,M_\\odot $ .",
"Adopting the latter value and keeping our baseline $f^{\\rm esc}_{912}(z)$ and $C_{\\rm HII}(z)$ shortens the fully ionized phase that now extends only up to $z \\sim 6$ (dotted red line in the main figure of Figure REF ), lessening the discrepancy with constraints on $Q_{\\rm HII}$ at the cost of $\\tau _{\\rm es}$ dropping almost $\\simeq 3\\,\\sigma $ below the best fit WMAP value and $2\\,\\sigma $ below the best fit value of [122].",
"This conclusion is unaffected by different choices for $C_{\\rm HII}(z)$ , illustrated in the upper left panel of Figure REF , as long as we keep our baseline $f^{\\rm esc}_{912}(z)$ .",
"This is because the production rate of ionizing photons (first term on the right-hand side of Equation (REF )) is always substantially larger than the recombination rate.",
"The minimum SFR density required to keep the universe fully ionized at the redshift $z$ is [86] $\\dot{\\rho }_\\star \\simeq 8.2 \\times 10^{-4} \\Big ( \\frac{C_{\\rm HII}}{f_{\\rm esc}} \\Big ) \\Big ( \\frac{\\Omega _{\\rm b} h^2}{0.022} \\Big )^2 \\Big ( \\frac{1+z}{7} \\Big )^3\\ M_\\odot \\ {\\rm yr}^{-1}\\ {\\rm Mpc}^{-3}.$ It is shown in Figure REF (grey area) for $3 \\lesssim C_{\\rm HII}/f_{\\rm esc} \\lesssim 30$ .",
"A similar figure was presented by [42].",
"A comparison of their green curve with our blue curve illustrates the difference between the UV luminosity density implied by our model for sources brighter than $M_{\\rm UV}=-18$ and that from the hydrodynamic simulations of [43].",
"A sort of tradeoff between the constraints on $Q_{\\rm HII}$ and those on $\\tau _{\\rm es}$ is obtained if the cooling rate of HII increases rapidly with decreasing $z$ for $z\\lesssim 8$ .",
"This might be the case if, for example, the clumping factor climbs in this redshift range, as in the $C_{-1}$ L6N256 no-reheating simulation by [120].",
"The drawbacks suffered by these no-reheating simulations were, however, pointed out by [120] and [45].",
"The tension between the determinations of $\\tau _{\\rm es}(\\leqslant z)$ and constraints on $Q_{\\rm HII}(z)$ has been repeatedly discussed in recent years [74], [51], [128].",
"Our conclusions are broadly consistent with the earlier ones, as can be seen by all models matching the observed high-$z$ UV LFs.",
"However, our model differs from the others because our LFs and the escape fraction as a function of redshift are physically grounded, while the quoted models are based on phenomenological fits to the data.",
"This means that our model is more constrained; for example, the extrapolations of the LFs outside the luminosity and redshift ranges covered by observations come out from our equations rather than being controlled by adjustable parameters.",
"In other words, we explore different parameter spaces.",
"As a result, we differ in important details such as the slope of the faint tail of the UV LF and its redshift dependence, the total number density of high-$z$ UV galaxies, and the redshift dependence of the escape fraction of ionizing photons.",
"The need for a redshift- or mass dependence of $f^{\\rm esc}_{912}$ has also been deduced by other studies [3], [96].",
"[96] combined data-constrained reionization histories and the evolution of the LF of early galaxies to find an empirical indication of a 2.6 times increase of the average escape fraction from $z = 6$ to $z = 8$ .",
"[3] argued that there are both theoretical and observations indications that $f^{\\rm esc}_{912}$ is higher at lower halo masses and proposed that a faint population of galaxies with host halo masses of $\\sim 10^{8-9}\\,M_\\odot $ dominated the ionizing photon budget at redshifts $z\\gtrsim 9$ due to their much higher escape fractions, again empirically estimated.",
"Our model provides a physical basis for the increase of the effective $f^{\\rm esc}_{912}$ with mass and redshift.",
"The present data do not allow us to draw any firm conclusion on the reionization history because they may be affected by substantial uncertainties.",
"Those uncertainties on data at $z=6$ –7 are discussed by [128].",
"Those on $\\tau _{\\rm es}$ are illustrated by the finding [122] that different corrections for the contamination by polarised foregrounds may lower the best fit value by about $1\\,\\sigma $ ."
],
[
"Conclusions",
"We have worked out a physical model for the evolution of the UV LF of high-$z$ galaxies and for the reionization history.",
"The LF is directly linked to the formation rate of virialized halos and to the cooling and heating processes governing the star formation.",
"For the low halo masses and young galactic ages of interest here it is not enough to take into account SN and AGN feedback, as usually done for halo masses $M_{\\rm vir}\\gtrsim 10^{11}\\,M_\\odot $ , because other heating processes, such as the radiation from massive low-metallicity stars, stellar winds, and the UV background, can contribute to reducing and eventually quenching the SFR.",
"We have modeled this by increasing the efficiency of cold gas removal and introducing a lower limit, $M_{\\rm crit}$ , to halo masses that can host active star formation.",
"Another still open issue is the production rate of UV photons per unit halo mass at high-$z$ , which is influenced by two competing effects.",
"On one side, the expected increase with redshift of the Jeans mass, hence of the characteristic stellar mass, entails a higher efficiency in the production of UV photons.",
"On the other side, more UV photons imply more gas heating, i.e., a decrease of the SFR.",
"We find that the observed UV LFs up to the highest redshifts are very well reproduced with the SFRs yielded by the model and the extinction law of Equation (REF ) for a production rate of UV photons corresponding to a [22] IMF.",
"The observed UV LFs (Figure REF ) constrain $M_{\\rm crit}$ to be $\\lesssim 10^{10}\\,M_\\odot $ , consistent with estimates from simulations.",
"Figure REF highlights several features of the model: i) dust extinction is higher for higher luminosities, associated to more massive halos which have a faster metal enrichment; ii) the higher feedback efficiency in less massive halos makes the slope of the faint end of the LF flatter than that of the halo formation rate; yet the former reflects to some extent the steepening with increasing $z$ of the latter; this has important implications for the sources of the ionizing background at high $z$ ; iii) the evolution of the LF from $z=2$ to $z=6$ is weak because the decrease with increasing redshift of the halo formation rate in the relevant range of halo masses is largely compensated by the increase of the star formation efficiency due to the faster gas cooling and by the increase of dust extinction with increasing halo mass.",
"Another key property of the model (Figure REF ) is the fast metal enrichment of the more massive galaxies that translates into a rapid increase of dust obscuration.",
"Therefore these galaxies show up mostly at far-IR/(sub-)millimeter wavelengths, a prediction successfully tested against observational data (Figures REF and REF ).",
"The model thus predicts a strong correlation between dust attenuation and halo/stellar mass for UV selected high-$z$ galaxies.",
"The ratio of dust-obscured to unobscured star formation has a broad maximum at $z\\simeq 2$ –3.",
"The decrease at lower redshifts is due to the decreasing amount of ISM in galaxies; at higher redshifts it is related to the fast decrease of the abundance of massive halos where the metal enrichment and, correspondingly, the dust extinction grow fast.",
"Similarly, good fits are obtained for the Ly$\\alpha $ line LFs (Figure REF ) that provide information on the production rate of ionizing photons and on their absorption by neutral interstellar hydrogen.",
"Further constraints on the attenuation by dust and HI are provided by recent measurements [103], [97] of the observed ratios of non-ionizing UV to Lyman-continuum flux densities for LAEs and LBGs.",
"These data have allowed us to derive a simple relationship between the optical depth for HI absorption and SFR.",
"Taking this relation into account, the model reproduces the very weak evolution of the Ly$\\alpha $ line LF between $z=2$ and $z=6$ , even weaker than in the UV.",
"The derived relationships linking the optical depths for absorption of ionizing photons by dust and HI to the SFR and, in the case of dust absorption, to the metallicity of the galaxies, imply higher effective escape fractions for galaxies with lower intrinsic UV luminosities or lower halo/stellar masses, and also a mild increase of the escape fraction with increasing redshift at fixed luminosity or halo/stellar mass.",
"Redshift- or mass-dependencies of the escape fraction were previously empirically deduced by, e.g., [3] and [96].",
"Our model provides a physical basis for these dependencies.",
"At this point we can compute the average injection rate of ionizing photons into the IGM as a function of halo mass and redshift.",
"To reconstruct the ionization history of the universe we further need the evolution of the clumping factor of the IGM, for which we have adopted, as our reference, the model $C_{\\rm HII,T_b, x_{\\rm HII}>0.95}$ by [45], but also considered alternative models, discussed in the literature.",
"With our recipe for the escape fraction of ionizing photons we find that galaxies already represented in the observed UV LFs, i.e., with $M_{\\rm UV}\\lesssim -18$ , hosted by halo masses $\\gtrsim 10^{10}\\,M_\\odot $ , can account for a complete ionization of the IGM up to $z\\simeq 6$ .",
"To get complete ionization up to $z\\simeq 7$ the population of star-forming galaxies at this redshift must extend in luminosity to $M_{\\rm UV}\\sim -13$ or fainter, in agreement with the conclusions of other analyses [128].",
"The surface densities of $M_{\\rm UV}\\sim -13$ galaxies would correspond to those of halo masses of $\\sim 10^{8.5}\\,M_\\odot $ , not far from the lower limit on $M_{\\rm crit}$ from hydrodynamical simulations.",
"A complete IGM ionization up to $z\\simeq 7$ is disfavoured by some (admittedly uncertain) data at $z\\simeq 6$ –7 collected by [128], that point to a fast decline of the ionization degree at $z\\gtrsim 6$ .",
"However, an even more extended ionized phase is implied by the determinations of electron scattering optical depths, $\\tau _{\\rm es}$ , from CMB experiments.",
"Our model adopting the critical halo mass $M_{\\rm crit} = 10^{8.5}\\ M_\\odot $ , yielding complete ionization up to $z\\simeq 7$ , gives a $\\tau _{\\rm es}$ consistent with determination by [123] and less than $2\\sigma $ below those by [57] and [123].",
"Raising $M_{\\rm crit}$ to $10^{10}\\ M_\\odot $ limits the fully ionized phase to $z \\lesssim 6$ and decreases $\\tau _{\\rm es}$ to a value almost $\\simeq 3\\,\\sigma $ below the estimates by [57] and [123] and $2\\,\\sigma $ below that by [122].",
"Since all these constraints on the reionization history are affected by substantial uncertainties, any firm conclusion is premature.",
"Better data are needed to resolve the issue.",
"We are grateful to the referee for a careful reading of the manuscript and many constructive comments that helped us substantially improving the paper.",
"We also acknowledge useful comments from G. Zamorani.",
"Z.Y.C.",
"acknowledges support from the joint PhD project between XMU and SISSA.",
"The work has been supported in part by ASI/INAF agreement n. I/072/09/0 and by PRIN 2009 “Millimeter and sub-millimeter spectroscopy for high resolution studies of primeval galaxies and clusters of galaxies”.",
"Figure: Evolution with redshift of the halo formation rate function.Figure: Left panels: Evolution with galactic age of the SFR, of the stellar mass, M star M_{\\rm star}, of the gas metallicity, Z g Z_{\\rm g}, and of the dust extinction, A 1350 A_{1350} (from top to bottom) for halo masses of M vir =10 9 M_{\\rm vir} = 10^{9} (dot-dashed lines), 10 10 10^{10} (solid lines), 10 11 10^{11} (dashed lines), and 10 12 M ⊙ 10^{12}\\ M_\\odot (dotted lines) virialized at z vir =6z_{\\rm vir} = 6.",
"The galactic age is measured from the virialization redshift, i.e., t=0t=0 for z=z vir z=z_{\\rm vir}.",
"Right panels: Evolution of the quantities on the left panels at fixed halo mass (M vir =10 10 M_{\\rm vir} = 10^{10} and 10 11 M ⊙ 10^{11}\\ M_\\odot , black and blue lines, respectively) for different redshifts: z vir =4z_{\\rm vir} = 4 (dot-dashed lines), 6 (solid lines), 8 (dashed lines), and 10 (dotted lines).Figure: Correlation of the intrinsic (extinction-corrected) rest-frame absolute UV magnitude, M 1350 int M_{1350}^{\\mathrm {int}}, with the color excess, E(B-V)E(B-V).",
"The open circles are the data by corrected for the different cosmology used here, the filled circles show the expectations of our model using the extinction law of Equation (), for halo masses in the range 10 10 M ⊙ M vir 4×10 12 M ⊙ 10^{10}\\, M_{\\odot }M_{\\rm vir} 4\\times 10^{12}\\, M_{\\odot } sampled in intervals ΔlogM vir =0.2\\Delta \\log M_{\\rm vir} = 0.2, and for ages in the range 4×10 6 t/ yr Δt burst (M vir ,z obs )4\\times 10^6t/{\\rm yr} \\Delta t_{\\mathrm {burst}}(M_{\\rm vir}, z_{\\rm obs}) (see Appendix of for an approximation of the duration of the star formation phase, Δt burst \\Delta t_{\\mathrm {burst}}, as a function of halo mass and redshift).",
"The relation E(B-V)≈A 1350 /11E(B-V)\\approx A_{1350}/11 by has been adopted.Figure: Average escape fractions of UV (f 1350 esc f^{\\rm esc}_{1350}, left panel) and Lyα\\alpha photons (f Ly α esc f^{\\rm esc}_{\\rm Ly\\alpha }, right panel), given by the model as a function of redshift, compared with observational estimates by , , , , , , and , for two bins of observed UV magnitudes: M 1350 obs ∈[-22,-20]M^{\\rm obs}_{1350} \\in [-22, -20] (blue dashed line and data points) and M 1350 obs ∈[-20,-18.3]M^{\\rm obs}_{1350} \\in [-20, -18.3] (red solid line and data point), roughly corresponding to LBGs and LAEs, respectively.Figure: Evolution with galactic age of the coefficient k UV k_{\\rm UV} (Equation ()) for a constant SFR, a IMF, and three gas metallicities: Z g =0.005Z_{\\rm g} = 0.005 (dashed line), 0.02 (dotted line), and 1 Z ⊙ Z_\\odot (solid line).",
"The far-UV/SFR calibration by for solar metallicity and a Kroupa IMF is also shown (filled circle at log(t/ yr )=8\\log (t/{\\rm yr})=8).",
"The horizontal blue line corresponds to the value adopted in the present paper.Figure: Average properties, weighted by the halo formation rate, of model galaxies at z obs =2z_{\\rm obs} = 2, 6, and 12 (dotted red lines and filled red squares, solid black line and black circles, dashed blue line and blue triangles, respectively) as a function of the observed UV luminosity (left panels) and of the observed Lyα\\alpha line luminosity without the IGM attenuation (right panels).",
"Larger symbols correspond to more numerous objects: the comoving number densities at the bin centers are 10 -5 10^{-5}, 10 -4 10^{-4}, 10 -3 10^{-3}, and 10 -2 dex -1 Mpc -3 10^{-2}\\,{\\rm \\,dex^{-1}\\,Mpc^{-3}}, respectively; in the bottom panels we can read out the corresponding luminosities.",
"Points with error bars show observational estimates.",
"The red open squares show the mean properties of 87 LBGs at z∼2.2z \\sim 2.2 , the black open circles those for 9 LBGs at z∼3.5z \\sim 3.5 .",
"The stellar mass-luminosity relation at z∼6z \\sim 6 given by and are also shown.",
"The legend for data symbols, given in the left A 1350 A_{1350} panel, applies to data in all panels.Figure: Gas metallicity versus stellar mass at various redshifts, specified in the upper left corner of each panel.",
"The green and blue dotted lines refer to model galaxies with halo masses of 10.5≲log(M vir /M ⊙ )≲11.510.5 \\lesssim \\log (M_{\\rm vir}/M_\\odot ) \\lesssim 11.5 and 11.5≲log(M vir /M ⊙ )≲13.311.5 \\lesssim \\log (M_{\\rm vir}/M_\\odot ) \\lesssim 13.3, respectively.",
"The limited extent of the blue dotted lines at z=1.4z=1.4 follows from the adopted lower limit to the considered virialization redshifts (z vir ⩾1.5z_{\\rm vir}\\geqslant 1.5), which translates into a lower limit to the stellar mass in massive halos at this redshift.",
"The black solid lines show the average mass-metallicity relation for model galaxies, weighted by the halo formation rate.",
"The black filled circles correspond to comoving number densities decreasing from 10 -2 10^{-2} to 10 -5 dex -1 Mpc -3 10^{-5}\\,{\\rm \\,dex^{-1}\\,Mpc^{-3}} in steps of one dex, with symbol size decreasing with the number density.",
"The bottom-right panel shows the evolution of the gas metallicity versus stellar mass relation from z obs =2z_{\\rm obs} = 2 to z obs =6z_{\\rm obs} = 6.",
"The data points are (see the legend within each panel) from , , , , and .Figure: Luminosity functions at 1350 A ˚{\\rm \\mathring{A}} at several redshifts, specified in the upper right corner of each panel.",
"The upper scale gives the UV magnitudes corresponding to the UV luminosities at 1350A ˚1350\\,{\\rm \\mathring{A}}.",
"The solid black lines show the predicted luminosity function neglecting extinction while the dot-dashed blue lines include the effect of extinction.",
"The low luminosity break corresponds to M crit =10 8.5 M ⊙ M_{\\rm crit}=10^{8.5}\\,M_\\odot .",
"The thin dotted and solid blue lines show the effect of increasing the minimum halo mass to 10 9.8 10^{9.8} and 10 11.2 M ⊙ 10^{11.2}\\,M_\\odot , respectively, including extinction.",
"The extinction affects mostly the highest luminosities, associated to the most massive objects which have the fastest chemical enrichment (see Figure ).",
"As illustrated by the bottom-right panel, showing the evolution of the luminosity function, its faint portion is predicted to steepen with increasing redshift.",
"The model implies a weak evolution of the luminosity function from z=2z=2 to z=6z=6.",
"The data are from , , , , , , , , , , , , , and , , .",
"Only the estimates by include an (uncertain) correction for dust extinction, based on the slope of the UV continuum.Figure: Comparison of the SFR functions yielded by the model at z=2z = 2 and 3 with those inferred from IR , , , , , and UV , , , luminosity functions.",
"The conversion of IR luminosities into SFRs was done using the calibration.",
"The SFR function from IR data can be directly compared to the SFR function yielded by the model (solid black line) because the star formation in these IR-bright galaxies is almost entirely dust-obscured and the contribution of older stars to dust heating is negligible.",
"The dot-dashed blue lines show the model SFR functions as determined from Equation () with k UV =1.0×10 28 ergs -1 Hz -1 M ⊙ -1 yrk_{\\rm UV}=1.0\\times 10^{28}\\,\\hbox{erg}\\,\\hbox{s}^{-1}\\,\\hbox{Hz}^{-1}\\,M_\\odot ^{-1}\\,\\hbox{yr}, applied to “observed” (i.e., attenuated by dust) UV luminosities, and therefore these curves can be directly compared with the observed UV data, uncorrected for attenuation.",
"The dot-dashed blue lines converge to the black lines at low SFRs, for which the dust attenuation is small.Figure: History of cosmic SFR density.",
"The solid black line shows the global value, which is the sum of the contributions of warm (dashed blue line) and cold (dotted red line) late-type galaxies and of proto-spheroidal galaxies with intrinsic M 1350 int ⩽-18M^{\\rm int}_{1350}\\leqslant -18 (dot-dashed orange line, that overlaps the solid black line for z>2z>2).",
"The SFR density of late-type galaxies was computed using the model; that of proto-spheroidal galaxies was computed with the present model, including halo masses log(M vir /M ⊙ )⩾8.5\\log (M_{\\rm vir}/M_\\odot ) \\geqslant 8.5.",
"The solid blue line shows the evolution, as given by our model, of the SFR density of galaxies with observed (i.e., attenuated by dust) magnitudes brighter than M 1350 obs =-18M^{\\rm obs}_{1350} = -18, already represented in the observed UV luminosity functions.",
"The gray region illustrates the minimum SFR densities required to keep the universe fully ionized if 3≲C HII /f esc ≲303 \\lesssim C_{\\rm HII}/f_{\\rm esc} \\lesssim 30 .Observational estimates of SFR densities from UV data are from , , , , , , , , , , , , , , , , , , and .",
"For completeness we also show SFR densities inferred from far-IR/sub-mm , , and radio data.Figure: Evolution with galactic age of the production rate of ionizing photons, N ˙ 912 int \\dot{N}^{\\rm int}_{912} (left y-scale), and of the intrinsic ratio of Lyman-continuum to UV luminosity, R int ≡L 912 int /L 1350 int R_{\\rm int}\\equiv L^{\\rm int}_{912}/L^{\\rm int}_{1350} (right y-scale) for a constant SFR, M ˙ ☆ =1M ⊙ yr -1 \\dot{M}_\\star = 1\\ M_\\odot \\ \\rm yr^{-1}, a IMF, and three metallicities: Z g =0.005Z_{\\rm g} = 0.005 (dashed line), 0.02 (dotted line), and 1 Z ⊙ Z_\\odot (solid line).",
"The chosen reference values, N ˙ 912 int =4.0×10 53 \\dot{N}^{\\rm int}_{912} = 4.0 \\times 10^{53} and R int =0.265R_{\\rm int}=0.265, are indicated by the upper and lower horizontal lines, respectively.Figure: Model cumulative Lyα\\alpha line luminosity functions at several redshifts, specified in the upper right corner of each panel corrected (solid black lines) and uncorrected (dot-dashed blue lines) for attenuation by the IGM computed following .",
"We have adopted a minimum halo mass of 10 8.5 M ⊙ 10^{8.5}\\,M_\\odot .",
"The dotted blue lines give the contributions of halo masses ⩾10 11 M ⊙ \\geqslant 10^{11}\\,M_\\odot .",
"The bottom-right panel illustrates the evolution of the Lyα\\alpha line luminosity function, without attenuation by the IGM.",
"The data are from , , , , , , , , , , , , , , , , , , , , and .",
"The data based on spectroscopic samples are shown by filled symbols and the references within panels are labeled “spec”.",
"Those based on photometric samples are shown by open symbols.",
"The label “DLF” associated to references means that the original papers gave the differential luminosity functions.Figure: Left panels: fractions of ionizing photons surviving dust, HI, and both absorptions (f 912 dust f^{\\rm dust}_{912}, f 912 HI f^{\\rm HI}_{912}, and f 912 esc f^{\\rm esc}_{912}, from top to bottom), weighted by the halo formation rate, yielded by our model for z obs =3z_{\\rm obs} = 3 (dot-dashed line), 6 (solid line), 9 (dashed line), and 12 (dotted line) as a function of the attenuated UV luminosity M 1350 obs M^{\\rm obs}_{1350}.",
"Right panels: escape fractions given by the model as a function of zz for two luminosity bins, i.e., M 1350 obs ∈[-22,-20]M^{\\rm obs}_{1350} \\in [-22, -20] (dashed blue line) and M 1350 obs ∈[-20,-18.3]M^{\\rm obs}_{1350} \\in [-20, -18.3] (solid red line).",
"Data in the bottom-right panel are from: at z≃3.1z \\simeq 3.1 (offset by Δz=0.2\\Delta z = 0.2 for readability; the open circle corresponds to the median value and the error bars extend to the minimum/maximum values); for LBGs (M UV ∈[-22,-20]M_{\\rm UV} \\in [-22, -20]; open blue diamond) and for LAEs (M UV ∈[-20,-18.3]M_{\\rm UV} \\in [-20, -18.3]; open red diamond) at z∼3z \\sim 3; at z∼2.85z \\sim 2.85 (the points are offset by Δz=-0.2\\Delta z = -0.2 for readability); ; for LBGs in the redshift range 6⩽z⩽106 \\leqslant z \\leqslant 10.Figure: Comoving emission rates of ionizing photons (〈N ˙ 912 esc 〉\\langle \\dot{N}^{\\rm esc}_{912} \\rangle , left yy-scale) and ionizing emissivities (ϵ LyC ≃〈N ˙ 912 esc 〉h P α\\epsilon _{\\rm LyC} \\simeq \\langle \\dot{N}^{\\rm esc}_{912} \\rangle h_{\\rm P} \\alpha for ϵ(ν)=ϵ LyC (ν/ν 912 ) -α \\epsilon (\\nu ) = \\epsilon _{\\rm LyC} (\\nu /\\nu _{912})^{-\\alpha } with α=2\\alpha = 2, right yy-scale) as a function of redshift.",
"The solid black line and the dotted red line correspond to critical halo masses of 10 8.5 10^{8.5} and 10 10 M ⊙ 10^{10}\\,M_\\odot , respectively.",
"Data points are from , , , , , and .Figure: Left panels: evolutionary laws for the IGM clumping factor C HII C_{\\rm HII} (upper panel) proposed in the literature and the corresponding recombination timescales t ¯ rec \\bar{t}_{\\rm rec} (lower panel).",
"Solid black line: C HII (z)=9.25-7.21log(1+z)C_{\\rm HII}(z) = 9.25 - 7.21 \\log (1 + z) ; dashed green line: C HII (z)=2.9[(1+z)/6] -1.1 C_{\\rm HII}(z) = 2.9 [(1+z)/6]^{-1.1} ; dotted red line: C HII (z)=3C_{\\rm HII}(z) = 3 ; triple-dot-dashed orange line: C HII (z)=26.2917exp(-0.1822z+0.003505z 2 )C_{\\rm HII}(z) = 26.2917 \\exp (-0.1822z+0.003505z^2) ; dot-dashed blue line: C HII (z)=min[C HII (z=6),exp(-0.47z+5.76)+1.29]C_{\\rm HII}(z) = \\min [C_{\\rm HII}(z=6), \\exp (-0.47z+5.76)+1.29], corresponding to the C -1 C_{-1} L6N256 no-reheating simulation by covering the range 6⩽z⩽206 \\leqslant z \\leqslant 20.Main figure: evolution with redshift of the volume filling factor Q HII Q_{\\rm HII} (left y-scale) and of the electron optical depth τ es \\tau _{\\rm es} (right y-scale).",
"The thick solid black lines correspond to the fiducial model with C HII (z)C_{\\rm HII}(z) by and M crit =10 8.5 M ⊙ M_{\\rm crit}=10^{8.5}\\,M_\\odot .",
"The dotted red lines correspond to the same model but with M crit =10 10 M ⊙ M_{\\rm crit} = 10^{10}\\ M_\\odot .",
"The dot-dashed blue lines show the results with the C HII (z)C_{\\rm HII}(z) by for M crit =10 8.5 M ⊙ M_{\\rm crit}=10^{8.5}\\,M_\\odot .",
"The observational constraints on the volume filling factor are from a collection of literature data made by .",
"The 9-year WMAP constraint on electron optical depth, τ es =0.089±0.014\\tau _{\\rm es} = 0.089 \\pm 0.014 , is represented by the gray region while the filled square and the filled circle with error bars represent the preliminary estimates by and , respectively."
]
] | 1403.0055 |
[
[
"Quantum Vision Transformers"
],
[
"Abstract We design and analyse quantum transformers, extending the state-of-the-art classical transformer neural network architectures known to be very performant in natural language processing and image analysis.",
"Building upon the previous work of parametrised quantum circuits for data loading and orthogonal neural layers, we introduce three quantum attention mechanisms, including a quantum transformer based on compound matrices.",
"These quantum architectures can be built using shallow quantum circuits and can provide qualitatively different classification models.",
"We performed extensive simulations of the quantum transformers on standard medical image datasets that showed competitive, and at times better, performance compared with the best classical transformers and other classical benchmarks.",
"The computational complexity of our quantum attention layer proves to be advantageous compared with the classical algorithm with respect to the size of the classified images.",
"Our quantum architectures have thousands of parameters compared with the best classical methods with millions of parameters.",
"Finally, we have implemented our quantum transformers on superconducting quantum computers and obtained encouraging results for up to six qubit experiments."
],
[
"Introduction",
"Quantum machine learning [4] uses quantum computation in order to provide novel and powerful tools that can be used to enhance the performance of classical machine learning algorithms.",
"One way to do this is to use efficient quantum methods for linear algebra both in more traditional learning algorithms, including similarity-based classification [13], [24], linear systems solvers [12], dimensionality reduction techniques [25], clustering algorithms [18], [1], [17], recommendation systems [21], [34], as well as in deep learning [20].",
"An alternative is to use parametrised quantum circuits as quantum neural networks that could potentially be more powerful by exploring a higher-dimensional optimization space [6], [2], [5] or by providing interesting properties, like orthogonality or unitarity [19], [16].",
"In this work, we focus on transformers, a recent neural network architecture [33] that has been proposed both for Natural Language Processing [10] and for visual tasks [9], and has provided state-of-the-art performance across different tasks and datasets [31].",
"At a high level, transformers are neural networks that use an attention mechanism that takes into account the global context while processing the entire input data element-wise.",
"In the case of visual analysis, for example, images are divided into smaller patches, and instead of simply performing patch-wise operations with fixed size kernels, a transformer learns attention coefficients per patch that weigh the attention paid to the rest of the image by each patch.",
"In the case of visual recognition or text understanding, the context of each element is vital, and the transformer can capture more global correlations between parts of the sentence or the image than convolutional neural networks without an attention mechanism [9].",
"Fig.REF shows a high-level representation of a Vision Transformer architecture, in line with [9].",
"The first part consists of dividing the images into patches, reformatted as vectors.",
"Then, a number of transformer layers are applied, one of which is described in Fig.REF .",
"Last, the classification at the end happens through a fully connected layer.",
"A key ingredient inside the transformer layer is the attention mechanism (Fig.REF ), which is trained to weigh each image patch in its global context.",
"While extracting information from each patch, the network simultaneously learns how much attention a patch should pay to the other patchess.",
"More details on classical transformers are given in Section .",
"In one related work, classical transformer architectures and attention mechanisms have been used to perform quantum tomography [7].",
"Moreover, a quantum-enhanced transformer for sentiment analysis has been proposed in [11], and a self-attention mechanism for text classification has been used in [26].",
"All these results use standard variational quantum circuits for the neural networks, and the attention coefficients are calculated classically.",
"A method for using a natively quantum attention mechanism for reinforcement learning has also been proposed in [30].",
"[36] performed semiconductor defect detection using quantum self-attention, also using standard variational quantum circuits.",
"We also note the proposal of [6] for variational circuits with similarities to convolutional neural networks for general purpose image classification.",
"Our approach differs from the above-mentioned approaches in three key aspects.",
"Firstly, we focus on quantum vision transformers and image classification tasks.",
"In addition to a quantum translation of the classical vision transformer, a novel and natively quantum method is proposed in this work.",
"This method provides what we call a compound transformer, which invokes Clifford Algebra operations that would be harder to compute classically.",
"Finally, the different parametrised quantum circuits proposed in our work are supported by linear algebraic tools to calculate gradients and other properties, making them much more Noisy Intermediate-Scale Quantum (NISQ)-friendly with proven scalability, in contrast to variational QC approaches which lack proof of scalability [28].",
"This advantage in scalability of our proposed parametrised quantum circuits is made possible by the use of a specific amplitude encoding for translating vectors as quantum states, and consistent use of hamming-weight preserving quantum gates instead of general quantum ansatz [5].",
"While we focus on vision transformers and benchmark our methods on vision tasks, our work could open the way for other applications, for example on textual data where transformers have been proven to be particularly efficient [10].",
"Figure: Representation of a Vision Transformer network architecture.",
"."
],
[
"Our Results",
"We provide now a more detailed description of our results.",
"We propose two types of quantum transformers, and apply the novel architectures to visual tasks.",
"The main ingredient in a transformer is the attention mechanism, shown in Fig.REF .",
"We focus on leveraging QC on this part of the architecture.",
"Other parts of the transformer architecture shown in Fig.REF and Fig.REF are derived from two previous works [13], [19].",
"In the first approach, coined the Orthogonal Transformer, we designed a quantum analogue for each of the two main components of a classical attention mechanism.",
"The classical attention mechanism (Fig.REF ) starts with fully connected linear layers, where each input, or patch, $x_i$ is a vector that corresponds to a part of the image and is multiplied by a weight matrix $V$ .",
"To perform this operation quantumly, we use an orthogonal quantum layer, where a parametrised quantum circuit consisting of Reconfigurable Beam Splitter ($RBS$ ) gates is applied to a unary amplitude encoding of the input (see Sections REF and REF for details).",
"An example of such an orthogonal layer has been defined in [19], and these results are extended in this work by definition of new architectures for these layers, including ones with only logarithmic depth.",
"The second step of the attention mechanism is the interaction between patches.",
"We define parametrised quantum circuits to learn the attention coefficients $A_{ij}$ by performing the linear-algebraic operation $x_i ^TW x_j$ for a trainable orthogonal matrix $W$ and all pairs of patches $x_i$ and $x_j$ .",
"After that, a simple non linearity is applied to obtain each output $y_i$ .",
"The two steps of the attention mechanism described above can be applied with separate quantum circuits.",
"In addition, a global quantum attention mechanism is also defined for forward inference after the matrices $V$ and $W$ have been trained.",
"For this, a quantum data loader for matrices is used, where for each patch of the input, the entire matrix whose rows are the patches re-scaled by the attention weights are loaded.",
"Once all patches are loaded in this attention-weighted superposition, the quantum circuit that applies $V$ on a patch is applied to provide a quantum version of an attention mechanism from which the classical output of the layer can be extracted (see also Section REF ).",
"In Section REF , a second approach is described that is a more natively quantum version of a transformer.",
"This architecture is termed a Compound Transformer, since it uses quantum layers that perform a linear algebraic operation that multiplies the input with a trainable second-order compound matrix.",
"The main idea is that all patches are loaded in superposition, and then a trainable orthogonal matrix is applied that gives different attention to each patch with different trainable weights, as is done in a classical transformer.",
"In order to benchmark the proposed methods, we apply them to a set of medical image classification tasks; datasets from MedMNIST were used.",
"This is a collection of 12 pre-processed open medical image datasets [37].",
"The collection has been standardized for classification tasks on 12 different imaging modalities, each with medical images of $28 \\times 28$ pixels.",
"The quantum transformers were trained on all 12 MedMNIST datasets, and achieved a very competitive level of accuracy while demonstrating a significant reduction in the number of model parameters with respect to the current benchmarks [37].",
"Detailed results can be found in Section REF .",
"Last, we performed a hardware experiment using two superconducting quantum computers provided by IBM, the 16-qubit ibmq_guadalupe and the 27-qubit ibm_hanoi.",
"We used different Transformer architectures and performed inference for the RetinaMNIST dataset using four, five and six qubit circuits.",
"The results were relatively close to the ones generated by the simulations, attesting that the hardware experiments succeeded for the given number of qubits.",
"Experiments with a higher number of qubits did not provide meaningful results.",
"Overall, we designed and analysed quantum Transformers and provided relevant benchmarks for their performance on medical image classification, both in simulations and quantum hardware.",
"We believe these quantum architectures can be widely used for other learning tasks, including natural language processing and other visual tasks.",
"For each of our quantum methods, we analyze the computation complexity of the quantum attention mechanisms which is lower compared to their classical counterpart.",
"In the simulations, our quantum transformers have reached comparable if not better levels of accuracy compared to the classical equivalent transformers, while using a similar or smaller number of trainable parameters, confirming our theoretical predictions.",
"The paper is organised as follows: We start with a description of classical Transformers in Section .",
"We then explain all the quantum tools we will use in Section , including data loaders for matrices and quantum orthogonal layers.",
"In Section , we define additional quantum methods and put everything together to define our two types of transformers, the Orthogonal Transformer and the Compound Transformer.",
"We also define an architecture without attention that we call an orthogonal patch-wise neural network.",
"Finally, in Section , we describe classification results on medical image tasks, both through simulations and quantum hardware experiments."
],
[
"Vision Transformers",
"In this section, we recall the details of a classical Vision Transformers, introduced by [9].",
"Some slight changes in the architecture have been made to ease the correspondence with quantum circuits later.",
"We also introduce important notations that will be reused in the quantum methods.",
"The transformer network starts by decomposing an image into patches and pre-processing the set of patches to map each one into a vector, as shown in Fig.REF .",
"The initial set of patches is enhanced with an extra vector of the same size as the patches, called class embedding.",
"This class embedding vector is used at the end of the network, to feed into a fully connected layer that yields the output (see Fig.REF ).",
"We also include one trainable vector called positional embedding, which is added to each vector.",
"At the end of this pre-processing step, we obtain the set of $n$ vectors of dimension $d$ , denoted $x_i$ to be used in the next steps.",
"Figure: Patch division part of the transformer network for an image split into four patches.",
"Note Class Embedding and Position Embedding are trainable vectors.",
".Next, feature extraction is performed using a transformer layer [33], [9] which is repeated $L$ times, as shown in Fig.REF .",
"Within the transformer layer, we first apply layer normalisation over all patches $x_i$ , and then apply the attention mechanism detailed in Fig.REF .",
"After this part, we obtain a state to which we add the initial input vectors before normalisation in an operation called residual layer, represented by the blue arrow in Fig.REF , followed by another layer normalisation.",
"After this, we apply a Multi Layer Perceptron (MLP), which consists of multiple fully connected linear layers for each vector that result in same-sized vectors.",
"Again, we add the residual from just before the last layer normalisation, which is the output of one transformer layer.",
"After repeating the transformer layer $L$ times, we finally take the vector corresponding to the class embedding, that is the vector corresponding to $x_0$ , in the final output and apply a fully connected layer of dimension ($d$ $\\times $ number of classes) to provide the final classification result (see Fig.REF ).",
"It is important to observe here that we only use the first vector outcome in the final fully connected layer to do the classification (therefore the name \"class embedding\").",
"Figure: Single transformer layer.",
".Looking inside the attention mechanism (see Fig.REF ), we start by using a fully connected linear layer with trainable weights $V$ to calculate for each patch $x_i$ the feature vector $Vx_i$ .",
"Then to calculate the attention coefficients, we use another trainable weight matrix $W$ and define the attention given by patch $x_i$ to patch $x_j$ as $x_i^TWx_j$ .",
"Next, for each patch $x_i$ , we get the final extracted features as the weighted sum of all feature vectors $Vx_j$ where the weights are the attention coefficients.",
"This is equivalent to performing a matrix multiplication with a matrix $A$ defined by $A_{ij} = x_i^TWx_j$ .",
"Note, in classical transformer architecture, a column-wise softmax is applied to all $A_{ij}$ and attention coefficients $A^{\\prime }_{ij} = softmax_j(A_{ij})$ is used instead.",
"Overall, the attention mechanism makes use of $2d^2$ trainable parameters, evenly divided between $V$ and $W$ .",
"Figure: The attention mechanism at the heart of the transformer layer.",
"Matrices VV and WW are trainable.In fact, the above description is a slight variant from the original transformers proposed in [33], where the authors used two trainable matrices to obtain the attention coefficients instead of one ($W$ ) in this work.",
"This choice was made to simplify the quantum implementation but could be extended to the original proposal using the same quantum tools.",
"Computational complexity of classical attention mechanism depends mainly on the number of patches $n$ and their individual dimension $d$ : the first patch-wise matrix multiplication with the matrix $V\\in \\mathbb {R}^{d\\times d}$ takes $O(nd^2)$ steps, while the subsequent multiplication with the large matrix $A^{\\prime }$ takes $O(n^2d)$ .",
"Obtaining $A^{\\prime }$ from $W$ requires $O(nd^2)$ steps as well.",
"Overall, the complexity is $O(nd^2 + n^2d)$ .",
"In classical deep learning literature, the emphasis is made on the second term, which is usually the most costly.",
"Note that a recent proposal [23] proposes a different attention mechanism as a linear operation that only has a $O(nd^2)$ computational complexity.",
"We compare the classical computational complexity with those of our quantum methods in Table REF .",
"These running times have an real impact on both training and inference, as they measure how the time to perform each layer scales with the number and dimension of the patches."
],
[
"Quantum Data Loaders for Matrices",
"In order to perform a machine learning task with a quantum computer, classical data (a vector, a matrix) needs to be loaded into the quantum circuit.",
"The technique we choose for this task is called amplitude encoding, which uses the classical scalar component of the data as amplitudes of a quantum state made of $d$ qubits.",
"In particular we build upon previous methods to define quantum data loaders for matrices, as shown in Fig.REF .",
"In previous work [13], we detailed how to perform such data loading for vectors.",
"[13] proposed three different circuits to load a vector $x\\in \\mathbb {R}^d$ using $d-1$ gates for a circuit depth ranging from $O(log(d))$ to $O(d)$ as desired (see Fig.REF ).",
"These data loaders use the unary amplitude encoding, where a vector $x = (x_1,\\cdots ,x_d)$ is loaded in the quantum state $\\mathinner {|{x}\\rangle } = \\frac{1}{\\left\\Vert x\\right\\Vert }\\sum _{i=1}^d x_i\\mathinner {|{e_i}\\rangle }$ where $\\mathinner {|{e_i}\\rangle }$ is the quantum state with all qubits in 0 except the $i^{th}$ one in state 1 (e.g.",
"$\\mathinner {|{0\\cdots 010\\cdots 0}\\rangle }$ ).",
"The circuit uses $RBS$ gates: a parametrised two-qubit gate given by the following unitary matrix: $ RBS(\\theta ) = \\left( \\begin{array}{cccc}1 & 0 & 0 & 0 \\\\0 & \\cos \\theta & \\sin \\theta & 0 \\\\0 & -\\sin \\theta & \\cos \\theta & 0 \\\\0 & 0 & 0 & 1 \\end{array} \\right)$ The $d-1$ parameters $\\theta _i$ of the RBS gates are classically pre-computed to ensure that the output of the circuit is indeed $\\mathinner {|{x}\\rangle }$ .",
"Figure: Three possible data loaders for dd-dimensional vectors (d=8d=8).",
"From left to right: the parallel, diagonal, and semi-diagonal circuit have respectively a circuit depth of log(d)log(d), dd, and d/2d/2.",
"The X gate represent the Pauli X gate, and the vertical lines represent RBSRBS gates with tunable parameters.In this work, a data loader for matrices is required.",
"Given a matrix $X \\in \\mathbb {R}^{n\\times d}$ , instead of loading a flattened vector, rows $X_i$ are loaded in superposition.",
"As shown in Fig.REF , on the top qubit register, we first load the vector $(\\left\\Vert X_1\\right\\Vert ,\\cdots ,\\left\\Vert X_n\\right\\Vert )$ made of the norms of each row, using a data loader for a vector and obtain a state $\\frac{1}{\\left\\Vert X\\right\\Vert }\\sum _{i=1}^n \\left\\Vert X_i\\right\\Vert \\mathinner {|{e_i}\\rangle }$ .",
"Then, on a lower register, we are sequentially loading each row $X_i \\in \\mathbb {R}^d$ .",
"To do so, we use vector data loaders and their adjoint, as well as CNOTs controlled on the $i^{th}$ qubit of the top register.",
"The resulting state is a superposition of the form: $\\mathinner {|{X}\\rangle } = \\frac{1}{\\left\\Vert X\\right\\Vert }\\sum _{i=1}^n\\sum _{j=1}^d X_{ij}\\mathinner {|{e_j}\\rangle }\\mathinner {|{e_i}\\rangle }$ Figure: Data loader circuit for a matrix X∈ℝ n×d X\\in \\mathbb {R}^{n\\times d}.",
"The top register uses nn qubits to load the norms of the rows.",
"The lower register uses dd qubits to load all rows sequentially, by applying the loader and their adjoint for each row X i X_i, with CNOTs controlled by the corresponding qubit ii of the top register.",
"Each loader on the lower register has depth O(logd)O(\\log d).One immediate application of data loaders that construct amplitude encodings is the ability to perform fast inner product computation with quantum circuits.",
"Applying the inverse data loader of $x_i$ after the regular data loader of $x_j$ effectively creates a state of the form $\\mathinner {\\langle {x_i,x_j}\\rangle }\\mathinner {|{e_1}\\rangle } + \\mathinner {|{G}\\rangle }$ where $\\mathinner {|{G}\\rangle }$ is a garbage state.",
"The probability of measuring $\\mathinner {|{e_1}\\rangle }$ , which is simply the probability of having a 1 on the first qubit, is $|\\mathinner {\\langle {x_i,x_j}\\rangle }|^2$ .",
"Techniques to retrieve the sign of the inner product have been developed in [27].",
"Loading a whole matrix in a quantum state is a powerful technique for machine learning.",
"However, for large matrices, it requires dee quantum circuits, especially in case of large $n$ , thus making this technique more suited for larger quantum computers."
],
[
"Quantum Orthogonal Layers",
"In this section, we outline the concept of Quantum Orthogonal Layers used in neural networks, which generalises the work in [19].",
"These layers correspond to parametrised circuits of $n$ qubits made of $RBS$ gates.",
"More generally, RBS gates preserve the number of ones and zeros in any basis state: if the input to a quantum orthogonal layer is a vector in unary amplitude encoding, the output will be another vector in unary amplitude encoding.",
"Similarly, if the input quantum state is a superposition of only basis states of hamming weight 2, so is the output quantum state.",
"This output state is precisely the result of a matrix-vector product, where the matrix is the unitary matrix of the quantum orthogonal layer, restricted to the basis used.",
"Therefore, for unary basis, we consider a $n \\times n$ matrix $W$ instead of the full $2^n \\times 2^n$ unitary.",
"Similarly for the basis of hamming weight two, we can restrict the unitary to a $n \\atopwithdelims ()2$ $\\times $ $n \\atopwithdelims ()2$ matrix.",
"Since the reduced matrix conserves its unitary property and has only real values, these are orthogonal matrices.",
"Each element $W_{ij}$ of the orthogonal matrix $W$ can be seen as the weight of all paths that map state $\\mathinner {|{e_j}\\rangle }$ to state $\\mathinner {|{e_i}\\rangle }$ , as explained in previous work [19].",
"More generally, we can think of such hamming weight preserving circuits with $n$ qubits as block-diagonal unitaries that act separately on $n+1$ subspaces, where the $k$ -th subspace is defined by all computational basis states with hamming weight equal to $k$ .",
"The dimension of these subspaces is equal to $n \\atopwithdelims ()k$ .",
"There exist many possibilities for building a Quantum Orthogonal Layer, each with different properties.",
"In [19], we proposed the Pyramid circuit shown in Fig.REF , composed of exactly $n(n-1)/2$ $RBS$ gates in pyramid-shaped layout.",
"This circuit requires only adjacent qubit connectivity, which is the case for most superconducting qubit hardware.",
"More precisely, the set of matrices that are equivalent to the Quantum Orthogonal Layers with pyramidal layout is exactly the Special Orthogonal Group, made of orthogonal matrices with determinant equal to $+1$ .",
"We have showed that by adding a final $Z$ gate on the last qubit would allow having orthogonal matrices with $-1$ determinant.",
"The pyramid circuit is therefore very general and cover all the possible orthogonal matrices of size $n \\times n$ .",
"Figure: Pyramid circuit on 8 qubits.",
"Each vertical bar is a two qubits RBSRBS gate with an independent parameter θ i \\theta _i.In this work, we introduce two new types of Quantum Orthogonal Layers: the butterfly circuit (Fig.REF ), and the $X$ circuit (Fig.REF ).",
"The $X$ circuit requiring smaller number of gates is the most suited for noisy hardware, since fewer gates imply less error, while retaining the property that there exists a path from every input qubit to every output qubit.",
"Given the smaller number of gates required, the $X$ circuit is less expressive since the set of possible orthogonal matrices is restrained, having fewer free parameters.",
"The butterfly circuit has the very interesting property of having logarithmic depth, a linear number of gates, retaining high level of expressivity.",
"It originated from the classical Cooley–Tukey algorithm [8], used for Fast Fourier Transform.",
"Note that this circuit requires the ability to apply the RBS gates on all qubit pairs.",
"In previous work [19], we have shown that there exists a method to compute the gradient of each parameter $\\theta _i$ in order to update them.",
"This backpropagation method for the pyramid circuit takes time $O(n^2)$ , corresponding to the number of gates, and provided a polynomial improvement in run time compared to the previsously known orthogonal neural network training algorithms [14].",
"The exact same method developed for the pyramid circuit can be used to perform quantum backpropagation on the new circuits introduced in this paper.",
"The run time also corresponds to the number of gates, which is lower for the butterfly and $X$ circuits.",
"See for full details on the comparison between the three types of circuits.",
"Table: Comparison of different Quantum Orthogonal Layer circuits with nn qubits.",
"NN stands for Nearest Neighbor connectivity.Figure: X Circuit"
],
[
"Quantum Transformers",
"Starting from the definition of classical transformers (Section ) and using the quantum tools described above, this section presents several proposals for quantum transformers.",
"Section REF begins, with a simple neural network architecture that serves as the basis for the transformer architectures.",
"In REF , quantum circuits for the computation of the attention coefficients are developed, and with this new tool, the quantum Orthogonal Transformer is proposed.",
"In REF , an efficient quantum attention mechanism suited for inference is proposed.",
"Finally, in REF , a different approach is taken to directly perform the full attention with a new kind of operation more native to quantum linear algebra: the compound matrix multiplication.",
"These last two proposals are very efficient but will require more qubits and larger depth to be implemented as they both use matrix data loading methods.",
"A comparison between these different quantum methods is provided in Table REF , which is applicable to both training and inference.",
"Table: Comparison of different quantum methods to perform the attention part of a transformer network.",
"nn and dd stand respectively for the number of patches and their individual dimension.",
"All Quantum Orthogonal layers are implemented using the butterfly circuits.",
"Note that the classical attention mechanisms require O(nd 2 +n 2 d)O(nd^2+n^2d) steps, while the computational complexity of the quantum architectures are reflect in the number of parametrised gates.",
"Both computational complexity and parameter counts are important: the first one for running time, the second one for expressivity and trainability of the network"
],
[
"Orthogonal Patch-wise Neural Network",
"First, a simple architecture named orthogonal patch-wise neural network is formulated, which forms the basis of the transformer architectures.",
"As seen in Fig.REF , the attention layer is first composed by a series of matrix-vector multiplications: each patch is multiplied by the same trainable matrix $V$ .",
"Building upon the tools developed in Section , one circuit per patch is used, as shown in Fig.REF .",
"Each circuit has $d$ qubits and each patch $x_i$ is encoded in a quantum state with a vector data loader.",
"We can use a Quantum Orthogonal Layer (e.g.",
"pyramid, butterfly) to perform a trainable matrix multiplication on each patch.",
"The Quantum Orthogonal Layer is equivalent to a matrix $V$ with a specific structure.",
"The output of each circuit is a quantum state encoding $V x_i$ , a vector which is retrieved through tomography.",
"The precision tomography depends on the number of measurements and will be assessed during the numerical simulations.",
"This network can be thought of as a transformer with a trivial attention mechanism, where each patch pays attention only to itself.",
"The computational complexity of this circuit is calculated as follows: from Section REF , a data loader with $d$ qubits has a complexity of $log(d)$ steps.",
"For the orthogonal quantum layer, as shown in Table REF , a butterfly circuit takes $log(d)$ steps as well, with $\\frac{d}{2}log(d)$ trainable parameters.",
"Overall, the complexity is $O(log(d))$ and the trainable parameters are $O(d \\log d)$ ."
],
[
"A Quantum Orthogonal Transformer",
"The missing part of the previous architecture is the attention mechanism: the second step described in Fig.REF , where all $V x_i$ are combined using attention coefficients $A_{ij}$ to obtain the list of final outputs $y_j$ .",
"The attention coefficients used here are defined as $A_{ij} = x_i ^TW x_j$ where $W$ is a trainable matrix.",
"As seen before, the quantum tools perform matrix-vector multiplication using Quantum Orthogonal Layers.",
"As shown in Fig.REF , loading any $x_j$ with a data loader, followed by a trainable Quantum Orthogonal Layer, is equivalent to encoding $W x_j$ .",
"Next, apply the inverse data loader of $x_i$ , which creates a state where the probability of measuring 1 on the first qubit is exactly $|x_i ^TW x_j|^2$ .",
"With classical post-processing, this is enough to retrieve all $A_{ij}$ .",
"It is also possible to classically apply a softmax column-wise and obtain $A^{\\prime }_{ij} = \\text{softmax}_j (A_{ij})$ or any other kind of non-linearity.",
"Note the square that appears in the circuit is one type of non-linearity.",
"Using this method, the values of $A$ are always positive, but during the training phase the coefficients are learned nonetheless.",
"Additional methods also exist to obtain the sign of the inner product [27].",
"The estimation of $A_{ij}$ is repeated each time with a different pair of patches and the same trainable Quantum Orthogonal Layer $W$ .",
"The computational complexity of this quantum circuit is similar to the previous one, with one more data loader.",
"One can now put everything together to have the complete quantum method for training an attention layer: a first set of quantum circuits presented in REF are implemented to obtain each $V x_j$ .",
"At the same time, each attention coefficient $|x_i ^TW x_j|^2$ is computed, and they are potentially post-processed column wise with softmax to obtain the $A^{\\prime }_{ij}$ .",
"The two parts can then be classically combined to compute each $y_i = \\sum _j A^{\\prime }_{ij}Vx_j$ .",
"This is the desired output that will be used in the rest of the network.",
"At the end, when the cost function is estimated, backpropagation is used to compute the gradient of each angle of the Quantum Orthogonal Layers (for matrices $V$ and $W$ ) and update them for the next iteration.",
"Overall, we presented a quantum Orthogonal Transformer, which is based on quantum transformer layers, whose attention layer has been exchanged for hamming weight preserving parametrised quantum circuits that are trained to provide the weight matrices $V$ and $W$ ."
],
[
"A Quantum Attention Mechanism",
"In the previous section, the output of the attention layer $y_i = \\sum _j A^{\\prime }_{ij}Vx_j$ was computed classically once the quantities $A^{\\prime }_{ij}$ and $Vx_j$ had been computed with the help of quantum circuits.",
"Here, the quantum attention mechanism is implemented more directly with a quantum circuit, which can be used during inference.",
"Consider that the matrices $V$ and $W$ have been trained before, and the attention matrix $A$ (or $A^{\\prime }$ ) is stored classically.",
"The goal is to compute each $y_i = \\sum _j A_{ij}Vx_j$ using a global quantum circuit.",
"For this, the use of the matrix data loader from Fig.REF is used.",
"In Fig.REF , we show how for each patch with index $i$ , we can use a quantum circuit whose qubits are split into two main registers.",
"On the top register, the vector $A_i$ is loaded via a vector data loader, i.e.",
"the $j^{th}$ column of the attention matrix $A$ (or $A^{\\prime }$ ) which is assumed to be normalized.",
"At this stage, the quantum state obtained is: $\\sum _j A_{ij}\\mathinner {|{e_j}\\rangle }\\mathinner {|{0}\\rangle }$ Now an operation is performed similar to the matrix data loader of Fig.REF : the data loader and its adjoint of vector $x_i$ are applied sequentially on the lower register, with CNOTs controlled on each qubit $i$ of the top register.",
"This gives the following quantum state: $\\begin{split}\\sum _j A_{ij} \\mathinner {|{e_j}\\rangle }\\mathinner {|{x_j}\\rangle }\\end{split}$ Another way to see this is that the matrix $X$ is loaded with all rows re-scaled according to the attention coefficients.",
"The last step consists of applying the Quantum Orthogonal Layer $V$ that has been trained before on the second register of the circuit.",
"As previously established, this operation performs matrix multiplication between $V$ and the vector encoded on the second register.",
"Since the $k^{th}$ element of the vector $V x_j$ can be written as $\\sum _q V_{kq}x_{jq}$ , we have: $\\begin{split}\\sum _j A_{ij} \\mathinner {|{e_j}\\rangle }\\mathinner {|{Vx_j}\\rangle } \\\\= \\sum _j A_{ij} \\mathinner {|{e_j}\\rangle } \\sum _k (\\sum _q V_{kq}x_{jq})\\mathinner {|{e_k}\\rangle } \\\\= \\sum _k \\sum _j A_{ij} (\\sum _q V_{kq}x_{jq}) \\mathinner {|{e_j}\\rangle }\\mathinner {|{e_k}\\rangle }\\end{split}$ Since $y_i = \\sum _j A_{ij} Vx_j$ , its $k^{th}$ element can be written $y_{ik} = \\sum _j A_{ij} (\\sum _q V_{kq}x_{jq})$ .",
"Therefore, the quantum state at the end of the circuit can be written as $\\mathinner {|{y_i}\\rangle } = \\sum _k y_{ik}\\mathinner {|{\\phi _k}\\rangle } \\mathinner {|{e_k}\\rangle }$ for some normalised states $\\mathinner {|{\\phi _k}\\rangle }$ which by performing tomography on the second register can give us the output vector $y_i$ .",
"This circuit is therefore a more direct method to compute each $y_i$ , and is repeated for each $y_i$ , using a different $A_i$ in the first loader.",
"As shown in Table REF , compared with the previous method, this method requires fewer quantum circuits to run, but each circuit requires more qubits and a deeper circuit.",
"To analyse the computational complexity: the first data loader on the top register has $n$ qubit and $ \\log n$ depth; the following $2n$ loaders on the bottom register have $d$ qubits and $2n \\log d$ depth; and the final quantum orthogonal layer $V$ implemented using a butterfly circuit, has a depth of $\\log d$ and $O(d \\log d)$ trainable parameters."
],
[
"A Quantum Compound Transformer",
"So far, we follow quite closely the steps of a classical transformer by using quantum linear algebra procedures to reproduce each step with a few deviations.",
"The same quantum tools can also be used in a more natively quantum fashion, while retaining the spirit of classical transformers, as shown in Fig.REF .",
"Figure: Quantum circuit to perform a compound transformer.",
"We use a matrix dataloader for XX (equivalent to Fig.)",
"and a quantum orthogonal layer for VV applied on both registers.This quantum circuit has two registers: the top one of size $n$ and the bottom one of size $d$ .",
"The full matrix $X \\in \\mathbb {R}^{n \\times d}$ is loaded into the circuit using the matrix data loader from Section REF with $n+d$ qubits.",
"This could correspond to the entire image, as every image can be split into $n$ patches of size $d$ each.",
"We then apply a Quantum Orthogonal Layer on the two registers at the same time.",
"As explained in Section REF , this hamming weight preserving quantum circuit should be seen as a multiplication with an orthogonal matrix of size $(n+d) \\times (n+d)$ , if the input state is a unary encoding.",
"In the case of the Compound Transformer, we are stepping out of this unary framework.",
"Here, the top qubits encode the norms of each vector using the matrix data loader, and the overall state over both registers is a superposition of states of hamming weight two (exactly two qubits are in state 1).",
"Note that not all states of hamming weight two, whose number is $\\binom{n+d}{2}$ , are part of this superposition, but only $n \\times d$ of them, which correspond to states with one 1 in the first $n$ qubits, and another 1 in the remaining $d$ qubits.",
"The operation performed in this circuit is a matrix-vector multiplication in a space of dimension $\\binom{n+d}{2}$ , where each dimension corresponds to a computational basis with hamming weight 2 [22].",
"Note that $V$ and $X$ are not a priori in this dimension.",
"$X$ as a $n \\times d$ vector need first be transformed into a vector of size $\\binom{n+d}{2}$ by adding zeros to all dimensions that correspond to computational basis vectors that have both 1s on the top $n$ qubits or bottom $d$ qubits.",
"Next, a matrix $\\mathcal {V}^{(2)}$ of size $\\binom{n+d}{2}\\times \\binom{n+d}{2}$ called the 2nd order compound matrix of $V$ can be computed.",
"Given a matrix $A \\in \\mathbb {R}^{n\\times n}$ , the $k^{th}$ -compound matrix $\\mathcal {A}^{(k)}$ for $k \\in [n]$ is the $\\binom{n}{k}$ dimensional matrix with entries $\\mathcal {A}^{(k)}_{IJ} = det(A_{IJ})$ where $I$ and $J$ are subsets of rows and columns of $A$ with size k. In the present case, the 2nd order compound matrix uses $k=2$ , and each entry of $\\mathcal {V}^{(2)}$ is the determinant of a $2\\times 2$ submatrix that corresponds to the intersection of 2 rows and 2 columns of $V$ .",
"Thus, simulating the application of the quantum circuit $V$ on an input state that corresponds to a $n \\times d$ matrix $X$ corresponds to performing a matrix-vector multiplication of dimension $\\binom{n+d}{2}$ , which takes $O((n+d)^4)$ time.",
"More generally, this compound matrix operation on an arbitrary input state of hamming weight $k$ is quite hard to perform classically, since all determinants must be computed, and a matrix-vector multiplication of size $\\binom{n+d}{k}$ needs to be applied.",
"These properties of the compound matrix operation are used to define a new quantum attention layer of dimension ${(n\\times d)} \\times {(n\\times d)}$ .",
"As shown in Fig.REF , the input to the circuit is a matrix $X \\in \\mathbb {R}^{n\\times d}$ which corresponds to the patches of an image.",
"The matrix data loader creates the state $\\mathinner {|{X}\\rangle }$ , and after applying the quantum orthogonal layer on all $n+d$ qubits, the resulting state is $\\mathinner {|{Y}\\rangle } = \\mathinner {|{\\mathcal {V}^{(2)}X}\\rangle }$ , where $\\mathcal {V}^{(2)}$ is the 2nd-order compound matrix of $V$ defined above.",
"As mentioned above, this state has dimension $\\binom{n+d}{2}$ , i.e.",
"there are exactly two 1s in the $(n+d)$ qubits, but one can postselect only the part of the state where there is exactly one qubit in state 1 on the top register and the other 1 on the lower register.",
"This way, $n\\times d$ output states are generated.",
"In other words, tomography is performed for a state of the form $\\mathinner {|{Y}\\rangle } = \\frac{1}{\\left\\Vert Y\\right\\Vert }\\sum _{i=1}^n\\sum _{j=1}^d y_{ij}\\mathinner {|{e_j}\\rangle }\\mathinner {|{e_i}\\rangle }$ which is used to conclude that this quantum circuit produces transformed patches $(y_1,\\cdots , y_n) \\in \\mathbb {R}^{n\\times d}$ .",
"For the complexity of this circuit we have: the matrix data loader, detailed in Fig.REF has depth of $\\log n + 2n \\log d$ ; the Quantum Orthogonal Layer applied on $n+d$ qubits has a depth $log(n+d)$ and $(n+d)log(n+d)$ trainable parameters implemented using butterfly circuit.",
"Overall, this simple circuit can replace both the patch-wise (REF ) and the attention layer (REF ) defined before, as a combined operation.",
"The use of compound matrix multiplication is different from usual transformers, but still share some interesting properties: patches are weighted in its global context and share gradients through the determinants.",
"One can say that the Compound Transformer operates in a similar spirit as the MLPMixer architecture presented in [32].",
"This state-of-the-art architecture used for image classification tasks mixes the different patches without using convolution or attention mechanisms.",
"The underlying mechanism operates in two steps by first mixing the patches and then extracting the patch-wise features using fully connected layers.",
"Similarly, the Compound Transformer performs both steps at the same time."
],
[
"Datasets",
"In order to benchmark our models, we used MedMNIST, a collection of 12 preprocessed, two-dimensional medical image open datasets [37], [38].",
"The collection has been standardised for classification tasks on 12 different imaging modalities, each with medical images of $28 \\times 28$ pixels.",
"We used our transformers to do simulations on all 12 MedMNIST datasets.",
"For the hardware experiments, we focused on one of them, RetinaMNIST.",
"The MedMNIST dataset was chosen for our benchmarking efforts due to its accessible size for simulations of the quantum circuits and hardware experiments, while being representative of one important field of computer vision application: classification of medical images."
],
[
"Simulations",
"First, simulations of our models are performed on the 2D MedMNIST datasets and demonstrate that the proposed quantum attention architecture reaches accuracy comparable to and at times better than the various standard classical models.",
"Next, the setting of our simulations are described and the results compared against those reported in the AutoML benchmark performed by the authors in [38]."
],
[
"Simulation setting MedMNIST",
"In order to benchmark the performance of the three different quantum transformer mechanisms described in this work, namely, Orthogonal Patch-wise from Section REF , Orthogonal Transformer from Section REF , and Compound Transformer from Section REF , the complete training procedure of the multiple quantum circuits that compose each network was simulated.",
"Moreover, two baseline methods have been included in the benchmark.",
"The first baseline is the Vision Transformer, described in Section ([9]), which is a classical neural network that has been applied to different image classification tasks.",
"The second baseline is the Orthogonal Fully-Connected Neural Network (OrthoFNN), a quantum method that has been trained on the RetinaMNIST dataset in [27].",
"To ensure comparable evaluations between the five neural networks, similar architectures were implemented for all five.",
"The chosen architecture comprises of three parts: pre-processing, features extraction, and post-processing.",
"The first part is classical and pre-processes the input image of size $28 \\times 28$ by extracting 16 patches of size $7 \\times 7$ .",
"We then map every patch to the dimension of the feature extraction part of the neural network, by using a fully connected neural network layer.",
"We have experimented with dimensions of 16, 32 and 64 for the size of these layers, but a dimension of 16 was providing accurate results and had the smallest number of parameters, so we report here those results.",
"Note that this first pre-processing fully connected layer is trained in conjunction to the rest of the architecture.",
"For the OrthoNN networks, used as our quantum baseline, patches of size 16 were extracted from the complete input image using a fully connected neural network layer of size $784 \\times 16$ .",
"This fully connected layer is trained in conjunction to the quantum circuits.",
"The second part of the common architecture transforms the extracted features by applying a sequence of layers, specific to every architecture, on the extracted patches.",
"Every sequence is made with 4 layers that maintain the dimension of the neural network.",
"Moreover, the same gate layout, the butterfly circuit, is used for all circuits that compose the quantum layers.",
"Finally, the last part of the neural network is classical, which linearly projects the extracted features and outputs the predicted label.",
"The JAX package [3] was used to efficiently simulate the complete training procedure of the five benchmark architectures.",
"The experimental hyperparameters used in [38] were replicated for our benchmark: every model is trained using the cross-entropy loss with the Adam optimiser [15] for 100 epochs, with batch size of 32 and a learning rate of $10^{-3}$ that is decayed by a factor of $0.1$ after 50 and 75 epochs."
],
[
"Simulation results MedMNIST",
"The 5 different neural networks were trained over 3 random seeds, and the best overall performance for each one of them was selected.",
"The evaluation procedure is similar to the AutoML benchmark in [37], [38], and the benchmark results are shown in Table REF where the area under receiver operating characteristic (ROC) curve (AUC) and the accuracy (ACC) are reported as evaluation metrics.",
"A full comparison with the classical benchmark provided by [37] is given in (Appendix , Table REF ).",
"Table: Resource analysis for the MedMNIST simulations shown in Table .",
"The number of required qubits, the total number of parametrised gates with fixed parameters, and with trainable parameters for each quantum network are shown.",
"These numbers are to be compared with the 512 trainable parameters per layer of the classical Vision Transformers.These simulations were performed for 16 patches per image with 16 features per patch and used the butterfly circuits as the orthogonal layer.",
"The Orthogonal Transformer uses two types of quantum circuits: one to perform the operation Vx i Vx_i and one to perform the operation x i T Wx j x_i^TWx_j, which respectively use the resources mentioned.",
"Note that the Compound Transformer scales more harshly with respect to number of gates required for data loading, but compensates with lowers the number of required circuits as a trade off.From Table REF , we observe that Quantum Orthogonal and Compound Transformer architectures outperform the Orthogonal Fully-Connected and Orthogonal Patch-wise neural networks most of the time.",
"This may be due to the fact that the latter do not rely on any mechanism that exchange information across the patches.",
"Second, all quantum neural networks provide very competitive performances compared to the AutoML benchmark and outperform their classical counterparts on 7 out of 12 MedMNIST datasets.",
"Moreover, comparisons can be made with regard to the number of parameters used by each architecture, in particular for feature extraction.",
"Table REF presents a resource analysis for the quantum circuits that were simulated, per layer.",
"It includes the number of qubits, the number of gates with trainable parameters, and the number of gates with fixed parameters used for loading the data.",
"The table shows that our quantum architectures have a small number of trainable parameters per layer.",
"The global count for each quantum method is as follows.",
"Orthogonal Patch-wise Neural Network: 32 parameters per circuit, 16 circuits per layer which use the same 16 parameters, and 4 layers, for a total of 128 trainable parameters.",
"Quantum Orthogonal Transformer: 32 parameters per circuit, 17 circuits which use the same 16 parameters and another 289 circuits which use another set of 16 parameters per layer, and 4 layers, for a total of 256 trainable parameters.",
"Compound Transformer: 80 parameters per circuit, 1 circuit per layer, and 4 layers, for a total of 320 trainable parameters.",
"These numbers are to be compared with the number of trainable parameters in the classical Vision Transformer that is used as a baseline.",
"As stated in Section , each classical attention layer requires $2d^2$ free parameters, which in the simulations performed here corresponds to 512, making the total trainable parameters used in attention layer in the classical Vision Transformer 2064.",
"Note again this resource analysis focuses on the attention layer of the each transformer network, and does not include parameters used for the preprocessing of the images (see Section REF ), as part of other transformer layers (Fig.REF ), and for the single layer used in the final classification (Fig.REF ), which are common in all cases.",
"More generally, performance of other classical neural network models provided by the authors of MedMNIST is compared to our approaches in Table REF found in the Appendix.",
"Some of these classical neural networks reach somewhat better levels of accuracy, but are known to use an extremely large number of parameters.",
"For instance, the smallest reported residual network has approximately a total number of $10^7$ parameters, and the automated machine learning algorithms train numerous different architectures in order to reach that performance.",
"Based on the results of the simulations in this section, quantum transformers are able to train across a number different of classification tasks, deliver performances that are highly competitive and sometimes better than the equivalent classical methods."
],
[
"Quantum Hardware Experiments",
"Quantum hardware experiments were performed on one specific dataset: RetinaMNIST.",
"It has 1080 images for training, 120 images for validation, and 400 images for testing.",
"Each image contains $28\\times 28$ RGB pixels.",
"Each image is classified into 1 of 5 classes (ordinal regression)."
],
[
"Hardware Description",
"The hardware demonstration was performed on two different superconducting quantum computers provided by IBM, with the smaller experiments performed on the 16-qubit ibmq_guadalupe machine (see Fig.REF ) and the larger ones on the 27-qubit ibm_hanoi machine.",
"Results are reported here from experiments with four, five and six qubits; experiments with higher numbers of qubits, which entails higher numbers of gates and depth, did not produce meaningful results.",
"Figure: Connectivity of the 16-qubit ibmq_guadalupe quantum computer.Figure: Connectivity of the 27-qubit ibm_hanoi quantum computer.Note that the main sources of noise are the device noise and the finite sampling noise.",
"In general, noise is undesirable during computations.",
"In the case of a neural network, however, noise may not be as troublesome: noise can help escape local minima [29], or act as data augmentation to avoid over-fitting.",
"In classical deep learning, noise is sometimes artificially added for these purposes [35].",
"Despite this, when the noise is too large, we also see a drop in the accuracy."
],
[
"Hardware Results",
"Hardware experiments were performed with four, five and six qubits to push the limits of the current hardware, in terms of both the number of qubits and circuit depth.",
"Three quantum proposals were run: the Orthogonal Patch-wise network (from Section REF ), the Quantum Orthogonal transformers (from Sections and REF ) and finally the Quantum Compound Transformer (from Section REF ).",
"Each quantum model was trained using a JAX-based simulator, and inference was performed on the entire test dataset of 400 images of the RetinaMNIST on the IBM quantum computers.",
"The first model, the Orthogonal Patch-wise neural network, was trained using 16 patches per image, 4 features per patch, and one $4\\times 4$ orthogonal layer, using a 4-qubit pyramid as the orthogonal layer.",
"The experiment used 16 different quantum circuits of 9 $RBS$ gates per circuit per image.",
"The result was compared with an equivalent classical (non-orthogonal) patch-wise neural network, and a small advantage in accuracy for the quantum native method could be reported.",
"The second model, the Quantum Orthogonal Transformer, used 4 patches per image, 4 features per patch, and an attention mechanism with one $4\\times 4$ orthogonal layer and trainable attention coefficients.",
"4-qubit pyramids were used as orthogonal layers.",
"The experiment used 25 different quantum circuits of 12 $RBS$ gates per circuit per image and 15 different quantum circuits of 9 $RBS$ gates per circuit per image.",
"The third set of experiments ran the Orthogonal Transformer with the quantum attention mechanism.",
"We used 4 patches per image, 4 features per patch, and a quantum attention mechanism that paid attention to only the neighbouring patch, thereby using a 5-qubit quantum circuit with the $X$ as the orthogonal layer.",
"The experiment used 12 different quantum circuits of 14 $RBS$ gates and 2 $CNOT$ s per circuit per image.",
"The last two quantum proposals were compared with a classical transformer network with a similar architecture and demonstrated similar level of accuracy.",
"Finally, the fourth experiment was performed on the ibmq_hanoi machine with 6 qubits, with the Compound Transformer, using 4 patches per image, 4 features per patch, and one orthogonal layer using the $X$ layout.",
"The hardware results were quite noisy with the X layer, therefore the same experiments were performed with a further-reduced orthogonal layer named the “\\Circuit\": half of a X Circuit (Fig.REF ) where only one diagonal of $RBS$ gates is kept, and which reduced the noise in the outcomes.",
"The experiment used 2 different quantum circuits of 18 $RBS$ gates and 3 $CNOT$ s per circuit per image.",
"Note that with the restriction to states with a fixed hamming weight, strong error mitigation techniques become available.",
"Indeed, as we expect to obtain only quantum superpositions of unary states or states with hamming weight 2 in the case of Compound Transformers, at every layer, every measurement can be processed to discard the ones that have a different hamming weight i.e.",
"states with more than one (or two) qubit in state $\\mathinner {|{1}\\rangle }$ .",
"This error mitigation procedure can be applied efficiently to the results of a hardware demonstration, and has been used in the results presented in this paper.",
"The conclusion from the hardware experiments is that all quantum proposals achieve state-of-the-art test accuracy, comparable to classical networks.",
"In particular, the quantum Compound methods on the simulator are notably more efficient than the classical networks, and they have no efficient classical equivalent.",
"However, the current hardware is often too noisy to achieve similar performance, even with a low number of qubits."
],
[
"Discussion",
"In this work, quantum approaches for training and forward inference of a vision transformer are presented, with each method tested on a set of image classification tasks.",
"These quantum networks reproduce, more or less closely, the steps that occur in the attention layer of the classical transformers but are quite different from each other.",
"Orthogonal Patchwise is the simplest approach and requires fewer qubits and lower circuit depth.",
"Orthogonal Transformer is the most similar to classical transformers.",
"Compound Transformer steps away from the classical architecture with a quantum native procedure, a linear algebraic operation which that cannot be efficiently done classically: multiplying a vector with a higher-dimensional compound matrix (see definition in Section REF ).",
"Inside all these quantum transformers are the Quantum Orthogonal Layers.",
"These trainable circuits can efficiently apply matrix multiplication on vectors encoded on specific quantum basis states.",
"We provide different types of circuits and analyse their advantage in terms of expressivity and complexity.",
"All circuits implement orthogonal matrix multiplication and can be trained using backpropagation detailed in [19].",
"Some circuits have fewer parameters and therefore represent a smaller expressivity but a faster running time.",
"In particular, a quantum computer with all-to-all qubit connectivity can implement the butterfly circuit, which has a logarithmic depth with respect to the equivalent matrix multiplication size.",
"As shown in Table REF , our quantum circuits show definite advantage in terms of computation complexity of the attention layer.",
"In particular, the Compound Transformer runs quadratically faster than the classical equivalent operation with respect to the dimension and the number of patches, without including preprocessing time.",
"The Compound Transformer also allows for the use of a single circuit instead of many repetitions, but might be more suited for larger quantum computers due its larger depth for data loading.",
"In addition to theoretical analysis, we performed extensive numerical simulation and quantum hardware experiments to test the quantum circuits on different datasets, and compared them with each other as well as to their classical baselines.",
"Our results show that our quantum circuits can classify the small images correctly, sometimes as well as or better than the state-of-the-art classical methods (see Table REF ).",
"Our quantum methods have the potential to address over-fitting issues by using a small number of parameters.",
"While the running time of the fully connected and attention layer has been theoretically proven to be advantageous, this is hard to observe in the current quantum computers due to the limited size, high level of noise, and latency of cloud access.",
"From our hardware experiments, it can be observed that results from the current hardware become too noisy as soon as the number of qubits or the size of the quantum circuit increased (see Table REF ).",
"Hence the exact number of parameters used to run the quantum circuits is expected to change with the availability of larger quantum computers, which will allow for larger quantum operations and for us to forgo classical preprocessing to downsize the input images.",
"The MedMNIST simulation results show that, in order to provide competitive results, a quantum transformer would need to utilise parametrised quantum circuits with 16 or 32 qubits and between 60 and 600 $RBS$ gates.",
"While the state of the hardware is yet insufficient, our simulation results show a promising level of performance that can be expected once the hardware technology matures.",
"Overall, our results are encouraging and show the benefit of using quantum circuits that can be trainable but also perform precise linear algebra operations.",
"This approach allows for much better control over the size of the Hilbert space that is explored by the model and provides models that are both expressive and trainable."
],
[
"Acknowledgements",
"The following members of the Roche pRED Quantum Computing Task Force also contributed to this work: Marielle van de Pol, Timothy Stitt, Betty Tai Lorentzen, Agnes Meyder, Detlef Wolf, Stanislaw Adaszewski, Clemens Wrzodek.",
"We also would like to thank all members of QC Ware who have contributed in the development of the quantum software that was used in this work.",
"We acknowledge the use of IBM Quantum services for this work.",
"The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team."
]
] | 2209.08167 |
[
[
"Optimal Scaling for Locally Balanced Proposals in Discrete Spaces"
],
[
"Abstract Optimal scaling has been well studied for Metropolis-Hastings (M-H) algorithms in continuous spaces, but a similar understanding has been lacking in discrete spaces.",
"Recently, a family of locally balanced proposals (LBP) for discrete spaces has been proved to be asymptotically optimal, but the question of optimal scaling has remained open.",
"In this paper, we establish, for the first time, that the efficiency of M-H in discrete spaces can also be characterized by an asymptotic acceptance rate that is independent of the target distribution.",
"Moreover, we verify, both theoretically and empirically, that the optimal acceptance rates for LBP and random walk Metropolis (RWM) are $0.574$ and $0.234$ respectively.",
"These results also help establish that LBP is asymptotically $O(N^\\frac{2}{3})$ more efficient than RWM with respect to model dimension $N$.",
"Knowledge of the optimal acceptance rate allows one to automatically tune the neighborhood size of a proposal distribution in a discrete space, directly analogous to step-size control in continuous spaces.",
"We demonstrate empirically that such adaptive M-H sampling can robustly improve sampling in a variety of target distributions in discrete spaces, including training deep energy based models."
],
[
"Introduction",
"The Markov Chain Monte Carlo (MCMC) algorithm is one of the most widely used methods for sampling from intractable distributions [1].",
"An important class of MCMC algorithms is Metropolis-Hastings (M-H) [2], [3], where new states are generated from a proposal distribution followed by a M-H test.",
"The efficiency for M-H algorithms depends critically on the proposal distribution.",
"For example, gradient based methods, such as the Metropolis Adjusted Langevin Algorithm (MALA) [4], Hamiltonian Monte Carlo (HMC) [5], and their variants [6], [7] substantially improve the performance of M-H algorithms in theory and in practice, compared to naive Random Walk Metropolis (RWM), by leveraging gradient information to guide the proposal distribution [8].",
"Despite many advances, progress in gradient based methods has generally focused on continuous spaces.",
"However, [9] recently proposed a general framework of locally balanced proposals (LBP) for discrete spaces, where a proposal distribution is designed to utilize probability changes between states.",
"Subsequently, [10] accelerated the sampler by using gradient information to approximate the probability change.",
"In empirical evaluations, similar to gradient based samplers in continuous spaces, LBP significantly outperforms RWM and other samplers in discrete spaces.",
"However, both [9] and [10] constrain the proposal distribution to lie within a 1-Hamming ball; i.e., only one site of the state variable is allowed to change per M-H step.",
"Such a restricted update reduces the efficiency of the sampler.",
"[11] noticed this problem and modified the proposal distribution to allow multiple sites to be changed per M-H step.",
"Although such larger updates significantly improve efficiency, [11] do not show how to determine the update size, leaving the number of sites updated in an M-H step as a hyperparameter to tune.",
"In continuous spaces, blackthe scale of the proposal distribution is known to be a critical hyperparameter for obtaining an efficient M-H sampler.",
"For example, consider a Gaussian proposal $\\mathcal {N}(x, \\sigma ^2)$ for modifying a current state $x$ blackwith scale $\\sigma $ .",
"If $\\sigma $ is too small, the Markov chain will converge slowly since its increments will be small.",
"Conversely, if $\\sigma $ is too large, the M-H test will reject too high a proportion of proposed updates.",
"A significant literature has studied optimal scaling for gradient based methods in continuous spaces [12], [13], [8], [14], showing that the optimal scaling can be adaptively tuned w.r.t.",
"the acceptance rate, independent of the target distribution.",
"Such results suggest a direction for solving the optimal scaling problem for LBP.",
"However, the underlying techniques for approximating a diffusion process cannot be directly applied to LBP given its discrete nature.",
"In this work, we consider an asymptotic analysis as the dimension of the discrete model, $N$ , converges to infinity.",
"Starting with a product distribution, we prove that the asymptotic efficiency of LBP in discrete spaces is $2R\\Phi (-\\frac{1}{2}\\lambda _1 R^\\frac{3}{2}/N)$ with an asymptotic acceptance rate of $2\\Phi (-\\frac{1}{2}\\lambda _1 R^\\frac{3}{2}/N)$ , blackwhere the scale $R$ represents the number of sites to update per M-H step.",
"Therefore, the asymptotically optimal scale of the proposal distribution is $R = O(N^\\frac{2}{3})$ with an asymptotically optimal acceptance rate of $0.574$ , independent of the target distribution.",
"Moreover, for RWM in a discrete space, we show that the asymptotic efficiency and acceptance rate are $2R\\Phi (-\\frac{1}{2}\\lambda _2 R^\\frac{1}{2})$ and $2\\Phi (-\\frac{1}{2}\\lambda _2 R^\\frac{1}{2})$ , respectively.",
"Hence, the asymptotically optimal scale is $O(1)$ and the asymptotically optimal acceptance rate is $0.234$ for RWM.",
"By comparing LBP and RWM at their respective optimal scales, it can be determined that LBP is $O(N^\\frac{2}{3})$ more efficient than RWM.",
"These asymptotically optimal acceptance rates are robust in the following respects.",
"First, although the initial derivation is established w.r.t.",
"product distributions, the result can be expanded to more general distributions.",
"Second, the efficiency is not sensitive around the optimal acceptance rate.",
"For example, whereas $0.574$ is the optimal acceptance rate for LBP, the algorithm retains high efficiency for acceptance rates between $0.5$ and $0.7$ .",
"Based on these observations, we propose an adaptive LBP (ALBP) algorithm that automatically tunes the update scale to suit the target distribution.",
"We validate these theoretical findings in a series of empirical simulations on the Bernoulli model, the Ising model, factorized hidden Markov models (FHMM) and restricted Boltzmann machines (RBM).",
"The experimental outcomes comport with the theory.",
"Moreover, we demonstrate that ALBP can automatically find near optimal scales for these distributions.",
"We also use ALBP to train deep energy based models (EBMs), finding that it reduces the MCMC steps needed in contrastive divergence training [15], [16], significantly improving the efficiency of the overall training procedure."
],
[
"Background",
"Metropolis-Hastings Algorithm Let $\\pi $ denote the target distribution.",
"Given a current state $x^{(n)}$ , a M-H sampler draws a candidate state $y$ from a proposal distribution $q(x^{(n)}, y)$ .",
"Then, with probability $\\min \\Big \\lbrace 1,\\, \\frac{\\pi (y)q(y, x^{(n)})}{\\pi (x^{(n)})q(x^{(n)}, y)}\\Big \\rbrace $ the proposed state is accepted and $x^{(n+1)} = y$ ; otherwise, $x^{(n+1)} = x^{(n)}$ .",
"In this way, the detailed balance condition is satisfied and the M-H sampler generates a Markov chain $x_0, x_1, ...$ that has $\\pi $ as its stationary distribution.",
"Locally Balanced Proposal.",
"The locally balanced proposal (LBP) is a special case of the pointwise informed proposal (PIP), which is a class of M-H algorithms for discrete spaces [9] using the proposal distribution $Q_g(x,y) \\propto g\\left(\\pi (y) / \\pi (x) \\right)$ such that $g$ is a scalar weight function.",
"[9] shows that the family of locally balancing functions $\\mathcal {G} = \\lbrace g: \\mathbb {R}_+ \\rightarrow \\mathbb {R}_+, g(t) = t g(\\frac{1}{t}), \\forall t > 0\\rbrace $ (e.g.",
"$g(t) = \\sqrt{t}$ or $\\frac{t}{t+1}$ ) is asymptotically optimal for PIP.",
"Hence, PIP with a locally balanced function for its weight function is referred to as LBP.",
"Despite having good proposal quality, PIP requires the weight $g(\\pi (z) / \\pi (x))$ to be calculated for all candidate states $z$ in the neighborhood of $x$ , which results in its high computational cost.",
"[10] propose to estimate the probability change by leveraging the gradient, improving the scalability of LBP.",
"Locally Balanced Proposal with Auxiliary Path.",
"[11] generalize LBP by introducing an auxiliary path sampler, which allows multiple sites to be updated per M-H step.",
"In particular, [11] sequentially selects the update indices without replacement, and uses these indices as auxiliary variables to keep the proposal distribution tractable while preserving the detailed balance condition.",
"Although this can achieve significant improvements in empirical performance, [11] manually tune the update size per M-H step, and leave the optimal scale problem open."
],
[
"Problem Statement",
"We establish asymptotic limit theorems for two M-H algorithms in discrete spaces: the locally balanced proposal (LBP) and random walk Metropolis (RWM).",
"Following previous work [12], [13], [14], [17], we conduct our analysis on a product probability measure $\\pi $ .",
"In particular, for a state space $\\mathcal {X} = \\lbrace 0, 1\\rbrace ^N$ , we consider a factored target distribution $\\pi ^{(N)}(x) = \\prod _{i=1}^N \\pi _i(x_i) = \\prod _{i=1}^N p_i^{x_i}(1-p_i)^{1-x_i}$ where each site is assumed to have a sufficiently large probability for being both 0 and 1; that is, for a fixed $ \\epsilon \\in (0, \\frac{1}{4})$ , we assume the target distribution belongs to: $\\mathcal {P}_\\epsilon := \\lbrace \\pi ^{(N)}: \\epsilon < p_j \\wedge (1 - p_j) < \\frac{1}{2} - \\epsilon , \\forall j = 1, ..., N, N \\ge 1\\rbrace $ blackwhere we denote $a \\wedge b = \\min \\lbrace a, b\\rbrace $ .",
"To measure the efficiency of the sampler, an ergodic estimate varies with the objective function considered.",
"Alternatively, we use a natural progress estimate: the expected jump distance (EJD).",
"Denote $P_\\theta $ as the transition kernel, $d(x, y)$ as the Hamming distance between $x$ and $y$ .",
"For a M-H sampler parameterized by $\\theta $ , its expected jump distance $\\rho (\\theta )$ and corresponding expected acceptance rate $a(\\theta )$ are $\\rho (\\theta ) = \\sum _{X, Y \\in \\mathcal {X}} \\pi (X) P_\\theta (X, Y) d(X, Y), \\quad a(\\theta ) = \\sum _{X, Y \\in \\mathcal {X}} \\pi (X) P_\\theta (X, Y) 1_{\\lbrace X\\ne Y\\rbrace }$ black In continuous space, the limit of sampling process is a diffusion process, whose efficiency is determined by the expected squared jump distance (ESJD) [8].",
"In discrete space, the limit of the sampling process is a jump process and EJD is the correct metric (see Appendix REF )."
],
[
"Locally Balanced Proposal",
"We consider the M-H sampler LBP-$R$ , where $R$ refers to flipping $R$ indices in each M-H step.",
"Given a current state $x$ , LBP-R calculates the weight $w_j$ for flipping index $j$ as in blackPIP.",
"Since we are considering a binary target distribution of the form (REF ), we have $w_j(x) = w_j(x_j) = g(\\pi _j(1-x_j) / \\pi _j(x_j)) $ where $g$ is a locally balanced function.",
"Following [11], LBP-R select indices $u_r$ with probability $\\mathbb {P}(u_r = j) \\propto w_j$ sequentially for $r = 1, ..., R$ , without replacement.",
"The new state $y$ is obtained by flipping indices $u_{1:R}$ of $x$ .",
"If we consider $u$ as an auxiliary variable, the accept rate $A(x, y, u)$ in the M-H acceptance test can be written as $A(x, y, u) = 1 \\wedge \\frac{\\pi (y)\\prod _{r=1}^R \\frac{w_{u_r}(y)}{W(y, u) + \\sum _{i=1}^r w_{u_i}(y)}}{\\pi (x)\\prod _{r=1}^R \\frac{w_{u_r}(x)}{W(x, u) + \\sum _{i=r}^R w_{u_i}(x)}}, \\quad \\text{ where } W(x, u) = \\sum _{i=1}^N w_i - \\sum _{r=1}^R w_{u_r} $ From theorem 1 in [11], the auxiliary sampler LBP-R satisfies detailed balance.",
"A M-H step of LBP-R is summarized in Algorithm REF .",
"1.5em [H] A M-H step of LBP-R and blueALBP Given current state $x^{(n)}$ , bluecurrent $R_t$ , initialize candidate set $\\mathcal {C} = \\lbrace 1, .., N\\rbrace $ $r = 1, ..., R$ black or $r = 1, ..., \\text{rounding}(R_t)$ Sample $u_r$ with $\\mathbb {P}(u_r=j) \\propto w_j(x^{(n)}) 1_{\\lbrace j \\in \\mathcal {C}\\rbrace }$ Pop $u_r$ out of the candidate set: $\\mathcal {C} \\leftarrow \\mathcal {C} \\backslash \\lbrace u_r\\rbrace $ Obtain $y$ by flipping indices $u_1, ..., u_R$ of $x^{(n)}$ .",
"rand(0,1) $< A(x^{(n)}, y, u)$$x^{(n+1)} = y$ else $x^{(n+1)} = x^{(n)}$ blue$t < T_\\text{warmup}$$R_{t+1} \\leftarrow R_t + (A(x^{(n)}, y, u) - 0.574)$ 1.5em"
],
[
"Optimal Scaling for Locally Balanced Proposal",
"We are now ready to state the first asymptotic theorem.",
"Theorem 3.1 For arbitrary sequence of target distributions $\\lbrace \\pi ^{(N)}\\rbrace _{N=1}^\\infty \\subset \\mathcal {P}_\\epsilon $ , the M-H sampler LBP-R with a locally balanced weight function $g$ obtains the following, if $R = {black}{\\lfloor } l N^\\frac{2}{3}{black}{\\rfloor }$ , $\\lim _{N\\rightarrow \\infty } a(R) - 2\\Phi \\left(-\\frac{1}{2}\\lambda _1 l^\\frac{3}{2}\\right) = 0 $ where $\\Phi $ is the c.d.f.",
"of standard normal distribution and $\\lambda _1$ only depends on $\\pi ^{(N)}$ $\\lambda _1^2 = \\lambda _1^2(\\pi ^{(N)}) = \\frac{\\sum _{j=1}^N p_j w_j(1) (w_j(0) - w_j(1))^2}{4( \\mathbb {E}_{x}[\\frac{1}{N}\\sum _{i=1}^N w_i(x_i)])^2 \\sum _{i=1}^N p_i w_i(1)}$ The definition of $\\lambda _1$ in (REF ) explains the motivation of restricting the target distributions in (REF ).",
"In fact, introducing the $\\epsilon $ gives upper and lower bounds of $\\lambda _1$ .",
"When all $p_j$ are arbitrarily close to $\\frac{1}{2}$ , $(w_j(0) - w_j(1))^2$ in numerator will be zero, so is $\\lambda _1$ .",
"As a result, the acceptance rate will always be 1.",
"Else, when all $p_j$ are arbitrarily close to 0 or 1, $\\mathbb {E}_{x}[\\frac{1}{N}\\sum _{i=1}^N w_i(x_i)]$ in denominator will be zero, and $\\lambda _1$ will be infinity.",
"As a result, the acceptance rate will always be 0.",
"So, we have to make the mild assumption in (REF ) to assure the following asymptotic result holds.",
"blackA more detailed discussion about $\\epsilon $ is given in Appendix REF .",
"Corollary 3.2 The optimal choice of scale for $R = l N^\\frac{2}{3}$ is obtained when the expected acceptance rate is $0.574$ , independent of the target distribution.",
"When $R = l N^\\frac{2}{3}$ , denote $z = \\lambda _1^\\frac{2}{3} l$ , we have: $\\rho (R) = a(R) R = {black}{ 2lN^\\frac{2}{3}\\left(\\Phi \\left(-\\frac{1}{2}\\lambda _1 l^\\frac{3}{2}\\right) + o(1)\\right)= \\Big (\\frac{N}{\\lambda _1}\\Big )^{\\frac{2}{3}} 2 z \\Phi \\Big (-\\frac{1}{2}z^\\frac{3}{2}\\Big ) + o\\left(N^\\frac{2}{3}\\right)}$ It means the optimal value of $z$ is independent of the target distribution $\\pi ^{(N)}$ .",
"As $\\Phi $ is known, we can numerically solve $z=1.081$ , and the corresponding expected acceptance rate is $a = 0.574$ ."
],
[
"Proof of Theorem ",
"Denote the current state as $x$ and a new state proposed in LBP-R as $y$ .",
"Consider the acceptance rate $A(x, y, u)$ in (REF ).",
"Using the fact that, if index $j$ is not flipped then $w_j(y) = w_j(x)$ , we have: $\\frac{\\pi (y)}{\\pi (x)} \\frac{\\prod _{r=1}^R w_{u_r}(y)}{\\prod _{r=1}^R w_{u_r}(x)}= \\frac{\\pi (y)}{\\pi (x)} \\frac{\\prod _{i=1}^N w_i(y)}{\\prod _{i=1}^N w_i(x)}= \\prod _{i=1}^N \\frac{ {\\pi _i(y_i)} / {\\pi _i(x_i)} g(\\pi _i(x_i) / \\pi _i(y_i))}{g(\\pi _i(y_i) / \\pi _i(x_i))} = 1 $ where (REF ) takes advantage of the property of a locally balanced function.",
"Hence, the acceptance rate $A(x, y, u)$ can be simplified to: $1 \\wedge \\exp \\left(\\sum _{r=1}^R \\log \\Big (\\frac{1 + \\sum _{i=r}^R w_{u_i}(x) / W(x, u)}{1 + \\sum _{i=1}^r w_{u_i}(y) / W(y, u)}\\Big ) \\right) $ From the definition in (REF ), we have $W(x, u) = W(y, u)$ .",
"Denote $i\\wedge j = \\min \\lbrace i, j\\rbrace $ and $i \\vee j = \\max \\lbrace i, j\\rbrace $ , we have the following approximation: Lemma 3.3 Define $W= \\mathbb {E}_{x,u}[W(x,u)]$ .",
"blackWe have: $\\lim _{N\\rightarrow 0} \\sum _{r=1}^R \\log (\\frac{1 + \\sum _{i=r}^R w_{u_i}(x) / W(x, u)}{1 + \\sum _{i=1}^r w_{u_i}(y) / W(y, u)}) - (A + B) = 0$ , where $A = & \\frac{1}{W}\\sum _{r=1}^R (R-r+1)w_{u_i}(x_{u_i}) - r w_{u_i}(y_{u_i}) \\\\B = & - \\frac{1}{2} \\frac{1}{W^2} \\sum _{i, j=1}^R \\Big [ i\\wedge j\\ w_{u_i}(x_{u_i})w_{u_j}(x_{u_j}) - (R - i \\vee j + 1) w_{u_i}(y_{u_i})w_{u_j}(y_{u_j})\\Big ] $ To analyze $A$ and $B$ , we reverse the order of $x$ and $u$ .",
"In particular, instead of first sampling $x \\sim \\pi (x)$ , then sampling $u \\sim p(x|u)$ , we use a reversed order where we first blackdetermine the indices $u$ , then the values of $x_u$ , and finally the values of $x_{-u}$ .",
"Lemma 3.4 The joint distribution $p(x, u) = \\pi (x) p(u|x)$ can be decomposed in the following form: $p(x, u) = \\prod _{r=1}^R p(u_r|u_{1:r-1}) \\ \\prod _{r=1}^R p(x_{u_r}|u, x_{u_{1:r-1}}) \\ p(x_{-u}|u, x_u)$ Denote $j \\notin u_{1:r-1}$ represents $j \\ne u_i$ for $i = 1, ..., r-1$ , the conditional probabilities are $p(u_r = j|u_{1:r-1}) = \\frac{p_j w_j(1) 1_{\\lbrace j \\notin u_{1:r-1}\\rbrace }}{\\sum _{i=1}^N p_i w_i(1) 1_{\\lbrace i \\notin u_{1:r-1}\\rbrace }} + O(N^{-\\frac{5}{2}}) \\\\p(x_j=1|u, x_{1:j-1}, u_r = j) = \\frac{1}{2} + r \\frac{w_j(0) - w_j(1)}{W} + O(N^{-\\frac{2}{3}}) $ With the conditional distribution in Lemma REF , we are able to give a concentration property of the term $B$ and show it is safe to ignore: Lemma 3.5 With a probability larger than $1 - O(\\exp (-N^\\frac{1}{2}))$ , $B = O\\big (N^{-\\frac{1}{12}}\\big )$ .",
"For term $A$ , we use martingale central limit theorem with convergence rate [18] to bound the Kolmogorov-Smirnov statistic.",
"Lemma 3.6 When $R = l N^\\frac{2}{3}$ , $\\lambda _1$ defined as (REF ), we have: $|\\mathbb {P}(\\frac{A - \\mu }{\\sigma } \\ge t) - \\Phi (t)| = O\\big ( N^{-\\frac{1}{12}}\\big ), \\quad \\mu = - \\frac{1}{2} \\lambda _1^2 l^3, \\quad \\sigma ^2 = \\lambda _1^2 l^3$ By (REF ), the expectation w.r.t.",
"$A$ asymptotically equals to the expectation on $\\mathcal {N}(\\mu , \\sigma ^2)$ .",
"The final step to prove Theorem REF is to exploit a property of the normal distribution.",
"Lemma 3.7 If $Z \\sim \\mathcal {N}(\\mu , \\sigma ^2)$ , then we have: $\\mathbb {E}[1\\wedge \\exp (Z)] = \\Phi \\Big (\\frac{\\mu }{\\sigma }\\Big ) + \\exp \\Big (\\mu + \\frac{\\sigma ^2}{2}\\Big ) \\Phi \\Big (-\\sigma - \\frac{\\mu }{\\sigma }\\Big )$ where $\\Phi $ is the c.d.f.",
"of the standard normal distribution.",
"By Lemma REF , REF , we have the expectation of (REF ), which is the expected accept rate, equals to: $\\mathbb {E}[a(R)] &= \\Phi \\Big (-\\frac{1}{2}\\lambda _1 l^\\frac{3}{2}\\Big ) + \\exp (0)\\Phi \\Big (-\\frac{1}{2}\\lambda _1 l^\\frac{3}{2}\\Big ) = 2\\Phi \\Big (-\\frac{1}{2}\\lambda _1 l^\\frac{3}{2}\\Big )$"
],
[
"Optimal Scaling for Random Walk Metropolis",
"We denote the Random Walk Metropolis in discrete space as RWM-$R$ , where $R$ refers to flipping $R$ indices in each M-H step.",
"Under the Bernoulli distribution, a site is more likely to stay at high probability position, so if we randomly flip a site, it is more likely to decrease its probability.",
"That is, intuitively, the acceptance rate will decrease exponentially as the scale $R$ increases.",
"Consequently, the optimal scale for RWM-$R$ should be $O(1)$ .",
"Though this is not a rigorous proof, the constant scaling indicates that it will be hard to directly prove an asymptotic theorem for RWM-$R$ .",
"To address this difficulty, we first restrict our target distribution to a smaller class of Bernoulli distributions $\\mathcal {P}^{(\\beta )}_\\epsilon \\subset \\mathcal {P}_\\epsilon $ , which is formally defined as follows.",
"For a fixed $\\epsilon \\in (0, \\frac{1}{4})$ and a fixed $\\beta > 0$ , define black $\\mathcal {P}^{(\\beta )}_\\epsilon := \\left\\lbrace \\pi ^{(N)}: \\frac{1}{2} - \\frac{1}{2N^\\beta } + \\frac{\\epsilon }{N^\\beta } < p_j \\wedge (1 - p_j) < \\frac{1}{2} - \\frac{\\epsilon }{N^\\beta } \\right\\rbrace $ When $N$ is large, each $p_j$ will be very close to $\\frac{1}{2}$ .",
"In this way, the acceptance rate will not drop too fast when $R$ is increased, and a non-constant $R$ will be permitted.",
"This enables us to prove: Theorem 3.8 For arbitrary sequence of target distributions $\\lbrace \\pi ^{(N)}\\rbrace _{N=1}^\\infty \\subset \\mathcal {P}_\\epsilon ^{(\\beta )}$ , the M-H sampler RWM-R obtains the following, if $R = l N^{2\\beta }$ , $\\lim _{N\\rightarrow \\infty } a(R) - 2\\Phi \\left(-\\frac{1}{2}\\lambda _2 l^\\frac{1}{2}\\right) $ where $\\Phi $ is the c.d.f.",
"of the standard normal distribution and $\\lambda _2$ only depends on $\\pi ^{(N)}$ .",
"$\\lambda _2^2 = \\lambda _2^2(\\pi ^{(N)}) = \\frac{2}{N}\\sum _{i=1}^N N^{2\\beta }(2p_i-1)\\log \\frac{p_i}{1 - p_i}$ Corollary 3.9 The optimal scale $R = l N^{2\\beta }$ is obtained when the expected acceptance rate is 0.234, independent of the target distribution.",
"The rate in Corollary REF is proved for arbitrary $\\beta >0$ .",
"If we let $\\beta $ decrease to 0, at $\\beta = 0$ the optimal scale for RWM-$R$ is $O(1)$ while the optimal acceptance rate is $0.234$ .",
"black Also, we can notice that $\\mathcal {P}^{(\\beta )}_\\epsilon $ converges to $\\mathcal {P}_\\epsilon $ when $\\beta $ decrease to 0 and we are able to show the optimal scale of RWM in $\\mathcal {P}_\\epsilon $ is $O(1)$ , see details in Appendix REF .",
"However, this limit is not mathematically rigorous, because Theorem REF and Corollary REF only hold asymptotically, such that a smaller $\\beta $ requires a larger $N$ .",
"Hence, when $\\beta $ decreases to 0, $N$ must approach infinity to satisfy the asymptotic theorem.",
"Although there is this minor gap in the analysis, the conclusion nevertheless aligns very well with different target distributions in the experiment section."
],
[
"Adaptive Algorithm",
"Given knowledge of the optimal acceptance rate, one can design an adaptive algorithm that automatically tunes the scale of the M-H samplers.",
"For this purpose, we use stochastic optimization [19], [20] to adjust the scaling parameter $R_t$ to ensure that the statistic $A_t = a_t - \\delta $ approaches 0, where $a_t$ is the acceptance probability for iteration $t$ and $\\delta $ is the target acceptance rate ($0.574$ for LBP and $0.234$ for RWM).",
"According to Theorem REF and Theorem REF , the acceptance rate is a decreasing function of the scaling $R_t$ .",
"Hence, we use the update rule: $R_{t+1} \\leftarrow R_t + \\eta _t A_t $ with step size $\\eta _t = 1$ .",
"We follow common practice and adapt the tunable MCMC parameters during a warmup phase before freezing them thereafter [21].",
"The computational cost for (REF ) is ignorable comparing the total cost of a M-H step.",
"The algorithm boxes for ALBP and ARWM are given in Appendix .",
"More advanced implementations are possible, but it is out of the focus in the paper.",
"We observe below that this simple approach is able to maintain the sampler robustly near the optimal acceptance rate."
],
[
"Related Work",
"Informed proposals for Metropolis-Hastings (M-H) algorithms have been extensively studied for continuous spaces [1].",
"The most famous algorithms are the Metropolis-adjusted Langevin algorithm (MALA) [4] and Hamiltonian Monte Carlo (HMC) [5].",
"MALA, HMC, and their variants [6], [7], [22], [23], [24], [25], [26], [27] use the gradient of the target distribution to guide the proposal distribution toward high probability regions, which brings substantial improvements in sampling efficiency compared to uninformed methods, such as random walk Metropolis (RWM) [2].",
"Informed proposals have also demonstrated recent success in discrete spaces.",
"[9] first gives a formal definition of the pointwise informed proposal (PIP) for discrete spaces, then proves that locally balanced proposals (LBP), using a family of locally balanced functions as the weight function in PIP, are asymptotically optimal for PIP.",
"Following this work, [28] extended the framework to Markov jump processes and introduced non-reversible heuristics to accelerate sampling.",
"[29] parameterize the locally balanced function and tune it by minimizing a mutual information.",
"[10] give a more scalable version of LBP for differentiable target distributions by estimating the probability change through the gradient.",
"Despite strong empirical results, the LBP method of [9] only flips one bit per M-H step, since PIP has to restrict the proposal distribution to a small neighborhood, e.g.",
"a 1-Hamming ball, due to its computational cost.",
"[11] generalize LBP to flip multiple bits in a single M-H step, gaining significant improvement in sampling efficiency.",
"However, the scaling of the proposal distribution in [11] was manually tuned and the optimal scaling problem was left open.",
"For continuous spaces, the optimal scaling problem for informed proposals has been well studied.",
"A significant literature has already shown that the scale can be tuned with respect to the optimal acceptance rate [8], e.g.",
"0.234 for RWM [12], 0.574 for MALA [13], 0.651 for HMC [14], and 0.574 for Barker [17], by decreasing the scale so that the Markov chain converges to a diffusion process.",
"However, such a technique is not directly applicable to LBP given its discrete nature.",
"[30] make an initial attempt on discrete space, however it assumes all dimensions satisfy independent, identical Bernoulli distribution.",
"In this work, we have established for the first time the optimal scale for LBP and RWM in discrete spaces."
],
[
"Experiments",
"The effectiveness of LBP has been extensively demonstrated in previous work, e.g.",
"[9], [10], [11], in comparison to other M-H samplers for discrete spaces, such as RWM, Gibbs sampling, the Hamming Ball sampler [31], and continuous relaxation based methods [32], [33], [34], [35].",
"Therefore, we focus on simulating LBP-$R$ , with weight function $g(t) = \\frac{t}{t+1}$ , and RWM-$R$ to validate our theoretical findings.",
"More experiments, including different weight functions and comparison between \"with\" and \"without\" replacement versions of LBP are given in Appendix .",
"Throughout the experiment section, we will use the gradient approximation [10].",
"That is to say, we estimate the change in probability of flipping index $i$ is estimated by: $\\tilde{d} x_i = \\exp ((1 - 2 x_i) (\\nabla \\log \\pi (x))_i)$ For the Bernoulli distribution, this is still exact and does not hinder the justification of the theoretical results.",
"For more complex models, this approximation makes the algorithms significantly more efficient.",
"In particular, the gradient approximation only requires two calls of the probability function and two calls of the gradient function.",
"Consequently, LBP with gradient approximation will take about twice time per update compared to RWM.",
"In our experiments, we observe that LBP and GWG takes $1.2 \\pm 0.2$ and $1.1 \\pm 0.1$ more time per update, respectively, than RWM, across all target distributions.",
"We therefore omit reporting the detailed run time for each method."
],
[
"Sampling from different target distributions",
"We consider four target distributions: the Bernoulli distribution, the Ising model, the factorial hidden Markov model (FHMM), and the restricted Boltzmann machine (RBM).",
"For each model, we consider three configurations: C1, C2, and C3 for smooth, moderate, and sharp target distributions.",
"To obtain performance curves, we first simulate LBP-1 and RWM-1 for an initial acceptance rate $a_{\\max }$ .",
"Then, we adopt $a_{\\max } - 0.02$ , ..., $a_{\\max } - 0.02k$ , ... as a target acceptance rate.",
"For each rate, we use the adaptive sampler to obtain an estimated scale $R$ , with which we simulate 100 chains and calculate the final real acceptance rate and efficiency.",
"In this way, we collect abundant data points to characterize the relationship between acceptance rate and efficiency to facilitate the following analyses.",
"Figure: Efficiency Curves on IsingBernoulli Distribution.",
"We validate our theoretical results on Bernoulli distribution.",
"The probability mass function is given in (REF ).",
"For each configuration, we simulate on domains with three dimensionalities: $N=100$ , 800, 6400.",
"The scatter plot for $N=800$ is reported in Figure REF .",
"We also estimate $\\lambda $ in (REF ) and (REF ) and plot the theoretical efficiency curve in (REF ) and (REF ).",
"From Figure REF , we can see that the simulation results align well with the theoretically predicted curves, and the optimal efficiencies were achieved at $0.574$ for LBP and $0.234$ for RWM for all configurations.",
"Ising Model.",
"The Ising model is a classical model in physics defined on a $p\\times p$ square lattice graph $(V_p, E_p)$ (details in Appendix REF ).",
"For each configuration, we simulate on three sizes $p = 20$ , 50, 100.",
"We report the results for $p=50$ in Figure REF .",
"For LBP, the optimal efficiencies are achieved at around $0.5$ , which is slightly less than $0.574$ , although these values are close.",
"Thus we can say that the asymptotically optimal acceptance rate for LBP still applies to the Ising model.",
"For RWM, $0.234$ perfectly matches the acceptance rate where the optimal efficiencies are obtained.",
"Factorial Hidden Markov Model The FHMM uses latent variables $x \\in \\mathcal {X} = \\lbrace 0, 1\\rbrace ^{L\\times K}$ to characterize time series data $y \\in \\mathbb {R}^L$ (details in Appendix REF ).",
"Given $y$ , we sample the hidden variables $x$ from the posterior $\\pi (x) = p(x|y)$ .",
"For each configuration, we simulate in three sizes $L=200$ , 1000, 4000.",
"We report the results for $L=1000$ in Figure REF .",
"One can observe that these results match the theoretical predictions very well.",
"Figure: Efficiency Curves on RBMRestricted Boltzmann Machine.",
"A RBM is a bipartite latent-variable model that defines a distribution over binary data $x \\in \\lbrace 0, 1\\rbrace ^N$ and latent data $z \\in \\lbrace 0, 1\\rbrace ^h$ (details in Appendix REF ).",
"We train an RBM on the MNIST dataset using contrastive divergence [15] and sample observable variables $x$ .",
"We report the results in Figure REF .",
"For LBP, although RBM is much more complex than a product distribution, its efficiency versus acceptance rate curves still match the theoretical predictions very well.",
"For RWM, even using $R=1$ will result in acceptance rates less than $0.234$ for all configurations.",
"Although we cannot check what the optimal value is, we still observe that efficiency is an increasing function of the acceptance rate when the acceptance rate is less than $0.234$ , as predicted by the theory.",
"Optimal Scaling and Efficiency.",
"We examine how optimal scaling $R$ for LBP, RWM and their relative efficiency ratio grow w.r.t.",
"the model dimension $N$ .",
"In figure REF , we can see that both the optimal scaling and efficiency ratio are linear in log-log plot and the slopes are close to $\\frac{2}{3}$ across Bernoulli, Ising, and FHMM.",
"The results matches the theories that the optimal scaling $R=O(N^\\frac{2}{3})$ for LBP, $R=O(1)$ for RWM, and the relative efficiency ratio LBP over RWM is $O(N^\\frac{2}{3})$ .",
"Figure: NO_CAPTION$R$ and Efficiency Ratio Table: Performance of the Samplers on Various Distributions"
],
[
"Adaptive Sampling",
"We have validated the theoretical findings regarding the optimal acceptance rates on various distributions.",
"In this section, we examine the performance of the adaptive sampler.",
"In addition to the expected jump distance (EJD), we also report the effective sample size (ESS) Computed using Tensorflow Probability.",
"We compare the adaptive sampler ALBP, ARWM with their single step version LBP-1, RWM-1, and grid search version GLBP, GRWM, where we tune the scaling $R$ by grid search.",
"We give the sampling results on Bernoulli model, Ising model, FHMM, and RBM with medium size and configuration C2 in table REF .",
"More results are given in Appendix .",
"We can see that the adaptive samplers are significantly more efficient than single step samplers, especially for LBP.",
"Also, the adaptive samplers can robustly achieve almost the same performance comparing to using grid search to find the optimal scaling."
],
[
"Training Deep Energy Based Models",
"Learning an EBM is a challenge task.",
"Given data sampled from a true distribution $\\pi $ , we maximize the likelihood of the target distribution $\\pi _\\theta (x) \\propto e^{-f_\\theta (x)}$ parameterized by $\\theta $ .",
"The gradient estimation requires samples from the current model, which is typically obtained via MCMC.",
"The speed of training an EBM is determined by how fast a MCMC algorithm can obtain a good estimate of the second expectation.",
"We evaluate adaptive samplers by learning deep EBMs.",
"Following the setting in [10], we train deep EBMs parameterized by Residual Networks [36] on small binary image datasets using PCD [16] with a replay buffer [37].",
"We compare two single step samplers and two adaptive samplers, where $\\text{LBP}_b$ uses $g(t) = \\frac{t}{t+1}$ as blackweight function and $\\text{LBP}_s$ uses $g(t) = \\sqrt{t}$ as weight function.",
"When we allow them to run enough iterations in PCD, they are able to train EBMs in same good quality.",
"To measure the efficiency of these samplers, we compare the minimum number of M-H steps needed in PCD in table REF .",
"We can see that adaptive samplers only need one half or even one fifth iterations compare to single step samplers.",
"We also present long-run samples from our trained models via $\\text{ALBP}_s$ in Figure REF .",
"Table: Minimum M-H Steps Needed for PCDFigure: Samples from deep EBMs trained by ALBP s \\text{ALBP}_s sampler."
],
[
"Discussion",
"In this paper, we have addressed the optimal scaling problem for the locally balanced proposal (LBP) in [11].",
"We verified, both theoretically and empirically, that the asymptotically optimal acceptance rate for LBP is $0.574$ , independent of the target distribution.",
"Moreover, knowledge of the optimal acceptance rate allows one to adaptively tune the neighborhood size for a proposal distribution in a discrete space.",
"We verified the theoretical findings on a diverse set of distributions, and demonstrated that adaptive LBP can improve sampling efficiency for learning deep EBMs.",
"We believe there is considerable room for future work that builds on these results.",
"For theoretical investigation, the theory established under a strong assumption that the target distribution is a product distribution, despite the results applies very well to more complicated distributions.",
"We believe the results still hold under a weaker assumption that the target distribution has no phase transition.",
"We also believe it is possible to design a HMC style sampler for discrete spaces in the framework of [11] by using LBP as a block for the auxiliary path.",
"For empirical investigation, many real-world problems involve probability models of discrete structured data, such as syntax trees for natural language processing [38], program synthesis [39], and graphical models for molecules [40].",
"Efficient discrete samplers should be able to accelerate both learning and inference with such models."
],
[
"A concentration of $W(x, u)$",
"Lemma A.1 Define $W = \\mathbb {E}_{x, u} [W(x, u)]$ .",
"We have: $\\mathbb {P}(|W(x, u) - W| > N^{\\frac{1}{2}} t) \\le 2 e^{-C_2 t^2}$ where $C_2$ is an absolute constant that only depends on the scalar $\\epsilon $ in (REF ).",
"Define a martingale $M_n$ , $n = 0, 1, ..., N + R$ .",
"blackWe let $M_0 = 0$ .",
"When $n \\le N$ , it has independent increment $M_n = \\sum _{i=1}^n w_i(x) - \\mathbb {E}[w_i(x)], \\quad n = 1, ..., N$ For $n > N$ , it is defined as $M_{N+r}& = M_{N+r-1} - w_{u_r}(x) + \\mathbb {E}[w_{u_r}(x)|M_1, ..., M_{N+r-1}] \\\\& = M_{N+r-1} - w_{u_r}(x) + \\frac{\\sum _{i\\notin u_{1:r-1}} w_i^2(x)}{\\sum _{i\\notin u_{1:r-1}} w_i(x)}$ where $i \\notin u_{1:r-1}$ means $i \\ne u_j$ for $j = 1, ..., r-1$ .",
"Since $p_i$ are controlled by $\\epsilon $ in (REF ), we can find a uniform bound $\\frac{1}{4C_1} = 2 \\sup _{\\epsilon < p < 1 - \\epsilon } g(\\frac{1- p}{p})$ For $1\\le n \\le N$ , we have $|M_n - M_{n-1}| = {black}{\\left|w_i(x) - \\mathbb {E}[w_i(x)]\\right|} \\le 2 \\max _{x, u} |w_i(x)| \\le \\frac{1}{4C_1}$ For $1 \\le r \\le R$ , we have $|M_{N+r} - M_{N+r-1}| = {black}{\\left|- w_{u_r}(x_{u_r}) + \\frac{\\sum _{i\\ne u_{1:r-1}} w_i^2(x_i)}{\\sum _{i\\ne u_{1:r-1}} w_i(x_i)}\\right|} \\le \\frac{1}{4C_1}$ Hence, we can apply the Azuma-Hoeffding inequality: $\\mathbb {P}(|W(x, u) - W| > {black}{t}N^{\\frac{1}{2}}) = \\mathbb {P}(|M_n - M_0| > t N^{\\frac{1}{2}}) \\le 2 e^{\\frac{-t^2 N}{2\\frac{1}{4C_1}(N+R)}} = 2 e^{-C_1 t^2}.$ Thus we prove the lemma.",
"The lemma indicates with high probability, for arbitrary $\\delta > 0$ $W(x, u) - W = o(N^{\\frac{1}{2} + \\delta })$ One observation of the proof is that, the concentration holds for arbitrary $0 \\le R \\le N$ .",
"For example, when $R = N$ , $W(x, u) \\equiv W \\equiv 0$ , the concentration is still valid."
],
[
"Lemma ",
"Using Taylor's series, we have $\\log (1 + \\sum _{i=r}^R w_{u_i}(x) / W(x, u)) = \\frac{\\sum _{i=r}^R w_{u_i}(x)}{W(x, u)} - \\frac{1}{2} (\\frac{\\sum _{i=r}^R w_{u_i}(x)}{W(x, u)})^2 + O(\\frac{R^3}{N^{3}})$ $\\log (1 + \\sum _{i=1}^r w_{u_i}(y) / W(x, u)) = \\frac{\\sum _{i=1}^r w_{u_i}(y)}{W(x, u)} - \\frac{1}{2} (\\frac{\\sum _{i=1}^r w_{u_i}(y)}{W(x, u)})^2 + O(\\frac{R^3}{N^3})$ Using Lemma REF and the property $W(x, u) = W(y, u)$ , with high probability, the first order term becomes to: $\\sum _{r=1}^R \\frac{\\sum _{i=r}^R w_{u_i}(x)}{W(x, u)} - \\frac{\\sum _{i=1}^r w_{u_i}(y)}{W(x, u)}& = \\sum _{r=1}^R \\frac{(R-r+1)w_{u_i}(x) - r w_{u_i}(y)}{W(x, u)} \\\\& = \\sum _{r=1}^R \\frac{(R-r+1)w_{u_i}(x) - r w_{u_i}(y)}{W} + O(\\frac{R^2}{N^{\\frac{3}{2}-\\delta }})$ Similarly, with high probability, the second order term becomes to: $& \\sum _{r=1}^R (\\frac{\\sum _{i=r}^R w_{u_i}(x)}{W(x, u)})^2 - (\\frac{\\sum _{i=1}^r w_{u_i}(y)}{W(x, u)})^2 \\\\= & \\frac{1}{W(x, u)^2} \\sum _{r=r}^R \\Big ( \\sum _{i, j=r}^R w_{u_i}(x) w_{u_j}(x) - \\sum _{i, j=1}^r w_{u_i}(y) w_{u_j}(y) \\Big ) \\\\= & \\frac{1}{W(x, u)^2} \\sum _{i=1}^R \\sum _{j=1}^R \\min \\lbrace i, j\\rbrace w_{u_i}(x) w_{u_j}(x) - (R - \\max \\lbrace i, j\\rbrace + 1) w_{u_i}(y) w_{u_j}(y) \\\\= & \\frac{1}{W^2} \\sum _{i=1}^R \\sum _{j=1}^R \\min \\lbrace i, j\\rbrace w_{u_i}(x) w_{u_j}(x) - (R - \\max \\lbrace i, j\\rbrace + 1) w_{u_i}(y) w_{u_j}(y) + o(\\frac{R^3}{N^{\\frac{5}{2}-\\delta }})$ Since $R = l N^\\frac{2}{3}$ , denote $i\\wedge j = \\min \\lbrace i, j\\rbrace , i\\vee j = \\max \\lbrace i, j\\rbrace $ , with high probability, we have $& \\sum _{r=1}^R \\log \\frac{1 + \\sum _{i=r}^R w_{u_i}(x_{u_i}) / W(x, u)}{1 + \\sum _{i=1}^r w_{u_i}(y_{u_i})/W(x,u)} \\\\=& \\frac{1}{W}\\sum _{r=1}^R (R-r+1)w_{u_i}(x) - r w_{u_i}(y) + o(N^{\\frac{1}{12}-\\delta }) \\\\& \\ \\ - \\frac{1}{2W^2} \\sum _{i=1}^R \\sum _{j=1}^R i\\wedge j w_{u_i}(x)w_{u_j}(x) - (R - i \\vee j + 1) w_{u_i}(y)w_{u_j}(y)$ blackSelect $0 < \\delta < \\frac{1}{12}$ , and the corresponding $t = N^\\delta $ , we have, for large enough $N$ , the above equation does not hold with probability exponentially small, and the term $o(N^{\\frac{1}{12}-\\delta })$ can be ignored.",
"Hence we prove the weak convergence."
],
[
"Proof for Lemma ",
"The distribution $p(u_r|u_{1:r-1})$ can be approximated using the following tricks.",
"First, using lemma REF , with high probability, we have: $\\mathbb {P}(u_r = i|u_{1:r-1})&= \\mathbb {E}_{x \\notin u_{1:r}} \\left[ \\frac{\\mathbb {P}(x_i=1) w_i(1)}{W(x_{-i}, x_i = 1, u_{1:r-1})} + \\frac{\\mathbb {P}(x_i=0) w_i(0)}{W(x_{-i}, x_i=0, u_{1:r-1})} \\right] \\\\&= \\frac{p_i w_i(1) + (1 - p_i) w_i(0)}{W} + O(N^{-\\frac{3}{2}})$ Derive the similar result for $\\mathbb {P}(u_r = j|u_{1:r-1})$ .",
"Since we have $R = l N^\\frac{3}{2}$ , for arbitrary $1 \\le r \\le R$ , we have $W$ has the same order as $N$ .",
"Using the property of locally balanced function, where $p_i w_i(1) = (1 - p_i) w_i(0)$ , we have $\\frac{\\mathbb {P}(u_1 = i)}{\\mathbb {P}(u_1 = j)}& = \\frac{p_i w_i(1)}{p_j w_j(1)} + O(N^{-\\frac{5}{2}})$ Then, we use the identity: $1&= \\sum _{i=1}^N \\mathbb {P}(u_1 = i) \\\\&= \\sum _{j=1}^N \\left(\\frac{p_i w_i(1)}{p_j w_j(1)} + O(N^{-\\frac{5}{2}})\\right) \\mathbb {P}(u_1 = j) \\\\&= \\left(\\frac{\\sum _{i=1}^N p_i w_i(1)}{p_j w_j(1)} + O(N^{-\\frac{3}{2}})\\right) \\mathbb {P}(u_1 = j)$ hence, we have for the first step $u_1$ : $\\mathbb {P}(u_1 = j) = \\frac{p_j w_j(1)}{\\sum _{i=1}^N p_i w_i(1)} + O(N^{-\\frac{5}{2}})$ Recursively use this trick, for $1 \\le r \\le R = l N^\\frac{2}{3}$ we have: $\\mathbb {P}(u_r = j|u_{1:r-1}) = \\frac{p_j w_j(1) 1_{\\lbrace j \\notin u_{1:r-1}\\rbrace }}{\\sum _{i=1}^N p_i w_i(1) 1_{\\lbrace i \\notin u_{1:r-1}\\rbrace }} + O(N^{-\\frac{5}{2}})$ Next, we calculate the conditional probability for $x$ .",
"To simplify the notation, we denote $\\mathbb {P}(x_j=1|u, u_r=j, x_{u_{1:j-1}})$ to represented index $j$ is selected at step $u_r$ , and not been selected in all previous steps $u_1, ..., u_{r-1}$ .",
"Also, we denote $W(x, u, s, t) = W(x, u) + \\sum _{k=s}^t w_{u_k}(x)$ In this way, the conditional probability for $x$ can be written as $&\\mathbb {P}(x_j=1|u, u_r=j, x_{u_{1:j-1}}) \\\\= & \\mathbb {E}[\\frac{\\pi _j(1)\\prod _{l=1}^{r-1} (1 - \\frac{w_j(1)}{W(x_{-j}, x_j=1, u, l, R)}) \\frac{w_j(1)}{W(x_{-j}, x_j=1, u, r, R)} }{\\sum _{v=0}^1 \\pi _j(v)\\prod _{l=1}^{r-1} (1 - \\frac{w_j(1)}{W(x_{-j}, x_j=v, u, l, R)}) \\frac{w_j(1)}{W(x_{-j}, x_j=v, u, r, R)} } |u, u_r=j, x_{u_{1:j-1}}] \\\\= & \\mathbb {E}[\\frac{\\prod _{l=1}^{r-1} (1 - \\frac{w_j(1)}{W(x_{-j}, x_j=1, u, l, R)}) \\frac{1}{W(x_{-j}, x_j=1, u, r, R)} }{\\sum _{v=0}^1 \\prod _{l=1}^{r-1} (1 - \\frac{w_j(1)}{W(x_{-j}, x_j=v, u, l, R)}) \\frac{1}{W(x_{-j}, x_j=v, u, r, R)} } |u, u_r=j, x_{u_{1:j-1}}]$ Since $R = l N^\\frac{2}{3}$ , according to lemma REF , with high probability we have: $\\frac{w_j(1)}{W(x_{-j}, x_j=v, u, l, R)} = \\frac{w_j(1)}{W + O(N^\\frac{1}{2}) + O(R)} = \\frac{w_j(1)}{W} + O(N^{-\\frac{4}{3}})$ Using this approximation, we have: $&\\mathbb {P}(x_j=1|u, u_r=j, x_{u_{1:j-1}}) \\\\= & \\mathbb {E}[\\frac{\\prod _{l=1}^{r-1} (1 - \\frac{w_j(1)}{W} + O(N^{-\\frac{4}{3}}) ) (\\frac{1}{W} + O(N^{-\\frac{4}{3}})) }{\\sum _{v=0}^1 \\prod _{l=1}^{r-1} (1 - \\frac{w_j(v)}{W} + O(N^{-\\frac{4}{3}}) ) (\\frac{1}{W} + O(N^{-\\frac{4}{3}}))} |u, u_r=j, x_{u_{1:j-1}}] \\\\= & \\mathbb {E}[\\frac{\\prod _{l=1}^{r-1} (1 - \\frac{w_j(1)}{W} + O(N^{-\\frac{4}{3}}) )}{\\sum _{v=0}^1 \\prod _{l=1}^{r-1} (1 - \\frac{w_j(v)}{W} + O(N^{-\\frac{4}{3}}) )} {black}{(1 + O(N^{-\\frac{2}{3}}))} |u, u_r=j, x_{u_{1:j-1}}] \\\\= & \\mathbb {E}[\\frac{1 - (r-1)\\frac{w_j(1)}{W} + (r-1)O(N^{-\\frac{4}{3}})}{(1 - (r-1)\\frac{w_j(0)}{W}) + (1 - (r-1)\\frac{w_j(1)}{W}) + (r-1) O(N^{-\\frac{4}{3}}) }|u, u_r=j, x_{u_{1:j-1}}] \\\\= & \\mathbb {E}[\\frac{1 - (r-1)\\frac{w_j(1)}{W}}{(1 - (r-1)\\frac{w_j(0)}{W}) + (1 - (r-1)\\frac{w_j(1)}{W})} + (r-1) O(N^{-\\frac{4}{3}})|u, u_r=j, x_{u_{1:j-1}}] \\\\= & \\frac{1}{2} + (r-1) \\frac{w_j(0) - w_j(1) }{ 4W} + (r-1) O(N^{-\\frac{4}{3}})$ Thus we prove the lemma."
],
[
"A Property for the conditional distribution of $u$",
"The following result shows that marginal distribution for $u_1$ is a good approximation of the conditional distribution.",
"Proposition A.2 For $N$ large enough, the conditional distribution for $u_r = j$ given $u_{1:r-1}$ can be approximated by the marginal distribution of $u_1$ $& p(u_r = j| u_{1:r-1}, j \\notin u_{1:r-1}) \\\\= & \\mathbb {E}_{u_{1:r-1}}\\Big [ \\frac{p_j w_j(1)}{\\sum _{i\\notin u_{1:r-1}} p_i w_i(1)} \\Big ] + O(N^{-\\frac{5}{2}})\\\\= & \\mathbb {E}_{u_{1:r-1}}\\Big [ \\frac{p_j w_j(1)}{\\sum _{i=1}^N p_i w_i(1)} +\\frac{p_j w_j(1) \\sum _{i=1}^N p_i w_i(1) (1 - 1_{\\lbrace i \\notin u_{1:r-1}\\rbrace })}{(\\sum _{i \\notin u_{1:r-1}} p_i w_i(1)) (\\sum _{i=1}^N p_i w_i(1))} \\Big ] + O(N^{-\\frac{5}{2}})\\\\= & p(u_1 = j) + O(\\frac{r}{N^2})$"
],
[
"Proof for Lemma ",
"We first calculate its expectation using the conditional distribution derived in lemma REF .",
"To simplify the notation, we denote $\\delta _w(i) = w_{u_i}(0) - w_{u_i}(1)$ for $i = 1, ..., R$ and $S(i, j, k, l) &= i \\wedge j w_{u_i}(k)w_{u_j}(l) - (R - i \\vee j + 1) w_{u_i}(1-k)w_{u_j}(1-l) \\\\P(i, k) &= \\frac{1}{2} - (-1)^k (i - 1) \\frac{\\delta _w(i)}{4W} + (i-1) O(N^{-\\frac{4}{3}})$ for $i, j = 1, ..., R$ , and $k, l = 0, 1$ .",
"Then we have $& -\\frac{1}{2W^{2}} \\sum _{i=1}^{R} \\sum _{j=1}^{R}[ i\\wedge j w_{u_{i}}(x_{u_{i}}) w_{u_{j}}(x_{u_{j}}) - (R - i\\vee j + 1) w_{u_i}(y_{u_i})w_{u_j}(y_{u_j}) | u]\\\\= & -\\frac{1}{2W^2} \\sum _{i, j=1}^R \\sum _{k=0}^1 \\sum _{l=0}^1 S(i, j, k, l)P(i, k)P(j, l) \\\\= & - \\frac{1}{2W^2} \\sum _{i, j=1}^R (R - (i + j) + 1)(w_{u_i}(0) + w_{u_i}(1))(w_{u_j}(0) + w_{u_j}(1)) + O(\\frac{R^2}{N}) \\\\= & - \\frac{1}{2W^2} \\sum _{i, j=1}^R (R - (i + j) + 1)(w_{u_i}(0) + w_{u_i}(1))(w_{u_j}(0) + w_{u_j}(1)) + O(\\frac{R^4}{N^3})$ The remaining expectation is with respect to $u$ .",
"From proposition REF , we know that the conditional expectation of $u_i$ can be estimated via the marginal distribution of $u_1$ .",
"In fact, when $R = l N^\\frac{2}{3}$ , we have: $& \\mathbb {E}[w_{u_r}(0) + w_{u_r}(1)|u_{1:r-1}] \\\\= & \\mathbb {E}[\\sum _{j=1}^N (w_j(1) + w_j(0)) (\\frac{p_jw_j(1)}{\\sum _{i=1}^N p_iw_i(1)} + O(\\frac{R}{N^2}) | u_{1:r-1}] \\\\= & \\mathbb {E}[w_{u_1}(0) + w_{u_1}(1)] + O(N^{-\\frac{4}{3}})$ and similarly, we have: $\\mathbb {E}[(w_{u_r}(0) + w_{u_r}(1))^2|u_{1:r-1}] = \\mathbb {E}[(w_{u_1}(0) + w_{u_1}(1))^2] + O(N^{-\\frac{4}{3}})$ Using these properties, we have $& \\mathbb {E}[\\sum _{i, j=1}^R (R - (i + j) + 1)(w_{u_i}(0) + w_{u_i}(1))(w_{u_j}(0) + w_{u_j}(1))] \\\\= & \\mathbb {E}[ \\mathbb {E}[\\cdots \\mathbb {E}[2\\sum _{i=1}^R \\sum _{j>i}^R (R - (i + j) + 1)(w_{u_i}(0) + w_{u_i}(1))(w_{u_j}(0) + w_{u_j}(1)) \\\\& \\quad + \\sum _{r=1}^R (R-2r+1) (w_{u_r}(0) + w_{u_r}(1))^2|u_{1:R-1}] \\cdots |u_1] ] \\\\= & \\mathbb {E} [2\\sum _{i=1}^R \\sum _{j>i}^R (R - (i + j) + 1)(w_{u_1}(0) + w_{u_1}(1))(w_{u_1}(0) + w_{u_1}(1))] + O(N^\\frac{2}{3}) \\\\& \\quad + \\mathbb {E}[\\sum _{r=1}^R (R-2r+1) (w_{u_1}(0) + w_{u_1}(1))^2] + O(1) \\\\= & (w_{u_1}(0) + w_{u_1}(1))^2 \\sum _{i, j=1}^N (R - (i+j) + 1) + O(N^\\frac{2}{3}) \\\\= & O(N^\\frac{2}{3})$ Hence, we prove that $\\mathbb {E}[-\\frac{1}{2W^{2}} \\sum _{i=1}^{R} \\sum _{j=1}^{R}[i \\wedge j w_{u_{i}}(x_{u_{i}}) w_{u_{j}}(x_{u_{j}}) - (R - i \\vee j + 1) w_{u_i}(y_{u_i})w_{u_j}(y_{u_j}) ] = O(N^{-\\frac{4}{3}})$ The expectation of the $B$ () is small.",
"To show it is save to ignore, we will prove the concentration property.",
"Consider a function of $x$ and $u$ : $F(x, u) = -\\frac{1}{2} \\frac{1}{W^{2}} \\sum _{i=1}^{R} \\sum _{j=1}^{R}[i \\wedge j w_{u_{i}}(x_{u_{i}}) w_{u_{j}}(x_{u_{j}}) - (R - i\\vee j + 1) w_{u_i}(y_{u_i})w_{u_j}(y_{u_j})$ where $y$ is obtained by flipping indices $u$ of $x$ .",
"For changing $x$ , we have: $|F(x_1, ..., x_j, ..., x_N, u_1, ..., u_R) - F(x_1, ..., x^{\\prime }_j, ..., x_N, u_1, ..., u_R)| \\le c_j$ where $c_j = 0$ if $j \\notin u$ or $c_j = O(\\frac{R^2}{N^2})$ if there exists $r$ and $u_r = j$ .",
"For chaning $u$ , we have $|F(x_1, ..., {black}{x_N}, u_1, ..., u_i, ... u_R) - F(x_1, ..., x_N, u_1, ..., {black}{u^{\\prime }_i, ..., u_R})| \\le d_i$ where $d_i = O(\\frac{R^2}{N^2})$ for $i = 1, ..., R$ .",
"By McDiarmid's inequality, we have: $\\mathbb {P}(|F(x, u) - \\mathbb {E}[F(x, u)] \\ge t \\frac{R^\\frac{5}{2}}{N^\\frac{7}{4}}) \\le 2 \\exp (-\\frac{2t^2 R^5 / {black}{N^\\frac{7}{2}}}{\\sum _{j=1}^N c^2_j + \\sum _{i=1}^R d^2_i}) \\lesssim \\exp (-2t^2N^\\frac{1}{2})$ Hence, $F(x, u)$ will concentrate to its expectation at scale $O(R^\\frac{5}{2} / N^\\frac{7}{4})$ .",
"Since $R = l N^\\frac{2}{3}$ , with probability larger than $1 - O(\\exp (-N^\\frac{1}{2}))$ , $B = O(N^{-\\frac{1}{12}})$ ."
],
[
"Lemma ",
"To show that $A$ weakly converges to a normal distribution, we use martingale central limit theorem.",
"Define a martingale $M_n$ , for $n=0, 1, ..., 2R$ .",
"When $n \\le R$ , we let the process $M_n = 0$ and the filter $F_n$ as the $\\sigma $ -algebra determined by $u_1, ..., u_n$ .",
"For $R + 1 \\le R + n \\le 2R$ , define $M_{R+n} = M_{R + n-1} + \\frac{1}{W}\\Big ( &(R - r + 1) w_{u_n}(x_n) - rw_{u_n}(1 - x_{u_n}) \\\\&- \\mathbb {E}[(R - r + 1) w_{u_n}(x_n) - rw_{u_n}(1 - x_{u_n})]\\Big )$ We first estimate the mean of the increment using the conditional probability derived in lemma REF .",
"If $n \\le R$ , the mean is obviously 0, else $& \\mathbb {E}[\\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W}|u_r = j] \\\\=& \\frac{(R - r + 1)w_j(1) - r w_j(0)}{W} (\\frac{1}{2} + r \\frac{w_j(0) - w_j(1)}{W} + O(\\frac{R}{N^\\frac{3}{2}} + \\frac{R^2}{N^2})) \\\\& + \\frac{(R - r + 1)w_j(0) - r w_j(1)}{W} (\\frac{1}{2} - r \\frac{w_j(0) - w_j(1)}{W} + O(\\frac{R}{N^\\frac{3}{2}} + \\frac{R^2}{N^2})) \\\\=& \\frac{1}{2}\\frac{R-2r+1}{W}(w_j(1) + w_j(0)) - \\frac{r(R+1)}{4W^2}(w_j(0) - w_j(1))^2 + O(\\frac{R^2}{N^\\frac{5}{2}} + \\frac{R^3}{N^3})$ Then we estimate the variance of $M_n - M_{n-1}$ .",
"We start with estimating the 2nd moment.",
"$& \\mathbb {E}[(\\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W})^2|u_r=j] \\\\=& (\\frac{(R - r + 1)w_j(1) - r w_j(0)}{W})^2 (\\frac{1}{2} + r \\frac{w_j(0) - w_j(1)}{W}) + O(\\frac{R}{N^\\frac{3}{2}} + \\frac{R^2}{N^2})) \\\\& + \\frac{((R - r + 1)w_j(0) - r w_j(1)}{W})^2 (\\frac{1}{2} - r \\frac{w_j(0) - w_j(1)}{W}) + O(\\frac{R}{N^\\frac{3}{2}} + \\frac{R^2}{N^2})) \\\\=& \\frac{1}{2}((R-r+1)^2 + r^2)\\frac{w_j^2(0) + w_j^2(1)}{W^2} - 2r(R-r+1)\\frac{w_j(0)w_j(1)}{W^2} + O(\\frac{R^3}{N^\\frac{7}{2}} + \\frac{R^4}{N^4})$ Then, we are able to calculate the variance: $& \\text{var} [\\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W}|u_r=j] \\\\= & \\mathbb {E}[(\\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W})^2|u_r=j] \\\\& - \\mathbb {E}^2[\\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W}|u_r=j] \\\\= & \\frac{(R+1)^2}{4}\\frac{w_j^2(0) + w_j^2(1)}{W^2} - \\frac{(R+1)^2}{2}\\frac{w_j(0)w_j(1)}{W^2} + O(\\frac{R^2}{N^\\frac{5}{2}} + \\frac{R^3}{N^3}) \\\\= & \\frac{(R+1)^2}{4W^2}(w_j(0)- w_j(1))^2 + O(\\frac{R^2}{N^\\frac{5}{2}} + \\frac{R^3}{N^3})$ We calculate the value of its mean $\\mu $ and variance $\\sigma ^2$ .",
"$\\mu & = \\mathbb {E}[\\sum _{r=1}^R \\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W}|u] \\\\& = \\sum _{r=1}^R \\frac{1}{2}\\frac{R-2r+1}{W}(w_{u_r}(1) + w_{u_r}(0)) - \\frac{r(R+1)}{4W^2}(w_{u_r}(0) - w_{u_r}(1))^2 \\\\\\sigma ^2& = \\sum _{r=1}^R \\text{var} [\\frac{(R - r + 1)w_{u_r}(x_{u_r}) - r w_{u_r}(y_{u_r})}{W}|u] \\\\& = \\sum _{r=1}^R \\frac{(R+1)^2}{4W^2}(w_{u_r}(0)- w_{u_r}(1))^2$ Define $\\mu _1 = \\mathbb {E}[w_{u_1}(1) + w_{u_1}(0)]$ .",
"For the first part in $\\mu $ , using proposition REF , we have $& \\mathbb {E}[\\sum _{r=1}^R \\frac{R-2r+1}{W} (w_{u_r}(1) + w_{u_r}(0))] \\\\= & \\mathbb {E}[\\sum _{r=1}^R \\frac{R-2r+1}{W} \\mu _1 + O(N^{-\\frac{5}{3}})] \\\\= & O(N^{-\\frac{2}{3}})$ Define $\\sigma _1^2 = \\mathbb {E}[(w_{u_1}(0) - w_{u_1}(1))^2]$ , From lemma REF , we have $\\mathbb {E}[(w_{u_r}(0) - w_{u_r}(1))^2] = \\sigma _1^2 + O(N^{-\\frac{4}{3}}), \\quad \\forall r = 1, ..., R$ for the second term in $\\mu $ , we have $\\sum _{r=1}^R - \\frac{r(R+1)}{4W^2}(w_{u_r}(0) - w_{u_r}(1))^2 = -\\frac{R(R+1)^2}{8W^2}\\sigma _1^2 + O(N^{-\\frac{4}{3}})$ for the variance $\\sigma ^2$ , we have: $\\sum _{r=1}^R \\frac{(R+1)^2}{4W^2}(w_{u_r}(0) - w_{u_r}(1))^2 = \\frac{R(R+1)^2}{4W^2}\\sigma _1^2 + O(N^{-\\frac{4}{3}})$ Finally, we will decouple $R$ with $W$ .",
"Specifically: $\\frac{1}{W^2} = \\frac{1}{\\mathbb {E}^2[\\sum _{k\\notin u} w_k(x_k)]} = \\frac{1}{\\mathbb {E}^2[\\sum _{k=1}^N w_k(x_k)]} + O(N^{-\\frac{8}{3}})$ Combine everything togetherblack, we have $\\mu &= -\\frac{R(R+1)^2}{8\\mathbb {E}^2[\\sum _{k=1}^N w_k(x_k)]}\\sigma _1^2 + O(N^{-\\frac{2}{3}}) \\\\\\sigma ^2 &= \\frac{R(R+1)^2}{4\\mathbb {E}^2[\\sum _{k=1}^N w_k(x_k)]}\\sigma _1^2 + O(N^{-\\frac{4}{3}})$ Since $R = l N^\\frac{2}{3}$ , we have the sum of the conditional variance is $O(1)$ and the reminder is $o(1)$ .",
"For a martingale, we need to check one more step.",
"We know $|M_n - M_{n-1}| = 0$ for $n \\le R$ .",
"For $n + R > R$ , we have: $|M_{R+n} - M_{R+n-1}|& = \\frac{1}{W}\\big ((R - r + 1) w_{u_r}(x) - rw_{u_r}(y) - \\mathbb {E}[(R - r + 1) w_{u_r}(x) - rw_{u_r}(y)]\\big ) \\\\& = O(\\frac{R}{N}) = O(N^{-\\frac{1}{3}})$ is uniformly bounded by a constant independent of $N$ and $R$ .",
"We denote $\\lambda _1^2 = \\frac{\\sum _{j=1}^N p_j w_j(1) (w_j(0) - w_j(1))^2}{4 \\mathbb {E}^2[\\frac{1}{N}\\sum _{k=1}^N w_k(x_k)] \\sum _{i=1}^N p_i w_i(1)}$ Then we can rewrite: $\\mu = -\\frac{1}{2}\\lambda _1^2 l^3 \\\\\\sigma ^2 = \\lambda _1^2 l^3$ By martingale central limit theorem, we have that: $\\frac{A - \\mu }{\\sigma } \\longrightarrow _\\text{dist.}",
"\\mathcal {N}(0, 1)$ Furthermore, we use the convergence rate in [18], we have: $L_{R, 2 \\delta } & \\equiv \\sum _{r=1}^{2R} E\\left(\\left|M_r - M_{r-1}\\right|^{2+2 \\delta }\\right) = O(\\frac{R^{3 + 2\\delta }}{N^{2 + 2\\delta }}) \\\\M_{R, 2 \\delta } & \\equiv \\mathbb {E}[|\\sum _{r=1}^{2R} \\mathbb {E}[(M_r - M_{r-1})^2|F_{r-1}] - 1 |^{1 + \\delta }] = O(\\frac{R^{4+4\\delta }}{N^{4+4\\delta }})$ Then we have the probability $|\\mathbb {P}(\\frac{A-\\mu }{\\sigma } \\le t) - \\Phi (t)| \\le D_R$ where $D_R \\le C_\\delta (L_{R, 2\\delta } + M_{R, 2\\delta })^\\frac{1}{3 + 2\\delta } = O(R / N^\\frac{2 + 2\\delta }{3 + 2\\delta }), \\quad \\forall \\delta > 0$ where $C_\\delta $ is an absolute constant that only depends on $\\delta $ .",
"We select $\\delta = \\frac{1}{2}$ , we have: $|\\mathbb {P}(\\frac{A-\\mu }{\\sigma } \\le t) - \\Phi (t)| \\le O(R / N^\\frac{3}{4})$ Since we consider $R= l N^\\frac{2}{3}$ , we prove the lemma."
],
[
"Proof of Lemma ",
"Assume $Z \\sim \\mathcal {N}(\\mu , \\sigma ^2)$ , then we have: $\\mathbb {E}\\min \\lbrace 1, e^{Z}\\rbrace & = \\int _{-\\infty }^0 e^z \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(z - \\mu )^2}{2\\sigma ^2}} dz + \\int _0^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(z - \\mu )^2}{2\\sigma ^2}} dz \\\\&= \\int _{-\\infty }^0 \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{z^2 - 2\\mu z + \\mu ^2 - 2 \\sigma ^2 z}{2\\sigma ^2}} dz + \\int _{-\\mu }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{z^2}{2\\sigma ^2}} dz \\\\&= \\exp (\\mu + \\frac{\\sigma ^2}{2}) \\int _{-\\infty }^0 \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{(z - (\\mu + \\sigma ^2))^2}{2\\sigma ^2}} dz + \\int _{-\\mu }^\\infty \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{z^2}{2\\sigma ^2}} dz \\\\&= \\exp (\\mu + \\frac{\\sigma ^2}{2}) \\int _{-\\infty }^{-\\mu -\\sigma ^2} \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{z^2}{2\\sigma ^2}} dz + \\int _{-\\infty }^\\mu \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{z^2}{2\\sigma ^2}} dz \\\\&= \\exp (\\mu + \\frac{\\sigma ^2}{2}) \\Phi (-\\frac{\\mu }{\\sigma } - \\sigma ) + \\Phi (\\frac{\\mu }{\\sigma })$ Specially, when $\\mu = -\\frac{1}{2} \\sigma ^2$ , we have: $\\mathbb {E}\\min \\lbrace 1, e^Z\\rbrace = 2\\Phi (-\\frac{1}{2}\\sigma )$"
],
[
"Proof for Theorem ",
"In RWM-R, the proposal distribution is uniform, hence we only need to consider the probability ratio in the acceptance rate.",
"Given current state $x$ and the picked indices $u$ , the proposed state $y$ is obtained by flipping indices $u$ of $x$ .",
"The acceptance rate is: $A(x, y, u)& = 1 \\wedge \\frac{\\pi (y)}{\\pi (x)} \\\\& = 1 \\wedge \\prod _{r=1}^R \\frac{\\pi _{u_r}(y)}{\\pi _{u_r}(x)} \\\\& = 1 \\wedge \\prod _{r=1}^R \\frac{p_{u_r}^{y_{u_r}} (1 - p_{u_r})^{1 - y_{u_r}}}{p_{u_r}^{x_{u_r}} (1 - p_{u_r})^{1 - x_{u_r}}} \\\\& = 1 \\wedge \\prod _{r=1}^R p_{u_r}^{1 - 2 x_{u_r}} (1 - p_{u_r})^{2 x_{u_r} - 1} \\\\& = 1 \\wedge \\exp (\\sum _{r=1}^R (1 - 2 x_{u_r})\\log \\frac{p_{u_r}}{1 - p_{u_r}})$ Define the martingale $M_n$ , $n=1, ..., 2R$ .",
"For $r=1, ..., R$ , we have $M_r = 0$ and the filtration $F_r$ is determined by the $\\sigma $ -algebra of $u_1, ..., u_R$ .",
"For $ R+1 \\le R + n \\le 2R$ , we have: $M_{R+n} = M_{R + n-1} + (1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}} - \\mathbb {E}[(1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}}]$ Hence, for $n \\le R$ , the increment is 0.",
"For $n + R > R$ , denote the mean of the increment is : $\\mathbb {E}[(1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}}] = (1 - 2p_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}}$ the variance of the increment is: $& \\mathbb {E}[(M_{R+j} - M_{R+j-1})^2|u, x_{1:j-1}] \\\\= & \\mathbb {E}[((1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}} - \\mathbb {E}[(1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}}])^2 ] \\\\= & \\mathbb {E}[((1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}})^2] - \\mathbb {E}^2[(1 - 2x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}}] \\\\= & (\\log \\frac{p_{u_n}}{1 - p_{u_n}})^2 - (1 - 2 p_{u_n})^2 (\\log \\frac{p_{u_n}}{1 - p_{u_n}})^2 \\\\= & 4p_j(1-p_{u_n}) (\\log \\frac{p_{u_n}}{1 - p_{u_n}})^2$ When $N$ is large, we have $p_{u_n} - \\frac{1}{2} = O(N^{-\\beta })$ , hence $& 4p_{u_n}(1-p_{u_n}) (\\log \\frac{p_{u_n}}{1 - p_{u_n}})^2 \\\\= & 4p_{u_n}(1-p_{u_n}) \\log (1 + \\frac{2p_{u_n}-1}{p_{u_n}}) \\log (\\frac{p_{u_n}}{1 - p_{u_n}}) \\\\= & 4(\\frac{1}{2} + O(N^{-\\beta })) (1-p_{u_n}) (\\frac{2p_{u_n}-1}{1 - p_{u_n}} + O(N^{-2\\beta }) ) \\log (\\frac{p_{u_n}}{1 - p_{u_n}}) \\\\= & 2(2p_{u_n} - 1) \\log (\\frac{p_{u_n}}{1 - p_{u_n}})(1 + O(N^{-\\beta }))$ is negative twice of the corresponding mean.",
"Since the indices $u$ are uniformly picked, the conditional distribution of $u_r$ is: $\\mathbb {P}(u_r = j | u_{1:r-1}) = \\frac{1_{\\lbrace j \\notin u_{1:r-1}\\rbrace }}{\\sum _{i=1}^N 1_{\\lbrace i \\notin u_{1:r-1}\\rbrace }} = \\frac{1}{N} + O(\\frac{R}{N^2})$ Hence, we have the mean is $\\mu &= \\mathbb {E}[\\sum _{r=1}^R (1 - 2 x_{u_n}) \\log \\frac{p_{u_n}}{1 - p_{u_n}}] \\\\& = \\mathbb {E}[R (1 - 2 x_{u_1}) \\log \\frac{p_{u_1}}{1 - p_{u_1}} + O(\\frac{R^2}{N^2})] \\\\& = \\frac{R}{N^{2\\beta }} \\frac{1}{N} \\sum _{i=1}^N N^{2\\beta }(1 - 2 p_i) \\log \\frac{p_i}{1 - p_i} + O(\\frac{R^2}{N^2})$ Similarly, we have the variance is: $\\sigma ^2 = \\mathbb {E}[\\sum _{r=1}^R 2(2 x_{u_n} - 1) \\log \\frac{p_{u_n}}{1 - p_{u_n}}] = \\frac{R}{N^{2\\beta }} \\frac{2}{N} \\sum _{i=1}^N N^{2\\beta }(2 p_i - 1) \\log \\frac{p_i}{1 - p_i} + O(\\frac{R^2}{N^2})$ When $R = O(N^{2\\beta }$ ), the variance is at a constant order.",
"For a martingale, we also need to check the increments are uniformly bounded.",
"When $n \\le R$ , the increment is always 0.",
"When $R + 1 \\le R + n \\le 2R$ , we have: $|M_{R+n} - M_{R+n-1}| = |(1 - 2 x_{u_n})\\log \\frac{p_{u_n}}{1 - p_{u_n}} - \\mathbb {E}[(1 - 2 x_{u_n})\\log \\frac{p_{u_n}}{1 - p_{u_n}}]| \\le C(\\epsilon )$ where $C(\\epsilon )$ is a constant only determined by $\\epsilon $ .",
"Hence, by martingale central limit theorem, we have the distribution of $M_{2R}$ converges to a normal distribution.",
"Denote $\\lambda _2^2 = \\frac{2}{N} \\sum _{i=1}^N N^{2\\beta }(2 p_i - 1) \\log \\frac{p_i}{1 - p_i}$ Then we can rewrite: $\\mu &= -\\frac{1}{2}\\lambda _2^2 \\frac{R}{N^{2\\beta }} \\\\\\sigma ^2 &= \\lambda _2^2 \\frac{R}{N^{2\\beta }}$ Denote $Z = \\sum _{r=1}^R (1 - 2 x_{u_r}) \\log \\frac{p_{u_r}}{1 - p_{u_r}}$ .",
"By martingale central limit theorem, we have $\\frac{Z - \\mu }{\\sigma } \\longrightarrow _\\text{dist.}",
"\\mathcal {N}(0, 1)$ Furthermore, using the convergence rate in [18], we have: $L_{R, 2 \\delta } & \\equiv \\sum _{r=1}^{2R} E\\left(\\left|M_r - M_{r-1}\\right|^{2+2 \\delta }\\right) = O(\\frac{R}{N^{(4 + 4\\delta )\\beta }}) \\\\M_{R, 2 \\delta } & \\equiv \\mathbb {E}[|\\sum _{r=1}^{2R} \\mathbb {E}[(M_r - M_{r-1})^2|F_{r-1}] - 1 |^{1 + \\delta }] = O(\\frac{R^{2+2\\delta }}{N^{2+2\\delta }})$ Then we have the probability $|\\mathbb {P}(\\frac{A-\\mu }{\\sigma } \\le t) - \\Phi (t)| \\le D_R$ where $D_R \\le C_\\delta (L_{R, 2\\delta } + M_{R, 2\\delta })^\\frac{1}{3 + 2\\delta } = O(R^\\frac{1}{3 + 2\\delta } / N^\\frac{4 + 4\\delta }{3 + 2\\delta }), \\quad \\forall \\delta > 0$ where $C_\\delta $ is an absolute constant that only depends on $\\delta $ .",
"We select $\\delta = \\frac{1}{2}$ , we have: $|\\mathbb {P}(\\frac{A-\\mu }{\\sigma } \\le t) - \\Phi (t)| \\le O(R^\\frac{1}{4} / N^\\frac{5}{4})$ Hence, the expectation w.r.t.",
"$\\sum _{r=1}^R (1 - 2 x_{u_r}) \\log \\frac{p_{u_r}}{1 - p_{u_r}}$ converges to the expectation w.r.t.",
"$\\mathcal {N}(-\\frac{1}{2}\\lambda _2^2 \\frac{R}{N^{2\\beta }}, \\lambda _2^2 \\frac{R}{N^{2\\beta }})$ Using lemma REF , we have the acceptance rate converges to: $a(R) = 2\\Phi (-\\frac{1}{2} \\lambda _2 \\frac{R^\\frac{1}{2}}{N^\\beta })$ In RWM-R, the distance between the current state $x$ and the proposed state $y$ is always $d(x, y) = R$ , hence we have: $\\rho (R) = R a(R) = 2R \\Phi (-\\frac{1}{2} \\lambda _2 \\frac{R^\\frac{1}{2}}{N^\\beta })$ When $R = \\omega (N^{2\\beta })$ , we can give a concentration property.",
"Since the selection of $u_r$ is a martingale, we can apply Azuma-Hoeffding inequality: $\\mathbb {P}(|M_{2R} - \\mu | > t \\lambda _2 R^\\frac{3}{4} / N^{\\frac{3}{2}\\beta })\\lesssim 2\\exp (- \\frac{2t^2 R^\\frac{3}{2} /N^{3\\beta }}{R N^{-2\\beta }}) = 2\\exp (-2t^2 R^\\frac{1}{2} / N^\\beta )$ Hence, When $N$ is sufficiently large, with probability larger than $1 - O(\\exp (-2t^2 R^\\frac{1}{2} / N^\\beta ))$ , we have: $\\sum _{r=1}^R (1 - 2 x_{u_r}) \\log \\frac{p_{u_r}}{1 - p_{u_r}} = -\\frac{1}{2}\\lambda _2^2 \\frac{R}{N^{2\\beta }} + {black}{O(\\frac{t R^\\frac{3}{4}}{N^{\\frac{3}{2}\\beta }})} = -\\frac{C}{2} \\lambda _2^2 \\frac{R}{N^{2\\beta }}$ For $C > 0$ independent with $N, R$ ."
],
[
"Proof for Corollary ",
"When $R = O(N^{2\\beta })$ , denote $z = R \\lambda _2^2 / N^{2\\beta }$ $& \\rho (R) = 2 R \\Phi (-\\frac{1}{2} \\lambda _2 \\frac{R^\\frac{1}{2}}{N^\\beta }) \\\\= & 2 (N^{2\\beta }/R) (R \\lambda _2^2 / N^{2\\beta }) \\Phi (-\\frac{1}{2} ( (R \\lambda _2^2 / N^{2\\beta })^\\frac{1}{2}) \\\\= & 2 (N^{2\\beta }/R) z \\Phi (-\\frac{1}{2} z^\\frac{1}{2})$ which means the optimal value of $z$ is independent of the target distribution.",
"As $\\Phi $ is known, we can numerically solve $z = 5.673$ .",
"Hence the corresponding expected acceptance rate $a = 0.234$ , independent with the target distribution, and the efficiency is $\\Theta (N^{2\\beta })$ .",
"When $R = \\omega (N^{2\\beta })$ , with probability $1 - O(\\exp (-2R / N^\\beta ))$ , the acceptance rate decrease exponentially fast, rendering $o(1)$ jump distance.",
"For the remaining probability $O(\\exp (-2R / N^\\beta ))$ , assuming all proposals are accepted, the efficiency is still bounded by: $R \\exp (-2R / N^\\beta ) = o(1)$ Hence, optimal efficiency is achieved when $R = O(N^{2\\beta })$ ."
],
[
"Expected Jump Distance as the Metric to Tune the Scale",
"black In this section, we want to convince the reader that the expected jump distance (EJD) is the correct metric to evaluate the efficiency for samplers in discrete space.",
"To simplify the derivation, we consider the distribution $\\pi ^{(N)}(x) = \\prod _{i=1}^N \\pi _i(x_i) = \\prod _{i=1}^N p^{x_i}(1-p)^{1-x_i}$ We can notice that, compared to the target distributions considered in the main text (REF ), we assume the target distribution is identical in each dimension.",
"black Let the LBP chain, with $R = lN^\\frac{2}{3}$ , being denoted as $\\lbrace x(1), x(2), ... \\rbrace $ .",
"Since all dimensions are identical, we only need to focus on the first dimension.",
"Denote $w_1 = g(\\frac{\\pi _1(x_1=0)}{\\pi _1(x_1=1)})$ and $w_0 = g(\\frac{\\pi _1(x_1=1)}{\\pi _1(x_1=0)})$ .",
"From Lemma.",
"REF , we can see that: $\\lim _{N\\rightarrow \\infty } \\frac{\\mathbb {P}(u, \\exists u_j = 1| x_1 = 0)}{\\mathbb {P}(u, \\exists u_j = 1| x_1 = 1)} = \\frac{w_0}{w_1}$ That's to say, the probability ratio of $x_1=0$ and $x_1=1$ being flipped equals to their weight ratio.",
"Then we compare the acceptance rate in M-H test.",
"From the proof of the main theorem REF , we know the acceptance rate is determined by the term $A$ defined in (REF ) $A = & \\frac{1}{W}\\sum _{r=1}^R (R-r+1)w_{u_i}(x_{u_i}) - r w_{u_i}(y_{u_i})$ We can see that, when the first dimension is flipped in proposal, the difference of $A$ is $O(N^{-\\frac{1}{3}})$ for $x_1=0$ and $x_1=1$ .",
"As a result, we have: $\\lim _{N\\rightarrow \\infty } \\frac{\\mathbb {P}(\\text{accept }|u, \\exists u_j=1, x_1 = 0)}{\\mathbb {P}(\\text{accept }|u, \\exists u_j=1, x_1 = 1)} = 1$ Now, we consider the one-dimensional process $Z^N_t = x_1(\\lfloor tN^\\frac{1}{3}\\rfloor )$ .",
"The identical assumption implies that, the frequency for a site, for example the first dimension, being selected is $l N^{-\\frac{1}{3}}$ .",
"We can easily see that when $N$ is large enough, $Z^N_t$ converges to a jump process $Z_t$ , whose generator we denote.",
"$Q = \\left[\\begin{array}{rr}-Q_{01} & Q_{01} \\\\Q_{10} & -Q_{10}\\end{array}\\right]$ From the derivation above, we know that $\\frac{Q_{01}}{Q_{10}} = \\lim _{N\\rightarrow \\infty } \\frac{\\sum _u \\mathbb {E}_{x_{2:N}}[\\mathbb {P}(u, \\exists u_j=1| x_1 = 0) \\mathbb {P}(\\text{accept }|u, \\exists u_j=1, x_1 = 0)]}{\\sum _u \\mathbb {E}_{x_{2:N}}[\\mathbb {P}(u, \\exists u_j=1| x_1 = 1) \\mathbb {P}(\\text{accept }|u, \\exists u_j=1, x_1 = 1)]} = \\frac{w_0}{w_1}$ Since the sketch of proof above shows that the ratio is independent with the parameter $l$ , we have the following important decomposition $Q = \\lambda (l) Q(p)$ where $Q(p)$ is a matrix only depends on $p$ and the locally balanced function $g$ selected, and $\\lambda (l)$ is a scalar only depends on the parameter $l$ .",
"black Since $Q(p)$ only depends on the target distribution, for any test functions $f(\\cdot )$ , the inverse auto-correlation of the jump process is proportional to $\\lambda (l)$ .",
"When we tune $l$ , the coefficient $\\lambda (l) = l \\cdot 2 \\Phi (- \\lambda _1 l^\\frac{3}{2})$ is the multiplication of the proposal frequency and the acceptance rate.",
"The value $\\lambda _1$ is defined in (REF ).",
"As a jump process, we don't have to analytically compute the value of $\\lambda (l)$ , as $\\lambda (l)$ is proportional to the expected jump distance (EJD).",
"So, we can tune $l$ by maximizing the EJD, without having to know the formulation of the target distribution.",
"blackRemark 1: The jump process in discrete space is different from the diffusion process in continuous space.",
"For diffusion process, its velocity is characterized by the ESJD.",
"But for jump process, its velocity is characterized by the EJD.",
"That's why Langevin algorithms tunes the step size via ESJD [13], but our LBP tunes the path length via EJD.",
"blackRemark 2: To simplify the derivation, we assume that the target distributions have identical marginals.",
"For target distributions with non-identical marginal distribution, different dimensions $i = 1, ..., N$ can have different velocity $\\lambda _i(l)$ .",
"But the sampling process will still converge to jump process, and we shall still use EJD to measure the efficiency."
],
[
"The Choice of $\\epsilon $ and the Optimal Acceptance Rate",
"black The convergence of (REF ) does not depend on the value of $\\epsilon $ in (REF ).",
"Based on the proof above, we can know (REF ) converges at the rate $O(N^{-\\frac{1}{12}})$ .",
"But the convergence of the optimal acceptance rate depends on the $\\epsilon $ .",
"We can first consider two extreme cases for intuition.",
"When all $p_i$ are close to $\\frac{1}{2}$ , $\\lambda _1$ in (REF ) will be close to 0 and the optimal acceptance rate will be close to 1; when all $p_i$ are close to 0 or 1, $\\lambda _1$ in (REF ) will be close to $\\infty $ and the optimal acceptance rate will be close to 0.",
"Hence, the main purpose to use fixed $\\epsilon $ is to give upper and lower bounds for $\\lambda _1$ in (REF ), such that the optimal acceptance rate can converge to $0.574$ as in Corollary REF .",
"black Next, we discuss how does the model dimension $N$ in (REF ) needed in terms of $\\epsilon $ to make sure the optimal convergence to $0.574$ .",
"When all $p_i$ have the extreme value determined by $\\epsilon $ , using locally balanced function $g(t) = \\sqrt{t}$ , we can consider the following two situations: All $|p_i - 0.5| = \\epsilon \\rightarrow 0$ .",
"Then we have: $\\lambda _1^2 = \\frac{\\sum _{i=1}^N \\sqrt{\\epsilon (1 - \\epsilon )} (\\sqrt{\\frac{\\epsilon }{1 - \\epsilon }} - \\sqrt{\\frac{1 - \\epsilon }{ \\epsilon }})^2}{4 \\epsilon (1 - \\epsilon ) \\sum _{i=1}^N \\sqrt{\\epsilon (1 - \\epsilon )}} \\approx \\frac{\\sum _{i=1}^N 0.5 \\cdot 4 \\epsilon ^2}{4 \\cdot \\sum _{i=1}^N 0.5} = \\epsilon ^2$ When the expected acceptance is $0.574$ , we need $\\lambda _1 l^\\frac{3}{2} = O(\\epsilon l^\\frac{3}{2})$ equals to a constant, which means $l$ has the same order as $\\epsilon ^{-\\frac{2}{3}}$ .",
"Since we have $R = l N^\\frac{2}{3} \\le N$ , we need $\\epsilon ^{-\\frac{2}{3}} = O(N^\\frac{1}{3})$ .",
"As a result, we requires $\\epsilon ^{-1} = O(N^{\\frac{1}{2}})$ , which basically means we need $N \\ge \\epsilon ^{-2}$ to have the optimal acceptance rate converges to $0.574$ .",
"All $0.5 - |p_i - 0.5| = \\epsilon \\rightarrow 0$ .",
"We have: $\\lambda _1^2 = \\frac{\\sum _{j=1}^N \\epsilon \\sqrt{\\frac{1 - \\epsilon }{\\epsilon }} (\\sqrt{\\frac{1 - \\epsilon }{\\epsilon }} - \\sqrt{\\frac{\\epsilon }{1 - \\epsilon }})^2 }{4 (\\sqrt{\\epsilon (1-\\epsilon )})^2 \\sum _{i=1}^N \\sqrt{\\epsilon (1 - \\epsilon )} } \\approx \\frac{\\sum _{j=1}^N \\epsilon ^\\frac{1}{2} \\epsilon ^{-1} }{4 \\epsilon \\sum _{j=1}^N \\epsilon ^{-\\frac{1}{2}}} = \\frac{1}{4}\\epsilon ^{-2}$ When the expected acceptance is $0.574$ , we need $\\lambda _1 l^\\frac{3}{2} = O(\\epsilon ^{-1} l^\\frac{3}{2})$ equals to a constant, which means $l$ has the same order as $\\epsilon ^{\\frac{2}{3}}$ .",
"Since we have $R = l N^\\frac{2}{3} \\ge 1$ , we have $l^{-1} = O(N^\\frac{2}{3})$ .",
"As a result, we requires $\\epsilon ^{-1} = O(N^{-1})$ , which basically means we need $N \\ge \\epsilon ^{-1}$ to have the optimal acceptance rate converges to $0.574$ .",
"So, both situations show that we need $N \\ge C^{\\prime } \\epsilon ^{-2}$ , for some constant $C^{\\prime }$ , to make sure the optimal acceptance rate converges.",
"In the main text, we assume $\\epsilon $ is a constant and it guarantees Corollary REF holds.",
"black We conduct extra numerical simulations to verify our results.",
"To simplify the experiments, we assume all dimensions have the same configurations: $p_i = p$ .",
"We report the size of $N$ needed to guarantee that the optimal acceptance rate is $0.574$ in Table REF and Table REF .",
"Table: NO_CAPTION"
],
[
"Optimal Scale of RWM",
"black When we assume the target distribution belongs to (REF ), the derivation of the optimal acceptance rate $0.234$ is no longer valid.",
"But we can still show the optimal scale is $R = O(1)$ by proving the acceptance rate decreasing exponentially fast.",
"black In particular, assume we use $R = l N^\\beta $ in RWM.",
"Then the acceptance rate can be written as: $A = \\min \\lbrace 1, A^{\\prime } = \\frac{\\pi (y)}{\\pi (x)} = \\frac{\\prod _{j=1}^R \\pi _{u_j}(y)}{\\prod _{j=1}^R \\pi _{u_j}(x)}\\rbrace $ Consider the martingale $M_j, j=0, 1, ..., R$ , such that $M_0 = 0$ and $M_j - M_{j-1} = \\log \\frac{\\pi _{u_j}(y)}{\\pi _{u_j}(x)} - \\mathbb {E}[\\log \\frac{\\pi _{u_j}(y)}{\\pi _{u_j}(x)}|u_{1:j-1}] = (1 - 2x_{u_j}) \\log \\frac{p_{u_j}}{1 - p_{u_j}}$ By assumption in (REF ), we know that $\\mathbb {E}[\\log \\frac{\\pi _{u_j}(y)}{\\pi _{u_j}(x)}|u_{1:j-1}]& = \\mathbb {E} [(1 - 2x_{u_j}) \\log \\frac{p_{u_j}}{1 - p_{u_j}}|u_{1:j-1}] = (1 - 2p_{u_j}) \\log \\frac{p_{u_j}}{1 - p_{u_j}} \\\\& \\le 2\\epsilon \\log \\frac{1 - 2 \\epsilon }{1 + 2 \\epsilon } < 0$ And we have $|M_j - M_{j-1}| \\le 2 \\left|(1 - 2 \\epsilon ) \\log \\frac{\\epsilon }{1 - \\epsilon }\\right| := K$ By Azuma-Hoeffding lemma, we have $\\mathbb {P}(|\\sum _{j=1}^R \\log \\frac{\\pi _{u_j}(y)}{\\pi _{u_j}(x)} - \\mathbb {E} [\\log \\frac{\\pi _{u_j}(y)}{\\pi _{u_j}(x)}] | \\ge R^\\frac{3}{4} t) \\le 2 e^{\\frac{-R^frac{1}{2} t^2}{K^2}}$ For $\\beta > 0$ , $R$ increases to infinity when $N \\rightarrow \\infty $ .",
"In this case, $\\log A^{\\prime }$ concentrates to a value $T \\le R\\cdot 2 \\epsilon \\log \\frac{1 - 2\\epsilon }{1 + 2\\epsilon }$ and $A^{\\prime }$ decreases exponentially fast.",
"Hence, the optimal scaling of RWM is $O(1)$ ."
],
[
"Adaptive Algorithm",
"We give the algorithm box for ALBP: Adaptive Locally Balanced Proposal [1] Initialize current state $x^{(1)}$ .",
"Initialize scale $R_1 = 1$ .",
"t=1, ..., T Initialize candidate set $\\mathcal {C} = \\lbrace 1, .., N\\rbrace $ .",
"$R \\leftarrow $ probabilistic rounding of $R_t$ r=1, ..., R Sample $u_r$ with $\\mathbb {P}(u_r=j) \\propto w_j(x^{(t)}) 1_{\\lbrace j \\in \\mathcal {C}\\rbrace }$ .",
"$\\mathcal {C} \\leftarrow \\mathcal {C} \\backslash \\lbrace u_r\\rbrace $ .",
"Obtain $y$ by flipping indices $u_1, ..., u_R$ of $x^{(t)}$ .",
"Compute acceptance rate $A = A(x^{(t)}, y, u)$ .",
"rand(0,1) $< A$ $x^{(t+1)} = y$ $x^{(t+1)} = x^{(t)}$ $t < T_\\text{warmup}$ $R_{t+1} \\leftarrow R_t + (A - 0.574)$ .",
"We give the algorithm box for ARWM: Adaptive Random Walk Metropolis [1] Initialize current state $x^{(1)}$ .",
"Initialize scale $R_1 = 1$ .",
"t=1, ..., T Initialize candidate set $\\mathcal {C} = \\lbrace 1, .., N\\rbrace $ .",
"$R \\leftarrow $ probabilistic rounding of $R_t$ Uniformly sample $u_1, ..., u_R$ .",
"Obtain $y$ by flipping indices $u_1, ..., u_R$ of $x^{(t)}$ .",
"Compute acceptance rate $A = A(x^{(t)}, y, u)$ .",
"rand(0,1) $< A$ $x^{(t+1)} = y$ $x^{(t+1)} = x^{(t)}$ $t < T_\\text{warmup}$ $R_{t+1} \\leftarrow R_t + (A - 0.234)$ ."
],
[
"Experiment Details",
"We consider five samplers: RWM: random walk Metropolis GWG($\\sqrt{t}$ ): LBP with replacement, same as algorithm REF except for skipping line 5, weight function $g(t) = \\sqrt{t}$ LBP($\\sqrt{t}$ ): LBP given in algorithm REF , weight function $g(t) = \\sqrt{t}$ GWG($\\frac{t}{t+1}$ ): LBP with replacement, same as algorithm REF except for skipping line 5, weight function $g(t) = \\frac{t}{t+1}$ LBP($\\frac{t}{t+1}$ ): LBP given in algorithm REF , weight function $g(t) = \\frac{t}{t+1}$ For each sampler, we first start simulating with $R = 1$ to get an initial acceptance rate $a_{\\max }$ .",
"Then we adopt $a_{\\max } - 0.02, a_{\\max } - 0.04, ..., a_{\\max } - 0.02k$ as the target acceptance rate, until $a_{\\max } - 0.02k < 0.03$ .",
"For each target acceptance rate $a_\\text{target}$ , we use our adaptive sampler to get an estimated scaling $R_\\text{target}$ .",
"Then we simulate 100 chains with scaling $R_\\text{target}$ to get the expected acceptance rate, expected jump distance, effective sample size $(a, d, e)$ .",
"To measure the performance of the adaptive sampler, we compare three versions for each sampler above.",
"In particular, for sampler X we have X-1, represents fixed scaling $R=1$ version of the sampler.",
"AX, represents the adaptive version of the sampler, whose target acceptance rate is selected as $0.234$ for RWM, and $0.574$ for else.",
"GX, represents the grid search version of the sampler, where we always use the best results among all simulations for different target acceptance rates we mentioned above."
],
[
"Simulation on Bernoulli Model",
"The density function for Bernoulli distribution is: $\\pi ^{(N)}(x) = \\prod _{i=1}^N \\pi _i(x_i) = \\prod _{i=1}^N p_i^{x_i}(1-p_i)^{1-x_i}$ We consider three configurations C1: $p_i$ is independently, uniformly sampled from $[0.25, 0.75]$ .",
"C2: $p_i$ is independently, uniformly sampled from $[0.15, 0.85]$ .",
"C3: $p_i$ is independently, uniformly sampled from $[0.05, 0.95]$ .",
"For each configuration, we simulate on three sizes: $N=100$ , sample Markov chain $x_{1:10000}$ , use $x_{1:5000}$ for burn in, use $x_{5001:10000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"$N=800$ , sample Markov chain $x_{1:40000}$ , use $x_{1:20000}$ for burn in, use $x_{20001:40000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"$N=6400$ , sample Markov chain $x_{1:100000}$ , use $x_{1:50000}$ for burn in, use $x_{50001:100000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"We give the scatter plot of $(a, d)$ and $(a, r)$ in figure REF .",
"And we examine the performance of our adaptive algorithm in table REF and table REF .",
"Figure: Simulation Results on Bernoulli ModelTable: Expected Jump Distance on Bernoulli ModelTable: Effective Sample Size on Bernoulli ModelTable: Running Time on Bernoulli Model"
],
[
"Simulation on Ising Model",
"Ising model is a classic model in physics defined on a $p\\times p$ square lattice graph $(V_p, E_p)$ .",
"That's to say, the nodes are indexed by $\\lbrace 1, ..., p\\rbrace ^2$ and an edge $((i, j), (k, l))$ exists if and only if one of the following condition holds: 2 $i = k, \\ j = l + 1$ $i = k, \\ j = l - 1$ $i = k + 1, \\ j = l$ $i = k - 1, \\ j = l$ The state space is $\\mathcal {X} = \\lbrace -1, 1\\rbrace ^{V_p}$ and the target distribution is defined as: $\\pi (x) \\propto \\exp \\Big (\\sum _{i \\in V_p} \\alpha _i x_i - \\lambda \\sum _{(i, j) \\in E_p} x_i x_j\\Big )$ Following [9], we consider three configurations C1: $\\alpha _v$ is independently and uniformly sampled from $(-0.2, 0.4)$ if $(v_1 - \\frac{p}{2})^2 + (v_2 - \\frac{p}{2})^2 \\le \\frac{p^2}{2}$ , else $\\alpha _v$ is independently and uniformly sampled from $(-0.4, 0.2)$ ; $\\lambda = 0.1$ .",
"C2: $\\alpha _v$ is independently and uniformly sampled from $(-0.3, 0.6)$ if $(v_1 - \\frac{p}{2})^2 + (v_2 - \\frac{p}{2})^2 \\le \\frac{p^2}{2}$ , else $\\alpha _v$ is independently and uniformly sampled from $(-0.6, 0.3)$ ; $\\lambda = 0.15$ .",
"C3: $\\alpha _v$ is independently and uniformly sampled from $(-0.4, 0.8)$ if $(v_1 - \\frac{p}{2})^2 + (v_2 - \\frac{p}{2})^2 \\le \\frac{p^2}{2}$ , else $\\alpha _v$ is independently and uniformly sampled from $(-0.8, 0.4)$ ; $\\lambda = 0.2$ .",
"For each configuration, we simulate on three sizes: $p=20$ , sample Markov chain $x_{1:10000}$ , use $x_{1:5000}$ for burn in, use $x_{5001:10000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"$p=50$ , sample Markov chain $x_{1:40000}$ , use $x_{1:20000}$ for burn in, use $x_{20001:40000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"$p=100$ , sample Markov chain $x_{1:100000}$ , use $x_{1:50000}$ for burn in, use $x_{50001:100000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"We give the scatter plot of $(a, d)$ and $(a, r)$ in figure REF .",
"And we examine the performance of our adaptive algorithm in table REF and table REF .",
"Figure: Simulation Results on Ising ModelTable: Expected Jump Distance on Ising ModelTable: Effective Sample Size on Ising ModelTable: Running Time on Ising Model"
],
[
"Simulation on FHMM",
"FHMM uses latent variables $x \\in \\mathcal {X} = \\lbrace 0, 1\\rbrace ^{L\\times K}$ to characterize time series data $y \\in \\mathbb {R}^L$ .",
"Denote $p(x)$ as the prior for hidden variables $x$ , and $p(y|x)$ for the likelihood: $p(x) &= \\prod _{l=1}^L p(x_{l, 1}) \\prod _{k=2}^K p(x_{l, k}|x_{l, k-1}) \\\\p(y|x) &= \\prod _{l=1}^L \\mathcal {N}(y_l; w^T x_l + b, \\sigma ^2)$ Specifically, we have $p(x_{l, 1}) = 0.1$ , $p(x_{l, k} = x_{l, k-1} | x_{l, k-1}) = 0.8$ independently $\\forall l = 1, ..., L$ and $\\forall k = 2, ..., K$ .",
"And we have all entries in $W$ and $b$ are independent Gaussian random variables.",
"We sample latent variables $x$ and sample $y \\sim p(y|x)$ .",
"Then we simulate our samplers to sample the latent variables $x$ from the posterior $\\pi (x) = p(x|y)$ .",
"We consider three configurations C1: $\\sigma ^2 = 2$ C2: $\\sigma ^2 = 1$ C3: $\\sigma ^2 = 0.5$ For each configuration, we simulate on three sizes: $L=200, K=5$ , sample Markov chain $x_{1:10000}$ , use $x_{1:5000}$ for burn in, use $x_{5001:10000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"$L=1000, K=5$ , sample Markov chain $x_{1:40000}$ , use $x_{1:20000}$ for burn in, use $x_{20001:40000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"$L=4000, K=5$ , sample Markov chain $x_{1:100000}$ , use $x_{1:50000}$ for burn in, use $x_{50001:100000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"We give the scatter plot of $(a, d)$ and $(a, r)$ in figure REF .",
"And we examine the performance of our adaptive algorithm in table REF and table REF .",
"Figure: Simulation Results on FHMMTable: Expected Jump Distance on FHMMTable: Effective Sample Size on FHMMTable: Running Time on FHMM"
],
[
"Simulation on RBM",
"RBM is a bipartite latent-variable model, defining a distribution over binary data $x \\in \\lbrace 0, 1\\rbrace ^N$ and latent data $z \\in \\lbrace 0, 1\\rbrace ^h$ .",
"Given parameters $W \\in \\mathbb {R}^{h \\times N}, b \\in \\mathbb {R}^N, c \\in \\mathbb {R}^h$ , the distribution of observable variables $x$ is obtained by marginalizing the latent variables $z$ : $\\pi (x) \\propto \\exp (b^T x) \\prod _{i=1}^h (1 + \\exp (W_i x + c_i))$ We train the RBM on MNIST dataset using contrastive divergence [15] in three configurations C1: $h = 100$ C2: $h = 400$ C3: $h = 1000$ For each configuration, we sample Markov chain $x_{1:40000}$ , use $x_{1:20000}$ for burn in, use $x_{20001:40000}$ to estimate expected acceptance rate, expected jump distance, effective sample size.",
"We give the scatter plot of $(a, d)$ and $(a, r)$ in figure REF .",
"And we examine the performance of our adaptive algorithm in table REF and table REF .",
"Figure: Simulation Results on RBMTable: Expected Jump Distance on RBMTable: Effective Sample Size on RBMTable: Running Time on RBM"
]
] | 2209.08183 |
[
[
"RISE-Based Adaptive Control with Mass-Inertia Parameter Estimation for\n Aerial Transportation of Multi-Rotor UAVs"
],
[
"Abstract This paper proposes an adaptive tracking strategy with mass-inertia estimation for aerial transportation problems of multi-rotor UAVs.",
"The dynamic model of multi-rotor UAVs with disturbances is firstly developed with a linearly parameterized form.",
"Subsequently, a cascade controller with the robust integral of the sign of the error (RISE) terms is applied to smooth the control inputs and address bounded disturbances.",
"Then, adaptive estimation laws for mass-inertia parameters are designed based on a filter operation.",
"Such operation is introduced to extract estimation errors exploited to theoretically guarantee the finite-time (FT) convergence of estimation errors.",
"Finally, simulations are conducted to verify the effectiveness of the designed controller.",
"The results show that the proposed method provides better tracking and estimation performance than traditional adaptive controllers based on sliding mode control algorithms and gradient-based estimation strategies."
],
[
"INTRODUCTION",
"In recent years, unmanned aerial vehicles (UAVs), especially multi-rotor UAVs, have been widely applied to military/civil transportation tasks such as parcel delivery [1], equipment deployment [2], and rescue missions [3].",
"These tasks share common characteristics that the mass-inertia parameters of UAVs vary from flight to flight, and a fine-tune of controllers to retain good tracking performances between missions is time-consuming [4].",
"Thus, the varying, namely, the uncertain mass-inertia parameters, can be problematic for the control of UAVs [5].",
"Meanwhile, these light and flexible UAVs are susceptible to external disturbances such as winds [6], which seriously degrades flight performances.",
"To maintain good flight performance of multi-rotor UAVs and better conduct transportation missions, researchers have proposed different control methods to reduce the influence of uncertain mass-inertia parameters and disturbances, such as PID control [7], [8], backstepping control [9], and sliding mode control (SMC) [10], just to enumerate a few.",
"These controllers incorporate mass-inertia changes caused by different loads into disturbances [11] and compensate for the overall disturbances via different methods.",
"For instance, [7] studied the stability of UAVs with dynamic load disturbances and improved the control performance by careful selection of control gains.",
"[10] proposed a non-singular terminal sliding mode controller with high-order sliding mode observers to address uncertain parameters and disturbances.",
"While incorporating mass-inertia uncertainties into disturbances is generally effective, improvements are possible.",
"In aerial transportation problems, the variable mass-inertia parameters of loads take a large proportion of the UAV self-parameters, but the ability of the above controllers to accommodate for such variation is moderate [12].",
"To tackle such problems, adaptive controllers with mass-inertia estimation have been considered.",
"These controllers provide mass-inertia estimation [13] to improve dynamic models while addressing external disturbances.",
"Hence, dynamic models can be corrected after UAVs take off with new loads, yielding better tracking performances [14], [15].",
"For instance, Mellinger et al.",
"[16] proposed an adaptive PID control method that estimates payload parameters during hover via the least-squares methods.",
"In [17], an adaptive cascade controller was designed with the estimation of external force and position of the center of mass.",
"[18] established a complete dynamics model of quadrotors and proposed an adaptive controller based on feedback linearization and mass-inertia estimation.",
"Bouadi et al.",
"[13] addressed a SMC algorithm with consideration of white Gaussian noise and mass-inertia uncertainties.",
"[19] designed a learning rate-based SMC controller with the estimation of the mass of variable loads for altitude control.",
"[20] proposed a finite-time sliding mode controller for disturbance rejection and an adaptive-tuning scheme for mass estimation.",
"In [21], no prior knowledge of uncertain parameters was required via adaptive estimation laws based on signum and saturation functions.",
"[22] designed a non-singular fast terminal sliding mode controller based on adaptive integral backstepping to overcome external disturbances and an adaptive estimation algorithm to estimate the variable mass of loads.",
"However, the methods mentioned can still be improved in two aspects.",
"Firstly, the parameter estimation performance is influenced by disturbances.The convergence of estimation error can not be guaranteed via traditional least-squares [16] or gradient-based [18] algorithms when external disturbance exists.",
"These methods are sensitive to disturbances and may trigger bursting phenomena, i.e., the estimated parameters may go to infinity, leading to the instability of the system [23].",
"In some improved gradient-based methods, the boundness of estimation error is retained, but the error convergence can not be achieved [17], [13].",
"In [19]$\\sim $[22], the estimation gradient is determined by high-order tracking errors, which makes the convergence speed easily interfered bu external disturbances.",
"Secondly, the performances of controllers are degraded in practical applications.",
"While SMC has been exploited in [21] to address the influence of environmental disturbances, the performance in practical application is not satisfactory enough.",
"The chattering phenomena of sliding mode controllers make the control input signal unreachable physically [19], which significantly degrades the control performances.",
"In [20], [22], such phenomena are restrained via continuous terms in sliding surfaces [22], but the rapid-changing amplitude of control input signals are still hard to achieve in physical systems.",
"Given the discussion above, this article proposes a new adaptive control method with mass-inertia estimation and disturbance rejection for aerial transportation tasks of multi-rotor UAVs.",
"A RISE term [24] is applied for smoothing control inputs of the controller and disturbance rejection.",
"A filter operation [25] is introduced to extract estimation errors exploited to guarantee the FT convergence of estimation errors theoretically.",
"Then adaptive estimation laws are designed based on the extracted history estimation errors.",
"The major contributions of our work are summarized as follows: An adaptive control method based on RISE terms is formulated with mass-inertia estimation.",
"The scheme guarantees the asymptotic convergence of tracking error and FT convergence of estimated parameters under disturbances and provides smooth control input signals achievable in practical applications.",
"The effectiveness of the proposed method is verified through comparative simulation results with MATLAB.",
"The rest of this article is organized as follows.",
"A mathematical model of the studied multirotor system is described in Section II.",
"Section III provides the cascade controller design and Section IV formulates the parameter update law.",
"Afterward, stability analysis is conducted in Section V. Section VI presents comparative simulation results.",
"Finally, conclusions are drawn in Section VII."
],
[
"DYNAMIC MODEL",
"To develop the dynamic model of UAV, the defination of frames is first given as is shown in the following figure.",
"The inertial frame (the earth frame) $\\lbrace E\\rbrace $ is fixed on the ground and the body fixed frame $\\lbrace B\\rbrace $ is chosen to coincide with the geometric center of the UAV.",
"Let $\\eta _1 \\triangleq [\\begin{matrix}x & y & z \\end{matrix}]^T \\in \\mathbb {R}^3$ denote the position of the origin of $\\lbrace B\\rbrace $ and $\\eta _2 \\triangleq [\\begin{matrix}\\phi & \\vartheta & \\psi \\end{matrix}]^T \\in \\mathbb {R}^3$ represent the three Euler angles roll, pitch and yaw in frame $\\lbrace E\\rbrace $ .",
"In order to simplify the model, the CoG is assumed to be fixed in $\\lbrace B\\rbrace $ when loads changes [19].",
"Ignoring the asymmetry of the multi-rotor UAV and according to the Newton-Euler formalism, the rigid body dynamics model used in the subsequent controller design and stability analysis are governed by ${\\left\\lbrace \\begin{array}{ll}m \\ddot{\\eta }_1 = F - \\left[ \\begin{matrix} 0 \\\\ 0 \\\\ mg\\end{matrix} \\right] - \\Delta _1 \\\\J \\ddot{\\eta }_2 = \\tau _B - \\dot{\\eta }_2 \\times J \\dot{\\eta }_2 - \\Delta _2\\end{array}\\right.",
"}$ where $m \\in \\mathbb {R}$ represents the unknown mass of the multirotor; $J \\in \\mathbb {R}^{3\\times 3}$ is a matrix representing unknown moment of inertia of the multirotor about the origin of $\\lbrace B\\rbrace $ .",
"Its non-diagonal elements are set to be zero due to the symmetry of the UAV.",
"$F \\in \\mathbb {R}^3$ denotes the multirotor force vector expressed in frame $\\lbrace E\\rbrace $ and $\\tau _B \\in \\mathbb {R}^3$ denotes the torque expressed in frame $\\lbrace B\\rbrace $ .",
"$\\Delta _1 \\in \\mathbb {R}^3$ and $\\Delta _2 \\in \\mathbb {R}^3$ are defined to express the unknown addictive nonlinear disturbances.",
"Equation (REF ) can be partly simplized and rewritten into a more compact form ${\\left\\lbrace \\begin{array}{ll}m \\ddot{\\eta }_1 + G + \\Delta _1 = F \\\\J \\ddot{\\eta }_2 +C(\\dot{\\eta }_2)\\dot{\\eta }_2 +\\Delta _2 = \\tau _B\\end{array}\\right.",
"}$ by defining $C(\\dot{\\eta }_2) \\triangleq -S(J\\dot{\\eta }_2)\\in \\mathbb {R}^{3 \\times 3}$ and $G \\triangleq [\\begin{matrix} 0 & 0 & mg \\end{matrix}]^T \\in \\mathbb {R}^3$ , where $S(\\cdot ) \\in \\mathbb {R}^{3 \\times 3}$ represents the skew symmetric matrix of a vector.",
"The rewritten equation (REF ) is still in a seperate form because the following controller and parameter update law design are developed in a cascade manner.",
"There are several properties and assumptions of the dynamics model which will be exploited in the subsequent development: Property 1.",
"Part of the dynamics equation (REF ) can be linearly parameterized as ${\\left\\lbrace \\begin{array}{ll}\\Psi _1 \\theta _1 \\triangleq m \\ddot{\\eta }_1 + G\\\\\\Psi _2 \\theta _2 \\triangleq J \\ddot{\\eta }_2 +C(\\dot{\\eta }_2)\\dot{\\eta }_2\\end{array}\\right.",
"}$ where $\\theta _1 \\in \\mathbb {R}$ and $\\theta _2 \\in \\mathbb {R}^{3 \\times 3}$ contain the unknown system mass-inertia parameters, $\\Psi _1(\\ddot{\\eta }_1) \\in \\mathbb {R}^3$ and $\\Psi _2(\\dot{\\eta }_2, \\ddot{\\eta }_2) \\in \\mathbb {R}^{3 \\times 3}$ are the regression matrices which contains known functions of measured acceleration, angular rate and angular acceleration respectively.",
"The above linearization can also be formulated with desired position and attitude vectors, yielding ${\\left\\lbrace \\begin{array}{ll}\\Psi _{1d} \\theta _1 \\triangleq m \\ddot{\\eta }_{1d} + G\\\\\\Psi _{2d} \\theta _2 \\triangleq J \\ddot{\\eta }_{2d} +C(\\dot{\\eta }_{2d})\\dot{\\eta }_{2d}\\end{array}\\right.",
"}$ where $\\Psi _{1d}(\\ddot{\\eta }_{1d}) \\in \\mathbb {R}^3$ and $\\Psi _{2d}(\\dot{\\eta }_{2d}, \\ddot{\\eta }_{2d}) \\in \\mathbb {R}^{3 \\times 3}$ are bounded desired regression matrices containing known functions of desired tracking vectors respectively.",
"Assumpition 1.",
"The regression matrices $\\Psi _{1d}$ and $\\Psi _{2d}$ defined above satisfy the PE condition described in [26], which can be easily fulfilled in our experiments.",
"And the condition is important for the parameter update law given later in the article.",
"Assumpition 2.",
"The nonlinear disturbances $\\Delta _1$ and $\\Delta _2$ and their first two-order time derivatives, i.e.",
"$\\dot{\\Delta }_i$ , $\\ddot{\\Delta }_i$ , $(i = 1,2)$ are bounded by known constants."
],
[
"CONTROL DESITN",
"The control objective is to design a controller which guarantees that the system tracks a desired trajectory $\\eta _{1d}$ and $\\psi _d$ despite the bounded disturbances and uncertain parameters in the dynamics model.",
"The desired trajectory $\\eta _{1d}$ and $\\psi _d$ are designed such that $\\eta ^{(i)}_{1d}(t)$ and $\\psi ^{(i)}_d(t)$ , $i = 0, 1, ...4$ exist and are bounded.",
"The controller illustrated in REF is constructed with a cascade structure consisting of an outer-loop controller and an inner-loop controller.",
"The outer-loop controller generates the thrust $F$ and desired roll, pitch angles to track the desired position trajectory and yaw angle.",
"The inner-loop controller is designed to generate the torque $T$ needed to track the desired yaw angle and the calculated roll and pitch angle trajectories.",
"Figure: Controller"
],
[
"Outer-Loop Controller",
"To quantify the control performance, tracking error $e_{o1} \\in \\mathbb {R}^3$ , and two auxiliary filtered tracking errors $e_{o2} \\in \\mathbb {R}^3$ and $r_{o} \\in \\mathbb {R}^3$ are defined as follows: $\\begin{aligned}& e_{o1} \\triangleq \\eta _{1d} - \\eta _1, \\\\& e_{o2} \\triangleq \\dot{e}_{o1} + k_{o1}e_{o1}, \\\\& r_{o} \\triangleq \\dot{e}_{o2} + k_{o2}e_{o2},\\end{aligned}$ where $k_{o1}$ , $k_{o2}$ $\\in \\mathbb {R}^+$ are designed constant control gains.",
"By substituting the errors in (REF ) and the linearized form REF into the first dynamic equation in (REF ), the open-loop error dynamics of outer-loop system can be developed as: $mr_{o} = \\Psi _{1d} \\theta _1 + S_1 + \\Delta _1 - F$ where the auxiliary function $S_1 \\in \\mathbb {R}^3$ is defined as $S_1 \\triangleq m(k_{o1} \\dot{e}_{o1} + k_{o2}\\dot{e}_{o2})$ The output force $F$ can be designed with an adaptive feedforward term and a RISE feedback term as $F \\triangleq \\Psi _{1d} \\hat{\\theta }_1 + \\mu _1$ In (REF ), $\\hat{\\theta }_1 \\in \\mathbb {R}$ denotes the adaptive estimate for the unknown parameter $\\theta _1$ whose implementation will be discussed with detail in Section IV later.",
"$\\mu _1 \\in \\mathbb {R}^3$ represents the RISE feedback term described in [27], which is designed as $\\begin{aligned}\\mu _1 \\triangleq &(k_{s1} +1)e_{o2} - (k_{s1} +1)e_{o2}(0) \\\\&+ \\int _0^t[(k_{s1} + 1)k_{o2}e_{o2}(\\tau ) + \\beta _o sgn(e_{o2}(\\tau ))] d\\tau \\end{aligned}$ where $k_{s1} \\in \\mathbb {R}^+$ and $\\beta _o \\in \\mathbb {R}^+$ are constant control gains and $sgn(\\cdot )$ represents the signum function.",
"Then the time derivative of the RISE term can be derived as $\\dot{\\mu }_1 = (k_{s1} + 1)r_{o} + \\beta _o sgn(e_{o2})$ Substituting equation (REF ) into (REF ), the closed-loop error dynamics of outer-loop system can be developed as $mr_o = \\Psi _{1d} \\tilde{\\theta }_1 + S_1 + \\Delta _1 - \\mu _1$ where $\\tilde{\\theta }_1 \\triangleq \\theta _1 - \\hat{\\theta }_1 \\in \\mathbb {R}$ denotes the parameter estimation error.",
"Equation (REF ) will be exploited in Section V to facilitate stability analysis of outer-loop controller.",
"The desired body attitude in frame $\\lbrace E\\rbrace $ then can be derived in the same manner as [28], by first calculating the desired z-axis direction of the body frame $\\lbrace B\\rbrace $ , which is alone the desired output force $F$ : ${\\bf z}_B = \\frac{F}{||F||}$ where $||\\cdot ||$ denotes the Euclidean norm.",
"$||F||$ will be nonzero to avoid free-falling.",
"Given the desired yaw angle $\\psi _d$ , a unit vector ${\\bf x}_C \\in \\mathbb {R}^3$ can be defined as ${\\bf x}_C \\triangleq [\\begin{matrix} -s\\psi _d & c\\psi _d & 0 \\end{matrix}]^T$ where $s \\psi _d$ and $c \\psi _d$ denotes $sin(\\psi _d)$ and $cos(\\psi _d)$ respectively.",
"Provided ${\\bf x}_C \\times {\\bf z}_B \\ne 0$ , the orientation of frame $\\lbrace B\\rbrace $ can be uniquely determined as $\\begin{aligned}& {\\bf x}_B = \\frac{{\\bf x}_C \\times {\\bf z}_B}{||{\\bf x}_C \\times {\\bf z}_B||} \\\\& {\\bf y}_B = {\\bf z}_B \\times {\\bf x}_B \\\\& R_B^E \\triangleq [\\begin{matrix} {\\bf x}_B & {\\bf y}_B &{\\bf z}_B\\end{matrix}]\\end{aligned}$ where ${\\bf x}_B$ and ${\\bf y}_B$ are x and y axes of frame $\\lbrace B\\rbrace $ respectively.",
"$R_B^E \\in \\mathbb {R}^{3 \\times 3}$ is the rotation matrix from the body-fixed frame $\\lbrace B\\rbrace $ to the inertial frame $\\lbrace E\\rbrace $ , given by $R_B^E =\\left[ \\begin{matrix}c\\psi c\\vartheta & c\\psi s\\vartheta s\\phi - s\\psi c\\phi & c\\psi s\\vartheta c\\phi + s\\psi s\\phi \\\\s\\psi c\\vartheta & s\\psi s\\vartheta s\\phi + c\\psi c\\phi & s\\psi s\\vartheta c\\phi - c\\psi s\\phi \\\\-s\\vartheta & c\\vartheta s\\phi & c\\vartheta c\\phi \\end{matrix} \\right]$ The desired roll angle $\\phi _d$ and pitch angle $\\theta _d$ can be calculated from $\\psi _d$ and $F$ via equations (REF ), (REF ), (REF ) and (REF )."
],
[
"Inner-Loop Controller",
"The realization of inner loop controller is similar to that of the outer controller.",
"First, tracking error $e_{i1} \\in \\mathbb {R}^3$ and auxiliary filtered errors $e_{i2}$ , $r_{i} \\in \\mathbb {R}^3$ are defined as $\\begin{aligned}& e_{i1} \\triangleq \\eta _{2d} - \\eta _2, \\\\& e_{i2} \\triangleq \\dot{e}_{i1} + k_{i1}e_{i1}, \\\\& r_{i} \\triangleq \\dot{e}_{i2} + k_{i2}e_{i2},\\end{aligned}$ where $k_{i1} \\in \\mathbb {R}$ and $k_{i2} \\in \\mathbb {R}$ are constant control gains.",
"By substituting the errors in (REF ) and the linearized form REF into the second equation in (REF ), the open-loop error dynamics of outer-loop system can be developed as: $Jr_{i} = \\Psi _{2d} \\theta _2 + S_2 + \\Delta _2 - \\tau _B$ where the auxiliary function $S_1 \\in \\mathbb {R}^3$ is defined as $\\begin{aligned}S_2 \\triangleq & J(k_{i1} \\dot{e}_{i1} + k_{i2}\\dot{e}_{i2}) + C(\\dot{\\eta }_2)\\dot{\\eta }_2 - C(\\dot{\\eta }_{2d})\\dot{\\eta }_{2d}\\end{aligned}$ The inner-loop control output $T$ can be designed with an adaptive feedforward term and a RISE feedback term as $\\tau _B \\triangleq \\Psi _{2d} \\hat{\\theta }_2 + \\mu _2$ In (REF ), $\\hat{\\theta }_2 \\in \\mathbb {R}$ denotes the adaptive estimate for the unknown parameter $\\theta _2$ ; $\\mu _2 \\in \\mathbb {R}^3$ represents the RISE feedback term and is designed similar to equation (REF ) as $\\begin{aligned}\\mu _2 \\triangleq &(k_{s2} +1)e_{i2} - (k_{s2} +1)e_{i2}(0) \\\\&+ \\int _0^t[(k_{s2} + 1)k_{i2}e_{i2}(\\tau ) + \\beta _i sgn(e_{i2}(\\tau ))] d\\tau \\end{aligned}$ and its time derivative similar to (REF ) as $\\dot{\\mu }_2 = (k_{s2} + 1)r_{i} + \\beta _i sgn(e_{i2})$ where $k_{s2} \\in \\mathbb {R}^+$ and $\\beta _i \\in \\mathbb {R}^+$ are constant control gains.",
"Substituting equation (REF ) into (REF ), the closed-loop error dynamics of outer-loop system can be developed as $Jr_i = \\Psi _{2d} \\tilde{\\theta }_2 + S_2 + \\Delta _2 - \\mu _2$ where $\\tilde{\\theta }_2 \\triangleq \\theta _2 - \\hat{\\theta }_2 \\in \\mathbb {R}^3$ denotes the parameter estimation error.",
"Equation (REF ) will be exploited in Section V to facilitate stability analysis of inner-loop controller."
],
[
"PARAMETER ESTIMATION",
"The parameter estimation is conducted in both of the control loops with the same error extraction process.",
"In the outer-loop, $\\hat{\\theta }_1$ is calculated and exploited to generate control outputs, $\\hat{\\theta }_2$ in the inner-loop respectively."
],
[
"Estimation In Outer-Loop",
"The estimation starts with the defination of two filtered auxiliary vectors $F_f$ , $\\Psi _{1f} \\in \\mathbb {R}^3$ as the solutions to the following equation ${\\left\\lbrace \\begin{array}{ll}\\begin{aligned}&\\alpha _1 \\dot{F}_f + F_f = F, \\quad F_f(0) = \\bf 0 \\\\&\\alpha _1 \\dot{\\Psi }_{1f} + \\Psi _{1f} = \\Psi _1, \\quad \\Psi _{1f} = \\bf 0\\end{aligned}\\end{array}\\right.",
"}$ where $\\alpha _1 \\in \\mathbb {R}^+$ is a designed constant.",
"Another filtered variable only used for analysis $\\Delta _{1f} \\in \\mathbb {R}^3$ is also defined as $\\alpha _1 \\dot{\\Delta }_{1f} + \\Delta _{1f} = \\Delta _1, \\quad \\Delta _{1f} = \\bf 0$ where $\\Delta _{1f}$ is bounded given that $\\Delta _1$ is bounded.",
"(REF ) and (REF ) acctually exert the same low-pass filter operation on both sides of the linearized dynamic model $\\Psi _1 \\theta _1 +\\Delta _1 = F$ Then a filtered form of the above equation can be expressed as $\\Psi _{1f} \\theta _{1} +\\Delta _1 = F_f$ To extract the estimation error $\\tilde{\\theta }_1$ , $P_1$ , $Q_1 \\in \\mathbb {R}$ are defined as ${\\left\\lbrace \\begin{array}{ll}\\begin{aligned}& P_1 \\triangleq \\int _0^t e^{-l_1(t-\\tau )}\\Psi _{1f}^T(\\tau ) \\Psi _{1f}(\\tau ) d \\tau + \\varrho _1\\\\& Q_1 \\triangleq \\int _0^t e^{-l_1(t-\\tau )}\\Psi _{1f}^T(\\tau ) F_f(\\tau ) d \\tau \\end{aligned}\\end{array}\\right.",
"}$ which are the solutions to the equation below ${\\left\\lbrace \\begin{array}{ll}\\begin{aligned}& \\dot{P}_1 = -l_1 P_1 + \\Psi _{1f}^T \\Psi _{1f}, \\quad P_1(0) = \\varrho _1\\\\& \\dot{Q}_1 = -l_1 Q_1 + \\Psi _{1f}^T F_f, \\quad Q_1(0) = 0\\end{aligned}\\end{array}\\right.",
"}$ where $l_1 \\in \\mathbb {R}^+$ is a designed constant, and $\\varrho _1 \\in \\mathbb {R}^+$ is a positive constant selected to ensure $P_1(0)$ is inversible at time $t = 0$ .",
"Such definition of $P_1$ yields the following property: Property 3.",
"$P_1$ is a positive variable satisfying $0 < \\varrho _1 < P_1$ .",
"Then $P_1^{-1}$ is globally invertible provided that the offset value $\\varrho _1$ is not selected as 0.",
"The proof of this property is similar to that of [25].",
"Similar to $P_1$ and $Q_1$ , $\\bar{\\Delta }_1 \\in \\mathbb {R}$ is defined as $\\bar{\\Delta }_1 \\triangleq - \\int _0^t e^{-l_1(t-\\tau )}\\Psi _{1f}^T(\\tau ) \\Delta _1 d \\tau + \\varrho _1 \\theta _1$ which is bounded by $||\\bar{\\Delta }_1||\\le \\xi _{\\Delta _1}$ , where $\\xi _{\\Delta _1} \\in \\mathbb {R}^+$ is a positive constant, since the regression vector $\\Psi _{1f}$ is locally bounded and $\\Delta _1$ is bounded.",
"Substituting the linearized form (REF ) into system dynamics (REF ), and substituting equation (REF ), (REF ) into (REF ), yields $Q_1 = P_1 \\theta _1 - \\bar{\\Delta }_1$ Then the estimation error is extracted by defining $H_1 \\in \\mathbb {R}$ as $H_1 \\triangleq P_1 \\hat{\\theta }_1 - Q_1$ which contains the estimation error $\\tilde{\\theta }_1$ as $H_1 = -P_1 \\tilde{\\theta }_1 + \\bar{\\Delta }_1$ is derived by substituting equation (REF ) into (REF ).",
"Based on the extracted estimation error above, the parameter update law can be designed as $\\dot{\\hat{\\theta }}_1 = -\\gamma \\left( \\gamma _1 H_1 +\\mathrm {sat}\\left(H_1\\right)\\right)$ where $\\gamma $ , $\\gamma _1 \\in \\mathbb {R}^+$ are positive learing gains and the saturation function $\\mathrm {sat}(\\cdot ): \\mathbb {R} \\rightarrow \\mathbb {R}$ is defined as $\\mathrm {sat}(x) = \\left\\lbrace \\begin{matrix}\\begin{aligned}&1, \\quad x>1 \\\\&x, \\quad |x|\\le 1 \\\\&-1, \\quad x<-1 \\\\\\end{aligned}\\end{matrix}\\right.$"
],
[
"Estimation In Inner-Loop",
"In the same manner as estimation in outer-loop, filtered auxiliary vectors $\\Psi _{2f} \\in \\mathbb {R}^{3\\times 3}$ , $R_f$ amd $\\Delta _{2f} \\in \\mathbb {R}^3$ are defined by the following differential equations: ${\\left\\lbrace \\begin{array}{ll}\\begin{aligned}&\\alpha _2 \\dot{\\tau }_{Bf} + \\tau _{Bf} = \\tau _B, \\quad \\tau _{Bf}(0) = \\bf 0 \\\\&\\alpha _2 \\dot{\\Psi }_{2f} + \\Psi _{2f} = \\Psi _2, \\quad \\Psi _{2f} = \\bf 0 \\\\&\\alpha _2 \\dot{\\Delta }_{2f} + \\Delta _{2f} = \\Delta _2, \\quad \\Delta _{2f} = \\bf 0\\end{aligned}\\end{array}\\right.",
"}$ where $\\alpha _2 \\in \\mathbb {R}^+$ is a designed constant.",
"For estimation error extraction, $P_2 \\in \\mathbb {R}^{3 \\times 3}$ , $Q_2 \\in \\mathbb {R}^3$ , and$\\bar{\\Delta }_2 \\in \\mathbb {R}^3$ are defined as ${\\left\\lbrace \\begin{array}{ll}\\begin{aligned}& P_2 \\triangleq \\int _0^t e^{-l_2(t-\\tau )}\\Psi _{2f}^T(\\tau ) \\Psi _{2f}(\\tau ) d \\tau + \\varrho _2 E_3\\\\& Q_2 \\triangleq \\int _0^t e^{-l_2(t-\\tau )}\\Psi _{2f}^T(\\tau ) \\tau _{Bf}(\\tau ) d \\tau \\\\& \\bar{\\Delta }_2 \\triangleq - \\int _0^t e^{-l_2(t-\\tau )}\\Psi _{2f}^T(\\tau ) \\Delta _2 d \\tau +\\varrho _2 E_3 \\theta _2\\end{aligned}\\end{array}\\right.",
"}$ where $l_2$ , $\\varrho _2 \\in \\mathbb {R}^+$ are designed positive constants and $E_3 \\in \\mathbb {R}^{3\\times 3}$ is the identity matrix.",
"$\\bar{\\Delta }_2$ is bounded by $||\\bar{\\Delta }_2|| \\le \\xi _{{\\Delta }_2}$ .",
"Simlar to $P_1$ , $P_2$ has the has the following property: Property 4.",
"$P_2$ is a positive definite matrix satisfying $0 < \\varrho _2 < \\lambda _m(P_2)$ where $\\lambda _m(P_2)$ is the minimum eigenvalue of $P_2$ .",
"And $P_2^{-1}$ is globally invertible.",
"Define $H_2 \\in \\mathbb {R}^3$ as $H_2 \\triangleq P_2 \\hat{\\theta }_2 - Q_2$ which yields $H_2 = -P_2 \\tilde{\\theta }_2 + \\bar{\\Delta }_2$ in the same manner as estimation in outer-loop.",
"The parameter update law for inner-loop can be designed as $\\dot{\\hat{\\theta }}_2 = -\\Gamma \\left(\\sigma _1 H_2 +\\sigma _1 \\frac{P_2^TH_2}{||P_2||} + \\sigma _2 \\frac{P_2^TH_2}{||P_2||\\cdot ||H_2||}\\right)$ where $\\Gamma \\in \\mathbb {R}^{3\\times 3}$ is a positive definite diagonal matrix, and $\\sigma _1$ , $\\sigma _2 \\in \\mathbb {R}^+$ are positive constants."
],
[
"STABILITY ANALYSIS",
"The stability analysis for the proposed methed is conducted in two parts: outer-loop and inner-loop.",
"Both controllers yields asymptotic convergence of tracking error and finite time convergence of estimation error."
],
[
"Inner-Loop Analysis",
"To facilitate stability analysis of inner-loop, the time derivative of equation (REF ) is exploited: $J\\dot{r}_i = \\tilde{N}_i +N_{{\\Delta }_i} - \\dot{\\mu }_2 - e_{i2}$ In equation (REF ), part of the equation is seperated into two unmeasurable auxiliary functions $\\tilde{N}_i$ , $N_{\\Delta _i}$ $\\in \\mathbb {R}^3$ which are upper-bounded by different terms.",
"The motivation for such operation has been discussed in [29].",
"Substituting equation (REF ) into equation (REF ), $\\tilde{N}_i$ and $N_{\\Delta _i}$ can be defined as $\\begin{aligned}& \\tilde{N}_i(t) \\triangleq \\dot{S}_2 + e_{i2} + N_i \\\\& N_{\\Delta _2} \\triangleq \\dot{\\Delta }_2\\end{aligned}$ where $N_i \\in \\mathbb {R}^3$ is another auxiliary function defined as $N_i \\triangleq \\dot{\\Psi }_{2d} \\tilde{\\theta }_2 - \\Psi _{2d}\\dot{\\hat{\\theta }}_2$ As is discussed in [27], $\\tilde{N}_i$ is upper bounded as follows: $||\\tilde{N}_i|| \\le \\rho (||z_i||)|| z_i||$ where the outer-loop error signal $z_i \\in \\mathbb {R}^{12}$ is defined as $z_i \\triangleq \\left[ \\begin{matrix} e_{i1}^T & e_{i2}^T & r_i^T & \\tilde{\\theta }_2 \\end{matrix}\\right]^T$ and $\\rho : \\mathbb {R}_{\\ge 0}\\rightarrow \\mathbb {R}_{\\ge 0}$ is a globally invertible, nondecreasing function.",
"From Assumption 1, $||N_{\\Delta _i}||$ and $||\\dot{N}_{\\Delta _i}||$ are bounded by positive constants: $||N_{\\Delta _i}|| \\le \\xi _{i}, \\quad ||\\dot{N}_{\\Delta _i}|| \\le \\dot{\\xi }_i$ Lemma 1.",
"Let the auxiliary function $L_i(t) \\in \\mathbb {R}$ be defined as follows: $L_i(t) \\triangleq r_i^T\\left(N_{\\Delta _i} - \\beta _i sgn(e_{i2})\\right) + C_i$ If the control gain $\\beta _i$ is selected to fulfill the following condition: $\\beta _i > \\xi _i + \\frac{1}{k_{i2}} \\dot{\\xi }_i$ and $C_i \\in \\mathbb {R}^+$ is defined as $C_i \\triangleq \\sigma _1 \\left( \\frac{1}{\\lambda _i} +\\frac{1}{\\varrho _2}\\right)\\xi _{\\Delta _2}^2 + \\sigma _2 \\frac{1}{\\varrho _2} \\xi _{\\Delta _2}$ where $\\lambda _i < \\varrho _2$ is a positive constant.",
"Then $W_i \\in \\mathbb {R}$ defined by the following differential equation is always positive: $\\begin{aligned}&\\dot{W}_i \\triangleq - \\dot{L}_i \\\\& W_i(0) \\triangleq \\beta _i |e_{i2}(0)| - e_{i2}(0)N_{\\Delta _i}(0)\\end{aligned}$ The proof of Lemma 1 is similar to that given in [23] and [27].",
"Theorem 1.",
"The inner-loop controller given in equation (REF ), (REF ), and (REF ) ensures that signal $z_i$ is regulated that $||z_i(t)|| \\rightarrow 0 $ as $t \\rightarrow \\infty $ provided that control gain $k_{s2}$ is selected sufficiently large, $k_{i1}, k_{i2} > \\frac{1}{2}$ , and $\\beta _i$ following the condition (REF ).",
"Proof.",
"Define an auxiliary vector $y \\in \\mathbb {R}^{13}$ as $y \\triangleq \\left[ \\begin{matrix} z_i^T &\\sqrt{W_i}\\end{matrix} \\right]^T$ and let $\\mathcal {D} \\subset \\mathbb {R}^{13}$ be a domain containing $y(t) = \\bf {0}$ .",
"Define a Lyapunov function candidate as $V_1(y,t) \\triangleq \\frac{1}{2} e_{i1}^Te_{i1} + \\frac{1}{2} e_{i2}^Te_{i2} + \\frac{1}{2}r_{i}^T J r_{i} + W_i + \\frac{1}{2} \\tilde{\\theta }_i^T\\Gamma ^{-1} \\tilde{\\theta _2}$ where $V_1(y,t): \\mathcal {D} \\rightarrow \\mathbb {R}$ is a positive definite, continuously differentiable function which satisfies $U_1(y)\\le V_1(y,t) \\le U_2(y)$ In equation (REF ), $U_1(y)$ , $U_2(y) \\in \\mathbb {R}$ are continuous positive definite functions which are defined as $\\begin{aligned}& U_1(y) \\triangleq c_1 ||y||^2 \\\\& U_2(y) \\triangleq c_2 ||y||^2\\end{aligned}$ where $c_1$ , $c_2 \\in \\mathbb {R}^+$ are defined as $\\begin{aligned}& c_1 \\triangleq \\frac{1}{2}min \\lbrace 1, \\underline{J}, {\\overline{\\Gamma }}^{-1} \\rbrace \\\\& c_2 \\triangleq \\frac{1}{2}max \\lbrace 1, \\overline{J}, {\\underline{\\Gamma }}^{-1} \\rbrace \\end{aligned}$ In (REF ), $\\overline{J}$ and $\\underline{J}$ indicate the maximum and minimum element of the diagonal matrix $J$ respectively.",
"The time derivative of $V_1(y,t)$ in (REF ) is expressed as $\\dot{V}_1 = e_{i1}^T \\dot{e}_{i1} + e_{i2}^T \\dot{e}_{i2} + r_i^T J \\dot{r}_i + \\dot{W}_i + \\tilde{\\theta }_2^T \\Gamma ^{-1} \\dot{\\tilde{\\theta }}_2$ Substituting equation (REF ) and (REF ) into the time derivative above, one has $\\begin{aligned}\\dot{V}_1 =& e_{i1}^T(e_{i2}-k_{i1}e_{i1}) + e_{i2}^T(r_i-k_{i2}e_{i2}) +\\dot{W}_i + \\tilde{\\theta }_1^T \\Gamma ^{-1} \\dot{\\tilde{\\theta }}_2 \\\\& + r_i^T(\\tilde{N}_i(t) + N_{\\Delta _i}-\\dot{\\mu }_2 - e_{i2})\\end{aligned}$ With the definition of $\\mu _2$ in (REF ) and $W_i$ in (REF ), some of the terms in (REF ) can be eliminated, which yields $\\begin{aligned}\\dot{V}_1 =& -k_{i1}e_{i1}^2 - k_{i2}e_{i2}^2 + e_{i1}^T e_{i2} - (k_{s2} + 1)r_i^2 + r_i^T \\tilde{N}_i(t) \\\\& - C_i + \\tilde{\\theta }_2^T \\Gamma ^{-1} \\dot{\\tilde{\\theta }}_2\\end{aligned}$ From equation (REF ) and the definition of $\\tilde{\\theta }_2$ , the time derivative of $\\tilde{\\theta }_2$ is expressed as $\\dot{\\tilde{\\theta }}_2 = \\Gamma \\left(\\sigma _1 H_2 +\\sigma _1 \\frac{P_2^TH_2}{||P_2||} + \\sigma _2 \\frac{P_2^TH_2}{||P_2||\\cdot ||H_2||}\\right)$ Then (REF ) is expressed as $\\begin{aligned}\\dot{V}_1 =& -k_{i1}e_{i1}^2 - k_{i2}e_{i2}^2 + e_{i1}^T e_{i2} - (k_{s2} + 1)r_i^2 + r_i^T \\tilde{N}_i(t) \\\\& - C_i + \\tilde{\\theta }_2^T \\left(\\sigma _1 H_2 +\\sigma _1 \\frac{P_2^TH_2}{||P_2||} + \\sigma _2 \\frac{P_2^TH_2}{||P_2||\\cdot ||H_2||}\\right) \\\\= & -k_{i1}e_{i1}^2 - k_{i2}e_{i2}^2 + e_{i1}^T e_{i2} - (k_{s2} + 1)r_i^2 + r_i^T \\tilde{N}_i(t) \\\\& - C_i -\\sigma _1\\tilde{\\theta }_2^T(P_2\\tilde{\\theta }_2 - \\bar{\\Delta }_2) - \\sigma _1 \\frac{(H_2 -\\bar{\\Delta }_2)^TH_2}{||P_2||}\\\\&- \\sigma _2 \\frac{(H_2 - \\bar{\\Delta }_2)^TH_2}{||P_2||\\cdot ||H_2||}\\end{aligned}$ By using Young's inequality and the bound of $\\tilde{N}_i(t)$ in (REF ),the following expressions are yielded $\\begin{aligned}& e_{i1}^T e_{i2} \\le \\frac{1}{2}(||e_{i1}||^2 + ||e_{i2}||^2) \\\\& r_i^T \\tilde{N}_i(t) \\le k_{s2}||r_i||^2 + \\frac{1}{4k_{s2}}\\rho ^2(||z_i||)||z_i||^2 \\\\&-\\sigma _1\\tilde{\\theta }_2^T(P_2\\tilde{\\theta }_2 - \\bar{\\Delta }_2) - \\sigma _1 \\frac{(H_2 -\\bar{\\Delta }_2)^TH_2}{||P_2||}-\\sigma _2 \\frac{(H_2-\\bar{\\Delta }_2)^TH_2}{||P_2||\\cdot ||H_2||} \\\\& \\le -\\sigma _1( \\varrho _2- \\lambda _i)||\\tilde{\\theta }_2||^2 + C_i\\end{aligned}$ Substituting (REF ), (REF ) is upper bounded as $\\begin{aligned}\\dot{V}_1 \\le & - (k_{i1} - \\frac{1}{2}) ||e_{i1}||^2 - (k_{i2} - \\frac{1}{2}) ||e_{i2}||^2 - ||r_i||^2 \\\\& + \\frac{1}{4k_{s2}}\\rho ^2(||z_i||)||z_i||^2 - \\sigma _1( \\varrho _2- \\lambda _i)||\\tilde{\\theta }_2||^2 \\\\\\le & -\\left(c_3 - \\frac{1}{4k_{s2}}\\rho ^2(||z_i||)\\right)||z_i||^2\\end{aligned}$ where $c_3 \\in \\mathbb {R}$ is defined as $c_3 \\triangleq min \\left\\lbrace k_{i1} - \\frac{1}{2}, k_{i2} - \\frac{1}{2},1,\\sigma _1(\\varrho _2- \\lambda _i)\\right\\rbrace $ $c_3$ is positive provided the definition of $\\lambda _i$ in Lemma 1.",
"The expression in (REF ) can be further upper bounded as $\\dot{V}_1 \\le -c_i ||y||^2, \\quad \\forall y \\in \\mathcal {D}_1$ for some positive constant $c_i$ .",
"Set $\\mathcal {D}_1 \\subset \\mathcal {D}$ is defined as $\\mathcal {D}_1 \\triangleq \\left\\lbrace y(t) \\in \\mathbb {R}^{13} \\mid ||y(t)|| \\le \\rho ^{-1}\\left(2\\sqrt{c_3k_{s2}}\\right) \\right\\rbrace $ The inequality (REF ) shows that $V_1(y,t) \\in \\mathcal {L}_{\\infty }$ in $\\mathcal {D}_1$ ; hence $e_{i1}$ , $e_{i2}$ , $r_i$ , and $\\tilde{\\theta }_2\\in \\mathcal {L}_{\\infty }$ in $\\mathcal {D}_1$ .",
"Similar to proof in [24], The attraction region $\\mathcal {R}_1 \\subset \\mathcal {D}_1$ as $\\mathcal {R}_1 \\triangleq \\left\\lbrace y(t) \\in \\mathbb {R}^{13} \\mid U_2(y) \\le c_1 \\left(\\rho ^{-1}\\left(2\\sqrt{c_3k_{s2}}\\right)\\right)^2 \\right\\rbrace $ Hence, $||y(t)|| \\rightarrow 0$ as $t\\rightarrow \\infty , \\forall y(0) \\in \\mathcal {R}_1$ , which further indicates that $||z_i(t)|| \\rightarrow 0$ as $t\\rightarrow \\infty , \\forall y(0) \\in \\mathcal {R}_1$ .",
"Corollary 1.",
"$P_2$ is upper bounded by $||P_2|| \\le \\xi _{p2}$ , where $\\xi _{p2} \\in \\mathbb {R}^+$ is a positive constant.",
"Proof.",
"From Theorem 1, $||e_{i1}|| \\rightarrow 0$ as $t \\rightarrow 0$ .",
"Since $||\\Psi _{2d}||$ is bounded in Property 1, the continuous function $\\Psi _{2f}$ is upper bounded by $||\\Psi _{2f}|| \\le \\zeta _2$ , where $\\xi _2 \\triangleq \\frac{1}{l_2}\\zeta _2^2 + ||\\varrho _2||\\in \\mathbb {R}^+$ is a positive constant.",
"From equation (REF ), $||P_2|| = e^{-l_2t}\\left\\Vert \\int _0^t e^{l_2\\tau )}\\Psi _{2f}^T(\\tau ) \\Psi _{2f}(\\tau ) d \\tau \\right\\Vert + \\left\\Vert \\varrho _2 E_3\\right\\Vert $ The norm of $P_2$ is upper bounded by $\\begin{aligned}||P_2|| & \\le e^{-l_2 t}\\zeta _2^2 \\int _0^te^{l_2 \\tau }d\\tau + \\varrho _2 \\\\& \\le \\frac{1}{l_2}\\zeta _2^2 + \\varrho _2\\end{aligned}$ The corollary is proved.",
"Theorem 2.",
"For error system (REF ) with the adaptive estimation law given in (REF ), the estimation error variable $P_2^{-1}H_2$ is regulated that $||P_2^{-1}H_2|| \\rightarrow 0 $ in a finite time $t_1$ if $\\Gamma $ , $\\sigma _1$ , $\\sigma _2$ and $\\varrho _2$ are properly selected (see the subsequent proof).",
"And the estimation error $\\tilde{\\theta }_2$ is guaranteed to converge to a compact set around zero in $t_i$ .",
"Proof.",
"Let $\\Xi \\subset \\mathbb {R}^3$ be a domain containing $||P_2^{-1}(t)H_2(t)|| = 0$ .",
"Define a Lyaponov candidate as $V_2(P_2^{-1}H_2) \\triangleq \\frac{1}{2}H_2^T P_2^{-1}P_2^{-1}H_2$ where $V_2(P_2^{-1}H_2): \\Xi \\rightarrow \\mathbb {R}_{\\ge 0}$ is a positive definite, continuously differentiable function satisfies a similar condition to (REF ).",
"The time derivative of $V_2$ is expressed as $\\dot{V}_2 = H_2^T P_2^{-1} \\frac{\\partial }{\\partial t} \\left(P_2^{-1} H_2\\right)$ From equation (REF ), one has $P_2^{-1}H_2 = -\\tilde{\\theta }_2 + P_2^{-1}\\bar{\\Delta }_2$ Exploiting $\\frac{\\partial }{\\partial t}P_2^{-1} = -P_2^{-1}\\dot{P}_2 P_2^{-1}$ , and substituting equation (REF ) and (REF ) into (REF ), $\\dot{V}_2$ can be expressed as $\\begin{aligned}\\dot{V}_2 =& - H_2^TP_2^{-1}\\left(\\Gamma \\sigma _1 H_2 + \\Gamma \\sigma _1 \\frac{P_2^T H_2}{||P_2||}\\right) \\\\&- H_2^TP_2^{-1}\\left( \\Gamma \\sigma _2 \\frac{P_2^T H_2}{||P_2||\\cdot ||H_2||} - \\Phi \\right)\\end{aligned}$ where $\\Phi \\in \\mathbb {R}^3$ is defined as $\\Phi \\triangleq - P_2^{-1}\\dot{P}_2P_2^{-1} \\bar{\\Delta }_2 + P_2^{-1} \\dot{\\bar{\\Delta }}_2$ From Assumption 2, Property 4, and Corollary 1, $\\Phi $ is verified to be bounded, and $\\dot{V}_2$ is upper bounded as $\\begin{aligned}\\dot{V}_2 \\le &-\\left(\\underline{\\Gamma } \\sigma _2 \\frac{1}{\\xi _{p2}} - \\frac{1}{\\varrho _2}\\left\\Vert \\Phi \\right\\Vert \\right)\\left\\Vert H_2\\right\\Vert - \\frac{2}{\\xi _{p2}}\\underline{\\Gamma } \\sigma _1 \\left\\Vert H_2\\right\\Vert ^2 \\\\\\le & -\\sqrt{2}\\varrho _2\\left(\\underline{\\Gamma } \\sigma _2 \\frac{1}{\\xi _{p2}} - \\frac{1}{\\varrho _2}\\left\\Vert \\Phi \\right\\Vert \\right)\\sqrt{V_2} - 4\\frac{\\varrho _2^2}{\\xi _{p2}}\\underline{\\Gamma } \\sigma _1 V_2\\end{aligned}$ If $\\sigma _2$ is selected sufficiently large and $\\Gamma $ , $\\varrho _2$ are selected to satisfy $\\underline{\\Gamma } \\sigma _2 \\frac{1}{\\xi _{p2}} - \\frac{1}{\\varrho _2}\\left\\Vert \\Phi \\right\\Vert > 0$ the expression in (REF ) can be further upper bounded as $\\dot{V}_2 \\le - c_{i1} \\sqrt{V_2}-c_{i2} V_2, \\quad \\forall P_2^{-1}H_2 \\in \\Xi _1$ where $c_{i1}$ , $c_{i2} \\in \\mathbb {R}^+$ are positive constants, and $\\Xi _1$ can be made arbitrarily large by increasing $\\sigma _2$ and select $\\Gamma $ , $\\varrho _2$ based on the design criteria in (REF ).",
"Similar to the proof of Theorem 1, an attraction region $\\mathcal {R}_{\\Xi } \\subset \\Xi _1$ exists that $||P_2^{-1}(t)H_2(t)|| \\rightarrow 0$ in $t_i \\le \\frac{2}{c_{i2}} ln\\left(1 + \\frac{c_{i2}}{c_{i1}}a_i\\right)$ , $\\forall P_2^{-1}(0)H_2(0) \\in \\mathcal {R}_{\\Xi }$ where $a_i \\in \\mathbb {R}^+$ is defined as $a_i \\triangleq \\frac{\\sqrt{2}}{2}\\left(||\\tilde{\\theta }_2(0)|| + \\frac{1}{\\varrho _2}\\xi _{\\Delta _2}\\right)$ .",
"From the definition of $H_2$ in equation(REF ), this further implies that $\\tilde{\\theta }_2$ converges to a compact set $\\mathcal {R}_i$ in $t_i$ , where $\\mathcal {R}_i \\subset \\mathbb {R}^3$ is defined as $\\mathcal {R}_i \\triangleq \\left\\lbrace \\tilde{\\theta }_2(t) \\in \\mathbb {R}^3 \\mid ||\\tilde{\\theta }_2|| \\le \\frac{1}{\\varrho _2}\\xi _{\\Delta _2}\\right\\rbrace $ This completes the proof.",
"Notice that though the FT convergence can be guaranteed by properly selecting $\\Gamma $ , $\\sigma _2$ , and $\\varrho _2$ while $\\sigma _1$ is selceted as zero, the selection of learning gain $\\sigma _1$ also influences the convergence rate of $||P_2^{-1}H_2||$ .",
"A large $\\sigma _1$ leads to a faster convergence.",
"However, it might also cause oscillations in the estimated parameters if $\\sigma _1$ is designed too large."
],
[
"Outer-Loop Analysis",
"The time derivative of equation (REF ) is calculated similar to (REF ) as $J\\dot{r}_o = \\tilde{N}_o +N_{\\Delta _1} - \\dot{\\mu }_1 - e_{o2}$ where auxiliary functions $\\tilde{N}_o$ , $N_{\\Delta _1}$ $\\in \\mathbb {R}^3$ are defined similar to (REF ).",
"The inner loop error signal $z_o \\in \\mathbb {R}^{10}$ is defined as $z_o \\triangleq \\left[ \\begin{matrix} e_{o1}^T & e_{o2}^T & r_o^T & \\tilde{\\theta }_1 \\end{matrix}\\right]^T$ The subsequent theorems can be proved.",
"Theorem 3.",
"The outer-loop controller given in equation (REF ), (REF ), and (REF ) ensures that signal $z_o$ is regulated that $||z_o(t)|| \\rightarrow 0 $ as $t \\rightarrow \\infty $ provided that control gain $k_{s1}$ is selected sufficiently large, $k_{o1}, k_{o2} > \\frac{1}{2}$ , and $\\beta _o$ following a condition similar to (REF ).",
"Theorem 4.",
"For error system (REF ) with the adaptive estimation law given in (REF ), the estimation error variable $P_1^{-1}H_1$ is regulated that $||P_1^{-1}H_1|| \\rightarrow 0 $ in a finite time $t_o$ satisfying $t_o \\le \\frac{2}{c_{o2}} ln\\left(1 + \\frac{c_{o2}}{c_{o1}}a_o\\right)$ if $\\gamma $ , $\\gamma _1$ , and $\\varrho _1$ are properly selected, where $a_o \\in \\mathbb {R}$ is defined as $a_o \\triangleq \\frac{\\sqrt{2}}{2}\\left(\\tilde{\\theta }_1(0) + \\frac{1}{\\varrho _1}\\xi _{\\Delta _1}\\right)$ .",
"And $\\tilde{\\theta }_1$ is guaranteed to converge to a compact set around zero in $t_o$ .",
"Proof of Theorem 3 and 4.",
"Noticing that $\\mathrm {sat}\\left(H_1\\right) \\le 1$ , the proof can be conducted in a similar method as that of the inner-loop controller."
],
[
"SIMULATION",
"In this section, the effectiveness of the designed controller is verified by simulations and experiments.",
"Comparative simulations are carried out between the proposed strategy and the traditional methods based on SMC and gradient algorithms in [13].",
"The results show that the proposed method yields better tracking error and estimation convergence.",
"Meanwhile, it generates smoother input signals for practical applications than SMC controllers.",
"The results indicate the robustness against disturbances and mass-inertia changes of the controller.",
"To verify the performance of the proposed control strategy, numrical simulations are conducted in MATLAB.",
"Table (REF ) shows the preset mass-inertia parameters of the UAV in simulation.",
"Table: True value of mass-inertia parametersThe control gains and learning gains of the proposed RISE-based adaptive controller with mass-inertia estimation (RISE-Emi) is selected as the following table (REF ).",
"And the learning gain matrix $\\Gamma $ is selected as $\\Gamma =\\left[ \\begin{matrix} 10^{-4} & 0 & 0 \\\\ 0 & 10^{-4} & 0 \\\\ 0 & 0 & 4.5\\times 10^{-3} \\\\\\end{matrix} \\right]$ Table: table: Control and learning gainsThe comparison is conducted between RISE-Emi and the adaptive sliding mode controller with gradient-based mass estimation (ASMC) proposed in [13].",
"We selcet the desired trajectry and yaw angle as $\\begin{aligned}&\\eta _{1d} = 2sin(t) \\cdot \\left[ \\begin{matrix} 1 & 1 & 1\\end{matrix}\\right]^T \\\\&\\psi _d = sin(1.1t)\\end{aligned}$ and add white noise disturbance to the dynamic model output of the system.",
"The result of mass-inertia estimation of RISE-Emi is show in Fig.",
"REF .",
"The initial estimation of mass is set $50\\%$ smaller than the real value, and the initial inertia estimations are about $100\\%$ , $100\\%$ and $50\\%$ larger respectively.",
"All 4 estimatied values converge to its truth finally.",
"Due to the different dynamic characters between yaw orientation and roll, pitch orientation, the estimation of $I_z$ overshoots for about $5\\%$ with the selected parameters.",
"And the convergence of mass is relatively slow because of the slower response of the outer loop compared to the inner loop.",
"Figure: Parameter estimation resultsMeanwhile, the estimation error is compared with the mass estimation of ASMC in Fig.",
"REF .",
"Initially, the estimated mass of the 2 methods converge at a similar speed.",
"Then, the estimation in RISE-Emi achieves the $2\\%$ bound faster, and gradually reaches the real value within about $10s$ , while keeps increasing at a large speed and saturates before convergence in ASME.",
"It is also noticable that because of the added white noise, the steady-state error can not be zero all the time.",
"However, the error caused by the noise in RISE-Emi is smaller than that of ASMC, which is shown in the sub-figure of Fig REF .",
"These results indicate the effectiveness of our method in mass-inertia estimation and its robustness against disturbances.",
"Figure: Comparison of estimation of massThen, the comparison of trajectory and attitude tracking errors are provided in Fig.",
"REF .",
"and Fig.",
"REF respectively.",
"The trajctory tracking errors increase greatly in the beginning mainly because of the imprecise initial estimation, which undermines the performance of the controllers.",
"Then, the tracking errors of the proposed method converge faster than ASMC, also yielding smaller steady-state errors in $x$ and $y$ directions.",
"And it is obvious that the scale of attitude tracking error in RISE-Emi is much smaller, which illustrate the disturbance rejection ability of our method.",
"Figure: Position tracking errorsFigure: Attitude tracking errorsFinally, the thurst output of the controller is compared in Fig.",
"REF .",
"The output of the proposed method is smoother than that of ASMC when disturbance exists, which is more physically achievable in practical applications.",
"Figure: Comparison of thrust"
],
[
"CONCLUSION",
"In this work, we have developed and validated an adaptive control strategy for UAVs in face of external disturbances and mass-inertia variation.",
"First, a dynamic model of multi-rotor UAVs with disturbances is derived with a linearly parameterized form.",
"Then, a cascade control law is designed based on this form with robust RISE terms.",
"Finally, mass-inertia estimation is conducted based on a filtering operation to improve the robustness against possible mass-inertia change.",
"Comparative simulations have shown that a better performance can be achieved with our method than the previously proposed method ASMC."
],
[
"ACKNOWLEDGEMENT",
"This work is motivated by Haoxuan Shan's work of a integrated quadruped-hexarotor system.",
"Gang Chen has also contributed to the idea and process of the research."
]
] | 2209.08209 |
[
[
"Combinatorics for Certain Skew Young Tableaux, Dyck Paths,\n Triangulations, and Dissections"
],
[
"Abstract We present combinatorial bijections and identities between certain skew Young tableaux, Dyck paths, triangulations, and dissections."
],
[
"Introduction",
"Bijections between Catalan objects are well understood.",
"For instance, see [11].",
"There are many generalizations of these objects and the bijections between them.",
"In [10], Stanley provides a bijection between certain standard Young tableaux and dissections of a polygon.",
"In [3], the authors provide a bijection between the same tableaux and certain Dyck paths.",
"Meanwhile, various papers consider a certain collection of skew Young tableaux—which may be seen as a generalization of the aforementioned tableaux—which are used to compute formulas for the ordinary and equivariant Kazhdan–Lusztig polynomial for uniform, sparse paving, and paving matroids [8], [9], [2], [6], [7].",
"Kazhdan-Lusztig polynomials for matroids were first defined in [1].",
"The primary goal of this paper is to generalize the bijection in [3], so that it involves the skew tableaux mentioned above, while simultaneously including bijections involving certain triangulations.",
"As a result of these bijections, properties about the skew tableaux will have implications for the Dyck paths and triangulation objects of interest.",
"Motivated by our findings, we then find a combinatorial bijection between the dissections in [10] and our triangulations.",
"In the next section, we will define relevant terminology for skew Young tableaux in subsection REF , Dyck paths in subsection REF , and then both dissections and triangulations in subsection REF .",
"Then in subsection REF , we discuss the main results and findings of this paper in detail.",
"In sections and , we provide the definitions for the maps involved in the main results."
],
[
"Acknowledgements:",
"The authors would like to thank Kyungyong Lee for his helpful input on this paper."
],
[
"Skew Young Tableaux and Nomincreasing Partitions",
"Definition 2.1 Let $\\lambda _1\\ge \\lambda _2\\ge \\cdots \\ge \\lambda _k$ be positive integers.",
"We say that $\\lambda =[\\lambda _1,\\lambda _2,\\dots ,\\lambda _k]$ is a partition of $n$ if $\\lambda _1+\\cdots +\\lambda _k=n$ .",
"The Young diagram of shape $\\lambda $ is represented by boxes that are left justified so that the $i$ th row has $\\lambda _i$ boxes.",
"A standard Young tableau is achieved by filling the boxes with the numbers so that each row strictly increases from left to right; each column increases from top to bottom; and if there are $n$ boxes, only the numbers 1 through $n$ are used.",
"See Figure REF below for an example of a Young diagram and standard Young tableaux.",
"Figure: The Young diagram and a standard Young tableaux of shape [7,4,2,2,1][7,4,2,2,1]Definition 2.2 Given partitions $\\mu =[\\mu _1,\\dots , \\mu _\\ell ]$ and $\\lambda =[\\lambda _1,\\dots , \\lambda _k]$ so that $\\mu _i\\le \\lambda _i$ for all $i$ , the skew Young diagram $\\lambda \\setminus \\mu $ is the set of squares from the diagram for $\\lambda $ that are not in the diagram for $\\mu $ .",
"As before, we define a skew Young tableau to be a skew Young diagram filled with numbers following the same rules described for standard Young tableau.",
"See Figure REF for an example of a skew Young tableaux.",
"Figure: A skew Young tableaux of shape λ∖μ\\lambda \\setminus \\mu where λ=[7,4,2,2,1]\\lambda =[7,4,2,2,1] and μ=[2,1,1]\\mu =[2,1,1].The authors in [9] introduce the notation $\\operatorname{Skyt}(a,i,b)$ to denote the skew Young tableaux of shape $[(i+1)^b,1^{a-2}]/[(i-1)^{b-2}]$ , where we write $x^t$ to denote $x,x,\\dots , x$ , where $x$ is written $t$ times.",
"These are precisely the skew tableaux we discussed in the introduction.",
"The diagram for the tableaux in $\\operatorname{Skyt}(a,i,b)$ is shown in Figure REF .",
"Figure: The diagram for Skyt(a,i,b)\\operatorname{Skyt}(a,i,b)."
],
[
"Dyck Paths",
"Definition 2.3 A Dyck path of semi-length $n$ is a string in $\\lbrace \\textsf {U},^{2n}$ so that the string has the same number of $\\textsf {U}$ 's and $^{\\prime }s (that is, $ n$ of each); and\\item the number of $U$^{\\prime }s is at least the number of $ 's in any initial segment of the word.",
"We will also often represent such a path visually using $(1,1)$ segments for $\\textsf {U}$ and $(-1,1)$ segments for $, as in Figure \\ref {fig:dyck_example}.$ Figure: The visual representation of the path corresponding to 𝖴 3 2𝖴 2 3\\textsf {U}^32\\textsf {U}^23.Definition 2.4 A long ascent is a maximal ascent of length at least 2.",
"A singleton is a maximal ascent of length 1.",
"Let $\\operatorname{Dyck}(n,\\ell ,s)$ be the Dyck paths of semi-length $n$ with $\\ell $ long ascents and $s$ singletons so that no singleton appears after the last long ascent.",
"Thus, the Dyck path in Figure REF is an element of $\\operatorname{Dyck}(8,2,3)$ ."
],
[
"Dissections and Triangulations",
"Throughout this section, we assume polygons with $n$ vertices have their vertices labeled 1 through $n$ in counter-clockwise order.",
"Definition 2.5 A dissection of a polygon $P$ is a way of adding chords between non-adjacent vertices so that no two chords intersect in the interior of the polygon.",
"Throughout, we let $\\operatorname{Dis}(n,i)$ be the set of all dissections of an $n$ -gon with $i$ chords.",
"Note that $i$ in $\\operatorname{Dis}(n,i)$ is at most $n-2$ .",
"The elements of $\\operatorname{Dis}(n,n-2)$ are the triangulations of an $n$ -gon.",
"Given a vertex $x$ in a triangulated polygon, a fan at $x$ is a maximal collection triangles all containing $x$ .",
"In this case, we call $x$ the origin of the fan.",
"A singular fan is a fan with only one triangle.",
"Let $e$ be a boundary edge of a fan $F$ at $x$ .",
"We are interested in being able to uniquely partition a triangulation into a collection of fans.",
"This leads to the following definition.",
"Definition 2.6 Let $T$ be a triangulation.",
"A fan decomposition is the the pair of sequences $(\\mathcal {F}(T),\\delta (T))$ , where $\\mathcal {F}(T)$ and $\\delta (T)$ are defined as follows: We let $\\mathcal {F}(T)$ be a sequence of fans defined recursively as follows.",
"Let $F$ be the fan at the vertex with the smallest label.",
"Delete this vertex and all edges incident with it in $T$ to obtain a sequence of triangulations $T_1,\\dots , T_k$ , arranged in counter clock-wise order so that $T_i\\cap T_{i+1}$ is just vertex.",
"If $T$ is just an edge, then $\\mathcal {F}(T)$ is the empty sequence, and otherwise $\\mathcal {F}(T)(F,\\mathcal {F}_1,\\dots , \\mathcal {F}_k)$ where $\\mathcal {F}_i=\\mathcal {F}(T_i)$ .",
"Let $x_j$ be the label of the origin of $F_j$ .",
"We let $\\delta (T)(d_1,\\dots , d_{k-1})$ , where $d_ix_{i+1}-x_i$ and $k$ is the number of fans in $\\mathcal {F}(T)$ .",
"One can think of $d_i$ as the number of edges between the origins of $F_i$ and $F_{i+1}$ when traveling along the boundary of $T$ counter-clockwise.",
"Example 2.7 Consider the triangulation $T$ in Figure REF .",
"Observe that $\\mathcal {F}(T)=(F_1,F_2,F_3,F_4,F_5)$ where $F_1$ is the size 1 fan at vertex 1, $F_2$ is the size 3 fan at vertex 2, $F_3$ is the size 1 fan at vertex 4, $F_4$ is the size 1 fan at vertex 5, and $F_5$ is the size 4 fan at vertex 7.",
"Thus, $\\delta (T)=(1,2,1,2)$ .",
"Figure REF shows the five fans, distinguishing them by thick boundary edges and different shades of orange in their interior.",
"The white vertices correspond to the origins of the fans.",
"Figure: A triangulation and its partition into fans.Remark 2.8 Observe that a fan decomposition uniquely determines $T$ .",
"That is, knowing the order and size of each fan along with the distance between origins of consecutive fans uniquely determine a triangulation.",
"Let $\\operatorname{Tri}(n,t,s)$ be the triangulations $T$ of an $n$ -gon so that $\\mathcal {F}(T)$ has $s+t$ fans so that precisely $s$ are singular and so that the last fan is not singular.",
"Thus, the triangulation in Figure REF is an element of $\\operatorname{Tri}(12,2,3)$ ."
],
[
"Main Results",
"We now may state the main results of this paper.",
"First, let us state [3], the result which we plan to generalize.",
"We state the result by referencing the object $\\operatorname{Skyt}(a,i,b)$ we defined above.",
"Proposition 2.9 ([3]) The tableaux in $\\operatorname{Skyt}(a,i,2)$ are in bijection with Dyck paths of length $2(a+2i)$ with $i+1$ peaks and no singletons.",
"In Section , we will provide explicit combinatorial maps which give us the following Theorem.",
"Theorem 2.10 The following objects are in bijection: $\\operatorname{Skyt}(a,i,b)$ ; $\\operatorname{Dyck}(a+b+2i-2,i+1,b-2)$ ; and $\\operatorname{Tri}(a+b+2i,i+1,b-2)$ .",
"The maps between the three objects are defined in section .",
"The following pairs of maps are mutual inverses: maps $\\operatorname{\\operatorname{\\mathtt {SD}}}$ and $\\operatorname{\\operatorname{\\mathtt {DS}}}$ ; maps $\\operatorname{\\operatorname{\\mathtt {ST}}}$ and $\\operatorname{\\operatorname{\\mathtt {TS}}}$ ; and maps $\\operatorname{\\operatorname{\\mathtt {TD}}}$ and $\\operatorname{\\operatorname{\\mathtt {DT}}}$ .",
"Note that this generalizes the result stated in Proposition REF , in addition to adding a triangulation interpretation.",
"With the original motivation for this paper in mind, we specialize Theorem REF to $b=2$ .",
"After incorporating the work of of [10] which provides a combinatorial bijection between $\\operatorname{Dis}(n+2,i)$ and $\\operatorname{Skyt}(n-i+1,i,2)$ , we have the following.",
"It is worth noting that this connection between $\\operatorname{Skyt}(a,i,b)$ and dissections of polygons has resurfaced recently in the work of Kazhdan-Lusztig polynomials for Matroids [1].",
"Compare the comments in [5] with the representation theoretic result [4] after setting $m=1$ and considering dimensions.",
"Corollary 2.11 The following objects are in bijection.",
"$\\operatorname{Dis}(n+2,i)$ .",
"$\\operatorname{Skyt}(n-i+1,i,2)$ .",
"$\\operatorname{Dyck}(n+i+1,i+1,0)$ .",
"$\\operatorname{Tri}(n+i+3,i+1,0)$ .",
"By specializing the maps involved in Theorem REF , we already have combinatorial bijections between the standard Young tableaux, Dyck paths, and triangulations in this theorem, though its important to note that are bijection between the standard Young tableaux and Dyck paths is precisely the proof of Proposition REF .",
"This leaves two pairs of objects with missing combinatorial bijections.",
"In section we demonstrate a bijection between the dissections and triangulations given in this corollary.",
"Using our bijection between Dyck paths and Triangulations, one can extend our work in section to give a bijection between the dissections and Dyck paths in Corollary REF , but we omit this interpretation from this paper.",
"For our final result, we recall the following Lemma in terms of Dyck paths and triangulations.",
"Lemma 2.12 [8] Let $a$ , $i$ , and $b$ be nonnegative integers.",
"Then $\\# \\operatorname{Skyt}(a,i,b)=\\#\\operatorname{Skyt}(b,i,a).$ One may apply Theorem REF to this Lemma in order to get the following.",
"Corollary 2.13 Let $n$ , $\\ell $ , $s$ be nonnegative integers.",
"Then $\\#\\operatorname{Dyck}(n,\\ell ,s)=\\#\\operatorname{Dyck}(n,\\ell ,n-s-2\\ell ).$ Let $n$ , $t$ , $s$ be nonnegative integers.",
"Then $\\#\\operatorname{Tri}(n,t,s)=\\#\\operatorname{Tri}(n,t,n-t-2\\ell -2).$ Although these equalities are naturally obtained with Theorem REF and Lemma REF , there is no known direct combinatorial bijection describing these equalities.",
"Hence we pose the following.",
"Problem 1 Find a direct combinatorial proof of Corollary REF which does not rely on using the skew tableaux or bijections given in this paper."
],
[
"Combinatorial Bijections",
"The following subsections describe maps going between any two of the objects given in Theorem REF .",
"For convenience, we identify maps according to where the map from and to by using S for skew Young tableaux, T for Triangulations, and D for Dyck paths.",
"For instance, map $\\operatorname{\\operatorname{\\mathtt {ST}}}$ represents the map from skew Young tableaux to traingulations, and $\\operatorname{\\operatorname{\\mathtt {TD}}}$ represents a map from triangulations to Dyck paths.",
"Examples are used to alleviate any ambiguity with our maps.",
"Before proceeding, however, we will point out a handy reinterpretation of the tableaux in $\\operatorname{Skyt}(a,i,b)$ .",
"Let $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ .",
"Let $X=\\lbrace x_1,x_2,\\dots , x_{i+b-1}\\rbrace $ be the set of values in the top $b-1$ rows so that $x_1<x_2<\\cdots < x_{i+b-1}$ .",
"If $x_j$ is in row $b-1$ , define $y_j$ to be the entry in the tableau directly below $x_j$ .",
"Then for $1\\le j<i+b-1$ let $A_{j}{\\left\\lbrace \\begin{array}{ll} \\lbrace x_j\\rbrace & \\text{if $x_j$ is in the first $b-2$ rows;}\\\\\\lbrace x_j\\rbrace \\cup \\left([y_j,y_{k}-1]\\setminus X\\right)& \\text{if $x_j$ is in row $b-1$ and $y_k$ is to the right of $y_j$,}\\end{array}\\right.",
"}$ where $[y_j,y_k-1]=\\lbrace y_j,y_j+1,y_j+2,\\dots , y_k-1\\rbrace $ .",
"Let $A_{i+b-1}\\lbrace x_j\\rbrace \\cup \\big ([x_j+1,a+b+2i-2]\\setminus X\\big )$ .",
"Note that $x_j$ is always the minimum of $A_j$ .",
"When $|A_j|>1$ , note the elements of $A_j$ are precisely the entries in row $b-1$ and $b$ in column $j$ along with all entries of column 1 which are between $y_j$ and $y_k$ .",
"See Figure REF .",
"Figure: We “push\" entries below row bb as far right as possible while maintaining the property that columns increase from top to bottom.",
"We have A 1 ={1}A_1=\\lbrace 1\\rbrace , A 2 ={2,3,8}A_2=\\lbrace 2,3,8\\rbrace , A 3 ={4}A_3=\\lbrace 4\\rbrace , A 4 ={5}A_4=\\lbrace 5\\rbrace , A 5 ={7,9,11,12,14}A_5=\\lbrace 7,9,11,12,14\\rbrace , A 6 ={10}A_6=\\lbrace 10\\rbrace , A 7 ={13,15}A_7=\\lbrace 13,15\\rbrace .The sequence $(A_1,\\dots , A_{i+b-1})$ has enough information to reconstruct $\\lambda $ .",
"Starting with $j=1$ , do the following.",
"If $|A_j|=1$ , let $x\\in A_j$ .",
"Then place $x$ in the highest possible position in the last column.",
"If $|A_j|>1$ , then let $x_j=\\min A_j$ and $y_j=\\min ( A_j\\setminus \\lbrace x_j\\rbrace )$ .",
"Place $x_j$ in row $b-1$ column $j$ and place $y_j$ in row $b$ column $j$ .",
"Place all remaining entries from $A_j$ —in increasing order—at the top most available position(s) in the first column.",
"Increase the value of $j$ by 1.",
"If $j<i+b-1$ , repeat these steps.",
"Otherwise, $\\lambda $ is filled and we are done.",
"Let $x_j$ and $y_j$ are defined in step (2) of the preceding procedure.",
"Pick an integer $j^{\\prime }$ minimally so that $j<j^{\\prime }$ and $|A_{j^{\\prime }}|>1$ .",
"Note that $(A_1,\\dots , A_{i+b-1})$ is an ordered partition of $[a+b+2i-2]$ so that $x_j<x_{j+1}$ and $y_j<y_{j^{\\prime }}$ , whenever $y_j$ and $y_{j^{\\prime }}$ exist.",
"These conditions guarantee that the rows of $\\lambda $ increase left-to-right.",
"Such a sequence is called a nomincreasing partition, as defined in [3].",
"To this end, we define the following.",
"Definition 3.1 Let $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ and define $A_1,\\dots , A_{i+b-1}$ for $\\lambda $ as above.",
"We define $\\operatorname{Nom}(\\lambda )$ to be the nomincreasing partition $\\operatorname{Nom}(\\lambda )(A_1,A_2,\\dots , A_{i+b-1}).$ Remark 3.2 One thing that will be useful to note for future reference is that for $\\operatorname{Nom}(\\lambda )=(A_1,\\dots , A_{i+b-1})$ , we always have $|A_{i+b-1}|>1$ .",
"This is preceisely because with the aforementioned choice of $x_1<\\cdots <x_{i+b-1}$ , we always have that $x_{i+b-1}$ is the entry in row $b-1$ column $i+1$ of $\\lambda $ .",
"In particular, this means given a nomincreasing sequence $(A_1,\\dots , A_{i+b-1})$ , even if $i+1$ of the $A_j$ satisfy $|A_j|>1$ and the remaining satisfy $|A_j|=1$ , there does not necessarily exists $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ so that $\\operatorname{Nom}(\\lambda )=(A_1,\\dots , A_{i+b-1})$ .",
"We need to additionally have $|A_{i+b-1}|>1$ .",
"Given this, though, such a $\\lambda $ must exist."
],
[
"Map $\\operatorname{\\operatorname{\\mathtt {SD}}}$",
"In this section, we define a map $\\operatorname{\\operatorname{\\mathtt {SD}}}$ from $\\operatorname{Skyt}(a,i,b)$ to $\\operatorname{Dyck}(a+b+2i-2, i+1, b-2)$ .",
"For simplicity, let $na+b+2i-2$ .",
"Note that given $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ , $n$ is the number of entries in $\\lambda $ and $t$ is the number of entries in the first $b-1$ rows of $\\lambda $ .",
"Definition 3.3 Let $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ .",
"We define $\\operatorname{\\operatorname{\\mathtt {SD}}}(\\lambda )$ , a certain lattice path, as follows.",
"Let $\\operatorname{Nom}(\\lambda )=(A_1,\\dots , A_{i+b-1})$ .",
"Let $x_j$ denote the minimum of $A_j$ .",
"Let $a_j \\#A_j$ .",
"Then let $\\operatorname{\\operatorname{\\mathtt {SD}}}(\\lambda )$ be the lattice path given by following string in $\\lbrace \\textsf {U},^{2n}$ : $\\textsf {U}^{a_1}{x_2-x_1}\\textsf {U}^{a_2}{x_3-x_2}\\cdots \\textsf {U}^{a_{t-1}}{x_{t}-x_{t-1}}\\textsf {U}^{a_{t}}{n-x_{t}+x_1}.$ Lemma 3.5 Given $\\lambda $ be a tableau in $\\operatorname{Skyt}(a,i,b)$ , $\\operatorname{\\operatorname{\\mathtt {SD}}}(\\lambda )$ is a Dyck path.",
"In particular, the lattice path $\\operatorname{\\operatorname{\\mathtt {SD}}}(\\lambda )$ is an element of $\\operatorname{Dyck}(n, i+1, b-2)$ .",
"Recall that given $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ , we can define have $\\operatorname{Nom}(\\lambda )=(A_1,\\dots , A_{i+b-1})$ where $|A_{i+b-1}|>1$ .",
"Recall that $x_j=\\min A_j$ is an entry in the top $b-1$ rows of $\\lambda $ .",
"In the string given in (REF ), $\\textsf {U}$ corresponds to an up step and $ correspond to a down step.Note that for any $ k$, we have $ [x1,xk]A1Ak$.Otherwise, there exists a $ w[x1,xk]$ so that $ wAj$ for some $ j>k$.",
"Then we have $ wxj>xkw$, a contradiction.$ Thus, for $k<t$ $\\sum _{j=1}^k (x_{j+1}-x_j)=x_{k+1}-x_1\\le \\sum _{j=1}^k a_j.$ Also, $\\sum _{j=1}^t (x_{j+1}-x_j)=x_{t}-x_1 + n-x_t+x_1 = n.$ Moreover, there are precisely $b-2$ of the $a_j$ so that $a_j=1$ , and there are $i$ of the $a_j$ so that $a_j>1$ .",
"Also, observe that $a_{i+b-1}>1$ , so the last ascent in $\\operatorname{\\operatorname{\\mathtt {SD}}}(\\lambda )$ is not a singleton.",
"Consequently, the constructed Dyck path is an element of $\\operatorname{Dyck}(n,i,b-2)$ .",
"Example 3.6 For the skew Young tableau $\\lambda $ in $\\operatorname{Skyt}(7,2,6)$ below, $x_1=1, x_2=2, x_3=4,x_4=5,x_5=7 , x_6=10,$ and $x_7=12$ .",
"As $x_2,$ $x_5$ and $x_7$ are the entries in row 5, $y_i$ is defined for $i=2,5,7$ .",
"We have $y_2=3$ , $y_5=11$ , and $y_7=13$ .",
"Thus, $A_1 =\\lbrace 1\\rbrace , A_2 =\\lbrace 2,3,6,8,9\\rbrace , A_3 =\\lbrace 4\\rbrace , A_4 =\\lbrace 5\\rbrace , A_5 =\\lbrace 7,11\\rbrace , A_6 =\\lbrace 10\\rbrace $ , and $A_7 =\\lbrace 12,13,14,15\\rbrace $ .",
"[scale=0.5, line width=1pt] (0,0) grid (1,-7); (1,0) grid (2,-2); (2,0) grid (3,-2); (2,4) grid (3,-2); (.5,-.5) node 2; (.5,-1.5) node 3; (.5,-2.5) node 6; (.5,-4.5) node 9; (.5,-3.5) node 8; (1.5,-.5) node 7; (1.5,-1.5) node 11; (.5,-5.5) node 14; (.5,-6.5) node 15; (2.5,-.5) node 12; (2.5,-1.5) node 13; (2.5,3.5) node 1; (2.5,2.5) node 4; (2.5,1.5) node 5; (2.5,.5) node 10; Thus $\\operatorname{\\operatorname{\\mathtt {SD}}}(\\lambda )$ is the Dyck path given by $\\textsf {U}^32{{}\\textsf {U}^43{{}\\textsf {U}^44.",
"}}$"
],
[
"Map $\\operatorname{\\operatorname{\\mathtt {DS}}}$",
"This subsection gives a map $\\operatorname{\\operatorname{\\mathtt {DS}}}$ from $\\operatorname{Dyck}(n,\\ell ,s)$ to $\\operatorname{Skyt}(n-s-2\\ell -2,\\ell -1,s+2)$ , which is the inverse of $\\operatorname{\\operatorname{\\mathtt {SD}}}.$ The reader can verify that they are indeed inverses.",
"Definition 3.7 Let $P$ be a Dyck path in $\\operatorname{Dyck}(n,\\ell ,s)$ .",
"We define $\\operatorname{\\operatorname{\\mathtt {DS}}}(P)$ , a skew tableau, as follows.",
"Given $P$ , label the down-steps, left to right, in increasing order, from 1 to $n$ .",
"Next, use the label on the down-step at each peak as the label for the up-step at the same peak.",
"Going through the ascents from left to right, greedily label the unlabeled up-steps from top to bottom using numbers from $[a+2i+b-2]$ not already appearing on any up-step.",
"Let $A_j$ be the labels appearing on the $j$ th ascent.",
"Now construct $\\operatorname{\\operatorname{\\mathtt {DS}}}(P)$ so that $\\operatorname{Nom}\\big (\\operatorname{\\operatorname{\\mathtt {DS}}}(P)\\big )=(A_1,\\dots , A_{\\ell +s})$ .",
"Lemma 3.8 Given a Dyck path $P$ in $\\operatorname{Dyck}(n,\\ell ,s)$ , the tableau $\\operatorname{\\operatorname{\\mathtt {DS}}}(P)$ is a skew Young tableau in $\\operatorname{Skyt}(n-s-2\\ell -2,\\ell -1,\\leavevmode {\\color {black}s+2})$ .",
"There are $\\ell +s$ ascents in $P$ .",
"Thus, precisely $\\ell $ of the $A_j$ satisfy $|A_j|>1$ , and precisely $s$ of the $A_j$ satisfy $|A_j|=1$ .",
"Next, notice that $(A_1,\\dots , A_{\\ell +s})$ is a nomincreasing sequence due to the greedy labeling of up steps of $P$ .",
"Also note that $|A_{s+\\ell }|>1$ since the last ascent in $P$ is not a singleton.",
"Since there are $n$ up steps in $P$ , we have $\\operatorname{\\operatorname{\\mathtt {DS}}}(P)\\in \\operatorname{Skyt}(n-s-2\\ell -2,\\ell -1,s+2)$ .",
"Example 3.9 Given a Dyck path in $\\operatorname{Dyck}(15,3,4)$ , we first label the down-steps, left to right, in increasing order.",
"[scale=0.5, line width=1pt] [color=black!40, thick] (-3,0)–(29,0); (-2,0) circle[radius=5pt]; [thick] (-2,0)–(-1,1); (-1,1) circle[radius=5pt]; [thick] (-1,1)–(0,0); (0,0) circle[radius=5pt]; in 0,1,...,2 [thick] (,)–(+1,+1); (+1,+1) circle[radius=5pt]; [thick] (3,3)–(4,2); (4,2) circle[radius=5pt]; [thick] (4,2)–(5,1); (5,1) circle[radius=5pt]; [thick] (5,1)–(6,2); (6,2) circle[radius=5pt]; [thick] (6,2)–(7,1); (7,1) circle[radius=5pt]; [thick] (7,1)–(8,2); (8,2) circle[radius=5pt]; [thick] (8,2)–(9,1); (9,1) circle[radius=5pt]; [thick] (9,1)–(10,0); (10,0) circle[radius=5pt]; [thick] (10,0)–(14,4); (11,1) circle[radius=5pt]; (12,2) circle[radius=5pt]; (13,3) circle[radius=5pt]; (14,4) circle[radius=5pt]; [thick] (14,4)–(17,1); (15,3) circle[radius=5pt]; (16,2) circle[radius=5pt]; (17,1) circle[radius=5pt]; [thick] (17,1)–(18,2); (18,2) circle[radius=5pt]; [thick] (18,2)–(20,0); (19,1) circle[radius=5pt]; (20,0) circle[radius=5pt]; [thick] (20,0)–(24,4); (21,1) circle[radius=5pt]; (22,2) circle[radius=5pt]; (23,3) circle[radius=5pt]; (24,4) circle[radius=5pt]; [thick] (24,4)–(28,0); (25,3) circle[radius=5pt]; (26,2) circle[radius=5pt]; (27,1) circle[radius=5pt]; (28,0) circle[radius=5pt]; (-.2,.8) node[color=red] 1; (3.8,2.8) node[color=red] 2; (4.8,1.8) node[color=red] 3; (6.8,1.8) node[color=red] 4; (8.8,1.8) node[color=red] 5; (9.8,.8) node[color=red] 6; (14.8,3.8) node[color=red] 7; (15.8,2.8) node[color=red] 8; (16.8,1.8) node[color=red] 9; (19,1.8) node[color=red] 10; (20,0.8) node[color=red] 11; (25.,3.8) node[color=red] 12; (26.,2.8) node[color=red] 13; (27.,1.8) node[color=red] 14; (28.,.8) node[color=red] 15; Then label the upstep of each peak: [scale=0.5, line width=1pt] [color=black!40, thick] (-3,0)–(29,0); (-2,0) circle[radius=5pt]; [thick] (-2,0)–(-1,1); (-1,1) circle[radius=5pt]; [thick] (-1,1)–(0,0); (0,0) circle[radius=5pt]; in 0,1,...,2 [thick] (,)–(+1,+1); (+1,+1) circle[radius=5pt]; [thick] (3,3)–(4,2); (4,2) circle[radius=5pt]; [thick] (4,2)–(5,1); (5,1) circle[radius=5pt]; [thick] (5,1)–(6,2); (6,2) circle[radius=5pt]; [thick] (6,2)–(7,1); (7,1) circle[radius=5pt]; [thick] (7,1)–(8,2); (8,2) circle[radius=5pt]; [thick] (8,2)–(9,1); (9,1) circle[radius=5pt]; [thick] (9,1)–(10,0); (10,0) circle[radius=5pt]; [thick] (10,0)–(14,4); (11,1) circle[radius=5pt]; (12,2) circle[radius=5pt]; (13,3) circle[radius=5pt]; (14,4) circle[radius=5pt]; [thick] (14,4)–(17,1); (15,3) circle[radius=5pt]; (16,2) circle[radius=5pt]; (17,1) circle[radius=5pt]; [thick] (17,1)–(18,2); (18,2) circle[radius=5pt]; [thick] (18,2)–(20,0); (19,1) circle[radius=5pt]; (20,0) circle[radius=5pt]; [thick] (20,0)–(24,4); (21,1) circle[radius=5pt]; (22,2) circle[radius=5pt]; (23,3) circle[radius=5pt]; (24,4) circle[radius=5pt]; [thick] (24,4)–(28,0); (25,3) circle[radius=5pt]; (26,2) circle[radius=5pt]; (27,1) circle[radius=5pt]; (28,0) circle[radius=5pt]; (-1.8,.8) node[color=blue] 1; (2.2,2.8) node[color=red] 2; (5.2,1.8) node[color=blue] 4; (7.2,1.8) node[color=blue] 5; (13.2,3.8) node[color=red] 7; (17.,1.8) node[color=blue] 10; (23.,3.8) node[color=red] 12; Now greedily label remaining up-steps.",
"[scale=0.5, line width=1pt] [color=black!40, thick] (-3,0)–(29,0); (-2,0) circle[radius=5pt]; [thick] (-2,0)–(-1,1); (-1,1) circle[radius=5pt]; [thick] (-1,1)–(0,0); (0,0) circle[radius=5pt]; in 0,1,...,2 [thick] (,)–(+1,+1); (+1,+1) circle[radius=5pt]; [thick] (3,3)–(4,2); (4,2) circle[radius=5pt]; [thick] (4,2)–(5,1); (5,1) circle[radius=5pt]; [thick] (5,1)–(6,2); (6,2) circle[radius=5pt]; [thick] (6,2)–(7,1); (7,1) circle[radius=5pt]; [thick] (7,1)–(8,2); (8,2) circle[radius=5pt]; [thick] (8,2)–(9,1); (9,1) circle[radius=5pt]; [thick] (9,1)–(10,0); (10,0) circle[radius=5pt]; [thick] (10,0)–(14,4); (11,1) circle[radius=5pt]; (12,2) circle[radius=5pt]; (13,3) circle[radius=5pt]; (14,4) circle[radius=5pt]; [thick] (14,4)–(17,1); (15,3) circle[radius=5pt]; (16,2) circle[radius=5pt]; (17,1) circle[radius=5pt]; [thick] (17,1)–(18,2); (18,2) circle[radius=5pt]; [thick] (18,2)–(20,0); (19,1) circle[radius=5pt]; (20,0) circle[radius=5pt]; [thick] (20,0)–(24,4); (21,1) circle[radius=5pt]; (22,2) circle[radius=5pt]; (23,3) circle[radius=5pt]; (24,4) circle[radius=5pt]; [thick] (24,4)–(28,0); (25,3) circle[radius=5pt]; (26,2) circle[radius=5pt]; (27,1) circle[radius=5pt]; (28,0) circle[radius=5pt]; (-1.8,.8) node[color=blue] 1; (2.2,2.8) node[color=red] 2; (1.2,1.8) node 3; (0.2,0.8) node 6; (5.2,1.8) node[color=blue] 4; (7.2,1.8) node[color=blue] 5; (13.2,3.8) node[color=red] 7; (12.,2.8) node 8; (11.,1.8) node 9; (10.,.8) node 11; (17.,1.8) node[color=blue] 10; (23.,3.8) node[color=red] 12; (22.,2.8) node 13; (21.,1.8) node 14; (20.,.8) node 15; Thus, we have $A_1=\\lbrace 1\\rbrace $ , $A_2=\\lbrace 2,3,6\\rbrace $ , $A_3=\\lbrace 4\\rbrace $ , $A_4=\\lbrace 5\\rbrace $ , $A_5=\\lbrace 7,8,9,11\\rbrace $ , $A_6=\\lbrace 10\\rbrace $ , and $A_7=\\lbrace 12,13,14,15\\rbrace $ .",
"The map $\\operatorname{\\operatorname{\\mathtt {DT}}}$ gives the tableau in Figure REF .",
"Figure: The construction of the skew tableaux in the final steps of the error\\operatorname{\\operatorname{\\mathtt {DS}}} map."
],
[
"Map $\\operatorname{\\operatorname{\\mathtt {DT}}}$",
"The inspiration for the following map $\\operatorname{\\operatorname{\\mathtt {DT}}}$ , a map from $\\operatorname{Dyck}(n,\\ell ,s)$ to $\\operatorname{Tri}(n+2,\\ell ,s)$ , comes from [11].",
"Definition 3.10 Let $P$ be a Dyck path in $\\operatorname{Dyck}(n,\\ell ,s)$ .",
"We define $\\operatorname{\\operatorname{\\mathtt {DT}}}(P)$ a certain type of triangulation, as follows.",
"A Dyck path in $\\operatorname{Dyck}(n,\\ell ,s)$ has the form $\\textsf {U}^{u_1}{d_1}\\textsf {U}^{u_2}{d_2}\\cdots \\textsf {U}^{u_{s+\\ell }}{d_{s+\\ell }},$ where $s$ is the number of singletons, $\\ell $ is the number of long ascents, and the $u_i$ and $d_i$ are positive integers.",
"Recall that the triangulation is determined by its fan decomposition.",
"Let $F_j$ be a fan with $u_j$ triangles.",
"Then $\\operatorname{\\operatorname{\\mathtt {DT}}}(P)$ is given by the fan decomposition $\\big ( (F_1,\\dots , F_{s+\\ell }), (d_1,\\dots , d_{s+\\ell })\\big )$ .",
"Lemma 3.11 If $P$ is a Dyck path in $\\operatorname{Dyck}(n,\\ell ,s)$ , then $\\operatorname{\\operatorname{\\mathtt {DT}}}(P)$ is a triangulation in $\\operatorname{Tri}(n+2,\\ell ,s)$ .",
"Given $P\\in \\operatorname{Dyck}(n,\\ell ,s)$ , note that that the number of triangles in $\\operatorname{\\operatorname{\\mathtt {DT}}}(P)$ is given by sum of sizes of $F_j$ : $\\sum _{j=1}^{s+\\ell }u_j=n$ As $\\operatorname{\\operatorname{\\mathtt {DT}}}(P)$ has $n$ triangles, the boundary must have $n+2$ edges.",
"Also note that since there is no singleton after the last long ascent in our Dyck path, the last fan $\\mathcal {F}(T)$ will not be a singleton fan.",
"Since $\\ell $ of the $u_j$ satisfy $u_j>1$ , our triangulation has $\\ell $ non-singular fans.",
"Similarly, since $s$ of the $u_j$ satisfy $u_j=1$ , our triangulation has $s$ singular fans.",
"Also, note that $u_{s+\\ell }>1$ , so $F_{s+\\ell }$ has more than one triangle.",
"Thus, we have constructed a triangulation in $\\operatorname{Tri}(n+2,\\ell ,s)$ , where the vertices are labeled as follows: label the origin of $F_1$ as 1, and then label the remaining vertices from 2 to $n$ in clock-wise order by starting at 1 and traveling along the boundary of the triangulation.",
"Example 3.12 Consider the path below.",
"[scale=0.7, line width=1pt] [color=black!40, thick] (-3,0)–(19,0); (-2,0) circle[radius=5pt]; [thick] (-2,0)–(-1,1); (-1,1) circle[radius=5pt]; [thick] (-1,1)–(0,0); (0,0) circle[radius=5pt]; in 0,1,...,2 [thick] (,)–(+1,+1); (+1,+1) circle[radius=5pt]; [thick] (3,3)–(4,2); (4,2) circle[radius=5pt]; [thick] (4,2)–(5,1); (5,1) circle[radius=5pt]; [thick] (5,1)–(6,2); (6,2) circle[radius=5pt]; [thick] (6,2)–(7,1); (7,1) circle[radius=5pt]; [thick] (7,1)–(8,2); (8,2) circle[radius=5pt]; [thick] (8,2)–(9,1); (9,1) circle[radius=5pt]; [thick] (9,1)–(10,0); (10,0) circle[radius=5pt]; [thick] (10,0)–(14,4); (11,1) circle[radius=5pt]; (12,2) circle[radius=5pt]; (13,3) circle[radius=5pt]; (14,4) circle[radius=5pt]; [thick] (14,4)–(18,0); (15,3) circle[radius=5pt]; (16,2) circle[radius=5pt]; (17,1) circle[radius=5pt]; (18,0) circle[radius=5pt]; Associated to this are five fans, given below.",
"We shade these fans differently so we may more easily keep track of them throughout.",
"regular polygon,regular polygon sides=3,minimum size=1cm,draw,rotate=180] (a); in 1,...,3circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,3 (a.corner 2) – (a.corner ) ; [white] (a.corner 3) circle (1.pt); $F_1$ regular polygon,regular polygon sides=5,minimum size=1cm,draw,rotate=-36] (a); [color=orange!15] (a.corner 1)–(a.corner 2)–(a.corner 3)–(a.corner 4)–(a.corner 5)–(a.corner 1); regular polygon,regular polygon sides=5,minimum size=1cm,draw,rotate=-36] (a); in 1,...,5circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,5 (a.corner 2) – (a.corner ) ; [white] (a.corner 2) circle (1.pt); $F_2$ regular polygon,regular polygon sides=3,minimum size=1cm,draw,rotate=180] (a); [color=orange!35] (a.corner 1)–(a.corner 2)–(a.corner 3)–(a.corner 1); regular polygon,regular polygon sides=3,minimum size=1cm,draw,rotate=180] (a); in 1,...,3circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,3 (a.corner 2) – (a.corner ) ; [white] (a.corner 3) circle (1.pt); $F_3$ regular polygon,regular polygon sides=3,minimum size=1cm,draw,rotate=180] (a); [color=orange!70] (a.corner 1)–(a.corner 2)–(a.corner 3)–(a.corner 1); regular polygon,regular polygon sides=3,minimum size=1cm,draw,rotate=180] (a); in 1,...,3circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,3 (a.corner 2) – (a.corner ) ; [white] (a.corner 3) circle (1.pt); $F_4$ regular polygon,regular polygon sides=6,minimum size=1cm,draw] (a); [color=orange!90] (a.corner 1)–(a.corner 2)–(a.corner 3)–(a.corner 4)–(a.corner 5)–(a.corner 6)–(a.corner 1); regular polygon,regular polygon sides=6,minimum size=1cm,draw] (a); in 1,...,6circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,6 (a.corner 2) – (a.corner ) ; [white] (a.corner 2) circle (1.pt); $F_5$ Below are the subsequent steps of attaching $F_j$ .",
"regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (a); [color=orange!15] (a.corner 1)–(a.corner 2)–(a.corner 3)–(a.corner 4)–(a.corner 5)–(a.corner 1); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (a); in 1,...,5circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,5 (a.corner 2) – (a.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=36] at (-.38,.51) (b); in 1,...,3circle,fill,inner sep=1.5pt] at (b.corner ) ; in 1,...,3 (b.corner 2) – (b.corner ) ; [white] (a.corner 2) circle (1.pt); [white] (b.corner 1) circle (1.pt); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (b); [color=orange!15] (b.corner 1)–(b.corner 2)–(b.corner 3)–(b.corner 4)–(b.corner 5)–(b.corner 1); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (b); in 1,...,5circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,5 (b.corner 2) – (b.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=36] at (-.38,.51) (a); in 1,...,3circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,3 (a.corner 2) – (a.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); [color=orange!35] (c.corner 1)–(c.corner 2)–(c.corner 3)–(c.corner 1); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); in 1,...,3circle,fill,inner sep=1.5pt] at (c.corner ) ; in 1,...,3 (c.corner 2) – (c.corner ) ; [white] (a.corner 1) circle (1.pt); [white] (b.corner 2) circle (1.pt); [white] (c.corner 2) circle (1.pt); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (b); [color=orange!15] (b.corner 1)–(b.corner 2)–(b.corner 3)–(b.corner 4)–(b.corner 5)–(b.corner 1); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (b); in 1,...,5circle,fill,inner sep=1.5pt] at (b.corner ) ; in 1,...,5 (b.corner 2) – (b.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=36] at (-.38,.51) (a); in 1,...,3circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,3 (a.corner 2) – (a.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); [color=orange!35] (c.corner 1)–(c.corner 2)–(c.corner 3)–(c.corner 1); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); in 1,...,3circle,fill,inner sep=1.5pt] at (c.corner ) ; in 1,...,3 (c.corner 2) – (c.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=192] at (.89,.05) (d); [color=orange!70] (d.corner 1)–(d.corner 2)–(d.corner 3)–(d.corner 1); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=192] at (.89,.05) (d); in 1,...,3circle,fill,inner sep=1.5pt] at (d.corner ) ; in 1,...,3 (d.corner 2) – (d.corner ) ; [white] (a.corner 1) circle (.9pt); [white] (b.corner 2) circle (1.pt); [white] (c.corner 2) circle (1.pt); [white] (d.corner 1) circle (1.pt); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (b); [color=orange!15] (b.corner 1)–(b.corner 2)–(b.corner 3)–(b.corner 4)–(b.corner 5)–(b.corner 1); regular polygon,regular polygon sides=5,minimum size=1.1cm,draw] (b); in 1,...,5circle,fill,inner sep=1.5pt] at (b.corner ) ; in 1,...,5 (b.corner 2) – (b.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=36] at (-.38,.51) (a); in 1,...,3circle,fill,inner sep=1.5pt] at (a.corner ) ; in 1,...,3 (a.corner 2) – (a.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); in 1,...,3circle,fill,inner sep=1.5pt] at (c.corner ) ; in 1,...,3 (c.corner 2) – (c.corner ) ; regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); [color=orange!35] (c.corner 1)–(c.corner 2)–(c.corner 3)–(c.corner 1); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=12] at (.61,-.2) (c); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=192] at (.89,.05) (d); in 1,...,3circle,fill,inner sep=1.5pt] at (d.corner ) ; in 1,...,3 (d.corner 2) – (d.corner ) ; [color=orange!70] (d.corner 1)–(d.corner 2)–(d.corner 3)–(d.corner 1); regular polygon,regular polygon sides=3,minimum size=.3cm,draw,rotate=192] at (.89,.05) (d); regular polygon,regular polygon sides=6,minimum size=1.25cm,draw,rotate=143] at (.6,.8) (e); [color=orange!90] (e.corner 1)–(e.corner 2)–(e.corner 3)–(e.corner 4)–(e.corner 5)–(e.corner 6)–(e.corner 1); regular polygon,regular polygon sides=6,minimum size=1.25cm,draw,rotate=143] at (.6,.8) (e); in 1,...,6circle,fill,inner sep=1.5pt] at (e.corner ) ; in 1,...,6 (e.corner 2) – (e.corner ) ; [white] (a.corner 1) circle (1.pt); [white] (b.corner 2) circle (1.pt); [white] (c.corner 2) circle (1.pt); [white] (d.corner 1) circle (1.pt); [white] (e.corner 2) circle (1.pt); Redrawn with vertex labels, we have the following, where the vertex labeled 1 is the origin of $F_1$ , the vertex labeled 2 is the origin of $F_2$ , and so on.",
"[scale=1] regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); [color=orange!1] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [thick] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [color=orange!15] (e.corner 2)–(e.corner 12)–(e.corner 7)–(e.corner 4)–(e.corner 3)–(e.corner 2); [thick] (e.corner 2)–(e.corner 12)–(e.corner 7)–(e.corner 4)–(e.corner 3)–(e.corner 2); [color=orange!35] (e.corner 4)–(e.corner 5)–(e.corner 7)–(e.corner 4); [thick] (e.corner 4)–(e.corner 5)–(e.corner 7)–(e.corner 4); [color=orange!70] (e.corner 6)–(e.corner 5)–(e.corner 7)–(e.corner 6); [thick] (e.corner 6)–(e.corner 5)–(e.corner 7)–(e.corner 6); [color=orange!100] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 12)–(e.corner 7); [thick] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 12)–(e.corner 7); regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; above] at (e.corner 1) 1; above] at (e.corner 2) 2; left] at (e.corner 3) 3; left] at (e.corner 4) 4; below] at (e.corner 5) 5; below] at (e.corner 6) 6; below] at (e.corner 7) 7; below] at (e.corner 8) 8; right] at (e.corner 9) 9; right] at (e.corner 10) 10; above] at (e.corner 11) 11; above] at (e.corner 12) 12; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 7) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 7) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 7) – (e.corner 11) ; (e.corner 7) – (e.corner 12) ; [white] (e.corner 1) circle (1.25pt); [white] (e.corner 2) circle (1.25pt); [white] (e.corner 4) circle (1.25pt); [white] (e.corner 5) circle (1.25pt); [white] (e.corner 7) circle (1.25pt);"
],
[
"Map $\\operatorname{\\operatorname{\\mathtt {TD}}}$",
"In this section, we construct a map $\\operatorname{\\operatorname{\\mathtt {TD}}}$ from $\\operatorname{Tri}(n,t,s)$ to $\\operatorname{Dyck}(n-2,t,s)$, which is the inverse to $\\operatorname{\\operatorname{\\mathtt {DT}}}$ .",
"Definition 3.13 Given $T\\in \\operatorname{Tri}(n,t,s)$ , we define $\\operatorname{\\operatorname{\\mathtt {TD}}}(T)$ , a certain lattice path, as follows.",
"Consider the fan decomposition $(\\mathcal {F}(T), \\delta (T)) = ((F_1,F_2,\\ldots , F_{t+s}), (d_1,\\ldots ,d_{t+s-1})$ .",
"Let $x_j$ denote the origin of $F_j$ and let $d_{t+s}$ be the number of boundary edges from $x_{t+s}$ to $x_1$ minus 2.",
"Now we identify the $x_j$ with its corresponding vertex in $T$ .",
"Letting $u_j$ be the number of triangles in $F_j$ , define $\\operatorname{\\operatorname{\\mathtt {TD}}}(T)$ to be $\\textsf {U}^{u_1}{d_1}\\textsf {U}^{u_2}{d_2}\\dots \\textsf {U}^{u_{t+s}}{d_{t+s}}.$ Lemma 3.14 If $T\\in \\operatorname{Tri}(n,t,s)$ , then the string $\\operatorname{\\operatorname{\\mathtt {TD}}}(T)$ is a dyck path in $\\operatorname{Dyck}(n-2,t,s)$ .",
"We claim $\\operatorname{\\operatorname{\\mathtt {TD}}}(T)$ is a valid Dyck path.",
"First, note that given any fan decomposition of a triangulation on $n$ vertices, the largest label for an origin is $n-1$ .",
"Thus, the largest number of boundary edges between the first and last origin (travelling counter-clockwise) is $n-2$ , which is precisely the number of triangles in the triangulation.",
"Thus, by studying the triangulation $T_k$ given by $\\mathcal {F}(T_k)=(F_1,F_2,\\dots , F_k)$ and the origins $x_1,\\dots , x_k$ , we see that $\\sum _{j=1}^k d_j\\le \\text{the number of triangles in $T_k$}=\\sum _{j=1}^k u_j.$ Also, note that $\\displaystyle \\sum _{j=1}^{t+s}u_j&=\\text{number of triangles in $T$}\\\\&=n-2\\\\&=\\text{number of boundary edges of $T$ minus 2}\\\\&=\\sum _{j=1}^{t+s}d_j.$ Hence this path is a Dyck path with semi length $n-2$ .",
"Finally, note that $u_{t+s}=\\#F_{t+s}\\ge 2$ since $F_{t+s}$ is the last fan in $\\mathcal {F}(T)$ .",
"Hence, we have constructed a path in $\\operatorname{Dyck}(n-2,t,s)$ .",
"Example 3.15 Suppose we start with the following triangulation.",
"[scale=1] regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; above] at (e.corner 1) 1; above] at (e.corner 2) 2; left] at (e.corner 3) 3; left] at (e.corner 4) 4; below] at (e.corner 5) 5; below] at (e.corner 6) 6; below] at (e.corner 7) 7; below] at (e.corner 8) 8; right] at (e.corner 9) 9; right] at (e.corner 10) 10; above] at (e.corner 11) 11; above] at (e.corner 12) 12; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 7) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 7) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 7) – (e.corner 11) ; (e.corner 7) – (e.corner 12) ; Below, we identify the origins of the fans in $\\mathcal {F}=(F_1,F_2,F_3,F_4,F_5)$ by using larger circles for such vertices.",
"[scale=.9] regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); [color=orange!1] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [thick] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [color=orange!15] (e.corner 2)–(e.corner 12)–(e.corner 7)–(e.corner 4)–(e.corner 3)–(e.corner 2); [thick] (e.corner 2)–(e.corner 12)–(e.corner 7)–(e.corner 4)–(e.corner 3)–(e.corner 2); [color=orange!35] (e.corner 4)–(e.corner 5)–(e.corner 7)–(e.corner 4); [thick] (e.corner 4)–(e.corner 5)–(e.corner 7)–(e.corner 4); [color=orange!70] (e.corner 6)–(e.corner 5)–(e.corner 7)–(e.corner 6); [thick] (e.corner 6)–(e.corner 5)–(e.corner 7)–(e.corner 6); [color=orange!100] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 12)–(e.corner 7); [thick] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 12)–(e.corner 7); regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); above] at (e.corner 1) 1; above] at (e.corner 2) 2; left] at (e.corner 3) 3; left] at (e.corner 4) 4; below] at (e.corner 5) 5; below] at (e.corner 6) 6; below] at (e.corner 7) 7; below] at (e.corner 8) 8; right] at (e.corner 9) 9; right] at (e.corner 10) 10; above] at (e.corner 11) 11; above] at (e.corner 12) 12; in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 7) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 7) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 7) – (e.corner 11) ; (e.corner 7) – (e.corner 12) ; [white] (e.corner 1) circle (1.25pt); [white] (e.corner 2) circle (1.25pt); [white] (e.corner 4) circle (1.25pt); [white] (e.corner 5) circle (1.25pt); [white] (e.corner 7) circle (1.25pt); Hence, $F_1$ , $F_3$ , and $F_4$ are all fans of size 1, $F_2$ is a fan of size 3, and $F_5$ is a fan of size 4.",
"The origins of these fans are identified in the image below.",
"Consequently, $u_1=u_3=u_4=1$ , $u_2=3$ , and $u_5=4$ .",
"Also note that $d_1=d_3$ and $d_2=d_4=2$ .",
"Hence we have the Dyck path $\\textsf {U}^32\\textsf {U}2\\textsf {U}^44,$ which visualized gives the following path.",
"[scale=0.6, line width=1pt] [color=black!40, thick] (-3,0)–(19,0); (-2,0) circle[radius=5pt]; [thick] (-2,0)–(-1,1); (-1,1) circle[radius=5pt]; [thick] (-1,1)–(0,0); (0,0) circle[radius=5pt]; in 0,1,...,2 [thick] (,)–(+1,+1); (+1,+1) circle[radius=5pt]; [thick] (3,3)–(4,2); (4,2) circle[radius=5pt]; [thick] (4,2)–(5,1); (5,1) circle[radius=5pt]; [thick] (5,1)–(6,2); (6,2) circle[radius=5pt]; [thick] (6,2)–(7,1); (7,1) circle[radius=5pt]; [thick] (7,1)–(8,2); (8,2) circle[radius=5pt]; [thick] (8,2)–(9,1); (9,1) circle[radius=5pt]; [thick] (9,1)–(10,0); (10,0) circle[radius=5pt]; [thick] (10,0)–(14,4); (11,1) circle[radius=5pt]; (12,2) circle[radius=5pt]; (13,3) circle[radius=5pt]; (14,4) circle[radius=5pt]; [thick] (14,4)–(18,0); (15,3) circle[radius=5pt]; (16,2) circle[radius=5pt]; (17,1) circle[radius=5pt]; (18,0) circle[radius=5pt]; The following map is an explicit interpretation of the map $\\operatorname{\\operatorname{\\mathtt {SD}}}$ composed with $\\operatorname{\\operatorname{\\mathtt {DT}}}$ without needing to bring up Dyck paths."
],
[
"Map $\\operatorname{\\operatorname{\\mathtt {ST}}}$",
"In this section, we construct a map $\\operatorname{\\operatorname{\\mathtt {ST}}}$ from $\\operatorname{Skyt}(a,i,b)$ to $\\operatorname{Tri}(a+b+2i,i+1,b-2)$ .",
"Definition 3.16 Given a tableau $\\lambda \\in \\operatorname{Skyt}(a,i,b)$, we define $\\operatorname{\\operatorname{\\mathtt {ST}}}(\\lambda )$ , a triangulation, as follows.",
"Let $d_j=x_{j+1}-x_j$ , where $x_1,\\dots , x_{i+b-1}$ are the entries in the top $b-1$ rows of $\\lambda $ so that $x_1<x_2<\\dots <x_{i+b-1}$ .",
"Let $\\operatorname{Nom}(\\lambda )=(A_1,\\dots , A_{i+b-1})$ .",
"For $1 \\le j\\le i+b-1$ , let $f_j=\\#A_j$ .",
"Let $F_j$ be a fan of size $f_j$ .",
"Then we define $\\operatorname{\\operatorname{\\mathtt {ST}}}(\\lambda )$ to be the triangulation whose fan decomposition is $(\\mathcal {F},\\delta )$ , where $\\mathcal {F}=(F_1,\\dots , F_{i+b-1})$ and $\\delta =(d_1,\\dots , d_{i+b-2})$ .",
"Lemma 3.17 Let $\\lambda \\in \\operatorname{Skyt}(a,i,b)$ .",
"Then $\\operatorname{\\operatorname{\\mathtt {ST}}}(\\lambda )\\in \\operatorname{Tri}(a+b+2i,i+1,b-2)$ .",
"Recall triangulations are uniquely determined by their fan decomposition, and thus $\\operatorname{\\operatorname{\\mathtt {ST}}}(\\lambda )$ is guaranteed to be a triangulation.",
"Note that the number of triangles in this triangulation is precisely the number of entries of $\\lambda $ , which is $a+b+2i-2$ .",
"Hence, the boundary of our constructed triangulation has $a+b+2i$ edges.",
"Recall that among $A_1,\\dots , A_{i+b-1}$ , precisely $i+1$ have cardinality larger than 1, and precisely $b-2$ have cardinality exactly 1.",
"Thus, our proposed fan decomposition for $\\operatorname{\\operatorname{\\mathtt {ST}}}(\\lambda )$ has precisely $i+1$ non singular fans and $b-2$ singular fans.",
"Finally, note that by construction, $f_{i+b-1}>1$ since $|A_{i+b-1}|>1$ by construction of $\\operatorname{Nom}(\\lambda )$ .",
"That is, the last fan appearing in $\\mathcal {F}$ is not singular.",
"All together, this verifies that we have constructed a triangulation in $\\operatorname{Tri}(a+b+2i, i+1, b-2)$ .",
"Example 3.18 Consider the following choice for $\\lambda $ .",
"decorations.pathreplacing centertableaux 1 4 2 5 7 369 8 10 Hence, $A_1=\\lbrace 1\\rbrace $ , $A_2=\\lbrace 2,3\\rbrace $ , $A_3=\\lbrace 4\\rbrace $ , $A_4=\\lbrace 5,6,8\\rbrace $ , and $A_5=\\lbrace 7,9,10\\rbrace $ .",
"We have $x_1=1$ , $x_2=2$ , $x_3=4$ , $x_4=5$ , and $x_5=7$ .",
"Thus, $d_1=1$ , $d_2=2$ , $d_3=1$ , and $d_4=2$ .",
"Also $f_1=1$ , $f_2=2$ , $f_3=1$ , $f_4=3$ , and $f_5=3$ .",
"This constructs the following triangulation.",
"[scale=1] regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); [color=orange!1] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [thick] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [color=orange!15] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [thick] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [color=orange!35] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [thick] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [color=orange!70] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [thick] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [color=orange!100] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); [thick] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); regular polygon,regular polygon sides=12,minimum size=4cm,draw,rotate=30] at (.74,.76) (e); in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; in 1,...,3left] at (e.corner ) ; in 4,...,7below] at (e.corner ) ; in 8,...,12right] at (e.corner ) ; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 12) ; (e.corner 5) – (e.corner 12) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 5) – (e.corner 11) ; (e.corner 7) – (e.corner 11) ; [white] (e.corner 1) circle (1.25pt); [white] (e.corner 2) circle (1.25pt); [white] (e.corner 4) circle (1.25pt); [white] (e.corner 5) circle (1.25pt); [white] (e.corner 7) circle (1.25pt); The following map is an explicit interpretation of the map $\\operatorname{\\operatorname{\\mathtt {TD}}}$ composed with $\\operatorname{\\operatorname{\\mathtt {DS}}}$ without needing to bring up Dyck paths."
],
[
"Map $\\operatorname{\\operatorname{\\mathtt {TS}}}$",
"In this section, we construct a map $\\operatorname{\\operatorname{\\mathtt {TD}}}$ from $\\operatorname{Tri}(n,t,s)$ to $\\operatorname{Skyt}(n-2-s-2t,t-1,s+2)$, which is the inverse to $\\operatorname{\\operatorname{\\mathtt {ST}}}$ .",
"Definition 3.19 Let $T\\in \\operatorname{Tri}(n,t,s)$ .",
"We define $\\operatorname{\\operatorname{\\mathtt {TS}}}(T)$ , a skew tableau, as follows.",
"We label the triangles in the triangulation in the following way: For each fan, label a triangle with the label of the fan's origin.",
"Then, greedily label other triangles with the unused vertices in the order that fans appear in $\\mathcal {F}(T)$ .",
"(Triangles within a single fan need not be labeled in any particular order.)",
"Let $A_j$ be the labels appearing in the fan $F_j$ .",
"Let $\\operatorname{\\operatorname{\\mathtt {TS}}}(T)$ be the skew diagram so that $\\operatorname{Nom}\\big (\\operatorname{\\operatorname{\\mathtt {TS}}}(T)\\big )=(A_1,\\dots , A_{t+s})$ .",
"Lemma 3.20 Given $T\\in \\operatorname{Tri}(n,t,s)$ , we have $\\operatorname{\\operatorname{\\mathtt {TS}}}(T) \\in \\operatorname{Skyt}(n-2-s-2t,t-1,s+2)$ .",
"Note that the number of $A_j$ so that $|A_j|=1$ is precisely the number of singleton fans in $T$ , which is $s$ .",
"Also, the number of $A_j$ so that $|A_j|>1$ is $t$ .",
"Next, notice that $(A_1,\\dots , A_{\\ell +s})$ is a nomincreasing sequence due to the greedy labeling of the triangles in $T$ , and we also have that $|A_{\\ell +s}|>1$ since $F_{\\ell +s}$ , the last fan appearing in $\\mathcal {F}(T)$ , must contain more than 1 triangle.",
"Finally, the number of triangles in $T$ is $n-2$ .",
"Thus, $\\operatorname{\\operatorname{\\mathtt {TS}}}(T)\\in \\operatorname{Skyt}(n-2-s-2t,t-1,s+2)$ .",
"Example 3.21 For example, given the triangulation in $\\operatorname{Tri}(12,3,2)$ below, label the vertices in counter-clockwise order.",
"[scale=1] regular polygon,regular polygon sides=12,minimum size=7cm,draw,rotate=30] at (.74,.76) (e); [color=orange!1] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [thick] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [color=orange!15] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [thick] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [color=orange!35] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [thick] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [color=orange!70] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [thick] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [color=orange!100] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); [thick] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); regular polygon,regular polygon sides=12,minimum size=7cm,draw,rotate=30] at (.74,.76) (e); in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; in 1,...,3left] at (e.corner ) ; in 4,...,7below] at (e.corner ) ; in 8,...,12right] at (e.corner ) ; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 12) ; (e.corner 5) – (e.corner 12) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 5) – (e.corner 11) ; (e.corner 7) – (e.corner 11) ; [white] (e.corner 1) circle (1.25pt); [white] (e.corner 2) circle (1.25pt); [white] (e.corner 4) circle (1.25pt); [white] (e.corner 5) circle (1.25pt); [white] (e.corner 7) circle (1.25pt); We now label a single triangle in each fan with the label of the corresponding origin.",
"[scale=.9] regular polygon,regular polygon sides=12,minimum size=7cm,draw,rotate=30] at (.74,.76) (e); [color=orange!1] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [thick] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [color=orange!15] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [thick] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [color=orange!35] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [thick] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [color=orange!70] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [thick] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [color=orange!100] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); [thick] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); regular polygon,regular polygon sides=12,minimum size=7cm,draw,rotate=30] at (.74,.76) (e); in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; above] at (e.corner 1) 1; above] at (e.corner 12) 12; in 2,...,3left] at (e.corner ) ; in 4,...,7below] at (e.corner ) ; in 8,...,11right] at (e.corner ) ; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 12) ; (e.corner 5) – (e.corner 12) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 5) – (e.corner 11) ; (e.corner 7) – (e.corner 11) ; t (0.1,4.3) 1; t (-1,3) 2; t (-1.3,.5) 4; t (0.8,1.5) 5; t (3.5,1.2) 7; [white] (e.corner 1) circle (1.25pt); [white] (e.corner 2) circle (1.25pt); [white] (e.corner 4) circle (1.25pt); [white] (e.corner 5) circle (1.25pt); [white] (e.corner 7) circle (1.25pt); We now label the remaining triangles greedily in the order that fans appear in $\\mathcal {F}(T)$ .",
"[scale=.9] regular polygon,regular polygon sides=12,minimum size=7cm,draw,rotate=30] at (.74,.76) (e); [color=orange!1] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [thick] (e.corner 1)–(e.corner 2)–(e.corner 12)–(e.corner 1); [color=orange!15] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [thick] (e.corner 2)–(e.corner 12)–(e.corner 4)–(e.corner 3)–(e.corner 2); [color=orange!35] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [thick] (e.corner 4)–(e.corner 5)–(e.corner 12)–(e.corner 4); [color=orange!70] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [thick] (e.corner 6)–(e.corner 5)–(e.corner 12)–(e.corner 11)–(e.corner 7)–(e.corner 6); [color=orange!100] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); [thick] (e.corner 7)–(e.corner 8)–(e.corner 9)–(e.corner 10)–(e.corner 11)–(e.corner 7); regular polygon,regular polygon sides=12,minimum size=7cm,draw,rotate=30] at (.74,.76) (e); in 1,...,12circle,fill,inner sep=1.5pt] at (e.corner ) ; above] at (e.corner 1) 1; above] at (e.corner 12) 12; in 2,...,3left] at (e.corner ) ; in 4,...,7below] at (e.corner ) ; in 8,...,11right] at (e.corner ) ; (e.corner 2) – (e.corner 12) ; (e.corner 2) – (e.corner 4) ; (e.corner 4) – (e.corner 12) ; (e.corner 5) – (e.corner 12) ; (e.corner 5) – (e.corner 7) ; (e.corner 6) – (e.corner 7) ; (e.corner 7) – (e.corner 10) ; (e.corner 7) – (e.corner 9) ; (e.corner 5) – (e.corner 11) ; (e.corner 7) – (e.corner 11) ; t (0.1,4.3) 1; t (-1,3) 2; t (-2.7,1.9) 3; t (-1.3,.5) 4; t (0.8,1.5) 5; t (0.8,-.9) 6; t (-0.2,-2.7) 8; t (3.5,1.2) 7; t (3.5,-.7) 9; t (3.2,-1.9) 10; [white] (e.corner 1) circle (1.25pt); [white] (e.corner 2) circle (1.25pt); [white] (e.corner 4) circle (1.25pt); [white] (e.corner 5) circle (1.25pt); [white] (e.corner 7) circle (1.25pt); Hence, $A_1=\\lbrace 1\\rbrace $ , $A_2=\\lbrace 2,3\\rbrace $ , $A_3=\\lbrace 4\\rbrace $ , $A_4=\\lbrace 5,6,8\\rbrace $ , and $A_5=\\lbrace 7,9,10\\rbrace $ .",
"This gives the following tableaux.",
"1 4 2 5 7 369 8 10"
],
[
"Dissections and Triangulations",
"In this section, we construct a combinatorial bijection between $\\operatorname{Dis}(n+2,i)$ and $\\operatorname{Tri}(n+i+1,i+1,0)$ as mentioned in the discussion following Corollary REF .",
"First, we define the map from $\\operatorname{Tri}(n+i+3,i+1,0)$ to $\\operatorname{Dis}(n+2,i)$ .",
"Definition 4.1 Let $T$ be a triangulation of an $(n+i+3)$ -gon with $i+1$ non-singular fans and no singular fans.",
"We know $\\mathcal {F}(T)$ is of the form $\\mathcal {F}(T)=(F_1,\\dots , F_{i+1})$.",
"Remove the internal diagonals of $F_j$ in $T$ for each $j$ , leaving us with exactly $i$ diagonals in $T$ .",
"Let $x_j$ be the origin of $F_j$ .",
"For each vertex $x_j$ , let $y_j$ be the immediate vertex that follows $x_j$ counterclockwise.",
"Note that it is possible to have $y_j=x_{j+1}$ .",
"Also, we always have that the $y_j$ is a vertex in $F_j$ , since $(x_j,y_j)$ must bound a triangle, and by definition this triangle is a part of $F_j$ .",
"Contract each edge $(x_j,y_j)$ , creating an $(n+2)$ -gon.",
"Note that the vertex labeled 1 will always be the origin of $F_1$ , so consequently we always contract $(1,2)$ .",
"Let 1 be the label of the new vertex after contracting this edge.",
"Relabel the vertices in increasing counterclockwise order, starting at the original vertex 1.",
"Since no fan of $T$ was singular, it must be that the contractions preserved all $i$ diagonals, giving us a dissection in $\\operatorname{Dis}(n+2,i)$ .",
"Now we define the map inverse to the one given above in definition REF , which is a map from $\\operatorname{Dis}(n+2,i)$ to $\\operatorname{Tri}(n+i+3,i+1,0)$ .",
"Definition 4.2 For the reverse map, let $D$ be a dissection of an $(n+2)$ -gon with $i$ chords, say $c_1,c_2,\\dots , c_i$ .",
"We assume, as with triangulations, that the vertices of $D$ are already labeled with the numbers 1 through $n+2$ .",
"We will describe a process which allows us to add new vertices and edges to $D$ .",
"Let $1^{\\prime }$ be a new vertex so that $(1^{\\prime },1)$ is an edge and $(1^{\\prime },2)$ is an edge.",
"Delete the edge $(1,2)$ .",
"If 1 was incident to more than one chords, shift all chords that do not form a triangle with the edge $(1,n+2)$ so that they are incident with $1^{\\prime }$ instead of 2.",
"Now, let $x$ be the next vertex counterclockwise to 1 incident to a chord.",
"(Note it may be that $x=1^{\\prime }$ .)",
"Proceed with the following procedure.",
"Let $c_{j_1},c_{j_2},\\dots , c_{j_k}$ be the list of chords incident with $x$ .",
"Let $z$ be the vertex immediately counterclockwise of $x$ .",
"Remove the edge $(x,z)$ and add a new vertex $x^{\\prime }$ along with edges $(x,x^{\\prime })$ and $(x^{\\prime },z)$ .",
"If $x$ is adjacent to exactly one chord, continue to step (5).",
"Otherwise, let $y$ be the vertex immediately clockwise to $x$ .",
"The edge $(y,x)$ bounds a closed region which contains exactly one chord $c_{j_\\ell }$ .",
"For each $c_{j_m}$ with $m\\ne \\ell $ , change its incidence with $x$ to an incidence with $x^{\\prime }$ .",
"If, after doing the prior step, we add a boundary edge to a region that we have already added a boundary edge to, undo the prior step and continue to the next step.",
"Move to the next vertex counterclockwise to $x$ incident to some chord, calling this new vertex $x$ .",
"(Note this new vertex may be the vertex $x^{\\prime }$ constructed in step (2).)",
"If $x$ is a vertex we have already visited before, terminate the procedure.",
"Otherwise, restart at step (1).",
"After doing this, observe that no region is a triangle.",
"Also observe that we added a single edge to the boundary for each region (hence the importance of step (4)), and so we now have an $(n+i+3)$ -gon.",
"Relabel the vertices, starting at the vertex labeled 1 and continuing counterclockwise.",
"We can decompose our new polygon into $i+1$ regions, labeled $P_1,P_2,\\dots , P_{i+1}$ .",
"Make each of these fans so that the origin of $P_j$ is the vertex with the minimum label of $P_j$ .",
"This gives us a triangulation of an $(n+i+3)$ -gon with $i+1$ non-singular fans and no singular fans.",
"Remark 4.3 There are a couple of things to keep in mind that may help justify why the maps given in Definitions REF to REF are mutual inverses.",
"The edges we contract going from a triangulation to a dissection are exactly the edges we add back going from a dissection to a triangulation.",
"This is because the origins of fans in triangulations are always chosen by the smallest vertex appearing in a fan, which appear sooner traveling counterclockwise around the polygons than vertices with larger labels.",
"The regions in a dissection are ultimately what become our fans for a triangulation, so we consequently always add an edge on the boundary of a dissection right after the vertex that would end up being the origin for a fan.",
"The chords of a dissection should be viewed as the parts of the triangulation that ultimately form the boundaries of the fans (along with the actual boundary of the polygon).",
"Hence, we can not expect two such chords to remain incident in the triangulation, as this would alter the number of fans.",
"See Figure REF below to see an illustration of the map from a triangulation to a dissection and Figure REF to see an illustration of the map of the other direction.",
"In this section, we construct a combinatorial bijection between $\\operatorname{Dis}(n+2,i)$ and $\\operatorname{Tri}(n+i+1,i+1,0)$ as mentioned in the discussion following Corollary REF .",
"First, we define the map from $\\operatorname{Tri}(n+i+3,i+1,0)$ to $\\operatorname{Dis}(n+2,i)$ .",
"Definition 4.1 Let $T$ be a triangulation of an $(n+i+3)$ -gon with $i+1$ non-singular fans and no singular fans.",
"We know $\\mathcal {F}(T)$ is of the form $\\mathcal {F}(T)=(F_1,\\dots , F_{i+1})$.",
"Remove the internal diagonals of $F_j$ in $T$ for each $j$ , leaving us with exactly $i$ diagonals in $T$ .",
"Let $x_j$ be the origin of $F_j$ .",
"For each vertex $x_j$ , let $y_j$ be the immediate vertex that follows $x_j$ counterclockwise.",
"Note that it is possible to have $y_j=x_{j+1}$ .",
"Also, we always have that the $y_j$ is a vertex in $F_j$ , since $(x_j,y_j)$ must bound a triangle, and by definition this triangle is a part of $F_j$ .",
"Contract each edge $(x_j,y_j)$ , creating an $(n+2)$ -gon.",
"Note that the vertex labeled 1 will always be the origin of $F_1$ , so consequently we always contract $(1,2)$ .",
"Let 1 be the label of the new vertex after contracting this edge.",
"Relabel the vertices in increasing counterclockwise order, starting at the original vertex 1.",
"Since no fan of $T$ was singular, it must be that the contractions preserved all $i$ diagonals, giving us a dissection in $\\operatorname{Dis}(n+2,i)$ .",
"Now we define the map inverse to the one given above in definition REF , which is a map from $\\operatorname{Dis}(n+2,i)$ to $\\operatorname{Tri}(n+i+3,i+1,0)$ .",
"Definition 4.2 For the reverse map, let $D$ be a dissection of an $(n+2)$ -gon with $i$ chords, say $c_1,c_2,\\dots , c_i$ .",
"We assume, as with triangulations, that the vertices of $D$ are already labeled with the numbers 1 through $n+2$ .",
"We will describe a process which allows us to add new vertices and edges to $D$ .",
"Let $1^{\\prime }$ be a new vertex so that $(1^{\\prime },1)$ is an edge and $(1^{\\prime },2)$ is an edge.",
"Delete the edge $(1,2)$ .",
"If 1 was incident to more than one chords, shift all chords that do not form a triangle with the edge $(1,n+2)$ so that they are incident with $1^{\\prime }$ instead of 2.",
"Now, let $x$ be the next vertex counterclockwise to 1 incident to a chord.",
"(Note it may be that $x=1^{\\prime }$ .)",
"Proceed with the following procedure.",
"Let $c_{j_1},c_{j_2},\\dots , c_{j_k}$ be the list of chords incident with $x$ .",
"Let $z$ be the vertex immediately counterclockwise of $x$ .",
"Remove the edge $(x,z)$ and add a new vertex $x^{\\prime }$ along with edges $(x,x^{\\prime })$ and $(x^{\\prime },z)$ .",
"If $x$ is adjacent to exactly one chord, continue to step (5).",
"Otherwise, let $y$ be the vertex immediately clockwise to $x$ .",
"The edge $(y,x)$ bounds a closed region which contains exactly one chord $c_{j_\\ell }$ .",
"For each $c_{j_m}$ with $m\\ne \\ell $ , change its incidence with $x$ to an incidence with $x^{\\prime }$ .",
"If, after doing the prior step, we add a boundary edge to a region that we have already added a boundary edge to, undo the prior step and continue to the next step.",
"Move to the next vertex counterclockwise to $x$ incident to some chord, calling this new vertex $x$ .",
"(Note this new vertex may be the vertex $x^{\\prime }$ constructed in step (2).)",
"If $x$ is a vertex we have already visited before, terminate the procedure.",
"Otherwise, restart at step (1).",
"After doing this, observe that no region is a triangle.",
"Also observe that we added a single edge to the boundary for each region (hence the importance of step (4)), and so we now have an $(n+i+3)$ -gon.",
"Relabel the vertices, starting at the vertex labeled 1 and continuing counterclockwise.",
"We can decompose our new polygon into $i+1$ regions, labeled $P_1,P_2,\\dots , P_{i+1}$ .",
"Make each of these fans so that the origin of $P_j$ is the vertex with the minimum label of $P_j$ .",
"This gives us a triangulation of an $(n+i+3)$ -gon with $i+1$ non-singular fans and no singular fans.",
"Remark 4.3 There are a couple of things to keep in mind that may help justify why the maps given in Definitions REF to REF are mutual inverses.",
"The edges we contract going from a triangulation to a dissection are exactly the edges we add back going from a dissection to a triangulation.",
"This is because the origins of fans in triangulations are always chosen by the smallest vertex appearing in a fan, which appear sooner traveling counterclockwise around the polygons than vertices with larger labels.",
"The regions in a dissection are ultimately what become our fans for a triangulation, so we consequently always add an edge on the boundary of a dissection right after the vertex that would end up being the origin for a fan.",
"The chords of a dissection should be viewed as the parts of the triangulation that ultimately form the boundaries of the fans (along with the actual boundary of the polygon).",
"Hence, we can not expect two such chords to remain incident in the triangulation, as this would alter the number of fans.",
"See Figure REF below to see an illustration of the map from a triangulation to a dissection and Figure REF to see an illustration of the map of the other direction."
]
] | 2209.08201 |
[
[
"A Decade of Code Comment Quality Assessment: A Systematic Literature\n Review"
],
[
"Abstract Code comments are important artifacts in software systems and play a paramount role in many software engineering (SE) tasks related to maintenance and program comprehension.",
"However, while it is widely accepted that high quality matters in code comments just as it matters in source code, assessing comment quality in practice is still an open problem.",
"First and foremost, there is no unique definition of quality when it comes to evaluating code comments.",
"The few existing studies on this topic rather focus on specific attributes of quality that can be easily quantified and measured.",
"Existing techniques and corresponding tools may also focus on comments bound to a specific programming language, and may only deal with comments with specific scopes and clear goals (e.g., Javadoc comments at the method level, or in-body comments describing TODOs to be addressed).",
"In this paper, we present a Systematic Literature Review (SLR) of the last decade of research in SE to answer the following research questions: (i) What types of comments do researchers focus on when assessing comment quality?",
"(ii) What quality attributes (QAs) do they consider?",
"(iii) Which tools and techniques do they use to assess comment quality?, and (iv) How do they evaluate their studies on comment quality assessment in general?",
"Our evaluation, based on the analysis of 2353 papers and the actual review of 47 relevant ones, shows that (i) most studies and techniques focus on comments in Java code, thus may not be generalizable to other languages, and (ii) the analyzed studies focus on four main QAs of a total of 21 QAs identified in the literature, with a clear predominance of checking consistency between comments and the code.",
"We observe that researchers rely on manual assessment and specific heuristics rather than the automated assessment of the comment quality attributes."
],
[
"Introduction",
"Software systems are often written in several programming languages [1], and interact with many hardware devices and software components [2], [3].",
"To deal with such complexity and to ease maintenance tasks, developers tend to document their software with various artifacts, such as design documents and code comments [4].",
"Several studies have demonstrated that high quality code comments can support developers in software comprehension, bug detection, and program maintenance activities [5], [6], [7].",
"However, code comments are typically written using natural language sentences, and their syntax is neither imposed by a programming language's grammar nor checked by its compiler.",
"Additionally, static analysis tools and linters provide limited syntactic support to check comment quality.",
"Therefore, writing high-quality comments and maintaining them in projects is a responsibility mostly left to developers [8], [9].",
"The problem of assessing the quality of code comments has gained a lot of attention from researchers during the last decade [10], [11], [12], [13], [14].",
"Despite the research community's interest in this topic, there is no clear agreement on what quality means when referring to code comments.",
"Having a general definition of quality when referring to code comments is hard, as comments are diverse in purpose and scope.",
"Problem Statement.",
"Maintaining high-quality code comments is vital for software evolution activities, however, assessing the overall quality of comments is not a trivial problem.",
"As developers use various programming languages, adopt project-specific conventions to write comments, embed different kinds of information in a semi-structured or unstructured form [15], [13], and lack quality assessment tools for comments, ensuring comment quality in practice is a complex task.",
"Even though specific comments follow all language-specific guidelines in terms of syntax, it is still challenging to determine automatically whether they satisfy other quality aspects, such as whether they are consistent or complete with respect to the code or not [16].",
"There are various such aspects, e.g., readability, content relevance, and correctness that should be considered when assessing comments, but tools do not support all of them.",
"Therefore, a comprehensive study of the specific attributes that influence code comment quality and techniques proposed to assess them is essential for further improving comment quality tools.",
"Previous mapping and literature review studies have collected numerous quality attributes (QAs) that are used to assess the quality of software documentation based on their importance and effect on the documentation quality.",
"Ding et al.",
"[17] focused specifically on software architecture and requirement documents, while Zhi et al.",
"[18] analyzed code comments along with other types of documentation, such as requirement and design documents.",
"They identified 16 QAs that influence the quality of software documentation.",
"However, the identified QAs are extracted from a body of literature concerning relatively old studies (i.e., studies conducted prior to the year blue2011) and are limited in the context of code comments.",
"For instance, only 10% of the studies considered by Zhi et al.",
"concern code comments.",
"Given the increasing attention that researchers pay to comment quality assessment, it is essential to know which QAs, tools and techniques they propose to assess code comment quality.",
"To achieve this objective, we perform an SLR on studies published in the last decade, i.e., blue2011-blue2020.",
"We review blue2353 studies and find blue47 to be relevant to assessing comment quality.",
"From these we extract the programming language, the types of analyzed comments, QAs for comments, techniques to measure them, and the preferred evaluation type to validate their results.",
"We observe that (i) most of the studies and techniques focus on comments in Java code, (ii) many techniques that are used to assess QAs are based on heuristics and thus may not be generalizable to other languages, (iii) a total of blue21 QAs are used across studies, with a clear dominance of consistency, completeness, accuracy, and readability, and (iv) several QAs are often assessed manually rather than with the automated approaches.",
"We find that the studies are typically evaluated by measuring performance metrics and surveying students rather than by performing validations with practitioners.",
"This shows that there is much room for improvement in the state of the art of comment quality assessment.",
"The contributions of this paper are: an SLR of a total of blue2353 papers, of which we review the blue47 most relevant ones, focusing on QAs mentioned and research solutions proposed to assess code comment quality, a catalog of blue21 QAs of which four QAs are often investigated, while the majority is rarely considered in the studies, and of which blue10 are new with respect to the previous study by Zhi et al.",
"[18], a catalog of methods used to measure these blue21 QAs in research studies, an overview of the approaches and tools proposed to assess comment quality, taking into account the types of comments and the programming languages they consider, a discussion of the challenges and limitations of approaches and tools proposed to assess different and complementary comment QAs, and a publicly available dataset including all validated data, and steps to reproduce the study in the replication package.https://doi.org/10.5281/zenodo.4729054 Paper structure.",
"The rest of the paper is organized as follows.",
"In sec:study-design we highlight our motivation and rationale behind each research question, and we present our methodology, including the different steps performed to answer our research questions.",
"In sec:results we report the study results.",
"We discuss the results in sec:discussion and their implications and future direction in sec:implication-future-work.",
"We highlight the possible threats to validity for our study in sec:Threats-to-validity.",
"Then sec:Related-work summarizes the related work, in relation to the formulated research questions.",
"Finally, sec:conclusion concludes our study, outlining future directions."
],
[
"Study Design",
"The main objective of our study is to present an overview of the state of the art in assessing the quality of code comments.",
"Specifically, we aim to highlight the QAs mentioned in the literature, and the techniques used so far to assess comment quality.",
"To this end, we carry out an SLR, following the widely accepted guidelines of Kitchenham et al.",
"[19] and Keele [20].",
"The first step in this direction is to specify the research questions related to the topic of interest [19].",
"The following steps focus on finding a set of relevant studies that are related to the research questions based on an unbiased search strategy."
],
[
"Research questions",
"Our goal is to foster research that aims at building code comment assessment tools.",
"To achieve this goal, we conduct an SLR, investigating the literature of the last decade to identify comment related QAs and solutions that address related challenges.",
"We formulate the following research questions: RQ1: What types of comments do researchers focus on when assessing comment quality?",
"Motivation: Comments are typically placed at the beginning of a file, usually to report licensing or author information, or placed preceding a class or function to document the overview of a class or function and its implementation details.",
"Depending on the specific type of comment used in source code and the specific programming language, researchers may use different techniques to assess them.",
"These techniques may not be generalizable to other languages.",
"For example, studies analyzing class comments in object-oriented programming languages may need extra effort to generalize the comment assessment approach to functional programming languages.",
"We, therefore, investigate the comment types researchers target.",
"RQ2: What QAs do researchers consider in assessing comment quality?",
"Motivation: QAs may solely concern syntactic aspects of the comments (e.g., syntax of comments), writing style (e.g., grammar), or content aspects (e.g., consistency with the code).",
"Researchers may use different terminology for the same QA and thus these terms must be mapped across studies to obtain a unifying view of them, for instance, if the accuracy QA is defined consistently across studies or another terminology is used for it.",
"We collect all the possible QAs that researchers refer to and map them, if necessary, following the methodology of Zhi et al.. Future studies that aim to improve specific aspects of comment quality evaluation can use this information to design their tools and techniques.",
"RQ3: Which tools and techniques do researchers use to assess comment QAs?",
"Motivation: Researchers may assess QAs manually, or may use sophisticated tools and techniques based on simple heuristics or complex machine learning (ML) to assess them automatically.",
"We aim to identify if there are clear winning techniques for this domain and collect various metrics and tools used for this purpose.",
"RQ4: What kinds of contribution do studies often make?",
"Motivation: Engineering researchers usually motivate their research based on the utility of their results.",
"Auyang clarifies that engineering aims to apply scientific methods to real world problems [21].",
"However, software engineering currently lacks validation [22].",
"With this question, we want to understand what types of solution researchers contribute to improving automatic comment quality assessment, such as metrics, methods, or tools.",
"This RQ can provide insight into specific kinds of solutions for future work.",
"RQ5: How do researchers evaluate their comment quality assessment studies?",
"Motivation: Researchers may evaluate their comment assessment approaches, e.g., by surveying developers, or by using a dataset of case studies.",
"However, how often they involve professional developers and industries in such studies is unknown."
],
[
"Search Strategy",
"After formulating the research questions, the next steps focus on finding relevant studies that are related to the research questions.",
"In these steps, we construct search keywords in subsection REF , choose the search timeline in subsection REF , collect sources of information in subsection REF , retrieve studies in subsection REF , select studies based on the inclusion/exclusion criteria in subsection REF , and evaluate the relevant studies to answer the research questions in subsection REF .",
"Figure: SLR stages to collect relevant papers"
],
[
"Search Keywords",
"Kitchenham et al.",
"recommended formulating individual facets or search units based on the research questions [19].",
"These search units include abbreviations, synonyms and other spellings, and they are combined using boolean operators.",
"Pettricrew et al.",
"suggested PIO (population, interventions, and outcome) criterion to define such search units [23].",
"The populations include terms related to the standards.",
"We first examine the definitions of documentation and comment in IEEE Standard Glossary of Software Engineering Terminology (IEEE Standard 610.12-1990) to collect the main keywords.",
"According to the definition, we identify the keywords comment, documentation, and specification and add them to the set $K_{1}$.",
"We further add frequently mentioned comment-related keywords, such as API, annotation, and summar to the set $K_{1}$.",
"The interventions include terms that are related to software methodology, tools, technology, or procedures.",
"With respect to quality assessment, we define the intervention keywords to be quality, assess, metric, measure, score, analy, practice, structur, study, or studied and add them to the set $K_{2}$.",
"Note that we add common variations of the words manually, for example, we add “summar” keyword to the set to cover both “summary” and “summarization”.",
"We do not use any NLP libraries to stem words due to two main reasons, (i) to reduce the noisy matches, and (ii) the words from the title and abstract of the papers are not preprocessed (stemmed or lemmatized), therefore stemming the keywords might not find the exact or prefix matches.",
"For example, using the porter stemming approach, the word “study” will be stemmed to “studi” and we might miss the papers with “study” word.",
"To avoid such cases, we add common variations of this word study and studied to our search keywords.",
"The outcomes include terms that are related to factors of significance to developers (e.g., reduced cost, reduced time to assess quality).",
"Since it is not a required unit to restrict the search scope, and our focus is on all kinds of quality assessment approaches, we exclude the outcomes in our search keywords.",
"However, to narrow down our search and exclude irrelevant papers, such those about code reviews or testing, or non-technical papers, we formulate another set of keywords, $K_{3}$.",
"In this set, we include code review, test, keynote, invited, and poster, to exclude entries of non-technical papers that were not filtered out using the heuristics on the number of pages.",
"Table: keywords selected according to PIO criterionHence, using the final set of keywords (also given in tab:PIO-keywords), we select a paper if its title and abstract match the keywords from $K_{1}$ and $K_{2}$ but not from $K_{3}$ where the prefix function is used to match the keywords in the paper."
],
[
"Timeline",
"We focus our SLR on the last decade (i.e., January blue2011-December blue2020) since Zhi et al.",
"investigated the works on software documentation quality — including code comments — from blue1971 to blue2011 [18].",
"Our results can thus be used to observe the evolution of comment quality assessment, but, more importantly, they naturally complement the existing body of knowledge on the topic.",
"We then proceed to the main steps i.e., retrieving the paper data, selecting venues, and identifying the relevant papers for our comment context."
],
[
"Data collection",
"Concretely, our data collection approach comprises three main steps, i.e., literature data collection, data selection, and data evaluation, which we sketch in fig:paper-selection-process and present in further detail as follows: We now describe how we automatically collect the data from the literature, explaining the rationale behind our selection of venues and our automatic keyword-based filtering to identify the likely relevant papers regarding comment quality assessment.",
"We justify the need for another step of data gathering based on the snowball approach in Section REF .",
"Finally, we present our criteria for the careful evaluation of the relevant papers in Sec REF .",
"Venue Selection.",
"Code comment analysis, generation, usage, and maintenance are of primary interest to the SE research community.",
"Thus, in order to systematically review the literature on the comment quality assessment, we start by focusing on the SE venues.",
"We use the latest 2020 updated version of the conference and journal database of the CORE ranking portal as a primary data source to identify all the potentially relevant SE venues.https://www.core.edu.au/conference-portal The portal provides assessments of major conferences and journals in the computing disciplines, and it is a well-established and regularly-validated registry maintained by the academic community.",
"We extract all ranked journals in SE (search code 803) from the CORE portalhttp://portal.core.edu.au/jnl-ranks/?search=803by=forsource=CORE2020sort=arankpage=1 accessed on 25 Mar, 2021 and all top conferences and workshops in the SE field (search code 4612).http://portal.core.edu.au/conf-ranks/?search=4612by=forsource=CORE2020sort=arankpage=1 accessed on 25 Mar, 2021 This process gives us an initial list of blue85 journal and blue110 conference venues.",
"We select in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 1; blue20.",
"software engineering (SE) conferences and journals from blue110.",
"candidate venues based on the likelihood of finding relevant papers in their proceedings.",
"We focus on A* and A conferences and journals, and add conferences of rank B or C if they are co-located with previously selected A* and A conferences to have venues, such as the IEEE/ACM International Conference on Program Comprehension (ICPC) or the IEEE International Workshop on Source Code Analysis and Manipulation (SCAM) that focus on source code comprehension and manipulation.",
"We prune venues that may not contain relevant contributions to source code comments.",
"Specifically, we exclude a venue if its ten years of proceedings contain fewer than five occurrences of the words documentation or comment.",
"This way, we exclude conferences, such as IEEE International Conference on Engineering of Complex Computer Systems (ICECCS), Foundations of Software Science and Computational Structures (FoSSaCS), and many others that primarily focus on other topics, such as verification or programming languages.",
"Thus, we reduce our dataset to 20 conferences and six journals, as shown in tab:venue-selection.",
"In tab:venue-selection, the column Type specifies whether a venue is a conference (C) or a journal (J), and the column Rank denotes the corresponding CORE rank of the venue as of April 2021.",
"The column Selection indicates the data collection phase in which the venue was first selected.",
"The column Papers per venue indicates the total number of papers selected from this venue, both during the direct search and the snowball search.",
"We consider only full papers (published in a technical track and longer than five pages) since they are likely to be an extended or mature version of the papers published in other tracks, such as NIER, ERA, or Poster."
],
[
"Data Retrieval",
"We retrieve in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 2; the proceedings from January blue2011 to December blue2020 of the selected venues from the DBLP digital library.",
"From each paper, we collect its metadata using the GitHub repositoryhttps://github.com/sbaltes/dblp-retriever, such as the title, authors, conference track (if present), its page length, and its Digital Object Identifier (DOI), directly from DBLP for a total of blue17554 publications.",
"For each paper, the DOI is resolved and its abstract is collected from the publisher webpage.",
"Table: Included Journals, Conferences, and Workshops."
],
[
"Keyword-based filtering",
"We apply in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 3; a keyword-based search (given in subsec:search-keywords) using a prefix function to the retrieved proceedings to select potentially relevant papers.",
"We account for possible upper- and lowercase letters in the keywords, and sometimes use variations of keywords (e.g., singular and plural forms).",
"Our filtering will get papers (whose title and abstract include keywords from $K_{1}$ and $K_{2}$ but not from $K_{3}$ ) that explicitly mention concepts we are interested in, e.g., “A Human Study of Comprehension and Code Summarization” from ICPC 2020 [24] is matched by keywords summar from $K_{1}$ in the title and quality from $K_{2}$ in the abstract, but will exclude papers not sufficiently close to our research subject, e.g., “aComment: mining annotations from comments and code to detect interrupt related concurrency bugs” from ICSE 2011 has two keywords comment and annotation from $K_{1}$ but none from the $K_{2}$.",
"The final set of keywords we use for filtering is the result of an iterative approach: we manually scan the full venue proceedings metadata to make sure the set of keywords did not prune relevant papers, and we refine the set of keywords during several iterative discussions.",
"This iterative approach gives us confidence that our keyword-based filtering approach does not lead to false negatives for the selected venues.",
"After applying the keyword-based filtering, we identify blue2043 studies as potentially-relevant papers from a total of blue17554, which we review manually."
],
[
"Data selection",
"We analyze [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 4; the blue2043 selected papers following the protocol where four authors or evaluators manually evaluate the papers based on the inclusion and exclusion criterion to ensure that they indeed assess comment quality.",
"Inclusion criteria The topic of the paper is about code comment quality.",
"The study presents a model/technique/approach to assess code comments or software documentation including code comments.",
"Exclusion criteria The paper is not in English.",
"It does not assess any form of quality aspects of comments e.g., content, style, or language used.",
"It is not published in a technical track.",
"It is a survey paper.",
"It is not a peer reviewed paper, or it is a pre-print.",
"It covers other documentation artifacts, i.e., not comments.",
"It is shorter than 5 pages."
],
[
"Manual analysis",
"The selected papers were equally divided among four evaluators (i.e., two Ph.D. candidates and two faculty members) based on years of publications so that each evaluator gets papers from all venues, e.g., the first author evaluate proceedings from 2011 to 2013.",
"We make sure that evaluators do not take decisions on papers they co-authored to avoid conflicts of interest.",
"Each evaluator has at least two years of experience in the domain of comment analysis.",
"Each paper is reviewed by three evaluators.",
"The evaluators follow a three-iteration-based process to evaluate the assigned papers.",
"In the first iteration, the first evaluator independently assesses the relevance of a paper based on the criteria by inspecting each paper's title and abstract, to make an initial guess, then inspecting its conclusion to reach the final decision.",
"In the next iteration, another evaluator reviews the paper and validates the previous decision by adding the label “agrees/disagrees with the first evaluator”.",
"With this process, every publication selected in the final set is reviewed by at least two researchers.",
"In case they do not agree, the third evaluator reviews it [25], and the final decision is taken based on the majority voting mechanism.",
"We decide, for instance, to include the study by Hata et al.",
"[26], even though it only talks about links in comments.",
"Though it does not explicitly describe any quality aspect of comments, it mentions the traceability of the links, which is a QA we consider in our study.",
"All studies considered in our SLR together with their evaluation (the agreement and disagreement for each study) are available in our replication package.",
"Thus, we reduce blue2043 papers to blue71 candidate papers (i.e., blue0.%) with a fair agreement according to Cohen's Kappa (k=0.36).",
"For all candidate papers, we read in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 5; their introduction, conclusion, and the study design (if needed), and discuss them amongst ourselves to ensure their relevance.",
"During this analysis process, some additional papers were found to be irrelevant.",
"For example, the study by Aghajani et al.",
"seems relevant based on the title and abstract, but does not really evaluate code comments, and we thus discarded it [27].",
"With this process, blue71.",
"papers in total were discarded, reducing the relevant paper set to blue30 papers."
],
[
"Data gathering for snowballing",
"To include further relevant papers that we might have missed with the venue-based approach, we perform in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 6; a forward and backward snowballing approach for the blue30 papers and retrieve a total of blue1624.",
"unique papers.",
"Table: NO_CAPTIONThe column Total reports the total number of references and citations collected.",
"The Unique column reports a total number of unique items (i.e., since relevant papers cover similar topics many references, and citations are shared across our set of studies).",
"Finally, the column Selected reports the total number of unique references and citations whose publication year falls within our time frame range, i.e., 2011-2020."
],
[
"Data selection from snowballing",
"We repeat in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 7; the same keyword-based filtering to these blue1624.",
"papers, as described in subsec:autom-data-coll.",
"As a result, blue311 papers were added for manual analysis.",
"We repeat in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 8; the three-iteration based manual analysis process and find blue39 additional candidate papers to analyze.",
"After the second round of discussion [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 9; we keep blue17 additional relevant papers.",
"We find a total of blue47 papers shown in tab:included-studies published in the venues shown in tab:venue-selection.",
"In tab:included-studies, the column Study ID indicates the ID assigned to each paper, the column Title presents the title of the paper, and the column Year indicates the years in which the paper is published.",
"To further ensure the relevance of our search strategy, we search our keywords on popular publication databases, such as ACM, IEEE Xplore, Wiley etc.",
"We search for our keywords in titles and abstracts.It is not possible to search the keywords in abstracts in Wiley.",
"We retrieve 13 144 results from IEEE Xplore, and 10 567 from ACM for the same timeline (2011-2020).",
"We inspect first 200 results (sorted by relevance criterion on the publisher webpage) from each of these databases.",
"We apply our inclusion and exclusion criterion to find the extent to which our venue selection criteria might have missed relevant papers.",
"Our results from ACM show that 19% of the these papers are already covered by our search strategy but only 5% of them fulfilled our inclusion criterion.",
"Nearly 81% of the papers are excluded due to their non-SE venue.",
"Among these papers, 80% are unrelated to the code comment quality aspect while 1% of papers (two papers) that are related to code comments are missed due to two main reasons, (i) the venue not being indexed in CORE2020, and (ii) the paper being from a non-technical track.",
"Similarly, the results from IEEE show that 30% of the papers are already covered by our search strategy but only 5% of them fulfilled the inclusion criterion.",
"Nearly 69% of the papers are excluded due to their non-SE venue and unrelated to code comment quality aspect.",
"We also find 1% papers that are relevant to our topic of interest but excluded due to the length criteria, specifically one of the paper is a poster paper and another is a short paper.",
"Table: Included studies"
],
[
"Data Evaluation",
"We work in step [baseline=(myanchor.base)] circle,fill=.,inner sep=1pt] (myanchor) 10; on the full versions of the blue47 relevant papers to identify the QAs and the approaches to assess comments.",
"In case we cannot retrieve the full PDF version of a paper, we use university resources to access it.",
"This affects only one paper by Sun et al., which requires payment to access the full version [41].",
"In case we cannot access a paper via any resource, we remove it from our list.",
"We find no such inaccessible study.",
"We report all papers in an online shared spreadsheet on Google Drive to facilitate their analysis collaboratively.",
"For each paper we extract common metadata, namely Publication year, Venue, Title, Authors, Authors' country, and Authors' affiliation.",
"We then extract various dimensions (described in the following paragraphs) formulated to answer all research questions."
],
[
"Data extraction for research questions",
"To answer RQ1 (What types of comments do researchers focus on when assessing comment quality?",
"), we record the Comment scope dimension.",
"It lists the scope of comments under assessment such as class, API, method (function), package, license, or inline comments.",
"In case the comment type is not mentioned, we classify it as “code comments”.",
"Additionally, we identify the programming languages whose comments are analyzed, and record this in the Language analyzed dimension.",
"Table: RQ2 QAs mentioned by Zhi et al.",
"(highlighted in bold) and other worksTo answer RQ2 (What QAs do researchers consider in assessing comment quality?",
"), we identify various QAs researchers mention to assess comment quality.",
"This reflects the various quality aspects researchers perceive as important to have high-quality comments.",
"tab:paper-fields-extraction-rq lists the QAs in the Quality attribute (QA) column and their brief summary in the Description column.",
"Of these QAs, several are mentioned by Zhi et al.",
"in their work [18], and are highlighted by the bold text compared to QAs mentioned in other works.",
"As Zhi et al.",
"considered various types of documentation, such as requirement and architectural documents, not all attributes fit exactly into our study.",
"For instance, the category “Format” includes the format of the documentation (e.g., UML, flow chart) in addition to the other aspects such as writing style of the document, use of diagrams etc.",
"Although the format of the documentation is not applicable in our case due to our comment-specific interest, we keep other applicable aspects (writing style, use of diagram) of this QA.",
"In addition to their QAs, we include any additional attribute mentioned in our set of relevant papers.",
"If a study uses different terminology but similar meaning to QAs in our list, we map such QAs to our list and update the list of possible synonyms as shown in the column Synonyms in tab:paper-fields-extraction-rq.",
"In case we cannot map a study to the existing QAs, we map it to the Other category.",
"For the cases where the studies do not mention any specific QA and mention comment quality analysis in general, we map the study to the list of existing QAs or classify it as Other based on their goal behind the quality analysis.",
"For example, Pascarella et al.",
"identify various information types in comments to support developers in easily finding relevant information for code comprehension tasks and to improve the comment quality assessment [13].",
"They do not mention any specific QA, but based on their study goal of finding relevant information easily, we map their study to the content relevance QA.",
"Similarly, we map other comment classification studies such as S06, S29, S33, and S41 to the content relevance attribute.",
"At the same time, the studies on linguistic anti-patterns (LAs) are mapped to the consistency attribute, given that LAs are practices that lead to lexical inconsistencies among code elements, or between code and associated comments [59], [34], [35].",
"Additionally, the studies that mention the negation of the QAs such as inconsistency, incorrectness, or incompleteness are mapped to their antonyms as consistency, correctness, or completeness, respectively to prevent duplication.",
"RQ3 (Which tools and techniques do researchers use to assess comment QAs?)",
"concerns various methods researchers use or propose to assess comment QAs, for instance, whether they use machine-learning based methods to assess comment quality or not.",
"Technique type.",
"This identifies whether the technique used to assess a QA is based on natural language processing (NLP), heuristics, static analysis, metrics, machine-learning (ML), or deep neural network (DNN) approaches.",
"The rationale is to identify which QAs are often assessed manually or using a specific automated approach.",
"For instance, if the study uses specific heuristics related to the programming environment to assess a QA, it is classified as heuristic-based technique, if it uses abstract syntax tree (AST) based static analysis approaches, then it is assigned to static analysis, and if it uses machine-learning or deep-learning-based techniques (including any or both of the supervised or unsupervised learning algorithms), then it is classified as ML-based, or DNN-based respectively.",
"A study can use mixed techniques to assess a specific QA and thus can be assigned to multiple techniques for the corresponding QA.",
"We often find cases where the studies do not use any automated technique to measure a QA and instead ask other developers to assess it manually, so we put such cases into the manual assessment category.",
"In case the study mentions a different technique, we extend the dimension values.",
"Metrics or tools.",
"This further elaborates specific metrics, or tools the studies propose or use to assess a QA.",
"A study can use an existing metric or can propose a new one.",
"Similarly, one metric can be used to assess multiple QAs.",
"We identify such metrics to highlight popular metrics amongst researchers.",
"RQ4 (What kinds of contribution do studies often make?)",
"captures the nature of the study and the type of contribution researchers use or propose to assess comment quality.",
"We first identify the nature of research of a study and then identify the type of contribution it provides.",
"This can reflect the kind of research often conducted to assess comment quality and the kind of contribution they make to support developers in assessing comment quality, for instance, what kind of solutions the Solution Proposal research often propose, such as a method, metric, model, or tool.",
"Table: Type of research approach studies use and type of contributions studies makeTo capture this information, we formulate the following dimensions: Research type.",
"This identifies the nature of the research approach used in the studies, such as empirical, validation, evaluation, solution proposal, philosophical, opinion, or experience paper [69], [70].",
"The dimension values are described in detail in tab:paper-research-contribution-type.",
"Paper contribution.",
"This dimension describes the type of contribution the study provides in terms of a method/technique, tool, process, model, metric, survey, or empirical results [70].",
"The dimension values are described in detail in tab:paper-research-contribution-type.",
"If we cannot categorize it into any of these, we mark it “Other”.",
"Tool availability.",
"This reflects whether the tool proposed in the study is accessible or not at the time of conducting our study.",
"González et al.",
"identified the reproducibility aspects characterizing empirical software engineering studies [71] in which availability of the artifact (the tool proposed in the study, or the dataset used to conduct the study) is shown as an important aspect to facilitate the replication and extension of the study.",
"Therefore, we record the availability of the proposed tool in this dimension and the availability of the dataset in the following dimension.",
"Dataset availability.",
"This reflects if the dataset used in the empirical study is accessible or not.",
"RQ5 (How do researchers evaluate their comment quality assessment studies?)",
"concerns how various kinds of research (Research type dimension described in the previous RQ), and various kinds of contribution (Paper contribution dimension) are evaluated in the studies.",
"For example, it helps us to observe that if a study proposes a new method/technique to assess comments, then the authors also conduct an experiment on open-source projects to validate the contribution, or they consult the project developers, or both.",
"We capture the type of evaluation in the Evaluation type dimension, and its purpose in Evaluation purpose.",
"The rationale behind capturing this information is to identify the shortcomings in their evaluations, e.g., how often the studies proposing a tool are validated with practitioners.",
"Evaluation type.",
"It states the type of evaluation the studies conduct to validate their approaches, such as conducting an experiment on open-source projects (Experiment), or surveying students, practitioners, or both.",
"For the automated approaches, we consider various performance metrics, also known as Information Retrieval (IR) metrics, that are used to assess the machine/deep learning-based models, such as Precision, Recall, F1 Measure, or Accuracy under the performance metrics.",
"In case the approach is validated by the authors of the work, we identify the evaluation type as Authors of the work.",
"Evaluation purpose.",
"It states the motivation of evaluation by authors such as evaluate the functionality, efficiency, applicability, usability, accuracy, comment quality in general, or importance of attributes.",
"As mentioned in subsec:data-selection, we analyze blue47 relevant papers in total.",
"Before answering our four RQs, we present a brief overview of the metadata (publishing venues) of the papers.",
"Figure: Relevant papers by yearsFigure: Relevant papers by countriestab:venue-selection highlights the publication venues of these papers.",
"Most studies were published in top-tier software engineering conferences (e.g., ICSE) and journals, especially the ones with a focus on empirical studies (e.g., EMSE).",
"This means that the SE community agrees that assessing comment quality is an important topic deserving of research effort.",
"fig:plot-papers-by-years shows the paper distribution over the past decade, indicating a clear trend of increasing interest of the SE research community in comment quality assessment.",
"fig:plot-papers-by-countries shows the author distribution of the selected papers by the institution.",
"For the timeline 1971-2011, we rely on the geographical statistics data from the replication package of our reference study by Zhi et al.",
"[18], while for the period 2011-2021, and we collect these statistics as follows.",
"For each paper, the primary affiliations of all authors are taken into account.",
"If people from different countries co-authored a paper, we calculate the proportion of a country's contribution for each paper so that each paper gets a total score of one to avoid over-representing papers.",
"For example, if five authors of a paper belong to Switzerland and one belongs to Spain, we assign 5/6 score for Switzerland and 1/6 for Spain for the paper.",
"Comparison with the previous data allows us to see the evolution of the field, with more even distribution of researchers nowadays and (unsurprising) rise of contributions from southeast Asia, specifically from China.",
"Finding 1.",
"The trend of analyzing comment quality has increased in the last decade (2011-2020), in part due to more researchers from southeast Asia working on the topic."
],
[
"RQ$_1$ : What types of comments do researchers focus on when assessing comment quality?",
"To describe the rationale behind code implementation, various programming languages use source code comments.",
"Our results show that researchers focus more on some programming languages compared to others as shown in fig:plot-comments-by-languages.",
"This plot highlights the types of comments on the y-axis; each stack in the bar shows the ratio of the studies belonging to a particular language.",
"For instance, the majority (87%) of the studies focus on code comments from Java, whereas only 15% of the studies focus on code comments from Python, and 10% of them focus on C# and C$++$ .",
"These results are in contrast to popular languages indicated by various developer boards, such as GitHub, Stack Overflow, or TIOBE.",
"For instance, the TIOBE index show Python and C languages more popular than Java.https://www.tiobe.com/tiobe-index/ verified on Sep, 2021 Similarly, the developer survey of 2019 and 2020 by Stack Overflow show that Java stands fifth after JavaScript, HTML/CSS, SQL, and Python among the most commonly used programming languages.https://insights.stackoverflow.com/survey/2020 We find only one study (S44) that seems to address the comment quality aspect in JavaScript.",
"Given the emerging trend of studies leveraging natural-language information in JavaScript code [72], [73], more research about comment quality may be needed in this environment.",
"It indicates that researchers need to analyze comments of other languages to verify their proposed approaches and support developers of other languages.",
"Finding 2.",
"87% of the studies analyze comments from Java while other languages have not yet received enough attention from the research community.",
"As code comments play an important role in describing the rationale behind source code, various programming languages use different types of comments to describe code at various abstraction levels.",
"For example, Java class comments should present high-level information about the class, while method comments should present implementation-level details [74].",
"We find that half of the studies (51% of the studies) focus on all types of comments whereas the other half focus on specific types of comments, such as inline, method, or TODO comments.",
"However, we also see in fig:plot-comments-by-languages that studies frequently focus on method comments and API documentation.",
"This proves the effort the research community is putting into improving API quality.",
"While some attention is given to often overlooked kinds of comments, such as license comments (S28,S33), TODO comments (S14), inline comments (S17), and deprecation comments (S45), no relevant paper seems to focus specifically on the quality of class or package comments.",
"Recently Rani et al.",
"studied the characteristics of class comments of Smalltalk in the Pharo environmenthttps://pharo.org/ and highlighted the contexts they differ from Java and Python class comments, and why the existing approaches (based on Java, or Python) need heavy adaption for Smalltalk comments [75], [76].",
"This may encourage more research in that direction, possibly for other programming languages.",
"Finding 3.",
"Even though 50% of the studies analyze all types of code comments, the rest focus on studying a specific type of comments such as method comments, or API comments, indicating research interest in leveraging a particular type of comment for specific development tasks.",
"Previous work by Zhi et al.",
"showed that a majority of studies analyze just one type of system [18].",
"In contrast, our findings suggest that the trend of analyzing comments of multiple languages and systems is increasing.",
"For example, 80% of the studies analyzing comments from Python and all studies analyzing comments from C$++$ also analyze comments from Java.",
"Only Pascarella et al.",
"(S42) and Zhang et al.",
"(S41) focus solely on Python [64], [63].",
"However, Zhang et al.",
"(S41) perform the comment analysis work in Python based on the Java study (S29) by Pascarella et al.",
"[63], [13].",
"Such trends also reflect the increasing use of polyglot environments in software development [77].",
"The “Other” label in fig:plot-comments-by-languages comprises language-agnostic studies, e.g., S16 or the studies considering less popular languages, e.g., S28 focuses on COBOL.",
"We find only one study (S44) that analyzes comments of six programming languages et al.",
"[26].",
"Finding 4.",
"The trend of analyzing multiple software systems of a programming language, or of several languages, shows the increasing use of polyglot environments in software projects."
],
[
"RQ$_2$ : Which QAs are used to assess code comments?",
"To characterize the attention that the relevant studies reserve to each QA over the past decade, fig:plot-qa-per-year shows all the QAs on the y-axis and the corresponding years on the x-axis.",
"Each bubble in the plot indicates both the number of papers by the size of the bubble and IDs of the studies.",
"Comparing the y-axis with the QAs in tab:paper-fields-extraction-rq demonstrates that our analysis finds new QAs with respect to the previous work of Zhi et al.",
"The blue10 additional QAs are: usefulness, use of examples, usability, references, preciseness, natural language quality, maintainability, visual models, internationalization, documentation technology, content relevance, conciseness, coherence, and availability.",
"However, not all QAs reported by Zhi et al.",
"for software documentation quality (highlighted in bold in tab:paper-fields-extraction-rq) are used in comment quality assessment.",
"In particular, we find no mention of trustworthiness, and similarity QAs even though previous works have highlighted the importance of both QAs to have high-quality documentation [78], [79], [80].",
"Also, Maalej et al.",
"showed in their study that developers trust code comments more than other kinds of software documentation [81], indicating the need to develop approaches to assess the trustworthiness of comments.",
"Finding 5.",
"Compared to the previous work by Zhi et al., we find 10 additional QAs researchers use to assess code comment quality.",
"Although several QAs received attention in 2013, the detailed analysis shows that there were mainly two studies (S02, S03) covering several QAs.",
"There is only one study published in 2014 (S05), while 2015 sees the first studies focusing on assessing comment quality.",
"One in particular, S26, attempts to cover multiple QAs.",
"The plot also shows which QAs receive the most attention.",
"A few QAs such as completeness, accuracy, content relevance, readability are often investigated.",
"The QA consistency is by far the one that receives constant and consistent attention across the years, with several in 2017 (S07, S08, S09, S29) and 2018 (S10, S11, S39, S42, S43).",
"Indeed, the problem of inconsistency has been studied from multiple points of view, such as inconsistency between code and comments that may emerge after code refactoring (S07), or the inconsistencies revealed by so-called linguistic antipatterns (S11, S37).",
"Unsurprisingly, the plot shows that up-to-dateness increasingly has received attention in the last three years of the decade, given that comments that are not updated together with code are also a cause of inconsistency (S15, S16).",
"A few attributes are rarely investigated, for instance the QAs investigated only by at most two studies over the past decade are format, understandability, spelling & grammar, organization, internationalization, documentation technology, coherence, conciseness, author related and accessibility.",
"More research would be needed to assess whether such attributes are intrinsically less important than others for comments according to practitioners.",
"Finding 6.",
"While QAs such as consistency and completeness are frequently used to assess comment quality, others are rarely investigated, such as conciseness and coherence.",
"Another aspect to analyze is whether researchers perceive the QAs as being the same or not.",
"For example, do all studies mean the same by consistency, conciseness, accuracy of comments?",
"We therefore collect the definition of each QA considered in the study.",
"We find that for various QAs researchers refer to the same QA but using different terminology.",
"We map such cases to the Synonyms column presented in tab:paper-fields-extraction-rq.",
"From this analysis we find that not all studies precisely define the QAs, or they refer to their existing definitions while evaluating comments using them.",
"For instance, the studies (S01, S04, S13, S17, S20, S29, S41) do not mention the specific QAs or their definition.",
"We put such studies, classifying comment content with the aim to improve comment quality, under content relevance.",
"On the other hand, in some studies researchers mention the QAs but not their definition.",
"For instance, S26 refers to various existing studies for the QA definitions but which QA definition is extracted from which study is not very clear.",
"Lack of precise definitions of QAs or having different definitions for the same QAs can create confusion among developers and researchers while assessing comment quality.",
"Future work needs to pay attention to either refer to the existing standard definition of a QA or define it clearly in the study to ensure the consistency and awareness across developer and scientific communities.",
"In this study, we focus on identifying the mention of QAs and their definition if given, and not on comparing and standardizing their definition.",
"Such work would require not only the existing definitions available in the literature for QAs but also collecting how researchers use them in practice, and what developers perceive from each QA for source code comments, which is out of scope for this work.",
"However, we provide the list of QAs researchers use for comment quality assessment to facilitate future work in mapping their definition and standardizing them for code comments.",
"Although each QA has its own importance and role in comment quality, they are not measured in a mutually exclusive way.",
"We find cases where a specific QA is measured by measuring another QA.",
"For example, accuracy is measured by measuring the correctness and completeness of comment, such as “the documentation is incorrect or incomplete and therefore no longer accurate documentation of an API.” (S24) Similarly, up-to-dateness is measured through consistency of comments (S40) or consistency is evaluated and improved using traceability (S31).",
"This indicates the dependency of various QAs on each other, and improving one aspect of comments can automatically improve other related aspects.",
"However, which techniques are used to measure which QAs is not yet known.",
"Finding 7.",
"Many studies miss a clear definition of the QAs they use in their studies.",
"This poses various challenges for developers and researchers, e.g., understanding what a specific QA means, mapping a QA to other similar QAs, and adapting the approaches to assess the QA to a certain programming environment."
],
[
"RQ$_3$ : Which tools and techniques do researchers use to assess comment QAs?",
"With respect to each QA, we first identify which techniques have been used to measure them.",
"We use the dimension Technique type to capture the type of techniques.",
"fig:plot-qas-technique-types shows that the majority of the QAs are measured by asking developers to manually assess it (manual assessment).",
"For instance, QAs such as coherence, format, organization, understandability, and usability are often assessed manually.",
"This indicates the need and opportunities to automate the measurement of such QAs.",
"A significant number of studies experimented with various automated approaches based on machine or deep learning approaches, but they focus on specific QAs and miss other QAs such as natural language quality, conciseness, correctness, traceability, coherence etc.",
"Similarly, another significant portion of studies uses heuristic-based approaches to measure various QAs.",
"The limitation of such heuristic-based approaches is their applicability to other software systems and programming languages.",
"More studies are required to verify the generalizability of such approaches.",
"Finding 8.",
"Manual assessment is still the most frequently-used technique to measure various QAs.",
"Machine learning based techniques are the preferred automated approach to asses QAs, but the majority of them focus on specific QAs, such as consistency, content relevance, and up-to-dateness, while ignoring other QAs.",
"We find that the majority of the machine learning-based approaches are supervised ML approaches.",
"These approaches require labeling the data and are therefore expensive in terms of time and effort.",
"To avoid the longer training time and memory consumption of ML strategies, Kallis et al.",
"used fastText to classify the issues reports on GitHub [82].",
"The fastText tool uses linear models and has achieved comparable results in classification to various deep-learning based approaches.",
"A recent study by Minaee et al.",
"shows that deep learning-based approaches surpassed common machine learning-based models in various text analysis areas, such as news categorization and sentiment analysis [83].",
"We also find some studies that use deep learning-based techniques partly (S06, S13, S20) along with machine learning techniques for a few QAs, such as assessing conciseness, spelling and grammar, and completeness.",
"However, there are still many QAs that are assessed manually and require considerable effort to support developers in automatically assessing comment quality.",
"Finding 9.",
"In the case of automated approaches to assess various QAs of comments, we observe that deep-learning based approaches are not yet explored even though various studies showed that they surpassed ML-based approaches in text analysis areas.",
"We see that machine learning-based approaches are used more often than deep-learning approaches, but whether it is due to their high accuracy, easy interpretation, or need for a small dataset is unclear and requires further investigation.",
"Table: Metrics and tools used for various quality attributes.Note: the description of each metric is given in tab:Metrics-descriptionTable: Description of each metric listed in tab:quality-attributes-metrics-toolsIn addition to identifying general techniques, we collect which metrics and tools have been used to measure various QAs.",
"tab:quality-attributes-metrics-tools shows various QAs in the column QAs, and metrics and tools used for each QA in the column Metrics, and Tools respectively.",
"The description of the collected metrics is presented in tab:Metrics-description.",
"We can see that out of 21, only 10 QAs have metrics defined for them.",
"A software metric is a function that takes some software data as input and provides a numerical value as an output.",
"The output provides the degree to which the software possesses a certain attribute affecting its quality [84].",
"To limit the incorrect interpretation of the metric, threshold values are defined.",
"However, the threshold value may change according to the type of comments analyzed, and the interpretation of the metric output may vary in turn.",
"We report threshold values, if present, for the collected metrics.",
"For readability QA, researchers were often found to be using the same metric (S08, S22, S39).",
"As developers spend significant amount of time reading code, including comments, having readable comment can help them in understanding code easier.",
"Yet readability remains a subjective concept.",
"Several studies, such as S08, S22, S39 identified various syntactic and textual features for source code and comments.",
"However, in context of code comments, they focus on the Flesch-Kincaid index method, which is typically used to assess readability of natural language text.",
"Since comments often consist of a mix of source code and natural language text, such methods can have disadvantages.",
"For example, developers can refer to the same code concept differently in comments, and they can structure their information differently.",
"Thus, formulating metrics that consider the special context of code comments can improve the the assessment of readability of comments.",
"Another popular metric is Consistency_1 used for assessing consistency between comments and code (S08, S22, S39).",
"This metric measures the overlap between the terms of method comments and method body.",
"These studies assume that the higher the overlap, better the readability of that code.",
"Similarly, metrics (coherence_1 , coherence_3, coherence_4) used for measuring the coherence QA suggest higher overlap between comments and code.",
"However, having too many overlapping words can defy the purpose of comments and can lead to redundant comments.",
"Using such metrics, a comment containing only rationale information about a method or class might be qualified as an incoherent or inconsistent comment whereas such comments can be very helpful in providing additional important information.",
"Although metrics can help developers easily estimate the quality of comments, their sensitivity towards various QAs can degrade comment quality overall.",
"More research is required to know the implication of given metrics on various QAs or combinations of QAs.",
"Finding 10.",
"Nearly 25% of the studies use metric-based methods to measure comment quality.",
"However, the metrics are defined or used for only 10 QAs out of blue21 QAs."
],
[
"Research types",
"As a typical development cycle can contain various research tasks, such as investigation of a problem, or validation of a solution, we collect which types of research are performed for the comment quality assessment domain, and what kinds of solutions researchers often contribute.",
"We categorize the papers according to the research type dimension and show its results in fig:plot-papers-research-approaches.",
"The results show that the studies often conduct validation research (investigating the properties of a solution) followed by the solution proposal (offering a proof-of-concept method or technique).",
"However, very few studies focus on evaluation research (investigating the problem or a technique implementation in practice).",
"We find only one study performing a replication study (S19).",
"Given the importance of research replicability in any field, future work needs to focus more on evaluating the proposed solution and testing their replicability in this domain."
],
[
"Paper contribution types",
"By categorizing the papers according to the paper contribution definition, fig:plot-papers-research-approaches and fig:plot-papers-approaches-evaluations show that over 44% of papers propose an approach (method/technique) to assess code comments.",
"A large part (blue1.%) of them are heuristics-based approaches, e.g., Zhou et al.",
"and Wang et al.",
"present such NLP based heuristics (S9, S13).",
"A few approaches rely on manual assessments.",
"As an example, consider how taxonomies assessing comment quality have emerged [14], [39].",
"Models are the second contribution by frequency, which makes sense considering the increasing trend of leveraging machine learning during the considered decade: blue1.% of the relevant papers proposing models are based on such approaches.",
"The label Empirical results comprises studies which mainly offer insights through authors' observations (e.g., S11, S15, S16, S19, S26).",
"Finally, given the important role that metrics have in software engineering [85], [86], it is valuable to look into metrics that are proposed or used to assess code comment quality as well.",
"For example, three studies (S18, S35, and S36) contribute metrics for completeness, accuracy, or coherence whereas other studies use existing established metrics, e.g., S08, S22, or S39 compute the readability of comments using the metric named the Flesch-Kincaid index."
],
[
"Tool availability",
"Previous work indicates the developers' effort in seeking tools to assess documentation quality, and highlights the lack of such tools [39].",
"In our study, we find that blue0.% of the studies propose tools to assess specific QAs, mainly for detecting inconsistencies between code and comments.",
"Of these studies proposing tools, blue1.% provide a link to them.",
"The lack of a direct link in the remaining blue100.% can hinder the reproducibility of such studies."
],
[
"Dataset availability",
"In terms of dataset availability, blue0.% of the studies provide a link to a replication package.",
"Of the remaining papers, some provide a link to the case studies they analyze (typically open-source projects) [28], build on previously existing datasets [61], or mention the reasons why they could not provide a dataset.",
"For instance, Garousi et al.",
"indicated the company policy as a reason to not to share the analyzed documentation in their case study [48].",
"Finding 11.",
"Nearly 50% of the studies still are lacking on the replicability dimension, with their respective dataset or tool often not publicly accessible."
],
[
"RQ$_5$ : How do researchers evaluate their comment quality assessment studies?",
"fig:plot-papers-approaches-evaluations shows how authors evaluate their contributions.",
"We see that code comment assessment studies generally lack a systematic evaluation, surveying only students, or conducting case studies on specific projects only.",
"Most of the time, an experiment is conducted without assessing the results through any kind of external expertise judgment.",
"Hence, only 30% of the relevant studies survey practitioners to evaluate their approach.",
"This tendency leads to several disadvantages.",
"First, it is difficult to assess the extent to which a certain approach may overfit specific case studies while overlooking others.",
"Second, approaches may be unaware of the real needs and interests of project developers.",
"Finally, the approaches may tend to focus too little on real-world software projects (such as large software products evolving at a fast pace in industrial environments).",
"Similarly, when a new method or technique or comment classification model is proposed, it is often assessed based on conventional performance metrics, such as Precision, Recall, or F1 (S02, S04, S07, S29, S41 etc.)",
"and rarely are the results verified in an industry setting or with practitioners.",
"Finding 12.",
"Many code comment assessment studies still lack systematic industrial evaluations for their proposed approaches, such as evaluating the metric, model, or method/technique with practitioners."
],
[
"Discussion",
"Below we detail our observations about state of the art in comment quality analysis together with implications and suggestions for future research."
],
[
"Comment Types",
"The analysis of the comment quality assessment studies in the last decade shows that the trend of analyzing comments from multiple languages and systems is increasing compared to the previous decade where a majority of the studies focus on one system [18].",
"It reflects the increasing use of polyglot environments in software development [77].",
"Additionally, while in the past researchers focused on the quality of code comments in general terms, there is a new trend of studies that narrow their research investigation to particular comment types (methods, TODOs, deprecation, inline comments), indicating the increasing interest of researchers in supporting developers in providing a particular type of information for program comprehension and maintenance tasks."
],
[
"Emerging QAs",
"Our analysis of the last decade of studies on code comment assessment shows that new QAs (coherence, conciseness, maintainability, understandability etc.",
"), which were not identified in previous work [18], are now being investigated and explored by researchers.",
"This change can be explained by the fact that while in the past researchers focused on the quality of code comments in general terms, in the last decade there has been a new trend of studies that narrow their research investigation to specific comment types (methods, TODOs, deprecation, inline comments) and related QAs."
],
[
"Mapping QAs",
"As a consequence of this shift of focus towards specific comment types, the same QAs used in prior studies can assume different definition nuances, depending on the kind of comments considered.",
"For instance, let us consider how the QA up-to-dateness, referred to in studies on code-comment inconsistency, assumes a different interpretation in the context of TODO comments.",
"A TODO comment that becomes outdated describes a feature that is not being implemented, which means that such a comment should be addressed within some deadline, and then removed from the code base (S14) when either the respective code is written and potentially documented with a different comment, or the feature is abandoned altogether.",
"At the same time, more research nowadays is conducted to understand the relations between different QAs."
],
[
"Mapping taxonomies",
"In recent years, several taxonomies concerning code comments have been proposed, however, all of them are characterized by a rather different focus, such as the scope of the comments (S02), the information embedded in the comment (S29, S41), the issues related to specific comment types (S06, S33, S40 ), as well as the programming language they belong to.",
"This suggests the need for a comprehensive code comment taxonomy or model that maps all these aspects and definitions in a more coherent manner to have a better overview of developer commenting practices across languages.",
"Rani et al.",
"adapted the code comment taxonomies of Java and Python (S29, S41) for class comments of Java and Python [76].",
"They mapped the taxonomies to Smalltalk class comments and found that developers write similar kinds of information in class comments across languages.",
"Such a mapping can encourage building language-independent approaches for other aspects of comment quality evaluation."
],
[
"Implication for Future studies",
"Besides the aspects discussed above, future studies on code comment assessment should be devoted to filling the gaps of the last decade of research as well as coping with the needs of developers interested in leveraging comment assessment tools in different program languages."
],
[
"Investigating specific comment types (RQ1)",
"Several works showed the importance of different types of comments to achieve specific development tasks and understanding about code.",
"Although, the trend of analyzing specific comment types has increased over the last decade, there are still comment types (e.g., class and package comments) that need more attention."
],
[
"Generalizing across languages (RQ1)",
"Given the preponderance of studies focusing on the Java language, and considering that statistics from various developer boards (StackOverflow, GitHub) suggest that there are other popular languages as well (e.g., Python and JavaScript), more studies on analyzing various types of comments in these languages are needed.",
"Interesting questions in this direction could concern the comparison of practices (e.g., given Python is often considered to be “self-explainable”, do developers write fewer comments in Python?)",
"and tools used to write code comments in different languages (e.g., popularity of Javadoc v.s.",
"Pydoc).",
"Similarly, whether various programming language paradigms, such as functional versus object-oriented languages, or statically-typed versus dynamic-typed languages, play a role in the way developers embed information in comments, or the way they treat comments, needs further work in this direction."
],
[
"Identifying QAs (RQ2)",
"Our results show various QAs, e.g., consistency, completeness, and accuracy that are frequently considered in assessing comment quality.",
"Additionally, various metrics, tools, and techniques that are proposed to assess them automatically.",
"Indeed, some QAs are largely overlooked in the literature, e.g., there is not enough research on approaches and automated tools that ensure that comments are accessible, trustworthy, and understandable, despite numerous studies suggesting that having good code comments brings several benefits."
],
[
"Standardizing QAs (RQ2)",
"We identify various QAs that researchers consider assessing comment quality.",
"Not all of these QAs are unique i.e., they have conceptual overlap (based on their definitions in tab:paper-fields-extraction-rq and measurement techniques in tab:quality-attributes-metrics-tools).",
"For example, the definition of up-to-datedness and consistency mention of keeping comments updated.",
"Similarly, the definition of coherence and similarity focus on the relatedness between code and comments.",
"In this study, we mainly focus on identifying various QAs from the literature and on extracting metrics, tools, and techniques to measure them.",
"Standardizing their definition can be an essential next step in the direction of comment quality assessment research.",
"Since not every study provides the definition of mentioned QAs, such a work will require surveying the authors to understand how they perceive various QAs and where they refer to for QAs definitions."
],
[
"Comment smells (RQ2)",
"Although there is no standard definition of good or bad comments, many studies indicate bloated comments (or non-informative comments), redundant comments (contain same information as in the code), or inconsistent comments (e.g., contain conflicting information compared to the code) as code or comment smells.",
"Arnaoudva et al.",
"identified various LAs that developers perceive as poor practices and should be avoided [59].",
"Still, what information is vital in comments is a subjective concept and can sometimes be contradictory.",
"For instance, Oracle's coding style guideline suggests including author information in class comments, whereas the Apache style guideline suggests removing it as it can be inferred from the version control system [87].",
"We find that researchers use the completeness QA to identify informative comments.",
"They define various metrics to assess the completeness of comments, as shown in tab:Metrics-description.",
"These metrics check the presence of specific information, such as summary, author, or exception information in class or method comments Future work can investigate the definition of good and bad comments by surveying various sources, such as documentation guidelines, researchers, and developers, and comparing the sources across to improve the understanding of high-quality comments.",
"Such work can inspire the development of more metrics and tools to ensure the adherence of comments to the standards."
],
[
"Automated tools and techniques (RQ3)",
"Finally, concerning techniques to assess comment quality, we observed that those based on AI, such as NLP and ML, were increasingly used in the past decade.",
"On the other hand, deep learning techniques do not yet seem to have gained a foothold within the community for assessing comment quality.",
"Since code comment generation is becoming more and more popular also due to such advanced techniques emerging, we envision that future work may study techniques and metrics to assess the quality of automatically generated code comments."
],
[
"Research evaluation (RQ4 and RQ5)",
"Scientific methods play a crucial role in the growth of engineering knowledge [88].",
"Several studies have indicated the weak validation in software engineering [22].",
"We also find that several studies propose solutions but do not evaluate their solution.",
"Also, various approaches were validated only by the authors of the work or by surveying students.",
"However, we need to do all steps as engineering science researchers do, empirically investigating the problems, proposing solutions, and validating those solutions.",
"In contrast to seven research types listed in tab:paper-research-contribution-type, we observe only limited types of research studies.",
"For example, we do not find any philosophical, opinion, or experience papers for the comment quality assessment domain even though this domain is more than a decade old now.",
"Philosophical papers sketch a new perspective of looking at things, conceptual frameworks, metrics etc.",
"Opinion papers present good or bad opinions of authors about something, such as different approaches to assess quality, using particular frameworks etc.",
"Similarly, experience papers often present insights about lessons learned or anecdotes by authors in using tools or techniques in practice.",
"Such papers help tool designers better shape their future tools."
],
[
"Threats to validity",
"We now outline potential threats to the validity of our study.",
"Threats to construct validity mainly concern the measurements used in the evaluation process.",
"In this case, threats can be mainly due to (i) the imprecision in the automated selection of relevant papers (i.e., the three-step search on the conference proceedings based on regular expressions), and to (ii) the subjectivity and error-proneness of the subsequent manual classification and categorization of relevant papers.",
"We mitigated the first threat by manually classifying a sample of relevant papers from a set of conference proceedings and compared this classification with the one recommended by the automated approach based on regular expressions.",
"This allowed us to incrementally improve the initial set of regular expressions.",
"To avoid any bias in the selection of the papers, we selected regular expression in a deterministic way (as detailed in the sec:study-design): We first examined the definition of documentation and comment in IEEE Standard Glossary of Software Engineering Terminology (IEEE Standard 610.12-1990) and identified the first set of keywords comment, documentation, and specification; we further added comment-related keywords that are frequently mentioned in the context of code comments.",
"Moreover, we formulated a set of keywords to discard irrelevant studies that presented similar keywords (e.g., code review comments).",
"To verify the correctness of the final set of keywords, we manually scanned the full venue proceedings metadata to make sure the set of keywords did not prune relevant papers.",
"This iterative approach allowed us to verify that our keyword-based filtering approach does not lead to false negatives for the selected venues.",
"We mitigated the second threat by applying multi-stage manual classification of conference proceedings, involving multiple evaluators and reviewers, as detailed in sec:study-design.",
"Threats to internal validity concern confounding factors that could influence our results and findings.",
"A possible source of bias might be related to the way we selected and analyzed the conference proceedings.",
"To deal with potential threats regarding the actual regular expressions considered for the selection of relevant studies, we created regular expressions that tend to be very inclusive, i.e., that select papers that are marginally related to the topic of interest, and we take a final decision only after a manual assessment.",
"Threats to external validity concern the generalization and completeness of results and findings.",
"Although the number of analyzed papers is large, since it involves studies spanning the last ten years of research, there is still the possibility that we missed some relevant studies.",
"We mitigate this threat by applying various selection criteria to select relevant conference proceedings, considering the well-established venues and communities related to code comment-related studies, as detailed in sec:study-design.",
"It is important to mention that this paper intentionally limits its scope in two ways, which threatens to the completeness of the study results and findings.",
"First of all, we mainly focus on research work investigating code comment quality without integrating studies from industry tracks of conference venues (as was done in previous studies thematically close to ours [17], [18]).",
"Second, we focus on those studies that involve manually written code comments in order to avoid auto-generated comments (already investigated in recent related work [89], [90]).",
"To further limit potential threats concerning the completeness of our study, we use the snowball approach to reach potentially relevant studies that we could have missed with our venue selection.",
"However, we support the argument of Garousi et al.",
"[91] who report that a multivocal literature review, with further replications, is desirable to make the overall interpretation of code comment quality attributes more complete for future work."
],
[
"Related Work",
"This section discusses the literature concerning (i) studies motivating the importance of quality attributes for software documentation, (ii) comment quality aspects, and (iii) recent SLRs discussing topics closely related to our investigation.",
"Important quality attributes for software documentation.",
"Various research works conducted surveys with developers to identify important quality attributes of good software documentation.",
"Forward and Lethbridge surveyed 48 developers, and highlighted developer concerns about outdated documentation [92].",
"Chen and Huang surveyed 137 project managers and software engineers [93].",
"Their study highlighted the typical quality problems developers face in maintaining software documentation: adequacy, complete, traceability, consistency, and trustworthiness.",
"Robillard et al.",
"conducted personal interviews with 80 practitioners and presented the important attributes for good documentation, such as including examples and usage information, complete, organized, and better design [94].",
"Similarly, Plosch et al.",
"surveyed 88 practitioners and identified consistency, clarity, accuracy, readability, organization, and understandability as the most important attributes [95].",
"They also indicated that developers do not consider documentation standards important (e.g., ISO 26514:2008, IEEE Std.1063:2001).",
"Sohan et al.",
"in their survey study highlighted the importance of examples in documentation [96].",
"The majority of the highlighted documentation quality attributes apply to code comments as well (as a type of software documentation).",
"However, which specific quality attributes (e.g., outdated, complete, consistent, traceable) researchers consider important to assess code comment quality and how these quality attributes are measured is yet to study.",
"Comment quality.",
"Evaluating comment quality according to various aspects has gained a lot of attention from researchers, for instance, assessing their adequacy [97] and their content quality [10], [11], analyzing co-evolution of comments and code [98], or detecting inconsistent comments [12], [14].",
"Several works have proposed tools and techniques for the automatic assessment of comment quality [10], [11], [99].",
"For instance, Khamis et al.",
"assessed the quality of inline comments based on consistency and language quality using a heuristic-based approach [10].",
"Steidl et al.",
"evaluated documentation comment quality based on four quality attributes, such as consistency, coherence, completeness, and usefulness of comments using a machine learning-based model [11].",
"Zhou et al.",
"proposed a heuristic and natural language processing-based technique to detect incomplete and incorrect comments [16].",
"These works have proposed various new quality attributes to assess comment quality, such as completeness, coherence, and language quality, that are not included in previous quality models.",
"However, a unifying overview of comment QAs and their assessment approaches is still missing.",
"Our paper complements these previous works by investigating comment QAs discussed in the last decade of research.",
"Previous SLRs on code comments and software documentation.",
"In recent years, SLRs have been conducted to investigate agile software development aspects in open-source projects [100], the usage of ontologies in software process assessment [101], and improvement aspects in DevOps process and practices [102].",
"Previous SLRs in the field investigated code comments and software documentation [17], [18], which are closely related to our work.",
"Specifically, Ding et al.",
"conducted an SLR to explore the usage of knowledge-based approaches in software documentation [17].",
"They identified twelve QAs.",
"They also highlighted the need to improve QAs, especially conciseness, credibility, and unambiguity.",
"Zhi et al.",
"have explored various types of software documentation to see which QAs impact it [18].",
"Both of the studies considered the timeline until 2011.",
"Additionally, they have not studied how the proposed comment quality assessment approaches are computed in practice for comments.",
"Inspired by these related studies, we focused specifically on the code comment aspect.",
"Song et al.",
"conducted a literature review on code comment generation techniques, and indicated the need to design an objective comment quality assessment model [89].",
"Complementarily, Nazar et al.",
"[90] presented a literature review in the field of summarizing software artifacts, which included source code comment generation as well as bug reports, mailing lists, and developer discussion artifacts.",
"Our work complements these previous studies since we mainly focus on manually written comments."
],
[
"Conclusion",
"In this work, we present the results of a systematic literature review on source code comment quality evaluation practices in the decade blue2011— blue2020.",
"We study blue47 publications to understand of effort of Software Engineering researchers, in terms of what type of comments they focus their studies on, what QAs they consider relevant, what techniques they resort to in order to assess their QAs, and finally, how they evaluate their contributions.",
"Our findings show that most studies consider only comments in Java source files, and thus may not generalize to comments of other languages, and they focus on only a few QAs, especially on consistency between code and comments.",
"Some QAs, such as conciseness, coherence, organization, and usefulness, are rarely investigated.",
"As coherent and concise comments play an important role in program understanding, establishing approaches to assess these attributes requires more attention from the community.",
"We also observe that the majority of the approaches appear to be based on heuristics rather than machine learning or other techniques and, in general, need better evaluation.",
"Such approaches require validation on other languages and projects to generalize them.",
"Though the trend of analyzing comments appearing in multiple projects and languages is increasing compared to the previous decade, as reported by Zhi et al., the approaches still need more thorough validation[18]."
],
[
"Acknowledgement",
"We gratefully acknowledge the financial support of the Swiss National Science Foundation for the project “Agile Software Assistance” (SNSF project No.",
"200020-181973, Feb 1, 2019 - Apr 30, 2022) and the Spanish Government through the SCUM grant RTI2018-102043-B-I00, and the Madrid Regional through the project BLOQUES.",
"We also acknowledge the Horizon 2020 (EU Commission) support for the project COSMOS (DevOps for Complex Cyber-physical Systems), Project No.",
"957254-COSMOS."
]
] | 2209.08165 |
[
[
"Selective Token Generation for Few-shot Natural Language Generation"
],
[
"Abstract Natural language modeling with limited training data is a challenging problem, and many algorithms make use of large-scale pretrained language models (PLMs) for this due to its great generalization ability.",
"Among them, additive learning that incorporates a task-specific adapter on top of the fixed large-scale PLM has been popularly used in the few-shot setting.",
"However, this added adapter is still easy to disregard the knowledge of the PLM especially for few-shot natural language generation (NLG) since an entire sequence is usually generated by only the newly trained adapter.",
"Therefore, in this work, we develop a novel additive learning algorithm based on reinforcement learning (RL) that selectively outputs language tokens between the task-general PLM and the task-specific adapter during both training and inference.",
"This output token selection over the two generators allows the adapter to take into account solely the task-relevant parts in sequence generation, and therefore makes it more robust to overfitting as well as more stable in RL training.",
"In addition, to obtain the complementary adapter from the PLM for each few-shot task, we exploit a separate selecting module that is also simultaneously trained using RL.",
"Experimental results on various few-shot NLG tasks including question answering, data-to-text generation and text summarization demonstrate that the proposed selective token generation significantly outperforms the previous additive learning algorithms based on the PLMs."
],
[
"Introduction",
"Recently, pretrained language models (PLMs) have shown great generalization ability when combined with large-scale data and big transformer-based models [7], [32], [23], [3], [38], [31], [40].",
"Therefore, transfer learning from PLMs has been popularly used for few-shot natural language generation (NLG) tasks with promising results.",
"In specific, the use of PLM for few-shot NLG can be categorized into three approaches: 1) prompt-based, 2) finetuning, and 3) additive learning.",
"Prompt-based approaches encode a task description and task-specific examples as a natural language prompt for few-shot text generation [32], [3], [45], [34], [24].",
"These approaches can take full advantage of the universal natural language understanding and generation capabilities of large-scale PLMs without further training of the main model, however, they have some limitations in dealing with a large domain shift from the pretraining corpus data, tuning suitable task-specific prompts, and covering an increased size of conditioning examples.",
"On the other hand, finetuning of the PLM is able to explicitly impart task-specific knowledge to the model and hence lift the above limitations [46], [42], [5].",
"However, these finetuned models are prone to overfitting when only a small amount of training data is available.",
"In order to alleviate such an overfitting problem, additive learning has been extensively exploited by incorporating task-specific adapters into the PLM [37], [17], [44].",
"Table: Generated answers from an instance of MS-MARCO QA dataset.",
"Two definitions about conflict are presented in bold text in the passage.",
"The answers are sampled from the models trained on 0.5%0.5\\% few-shot subset data.",
"The proposed selective token generation (STG) produces the first two words (highlighted in red) by the task-specific adapter while the others by the PLM.In general, task-specialized adapters for few-shot NLG are trained by maximum likelihood estimation (MLE) or reinforcement learning (RL).",
"While MLE is efficient in learning, it suffers from the exposure bias problem due to the difference in the training and inference mechanisms [16], and this problem can be severe with limited training data.",
"One solution is RL, capable of resolving this exposure bias problem by sequential output sampling during training [33], [19], [36].",
"However, the exponentially large space of output sequences restricts the use of RL since it leads to high variance and unstable training which is more serious in the few-shot setting.",
"More importantly, the existing additive learning generally produces the whole output sequence by its own task-specific adapter, which leads to a fundamental limitation in maintaining the knowledge of the PLM and the strong generation ability.",
"An example of this limitation from our empirical observation on the task of question and answering is shown in Table REF .",
"In this case, a passage that contains two definitions (super-scripted and bolded) about conflict is given with a query that asks about the psychological meaning of conflict.",
"Without the knowledge of who ColmanA psychologist, https://en.wikipedia.org/wiki/Peter_T._Coleman_(academic) is, it can be hard to answer since the word psychology in the query does not appear in the passage.",
"Here, the PLM repeats the given query as its generated answer due to the lack of domain adaptation while the added adapter incorrectly outputs not the psychological meaning but the general meaning of conflict.",
"This is because most queries in this few-shot training data ask a general meaning of a concept, and therefore the adapter is overfitted to this pattern (more examples are described in Section REF ).",
"Note that the PLM generates the correct answer if the proper conditioning text (the meaning of conflict is) is provided.",
"Motivated by these observations, in this work, we propose a novel RL-based selective token generation (STG) between the task-general PLM and the task-specific adapter.",
"The selection of this output token generator enables to explicitly maintain a general prior knowledge from the frozen PLM and the adapter to focus only on the task-relevant parts in sequence generation.",
"Note that the proposed algorithm is different from previous selective generation algorithms such as copy mechanism [14] in that STG selects a generator rather than existing tokens in a given passage.",
"In few-shot learning, the proposed partial token generation makes the task-specific adapter more resilient to overfitting and furthermore reduces the overall output space which leads to stable RL training.",
"Here, in order to make the two token generators (policies) complement each other as well as to realize the robust output selection at the token level on the fly, we exploit a separate token-level policy selector.",
"Note that both the policy selector and the task-specific adapter are simultaneously learned by the RL algorithm.",
"Experimental results on various few-shot NLG tasks show that the proposed selective token generation outperforms the previous PLM-based additive learning algorithms with the comprehensive (non-selective) token generation.",
"Our main contributions can be summarized as follows.",
"A novel selective token generation between the PLM and the task-specific adapter is proposed for few-shot NLG.",
"RL is applied to train both the policy selector and the task-specific adapter that is complementary to the PLM in text generation.",
"Extensive empirical validation on few-shot NLG tasks demonstrates that the proposed selective token generation performs better in comparison to the previous PLM-based additive learning algorithms."
],
[
"Natural Language Generation",
"The goal of NLG is to generate a text sequence ${\\bf y} = [y_0, ..., y_T]$ for a given task, where $y_t$ is the $t$ th output token from a vocabulary $\\mathcal {V}$ , and $T$ is the output sequence length.",
"For this generation, we aims to model the distribution of $\\bf y$ that is autoregressively factorized as $p_{\\theta }({\\bf y}) = \\prod _{t=0}^{T} p_{\\theta }(y_t | {\\bf y}_{<t})$ , where $\\theta $ denotes the model parameters and ${\\bf y}_{<t} = [y_0, ..., y_{t-1}]$ .",
"Here, the conditional distribution to sample a token for each step, $p_{\\theta }(y_t | {\\bf y}_{<t})$ , is defined by the softmax function on the output logits $f_{\\theta }(y_t | {\\bf y}_{<t})$ .",
"Note that in general, the language generation is conditioned on input context according to a given task.",
"Here, we encode the conditioning context by the same sequential model for generating an output sequence, and for simplicity we omit it.",
"Figure: Text generation processes of Non-STG and STG are described.",
"In the Non-STG, every token is sampled from the task-specific policy π a \\pi _a (Left).",
"On the other hand, in the proposed STG, each token is selectively sampled from either the PLM policy π LM \\pi _{LM} or the test-specific policy π a \\pi _a where the selection is performed by the selection policy π s \\pi _s (Right).",
"Symbols with dashed line represent learnable models."
],
[
"Additive Learning for Few-shot Generation",
"To effectively leverage the general linguistic knowledge, $\\theta $ is first initialized by the PLM parameters, $\\theta _{LM}$ , for NLG.",
"Given $N$ task-specific training instances, ${\\mathcal {D}}=\\lbrace {\\bf y}^{n*}\\rbrace _{n=1}^{N}$ , where ${\\bf y}^{n*}$ is the $n$ th ground-truth output sequence, directly finetuning $\\theta _{LM}$ using ${\\mathcal {D}}$ can incur the severe overfitting problem when $N$ is small in the few-shot scenario.",
"Therefore, we add the task-specific adapter, $g_{\\theta _{a}}$ parameterized by $\\theta _{a}$ , on top of the PLM, and optimize only $\\theta _{a}$ [44], [37].",
"In specific, we reformulate $f(\\cdot | {\\bf y}_{<t}; \\theta ) = W^T h({\\bf y}_{<t}; \\theta _{h})$ where $W \\in {R}^{H \\times |\\mathcal {V}|}$ and $h \\in {R}^{H}$ denote the weight matrix and the penultimate representations, respectively, and $\\theta = \\lbrace W, \\theta _{h}\\rbrace $ .",
"Then, we define the task-specific conditional distribution as follows: $ p (y_t | {\\bf y}_{<t}; \\theta _{LM}, \\theta _{a}) =\\sigma \\bigg ( {W_{LM}}^T h_{LM} ({\\bf y}_{<t}) \\\\+ {W_{a}}^T g \\big ( h_{LM} ({\\bf y}_{<t}); \\theta _{g} \\big ) \\bigg ),$ where $h_{LM} ({\\bf y}_{<t}) = h({\\bf y}_{<t}; \\theta _{h, LM})$ , $\\theta _{a} = \\lbrace W_a, \\theta _{g}\\rbrace $ and $\\sigma $ is the softmax function.",
"Here, the summation of the PLM logits and the adapter logits is motivated by auxiliary trainingAlthough the auxiliary training is particularly designed for maximizing the likelihood of the target task output, it also can take an advantage for RL since the adapter logits are nearly zero before training is advanced.",
"Namely, it lets the task-specific conditional distribution start learning from the distribution of PLM, not a uniform distribution.",
"[44].",
"It is noted that in our additive learning $\\theta _{a}$ is updated while $\\theta _{LM}$ is kept frozen.",
"Hence, in the following we omit $\\theta _{LM}$ such that $p_{\\theta _{a}}(y_t | {\\bf y}_{<t}) = p (y_t | {\\bf y}_{<t}; \\theta _{LM}, \\theta _{a})$ for simplicity."
],
[
"Reinforcement Learning (RL)",
"As an alternative to MLE, RL is able to overcome the exposure bias problem of MLE by sequence-level sampling from the model distribution during training [33] and allows to leverage the target-specific sequence-level objectives such as BLEU [41], [15].",
"In order to use RL for our additive learning, we reformulate our text generation as an RL problem: at each time step $t$ , the agent takes the current state ${\\bf s}_t = {\\bf y}_{<t}$ as an input and performs an action $a_t$ that outputs a token $y_t$ by a policy $\\pi _{\\theta } (a_t|{\\bf s}_t)$ corresponding to $p_{\\theta }(y_t | {\\bf y}_{<t})$ .",
"Then, the agent receives a reward $r_t = r({\\bf s}_t, a_t)$ and deterministically transitions to the next state ${\\bf s}_{t+1}$ .",
"Here, note that the token-level intermediate reward $r_{t} = 0, \\forall t < T$ when we use the delayed reward associated with the sequence-level evaluation metric between the two full sequences, $\\bf y$ and ${\\bf y}^*$ .",
"Let $\\tau = \\lbrace ({\\bf s}_t, a_t, r_t)\\rbrace _{t=0}^{T}$ be the trajectory generated by $\\pi _{\\theta }$ .",
"The RL objective for the optimal agent is to maximize the expected sum of future discounted rewards ${\\mathbb {E}}_{\\tau \\sim \\pi _{\\theta }} [ \\sum _{t=0}^{T} \\gamma ^t r_t ]$ , where $\\gamma \\in [0,1]$ is the discount factor.",
"We employ an actor-critic algorithm [2] which requires the additional critic network to estimate the value of a state, $V^{\\pi }({\\bf s}_t) = {E}_{\\pi }[\\sum _{t^{\\prime }=t}^{T} \\gamma ^{t^{\\prime }-t} r_{t^{\\prime }}|{\\bf s}_t] = \\sum _{a_t} \\pi (a_t|{\\bf s}_t)Q^{\\pi }({\\bf s}_t, a_t)$ where the state-action value function $Q^{\\pi }({\\bf s}_t, a_t) = {E}_{\\pi }[\\sum _{t^{\\prime }=t}^{T} \\gamma ^{t^{\\prime }-t} r_{t^{\\prime }}|{\\bf s}_t, a_t] = r_t + V^{\\pi }({\\bf s}_{t+1})$ .",
"We use the policy gradient loss to learn the policy parameters $\\theta $ : ${\\mathcal {L}} = - \\sum _{t=0}^{T} A^{\\pi _{\\theta }}({\\bf s}_t, a_t) \\log \\pi _{\\theta }(a_t | {\\bf s}_{t}),$ where $A^{\\pi _{\\theta }}({\\bf s}_t, a_t) = Q^{\\pi _{\\theta }}({\\bf s}_t, a_t) - V^{\\pi _{\\theta }}({\\bf s}_t)$ is the advantage function."
],
[
"Selective Token Generation",
"Instead of generating all tokens in an output sequence from the single task-specific policy, $\\pi _a = \\pi _{\\theta _a}(a_t|{\\bf s}_t)$ , at each time step $t$ , we sample an output token $y_t$ selectively from either the PLM policy $\\pi _{LM} = \\pi _{\\theta _{LM}}(a_t|{\\bf s}_t)$ or the task-specific policy $\\pi _{a}$ : $y_t = a_t \\sim \\big ( \\mathbb {1}_t[\\text{$\\pi _{LM}$ is selected}] \\pi _{LM}(a_t|{\\bf s}_t) \\\\+ (1-\\mathbb {1}_t[\\text{$\\pi _{LM}$ is selected}]) \\pi _{a}(a_t|{\\bf s}_t) \\big ),$ where $\\mathbb {1}_t[\\cdot ]$ is the indicator function (at $t$ ) that equals 1 if it is true and 0 otherwise.",
"This output token selection allows to explicitly utilize a general linguistic knowledge from the PLM without catastrophic forgetting in few-shot learning.",
"Also, the task-specific policy can focus on generating only the task-relevant parts, which enables more effective few-shot training with a reduced search space.",
"Now we need to determine how to select the proper policy at each step on the fly as well as to make the task-specific policy complementary to the PLM policy.",
"For this, we exploit a separate token-level policy selector.",
"The proposed policy selector $\\pi _{s}(i_t|{\\bf s}_t; \\theta _s)$ with the parameters $\\theta _s$ , where $i_t \\in \\lbrace 0, 1\\rbrace $ , is an another policy that stochastically decides a policy to generate $a_t$ for ${\\bf s}_t$ .",
"Namely, a token sample $y_t$ is generated by the following process: $i_t &\\sim & \\pi _{s}(i_t|{\\bf s}_t), \\\\y_t &=&{\\left\\lbrace \\begin{array}{ll}a_t\\sim \\pi _{LM}(a_t|{\\bf s}_t) & \\text{if} \\ i_t = 0, \\\\a_t\\sim \\pi _{a}(a_t|{\\bf s}_t) & \\text{if} \\ i_t = 1.\\end{array}\\right.",
"}$ This process can be considered as a token generation from a hierarchical policy $\\pi _h(a_t|{\\bf s}_t; \\theta _s, \\theta _{LM}, \\theta _a)$ where the policy selector represents the upper-level prior for the preference of the low-level policy.",
"Therefore, the value function of this hierarchical policy can be formulated as $&V^{\\pi _h}& = {\\mathbb {E}}_{\\pi _h}[\\sum _{t^{\\prime }=t}^{T} \\gamma ^{t^{\\prime }-t} r_{t^{\\prime }}|{\\bf s}_t] \\\\&=& \\pi _{s}(0_t|{\\bf s}_t) \\sum _{a_t} \\pi _{LM}(a_t|{\\bf s}_t)Q^{\\pi _{h}}({\\bf s}_t, a_t) \\nonumber \\\\&+& \\nonumber \\pi _{s}(1_t|{\\bf s}_t) \\sum _{a_t} \\pi _{a}(a_t|{\\bf s}_t)Q^{\\pi _{h}}({\\bf s}_t, a_t), \\nonumber \\\\$ and $A^{\\pi _h}({\\bf s}_t, a_t) = Q^{\\pi _{h}}({\\bf s}_t, a_t) - V^{\\pi _h}({\\bf s}_t)$ .",
"We denote $i_t=0$ and $i_t=1$ as $0_t$ and $1_t$ respectively.",
"Here, it is noted that a single critic network is used for the hierarchical policy since $i_t$ does not affect ${\\bf s}_t$ .",
"Given a sample trajectory $\\lbrace ({\\bf s}_t, i_t, a_t, r_t)\\rbrace _{t=0}^T$ , the loss for optimizing $\\theta _s$ and $\\theta _a$ is $ {\\mathcal {L}} = - \\sum _{t=0}^{T} A^{\\pi _h}({\\bf s}_t, a_t)\\bigg ( \\mathbb {1}[0_t] \\mathcal {L}_{LM}+ \\mathbb {1}[1_t] \\mathcal {L}_{a}\\bigg ),$ where ${\\mathcal {L}_{LM}} =& \\log sg[\\pi _{LM}(a_t | {\\bf s}_{t})] + \\log \\pi _{s}(i_t | {\\bf s}_{t}), \\nonumber \\\\{\\mathcal {L}_{a}} =& \\log \\pi _{a}(a_t | {\\bf s}_{t}) + \\log \\pi _{s}(i_t | {\\bf s}_{t})$ and sg stands for the stop-gradient operator.",
"Similar to $\\pi _a$ , $\\pi _s$ makes use of the PLM representations and the task-specific adapter such that $ \\pi _s(i_t | {\\bf s}_t; \\theta _s) = \\sigma \\bigg ( m \\Big ( g \\big ( h_{LM} ({\\bf s}_{t})\\big ); \\theta _s \\Big ) \\bigg ),$ where $\\sigma $ is the softmax function and $m$ is the selector module.",
"Figure REF depicts the overall text generation process by the proposed selective token generation (STG) in comparison to the previous non-selective token generation (Non-STG).",
"Here, note that since all policies in STG share the same PLM representations, the increased computational cost by STG over Non-STG is negligible.",
"The use of the separated policy selector that is simultaneously trained with the task-specific policy allows the task-specific policy to be complementary to the PLM policy.",
"Especially, this cooperative ensemble learning can be realized by our RL algorithm that performs sequential sampling from the model during training.",
"The advantages of STG are as follows: (1) STG makes use of the PLM not at the feature level but the output distribution level in text generation.",
"In our few-shot learning this is beneficial in explicitly retaining strong linguistic and world knowledge from the PLM.",
"(2) STG resolves the exponentially large search space $|\\mathcal {V}|^{T}$ since the frozen PLM chooses a token when it is selected, and therefore the search space of the generator is approximately decreased from $|\\mathcal {V}|^{T}$ to $|\\mathcal {V}|^{T-\\overline{T}_{PLM}}$ where $\\overline{T}_{PLM}$ is the average length of sequences generated by PLM.",
"(3) STG is efficient in credit assignment.",
"The loss function of STG (Equation REF ) intuitively shows that the gradient to the task-specific policy $\\pi _a$ associated with producing $a_t$ will depend on the selector's action (i.e.",
"$i_t=1$ ).",
"Hence, unlike Non-STG, $\\pi _a$ of STG knows which token is used as a task-specific token and contributed to the reward (see Figure REF for an illustration).",
"It is noted that although the STG also can be trained by MLE, it can be easily collapsed to select only a task-specific policy irrespective of a given content.",
"We analyze the MLE version of STG in Appendix .",
"Figure: A simple schematic illustration of Non-STG and STG.",
"Non-STG(RL): the whole sequence of target is generated from the task-specific policy π a \\pi _a so the right sub-sequence AB is also penalized from the delayed feedback.",
"STG: the third token is sampled from π a \\pi _a and the model lets the other tokens (highlighted with cyan) generated from the PLM's policy π LM \\pi _{LM} which generates a next letter of the previous alphabet input.",
"Here, π a \\pi _a will be penalized at the third token."
],
[
"Experiments",
"We evaluate our method against additive learning baselines on Data-to-Text, Question Answering and Text Summarization tasks which are widely used in few-shot NLG."
],
[
"Baseline",
"PLM.",
"In our experiments, we assume that the PLM works to some extent for a given task.",
"However, the naive PLM usually does not satisfy it for a new task unseen during training.",
"Hence, we finetuned GPT-2We make use of GPT-2 with 345M parameters as the initial checkpoint.",
"We follow the training details in the previous works [30], [20] for each task.",
"[32] with MLE for few epochs and used it as the PLM.",
"Fine-tuning the PLM with MLE is most commonly used for task adaptation and thus it can also be a strong baseline.",
"This fine-tuning phase accelerates the learning of the adapter.",
"This is particularly when the adaptation requires to cover the large domain shift.",
"Severe performance degradation was observed for all the tasks when we skipped this fine-tuning.",
"Non-STG.",
"This method stands for Non-Selective Token Generation which uses the above the PLM as an encoder (frozen) and the adapter (additional layer to be trained).",
"We use two objectives, MLE and RL, for additive learning.",
"These will be denoted as Non-STG-MLE and Non-STG-RL, respectively.",
"Table: Data-to-Text performance on FewShotWOZ dataset.STG-Naive Ensemble.",
"We believe that the proposed generation encourages the task-specific policy ($\\pi _a$ ) to complement the PLM's policy ($\\pi _{LM}$ ) with a proper selection of the selector through the joint training.",
"To investigate this, we evaluate against two different naive ensembles of the policies, $\\pi _{a}$ trained from Non-STG and $\\pi _{LM}$ of the PLM.",
"These ensemble schemes are as follows: NE($max$ ): $\\pi _{max} = \\sigma (Max(\\pi _{a}, \\pi _{LM}))$ NE($mix$ ): $\\pi _{mix} = (\\pi _{a} +\\pi _{LM})/2$ We also evaluate another naive ensemble strategy NE($random$ ) that randomly selects a token policy at each step between $\\pi _{a}$ and $\\pi _{LM}$ , however it shows lower performances than the others."
],
[
"Implementation",
"Adapter.",
"The task-specific adapter $g$ in Section REF is implemented by a LSTM to encode the dynamics of the representation vector $h_{LM}$ .",
"We found that the use of MLP was not good in the sense of performance.",
"Selector.",
"We use a 2-layer MLP with ReLU activation for $m$ of Equation REF .",
"Table: Averaged performances for Question Answering on various few-shot subset data of MS-MARCO.Figure: Averaged performance gains against the PLM for Question Answering on various few-shot subset data of MS-MARCO.",
"The x-axis represents the size of the subset data and the shaded area represents a range of standard deviation over 3 randomly sampled subset data with different random seeds.",
"STG provides significantly larger gains compared to Non-STGs on BLEU (Left) and ROUGE-L (Right).Reinforcement Learning.",
"We employ Actor-Critic method [21], [12] for RL.",
"The agents (i.e.",
"selector and generator) receive a reward after generating a sentence.",
"Here, we use different reward functions according to tasks.",
"We use delexicalised BLEU for Data-to-Text following [30], Averaged score of BLEU and ROUGE-L for Question Answering and ROUGE-L for Text Summarization following [29] as the reward function.",
"Token Sampling.",
"During the training, $i_t \\in \\lbrace 0, 1\\rbrace \\sim \\pi _s$ is first sampled, and then we use either $\\pi _{LM}$ of the PLM for $i_t=0$ or the task-specific policy $\\pi _a$ for $i_t=1$ to sample the $t$ th token.",
"During the evaluation, any decoding strategy, such as a beam search, can be used with the mixture of policies $\\pi _{h}(\\cdot ) = \\pi _s(0_t)\\pi _{LM}(\\cdot ) + \\pi _s(1_t)\\pi _a(\\cdot )$ .",
"We use the beam search decoding with a sample size of $k=3$ for Text Summarization and $top_p=0.9$ decoding for both Data-to-Text ($k=10$ ) and Question Answering ($k=3$ )."
],
[
"Data-to-Text",
"Data-to-Text is a task that transforms structured data such as graphs or tables into natural language.",
"Recent works [27], [30], [18] show that the PLM can be adapted successfully to this task by taking a serialized form of data as an input without a carefully designed model to encode the structured data.",
"Here, we perform experiments on FewShotWOZ [30] dataset.",
"The evaluation is conducted on the topics which are availablehttps://github.com/pengbaolin/SC-GPT.",
"Only 50 instances for each topic are available for training and 129, 78, 1379, and 680 testing instances for Restaurant, Hotel, Laptop, and TV, respectively.",
"The models are evaluated by measuring fluency and informativeness using BLEU score and ERR (slot ERror Rate), respectively.",
"Table REF shows the obtained results."
],
[
"Long Answer Question Answering",
"We consider Long Answer Question Answering (QA) task on MS-MARCO [28] dataset.",
"In this task, a passage and a query are given, and the model generates an answer with respect to the query by referring to the passage.",
"Here, we randomly sample various sizes of (50, 100, 500, 1,000 $\\approx 1\\%$ , and 2,000) subset data from the train dataset.",
"We also sample a validation and a test set, which contains 500 and 12,000 instances, respectively, from the dev dataset.",
"We repeat this test three times with different random seeds and thus perform experiments on total nine subsets.",
"The models are evaluated by measuring BLEU and ROUGE-L (denoted as R-L).",
"We report averaged performances over the three runs and averaged performance gain against the PLM in Table REF and Figure REF , respectively."
],
[
"Text Summarization",
"We consider the problem of abstractive summarization for long text generation.",
"Here, we randomly sample various sizes of (50, 100, 300, 1,500, and 3,000 $\\approx 1\\%$ ) subset data from CNN/Daily Mail [35].",
"We repeat this test three times for each size of few-shot as in above QA task.",
"ROUGE [26] is commonly used to evaluate n-grams recall of the summaries with gold references.",
"The models are evaluated by measuring ROUGE-1, ROUGE-2, and ROUGE-L (denoted as R1, R2, and R-L, respectively).",
"We report averaged performances over the three runs and averaged performance gain against the PLM in Table REF and Figure REF , respectively.",
"Table: Averaged performances for Text Summarization on various few-shot subset data of CNN/DM.Figure: Averaged performance gains against the PLM for Text Summarization on various few-shot subset data of CNN/DM.",
"The x-axis represents the size of the subset data and the shaded area represents a range of standard deviation over 3 randomly sampled subset data with different random seeds.",
"STG provides significantly larger gains compared to Non-STGs on ROUGE-1 (Left), ROUGE-1 (Middle), and ROUGE-L (Right)."
],
[
"Result",
"In most cases, additive learning improves the performances over the PLM.",
"However, they do not always guarantee a performance improvement.",
"For example, the ERR score of the PLM on Laptop shows a better result except for STG and NE($mix$ )-RL (see Table REF ) and the Non-STGs trained on $1,000 \\approx 1\\%$ few-shot subset of MS-MARCO do not outperform the PLM (see Table REF ).",
"Data-to-Text.",
"As shown in Table REF , we can observe that the Non-STGs do not outperform the PLM even though it has more neural units and takes more training time.",
"The models trained on the RL objective show better performances for the ERR (lower is better).",
"Interestingly, NE($mix$ ) methods show strong improvements for the BLEU which measures the fluency of sentence but obvious degeneration for the ERR which measures the rate of missing information from the given data.",
"These results suggest that the PLM is much more capable of task-general knowledge than the task-specific generator (i.e.",
"$\\pi _a$ ) trained on few-shot dataset, which ensures our motivation of jointly training the policy selector and the task-specific generator is valid.",
"Note that while other methods show some trade-off between BLEU and ERR, only STG shows improvements on both metrics for all topics in the dataset.",
"Question Answering.",
"As shown in Table REF , STG shows significantly better performances than the other methods.",
"Notably, NE($mix$ ) show good performances as much as STG especially where the training data size $\\ge 1,000$ .",
"It obviously suggests that the PLM can be a complementary model to the additional model.",
"Therefore, in this context, it can be lost of the prior knowledge of the PLM even if the additional model has been built over the feature space of the PLM.",
"In addition, we can expect that STG would be more beneficial on the small number of samples for this kind of tasks which depend on the PLM's ability like common sense knowledge.",
"As shown in Figure REF , STG shows strong improvements compared to Non-STG-RL especially where the training data size $\\le 500$ .",
"Summarization.",
"As shown in Table REF , STG shows significantly larger gains than Non-STGs, and their naive ensembles with the PLM in every score metric and training data size.",
"Similar to QA, STG shows improvements compared to Non-STGs especially where the training data size $\\le 300$ as shown in Figure REF .",
"However in contrast to the QA task, the improvement may seem limited for all models including STG.",
"We think that the adapters used in this study may not be suitable for this particular task which requires to understand the long context and compress it into a summary.",
"It may need the use of lower-level features or more parameters to adapt to such tasks.",
"We discuss this limitation in Section .",
"Overfitting in Non-STGs.",
"In the example as shown in Table REF the answer of STG, which is close to the ground truth, is generated by the PLM policy $\\pi _{LM}$ after some sequence of tokens (conflict is) that are sampled from the task-specific policy $\\pi _a$ .",
"The Non-STGs generate general meaning which is not intended.",
"We can find such examples for the other tasks in Appendix : In Data-To-Text, as shown in the last example of Table REF , Non-STG generates nicam stereo which is not appeared in the given data.",
"This is due to that nicam stereo was appeared 7 times (7/50, $14\\%$ ) in training data.",
"In Summarization, as shown in the first example of Table REF , Non-STGs only consider the forepart of the given article.",
"Since the most of the major information is appeared in the forepart in News data, Non-STGs can be easily overfitted to generate the text according to such a pattern.",
"Hence, we claim that Non-STG is easily exposed to learning patterns of typical answering, but STG resolves this issue since it can be fully accessible to the knowledge of the PLM."
],
[
"Related Work",
"Recently, prompt-based in-context learning with an extremely large PLM shows impressive few-shot generation performances [32], [3].",
"[34] propose manually designed natural language prompts for improved few-shot text summarization and headline generation.",
"[10] conduct zero-shot learning for question generation from knowledge graphs, however they require a large amount of in-domain training data for their transfer learning.",
"[5] directly finetune the pretrained GPT-2 with a small amount of serialized attribute-value pairs for table-to-text generation.",
"[13] further apply multiple tasks to effectively leverage the structured information of tables.",
"In contrast to these approaches, our proposed method utilizes RL-based additive learning for few-shot text generation.",
"Applying RL for text generation has been widely used to mitigate the exposure bias problem of MLE as well as to directly optimize task-relevant evaluation metrics.",
"[33] use the REINFORCE algorithm for text summarization and machine translation while [2] use the actor-critic algorithm for machine translation.",
"However, they require pretraining using MLE.",
"[8] propose softmax policy gradient to remove the MLE-based pretraining.",
"However, it requires various techniques for effective training.",
"[39] propose an entropy-regularized policy optimization that subsumes many of the previous training algorithms.",
"Our proposed method is different from these methods in that we apply RL for more difficult few-shot generative modeling.",
"The use of RL training in PLM has been explored in many works.",
"[6] propose a controllable text generation which uses discriminators to guide generation of the PLM.",
"This approach assumes that constant classes like topics or preferences are available.",
"[22] use a PLM as a caption generator for given image.",
"In their referential game, the generator is rewarded by a kind of discriminator that responses a signal to the generator whether the corresponding caption is correct or not.",
"Various methods take into account the RL tasks with large action spaces like NLG.",
"[9] consider only actions in a cluster around the latent state of action obtained from a given state.",
"[4] define the action embedding as a distribution with semantic of action and use a deterministic policy to take an action.",
"[11], [43] devise a method of incorporating the process of directly removing unnecessary actions according to the state in the RL problem.",
"Unlike these approaches, we use the hierarchical policy that reduces the sequential action space."
],
[
"Limitations & Future work",
"Adapter.",
"In this study, we aim to propose a new generation framework for few-shot natural language generation tasks.",
"In particular, a relatively naive neural adapter which utilizes only the top layer of the PLM is used in this paper, and thus it may lead to limited improvements as shown in the experimental results on the summarization task.",
"Fortunately, there are several neural architectures [17], [25], [1] for efficient task adaptation, and we believe that such adapters also make STG more efficient for covering a large domain shift and scaling.",
"The study on the architectures of the adapters will be conducted in future works.",
"Efficient exploration.",
"The fundamental limitation in STG is a high dependency on PLM; When STG has a sufficient powerful PLM, the selector does not select the additional adapter and it is thus nothing more than the PLM.",
"We can find such phenomenon in some examples in Table REF and REF in Appendix.",
"On the other hand, when STG has an extremely poor PLM, the selector selects the adapter always and it is thus equivalent to Non-STG.",
"Therefore, in the perspective of exploration of RL the STG needs balanced selections between the PLM and the adapter.",
"Furthermore, the use of RL objective requires more training time than the methods which use MLE objective such as Prefix-Tuning [25] due to the auto-regressive sequence sampling during training.",
"Therefore, an analysis on efficient exploration of STG is important for future works."
],
[
"Conclusion",
"In this work, we propose to exploit a selective token generation between the pretrained language model and the task-specific adapter with RL-based additive learning for the tasks of few-shot natural language generation.",
"In particular, we devise a trainable policy selector at the token level and jointly learn it with the task-specific policy.",
"The proposed policy selector and RL algorithm make the two policies complementary to each other and lead to robust few-shot generative modeling.",
"Experimental results on various tasks of few-shot text generation show that the proposed selective token generation along with RL-based additive learning consistently and significantly improves the performances with less overfitting."
],
[
"Training Settings",
"In our experiments all the models of additive learning, Non-STG and STG, are used the same architecture and hyper-parameters (except whether to use pre-training) for training as described in Table REF .",
"We found that pre-training the addtional layer of Non-STG-RL with MLE helps the performance improvements.",
"On the other hand, STG without pre-training shows better performances.",
"We use the training data for each topic of the task of Data-to-Text as their validation data."
],
[
"STG-MLE",
"Here, we evaluate the MLE version of STG (denoted as STG-MLE) which is trained by MLE for the mixture policy $\\pi _{h}(\\cdot ) = \\pi _s(i_t=0)\\pi _{LM}(\\cdot ) + \\pi _s(i_t=1)\\pi _a(\\cdot )$ similar to copy mechanism [14].",
"In few-shot training, the explicit use of PLM logits can efficiently reduce the fine-tuning loss especially when the adapter is light since the adapter can focus only on the task-relevant part in generation.",
"STG-RLWe add \"-RL\" to the STG to distinguish with STG-MLE in this context.",
"learns to do this naturally by stochastic policy sampling if the policy selector is initialized to perform uniform sampling.",
"On the other hand, STG-MLE can be easily collapsed to select only a task-specific policy (i.e.",
"$i_t=1$ ).",
"This is because the gradient flows the additional model only and, unlike STG-RL, there is no chance to exploit diverse paths during training in the teacher forcing manner.",
"As shown in Figure REF , the score of STG-MLE starts from the same point of STG-RL but it collapsed to Non-STG-MLE."
],
[
"Learning Curve",
"It is well known that the RL-tuning resolves the exposure bias of MLE-tuning.",
"We can expect that an additive learner of MLE would be affected by the exposure bias as well, and the RL objective for additive learning resolves it.",
"Here, we present some learning curvesThe curve for Data-to-Text is not presented since there is no actual validation set.",
"obtained from training in our experiments.",
"As shown in Figure REF , the learners of MLE seem to have overfitting (in terms of Perplexity, PPL) and exposure bias (in terms of Score).",
"On the other hand, the learners of RL were less effected by the problems.",
"We can find that the STGs (denoted STG-RL) are superior to the others from the perspective of the score."
],
[
"Effectiveness of Selector",
"Here, we investigate the effectiveness of the selector $\\pi _s$ of the STG.",
"We compare Fixed Selection against the Dynamic selection.",
"In the fixed selection, the probability of selecting the PLM's policy $\\pi _{LM}$ is fixed to $\\pi _{s}(i_t=0|s_t) = 1 - \\pi _{s}(i_t=1|s_t)$ .",
"We measure the performance with respect to $\\pi _{s}(i_t=1|s_t) = c$ where $c$ is a constant.",
"The selection will be uniformly random when $c=0.5$ , and when $c=0$ , the performance will be equivalent to the performance of the PLM without additive learning.",
"Figure REF shows that the input-dependent dynamic selection by our STG outperforms the fixed selection with any $c$ .",
"We can find that how $\\pi _s$ works for each task.",
"For instance, in QA task, the first few tokens of an answer may decide the quality of generation (i.e.",
"\"yes\" or \"no\" in binary QA).",
"Therefore, an optimal strategy of the STG might be producing the first few tokens sampled from the task-specific $\\pi _{a}$ and the remaining tokens from the PLM $\\pi _{LM}$ .",
"The curve supports this interpretation since the score is decreased as $c$ is close to 1.",
"Our STG learns such a strategy as shown from the generated answers in Table REF and REF .",
"In Data-to-Text, the BLEU score is increased as $c$ is close to 1 while the ERR score is decreased.",
"This fact supports the results of NE($mix$ ) models as discussed in Section REF .",
"The $\\pi _s$ learns to balance between the BLEU and ERR."
],
[
"Generated Sentence Examples",
"Here, we show generated sentence examples for each task (see Table REF for Data-to-Text, Table REF and Table REF for Question Answering and Table REF and Table REF for Summarization.).",
"The tokens sampled from the task-specific policy $\\pi _a$ are presented in red."
]
] | 2209.08206 |
[
[
"OysterNet: Enhanced Oyster Detection Using Simulation"
],
[
"Abstract Oysters play a pivotal role in the bay living ecosystem and are considered the living filters for the ocean.",
"In recent years, oyster reefs have undergone major devastation caused by commercial over-harvesting, requiring preservation to maintain ecological balance.",
"The foundation of this preservation is to estimate the oyster density which requires accurate oyster detection.",
"However, systems for accurate oyster detection require large datasets obtaining which is an expensive and labor-intensive task in underwater environments.",
"To this end, we present a novel method to mathematically model oysters and render images of oysters in simulation to boost the detection performance with minimal real data.",
"Utilizing our synthetic data along with real data for oyster detection, we obtain up to 35.1% boost in performance as compared to using only real data with our OysterNet network.",
"We also improve the state-of-the-art by 12.7%.",
"This shows that using underlying geometrical properties of objects can help to enhance recognition task accuracy on limited datasets successfully and we hope more researchers adopt such a strategy for hard-to-obtain datasets."
],
[
"INTRODUCTION",
"Oyster reefs are filter feeders and they provide crucial benefits for the benthic marine ecosystem(s) such as increasingthe richness for a variety of species, providing living habitat, food, and protection for numerous marine species.",
"However, the standing stocks for the oysters near the Chesapeake Bay[1] and North Sea[2] have dropped significantly due to over-fishing, global warming and the effect of diseases across the 19th century.",
"To tackle this devastating ecological problem, massive efforts are being carried out to restore oyster habitats across the United States [3], [4], [5], [6] and Europe [2].",
"One of the core challenges to advance, improve and adapt the restoration process is monitoring of the progress of oyster restorations effectively.",
"Beck et al.",
"[7] proposed to standardize monitoring metrics, units, and performance criteria for the evaluation of the oyster reefs.",
"A set of environmental variables including water salinity, temperature, and dissolved oxygen are being monitored to determine the well-being of oyster habits.",
"For the oyster reefs, universal parameters such as “reef areal dimensions, reef height, oyster density, and oyster size-frequency distribution” are monitored and reported in the literature [7], [2], [4], [5], [6].",
"Figure: Each row left to right: Input image, output of the network when trained using only real data, output of the network (which we call OysterNet) when trained using real data augmented with our synthetic data.",
"Yellow represents the oyster segmentation ground truth and the blue is the predicted segmentation result.",
"Notice how the number of false positives and false negatives drop significantly when the training data is augmented with our synthetic data, All the images in this paper are best seen in color on a computer screen at 200% zoom.However, these metrics rely on recognizing and counting oysters, which is currently largely done by expert manual labor.",
"Such a process is slow, time-consuming, and has poor scalability.",
"Using such a manual approach, the oyster reefs can only be monitored within a restricted area with few samples, e.g., 100 oysters per sample site [5].",
"Furthermore, the material for the oyster's surface is similar to the seabed sediment (See Fig.",
"REFa, Fig.",
"REFd,).",
"And it is significantly different from the washed oysters in Fig.",
"REFa, which makes it very challenging to train new people or algorithms to perform oyster counting.",
"To streamline the process of oyster mapping, the goal is to utilize the advancements in robotics and artificial intelligence that can enable us to gather images from underwater Remotely Operated Vehicles (ROVs) and then automate the oyster detection and density calculation.",
"The central part of this process is to build an oyster detection system.",
"In this work, we present a mathematical model to generate oyster models and further use Generative Adversarial Networks to enable sim-to-real transfer.",
"To the best of our knowledge, this is the first attempt to geometrically model oysters.",
"The contributions of this paper are as follows: We propose a novel mathematical model for the 3D shape of oysters.",
"We simplify the geometric model of an oyster for the projection on the image plane which is used to generate photorealistic synthetic oyster images.",
"These images are used to train a deep segmentation network OysterNet for oysters that achieves the new state-of-the-art.",
"We open-source our oyster generation model and dataset associated with this work to accelerate further research.",
"Figure: An overview of our approach: The proposed geometric model is used to generate synthetic images which are further fed into a Generative Adversarial Network to enable sim-to-real transfer (domain adaptation) by generating photorealistic oyster images.",
"We then combine the synthetic data with real data to train a OysterNet for oyster detection.The rest of this paper is organized as follows: First, we place this work in the context of previous works in Sec. .",
"Then, we describe the proposed geometric model of the oyster which is used to create realistic images in Sec.",
".We then present extensive quantitative and qualitative evaluations of our approach in Sec. .",
"Finally, we conclude the paper in Sec.",
"with parting thoughts on future work."
],
[
"RELATED WORK",
"Automation and robotics are becoming an integral part of many applications, particularly in the marine domain for environmental monitoring[8], [9], [10].",
"The marine domain poses many additional challenges on top of the classical ones faced by robotics.",
"Some of the commonly encountered problems are image visibility distortions caused by the water or sediment, and the difficulty of acquiring data for developing recognition or planning methods that are driven by the information.",
"This is especially true for oyster monitoring.",
"Systematic monitoring of underwater ecosystem requires reliable autonomous navigation.",
"However, in underwater environments, navigation is challenging not only due to the dynamics of the systems but also due to a lack of adequate spatial awareness.",
"To overcome these challenges vision-only based methods have been developed in the literature, that use human-labeled data to learn navigation commands for surveying coral reefs [11] and shipwrecks [12].",
"It is important to point out that these methods heavily depend on the quality of data and are hence require extensive data collection for a high degree of robustness.",
"Sadrfaridpour et al.",
"[13] recently collected an underwater dataset and used Mask-R-CNN[14] for oyster detection.",
"However, this dataset is relatively small and is not collected in the ocean/sea bed but rather on an oyster farm.",
"The oysters are stacked and are very dense in the dataset which lead to poor detection results for oyster reefs in the real world.",
"There is a lack of variety of oysters and environment in Sadrfaridpour's dataset for a more robust detection result when deployed in the wild.",
"Instead of collecting large datasets for detection tasks, we follow the conceptual approach proposed by Sanket et al.",
"[15] which is to geometrically model the object under consideration to generate an enormous amounts of data synthetically.",
"In particular, we model the 3D shape of oysters to create an underwater dataset for oyster detection.",
"Alternatively, Generative Adversarial Networks (GANs) has been also widely used to generate underwater datasets.",
"Joshi et al.",
"[16] utilized a GAN to create images for pose estimation of an autonomous underwater vehicle (AUV).",
"However, this method also required a large number of ground truth data samples for generating all possible pose images.",
"To tackle the visibility distortions caused by the water or sediment, Li et al.",
"[17] proposed a system that restores underwater images into in-air images.",
"Moreover, Wang [18] also proposed an approach for real-world underwater color restoration and dehazing by utilizing GANs.",
"With the ultimate goal of developing an autonomous information-driven oyster monitoring system, we acknowledge the need of good oyster prediction methods.",
"Moreover due to the lack of large samples of oyster detests and the lack of literature to address this problem we are seeking an alternative path for generating a large oyster dataset by looking into the mathematical model of oysters.",
"To the best of our knowledge, we are the first to propose a geometric model of oysters and use that model to generate synthetic data which is described next."
],
[
"Synthetic Oyster Generation",
"In this work, we propose a novel method for modeling oysters which are then used to generate a large dataset of oyster reef images.",
"In this section, we will talk about our proposed method (which is summarised in Fig.",
"REF ).",
"First, we will describe the process of modeling oysters.",
"Next, we will talk about how this model can be used to generate photorealistic images of oyster reefs using domain adaptation.",
"Last but not least, we train a semantic segmentation model using the OysterNet for detecting the oysters in the real world.",
"We present each sub-part in the following sub-sections."
],
[
"Geometric Modelling Of Oysters",
"In order to model the geometry of an oyster, we first 3D scanned ten washed oysters (sample image example shown in REFa and scanned model shown in REFb) which were used to build a mathematical model.",
"Furthermore, the parameters in the proposed mathematical model can be adjusted to generate models of various oysters.",
"We will describe the mathematical model next.",
"It is important to note that the geometrical shape of an oyster is extremely complex, with each one having a different shape and thickness.",
"Oysters grow following a general oyster shape but expand somewhat randomly along their margins.",
"In the first step, we will model our oysters in 3D.",
"Each oyster can be approximated as a series of mathematical functions in a stratified manner (akin to the way a 3D printer prints).",
"We start with each `horizontal' layer (slice of the cross-section of the oyster) by looking at the top view of our scanner oysters.",
"We call this the perimeter of the oyster.",
"We noticed that the perimeter can be easily modeled using two cubic B-splines.",
"Let the $n+1$ control points for the splines be $\\lbrace c_0, ... , c_n\\rbrace $ and $m + 1$ knot vectors be ${\\lbrace t_0, ..., t_m\\rbrace }$ , then the spline curve $S(t)$ of degree $k$ is given by $S(t) = \\sum _{i=0}^{n} c_i B_{i,k}(t) = 1,$ where, $B_{i,k}(t)$ denotes the basic function of degree $k$ and is computed recursively as $B_{i,0}(t) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{if } t_{i+1}\\ge t\\ge t_{i}\\\\0, & \\text{otherwise},\\end{array}\\right.",
"}$ $B_{i,k}(t) = \\frac{t-t_i}{t_{i+k}-t_i}B_{i,k-1}(t)+ \\frac{t_{i+k+1}-t}{t_{i+k+1}-t_{i+1}}B_{i+1,k-1}(t).$ In our case here, $m = n + k + 1$ and we selected $k=3$ for a cubic spline.",
"Particularly, we utilize two cubic B-splines: one for the top half of the shell, and one for the bottom of the shell (See Figs.",
"REFc and REFd).",
"We will call this perimeter model (Eqs.",
"REF , REF and REF ) as the 2D model since it models only a single layer of the oyster.",
"Now, we want to extend the 2D model to 3D in the stratified manner we described before.",
"However, having a high resolution in depth is computationally prohibitive due to all the nooks and crannies (high-frequency edges) on the oyster shell (See Fig.",
"REFb).",
"In order to make this computation tractable, we simplify the shape of the oyster by assuming that it is smooth since the variation on the shell is much smaller than the distance to the oyster.",
"It is important to highlight that the visual changes these high-frequency edges created are approximated by the visual textures we place on our generated oysters.",
"The 3D model of the oyster (Fig.",
"REFe) follows the same perimeter model from before but also adds changes to depth.",
"Let $c^\\alpha _0, ... , c^\\alpha _n$ denote the control points for $\\alpha $ -th layer of the 3D oyster.",
"And the knot vectors for the $\\alpha $ -th layer be${\\lbrace t^\\alpha _0,...,t^\\alpha _m\\rbrace }$ .",
"We define the ${c^\\alpha _n}$ ,$t^\\alpha _m$ as follows $\\vspace{-8.53581pt}{c^\\alpha _n} ={c_n} + X_1$ ${t^\\alpha _m} = t_m + X_2$ where $X1$ and $X2$ are the Gaussian Noise used to model the high frequency edges of the oyster's perimeter.",
"Formally, $X1 \\sim \\mathcal {N}(\\mu _1,\\,\\sigma _1^{2})$ and $X2 \\sim \\mathcal {N}(\\mu _2,\\,\\sigma _2^{2})$ .",
"We can then substitute $c_n,t_m$ with ${c^\\alpha _n},{t^\\alpha _m}$ in Eqs.",
"REF to REF to get the spline curve $S^\\alpha (t)$ for every single layer.",
"For each spline curve $S^\\alpha (t)$ , we can use $[x^\\prime _\\alpha $ , $y^\\prime _\\alpha ]^T$ to represent spline curve points as follows: $[x_\\alpha , y_\\alpha ]^T = [x^\\prime _\\alpha , y^\\prime _\\alpha ]^Tr^\\alpha $ where $r$ is the in-growth rate as a fixed number.",
"Finally, the growth rate of the oyster is defined as follows: $z_\\alpha = \\alpha d,$ where $d$ is the depth at a fixed number here.",
"Now we can use all the $\\alpha $ layers with $[x_\\alpha ,y_\\alpha ,z_\\alpha ]^T$ points to generate 3D model (Fig.",
"REFf) by using pyvista [19] data visualizer.",
"By varying the parameters $\\Theta =\\lbrace \\sigma _1, \\sigma _2, \\mu _1, \\mu _2, \\alpha \\rbrace $ we obtain different oyster shapes (Fig.",
"REF ).",
"Further, we use image textures from real oysters collected in the Chesapeake Bay to be warped on the generated 3D model (See Fig.",
"REFg).",
"Figure: (a) One of the synthetic models generated from Sec.",
"-A), (b) Synthetic model with real oyster texture added, (c) image of an oyster farm with 50 synthetic oysters generated in Blender TM ^{\\text{TM}}, (d) masks for oysters in (c).In the next section, we will talk about how the actual images are rendered from the 3D models we just constructed.",
"A 3D model is not sufficient for creating images for oyster segmentation.",
"We utilized the Blender$^{\\text{TM}}$ [20] game engine to simulate the oysters on a seabed.",
"We rendered 13K synthetic oysters with different 3D models (we will just call it the synthetic model for simplicity) by varying parameters $\\Theta $ as described in Sec.",
".",
"Then we used these synthetic 3D models for synthetic data generation.",
"First, we employed 14 real oyster texture images and applied them randomly to all the synthetic models (Fig.",
"REFa).",
"A sample of the synthetic model with applied texture image can be seen in Fig.",
"REFb.",
"To generate an simulated oyster reef image, we placed the oysters with random poses onto a flat surface with the seabed textures as shown in Fig REFc.",
"Images of the oyster reef are then rendered in Blender$^{\\text{TM}}$ along with segmentation masks (Fig.",
"REFd).",
"However, Fig.",
"REFc is not photorealistic and does not look like a real underwater image which will lead to a poor detection performance when deployed.",
"We describe how we use a Generative Adversarial Network (GAN) to perform sim-to-real domain adaptation such that our images are photorealistic such that they can generalize to the real world after being trained in simulation."
],
[
"Domain Adaptation",
"To render a realistic oyster image, we employ contrastive unpaired translation (CUT) [21] for unpaired image-to-image translation by learning functions to map from the Blender$^{\\text{TM}}$ [20] domain $R$ to the target domain $T$ .",
"The overall loss function consists of three parts: GAN loss, PatchNCE loss, and ExternalNCE loss which are described next.",
"GAN Loss: Generators $G$ and $F$ are used to transfer domains: $G : R \\rightarrow T$ and $F : T \\rightarrow R$ .",
"In order to improve image-to-image translation in CUT, ensuring the reconstructed images $F(G(R)) \\approx R$ is necessary.",
"Thus, we want to minimize the adversarial loss.",
"Adversarial loss is defined by the following two components: discriminator loss and generator loss.",
"Where discriminator loss is to minimize the loss from misclassification between real and fake samples.",
"As for the generator loss, the goal is to maximize the discriminator’s probabilities of being real.",
"PatchNCE loss: We first break the function $G$ into two parts, one encoder, and one decoder.",
"$\\hat{T} = G(R) = G_{dec}(G_{enc}(R))$ .",
"$G_{enc}$ is used for image translation.",
"Therefore, a patch of the input images can be represented as the feature stack of each layer, and the spatial location of the feature stack from the encoder $G_{enc}$ .",
"We want to ensure the cross-entropy between these feature stacks from different layers and spatial locations is minimized.",
"Then the PatchNCE loss is introduced where NCE represents Noise Contrastive Estimation.",
"ExternalNCE loss: Not only do we want to minimize PatchNCE loss, but we can also use the image patches from the rest of the dataset while training.",
"A random negative image from the dataset is encoded and is used to define externalNCE loss.",
"We refer the readers to [21] for a detailed description of the Loss function and network.",
"In our proposed system, the target domain is the realistic environment $T_{real}$ ."
],
[
"Details of Training the CUT",
"We want the CUT network to be able to capture the synthetic oyster to real oyster translation.",
"So, we extract 8959 images of single oysters from the synthetic data that we generated as $R$ and we also extract 957 samples of single oysters from the real underwater images as $T_{real}$ .",
"In the training phase, we learn two mapping functions: $G : R \\rightarrow T$ and $F : T \\rightarrow R$ .",
"Once the CUT is learned, we perform inference on our synthetic images to make them look photorealistic (See Fig.",
"REF ) by performing domain transfer which is further used to train our oyster detection network OysterNet.",
"In this work, we train CUT for 153 epochs to obtain our generator $G$ ."
],
[
"Experiments And Results",
"First, we describe the datasets used in this section.",
"To validate our generated oyster model, we then compare the results obtained by two different segmentation networks and two different datasets.",
"Lastly, we experiment with how different values of the oyster model parameters affect the segmentation results."
],
[
"Description of Datasets",
"Rendered/Synthetic Dataset: contains images obtained by randomly rendering generated 3D oyster models on the seabed using Blender$^{\\text{TM}}$ .",
"Rather than just laying the oysters on the seabed, we simulate the randomization pose of the oysters in various realistic positions.",
"This randomness in oyster pose makes the neural network more robust for recognizing the oyster in different poses.",
"After the images are generated from Blender$^{\\text{TM}}$ , we used CUT, as described in Sec.",
"-B, to create photo-realistic images.",
"The rendered dataset contains 4800 images.",
"Figure: BlueROV collecting data for our real dataset.",
"The inset shows a sample image captured which is a part of our real dataset.Real Dataset: contains images from Harris Creek taken by Chesapeake Bay Foundation by driving a BlueROV as shown in Fig.",
"REF .",
"We labeled the images by hand.",
"We have 29925 images of size $256\\times 256$ .",
"There are over 3900 oysters labeled in the dataset.",
"Unlike some other datasets for underwater object detection, oysters are really hard to recognize, even for humans.",
"So, we worked with experts at the Chesapeake Bay Foundation to obtain this dataset."
],
[
"Experimental Results",
"OysterNet adapts UNet [22] as its backbone.",
"We trained OysterNet with a learning rate of 0.001 with decay.",
"We use Adam optimizer coupled to the Jaccard loss [23] loss function.",
"We used a batch size is 32 and trained the network for 100 epochs.",
"We want to verify our hypothesis that the synthetic oyster data we generated can help improve the oyster semantic segmentation.",
"[13] obtained the best results for oyster detection using the FPN[24] network.",
"This is the model we use which we denote as the DCO (Detecting and Counting Oysters) method.",
"In our observation UNet [22] architecture performed better using our approach which refer to it as OysterNet (“Ours”) in Table REF .",
"We want to use the minimal amount of real data for training as they are hard to obtain, expensive, and labor intensive.",
"To this end, we use only 25% of the real dataset we used as training which we will denote as $O_{real}$ .",
"We test our method on the remaining 75% of the real dataset that we call $O$ in the Test Data.",
"We denote all the images in our generated synthetic dataset as $O_{syn}$ which will be used for training with/without real data.",
"Formally, $O_{syn\\_and\\_real}$ is the combination of $O_{real}$ and $O_{syn}$ .",
"In this experiment, we perform training on $O_{real}$ (our real dataset) and test it on the $O$ (our held-out real dataset) with both OysterNet and DCO methods.",
"The Intersection over Union (IoU) scores are 18.16% and 18.88% respectively which serves as the baseline for oyster segmentation results for our dataset.",
"Both the methods perform similarly in this case.Then we want to observe the results when trained only on the generated synthetic dataset ($O_{syn}$ ).",
"We train on $O_{syn}$ and test on $O$ with both methods.",
"The IoU score is 7.45% and 6.47% respectively which is lower than our baseline.",
"The network has learned to recognize oysters in the synthetic domain but the sim-to-real domain transfer is still not as desired.",
"To this end, we add a small amount of real data to the synthetic data and use it for training ($O_{syn\\_and\\_real}$ ).",
"We achieve a state-of-the-art IoU Score of 24.54% against the expert human labeled ground truth which is 35.1% better than just using real dataset for training and 12.7% better than DCO when trained on synthetic augmented real data.",
"Table: Comparison of Semantic Segmentation Results with the State-of-the-art.Table: Ablation Studies Of The Proposed Oyster Geometric Model (Also see Fig.",
").The results are tabulated in Table REF .",
"It shows that with the added synthetic dataset, we achieve a significant $\\sim $ 35.1% accuracy improvement over just using the real dataset for training.",
"As we can see from Figs.",
"REFb and REFe, the network predicts mostly sediments as oysters when trained using only real data.",
"This leads to a lot of false positives and false negatives.",
"However, when the network is trained using real data augmented with our synthetic data (which we call OysterNet), there is a significant decrease in false predictions as we can see from Figs.",
"REFc and REFf.",
"Since the generated dataset affects accuracy significantly, we ablate on how different parameters in our proposed geometric model affect performance in the next section."
],
[
"Ablation Studies",
"We varied the parameters $\\Theta $ that controlled amount of high-frequency components and the number of layers to generate oyster models.",
"We also varied the size of the oyster when rendering it in the simulation.",
"When not specified, the parameters are set as $\\mu _1=150$ , $\\mu _2=150$ , $\\sigma _1=150$ , $\\sigma _2=15$ , $\\alpha \\in [15,20]$ , and $ Scale \\in [25,30]$ .",
"Only one parameter is varied at a time while keeping others fixed to the values specified.",
"No two oysters are the same with respect to size and height, therefore we chose different ranges for $\\alpha $ and Scale.",
"We see from Table REFa that IoU Score increases with mean of the noise and then decreases.",
"Figure: Qualitative demonstration of how different parameters of our oyster model affect the shape of the oyster.",
"The parameters are in the same order as the experiments in Table .This is because the noise increases, and the variety of the oysters increases which would benefit the detection of the oysters.",
"However, when the noise is too large ($\\mu _1 > 150$ ), the oyster generated no longer looks like an oyster (See Fig.",
"REF ).",
"The IoU Score drops significantly when $\\mu _1 = 250$ .",
"A similar trend is observed with $\\mu _2$ and $\\sigma _1$ with the IoU Score peaking around 150 (Table REFb and REFc ).",
"We also observe that with increase in standard deviation ($\\sigma _2$ ) for $X_2$ (Table REFd), the detection accuracy drop slightly.",
"By varying the number of layers ($\\alpha $ ) for the oysters, we notice that the change in the IoU Score is relatively minimal.",
"Since the image that we generated is in 2D, the parameter for layers ($\\alpha $ ) does not have a significant effect (Table REFe) as we mostly observe oysters from far away and in mostly top view.",
"In Table REFf, we observe that the scale of the oyster has to match the size of the oyster in the image to get the optimal result.",
"The accuracy drops when the scale of the oyster is either too large or too small.",
"The IoU Score difference between the scale values in the ranges of $20-25$ and $25-30$ is only $0.18\\%$ (Tables REFf)."
],
[
"CONCLUSIONS",
"In this work, we first model the geometry of oyster shells and render the oyster images in a game engine.",
"Then we perform an image-to-image transformation from the simulation domain to the real-world domain.",
"With the help of the generated synthetic dataset, when augmented to the real dataset we showed an improvement in the semantic segmentation IoU score for the oysters by 35.1% over just using real data for training and 12.7% over the current state-of-the-art.",
"These results highlight that for data-critical applications when collecting real images is challenging, it is possible to model the images using the underlying geometry of the object to create photorealistic images that will improve object detection drastically.",
"To the best of our knowledge, this is the first attempt to model 3D structure of oysters.",
"Being the first work in this field, there are many directions and possible improvements that can be made to our OysterNet framework to increase the accuracy of semantic segmentation.",
"One possible improvement will be to model both shells of the oysters instead of a single shell.",
"As for the detection phase, new network architectures can be explored to tackle this problem.",
"Finally, more models of the oysters along with additional shape noise could be utilized in the future for a more robust detection."
],
[
"ACKNOWLEDGMENT",
"We would like to thank to Patrick Beall and Doug Myers from Chesapeake Bay Foundation for annotation some of the images for us.",
"We would also like to thank Allen Pattillo and Chahat Deep Singh from University of Maryland for proofreading our manuscript."
]
] | 2209.08176 |
[
[
"Joint Network Topology Inference via a Shared Graphon Model"
],
[
"Abstract We consider the problem of estimating the topology of multiple networks from nodal observations, where these networks are assumed to be drawn from the same (unknown) random graph model.",
"We adopt a graphon as our random graph model, which is a nonparametric model from which graphs of potentially different sizes can be drawn.",
"The versatility of graphons allows us to tackle the joint inference problem even for the cases where the graphs to be recovered contain different number of nodes and lack precise alignment across the graphs.",
"Our solution is based on combining a maximum likelihood penalty with graphon estimation schemes and can be used to augment existing network inference methods.",
"The proposed joint network and graphon estimation is further enhanced with the introduction of a robust method for noisy graph sampling information.",
"We validate our proposed approach by comparing its performance against competing methods in synthetic and real-world datasets."
],
[
"Proofs of Theoretical Results",
"Proof of Lemma 1.",
"If $|{\\mathbf {s}}_i|\\rightarrow \\infty $ ($|{\\mathbf {w}}_i|\\rightarrow \\infty $ ), then $|{\\mathbf {p}}_i|\\rightarrow \\infty $ ($|{\\mathbf {v}}_i|\\rightarrow \\infty $ ) and ${\\mathbf {p}}_i\\notin \\lbrace 0,1\\rbrace $ (${\\mathbf {v}}_i\\notin [0,1]$ ), so $f({\\mathbf {p}},{\\mathbf {v}})\\rightarrow \\infty $ .",
"Thus, $\\phi ({\\mathbf {s}},{\\mathbf {p}},{\\mathbf {t}},{\\mathbf {w}},{\\mathbf {v}})\\rightarrow \\infty $ when $\\Vert ({\\mathbf {s}},{\\mathbf {p}},{\\mathbf {t}},{\\mathbf {w}},{\\mathbf {v}})\\Vert \\rightarrow \\infty $ .",
"For ${\\mathbf {t}}\\in \\text{dom}(\\phi )$ , we have that ${\\mathbf {t}}\\in [\\epsilon ,1-\\epsilon ]^{L_K}$ .",
"Then, we conclude that $\\phi ({\\mathbf {s}},{\\mathbf {p}},{\\mathbf {t}},{\\mathbf {w}},{\\mathbf {v}})$ is coercive over the feasible set ${\\mathcal {F}}$ .",
"$\\blacksquare $ Proof of Lemma 2.",
"We have the following bound for the gradient of $\\Gamma ({\\mathbf {s}},{\\mathbf {t}})$ with respect to $s_i$ as $&\\left| \\log \\left(\\frac{t_i}{1-t_i}\\right) - \\log \\left(\\frac{\\hat{t}_i}{1-\\hat{t}_i}\\right) \\right|^2 =\\left| \\log \\left(\\frac{t_i(1-\\hat{t}_i)}{\\hat{t}_i(1-t_i)}\\right) \\right|^2&\\nonumber \\\\&\\qquad \\qquad \\qquad \\le \\left| \\frac{t_i(1-\\hat{t}_i)}{\\hat{t}_i(1-t_i)} - 1 \\right|^2\\le \\frac{1}{\\hat{t}_i^2(1-t_i)^2}|t_i-\\hat{t}_i|^2&\\nonumber \\\\&\\qquad \\qquad \\qquad \\le \\frac{1}{\\epsilon ^4}|t_i-\\hat{t}_i|^2.&\\nonumber $ We can then bound the difference of the gradient $\\nabla _{{\\mathbf {s}}} g({\\mathbf {s}},{\\mathbf {t}},{\\mathbf {w}})$ as $&\\Vert \\nabla _{{\\mathbf {s}}} g({\\mathbf {s}},{\\mathbf {t}},{\\mathbf {w}})-\\nabla _{{\\mathbf {s}}} g(\\hat{{\\mathbf {s}}},\\hat{{\\mathbf {t}}},\\hat{{\\mathbf {w}}})\\Vert _2^2&\\nonumber \\\\&~\\quad \\le \\lambda _1^2\\Vert \\Psi ^\\top \\Psi \\Vert _F^2\\Vert {\\mathbf {s}}-\\hat{{\\mathbf {s}}}\\Vert _2^2+ \\left(1/\\epsilon ^4 + \\lambda _1^2\\Vert \\Psi ^\\top \\Vert _F^2\\right)\\Vert {\\mathbf {t}}-\\hat{{\\mathbf {t}}}\\Vert _2^2&\\nonumber \\\\&~\\quad \\le \\lambda _1^2G_s^s\\Vert {\\mathbf {s}}-\\hat{{\\mathbf {s}}}\\Vert _2^2 + \\left(1/\\epsilon ^4 + \\lambda _1^2G_s^t\\right)\\Vert {\\mathbf {t}}-\\hat{{\\mathbf {t}}}\\Vert _2^2,&\\nonumber $ where $G_s^s = \\Vert \\Psi ^\\top \\Psi \\Vert _F^2$ and $G_s^t = \\Vert \\Psi ^\\top \\Vert _F^2$ .",
"Similarly, for the gradient of $g({\\mathbf {s}},{\\mathbf {t}},{\\mathbf {w}})$ with respect to ${\\mathbf {w}}$ , we have $&\\Vert \\nabla _{{\\mathbf {w}}} g({\\mathbf {s}},{\\mathbf {t}},{\\mathbf {w}})-\\nabla _{{\\mathbf {w}}} g(\\hat{{\\mathbf {s}}},\\hat{{\\mathbf {t}}},\\hat{{\\mathbf {w}}})\\Vert _2^2&\\nonumber \\\\&~~\\quad \\le \\lambda _2^2\\Vert \\Sigma ^\\top \\Sigma \\Vert _F^2\\Vert {\\mathbf {w}}-\\hat{{\\mathbf {w}}}\\Vert _2^2+ \\lambda _2^2\\Vert \\Sigma ^\\top \\Vert _F^2\\Vert {\\mathbf {t}}-\\hat{{\\mathbf {t}}}\\Vert _2^2&\\nonumber \\\\&~~\\quad \\le \\lambda _2^2G_w^w\\Vert {\\mathbf {w}}-\\hat{{\\mathbf {w}}}\\Vert _2^2 + \\lambda _2^2G_w^t\\Vert {\\mathbf {t}}-\\hat{{\\mathbf {t}}}\\Vert _2^2,&\\nonumber $ where $G_w^w = \\Vert \\Sigma ^\\top \\Sigma \\Vert _F^2$ and $G_w^t = \\Vert \\Sigma ^\\top \\Vert _F^2$ .",
"We bound the gradient difference with respect to ${\\mathbf {t}}$ with fixed ${\\mathbf {s}}$ and ${\\mathbf {t}}$ as $&\\Vert \\nabla _{{\\mathbf {t}}} g({\\mathbf {s}},{\\mathbf {t}},{\\mathbf {w}})-\\nabla _{{\\mathbf {t}}} g({\\mathbf {s}},{\\mathbf {t}},\\hat{{\\mathbf {w}}})\\Vert _2^2&~\\le ~& \\lambda _2^2\\Vert \\Sigma \\Vert _F^2\\Vert {\\mathbf {w}}-\\hat{{\\mathbf {w}}}\\Vert _2^2&\\nonumber \\\\&&~\\le ~& \\lambda _2^2G_w^t\\Vert {\\mathbf {w}}-\\hat{{\\mathbf {w}}}\\Vert _2^2,&\\nonumber $ with $G_w^t$ defined previously.",
"The bounds for part (b) are found similarly, where $&\\Vert \\nabla _{{\\mathbf {s}}} h({\\mathbf {s}},{\\mathbf {w}})-\\nabla _{{\\mathbf {s}}} h(\\hat{{\\mathbf {s}}},\\hat{{\\mathbf {w}}})\\Vert _2^2&~\\le ~& \\alpha ^2\\Vert {\\mathbf {M}}^\\top {\\mathbf {M}}\\Vert _F^2\\Vert {\\mathbf {s}}-\\hat{{\\mathbf {s}}}\\Vert _2^2&\\nonumber \\\\&&~\\le ~& \\alpha ^2H_s\\Vert {\\mathbf {s}}-\\hat{{\\mathbf {s}}}\\Vert _2^2,&\\nonumber $ $&\\Vert \\nabla _{{\\mathbf {w}}} h({\\mathbf {s}},{\\mathbf {w}})-\\nabla _{{\\mathbf {w}}} h(\\hat{{\\mathbf {s}}},\\hat{{\\mathbf {w}}})\\Vert _2^2&~\\le ~& \\beta ^2\\Vert \\Phi ^\\top \\Phi \\Vert _F^2\\Vert {\\mathbf {w}}-\\hat{{\\mathbf {w}}}\\Vert _2^2&\\nonumber \\\\&&~\\le ~& \\beta ^2H_w\\Vert {\\mathbf {w}}-\\hat{{\\mathbf {w}}}\\Vert _2^2,&\\nonumber $ where $H_s = \\Vert {\\mathbf {M}}^\\top {\\mathbf {M}}\\Vert _F^2$ and $H_w = \\Vert \\Phi ^\\top \\Phi \\Vert _F^2$ .",
"$\\blacksquare $ Proof of Lemma 3.",
"Since ${\\mathbf {s}}^j$ and ${\\mathbf {w}}^j$ are optimal with respect to (15a) and (15d) at iteration $j$ , we have that $0 = \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^j,{\\mathbf {w}}^{j-1}) + \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^j,{\\mathbf {t}}^{j-1},{\\mathbf {w}}^{j-1}) + \\rho _1{\\mathbf {u}}_1^{j-1} + \\rho _1({\\mathbf {s}}^j-{\\mathbf {p}}^{j-1})$ and $0 = \\nabla _{{\\mathbf {w}}}h({\\mathbf {s}}^j,{\\mathbf {w}}^j) + \\nabla _{{\\mathbf {w}}}g({\\mathbf {s}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j) + \\rho _2{\\mathbf {u}}_2^{j-1} + \\rho _2({\\mathbf {w}}^j-{\\mathbf {v}}^{j-1})$ .",
"We also have that ${\\mathbf {u}}_1^j = {\\mathbf {u}}_1^{j-1} + {\\mathbf {s}}^j-{\\mathbf {p}}^j$ and ${\\mathbf {u}}_2^j = {\\mathbf {u}}_2^{j-1} + {\\mathbf {w}}^j-{\\mathbf {v}}^j$ .",
"Then, parts (a) and (b) follow.",
"Next, by Lemma 2 and parts (a) and (b), we can show that (c) holds by $&\\Vert {\\mathbf {u}}_1^j-{\\mathbf {u}}_1^{j+1}\\Vert _2^2 \\le \\frac{1}{\\rho _1^2}\\Vert \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^j)-\\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^j,{\\mathbf {w}}^{j-1})\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\frac{1}{\\rho _1^2}\\Vert \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^j,{\\mathbf {w}}^j)-\\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^j,{\\mathbf {t}}^{j-1},{\\mathbf {w}}^{j-1})\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\Vert ({\\mathbf {p}}^{j-1}-{\\mathbf {p}}^j) - ({\\mathbf {p}}^j-{\\mathbf {p}}^{j+1})\\Vert _2^2&\\nonumber \\\\&\\quad \\le \\frac{\\lambda _1^2G_s^s+\\alpha ^2H_s}{\\rho _1^2}\\Vert {\\mathbf {s}}^j-{\\mathbf {s}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\frac{1/\\epsilon ^4+\\lambda _1^2G_s^t}{\\rho _1^2}\\Vert {\\mathbf {t}}^{j-1}-{\\mathbf {t}}^j\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\Vert {\\mathbf {p}}^{j-1}-{\\mathbf {p}}^j\\Vert _2^2 + \\Vert {\\mathbf {p}}^j-{\\mathbf {p}}^{j+1}\\Vert _2^2&\\nonumber $ and (d) holds by $&\\Vert {\\mathbf {u}}_2^j-{\\mathbf {u}}_2^{j+1}\\Vert _2^2 \\le \\frac{1}{\\rho _2^2}\\Vert \\nabla _{{\\mathbf {w}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^{j+1})-\\nabla _{{\\mathbf {w}}}h({\\mathbf {s}}^j,{\\mathbf {w}}^j)\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\frac{1}{\\rho _2^2}\\Vert \\nabla _{{\\mathbf {w}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})-\\nabla _{{\\mathbf {w}}}g({\\mathbf {s}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j)\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\Vert ({\\mathbf {v}}^{j-1}-{\\mathbf {v}}^j) - ({\\mathbf {v}}^j-{\\mathbf {v}}^{j+1})\\Vert _2^2&\\nonumber \\\\&\\quad \\le \\frac{\\lambda _2^2G_w^w+\\beta ^2H_w}{\\rho _2^2}\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{j+1}\\Vert _2^2 + \\frac{\\lambda _2^2G_w^t}{\\rho _2^2}\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\Vert {\\mathbf {v}}^{j-1}-{\\mathbf {v}}^j\\Vert _2^2 + \\Vert {\\mathbf {v}}^j-{\\mathbf {v}}^{j+1}\\Vert _2^2,&\\nonumber $ verifying the desired upper bounds.",
"$\\blacksquare $ Proof of Property 1.",
"We first show that $\\mathfrak {L}_{\\rho }$ is monotonically decreasing for all $j\\in \\mathbb {N}$ , which will lead to the sequence $\\lbrace {\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j\\rbrace $ being bounded.",
"By the optimality of each primal update (15a)-(15e), we first have that $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad \\qquad \\ge \\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j_1},{\\mathbf {p}}^{j_2},{\\mathbf {t}}^{j_3},{\\mathbf {w}}^{j_4},{\\mathbf {v}}^{j_5},{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber $ for $j_1\\ge j_2\\ge j_3\\ge j_4\\ge j_5$ and $j_i\\in \\lbrace j,j+1\\rbrace $ for all $i$ .",
"Furthermore, it is clear that $\\mathfrak {L}_{\\rho }({\\mathbf {s}},{\\mathbf {p}},{\\mathbf {t}},{\\mathbf {w}},{\\mathbf {v}},{\\mathbf {u}}_1,{\\mathbf {u}}_2)$ is strongly convex with respect to ${\\mathbf {s}}$ and ${\\mathbf {w}}$ , and $\\mathfrak {L}_{\\rho }({\\mathbf {s}},{\\mathbf {p}},{\\mathbf {t}},{\\mathbf {w}},{\\mathbf {v}},{\\mathbf {u}}_1,{\\mathbf {u}}_2)-f({\\mathbf {p}},{\\mathbf {v}})$ is strongly convex with respect to ${\\mathbf {p}}$ and ${\\mathbf {v}}$ .",
"Specifically, we have that $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad -\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)\\ge \\frac{\\rho _1}{2}\\Vert {\\mathbf {s}}^j-{\\mathbf {s}}^{j+1}\\Vert _2^2,\\nonumber $ where the inequality results from the optimality of ${\\mathbf {s}}^{j+1}$ , that is, $\\nabla _{{\\mathbf {s}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)=0$ .",
"Analogously for ${\\mathbf {w}}$ , we have $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad \\qquad -\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad \\ge \\frac{\\rho _2}{2}\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{j+1}\\Vert _2^2.\\nonumber $ By the optimality of ${\\mathbf {p}}^{j+1}$ , we have that $0\\in \\partial _{{\\mathbf {p}}}f({\\mathbf {p}}^{j+1},{\\mathbf {v}}^{j}) - \\rho _1{\\mathbf {u}}_1^{j+1}$ , so $\\rho _1{\\mathbf {u}}_1^{j+1}\\in \\partial _{{\\mathbf {p}}}f({\\mathbf {p}}^{j+1},{\\mathbf {v}}^{j})$ .",
"By the definition of the subgradient, we have that $\\langle \\rho _1{\\mathbf {u}}_1^{j+1},{\\mathbf {p}}^{j+1}\\rangle \\ge \\langle \\rho _1{\\mathbf {u}}_1^{j+1},{\\mathbf {p}}\\rangle $ for any ${\\mathbf {p}}\\in \\lbrace 0,1\\rbrace ^{L_K}$ .",
"It then follows that $\\rho _1\\langle {\\mathbf {u}}_1^{j+1},{\\mathbf {p}}^{j+1}-{\\mathbf {p}}^{i}\\rangle \\ge 0$ for all $i\\in \\mathbb {N}$ .",
"By strong convexity of $\\mathfrak {L}_{\\rho }({\\mathbf {s}},{\\mathbf {p}},{\\mathbf {t}},{\\mathbf {w}},{\\mathbf {v}},{\\mathbf {u}}_1,{\\mathbf {u}}_2)-f({\\mathbf {p}},{\\mathbf {v}})$ with respect to ${\\mathbf {p}}$ , $&\\rho _1\\langle {\\mathbf {u}}_1^{j+1},{\\mathbf {p}}^{j+1}-{\\mathbf {p}}^j \\rangle &~\\ge ~&\\rho _1\\langle {\\mathbf {u}}_1^{j+1}-{\\mathbf {u}}_1^j,{\\mathbf {p}}^{j+1}-{\\mathbf {p}}^j \\rangle &\\nonumber \\\\&&~\\ge ~&\\rho _1\\Vert {\\mathbf {p}}^j-{\\mathbf {p}}^{j+1}\\Vert _2^2.&\\nonumber $ By an analogous process for ${\\mathbf {v}}$ , we can also show that $\\rho _2\\langle {\\mathbf {u}}_2^{j+1},{\\mathbf {v}}^{j+1}-{\\mathbf {v}}^j \\rangle \\ge \\rho _2\\Vert {\\mathbf {v}}^j-{\\mathbf {v}}^{j+1}\\Vert _2^2$ .",
"Finally, since $\\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^j)=0$ by optimality of ${\\mathbf {t}}^{j+1}$ , we have $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad \\qquad -\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad = g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j},{\\mathbf {w}}^{j})-g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j})&\\nonumber \\\\&\\qquad = \\Gamma ({\\mathbf {s}}^{j+1},{\\mathbf {t}}^j) - \\Gamma ({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1})&\\nonumber \\\\&\\qquad \\qquad - \\langle \\nabla _{{\\mathbf {t}}}\\Gamma ({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1}),{\\mathbf {t}}^j-{\\mathbf {t}}^{j+1} \\rangle &\\nonumber \\\\&\\qquad \\qquad + \\langle \\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j}),{\\mathbf {t}}^j-{\\mathbf {t}}^{j+1} \\rangle &\\nonumber \\\\&\\qquad \\qquad + \\frac{\\lambda _1+\\lambda _2}{2}\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\qquad = \\Gamma ({\\mathbf {s}}^{j+1},{\\mathbf {t}}^j) - \\Gamma ({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1})&\\nonumber \\\\&\\qquad \\qquad - \\langle \\nabla _{{\\mathbf {t}}}\\Gamma ({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1}),{\\mathbf {t}}^j-{\\mathbf {t}}^{j+1} \\rangle &\\nonumber \\\\&\\qquad \\qquad + \\frac{\\lambda _1+\\lambda _2}{2}\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\qquad \\ge \\left(\\frac{\\lambda _1+\\lambda _2}{2} - \\frac{1}{2\\epsilon ^4}\\right)\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2,&\\nonumber $ where the inequality results since $\\Gamma ({\\mathbf {s}},{\\mathbf {t}})$ is $1/\\epsilon ^4$ -smooth.",
"Combining the previously derived lower bounds and Lemma 3 for the full update of $\\mathfrak {L}_{\\rho }$ , we have $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\qquad \\qquad -\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})&\\nonumber \\\\&\\qquad \\ge \\left(\\frac{\\rho _1}{2}-\\frac{\\lambda _1^2G_s^s+\\alpha ^2H_s}{\\rho _1}\\right)\\Vert {\\mathbf {s}}^j-{\\mathbf {s}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\left(\\frac{\\rho _2}{2}-\\frac{\\lambda _2^2G_w^w+\\beta ^2H_w}{\\rho _2}\\right)\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\left(\\frac{\\lambda _1+\\lambda _2}{2}-\\frac{1}{2\\epsilon ^4}-\\frac{\\lambda _2^2G_w^t}{\\rho _2}\\right)\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad - \\frac{\\epsilon ^{-4}+\\lambda _1^2G_s^t}{\\rho _1}\\Vert {\\mathbf {t}}^{j-1}-{\\mathbf {t}}^j\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\frac{\\rho _1}{2}\\Vert {\\mathbf {p}}^j-{\\mathbf {p}}^{j+1}\\Vert _2^2- \\frac{\\rho _1}{2}\\Vert {\\mathbf {p}}^{j-1}-{\\mathbf {p}}^j\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\frac{\\rho _2}{2}\\Vert {\\mathbf {v}}^j-{\\mathbf {v}}^{j+1}\\Vert _2^2- \\frac{\\rho _2}{2}\\Vert {\\mathbf {v}}^{j-1}-{\\mathbf {v}}^j\\Vert _2^2.&\\nonumber $ Summing over all iterates up to a given $j\\in \\mathbb {N}$ , we have the following difference $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{0},{\\mathbf {p}}^{0},{\\mathbf {t}}^{0},{\\mathbf {w}}^{0},{\\mathbf {v}}^{0},{\\mathbf {u}}_1^{0},{\\mathbf {u}}_2^{0})&\\nonumber \\\\&\\quad \\quad -\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)&\\nonumber \\\\&\\quad \\ge \\sum _{i=0}^j \\left(\\frac{\\rho _1}{2}-\\frac{\\lambda _1^2G_s^s+\\alpha ^2H_s}{\\rho _1}\\right)\\Vert {\\mathbf {s}}^{i}-{\\mathbf {s}}^{i+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\quad + \\sum _{i=0}^j \\left(\\frac{\\rho _2}{2}-\\frac{\\lambda _2^2G_w^w+\\beta ^2H_w}{\\rho _2}\\right)\\Vert {\\mathbf {w}}^{i}-{\\mathbf {w}}^{i+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\quad + \\sum _{i=0}^j \\bigg (\\frac{\\lambda _1+\\lambda _2}{2}-\\frac{1}{2\\epsilon ^4} \\bigg .&\\nonumber \\\\&\\quad \\qquad \\qquad \\bigg .",
"-\\frac{\\epsilon ^{-4}+\\lambda _1^2G_s^t}{\\rho _1}-\\frac{\\lambda _2^2G_w^t}{\\rho _2}\\bigg )\\Vert {\\mathbf {t}}^i-{\\mathbf {t}}^{i+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\quad + \\sum _{i=0}^j \\frac{\\rho _1}{2}\\Vert {\\mathbf {p}}^{i}-{\\mathbf {p}}^{i+1}\\Vert _2^2+ \\sum _{i=0}^j \\frac{\\rho _2}{2}\\Vert {\\mathbf {v}}^{i}-{\\mathbf {v}}^{i+1}\\Vert _2^2,&$ where $\\rho _1$ and $\\rho _2$ are chosen to be large enough so that coefficients of $\\Vert {\\mathbf {s}}^{i}-{\\mathbf {s}}^{i+1}\\Vert _2^2$ and $\\Vert {\\mathbf {w}}^{i}-{\\mathbf {w}}^{i+1}\\Vert _2^2$ are nonnegative, and $\\lambda _1$ and $\\lambda _2$ are large enough so that the coefficient of $\\Vert {\\mathbf {t}}^i-{\\mathbf {t}}^{i+1}\\Vert _2^2$ is nonnegative.",
"Then, we can guarantee that $\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)$ is upper bounded by $\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{0},{\\mathbf {p}}^{0},{\\mathbf {t}}^{0},{\\mathbf {w}}^{0},{\\mathbf {v}}^{0},{\\mathbf {u}}_1^{0},{\\mathbf {u}}_2^{0})$ .",
"Thus, $\\phi ({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j)$ , $\\Vert {\\mathbf {s}}^j-{\\mathbf {p}}^j\\Vert _2^2$ , and $\\Vert {\\mathbf {w}}^j-{\\mathbf {v}}^j\\Vert _2^2$ are also upper bounded.",
"It then follows that $\\lbrace {\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j\\rbrace $ is bounded due to Lemma 1.",
"By Lemma 3, we conclude that $\\lbrace {\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j\\rbrace $ is also bounded.",
"Finally, to show that $\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)$ is lower bounded for all $j\\in \\mathbb {N}$ , let us introduce ${\\mathbf {w}}^{\\prime }={\\mathbf {v}}^j$ .",
"We have that $g({\\mathbf {s}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^{\\prime }) + h({\\mathbf {s}}^j,{\\mathbf {w}}^{\\prime }) + f({\\mathbf {p}}^j,{\\mathbf {v}}^j)$ is greater than $\\min _{{\\mathbf {w}}} g({\\mathbf {s}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}) + h({\\mathbf {s}}^j,{\\mathbf {w}}) + f({\\mathbf {p}}^j,{\\mathbf {w}})$ , which is lower bounded by Lemma 1.",
"We then have that $&\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)= g({\\mathbf {s}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j)&\\nonumber \\\\&\\qquad \\qquad + h({\\mathbf {s}}^j,{\\mathbf {w}}^j) + f({\\mathbf {p}}^j,{\\mathbf {v}}^j)&\\nonumber \\\\&\\qquad \\qquad + \\rho _1 \\langle {\\mathbf {u}}_1^j, {\\mathbf {s}}^j-{\\mathbf {p}}^j \\rangle + \\frac{\\rho _1}{2}\\Vert {\\mathbf {s}}^j-{\\mathbf {p}}^j\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\rho _2 \\langle {\\mathbf {u}}_2^j, {\\mathbf {w}}^j-{\\mathbf {v}}^j \\rangle + \\frac{\\rho _2}{2}\\Vert {\\mathbf {w}}^j-{\\mathbf {v}}^j\\Vert _2^2&\\nonumber \\\\&\\qquad \\ge g({\\mathbf {s}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^{\\prime }) + h({\\mathbf {s}}^j,{\\mathbf {w}}^{\\prime }) + f({\\mathbf {p}}^j,{\\mathbf {v}}^j)&\\nonumber \\\\&\\qquad \\qquad + \\frac{\\rho _2-\\lambda _2^2G_w^w-\\beta ^2H_w}{2}\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{\\prime }\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\rho _1 \\langle {\\mathbf {u}}_1^j, {\\mathbf {s}}^j-{\\mathbf {p}}^j \\rangle + \\frac{\\rho _1}{2}\\Vert {\\mathbf {s}}^j-{\\mathbf {p}}^j\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\rho _2\\langle {\\mathbf {u}}_2^j-{\\mathbf {u}}_2^{j-1},{\\mathbf {v}}^{j-1}-{\\mathbf {v}}^j\\rangle > -\\infty ,&\\nonumber $ where the inner product terms are lower bounded since $\\lbrace {\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j\\rbrace $ is bounded for all $j\\in \\mathbb {N}$ .",
"Thus, $\\mathfrak {L}_{\\rho }({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)$ is lower bounded.",
"$\\blacksquare $ Proof of Property 2.",
"The upper bound in Property 2 follows directly from the bound (REF ) in the previous proof.",
"$\\blacksquare $ Proof of Property 3.",
"Since the objective consists of terms containing $({\\mathbf {s}}^j,{\\mathbf {p}}^j,{\\mathbf {t}}^j,{\\mathbf {w}}^j,{\\mathbf {v}}^j,{\\mathbf {u}}_1^j,{\\mathbf {u}}_2^j)$ , we need only show that each block of $\\partial \\mathfrak {L}_{\\rho }$ can be bounded with respect to a controllable constant related to $\\rho $ or $\\lambda $ .",
"First, we consider the gradient with respect to ${\\mathbf {s}}$ , $&\\nabla _{{\\mathbf {s}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})&\\nonumber \\\\&\\qquad = \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^{j+1})+ \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})&\\nonumber \\\\&\\qquad \\qquad + \\rho _1({\\mathbf {s}}^{j+1}-{\\mathbf {p}}^{j+1}+{\\mathbf {u}}_1^{j+1})&\\nonumber \\\\&\\qquad = \\rho _1({\\mathbf {u}}_1^{j+1}-{\\mathbf {u}}_1^j) + \\rho _1({\\mathbf {p}}^j-{\\mathbf {p}}^{j+1})&\\nonumber \\\\&\\qquad \\qquad + \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^{j+1})+ \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})&\\nonumber \\\\&\\qquad \\qquad - \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^j)- \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^j,{\\mathbf {w}}^j),&\\nonumber $ whose magnitude can be bounded as $&\\Vert \\nabla _{{\\mathbf {s}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})\\Vert _2^2&\\nonumber \\\\&\\quad \\le \\rho _1^2\\Vert {\\mathbf {u}}_1^j-{\\mathbf {u}}_1^{j+1}\\Vert _2^2 + \\rho _1^2\\Vert {\\mathbf {p}}^j-{\\mathbf {p}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\Vert \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^{j+1})- \\nabla _{{\\mathbf {s}}}h({\\mathbf {s}}^{j+1},{\\mathbf {w}}^j) \\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\Vert \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})- \\nabla _{{\\mathbf {s}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^j,{\\mathbf {w}}^j)\\Vert _2^2,&\\nonumber $ and, by Lemmas 2 and 3, this results in the bound $&\\Vert \\nabla _{{\\mathbf {s}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})\\Vert _2^2&\\nonumber \\\\&\\quad \\le (\\lambda _1^2G_s^s+\\alpha ^2H_s)\\Vert {\\mathbf {s}}^j-{\\mathbf {s}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + (1/\\epsilon ^4 + \\lambda _1^2G_s^t)\\left(\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2+ \\Vert {\\mathbf {t}}^{j-1}-{\\mathbf {t}}^j\\Vert _2^2\\right)&\\nonumber \\\\&\\quad \\qquad + 2\\rho _1^2\\Vert {\\mathbf {p}}^j-{\\mathbf {p}}^{j+1}\\Vert _2^2+ \\rho _1^2\\Vert {\\mathbf {p}}^{j-1}-{\\mathbf {p}}^j\\Vert _2^2.&\\nonumber $ By an analogous derivation, we can obtain the following bound with respect to ${\\mathbf {w}}$ as $&\\Vert \\nabla _{{\\mathbf {w}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})\\Vert _2^2&\\nonumber \\\\&\\quad \\le (\\lambda _2^2G_w^w+\\beta ^2H_w)\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{j+1}\\Vert _2^2+ \\lambda _2^2G_w^t\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + 2\\rho _2^2\\Vert {\\mathbf {v}}^j-{\\mathbf {v}}^{j+1}\\Vert _2^2+ \\rho _2^2\\Vert {\\mathbf {v}}^{j-1}-{\\mathbf {v}}^j\\Vert _2^2.&\\nonumber $ We recall that $\\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^j)=0$ , so we have $&\\nabla _{{\\mathbf {t}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})&\\nonumber \\\\&\\qquad = \\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})&\\nonumber \\\\&\\qquad = \\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})-\\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^j),&\\nonumber $ and thus with Lemma 2 we can write $&\\Vert \\nabla _{{\\mathbf {t}}}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})\\Vert _2^2&\\nonumber \\\\&\\qquad = \\Vert \\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1})-\\nabla _{{\\mathbf {t}}}g({\\mathbf {s}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^j)\\Vert _2^2,&\\nonumber \\\\&\\qquad \\le \\lambda _2^2G_w^t\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{j+1}\\Vert _2^2.&\\nonumber $ Figure: Performance analysis in synthetic networks sampled from different latent point sets in the same graphon.",
"(a) Recovery error for K=3K=3 graphs of the same size N=30N=30 as a function of the number of observed signals.",
"(b) Recover error for K=3K=3 probability matrices as subsets of the generating graphon at the latent sample points.",
"(c) Recover error for the graphon.The bounds for ${\\mathbf {u}}_1$ and ${\\mathbf {u}}_2$ are obtained similarly, where $&\\nabla _{{\\mathbf {u}}_1}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})&\\nonumber \\\\&\\qquad = \\rho _1({\\mathbf {s}}^{j+1}-{\\mathbf {p}}^{j+1})= \\rho _1({\\mathbf {u}}_1^{j+1}-{\\mathbf {u}}_1^j)&\\nonumber $ which is bounded via Lemma 3 as $&\\Vert \\nabla _{{\\mathbf {u}}_1}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})\\Vert _2^2&\\nonumber \\\\&\\qquad \\le (\\lambda _1^2G_s^s+\\alpha ^2H_s)\\Vert {\\mathbf {s}}^j-{\\mathbf {s}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + (1/\\epsilon ^4+\\lambda _1^2G_s^t)\\Vert {\\mathbf {t}}^{j-1}-{\\mathbf {t}}^j\\Vert _2^2&\\nonumber \\\\&\\qquad \\qquad + \\rho _1^2\\Vert {\\mathbf {p}}^j-{\\mathbf {p}}^{j+1}\\Vert _2^2+ \\rho _1^2\\Vert {\\mathbf {p}}^{j-1}-{\\mathbf {p}}^j\\Vert _2^2,&$ thus, similarly, we have the bound with respect to ${\\mathbf {u}}_2$ as $&\\Vert \\nabla _{{\\mathbf {u}}_2}\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})\\Vert _2^2&\\nonumber \\\\&\\quad \\le (\\lambda _2^2G_w^w+\\beta ^2H_w)\\Vert {\\mathbf {w}}^j-{\\mathbf {w}}^{j+1}\\Vert _2^2+ \\lambda _2^2G_w^t\\Vert {\\mathbf {t}}^j-{\\mathbf {t}}^{j+1}\\Vert _2^2&\\nonumber \\\\&\\quad \\qquad + \\rho _2^2\\Vert {\\mathbf {v}}^j-{\\mathbf {v}}^{j+1}\\Vert _2^2+ \\rho _2^2\\Vert {\\mathbf {v}}^{j-1}-{\\mathbf {v}}^j\\Vert _2^2.&$ For ${\\mathbf {p}}$ , we first observe the subdifferential $&\\partial _{{\\mathbf {p}}} \\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j+1},{\\mathbf {p}}^{j+1},{\\mathbf {t}}^{j+1},{\\mathbf {w}}^{j+1},{\\mathbf {v}}^{j+1},{\\mathbf {u}}_1^{j+1},{\\mathbf {u}}_2^{j+1})&\\nonumber \\\\&\\qquad = \\partial _{{\\mathbf {p}}}f({\\mathbf {p}}^{j+1},{\\mathbf {v}}^{j+1})-\\rho _1({\\mathbf {s}}^{j+1}-{\\mathbf {p}}^{j+1}+{\\mathbf {u}}_1^{j+1})&\\nonumber \\\\&\\qquad = \\partial _{{\\mathbf {p}}}f({\\mathbf {p}}^{j+1},{\\mathbf {v}}^{j+1})-\\rho _1({\\mathbf {s}}^{j+1}-{\\mathbf {p}}^{j+1}+{\\mathbf {u}}_1^j)&\\nonumber \\\\&\\qquad \\qquad +\\rho _1({\\mathbf {u}}_1^j-{\\mathbf {u}}_1^{j+1}),&\\nonumber $ and by the optimality of the ${\\mathbf {p}}$ update, $0\\in \\partial _{{\\mathbf {p}}}f({\\mathbf {p}}^{j+1},{\\mathbf {v}}^{j+1})-\\rho _1({\\mathbf {s}}^{j+1}-{\\mathbf {p}}^{j+1}+{\\mathbf {u}}_1^j)$ , so we have the subgradient $&{\\mathbf {d}}_{{\\mathbf {p}}} = \\rho _1({\\mathbf {u}}_1^j-{\\mathbf {u}}_1^{j+1}) \\in \\partial _{{\\mathbf {p}}}\\mathfrak {L}_{\\rho },&\\nonumber $ which does not depend on ${\\mathbf {p}}$ .",
"Note that $\\Vert {\\mathbf {d}}_{{\\mathbf {p}}}\\Vert _2^2$ is upper bounded as in (REF ), and similarly $\\Vert {\\mathbf {d}}_{{\\mathbf {v}}}\\Vert _2^2$ is upper bounded as in (REF ).",
"Thus, we have that the subgradient of $\\mathfrak {L}_{\\rho }$ is bounded with respect to the iterative updates.",
"$\\blacksquare $ Proof of Property 4.",
"Since the subsequence $\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j_a},{\\mathbf {p}}^{j_a},{\\mathbf {t}}^{j_a},{\\mathbf {w}}^{j_a},{\\mathbf {v}}^{j_a},{\\mathbf {u}}_1^{j_a},{\\mathbf {u}}_2^{j_a})$ is lower bounded and asymptotically monotonically nonincreasing, the sequence is convergent.",
"Then, we have that the limit $\\lim _{a\\rightarrow \\infty }\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j_a},{\\mathbf {p}}^{j_a},{\\mathbf {t}}^{j_a},{\\mathbf {w}}^{j_a},{\\mathbf {v}}^{j_a},{\\mathbf {u}}_1^{j_a},{\\mathbf {u}}_2^{j_a})\\ge \\mathfrak {L}_{\\rho }({\\mathbf {s}}^{*},{\\mathbf {p}}^{*},{\\mathbf {t}}^{*},{\\mathbf {w}}^{*},{\\mathbf {v}}^{*},{\\mathbf {u}}_1^{*},{\\mathbf {u}}_2^{*})$ .",
"The only potentially discontinuous terms in $\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j_a},{\\mathbf {p}}^{j_a},{\\mathbf {t}}^{j_a},{\\mathbf {w}}^{j_a},{\\mathbf {v}}^{j_a},{\\mathbf {u}}_1^{j_a},{\\mathbf {u}}_2^{j_a})-\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{*},{\\mathbf {p}}^{*},{\\mathbf {t}}^{*},{\\mathbf {w}}^{*},{\\mathbf {v}}^{*},{\\mathbf {u}}_1^{*},{\\mathbf {u}}_2^{*})$ would be $f({\\mathbf {p}}^{j_a},{\\mathbf {v}}^{j_a})-f({\\mathbf {p}}^{*},{\\mathbf {v}}^{*})$ .",
"However, since $f({\\mathbf {p}}^j,{\\mathbf {v}}^j)=0$ for all $j\\in \\mathbb {N}$ due to the projections in update steps (15b) and (15e), we also have that $f({\\mathbf {p}}^{j_a},{\\mathbf {v}}^{j_a})=0$ for all $a\\in \\mathbb {N}$ , so $f({\\mathbf {p}}^{j_a},{\\mathbf {v}}^{j_a})-f({\\mathbf {p}}^{*},{\\mathbf {v}}^{*})\\le 0$ .",
"Thus, we have that $\\lim _{a\\rightarrow \\infty }\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{j_a},{\\mathbf {p}}^{j_a},{\\mathbf {t}}^{j_a},{\\mathbf {w}}^{j_a},{\\mathbf {v}}^{j_a},{\\mathbf {u}}_1^{j_a},{\\mathbf {u}}_2^{j_a})=\\mathfrak {L}_{\\rho }({\\mathbf {s}}^{*},{\\mathbf {p}}^{*},{\\mathbf {t}}^{*},{\\mathbf {w}}^{*},{\\mathbf {v}}^{*},{\\mathbf {u}}_1^{*},{\\mathbf {u}}_2^{*})$ , and thus the subsequence converges to a limit point as $a\\rightarrow \\infty $ .",
"$\\blacksquare $ Figure: Comparison of estimated probability matrices.",
"The image shows pixel pictures of probability matrices whose entries represent estimated edge probability for each node pair in the networks.Figure: Performance analysis in weighted synthetic networks sampled from the same latent point sets in the same graphon.",
"(a) Recovery error for K=3K=3 graphs as a function of the number of observed signals.",
"(b) Recovery error for K=3K=3 probability matrices as subsets of the generating graphon at the latent sample points.",
"(c) Recovery error for the estimated graphon."
],
[
"Additional Numerical Experiments",
"We include additional results to empirically illustrate the effects of jointly inferring the networks, probability matrices, and graphon.",
"We also demonstrate the practicality of our proposed method for weighted networks.",
"Joint estimation of networks, probability matrices, and graphon.",
"Observing the empirical results provides us with insight into how each of the three object types—networks, probability matrices, and graphons—interact when jointly inferred.",
"In Fig.",
"REF , we consider estimating multiple networks of the same size sampled from the same graphon at different latent points.",
"While all proposed augmentations achieve significant improvement in network recovery, jointly estimating the graphon achieves the greatest improvement for both separate and joint network inference in Fig.",
"REF (a).",
"We also include recovery error of the probability matrices in Fig.",
"REF (b), which demonstrates that joint inference of the networks and graphon in (4) can achieve significantly more accurate probability matrix estimates than joint inference of the networks and the probability matrices in (3).",
"Joint graphon estimation not only assumes the correct signal model, where nodes across networks are not aligned, but also ensures that closer probability matrix entries are more similar, in accordance with the smooth graphon assumption.",
"Fig.",
"REF demonstrates the influence of the smooth graphon assumption on the estimated probability matrices, where we show a comparison of the inferred probability matrices from our proposed methods.",
"For joint network and probability matrix inference (3), no dependence is assumed between the edge probabilities in the probability matrices, whereas joint network and graph inference (4) relates edges between nodes based on their assignments in the latent sample space.",
"Finally, graphon recovery error is shown in Fig.",
"REF (c).",
"The joint inference method augmented with graphon estimation in red assumes that nodes are aligned across networks, an incorrect assumption given that networks are sampled from different latent point sets.",
"Thus, the augmented separate inference method exhibits superior graphon estimates.",
"Estimation of weighted networks.",
"While our method focuses on learning the support of the estimated networks, there are many examples of network applications that require additional features such as edge weights.",
"To this end, we combine our proposed method in a two-phase approach with another process that obtains node or edge features from network structure.",
"Indeed, graph-based learning is often approached in three steps: obtaining the network structure, assigning weights to the edges, and performing downstream learning tasks [1].",
"We include results in Fig.",
"REF using this two-phase approach, where we first estimate the network structures via our proposed method, then we obtain edge weights via similarity-based weights.",
"In particular, we apply the commonly used Gaussian edge weighting scheme [1].",
"For two nodes $i$ and $j$ in the $k$ -th network $\\mathcal {G}^{(k)}$ , if the edge $(i,j)$ exists, then we assign the edge weight ${W}_{ij}^{(k)}$ as $W_{ij}^{(k)} = \\exp \\left\\lbrace \\frac{-\\Vert \\mathbf {X}^{(k)}_i-\\mathbf {X}^{(k)}_j\\Vert _2^2}{2\\sigma ^2}\\right\\rbrace ,$ where $\\sigma ^2$ is the variance of the Gaussian kernel, and $\\mathbf {X}_i^{(k)}$ denotes the vector of all graph signal values at the the $i$ -th node of the $k$ -th network.",
"Even when the underlying networks are weighted, implementing joint estimation of the probability matrix proves effective at improving graph estimation.",
"The combination of inferring network connectivity and estimating edge weights is not only empirically feasible but also straightforward in implementation, as there exist several strategies to obtain graph characteristics given network structure [1]."
]
] | 2209.08223 |
[
[
"Anisotropic resistivity tensor from disk geometry magneto-transport"
],
[
"Abstract Magneto-transport measurements on two dimensional van der Waals heterostructures have recently shown signatures of uniaxial anisotropy.",
"Such measurements are almost exclusively performed in a Hall bar geometry which makes it difficult to extract the full resistivity tensor.",
"The goal of this paper is to theoretically analyze anisotropic magneto-transport in a homogeneous disk geometry and to provide a closed form expression for the electrical potential anywhere on the disk if the current source and drain are located somewhere on the circumference.",
"This expression can then be used to experimentally extract the full resistivity tensor."
],
[
"Introduction",
"Two dimensional van der Waals (vdW) heterostructures host a broad range of interesting physical phenomena[1], including anisotropic magnetotransport.",
"With rare exceptions[2], [3], the transport measurements are performed in a Hall bar geometry, making it difficult to extract the full resistivity tensor particularly if the transport principal axis is misaligned with the current flow.",
"For example, the heterostructures can be subject to an unintentional strain, in which case the misalignment is not directly controlled in an experiment.",
"Moreover, the orientation of the transport principal axis can be carrier concentration (filling) dependent as was recently shown [4] in numerical solutions of the Boltzman equation for twisted bilayer graphene subject to heterostrain, even if the strain tensor and the transport relaxation time are momentum and filling independent.",
"For open Fermi surfaces, the magneto-resistance is expected to grow with the magnetic field $B$ without saturation along one of the principal axis, but to saturate with increasing $B$ along the perpendicular principal axis[5].",
"Direct measurement of the full anisotropic resistivity tensor in the vdW heterostructures as a function of filling and $B$ would therefore help in understanding the complex transport phenomena in these materials.",
"In this paper we analyze the solution to the magneto-transport equations in a uniform disk of radius $a$ .",
"Inside the disk the anisotropic conductivity tensor is assumed to be homogeneous, while outside the disk there is no conduction.",
"Thus, $\\sigma =D(x,y)\\left(\\sigma _+\\hat{\\mathbf {x}}\\hat{\\mathbf {x}}+\\sigma _-\\hat{\\mathbf {y}}\\hat{\\mathbf {y}}+\\sigma _H\\left(\\hat{\\mathbf {x}}\\hat{\\mathbf {y}}-\\hat{\\mathbf {y}}\\hat{\\mathbf {x}}\\right)\\right),$ where $D(x,y)=\\Theta (a^2-x^2-y^2)$ and $\\Theta $ is the Heaviside step function, restricting the conduction to the interior of the circle.",
"Without loss of generality, we also assume that the principal axes are aligned with the $x$ and $y$ axes of the coordinate system and adopt the dyadic product to represent the conductivity tensor.",
"Here $\\sigma _{\\pm }$ are the two components of the longitudinal conductivity along the principal axes and $\\sigma _H$ is the Hall conductivity.",
"We express the longitudinal conductivities as $\\sigma _\\pm =\\bar{\\sigma }\\pm \\Delta \\sigma $ and without loss of generality take the $x$ -axis to be along the axis with larger resistivity i.e.",
"$\\Delta \\sigma /\\bar{\\sigma }<0$ .",
"The analysis below then provides an expression in the form of a rapidly convergent series which can be used to extract the resistivity tensor for a point current source/drain at $\\mathbf {r}_{S,D}=a(\\cos \\theta _{A,B},\\sin \\theta _{A,B})$ .",
"The expression for the electrical potential at $x,y$ reads $&&V(x,y;\\mathbf {r}_S,\\mathbf {r}_D)=\\nonumber \\\\&&\\frac{I}{\\pi }\\frac{\\sqrt{\\sigma _+\\sigma _-}}{\\sigma _+\\sigma _-+\\sigma ^2_H}\\left(\\sum _{n=0,2,4,\\ldots }^{\\infty }\\ln \\frac{\\left|1+e^{-2i\\theta _B}\\Omega ^{2+4n}-e^{-i\\theta _B}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|}{\\left|1+e^{-2i\\theta _A}\\Omega ^{2+4n}-e^{-i\\theta _A}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|}+\\sum _{n=1,3,5,\\ldots }^{\\infty }\\ln \\frac{\\left|1+e^{2i\\theta _B}\\Omega ^{2+4n}-e^{i\\theta _B}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|}{\\left|1+e^{2i\\theta _A}\\Omega ^{2+4n}-e^{i\\theta _A}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|}\\right)\\nonumber \\\\&+&\\frac{I}{\\pi }\\frac{\\sigma _H}{\\sigma _+\\sigma _-+\\sigma ^2_H}\\left(\\sum _{n=0,2,4,\\ldots }^{\\infty }\\arg \\left(1+e^{-2i\\theta _B}\\Omega ^{2+4n}-e^{-i\\theta _B}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)-\\arg \\left(1+e^{-2i\\theta _A}\\Omega ^{2+4n}-e^{-i\\theta _A}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)\\right.\\nonumber \\\\&+&\\left.\\sum _{n=1,3,5,\\ldots }^{\\infty }\\arg \\left(1+e^{2i\\theta _B}\\Omega ^{2+4n}-e^{i\\theta _B}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)-\\arg \\left(1+e^{2i\\theta _A}\\Omega ^{2+4n}-e^{i\\theta _A}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)\\right).$ where the $x,y$ position enters via the complex variable $Z=X+iY=\\frac{x}{\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}+i\\frac{y}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}$ , and the parameters $\\alpha _{+}=\\frac{a}{2}\\left(\\frac{1}{\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}+\\frac{1}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}\\right)$ and $\\Omega =\\sqrt{\\frac{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}-\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}+\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}}$ .",
"The function $\\arg $ is the argument of a complex number.",
"Note that because $|\\Delta \\sigma |<\\bar{\\sigma }$ , the parameter $0\\le \\Omega <1$ and therefore the above sum converges (the convergence is rapid unless $\\Omega $ is very close to 1).",
"Illustrative contour plots of $V(x,y;\\mathbf {r}_S,\\mathbf {r}_D)$ for several parameters are shown in the Fig.REF .",
"Although the above expression is obtained for a point current source/drain, the linearity of the differential equation whose solution it is allows direct determination of the formula for multiple point, as well as spatially extended, current sources/drains.",
"Such formula is presented in the discussion section.",
"Figure: Equipotential contours computed using Eq.",
"() for isotropic conductivity tensor (a) for σ H =0\\sigma _H=0 and (b) for σ H =0.4σ ¯\\sigma _H=0.4\\bar{\\sigma }; in each case the point source is at θ A =π/2\\theta _A=\\pi /2 and the point drain at θ B =-π/2\\theta _B=-\\pi /2.",
"Equipotential contours for anisotropic conductivity tensor for σ H =0\\sigma _H=0 and Δσ=-0.7σ ¯\\Delta \\sigma =-0.7\\bar{\\sigma } for point drain at θ B =-π/2\\theta _B=-\\pi /2 and (c) point source at θ A =π/2\\theta _A=\\pi /2 and (d) θ A =π/4\\theta _A=\\pi /4; ten terms in the sum were kept."
],
[
"Analysis",
"The starting assumption is that Ohm's law holds, i.e.",
"$\\mathbf {j}&=&\\sigma \\cdot \\mathbf {E}=-\\sigma \\cdot \\nabla V,$ where $\\mathbf {j}$ is the current density, $\\mathbf {E}$ is the electric field and $V$ is the electrical potential, all of which are assumed to be position dependent.",
"For an idealized point current source and drain, the continuity equation gives $\\nabla \\cdot \\mathbf {j}&=&I\\left(\\delta (\\mathbf {r}-\\mathbf {r}_A)-\\delta (\\mathbf {r}-\\mathbf {r}_B)\\right),$ where $I$ is the current, its source is at $\\mathbf {r}_A$ , its drain at $\\mathbf {r}_B$ , and $\\delta (\\mathbf {r})$ is the Dirac delta function.",
"Combining Eqs.",
"(REF -REF ) gives $-\\frac{\\partial }{\\partial x}\\left(D\\sigma _+\\frac{\\partial V}{\\partial x}\\right)-\\frac{\\partial }{\\partial y}\\left(D\\sigma _-\\frac{\\partial V}{\\partial y}\\right)-\\frac{\\partial }{\\partial x}\\left(D\\sigma _H\\frac{\\partial V}{\\partial y}\\right)+\\frac{\\partial }{\\partial y}\\left(D\\sigma _H\\frac{\\partial V}{\\partial x}\\right)=I\\left(\\delta (\\mathbf {r}-\\mathbf {r}_A)-\\delta (\\mathbf {r}-\\mathbf {r}_B)\\right).$ The solution to the above inhomogeneous linear partial differential equation gives $V$ as a function of $\\mathbf {r}$ .",
"Expressing the longitudinal conductivities as $\\sigma _\\pm &=&\\bar{\\sigma }\\pm \\Delta \\sigma ,$ it will be convenient to rescale the coordinate axes according to $X&=&\\frac{x}{\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}},\\\\Y&=&\\frac{y}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}},$ so that Eq.",
"(REF ) becomes $&&-\\left(\\frac{\\partial }{\\partial X}\\left(D\\frac{\\partial V}{\\partial X}\\right)+\\frac{\\partial }{\\partial Y}\\left(D\\frac{\\partial V}{\\partial Y}\\right)\\right)-\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\frac{\\partial }{\\partial X}\\left(D\\frac{\\partial V}{\\partial Y}\\right)-\\frac{\\partial }{\\partial Y}\\left(D\\frac{\\partial V}{\\partial X}\\right)\\right)=\\nonumber \\\\&&\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\delta (X-X_A)\\delta (Y-Y_A)-\\delta (X-X_B)\\delta (Y-Y_B)\\right).$ The new domain, specified by $D\\left(\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}X,\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}Y\\right)$ , is given by $\\Theta (a^2-\\left(1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)X^2-\\left(1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)Y^2)$ , i.e.",
"it is an ellipse.",
"If $\\Delta \\sigma /\\bar{\\sigma }>0$ , the ellipse is elongated along the $Y$ -direction, if $\\Delta \\sigma /\\bar{\\sigma }<0$ , then the ellipse is elongated along the $X$ -direction.",
"Without loss of generality we can choose the $x$ -axis to be along the axis with larger resistivity, i.e.",
"it will be assumed from now on that $\\Delta \\sigma /\\bar{\\sigma }<0.$ The equation (REF ) can be expressed using complex coordinates $Z=X+iY,$ when, after some simplification, it becomes $&&-2\\left(\\frac{\\partial }{\\partial Z}\\left(D\\frac{\\partial V}{\\partial \\bar{Z}}\\right)+\\frac{\\partial }{\\partial \\bar{Z}}\\left(D\\frac{\\partial V}{\\partial Z}\\right)\\right)-\\frac{2i\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\frac{\\partial }{\\partial \\bar{Z}}\\left(D\\frac{\\partial V}{\\partial Z}\\right)-\\frac{\\partial }{\\partial Z}\\left(D\\frac{\\partial V}{\\partial \\bar{Z}}\\right)\\right)=\\nonumber \\\\&&\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\delta (X-X_A)\\delta (Y-Y_A)-\\delta (X-X_B)\\delta (Y-Y_B)\\right).$ To avoid confusion, the right-hand-side is kept in terms of the real and imaginary parts of $Z$ .",
"This form makes it clear that inside the ellipse where $D=1$ , the solution can be written in terms of a sum of a function of $Z$ and a function of $\\bar{Z}$ .",
"The boundary conditions are determined from the right hand side and the derivatives of the boundary function $D$ ."
],
[
"Zhukovsky conformal mapping of the ellipse to annulus",
"It will be convenient to perform a conformal map transforming the boundary of the ellipse to the boundary of the circle.",
"This can be done using the Zhukovsky transformation $Z&=&\\alpha _+ w+\\frac{\\alpha _-}{w},\\\\w&=&u+iv,$ where $u(X,Y)$ and $v(X,Y)$ are purely real.",
"To determine the coefficients $\\alpha _+$ and $\\alpha _-$ we demand that $\\left(1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)X_0^2+\\left(1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)Y_0^2&=&a^2,$ implies $u_0^2+v_0^2=1,$ i.e.",
"if $X_0$ and $Y_0$ lie on the ellipse, then $u_0$ and $v_0$ are forced to lie on the unit circle.",
"From Eq.",
"(REF ), we have $X_0+iY_0&=&\\alpha _+(u_0+iv_0)+\\alpha _-(u_0-iv_0),$ because, being on unit circle, $1/(u_0+iv_0)=u_0-iv_0$ .",
"Therefore, $X_0&=&(\\alpha _++\\alpha _-)u_0,\\\\Y_0&=&(\\alpha _+-\\alpha _-)v_0.$ So, from (REF ) $&&\\left(1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)(\\alpha _++\\alpha _-)^2u^2_0+\\left(1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)(\\alpha _+-\\alpha _-)^2v^2_0=a^2,\\nonumber \\\\$ which implies $\\alpha _{\\pm }=\\frac{a}{2}\\left(\\frac{1}{\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}\\pm \\frac{1}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}\\right).$ This fixes the conformal map.",
"Having established that the ellipse in the $(X,Y)$ -plane maps onto the unit circle in the $(u,v)$ -plane, we wish to know where does the interior of the ellipse map.",
"To this end, seek such $w=\\Omega $ that would give $\\alpha _+ \\Omega &=&\\frac{\\alpha _-}{\\Omega }\\in \\Re e,$ for $\\Delta \\sigma /\\bar{\\sigma }<0$ .",
"This gives $\\Omega &=&\\sqrt{\\frac{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}-\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}+\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}}.$ So, letting $w=\\Omega e^{i\\phi }$ where $\\phi $ is the polar angle in the $u,v$ -plane and using (REF ) results in $\\alpha _+ \\Omega e^{i\\phi }+\\frac{\\alpha _-}{\\Omega e^{i\\phi }}=a\\frac{\\sqrt{-2\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}{\\sqrt{1-\\left(\\frac{\\Delta \\sigma }{\\bar{\\sigma }}\\right)^2}}\\cos \\phi .$ This means that the circle of radius $\\Omega $ in $u,v$ -plane maps onto the line segment connecting the foci of the ellipse in the $X,Y$ -plane.",
"For $\\Delta \\sigma /\\bar{\\sigma }<0$ , the foci lie on the x-axis.",
"Therefore, the ellipse in $X,Y$ -plane, including its interior, maps onto an annulus in the $u,v$ -plane with the outer radius 1 and the inner radius $\\Omega $ as illustrated in the Figure REF .",
"Figure: For Δσ/σ ¯<0\\Delta \\sigma /\\bar{\\sigma }<0, (a) the interior of the circular device with radius a maps onto the interior of the ellipse after the coordinate rescaling.",
"(b) The interior of the ellipse in the X,YX,Y-plane conformally maps onto the annulus in the u,vu,v-plane with unit outer radius and inner radius set by Ω\\Omega .Because $Z=f(w)$ i.e.",
"$Z$ is a function of $w$ , $w$ is in turn a function of $Z$ , i.e.",
"$w=g(Z)$ , the left hand side of the differential equation can be written as $&&-2\\left(\\frac{\\partial }{\\partial Z}\\left(D\\frac{\\partial V}{\\partial \\bar{Z}}\\right)+\\frac{\\partial }{\\partial \\bar{Z}}\\left(D\\frac{\\partial V}{\\partial Z}\\right)\\right)-\\frac{2i\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\frac{\\partial }{\\partial \\bar{Z}}\\left(D\\frac{\\partial V}{\\partial Z}\\right)-\\frac{\\partial }{\\partial Z}\\left(D\\frac{\\partial V}{\\partial \\bar{Z}}\\right)\\right)=\\nonumber \\\\&&\\frac{\\partial w}{\\partial Z}\\frac{\\partial \\bar{w}}{\\partial \\bar{Z}}\\left(-\\left(\\frac{\\partial }{\\partial u}D\\frac{\\partial V}{\\partial u}+\\frac{\\partial }{\\partial v}D\\frac{\\partial V}{\\partial v}\\right)+\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\frac{\\partial D}{\\partial v}\\frac{\\partial V}{\\partial u}-\\frac{\\partial D}{\\partial u}\\frac{\\partial V}{\\partial v}\\right)\\right).$ Using Cauchy-Riemann conditions, it can be readily shown that $\\frac{\\partial w}{\\partial Z}\\frac{\\partial \\bar{w}}{\\partial \\bar{Z}}=J\\left(\\frac{u,v}{X,Y}\\right),$ where $J\\left(\\frac{u,v}{X,Y}\\right)$ is the Jacobian determinant.",
"Eq.",
"(REF ) therefore gives $-\\left(\\frac{\\partial }{\\partial u}D\\frac{\\partial V}{\\partial u}+\\frac{\\partial }{\\partial v}D\\frac{\\partial V}{\\partial v}\\right)+\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\frac{\\partial D}{\\partial v}\\frac{\\partial V}{\\partial u}-\\frac{\\partial D}{\\partial u}\\frac{\\partial V}{\\partial v}\\right)=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\delta (X-X_A)\\delta (Y-Y_A)-\\delta (X-X_B)\\delta (Y-Y_B)}{J\\left(\\frac{u,v}{X,Y}\\right)}.$ But, by the properties of the Dirac $\\delta $ function under coordinate transformation, it follows that $&&-\\left(\\frac{\\partial }{\\partial u}D\\frac{\\partial V}{\\partial u}+\\frac{\\partial }{\\partial v}D\\frac{\\partial V}{\\partial v}\\right)+\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\frac{\\partial D}{\\partial v}\\frac{\\partial V}{\\partial u}-\\frac{\\partial D}{\\partial u}\\frac{\\partial V}{\\partial v}\\right)=\\nonumber \\\\&&\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\delta (u-u_A)\\delta (v-v_A)-\\delta (u-u_B)\\delta (v-v_B)\\right),$ where $D=\\Theta \\left(1-u^2-v^2\\right)$ .",
"Now, because $X_{A,B},Y_{A,B}$ lie on the ellipse, $u_{A,B},v_{A,B}$ must lie on the unit circle.",
"From Eq.",
"REF $\\frac{x_{A,B}}{\\sqrt{1+\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}&=&\\left(\\alpha _++\\alpha _-\\right)u_{A,B}\\Rightarrow x_{A,B}=a u_{A,B},\\\\\\frac{y_{A,B}}{\\sqrt{1-\\frac{\\Delta \\sigma }{\\bar{\\sigma }}}}&=&\\left(\\alpha _+-\\alpha _-\\right)v_{A,B}\\Rightarrow y_{A,B}=a v_{A,B}.$"
],
[
"Polar coordinates in the $u,v$ -plane",
"Switching to polar coordinates in the $u,v$ -plane $\\rho &=&\\sqrt{u^2+v^2},\\\\\\phi &=&\\tan ^{-1}\\frac{v}{u},$ gives $\\frac{\\partial }{\\partial u}&=&\\cos \\phi \\frac{\\partial }{\\partial \\rho }-\\frac{\\sin \\phi }{\\rho }\\frac{\\partial }{\\partial \\phi },\\\\\\frac{\\partial }{\\partial v}&=&\\sin \\phi \\frac{\\partial }{\\partial \\rho }+\\frac{\\cos \\phi }{\\rho }\\frac{\\partial }{\\partial \\phi }.$ Therefore, the derivatives of the boundary function are $\\frac{\\partial D}{\\partial u}&=&-\\cos \\phi \\delta (\\rho -1),\\\\\\frac{\\partial D}{\\partial v}&=&-\\sin \\phi \\delta (\\rho -1),$ and the differential equation (REF ) becomes $&&-\\left(\\frac{\\partial }{\\partial \\rho }\\left(D\\rho \\frac{\\partial V}{\\partial \\rho }\\right)+\\frac{D}{\\rho }\\frac{\\partial ^2 V}{\\partial \\phi }\\right)+\\frac{\\sigma _H\\delta (\\rho -1)}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\partial V}{\\partial \\phi }=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\delta (\\rho -1)\\left(\\delta (\\phi -\\theta _A)-\\delta (\\phi -\\theta _B)\\right).$"
],
[
"Homogeneous solution and the boundary conditions",
"A general solution of Eq.",
"(REF ) for $\\rho <1$ where the terms containing $\\delta (\\rho -1)$ vanish can be written as $V(\\rho ,\\phi )&=&\\sum _{m=1}^{\\infty }\\left(A_{|m|}\\left(\\frac{\\rho ^m}{\\Omega ^m}+\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\cos m\\phi +B_{|m|}\\left(\\frac{\\rho ^m}{\\Omega ^m}-\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\sin m\\phi \\right).$ This form satisfies the homogeneous differential equation and is continuous and differentiable across the line cut joining the foci.",
"To see this, notice that the points on the circle of radius $\\Omega $ in the $u,v$ -plane map onto the line segment joining the foci $X\\in (-2\\alpha _+\\Omega ,2\\alpha _+\\Omega )$ , $Y=0$ , as we saw in the Eq.REF .",
"Therefore, the points on the inner circle in the $u,v$ plane which are related by the mirror reflection about the $v=0$ axis should be identified as the same points.",
"In other words, $\\rho =\\Omega $ and $\\phi $ , and $\\rho =\\Omega $ and $-\\phi $ map onto the same physical point in the $X,Y$ and therefore $x,y$ plane.",
"We therefore want the potential at $\\Omega ^+$ and $\\phi $ to either be the same at $-\\phi $ which is accomplished by $\\left(\\frac{\\rho ^m}{\\Omega ^m}+\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\cos m\\phi $ , or we want it to vanish at $\\Omega ^+$ with a continuous slope.",
"Vanishing at $\\Omega $ is accomplished by $\\frac{\\rho ^m}{\\Omega ^m}-\\frac{\\Omega ^m}{\\rho ^{m}}$ , and the reason why only $\\sin m\\phi $ can multiply it is that multiplying it by $\\cos m\\phi $ would introduce a cusp across the line segment.",
"Integrating both sides of Eq.REF over an infinitesimal interval straddling $\\rho =1$ gives the boundary condition $\\frac{\\partial V}{\\partial \\rho }|_{\\rho =1}+\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\partial V}{\\partial \\phi }|_{\\rho =1}=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\left(\\delta (\\phi -\\theta _A)-\\delta (\\phi -\\theta _B)\\right).$ Substituting Eq.REF into the above results in $\\frac{\\partial V(\\rho ,\\phi )}{\\partial \\rho }|_{\\rho =1}&=&\\sum _{m=1}^{\\infty }\\left(mA_{|m|}\\left(\\frac{1}{\\Omega ^m}-\\Omega ^m\\right)\\cos m\\phi +mB_{|m|}\\left(\\frac{1}{\\Omega ^m}+\\Omega ^m\\right)\\sin m\\phi \\right),\\\\\\frac{\\partial V(\\rho ,\\phi )}{\\partial \\phi }|_{\\rho =1}&=&\\sum _{m=1}^{\\infty }\\left(-mA_{|m|}\\left(\\frac{1}{\\Omega ^m}+\\Omega ^m\\right)\\sin m\\phi +mB_{|m|}\\left(\\frac{1}{\\Omega ^m}-\\Omega ^m\\right)\\cos m\\phi \\right),$ and the differential equation (REF ) becomes $&&\\sum _{m=1}^{\\infty }\\left(mA_{|m|}\\left(\\frac{1}{\\Omega ^m}-\\Omega ^m\\right)\\cos m\\phi +mB_{|m|}\\left(\\frac{1}{\\Omega ^m}+\\Omega ^m\\right)\\sin m\\phi \\right)+\\nonumber \\\\&+&\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\sum _{m=1}^{\\infty }\\left(-mA_{|m|}\\left(\\frac{1}{\\Omega ^m}+\\Omega ^m\\right)\\sin m\\phi +mB_{|m|}\\left(\\frac{1}{\\Omega ^m}-\\Omega ^m\\right)\\cos m\\phi \\right)=\\nonumber \\\\&&\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\sum _{m=1}^{\\infty }\\left(\\cos m\\phi \\left(\\cos m\\theta _{A}-\\cos m\\theta _{B}\\right)+\\sin m\\phi \\left(\\sin m\\theta _{A}-\\sin m\\theta _{B}\\right)\\right),$ where the following identity was used for the right hand side $\\delta \\left(\\phi -\\theta _{A,B}\\right)&=&\\frac{1}{2\\pi }\\sum _{m=-\\infty }^{\\infty }e^{im\\phi }e^{-im\\theta _{A,B}}\\\\&=&\\frac{1}{2\\pi }+\\frac{1}{\\pi }\\sum _{m=1}^{\\infty }\\left(\\cos m\\phi \\cos m\\theta _{A,B}+\\sin m\\phi \\sin m\\theta _{A,B}\\right).$ Matching the coefficients of $\\cos m\\phi $ and $\\sin m\\phi $ and solving for $A_{|m|}$ and $B_{|m|}$ gives $&&A_{|m|}=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\frac{1}{1+\\frac{\\sigma ^2_H}{\\sigma _+\\sigma _-}}\\frac{1}{m}\\left(\\frac{\\cos m\\theta _{A}-\\cos m\\theta _{B}}{\\Omega ^{-m}-\\Omega ^m}-\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\sin m\\theta _{A}-\\sin m\\theta _{B}}{\\Omega ^{-m}+\\Omega ^m}\\right)\\\\&&B_{|m|}=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\frac{1}{1+\\frac{\\sigma ^2_H}{\\sigma _+\\sigma _-}}\\frac{1}{m}\\left(\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\cos m\\theta _{A}-\\cos m\\theta _{B}}{\\Omega ^{-m}-\\Omega ^m}+\\frac{\\sin m\\theta _{A}-\\sin m\\theta _{B}}{\\Omega ^{-m}+\\Omega ^m}\\right).$ Thus, $&&V(\\rho ,\\phi )=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\frac{1}{1+\\frac{\\sigma ^2_H}{\\sigma _+\\sigma _-}}\\sum _{m=1}^{\\infty }\\frac{1}{m}\\left(\\frac{\\cos m\\theta _{A}-\\cos m\\theta _{B}}{\\Omega ^{-m}-\\Omega ^m}-\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\sin m\\theta _{A}-\\sin m\\theta _{B}}{\\Omega ^{-m}+\\Omega ^m}\\right)\\left(\\frac{\\rho ^m}{\\Omega ^m}+\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\cos m\\phi \\nonumber \\\\&+&\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\frac{1}{1+\\frac{\\sigma ^2_H}{\\sigma _+\\sigma _-}}\\sum _{m=1}^{\\infty }\\frac{1}{m}\\left(\\frac{\\sin m\\theta _{A}-\\sin m\\theta _{B}}{\\Omega ^{-m}+\\Omega ^m}+\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{\\cos m\\theta _{A}-\\cos m\\theta _{B}}{\\Omega ^{-m}-\\Omega ^m}\\right)\\left(\\frac{\\rho ^m}{\\Omega ^m}-\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\sin m\\phi \\nonumber \\\\&-&(A\\rightarrow B).$"
],
[
"Summing over the angular momenta",
"The sum over $m$ converges slowly.",
"In order to convert it into a rapidly convergent sum, we first Taylor expand the denominators involving $\\Omega ^m$ and $\\Omega ^{-m}$ , in powers of $\\Omega $ as $&&V(\\rho ,\\phi )=\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\frac{1}{1+\\frac{\\sigma ^2_H}{\\sigma _+\\sigma _-}}\\sum _{n=0}^{\\infty }\\sum _{m=1}^{\\infty }\\frac{\\left(\\Omega ^{1+2n}\\right)^m}{m}\\left(\\cos m\\theta _{A}-(-1)^n\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\sin m\\theta _{A}\\right)\\left(\\frac{\\rho ^m}{\\Omega ^m}+\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\cos m\\phi \\nonumber \\\\&+&\\frac{I}{\\sqrt{\\sigma _+\\sigma _-}}\\frac{1}{\\pi }\\frac{1}{1+\\frac{\\sigma ^2_H}{\\sigma _+\\sigma _-}}\\sum _{n=0}^{\\infty }\\sum _{m=1}^{\\infty }\\frac{\\left(\\Omega ^{1+2n}\\right)^m}{m}\\left((-1)^n\\sin m\\theta _{A}+\\frac{\\sigma _H}{\\sqrt{\\sigma _+\\sigma _-}}\\cos m\\theta _{A}\\right)\\left(\\frac{\\rho ^m}{\\Omega ^m}-\\frac{\\Omega ^m}{\\rho ^{m}}\\right)\\sin m\\phi \\nonumber \\\\&-&(A\\rightarrow B).$ Then the resulting sum over $m$ is related to the geometric series by integration, and since $\\Omega <1$ , the sum over $n$ will be rapidly convergent.",
"Therefore, for $C=A,B$ , we have $\\sum _{m=1}^\\infty e^{im\\theta _C}e^{im\\phi }\\frac{\\Omega ^{m(1+2n)}}{m}\\left(\\frac{\\rho ^m}{\\Omega ^m}\\pm \\frac{\\Omega ^m}{\\rho ^m} \\right)&=&-\\ln \\left(1-e^{i\\theta _C}\\frac{w}{\\Omega }\\Omega ^{1+2n}\\right)\\mp \\ln \\left(1-e^{i\\theta _C}\\frac{\\Omega }{\\bar{w}}\\Omega ^{1+2n}\\right).$ Adding longitudinal and Hall contributions finally gives Eq.",
"(REF )."
],
[
"Discussion",
"Because the differential equation is linear, it is straightforward to generalize the expression derived above to the case with multiple sources and drains.",
"In such a case the continuity equation reads $\\nabla \\cdot \\mathbf {j}=\\sum _{j=1}^{n_S}I^{S}_j\\delta (\\mathbf {r}-\\mathbf {r}_{A,j})-\\sum _{j=1}^{n_D}I^{D}_j\\delta (\\mathbf {r}-\\mathbf {r}_{B,j}),$ where $n_{S}$ is the number of point sources and $n_D$ is the number of point drains, and $\\sum _{j=1}^{n_S}I^{S}_j=\\sum _{j=1}^{n_D}I^{D}_j=I$ .",
"The resulting expression is $&&V(x,y;\\lbrace \\mathbf {r}_{A,j}\\rbrace ,\\lbrace \\mathbf {r}_{B,j}\\rbrace )=\\nonumber \\\\&&\\frac{1}{\\pi }\\frac{\\sqrt{\\sigma _+\\sigma _-}}{\\sigma _+\\sigma _-+\\sigma ^2_H}\\sum _{j=1}^{n_D}I^{D}_j\\left(\\sum _{n=0,2,4,\\ldots }^{\\infty }\\ln \\left|1+e^{-2i\\theta _{B,j}}\\Omega ^{2+4n}-e^{-i\\theta _{B,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|+\\sum _{n=1,3,5,\\ldots }^{\\infty }\\ln \\left|1+e^{2i\\theta _{B,j}}\\Omega ^{2+4n}-e^{i\\theta _{B,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|\\right)\\nonumber \\\\&-&\\frac{1}{\\pi }\\frac{\\sqrt{\\sigma _+\\sigma _-}}{\\sigma _+\\sigma _-+\\sigma ^2_H}\\sum _{j=1}^{n_S}I^{S}_j\\left(\\sum _{n=0,2,4,\\ldots }^{\\infty }\\ln \\left|1+e^{-2i\\theta _{A,j}}\\Omega ^{2+4n}-e^{-i\\theta _{A,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|+\\sum _{n=1,3,5,\\ldots }^{\\infty }\\ln \\left|1+e^{2i\\theta _{A,j}}\\Omega ^{2+4n}-e^{i\\theta _{A,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right|\\right)\\nonumber \\\\&+&\\frac{1}{\\pi }\\frac{\\sigma _H}{\\sigma _+\\sigma _-+\\sigma ^2_H}\\left(\\sum _{n=0,2,4,\\ldots }^{\\infty }\\sum _{j=1}^{n_D}I^{D}_j\\arg \\left(1+e^{-2i\\theta _{B,j}}\\Omega ^{2+4n}-e^{-i\\theta _{B,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)-\\sum _{j=1}^{n_S}I^{S}_j\\arg \\left(1+e^{-2i\\theta _{A,j}}\\Omega ^{2+4n}-e^{-i\\theta _{A,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)\\right.\\nonumber \\\\&+&\\left.\\sum _{n=1,3,5,\\ldots }^{\\infty }\\sum _{j=1}^{n_D}I^{D}_j\\arg \\left(1+e^{2i\\theta _{B,j}}\\Omega ^{2+4n}-e^{i\\theta _{B,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)-\\sum _{j=1}^{n_S}I^{S}_j\\arg \\left(1+e^{2i\\theta _{A,j}}\\Omega ^{2+4n}-e^{i\\theta _{A,j}}\\frac{Z}{\\alpha _+}\\Omega ^{2n}\\right)\\right).$ The expression for extended source/drain can be found by treating $I^{S/D}_j$ as infinitesimal and then converting the Riemann sum into an integral.",
"The obtained expression can now be used to fit measurements with multiple voltage probes for arbitrary current source and drain placed on the perimeter of the disk.",
"I wish to express sincere gratitude to Prof. J.I.A.",
"Li for sharing their unpublished results and for his encouragement to publish this article.",
"I would also like to thank Prof. Jian Kang for going over the calculations in the manuscript.",
"O. V. is supported by NSF DMR-1916958 and is partially funded by the Gordon and Betty Moore Foundation's EPiQS Initiative Grant GBMF11070, National High Magnetic Field Laboratory through NSF Grant No.",
"DMR-1157490 and the State of Florida."
]
] | 2209.08208 |
[
[
"Engineering Chemo-Mechanical Properties of Zn Surfaces via Alucone\n Coating"
],
[
"Abstract Aqueous zinc (Zn)-ion batteries (AZIB) are promising candidates for the next-generation energy store systems due to their high capacity and low cost.",
"Despite their nominal performance, Zn anodes tend to rapidly develop dendrite and fracture, leading to substantial capacity loss and cycling stability failure.",
"Well-controlled coating using organic-inorganic hybrid molecules is highly promising to substantially improve their chemo-mechanical stability without compromising their performance.",
"We herein present a critical assessment of the chemical and mechanical stability of alucone-coated Zn surfaces using first-principles simulations.",
"Negative adsorption energies indicate strong cohesive strengths between alucone and the selected Zn surfaces.",
"Energetically favorable alucone coatings are further verified by charge transfer at interfaces as seen through Bader charge analysis.",
"Negative surface stress profiles at alucone coated interface are mostly responsible for surface reconstruction.",
"The contributions of surface elastic constants are dependent on the selection of slip planes and the thickness of the thin film.",
"By considering plane stress conditions, we calculate the mechanical properties which indicate the ductility of the alucone-coated basal thin film."
],
[
"Introduction",
"Fossil-fuel consumption in transportation technologies is the second largest source of $\\mathrm {CO_{2}}$ emission, and expected to increase by 28% by 2040 [1].",
"Greener energy applications highly depends on the development of well-performing, reliable, and cost-effective energy storage systems such as batteries, fuel cell, and supercapacitors [2], [3], [4].",
"Regardless of any battery design, electrodes are the main components that determine the bottleneck of battery capacity, power output, cyclic stability, and lifetimes of energy storage systems [5], [6].",
"Thus, fabrication and surface engineering of electrode materials are the most vital aspect of the next-generation energy storage systems [5].",
"Aqueous zinc-ion batteries (AZIBs) have attracted considerable interest due to their exceptional storage capacity of $\\sim 5854$ mAh$\\cdot $ cm$^{-1}$ ($\\sim 820$ mAh$\\cdot $ g$^{-1}$ ) compared to conventional Li batteries with a storage capacity of $\\sim 2042$ mAh$\\cdot $ cm$^{-1}$ [7].",
"AZIBs generally consist of a Zn anode and an (in)organic cathodes such as MnO$_{2}$ , an aqueous electrolyte (mildly acidic pH) such as ZnSO$_{4}$ , and a separator such as a ceramic separator[8].",
"Zn is non-toxic and abundant in Earth's crust [6].",
"It can be oxidized to Zn$^{2+}$ during stripping (discharging) without forming any intermediate phases [9] under mildly acidic conditions (pH = $4-6$ ) which promote the reversible Zn stripping/plating (discharging/charging) process [10], [11], [12].",
"It also has a redox potential is $-0.76$ V for Zn, for which the hydrogen-evolution reaction provides an over-potential suitable for effective battery operation [10].",
"Thus, AZIBs are highly favorable for flexible and wearable batteries as they are safe, sustainability, cost-effective, environmentally friendly, and provide reliable power output [13], [14].",
"Despite their highly promising premise, AZIBs suffer two major drawbacks, limiting their widespread commercial applications[15].",
"The first drawback is stability issues in $\\mathrm {MnO_{2}}$ cathodes.",
"The second one is dendrite formation on the Zn anode surface due to irregular plating/stripping of Zn-ions during the charging/discharging cycles.",
"This commonly leads to a short circuit between the cathode and anode [16].",
"A suggested strategy to overcome rapid dendrite formation in Zn anodes is nano-structured coating [11].",
"For this purpose, atomic layer deposition (ALD) [17] and molecular layer deposition (MLD) [18] are the state-of-art techniques to achieve well-controlled thickness and structure [19], superior conformity [20], [21], and uniformity [22].",
"It has been shown that ALD and MLD are more reliable for surface engineering compared to more conventional coating techniques such as physical vapor deposition (PVD) or chemical vapor deposition (CVD) [19], [23].",
"Furthermore, ALD/MLD allow tuning compositions easily where the deposition temperatures are relatively low [19].",
"Thus, they became quite popular in the field of surface engineering for energy storage systems, which require high-level accuracy and tunability in surface construction [24], [25].",
"While ALD is mostly limited to inorganic coating agents, MLD provides more diverse coating agents such as hybrid organic-inorganic coating at the expense of a more complex coating process [19].",
"However, MLD opens up virtually limitless possibilities in terms of coating agents, enabling superior control over thermodynamic and mechanical properties at interfaces [26].",
"There have been previous works on the nano-structured coating of Zn anodes such as the modified polyamide coating [27], the ultra-thin TiO$_2$ coating using ALD [28] and the drop-casting of nanoporous CaCO$_3$ [29].",
"Despite improving mechanical stability, these ceramic and polymeric coatings severely hinder the performance of Zn anodes by significantly increasing surface resistance, leading to poor Zn ions diffusion [15].",
"With that in mind, an effective coating agent is expected to satisfy the four major conditions: (i) restricting free diffusion of Zn ions by electrostatic interaction or by physical restriction, (ii) facilitating ionic conduction only and restricting electronic conduction, (iii) keeping the Zn electrode stable in an electrolyte, (iv) maintaining sufficient toughness and rigidity to adjust to the volume change throughout the charging/discharging process.",
"Thus, it is quite challenging to select and deposit an effective and cost-efficient coating material that suppresses the dendrite formation while maintaining battery performance [15].",
"Aluminum alkoxide (alucone) [30] is a promising candidate which has been already used stabilize sulfur (S) cathodes and silicon (Si) anodes in Li–S[31] and Li–ion batteries [32], respectively.",
"It is produced by chemical reactions of organic alcohols and trimethylaluminum (TMA) precursors during the MLD process.",
"Alucone possibly suppresses dendrite formation while not hindering charge transfer in charging/discharging cycles by providing flexible gas-diffusion barriers, superior corrosion resistance, and wettability for Zn anodes [33], [34].",
"Thus, it is crucial to understand the effects of alucone coating on the thermodynamic, chemical, and mechanical properties of Zn anodes for effective and safe AZIB applications.",
"However, it is a challenging task to experimentally study underlying mechanisms as well as limited only to a small number of coating agents and samples.",
"First-principles simulations offer powerful and feasible tools to study surface chemistry and properties of complex structures [35], [36].",
"Tran et al.",
"have calculated surface energies of 70 elements using high-throughput first-principles approaches [37].",
"Furthermore, they are also highly accurate to capture charge transfer at the interface [38].",
"For instance, Lawson et al.",
"have studied the interaction between Al$_2$ O$_3$ , HfO$_2$ and MgO surfaces and MoS$_2$ coating during the first cycle of ALD.",
"In this work, we present an accurate first-principles investigation of thermal and chemical stability and mechanical properties of alucone-coated Zn surfaces.",
"Crystal and electronic structures are optimized for the (0 0 0 1) and (1 -1 0 0) surfaces of the hexagonal closed-packed (HCP) Zn with and without single-layer alucone to calculate adsorption energies and analyze charge transfer at the interfaces.",
"It is shown that the alucone-coated surfaces are energetically and chemically stable.",
"Using the pristine bulk Zn, its surfaces, and thin films, we approximate the elastic properties at the interfaces of the alucone-coated surfaces.",
"The mechanical properties analysis indicates the ductility of alucone-coated basal thin film however these properties are direction-dependent.",
"Finally, we provide crucial figures-of-merits (FOMs) for substrate + coating selection based on their individual electric properties."
],
[
"Theoretical and Computational Methodology",
"Kohn-Sham density-functional theory (KS-DFT) [39], [40] using semi-local exchange-correlation functionals [40] provides feasible and accurate tools for studying ground-sate properties of solids.",
"Beyond structural and electronic properties, the approximate KS-DFT is also suitable to expediently calculate elastic properties using Hooke's law.",
"In this work, its plane-wave-based implementation within the Vienna Ab-Initio Simulation Package (VASP) [41], [42], [43] using the projector augmented-wave (PAW) pseudo-potentials [44].",
"The exchange-correlation functional was expressed by the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [45].",
"The total energy is minimized in the DFT calculation for the occupation of Kohn-Sham's self-consistent scheme.",
"In this study, partial wave occupancies were computed by Gaussian smearing, and the width of smearing was $\\sigma $ =0.026 eV which corresponds to 300 K. Electronic energies were computed with the tolerance of $\\mathrm {10^{-6}}$ eV using a self-consistent-field (SCF).",
"One of the key features of a crystal/surface is to determine a relaxed geometry, that is to find the geometry with the lowest energy using the geometry relaxation test.",
"As a result, the unit cell of the crystal goes through relaxation in all degrees of freedom (ionic positions, cell shape, and volume) to find the lowest energy model.",
"The conventional convergence criterion for kinetic-energy cutoff $\\mathrm {{\\it {E}}_{cut}}$ is to attain a change of less than 1 meV/atom in total energy for raising the $\\mathrm {{\\it {E}}_{cut}}$ .",
"The PAW pseudopotential was exercised to report the core electrons whereas the valence orbitals were illustrated using a cutoff kinetic energy of 360 eV with plane-wave basis set as it fulfills the convergence criterion.",
"The Zn unit cell was optimized with a 15 x 15 x 15 k-points mesh to formulate the Zn surface after the convergence test, such as the entropy< 1 meV per atom.",
"A Zn (0001) surface with six Zn layers was constructed using the optimized Zn unit cell (Supporting Information (SI) Fig.",
"REF ).",
"A 15 x 15 x 1 Monkhorst–Pack k-points mesh for the Zn surface model was employed to produce the plane wave basis set.",
"Figure: Schematic illustration of the molecular layer deposition of alucone on a substrate.In Fig.",
"REF , the MLD of a single-layer alucone on a substrate is schematically shown.",
"It is a two-step process for depositing single layer of alucone [46].",
"Alucone is the general name of polymeric aluminum alkoxide compounds which have a general sequence of as $\\mathrm {\\cdots Al-O-R-O-Al \\cdots }$ .",
"The radical R is an organic molecule.",
"Alucone is deposited on a substrate by first pulsing TMA on to the hydroxylated substrate, purging CH$_4$ as by-product.",
"At the second stage, the activated surface reacts with ethylene glycol (EG) while purging additional CH$_4$ .",
"These two step deposition process can be chemically expressed as [47], [46] -OH* + Al(CH3)3 = -OAl(CH3)*2+CH4 -AlCH*3 + OHCH2CH2OH = AlOCH2CH2OH*+CH4, where the asterisks indicate the surface species.",
"By repeating this two-step process, the thickness of alucone coating can be linearly increased.",
"In this work, we only focus on the Zn surfaces which are fully coated with a single layer of alucone."
],
[
"Van der Waals Interactions",
"London dispersion forces, generally mentioned as van der Waals (vdW) forces, develops a weak instantaneous interaction that varies as $\\sim 1/r^{6}$ decay [48].",
"These interactions form due to the charge fluctuation between dipoles.",
"However, the contribution of vdW interactions is absent in standard DFT exchange-correlation (XC) functionals.",
"This limitation puts an error bar on the calculation of the adsorption and cohesive energy of weakly bonded molecules on surfaces.",
"In this study, we used DFT-D2 semi-empirical correction method suggested by Grimme [49], which adds a correction term ($\\mathrm {{\\it {E}}_{disp}}$ ) to the total energy after every self-consistent cycle[50].",
"E = EKS+Edisp Edisp = -S6iN-1 j=i+1NC6ij(Rij)6 Fdamp(Rij) Fdamp(Rij)=11+ed(RijR0i+R0j-1) C6ij = C6iC6j where ${E}_{\\textrm {disp}}$ is dispersion term, $F_{\\textrm {damp}}$ is damping function, $ C_{6}^{ij}$ is dispersion coefficient, $S_{6}$ is the global scaling factor, $R_{0}^{i}$ is the vdW radius of atom i, $R^{ij}$ is the distance between atom number i and j, $C_{6}^{i}$ is the atomic parameter, and d is a damping parameter."
],
[
"Energetic and Chemical Stability",
"Surface energy ($\\gamma $ ) is a central quantity due to its analytical relation to adsorption energy and surface elastic properties.",
"It is defined as the energy required to separate a bulk solid into two pieces.",
"Despite its simple definition, it is quite challenging to obtain $\\gamma $ within experimental methods.",
"For instance, it generally requires estimating surface tension for metals at their melting temperatures [51], [52].",
"Thus, it is generally calculated using first-principles simulations.",
"Conventionally, it is given by [53] = 12A [Eslab-N Ebulk], where $E_\\textrm {slab}$ is the ground-state energy of the slab with the surface area of $A$ and consisting of $N$ atoms, and $E_\\textrm {bulk}$ is the ground-state energy of the bulk per atom.",
"In practice, $E_\\textrm {slab}$ and $E_\\textrm {bulk}$ are obtained through separate simulations which may lead to some convergence problems [53], [54].",
"Thus, there are some modified expressions available [54].",
"In this work, Eq.",
"(REF ) works without any convergence problem and with similar accuracy with more complex approaches (see SI for further details and comparison of various approaches).",
"While the surface energy is crucial to assess the energetic stability of a pristine surface, the adsorption energy ($E_\\mathrm {ads}$ ) is used to assess whether an absorbate (molecule) is likely to be chemically attached to an adsorbent (surface), given by [55], [56]: Eads = Esurf+mol - (Esurf+Emol), where $E_\\mathrm {surf+mol}$ and $E_\\mathrm {surf}$ are the total energy of the slab with and without the absorbate, respectively.",
"$E_\\mathrm {mol}$ is the total energy of the absorbate in the vacuum.",
"In practice, a surface has to be constructed as a thick slab with large vacuums on both sides as a semi-infinite structure is not defined in plane-wave-based implementations of the approximate KS-DFT due to mismatching boundary conditions [53].",
"We defined the model with a periodic and free boundary conditions for [11-20], [2-1-10] and [0001] directions, respectively, which is shown in Fig.",
"REF .",
"For prismatic systems, periodic boundary conditions are in (1/3)[11-20] and [0001] direction and free boundary in [1-100] direction.",
"Figure: Thin film model.In the case of full-coverage coating, Eq.",
"(REF ) can be expressed either per absorbate or per unit surface area.",
"Eq.",
"(REF ) serves as a FOM for the energetic stability of a coated surface.",
"Addition to that, the charge transfer between them indicates the chemical stability due to chemical bonding.",
"Although, the charge transfer is far more complex, the charge-density difference given by [57] = surf+mol - (surf+mol), where $\\rho _\\mathrm {surf+mol}$ and $\\rho _\\mathrm {surf}$ are the charge densities of the coated and pristine surfaces, respectively, and $\\rho _\\mathrm {mol}$ is the charge density of the molecular thin film.",
"The three charge densities are required to be computed on appropriately matching and relative position at a common coordinate system."
],
[
"Surface Mechanical Properties",
"The surface-stress profile is crucial to assess the mechanical stability of a surface.",
"The strain-dependent surface stress is commonly given by [58] () = + , where $\\epsilon $ is the infinitesimal strain.",
"On the other hand, the total energy of the deformed surface with an area $A$ due to the infinitesimal strain can be defined as ES=A().",
"At the zero-strain condition ($\\tau (0)=\\tau ^0$ ), the components of the surface stress tensor given in Eq.",
"(REF ) can be expressed as ij0 = [1AESij]=0 for a small infinitesimal strain of $\\epsilon $ .",
"Usually, we refer to stress at zero strain condition ($\\tau ^{0}_{ij}$ ) as surface stress.",
"Using Eq.",
"(REF ), the fourth-order surface elastic tensor is defined as [58] CSijkl = ijkl|=0 =[2ikjl+ijkl+2 ij kl]=0.",
"In practice, $\\gamma $ is calculated for a small set of $\\epsilon $ , and it requires accurate interpolation to compute the partial derivations.",
"We used three interpolation techniques, namely the finite-difference method (FDM) [59], the local-maximum entropy(LME) [60] and the higher-order LME scheme (HOLMES) [61] to ensure accuracy and compare their respective performance (see SI for further details).",
"The surface elastic constants can be notably dissimilar from their corresponding bulk values [62] and the elastic response of nano-structures relies on it.",
"Furthermore, it is difficult to compute the surface elastic constants experimentally [58], [63].",
"Streitz et al.",
"first mentioned the impact of surfaces on the multilayers and biaxial modulus of thin films using a scaling relation but did not fully formulate the surface elastic constants [63].",
"To illustrate the modification in elastic constants because of the exposed surface (thin-film approximation (TFA)), surface elastic constants are added with second-order bulk elastic constants considering the plane-stress condition [64].",
"Using Hooke's law for orthotropic materials, the plane-stress condition can be expressed as [64] 1 2 3= C11plane C12plane C13plane C21plane C22plane C23plane C31plane C32plane C33plane 1 2 3 , which provide the relations between the plane stresses and the bulk elastic constants for an orthotropic elastic body, given by [64], [65], [66] $C_{11}^{\\textrm {plane}} = (C_{11}-\\frac{C_{13}^2}{C_{33}}) \\nonumber \\\\C_{22}^{\\textrm {plane}} = (C_{22}-\\frac{C_{23}^2}{C_{33}}) \\nonumber \\\\C_{12}^{\\textrm {plane}} = C_{21}^{\\textrm {plane}} = (C_{12}-\\frac{C_{12}C_{13}}{C_{33}}) \\nonumber \\\\C_{33}^{\\textrm {plane}} = C_{66} \\nonumber \\\\C_{13}^{\\textrm {plane}} = C_{23}^{\\textrm {plane}} = C_{31}^{\\textrm {plane}} = C_{32}^{\\textrm {plane}} = 0,$ where $C_{ij}$ are components of the elastic tensor of the bulk crystals using Voigt's notation.",
"Finally, the second-order effective elastic constants can be given by [67], [64] Cijfilm = Cijplane+ 2LzCSij, where $L_z$ is the film thickness."
],
[
"Figures of Merits for Mechanical Performance",
"Elastic constants are microscopic quantities that are well-defined.",
"Other technologically relevant mechanical quantities can serve as a FOM to assess the quality of the coated surface.",
"It is particularly crucial to determine the hardness, and fracture toughness of nanostructures used in devices such as batteries to ensure safety and durability.",
"The Vickers hardness ($H_\\mathrm {V}$ ) is the industry-standard FOM to assess wear resistance.",
"There is no robust theoretical model as its measurement is particularly dependent on morphology, impurity of samples, choice of indenter, and direction of acting forces relative to crystallographic orientation [68].",
"However, there are some semi-empirical approximations such as [69], [70] HVTeter=0.151G, and HVTian = 0.92(G/B)1.137G0.708, where $B$ and $G$ are the bulk and shear moduli, respectively.",
"While the former tends to overestimate, the latter tends to underestimate.",
"Thus, we used their arithmetic averaging such as $H_\\mathrm {V} = 0.5 (H_\\mathrm {V}^\\mathrm {Teter} + H_\\mathrm {V}^\\mathrm {Tian}$ ).",
"Another crucial FOM is the fracture toughness ($K_\\mathrm {IC}$ ), which restrains the crack propagating [71], [72], [73].",
"Niu et al.",
"has proposed the semi-empirical formula, given by [73], [74] KIC = V016G(BG)12, where $V_0$ is the volume of the unstrained structure.",
"The development of stress in the thin-film deposition process is one of the least explored fundamental aspects of managing the dendrite growth problem [75].",
"Micro-structural evolution phenomena in materials such as thin film deposition are stress-driven.",
"Therefore, it is important to check whether notable stress exists during alucone deposition and if it is the source of Zn dendrite growth or surface wrinkling.",
"The minimum membrane strain for the beginning of wrinkling is specified by [75] m = -14[3Esub(1-tf2)Etf(1-sub2)]23 where $E_\\mathrm {sub}$ and $E_\\mathrm {tf}$ are the elastic moduli of the substrate and thin film, respectively, and $\\nu _\\mathrm {sub}$ and $\\nu _\\mathrm {tf}$ are Poisson's ratios of the substrate and thin film."
],
[
"Results and Discussion",
"Second-order elastic properties of Bulk Zn have been computed and verified with experimental data as shown in Table REF .",
"For Zn thin film, we choose a model of four fixed layers and two free layers which have been justified by calculating surface layer relaxation as shown in Fig.",
"REF and Fig.",
"REF ."
],
[
"Structural Optimization of Alucone-Coated Zn Surface",
"The Al-O bond length varies from 1.7 - 1.9 Å depending on the bonding configuration.",
"The C-O, C-H, and C-C bond lengths are approximately 1.1 Å, 1.1 Å, and 1.4 Å, respectively.",
"Through a series of extensive convergence tests (5, 10, 20, 30 Å), it was verified that 10 Å vacuum were adequate to confirm that the surface energies were converged with variation less than 1 $\\mathrm {meV/\\mathring{A}^2}$ along with avoiding inter-slab interaction.",
"The Van der Waals energy was approximately 5 percent with respect to the total ground state energy (Fig.",
"REF ).",
"Figure: (a) Optimized geometry of oxygen deposition in Zn (0001) surface using binding sites of O atom in the middle of 6 Zn atoms (hollow-hexagonal close-packed (hcp)), (b) DFT optimized alucone thin film structure on hydroxylated Zn (0001) surface, and (c) (c) DFT optimized alucone thin film structure on hydroxylated Zn (1-100) surface."
],
[
"Chemical Stability Analysis",
"From Table REF , all three adsorption energies are negative for the adsorption reactions mentioned in equation REF .",
"Znslab+ (n2)O2 = Znslab-nO(n=1) Znslab+ (n2)H2 + (n2)O2 = Znslab-nOH (n=1) 6Znslab-OH+ 6Al(CH3)3 + 9OHCH2CH2OH = 6Znslab-O-AlOCH2CH2OH+15CH4 The negative adsorption energies for Zn-terminated surfaces indicate an exothermic process and the higher the adsorption energy indicates stronger interactions between adsorbates and the adsorbent.",
"Therefore, the adsorption of alucone on hydroxylated Zn (0001) surface is thermodynamically stable.",
"Furthermore, the range of the adsorption energies (-2.6 eV) indicate strong chemical adsorption of alucone group on the surface [76].",
"Table: Adsorption energies of dffierent absorbate in Zn absorbant.Figure: Differential charge density for the hydroxylated Zn with alucone.Here, a, b and c are along [11-20], [2-1-10] and [0001] direction, respectively.",
"(a) The yellow and blue regions in the isosurface show a gain and loss of electrons, respectively.",
"(b) A slice along the ab plane where the red regions indicate the electron gains in oxygen on the Zn surface.The plotted charge density difference indicates the covalent nature that develops in between the absorbate and hydroxylated substrate.Fig.",
"REF maps the position of the redistributed electron densities for the MLD alucone adsorption on the hydroxylated Zn (0001) surface.",
"Here, the loss of electrons is presented using blue zones whereas yellow zones point to electrons gain.",
"Due to the adsorption of alucone, the hydroxylated Zn surface displays a charge density increment (yellow isosurfaces) at the Zn-O bonds and a charge density reduction (blue isosurfaces) between the Al-O bonds (Fig.",
"REF (a)).",
"From the 2D slice along the ab plane (along with O atoms of the hydroxylated Zn surface) in Fig.",
"REF (b), the hydroxylated Zn surface displays a sharp difference in charge density.",
"Due to alucone deposition, there is a drastic rise in the charge density surrounding the hydroxylated surface O atoms (red regions).",
"This sharp charge density increment indicates the covalent bonding behavior or chemisorption during the MLD deposition process.",
"Table: Bader net atomic charges.Unlike charge-density difference, which is a qualitative analysis technique, Bader charge analysis can present the extent of the chemical reaction between atoms (hydroxyl groups on precursor decomposition) quantitatively [77].",
"Table REF summarizes the net Bader atomic charges for each atom.",
"From Table REF , it is logical that chemisorption dominates the adsorption process due to charge transfer between absorbate and absorbent.",
"The sum of the Bader charges of O and H atom has to equal that of the central Al atom for complete physisorption.",
"In that case, only weak van der Waals forces could be applicable between the precursor and the surface.",
"However, for the alucone coated hydroxylated Zn (0001) surface, we observed the opposite trend here.",
"The Al atom has more positive (loss of electron) net atomic Bader charges after adsorption while the charges increase for O atoms.",
"Therefore, the Al-hydroxylated Zn surface bonds get stronger which indicates chemisorption due to the changes in the surface electronic properties."
],
[
"Mechanical Stability and Properties",
"The surface energy of alucone-coated Zn (Fig.",
"REF and Fig.",
"REF ) and hydroxylated Zn surface is negative with the strain which is possible for multi-component systems.",
"But surface energy in a single-component stable solid such as Zinc is positive (Fig.",
"REF ) [78], [79].",
"That is because a clean solid surface's interfacial energy stays in equilibrium with its own vapor and therefore no chemical effects for changing the Gibbs free energy.",
"The change in Gibbs free energy of the substrate per unit surface area at constant temperature and pressure defines the surface energy of the system.",
"However, for multi-component system (such as Alcuone on hydroxylated Zn surface) chemical effects must be taken into consideration which changes the solid's surface energy.",
"Furthermore, adsorption is an exothermic process on solid surfaces which reduces the surface energy of the solid.",
"Large adsorption energies as shown in Table REF also lead to negative surface energies.",
"From Fig.",
"REF , surface energy raises with tensile strain but decreases by compressive strain.",
"This phenomenon leads to the roughness of the strained layer which altered the surface energy [80].",
"Therefore, roughness has been promoted by compressive strain whereas it has been inhibited by tensile strain which is also observed by Xie et al.",
"[81], [80].",
"Table: List of surface stress (τ)(\\tau ) ( meV /Å 2 \\mathrm {meV/\\textrm {Å}^{2}}) and surface energy (γ)(\\gamma ) ( meV /Å 2 \\mathrm {meV/\\textrm {Å}^{2}}) of zinc, oxygen deposited zinc and alucode deposited zinc thin films.From our optimized alucone thin film, there is a lattice mismatch with the substrate which rises compressive strain that induces surface reconstruction.",
"The Zn metal surface observes an increased atom number than bulk due to surface reconstruction and hence exhibits negative surface stresses as shown in Table REF .",
"However, negative surface stress makes a surface unstable[82], unless its absolute magnitude of it is small [83].",
"Impacts of surface instabilities due to large surface stress can be manifold.",
"Firstly, compressive (negative) surface stress can result in buckling of the surface which is well known for the unstable elastic continuum model [84].",
"Secondly, atoms' density within the surface layer may change due to this instability.",
"From an atomistic viewpoint, there are three effects related to the variation in the density of surface atoms due to the surface reconstructions [85], [83].",
"Due to inter-atomic interactions, the natural bond lengths change compared to bulk systems.",
"During reconstruction, the extra energy cost is associated with the atom transfer from or to the surface layers.",
"Finally, the surface and substrate atoms' interaction changes due to possible disturbance of the surface-substrate bonding.",
"Table: List fourth-order surface elastic-stiffness coefficients (C ijkl S )(C^{\\textrm {S}}_{ijkl}) ( eV /Å 2 \\mathrm {eV/\\textrm {Å}^{2}}) of zinc, hydroxylated deposited zinc and alucode deposited zinc surface.Point to be noted that $C^{\\textrm {S}}_{ijkl}$ not necessarily is always positive like the bulk Zn elastic tensor (Table REF ).",
"It seemingly contradicts the postulates of basic thermodynamics which assures the stability of the solid on the bulk elastic modulus tensors' positive definiteness.",
"However, it is not applicable for the surface elastic tensor $C^{\\textrm {S}}_{ijkl}$ .",
"The reason behind this is that a surface cannot remain individually without the bulk and therefore, only the combined bulk-surface model requires to fulfill the positive definiteness criterion [58].",
"It should be pointed out that the surface elastic constants ($C^{\\textrm {S}}_{ijkl}$ ) have dimensions of force$\\cdot $ length$^{-1}$ (e.g., N/m or eV/Å$^{2}$ ), which is dissimilar to the bulk elastic constants, as $C^{\\textrm {S}}_{ijkl}$ exists on a two-dimensional surface.",
"It is well accepted that the easiest slip for HCP materials such as zinc is along with the basal slip system [11-20](0001), whereas slip is notably harder along other plane directions (such as prismatic and pyramidal planes) due to higher-order stress requirements [86].",
"In our study, we modeled prismatic (1-100) surfaces to understand the impact of plane orientation on surface properties.",
"From Table REF , it is clear that crystalline orientation plays a significant role to determine the surface elastic constants, which contributes to the anisotropic elastic deformation of the system."
],
[
"Mechanical Figures of Merits of Alucone-Coated Thin Films",
"A film thickness of $L_{z}=50$ nm (see Eq.",
"(REF )) was used to calculate the mechanical properties of different thin films, which satisfies the positive definiteness condition of all elastic modulus tensors (Fig.",
"REF ).",
"From Table REF , $C_{ij}^{\\textrm {film}}$ shows the thin film's stiffness level to the bulk material.",
"For bulk Zn, planar elastic constants are less than the bulk material which indicates that the bare Zn thin film is softer than its bulk counterpart.",
"Table: List of Young's modulus (E) (GPa), Shear modulus (G) (GPa), Bulk modulus (B) (GPa), Poisson's ratio (ν)(\\nu ) and Pugh ratio (k) of different thin films.The Poisson's and Pugh's ratios can be evaluated as the benchmark of ductility.",
"Dimensionless quantities like Poisson's and Pugh's ratio suggest a direct method of anticipating ductility from first principles as they are determined from bulk (B) and shear modulus (G).",
"From Table REF , alucone-coated hydroxylated basal Zn thin film has a Pugh's ratio close to 1.75 [87] and Poisson's ratio close to 0.26 [87], suggesting a good ductility of the thin film.",
"However, for the prismatic surface (1-100), the Pugh and Poisson's ratio are less than the cutoff values for ductility, due to the increase of G and decrease of B compared to the (0001) surface.",
"This observation suggests that the coating could lead to an increased anisotropic deformation mode in Zn, facilitating twinning as in other hcp metals[86].",
"These thin film properties are size-dependent as the contribution of surface elastic constants on thin film elastic constants decreases exponentially with increasing film thickness (Fig.",
"REF ).",
"Alucone-coated Zn met all possible necessary and sufficient elastic stability criteria: $\\textrm {det(C}_{ij})>0$ ; $\\lambda \\mathrm {_{1}}>0,\\lambda \\mathrm {_{2}}>0,\\lambda \\mathrm {_{3}}>0$ ; leading principle minors $>0$ and trailing minors $>0$ [88], [89].",
"Here, $\\lambda $ indicates the eigenvalues of second order elastic constant matrix $\\mathrm {C}$ .",
"From Table REF , the estimated hardness ($H_\\mathrm {V}$ ) of the alucone-coated thin film in the basal surface (0001) is less than the uncoated thin film.",
"However, the hardness of the prismatic surface (1-100) is higher than the basal surface.",
"From Table REF , fracture toughness of alucone deposited Zn (0001) thin film was estimated to be $\\sim 0.5$ MPa$\\cdot $ m$\\mathrm {^{1/2}}$ .",
"From previous experimental measurements with a fracture mechanics model using crack density versus applied tensile strain, the $\\mathrm {{\\it {K}}_{IC}}$ of alucone films on polyethylene naphthalate (PEN) substrate was determined to be 0.16-0.18 MPa$\\cdot $ m$\\mathrm {^{1/2}}$[90], which is in the same order of magnitude as our prediction.",
"In the case of surface wrinkling, the compressive stress in thin film deposition is generated by mechanical force and temperature mismatch.",
"The membrane strain has to be small for surface wrinkling and for 1D wrinkling in alcuone deposited Zn thin film, we observed a membrane strain of 0.5-0.6 %.",
"Table: List of Vickers hardness, H V H_\\mathrm {V} (GPa), fracture toughness, K IC \\mathrm {K_{\\textrm {IC}}} (MPa·\\cdot m 1/2 \\mathrm {^{1/2}}) and threshold membrane strain (ϵ m \\epsilon _{\\textrm {m}} in percentage) of zinc, hydroxylated zinc, and alucode deposited zinc thin films.The development of in-plane compressive stress is one of the main causes of dendrite formation in the thin film deposition technique [91].",
"Two cases are possible due to this compressive stress.",
"First case: when compressive deposition stress develops in alucone film, it will be transported to the Zn substrate.",
"As a consequence, it causes Zn to wrinkle above a threshold membrane strain.",
"Second case: the developed stress cannot be relaxed and eventually dendrites grow in the Zn substrate.",
"The compressive stress in MLD alucone thin film due to wrinkling is evaluated to be $\\sigma _{\\textrm {m}}=E_{\\textrm {alucone}}*\\epsilon _{\\textrm {m}}$ [75] by employing our calculated $\\mathrm {{\\it {E}}_{alucone}}$ .",
"This stress is well below the microsized alucones' yield strength and the compressive residual stress level [92].",
"The above estimation suggests that the alucone-coated Zn anode is dendrite free and likely to have wrinkles."
],
[
"Conclusion",
"Using a combination of chemo-mechanical analysis, we explored the synergy between alucone coating's stability onto Zn.",
"We analyzed the mechanistic insights into the alucone's chemo-mechanical stability due to the molecular layer deposition technique.",
"We studied the alucone precursor and hydroxylated Zn surface interactions by quantifying their charge density difference, Bader charges, and adsorption energies.",
"Coupling the high adsorption energies with the charge transfer analysis, namely Bader charge analysis (quantitative) and charge density difference (qualitative), demonstrated that chemisorption is the dominant mechanism of MLD alucone deposition.",
"We found out the presence of compressive stress and wrinkling formation during coating deposition and its role in the stability of alucone coating.",
"A critical analysis of the surface stress and surface energy level predicted surface reconstruction which induced residual stress during alucone coating on the Zn substrate.",
"The common fact for the thin film in different Zn planes (0001)/(1-100) was that all the coated surfaces have negative surface elastic constants and Young modulus was less than that of the bulk counterpart.",
"Moreover, our study offers a systematic framework to analyze the chemo-mechanical stability of coatings composed of a new class of material for the development of dendrite-free electrodes for other metallic electrode systems.",
"The distribution and effects of the residual stresses imposed on the Zn anodes by the coating deposition need to be extensively studied in the future to develop strategies to utilize dendrite mitigation techniques for feasible AZIBs."
],
[
"Acknowledgement",
"We acknowledge the support of New Frontiers in Research Fund (NFRFE-2019-01095) and from the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grant under Award Application Number 2016-06114.",
"M.G.",
"gratefully acknowledges the financial support from the Department of Mechanical Engineering at UBC through the Four Years Fellowship.",
"This research was supported through high-performance computational resources and services provided by Advanced Research Computing at the University of British Columbia and Digital Research Alliance of Canada.",
"Supporting Information"
],
[
"Determining Fixed and Free Layers of Zn Thin Film Model",
"From Fig.",
"REF (a, b, c), the computational analysis indicated that relaxations primarily existed in the z-direction.",
"Surface atoms do not reconstruct in x and y-direction as negligible in-plane movements were noticed.",
"The level of relaxation was found by computing the distance between the $i\\: \\&\\: j$ layer shifts which we presented here as interlayer distance ($d_{ij}$ ) .",
"The upwards shifting of atoms towards the surface results in a positive value, whereas a downwards shifting in the direction of the bulk results in a negative value of $d_{ij}$ .",
"We found that the hexagonal close-packed (hcp) Zn metal exhibits large inward layer relaxation shown in Fig.",
"REF (c,d).",
"Table: Calculated surface layer relaxations (d ij d_{ij} in percent) as a percentage of the interlayer distance of bulk for the top four layers of Zn (0001) thin film.In Table REF , the layer relaxation is interpreted as [93] - $d_{ij} = (\\lambda _{{ij}}^{\\textrm {s}} - \\lambda _{ij}^{\\textrm {b}}) / \\lambda _{ij}^{\\textrm {b}}$ where $\\lambda _{ij}^{\\textrm {b}}$ and $\\lambda _{ij}^{\\textrm {s}}$ are the bulk and surface interlayer spacing, respectively.",
"The relaxations in multilayer Zn metal indicate that the relaxation reduces in the z-direction with the depth from the surface.",
"Our calculations indicate the movement of the top four layers with an 8-layer Zn thin film model.",
"The distance the first four layers shifted was -0.4, -0.5, -0.3, and -0.3 pm, respectively.",
"The contraction noted in our calculations for all four free layers of the zinc surfaces, can be interpreted using the electrostatic model [94].",
"This model expresses that the outermost surface atoms followed an inward movement for reorganizing themselves to acquire a consistent electron density parallel to the Zn surface.",
"As a consequence, atoms in the lower layers rearranged their positions to adjust neighboring displacements.",
"The contraction phenomenon of the free atomic layers can also be explained using the idea of the bond-order/bond-length.",
"The surface atomic bonds are broken during relaxation and electrons relocate to form a stronger bond which results in a small bond length in the top two free layers of the Zn surface model [94].",
"This can be validated by the relaxations of the lower layers of our model where relaxation decreases and swiftly advance the bulk spacings (-5.1% to -3.3%), indicating the model is acceptable for the study of zinc surface with two free surface layers.",
"Figure: The relative movement of free surface layers with respect to lattice parameters in fractional co-ordinates in (a) x, (b) y, and (c) z, and (d) Relaxation estimation, d ij d_{ij} (i,j=1,2,3,...\\ldots ) as a percentage of the bulk interlayer distance for the Zn (0001) surface.Similar to Fig.",
", positive and negative value specifies expansion and contraction, respectively.Here, we varied the number of fixed layers to determine optimal number of fixed layers fixing 2 free layers.For improving the computational efficiency it is necessary to find the minimum number of fixed layers which will be acceptable for studying the Zn surface.",
"Fig.",
"REF shows similar results to Fig.",
"REF that surface reconstruction occurs only in the z-direction.",
"The degree of relaxation $d_{ij}$ presents the shifting of the two free layers due to changing the number of fixed layers.",
"We find that the movement of the free layers is highest for two fixed layers as shown in Fig.",
"REF (d).",
"All other fixed layers cases display similar second free slab movement and we choose four fixed layers for studying the Zn surface with 6-layer (2 free + 4 fixed) model.",
"Figure: Optimized geometry of different structures (a) Zn (0001), (b) hydroxylated Zn (0001), (c) prismatic Zn (1-100), and (d) prismatic hydroxylated Zn (1-100) thin film."
],
[
"Surface Energy",
"Surface energy ($\\gamma $ ) is a basic physical criterion of metallic surface, it is essential to comprehend a wide range of surface events, such as adsorption, surface corrosion, surface segregation growth rate, etc).",
"As discussed in the main article, there are several approaches to evaluating surface energy to solve the problem of surface energy divergence in separate bulk and surface simulations.",
"Standard approach - The surface energy ($\\gamma $ ) can be determined by the formula - $ \\begin{split}\\gamma (N) = \\frac{1}{2.A} (E_{\\textrm {slab}}(N)-N.E_{\\textrm {bulk}})\\end{split}$ Here, $\\mathrm {{\\it {E}}_{bulk}}$ and $\\mathrm {{\\it {E}}_{slab}}$ are the bulk energy per atom and slabs' total energy, respectively.",
"A is the surface area, and N represents the total atoms count in the surface model.",
"The denominator of Eq.",
"(REF ) has a factor of 2 due to the two surfaces of the slab [53].",
"However, it is noticed that the standard approach goes through divergence problems due to the origination of $\\mathrm {{\\it {E}}_{slab}}$ and $\\mathrm {{\\it {E}}_{bulk}}$ from two distinct sets of simulations with probable incompatibilities in the numerical calculations.",
"Boettger relation- Boettger estimated the value for $\\mathrm {{\\it {E}}_{bulk}}$ in Eq.",
"(REF ) by [54] $\\begin{split}\\frac{\\Delta E}{\\Delta N} = \\frac{E_{\\textrm {slab}}(N) - E_{\\textrm {slab}}(N-1)}{N_{2}-N_{1}} \\approx E_{\\textrm {bulk}}\\end{split}$ Here, $\\mathrm {{\\it {E}}_{bulk}}$ is calculated as the enhancement of $\\mathrm {{\\it {E}}_{slab}}$ by attaching one layer to the slab.",
"This method has the advantage of avoiding separate bulk calculations for energy value though it involves an extra computational effort.",
"Linear fit method- This method can be applied on Eq.",
"(REF ) for large N which approaches convergence quickly than the previous two methods.",
"$\\begin{split}E_{\\textrm {slab}}(N) \\approx 2.",
"A.\\gamma + N. E_{\\textrm {bulk}} \\\\or, \\gamma = \\frac{E_{\\textrm {slab}}(N) - N. E_{\\textrm {bulk}}}{A}\\end{split}$ The target here is to set a straight line to the $\\mathrm {{\\it {E}}_{slab}}$ versus N data set and then utilize the slope of the line as the bulk energy [53].",
"Figure: Surface energies with number of layers by different methods: (a) Linear fit, and standard relation, (b) Boettger relation in eV /A ˚ 2 \\mathrm {eV/\\mathring{A}^2} as a function of number of layers.From the current case studies presented in Fig.",
"REF , it is inferred that the computed values of surface energy are reasonably converged with film thicknesses of 3 or more atomic layers whereas 6 atomic layers can be judged as adequately thick.",
"In our study, both linear fit and standard approach show similar convergence patterns while analyzing the number of layers.",
"Secondly, the Boettger method also performs fairly well in given conditions on increasing the step size.",
"This method was already used in the prior analysis, where so-called quantum size effects (QSEs) in the form of oscillating patterns for $\\gamma $ were found similar to Fig.",
"REF .",
"It was noticed that increasing the step width, that is $\\mathrm {{\\it {N}}_{2}-{\\it {N}}_{1} = 2, 3,}$ etc lessens these oscillating patterns [54].",
"Therefore, we also examined this option for our Zn (0001) surface and observed similar patterns.",
"Furthermore, the standard approach, which applies an independent bulk energy simulation exhibits an impressive convergence and didn't diverge with increasing slab layers number.",
"All three methods give average surface energy of 20 $\\mathrm {meV/\\mathring{A}^2}$ with a deviation of less than $\\mathrm {\\pm 1.5\\:meV/\\mathring{A}^2}$ for 6 mono-crystalline layers.",
"Furthermore, our model's surface energy is consistent with the number of layers and with the range of previous computational studies on Zn(0001) 22 $\\mathrm {meV/\\mathring{A}^2}$ [37]."
],
[
"Van der Waals Interactions",
"In this study, on average the effect of dispersion force is -29.6 eV as displayed in Fig.",
"REF which is nearly 5% of total energy."
],
[
"Bulk Elastic Properties",
"In this study, the elastic constants for bulk Zn are determined to verify the ab initio calculations.",
"For a hexagonal close-packed (hcp) crystal, the five independent second-order elastic constants are $C_{ij}: C_{11},C_{12},C_{13},C_{33},$ and $C_{44}$ [95].",
"Here, the convention to denote the indices i and j in $C_{ij}$ is xx=1, yy=2, zz=3 for the compression components, and as yz=4, zx=5, xy=6 for the shear components.",
"$ C_{ij}=\\begin{pmatrix}C_{11} & C_{12} & C_{13} & 0 & 0 & 0\\\\C_{12} & C_{11} & C_{13} & 0 & 0 & 0\\\\C_{12} & C_{13} & C_{33} & 0 & 0 & 0\\\\0 & 0 & 0 & C_{44} & 0 & 0\\\\0 & 0 & 0 & 0 & C_{44} & 0\\\\0 & 0 & 0 & 0 & 0 & \\frac{C_{11}-C_{12}}{2}\\end{pmatrix}$ The second-order elastic constants shown here can only characterize the linear elastic responses.",
"On the other hand, to model the nonlinear elastic response of a system, we need higher $\\mathrm {(>2)}$ order elastic constants [96].",
"By determining the eigenvalues of the stiffness matrix, sufficient and necessary criteria for elastic stability in the hexagonal crystal are as follows- [88] $C_{11} > |C_{12}|, \\;\\;2C_{13}^2 < (C_{11}+C_{12})C_{33}, \\;\\; C_{44}> 0, \\:C_{66} > 0$ Two closely linked approaches, namely the stress-strain method, and the energy density method (EDM), are available in determining the elastic properties from the first principles technique.",
"EDM computes the total energy of a crystal as a function of its pressure or volume and the second-order expansions of this energy with regard to the applied lattice strains define the elastic constants [97].",
"After applying a small Lagrangian strain $\\epsilon $ as shown in Table REF on unit cell, we used Taylor expansion of the strain energy density $(\\Delta {E}/V_0)$ in terms of the strain tensor.",
"$E(V,\\epsilon )=E(V_0)+V\\sum _{i,j=1}^6\\sigma _i\\epsilon _i+\\frac{V}{2}\\sum _{i,j=1}^6C_{ij}\\epsilon _i\\epsilon _j+...$ $\\frac{\\Delta {E}}{V_0}=\\frac{1}{2!",
"}\\sum _{i,j=1}^6 C_{ij}\\epsilon _{i}\\epsilon _{j}+O(\\epsilon ^3)$ where, $C_{ij}$ denotes the elastic constants using Voigt notation, and $V_{0}$ stands for the unit cell's volume [98].",
"Therefore, $C_{ij}$ can be obtained by fitting the total energies calculated under applied strains to a parabola near the minimum energy.",
"The stress-strain method computes stress values triggering different strains in the crystal [97].",
"In the present work, we used EDM to determine mechanical properties of thin film over Zn anode using ab-initio density functional theory (DFT).",
"Table: List of applied strain modes to computed elastic constants based on energy-strain approach for hexagonal system.",
"The Voigt-Reuss-Hill (VRH) approximations can be used for anisotropic single-crystal system to estimate the mechanical properties such as shear (G), bulk (B), Young's modulus (E) and the Pugh ratio $(\\gamma )$ in terms of an isotropic polycrystalline system [99], [100], [101].",
"Firstly, using Voigt approximation, the upper limit (Voigt) of G and B are computed from $(C_{ij} )$ as $B_{\\textrm {V}} = \\frac{(C_{11}+C_{22}+C_{33})+2(C_{12}+C_{23}+C_{31})}{9} \\\\G_{\\textrm {V}} = \\frac{(C_{11}+C_{22}+C_{33})-(C_{12}+C_{23}+C_{31})+3(C_{44}+C_{55}+C_{66})}{9}$ Secondly, using Reuss approximation, the lower limit of B and G are computed as, $B_{\\textrm {R}}=\\frac{1}{(S_{11}+S_{22}+S_{33})+2(S_{12}+S_{23}+S_{31})} \\\\G_{\\textrm {R}}=\\frac{15}{4(S_{11}+S_{22}+S_{33})-4(S_{12}+S_{23}+S_{31})+3(C_{44}+C_{55}+C_{66})}$ where the compliance tensor matrix, $S_{ij}$ is specified as $ S_{ij} = C_{ij}^{-1}$ .",
"Finally, B and G can be expressed using the Hill approximation as the average of Voigt and Reuss limit as, $B =\\frac{B_{\\textrm {V}}+B_{\\textrm {R}}}{2} \\;\\;and\\;\\; G =\\frac{G_{\\textrm {V}}+G_{\\textrm {R}}}{2}$ Using the calculated B and G from VRH approximations, Young's modulus (E), Poisson's ratio $(\\nu )$ , and Pugh's ratio $(k)$ are defined as, $E = \\frac{9GB}{(G+3B)}, \\:\\: \\nu = \\frac{3B-2G}{2(G+3B)}, \\:\\: k = \\frac{B}{G}$ From Table REF , we computed the elastic properties of bulk Zn and compared them with experimental studies.",
"The benchmark of ductility, namely Poisson's ratio, and Pugh's ratio, indicates that bulk Zn is below the ductility cutoff which is 0.26 for $\\nu $ and 1.75 for k [87].",
"Table: List of second-order elastic-stiffness coefficients (C ij )(C_{ij}) (GPa), Bulk modulus (B) (GPa), Shear modulus (G) (GPa), Young's modulus (E) (GPa), Poisson's ratio (ν)(\\nu ) and Pugh ratio (k) of zinc."
],
[
"Interpolating the Strain-Dependent Surface Energy",
"In this study, we used the Finite difference method (FDM) [59], Local Maximum Entropy (LME) [60] and the Higher Order LME Scheme (HOLMES) [61], to calculate 1st and 2nd order derivatives from strain-energy data.",
"LME and HOLMES are both mesh-free interpolation schemes designed to work over unstructured data, and whose shape function's width is governed by a parameter $\\xi $ which corresponds to the rate of Gaussian decay.",
"From Table REF , all three approaches provide data in a similar range and we used derivatives from HOLMES approach to evaluate elastic properties.",
"For Finite Differences, $\\mathrm {O(h^{2}})$ and $\\mathrm {O(h^{4}})$ central difference schemes are performed in 2D for mixed derivatives whereas arbitrary order central differences are used in one dimension.",
"In general, the agreement between different orders of accuracy deviates due to approximation errors in data.",
"LME, which is a smoothing interpolation technique, provides results that seem more robust given the approximation error.",
"Locality parameters ($\\xi $ ) of 1.8 and 4.0 were tested.",
"In general, the larger value catches local features better, but results are reasonable either way.",
"This is an $\\mathrm {O(h^{2}})$ method for Hessian which is not applicable near the boundary of the domain.",
"HOLMES is a higher-order smoothing interpolation technique where $\\xi $ of 0.8 to 4.0 has been tested.",
"HOLMES delivers the best results compared to the other two methods as it used an $\\mathrm {O(h^{2}})$ hessian approximation to avoid over-fitting with smoothing interpolation to compromise robustness and higher order accuracy.",
"However, it is not applicable near the boundary of the domain."
],
[
"Size Dependency of Elastic Properties",
"It is seen from Fig.",
"REF that the computed Zn thin film modulus differs in a definite way when plotted with film thickness.",
"It is because of the size dependency of elastic properties of nano-structured thin films.",
"The existence of the free surfaces in a thin film is the underlying cause for the above responses [105].",
"As the thin film is stretched or compressed, strain energy is accumulated both on the surface and in the bulk.",
"In the context of an equilibrium viewpoint, both bulk stresses and surface stresses contribute to the force across the thin film's cross-section.",
"As the effects of surface stress increase significantly with the decreasing thickness of the plate, the thin film elastic constants change exponentially with the decreasing thickness."
]
] | 2209.08184 |
[
[
"Lossless SIMD Compression of LiDAR Range and Attribute Scan Sequences"
],
[
"Abstract As LiDAR sensors have become ubiquitous, the need for an efficient LiDAR data compression algorithm has increased.",
"Modern LiDARs produce gigabytes of scan data per hour and are often used in applications with limited compute, bandwidth, and storage resources.",
"We present a fast, lossless compression algorithm for LiDAR range and attribute scan sequences including multiple-return range, signal, reflectivity, and ambient infrared.",
"Our algorithm -- dubbed \"Jiffy\" -- achieves substantial compression by exploiting spatiotemporal redundancy and sparsity.",
"Speed is accomplished by maximizing use of single-instruction-multiple-data (SIMD) instructions.",
"In autonomous driving, infrastructure monitoring, drone inspection, and handheld mapping benchmarks, the Jiffy algorithm consistently outcompresses competing lossless codecs while operating at speeds in excess of 65M points/sec on a single core.",
"In a typical autonomous vehicle use case, single-threaded Jiffy achieves 6x compression of centimeter-precision range scans at 500+ scans per second.",
"To ensure reproducibility and enable adoption, the software is freely available as an open source library."
],
[
"INTRODUCTION",
"LiDAR sensors are rapidly becoming commonplace in applications ranging from infrastructure monitoring to autonomous cars and drones.",
"At the same time, LiDAR sensors are rapidly improving in range, precision, spatial resolution, and scan rate.",
"Additional outputs, including multi-return range, intensity, reflectivity, and ambient light, result in a flood of LiDAR data that must be processed, transmitted, and stored, consuming scarce resources of bandwidth, memory, time, and power.",
"Fast, lossless LiDAR compression is needed to address the deluge (Fig.",
"REF ).",
"Figure: The Jiffy algorithm uses techniques adapted from image encoding and database compression to achieve lossless compression of LiDAR range scan sequences by a factor of 12-18x vs. uncompressed 32-bit Cartesian point cloud format, or 4-6x vs. uncompressed 32-bit range image format in applications including autonomous driving, infrastructure monitoring, and handheld & drone mapping.General purpose algorithms have been used to compress heterogeneous sensor data losslessly, but without prior knowledge of the data, they are limited.",
"Algorithms specifically designed for LiDAR scan data have been proposed to exploit the structure of LiDAR data.",
"These algorithms have primarily focused on lossy compression of LiDAR range scans by reusing concepts from image, video, or point cloud compression.",
"Thus far, they have proven too slow, too lossy, or too ineffective for wide adoption.",
"This work leverages the sparsity and spatiotemporal coherence of LiDAR scan sequences to develop Jiffy— a fast, lossless codec for high-resolution, multiple-attribute LiDAR scans (Fig.",
"REF ).",
"Single-threaded performance is ensured by maximizing use of single-instruction-multiple-data (SIMD) instructions.",
"In multi-attribute benchmarks from real-world use cases, Jiffy consistently achieves best-in-class lossless compression with single-CPU throughput in excess of 65 million points per second.",
"Figure: The Jiffy algorithm losslessly compresses high bit-depth, multi-attribute LiDAR scan data.",
"Incoming scans are quantized to the precision of the sensor.",
"A bitmask locating zero-valued samples is extracted from the scan and compressed separately (green) from the remaining scan data (blue).",
"A trial compression heuristic selects an encoding method: intra-scan (I-scan) coding is selected when the previous scan differs significantly from the current scan, otherwise, predicted-scan (P-scan) coding is selected."
],
[
"RELATED WORK",
"Some of the earliest general purpose, lossless compression algorithms, Bzip2[1] and DEFLATE (Zlib)[2] have been used to compress binary files for decades.",
"DEFLATE forms the basis for many application-specific file formats (e.g.",
"ZIP[3], DOCX[4], PNG[5], OpenEXR[6]).",
"These early compression algorithms were developed before memory became a bottleneck, before many-core computers became common, and before standard SIMD instruction sets were widely available.",
"Recently, new algorithms have been designed to take advantage of the hardware in modern CPUs.",
"LZ4 is a fast dictionary matching algorithm that runs at GB/sec throughput, but produces modest compression ratios[7].",
"ZStandard (Zstd) leverages recent developments in Asymmetric Numeral Systems (ANS) and SIMD implementation to achieve improved compression at much faster speeds[8].",
"The popular Robotics Operating System (ROS) initially adopted Bzip2 as its data logging compression codec, and later added LZ4 as a faster option[9].",
"Both LZ4 and Bzip2 have been replaced by Zstd in ROS2[10].",
"These general purpose compressors are convenient for compressing heterogeneous sensor data because they do not require a specific model of the data, but without a model of the data, their compression ratios are often limited.",
"Point cloud compression algorithms like Draco [11] or PCD [13] have been employed to compress LiDAR-derived point clouds and attributes.",
"Google's Draco triangle mesh codec treats point clouds as degenerate meshes with no edges or faces.",
"Floating-point vertices are quantized as integers, then ANS coding is applied, similar to Zstd.",
"The PCD file format groups coordinates by axis, then applies fast LZF compression.",
"The Motion Picture Experts Group (MPEG) has developed two codecs, G-PCC and V-PCC, for compressing point clouds[18].",
"The Geometry-Point Cloud Compressor (G-PCC) is based on voxel- and tree-based decomposition of 3D space and is best-suited for static point clouds.",
"The Video-Point Cloud Compressor (V-PCC) tries to reuse the HEVC video codec.",
"V-PCC searches for planar regions of a point cloud and projects geometry and attribute information onto them.",
"It then packs planar segments into video frames and applies HEVC.",
"Deep point cloud compression using hardware-intensive neural networks is an active area of research.",
"A more complete review of point cloud compression methods can be found in Pereira[16].",
"The range image representation is a method of representing LiDAR point measurements in spherical coordinates.",
"Conversion from Cartesian XYZ coordinates to the range image representation can mean 3x compression of a LiDAR scan.",
"Azimuth and altitude angles are implied by the x,y position of a sample, leaving an x,y array of polar magnitudes (ranges).",
"Conversion to range images enables the use of image and video compression codecs.",
"Tu et al.",
"applied JPEG2000 to LiDAR range images[20].",
"Ahn et al.",
"developed a lossless codec for stationary survey scanners[19].",
"Their method recursively subdivides the image into patches and applies a variety of predictors to each patch to achieve reasonable compression at the expense of speed.",
"Nenci et al.",
"used many instances of H.264 video codecs to compress LiDAR sequences[17].",
"They achieved 10x lossless compression of Microsoft Kinect range images at 25 Hz using 8 parallel video streams, but their method does not scale to long range sensors.",
"Their method requires 391 concurrent video streams to encode a 100 meter range LiDAR scan with 1 mm resolution.",
"Researchers interested in database compression have developed integer compression algorithms capable of losslessly compressing billions of unsigned 32-bit integers per second at compression ratios competitive with general purpose compressors like Zstd.",
"Goldstein et al.",
"[21] developed Frame Of Reference (FOR) compression, which groups values into frames (e.g.",
"128 integers).",
"The minimum value in the frame is identified, then the values in that frame are re-coded as an offset from that minimum.",
"Zukowski et al.",
"[22] proposed Patched-FOR (PFOR), which stores values exceeding a thresholded number of bits in a separate list.",
"Lemire et al.",
"[23] implemented SIMD-FastPFOR (SIMDPFOR)— PFOR using SIMD acceleration."
],
[
"Standardizing LiDAR Input",
"LiDAR scanners produce geometry and attribute information in formats that differ by manufacturer.",
"Some sensors output range data as an ordered stream of floating point Cartesian coordinates.",
"Others represent ranges as an integer number of millimeters and natively output range images.",
"Signal, reflectivity and other attributes may be returned as floats, as n-byte unsigned integers, or not at all.",
"Jiffy expects all scan types in a canonical 2D format.",
"Cartesian points must be converted to spherical coordinates, sorted by altitude and azimuth, then quantized to 1, 2 or 4 byte unsigned integers.",
"Invalid or out-of-range measurements are set to a sentinel value of zero.",
"The user-provided quantization precision is stored in the compressed bitstream and used to restore the original representation of a quantized scan.",
"Attribute values are similarly arranged and quantized."
],
[
"Quantization",
"Quantization precision is an important parameter of the Jiffy encoder.",
"Range measurements are quantized losslessly by default, but lossy quantization can be used to improve compression.",
"Pereira[16] defines quantization as lossless when the quantization precision is less than or equal to the resolution of the sensor.",
"By default, the Jiffy encoder quantizes range measurements using the sensor resolution—one millimeter for all sensors evaluated here (Table REF ).",
"In some applications the sensor resolution is orders of magnitude more precise than is necessary (e.g.",
"a streaming visualizer), and aggressive quantization can be used to substantially improve compression.",
"It is straightforward to set the quantization precision, and the maximum quantization error is always less than half of the precision."
],
[
"Intra-Scan (I-Scan) Compression",
"Delta compression exploits spatial redundancy by subtracting neighboring samples within a scanline.",
"Jiffy subtracts each sample in a scanline from its left-side neighbor and compresses the results.",
"Delta encoding is effective on range data because the difference between adjacent samples is typically much smaller than the samples themselves.",
"However, because Jiffy encodes 'out of range' (too far or too close) with a zero-valued sentinel, transitions into- and out-of-range can produce very large deltas, reducing compression.",
"To eliminate the large deltas caused by these transitions, Jiffy creates a binary mask of one bit per sample.",
"The mask records a zero bit for in-range samples and a one for out-of-range samples.",
"The bitmask is packed into bytes and compressed using Zstd.",
"After creating the out-of-range bitmask, all out-of-range samples are removed from the 2D scan, creating a flattened 1D vector.",
"The 1D vector can now be delta-encoded without introducing large differences caused by transitions into and out of range.",
"Masking of zero values is not required for attribute scan types since they do not use zero to indicate valid or invalid measurements.",
"However, Jiffy applies zero masking to all scan types, producing a positive effect on compression ratio.",
"The SIMD-PFOR algorithm is designed to compress unsigned integers, but delta encoding produces many negative values.",
"Signed integers can be cast to unsigned and sent directly to SIMD-PFOR for compression, but negative integers cast to extremely large unsigned values because two's-complement sign extension sets all of their highest bits.",
"Since SIMD-PFOR is designed to pack small unsigned integers with many leading zeros in the upper bits, this leads to poor compression.",
"Jiffy uses ZigZag encoding to represent negative differences without the problems caused by sign extension (Equation REF ).",
"$\\text{ZigZag}(x) =2\\vert x\\vert + \\begin{cases*}1 & if x < 0 \\\\0 &otherwise\\end{cases*}$ Jiffy performs standalone (Intra-scan or I-scan) compression of individual LiDAR scans by masking and removing out-of-range samples, delta encoding the remaining values, ZigZag encoding the deltas, and SIMD-PFOR compressing the result.",
"I-scan decoding is performed by a straightforward reversal of the encoding procedure."
],
[
"Predicted-Scan (P-Scan) Compression",
"When LiDAR sensors are stationary or moving slowly, successive scans exhibit a high degree of similarity that can be exploited to improve compression.",
"Stationary infrastructure monitoring LiDARs typically overlook a largely static scene.",
"Autonomous vehicles are frequently stationary, waiting for passengers or idling in traffic.",
"In these situations, it is redundant to encode near-identical scans of a static scene.",
"Jiffy provides an inter-scan compression mode called Predicted-scan (P-scan).",
"The 'predicted' nomenclature is borrowed from MPEG video compression naming conventions.",
"Like Intra-scan encoding, a compressed bitmask is created to represent zero-valued samples in the input scan.",
"In predicted-scan (P-scan) mode, the bitmask for the current frame is XORed with the bit mask from the previous frame to remove redundancy from successive bitmasks.",
"The XORed mask result is compressed using Zstd and appended to the output bitstream.",
"Jiffy's P-scan compression mode uses temporal differencing to avoid encoding redundant scans.",
"After masking the current frame and the previous frame with the current bitmask, the scan differences are encoded using the I-scan compression method."
],
[
"Automatic I/P Mode Selection",
"For some applications, I-scan only or P-scan only encoding might make sense.",
"For example, a fast-moving drone could use I-scan only encoding for best results, whereas a pole-mounted monitoring sensor would benefit from using P-scan only.",
"However, applications such as a car in stop-start traffic, a racing drone sitting on its launch pad, or a tree branch waving in front of a static monitor sensor should also be considered.",
"These cases, and in fact, most use-cases benefit from the ability to adaptively select a compression mode on a per-scan basis.",
"The Jiffy encoder could employ brute force to select a compression mode.",
"Because I- and P-scan compression modes are extremely fast, it might be reasonable to encode each scan using both methods and then select the better result.",
"Although brute force trial compression ensures the most effective mode is used, it also reduces throughput by 50%.",
"As explained in subsection REF , a trial compression of a few scanlines in each scan (excluding mask compression) proves to be successful in selecting the most effective compression mode for the full scan.",
"Figure: The Jiffy codec compresses LiDAR range scans by a factor of 4.4 at a rate of 660 scans per second on a single core of a 12th Gen Intel i7-1250U processor.",
"The same codec is pareto-optimal for compression of 16-bit per pixel intensity (Signal) and ambient infrared (NearIR) scan data and is near-optimal for compression of 8-bit per pixel reflectivity data.",
"For sparse second-return data, the Jiffy codec operates at 1000 scans per second and compresses at 35-85x compression ratios.",
"Results presented here are averaged across three autonomous vehicle datasets collected using Ouster OS0-128, OS1-128, and OS2-128 LiDAR sensors."
],
[
"Software and System",
"All code, including benchmarks, compatible datasets, implementations of competing codecs, and unit tests, are freely available[3].",
"All codecs and benchmarks are implemented in python3.",
"Performance critical code relies on python3 bindings to fast C/C++ libraries such as numpy, Zstd, SIMD-PFOR, zlib, lz4, etc.",
"Benchmarks were performed on a laptop containing a recent 12th Gen Intel i7-1250U processor.",
"Sufficient cooling and power minimize potential methodological problems with benchmarking on a laptop.",
"The benchmark process is pinned to CPU 0 (a performance core) and configured to use the performance CPU governor.",
"Sensors used are summarized in Table REF ."
],
[
"Datasets",
"Benchmarks for several LiDAR use cases are employed: autonomous driving, infrastructure monitoring, handheld mapping, and drone flying.",
"To simplify evaluation, all datasets were constructed using Ouster OS0, OS1, or OS2 LiDAR sensors.",
"The autonomous driving and infra- structure monitoring datasets were supplied by the sensor manufacturer[24].",
"The handheld mapping dataset is extracted from the park sequence of the multi-camera Newer College dataset[25], and the drone datasets are derived from the nya_02 sequence of the NTU VIRAL dataset[26].",
"The NTU drone dataset includes simultaneous scans from one vertically-mounted and one horizontally-mounted OS0-16 LiDAR sensor.",
"Because their compression characteristics differ significantly, the horizontal- and vertical-mounted drone LiDAR scans are treated as separate use cases.",
"The autonomous driving datasets were collected using the short-range, medium-range, and long-range sensors, detailed in Table REF .",
"Results for the codec comparisons in subsection REF are averaged across all three sequences.",
"All datasets contain range scans with single-precision floating point samples.",
"The autonomous driving datasets additionally contain signal, reflectivity, near-infrared, and second-return scan types.",
"The second-return range2 scan type contains 32-bit ranges.",
"The signal scan type and its second-return equivalent, signal2, record the 16-bit unsigned integer intensity of the LiDAR return pulse.",
"The reflectivity and reflectivity2 scan types record an 8-bit unsigned integer estimate of surface reflectivity for every sample in a scan.",
"The near-infrared scan type is a 16-bit unsigned integer measurement of ambient light.",
"Dataset features are summarized in Table REF .",
"Table: A Summary of Benchmarked LiDAR SensorsTable: A Summary of Benchmarked Datasets"
],
[
"Compression Results vs. Existing Lossless Algorithms",
"A comparison of encoding speed and compression ratio for the Jiffy codec and competing lossless methods was performed using the autonomous vehicle data sets[24].",
"The mean speed and mean compression ratio over all range scans are presented in Fig.",
"REF .",
"Lz4 is consistently fastest, but its compression ratio ranks among the worst.",
"The Jiffy codec is the clear winner in both speed and compression ratio for the first-return 'range' scan types and the second-return 'range2' and 'signal2' scan types.",
"These scan types exhibit the most sparsity and the highest dynamic range, so they benefit most from the zero bitmasking and SIMD-PFOR algorithms.",
"Horizontal delta encoding provides compression gains due to the low spatial frequency content of this scan type, and adaptive coding of I- and P-scans also contributes to higher compression ratios when the vehicle comes to a stop.",
"Reflectivity (first- and second- returns) were the most challenging for Jiffy, as these scan types had the smallest dynamic range (8 bits) coupled with with the highest spatial frequency content.",
"The high frequency content counters the benefit of delta encoding, and coupling that deficit with the 8-bit dynamic range provides little opportunity for the SIMD-PFOR codec to effectively pack the data.",
"The Zlib codec was a close competitor for these scan types.",
"The Jiffy compression ratio for the nearIR 16-bit scan type was very competitive with the Bzip2 and Lzma codecs at a 7x to 21x higher compression speed."
],
[
"Compression Results by Use Case",
"Compression performance was tested for five unique usage scenarios: autonomous driving, with short- medium- and long-range sensors, infrastructure monitoring with a stationary long range sensor, a short range walking scan using a handheld sensor without motion stabilization, and a drone equipped with two short range sensors, one scanning horizontally and one vertically (Fig.",
"REF ).",
"For the autonomous driving scenario at the default 1 mm precision, the short range sensor had the best compression ratio (4x), and the long range sensor had the lowest (2.6x).",
"This result clearly demonstrates the efficient bit packing of the SIMD-PFOR algorithm, because the short range sensor data uses fewer of the original 32 bits per sample than the medium- and long-range sensors.",
"The drone scenario clearly demonstrates the effect of the zero bitmask compression.",
"The horizontally- and vertically-mounted sensors have the same dynamic range, but the vertically mounted sensor data has a higher degree of sparsity relative to the horizontally mounted sensor, so it achieves a higher compression ratio (3.3x horizontal vs. 4.0 vertical).",
"The infrastructure scenario clearly demonstrates the coding gain of the temporal (P-Scan) compression.",
"This scenario is using the long-range sensor, yet it exhibits a compression ratio that rivals the best-compressed scenarios using the short range sensor, drone(vertical), walking, and driving."
],
[
"Validation of Spatiotemporal Encoder Selection Heuristic",
"Testing the efficacy and error rate of the I/P scan selection heuristic was performed by comparing the predicted optimal compression method (I or P), to the actual optimum compression method for each of 22,877 range scans in all test datasets.",
"The actual optimal method was determined by comparing the relative results of compressing the scans using an I-scan only compressor and a P-scan only compressor.",
"Using just 4 test lines in the heuristic test, the algorithm correctly predicted the optimal compression method for 96% of the 22,877 scans tested, with 0.5% sub-optimal I-scans and 3.5% sub-optimal P-scans."
],
[
"Compression / Decompression Speed",
"Jiffy encode/decode speed was tested using three autonomous driving datasets.",
"All three datasets use high data rate sensors that produce 128 $\\times $ 1024 scans at 10 Hz.",
"Benchmark results for each scan type are presented in Table REF .",
"On a single CPU core, the Jiffy codec sustains a mean encoding(decoding) rate of more than 1300(2100) scans/sec over all scan types.",
"On a single thread, Jiffy can encode(decode) all seven scan types at a rate of 120 Hz, twelve times faster than the sensor's 10 Hz output rate.",
"Table: Jiffy Compression/Decompression Rates by Scan TypeFigure: Jiffy quantizes LiDAR range measurements prior to lossless compression.",
"For applications where some error is acceptable, aggressive quantization can be traded for improved compression.",
"Doubling the quantization precision P q P_q decreases the compressed output by approx.",
"one bit per measurement.",
"Max.",
"roundtrip error is upper bounded by 0.5P q 0.5 P_q."
],
[
"Effect of Quantization on Compression Ratio",
"Jiffy quantizes LiDAR range data with a precision of one millimeter by default.",
"For less precise sensors, or in circumstances where quantization errors are acceptable, more aggressive quantization can be used to improve compression.",
"Increased quantization leads to a predictable improvement in compression performance with bounded absolute errors (Fig.",
"REF ).",
"Doubling the quantization precision removes one bit of information from each range measurement.",
"This enables Jiffy to compress each range measurement using at least one less bit and produces a corresponding improvement in compression ratio.",
"Because Jiffy compression of quantized scan data is lossless, the decoded error will not exceed half of the quantization precision setting."
],
[
"Ablation Study of the Jiffy Algorithm",
"An ablation study of the Jiffy encoder using an infrastructure monitoring LiDAR dataset[24] demonstrates the compression achieved by each component of the Jiffy algorithm (Table REF ).",
"The entire dataset is encoded using bare SIMD-PFOR compression, resulting in a 3.2x compression ratio.",
"Adding horizontal delta-encoding along the scanlines and compressing the residuals using SIMD-PFOR predictably lowers the ratio to 2.2x, due to the effect of negative integers on SIMD-PFOR.",
"The ZigZag transform resolves this problem, boosting the ratio to 3.8x.",
"The out-of-range mask encodes zero values in the range scan using much less than one bit per sample.",
"The bitmask also allows the removal of zero values from the delta-encoded measurement stream.",
"This effectively removes large residuals at the boundaries between out-of-range and in-range values.",
"With masking, delta encoding, the ZigZag transform, and SIMD-PFOR compression, the dataset is compressed by a factor of 5.4x.",
"Adding the final algorithmic improvement, P-scan prediction, completes the Jiffy encoding algorithm, achieving a final dataset compression ratio of 6.2x.",
"Table: Performance of Ablated Jiffy Codecs Applied to the Infrastructure Monitoring LiDAR Dataset"
],
[
"CONCLUSIONS",
"The Jiffy codec successfully addresses the need for a fast, lossless LiDAR compression algorithm.",
"In multiple-attribute benchmarks from a variety of use cases, Jiffy achieved best-in-class lossless compression at a rate of hundreds of scans per second on a single CPU core.",
"This is more than 10x faster than the uncompressed sensor output rate for all of the sensors tested.",
"The ability to control the quantization of range data is an effective and intuitive way to influence the compression ratio with a predictable error ceiling.",
"The techniques introduced here could be easily adapted to compress high dynamic range depth images and videos produced by stereo cameras or solid-state LiDAR sensors.",
"Further research could investigate compression of per-pixel velocity measurements produced by continuous-wave LiDAR sensors or of derived perception data such as per-pixel segmentation labels.",
"Lossy compression is also of considerable interest, but research into lossy algorithms should also quantify the downstream impact of compression artifacts on perception and localization algorithms."
]
] | 2209.08196 |
[
[
"Neural Implicit Surface Reconstruction using Imaging Sonar"
],
[
"Abstract We present a technique for dense 3D reconstruction of objects using an imaging sonar, also known as forward-looking sonar (FLS).",
"Compared to previous methods that model the scene geometry as point clouds or volumetric grids, we represent the geometry as a neural implicit function.",
"Additionally, given such a representation, we use a differentiable volumetric renderer that models the propagation of acoustic waves to synthesize imaging sonar measurements.",
"We perform experiments on real and synthetic datasets and show that our algorithm reconstructs high-fidelity surface geometry from multi-view FLS images at much higher quality than was possible with previous techniques and without suffering from their associated memory overhead."
],
[
"Introduction",
"Imaging or forward-looking sonar (FLS) is an extensively used sensor modality by Autonomous Underwater Vehicles (AUV).",
"The key motivation for using FLS sensors is their ability to provide long-range measurements, unlike optical cameras whose range is severely limited in turbid water—a common situation in the field.",
"Their versatility has resulted in their incorporation as a core sensor modality in applications including robotic path planning [1], [2], localization [3], [4], [5], [6], [7], and the automation of tasks potentially dangerous or mundane for humans such as underwater inspection [8] and mapping [9], [10], [11], [12].",
"FLS outputs 2D image measurements of 3D structures by emitting acoustic pulses and measuring the energy intensity of the reflected waves.",
"While the sonar resolves azimuth and range, the elevation angle is ambiguous, and an object at a specific range and azimuth can be located anywhere along the elevation arc.",
"Hence, the task of 3D reconstruction using FLS measurements can be equivalently framed as the task of resolving the elevation ambiguity from the image readings.",
"Existing algorithms for 3D reconstruction from FLS measurements can be grouped into geometry-based, physics- based and, more recently, learning-based methods.",
"How- ever, most existing approaches either place restrictions on the robotic/sensor setup (elevation aperture, motion of the vehicle, etc.",
"); rely on volumetric grids that are prohibitively expensive for large scenes or scenes with fine-grained geometry; or, specific to learning approaches, require the use of large labeled training sets that are difficult to collect in underwater environments.",
"To address these shortcomings, we approach the problem of underwater FLS-based 3D reconstruction through the lens of differentiable rendering and leverage the representational power of neural networks to encode the imaged object as an implicit surface.",
"Our overall reconstruction approach comprises the following components: A differentiable volumetric renderer that models the propagation of acoustic spherical wavefronts.",
"A representation of 3D surfaces as zero-level sets of neural implicit functions.",
"A regularized rendering loss for 3D reconstruction using imaging sonars.",
"To the best of our knowledge, this work is the first to introduce a physics-based volumetric renderer suitable for dense 3D acoustic reconstruction using wide-aperture imag- ing sonars.",
"We evaluate our approach against different unsupervised methods on simulated and real-world datasets, and show that it outperforms previous state of the art.",
"We will open-source our code together with different datasets$^2$ .",
"Different methods have been introduced to produce both sparse [13], [14], [10], [11], [15], [5] and dense 3D reconstructions using FLS.",
"The focus of this work is on dense object-level 3D reconstruction.",
"A number of algorithms enforced assumptions or constraints on the physical system to obtain reliable 3D models.",
"Teixeira et al.",
"[16] successfully reconstructed a 3D map of a ship hull by leveraging probabilistic volumetric techniques to create submaps which are later aligned using Iterative Closest Point (ICP).",
"However, the sonar aperture was set to $1^{}$ and all detected points were assumed to lie on the zero-elevation plane which leads to reconstruction errors and prohibits extending the method to larger apertures.",
"A line of work [17], [18], [19], [20] uses two complementary sensors (imaging and profiling sonars) and performs sensor fusion to disambiguate the elevation angle.",
"In our work, we restrict our setup to a single imaging sonar.",
"Westman et al.",
"[21] proposed a method to reconstruct specific points on surfaces (aka.",
"Fermat Paths).",
"However, it places constraints on the vehicle's motion as it needs a view ray perpendicular to the surface at each surface point and hence, requires a large number of images collected from specific views.",
"Another set of methods uses generative models to obtain dense 3D reconstructions.",
"Aykin et al.",
"[22], [23] attempt to estimate the elevation angle of each pixel by leveraging information from both object edges and shadows which restricts the object to be on the seafloor.",
"Westman et al.",
"[24] further extended the idea to do away with the seafloor assumption but still required estimates of object edges.",
"Negahdaripour et al.",
"[25] proposed an optimization-based algorithm to refine an initial 3D reconstruction obtained using space carving by encouraging consistency between the actual sonar images and the images produced by the generative model.",
"However, generative methods generally rely on assumptions of the surface reflectivity proprieties and on 3D estimates of object edges which makes them impractical in real scenarios.",
"Various volumetric methods have also been proposed.",
"Wang et al.",
"[26] introduced an inverse sonar sensor model to update the occupancy in a voxel grid and later extended it to handle errors in pose estimates by aligning local submaps using graph optimization [27].",
"Although these methods, as probabilistic frameworks, can be more robust compared to space carving techniques [22], [9], they consider each voxel independently and ignore inherent surface constraints.",
"Guerneve et al.",
"[28] frame the problem of 3D volumetric reconstruction as a blind deconvolution with a spatially varying kernel which can be reformulated as a constrained linear least squares objective.",
"However, the method makes a linear approximation to the vertical aperture and places restrictions on the motion of the sonar limiting its practical application.",
"Westman et al.",
"[29] noted the equivalence of 3D sonar reconstruction to the problem of Non-Line-of-Sight (NLOS) imaging.",
"It introduced a regularized least square objective and solved it using the alternating direction method of multipliers (ADMM).",
"All aforementioned volumetric methods, however, share similar limitations since extracting high-fidelity surfaces from volumetric grids is difficult.",
"These approaches can also be computationally expensive for larger scenes or a fine discretization of volumes.",
"More recently, learning-based methods were proposed to resolve the elevation ambiguity.",
"DeBortoli et al.",
"[30] proposed a self-supervised training procedure to fine-tune a Convolutional Neural Network (CNN) trained on simulated data with ground truth elevation information.",
"Wang et al.",
"[31] use deep networks to transfer the acoustic view to a pseudo frontal view which was shown to help with estimating the elevation angle.",
"However, these methods are limited to simple geometries or require collecting a larger dataset of real elevation data.",
"Arnold et al.",
"[32] propose training a CNN to predict the signed distance and direction to the nearest surface for each cell in a 3D grid.",
"However, the method requires ground truth Truncated Signed Distance Field (TSDF) information which can be difficult to obtain.",
"In this work, we propose a physics-based renderer which uses raw FLS images and known sonar pose estimates to represent objects as zero-level sets of neural networks.",
"It does not require hand-labeled data for training nor does it place restrictions on the setup or environment (voxel size, need for reflectance information, etc.)"
],
[
"Neural Implicit Representation",
"NeRF [33] introduced a volume rendering method to learn a density field aimed at novel view synthesis.",
"It samples points along optical rays and predicts an output color which is then checked against that of the ground truth pixel.",
"IDR [34] introduced a surface rendering technique that contrary to the volume rendering technique of NeRF, only considers a single point intersection on a surface.",
"Hence, it fails to properly capture areas of abrupt changes in the scene.",
"NeuS [35] leveraged the volume rendering technique of NeRF to perform 3D surface reconstructions and showed impressive results against state-of-the-art neural scene representation methods for scenes with severe occlusions and complex structures.",
"NeTF [36] applied ideas from NeRF to the problem of NLOS imaging which was shown in [29] to have close similarity to FLS 3D reconstruction.",
"All these methods focus on 3D reconstruction using optical sensors, either intensity or time-of-flight based.",
"Our focus is on learning surfaces from acoustic sonar images."
],
[
"Image Formation Model",
"An FLS 2D image $\\mathcal {I}$ comprises pixels corresponding to discretized range and azimuth $(r_i, \\theta _i)$ bins.",
"Each pixel value is proportional to the sum of acoustic energy from all reflecting points $\\lbrace \\mathbf {P}_i = (r_i, \\theta _i, \\phi _i); \\phi _{\\mathrm {min}} \\le \\phi _i \\le \\phi _{\\mathrm {max}} \\rbrace $ , $\\phi _i$ being the elevation angle (Fig.",
"REF ).",
"However, the elevation angle $\\phi _i$ is lost since each column $\\theta _i$ of an FLS image is the projection onto the $z=0$ plane of a circular sector $\\pi _i$ constrained to the sonar vertical aperture $(\\phi _{\\mathrm {min}}, \\phi _{\\mathrm {max}})$ and containing the $z$ axis (Fig.",
"REF ).",
"Figure: (a) Sound propagates as spherical wavefronts.",
"An acoustic ray is defined as the ray starting at the acoustic source and terminating at the wavefront (figure inspired by the Discovery of Sound in the Sea project ) (b) Each image column θ i \\theta _i is the projection of the circular arc π i \\pi _i onto the plane z=0z=0.",
"(c) Example of a sonar image.",
"Each pixel at (r,θ)(r, \\theta ) corresponds to the intensity reading of all points along the elevation arc.We now present our rendering equation.",
"Imaging sonars are active sensors that emit a pulse of sound and measure the strength of the reflected acoustic energy.",
"Let $\\textit {E}_e$ be the emitted acoustic energy by the sonar.",
"Now, consider a unique infinitesimal reflecting patch $\\mathcal {P}_i$ “illuminated\" by the acoustic wave and located on the arc $\\mathcal {A}(\\phi ) \\in \\pi _i$ which passes through $(r_i, \\theta _i, 0)$ (Fig.",
"REF ).",
"The reflected acoustic energy at $\\mathcal {P}_i$ and received by the sonar can be approximated as: Er(ri, i, i)=ri-ri+ Eer2 e-0ri (r', i, i ) dr'T (r, i, i ) rdr where $2\\epsilon $ is the patch thickness, $\\sigma $ is the particle density at $\\mathcal {P}_i$ , and the factor $\\frac{1}{r^2}$ accounts for spherical spreading on both the transmit and receive paths.",
"$T$ is the transmittance, corresponding to exponential attenuation of a wave due to particle absorption—equivalently, the probability that the acoustic wave travels between two points unoccluded.",
"We note that, when the sonar emitter and receiver are collocated, this probability is identical during the transmit (sonar to $\\mathcal {P}_i$ ) and return ($\\mathcal {P}_i$ to sonar) paths; thus, transmittance is accounted for only once for both paths.",
"This is analogous to the effect of coherent backscattering in optical wave propagation with collocated emitter and receiver [38].",
"Figure: 1) All points 𝐏=(r,θ,φ)\\mathbf {P} = (r, \\theta , \\phi ) on the arc are projected onto the z=0z=0 elevation plane.",
"2) An example of an infinitesimally small patch on the arc 𝒫\\mathcal {P} is shown in yellow.",
"3) Illustrating our sampling scheme: sampled pixels are colored in blue.",
"Sampled points on the arc are shown in black.",
"For each point on the arc, we construct the acoustic ray (green arrow) and sample points on each ray (green points).Now consider a surface composed of many such patches.",
"The received energy by the sonar is simply the sum of the reflected energy by all patches $\\lbrace \\mathcal {P}_i\\rbrace \\in \\mathcal {A}(\\phi )$ which approximate the surface.",
"Hence, we arrive at the following image formation model: I(ri, i) = minmax ri-ri+ Eer2 e-0ri (r', i, ) dr' (r, i, ) rdrd = minmax ri-ri+ Eer T(r, i, ) (r, i, ) drd.",
"Note that although sound propagation through liquids is fundamentally different from that of light (longitudinal vs. transverse waves), different geometric acoustic modeling techniques still borrowed heavily from graphics and ray optics [39].",
"These methods fundamentally rely on Huygen's principle of sound travel through mediums which approximates the spherical wavefront as many energy-carrying particles travelling at the speed of sound.",
"Hence, analogous to the concept of a light ray, we view an acoustic ray as the ray starting at the sonar acoustic center and ending at $\\mathcal {P}_i$ (Fig.",
"REF )."
],
[
"Rendering Procedure",
"Similarly to Yariv et al.",
"[34] and Wang et al.",
"[35], we represent the sonar-imaged surface using two multi-layer perceptrons (MLPs): a neural implicit surface, $\\mathbf {N}$ , which maps a spatial coordinate $\\mathbf {x}=(r, \\theta , \\phi )$ to its signed distance; and a neural renderer, $\\mathbf {M}$ , which outputs the outgoing radiance at $\\mathbf {x}$ .",
"Once the surface $\\mathcal {S}$ is learned, we can extract it as the zero level set of $\\mathbf {N}$ : S = {$\\mathbf {x}$ R3: N($\\mathbf {x}$ ) = 0 }.",
"To train our network using the rendering loss (Eq.",
"REF ), we leverage the following equation from Wang et al.",
"[35] to estimate the value of the density $\\sigma (\\mathbf {x})$ from the SDF: ($\\mathbf {x}$ ) = ( -d s (N($\\mathbf {x}$ )) d$\\mathbf {x}$ s (N($\\mathbf {x}$ )) , 0 ) where $\\Phi _s(\\tau ) \\equiv (1 + e^{-s\\tau })^{-1}$ is the sigmoid function used as a smooth approximator of the occupancy indicator function $\\mathcal {O}(\\mathbf {x}) \\equiv \\mathbf {1}[\\mathbf {N}(\\mathbf {x}) \\ge 0] $ ."
],
[
"Sampling Procedure",
"Existing work that targets optical cameras leverages ray optics where sampling points along a ray originating at some pixel is sufficient to approximate the rendering loss.",
"On the contrary, our rendering loss in Eq.",
"REF requires producing point samples along the arc at $p_i=(r_i, \\theta _i)$ as well as samples along each acoustic ray.",
"To obtain a balanced dataset of zero and non-zero intensity samples when processing an image, we sample $\\mathbf {N}_{\\mathcal {P}^1}$ random image pixels as well as $\\mathbf {N}_{\\mathcal {P}^2}$ pixels with an intensity $I(r_i, \\theta _i)$ greater than a threshold.",
"Let $\\mathcal {P}$ be the set of sampled pixels.",
"For each pixel $p_i \\in \\mathcal {P}$ , we use stratified sampling to obtain samples along the arc.",
"We discretize the elevation range $[-\\phi _{\\text{min}}, \\phi _{\\text{max}}]$ into $\\mathbf {N}_{\\mathcal {A}}$ equally spaced angles.",
"Hence, the difference between two consecutive angles is $\\Delta \\phi = \\frac{\\phi _{\\text{max}} - \\phi _{\\text{min}}}{\\mathbf {N}_{\\mathcal {A}}}$ .",
"We perturb these angles by adding $\\mathbf {N}_{\\mathcal {A}}$ randomly generated noise values $ \\sim \\text{Uniform}(0, 1) \\Delta \\phi $ to obtain a set of points $\\mathbf {A}_p = \\lbrace \\mathbf {P}_p=(r_i, \\theta _i, \\phi _{\\mathbf {P}_p})) \\rbrace $ on the arc.",
"For each sampled point $\\mathbf {P}_p$ , we first construct the acoustic ray $\\mathcal {R}_{\\mathbf {P}_p} $ which starts at the acoustic center of the sonar and terminates at $\\mathbf {P}_p$ and then sample $\\mathbf {N}_{\\mathcal {R}} - 1$ points along each ray.",
"Specifically, we first sample $\\mathbf {N}_{\\mathcal {R}} - 1$ range values $r^{\\prime }$ such that $r^{\\prime } < r$ and $r^{\\prime }=i\\epsilon _r $ for some $i>0$ ($\\epsilon _r$ being the sonar range resolution).",
"We obtain the set of points $\\mathbf {R}_{\\mathbf {P}_p} = \\lbrace \\mathbf {p} = (r^{\\prime }, \\theta , \\phi _{\\mathbf {P}_p}) \\rbrace $ .",
"The points $\\mathbf {R}_{\\mathbf {P}_p} \\cup \\mathbf {A}_p$ constitute a set of $\\mathbf {N}_{\\mathcal {R}}$ points along the ray ($\\mathbf {N}_{\\mathcal {R}} - 1$ points along the ray + exactly 1 point on the arc).",
"Finally, we perturb the range value of all points by adding uniformly distributed noise $ \\sim \\text{Uniform}(0, 1) \\epsilon _r$ (Fig.",
"REF ).",
"Figure: Sampling along radius rr.",
"We first sample range bins andthen sample one point in each bin (green points).",
"This is the set 𝐑 𝐏 p \\mathbf {R}_{\\mathbf {P}_p}.",
"The black point is the perturbed point on the arc 𝐏 p \\mathbf {P}_p.Note that the points $\\mathbf {R}_{\\mathbf {P}_p} \\cup \\mathbf {A}_p$ are expressed in spherical coordinates in the local sonar coordinate frame and hence need to be re-expressed in a global reference frame common to all sonar poses.",
"We first transform the points to Cartesian coordinates: x = r () () y = r () () z = r () and then transform to world frame by multiplying with the sonar to world transform $T_W^{\\text{sonar}} = \\begin{bmatrix}R_W^{\\text{sonar}} & t_W^\\text{{sonar}} \\\\\\mathbf {0}^T & 1\\end{bmatrix}$$.$ The resulting set of points expressed in world frame $\\mathbf {R}^W_{\\mathbf {P}_p} \\cup \\mathbf {A}^W_p$ are used as inputs to the SDF neural network $\\mathbf {N}$ .",
"Finally, the direction of each ray is defined by the unit vector D($\\mathbf {P}_p$ ) = TWsonar$\\mathbf {P}_p$ - tWsonar |TWsonar$\\mathbf {P}_p$ - tWsonar| Let $\\mathbf {X}$ be the set of all sampled points across all pixels, arcs and rays.",
"This is the input batch to the neural network."
],
[
"Discretized Image Formation Model",
"The discrete counterpart of the image formation model in Eq.",
"REF is: I(r, ) = $\\mathbf {P}_p$ Ap 1r$\\mathbf {P}_p$ T[$\\mathbf {P}_p$ ] [$\\mathbf {P}_p$ ] M($\\mathbf {P}_p$ ), where: $\\mathcal {A}_p$ is the arc located at $(r, \\theta )$ , $r_{\\mathbf {P}_p}$ is the range of the disturbed point $\\mathbf {P}_p$ on the arc, $\\mathbf {M}(\\mathbf {P}_p)$ is the predicted intensity at $\\mathbf {P}_p$ by the neural renderer, [pi] = 1 - ( - pipi+1 (p)dp ) is the discrete opacity [35] at a point $\\mathbf {p}_i$ ($\\mathbf {p}_i$ and $\\mathbf {p}_{i+1}$ being consecutive samples along the ray) which was further shown to equal: [pi] = (s(N(pi))-s(N(pi+1))s(N(pi)), 0).",
"Finally, T[$\\mathbf {P}_p$ ] = p1 R$\\mathbf {P}_p$ (1- [p1]) is the discrete transmittance value at $\\mathbf {P}_p$ (the endpoint of the ray).",
"This is the product of one minus the opacity values $\\alpha $ of all points on the acoustic ray excluding the $\\alpha $ at $\\mathbf {P}_p$ ."
],
[
"Training Loss",
"Our loss function is constituted of three terms: the intensity loss in addition to eikonal and $\\ell _1$ regularization terms.",
"The intensity loss Lint 1NP1 + NP2 p P||I(p) - I(p)||1, encourages the predicted intensity to match the intensity of the raw input sonar images.",
"The eikonal loss [40] Leik 1NRNA (NP1 + NP2) $\\mathbf {x}$ X (||N($\\mathbf {x}$ )||2 - 1)2, is an implicit geometric regularization term used to regularize the SDF encouraging the network to produce smooth reconstructions.",
"Finally, we draw inspiration from the NLOS volumetric albedo literature [41], [29], and add the $\\ell _1$ loss term Lreg 1NRNA (NP1 + NP2) $\\mathbf {x}$ X || [$\\mathbf {x}$ ]||1, to help produce favorable 3D reconstructions when we use sonar images from a limited set of view directions.",
"Hence, our final training loss term is: L = Lint + eik Leik + reg Lreg."
],
[
"Network Architecture",
"We model $\\mathbf {N}$ and $\\mathbf {M}$ as two MLPs each with 4 hidden layers of size 64 (Fig REF ).",
"We additionally apply positional encoding to the input spatial coordinates and use weight normalization similar to IDR.",
"While existing works that use optical cameras typically rely on larger networks to successfully learn high-frequency color and texture information, we found the proposed architecture to have sufficient capacity to learn different shapes from FLS images.",
"Decreasing the size of the network was especially important to handle GPU memory overhead during training caused by the added sampling dimension (arcs)."
],
[
"Evaluation",
"As our comparison metric, we use the mean and root mean square (RMS) Hausdorff distance defined as: $d_H(\\mathcal {M}_1, \\mathcal {M}_2) = \\max ( \\max _{\\mathbf {p} \\in \\mathcal {M}_1} \\min _{\\mathbf {q} \\in \\mathcal {M}_2} || p - q ||_2 , \\\\\\max _{\\mathbf {q} \\in \\mathcal {M}_2} \\min _{\\mathbf {p} \\in \\mathcal {M}_1} || p - q ||_2 )$ $\\mathcal {M}_1 $ and $\\mathcal {M}_2$ being respectively the ground truth (GT) and reconstructed meshes.",
"We evaluate our method against Back-Projection (BP) and ADMM, two state-of-the-art optimization-based methods for unsupervised object-centric 3D reconstruction using imaging sonar.",
"We use the implementation of Westman et al. [29].",
"BP is similar to the occupancy grid mapping method (OGM) as it uses the inverse sensor model to update the voxel occupancy while, however, ignoring the correlation between grid cells.",
"We note that both ADMM and BP generate a density field $\\mathbf {F}(\\sigma )$ .",
"Hence, for each possible density $\\sigma $ (i.e., $\\sigma \\in [0, 1]$ ), we extract a surface using Marching Cubes and report the metrics based on the best $\\sigma $ value.",
"The mesh quality generated by ADMM also depends on the regularization weight terms which we empirically tuned for each object.",
"With our approach, extracting the zero-level set of $\\mathbf {N}(x)$ directly generates a high-quality mesh.",
"However, for the purpose of metric generation, we also try different level-sets near zero: $\\mathcal {S} = \\lbrace \\mathbf {x}\\!\\in \\mathbb {R}^3\\!",
": \\mathbf {N}(\\mathbf {x}) = s \\ | \\ s \\in [-0.1, 0.1] \\rbrace $ .",
"We run our experiments on a system with an NVIDIA RTX 3090 GPU, an Intel Core i9-10900K, and 32GB of RAM.",
"Our network training time until convergence is $\\sim 6$ hours."
],
[
"Simulation",
"We use HoloOcean [43], [44], an underwater simulator to collect datasets of different objects of various shapes and sizes.",
"We use the simulator's default noise parameters; namely a multiplicative noise $w^{\\text{sm}} \\sim \\mathcal {N}(0, 0.15)$ and additive noise $w^{\\text{sa}} \\sim \\mathcal {R}(0.2)$ (where $\\mathcal {R}$ is the Rayleigh distribution and parameters are in units of normalized pixel intensity in the range $[0, 1]$ ).",
"We also enable the simulation of multipath effects.",
"The maximum range of the sonar was set to 8m.",
"Before feeding the raw data to the three algorithms, we perform minimal filtering of speckle noise in the images by zeroing out pixels whose intensities are less than a threshold.",
"After generating the meshes, we align them to the GT using ICP and report in Table REF the mean and RMS Hausdorff distance to the GT for different objects and two sonar vertical apertures (14 and 28).",
"Figure REF shows example 3D reconstructions obtained using each algorithm.",
"Figure: 3D reconstructions generated by each method using simulated data from HoloOcean with a 14 elevation angle.",
"Qualitatively, our method outputs more faithful 3D reconstructions compared to ADMM and BP.We see that our method produces more accurate reconstructions compared to the baselines in terms of 3D reconstruction accuracy and mesh coverage.",
"The neural network implicit regularization combined with the eikonal loss favors learning smooth surfaces while avoiding bad local minimas when the input images potentially do not contain enough information to completely constrain and resolve the elevation of every 3D point in space.",
"For large objects (specified by an asterisk in the table), we decreased the grid voxel resolution of the baseline methods by one-half (increased the voxel size from the default value of 0.025m to 0.05m) to prevent the system from running out of memory (OOM): the ADMM and BP baselines do not leverage stochastic updates and hence, need to construct the optimization objective by processing all images in one go.",
"This leads to memory overhead for larger objects, objects that require a fine discretization of the volume, or in the presence of a large number of non-zero pixel intensities A re-implementation of the baselines which solves the optimization problem using stochastic updates can help dealing with OOM errors..",
"In contrast, we train our renderer on a different subset of images in every iteration and use stochastic updates (the Adam optimizer) to optimize the function which significantly reduces memory requirements.",
"Table: Size (W×L×H\\text{W} \\times \\text{L} \\times \\text{H}), root mean square (RMS) and mean Hausdorff distance errors (all in meters) for different simulated objects.",
"For certain objects (*), we increased the voxel size from 0.0250.025m to 0.050.05m to prevent OOM errors with the baseline methods."
],
[
"Water Tank Experiments",
"We evaluate our method on real-world datasets of a test structure (Fig.",
"REF ) submerged in a test tank (Fig.",
"REF ) using a SoundMerics DIDSON imaging sonar mounted on a Bluefin Hovering Autonomous Underwater Vehicle (HAUV).",
"The sonar can achieve three different elevation apertures ($1, 14, 28$ ).",
"We test our method on three different datasets, one for each feasible aperture.",
"The vehicle uses a high-end IMU and a Doppler Velocity Log (DVL) to provide accurate vehicle pose information (i.e., minimal drift for the duration of data capture).",
"Fig.",
"REF shows the RMS and mean Hausdorff distance error of the three methods.",
"The quality of the mesh generated by ADMM and BP depends on the selected marching cubes threshold $\\sigma $ .",
"Hence, we plot the metrics generated using different $\\sigma $ s and report the best value.",
"With our method, we can extract the zero-level set of $\\mathbf {N}$ directly alleviating the need for a postprocessing step for surface generation.",
"Since the structure is submerged and lying at the bottom of the test tank (and hence, no sonar image captures the backside of the object), we limit the matching distance of the Hausdorff metric to $0.15\\text{m}, 0.2\\text{m}, \\text{ and } 0.25\\text{m}$ for the $1,14, \\text{ and } 28$ apertures respectively.",
"We see that our method generates higher quality reconstructions especially when using larger apertures: With 14, our method achieves an (RMS, Mean)=$(0.058\\text{m}, 0.040\\text{m})$ while BP and ADMM are respectively at $(0.077\\text{m}, 0.063\\text{m})$ and $(0.069\\text{m}, 0.052\\text{m})$ .",
"Similarly for a 28 aperture, our method achieves a lower (RMS, Mean) = $(0.072\\text{m}, 0.055\\text{m})$ compared to BP $(0.104\\text{m}, 0.079\\text{m})$ and ADMM $(0.091\\text{m}, 0.070\\text{m})$ .",
"Fig.",
"REF shows the resulting meshes for each method.",
"While all three methods perform well with a 1 aperture, the difference in reconstruction quality becomes visually more apparent as the aperture angle increases.",
"With a 14 aperture, we begin to lose the main feature of the object with ADMM and BP: the hole, short piling and crossbar are not easily discernible.",
"When the aperture is increased to 28, both baseline methods perform poorly: the hole, crossbar, and short piling are lost.",
"On the other hand, our proposed method successfully captures the major components of the structure for all three different apertures (a base, two vertical pilings, and a crossbar).",
"Figure: Plots showing the root mean square (RMS) and mean Hausdorff distance in meters for all three methods on the real datasets (1,14,1, 14, and 28 elevation apertures).",
"To easily compare against the baselines, we add the constant green dashed line to report our method's metrics.",
"Note however that our algorithm does not depend on the σ\\sigma values in the xx axis."
],
[
"Conclusion and Future Work",
"We proposed an approach for reconstructing 3D objects from imaging sonar which represents imaged surfaces as zero-level sets of neural networks.",
"We performed experiments on simulated and real datasets with different elevation apertures and showed that our method outperforms current state-of-the-art techniques for unsupervised 3D reconstruction using FLS in terms of reconstruction accuracy.",
"While existing volumetric methods can suffer from memory overhead as well as require a separate step to extract meshes from volumetric grids (a process often difficult and prone to error), our method allows for easy surface extraction from implicit representations and uses stochastic updates to lessen the computational requirements.",
"Our algorithm has some limitations, all of which create opportunities for future work.",
"First, we currently focus on single-object reconstruction but plan to expand our method to large-scale reconstruction of marine environments at the scale of harbors by taking inspiration from techniques such as Block-Nerf [45].",
"Second, our method is currently mostly suited for offline 3D reconstructions but using techniques such as Instant-NGP [46] and Relu-Fields [47] can bring it to real-time performance needed for robotic navigation applications.",
"Finally, all of our experiments now use only sonar but underwater robots are typically equipped with other sensors such as optical cameras.",
"Hence, another interesting direction from future work is to fuse multi-modal sensor inputs (acoustic and optical) where, for example, optical cameras are used to obtain high resolution models of specific interest areas in the scene while a sonar, with longer range, is used elsewhere.",
"45"
]
] | 2209.08221 |
[
[
"An Asymptotic-Preserving and Energy-Conserving Particle-In-Cell Method\n for Vlasov-Maxwell Equations"
],
[
"Abstract In this paper, we develop an asymptotic-preserving and energy-conserving (APEC) Particle-In-Cell (PIC) algorithm for the Vlasov-Maxwell system.",
"This algorithm not only guarantees that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model, but also preserves Gauss's law and energy conservation at the same time, thus it is promising to provide stable simulations of complex plasma systems even in the quasi-neutral regime.",
"The key ingredients for achieving these properties include the generalized Ohm's law for electric field such that the asymptotic-preserving discretization can be achieved, and a proper decomposition of the effects of the electromagnetic fields such that a Lagrange multiplier method can be appropriately employed for correcting the kinetic energy.",
"We investigate the performance of the APEC method with three benchmark tests in one dimension, including the linear Landau damping, the bump-on-tail problem and the two-stream instability.",
"Detailed comparisons are conducted by including the results from the classical explicit leapfrog and the previously developed asymptotic-preserving PIC schemes.",
"Our numerical experiments show that the proposed APEC scheme can give accurate and stable simulations both kinetic and quasi-neutral regimes, demonstrating the attractive properties of the method crossing scales."
],
[
"Introduction",
"Accurate simulation of collisionless plasma plays an important role in a broad range of applications, such as astrophysics, nuclear physics, inertial confinement fusion and material processing [26], [34], [20].",
"When classical continuum fluid models such as the magnetohydrodynamic equations fail to characterize the behavior of the plasmas under thermal or chemical non-equilibrium conditions, the Vlasov-Maxwell (VM) system serves as a viable kinetic description under such circumstances.",
"There has been a longstanding interest among mathematicians and physicists in designing numerical algorithms that inherent some intrinsic properties of the Vlasov-Maxwell equations [40], [39], [13], [14], [17], [33], [46], such as the preservation of conservation laws of total energy, momentum and charge [11], [12], [49], [37], [48], [44], [41], [4], mimicking the asymptotic transition among multiple scales at the discrete level [32], [22], [21], [4], [28], [42], or the exploration of modern exascale computer architectures for high performance computing [25].",
"Despite the simplicity in its formulation, the VM system is notoriously difficult to solve numerically due to the high dimensionality of the Vlasov equation.",
"In real simulations, if one adopts grid-based numerical methods to solve the six-dimensional Vlasov equation, the computational cost will be prohibitively expensive due to “the curse of dimensionality”.",
"One well-established way to tackle this issue is the Particle-In-Cell (PIC) method [35], [30], [5], [19], [18], [38], [3], in which Newton's second law of motion for a sequence of macro particles is solved in stead of the Vlasov equation.",
"The charge and current densities in the source term of the Maxwell equations are then calculated through a particle-to-grid assignment technique.",
"Once the electromagnetic fields are updated, they can be interpolated back to evaluate the Lorentz force to update the velocities and positions of the macro particles.",
"Besides the high dimensionality of the Vlasov equation, another extremely challenging issue for numerically simulating the VM system is the existence of multi-scale transitions [9], [8].",
"For instance, non-neutral and quasi-neutral regions could coexist in the domain of interest and evolve with time together.",
"To be more specific, there are two typical physical parameters of plasma, i.e.",
"the Debye length and the electron plasma period [10].",
"The former characterizes the average distance between ions and electrons, and the latter represents the oscillation period of the electrons.",
"When both parameters are relatively small compared with the typical macroscopic scales of the system, the corresponding system is called the quasi-neutral limit.",
"When quasi-neutral areas exist and the kinetic description is applied, one needs to adopt very small spatial and temporal steps in order to obtain accurate and stable simulations.",
"This is time consuming and practically prohibitive.",
"In order to overcome this numerical obstacle and accelerate the calculation where non-neutral and quasi-neutral areas coexist, the asymptotic-preserving (AP) method has been widely used in plasma physics [21], [22], [23], [16], [24], [47], [15], [1].",
"In Degond et al.",
"[21], a reformulated VM system is proposed, which unifies the models of non-neutral and quasi-neutral limit in a single set of equations by a generalized Ohm's law, allowing for a smooth transition from one model to the other.",
"The PIC method designed based on this reformulated VM system has proven to be stable in both numerically resolved and under-resolved cases, which makes it possible to simulate multiscale and complex systems with much less computational cost.",
"However, when one approximates the probability distribution function in the reformulated VM system via macro particles, the generalized Ohm's law and Newton's second law for particle motion both exist in the resultant particle-Maxwell system which can be viewed as a continuous and kinetic description of the same set of equations.",
"This redundancy destroys the energy conservation of the reformulated particle-Maxwell system.",
"To the best of our knowledge, there is no asymptotic-preserving schemes for the PIC discretization that also preserve the energy conservation law.",
"With the help of the reformulated VM model [21], in this paper we develop an asymptotic-preserving and energy-conserving scheme that are efficient in both quasi-neutral and non-neutral systems.",
"The main idea draws inspiration from the Lagrange multiplier method proposed by Antoine et al.",
"[2].",
"However, a direct application of this technique will destroy the discrete Gauss' law, which makes the side effects outweigh the benefits.",
"In this work, we properly introduce a nonlocal Lagrange parameter at each time step to correct the kinetic energy, without destroying the physical properties, such as the charge conservation and the asymptotic-preserving nature of the discrete algorithm.",
"The Lagrange parameter is solved through a scalar quadratic equation and the computational complexity of this APEC scheme is almost the same as the original AP algorithm.",
"We perform many numerical results to demonstrate the attractive feature of the scheme in this paper.",
"The organization of this paper is as follows.",
"In Section , we introduce the classical VM system and the reformulated Vlasov-Maxwell system proposed in Ref.",
"[21].",
"In Section , the original AP scheme is recast into an operator-splitting framework and the APEC scheme is proposed.",
"Specifically, we show the detailed steps of the APEC scheme for the Vlasov-Poisson system.",
"The numerical tests are performed with three classical plasma problems in the electrostatic limit, including the linear Landau damping, the bump-on-tail problem and the two-stream instability in Section , together with a detailed analysis of these results.",
"Finally, concluding remarks are made in Section ."
],
[
"The Vlasov-Maxwell system and its reformulated form",
"In this section, we will introduce the classical VM system and review the reformulated VM system [21] which is consistent with the quasi-neutral limit.",
"Then, we will briefly show the PIC method used to approximate the continuous Vlasov equation and the energy-conserving properties of the related models."
],
[
"The Vlasov-Maxwell system",
"Consider the collisionless plasma where the density distribution function $f_s(\\mathbf {x},\\mathbf {v},t)$ of species $s$ satisfies the Vlasov equation.",
"The electric field $\\mathbf {E}$ and magnetic field $\\mathbf {B}$ satisfy the Maxwell equations where the source terms, namely, the current density $\\mathbf {J}$ and electric charge density $\\rho $ , are calculated by the density distributions of all species.",
"The whole Vlasov-Maxwell system is given by, $&\\displaystyle \\partial _{t}f_s(\\mathbf {x},\\mathbf {v},t) +\\mathbf {v} \\cdot \\nabla _{\\mathbf {x}}f_s(\\mathbf {x},\\mathbf {v},t) +\\frac{q_se}{m_s}(\\mathbf {E}(\\mathbf {x},t)+\\mathbf {v} \\times \\mathbf {B}(\\mathbf {x},t))\\cdot \\nabla _{\\mathbf {v}}f_s(\\mathbf {x},\\mathbf {v},t)=0,\\\\&\\displaystyle \\frac{1}{c^{2}}\\partial _{t}\\mathbf {E}(\\mathbf {x},t)-\\nabla \\times \\mathbf {B}(\\mathbf {x},t)=-\\mu _{0} \\mathbf {J}(\\mathbf {x},t),\\\\&\\displaystyle \\partial _{t} \\mathbf {B}(\\mathbf {x},t)+\\nabla \\times \\mathbf {E}(\\mathbf {x},t)=0,\\\\&\\displaystyle \\nabla \\cdot \\mathbf {E}(\\mathbf {x},t)=\\frac{\\rho (\\mathbf {x},t)}{\\epsilon _{0}},\\\\&\\displaystyle \\nabla \\cdot {\\mathbf {B}(\\mathbf {x},t)}=0,$ where $(\\mathbf {x},\\mathbf {v},t)\\in \\Omega _{\\mathbf {x}} \\times \\Omega _{\\mathbf {v}} \\times \\mathbb {R}^+ $ are spatial, velocity and temporal variables, $m_s$ and $q_s$ are the mass and valence of species $s$ , $e$ is the elementary charge, $c$ is the speed of light, $\\epsilon _0$ and $\\mu _0$ are the permittivity and permeability in the vacuum.",
"The electron number density $n_s(\\mathbf {x},t)$ , the electric charge density $\\rho (\\mathbf {x},t)$ , the current density $\\mathbf {J}(\\mathbf {x},t)$ and the stress tensor $\\mathcal {S}_s(\\mathbf {x},t)$ are defined as follows: $n_s =\\int _{\\Omega _{\\mathbf {v}}} f_sd\\mathbf {v}, ~~\\rho =\\sum \\limits _s q_se n_s, ~~\\mathbf {J}=\\sum \\limits _{s}q_{s}e\\int _{\\Omega _{\\mathbf {v}}}f_{s}\\mathbf {v} d\\mathbf {v}, ~~\\mathcal {S}_s =\\int _{\\Omega _{\\mathbf {v}}}f_s\\mathbf {v}\\otimes \\mathbf {v}d\\mathbf {v}.$"
],
[
"The reformulated Vlasov-Maxwell system",
"We briefly describe the non-dimensionalization of physical variables and parameters by using the assumptions of magneto-hydrodynamic (MHD) model to derive the dimensionless form [27], [6], [21].",
"Let $x_0$ denote a typical length scale, $t_0$ a time scale, $v_0$ a velocity scale, $n_0$ a density scale, $T_0$ a temperature scale, $E_0$ and $B_0$ the electric field and magnetic field scales, respectively.",
"Under the MHD assumptions, one has $ex_0E_0=m_ev_0^{2}$ , $v_0B_0=E_0$ , $v_0 \\ll c$ , and $v_{0}=v_{0,th}$ , with $v_{0,th}$ being the thermal velocity, $m_e$ being the mass of electrons.",
"Especially, one sets the dimensionless ratio $v_0/c$ to be equal to the dimensionless Debye length $\\lambda =\\lambda _D/x_0=\\sqrt{m_e\\epsilon _0 v_{0,th}^2/(e^2n_0)}/x_0$ , that is, $v_0/c =\\lambda $ .",
"By consistently normalizing the physical variables and parameters based on the normalization constants given in Table REF , the resultant dimensionless problem takes the form: $&\\displaystyle \\partial _{t}f_s+\\mathbf {v}\\cdot \\nabla f_s+\\frac{q_s}{m_s}(\\mathbf {E}+\\mathbf {v}\\times \\mathbf {B})\\cdot \\nabla _{\\mathbf {v}}f_s=0,\\\\&\\displaystyle \\lambda ^{2}\\partial _{t}\\mathbf {E}-\\nabla \\times \\mathbf {B}=- \\mathbf {J},\\\\&\\displaystyle \\partial _{t}\\mathbf {B}+\\nabla \\times \\mathbf {E}=0,\\\\&\\displaystyle \\lambda ^{2}\\nabla \\cdot \\mathbf {E}=\\rho ,\\\\&\\displaystyle \\nabla \\cdot \\mathbf {B}=0.$ This system (REF ) obeys the energy conservation law as described in Theorem REF .",
"Table: Normalization of variables and simulation parametersTheorem 1 (Energy conservation law of the VM system.)",
"Under natural or periodic boundary conditions, the total energy of the VM system is preserved.",
"Namely, the following relation holds, $\\frac{\\text{d}}{\\text{d}t}\\left(\\frac{1}{2}\\sum _s\\int _{\\Omega _{\\mathbf {x}}}\\int _{\\Omega _{\\mathbf {v}}}f_s\\mathbf {v}^2d\\mathbf {x}d\\mathbf {v}+\\frac{\\lambda ^{2}}{2}\\int _{\\Omega _{\\mathbf {x}}} |\\mathbf {E}(\\mathbf {x})|^{2}d\\mathbf {x}+\\frac{1}{2}\\int _{\\Omega _{\\mathbf {x}}}|\\mathbf {B}(\\mathbf {x})|^{2}d\\mathbf {x}\\right)=0.",
"$ For this dimensionless form, the system is said to be in the quasi-neutral limit if one takes the Debye length $\\lambda \\rightarrow 0$ .",
"In this case, the Gauss law degenerates to $\\rho (\\mathbf {x},t) =0$ .",
"By a combination of Faraday's law and Ampère's law, one obtains $\\nabla \\times \\nabla \\times \\mathbf {E} = -\\partial _t \\mathbf {J},$ which is not well-posed since the irrotational part of $\\mathbf {E}$ is undetermined.",
"In order to deal with the degeneracy of the quasi-neutral limit, Degond et al.",
"[21] proposed to combine the generalized Ohm's law, which is obtained by taking the first order momentum of the Vlasov equation with respect to the velocity variable and incorporating with the Maxwell system to obtain a reformulated system.",
"This new system can degrade to the quasi-neutral limit system without degeneracy when $\\lambda $ takes the zero limit.",
"Here, we adopt this reformulated model and readers can refer to [21] for more details.",
"The reformulated system reads, $&\\displaystyle \\partial _{t}f_s+\\mathbf {v}\\cdot \\nabla f_s+\\frac{q_s}{m_s}(\\mathbf {E}+\\mathbf {v}\\times \\mathbf {B})\\cdot \\nabla _{\\mathbf {v}}f_s=0,\\\\&\\displaystyle \\lambda ^{2}\\partial _{t}^2 \\mathbf {E} +n_s\\mathbf {E} +\\nabla \\times (\\nabla \\times \\mathbf {E}) = \\mathbf {J}\\times \\mathbf {B}-\\nabla \\cdot \\mathcal {S}, \\\\&\\displaystyle \\partial _{t}\\mathbf {B}+\\nabla \\times \\mathbf {E}=0,\\\\&\\displaystyle \\lambda ^{2}\\nabla \\cdot \\mathbf {E}=\\rho ,\\\\&\\displaystyle \\nabla \\cdot \\mathbf {B}=0.$ The reformulated VM system (REF ) is equivalent to the original one (REF ), thus it also obeys the energy conservation law (REF ).",
"Nevertheless, once the density distribution function $f_s(\\mathbf {x},\\mathbf {v},t)$ is approximated by the macro particles, the Vlasov equation will be replaced by Newton's equations of motion for these particles and the generalized Ohm's law becomes an redundant equation in the system of the particle motion.",
"Thus, though the original particle-Maxwell system satisfies an energy conservation law, the reformulated particle-Maxwell system may violate the conservation law.",
"This will be discussed below."
],
[
"The reformulated particle-Maxwell system",
"In the PIC method, the distribution function $f_s$ of species $s$ is discretized as a summation of $N_{s}$ computational (or macro) particles, $f_{s}(\\mathbf {x},\\mathbf {v},t)=\\sum _{p=1}^{N_s}w_{p}S(\\mathbf {x}-\\mathbf {x}_{s, p}(t))\\delta (\\mathbf {v}-\\mathbf {v}_{s,p}(t)),$ where $\\delta $ is the Dirac-delta function, $S$ is the shape function of spatial variable $\\mathbf {x}$ , and $\\mathbf {x}_{s,p}$ and $\\mathbf {v}_{s,p}$ are the position and velocity of the $p$ th particle of species $s$ , respectively.",
"$w_{p}$ is the weight of the macro particle and hereafter it is assumed to be a constant.",
"In common PIC algorithms, the shape functions are usually assumed to be B-spline functions [25], [36], Gaussians, cut-off cosines or polynomials [31], [43], which are symmetric and compactly supported, and satisfy the property $\\int _{\\Omega _{\\mathbf {x}}}S(\\mathbf {x})d\\mathbf {x}=1$ .",
"Under the macro-particle approximation (REF ), if one takes the first order momentum of the Vlasov equation in Eq.",
"(REF ) with respect to the velocity field and uses the properties of the shape function, the following system of Newton's equations of motion for the macro particles [5], [19], [30] can be obtained, $&\\displaystyle \\frac{d\\mathbf {v}_{s,p}}{dt}=\\frac{q_{s}m_e}{m_{s}}[\\mathbf {E}(\\mathbf {x}_{s,p})+\\mathbf {v}_{s,p}\\times \\mathbf {B}(\\mathbf {x}_{s,p})], \\\\&\\displaystyle \\frac{d \\mathbf {x}_{s,p}}{d t}=\\mathbf {v}_{s,p},$ for $p=1, \\cdots , N_s$ , where $\\mathbf {E}(\\mathbf {x}_{s,p})$ and $\\mathbf {B}(\\mathbf {x}_{s,p})$ will be calculated by the interpolation of electromagnetic fields calculated from the Maxwell equations.",
"Here, for simplicity we assume that there is only a species of particles in the system $s=1$ and the ions are taken into account as background, if no ambiguity is introduced.",
"As a consequence, the original particle-Maxwell system takes the form: $&\\displaystyle \\frac{d\\mathbf {v}_{p}}{dt}=-\\mathbf {E}(\\mathbf {x}_{p})-\\mathbf {v}_{p}\\times \\mathbf {B}(\\mathbf {x}_{p}),\\\\&\\displaystyle \\frac{d \\mathbf {x}_{p}}{d t}=\\mathbf {v}_{p},\\\\&\\displaystyle \\lambda ^{2}\\partial _{t}\\mathbf {E}-\\nabla \\times \\mathbf {B}=- \\mathbf {J},\\\\&\\displaystyle \\partial _{t}\\mathbf {B}+\\nabla \\times \\mathbf {E}=0,\\\\&\\displaystyle \\lambda ^{2}\\nabla \\cdot \\mathbf {E}=1-n,\\\\&\\displaystyle \\nabla \\cdot \\mathbf {B}=0,$ with $p=1,\\cdots , N$ and $n$ being the density of electrons.",
"It is not difficult to prove that this standard particle-Maxwell model is energy-conserving, shown in Theorem REF .",
"Theorem 2 (Energy conservation law of the particle-Maxwell system) Under natural or periodic boundary conditions, the total energy of the particle-Maxwell system is preserved.",
"$\\frac{\\text{d}}{\\text{d}t}\\left(\\sum \\limits _{p=1}\\limits ^N\\frac{1}{2}w_p\\mathbf {v}_{p}^{2}+\\frac{\\lambda ^{2}}{2}\\int _{\\Omega _{\\mathbf {x}}} |\\mathbf {E}(\\mathbf {x})|^{2}d\\mathbf {x}+\\frac{1}{2}\\int _{\\Omega _{\\mathbf {x}}}|\\mathbf {B}(\\mathbf {x})|^{2}d\\mathbf {x}\\right)=0.$ However, as mentioned in the previous section, the above system is not well-posed at the quasi-neutral limit of $\\lambda \\rightarrow 0.$ By taking the first-order momentum of the Vlasov equation in (REF ) with respect to the velocity field and using the definitions in (REF ), one arrives at the generalized Ohm's law, $\\partial _t \\mathbf {J} = \\nabla \\cdot \\mathcal {S} +n\\mathbf {E} - \\mathbf {J}\\times \\mathbf {B}$ Combing the Faraday's law and the Ampère's law together with equation (REF ), one finally arrives at the reformulated particle-Maxwell system: $&\\displaystyle \\frac{d\\mathbf {v}_{p}}{dt}=-\\mathbf {E}(\\mathbf {x}_{p})-\\mathbf {v}_p\\times \\mathbf {B}(\\mathbf {x}_p),\\\\&\\displaystyle \\frac{d \\mathbf {x}_{p}}{d t}=\\mathbf {v}_{p}, ~~~~~~~~ p=1,\\cdots ,N,\\\\[5pt]&\\displaystyle \\lambda ^{2}\\partial _{t}^2 \\mathbf {E} +n\\mathbf {E} +\\nabla \\times (\\nabla \\times \\mathbf {E}) = \\mathbf {J}\\times \\mathbf {B}-\\nabla \\cdot \\mathcal {S},\\\\&\\displaystyle \\partial _{t}\\mathbf {B}+\\nabla \\times \\mathbf {E}=0,\\\\&\\displaystyle \\lambda ^{2}\\nabla \\cdot \\mathbf {E}=1-n,\\\\&\\displaystyle \\nabla \\cdot \\mathbf {B}=0.$ Once one approximates the Vlasov equation by Newton's equations of motion, the resulting reformulated particle-Maxwell system is not equivalent to the original particle-Maxwell system any more, and the energy conservation for this reformulated system cannot be simply proved anymore.",
"Remark 1 One can conjecture the existence of the energy conservation law under these boundary conditions for finite $\\lambda $ due to the equivalence between the reformulated Vlasov-Maxwell system and the standard Vlasov-Maxwell system.",
"Under natural or periodic boundary conditions, it can be shown that the total energy of the reformulated particle-Maxwell system satisfies the following balance equation, $\\frac{d^{2}}{dt^{2}}\\left[\\sum \\limits _{p=1}\\limits ^{N_{p}}\\frac{1}{2}w_{p}{\\mathbf {v}}_{p}^{2}+\\frac{\\lambda ^{2}}{2} \\int _{\\Omega _{\\mathbf {x}}}|{\\mathbf {E}}|^{2}d\\mathbf {x}\\right]=\\int _{\\Omega _{\\mathbf {x}}}\\mathbf {J}\\cdot \\partial _{t}{\\mathbf {E}}d{\\mathbf {x}}+\\lambda ^{2}\\int _{\\Omega _{\\mathbf {x}}}\\left( \\partial _{t}{\\mathbf {E}}\\right)^2 d\\mathbf {x}-\\int _{\\Omega _{\\mathbf {x}}}\\left( \\partial _{t}\\mathbf {B}\\right)^2 d\\mathbf {x}.$ A rigorous proof of energy conservation remains an open problem.",
"It is then reasonable to make attempt to develop a numerical scheme started from the reformulated particle-Maxwell system and make it preserve the Gauss law and total energy at the same time, achieving both the asymptotic preserving and energy conserving.",
"In our scheme, the violation of Gauss law will be corrected by the Boris correction.",
"The energy conservation will be satisfied through the Lagrange multiplier method which is illustrated in following section."
],
[
"AP and APEC schemes",
"Before introducing the numerical scheme, we should emphasize that the reformulated Maxwell system integrates the generalized Ohm's law, the Faraday equation and the Ampère equation to derive a new equation Eq.",
"(REF ) which avoids the degeneracy of the quasi-neutral limit [21].",
"In this section, we design the APEC algorithm from the classical Maxwell equations and the generalized Ohm's law.",
"Then we will prove that this numerical algorithm is the discretization of the reformulated Vlasov Maxwell system (REF )."
],
[
"The AP scheme",
"We briefly review the first-order AP algorithm developed in Ref.",
"[21] for the reformulated VM system and recast it into a semi-Lagrangian operator-splitting framework such that higher-order schemes can be developed straightforwardly.",
"We start with the generalized Ohm's law (REF ) and use the operator splitting technique to split the above equation into the following two subproblems: $& \\partial _t \\mathbf {J}=\\nabla \\cdot \\mathcal {S} - \\mathbf {J}\\times \\mathbf {B}, \\\\& \\partial _t \\mathbf {J}=n\\mathbf {E}.$ By the definitions of $\\mathbf {J}$ and $\\mathcal {S}$ in Eq.",
"(REF ), using macro particles to approximate $f(\\mathbf {x},\\mathbf {v},t)$ , and taking integral of Eq.",
"(REF ) over the spatial domain, one arrives at a sequence of ODEs for the macro particles $&\\displaystyle \\frac{d \\mathbf {x}_p}{dt}=\\mathbf {v}_p,\\\\&\\displaystyle \\frac{d \\mathbf {v}_p}{dt}=-\\mathbf {v}_p \\times \\mathbf {B}(\\mathbf {x}_p).$ Denote $\\mathbf {X}=(\\mathbf {x}_1,\\cdots , \\mathbf {x}_N)^T$ and $\\mathbf {V}=(\\mathbf {v}_1,\\cdots , \\mathbf {v}_N)^T$ .",
"Given the values of the $m$ th time step, one can discretize Eqs.",
"(REF ) and Eq.",
"() by a first-order scheme as follows: $&\\displaystyle \\frac{{\\mathbf {x}}_{p}^{m+1,\\ast }-{\\mathbf {x}}_{p}^{m}}{\\Delta t}={\\mathbf {v}}_{p}^{m+1,\\ast },\\\\&\\displaystyle \\frac{{\\mathbf {v}}_{p}^{m+1,\\ast }-{\\mathbf {v}}_{p}^{m}}{\\Delta t}=-{\\mathbf {v}}_{p}^{m}\\times {\\mathbf {B}}^{m}(\\mathbf {x}_{p}^{m}),$ and $\\frac{\\widetilde{\\mathbf {J}}^{m+1}- \\mathbf {J}^{m+1,\\ast }}{\\Delta t}=n^m\\widetilde{\\mathbf {E}}^{m+1},$ where $n^m=n(\\mathbf {X}^m)$ and ${\\mathbf {J}}^{m+1,\\ast } = \\mathbf {J}({\\mathbf {X}}^{m+1,\\ast },\\mathbf {V}^{m+1,\\ast })$ are defined through Eq.",
"(REF ).",
"The field quantities $\\mathbf {E}$ and $\\mathbf {B}$ are obtained by discretizing Ampère's and Faraday's laws in a semi-implicit manner: $&\\displaystyle \\lambda ^{2}\\frac{\\widetilde{\\mathbf {E}}^{m+1}-\\mathbf {E}^{m}}{\\Delta t}-\\nabla \\times \\mathbf {B}^{m+1}=-\\widetilde{\\mathbf {J}}^{m+1},\\\\&\\displaystyle \\frac{\\mathbf {B}^{m+1}-\\mathbf {B}^{m}}{\\Delta t}+\\nabla \\times \\widetilde{ \\mathbf {E}}^{m+1}=0,$ which can be reformulated as the following coupled system, $ \\begin{aligned}\\begin{pmatrix}{\\lambda ^{2}}/{\\Delta t}+{\\Delta t}n^m & -\\nabla \\times \\\\\\nabla \\times & {1}/{\\Delta t}\\end{pmatrix}\\begin{pmatrix}\\widetilde{\\mathbf {E}}^{m+1}\\\\\\mathbf {B}^{m+1}\\end{pmatrix}=\\begin{pmatrix}{\\lambda ^{2}}/{\\Delta t} & 0\\\\0 & {1}/{\\Delta t}\\end{pmatrix}\\begin{pmatrix}\\mathbf {E}^{m}\\\\\\mathbf {B}^{m}\\end{pmatrix}+\\begin{pmatrix}-\\mathbf {J}^{m+1,\\ast }\\\\0\\end{pmatrix}.\\end{aligned}$ This is a linear system, to be solved in each time step to obtain $\\widetilde{\\mathbf {E}}^{m+1}$ and $\\mathbf {B}^{m+1}$ .",
"In the second step, one updates the electric field $\\mathbf {E}^{m+1}$ via the Boris correction [7] for preserving Gauss's law.",
"One introduces an auxiliary variable $P$ such that $\\mathbf {E}^{m+1}=\\widetilde{\\mathbf {E}}^{m+1}-\\nabla P.$ By the charge conservation law, one has $ \\partial _t n =\\nabla \\cdot \\mathbf {J},$ which can be discretized as $\\frac{{n}^{m+1}-n^m}{\\Delta t}=\\nabla \\cdot {\\mathbf {J}}^{m+1}.$ Here, ${\\mathbf {J}}^{m+1}$ is obtained by discretizing Eq.",
"() by $\\frac{ {\\mathbf {J}}^{m+1}- \\mathbf {J}^{m+1,\\ast }}{\\Delta t}=n^m {\\mathbf {E}}^{m+1},$ where ${\\mathbf {J}}^{m+1,\\ast }$ can be obtained upon ${\\mathbf {X}}^{m+1,\\ast }$ and $\\mathbf {V}^{m+1,\\ast }$ once they are given.",
"Note that the form of ${\\mathbf {J}}^{m+1}$ is different from $\\widetilde{\\mathbf {J}}^{m+1}$ in Eq.",
"(REF ), which is crucial for the numerical algorithm to remain non-degenerate when the Debye length $\\lambda $ tends to 0.",
"Inserting Eq.",
"(REF ) into Gauss's law $\\lambda ^2 \\nabla \\cdot \\mathbf {E}^{m+1} = 1- n^{m+1},$ one arrives at a Poisson equation with variable coefficients for variable $P$ : $-\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^{2}}{\\Delta t^{2}}+n^m\\right)\\nabla P\\right]=\\frac{1-n^m}{\\Delta t^{2}}-\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^{2}}{\\Delta t^{2}}+ n^m\\right)\\widetilde{\\mathbf {E}}^{m+1} \\right] -\\frac{1}{\\Delta t}\\nabla \\cdot {\\mathbf {J}}^{m+1,\\ast }.$ Taking divergence on both sides of Eq.",
"(REF ) and inserting the resultant into the above equation, one finally obtains $-\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^{2}}{\\Delta t^{2}}+n^m\\right)\\nabla P\\right]=\\frac{1-n^m}{\\Delta t^{2}}-\\frac{\\lambda ^{2}}{\\Delta t^{2}}\\nabla \\cdot \\mathbf {E}^{m}.$ The correction $P$ is then determined by solving this equation.",
"Finally, one advances the Newton's equations of motion to obtain the positions and velocities of macro particles at the $t^{m+1}$ by using the following scheme, $&\\displaystyle \\frac{\\mathbf {v}_{p}^{m+1}-\\mathbf {v}_{p}^{m}}{\\Delta t}=- \\mathbf {E}^{m+1}(\\mathbf {x}_{p}^{m})-\\frac{\\mathbf {v}^{m+1}_p +\\mathbf {v}^{m}_p}{2}\\times \\mathbf {B}^{m}(\\mathbf {x}_{p}^{m}), \\\\&\\displaystyle \\frac{\\mathbf {x}_{p}^{m+1}-\\mathbf {x}_{p}^{m}}{\\Delta t}=\\mathbf {v}_{p}^{m+1},$ for $p=1, \\cdots , N$ , where the fields at the particle positions are obtained by the interpolation using the shape function."
],
[
"The APEC scheme with Lagrange multiplier",
"We now proceed to construct an APEC algorithm based on the above AP scheme.",
"The main idea for achieving the energy conservation is the decomposion of the electric force into two contributions and introducing a Lagrange multiplier to adjust the kinetic energy in the current step so that the total energy is conserved.",
"The corrected velocities in the current step will then influence the computation of the electromagnetic fields.",
"The first step for the APEC scheme resembles that of the AP scheme in the previous section, i.e.",
"$\\widetilde{\\mathbf {J}}^{m+1}$ is computed using Eqs.",
"(REF )-(REF ).",
"We then solve Eq.",
"(REF ).",
"To be convenient, we split the electric and magnetic fields into two contributions, $\\widetilde{\\mathbf {E}}^{m+1}=\\widetilde{\\mathbf {E}}^{m+1}_1+\\widetilde{\\mathbf {E}}_2^{m+1}$ and $ {\\mathbf {B} }^{m+1}= {\\mathbf {B}}^{m+1}_1+ {\\mathbf {B}}_2^{m+1}$ , and they are governed by: $\\begin{aligned}\\begin{pmatrix}{\\lambda ^{2}}/{\\Delta t}+{\\Delta t}n^m & -\\nabla \\times \\\\\\nabla \\times & {1}/{\\Delta t}\\end{pmatrix}\\begin{pmatrix}\\widetilde{\\mathbf {E}}_1^{m+1}\\\\\\mathbf {B}^{m+1}_1\\end{pmatrix}=\\begin{pmatrix}{\\lambda ^{2}}/{\\Delta t} & 0\\\\0 & {1}/{\\Delta t}\\end{pmatrix}\\begin{pmatrix}\\mathbf {E}^{m}\\\\\\mathbf {B}^{m}\\end{pmatrix},\\end{aligned}$ and $\\begin{aligned}\\begin{pmatrix}{\\lambda ^{2}}/{\\Delta t}+{\\Delta t}n^m & -\\nabla \\times \\\\\\nabla \\times & {1}/{\\Delta t}\\end{pmatrix}\\begin{pmatrix}\\widetilde{\\mathbf {E}}_2^{m+1}\\\\\\mathbf {B}^{m+1}_2\\end{pmatrix}=\\begin{pmatrix}-\\mathbf {J}^{m+1,\\ast }\\\\0\\end{pmatrix}.\\end{aligned}$ After obtaining $\\lbrace \\widetilde{\\mathbf {E}}^{m+1}_1,\\widetilde{\\mathbf {E}}^{m+1}_2 \\rbrace $ and $\\lbrace {\\mathbf {B}}^{m+1}_1,{\\mathbf {B}}^{m+1}_2 \\rbrace $ , one uses Boris correction Eq.",
"(REF ) to update the electric field.",
"Here, the main difference from the previous AP scheme is that we have two parts of the corrected electric field $\\mathbf {E}^{m+1}$ , $\\mathbf {E}^{m+1}_1 = \\widetilde{\\mathbf {E}}_1^{m+1} ,~\\hbox{and}~\\mathbf {E}^{m+1}_2 = \\widetilde{\\mathbf {E}}_2^{m+1}-\\nabla P.$ The position and velocity of each macro particle is also split into two contributions according to the forces induced by $\\lbrace \\widetilde{\\mathbf {E}}^{m+1}_1,\\widetilde{\\mathbf {E}}^{m+1}_2 \\rbrace $ and $\\lbrace {\\mathbf {B}}^{m+1}_1,{\\mathbf {B}}^{m+1}_2 \\rbrace $ .",
"The discretization schemes become, $&\\displaystyle \\frac{\\mathbf {v}_{p,1}^{m+1}-\\mathbf {v}_{p}^{m}}{\\Delta t}=- \\mathbf {E}^{m+1}_1(\\mathbf {x}_{p}^{m})-\\frac{\\mathbf {v}^{m+1}_{p,1} +\\mathbf {v}^{m}_{p}}{2}\\times \\mathbf {B}^{m}(\\mathbf {x}_{p}^{m}),\\\\&\\displaystyle \\frac{\\mathbf {x}_{p,1}^{m+1}-\\mathbf {x}_{p}^{m}}{\\Delta t}=\\mathbf {v}_{p,1}^{m+1},$ and $&\\displaystyle \\frac{\\mathbf {v}_{p,2}^{m+1}-\\mathbf {0}}{\\Delta t}=- \\mathbf {E}^{m+1}_2(\\mathbf {x}_{p}^{m})-\\frac{\\mathbf {v}^{m+1}_{p,2} }{2}\\times \\mathbf {B}^{m}(\\mathbf {x}_p^{m}),\\\\&\\displaystyle \\frac{\\mathbf {x}_{p,2}^{m+1}-\\mathbf {0}}{\\Delta t}=\\mathbf {v}_{p,2}^{m+1}.$ Finally, one corrects the velocities of the macro particles by introducing a Lagrange multiplier.",
"In each step, we determine a scalar constant $\\xi ^{m+1}$ to correct the velocities of all particles $\\mathbf {v}_{p}^{m+1} = \\mathbf {v}_{p,1}^{m+1}+\\xi ^{m+1} \\mathbf {v}_{p,2}^{m+1},$ for $p=1,\\cdots ,N$ , such that the total energy is conserved.",
"By the energy conservation law, $\\xi ^{m+1}$ satisfies, $ \\begin{aligned}& \\int (\\lambda ^2 |\\mathbf {E}^{m+1}|^2+|\\mathbf {B}^{m+1}|^2 ) + \\sum _p w_p\\Big ( |\\mathbf {v}_{p,1}^{m+1}|^2 +2 \\mathbf {v}_{p,1}^{m+1}\\cdot \\mathbf {v}_{p,2}^{m+1}\\xi ^{m+1}+ |\\mathbf {v}_{p,2}^{m+1}|^2 (\\xi ^{m+1})^2 \\Big ) \\\\&= \\lambda ^2 \\int |\\mathbf {E}^{0}|^2 + \\int |\\mathbf {B}^{0}|^2+ \\sum _p w_p |\\mathbf {v}_{p}^{0}|^2:=2W_0,\\end{aligned}$ with $W_0$ being the initial total energy.",
"This is a quadratic equation for the scalar value $\\xi ^{m+1}$ , and the exact solution close to 1 is adopted in our calculations.",
"One then updates the velocity field by equation (REF ) and the electromagnetic fields and the positions of the macro particles by $\\mathbf {E}^{m+1}=\\mathbf {E}^{m+1}_1 +\\mathbf {E}^{m+1}_2, \\mathbf {B}^{m+1} = \\mathbf {B}_1^{m+1}+\\mathbf {B}_2^{m+1}$ and $\\mathbf {x}_p^{m+1} = \\mathbf {x}_{p,1}^{m+1} +\\mathbf {x}_{p,2}^{m+1}.$ Note that the fields and particle positions remain unchanged in order to preserve Gauss's law.",
"Remark 2 The Boris correction is equivalent to a Lagrange multiplier method to enforce Gauss's law [3], where the correction field $P$ is the multiplier.",
"Here in our APEC scheme, the Lagrange multiplier $\\xi $ to enforce the energy conservation is a scalar constant.",
"It satisfies a quadratic equation and can be easily determined by an explicit formula.",
"It is noted that the Gauss law is preserved during the correction of the total energy.",
"It is essential in order that the PIC scheme can work accurately.",
"The steps of the APEC PIC scheme are summarized in Algorithm REF .",
"APEC PIC algorithm for Vlasov-Maxwell equations loaalgorithmAPEC PIC algorithm for Vlasov-Maxwell equations [1] Initialization: Given $\\Delta t$ , $\\Delta \\mathbf {x}$ and total time $T$ , assign velocities and positions of all particles and magnetic field.",
"Calculate initial charge densities through the particle-to-grid assignment.",
"Solve the Poisson equation to obtain Gauss-Law-satisfying initial electric field.",
"While $t^{m+1} < T$ Calculate $\\mathbf {J}^{m+1,\\ast }$ by the particle-to-grid assignment, and then solve the two splitting parts of $\\widetilde{\\mathbf {E}}^{m+1}$ and $\\mathbf {B}^{m+1}$ by Eqs.",
"(REF ) and (REF ).",
"Calculate $P$ with the Boris correction by Eq.",
"(REF ), and then obtain the two parts, $\\mathbf {E}_1^{m+1} = \\widetilde{\\mathbf {E}}_1^{m+1}$ and $\\mathbf {E}_2^{m+1} = \\widetilde{\\mathbf {E}}_2^{m+1} -\\nabla P$ .",
"Interpolate the electromagnetic fields on macro particles to get $\\mathbf {E}^{m+1}_1(\\mathbf {x}_p^m)$ , $\\mathbf {E}^{m+1}_2(\\mathbf {x}_p^m)$ , $\\mathbf {B}^{m+1}_1(\\mathbf {x}_p^m)$ and $\\mathbf {B}^{m+1}_2(\\mathbf {x}_p^m)$ .",
"Evolve velocities and positions of particles by Eqs.",
"(REF ) and (REF ).",
"Sum up two contributions of the electromagnetic fields and particle positions.",
"Update the velocities of macro particles by, $\\mathbf {v}_{p}^{m+1} = \\mathbf {v}_{p,1}^{m+1}+\\xi ^{m+1} \\mathbf {v}_{p,2}^{m+1},$ where parameter $\\xi ^{m+1}$ is determined by solving Eq.",
"(REF ).",
"Calculate the charge densities $1-n^{m+1}$ through the particle-to-grid assignment.",
"$m = m+1$ .",
"End While The AP particle scheme in Section REF is consistent with the reformulated particle-Maxwell system in both the non-neutral and the quasi-neutral regimes [21].",
"Similarly, our APEC algorithm also starts from the particle-Maxwell system and the current is calculated from the generalized Ohm's law.",
"Theorem REF shows that the APEC scheme is consistent with the reformulated particle-Maxwell system with the Boris correction for both finite $\\lambda $ and at the quasi-neutral limit $\\lambda \\rightarrow 0$ .",
"Theorem 3 The APEC scheme developed in Section REF is consistent with the reformulated particle-Maxwell system with the Boris correction: $&\\displaystyle \\frac{d\\mathbf {v}_{p}}{dt}=-\\mathbf {E}(\\mathbf {x}_p)-\\mathbf {v}_p \\times \\mathbf {B}(x_p),\\\\&\\displaystyle \\frac{d\\mathbf {x}_{p}}{d t}=\\mathbf {v}_{p},\\\\&\\displaystyle \\lambda ^{2}\\partial _{t}^2 \\widetilde{\\mathbf {E}} +n\\widetilde{\\mathbf {E}} +\\nabla \\times (\\nabla \\times \\widetilde{\\mathbf {E}}) = \\mathbf {J}\\times \\mathbf {B}-\\nabla \\cdot \\mathcal {S}, \\\\&\\displaystyle \\partial _{t}\\mathbf {B}+\\nabla \\times \\mathbf {E}=0,\\\\&\\displaystyle -\\lambda ^2 \\partial _t^2 \\Delta P - \\nabla \\cdot (n\\nabla P) = -\\lambda ^2 \\partial _t^2 \\nabla \\cdot \\widetilde{\\mathbf {E}} -\\nabla ^2:\\mathcal {S}-\\nabla \\cdot (n\\widetilde{\\mathbf {E}})+\\nabla \\cdot (\\mathbf {J}\\times \\mathbf {B}),\\\\&\\displaystyle \\nabla \\cdot \\mathbf {B}=0,\\\\&\\displaystyle \\mathbf {E} = \\widetilde{\\mathbf {E}}-\\nabla P.$ with $p=1,\\ldots ,N$ , for both finite $\\lambda $ and the quasi-neutral limit $\\lambda \\rightarrow 0$ .",
"Since Eqs.",
"(REF )(df) are the same as those in the standard particle-Maxwell system, we only need to prove that the APEC algorithm is the discretization of Eqs.",
"(REF )(abce).",
"The current $\\mathbf {J}^{m+1,\\ast }$ can be equivalently written as $ \\mathbf {J}^{m+1,\\ast } = \\mathbf {J}^m+\\Delta t \\nabla \\cdot \\mathcal {S}^m -\\Delta t \\mathbf {J}^m \\times \\mathbf {B}^m,$ which was shown by Degond et al.",
"[21] (the moment form therein).",
"This expression will be used below.",
"In the APEC scheme, the electromagnetic fields are the summation of the two contributions, $\\widetilde{\\mathbf {E}}^{m+1} = \\widetilde{\\mathbf {E}}_1^{m+1}+\\widetilde{\\mathbf {E}}_2^{m+1}$ , and $\\mathbf {B}^{m+1}=\\mathbf {B}_1^{m+1}+\\mathbf {B}_2^{m+1}$ .",
"The electromagnetic fields from Eqs.",
"(REF ) and (REF ) are equal to those solutions from Eq.",
"(REF ).",
"By Eq.",
"(REF )(b), one has $\\mathbf {B}^{m+1} = \\mathbf {B}^m -\\Delta t \\nabla \\times \\widetilde{\\mathbf {E}}^{m+1}$ .",
"Substituting this expression into the Ampère equation and using Eqs.",
"(REF ) and (REF ), one obtains, $\\frac{\\lambda ^2( \\widetilde{\\mathbf {E}}^{m+1} -\\mathbf {E}^m)}{\\Delta t^2} = \\frac{\\nabla \\times \\mathbf {B}^m - \\mathbf {J}^m}{\\Delta t} - \\nabla \\times \\nabla \\times \\widetilde{\\mathbf {E}}^{m+1}-n^m \\widetilde{\\mathbf {E}}^{m+1} -\\nabla \\cdot \\mathcal {S}^m +\\mathbf {J}^m \\times \\mathbf {B}^m.$ Since the Faraday equation has the following discretization, $\\frac{\\lambda ^2 (\\mathbf {E}^m - \\mathbf {E}^{m-1})}{\\Delta t} = \\nabla \\times \\mathbf {B}^m -\\mathbf {J}^m,$ Eq.",
"(REF ) can be further expressed as, $\\frac{\\lambda ^2( \\widetilde{\\mathbf {E}}^{m+1} -2\\mathbf {E}^m +\\mathbf {E}^{m-1})}{\\Delta t^2} + \\nabla \\times \\nabla \\times \\widetilde{\\mathbf {E}}^{m+1} +n^m\\widetilde{\\mathbf {E}}^{m+1} +\\nabla \\cdot \\mathcal {S}^m -\\mathbf {J}^m\\times \\mathbf {B}^m =0.$ Apparently, this equation is the discretization of Eq.",
"(REF ) which is the reformulated Ampère law.",
"The Boris correction $P$ satisfies, $-\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^2}{\\Delta t^2} +n^m\\right)\\nabla P\\right] = \\frac{1-n^m}{\\Delta t^2} -\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^2}{\\Delta t^2}+n^m\\right) \\widetilde{\\mathbf {E}}^{m+1}\\right] -\\frac{1}{\\Delta t} \\nabla \\cdot \\mathbf {J}^{m+1,\\ast }.$ If one inserts Eq.",
"(REF ) into this elliptic equation, one derives, $-\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^2}{\\Delta t^2} +n^m\\right)\\nabla P\\right] =& \\frac{1-n^m-\\lambda ^2 \\nabla \\cdot \\widetilde{\\mathbf {E}}^{m+1}}{\\Delta t^2} -\\nabla \\cdot \\left(n^m\\widetilde{\\mathbf {E}}^{m+1}\\right)\\nonumber \\\\&-\\frac{1}{\\Delta t}\\nabla \\cdot \\mathbf {J}^m -\\nabla ^2:\\mathcal {S}^m +\\nabla \\cdot \\left(\\mathbf {J}^m\\times \\mathbf {B}^m\\right).$ The corrected electric field at the $m$ th step satisfies the Gauss law and Ampère equation, i.e., $1-n^m &= \\lambda ^2 \\nabla \\cdot \\mathbf {E}^m,\\\\-\\frac{1}{\\Delta t}\\nabla \\cdot \\mathbf {J}^m &= \\frac{\\lambda ^2}{\\Delta t^2}\\nabla \\cdot (\\mathbf {E}^m -\\mathbf {E}^{m-1}).$ With these two expressions, Eq.",
"(REF ) can be rearranged as $-\\nabla \\cdot \\left[\\left(\\frac{\\lambda ^2}{\\Delta t^2} +n^m\\right)\\nabla P\\right] =& - \\frac{\\lambda ^2}{\\Delta t^2}\\left(\\nabla \\cdot \\widetilde{\\mathbf {E}}^{m+1} -2\\nabla \\cdot \\mathbf {E}^m +\\nabla \\cdot \\mathbf {E}^{m-1} \\right)-\\nabla \\cdot \\left(n^m\\widetilde{\\mathbf {E}}^{m+1}\\right)\\nonumber \\\\&-\\frac{1}{\\Delta t}\\nabla \\cdot \\mathbf {J}^m -\\nabla ^2:\\mathcal {S}^m +\\nabla \\cdot \\left(\\mathbf {J}^m\\times \\mathbf {B}^m\\right).$ This is the discretization of Eq.",
"().",
"The particle velocities $\\mathbf {v}_{p,1}^{m+1}$ , $\\mathbf {v}_{p,2}^{m+1}$ and positions $\\mathbf {x}_{p,1}^{m+1}$ , $\\mathbf {x}_{p,2}^{m+1}$ are updated with two parts of the electric fields from Eqs.",
"(REF ) and (REF ).",
"This splitting operation is consistent with the discretization of the un-splitting motion equations.",
"The additional advantage of this splitting technique, i.e., $\\mathbf {v}_p^{m+1} = \\mathbf {v}_{p,1}^{m+1} +\\xi ^{m+1} \\mathbf {v}_{p,2}^{m+1}$ , is the energy conservation being preserved exactly.",
"Moreover, $\\mathbf {v}_{p,2}^{m+1}$ and $\\mathbf {x}_{p,2}^{m+1}$ in Eq.",
"(REF ) vanish at the $\\Delta t\\rightarrow 0$ limit, implying that a finite $\\xi ^{m+1}$ does not affect the consistency of the discretization for Newton's equations of motion.",
"Therefore, the whole algorithm is first-order consistent with the reformulated particle-Maxwell system with the Boris correction, and the energy is preserved exactly at the same time."
],
[
"Numerical results",
"We perform numerical results of the APEC PIC method.",
"We focus on numerical solutions of the one-dimensional electrostatic model of two species with ions being motionless treated as the uniform background and only electrons present in the model.",
"At the electrostatic limit, the magnetic field is disregarded $\\mathbf {B}=0$ and the Faraday equation $\\partial _{t} {\\mathbf {B}}+\\nabla \\times {\\mathbf {E}}=0$ reduces to $\\nabla \\times {\\mathbf {E}}=0$ , which indicates that the electric field is irrotational.",
"The electric field can be solved through the gradient of the electric potential, $\\mathbf {E}=-\\nabla \\phi $ .",
"Thus, in 1D, the original Vlasov-Maxwell system is reduced to the Vlasov-Poisson system, $&\\displaystyle \\partial _{t}f+ {v} \\partial _{x}f- {E} \\partial _{v}f=0,\\\\&\\displaystyle - \\lambda ^2 \\partial _{x} \\phi =1-n.$ Correspondingly, the particle-Poisson equations read, $&\\displaystyle \\frac{d x_{p}}{dt}=v_{p},\\\\&\\displaystyle \\frac{d v_{p}}{d t}=-E(x_{p}), p=1,\\cdots , N,\\\\&\\displaystyle - \\lambda ^2 \\partial _{x}^{2} \\phi =1-n.$ With periodic or homogeneous Dirichlet boundary condition of the Poisson equation, the particle-Poisson system has the following conservation law of energy, $\\frac{d}{dt}\\left[\\sum \\limits _{p=1}\\limits ^{N_{p}}\\frac{1}{2}w_{p}v_{p}^{2}+\\frac{\\lambda ^{2}}{2}\\int _{\\Omega _{x}}E(x)^{2}dx\\right]=0.$ The APEC PIC algorithm for the 1D Vlasov-Poisson equations can be obtained by following Algorithm REF with the magnetic field set to be zero.",
"We use central finite-difference scheme to solve the Poisson equation with the space domain $[a, b]$ being equidistantly divided into $N_x$ cells.",
"We use the fourth-order B-spline function as the shape function [25].",
"For the enforcement of the energy conserving, the quadratic equation of $\\xi $ is simplified to, $\\bar{A}\\xi ^{2}+\\bar{B}\\xi +\\bar{C}=2W_0$ , with coefficients $&\\bar{A}=w_p \\sum _p (v_{p,2}^{m+1})^2,~\\bar{B} = 2\\sum _p w_pv_{p,1}^{m+1} v_{p,2}^{m+1}, ~\\hbox{and}, \\nonumber \\\\&\\bar{C} = \\lambda ^2 \\sum _i ({E}_i^{m+1})^2 \\Delta x+w_p \\sum _p(v_{p,1}^{m+1})^2.$ This equation has two roots and we take the one close to 1 as the Lagrange multiplier.",
"If there are no real roots, we set $\\xi =1$ which reduces to the AP scheme described in Section REF .",
"We perform numerical calculations for three benchmark problems including the linear Landau damping, the bump-on-tail problem and the two-stream instability.",
"Besides the APEC scheme, we also do the calculation with the AP particle scheme shown in Section REF and the classical explicit scheme [5] for a comparison study.",
"The classical explicit scheme is built on the leapfrog scheme both the particle motion and the Maxwell equations together with a Boris correction for the enforcement of Gauss law.",
"However, this explicit algorithm is not exactly energy conserving, despite that the energy variation will be small for smaller temporal and spatial steps [37].",
"Algorithm displays the classical explicit PIC scheme for Vlasov-Poisson equations.",
"The classical explicit PIC scheme for Vlasov-Poisson equations loaalgorithmThe classical explicit PIC scheme for Vlasov-Poisson equations [1] Initialization: Given $\\Delta t$ , $\\Delta \\mathbf {x}$ and total time $T$ , assign velocities and positions of all particles.",
"Calculate initial charge densities through the particle-to-grid assignment.",
"Solve the Poisson's equation to obtain Gauss-Law-satisfying initial electric field.",
"While $t^{m+1} < T$ Update particle velocities by $ \\mathbf {v}_p^{m+1/2}-\\mathbf {v}_p^{m-1/2}=-\\Delta t\\mathbf {E}^m(\\mathbf {x}_p^m)$ and positions by $ \\mathbf {x}_p^{m+1} -\\mathbf {x}_p^m=\\Delta t \\mathbf {v}_p^{m+1/2}$ .",
"Calculate current and charge densities on grid sites, $\\mathbf {J}^{m+1}=\\mathbf {J}(\\mathbf {X}^{m+1}, \\mathbf {V}^{m+1/2})$ and $n^{m+1}=n(\\mathbf {X}^{m+1})$ , through the particle-to-grid assignment.",
"Update the electric field by $\\lambda ^2(\\widetilde{\\mathbf {E}}^{m+1} -\\mathbf {E}^m)=- \\Delta t \\mathbf {J}^{m+1}$ and obtain the Boris correction $\\nabla P$ through $\\lambda ^2 \\Delta P=\\lambda ^2 \\nabla \\cdot \\widetilde{\\mathbf {E}}^{m+1}-(1-n^{m+1})$ .",
"Correct the electric field, $\\mathbf {E}^{m+1} =\\widetilde{\\mathbf {E}}^{m+1}-\\nabla P$ .",
"$m=m+1$ .",
"End While"
],
[
"Landau damping",
"To investigate our APEC algorithm, we first test the performance with a classical experiment in plasma physics, the linear Landau damping [21], [11], [29], [4].",
"The uniformly distributed electrons are slightly disturbed with an initial distribution function, $f_{0}(x,v)=\\left[1+\\alpha \\cos \\left(\\frac{x}{2}\\right)\\right]\\frac{1}{\\sqrt{2\\pi }}e^{-\\frac{v^{2}}{2}},$ where $\\alpha $ is a small constant and we take $\\alpha =0.05$ in the test.",
"Ions are motionless and form a uniform background.",
"The space domain is $[0,4\\pi ]$ and the dimensionless parameter takes $\\lambda =1$ .",
"The Poisson equation is solved with the periodic boundary condition.",
"Figure REF (a) displays the results of the classical explicit, the AP and the APEC algorithms for the resolved case with $N_x=250$ , $\\Delta t = 0.05$ and $N = 1\\times 10^6$ .",
"All PIC algorithms capture the correct damping rate of the perturbed plasma.",
"The results of APEC and explicit algorithms give almost the same rate as the theoretical rate (the dotted line) during the damping period, while the AP algorithm is slightly overdamping.",
"Figure REF (b) indicates the change of the total energy ($\\Delta W= W-W_0$ ) during the simulations and in this case $W_0=6.3176$ .",
"One can observe that the energy of the proposed APEC algorithm is exactly preserved, but the results of AP and the classical explicit methods monotonically decrease with time with different dissipative rates.",
"Figure: Resolved case of the Landau damping calculated by the classical explicit, the AP and the APEC schemes.",
"(a) Electric energy with time; (b) Total energy change with time.For practical applications, it is important to validate the effectiveness of the PIC methods for simulations with large time steps.",
"Figure REF presents the results of the under-resolved case for the three PIC schemes with the same $\\lambda $ , $N_x$ and $N$ but a large time step $\\Delta t=2$ .",
"Apparently, both the AP and APEC algorithms are stable while the explicit algorithm is unstable when the time step is too large.",
"The results of the AP-type schemes are consistent with those of energy-conserving schemes [11].",
"Figure: Under-resolved case of the Landau damping calculated by the classical explicit, the AP and the APEC schemes.",
"(a) Electric energy with time; (b) Total energy with time."
],
[
"Bump-on-tail instability",
"In the second benchmark test, we consider the bump-on-tail instability problem, where the velocity distribution has multiple peaks.",
"The initial density of the Vlasov-Poisson equations is given by, $f_0(x,v) =g(v)\\left[1+\\delta \\cos (\\kappa x)\\right],$ where function $g$ is the summation of two normal distributions with different means and variances, $\\displaystyle g(v) = Ce^{-v^2/2} +\\alpha e^{- (v-v_d)^2/ (2v_t^2)},$ with constant $C$ being determined to satisfy $\\int g(v) dv=1$ .",
"Here we use the similar settings as in Ref.",
"[22] with $\\Omega _x= (0,20\\pi )$ , $\\delta =0.04$ , $\\kappa =0.3$ , $v_d=4.5$ , $v_t=0.5$ and $\\alpha =2/9$ .",
"The Poisson equation is endowed with a homogeneous Dirichlet boundary condition.",
"For the resolved case, we set the Debye length $\\lambda =1$ , the plasm period $\\tau _p=\\lambda $ , the total time $T=200$ , the number of macro particles $N=1\\times 10^5$ and the grid parameters $N_x=500$ and $\\Delta t=0.01$ .",
"Figure REF displays the electric energy and the total energy change with time evolution.",
"One can observe similar evolutions of the electric energy for the three PIC algorithms, which are consistent with literature results [22], [11], [4].",
"Specifically, the instability increases rapidly from $t = 10$ and the electric energy attains the same maximum value at $t = 20$ for all numerical schemes.",
"The initial total energy of this system is $W_0= 92.0822$ .",
"Panel (b) of the figure shows that the total energy of the APEC method is conserved for the simulation time up to $T=200$ , but the AP scheme dissipates the energy to $\\sim 95\\%$ of the initial energy at time $T$ .",
"The explicit scheme performs well in the energy conservation for this resolved case and the total energy has a small oscillation around $W_0$ .",
"In Figure REF , the velocity distributions for different times are displayed.",
"Obviously, the results of the three PIC algorithms are nearly the same, and in agreement with those in literature [4].",
"There are two peaks at earlier times and then the smaller one gradually disappeared from $t=10$ when the instability increases.",
"Figure: Resolved case of the bump-on-tail instability calculated by the classical explicit, the AP and the APEC schemes.",
"(a) Electric energy with time; (b) Total energy change with time.Figure: Velocity distributions at different times for the resolved case of the bump-on-tail instability.",
"(a) the classical explicit algorithm; (b) the AP algorithm; and (c) the APEC algorithm.The results of the under-resolved case are shown in Figure REF , where we set the simulation parameters as the Debye length $\\lambda =0.1$ and the number of macro particles $N=1\\times 10^6$ .",
"The grid parameters are $N_x=20$ and $\\Delta t= 0.4$ , which satisfy $\\Delta x>\\lambda $ and $\\Delta t>\\tau _p$ .",
"In this setup, the initial total energy is $W_0=106.1615$ .",
"As expected, the classical explicit algorithm is unstable and blows up in this case due to the large time step, while both AP and APEC algorithms provide the stable results.",
"Again, the total energy of the APEC algorithm is conserved for the whole simulation time for this under-resolved case.",
"With the same setup, the velocity distributions at different times are displayed in Figure REF .",
"One can see that the explicit algorithm heats up the system quickly such that the macro particles with smaller velocities vanish soon.",
"One can see that the AP and APEC algorithms give similar results, in consistent with those in Refs.",
"[4], [22].",
"Since the APEC scheme has no energy dissipation, the number of the macro particles with large velocities ($v_p>8$ ) is slightly bigger that the results of the AP scheme.",
"This is in agreement with our intuition.",
"Additionally, in order to validate the APEC method with different time steps, we show in Figure REF the results of parameters $\\lambda =0.1$ and $N_x=20$ for $\\Delta t=0.01, 0.1$ and $0.2$ , where the electric energy with the time evolution is calculated.",
"One can clearly observe that the APEC algorithm is stable and the electric energy curve has similar behavior for different time steps.",
"Figure: Under-resolved case of the bump-on-tail instability calculated by the classical explicit, the AP and the APEC schemes.",
"(a) Electric energy with time; (b) Total energy with time.Figure: Velocity distributions at different times for the under-resolved case of the bump-on-tail instability.",
"(a) the classical explicit algorithm; (b) the AP algorithm; and (c) the APEC algorithm.Figure: Electric energy with time by the APEC algorithm for the bump-on-tail instability with different time steps Δt=0.01,0.1\\Delta t=0.01, 0.1 and 0.20.2."
],
[
"Two-stream instability",
"The third benchmark test is the two-stream instability which has been often studied in plasma literature [29], [11], [45].",
"The initial density distribution is $f_0(x,v) =\\frac{1}{2\\sqrt{2\\pi }\\sigma }\\left[e^{-(v+v_b)^2/2\\sigma ^2} + e^{-(v-v_b)^2/2\\sigma ^2}\\right] \\left( 1+\\alpha \\cos x\\right),$ with $v_b=\\sqrt{3}/2$ being the beam speed, $x\\in (0,2\\pi )$ and $\\sigma =0.008$ .",
"For the resolved case, we set $\\lambda =0.5$ , $\\alpha =0.005$ , $N_x=64$ , $N=1\\times 10^5$ and $\\Delta t=0.02$ .",
"The initial total energy is $2.3565$ .",
"The periodic boundary condition is used for the Poisson equation.",
"The results are displayed in Figure REF .",
"Again, all the methods give similar results and have similar growing rates on the electric energy at the begin.",
"However, when $t$ is about 9, the total energies change significantly for the results predicted by the explicit and the AP algorithms due to the interaction of plasmas of opposite speeds.",
"During the interaction and the following run, the APEC method conserves the energy and shows the best performance among the algorithms.",
"Without the energy conservation, the AP algorithm has dissipation.",
"Interestly, the explicit method dissipates the energy during the interaction, but returns to the initial energy at later time.",
"Figure REF presents the distributions of the macro particles in the phase space for three snapshots.",
"Clearly, during the growth period of the instability with the same rate shown in Figure REF (a), the three PIC algorithms predict similar results.",
"But after the instability getting saturated at $t\\approx 10$ , the particle distributions of the three methods become pretty different in some portions of the phase space.",
"Figure: Resolved case of the two-stream instability calculated by the classical explicit, the AP and the APEC schemes.",
"(a) Electric energy with time; (b) Total energy with time.Figure: Phase-space distributions of the macro particles at time T=10,15T=10, 15 and 20 for the two-stream instability.",
"(abc) the classical explicit algorithm; (def) the AP algorithm; and (ghi) the APEC algorithm.We then consider the under-resolved case by setting $\\lambda =0.005$ , $N_x=64$ and $\\Delta t=0.1$ .",
"The other parameters are the same as the resolved case.",
"Figures REF displays the results of the electric energy and the total energy evolutions with time.",
"Figure REF shows the APEC results for different time steps.",
"Here, the initial total energy is $W_0=5.021160162244403$ .",
"Similar performance as the bump-on-tail instability can be observed for the three algorithms, i.e., the AP-type methods present stable simulations, but the explicit method is unstable.",
"One can see from Figure REF (b) that the energy charge of the APEC algorithm is at the machine precision, demonstrating the promising performance of this algorithm.",
"Figure: Under-resolved case of the two-stream instability calculated by the classical explicit, the AP and the APEC schemes.",
"(a) Electric energy with time; (b) Total energy with time.Figure: The APEC algorithm for the two-stream instability with different time steps Δt=0.005,0.01\\Delta t=0.005, 0.01 and 0.050.05.",
"(a) Electric energy with time; (b) Total energy change with time."
],
[
"Conclusion",
"We have proposed an asymptotic-preserving and energy-conserving PIC algorithm for the Vlasov-Maxwell system.",
"The algorithm can give accurate numerical solutions on both non-neutral and quasi-neutral regimes and the total energy is preserved exactly.",
"In the quais-neutral regimes, the APEC method is still stable with large time steps and can capture the main mechanism of the system while the reference explicit scheme will blow up.",
"Several numerical tests are performed to demonstrate the attractive performance of the new algorithm.",
"In future, we will extend the PIC algorithm to high dimensional Vlasov-Maxwell systems for more complex plasma applications."
],
[
"Acknowledgement",
"This work is funded by the Strategic Priority Research Program of Chinese Academy of Sciences (grant Nos.",
"XDA25010402, XDA25010403, XDA250050500).",
"L. Ji acknowledges the support from China Postdoctoral Science Foundation No.",
"2021M702141.",
"Z. Yang acknowledges the support from the NSFC (No.",
"12101399) and the Shanghai Sailing Program (No.",
"21YF1421000).",
"Z. Xu acknowledges the support from the NSFC (grant No.",
"12071288).",
"D. Wu acknowledges the support from the NSFC (grant No.",
"12075204) and the Shanghai Municipal Science and Technology Key Project (No.",
"22JC1401500).",
"S. Jin acknowledges the support from the NSFC (grant No.",
"12031013)."
],
[
"Data availability",
"The data that support the findings of this study are available from the corresponding author upon reasonable request."
]
] | 2209.08227 |
[
[
"Reasoning about Dependence, Preference and Coalitional Power"
],
[
"Abstract This paper presents a logic of preference and functional dependence (LPFD) and its hybrid extension (HLPFD), both of whose sound and strongly complete axiomatization are provided.",
"The decidability of LPFD is also proved.",
"The application of LPFD and HLPFD to modelling cooperative games in strategic and coalitional forms is explored.",
"The resulted framework provides a unified view on Nash equilibrium, Pareto optimality and the core.",
"The philosophical relevance of these game-theoretical notions to discussions of collective agency is made explicit.",
"Some key connections with other logics are also revealed, for example, the coalition logic, the logic functional dependence and the logic of ceteris paribus preference."
],
[
"Introduction",
"Dependence, preference and coalitional power are three key concepts in game theory.",
"There have been a lot of logical works on analyzing these three notions.",
"To name but a few, for dependence, the dependence logic [26] has been studied in various ways (c.f.",
"[10]) and a simple logic of functional dependence is recently proposed in [3]; for coalitional power, the coalition logic [18] and the alternating-time temporal logic (ATL) [1], [13] are representative; for preference, good surveys can be found in [14] and [16].",
"Despite not being explicitly emphasized, the concept of dependence permeates the analyses of the other two concepts, for example, in [18] and [6].",
"However, as far as we know, there is hardly any logic explicitly modeling all of these three concepts, especially making dependence the hub to which the other two concepts join.",
"In this paper, we provide such a logic, which characterizes the interaction between the three concepts.",
"Moreover, we show that by making the role of dependence explicit, our logical analysis leads to a unified view of several key concepts in game theory, namely Nash equilibrium, Pareto optimality and the core.",
"We also explore a philosophical implication about collective agency of our logical analysis.",
"We take the stability of a group to be an essential aspect of what makes it a coalition.",
"Instead of focusing on intentionality as in the philosophical literature [20], we elaborate on our understanding in a game theoretical context.",
"Our main work in this paper centers on introducing preference into the logic of functional dependence [3] by adding preference relations in the original semantic model and a new modal operator in the original language for the intersection of different kinds of relations, including equivalence relations, preorders and strict preorders.",
"By taking a game theoretic interpretation of the semantic setting, the new operator enables us to express not only Nash equilibrium but also Pareto optimality.",
"While Nash equilibrium is taken to be a benchmark for modern logics of games and many logics have been demonstrated to be able to express it (see [6] and the reference in it), Pareto optimality as an equally important notion in game theory For example, the prisoners' dilemma is the divergence between Nash equilibrium and Pareto optimality.",
"seems to receive less attention in logical literature than Nash equilibrium.",
"As shown in this paper, to express Pareto optimality, the new modal operator is critical.",
"In fact, given the operator, we can express a relativized version of Nash equilibrium and Pareto optimality, that is, “given the current strategies of some players, the current strategy profile of the other players would be a Nash equilibrium/Pareto optimality.\"",
"Moreover, by taking dependence relation into consideration, our logic shows that Nash equilibrium can be defined by Pareto optimality.",
"As Pareto optimality is seldom studied by logicians, compared to the non-cooperative game theory, the cooperative game theory [19] seems not very salient to logicians either.",
"The review on modal logic for games and information [27] is exclusively about non-cooperative game theory; the book [5] touches on few issues on cooperative game theory either.",
"The only exception we know is the work in [29], where two different logics are proposed to reason about cooperative games.",
"We will demonstrate that our logic of preference and functional dependence (LPFD) can also be adapted to model a qualitative version of cooperative games in strategic and coalitional forms [19].",
"We will also show that a hybrid extension of LPFD can express the core, an essential solution concept in the cooperative games analogous to Nash equilibrium in the non-cooperative games.",
"The core characterizes a coalition's stability as a state where none of its subcoalitions has any incentive to deviate even if they can.",
"The three concepts, dependence, preference and coalitional power, crystallize in the core.",
"Through the lens framed by the three concepts, a unified view of the core, Nash equilibrium and Pareto optimality is revealed by our logics.",
"In addition to the logics and their application to a unified analysis of key game theoretical concepts, our contributions include several technical results about the logics themselves.",
"We provide a sound and strongly complete axiomatization respectively for LPFD and its hybrid extension (HLPFD).",
"Moreover, we also prove that the satisfiability problem of LPFD is decidable.",
"While the proof for the completeness result of HLPFD is standard, the completeness of LPFD is much harder to prove and requires new techniques.",
"Our proof modifies the classical unraveling method [7] and combines it with a special way of selecting the tree branches.",
"is summarized as follows.",
"The background on the logic of functional dependence (LFD) are presented in Section .",
"In the same section, we show how LFD can be used to analyze games in strategic form, especially the notion of coalitional effectiveness as modeled in [18].",
"In Section , we introduce the logic of preference and functional dependence and show how it can naturally express Nash equilibrium and Pareto optimality.",
"Section contains sound and strongly complete axiomatization of LPFD and its hybrid extension and the decidability of LPFD's satisfiability problem.",
"For those who are not interested in the proof details, Section REF and Section REF can be safely skipped.",
"In Section , we turn to our modelling of cooperative games in strategic and coalitional forms in LPFD and analyze the core.",
"In Section , we show how the core can be relevant to philosophical discussions of collective agency.",
"Before conclusion, we compare our work with the logical works in [29] and [6]."
],
[
"Notations",
"The following notations will be used throughout this paper.",
"Let $A$ and $B$ be sets.",
"Let $B^A$ denote the set of mappings from $A$ to $B$ .",
"Let $\\mathcal {P}^{<\\aleph _0}(A)$ denote the set of all finite subsets of $A$ .",
"We write $B\\subseteq _{\\aleph _0}A$ if $B\\in \\mathcal {P}^{<\\aleph _0}(A)$ .",
"For each string $\\vec{x}=(x_i:i\\in I)$ , we write $\\mathsf {set}(\\vec{x})$ for the set $\\lbrace x_i:i\\in I\\rbrace $ .",
"In this section, we introduce LFD and make a first demonstration of its relevance to games."
],
[
"LFD Interpreted in Games",
"LFD starts with a set of variables $\\mathsf {V}$ and a domain of objects $O$ .",
"We take $\\mathsf {V}$ as the set of players in a game and $O$ as the set of actions or strategies each player can take in the game.",
"Then a set of admissible assignments of actions to players $A\\subseteq O^\\mathsf {V}$ can be collected to represent possible strategy profiles of the game.",
"In addition, a relational vocabulary $(\\mathsf {V},\\mathsf {Pred},\\mathsf {ar})$ is given to describe these possible strategy profiles, where $\\mathsf {Pred}$ is a set of predicate symbols and $\\mathsf {ar}: \\mathsf {Pred}\\rightarrow \\mathbb {N}$ is an arity map, associating to each predicate $P\\in \\mathsf {Pred}$ a natural number $\\mathsf {ar}(P)$ .",
"In what follows, if there is no other explanation, the vocabulary $(\\mathsf {V},\\mathsf {Pred},\\mathsf {ar})$ is the one such that $|\\mathsf {V}|=\\aleph _0$ and $|\\lbrace P\\in \\mathsf {Pred}:\\mathsf {ar}(P)=n\\rbrace |=\\aleph _0$ for each $n\\in \\omega $ .",
"[Dependence models] A model is a pair $M=(O,I)$ , where $O$ is a non-empty set of actions and $I$ is a mapping that assigns to each predicate $P\\in \\mathsf {Pred}$ a subset of $O^{\\mathsf {ar}(P)}$ .",
"A dependence model $\\mathbf {M}$ is a pair $\\mathbf {M} = (M,A)$ , where $M = (O,I)$ is a model and $A\\subseteq O^\\mathsf {V}$ is a set of strategy profiles.",
"For each $X\\subseteq _{\\aleph _0}\\mathsf {V}$ , we define a binary relation $=_X\\subseteq A\\times A$ such that $a=_Xa^{\\prime }$ if and only if $a{\\upharpoonright }X=a^{\\prime }{\\upharpoonright }X$ , i.e., the action of $x$ in $a$ is the same as her action in $a^{\\prime }$ for each $x\\in X$ .",
"In a dependence model, when $A\\ne O^\\mathsf {V}$ , some strategy profiles are missing.",
"This gives rise to dependence between players' actions.",
"Suppose a strategy profile $s$ for two players $x$ and $y$ is not in $A$ .",
"Then $x$ and $y$ cannot act according to $s$ simultaneously.",
"In some sense this form of dependence is weak because it does not differentiate between different types of dependence, for example, correlation and causation.",
"However, the other side of the same coin is its generality which is helpful for capturing some common properties of different types of dependence.",
"For further explanation of how and what kinds of dependence can be captured in a dependence model, we refer readers to [3].",
"Differently, the standard setting of strategic form games usually contains all possible strategy profiles, namely $O^\\mathsf {V}$ .",
"This difference plays an essential role in making valid one of the axioms of the coalition logic, namely superadditivity, as we will explain in detail.",
"For now, we turn to the syntax and semantics of LFD.",
"To capture functional dependence, LFD uses two operators $\\mathbb {D}$ and $D$ in its language.",
"The language $\\mathcal {L}$ of LFD is given by $\\varphi :: = P\\vec{x}\\mid D_X y \\mid \\lnot \\varphi \\mid \\varphi \\wedge \\varphi \\mid \\mathbb {D}_X \\varphi $ where $P\\in \\mathsf {Pred}$ , $\\vec{x}= (x_1,\\ldots , x_n)$ is a finite string of players of length $n = \\mathsf {ar}(P)$ , $X\\subseteq _{\\aleph _0} \\mathsf {V}$ is a finite set of players and $y\\in \\mathsf {V}$ is a player.",
"$\\mathbb {D}_X \\varphi $ says that whenever the players in $X$ take their current actions, $\\varphi $ is the case; $D_X y$ says that whenever the players in $X$ take their current actions, $y$ also takes its current action.",
"Truth of a formula $\\varphi \\in \\mathcal {L}$ in a dependence model $\\mathbf {M} = (M,A)$ at a strategy profile $a\\in A$ is defined as follows: Table: NO_CAPTIONNote that $=_X$ is an equivalence relation on $A$ and $a=_\\emptyset a^{\\prime }$ holds for all $a,a^{\\prime }\\in A$ .",
"So $\\mathbb {D}_\\emptyset $ is a universal operator and we define $\\begin{sideways}\\begin{sideways}\\forall \\end{sideways}\\end{sideways}\\varphi := \\mathbb {D}_\\emptyset \\varphi $ and $\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\varphi := \\lnot \\begin{sideways}\\begin{sideways}\\forall \\end{sideways}\\end{sideways}\\lnot \\varphi $ ."
],
[
"Effective Function and Coalition Logic in LFD",
"We have introduced the basics of LFD and interpreted it in the setting of games in strategic form.",
"In this subsection, we continue to explore the potential of this game theoretic perspective on LFD.",
"In particular, we show how the notion of coalitional effectiveness as modeled in [18] can be characterized in LFD.",
"The coalitional effectiveness that the coalition logic aims to reason about is formally characterized by an effectivity function $E_G$ .",
"Based on this effectivity function, the main operator of the coalition logic $[C]\\varphi $ is defined, expressing that the set of agents $C$ can force $\\varphi $ to be the case at their current state.",
"The effective function, when adapted in a dependence model $\\mathbf {M} = (M,A)$ , can be defined as $E_\\mathbf {M}: \\mathcal {P}^{<\\aleph _0}(\\mathsf {V})\\rightarrow \\mathcal {P}(\\mathcal {P}(A))$ satisfying $ S\\in E_\\mathbf {M}(X) \\text{ iff } \\exists a\\in A, \\forall a^{\\prime }\\in A \\text{ if } a^{\\prime } =_X a \\text{ then } a^{\\prime }\\in S\\hspace{5.0pt}.$ Here, $S\\in E_\\mathbf {M}(X)$ means that the coalition $X$ can force the game to be in $S$ .",
"We can express $S\\in E_\\mathbf {M}(X)$ in LFD as $\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X\\varphi $ assuming that $S = \\llbracket \\varphi \\rrbracket $ , because $\\mathbf {M}\\models \\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X\\varphi \\text{ iff } \\llbracket \\varphi \\rrbracket \\in E_\\mathbf {M}(X)\\hspace{5.0pt}.$ The operator $[C]\\varphi $ in the coalition logic essentially has the same semantic meaning despite being interpreted in the neighborhood semantics.",
"We will not go into a detailed comparison between LFD and the coalition logic, but only point out a substantial difference between $\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X\\varphi $ and $[C]\\varphi $ with regard to the characteristic axiom of the coalition logic, superadditivity: $([C_1]\\varphi _1\\wedge [C_2]\\varphi _2)\\rightarrow [C_1\\cup C_2](\\varphi _1\\wedge \\varphi _2) \\text{ where } C_1\\cap C_2 = \\emptyset \\hspace{5.0pt}.$ Superadditivity fails for $\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X\\varphi $ , because in a dependence model $A$ is not required to be $O^\\mathsf {V}$ as we have noted after Definition REF .",
"In fact, the following proposition holds, which reveals that dependence between the players' actions invalidates superadditivity of coalitional effectiveness.",
"Let $\\mathcal {M}$ be a class of dependence models.",
"Superadditivity for $\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X$ , $(\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X\\varphi _1\\wedge \\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_Y\\varphi _2)\\rightarrow \\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_{X\\cup Y}(\\varphi _1\\wedge \\varphi _2) \\text{ where } X\\cap Y = \\emptyset $ , is valid in $\\mathcal {M}$ if and only if , $\\lbrace a{\\upharpoonright }X:a\\in A\\rbrace =O^X$ for all $X\\subseteq _{\\aleph _0}\\mathsf {V}$ and $((O,I),A)\\in \\mathcal {M}$ .",
"As the readers who are familiar with the coalition logic can verify, except for superadditivity, its other axioms are all valid for $\\begin{sideways}\\begin{sideways}\\exists \\end{sideways}\\end{sideways}\\mathbb {D}_X$ in LFD.",
"In this sense, LFD provides a suitable framework for analyzing and understanding the relationship between dependence and coalitional effectiveness in games.",
"However, as a framework for reasoning about other aspects of games, LFD and the coalition logic are both in want of a key element, namely the players' preference.",
"In the next section, we extend LFD with the players' preference relations, study the resulted logic and show how it can capture key concepts in game theory.",
"In Section , the issue of coalitional power will come back and manifest itself in our analysis of cooperative games in strategic and coalitional forms."
],
[
"Logic of Preference and Functional Dependence",
"In this section, we extend LFD to LPFD."
],
[
"Syntax and Semantic for LPFD",
"[Syntax] The language $\\mathcal {L}^\\preceq $ of LPFD is given by: $\\mathcal {L}^\\preceq \\ni \\varphi ::= P\\vec{x}\\mid D_Xy \\mid \\lnot \\varphi \\mid \\varphi \\wedge \\varphi \\mid \\llbracket {X,Y,Z}\\rrbracket \\varphi $ which only differs from the language of LFD in the new operator $\\llbracket {X,Y,Z}\\rrbracket \\varphi $ .",
"In $\\mathcal {L}^\\preceq $ , $\\mathbb {D}_X\\varphi $ is defined as $\\llbracket {X,\\emptyset ,\\emptyset }\\rrbracket \\varphi $ .",
"We define $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi :=\\lnot \\llbracket {X,Y,Z}\\rrbracket \\lnot \\varphi $ and $D_XY:=\\bigwedge _{y\\in Y}D_Xy$ for each $Y\\subseteq _{\\aleph _0}\\mathsf {V}$ .",
"$\\llbracket {X,Y,Z}\\rrbracket $ is an operator for ceteris paribus group preference, which is semantically interpreted as follows.",
"[PD-models]A preference dependence model (PD-model) is a pair $\\mathbb {M}=(\\mathbf {M},\\preceq )$ in which $\\mathbf {M}=(M,A)$ is a model and $\\preceq :V\\rightarrow \\mathcal {P}(A\\times A)$ is a mapping assigning to each $x\\in \\mathsf {V}$ a pre-order $\\preceq _x$ on $A$ .",
"For each $x\\in \\mathsf {V}$ , we define the binary relation $\\prec _x=\\lbrace (a,b)\\in \\preceq _x:(b,a)\\notin \\preceq _x\\rbrace $ .",
"For all $a,b\\in A$ , we write $a\\preceq _Xb$ ($a\\prec _Xb$ ) if $a\\preceq _xb$ ($a\\prec _xb$ ) for each $x\\in X$ .",
"We write $s\\simeq _X t$ if $s\\preceq _X t$ and $t\\preceq _X s$ .",
"Truth of PD-formulas of the form $P\\vec{x},D_Xy,\\lnot \\varphi $ or $\\varphi \\wedge \\psi $ is defined as in Definition REF .",
"For formulas of the form $\\llbracket {X,Y,Z}\\rrbracket \\varphi $ , we say $\\llbracket {X,Y,Z}\\rrbracket \\varphi $ is true at $a$ in $\\mathbb {M}$ , notation: $\\mathbb {M},a \\models \\llbracket {X,Y,Z}\\rrbracket \\varphi $ , if $\\mathbb {M},a^{\\prime } \\models \\varphi $ for all $a^{\\prime }\\in A$ satisfying $a =_{X} a^{\\prime }$ , $a \\preceq _{Y} a^{\\prime }$ and $a\\prec _{Z} a^{\\prime }$ .",
"A formula $\\varphi \\in \\mathcal {L}^\\preceq $ is valid if $\\mathbb {M},a\\models \\varphi $ for all PD-model $\\mathbb {M}=(M,A,\\preceq )$ and $a\\in A$ .",
"Let $\\mathsf {LPFD}$ denote the set of all valid formulas in $\\mathcal {L}^\\preceq $ .",
"Note that $\\llbracket \\emptyset ,\\lbrace x\\rbrace ,\\emptyset \\rrbracket \\varphi $ and $\\llbracket \\emptyset ,\\emptyset ,\\lbrace x\\rbrace \\rrbracket \\varphi $ are standard modal operators defined on $\\preceq _x$ and $\\prec _x$ respectively.",
"Thus $\\llbracket X,Y,Z\\rrbracket \\varphi $ is in fact a standard modal operator defined on the intersection of the relations $=_X$ , $\\preceq _{Y}$ and $\\prec _{Z}$ .",
"There are two types of interdependence between players in a game captured by LPFD.",
"The first type, which comes from restricting what a player can do, is captured by the operators $\\mathbb {D}_X$ and $D_X$ ; the second type, captured by $\\llbracket X,Y,Z\\rrbracket $ , concerns how the preferences of the players in $Y$ and $Z$ depend on the actions of the players in $X$ .",
"There is a close connection between LPFD and the work in [6] on ceteris paribus preference.",
"We will discuss this connection in Section 8.",
"Next, we show how some key game theoretical notions can be expressed in LPFD."
],
[
"Pareto Optimality and Nash Equilibrium in LPFD",
"Having laid out the basics of LPFD, we turn to questions concerning expressing and reasoning about Pareto optimality and Nash equilibrium in LPFD.",
"One important assumption we will adopt is that the group of players $\\mathsf {V}$ has to be finite.",
"In LPFD, there is no such restriction on $\\mathsf {V}$ .",
"However, it is worth noting that in the language of LPFD, all subscripts in the two operators need to be finite.",
"So to express something like $\\llbracket -X,\\emptyset ,X\\rrbracket \\varphi $ in LPFD where $-X := V-X$ , which is frequently referred to in game theory, we have to ensure that $X$ and $-X$ are both finite.",
"We start with recalling what Nash equilibrium and weak/strong Pareto optimality mean.",
"Let $\\mathbb {M}$ be a PD-model and $X\\subseteq \\mathsf {V}$ .",
"Given that the players in $-X$ have acted according to the strategy profile $s\\in A$ , $s$ is a Nash equilibrium for $X$ if for all $x\\in X$ there is no $t=_{-\\lbrace x\\rbrace } s$ such that $s\\prec _x t$ ; $s$ is strongly Pareto optimal for $X$ if there is no $t=_{-X} s$ such that (a) for all $x\\in X$ , $s\\preceq _x t$ and (b) there is one $x\\in X$ such that $s\\prec _x t$ ; $s$ is weakly Pareto optimal for $X$ if there is no $t=_{-X} s$ such that for all $x\\in X$ , $s\\prec _x t$ .",
"Note that such a way of defining the notions of Nash equilibrium, weak and strong Pareto optimality in a PD-model applies to all subgroups of $\\mathsf {V}$ rather than only the whole group of players $\\mathsf {V}$ .",
"It is relatively easy to get how Nash equilibrium and weak Pareto optimality can be expressed in LPFD, as the following fact shows.",
"Fact 1 Let $\\mathbb {M}=(M,A,\\preceq )$ be a PD-model and $s\\in A$ .",
"Then $s$ is a Nash equilibrium for $X\\subseteq \\mathsf {V}$ given that the players in $-X$ have acted according to $s$ , if and only if, $\\mathbb {M}, s\\models \\bigwedge _{x\\in X}\\llbracket -\\lbrace x\\rbrace ,\\emptyset ,\\lbrace x\\rbrace \\rrbracket \\bot $ ; $s$ is weakly Pareto optimal for $X\\subseteq \\mathsf {V}$ given that the players in $-X$ have acted according to $s$ , if and only if, $\\mathbb {M}, s\\models \\llbracket -X,\\emptyset ,X\\rrbracket \\bot $ .",
"In the case of weak Pareto optimality, because the truth condition of the operator $\\llbracket -X,\\emptyset ,X\\rrbracket $ depends on what formulas are satisfied on all elements in the set $\\lbrace t\\in A\\mid s=_{-X} t, s\\prec _X t\\rbrace $ , if it is an empty set and thus $\\bot $ can be vacuously satisfied on all elements in it, then $s$ is weakly Pareto optimal for $X$ .",
"To express strong Pareto optimality in LPFD, we need to express the following model theoretical fact, namely, the set $\\lbrace t\\in A\\mid s=_{-X} t, s\\preceq _X t \\text{ and }t_X s\\rbrace $ $= \\bigcup _{x\\in X}\\lbrace t\\in A\\mid s=_{-X} t, s\\preceq _{X-\\lbrace x\\rbrace } t,s \\prec _{x} t\\rbrace $ is empty.",
"Since $s\\models \\llbracket -X,X-\\lbrace x\\rbrace , \\lbrace x\\rbrace \\rrbracket \\bot $ iff $\\lbrace t\\in A\\mid s=_{-X} t, s\\preceq _{X-\\lbrace x\\rbrace } t,s\\prec _{x} t\\rbrace = \\emptyset $ , we can define strong Pareto optimality as follows.",
"Fact 2 In a PD-model $\\mathbb {M}$ , $s$ is strongly Pareto optimal for $X\\subseteq \\mathsf {V}$ given that the players in $-X$ have acted according to $s$ iff $\\mathbb {M}, s\\models \\bigwedge _{x\\in X}\\llbracket -X,X-\\lbrace x\\rbrace , \\lbrace x\\rbrace \\rrbracket \\bot $ .",
"To facilitate our discussion, we define weak and strong Pareto optimality and Nash equilibrium in LPFD as $\\operatorname{\\mathsf {wPa}}X &:= \\llbracket -X,\\emptyset ,X\\rrbracket \\bot \\\\\\operatorname{\\mathsf {sPa}}X &:= \\bigwedge _{x\\in X}\\llbracket -X,X-\\lbrace x\\rbrace , \\lbrace x\\rbrace \\rrbracket \\bot \\\\\\operatorname{\\mathsf {Na}}X &:= \\bigwedge _{x\\in X} \\llbracket -\\lbrace x\\rbrace ,\\emptyset ,\\lbrace x\\rbrace \\rrbracket \\bot $ An easy but important observation is that Nash equilibrium is a special case of Pareto optimality.",
"$\\operatorname{\\mathsf {Na}}X = \\bigwedge _{x\\in X} \\operatorname{\\mathsf {sPa}}\\lbrace x\\rbrace = \\bigwedge _{x\\in X} \\operatorname{\\mathsf {wPa}}\\lbrace x\\rbrace $ ."
],
[
"Calculus of LPFD and its Hybrid Extension",
"In this section, a Kripke style semantics of LPFD shall be introduced.",
"It is proved to be equivalent to the standard semantics in Section REF .",
"The new semantics provides us with a modal view, which facilitates our calculus $\\mathsf {C}_\\text{LPFD}$ and the proof of its soundness and strongly completeness.",
"We show that LPFD is decidable while it lacks the finite model property.",
"Moreover, we extend it with nominals and give also a sound and complete calculus $\\mathsf {C}_\\text{HLPFD}$ .",
"In Section , this hybrid extension will be useful in expressing a key game theoretic concept."
],
[
"Kripke Style Semantics",
"In this part, we introduce the Kripke style semantics for LPFD and show that it is equivalent to the standard semantics.",
"A relational PD-frame (RPD-frame) is a pair $\\mathfrak {F}=(W,\\sim ,\\le )$ , where $W$ is a non-empty set, $\\sim :V\\rightarrow \\mathcal {P}(W\\times W)$ and $\\le :V\\rightarrow \\mathcal {P}(W\\times W)$ are maps such that $\\sim _x$ is an equivalence relation and $\\le _x$ is a pre-order for all $x\\in \\mathsf {V}$ .",
"For all $x\\in \\mathsf {V}$ and $X,Y,Z\\subseteq _{\\aleph _0}\\mathsf {V}$ , let $<_x=\\lbrace (w,u)\\in \\le _x:(u,w)\\notin \\le _x\\rbrace $ and $R(X,Y,Z)=\\bigcap _{x\\in X}\\sim _x\\cap \\bigcap _{y\\in Y}\\le _y\\cap \\bigcap _{z\\in Z}<_z.$ A relational PD-model (RPD-model) is a pair $\\mathfrak {M}=(\\mathfrak {F},V)$ where $\\mathfrak {F}=(W,\\sim ,\\le )$ is a RPD-frame and $V$ is a valuation associating to each formula of the form $P\\vec{x}$ a subset $V(P\\vec{x})$ of $W$ .",
"The valuation $V$ is required to satisfy the following condition for all $w,u\\in W$ and $P\\in \\mathsf {Pred}$ : if $w\\sim _{\\mathsf {set}(\\vec{x})}u$ , then $w\\in V(P\\vec{x})$ if and only if $u\\in V(P\\vec{x})$ .",
"(Val) Truth of a formula $\\varphi \\in \\mathcal {L}^\\preceq $ in $\\mathfrak {M}=(W,\\sim ,\\le ,V)$ at $w\\in W$ is defined as follows: Table: NO_CAPTIONValidity is defined as usual.",
"Let $\\mathsf {RLPFD}$ denote the set of all valid $\\mathcal {L}^\\preceq $ -formulas.",
"For each $\\varphi \\in \\mathcal {L}^\\preceq $ , if $\\varphi $ is satisfied by some PD-model $\\mathbb {M}$ , then it is satisfied by some RPD-model.",
"Let $\\mathbb {M}=(O,I,A,\\preceq )$ be a PD-model.",
"We define the RPD-model $rel(\\mathbb {M})=(W,\\sim ,\\le ,V)$ by $W:=A$ , $V(P\\vec{x}):=\\lbrace a\\in A:a(\\vec{x})\\in I(P)\\rbrace $ and $\\sim _x:=(=_x)$ , $\\le _x:=\\preceq _x$ for each $x\\in \\mathsf {V}$ .",
"It is clear that for each $\\varphi \\in \\mathcal {L}^\\preceq $ and $a\\in A$ , $\\mathbb {M},a\\models \\varphi $ if and only if $rel(\\mathbb {M}),a\\models \\varphi $ .",
"Then we are done.",
"Let $\\mathfrak {M}=(W,\\sim ,\\le ,V)$ be a RPD-model.",
"Then we define the PD-model $dp(\\mathfrak {M})=(O,I,A,\\preceq )$ induced by $\\mathfrak {M}$ as follows: $O=\\lbrace (x,|w|_x):x\\in \\mathsf {V},w\\in W\\text{ and } |w|_x=\\lbrace v\\in W:w\\sim _x v\\rbrace \\rbrace $ .",
"$A=\\lbrace w^*:w\\in W\\rbrace $ , where $w^*(x)=(x,|w|_x)$ for each $x\\in \\mathsf {V}$ .",
"$\\preceq _x=\\lbrace (w^*,v^*):w\\le _xv\\rbrace $ for each $x\\in \\mathsf {V}$ .",
"$I$ is the interpretation maps each n-ary predicate $P$ to the set $I(P)=\\lbrace w^*(\\vec{x}):w\\in W,\\vec{x}\\in \\mathsf {V}^n\\text{ and }w\\in V(P\\vec{x})\\rbrace .$ $I(P)$ is well-defined for each predicate $P$ since $x=y$ and $w\\sim _xv$ whenever $w^*(x)=v^*(y)$ .",
"It is clearly that $\\prec _x=\\lbrace (w^*,v^*):w<_xv\\rbrace $ for all $x\\in \\mathsf {V}$ .",
"Let $\\mathfrak {M}$ be a RPD-model and $dp(\\mathfrak {M})$ the PD-model induced by $\\mathfrak {M}$ .",
"Then for each $w$ in $\\mathfrak {M}$ and formula $\\varphi \\in \\mathcal {L}^\\preceq $ , $\\mathfrak {M},w\\models \\varphi $ if and only if $dp(\\mathfrak {M}),w^*\\models \\varphi $ .",
"By Proposition REF and Proposition REF , we have $\\mathsf {RLPFD}=\\mathsf {LPFD}$ immediately."
],
[
"Hilbert-style Calculus ${\\mathsf {C}_{\\mathrm {LPFD}}}$",
"In this part, we present a calculus ${\\mathsf {C}_{\\mathrm {LPFD}}}$ of LPFD and show that ${\\mathsf {C}_{\\mathrm {LPFD}}}$ is sound, by which some key axioms are semantically explained.",
"Axioms and rules for classical propositional logic; from $\\varphi $ infer $\\llbracket {X,Y,Z}\\rrbracket \\varphi $ ; $\\llbracket {X,Y,Z}\\rrbracket (\\varphi \\rightarrow \\psi )\\rightarrow (\\llbracket {X,Y,Z}\\rrbracket \\varphi \\rightarrow \\llbracket {X,Y,Z}\\rrbracket \\psi )$ ; Axioms for preference relations: $\\llbracket {X,Y,\\varnothing }\\rrbracket \\varphi \\rightarrow \\varphi $ ; $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X\\cap X^{\\prime }, Y\\cap Y^{\\prime },(Z\\cap Y^{\\prime })\\cup (Z\\cap Z^{\\prime })\\cup (Y\\cap Z^{\\prime })\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ ; $\\llbracket {X,Y,Z}\\rrbracket \\varphi \\rightarrow \\llbracket {X^{\\prime },Y^{\\prime },Z^{\\prime }}\\rrbracket \\varphi $ , provided $X\\subseteq X^{\\prime }$ , $Y\\subseteq Y^{\\prime }$ and $Z\\subseteq Z^{\\prime }$ .",
"$\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y\\cup Z,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ ; $(\\varphi \\wedge \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi )\\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}(\\psi \\wedge \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi )\\vee \\bigvee _{y\\in Y}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\cup \\lbrace y\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi $ .",
"Axioms and rules for dependence: $D_XX$ ; $\\varphi \\rightarrow \\mathbb {D}_X\\varphi $ , provided $\\varphi \\in \\mathsf {Atom}(X)=\\lbrace P\\vec{x}:\\mathsf {set}(\\vec{x})\\subseteq X\\rbrace \\cup \\lbrace D_{Y}z:Y\\subseteq X\\rbrace $ ; $D_XS\\wedge D_ST\\rightarrow D_XT$ ; $D_XS\\wedge \\llbracket {S,Y,Z}\\rrbracket \\varphi \\rightarrow \\llbracket {X,Y,Z}\\rrbracket \\varphi $ .",
"In what follows, we write $\\mathsf {C}$ for ${\\mathsf {C}_{\\mathrm {LPFD}}}$ if there is no danger of confusion.",
"[Soundness] For each $\\varphi \\in \\mathcal {L}^\\preceq $ , $\\vdash _\\mathsf {C}\\varphi $ implies $\\varphi \\in \\mathsf {LPFD}$ .",
"We take (Ord,b) and (Ord,e) as two examples, showing their validity and giving some intuitions.",
"Other axioms and rules can be easily checked to be valid.",
"Let $\\mathfrak {M}=(W,\\sim ,\\le ,V)$ be a RPD-model and $w\\in W$ a point.",
"For (Ord,b), it characterizes some kind of generalized transitivity.",
"Suppose $\\mathfrak {M},w\\models \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ .",
"Then there are points $u,v\\in W$ such that $u\\in R(X,Y,Z)(w)$ , $v\\in R(X^{\\prime },Y^{\\prime },Z^{\\prime })$ and $\\mathfrak {M},v\\models \\varphi $ .",
"Let $T=(Z\\cap Y^{\\prime })\\cup (Z\\cap Z^{\\prime })\\cup (Y\\cap Z^{\\prime })$ .",
"It is obvious that $w\\sim _{X\\cap X^{\\prime }}v$ and $w\\le _{Y\\cap Y^{\\prime }}$ hold.",
"It suffices to show that $w<_{T}v$ .",
"Suppose $x\\in Z\\cap Y^{\\prime }$ .",
"Then $w\\le _xu$ , $u\\lnot \\le _xw$ and $u\\le _xv$ .",
"By the transitivity of $\\le _x$ , we see $w\\le _xv$ and $v\\lnot \\le _xw$ , i.e., $w<_xv$ .",
"Similarly, we see $w<_xv$ whenever $x\\in Y\\cap Z^{\\prime }$ or $x\\in Z\\cap Z^{\\prime }$ .",
"Hence $\\mathfrak {M},w\\models (Ord,b)$ .",
"For (Ord,e), it characterizes to some degree the definition of $<$ .",
"Suppose $\\mathfrak {M},w\\models \\varphi \\wedge \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi $ .",
"Then there is a point $u\\in R(X,Y,Z)(w)$ such that $\\mathfrak {M},u\\models \\psi $ .",
"If $u\\le _Yw$ , then clearly $\\mathfrak {M},u\\models \\psi \\wedge \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ , which entails $\\mathfrak {M},w\\models \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}(\\psi \\wedge \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi )$ .",
"Suppose $u\\lnot \\le _Yw$ .",
"Then there is $y\\in Y$ such that $u\\lnot \\le _yw$ and so $w<_yu$ .",
"Recall that $u\\in R(X,Y,Z)(w)$ , we obtain $u\\in R(X,Y,Z\\cup \\lbrace y\\rbrace )$ and so $\\mathfrak {M},w\\models \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\cup \\lbrace y\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi $ .",
"Hence $\\mathfrak {M},w\\models (Ord,e)$ ."
],
[
"Strong Completeness of ${\\mathsf {C}_{\\mathrm {LPFD}}}$",
"For the proof of completeness, a special kind of unraveling method is used.",
"The main reason we take such a method is that the `canonical model' need not be an RPD-model, and modification is needed.",
"To construct an RPD-model satisfying some given consistent set of formulas, we first pick out those so-called saturated formulas, which are sufficient to determine the preference relations in the model.",
"Then we take `paths' as the domain of the desired model instead of using just maximal consistent sets, which helps us deal with the intersections of relations.",
"The relations in this model are closures of some `one-step' relations, which help solve the problems that arise from dependence formulas.",
"With such a model, we prove the Truth Lemma and so the Completeness Theorem.",
"To define a model for some satisfiable set of formulas $\\Gamma $ , we first define the canonical quasi-frame and investigate some properties of it: [Canonical Quasi PD-Frame] Let $\\Delta $ be a set of $\\mathcal {L}^\\preceq $ -formulas.",
"We say that $\\Delta $ is consistent if $\\Delta \\lnot \\vdash \\bot $ .",
"We say that $\\Delta $ is a maximal consistent set (MCS) if $\\Delta $ is consistent and every proper extension of $\\Delta $ is not consistent.",
"The canonical Quasi PD-frame $\\mathfrak {F}^q=(W^q,R^q)$ of $\\mathsf {C}$ is defined as follows: $W^q$ is the set of all MCSs; for all $X,Y,Z\\subseteq _{\\aleph _0} \\mathsf {V}$ , we define $R^q(X,Y,Z)\\subseteq W^q\\times W^q$ by: $wR^q(X,Y,Z)u$ if and only if $\\lbrace \\varphi \\in \\mathcal {L}^\\preceq :\\llbracket {X,Y,Z}\\rrbracket \\varphi \\in w\\rbrace \\subseteq u$ .",
"For all $\\Delta _1,\\Delta _2,\\Delta _3\\in W^q$ and $X,Y,Z\\subseteq _{\\aleph _0}\\mathsf {V}$ : $R^q(X,Y,\\varnothing )$ is reflexive; If $\\Delta _1 R^q(X,Y,Z)\\Delta _2$ , then $\\Delta _1 R^q(X^{\\prime },Y^{\\prime },Z^{\\prime })\\Delta _2$ for all $X^{\\prime }\\subseteq X$ , $Y^{\\prime }\\subseteq Y\\cup Z$ and $Z^{\\prime }\\subseteq Z$ ; For all $Z^{\\prime }\\subseteq _{\\aleph _0}\\mathsf {V}$ , if $Z,Z^{\\prime }\\subseteq Y$ , $\\Delta _1 R^q(X,Y,Z)\\Delta _2$ and $\\Delta _2 R^q(X,Y,Z^{\\prime })\\Delta _3$ , then $\\Delta _1 R^q(X,Y,Z\\cup Z^{\\prime })\\Delta _3$ ; If $D_XS\\in \\Delta _1$ and $\\Delta _1 R^q(X,Y,Z)\\Delta _2$ , then $\\Delta _1 R^q(S,Y,Z)\\Delta _2$ and $D_XS\\in \\Delta _2$ .",
"(1) follows form axiom (Ord,a), (2) follows from Axiom (Ord,c,d), (3) follows from axiom (Ord,b) and (4) follows from axiom (Dep,b,d) immediately.",
"Let $\\Sigma $ be a MCS and $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Sigma $ .",
"We say that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ is a saturated formula in $\\Sigma $ if $\\bigvee _{y\\in Y}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\cup \\lbrace y\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\notin \\Sigma $ and $Y\\cap Z=\\varnothing $ .",
"Let $S(\\Sigma )$ denote the set of all saturated formulas in $\\Sigma $ .",
"Let $\\Sigma \\in W^q$ be a MCS, $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Sigma $ and $S=Y\\cup Z$ .",
"Then there is $T\\in \\mathcal {P}(\\mathsf {V})$ such that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,T,(Y\\cup Z)\\setminus T\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in S(\\Sigma )$ .",
"The proof proceeds by induction on the size $n$ of $Y\\setminus Z$ .",
"When $n=0$ , one obtains $Z=Y\\cup Z$ .",
"By axiom (Ord,c), $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,\\varnothing ,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Sigma $ .",
"Note that $\\bigvee \\varnothing =\\bot \\notin \\Sigma $ , $\\varnothing $ is the desired set.",
"Suppose $n>0$ and $Y\\setminus Z=\\lbrace y_0,\\cdots ,y_{n-1}\\rbrace $ .",
"If $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\cup \\lbrace y_i\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\notin \\Sigma $ for any $i<n$ , then $T=Y\\setminus Z$ satisfies the requirement.",
"Suppose $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\cup \\lbrace y_i\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Sigma $ for some $i<n$ .",
"Then we see $|Y\\setminus (Z\\cup \\lbrace y_i\\rbrace )|<n$ and by induction hypothesis, there is $T\\in \\mathcal {P}(\\mathsf {V})$ such that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,T,(Y\\cup Z)\\setminus T\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in S(\\Sigma )$ .",
"Since $Y\\cup Z=Y\\cup (Z\\cup \\lbrace y_i\\rbrace )$ , $T$ satisfies the requirement.",
"Let $\\Sigma \\in W^q$ and $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in S(\\Sigma )$ .",
"Then there is a MCS $\\Delta \\in R^q(X,Y,Z)(\\Sigma )$ such that $\\varphi \\in \\Delta $ and $\\Delta R^q(X,Y,\\varnothing )\\Sigma $ .",
"We write $\\Box $ for $\\llbracket {X,Y,Z}\\rrbracket $ and $\\blacklozenge $ for $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}$ in this proof.",
"It is sufficient to show that $\\Delta _0=\\lbrace \\psi :\\Box \\psi \\in \\Sigma \\rbrace \\cup \\lbrace \\blacklozenge \\gamma :\\gamma \\in \\Sigma \\rbrace \\cup \\lbrace \\varphi \\rbrace $ is consistent.",
"Otherwise, there are formulas $\\Box \\psi _1,\\cdots ,\\Box \\psi _n,\\gamma _1,\\cdots ,\\gamma _m,\\in \\Sigma $ such that $\\vdash \\psi _1\\wedge \\cdots \\wedge \\psi _n\\wedge \\blacklozenge \\gamma _1\\wedge \\cdots \\wedge \\blacklozenge \\gamma _m\\wedge \\varphi \\rightarrow \\bot .$ Let $\\gamma =\\gamma _1\\wedge \\cdots \\wedge \\gamma _m$ and $\\psi =\\psi _1\\wedge \\cdots \\wedge \\psi _n$ .",
"Clearly, $\\gamma \\in \\Sigma $ .",
"By axiom (Nec) and (K), we have $\\vdash \\blacklozenge \\gamma \\rightarrow (\\blacklozenge \\gamma _1\\wedge \\cdots \\wedge \\blacklozenge \\gamma _m$ ).",
"Thus $\\vdash \\psi \\wedge \\blacklozenge \\gamma \\wedge \\varphi \\rightarrow \\bot $ , which entails $\\vdash \\Box \\psi \\rightarrow \\lnot \\Diamond (\\varphi \\wedge \\blacklozenge \\gamma )$ .",
"Note that $\\Box \\psi \\in \\Sigma $ , we have $\\lnot \\Diamond (\\varphi \\wedge \\blacklozenge \\gamma )\\in \\Sigma $ .",
"Since $\\gamma \\wedge \\Diamond \\varphi \\in \\Sigma $ and $(\\gamma \\wedge \\Diamond \\varphi )\\rightarrow \\Diamond (\\varphi \\wedge \\blacklozenge \\gamma )\\vee \\bigvee _{y\\in Y}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\cup \\lbrace y\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ is an instant of axiom (Ord,e), we obtain $\\bigvee _{y\\in {Y}}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,{Y},{Z}\\cup \\lbrace y\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\in \\Sigma $ , which contradicts that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in S(\\Sigma )$ .",
"With the help of Lemma REF and Lemma REF , we are now able to define the paths in $W^q$ , which constitute the domain of our desired model.",
"A path in $W^q$ is a sequence $\\pi =\\langle \\Sigma _0,\\psi _0,\\cdots ,\\Sigma _{n-1},\\psi _{n-1},\\Sigma _n \\rangle $ in which the following conditions hold for all $i<n$ : $\\psi _i=\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X_i,Y_i,Z_i\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi _i\\in S(\\Sigma _i)$ is a saturated formula in $\\Sigma _i\\in W^q$ ; $\\varphi _i\\in \\Sigma _{i+1}\\in W^q$ , $\\Sigma _{i+1}R^q(X_i,Y_i,\\varnothing )\\Sigma _i$ and $\\Sigma _i R^q(X_i,Y_i,Z_i)\\Sigma _{i+1}$ .",
"We denote $\\Sigma _0$ by $\\mathrm {start}(\\pi )$ , $\\Sigma _n$ by $\\mathrm {last}(\\pi )$ and the set of all paths by $\\mathrm {Path}$ .",
"In what follows, let $\\Gamma $ be some fixed consistent set.",
"Without loss of generality, suppose $\\Gamma $ is a MCS.",
"We now construct a model for $\\Gamma $ .",
"[$\\Gamma $ -Canonical PD-model] The $\\Gamma $ -canonical PD-model $\\mathfrak {M}^c_\\Gamma =(\\mathfrak {F}^c_\\Gamma ,V^c)$ , in which $\\mathfrak {F}^c_\\Gamma =(W^c_\\Gamma ,\\le ^c,\\sim ^c)$ , is defined as follows: $W^c_\\Gamma =\\lbrace \\pi \\in \\mathrm {Path}:\\mathrm {start}(\\pi )=\\Gamma \\rbrace $ , and we write $W^c$ for $W^c_\\Gamma $ in what follows; for all $y\\in \\mathsf {V}$ and $\\pi ,\\pi ^{\\prime }\\in W^c$ , $\\pi \\le _y\\pi ^{\\prime }$ iff one of the following holds: $\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ and $y\\in Y\\cup Z$ ; $\\pi =\\langle \\pi ^{\\prime },\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ and $y\\in Y$ ; $\\pi =\\pi ^{\\prime }$ .",
"Let $\\le ^c_y$ be the transitive closure of $\\le _y$ .",
"for all $s\\in \\mathsf {V}$ and $\\pi ,\\pi ^{\\prime }\\in W^c$ , $\\pi \\rightharpoonup _s\\pi ^{\\prime }$ if and only if $\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ and $D_Xs\\in \\mathrm {last}(\\pi )$ .",
"Let $\\rightleftharpoons _s$ be the reflexive-symmetric closure of $\\rightharpoonup _s$ .",
"Let $\\sim ^c_s$ be the transitive closure of $\\rightleftharpoons _s$ .",
"for all $P\\vec{x}\\in \\mathcal {L}$ , $V^c(P\\vec{x})=\\lbrace \\pi \\in W^c: P\\vec{x}\\in \\mathrm {last}(\\pi )\\rbrace $ .",
"For all $X,Y,Z\\subseteq _{\\aleph _0}\\mathsf {V}$ , the binary relations $R^c(X,Y,Z)$ , $\\sim ^c_X$ , $\\le ^c_Y$ and $<^c_Z$ are defined in the natural way.",
"By Axiom (Dep,a), $D_XX$ always holds.",
"Thus for each $\\pi \\in W^c_\\Gamma $ and $\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Delta \\rangle $ , we have $\\pi R^c(X,Y,Z)\\pi ^{\\prime }$ .",
"To characterize the structure of $W^c$ , we define $T\\subseteq W^c\\times W^c$ as follows: $\\pi T\\pi ^{\\prime }$ if and only if $\\pi ^{\\prime }$ is of the form $\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ .",
"It is clear that $(W^c,T)$ is a tree.",
"Then for all $\\pi ,\\pi ^{\\prime }\\in W^c$ , there is a shortest $T$ -sequence $\\langle \\pi _0,\\cdots ,\\pi _n \\rangle $ such that $\\pi =\\pi _0$ , $\\pi ^{\\prime }=\\pi _n$ and for all $i<n$ , $\\pi _i T\\pi _{i+1}$ or $\\pi _{i+1}T\\pi _i$ .",
"We denote the shortest sequence by $T^\\pi _{\\pi ^{\\prime }}$ .",
"Fact 3 Let $\\pi ,\\pi ^{\\prime }\\in W^c$ , $T^\\pi _{\\pi ^{\\prime }}=\\langle \\pi _0,\\cdots ,\\pi _n \\rangle $ and $y,s\\in \\mathsf {V}$ .",
"Then $\\pi \\sim ^c_s\\pi ^{\\prime }$ iff $\\pi _i\\rightleftharpoons _s\\pi _{i+1}$ for all $i<n$ .",
"$\\pi \\le ^c_y\\pi ^{\\prime }$ iff $\\pi _i\\le _y\\pi _{i+1}$ for all $i<n$ .",
"Since $\\rightleftharpoons _s,\\le _y\\subseteq (T\\cup T^{-1})$ , $\\sim ^c_s$ is the transitive closure of $\\rightleftharpoons _s$ and $\\le ^c_y$ the transitive closure of $\\le _y$ , the proof can be done by induction on $n$ easily.",
"In what follows, we show that the relations $R^c(X,Y,Z)$ are consistent with the relations $R^q(X,Y,Z)$ .",
"Let $\\pi ,\\pi ^{\\prime }\\in W^c$ , $X,Y,Z\\subseteq \\mathsf {V}$ , $\\pi \\rightleftharpoons _X\\pi ^{\\prime }$ , $\\pi \\le _Y\\pi ^{\\prime }$ and $\\pi <_Z\\pi ^{\\prime }$ .",
"Then $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ .",
"if $D_XS\\in \\mathrm {last}(\\pi )$ , then $\\pi \\rightleftharpoons _S\\pi ^{\\prime }$ .",
"Suppose $\\pi \\rightleftharpoons _X\\pi ^{\\prime }$ , $\\pi \\le _Y\\pi ^{\\prime }$ and $\\pi <_Z\\pi ^{\\prime }$ .",
"Then we have three cases: $\\pi =\\pi ^{\\prime }$ .",
"Then $Z=\\varnothing $ .",
"By Proposition REF (1), $R^q(X,Y,\\varnothing )$ is reflexive and $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ .",
"$\\pi =\\langle \\pi ^{\\prime },\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi ,\\Delta \\rangle $ .",
"Then $Z=\\varnothing $ , $Y\\subseteq Y^{\\prime }$ and $D_{X^{\\prime }}X\\in \\mathrm {last}(\\pi ^{\\prime })$ .",
"Clearly, $\\mathrm {last}(\\pi ^{\\prime })R^q(X^{\\prime },Y^{\\prime },Z^{\\prime })\\mathrm {last}(\\pi )$ .",
"by Proposition REF (4), $D_{X^{\\prime }}X\\in \\mathrm {last}(\\pi )$ .",
"Recall that one has $\\mathrm {last}(\\pi )R^q(X^{\\prime },Y^{\\prime },\\varnothing )\\mathrm {last}(\\pi ^{\\prime })$ , by Proposition REF (2,4), we see $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ .",
"$\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi ,\\Delta \\rangle $ .",
"Then $Z\\subseteq Z^{\\prime }$ , $Y\\subseteq Y^{\\prime }\\cup Z^{\\prime }$ and $D_{X^{\\prime }}X\\in \\mathrm {last}(\\pi )$ .",
"Note that $\\mathrm {last}(\\pi )R^q(X^{\\prime },Y^{\\prime },Z^{\\prime }) \\mathrm {last}(\\pi ^{\\prime })$ , by Proposition REF (2,4), we see $\\mathrm {last}(\\pi ^{\\prime }) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ .",
"Hence $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ and (1) holds.",
"For (2), suppose $D_XS\\in \\mathrm {last}(\\pi )$ .",
"Then we have also three cases: $\\pi =\\pi ^{\\prime }$ .",
"Note that $\\rightleftharpoons _S$ is reflexive, $\\pi \\rightleftharpoons _S\\pi ^{\\prime }$ .",
"$\\pi \\rightharpoonup _X\\pi ^{\\prime }$ .",
"Then $\\pi ^{\\prime }$ is of the form $\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi ,\\Delta \\rangle $ and $D_{X^{\\prime }}X\\in \\mathrm {last}(\\pi )$ .",
"By axiom (Dep,c), $D_{X^{\\prime }}S\\in \\mathrm {last}(\\pi )$ .",
"Thus $\\pi \\rightharpoonup _S\\pi ^{\\prime }$ .",
"$\\pi ^{\\prime }\\rightharpoonup _X\\pi $ .",
"Then $\\pi $ is of the form $\\langle \\pi ^{\\prime },\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi ,\\Delta \\rangle $ and $D_{X^{\\prime }}X\\in \\mathrm {last}(\\pi ^{\\prime })$ .",
"By (1), $D_XS\\in \\mathrm {last}(\\pi ^{\\prime })$ .",
"By axiom (Dep,c), $D_{X^{\\prime }}S\\in \\mathrm {last}(\\pi )$ .",
"Thus $\\pi ^{\\prime }\\rightharpoonup _S\\pi $ .",
"Hence $\\pi \\rightleftharpoons _S\\pi ^{\\prime }$ and (2) holds.",
"Let $\\pi ,\\pi ^{\\prime }\\in W^c$ , $X,Y,Z\\subseteq _{\\aleph _0} \\mathsf {V}$ and $\\pi R^c(X,Y,Z)\\pi ^{\\prime }$ .",
"Then $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ .",
"$D_XS\\in \\mathrm {last}(\\pi )$ implies $\\pi R^c(S,Y,Z)\\pi ^{\\prime }$ .",
"Suppose $\\pi R^c(X,Y,Z)\\pi ^{\\prime }$ .",
"Then $\\pi \\sim ^c_X\\pi ^{\\prime }$ , $\\pi \\le ^c_{Y\\cup Z}\\pi ^{\\prime }$ and $\\pi <^c_Z\\pi ^{\\prime }$ .",
"Let $T^\\pi _{\\pi ^{\\prime }}=\\langle \\pi _0,\\cdots ,\\pi _n \\rangle $ .",
"By Fact REF , for all $i<n$ , $\\pi _i\\rightleftharpoons _X\\pi _{i+1}$ and $\\pi _i\\le _{Y\\cup Z}\\pi _{i+1}$ .",
"Moreover, for each $z\\in Z$ , there is $i_z\\in n$ such that $\\pi _{i_z}<_{z}\\pi _{i_z+1}$ .",
"Then by Lemma REF (1), $\\mathrm {last}(\\pi _i) R^q(X,Y\\cup Z,\\varnothing )\\mathrm {last}(\\pi _{i+1})$ for all $i\\in n$ and for all $z\\in Z$ , $\\mathrm {last}(\\pi _{i_z}) R^q(X,Y\\cup Z,\\lbrace z\\rbrace )\\mathrm {last}(\\pi _{i_z+1})$ .",
"Then by Proposition REF (2,3), we see $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ and (1) holds.",
"Suppose $D_XS\\in \\mathrm {last}(\\pi )$ .",
"Note that $\\pi \\sim ^c_X\\pi _i$ for all $i\\le n$ , by (1), $D_XS\\in \\mathrm {last}(\\pi _i)$ for all $i\\le n$ .",
"Then by Lemma REF (2), $\\pi _i\\rightleftharpoons _S\\pi _{i+1}$ for all $i\\in n$ , which entails $\\pi \\sim ^c_S\\pi ^{\\prime }$ .",
"The final step is to show that $\\mathfrak {M}^c$ is a PD-model in which $\\Gamma $ is satisfiable.",
"$\\mathfrak {M}^c$ is a PD-model.",
"It suffices to show that $V^c$ satisfies (Val).",
"Let $\\pi ,\\pi ^{\\prime }\\in W^c$ be points such that $\\pi \\sim ^c_X\\pi ^{\\prime }$ .",
"By Lemma REF , $\\mathrm {last}(\\pi ) R^q(X,\\varnothing ,\\varnothing )\\mathrm {last}(\\pi ^{\\prime })$ .",
"Assume $P\\vec{x}\\in \\mathrm {last}(\\pi )$ , then by axiom (Dep,b), $\\mathbb {D}_XP\\vec{x}\\in \\mathrm {last}(\\pi )$ , which entails $P\\vec{x}\\in \\mathrm {last}(\\pi ^{\\prime })$ .",
"Similarly, we can verify that $P\\vec{x}\\in \\mathrm {last}(\\pi ^{\\prime })$ implies $P\\vec{x}\\in \\mathrm {last}(\\pi )$ .",
"Thus $V^c$ satisfies (Val) and so $\\mathfrak {M}^c_\\Gamma $ is a PD-model.",
"[Truth Lemma] For each formula $\\varphi \\in \\mathcal {L}^\\preceq $ and path $\\pi \\in W^c$ , $\\mathfrak {M}^c,\\pi \\models \\varphi $ if and only if $\\varphi \\in \\mathrm {last}(\\pi )$ .",
"The proof proceeds by induction on the complexity of $\\varphi $ .",
"The case when $\\varphi $ is of the form $P\\vec{x}$ is trivial.",
"The Boolean cases are also trivial.",
"Let $\\varphi $ be of the form $D_Xs$ .",
"Suppose $D_Xs\\in \\mathrm {last}(\\pi )$ .",
"Let $\\pi ^{\\prime }\\in W^c$ such that $\\pi \\sim ^c_X\\pi ^{\\prime }$ .",
"By Lemma REF , $\\mathrm {last}(\\pi ) R^q(X,\\varnothing ,\\varnothing )\\mathrm {last}(\\pi ^{\\prime })$ .",
"Then by Proposition REF (2,4), $\\pi \\sim ^c_s\\pi ^{\\prime }$ .",
"Thus $\\mathfrak {M}^c,\\pi \\models D_Xs$ .",
"Suppose $D_Xs\\notin \\mathrm {last}(\\pi )$ .",
"Let $\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\top ,\\mathrm {last}(\\pi ) \\rangle $ .",
"Then $\\pi \\lnot \\rightharpoonup _s\\pi ^{\\prime }$ and so $\\pi \\lnot \\rightleftharpoons _s\\pi ^{\\prime }$ .",
"Clearly, $T^\\pi _{\\pi ^{\\prime }}=\\langle \\pi ,\\pi ^{\\prime } \\rangle $ .",
"By Fact REF , $\\pi \\lnot \\sim ^c_s\\pi ^{\\prime }$ .",
"Note that $\\pi \\sim ^c_X\\pi ^{\\prime }$ , we see $\\mathfrak {M}^c,\\pi \\lnot \\models D_Xs$ .",
"Let $\\varphi =\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\psi $ .",
"Suppose $\\mathfrak {M}^c,\\pi \\models \\varphi $ .",
"Then there is $\\pi ^{\\prime }\\in R^c(X,Y,Z)\\pi $ such that $\\mathfrak {M}^c,\\pi ^{\\prime }\\models \\psi $ .",
"By induction hypothesis, $\\psi \\in \\mathrm {last}(\\pi ^{\\prime })$ .",
"By Lemma REF , $\\mathrm {last}(\\pi ) R^q(X,Y,Z)\\mathrm {last}(\\pi ^{\\prime })$ .",
"Then $\\varphi \\in \\mathrm {last}(\\pi )$ .",
"Suppose $\\varphi \\in \\mathrm {last}(\\pi )$ .",
"Without loss of generality, assume that $\\varphi \\in S(\\mathrm {last}(\\pi ))$ .",
"Then by Lemma REF , there is a $\\Delta $ such that $\\pi ^{\\prime }=\\langle \\pi ,\\varphi ,\\Delta \\rangle $ is a path with $\\psi \\in \\mathrm {last}(\\pi ^{\\prime })$ .",
"By induction hypothesis, $\\mathfrak {M}^c,\\pi ^{\\prime }\\models \\psi $ .",
"Note that $\\pi R^c(X,Y,Z)\\pi ^{\\prime }$ , we have $\\mathfrak {M}^c,\\pi \\models \\varphi $ .",
"For each $\\Gamma \\subseteq \\mathcal {L}^\\preceq $ , if $\\Gamma $ is consistent, then $\\Gamma $ is satisfiable."
],
[
"Properties of LPFD",
"In this part, we prove that LPFD lacks the finite model property.",
"The decidability of LPFD shall also be shown.",
"LPFD lacks the finite model property; that is, some formula $\\varphi \\in \\mathcal {L}^\\preceq $ is only satisfiable in infinite RPD-models.",
"Let $\\varphi =\\lnot (\\llbracket {\\varnothing ,\\varnothing ,\\lbrace z\\rbrace }\\rrbracket \\bot \\vee \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\varnothing ,\\lbrace z\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\llbracket {\\varnothing ,\\varnothing ,\\lbrace z\\rbrace }\\rrbracket \\bot )$ .",
"Note that for each PD-frame $\\mathfrak {F}=(W,\\sim ,\\le )$ and $z\\in \\mathsf {V}$ , $<_z$ is irreflexive and transitive.",
"Thus for each finite PD-frame $\\mathfrak {G}$ , we have $\\mathfrak {G}\\models \\llbracket {\\varnothing ,\\varnothing ,\\lbrace z\\rbrace }\\rrbracket \\bot \\vee \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\varnothing ,\\lbrace z\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\llbracket {\\varnothing ,\\varnothing ,\\lbrace z\\rbrace }\\rrbracket \\bot $ .",
"Clearly, $\\varphi $ is satisfiable in $(\\omega ,\\sim ,\\le )$ , where $\\le _z$ is the usual $\\le $ relation on $\\omega $ .",
"In what follows, let $\\alpha $ be some fixed formula, $\\mathsf {V}_\\alpha $ the set of variables occur in $\\alpha $ and $\\mathsf {Pred}_\\alpha $ the set of predicates occur in $\\alpha $ .",
"Without loss of generality, we assume that the modal depth of $\\alpha $ is not 0.",
"Then we define $\\text{Vo}=(V_\\alpha ,\\mathsf {Pred}_\\alpha ,ar{\\upharpoonright }V_\\alpha )$ as the vocabulary restricted to $\\alpha $ .",
"Let $\\mathcal {L}_\\alpha $ be the fragment of $\\mathcal {L}^\\preceq $ based on $\\text{Vo}$ , in which every formula is of modal degree no more than $\\alpha $ .",
"It can be easily verified that up to modal equivalence, $\\mathcal {L}_\\alpha $ contains only finitely many formulas.",
"A set $\\Gamma $ of $\\mathcal {L}_\\alpha $ -formulas is said to be a $\\mathcal {L}_\\alpha $ -maximal consistent set if $\\Gamma \\lnot \\vdash \\bot $ and $\\Gamma ^{\\prime }\\vdash \\bot $ for all $\\Gamma ^{\\prime }$ such that $\\Gamma \\subsetneq \\Gamma ^{\\prime }\\subseteq \\mathcal {L}_\\alpha $ .",
"Let $\\mathrm {MCS}_\\alpha $ denote the set of all $\\mathcal {L}_\\alpha $ -maximal consistent sets.",
"For all $X,Y,Z\\subseteq V_\\alpha $ and $\\Delta ,\\Sigma \\in \\mathrm {MCS}_\\alpha $ , we write $\\Delta R^p_{\\alpha }(X,Y,Z)\\Sigma $ if $\\lbrace \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X\\cap X^{\\prime },Y\\cap Y^{\\prime },(Z\\cap Y^{\\prime })\\cup (Z^{\\prime }\\cap Y)\\cup (Z\\cap Z^{\\prime })\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\mathcal {L}_\\alpha :\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Sigma \\rbrace \\subseteq \\Delta $ .",
"One may find that the definition of $R^p_{\\alpha }(X,Y,Z)$ is modified from the Lemmon filtration.",
"Given that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\mathcal {L}_\\alpha $ and $\\Delta R^p_{\\alpha }(X,Y,Z)\\Sigma $ , we see $\\varphi \\in \\Sigma $ implies $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Sigma $ and so $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Delta $ .",
"Then we have the following proposition: For all $\\Delta _1,\\Delta _2,\\Delta _3\\in \\mathrm {MCS}_\\alpha $ and $X,Y,Z\\subseteq V_\\alpha $ , $R^p_{\\alpha }(X,Y,\\varnothing )$ is reflexive; If $\\Delta _1 R^p_{\\alpha }(X,Y,Z)\\Delta _2$ , then $\\Delta _1 R^p_{\\alpha }(X^{\\prime },Y^{\\prime },Z^{\\prime })\\Delta _2$ for all $X^{\\prime }\\subseteq X$ , $Y^{\\prime }\\subseteq Y\\cup Z$ and $Z^{\\prime }\\subseteq Z$ ; If $D_XS\\in \\Delta _1$ and $\\Delta _1 R^p_{\\alpha }(X,Y,Z)\\Delta _2$ , then $\\Delta _1 R^p_{\\alpha }(S,Y,Z)\\Delta _2$ and $D_XS\\in \\Delta _2$ For all $Z^{\\prime }\\subseteq _{\\aleph _0}V_\\alpha $ , if $Z,Z^{\\prime }\\subseteq Y$ , $\\Delta _1 R^p_{\\alpha }(X,Y,Z)\\Delta _2$ and $\\Delta _2 R^p_{\\alpha }(X,Y,Z^{\\prime })\\Delta _3$ , then $\\Delta _1 R^p_{\\alpha }(X,Y,Z\\cup Z^{\\prime })\\Delta _3$ .",
"(1) and (2) are trivial.",
"For (3), $\\Delta _1 R^p_{\\alpha }(S,Y,Z)\\Delta _2$ follows from axiom (Dep,d).",
"Recall that the modal depth of $\\alpha $ is not 0, we see $\\mathbb {D}_XD_XS\\in \\Delta _1$ and so $D_XS\\in \\Delta _2$ .",
"For (4), suppose $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X_0,Y_0,Z_0\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Delta _3$ .",
"Then $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X\\cap X_0,Y\\cap Y_0,(Z^{\\prime }\\cap Y_0)\\cup (Z_0\\cap Y)\\cup (Z^{\\prime }\\cap Z_0)\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Delta _2$ .",
"Recall that $Z,Z^{\\prime }\\subseteq Y$ , it follows that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X\\cap X_0,Y\\cap Y_0,((Z\\cup Z^{\\prime })\\cap Y_0)\\cup (Z_0\\cap Y)\\cup ((Z\\cup Z^{\\prime })\\cap Z_0)\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Delta _1.$ Thus $\\Delta _1 R^p_{\\alpha }(X,Y,Z\\cup Z^{\\prime })\\Delta _3$ and (4) holds.",
"[$\\mathcal {L}_\\alpha $ -Pre-model] An $\\mathcal {L}_\\alpha $ -pre-model is a set $F$ of $\\mathcal {L}_\\alpha $ -MCSs such that for all $X,Y,Z\\subseteq V_\\alpha $ and $\\Delta \\in F$ , the following statement holds: If $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ is a saturated formula in $\\Delta $ , then there is $\\Sigma \\in F$ such that $\\Delta R^p_{\\alpha }(X,Y,Z)\\Sigma $ , $\\varphi \\in \\Sigma $ and $\\Sigma R^p_{\\alpha }(X,Y,\\varnothing )\\Delta $ .",
"We say $\\varphi $ is satisfied in $F$ if there is some $\\Delta \\in F$ such that $\\varphi \\in \\Delta $ .",
"For each satisfiable $\\varphi \\in \\mathcal {L}^\\preceq $ , $\\varphi $ is satisfied in some pre-model.",
"Let $\\mathfrak {M}=(W,\\sim ,\\le ,V)$ be a RPD-model and $w\\in W$ such that $\\mathfrak {M},w\\models \\varphi $ .",
"Then we define $F_\\mathfrak {M}=\\lbrace \\Delta _w:w\\in \\mathfrak {M}\\text{ and }\\Delta _w=\\lbrace \\varphi \\in \\mathcal {L}_\\alpha :\\mathfrak {M},w\\models \\varphi \\rbrace \\rbrace $ .",
"It suffices to show that $F_\\mathfrak {M}$ satisfies (†).",
"Suppose $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ is a saturated formula in $\\Delta _w$ .",
"Then $\\mathfrak {M},w\\models \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ and there is $u\\in R(X,Y,Z)(w)$ such that $\\mathfrak {M},u\\models \\varphi $ .",
"Note that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ is a saturated formula, we have $w\\in R(X,Y,\\varnothing )(u)$ .",
"Then it is not hard to verify that $\\Delta _w R^p_{\\alpha }(X,Y,Z)\\Delta _u$ and $\\Delta _u R^p_{\\alpha }(X,Y,\\varnothing )\\Delta _w$ .",
"Recall that $\\varphi \\in \\Delta _u$ , we see that (†) holds for $F_\\mathfrak {M}$ .",
"[Induced Model] Let $F$ be a pre-model.",
"An $F$ -path is a tuple $\\langle \\Sigma _0,\\psi _0,\\cdots ,\\Sigma _{n-1},\\psi _{n-1},\\Sigma _n \\rangle $ where the following conditions hold for all $i<n$ : $\\psi _i=\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X_i,Y_i,Z_i\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi _i$ is a saturated formula in $\\Sigma _i\\in F$ ; $\\varphi _i\\in \\Sigma _{i+1}\\in F$ , $\\Sigma _{i+1}R^p_{\\alpha }(X_i,Y_i,\\varnothing )\\Sigma _i$ and $\\Sigma _i R^p_{\\alpha }(X_i,Y_i,Z_i)\\Sigma _{i+1}$ .",
"The RPD-model $\\mathfrak {M}^F=(W^F_\\Gamma ,\\le ^F,\\sim ^F,V^F)$ induced by $\\Gamma \\in F$ is defined by: $W^F_\\Gamma $ is the set of all paths in $F$ begins with $\\varphi $ .",
"for all $y\\in V_\\alpha $ and $\\pi ,\\pi ^{\\prime }\\in W^F_\\Gamma $ , $\\pi \\le _y\\pi ^{\\prime }$ iff one of the following holds: $\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ and $y\\in Y\\cup Z$ ; $\\pi =\\langle \\pi ^{\\prime },\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ and $y\\in Y$ .",
"Let $\\le ^F_y$ be the reflexive-transitive closure of $\\le _y$ .",
"for all $s\\in V_\\alpha $ and $\\pi ,\\pi ^{\\prime }\\in W^F_\\Gamma $ , $\\pi \\rightharpoonup _s\\pi ^{\\prime }$ if and only if $\\pi ^{\\prime }=\\langle \\pi ,\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi ,\\Sigma \\rangle $ and $D_Xs\\in \\mathrm {last}(\\pi )$ .",
"Let $\\sim ^F_s$ the reflexive-symmetric-transitive closure of $\\rightleftharpoons _s$ .",
"for all $P\\vec{x}\\in \\mathcal {L}_\\alpha $ , $V^F(P\\vec{x})=\\lbrace \\pi \\in W^F: P\\vec{x}\\in \\mathrm {last}(\\pi )\\rbrace $ .",
"One may notice now that the construction of the desired model is almost the same as the one we used in the proof of Completeness Theorem.",
"And similar to the proof of Completeness Theorem, with the help of Fact REF and the definition of pre-models, we can verify that the following lemma holds: [Truth Lemma] For each formula $\\varphi \\in \\mathcal {L}_\\alpha $ and path $\\pi \\in W^F$ , $\\mathfrak {M}^F,\\pi \\models \\varphi $ if and only if $\\varphi \\in \\mathrm {last}(\\pi )$ .",
"As a consequence, for each $\\varphi \\in \\mathcal {L}^\\preceq $ , $\\varphi $ is satisfiable if and only if $\\varphi $ is satisfied in some $\\mathcal {L}_\\varphi $ -pre-model.",
"Recall that up to modal equivalence, $\\mathcal {L}_\\varphi $ contains finitely many formulas, $\\mathrm {MCS}_\\varphi $ is finite for each $\\varphi \\in \\mathcal {L}^\\preceq $ , we obtain the following theorem: The satisfiability problem of LPFD is decidable."
],
[
"The Hybrid Extension of LPFD",
"In this subsection, we extend LPFD with nominals.",
"By a vocabulary with nominals we mean a tuple $(\\mathsf {V},\\mathsf {Pred},\\mathsf {Nom},\\mathsf {ar})$ where $(\\mathsf {V},\\mathsf {Pred},\\mathsf {ar})$ is a vocabulary and $\\mathsf {Nom}=\\lbrace i_k:k\\in \\omega \\rbrace $ a denumerable set of nominals.",
"The language $\\mathcal {L}^\\preceq _\\mathsf {Nom}$ with nominals is given by: $\\mathcal {L}^\\preceq _\\mathsf {Nom}\\ni \\varphi ::= P\\vec{x}\\mid D_Xy \\mid i \\mid \\lnot \\varphi \\mid \\varphi \\wedge \\varphi \\mid \\llbracket {X,Y,Z}\\rrbracket \\varphi ,$ which only differs from the language of LPFD in those nominals.",
"We modify the valuation $V$ in a RPD-model $\\mathfrak {M}=(W,\\sim ,\\le ,V)$ correspondingly such that $V{\\upharpoonright }\\mathsf {Nom}$ is a partial function from $\\mathsf {Nom}$ to $W$ .",
"The resulted RPD-models are called RPDN-models.",
"The semantic truth of the nominals in an RPDN-model is defined as follows: Table: NO_CAPTIONAs usual, we call LPFD with nominals `hybrid LPFD', abbreviated to HLPFD.",
"Let $\\mathsf {Nom}$ be a fixed set of nominals.",
"We present here the calculus $\\mathsf {C}_\\mathsf {Nom}$ for HLPFD and show its soundness and completeness.",
"Let $X,Y,Z\\in \\mathcal {P}^{<\\aleph _0}(\\mathsf {V})$ , $\\varphi ,\\psi \\in \\mathcal {L}^\\preceq _\\mathsf {Nom}$ , $i,j\\in \\mathsf {Nom}$ , $P\\in \\mathsf {Pred}$ and $v\\in \\mathsf {V}$ .",
"The axioms and rules of $\\mathsf {C}_\\mathrm {HLPFD}$ are as follows: Axioms and rules for classical propositional logic; from $\\varphi $ infer $\\llbracket {X,Y,Z}\\rrbracket \\varphi $ ; $\\llbracket {X,Y,Z}\\rrbracket (\\varphi \\rightarrow \\psi )\\rightarrow (\\llbracket {X,Y,Z}\\rrbracket \\varphi \\rightarrow \\llbracket {X,Y,Z}\\rrbracket \\psi )$ ; $\\varphi \\rightarrow \\mathbb {D}_X\\varphi $ , provided $\\varphi \\in \\mathsf {Atom}(X)=\\lbrace P\\vec{x}:\\mathsf {set}(\\vec{x})\\subseteq X\\rbrace $ ; $@_i\\varphi \\rightarrow \\llbracket {\\varnothing ,\\varnothing ,\\varnothing }\\rrbracket (i\\rightarrow \\varphi )$ , provided $i\\in \\mathsf {Nom}$ ; from $i\\rightarrow \\varphi $ infer $\\varphi $ , provided that $i\\notin \\varphi $ , i.e., $i$ does not occur in $\\varphi $ ; from $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\rightarrow @_j\\varphi $ infer $@_i\\llbracket {X,Y,Z}\\rrbracket \\varphi $ , provided $i\\ne j$ and $j\\notin \\varphi $ ; Axioms and rules for $\\llbracket {\\ }\\rrbracket -D$ interaction: $D_Xs\\wedge \\llbracket {\\lbrace s\\rbrace ,\\varnothing ,\\varnothing }\\rrbracket \\varphi \\rightarrow \\llbracket {X,\\varnothing ,\\varnothing }\\rrbracket \\varphi $ ; $i\\wedge \\lnot D_Xs\\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\llbracket {s,\\varnothing ,\\varnothing }\\rrbracket \\lnot i$ .",
"Axioms for the preference orders: $\\llbracket {X,Y,\\varnothing }\\rrbracket \\varphi \\rightarrow \\varphi $ ; $\\varphi \\rightarrow \\llbracket {\\lbrace v\\rbrace ,\\varnothing ,\\varnothing }\\rrbracket \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\lbrace v\\rbrace ,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ ; $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X\\cap X^{\\prime }, Y\\cap Y^{\\prime },(Z\\cap Y^{\\prime })\\cup (Z\\cap Z^{\\prime })\\cup (Y\\cap Z^{\\prime })\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi $ ; $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\varnothing ,\\lbrace v\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\leftrightarrow @_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\lbrace v\\rbrace ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\wedge @_j\\lnot \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\lbrace v\\rbrace ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}i$ , provided $i,j\\in \\mathsf {Nom}$ ; $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}i\\wedge \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X^{\\prime },Y^{\\prime },Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}i\\leftrightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X\\cup X^{\\prime },Y\\cup Y^{\\prime },Z\\cup Z^{\\prime }\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}i$ , provided $i\\in \\mathsf {Nom}$ .",
"Comparing $\\mathsf {C}_\\mathsf {Nom}$ with $\\mathsf {C}$ , in addition to the standard axioms and rules for nominals, axioms (Ord,4,5) and (DD,2) are new, which characterize RPD-models in a more refined way.",
"Note also that some old axioms in $\\mathsf {C}$ are presented in $\\mathsf {C}_\\mathsf {Nom}$ in a different way.",
"For example, axiom (DD,1) in $\\mathsf {C}_\\mathsf {Nom}$ are bottom-up versions of axioms (Dep,d) in $\\mathsf {C}$ .",
"With the above mentioned changes in $\\mathsf {C}_\\mathsf {Nom}$ due to the addition of nominals, the completeness of $\\mathsf {C}_\\mathsf {Nom}$ can be proved by directly using the canonical model, which is a standard method and relatively routine.",
"So we relegate the details of the following theorem's proof in the appendix.",
"$\\mathsf {C}_\\mathsf {Nom}$ is sound and strongly complete.",
"Note that the equivalence between PD-models with nominals (where $I(i) \\in A$ for $i\\in \\mathsf {Nom}$ ) and RPDN-models with respect to $\\mathcal {L}^\\preceq _\\mathsf {Nom}$ can be established as in Section REF .",
"So for the class of PD-models with nominals we also have the soundness and strong completeness of $\\mathsf {C}_\\mathsf {Nom}$ .",
"As for the decidability of HLPFD, we cannot prove it by directly following the strategy used in the proof of LPFD's decidability.",
"We will not attack this problem in this paper but rather leave it for future work.",
"Coalition logic is proposed to reason about coalitional effectiveness in games in strategic form.",
"However, in non-cooperative games, each player plays separately rather than as an integral part of a coalition.",
"The so-called coalitional effectiveness is essentially the effectiveness of an agglomeration of actions.",
"Should there be any difference between a coalitional action and an agglomeration of actions?",
"This is a key issue in the philosophical analysis of collective agency [20].",
"In this section, we provide a game theoretical perspective on this issue by modelling cooperative games in strategic and coalitional form [19] in LPFD and characterizing one of its solution concepts, the core, in HLPFD."
],
[
"Cooperative Games in LPFD",
"Different from non-cooperative games, in cooperative games in strategic and coalitional form [19], players can not only act individually but also choose to join a coalition and act as a part of the coalition.",
"In such games, the players in a coalition can do something together in agreement rather than separately.",
"So coalitional actions and power are different from an agglomeration of actions and its effectiveness.",
"This difference is essential to our game theoretical perspective on collective agency, which we will elaborate on in Section .",
"In this part, we propose a framework based on LPFD to represent cooperative games and make the difference explicit.",
"For simplicity, we restrict ourselves to the case where $\\mathsf {V}$ is finite.",
"We use $N = \\lbrace 1,2,\\ldots ,n\\rbrace $ rather than $\\mathsf {V}$ to indicate the finiteness of the players.",
"We first specify what constitutes $O$ and $A$ .",
"To explicitly model coalitions as a different part of each player's choices from strategies, we distinguish between the terms “strategy”(or equivalently “actions”) and “choices”.",
"[Players' Strategies] Let $\\Sigma $ be the set of all possible strategies of all players and $\\Sigma ^{<N} := \\bigcup _{i=1}^n \\Sigma ^i$ be the set of all strategy sequences of length at most $n$ .",
"[Players' Choices and Choices Merging] The set of the players' choices is defined as follows: $O := \\lbrace f: I\\rightarrow \\Sigma \\mid I\\subseteq N\\rbrace \\hspace{5.0pt}.$ For $f,f^{\\prime }\\in O$ with $\\mathsf {dom}( f )\\cap \\mathsf {dom}( f^{\\prime } ) = \\emptyset $ , $f\\oplus f^{\\prime }:= f\\cup f^{\\prime }$ .",
"For example, given three players $N = \\lbrace 1,2,3\\rbrace $ and the players' possible strategies in $\\Sigma = \\lbrace \\alpha ,\\beta \\rbrace $ , $f = \\lbrace (1,\\alpha ),(3,\\beta )\\rbrace \\in O$ denotes a possible choice of the players 1 and 3 as a coalition; $f^{\\prime } = \\lbrace (2,\\alpha )\\rbrace \\in O$ denotes a possible choice of the player 2.",
"Then $f\\oplus f^{\\prime } = \\lbrace (1,\\alpha ),(2,\\alpha ),(3,\\beta )\\rbrace $ .",
"In a PD-model, there is no requirement on $A\\subseteq O^N$ .",
"This is not the case any longer when the players' choices concern forming coalitions.",
"We impose three conditions on a realizable choice profile.",
"First of all, a player cannot choose to form a coalition she is not in.",
"Second, a player cannot choose to form a coalition without the others in the coalition making the same choice.",
"Third, once a coalition forms, it acts as a whole, which means that its members act according to a unique strategy sequence.",
"This strategy sequence can be seen as a collective plan which is made effective by common consent.",
"To make the definition of realizable choice profiles precise, we make use of the following notations.",
"Notation $\\Pi (N)$ is the set of all partitions of $N$ .",
"A partition of $N$ is a set of non-empty subsets of $N$ whose union is $N$ and which do not intersect each other.",
"Given $a\\in O^N$ , $a_i$ denotes the $i$ th element of $a$ , which is a function; $a_\\mathsf {rng} := \\lbrace a_i\\in O\\mid i\\in N\\rbrace $ ; $a_\\mathsf {dom} := \\lbrace \\mathsf {dom}( a_i )\\subseteq N\\mid i\\in N\\rbrace $ ; [Realizable Choice and Strategy Profiles] A choice profile $a\\in O^N$ is realizable if and only if it satisfies the following three conditions: $i\\in \\mathsf {dom}( a_i )$ ; $a_\\mathsf {dom}\\in \\Pi (N)$ $\\mathsf {dom}( a_i ) = \\mathsf {dom}( a_j )$ implies that $a_i = a_j$ for all $i,j\\in N$ .",
"Let $\\Xi $ denote the set of all realizable choice profiles.",
"Let $a_\\mathsf {merge} := \\bigoplus _{f\\in a_\\mathsf {rng}} f$ for $a\\in \\Xi $ .",
"Given $A\\subseteq \\Xi $ , the set of all realizable strategy profiles of a partition $\\pi \\in \\Pi (N)$ in $A$ is $\\sigma _A(\\pi ) := \\lbrace a_\\mathsf {merge}\\mid a\\in A \\text{ and } a_\\mathsf {dom} = \\pi \\rbrace \\hspace{5.0pt}.$ When there is no danger of ambiguity, we will leave out the subscript $A$ .",
"Having defined $O$ and $\\Xi $ , we define a class of PD-models we will work with.",
"[Coalition-preference-dependence (CPD) models] A coalition-preference-dependence model is a PD-model $\\mathbb {M} = ((O,I),A)$ in which $O$ is defined in Definition REF and $A$ and $\\preceq _i$ satisfy the following conditions: $A\\subseteq \\Xi $ ; $\\lbrace a_\\mathsf {dom}\\mid a\\in A\\rbrace = \\Pi (N)$ ; if $\\pi \\in \\Pi (N)$ is finer than $\\pi ^{\\prime }\\in \\Pi (N)$ ,That is, for all $X\\in \\pi $ there is $X^{\\prime }\\in \\pi ^{\\prime }$ such that $X\\subseteq X^{\\prime }$ .",
"then $\\sigma _A(\\pi )\\subseteq \\sigma _A(\\pi ^{\\prime })$ ; if $a_\\mathsf {merge} = a^{\\prime }_\\mathsf {merge}$ , then $a\\simeq _i a^{\\prime }$ for all $i\\in N$ ; $\\preceq _i$ is total for all $i\\in N$ .",
"The first condition says that $A$ should contain realizable choice profiles.",
"The second condition says that the players can form coalitions according to all possible partitions of $N$ .",
"The third condition requires bigger coalitions to have no less strategies than smaller coalitions.",
"The fourth condition requires that the players' preference relations depend directly on strategy profiles.",
"The players' choices of coalitions can only influence the players' preferences by affecting their strategies.",
"The last condition requires the players' preference relations to be total, which is a standard assumption in game theory.",
"The following example illustrates our notations and the CPD-models.",
"Let $N = \\lbrace 1,2,3\\rbrace $ and $\\Sigma = \\lbrace \\alpha ,\\beta ,\\gamma \\rbrace $ .",
"$A$ is given in Table REF .",
"Table: AA in Example According to our notation, $a_\\mathsf {merge} = a^{\\prime }_\\mathsf {merge} = a_\\mathsf {merge}^{3\\prime } = a_\\mathsf {merge}^{5\\prime } = a_\\mathsf {merge}^{7\\prime } = \\lbrace (1,\\alpha ),(2,\\beta ),(3,\\alpha )\\rbrace $ ; $\\sigma (\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,\\lbrace 3\\rbrace \\rbrace ) = \\lbrace \\lbrace (1,\\alpha ),(2,\\beta ),(3,\\alpha )\\rbrace \\rbrace $ and $\\sigma (\\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 3\\rbrace \\rbrace ) = \\lbrace \\lbrace (1,\\alpha ),(2,\\beta ),(3,\\alpha )\\rbrace , \\lbrace (1,\\alpha ),(2,\\gamma ),(3,\\beta )\\rbrace \\rbrace $ .",
"As the readers can verify, all the requirements of a CPD-model concerning $A$ are satisfied here.",
"For example, $\\sigma (\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,\\lbrace 3\\rbrace \\rbrace )\\subseteq \\sigma (\\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 3\\rbrace \\rbrace )\\subseteq \\sigma (\\lbrace N\\rbrace )$ .",
"To make sure $\\preceq _i$ satisfy the requirements, $a\\simeq _i a^{\\prime }\\simeq _i a^{3\\prime }\\simeq _i a^{5\\prime }\\simeq _i a^{7\\prime }$ needs to be the case.",
"As can be easily spotted in the above example, coalitions are explicitly incorporated into the players' choices in the CPD-models.",
"Once a coalition forms, the players in it act as a whole.",
"Moreover, a coalition could possibly do more than its constituent parts.",
"The coalition partition formed in a game directly affects each player's strategy.",
"Hence it has a substantial influence on the final outcome of the game.",
"Can the language of LPFD express what partition is formed in a realizable choice profile?",
"The following proposition gives a partially positive answer.",
"Let $\\mathbb {M}=((M,A),\\le )$ be a CPD-model with $\\mathbb {M},a^{\\prime }\\models \\lnot D_X(-X)$ for all $a^{\\prime }\\in A$ satisfying $a^{\\prime }_\\mathsf {dom} = \\lbrace X,-X\\rbrace $ .",
"Then for all $a\\in A$ and non-empty subset $X\\subseteq N$ , the following two are equivalent: $X\\in a_\\mathsf {dom}$ ; $\\mathbb {M},a\\models \\bigwedge _{i\\in X} D_iX\\wedge \\bigwedge _{j\\notin X} \\lnot D_X j$ .",
"From 1 to 2.",
"Assume $X\\in a_\\mathsf {dom}$ .",
"Suppose $a^{\\prime }\\in A$ and $a =_i a^{\\prime }$ for some $i\\in X$ .",
"Then $X\\in a^{\\prime }_\\mathsf {dom}$ .",
"Since $A\\subseteq \\Xi $ , $a_i = a_j$ and $a^{\\prime }_i = a^{\\prime }_j$ for all $i,j\\in X$ .",
"Note that $a =_i a^{\\prime }$ for some $i\\in X$ , we see $a_j = a_i = a^{\\prime }_i = a^{\\prime }_j$ for all $j\\in X$ , i.e.",
"$a =_X a^{\\prime }$ .",
"Thus $\\mathbb {M},a\\models D_iX$ .",
"By the arbitrariness of $i\\in X$ , we see $\\mathbb {M},a\\models \\bigwedge _{i\\in X} D_iX$ .",
"When $X=N$ , we see that $\\bigwedge _{j\\notin X} \\lnot D_X j$ is $\\top $ and $\\mathbb {M},a\\models \\bigwedge _{j\\notin X} \\lnot D_X j$ .",
"Suppose $X\\ne N$ .",
"Take an arbitrary $j\\notin X$ .",
"Then we have the following cases: $a_\\mathsf {dom}\\ne \\lbrace X,-X\\rbrace $ .",
"Let $\\pi = \\lbrace X,-X\\rbrace $ .",
"Note that $\\sigma _A(a_\\mathsf {dom})\\subseteq \\sigma _A(\\pi )$ , there must be $b\\in A$ such that $b_\\mathsf {dom} = \\pi $ and $a_\\mathsf {merge} = b_\\mathsf {merge}$ .",
"Then it must be the case that $\\mathsf {dom}( b_j )= -X\\ne \\mathsf {dom}( a_j )$ and so $a \\ne _j b$ .",
"$a_\\mathsf {dom}=\\lbrace X,-X\\rbrace $ .",
"Since $\\mathbb {M},a\\models \\lnot D_X(-X)$ , there must be $b\\in A$ such that $a =_X b$ and $a\\ne _{-X} b$ .",
"If $a_\\mathsf {dom} \\ne b_\\mathsf {dom}$ , then $\\mathsf {dom}( a_j )= -X\\ne \\mathsf {dom}( b_j )$ and so $a\\ne _j a^{\\prime }$ .",
"Suppose $a_\\mathsf {dom} = b_\\mathsf {dom}$ .",
"Then $a_k$ are all the same for $k\\in -X$ and $a^{\\prime }_h$ are all the same for $h\\in -X$ .",
"Since $a\\ne _{-X} a^{\\prime }$ , we see $a_j \\ne a^{\\prime }_j$ .",
"Hence $\\mathbb {M},a\\models \\lnot D_X j$ .",
"By the arbitrariness of $j$ , we see $\\mathbb {M},a\\models \\bigwedge _{j\\notin X} \\lnot D_X j$ .",
"From 2 to 1.",
"Assume that $X\\notin a_\\mathsf {dom}$ and $\\mathbb {M},a\\models \\bigwedge _{i\\in X} D_iX\\wedge \\bigwedge _{j\\notin X} \\lnot D_X j$ .",
"Let $x\\in X$ .",
"$X\\subsetneq \\mathsf {dom}(a_x)$ .",
"Then there is $j\\in \\mathsf {dom}(a_x)\\setminus X$ such that $a_j = a_i$ for all $i\\in \\mathsf {dom}(a_x)$ .",
"So for all $a^{\\prime } =_X a$ , $a^{\\prime }_j = a_i = a^{\\prime }_i$ for all $i\\in X$ .",
"Then we have $\\mathbb {M},a\\models D_X j$ where $j\\notin X$ .",
"Contradiction!",
"Otherwise, there is $j\\in X\\setminus \\mathsf {dom}(a_x)$ .",
"Since $\\mathbb {M},a\\models D_xX$ , we see $\\mathbb {M},a\\models D_{\\mathsf {dom}(a_x)}j$ .",
"By the direction we have proved above, $\\mathbb {M},a\\models \\lnot D_{\\mathsf {dom}(a_x)}j$ , which is a contradiction.",
"The assumption of the above proposition that $\\mathbb {M},a\\models \\lnot D_X(-X)$ for all $a\\in A$ satisfying $a^{\\prime }_\\mathsf {dom} = \\lbrace X,-X\\rbrace $ requires that no coalition can completely decides what its complementary coalition chooses to do.",
"If $X$ can completely control what $-X$ chooses, then the division of $X$ and $-X$ is senseless, because $\\mathbb {M},a\\models D_X N$ follows from $\\mathbb {M},a\\models D_X -X$ .",
"As the readers can verify, the CPD-model in Example REF does not satisfy the assumption at $a^{\\prime \\prime },a^{4\\prime },a^{6\\prime }$ .",
"To avoid vacuous coalitions division, we will work with the CPD-models with the above assumption.",
"[Real CPD-models] A real CPD-model (RCPD-model) $\\mathbb {M}$ is a CPD-model that satisfies the assumption that $\\mathbb {M},a\\models \\lnot D_X(-X)$ for all $a\\in A$ satisfying $a_\\mathsf {dom} = \\lbrace X,-X\\rbrace $ .",
"In a RCPD-model, $\\bigwedge _{i\\in X} D_iX\\wedge \\bigwedge _{j\\notin X} \\lnot D_X j$ expresses that $X$ is in the coalition partition.",
"We will use the abbreviation $p_X := \\bigwedge _{i\\in X} D_iX\\wedge \\bigwedge _{j\\notin X} \\lnot D_X j$ for convenience in the next section, where we demonstrate that LPFD as presented in this section provides a useful scaffolding for approaching several issues on coalitions."
],
[
"The Core in HLPFD",
"Having set up the LPFD framework for representing cooperative games in strategic and coalitional games, in this part, we show that the core, an important solutions concept in the cooperative game theory, can be expressed in HLPFD.",
"Moreover, by considering functional dependence explicitly, we generalize the core and show how it is related to Nash equilibrium and Pareto optimality.",
"Just as Nash equilibrium in non-cooperative games captures stability of a strategy profile, the concept of the core, as a basic solution concept in cooperative games, also captures stability of a strategy profile in cooperative games.",
"The difference is that the core takes the stability of a coalition into consideration.",
"There are other notions for characterizing stability in cooperative games, for example, stable set, bargaining set and so on.",
"In this paper, we focus on the core.",
"The concept of the core is formulated in CPD-models as follows.",
"The definition of the core can vary in different settings.",
"Our definition is based on [9], which is a relatively general version.",
"[Core in CPD-Model] Given a CPD-model $\\mathbb {M}$ , a choice profile $a\\in A$ is in the core of $\\mathbb {M}$ if and only if $a_\\mathsf {dom} = \\lbrace N\\rbrace $ ; and there is no $X\\subseteq N$ and $a^{\\prime }\\in A$ such that $X\\in a^{\\prime }_\\mathsf {dom}$ ; and for all $a^{\\prime \\prime } =_X a^{\\prime }$ and all $i\\in X$ , $a \\prec _i a^{\\prime \\prime }$ .",
"Let $Co_\\mathbb {M}$ denote the core of $\\mathbb {M}$ .",
"If $N$ arrives at a choice profile $a$ , which is in the core, then no $X\\subset N$ has any incentive to deviate from the coalition $N$ , because forming the coalition $X$ cannot guarantee all players in $X$ end up with a better outcome.",
"Coalitional power plays a key role in the basic idea of core, because whether $X$ has any incentive to deviate depends on whether $X$ as a coalition can force a choice profile that all of its members prefer to the current choice profile.",
"Note that according to the definition of the core, if $X = N$ , there is no other choice profile with the coalition partition $\\lbrace N\\rbrace $ which is strictly preferred by every player in $N$ .",
"Namely, $a$ is weakly Pareto optimal among the choice profiles with the coalition partition $\\lbrace N\\rbrace $ .",
"In fact, the following proposition holds.",
"Given a CPD-model $\\mathbb {M}$ , if a choice profile $a\\in A$ is in the core of $\\mathbb {M}$ then $a$ is weakly Pareto optimal.",
"Since $\\mathbb {M}$ satisfies the condition that $\\sigma _A(\\pi )\\subseteq \\sigma _A(\\lbrace N\\rbrace )$ for all $\\pi \\in \\Pi (N)$ , by the fourth condition of Definition REF , the weak Pareto optimality of $a$ within the choice profiles having $\\lbrace N\\rbrace $ as their coalition partition can be generalized trivially to all choice profiles.",
"The following example illustrates the concept of core and how it differs from Nash equilibrium and Pareto optimality.",
"Let $N = \\lbrace 1,2\\rbrace $ and $\\Sigma = \\lbrace \\alpha ,\\beta \\rbrace $ .",
"$A$ and the preference relations are given in Table REF .",
"The preference relations are given in the form of a pair of ordinal utilities where the first element is for player 1 and the second for player 2.",
"Table: AA in Example Readers familiar with game theory can recognize that without the last four rows the table represents the prisoners' dilemma.",
"$a^{3\\prime }$ is a Nash equilibrium but $a$ is not as in the original prisoners' dilemma.",
"Now our coalitional version allows player 1 and player 2 to form a coalition by whatever means, for example, a binding agreement or switching to the mode of team reasoning simultaneously.",
"So there are four extra profiles in which both players explicitly choose to join the coalition.",
"Among these four extra profiles, $a^{4\\prime }$ is the only element in the core.",
"Thus it is both Pareto optimal and a Nash equilibrium.",
"Note that in the example $\\lbrace 1,2\\rbrace $ as a coalition does not expand what each of the players can choose, namely $\\sigma (\\lbrace 1,2\\rbrace ) = \\sigma (\\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace \\rbrace )$ .",
"But it still makes some difference.",
"This difference is brought about by something collective as clearly reflected in our example.",
"The core captures this collective element in the example.",
"Next, we show that the core can be expressed in HLPFD with respect to the class of RCPD-models (Definition REF ) with nominals.",
"Given a RCPD-model $\\mathbb {M}$ with nominals $\\mathsf {Nom}$ , the current choice profile $a$ with name $i$ , i.e., $a = I(i)\\in A$ , is in the core of $\\mathbb {M}$ , if and only if $\\mathbb {M},a\\models i\\wedge p_N\\wedge \\bigwedge _{\\emptyset \\ne X\\subseteq N} \\begin{sideways}\\begin{sideways}\\forall \\end{sideways}\\end{sideways}(p_X\\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,\\emptyset ,\\emptyset \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\bigvee _{x\\in X}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\emptyset ,\\lbrace x\\rbrace ,\\emptyset \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}i).$ In fact, as in the case of Nash equilibrium and Pareto optimality, we can also have a relativized version of the core as follows $\\mathsf {Core}_X i := i\\wedge p_X\\wedge \\bigwedge _{\\emptyset \\ne C\\subseteq X} \\mathbb {D}_{-X} (p_C\\rightarrow \\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}-X\\cup C,\\emptyset ,\\emptyset \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\bigvee _{c\\in C}\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}-X,\\lbrace c\\rbrace ,\\emptyset \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}i)$ Note that when taking $X = N$ , we get the original definition of the core as expressed in Proposition REF .",
"The relativized version of the core enables us to express some interesting relationships between coalitions.",
"For example, $\\mathsf {Core}_X i\\wedge \\mathsf {Core}_{-X} i$ which says that in the current choice profile $i$ , both $X$ and $-X$ form coalitions and are in their relativized cores.",
"More generally, we can define the following concept: $\\mathsf {Core}_\\pi i := \\bigwedge _{X\\in \\pi } \\mathsf {Core}_X i$ where $\\pi $ is a partition of $N$ .",
"It characterizes the stability of a collection of coalitions at a choice profile $i$ .",
"The core is a special case of it where $\\pi = \\lbrace N\\rbrace $ .",
"Moreover, Nash equilibrium $\\operatorname{\\mathsf {Na}}N$ is also a special case of it where $a_\\mathsf {dom} =\\pi = \\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,\\ldots ,\\lbrace n\\rbrace \\rbrace $ .",
"Given a RCPD-model $\\mathbb {M}$ with nominals $i\\in \\mathsf {Nom}$ , and $a\\in A$ with $a_\\mathsf {dom} =\\pi = \\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,\\ldots ,\\lbrace n\\rbrace \\rbrace $ , $\\mathbb {M},a\\models \\mathsf {Core}_\\pi i \\leftrightarrow (i\\wedge \\operatorname{\\mathsf {Na}}N)\\hspace{5.0pt}.$ As a corollary to this proposition, we see that unlike the core $\\mathsf {Core}_\\pi i$ does not necessarily imply the weak Pareto optimality of $i$ for $N$ .",
"But the following generalization of Proposition REF holds.",
"Given a RCPD-model $\\mathbb {M}$ with nominals $i\\in \\mathsf {Nom}$ and $a\\in A$ , $\\mathbb {M},a\\models \\mathsf {Core}_\\pi i \\rightarrow \\bigwedge _{X\\in \\pi }\\operatorname{\\mathsf {wPa}}X\\hspace{5.0pt}.$ Therefore, in the sense of the above two theorems, our generalization of the core can be seen as a notion that unifies the core, Nash equilibrium and Pareto optimality."
],
[
"Stability, Coalitional Power and Collective Agency",
"In this section, we show how the CPD-models can help clarify issues on collective agency and explore some philosophical implications from the game-theoretical perspective on collective agency in CPD-models.",
"Philosophical discussions about collective agency have flourished in recent decades.",
"Despite disagreements on the detailed definition of collective agency, most theories share the idea that joint actions by a group with collective agency are more than simply a coordination or cooperation between its members (cf.",
"[8]; [11]; [15]; [22]; [24]; [25]).",
"Nevertheless, the conundrum is where does this essential difference lie.",
"Gilbert [11], Searle [22], and Tuomela [25] admit an irreducible concept of a collective in a methodological sense.",
"In Gilbert, it is a unique type of commitment of will: “joint commitment”; in Searle, it is a special kind of intention: “we-intention\"; and in Tuomela, it is a complex of “we-intention\" and a particular form of attitude: “we-mode.\"",
"They all try to start from an irreducible concept of a collective to capture the extras of joint actions.",
"In a similar sense, List and Pettit [15] emphasize that a group with agency must have a procedure to ensure that its decision-making process meets the necessary functional conditions of an agent, such as manifest rationality; Tuollefsen [24] highlights that a group with agency must contain stable structures in order to conform to the basic phenomena that can be reasonably explained by the observer.",
"Even for Bratman [8], who famously argues that we can explain collective agency without any irreducible concept of a collective, he still claims the critical role of the interdependent relations and the mesh of individual plans between members in forming collective intentions.",
"As Bratman emphasized, we also pay special attention to the critical role of interdependence in forming a collective agent.",
"Instead of intentionality, following the standard game-theoretical approach, individual preferences and choices are the starting point of our analysis, from which we explore the stability of the interdependence as embodied in the core.",
"In CPD-models, coalitions are taken explicitly as a part of each individual player's choices.",
"That is, each player chooses which coalitions to join.",
"This makes it possible to distinguish between a group action and a set of individual actions.",
"In Example REF , although $a^{4\\prime }$ and $a$ have the same strategy profile, namely $a^{4\\prime }_\\mathsf {merge} = a_\\mathsf {merge}$ , acting together ($a^{4\\prime }_\\mathsf {dom} = \\lbrace N\\rbrace $ ) or acting individually ($a_\\mathsf {dom} = \\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace \\rbrace $ ) make two totally different choice profiles.",
"However, condition 4 in Definition REF stipulates that once their strategy profile keeps the same, no players would prefer one to the other.",
"This means that $a^{4\\prime }$ and $a$ make no difference to both players' preference relations.",
"Then what can the difference between $a^{4\\prime }$ and $a$ bring about to the players?",
"The critical observation is that $a^{4\\prime }$ is in the core while $a$ is not even a Nash equilibrium.",
"That is, although their strategy profiles are the same, one as a joint action of the group is stable (in the sense of the core) while the other as cooperation of two parties is not stable (in the sense of the Nash equilibrium).",
"This suggests that the stability of acting together is essential for understanding collective agency.",
"To elaborate on the above claim, we first make the following clarification about condition 3 in Definition REF .",
"It does not require that by choosing to join the same coalition together, the players in the coalition should have more strategies than they have when acting separately, but only no less than.",
"We leave it open whether it is a strict inclusion ($\\subset $ ) or an equation ($=$ ).",
"As we can see in Example REF , the coalition $\\lbrace 1,2\\rbrace $ does not have extra strategies.",
"The special status of $a^{4\\prime }$ does not rely on having a strategy profile that cannot be realized by the players separately.",
"The comparison between $a^{4\\prime }$ and $a$ highlights the role of the coalition in making the cooperation stable.",
"It reveals that collective agency should be a binding power that makes a coalition and its joint action stable.",
"This binding power may come from different sources and be present in different forms over which various theories on collective agency debate.",
"No matter which source it comes from and which form it takes, the binding power should come with the stability of what it binds together.",
"Regarding stability, we share the same spirit with [12]; [23]; [24]; [25], in which they also directly or indirectly take stability as a condition for the formation of a collective agent.",
"Moreover, suppose we further abstractly understand the concept of the core as a specific pattern for inter-sub-coalition relations within a coalition.",
"In that case, our interpretation highlights the understanding of collective agency as a relatively stable state of relations rather than an imagined conceptual entity.",
"In this sense, we are in line with the call for a relationalist account (cf.",
"[2]; [17]; [21]; [28]).",
"Game theory, especially the cooperative game theory, is a powerful tool for analyzing the kind of stability we consider essential for collective agency.",
"The concept of the core is not the only solution concept in cooperative game theory.",
"A lot of other solution concepts have been proposed, taking different issues related to coalitional stability into consideration.",
"Abstracting and logically fusing these concepts into a unified framework will bring more insights into the philosophical discussion of collective agency.",
"Our analysis by CPD-models serves as a first attempt to make this connection explicit by testing collective agency in games."
],
[
"Related Works",
"Before conclusion, we compare our work with two closely related works, the modal coalitional game logic (MCGL) in [29]There are two logics in [29].",
"MCGL is the second one.",
"The first one is more customized and limited than the second one.",
"For example, it only considers finite games where both players and states need to be finite.",
"and the logic of ceteris paribus preference (LCP) in [6].",
"All the three works involve the modal way of modeling preference, that is, using modal operators for characterizing preorders.",
"Of the three works, as regards to basic modal operators for preference, LCP is the simplest one.",
"Given a preorder $\\preceq $ in its semantic model, it only includes one modal operator for $\\preceq $ and one for $\\prec $ .",
"MCGL concerns a multi-agent setting where for each agent there is a preorder.",
"Besides modal operators for individual agents, MCGL includes group operators, one for the intersections of a set of preorders and one for the intersection of a set of strict preorders.",
"It also includes modal operators for the inverse of the preorders and a difference operator.",
"Nevertheless, it does not have any operator for the intersection of strict and non-strict preorders.",
"Our logic has such operators and we show that they are critical for expressing strong Pareto optimality.",
"Next, with each of these two other logics, the comparison will focus on different aspects.",
"Comparison with [29] on different formulations of the core It is shown in [29] that MCGL can express not only the core in coalitional games but also the stable set and the bargaining set.",
"However, the setting they adopt for representing coalitional games is not general enough to model the coalitional games formalized by the CPD-models.",
"The limitation is due to their way of defining the coalitional effective function or the characteristic function as they call it.",
"In a CPD-model $\\mathbb {M}$ , their characteristic function can be understood as $V: 2^N\\setminus \\lbrace \\emptyset \\rbrace \\rightarrow \\mathcal {P}(A)$ , a function assigning a set of choice profiles to each coalition.",
"Their formulation of the core only requires that the current choice profiles are strictly preferred to all the choice profiles in $V(X)$ for all $X\\subseteq N$ .",
"But in our formulation of the core in Definition REF , what matters is the following set for each $X\\subseteq N$ $E(X) := \\lbrace a(X)\\subseteq A\\mid a\\in A \\text{ and } X\\in a_\\mathsf {dom}\\rbrace $ where $a(X) := \\lbrace a^{\\prime }\\in A\\mid a =_X a^{\\prime }\\rbrace $ .",
"$E: 2^N\\rightarrow \\mathcal {P}(\\mathcal {P}(A))$ is a function assigning to each coalition a set of sets of choices profiles.",
"This is in line with the coalitional effective function defined in subsection REF with only one difference, namely $E(X)$ here is not upward closed.",
"Our formulation of the core requires a comparison between the current choice profile and each of the set in $E(X)$ .",
"Note that the compartmentalization of what a coalition $X$ can enforce as $E(X)$ formalizes it is essential for our formulation of the core, because what a coalition $X$ can enforce depends on what $X$ would do.",
"This subtlety is not captured by the characteristic function in [29].",
"Comparison with [6] on different ways of characterizing dependence We have seen that in LPFD variables are taken to partition the space of possible assignments according to their possible values.",
"The dependence relation is the relation between different partitions.",
"In LCP, what partitions the space of possible states are all possible sets of formulas of its base language.",
"If we think of a formula as a binary variable with its values 0 or 1, then the operators $[\\Gamma ]$ , $[\\Gamma ]^{\\preceq _x}$ and $[\\Gamma ]^{\\prec _x}$ in LCP correspond to our operators $\\llbracket \\Gamma ,\\emptyset ,\\emptyset \\rrbracket $ , $\\llbracket \\Gamma ,x,\\emptyset \\rrbracket $ and $\\llbracket \\Gamma ,\\emptyset ,x\\rrbracket $ respectively.",
"This raises an interesting question: if we only allow binary variables, what is the difference between using variables (as in LFD) and formulas (as in LCP) to capture the functional dependence between variables?",
"Furthermore, do we really lose anything in LFD if we only allow binary variables?",
"A systematic study of these two questions would require future work."
],
[
"Conclusion and More Future Work",
"We have proposed two logics by extending LFD and studying their axiomatizations and other properties.",
"We have also demonstrated how our logics can help reason about the notions of dependence, preference and coalitional power in a game theoretical setting and provide a unified view on three key concepts in game theory, i.e., Nash equilibrium, Pareto optimality and the core.",
"On the basis of the two logics, we bring novel insights to the general discussion on collective agency, where we consider agency of a collective as a stable state that is constituted by each member's preference and the interdependency between them.",
"More work on collective agency from a cooperative-game-theoretical perspective needs to be done as we have instigated.",
"The connection between LFD and the coalition logic we have revealed indicates that it may be fruitful to explore the relationship between LPFD and ATL [13].",
"Some work has been done on exploring the temporal dimension of dependence [4].",
"Further work in these directions could make a logical analysis of extensive games more full-fledged."
],
[
"Strong Completeness of $\\mathsf {C}_\\mathsf {Nom}$",
"Let $\\Gamma $ be a $\\mathsf {C}_\\mathsf {Nom}$ -consistent set and $\\mathsf {Nom^{\\prime }}=\\mathsf {Nom}\\cup \\lbrace j_n:n\\in \\omega \\rbrace $ .",
"Then $\\Gamma $ can be extended to a maximal $\\mathsf {C}_\\mathsf {Nom^{\\prime }}$ -consistent set $\\Gamma ^+$ of formulas satisfying the following conditions: $\\Gamma ^+\\cap \\mathsf {Nom^{\\prime }}\\ne \\varnothing $ ; For all $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Gamma $ , there is a nominal $j\\in \\mathsf {Nom^{\\prime }}$ such that $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\wedge @_j\\varphi \\in \\Gamma $ .",
"The proof of Lemma REF is standard.",
"Fact 4 Let $\\Gamma $ be a named and pasted maximal $\\mathsf {C}_{\\mathsf {Nom}}$ -consistent set.",
"For each $i\\in \\mathsf {Nom}$ such that $@_i\\top \\in \\Gamma $ , let $\\Delta _i=\\lbrace \\varphi :@_i\\varphi \\in \\Gamma \\rbrace $ .",
"Then for all $i,j\\in \\mathsf {Nom}$ , $\\Delta _i$ is a maximal $\\mathsf {C}_\\mathsf {Nom}$ -consistent set.",
"$i\\in \\Delta _j$ if and only if $\\Delta _i=\\Delta _j$ .",
"Given a named and pasted maximal $\\mathsf {C}_\\mathsf {Nom}$ -consistent set $\\Gamma $ , we define the canonical model $\\mathfrak {M}_\\Gamma =(W_\\Gamma ,\\sim _\\Gamma ,\\le _\\Gamma ,V_\\Gamma )$ for $\\Gamma $ as follows: $W_\\Gamma =\\lbrace \\Delta _i:@_i\\top \\in \\Gamma \\text{ and }\\Delta _i=\\lbrace \\varphi :@_i\\varphi \\in \\Gamma \\rbrace \\rbrace $ ; for each $v\\in \\mathsf {V}$ , $\\Delta _i\\sim _v\\Delta _j$ if and only if $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\lbrace v\\rbrace ,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Gamma $ ; for each $v\\in \\mathsf {V}$ , $\\Delta _i\\le _v\\Delta _j$ if and only if $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\lbrace v\\rbrace ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Gamma $ ; $V(P\\vec{x})=\\lbrace \\Delta _i:@_iP\\vec{x}\\in \\Gamma \\rbrace $ and $V(i)=\\Delta _i$ .",
"Let $\\Gamma $ be a named and pasted maximal $\\mathsf {C}_\\mathsf {Nom}$ -consistent set.",
"Then $\\mathfrak {M}_\\Gamma =(W,\\sim ,\\le ,V)$ is an RDPN-model.",
"Let $v\\in \\mathsf {V}$ .",
"By axiom (Ord,1,2,3), $\\sim _v$ is a pre-order and $\\le _v$ is an equivalence relation.",
"Then $(W,\\sim ,\\le )$ is an RPD-frame.",
"Note that $V(i)\\in W$ for each $i\\in \\mathsf {Nom}\\cap \\mathrm {dom}(V)$ .",
"To show that $\\mathfrak {M}_\\Gamma $ is a RPDN-model, it suffices to show that $V$ satisfies (Val).",
"Let $\\vec{x}=(x_1,\\cdots ,x_n)$ .",
"Suppose $\\Delta _i\\sim _{\\mathsf {set}(\\vec{x})}\\Delta _j$ and $\\Delta _i\\in V(P\\vec{x})$ .",
"Then $P\\vec{x}\\in \\Delta _i$ .",
"By (Dep), $\\mathbb {D}_XP\\vec{x}\\in \\Delta _i$ , which entails $P\\vec{x}\\in \\Delta _j$ .",
"Let $\\Gamma $ be a named and pasted maximal $\\mathsf {C}_\\mathsf {Nom}$ -consistent set, $\\mathfrak {M}_\\Gamma =(W,\\sim ,\\le ,V)$ , $i\\in \\mathsf {Nom}$ and $\\Delta _i\\in W$ .",
"Then If $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Delta _i$ , then $\\Delta _iR(X,Y,Z)\\Delta _j$ ; If $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Gamma $ , then there is $j\\in \\mathsf {Nom}$ with $\\varphi \\in \\Delta _j$ and $\\Delta _iR(X,Y,Z)\\Delta _j$ .",
"$D_Xs\\in \\Delta _i$ if and only if $\\mathfrak {M}_\\Gamma ,\\Delta _i\\models D_Xs$ .",
"For all $\\varphi \\in \\mathcal {L}_\\mathsf {Nom}$ , $\\varphi \\in \\Delta _i$ if and only if $\\mathfrak {M}_\\Gamma ,\\Delta _i\\models \\varphi $ .",
"For (1), suppose $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Delta _i$ .",
"By axiom (Ord,5), we see $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\lbrace x\\rbrace ,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j,$ $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\lbrace y\\rbrace ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j,$ $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\varnothing ,\\varnothing ,\\lbrace z\\rbrace \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Delta _i$ for all $x\\in X$ , $y\\in Y$ and $z\\in Z$ , which entails by axiom (Ord,4) that $\\Delta _i\\sim _X\\Delta _j$ , $\\Delta _i\\le _Y\\Delta _j$ and $\\Delta _i<_Z\\Delta _j$ .",
"Thus $\\Delta _iR(X,Y,Z)\\Delta _j$ .",
"For (2), suppose $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\varphi \\in \\Gamma $ .",
"Since $\\Gamma $ is pasted, there is $j\\in \\mathsf {Nom}$ such that $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\wedge @_j\\varphi \\in \\Gamma $ .",
"Thus $\\varphi \\in \\Delta _j$ and $\\Delta _iR(X,Y,Z)\\Delta _j$ .",
"For (3), suppose $D_Xs\\in \\Delta _i$ and $\\Delta _i\\sim _X\\Delta _j$ .",
"We show that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\lbrace s\\rbrace ,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Delta _i$ .",
"Assume $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}\\lbrace s\\rbrace ,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\notin \\Delta _i$ .",
"Then by axiom (DD,1), we see $\\mathbb {D}_X\\lnot j\\in \\Delta _i$ , which contradicts to $\\Delta _i\\sim _X\\Delta _j$ .",
"Thus $\\mathfrak {M}_\\Gamma ,\\Delta _i\\models D_Xs$ .",
"Suppose $D_Xs\\notin \\Delta _i$ .",
"Then $i\\wedge \\lnot D_Xs\\in \\Delta _i$ .",
"By axiom (DD,2), we see $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\mathbb {D}_s\\lnot i\\in \\Gamma $ .",
"Since $\\Gamma $ is pasted, there is $j\\in \\mathsf {Nom}$ such that $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,\\varnothing ,\\varnothing \\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\wedge @_j\\mathbb {D}_s\\lnot i\\in \\Gamma $ .",
"Thus $\\Delta _i\\sim _X\\Delta _j$ and $\\Delta _i\\lnot \\sim _s\\Delta _j$ .",
"Note that $\\sim _s$ is symmetric, $\\Delta _j\\lnot \\sim _s\\Delta _i$ .",
"Thus $\\mathfrak {M}_\\Gamma ,\\Delta _i\\lnot \\models D_Xs$ .",
"For (4), the proof proceeds by induction on the complexity of $\\varphi $ .",
"The case when $\\varphi =D_Xs$ follows from (3) immediately.",
"The case $\\varphi =P\\vec{x}$ or $\\varphi \\in \\mathsf {Nom}$ is trivial.",
"The Boolean cases are also trivial.",
"Let $\\varphi =\\llbracket {X,Y,Z}\\rrbracket \\psi $ .",
"Assume $\\llbracket {X,Y,Z}\\rrbracket \\psi \\notin \\Delta _i$ .",
"Then $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\lnot \\psi \\in \\Delta _i$ and so $@_i\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\lnot \\psi \\in \\Gamma $ .",
"By (2), $\\lnot \\psi \\in \\Delta _j$ for some $\\Delta _j\\in R(X,Y,Z)(\\Delta _i)$ .",
"Then $\\psi \\notin \\Delta _j$ and by induction hypothesis, $\\mathfrak {M}_\\Gamma ,\\Delta _j\\lnot \\models \\psi $ , which entails $\\mathfrak {M}_\\Gamma ,\\Delta _i\\lnot \\models \\llbracket {X,Y,Z}\\rrbracket \\psi $ .",
"Assume that $\\mathfrak {M}_\\Gamma ,\\Delta _i\\lnot \\models \\llbracket {X,Y,Z}\\rrbracket \\psi $ .",
"Then there is $\\Delta _j\\in R(X,Y,Z)(\\Delta _i)$ such that $\\mathfrak {M}_\\Gamma ,\\Delta _j\\lnot \\models \\psi $ .",
"By induction hypothesis, $\\psi \\notin \\Delta _j$ and so $\\lnot \\psi \\wedge j\\in \\Delta _j$ .",
"Note that $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}j\\in \\Delta _i$ , we see $\\langle \\hspace{-5.0pt}\\langle \\hspace{1.111pt}X,Y,Z\\hspace{1.111pt}\\rangle \\hspace{-5.0pt}\\rangle \\hspace{1.38885pt}\\lnot \\psi \\in \\Delta _i$ , which entails $\\llbracket {X,Y,Z}\\rrbracket \\psi \\notin \\Delta _i$ .",
"Theorem.",
"$\\mathsf {C}_\\mathsf {Nom}$ is sound and strongly complete.",
"Soundness is not hard to verify.",
"Let $\\Gamma ^-\\subseteq \\mathcal {L}_\\mathsf {Nom}$ be any consistent set of formulas.",
"By Lemma REF , $\\Gamma ^-$ can be extended to a named and pasted maximal $\\mathbf {LPFD_{Nom^{\\prime }}}$ -consistent set $\\Gamma $ .",
"By Lemma REF , the triple $\\mathfrak {M}_\\Gamma =(W_\\Gamma ,R_\\Gamma ,V_\\Gamma )$ defined in Definition REF is a DP-model with nominals.",
"By Lemma REF , we see $\\mathfrak {M}_\\Gamma \\models \\Gamma ^-$ .",
"Then $\\mathfrak {M}_\\Gamma {\\upharpoonright }\\mathsf {Nom}$ is a RPDN-model satisfying $\\Gamma ^-$ ."
]
] | 2209.08213 |
[
[
"Linear TreeShap"
],
[
"Abstract Decision trees are well-known due to their ease of interpretability.",
"To improve accuracy, we need to grow deep trees or ensembles of trees.",
"These are hard to interpret, offsetting their original benefits.",
"Shapley values have recently become a popular way to explain the predictions of tree-based machine learning models.",
"It provides a linear weighting to features independent of the tree structure.",
"The rise in popularity is mainly due to TreeShap, which solves a general exponential complexity problem in polynomial time.",
"Following extensive adoption in the industry, more efficient algorithms are required.",
"This paper presents a more efficient and straightforward algorithm: Linear TreeShap.",
"Like TreeShap, Linear TreeShap is exact and requires the same amount of memory."
],
[
"Introduction",
"Machine learning in the industry has played more and more critical roles.",
"For both business and fairness purposes, the need for explainability has been increasing dramatically.",
"As one of the most popular machine learning models, the tree-based model attracted much attention.",
"Several methods were developed to improve the interpretability of complex tree models, such as sampling-based local explanation model LIME[9], game-theoretical based Shapley value[10], etc.",
"Shapley value gained particular interest due to both local and globally consistent and efficient implementation: TreeShap[5].",
"With the broad adoption of Shapley value, the industry has been seeking a much more efficient implementation.",
"Various methods like GPUTreeShap[6] and FastTreeShap[12] were proposed to speed up TreeShap.",
"GPUTreeShap primarily focuses on utilizing GPU to perform efficient parallelization.",
"And FastTreeShap improves the efficiency of TreeShap by utilizing caching.",
"All of them are empirical approaches lacking a mathematical foundation and are thus making them hard to understand.",
"We solve the exact Shapley value computing problem based on polynomial arithmetic.",
"By utilizing the properties of polynomials, our proposed algorithm Linear TreeShap can compute the exact Shapley value in linear time.",
"And there is no compromise in memory utilization."
],
[
"Contrast with previous result",
"We compare the running time of our algorithm with previous results for a single tree in Table REF , since all current algorithms for ensemble of trees are applying the same algorithm to each tree individually.",
"Let $S$ be the number of samples to be explained, $N$ the number of features, $L$ the number of leaves in the tree, and $D$ is the maximum depth of the tree.",
"For simplicity, we assume every feature is used in the tree, and therefore $N=O(L)$ .",
"Also, $D\\le L$ .",
"Figure: NO_CAPTION"
],
[
"Notation & Background",
"Elementary symmetry polynomials are widely used in our paper.",
"$[x]$ denotes the set of polynomials with coefficient in $$ .",
"$[x]_d$ are polynomials of degree no larger than $d$ .",
"We use $\\odot $ for polynomial multiplication.",
"For two polynomials $a$ and $b$ , $\\left\\lfloor \\frac{a}{b}\\right\\rfloor $ is the quotient of the polynomial division $a/b$ .",
"For two vectors $x,y\\in ^d$ , $\\langle x, y \\rangle $ is the inner product of $x$ and $y$ .",
"We abuse the notation, so when a polynomial appears in the inner product, we take it to mean the coefficient vector of the polynomial.",
"Namely, if $p,q$ are both polynomials of the same degree, then $\\langle p, q\\rangle $ is the inner product of their coefficient vectors.",
"We use $\\cdot $ for matrix multiplication.",
"We refer to $x \\in X \\subset ^m$ as an instance and $f: X \\rightarrow $ as the fitted tree model in a supervised learning task.",
"Here, $m$ denotes the number of all features, $M$ is the set of all features, and $|.|$ is the cardinality operation, namely $|M| = m$ .",
"We denote $x[i]$ as the value of feature $i$ of instance $x$ .",
"We have to start with some common terminologies because our algorithm is closely involved with trees.",
"A rooted tree $T=(V, E)$ is a directed tree where each edge is oriented away from the root $r\\in V$ .",
"For each node $v$ , $P_v$ is the root to $v$ path, i.e.",
"the set of edges from the root to $v$ .",
"$L(v)$ is the set of leaves reachable from $v$ .",
"$L(T)=L(r)$ is the set of leaves of $T$ .",
"$T$ is a full binary tree if every non-leaf node has two children.",
"If an edge $e$ goes from $u$ to $v$ , then $u$ and $v$ are the tail and head of $e$ , respectively.",
"We write $h(e)$ for the head of $e$ .",
"A tree is weighted, if there is an edge weight $w_e$ for each edge $e$ .",
"It is a labeled tree if each edge also has a label $\\ell _e$ .",
"For a labeled tree $T$ , let $E_i$ be all the edges of the tree with label $i$ .",
"Similarly, define $P_{i,v} = P_v \\cap E_i$ , the set of edges in the root to $v$ path that has label $i$ .",
"The last edge of any subset of a path is the edge furthest away from the root.",
"For our purpose, a decision tree is a weighted labeled rooted full binary tree.",
"There is a corresponding decision tree for the fitted tree model $T_f$ .",
"The internal nodes of the tree are called the decision nodes, and the leaves are called the end nodes.",
"Every decision node has a label of feature $i$ , and every end node contains a prediction $v$ .",
"The label of each edge is the feature of the head node of the edge.",
"We will call the label on the edge of the feature.",
"Every edge $e$ contains weight $w_e$ that is the conditional probability based on associated splitting criteria during training.",
"When predicting a given instance, $x$ , decision tree model $f$ sends the instance to one of its leaves according to splitting criteria.",
"We draw an example decision tree in Fig REF .",
"Each leaf node is labeled with id and prediction value.",
"Every decision node is labeled with the feature.",
"We also associate each edge with conditional probability $w$ and splitting criteria of features in the parenting node.",
"To represent the marginal effect, we use $f_S(x): X \\rightarrow , S \\subseteq M$ to denote the prediction of instance $x$ of the fitted tree model using only the features in active set $S$ , and treat the rest of features of instance $x$ as missing.",
"Using this representation, the default prediction $f(x)$ is a shorthand for $f_M(x)$ .",
"The Shapley value of a decision tree model $f$ is the function $\\phi (f,i): X \\rightarrow $ , $\\begin{split}\\phi (f, i)(x) = \\frac{1}{|M|} \\sum _{S \\subseteq M \\setminus \\lbrace i\\rbrace } \\frac{1}{\\binom{|M|-1}{|S|}} f_{S \\cup \\lbrace i \\rbrace }(x) - f_{S}(x).\\end{split}$ The Shapley value $\\phi (f, i)(x)$ quantifies the marginal contribution of feature $i$ in the tree model $f$ when predicting instance $x$ .",
"The problem of computing the Shapley values is the following algorithmic problem.",
"The Tree Shapley Value ProblemInput: A decision tree $T_f$ for function $f:X\\rightarrow $ over $m$ features, and $x\\in X$ .Output: The vector $(\\phi (f,1)(x),\\ldots ,\\phi (f,m)(x))$ .",
"Meanwhile, decision nodes cannot split instances with missing feature values.",
"A common convention is to use conditional expectation.",
"When a decision node encounters a missing value, it redirects the instance to both children and returns the weighted sum of both children's predictions.",
"The weights differ between decision nodes and are empirical instance proportions during training: $w_l, w_r$ .",
"Here $w_l + w_r = 1$ and $0 < w_l < 1$ .",
"A similar approach is also used in both Treeshap[5], and C4.5 [7] to deal with missing values.",
"Any instance would result in a single leaf when there is no missing feature.",
"In contrast, an instance might reach multiple different leaves with a missing feature.",
"Here, we use an example instance $x=$(temperature: 20, cloudy: no, wind speed: 6) with tree $f$ in Fig REF to show the full process of Shapley value computing.",
"By following the decision nodes of $T_f$ , the prediction $f(x)$ is leaf $C$ : 0.4 chance of raining.",
"Now we compute the importance/Shapley value of feature (temperature: 18) among $x$ for getting prediction of 0.4 chance of raining.",
"The importance/Shapley value of feature (temperature: 18) is $\\begin{split}& \\phi (f, \\textbf {temperature})(x) = \\frac{1}{3}(\\frac{1}{\\binom{2}{2}}( f_{\\lbrace \\textbf {temperature, cloudy, wind speed} \\rbrace }(x) - f_{\\lbrace \\textbf {cloudy, wind speed} \\rbrace }(x)) \\\\& + \\frac{1}{\\binom{2}{1}}( f_{\\lbrace \\textbf {temperature, wind speed}\\rbrace }(x) - f_{\\lbrace \\textbf {wind speed}\\rbrace }(x) + f_{\\lbrace \\textbf {temperature, cloudy}\\rbrace }(x) - f_{\\lbrace \\textbf {cloudy}\\rbrace }(x)) \\\\& +\\frac{1}{\\binom{2}{0}}(f_{\\lbrace \\textbf {temperature}\\rbrace }(x) - f_{\\emptyset }(x)))\\end{split}$ To elaborate more, a term $f_{\\lbrace \\textbf {cloudy, wind speed} \\rbrace }(x)$ with $x=$(temperature: 20, cloudy: no, wind speed: 6) is equivalent to $f($cloudy: no, wind speed: 6$)$ .",
"When traversal first decision node: temperature, value to current feature is considered as unspecified.",
"We sum over children leaves with empirical weights and get $0.5\\cdot D + 0.5\\cdot C$ as prediction.",
"On the other hand, decision tree $f$ can be linearized into decision rules [8].",
"A decision rule can be seen as a decision tree with only a single path.",
"A decision rule $R^v: X \\rightarrow $ for a leaf $v$ can be constructed via starting from root node, following all the conditions along the path to the leaf $v$ .",
"We use $F(R)$ to represent the set of all features specified in decision rule $R$ , namely, $F(R) = \\lbrace i | P_{i,v} \\ne \\emptyset \\rbrace $ .",
"The linearization of the decision tree $f$ to decision rules is the relation $f(x) = \\sum _{v\\in L(T_f)} R^v(x)$ .",
"Example tree in Fig REF can be linearized into 4 rules: $R^A$ : if (temperature $>$ 19) and (is cloudy) then predict 0.7 chance of rain else predict 0 chance of rain $R^B$ : if (temperature $>$ 19) and (is not cloudy) and (wind speed $>$ 8) then predict 0.6 chance of rain else predict 0 chance of rain $R^C$ : if (temperature $>$ 19) and (is not cloudy) and (wind speed $\\le $ 8) then predict 0.4 chance of rain else predict 0 chance of rain $R^D$ : if (temperature $\\le $ 19) then predict 0.5 chance of rain else predict 0 chance of rain For a decision rule $R$ , we also introduce prediction with active set $S$ , $R_S: X \\rightarrow $ .",
"When features are missing, leaf value further weighted by their conditional probability is provided as the prediction.",
"Here we introduce the definition recursively.",
"First, we define the prediction of rule $R$ associated with leaf value $\\mathcal {V}$ with empty input: $\\begin{split}R_{\\emptyset }^v = R_{\\emptyset }^v(x) = \\mathcal {V} \\prod _{e \\in P_v} w_e\\end{split}$ Where $w_e$ is the conditional probability/proportion of instances, when splitting by decision node at the source of edge $e$ , the proportion of instances belong to the current edge.",
"$\\mathcal {V}$ is the prediction of the leaf node $v$ that defines that decision rule.",
"We say $x \\in \\pi _i(R)$ , if $x[i]$ is satisfied by every splitting criteria concerning feature $i$ in decision rule $R$ .",
"For a given instance $x$ and leaf $v$ , we use a new variable $q_{i,v}(x)$ to denote the marginal prediction of $R^v$ when adding feature $i$ to active set $S$ .",
"$\\begin{split}q_{i,v}(x) := {\\left\\lbrace \\begin{array}{ll}\\prod _{e \\in P_{i,v} } \\frac{1}{w_{e}} & x \\ \\in \\pi _i(R^v)\\\\0 & x \\ \\notin \\ \\pi _i(R^v)\\end{array}\\right.",
"}\\end{split}$ The empty product equals 1, hence if $P_{i,v}=\\emptyset $ , $q_{i,v}(x)=1$ .",
"We omit the super/subscript $v$ if there is no ambiguity on the leaf node.",
"So, with $i \\notin S$ , we can write: $R_{ \\lbrace i\\rbrace \\cup S}(x) = q_i(x) R_S(x)$ Since $\\emptyset $ is a subset of any set $S$ , we can get $R_S(x)$ via products of weights: $R_S(x) = R_{\\emptyset } \\prod _{j \\in S} q_j(x)$ With $R_S$ , $f_S$ can also be linearized into the sum of rule predictions: $f_S(x) = \\sum _{v \\in L(T_f)} R_S^v(x).$"
],
[
"Some special functions and their properties",
"Definition 2.1 Define the reciprocal binomial polynomial to be $B_d(x) = \\sum _{i=0}^d {\\binom{d}{i}}^{-1} x^i$ .",
"Definition 2.2 The function $\\psi _d : [x]_d \\rightarrow \\mathbb {R} $ is defined as $\\psi _d(A) := \\frac{\\langle A, B_d \\rangle }{d+1}.$ We write $\\psi (p)=\\psi _d(p)$ where $d$ is the degree of $p$ .",
"The function $\\psi _d$ has two nice properties: additive for same degree polynomial and \"scale\" invariant when multiplying binomial coefficient.",
"Proposition 2.1 Let $p,q\\in [x]_d$ , and $k\\in $ .",
"Additivity: $\\psi _d(p) + \\psi _d(q) = \\psi _d(p+q)$ .",
"Scale Invariant: $\\psi (p\\odot (1+y)^k) = \\psi (p)$ ."
],
[
"Summary polynomials and their relation to Shapley value",
"Consider we have a function $f$ represented by a decision tree $T_f$ .",
"We want to explain a particular sample $x$ , therefore in the later sections, we abuse the notation and let $g$ to mean $g(x)$ whenever $g:X\\rightarrow $ , e.g $q_{i,v} = q_{i,v}(x)$ .",
"In order to not confuse the readers, the polynomials we are constructing always have the formal variable $y$ .",
"Since tree prediction can be linearized into decision rules, and the Shapley value also has Linearity property, we decompose the Shapley value of a tree as the sum of the Shapley value of decision rules.",
"$\\begin{split}\\phi (f, i) = \\sum _{v \\in L(T_f)} \\phi (R^v,i)\\end{split}$ Now, for each decision rule, we define a summary polynomial.",
"Definition 2.3 For a decision tree $T_f$ and an instance $x$ .",
"For a decision rule associated with leaf $v$ in $T_f$ , the summary polynomial $G_v$ is defined as $G_v(y) = R^v_{\\emptyset } \\prod _{j \\in F(R^v)} (q_{j,v} + y)$ Next, we study the relationship between the summary polynomial and the Shapley value of corresponding decision rule.",
"Lemma 2.2 Let $v$ be a leaf in $T_f$ , then $\\phi (R^v,i) = (q_{i,v}-1)\\psi \\left(\\frac{G_v}{q_{i,v}+y}\\right).$ Since everything involved in the proof is related to the leaf $v$ , we drop $v$ from the super/subscripts for simplicity.",
"First, we simplify the Shapley value of rule $R$ into: $\\begin{split}\\phi (R,i) = \\frac{1}{m} \\sum _{S \\subset M \\setminus \\lbrace i\\rbrace } \\frac{1}{\\binom{m-1}{|S|}} R_{S \\cup \\lbrace i\\rbrace } - R_S= \\frac{R_{\\emptyset } (q_i-1)}{m} \\sum _{S \\subset M \\setminus \\lbrace i\\rbrace } \\frac{1}{\\binom{m-1}{|S|}} \\prod _{j \\in S} q_j\\end{split}$ When feature $i$ does not appear in $R$ , $q_i-1$ returns 0, thus Shapley value on feature $i$ from rule $R$ is 0.",
"Let $|F(R)|=d$ , the number of features in $R$ .",
"The Shapley value of $R$ further reduces to: $\\begin{split}\\phi (R,i)= \\frac{R_{\\emptyset }(q_i-1)}{d} \\sum _{k=0}^{d-1} \\frac{1}{\\binom{d-1}{k}} \\sum _{S \\subset F(R) \\setminus \\lbrace i\\rbrace }^{|S| = k} \\prod _{j \\in S} q_j\\end{split}$ We observe that $R_{\\emptyset } \\sum _{S \\subset F(R) \\setminus \\lbrace i\\rbrace }^{|S| = k} \\prod _{j \\in S} q_j$ is precisely the coefficient of $y^k$ in $\\frac{G}{q_i+y}$ .",
"We obtain the weighted sum of all subsets' decision rule prediction using the inner product: $R_{\\emptyset } \\sum _{S \\subset F(R) \\setminus \\lbrace i\\rbrace } {1}/{\\binom{d-1}{|S|}} \\sum _{S \\subset F(R) \\setminus \\lbrace i\\rbrace }^{|S| = k} \\prod _{j \\in S} q_j = \\langle \\frac{G}{q_i+y}, B_{d-1} \\rangle $ Shapley value for $R$ has a compact form as shown in Eq.REF .",
"$\\begin{aligned}\\phi (R,i) &= \\frac{R_{\\emptyset } (q_i-1)}{d} \\sum _{S \\subset F(R) \\setminus \\lbrace i\\rbrace } \\frac{1}{\\binom{d-1}{|S|} } \\prod _{j \\in S} q_j\\\\&= \\frac{(q_i-1)}{d} \\langle \\frac{G}{q_i+y}, B_{d-1} \\rangle \\\\&= (q_i-1)\\psi \\left(\\frac{G}{q_i + y}\\right)\\end{aligned}$"
],
[
"Computations",
"Even though we have simplified the Shapley value of a decision rule in a compact form using polynomials, it is still not friendly in computational complexity.",
"In particular, the values $q_{i,v}$ are flat aggregated statistics and do not necessarily share terms in-between different rules.",
"This makes it difficult to share intermediate results across different rules.",
"To benefit from the fact that decision rules overlap, we develop an edge-based polynomial representation.",
"For every edge $e$ with feature $i$ , we use $e^{\\uparrow }$ to denote its closest ancestor edge that shares the same feature.",
"In cases such edge does not exist, $e^{\\uparrow }=\\bot $ .",
"We also use $x \\in \\pi _u$ to represent the $x[i]$ is satisfied by all splitting criteria on feature $i$ associated with all edges in $P_{i,u}$ .",
"$\\begin{split}p_{e} := {\\left\\lbrace \\begin{array}{ll}\\prod _{e^{\\prime } \\in P_{i,h(e)}} \\frac{1}{w_{e^{\\prime }}} & x \\ \\in \\pi _{h(e)}\\\\0 & x \\ \\notin \\ \\pi _{h(e)}\\end{array}\\right.",
"}\\end{split}$ We also define additionally that $p_{\\bot }=1$ .",
"If $e$ is the last edge in $P_{i,v}$ then $p_e = q_{i,v}$ .",
"This is the key to avoid $q_{i,v}$ completely, and instead switch to $p_e$ .",
"Our analysis will make sure that any $p_e$ that does not correspond to $q_{i,v}$ for any $v$ and $i$ gets cancelled out.",
"We show a relation between the Shapley value of a decision rule and the newly defined $p_e$ 's.",
"Consider an operation $\\oplus _{d_1,d_2} : [x]_{d_1} \\times [x]_{d_2} \\rightarrow [x]_{\\max (d_1, d_2)}$ .",
"The subscript is omitted when $d_1,d_2$ is implicit.",
"$G^1 \\oplus G^2 := G^1 + G^2 \\odot (1+y)^{d_1-d_2},$ We extend the summary polynomial to all nodes in the tree.",
"Let $G_u=\\bigoplus _{v \\in L(u)} G^v$ .",
"Denote $d_e$ as the degree of $G_u$ , where $h(e)=u$ .",
"Proposition 2.3 Let $v$ be a leaf in $T_f$ , and $d_v$ be the degree of $G_v$ then $\\phi (R^v, i) = \\sum _{e \\in P_{i,v}} (p_e - 1) \\psi \\left(\\left\\lfloor \\frac{G_v\\odot (y+1)^{d_e-d_v} }{y+p_e}\\right\\rfloor \\right) - (p_{e^\\uparrow } - 1) \\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{d_{e^{\\uparrow }}-d_v} }{y+p_{e^\\uparrow }}\\right\\rfloor \\right).$ Let $e^*$ be the last edge of $P_{i,v}$ .",
"We note a few facts.",
"$p_{e^*} = q_{i,v}$ , $y+q_{i,v}$ divides $G_v$ , and the sum is a telescoping sum.",
"Put them together.",
"$\\begin{aligned}&\\sum _{e \\in P_{i,v}} (p_e - 1) \\psi \\left(\\left\\lfloor \\frac{G_v\\odot (y+1)^{d_e-d_v} }{y+p_e}\\right\\rfloor \\right) - (p_{e^\\uparrow } - 1) \\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{d_{e^{\\uparrow }}-d_v} }{y+p_{e^\\uparrow }}\\right\\rfloor \\right)\\\\=& (p_{e^*} - 1) \\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{d_{e^*}-d_v} }{y+p_{e^*}}\\right\\rfloor \\right)\\\\=& (q_{i,v}-1)\\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{d_{e^*}-d_v} }{y+q_{i,v}}\\right\\rfloor \\right)\\\\=& (q_{i,v}-1)\\psi \\left(\\frac{G_v }{y+q_{i,v}} \\odot (y+1)^{d_{e^*}-d_v}\\right) \\\\=& (q_{i,v}-1)\\psi \\left(\\frac{G_v }{y+q_{i,v}}\\right)\\\\=&\\phi (R^v, i)\\end{aligned}$ The following theorem establishes the relation between Shapley values, the summary polynomials at each node, and $p_e$ for each edge $e$ .",
"Theorem 2.4 (Main) $\\phi (f,i) = \\sum _{e\\in E_i} (p_e - 1)\\psi \\left( \\left\\lfloor \\frac{G_{h(e)}}{y+p_e}\\right\\rfloor \\right) - (p_{e^{\\uparrow }} - 1)\\psi \\left(\\left\\lfloor \\frac{G_{h(e)} \\odot (y+1)^{d_{e^{\\uparrow }}-d_e}}{y+p_{e^{\\uparrow }}}\\right\\rfloor \\right)$ Based on linearity of Shapley Value, $\\phi (f,i) = \\sum _{v \\in L(T_f)} \\phi (R^v, i)$ .",
"For each rule $R^v$ , we can scale their summary polynomial $G_v$ to the degree of $G_{h(e)}$ .",
"Based on Proposition REF , $\\phi (f,i) = \\sum _{v \\in L(T_f)} \\sum _{e \\in P_{i,v}} (p_e - 1) \\psi \\left(\\left\\lfloor \\frac{G_v\\odot (y+1)^{d_e-d_v} }{y+p_e}\\right\\rfloor \\right) - (p_{e^\\uparrow } - 1) \\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{(d_e - d_v)+ (d_{e^{\\uparrow }}-d_e)} }{y+p_{e^\\uparrow }}\\right\\rfloor \\right)$ Observe that for any $(e,v)$ pair, we have $e\\in E_i$ and $v\\in L(h(e))$ if and only if $v\\in L(T_f)$ and $e\\in P_{i,v}$ .",
"Hence, can order the summation by summing through the edges.",
"$\\phi (f,i) = \\sum _{e\\in E_i} \\sum _{v \\in L(h(e))} (p_e - 1) \\psi \\left(\\left\\lfloor \\frac{G_v\\odot (y+1)^{d_e-d_v} }{y+p_e}\\right\\rfloor \\right) - (p_{e^\\uparrow } - 1) \\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{(d_e - d_v)+ (d_{e^{\\uparrow }}-d_e)} }{y+p_{e^\\uparrow }}\\right\\rfloor \\right)$ Observe that at each edge $e$ , all summary polynomial $G$ is scaled to the same degree $d_e$ .",
"According to Proposition REF , we can add the summary polynomials before evaluate using $\\psi (.",
")$ .",
"Now, focus on the first part of the sum.",
"$\\begin{aligned}\\sum _{e\\in E_i} \\sum _{v \\in L(h(e))} ((p_e - 1) \\psi (\\lfloor \\frac{ G_v \\odot (y+1)^{d_e-d_v} }{y+p_e} \\rfloor )&= \\sum _{e\\in E_i} (p_e - 1)\\psi \\left(\\sum _{v \\in L(h(e))} \\left\\lfloor \\frac{G_v \\odot (y+1)^{d_e-d_v} }{y+p_e} \\right\\rfloor \\right)\\\\&=\\sum _{e\\in E_i} (p_e - 1)\\psi (\\lfloor \\frac{ \\sum _{v \\in L(h(e))} G_v \\odot (y+1)^{d_e-d_v} }{y+p_e} \\rfloor )\\\\&=\\sum _{e\\in E_i} (p_e - 1)\\psi (\\lfloor \\frac{ \\bigoplus _{v \\in L(h(e))} G_v }{y+p_e} \\rfloor )\\\\&=\\sum _{e\\in E_i} (p_e - 1)\\psi \\left( \\left\\lfloor \\frac{G_{h(e)}}{y+p_e}\\right\\rfloor \\right)\\end{aligned}$ Using the exact same proof, we can also obtain $\\sum _{e\\in E_i} \\sum _{v \\in L(h(e))}(p_{e^\\uparrow } - 1) \\psi \\left(\\left\\lfloor \\frac{G_v \\odot (y+1)^{(d_e - d_v)+ (d_{e^{\\uparrow }}-d_e)} }{y+p_{e^\\uparrow }}\\right\\rfloor \\right) = \\sum _{e\\in E_i} (p_{e^{\\uparrow }} - 1)\\psi \\left(\\left\\lfloor \\frac{G_{h(e)} \\odot (y+1)^{d_{e^{\\uparrow }}-d_e}}{y+p_{e^{\\uparrow }}}\\right\\rfloor \\right)$"
],
[
"Linear TreeSHAP and complexity analysis",
"By Theorem REF , we can obtain an algorithm in two phases.",
"Efficiently compute the summary polynomial on each node(Algorithm REF ) and then evaluates for $\\phi (f, i)$ directly(Algorithm REF ).",
"Both parts of the algorithm are straightforward, basically computing directly through definition and tree traversal.",
"The final values of $S[i]$ is the desired value $\\phi (f,i)(x)$ after running Algorithm REF .",
"To analyze the running time, one can see each node is visited a constant number of times.",
"The operations are polynomial addition, multiplication, division, inner product, or constant-time operations.",
"In general, all those polynomial operations takes $O(D\\log D)$ time for degree $D$ polynomial [1].",
"This shows the total running time is $O(LD\\log D)$ .",
"However, we never need the coefficients of the polynomials.",
"So we can improve the running time by storing the summary polynomials in a better-suited form, the multipoint interpolation form.",
"Namely, we evaluate the polynomials $G$ on a set of predefined unique points $Y = (y_0, y_1, y_2, \\cdots , y_D) \\in \\mathbb {R}^{D+1}$ , and store $G(Y) = (G(y_0),\\ldots ,G(y_D))$ instead.",
"In this form, addition, product and division takes $O(D)$ time [2].",
"The evaluation function $\\psi (G)$ also takes $O(D)$ time but needs more explanation.",
"Denote $\\mathcal {V}(Y) \\subset ^{D+1 \\times D+1 }$ as the Vandermonde matrix of $Y$ , where $v_{i,j} \\in \\mathcal {V}(Y) = y_i^j $ is the $j$ th power of $y_i$ .",
"Lemma 2.5 Let $p,q\\in R[x]_d$ , and its coefficients $A$ and $B$ , respectively, then we have $\\langle p, q\\rangle = \\langle A, B \\rangle = \\langle p(Y), \\mathcal {V}(Y)^{-1} B \\rangle .$ Polynomial evaluation can be consider as inner product of coefficient and Vandermonde matrix of input $Y$ .",
"Namely $p(Y) = A \\cdot \\mathcal {V}(Y)$ .",
"Therefore $\\langle p(Y), \\mathcal {V}(Y)^{-1}B \\rangle = \\langle A \\cdot \\mathcal {V}(Y), \\mathcal {V}(Y)^{-1}B \\rangle = \\langle A , \\mathcal {V}(Y)\\cdot \\mathcal {V}(Y)^{-1}B \\rangle = \\langle A , B \\rangle $ completes the proof.",
"In order to compute the inner products $\\langle G, B_d\\rangle $ in $O(D)$ time, we have to precompute $N_d = \\mathcal {V}(Y)^{-1}C_d$ , where $C_d$ is the coefficient of $B_d$ , for all $0\\le d\\le D$ .",
"This can be done simply in $O(D)$ time for each $d$ , so a total of $O(D^2)$ time.",
"By storing the polynomial in interpolation form, all our polynomial operations on each node take $O(D)$ time.",
"Therefore the total running time is $O(LD)$ .",
"Other than the summary polynomials, the algorithm uses constant space to store information on nodes and edges.",
"Each summary polynomial takes $O(D)$ space to store.",
"Therefore the algorithm takes $O(LD)$ space.",
"Nevertheless, we can save space by realizing the algorithms only need a single top-down and a single bottom-up step.",
"By joining two steps into one, the algorithm consumes the summary polynomials on the spot.",
"Hence the total space usage will be bounded by $O(D)$ times the stack size, bounded by $D$ , the depth of the tree.",
"The final total space usage is improved to $O(D^2)$ ."
],
[
"Remark",
"Even though $Y$ can be arbitrarily chosen based on the maximum depth of the tree $D$ , it is shown that Chebyshev points are near-optimal in numerical stability [11].",
"In our Linear TreeShap implementation, we used the Chebyshev points of the second kind."
],
[
"Experiments",
"We run an experiment on both regression dataset adult and classification dataset conductor(summary in Table REF ) to compare both our method and two popular algorithms, TreeShap and Fast TreeShap.",
"We explain Trees with depths ranging from 2 to 18.",
"And to align the performance across different depths of trees, we plot the ratio between the time of Tree Shap and the time of all methods in Figure REF .",
"We run every algorithm on the same test set 5 times to get both average speeds up and the error bar.",
"And for fair comparison purposes, all methods are limited to using a single core.",
"Linear TreeShap is the fastest among all setups.",
"And due to heavy memory usage, Fast TreeShap V2 falls back to V1 when tree depth reaches 18 for the dataset conductor.",
"Since the degree of the polynomial is bounded both by the depth of the tree also the number of unique features per decision rule, with deeper depth, dataset Conductor has much more speed up gains thanks to a higher number of features.",
"We can conclude that the Linear TreeShap is more efficient than all state-of-the-art Shapley value computing methods in both theory and practice."
],
[
"Checklist",
" For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?",
"Did you describe the limitations of your work?",
"Did you discuss any potential negative societal impacts of your work?",
"Have you read the ethics review guidelines and ensured that your paper conforms to them?",
"If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?",
"Did you include complete proofs of all theoretical results?",
"If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?",
"Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?",
"Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?",
"Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?",
"If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...",
"If your work uses existing assets, did you cite the creators?",
"Did you mention the license of the assets?",
"Did you include any new assets either in the supplemental material or as a URL?",
"Did you discuss whether and how consent was obtained from people whose data you're using/curating?",
"Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?",
"If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?",
"Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?",
"Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?"
]
] | 2209.08192 |
[
[
"ChemNLP: A Natural Language Processing based Library for Materials\n Chemistry Text Data"
],
[
"Abstract Natural language processing (NLP) has an immense potential to aid materials design processes.",
"While there have been several advancements in this field, a complete and integrated framework with well-curated dataset and tools to apply NLP is still needed.",
"In this work, we present the ChemNLP library and an accompanying web-app that can be used to analyze important materials chemistry information.",
"We use the publicly available arXiv dataset that has been collected over 34 years and contains ~1.8 million articles.",
"First, we analyze the article publication trend, categorizations, and common phrases in the arXiv dataset.",
"Then, we develop a user-friendly, interactive web-app to retrieve articles for a given chemical compound.",
"Furthermore, we demonstrate the effectiveness of the proposed framework to accelerate the identification of superconducting materials.",
"We determine the overlap between density functional theory and text-based databases for superconductors.",
"Finally, we perform machine learning based clustering and classification tasks to quickly categorize scholarly articles given article title information with accuracy up to 81.2 %.",
"ChemNLP is available at the websites: https://github.com/usnistgov/chemnlp and https://jarvis.nist.gov/jarvischemnlp."
],
[
"Introduction",
"The number of scholarly articles available on the web is estimated to be more than 100 million [1], [2].",
"It is an overwhelming task to perform a specific scientific query and extract meaningful information from such a large corpus.",
"Natural language processing (NLP) is a subfield of artificial intelligence and linguistics to make computers understand the statements or words written in human languages and perform useful tasks [3], [4].",
"NLP can be used on scholarly articles for several applications such as text summarization [5], topic modeling [6], machine translation [7], speech recognition [8], lemmatization, part of speech tagging [9], grammatical error correction [10], scholarly citation network analysis [11], named entity linking [12], text to text and text to image generation etc [13], [14], [15], [16].",
"Several web-tools such as Web of science, Scopus, Google scholar, Microsoft academic, Crossref, and PubMed etc.",
"are using NLP to extract and analyze information from scholarly articles [17], [18], [19], [20], [21] .",
"However, scientific literature, especially chemistry and materials science data, contains numerous technical terms (such as chemical names, methodologies, and instrumental techniques) that are difficult to process using conventional NLP tools.",
"Luckily, there have been several advancements for applying NLP techniques for chemistry and materials science.",
"One of the pioneer works for applying NLP for materials chemistry was carried out by Cole et al.",
"[22], who demonstrated the application of NLP for magnetic and battery materials [23], [24] using ChemDataExtractor.",
"Other popular NLP for chemistry and materials tools include ChemicalTagger, ChemListem, ChemSpot, MaterialsParser, OSCAR4 details of which can be found elsewhere [25].",
"In addition to the magnetic and battery materials, similar works have been also performed for numerous other material classes such as metal organic frameworks [26], Mott insulator transition materials [27], glasses [28] and polymers [29] etc.",
"Other applications of NLP for materials data involve using Long short-term memory (LSTM) and transformer-based models to extract various categories of information, and in particular materials synthesis information from text sources [30], [31], [32], [33], [34], [35].",
"A detailed review of application of NLP for materials can be found in refs.",
"[25], [36], [37].",
"Nevertheless, application of NLP for materials applications is still an active area of development and there are a number of pain points that make NLP for materials data difficult.",
"Some of these challenges are: 1) restricted-access of full-text and pay-walls, 2) standard dataset and tools to benchmark NLP techniques for materials chemistry analysis , 3) example applications with both materials science and NLP domain knowledge, 4) resolving dependencies between words and phrases across multiple sentences and paragraphs and cross-domains.",
"In this work, we present ChemNLP library that 1) provides a curated and open-access arXiv dataset (https://arxiv.org/) that can be directly used for NLP tasks, 2) share and illustrate software tools that can be used to visualize, analyse and perform NLP tasks for materials chemistry specific text data, and 3) develop a user-interface to search chemistry data within available literature.",
"Although, in this work we primarily use arXiv dataset, the tools can be used for other infrastructures as well."
],
[
"Methods",
"We use JARVIS-Tools to obtain the arXiv dataset [38] (https://jarvis-tools.readthedocs.io/en/master/databases.html).",
"ArXiv is a collaboratively funded, community-supported resource and maintained and operated by Cornell University.",
"Originally this dataset was obtained from the Kaggle competition dataset and then converted into JARVIS-Tools data format.",
"The JARVIS-Tools provides scripts and workflows for running and analyzing various simulations.",
"In the present version, the arXiv has 1796911 pre-print articles collected over 34 years, hosting literature from scientific fields including Physics, Mathematics, and Computer Science.",
"Each pre-print in arXiv contains text, figures, authors, citations, categories, and other metadata which are of immense importance for NLP applications.",
"These metadata are entered by users while uploading their manuscripts in arXiv.",
"We use Pandas, NumPy, Matplotlib and Seaborn python libraries [39] for initial analysis of the arXiv dataset.",
"We visualize the frequency of one- and two-order n-grams using the WordCloud library [40].",
"Here, \"n-\" refers to consecutive words e.g.",
"uni-grams (single words such as magnetic) and bi-grams (two consecutive words such as two-dimensional).",
"We use ChemDataExtractor [22] to identify chemical names from abstracts in condensed matter category articles.",
"ChemDataExtractor is a toolkit for the automated extraction of chemical entities and their associated properties, measurements, and relationships from scientific documents that can be used to populate structured chemical databases.",
"Using the chemical information, we develop a web-app using JARVIS-Tools and Configurable Data Curation System (CDCS) which is primarily based on Django-python and java-script libraries.",
"The arXiv dataset consists of multiple materials classes, and it is beyond the scope of this work to analyze NLP applications for each material class.",
"However, we analyze the superconducting material information in the dataset and compare with recently developed JARVIS-SuperconDB dataset [41] to identify new and common chemical formulae.",
"Such analysis can be highly important and useful for materials discovery applications.",
"Usually machine learning algorithms require fixed-length input numerical vectors for supervised as well as unsupervised learning tasks.",
"Some of the popular and simple method of feature extraction with text data are bag of words (BOW), term frequency-inverse document frequency (TF-IDF) and Word2Vec.",
"We use TF-IDF in this work.",
"TF-IDF is a numerical statistic that reflects importance of a word in a document.",
"The TF-IDF value increases proportionally to the number of times a word appears in the document and is offset by the number of documents in the corpus that contain the word, which helps to adjust for the fact that some words appear more frequently in general.",
"Term frequency (TF) indicates the significance of a particular term within the overall document.",
"Term frequency, $TF(t,d)$ , is the relative frequency of term $t$ within document $d$ : $TF(t,d)= \\frac{f_{td}}{\\sum _{t^{^{\\prime }} \\in d} f_{t^{^{\\prime }}d}}$ where $f_{td}$ is the raw count of a term in a document, i.e., the number of times that term $t$ occurs in document $d$ .",
"The denominator is simply the total number of terms in document $d$ .",
"Hence, TF can be considered as the probability of finding a word in a document.",
"The inverse document frequency (IDF) is a measure of how much information a word provides, i.e., if it’s common or rare across all documents.",
"It is used to calculate the weight of rare words across all documents in the corpus.",
"The words that occur rarely in the corpus have a high IDF score and vice versa.",
"IDF calculated as the logarithmically scaled inverse fraction of the documents that contain the word (obtained by dividing the total number of documents by the number of documents containing the term, and then taking the logarithm of that quotient): $IDF(t,D)= \\log \\frac{N}{|d \\in D:t \\in d|}$ where N is the total number texts/documents in the corpus $N=|D|$ .",
"Now, combining the above, the TFIDF is given as: $TFIDF(t,d,D)= TF(t,d)*IDF(t,D)$ TF-IDF is one of the most popular schemes today and is used for both the clustering and supervised classification tasks adopted in this work.",
"For clustering analysis, we use t-distributed stochastic neighbor embedding (t-SNE), which is a statistical method for visualizing high-dimensional data in a two- or three-dimensional map.",
"The t-SNE plot was generated with the help of Scikit-learn library [42].",
"For the supervised machine learning classification task, we use Scikit-learn library as well.",
"There are a number of algorithms for supervised classification in Scikit-learn.",
"We compare the results of four algorithms: random forest, linear support vector machine, multinomial naive Bayes and logistic regression algorithms.",
"The random forest algorithm is a type of supervised machine learning method based on ensemble learning.",
"Ensemble learning is a based on joining different types of algorithms or same algorithm multiple times to form a more powerful prediction model.",
"Support vector machine (SVM) finds a hyperplane in an N-dimensional space (where N is the number of features) that distinctly classifies the data points.",
"Support vectors are data points that are closer to the hyperplane and influence the position and orientation of the hyperplane.",
"A Naive Bayes classifier is a probabilistic machine learning model based on Bayes theorem which can be used for classification tasks.",
"Logistic Regression is a classification technique which uses a logistic function to model the dependent variable.",
"We calculate the classification accuracy as: $Accuracy=\\frac{TP+TN}{TP + FN + FP + TN}$ where, TP, TN, FN, FP are True Positive, True Negative, False Negative, and False Positive instances respectively."
],
[
"Results and discussion",
"We obtained arXiv dataset which contains papers from from 1986 to 2020 and show the yearly publication analysis in Fig.",
"REF .",
"A simple polynomial fit provides the relationship $y=ax^2+bx+c$ with $a=138.83, b=-223.12, c=1677.58$ which can be used to predict total number of publications in future.",
"Figure: Most common first name/last names in the arXiv dataset.",
"The size of the bubbles are proportional to the frequency or occurrence of corresponding words.In addition to publication year information, the arXiv dataset also provides the author details.",
"In Fig.",
"REF , we visualize the most common first or last name in the entire database using bubble charts.",
"A packed bubble chart displays data in a cluster of circles and the size of the bubbles are proportional to the frequency or occurrence of corresponding words.",
"Interestingly, we find that some of the top 10 common first/last names are: Wang (71835), Li (64812), Zhang (64492), Liu (49169), David (48238), Chen (46239), Michael (44264), Yang (32909), Thomas (32470), and Daniel (29899).",
"In future work, we can analyze the citation analysis where author list, their connectivity and occurrence information will be useful.",
"Figure: Several categories of scholarly articles in the arXiv dataset.",
"a) overall categories, b) condensed matter sub-categories.In Fig.",
"REF , we show several categories of scholarly articles in the arXiv dataset.",
"Such categorizations are possible because of 167 taxonomy categories data available in the dataset.",
"The details of the taxonomy can be found at https://arxiv.org/category_taxonomy.",
"We note these categories are based on the available data in arXiv only.",
"We choose the first category of the article which has multiple sub-categories.",
"In Fig.",
"REF a we notice that most of the articles belong to Physics, Mathematics and Computer science.",
"The number of articles in Physics, Mathematics, Computer science, Statistics, Quantitative Biology, Electrical Engineering, Quantitative Finance and Economics are 1042227, 425745, 209068, 72058, 24720, 13064, 8920, and 1109 respectively.",
"Furthermore, we visualize the condensed matter physics categories in Fig.",
"REF b.",
"The cond-mat.mtrl-sci, cond-mat.mes-hall, cond-mat.str-el, cond-mat.stat-mech, cond-mat.supr-con, cond-mat.soft, cond-mat.quant-gas, cond-mat.other, and cond-mat.dis-nn categories have 30107, 29751, 22375, 17359, 14697, 10939, 5041, 3930 and 3728 articles respectively.",
"We find that cond-mat.mtrl-sci or materials science category has the higher number of entries.",
"This category is based on techniques, synthesis, characterization, structure, structural phase transitions, mechanical properties, phonons, defects, adsorbates, and interfaces.",
"cond-mat.mtrl-sci is followed by cond-mat.mes-hall i.e.mesoscale and nanoscale Physics fields.",
"This branch deals with semiconducting nanostructures: quantum dots, wires, and wells, single electronics, spintronics, 2D electron gases, quantum Hall effect, nanotubes, graphene,and plasmonic nanostructures.",
"The next set of branches are :cond-mat.str-el or strongly correlated electrons (dealing with quantum magnetism, non-Fermi liquids, spin liquids, quantum criticality, charge density waves, metal-insulator transitions), cond-mat.supr-con or superconductivity (dealing with superconductivity: theory, models, experiment, and superflow in helium), cond-mat.soft or soft condensed matter (dealing with membranes, polymers, liquid crystals, glasses, colloids, granular matter),cond-mat.quant-gas (ultracold atomic and molecular gases, Bose-Einstein condensation, Feshbach resonances, spinor condensates, optical lattices, quantum simulation with cold atoms and molecules, macroscopic interference phenomena), cond-mat.other (work in condensed matter that does not fit into the other cond-mat classifications) and cond-mat.dis-nn or disordered systems and neural networks (dealing with glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; and neural networks).",
"Figure: Word-cloud charts for different words in major condensed matter article titles.Next, in Fig.",
"REF we show word-cloud charts for different words in the condensed matter articles' titles.",
"A word cloud is a collection, or cluster, of words depicted in different sizes.",
"The bigger and bolder the word appears, the more often it's mentioned within a given text and the more importance it holds.",
"We find that first-principle, electronic structure, graphene, thin film, surface, carbon nanotube, two dimensional, magnetic etc.",
"are some of the most common words in this subtopic.",
"Similarly, Josephson junction, superconductivity, d wave, single crystal etc.",
"are the most common words in the super-con category.",
"Polymer, diffusion and fluid words are common ones in the soft category.",
"Words like quantum-dot, quantum-well, spin-orbit, Hall-effect, topological insulator, quantum hall are common in mesh-hall category.",
"The stat-mech category has phase-transition, fluctuation, dynamic, Thermodynamic etc.",
"as some of the common words.",
"The words like Hubbard-model, density-wave, spin-liquid etc.",
"are common in str-el category.",
"Words such as disorder, spin glass, network etc.",
"are common in dis-nn category.",
"Interestingly, some words such as two-dimensional occur in all the categories showing such class are one of the highly investigated materials.",
"Similarly, the words such as magnetic and superconductivity occur in multiple categories showing such class of materials are investigated by experts in multiple domains.",
"Figure: A snapshot of the web-app that can be used to find articles containing periodic table elements.",
"This is based on arXiv condensed matter article abstracts.",
"The web-app is available at : https://jarvis.nist.gov/jarvischemnlp.Next, in Fig.",
"REF we show a snapshot of the web-app that can be used to find articles containing periodic table elements.",
"For instance, as we click on the elements Al, Ga and N and click Search button it returns 36 entries as results which can be used to find details of respective articles.",
"Similarly, if element combinations such as Mo and S are selected, 1374 results are returned showing that articles with Mo-S compounds are much larger than Al-Ga-N. For this work, we collected all the abstracts in condensed matter Physics articles and attempted to find chemical formula with a combination of ChemDataExtractor and JARVIS-Tools packages.",
"We found 37944 articles ($\\sim $ 30 % of condensed matter Physics entries) with a chemical formula in abstracts using the above approach.",
"The number of unique combinations of elements (such as Al-Ga-N, Mo-Te, and Cu-Ga-In-S-Se etc.)",
"that can be searched using the app are 6295.",
"Figure: Venn diagram for chemical formula available in arXiv cond-mat.supr-con and JARVIS-SuperconDB.Next, we demonstrate a simple application of ChemNLP for discovery and design of superconductors.",
"Superconductors are class of materials with vanishing electrical resistance under a characteristic temperature called the superconducting transition temperature.",
"Superconductors can be both phonon and non-phonon mediated.",
"Recently, we developed a phonon-mediated superconducting transition temperature database using density functional theory with more than 1000 materials (JARVIS-SuperconDB) [41] in the JARVIS-DFT database.",
"The JARVIS-SuperconDB was based on the Debye temperature, electronic density of states at the Fermi-level, and subsequent McMillan-Allen-Dynes based formulation.",
"We obtained the chemical formula from the articles in cond-mat.supr-con abstracts and that from the JARVIS-SuperconDB and plot a Venn diagram in Fig.",
"REF .",
"Note that we compare based on chemical formula only ignoring the crystal structure information.",
"Interestingly, we find only 43 materials common in these two sets including well-known materials such as MgB$_2$ , Nb, NbN, HfN, Nb, Al, TiN, and VRu etc.",
"There are 635 chemical formula with DFT $T_C \\ge 1K$ which we didn't find in the arXiv dataset and 1071 formula were present in the arXiv dataset only.",
"There are many unconventional/non-phonon mediated superconductors such as Yttrium barium copper oxide, and high pressure as well as doped superconductors such as NaFe1-xCoxAs which are not present in current JARVIS-SuperconDB.",
"Additionally, several novel superconductors predicted such as KB$_6$ , Ru$_3$ NbC, V$_3$ Pt, ScN, LaN$_2$ which are not available in literature to the best of our knowledge.",
"Therefore, NLP combined with traditional materials design motivates us in our further screening of superconductors.",
"In JARVIS-DFT, there are many other properties such as thermoelectric, magnetic, dielectric, piezoelectric, topological, elastic, thermodynamic, vibrational, nuclear and low-dimensional properties on which similar strategies could be applied but its currently beyond the scope of the present paper.",
"Figure: A t-SNE visualtion of the cond.mat articles in the arXiv dataset.After gathering insights into the dataset, we can now apply AI techniques to the arXiv dataset.",
"AI techniques can be unsupervised (e.g., clustering), supervised (i.e., the target or ground truth data such as in classification and regression is known), and generative (i.e., aim to learn underlying distributions).",
"In Fig.",
"REF , we apply clustering based t-distributed stochastic neighbor analysis (t-SNE).",
"The t-SNE reveals local structure in high-dimensional data, placing points in the low-dimensional visualization close to each other with high probability if they have similar high-dimensional feature vectors.",
"First, we stem the paper titles from the condensed matter physics articles, and get \"Term Frequency-Inverse Document Frequency\" (TF-IDF) of a given word stem.",
"The TF-IDF is a product of the relative frequency with which a term appears in a single document and the log of the total number of papers in the document pool divided by the number of documents in which a term appears.",
"Then we perform truncated singular value decomposition (TruncatedSVD) for sparse data to reduce the dimensionality of the embedding space (128 size).",
"Furthermore, we perform TSNE to reduce embedding space to 2-dimension.",
"The t-SNE plot thus obtained is shown in Fig.",
"REF with the marker colors indicate the article category of each article.",
"We show the cluster of super-con, mes-hall, dis-nn and stat-mech while matrl-sci category seems to overlap with other classes as well.",
"The plot suggests the article data is well-distributed and we can cluster articles using title words only.",
"Figure: Model accuracy comparisons for classifying cond.mat articles in the arXiv dataset using bag of words and several machine learning models.Now, we apply classification of 137927 arXiv article titles in the cond-mat field using supervised machine learning (ML) methods.",
"ML algorithms cannot directly process text data so we convert the texts to numerical vectors using bag of words model and Term Frequency, Inverse Document Frequency (TF-IDF) as available in the Scikit-learn package.",
"After converting the titles in the dataset to numerical representation, we apply a few well-known ML algorithms such as random-forest, linear support vector machine, multinomial naive Bayes, and logistic regression.",
"We use 3-fold cross-validation strategy and show the accuracy of the models in Fig.",
"REF .",
"The mean accuracies of these models are: 68 %, 72 %, 70 %, and 72 % for random-forest, linear support vector machine, multinomial naive Bayes, and logistic regression models respectively.",
"Here, the accuracies of the four models are comparable, but logistic regression provides the highest accuracy model.",
"As a baseline, the random guessing/baseline model has an accuracy of $1/9 = 11.11 \\%$ (for nine classes), hence the ML models are more than 6 times accurate than a random guessing model.",
"Figure: Confusion matrix for classifying cond-mat articles with logistic regression model.The above results are based on titles only.",
"In addition to titles, the arXiv dataset also contains the abstracts' text, and we investigated whether the inclusion of abstracts improves the model accuracy.",
"We used the linear support vector machine for this task and split the condensed matter article dataset in 80:20 ratio for training and testing purposes.",
"We find that the model accuracy for title only, abstract only and title with abstract together are 74.3 %, 80.1 %, and 81.2 %, respectively as shown in Fig.",
"REF .",
"The results indicate that by including abstract along with title we can enhance the model accuracy by 6.9 % which is interesting.",
"Figure: Confusion matrix for classifying cond-mat articles with linear support vector machine model.We show the classification confusion matrix for the combined title and abstract model in Fig.",
"REF .",
"The confusion matrix allows us to interpret and analyze the detailed accuracy of the model for each class rather than just a global accuracy value.",
"Ideally, a perfect classifier would result in a confusion matrix with diagonal entries only with a value of 100 % .",
"We find that a vast majority of the predictions end up on the diagonal (predicted label = actual label).",
"This is especially true for the supr-con, mes-hall and matrl-sci subfields with the supr-con class showcasing the highest value at 90.6 %, which is interesting.",
"The lowest accuracy was achieved for the cond-mat.other subfield, which significantly overlaps with other categories such as mes-hall and quant-gas.",
"The dis-nn sub-category also overlaps with stat-mech su-category.",
"Beyond these two categories (other and dis-nn), we obtain over 80 % for all the sub-categories in the confusion matrix, demonstrating the promise of accurate classification through machine learning.",
"In summary, we have developed a ChemNLP package and web-app that can be used to analyze important materials chemistry information using the publicly-available arXiv dataset comprised of $\\sim $ 1.8 million articles.",
"We analyzed the article publications, categories, and their common trends in this dataset.",
"Then, we developed an interactive web-app to isolate articles with specific chemistry.",
"To demonstrate the application of our framework, we apply this tool to help identify new superconducting materials.",
"Finally, we perform machine learning based clustering and classification tasks to automatically and rapidly categorize scholarly articles in several materials categories.",
"We believe the software tool and web-app developed in this work will be valuable to materials science community.",
"K.C.",
"thanks Jacob Collard and Talapady Bhat at NIST for helpful discussion.",
"Contributions from K.C.",
"were supported by the financial assistance award 70NANB19H117 from the U.S. Department of Commerce, National Institute of Standards and Technology."
]
] | 2209.08203 |
[
[
"Understanding the Impact of Image Quality and Distance of Objects to\n Object Detection Performance"
],
[
"Abstract Deep learning has made great strides for object detection in images.",
"The detection accuracy and computational cost of object detection depend on the spatial resolution of an image, which may be constrained by both the camera and storage considerations.",
"Compression is often achieved by reducing either spatial or amplitude resolution or, at times, both, both of which have well-known effects on performance.",
"Detection accuracy also depends on the distance of the object of interest from the camera.",
"Our work examines the impact of spatial and amplitude resolution, as well as object distance, on object detection accuracy and computational cost.",
"We develop a resolution-adaptive variant of YOLOv5 (RA-YOLO), which varies the number of scales in the feature pyramid and detection head based on the spatial resolution of the input image.",
"To train and evaluate this new method, we created a dataset of images with diverse spatial and amplitude resolutions by combining images from the TJU and Eurocity datasets and generating different resolutions by applying spatial resizing and compression.",
"We first show that RA-YOLO achieves a good trade-off between detection accuracy and inference time over a large range of spatial resolutions.",
"We then evaluate the impact of spatial and amplitude resolutions on object detection accuracy using the proposed RA-YOLO model.",
"We demonstrate that the optimal spatial resolution that leads to the highest detection accuracy depends on the 'tolerated' image size.",
"We further assess the impact of the distance of an object to the camera on the detection accuracy and show that higher spatial resolution enables a greater detection range.",
"These results provide important guidelines for choosing the image spatial resolution and compression settings predicated on available bandwidth, storage, desired inference time, and/or desired detection range, in practical applications."
],
[
" We are thrilled that all reviewers are supportive of our comprehensive study and well-designed experiments.",
"As pointed out by Reviewer 1, the study of the influence on the image quality (spatial resolution and amplitude resolution) and the distance of objects in the image is comprehensive.",
"Reviewer 2 states that the comprehensive analysis of such a trade-off may help a better understanding of the importance of image resolution.",
"Reviewer 3 mentioned that the paper presents not only theoretical proves but also shows experimental evidence of spatial resolution and object distance's dependency on the detection accuracy.",
"We also appreciate that all reviewers agree on the novelty and practicality of our model.",
"We appreciate that Reviewer 1 agrees that this variant of YOLOv5 is novel.",
"It's nice to hear from Reviewer 2 that a good algorithm to tackle multi-resolution detection will help promote the application of such a detection model.",
"We are glad to hear that reviewer 3 emphasizes this paper solves a very practical problem that exists with modern object detection methods like YOLO, SSD, and more; and the new model can be helpful in real-world scenarios as it can help us decide the image spatial resolution and compression settings based on available bandwidth, storage, desired inference time, and/or desired detection range.",
"We are also grateful that all reviewers pointed out the mishandled figures and errors in grammar and words, and we will proofread the revised version.",
"We address the individual comments raised below and will add them to our revised paper.",
"To Reviewer #1 Q1: Question about adding more related work regarding object detection A1: We also think it is reasonable to add more related work to give credit to existing object detection methods.",
"We will add more related work about one-stage and two-stages detection methods and the vision transformer-based methods.",
"To Reviewer #2 Q2: Question about equipping our method on other detection models and adding more experiments compared with other methods.",
"A2: We also think it will be interesting to study equipping our method on other models or compare our proposed method with others.",
"However, in this paper, we didn't explore that direction because we would like to highlight the pivotal point of our main contributions is to explore the impact of spatial and amplitude resolution, as well as object distance, on object detection accuracy and computational cost.",
"Therefore, we concentrate more on the study of utilizing our proposed resolution-adaptive variant of the YOLOv5 model (RA-YOLO) in handling images of varying resolutions over a large range, rather than extensive tests on other detection models to beat the SOTA performance.",
"We believe our experiments of speed vs accuracy trade-offs (with and without resolution adaptive architecture), which were evaluated on the datasets that contain images compressed to different spatial and amplitude resolutions, are sufficient to confirm the correctness and effectiveness of our approach.",
"Moreover, regarding the reviewer 2's argument about equipping our method on other detection models, we confirm that this proposed simple but effective resolution-adaptive framework can work with different detection backbones, which is actually one of the key advantages of our method.",
"We can easily equip our proposed resolution adapative architecture on existing object detection models such as SSD, R-CNNand Fast R-CNN.",
"We anticipate a broader impact that our peers from the community can benefit from our future released codes and test on a broad range of detection methods.",
"Furthermore, due to the page limits of a conference paper, we are unable to report results on other object detection models or extensive range of datasets.",
"Q3: Question about logical and motivation of our paper.",
"A3: In this paper, we aim to explore the influence of image quality (spatial resolution and amplitude resolution), and the distance of objects in the image affects detection accuracy and computational cost.",
"We would like to highlight the motivation of our proposed RA-YOLO is stated in Section 1.",
"In short, the existing object detection models are not designed to operate on images over a large resolution range, rather separate models are optimized for different spatial resolutions.",
"We propose a resolution-adaptive variant of the YOLOv5 architecture (called RA-YOLO), which varies the number of scales in the feature pyramid based on the spatial resolution of the input image.",
"Q4: Question about details for reproducing this work.",
"A4: Special thanks to the reviewer for bringing our attention to add more implementation details to reproduce our work.",
"As for the hyper parameters setting, we set the learning rate as 0.01 and we use Stochastic gradient descent(SGD) as our optimizer.",
"The training epoch is set as 250.",
"We will add more detailed description about implementation details in Section 2.2.",
"Also, We will release the code after the paper is accepted.",
"Q5: Question about imbalance of our dataset A5: We apologize for the confusion.",
"We described the dataset preparation in Section 3.3 and experimental setting in Section 3.5.",
"For the middle-resolution (low-resolution) images, we select 700 images from each city in the Eurocity (Eurocity Down1.42) training set as training data and 74 images from each city in the Eurocity (Eurocity Down1.42) validation set as testing data.",
"Notice that we mentioned our dataset contains images from 31 cities and exact 700 images from each city are selected for the training.",
"Similarly, 74 images for each city are selected to form the testing dataset.",
"So there are about 31*700 = 22k images in our middle-resolution training set and 31*74 = 2.2k images in our middle-resolution testing set.",
"We respectively repeat the process to generate the dataset for high-resolution images, middle-resolution images, and low-resolution images.",
"Therefore, the dataset should be considered balanced.",
"We also rephrased this part in the paper, and hope it will be clearer.",
"To Reviewer #3 Q6: Question about releasing code and dataset A6: We would like to express our gratitude to the reviewer for the thoughtful and positive comments.",
"We will release the code and the dataset after the paper is accepted."
]
] | 2209.08237 |
[
[
"Gradient Properties of Hard Thresholding Operator"
],
[
"Abstract Sparse optimization receives increasing attention in many applications such as compressed sensing, variable selection in regression problems, and recently neural network compression in machine learning.",
"For example, the problem of compressing a neural network is a bi-level, stochastic, and nonconvex problem that can be cast into a sparse optimization problem.",
"Hence, developing efficient methods for sparse optimization plays a critical role in applications.",
"The goal of this paper is to develop analytical techniques for general, large size sparse optimization problems using the hard thresholding operator.",
"To this end, we study the iterative hard thresholding (IHT) algorithm, which has been extensively studied in the literature because it is scalable, fast, and easily implementable.",
"In spite of extensive research on the IHT scheme, we develop several new techniques that not only recover many known results but also lead to new results.",
"Specifically, we first establish a new and critical gradient descent property of the hard thresholding (HT) operator.",
"Our gradient descent result can be related to the distance between points that are sparse.",
"Also, our gradient descent property allows one to study the IHT when the stepsize is less than or equal to 1/L, where L>0 is the Lipschitz constant of the gradient of an objective function.",
"Note that the existing techniques in the literature can only handle the case when the stepsize is strictly less than 1/L.",
"By exploiting this we introduce and study HT-stable and HT-unstable stationary points and show no matter how close an initialization is to a HT-unstable stationary point (saddle point in sparse sense), the IHT sequence leaves it.",
"Finally, we show that no matter what sparse initial point is selected, the IHT sequence converges if the function values at HT-stable stationary points are distinct."
],
[
"Introduction",
"Solving sparse problems has gained increasing attention in the fields of statistics, finance, and engineering.",
"These problems emerge in statistics as variable selection in linear regression problems [23], [64], [15], [20], mixed-integer programs [8], [35], [19], portfolio optimization in finance [9], [12], compressed sensing in signal processing [26], [22], and compressing deep neural networks in machine learning [16], [40], [27], just to name a few.",
"Due to the use of $\\ell _0$ -(pseudo) norm$\\ell _0$ is not mathematically a norm because for any norm $\\Vert \\cdot \\Vert $ and $\\alpha \\in \\mathbb {R}$ , $\\Vert \\alpha \\mathbf {\\theta } \\Vert = \\vert \\alpha \\vert \\Vert \\mathbf {\\theta }\\Vert $ , while $\\Vert \\alpha \\mathbf {\\theta } \\Vert _0 = \\vert \\alpha \\vert \\Vert \\mathbf {\\theta }\\Vert _0$ if and only if $\\vert \\alpha \\vert = 1$, these problems are discontinuous and nonconvex.",
"The $\\ell _0$ -norm case have been addressed by the hard thresholding (HT) techniques specially the iterative HT (IHT) scheme [6], [3], [37], [62].",
"The Lasso-type, Basic Pursuit(BP)-type, and BP denoising(BPDN)-type problems consider $\\ell _1$ -norm as a convex approximation of $\\ell _0$ -norm [52], [41].",
"Nonconvex approximation of $\\ell _0$ -norm as $\\ell _p$ -(pseudo) norm ($0<p<1$ ) has also been studied well [13], [25], [31], [54], [59], [56].",
"Sparse optimization problems can also be formulated as mixed-integer programs [11].",
"Intrinsic combinatorics involved in sparse optimization problems makes it an NP-hard problem (even for a quadratic loss [18], [42]) so it is difficult to find a global minimizer.",
"However, greedy algorithms have developed to find local minimizers.",
"To this end, following the ideas of matching pursuit (MP) and orthogonal MP (OMP) [39], [51] as greedy algorithms, numerous other greedy algorithms have been developed such as stagewise OMP (StOMP) [21], regularized OMP (ROMP) [44], [45], Compressive Sampling MP (CoSaMP) [43], and Gradient Support Pursuit (GraSP) [1].",
"It should be noted that sparse optimization is not restricted to finding a sparse vector.",
"For example [24], [28], [17], finding a low-rank matrix is considered.",
"The problem of finding a low-rank matrix is a counterpart to finding a sparse vector when it comes to applications dealing with matrices.",
"In addition to devising algorithms for solving sparse optimization problems, developing first and second order optimality conditions have also been addressed well [48], [2], [4], [33], [38], [47], [10].",
"The general sparse optimization problem is the following: $\\quad \\begin{array}{l}\\min f(\\mathbf {x}) \\\\\\text{s.t.",
"}C_s \\cap \\mathcal {X}\\end{array}$ where $C_s=\\lbrace \\mathbf {x} \\in \\mathbb {R}^n \\mid \\Vert \\mathbf {x}\\Vert _0 \\le s\\rbrace $ (sparsity constraint) is the union of finitely many subspaces of dimension $s$ such that $1 \\le s<n$ , $\\mathcal {X}$ is a constraint set in $\\mathbb {R}^n$ , and the objective function $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is lower bounded and continuously differentiable, i.e., $C^1$ .",
"In this paper we address a special case of Problem (REF ) where $\\mathcal {X}=\\mathbb {R}^n$ as follows: $(\\text{P}):\\quad \\begin{array}{l}\\min f(\\mathbf {x}) \\\\\\text{s.t.",
"}\\mathbf {x} \\in C_s\\end{array}$ To address Problem (REF ) the following fundamental questions arise: What are the necessary/sufficient conditions for a local/global minimizer of Problem (REF )?",
"What are the characteristics of accumulation points of algorithms solving Problem (REF )?",
"Under what condition(s) does an accumulation point become a local/global minimizer?",
"If an accumulation point is a local/global minimizer, what is the rate of convergence?",
"[t] The iterative hard thresholding (IHT) [1] $\\mathbf {x}^0\\in \\mathbb {R}^n$ such that $\\Vert \\mathbf {x}^0\\Vert _0\\le s$ and stepsize $\\gamma >0$ .",
"$\\mathbf {x}^{k+1} \\in H_s(\\mathbf {x}^k-\\gamma \\nabla f(\\mathbf {x}^k))$ for $k=0,1, \\dots $ By considering the IHT algorithm, we will answer the above questions.",
"This algorithm has been extensively studied in the literature.",
"It was originally devised for solving compressed sensing problems in 2008 [6], [7].",
"Since then, there has been a large body of literature studying the IHT-type algorithms from different standpoints.",
"For example, [3], [37], [38], [49], [62] consider convergence of iterations, [29], [36] study the limit of the objective function value sequence, [35], [63] address duality, [61], [58] extend it to Newton's-type IHT, [14], [32], [34], [60] consider the stochastic version, [5], [30], [53], [57] address accelerated IHT, and [55], [1] solve logistic regression problem using the IHT."
],
[
"Summary of Contributions",
"By considering the IHT Algorithm for Problem (REF ), we develop the following results: We establish a new critical gradient descent property of the hard thresholding (HT) operator that has not been found in the literature.",
"Our gradient descent result can be related to the distance between points that are sparse.",
"However, the distance between sparse points cannot provide any information about the gradient in the sparse setting.",
"To the best of our knowledge, the other way around (the gradient to the distance) has not been shown so far in the literature.",
"This property allows one to study the IHT when the stepsize is less than or equal to $1/L$ , where $L > 0$ is the Lipschitz constant of the gradient of an objective function.",
"Note that the existing techniques in the literature can only handle the case when the stepsize is strictly less than $1/L$ .",
"As an example, one can refer to [36] that needs the stepsize to be greater than or equal to $1/L$ .",
"We introduce the notion of HT-stable/unstable stationary points.",
"Using them we establish the escapability property of HT-unstable stationary points (saddle point the sparse sense) and local reachability property of strictly HT-stable stationary points.",
"We provide a video of 4000 independent runs where the IHT algorithm is initialized very close to a HT-unstable stationary point and show the sequences escape them.",
"We also show that the IHT sequence converges globally under a new assumption that has not been found in the literature.",
"In addition, Q-linearly convergence of the IHT algorithm towards a local minimum when the objective function is both RSS and restricted strictly convex is shown.",
"According to our results, we address (Q1) and (Q2) by establishing a new gradient descent property of the hard thresholding (HT) operator and introducing the notion of HT-stable/unstable stationary points.",
"By considering RSS, restricted strictly convex, and RSC properties we address (Q3) and (Q4).",
"Table REF is provided to compare our results with those in the literature.",
"It shows what has been done chronologically and demonstrates our results."
],
[
"Related work",
"To answer (Q1) [3] introduces $L$ -stationarity property as a necessary condition for an optimal solution of Problem (REF ).",
"The $L$ -stationarity property is defined when the gradient of the objective function is Lipschitz.",
"Also, [3] addresses (Q2) by showing any accumulation point of the IHT algorithm is $L$ -stationary.",
"Lu in [37] restricts the objective function to be convex and shows that the IHT sequence converges to a local minimum when the objective function is regularized by $\\ell _0$ -norm and $\\mathcal {X}$ is a box constraint.",
"Jain et al.",
"[29] put more restriction on the objective value function and show that the objective value function sequence generated by the IHT algorithm converges to a value attained under a more restricted sparsity constraint.",
"The restrictions used in [29] are Restricted Strong Smoothness (RSS) and Restricted Strong Convexity (RSC).",
"The RSS and RSC properties are introduced by [46] and first used by [1] for sparsity optimization problems.",
"Currently, they have become standard restrictions for analyzing sparsity optimization problems.",
"Under RSS and RSC properties for the objective function, one is able to address (Q3) and (Q4).",
"Finding a closed-form expression for $P_{C_s \\cap \\mathcal {X}}$ when $\\mathcal {X}$ is an arbitrary set is difficult.",
"However, [4] shows orthogonal projection of a point onto $C_s \\cap \\mathcal {X}$ can be efficiently computed when $\\mathcal {X}$ is a symmetric closed convex set.",
"In this context, two types of sets are of interest: nonnegative symmetric sets and sign free sets.",
"To address (Q1) in a more generalized setting, Beck and Hallak [4] characterize $L$ -stationary points of Problem (REF ) when $\\mathcal {X}$ is either nonnegative symmetric set or sign free.",
"Also, Lu in [38] considers the same setting as [4] and introduces a new optimality condition that is stronger than $L$ -stationary.",
"He devises a Nonmonotone Projected Gradient (NPG) algorithm and shows an accumulation of the NPG sequence is the global optimal of Problem (REF ).",
"Pan et al.",
"[48] consider Problem (REF ) when $\\mathcal {X}=\\mathbb {R}^n_{+}$ .",
"They develop an Improved IHT algorithm (IIHT) that employs the Armijo-type stepsize rule.",
"They show when the objective function is RSS and RSC, the IIHT sequence converges to a local minimum.",
"A recent work by Zhou et al.",
"[62] develops Newton Hard-Thresholding Pursuit (NHTP) for solving problem (REF ).",
"They show that when accumulation points of the NHTP sequence are $L$ -stationary and are isolated, the sequence converges with a locally Q-quadratic rate.",
"Table REF compares current results in the literature.",
"Table: Comparison of results for the deterministic IHT-type algorithms."
],
[
"Definitions",
"We provide some definitions that will be used throughout the paper.",
"These definitions are the HT operator (HTO) and HTO inequality, RSS and RSC functions.",
"Definition 1 (Restricted Strong Smoothness (RSS)) A differentiable function $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is said to be restricted strongly smooth with modulus $L_s>0$ or is $L_s$ -RSS if $f(\\mathbf {y}) \\le f(\\mathbf {x}) + \\langle \\nabla f(\\mathbf {x}) , \\mathbf {y}-\\mathbf {x} \\rangle + \\frac{L_{s}}{2}\\Vert \\mathbf {y}-\\mathbf {x}\\Vert _2^2 \\quad \\forall \\mathbf {x},\\mathbf {y} \\in \\mathbb {R}^n \\text{ such that } \\Vert \\mathbf {x}\\Vert _0 \\le s,\\Vert \\mathbf {y}\\Vert _0\\le s.$ Definition 2 (Restricted Strong Convexity (RSC)) A differentiable function $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is said to be restricted strongly convex with modulus $\\beta _s>0$ or is $\\beta _s$ -RSC if $f(\\mathbf {y})\\ge f(\\mathbf {x}) + \\langle \\nabla f(\\mathbf {x}) , \\mathbf {y}-\\mathbf {x} \\rangle + \\frac{\\beta _s}{2}\\Vert \\mathbf {y}-\\mathbf {x}\\Vert _2^2\\quad \\forall \\mathbf {x},\\mathbf {y} \\in \\mathbb {R}^n\\text{ such that } ||\\mathbf {x}||_0 \\le s,||\\mathbf {y}||_0\\le s.$ Definition 3 (The HT operator) The HT operator $H_s(\\cdot )$ denotes the orthogonal projection onto multiple subspaces of $\\mathbb {R}^n$ with dimension $1 \\le s<n$ , that is, $H_s(\\mathbf {x}) \\in \\arg \\min _{\\Vert \\mathbf {z}\\Vert _0\\le s }\\Vert \\mathbf {z}-\\mathbf {x}\\Vert _2.$ Claim 1 The HT operator keeps the $s$ largest entries of its input in absolute values.",
"For a vector $\\mathbf {x} \\in \\mathbb {R}^n$ , $\\mathcal {I}^{\\mathbf {x}}_s \\subset \\lbrace 1,\\dots , n\\rbrace $ denotes the set of indices corresponding to the first $s$ largest elements of $\\mathbf {x}$ in absolute values.",
"For example $H_2([1,-3,1]^{\\top })$ is either $[0,-3,1]^{\\top }$ or $[1,-3,0]^{\\top }$ where $\\mathcal {I}^{\\mathbf {y}}_2=\\lbrace 2,3\\rbrace $ and $\\mathcal {I}^{\\mathbf {y}}_2=\\lbrace 1,2\\rbrace $ , respectively.",
"Therefore, the output of it may not be unique.",
"This clearly shows why HTO is not a convex operator and why there is an inclusion in (REF ) not an inequality."
],
[
"Results",
"We consider solving Problem (REF ) using the IHT Algorithm and develop results on the HT operator.",
"Using them, the behavior of the IHT sequence generated by Algorithm is characterized.",
"Towards this end, statements of the main results are provided and all the technical proofs are postponed to the Appendix for the reviewers."
],
[
"Gradient descent property",
"First, we establish a new and critical gradient descent property of the hard thresholding (HT) operator.",
"Theorem 1 Let $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be a differentiable function that is $L_{s}$ -RSS, $\\mathbf {y} \\in H_s(\\mathbf {x}-\\gamma \\nabla f(\\mathbf {x}))$ with any $\\mathcal {I}_s^{\\mathbf {y}}$ and $0<\\gamma \\le \\frac{1}{L_s}$ , and $\\mathbf {x}$ be a sparse vector such that $\\Vert \\mathbf {x}\\Vert _0\\le s$ with any $\\mathcal {I}_s^{\\mathbf {x}}$ .",
"Then, $\\frac{\\gamma }{2}(1-L_{s}\\gamma )\\Vert \\nabla _{\\mathcal {I}_s^{\\mathbf {x}} \\cup \\mathcal {I}_s^{\\mathbf {y}}}f(\\mathbf {x})\\Vert _2^2 \\le f(\\mathbf {x}) - f(\\mathbf {y})$ where $\\nabla _{\\mathcal {I}_s^{\\mathbf {x}} \\cup \\mathcal {I}_s^{\\mathbf {y}}}f(\\mathbf {x})$ is the restriction of the gradient vector to the union of the index sets $\\mathcal {I}_s^{\\mathbf {x}}$ and $ \\mathcal {I}_s^{\\mathbf {y}}$ .",
"Theorem REF provides a lower bound on the difference between the current function value evaluated at $\\mathbf {x}$ and the one evaluated at the updated point provided by the HTO, i.e., $\\mathbf {y}$ .",
"Note that, $\\mathbf {y}$ may not be a unique vector that has $s$ nonzero elements.",
"Nonetheless, as stated in Theorem REF , Inequality (REF ) holds for any $\\mathbf {y}$ that might be the output of the HTO.",
"As one clearly see, the descent can only be characterized by looking at the entries of the gradient that are restricted to the union of the $s$ largest elements in both $\\mathbf {x}$ and $\\mathbf {y}$ .",
"The rest of the gradient can be ignored.",
"Since one may be interested in characterizing the descent using the distance between $\\mathbf {x}$ and $\\mathbf {y}$ , we provide the following corollary.",
"Corollary 1 Assume all the assumptions in Theorem REF hold, then, $\\frac{1-L_{s}\\gamma }{6\\gamma }\\Vert \\mathbf {y}-\\mathbf {x}\\Vert _2^2\\le \\frac{\\gamma }{2}(1-L_{s}\\gamma )\\Vert \\nabla _{\\mathcal {I}_s^{\\mathbf {x}} \\cup \\mathcal {I}_s^{\\mathbf {y}}}f(\\mathbf {x})\\Vert _2^2 \\le f(\\mathbf {x}) - f(\\mathbf {y})$ The above result shows the superiority of our gradient result because our gradient result can be related to the distance of points that are sparse.",
"However, the distance between sparse points cannot provide any information about the gradient.",
"To the best of our knowledge, the other way around (the gradient to the distance) has not been shown so far in the literature.",
"Algebraically speaking, characterizing a descent of the function value solely with the information of the current iterate, i.e., $\\mathbf {x}$ , is of more interest.",
"To this end, we provide another corollary to Theorem REF that ties the descent to $\\mathbf {x}$ only.",
"Corollary 2 Assume all the assumptions in Theorem REF hold.",
"Then, the norm of the gradient restricted to any $\\mathcal {I}_s^{\\mathbf {x}}$ can be bounded as follows: $\\frac{\\gamma }{2}\\Vert \\nabla _{\\mathcal {I}_s^{\\mathbf {x}}}f(\\mathbf {x})\\Vert _2^2 \\le f(\\mathbf {x}) - f(\\mathbf {y})$ By this point, we have shown that applying the HTO once, can result in smaller function value provided the gradient over the $s$ largest entries of $\\mathbf {x}$ are nonzero.",
"This can be utilized to show the sequence generated by the IHT algorithm is nonincreasing.",
"Specially, if the generated sequence has an accumulation point, the objective value function sequence converges to the objective value of the accumulation point.",
"A sequence may not converge but it may have an accumulation point.",
"For example $1,-1,1,-1,\\dots $ is not a convergent sequence but it has two accumulation points.",
"Corollary 3 Let $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be a bounded below differential function that is $L_{s}$ -RSS and $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ be the IHT sequence $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ with $0<\\gamma \\le \\frac{1}{L_s}$ .",
"Then, $\\Big (f(\\mathbf {x}^{k})\\Big )_{k\\ge 0}$ is nonincreasing and converges.",
"Also, if $\\mathbf {x}^*$ is an accumulation point of $\\big (\\mathbf {x}^{k}\\big )_{k\\ge 0}$ then $\\Big (f(\\mathbf {x}^{k})\\Big )_{k\\ge 0}\\rightarrow f(\\mathbf {x}^*)$ .",
"Next, we look at basic stationary points of Problem (REF ) and show their properties."
],
[
"Optimality condition based on the HT properties",
"In this subsection, we will show that not all basic stationary points of Problem (REF ) are reachable when the IHT algorithm is run.",
"To do so, the notion of HT stationary points are introduced as follows."
],
[
"Definition 4 For a given constant $\\gamma >0$ , we say that a sparse vector $\\mathbf {x}^* \\in C_s$ is HT-stable stationary point of Problem (REF ) associated with $\\gamma $ if $\\nabla _{\\text{supp}(\\mathbf {x}^*)} f(\\mathbf {x}^*)=0$ , and $\\min \\Big (|x^*_i|:i\\in \\mathcal {I}^{{\\mathbf {x}^*}}_s\\Big )\\ge \\gamma \\max \\Big (|\\nabla _j f(\\mathbf {x}^*)|:j \\notin \\text{supp}(\\mathbf {x}^*)\\Big )=\\gamma \\Vert \\nabla _{({\\text{supp}(\\mathbf {x}^*)})^c} f(\\mathbf {x}^*)\\Vert _{\\infty }.$ (Note that $\\min \\Big (|x^*_i|:i\\in \\mathcal {I}^{\\mathbf {x}^*}_s\\Big )$ is unique and does not depend on the choice $\\mathcal {I}^{\\mathbf {x}^*}_s$ .)",
"If $\\nabla _{\\text{supp}(\\mathbf {x}^*)} f(\\mathbf {x}^*)=0$ but (REF ) fails, we say that $\\mathbf {x}^*$ is HT-unstable stationary point with $\\gamma $ .",
"Moreover, if $\\nabla _{\\text{supp}(\\mathbf {x}^*)} f(\\mathbf {x}^*)=0$ and (REF ) holds strictly, namely, $\\min \\Big (|x^*_i|:i\\in \\mathcal {I}^{\\mathbf {x}^*}_s\\Big )>\\gamma \\max \\Big (|\\nabla _j f(\\mathbf {x}^*)|:j \\notin \\text{supp}(\\mathbf {x}^*)\\Big )$ we say that $\\mathbf {x}^*$ is a strictly HT-stable stationary point associated with $\\gamma $ .",
"Note that when $\\Vert \\mathbf {x}^*\\Vert _0=s$ , $\\mathcal {I}^{\\mathbf {x}^*}_s$ is unique and equals $\\text{supp}(\\mathbf {x}^*)$ such that $\\text{supp}(\\mathbf {x}^*)$ in the above definition can be replaced by $\\mathcal {I}^{\\mathbf {x}^*}_s$ .",
"Moreover, if $\\mathbf {x}^*$ is a strictly HT-stable stationary point, then we must have $\\mathcal {I}^{\\mathbf {x}^*}_s=\\text{supp}(\\mathbf {x}^*)$ (or equivalently $\\Vert \\mathbf {x}^*\\Vert _0=s$ ) because otherwise, $0=\\min \\Big (|x^*_i|:i\\in \\mathcal {I}^{{\\mathbf {x}^*}}_s\\Big )>\\gamma \\Vert \\nabla _{({\\text{supp}(\\mathbf {x}^*)})^c} f(\\mathbf {x}^*)\\Vert _{\\infty }$ which is impossible.",
"Remark 1 As stated in the Definition REF , a basic stationary point is a point whose gradient is zero over the nonzero elements.",
"For example, suppose $\\tilde{\\mathbf {x}}=[0, 4, 0, 2]^{\\top } \\in \\mathbb {R}^4$ is a basic stationary point.",
"Then $\\nabla f(\\tilde{\\mathbf {x}})=[c_1, 0, c_2, 0]^{\\top }$ where $c_1,c_2$ are scalars.",
"The main idea of the HT-stable stationary point is that it has to be a basic stationary point.",
"In other words $\\tilde{\\mathbf {x}}$ can be a basic stationary point but not a HT-stable stationary point.",
"This is the analogue of the non-sparse optimization where a point $\\hat{\\mathbf {x}}$ whose gradient is zero, i.e., $\\nabla f(\\hat{\\mathbf {x}})=0$ may not be necessary a local or global minimizer.",
"It can be a saddle point.",
"Remark 2 The main message of Definition REF is the following: “only by looking at the gradient restricted to the nonzero entries of a basic feasible point, one cannot say whether it is a local minimizer of Problem (REF ) or not”.",
"An HT-stable stationary point associated with $\\gamma $ is equivalent to the $\\frac{1}{\\gamma }$ -stationary point of Problem (REF ) defined in [3].",
"Thus, by [3], $\\mathbf {x}^*$ is a HT-stable point if and only if $\\mathbf {x}^* \\in H_s(\\mathbf {x}^* -\\gamma \\nabla f(\\mathbf {x}^*))$ .",
"The notion of a HT-unstable stationary point is novel and is a key point for proving Theorem 2.",
"Theorem 2 is the foundation for the proof of part b) of Theorem 3 as well as Theorem 4 which characterizes the accumulation point of the IHT sequence.",
"In addition, we have introduced another stationary point, namely strictly HT-stable stationary which is a crucial concept for local convergence of the IHT sequence.",
"In the following, we present a result that characterizes a HT-unstable stationary point.",
"In essence, the following result shows that there always exists a neighborhood around a sparse HT-unstable stationary point whose gradient is zero over the nonzero elements and one can decrease the function value by going towards the direction of any nonzero coordinates.",
"Theorem 2 Suppose $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ is $C^1$ and $L_s$ -RSS.",
"Given any $0<\\gamma \\le \\frac{1}{L_s}$ , if a vector $\\tilde{\\mathbf {x}}\\in C_s$ is such that $\\nabla _{\\text{supp}(\\tilde{\\mathbf {x}})}f(\\tilde{\\mathbf {x}})=0$ and $\\min \\big ( |\\tilde{x}_i|: i \\in \\mathcal {I}^{\\tilde{\\mathbf {x}}}_s\\big )<\\gamma \\Vert \\nabla _{(\\text{supp}(\\tilde{\\mathbf {x}}))^c}f(\\tilde{\\mathbf {x}})\\Vert _{\\infty }$ for some $\\mathcal {I}^{\\tilde{\\mathbf {x}}}_s$ , then there exist a constant $\\nu > 0$ and a neighborhood $\\mathcal {N}$ of $\\tilde{\\mathbf {x}}$ such that $f(\\mathbf {y}) \\le f(\\mathbf {x})-\\nu $ for any $\\mathbf {x} \\in \\mathcal {N}\\cap C_s$ and any $\\mathbf {y} \\in H_s(\\mathbf {x}-\\gamma \\nabla f(\\mathbf {x})).$ The above result leads to the following necessary optimality conditions for a global minimizer of Problem (REF ) in terms of hard thresholding operator $H_s$ .",
"For the case where $\\gamma =1/L_s$ , i.e., part b), one needs to use Theorem 2.",
"Indeed, to the best of our knowledge, no proof has not been found for it in the literature.",
"Essentially, establishing the result in part b) is one of our contributions.",
"For the case $\\gamma < 1/L_s$ it is proven that $\\mathbf {x}^*=H_s(\\mathbf {x}^*-\\gamma \\nabla f(\\mathbf {x}^*))$ .",
"Note that the condition for $0 < \\gamma < \\frac{1}{L_s}$ has been obtained in [3] without using gradient properties of the HT operator.",
"Theorem 3 Suppose $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ is $L_s$ -RSS and $\\mathbf {x}^*$ is a global minimizer.",
"Then, $\\mathbf {x}^*$ is a HT-stable (or $\\frac{1}{\\gamma }$ -) stationary point for any $0< \\gamma \\le \\frac{1}{L_s}$ .",
"Particularly, the following hold: ) For any $0<\\gamma < \\frac{1}{L_s}$ , $\\mathbf {x}^*=H_s(\\mathbf {x}^*-\\gamma \\nabla f(\\mathbf {x}^*))$ . )",
"For $\\gamma =\\frac{1}{L_s}$ , $\\mathbf {x}^* \\in H_s(\\mathbf {x}^*-\\gamma \\nabla f(\\mathbf {x}^*))$ .",
"The following result shows that any accumulation point of an IHT sequence must be a HT-stable stationary point.",
"Theorem 4 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ be $L_s$ -RSS and $C^1$ function.",
"Suppose $f$ is bounded below on $C_s$ .",
"Consider an IHT sequence $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ associated with an arbitrary $\\gamma \\in (0, \\frac{1}{L_s}]$ , and let $\\mathbf {x}^*$ be an accumulation point of $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ .",
"Then, $\\mathbf {x}^*$ is a HT-stable stationary point of Problem (REF ).",
"Remark 3 The above theorem shows that any accumulation point of an IHT sequence is a HT-stable stationary point of Problem (REF ).",
"Since each HT-stable stationary point must be a basic stationary point, one can observe that any accumulating point $\\mathbf {x}^*$ of an IHT sequence must satisfy $\\nabla _{\\text{supp}(\\mathbf {x}^*)} f(\\mathbf {x}^*)=0$ when $\\Vert \\mathbf {x}^*\\Vert _0=s$ , or $\\nabla f(\\mathbf {x}^*)=0$ when $\\Vert \\mathbf {x}^*\\Vert _0<s$ .",
"The following result pertains to the objective function values of HT-stable and HT-unstable stationary points.",
"Corollary 4 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ be $L_s$ -RSS and $C^1$ function.",
"Suppose that every (nonempty) sub-level set of $f$ contained in $C_s$ is bounded, i.e., for any $\\alpha \\in \\mathbb {R}$ , $\\lbrace x \\in C_s | f(\\mathbf {x})\\le \\alpha \\rbrace $ is bounded (and closed).",
"Consider an arbitrary $\\gamma \\in (0, \\frac{1}{L_s}]$ .",
"For any HT-unstable stationary point $\\mathbf {x}^*$ associated with $\\gamma $ , there exists a HT-stable stationary point $\\tilde{\\mathbf {x}}^*$ associated with $\\gamma $ such that $f(\\mathbf {x}^*)>f(\\tilde{\\mathbf {x}}^*)$ .",
"Based on the above corollary, it is easy to see that if there are finitely many HT-unstable stationary points (happens when the function is RSC), then there is a HT-stable stationary point $\\tilde{\\mathbf {x}}^*$ such that $f(\\mathbf {x}^*)>f(\\tilde{\\mathbf {x}}^*)$ for any HT-unstable stationary point $\\mathbf {x}^*$ .",
"The following result provides sufficient conditions for the convergence of an IHT sequence.",
"Corollary REF aims to remove any restrictions on the initial condition.",
"This corollary shows that no matter what initial condition in $C_s$ is selected, the IHT sequence will converge to a HT-stable stationary point.",
"Note that we say a HT-stable/unstable stationary point $\\mathbf {x}^*$ associated with $\\gamma \\in (0, \\frac{1}{L_s}]$ is isolated if there exists a neighborhood $\\mathcal {N}$ of $\\mathbf {x}^*$ such that $\\mathcal {N}$ does not contain any HT stationary point other than $\\mathbf {x}^*$ .",
"Corollary 5 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ be $L_s$ -RSS and $C^1$ function.",
"Suppose that every (nonempty) sub-level set of $f$ contained in $C_s$ is bounded.",
"Consider an arbitrary $\\gamma \\in (0,\\frac{1}{L_s}]$ .",
"Assume that : For any two distinct HT-stable stationary points $\\mathbf {x}^*$ and $\\mathbf {y}^*$ associated with $\\gamma $ , $f(\\mathbf {x}^*) \\ne f(\\mathbf {y}^*)$ .",
"Then, for any $\\mathbf {x}^0 \\in C_s$ , the IHT sequence $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ converges to a HT-stable stationary point associated with $\\gamma $ .",
"This convergence results also hold under the following assumption: : when $0<\\gamma < \\frac{1}{L_s}$ , each HT-stable stationary point associated with $\\gamma $ is isolated.",
"The following corollary shows that any IHT sequence always “escape\" from a HT-unstable stationary point.",
"Corollary 6 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ be $L_s$ -RSS and $C^1$ function.",
"Suppose $f$ is bounded below on $C_s$ .",
"For any given $\\gamma \\in (0, \\frac{1}{L_s}]$ and any HT-unstable stationary point $\\mathbf {x}^*$ associated with $\\gamma $ , there exists a neighborhood $\\mathcal {N}$ of $\\mathbf {x}^*$ such that for any IHT sequence starting from any $\\mathbf {x}^0 \\in C_s$ , there exists $N \\in \\mathbb {N}$ such that $\\mathbf {x}^k \\notin \\mathcal {N} \\cap C_s$ for all $k \\ge N$ .",
"The next result shows the attraction towards a strictly HT-stable stationary point in a neighborhood of such a stationary point.",
"In what follows, for each index subset $\\mathcal {J}$ with $|\\mathcal {J}|=s$ , a subspace $\\mathcal {S_J}:=\\lbrace \\mathbf {x} \\in \\mathbb {R}^n \\mid \\mathbf {x}_{\\mathcal {J}^c}\\rbrace $ associated with $\\mathcal {J}$ is defined.",
"Clearly, $C_s$ is the union of $\\mathcal {S_J}$ 's for all $\\mathcal {J}$ 's with $|\\mathcal {J}|=s$ .",
"Proposition 1 Let $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ be $L_s$ -RSS and $C^1$ function.",
"Suppose $f$ is bounded below on $C_s$ and $f$ is strictly convex on $\\mathcal {S_J}$ for any index subset $\\mathcal {J}$ with $|\\mathcal {J}|=s$ .",
"Let $\\mathbf {x}^*$ be a strictly HT-stable stationary point associated with any given $\\gamma \\in (0, \\frac{1}{L_s}]$ .",
"Then there exists a neighborhood $\\mathcal {B}$ of $\\mathbf {x}^*$ such that for every $\\mathbf {x}^0 \\in \\mathcal {B} \\cap C_s$ , the IHT sequence $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ converges to $\\mathbf {x}^*$ .",
"Moreover, if $f$ is strongly convex on $\\mathcal {J}$ for every index subset $\\mathcal {J}$ with $|\\mathcal {J}|=s$ , then for $\\mathbf {x}^0 \\in \\mathcal {B} \\cap C_s$ , the IHT sequence $\\big (\\mathbf {x}^k\\big )_{k\\ge 0}$ Q-linearly converges to $\\mathbf {x}^*$ .",
"Next, we provide an example to show the escapability property of HT-unstable points."
],
[
"Simulation",
"To elaborate on theoretical results including Corollary REF , Theorem REF , the notion of HT-stationary points, Corollary REF which shows escapability property of HT-unstable stationary points, and Proposition REF which shows Reachability to HT-stable stationary points, we use a quadratic function $f(\\mathbf {x})=\\frac{1}{m}\\sum _{i=1}^m(A_{i\\bullet }\\mathbf {x}-y_i)^2=\\frac{1}{m}\\Vert A\\mathbf {x}-\\mathbf {b}\\Vert ^2$ where $\\mathbf {A} \\in \\mathbb {R}^{m\\times n}$ , $A_{i\\bullet }$ is the $i$ -th row of $A$ , $\\mathbf {x} \\in \\mathbb {R}^n$ is the optimization variable, and $\\mathbf {y} \\in \\mathbb {R}^m$ is the target.",
"This function is both RSS and RSC so both Corollary REF and Proposition REF follow.",
"To better visualize the process, we let $m=n=4$ and $s=2$ .",
"Therefore, there are six HT-stationary points where the gradient over the nonzero elements is zero.",
"We use Pytorch [50] to select the matrix $A$ and $\\mathbf {y}$ .",
"By setting the random seed to be 45966 we draw a $4\\times 4$ matrix $A$ whose elements are standard normal.",
"Keeping the same seed, we generate $\\mathbf {y}$ .",
"The following would be $A$ and $\\mathbf {y}$ : $A=\\begin{bmatrix}-1.0655 & 0.2249 & -0.0897 & 0.1876 \\\\1.1627 & -1.1229 & -0.0823 & -0.3059 \\\\-0.2011 & 0.5342 & -0.0551 & -1.3459 \\\\0.2308 & -0.6404 & -0.7468 & 0.0378\\end{bmatrix},\\quad \\mathbf {y}=\\begin{bmatrix}-1.7861\\\\ -0.3556\\\\ -0.1881\\\\ 0.3896\\end{bmatrix}$ The restricted Lipschitz constant, i.e., $L_s$ , for the above quadratic function is $\\frac{2}{m}\\times \\lambda _{max}(A^{\\top }A)$ where $\\lambda _{max}$ is the maximum eigenvalue of $A^{\\top }A$ .",
"Thus, for the above choice of $A, \\mathbf {y}$ , the maximum allowable stepsize is $\\gamma =\\frac{1}{L_s}=0.06$ .",
"Once, $\\gamma $ is fixed, one can determine stability of each stationary point.",
"The following are HT-stationary points along with their stability status as well as the gradient status of each HT-stationary point.",
"As you can see, the gradient corresponding to nonzero elements in HT-stationary point are zero: $\\begin{bmatrix}No.",
"& x_1 & x_2 & x_3 & x_4 & g_1 & g_2 & g_3 & g_4 & HT-stability\\\\1 & 1.3474 & 1.0331 & 0 & 0 & 0 & 0 & 0.2060 & -0.3916 & \\text{strictly HT-stable}\\\\2 & 0.6278 & 0 & 0.0177 & 0 & 0 & -0.3843 & 0 & -0.1070 & \\text{HT-unstable}\\\\3 & 0.6387 & 0 & 0 & 0.1123 & 0 & -0.4189 & -0.0029 & 0 & \\text{strictly HT-stable}\\\\4 & 0 & -0.1758 & 0.0008 & 0 & -0.6506 & 0 & 0 & 0.0106 & \\text{HT-unstable}\\\\5 &0 & -0.1776 & 0 & -0.0113 & -0.6473 & 0 & -0.0010 & 0 & \\text{HT-unstable}\\\\6 & 0 & 0 & -0.1608 & 0.0259 & -0.7994 & 0.1297 & 0 & 0 & \\text{HT-unstable}\\end{bmatrix}$ where $x_1, x_2, x_3, x_4$ are four coordinates of each HT-stationary point and where $g_1, g_2, g_3, g_4$ are the four gradient entries corresponding to each HT-stationary point.",
"Since HT-stationary points are vectors in $\\mathbb {R}^4$ , there is no way to show all of them on one 2-d plane.",
"Thus, we use six 2-d plains where each plane shows only two coordinates of HT-stationary points.",
"On each 2-d plain we have 6 different points, each one associated with one of the HT-stationary points shown in a particular 2-d plain with specified coordinates.",
"In Figure REF the points with red stars are HT-unstable ones, and the blue ones are the HT-stable ones.",
"For example, the first 2-d plain (first row-first column) including coordinates $x_1-x_2$ shows the $x_1,x_2$ coordinates of all of the six HT-stationary points.",
"On the first row-first column 2-d plain, the first HT-stationary point is more distinct because it is the only one that has two nonzeros elements associated with $x_1-x_2$ coordinates.",
"We also can see three points with $x_2=0$ , two of which are HT-unstable points and one is HT-unstable one.",
"This is more clear, if one looks at the column $x_2$ in HT-stationary points matrix above.",
"Also, it is clear that we have three HT-unstable points with $x_1=0$ on the first 2-d plane.",
"Figure: Illustration of 4,000 initialization close to four HT-unstable stationary points.We perturb nonzero coordinates of all HT-unstable points with a normal random noise with mean zero and standard deviation of $\\sigma =0.5$ to create 4,000 different initialization points.",
"These points create four clouds around HT-unstable points which are shown in Figure REF .",
"Figure: Illustration of 4,000 IHT sequences initialized close to four HT-unstable stationary points after 400 steps (please refer to the video of all 400 steps).Then we run the IHT algorithm for 400 steps.",
"After 300 steps, all of these initializations escape from those HT-unstable points and converge to either of the HT-stable stationary points on $x_1-x_2$ or $x_1-x_4$ 2-d planes.",
"In fact, these two HT-stable stationary points are sparse local minimizers.",
"Figure REF shows the 300-th step of IHT algorithm.",
"There is a video in the supplementary materials that shows 400 steps of applying IHT algorithm for 4,000 different runs.",
"These numerical results corroborate our theoretical results as expected.",
"By looking at the video, one can easily see escapability property of HT-unstable stationary points, and Reachability to HT-stable stationary points."
],
[
"Conclusion",
"This paper provide theoretical results that help to understand the IHT algorithm.",
"These theoretical results include a critical gradient descent property of the hard thresholding (HT) operator which is used to show the sequence of the IHT algorithm is decreasing and by doing it over and over we get smaller objective value.",
"This property also allows one to study the IHT algorithm when the stepsize is less than or equal to $1/L_s$ , where $L_s>0$ is the Lipschitz constant of the gradient of an objective function.",
"We introduced different stationary points including HT-stable and HT-unstable stationary points and show no matter how close an initialization is to a HT-unstable stationary point, the IHT sequence leaves it.",
"We provided a video of 4000 independent runs where the IHT algorithm is initialized very close to a HT-unstable stationary point and showed the sequences escape them.",
"This property is used to prove that the IHT sequence converges to a HT-stable stationary point.",
"Also, we established a condition for a HT-stable stationary that is a global minimizer with respect to $\\gamma =1/L_s$ .",
"Finally, we showed the IHT sequence always converges if the function values of HT-stable stationary points are distinct, this is a new assumption that has not been found in the literature."
]
] | 2209.08247 |
[
[
"Quantum Non-Demolition Measurement on the Spin Precession of\n Laser-Trapped $^{171}$Yb Atoms"
],
[
"Abstract Quantum non-demolition (QND) measurement enhances the detection efficiency and measurement fidelity, and is highly desired for its applications in precision measurements and quantum information processing.",
"We propose and demonstrate a QND measurement scheme for the spin states of laser-trapped atoms.",
"On $^{171}$Yb atoms held in an optical dipole trap, a transition that is simultaneously cycling, spin-state selective, and spin-state preserving is created by introducing a circularly polarized beam of control laser to optically dress the spin states in the excited level, while leaving the spin states in the ground level unperturbed.",
"We measure the phase of spin precession of $5\\times10^{4}$ atoms in a bias magnetic field of 20 mG.",
"This QND approach reduces the optical absorption detection noise by $\\sim$19 dB, to a level of 2.3 dB below the atomic quantum projection noise.",
"In addition to providing a general approach for efficient spin-state readout, this all-optical technique allows quick switching and real-time programming for quantum sensing and quantum information processing."
],
[
"Introduction",
"The nuclear and electronic spin states of atoms, with the advantage of having long coherence times, are of great importance in precision measurement and quantum information experiments.",
"Measurements on atomic spin states are performed to realize magnetometers [1], [2], [3], optical clocks [4], [5], [6], [7], and in experiments that search for permanent electric dipole moment [8], [9], [10], [11], [12], [13], [14], [15].",
"As an information carrier, the spin state is also widely used in quantum information processing [16], [17], [18], [19], [20], [21], [22], [23] and quantum simulation [24], [25].",
"For these applications, it is often crucial to measure the population in each spin state with a high efficiency in order to reduce the statistical errors in precision measurements and to enhance the readout fidelities in quantum information experiments.",
"The populations in spin states can be determined either dispersively by measuring the state-dependent phase shift of an off-resonant laser beam, or dissipatively by measuring the absorption of or the fluorescence induced by near-resonant light.",
"In the recent quantum computing experiments, it is challenging to achieve high fidelity of spin selectivity in the qubit-state readout [21], [20], [23].",
"The detection efficiency is often limited by measurement-induced spin flipping, upon which the quantum state is demolished prematurely.",
"For example, probing the states of a spin-1/2 system on non-cycling transitions, as shown in Figs.",
"REF (a) and REF (b), induces a spin flip with just a few excitation–emission cycles [10], [16].",
"Alternatively, in a quantum non-demolition (QND) measurement [26], [27], [28], the state is preserved under repeated excitation–emission cycles, thus the signal-to-noise ratio of state detection can be greatly enhanced.",
"For this reason, QND measurements have attracted increasing interests and have been successfully demonstrated on various systems achieving improved measurement fidelity in quantum information processing [29], [30], [31] and higher precision in quantum measurements [32], [33], [34].",
"Several QND strategies for spin-state detection have been demonstrated.",
"For example, a magnetic field can be applied to lift the degeneracy among transitions of different magnetic sublevels [35], under the condition that the induced Zeeman splittings are much larger than the transition linewidth.",
"While this method works efficiently, the required magnetic field and its on-and-off switching can disturb the spin states, resulting in decoherence and loss of sensitivity.",
"Another strategy, more suitable for solid-state systems, is to modify the density of states and induce a state-dependent spontaneous relaxation rate via the Purcell effect [36].",
"Recently, such a scheme is implemented for single rare-earth ions embedded in a nano-photonic cavity, boosting the transition cyclicity by several orders of magnitude [37].",
"In this work, we present a theoretical analysis and an experimental demonstration of a QND approach to probe spin states and measure the phase of spin precession via optical excitation.",
"By applying an ancillary control laser to shift the excited states via the ac-Stark effect, while leaving the spin states in the ground level unperturbed, the chosen optical transition can simultaneously become cycling, spin-selective, and spin-preserving.",
"AC Stark shift has been successfully employed in many applications, including state-selective manipulation of atomic internal states [38], [39], [40], site-selective addressing in atom array [41], [42], [43], [44], [45], and narrow-line Sisyphus cooling [46], [47], [48], [49].",
"Our approach is demonstrated on $^{171}$ Yb atoms in an optical dipole trap (ODT) whose wavelength satisfies the magic condition for the probe transition [50].",
"QND measurements are performed on spin precession, demonstrating a reduction in optical noise by $\\sim $ 19 dB.",
"This all-optical approach of QND measurement avoids the need to switch and shield any control magnetic fields.",
"Its principle can be applied to many different atomic systems and is compatible with general cold-atom experiments in precision measurements and quantum information science.",
"This novel method was introduced in Ref.",
"[51], where it was used in a measurement of the electric dipole moment of $^{171}$ Yb."
],
[
"Principle",
"The principle of optical pumping and spin-state detection is often explained with the simple case of $F=1/2\\leftrightarrow F=1/2$ [Fig.",
"REF (a)].",
"Throughout this paper, the quantization direction is chosen to be along the common $\\hat{k}$ vector of the polarization beam, probe beam and control beam.",
"A laser beam of resonant frequency and $\\sigma ^{+}$ polarization excites the $m_{F}=-1/2$ state in the ground level, but not the $m_{F}=+1/2$ “dark state”.",
"In this case, an atom in $|g;1/2,-1/2\\rangle $ absorbs and emits on average only three photons before dropping into $|g;1/2,+1/2\\rangle $ , thus limiting the fidelity of state detection.",
"For the case of a different transition, $F=1/2\\leftrightarrow F=3/2$ [Fig.",
"REF (b)], even though an atom in $|g;1/2,+1/2\\rangle $ can be probed repeatedly on the $|g;1/2,+1/2\\rangle $ $\\leftrightarrow $ $|e;3/2,+3/2\\rangle $ cycling transition, an atom in $|g;1/2,-1/2\\rangle $ absorbs and emits on average only 1.5 photons before a spin flip occurs.",
"For a QND measurement, we need a transition that is simultaneously cycling, spin selective and spin preserving.",
"We propose a QND measurement scheme based on the optical dressing effect.",
"Consider a ladder-type atomic system with three levels: the ground level $|g\\rangle $ , the excited level $|e\\rangle $ , and the excited level $|c\\rangle $ [Fig.",
"REF (c)] ($|c\\rangle $ can be either higher or lower than $|e\\rangle $ in energy).",
"Their angular momenta are 1/2, 3/2, and 3/2, and spontaneous decay rates are 0, $\\Gamma _{e}$ , and $\\Gamma _{c}$ , respectively.",
"The control beam, on resonance with the $|e\\rangle \\leftrightarrow |c\\rangle $ transition, dresses the $|e\\rangle $ state with Rabi frequency $\\Omega _{c}$ .",
"The optical dressing is on all Zeeman states $|e; 3/2, m_F\\rangle $ , except for the stretched state $|e; 3/2, +3/2\\rangle $ .",
"The dressed Zeeman states are shifted by $\\pm \\Omega _{c}/2$ to form Aulter-Towns doublets [52] and the stretched state is protected by the angular momentum selection rules.",
"Such a difference among the Zeeman states is essential for spin-selective detection of the spin state.",
"To probe the nuclear spin state, the probe beam resonantly drives the $|g\\rangle \\leftrightarrow |e\\rangle $ transition with Rabi frequency $\\Omega _{p}$ .",
"In the condition where $\\Omega _{p}\\ll \\Omega _{c},\\Gamma _{e}$ , the optical transition strength to the dressed Zeeman states is reduced by a factor of $\\sim \\Omega _{c}^{2} / (\\Gamma _{e}\\Gamma _{c})$ .",
"The $|g; 1/2, -1/2\\rangle $ state then becomes a “dark state” and the rate of spin flip is reduced by the same factor.",
"With modest intensity of the control beam, the reduction factor can be on the order of $10^{3}$ .",
"The atoms in $|g; 1/2, +1/2\\rangle $ can be excited to the unaffected stretched state repeatedly, and optical detection of the nuclear spin state with cycling transition is realized.",
"It is important to note that the control beam can be switched on and off at a rate much faster than the spin precession rate, and does not affect the spin states in the ground level."
],
[
"Experimental setup",
"We have implemented the QND measurement on the spin states of $^{171}$ Yb (I = 1/2) atoms in the ground level.",
"The three levels $6s^2\\,{}^1S_0\\ (F=1/2)$ , $6s6p\\,{}^3P_1\\ (F=3/2)$ and $6s8s\\,{}^3S_1\\ (F=3/2)$ form a ladder system as shown in Fig.",
"REF (e).",
"The QND measurement employs the following laser beams and transitions: The probe beam at 556 nm, tuned to the resonance of $6s^2\\,{}^1S_0\\leftrightarrow 6s6p\\,{}^3P_1$ ($\\Gamma _{e}/2\\pi =182$ kHz), is supplied by a frequency-doubled diode laser; the control beam at 423 nm, tuned to the resonance of $6s6p\\,{}^3P_1\\leftrightarrow 6s8s\\,{}^3S_1$ ($\\Gamma _{c}=2\\pi \\times 1.7$ MHz), is supplied by a frequency-doubled Ti:Sapphire laser; the polarization beam at 399 nm, tuned to the resonance of $6s^2\\,{}^1S_0\\leftrightarrow 6s6p\\,{}^1P_1$ ($\\Gamma _{399}/2\\pi =28$ MHz), is supplied by a frequency-doubled diode laser.",
"The probe beam, the control beam, and the polarization beam all have the same circular polarization (e.g., $\\sigma ^{+}$ ), and all co-propagate with the stationary ODT beam along the $z$ direction [Fig.",
"REF (d)].",
"The absorption of the probe beam by the trapped atoms is imaged onto a CMOS camera.",
"To prepare the atomic ensemble, $^{171}$ Yb atoms are loaded into a two-stage magneto-optical trap (MOT): The first-stage MOT is operated on the strong transition (same as the polarization transition) to efficiently capture the atoms from a Zeeman slower; the second-stage MOT on the narrow-linewidth intercombination transition (same as the probe transition) cools the atoms to 20 $\\mu $ K. The cold atoms are then handed over to a movable ODT.",
"More details of the apparatus are given in Ref [50].",
"The atoms are carried into a neighboring science chamber by translating the focal point of the movable ODT along the $y$ direction [Fig.",
"REF (d)] and, finally, handed over to a stationary ODT pointed in the $z$ direction.",
"The two ODTs are provided by two separate fiber lasers.",
"We prepare $10^{3}-10^{4}$ $^{171}$ Yb atoms in the ODT for measurements.",
"A $\\cos \\theta $ coil [53] inside magnetic shields generates a uniform B field ($\\sim $ 20 mG) in the $x$ direction to drive spin precession.",
"The stationary ODT in the science chamber has a waist of $50\\ \\mathrm {\\mu m}$ and a Rayleigh length of $\\sim $ 4 mm, is linearly polarized in the $y$ direction, and has a power of 35 W and a wavelength of 1035.84 nm.",
"This wavelength meets the magic condition for the probe transition so that the probe remains effective despite of the deep trapping potential of 200 $\\mu $ K [50].",
"The probe linewidth is measured to be $\\sim $ 400 kHz, reflecting Doppler broadening at 100 $\\mu $ K. The vector light shift of the spin states introduced by the linearly polarized ODT beam is negligible (< 1 mHz).",
"The control beam is focused on the atoms with a beam waist of about 300 $\\mu $ m. The parameters for the control beam is determined by measuring the induced light shifts of the probe transition (Appendix ).",
"At the control beam power of 40 mW ($\\Omega _{c}\\sim 2\\pi \\times 40$ MHz), the spin-flip rate in the ground level is reduced by a factor of $\\Omega _{c}^{2}/(\\Gamma _{e}\\Gamma _{c})\\sim 10^{3}$ .",
"Since the control beam is far detuned from any transitions that connect to the ground level, its effects on spin precession are negligible: the scalar light shift of $|g\\rangle $ induced by the control beam is at $\\sim $ kHz, much less than $\\Gamma _{e}$ ; the vector light shift is at $\\sim $ mHz, much less than the precession frequency of 15 Hz."
],
[
"Phase measurement of spin precession",
"We demonstrate the advantage of the QND approach with phase measurements of the spin precession of $^{171}$ Yb atoms.",
"The timing sequence for the QND measurement is shown in Fig.",
"REF (a).",
"Initial spin polarization is produced by a 2 ms pulse of the polarization beam (“Pol.",
"1”, $I/I_{s}=3\\times 10^{-4}$ ).",
"The spin polarized atoms precess about the bias magnetic field ($\\sim $ 20 mG) at a Larmor frequency of $\\sim $ 15 Hz.",
"After a given precession time, chosen to be 1 s in this study, an overlapping pulse of both the probe beam (0.4 ms, $I/I_{s}=0.25$ ) and control beam, named “Probe 1+”, is applied for a spin projection measurement.",
"The population in $|g; 1/2, +1/2\\rangle $ ($\\rho _{+}$ ) is measured, while the population in $|g; 1/2, -1/2\\rangle $ ($\\rho _{-}$ ) remains unchanged because its excitation is suppressed by the presence of the control beam.",
"Half of a period ($T_{p}/2$ ) later, the precession swaps the populations of $|g; 1/2, +1/2\\rangle $ and $|g; 1/2, -1/2\\rangle $ states, and “Probe 1-” is fired to measure the original $\\rho _{-}$ prior to swapping.",
"The probe pulses are repeated, each with a $T_{p}/2$ delay from the previous pulse [Fig.",
"REF (a)].",
"The Bloch vector $S_{z}$ can be calculated as $S_{z}=\\rho _{+}-\\rho _{-}=\\frac{N_{+}-N_{-}}{N_{+}+N_{-}},$ where $\\rho _{+}+\\rho _{-}=1$ , $N_{+}$ and $N_{-}$ are the number of atoms in $|g;\\,1/2,\\,+1/2\\rangle $ and $|g;\\,1/2,\\,-1/2\\rangle $ derived from absorption images of the probe pulses.",
"For each pulse, “Probe 1+” or “Probe 1-”, an absorption image taken by the CMOS camera is compared to a background image taken without atoms to derive an optical depth value at each pixel.",
"Both $\\rho _{+}$ and $\\rho _{-}$ are measured under the same set of probe conditions, with only $T_{p}/2$ apart in timing.",
"In this way of calculating $S_{z}$ , many common-mode imperfections in the laser and detector parameters are suppressed.",
"For comparison, we have also conducted phase measurements based on the non-QND optical pumping method.",
"Here, the normalization, polarization and probe all use the same laser tuned to the resonance of $6s^2\\,{}^1S_0\\ (F=1/2) \\leftrightarrow 6s6p\\,{}^1P_1\\ (F=1/2)$ at 399 nm, and no control beam is needed.",
"A different timing sequence is used [Fig.",
"REF (c)].",
"A 0.2 ms normalization pulse [“Norm.” in Fig.",
"REF (c)], fired at $T_{p}/2$ after the initial polarization pulse (“Pol.",
"1”), measures the total population $N_{+}+N_{-}$ .",
"Afterwards the atoms need to be repolarized with “Pol.",
"2”.",
"Following the free precession time, a 0.2 ms “Probe” pulse ($I/I_{sat}=3\\times 10^{-4}$ ) is applied to measures $\\rho _{-}$ .",
"In this non-QND approach, the probe causes spin flips and can only be applied briefly before the spin information is lost.",
"The Bloch vector evolves as $S_{z}=P_{z}\\cos (2\\pi f t+\\phi _{0})$ , where $P_{z}$ is the degree of spin polarization, $f$ is the Larmor precession frequency and $\\phi _{0}$ is the initial phase.",
"The sinusoidal precession signal, shown in Fig.",
"REF (d) and REF (e) are obtained by measuring $S_{z}$ at different precession times around 1 s. The measurement uncertainties in the QND approach are significantly reduced in comparison with those in non-QND approach.",
"A key requirement of the QND approach is that the Bloch vector can be repeatedly measured.",
"As shown in Fig.",
"REF (f), $S_{z,1}$ , measured by “Probe 1”, is highly correlated with $S_{z,2}$ , measured by “Probe 2”."
],
[
"Measurement Uncertainty",
"The Larmor precession phase is determined in the $S_{z}$ measurements, with the highest sensitivity occurring at the points of $\\rho _{+}=\\rho _{-}$ .",
"In the QND approach, the variance of $S_{z}$ can be expressed as (see Appendix REF ), $\\sigma ^2_{S_{z}}=\\sigma ^2_{\\textbf {op}}+\\frac{1}{N_{a}}\\simeq \\frac{4\\overline{p}}{(N_{a}\\bar{n})^{2}\\epsilon }+\\frac{1}{N_{a}},$ where the first term, denoted $\\sigma ^2_{\\textbf {op}}$ , describes the optical noise arising from the fluctuations of both the incoming and the absorbed photons.",
"The second term is the atomic quantum projection noise.",
"All variables in Eq.",
"(REF ) are defined in Table REF .",
"Results of Eq.",
"(REF ) are approximation for weak absorption cases when $N_{a}\\bar{n}/\\overline{p}\\ll 1$ .",
"The numerical factor “4” is the total number of images used in the measurement sequence combining “Probe 1+” and “Probe 1-”, with each containing both the absorption and background images.",
"The atomic quantum projection noise is induced by the detection pulse “Probe 1+”, after which all other detection pulses do not contribute any more quantum projection noise.",
"Detailed derivation for both QND and non-QND cases are given in Appendix .",
"In principle, the excitation cycle in the QND measurement can be repeated indefinitely until the optical noise becomes negligible compared to $1/N_{a}$ .",
"In actual experiments, however, the number of excitation cycles is limited by atom losses due to either heating or imperfection in the near-cycling transition.",
"Consider a pulse of $\\bar{p}$ photons shot on the atomic clouds through a region of interest of area $A_{\\mathrm {ROI}}$ , it induces an average number of excitation cycles ($\\bar{n}$ ).",
"The two quantities are related as $\\exp (-b_{l}\\overline{p}\\frac{A_{\\mathrm {abs}}}{A_{\\mathrm {ROI}}})+\\bar{n}b_{l}=1,$ where the first term characterizes the probability for the atom to survive in the bright state, and the second term for the atom to either escape the trap or decay into a dark state.",
"The variables in the equation above are defined in Table REF .",
"The average number of excitation cycles ($\\bar{n}$ ) is affected by the loss branching ratio ($b_{l}$ ) that takes into account both loss mechanisms.",
"In the QND measurement of this study, the atoms are lost from the trap after an average of 80 excitation cycles due to heating ($b_{l}=1/80$ ).",
"For comparison, in the non-QND measurement, the spin state is demolished after an average of 3 excitation cycles due to optical pumping ($b_{l}=1/3$ ).",
"This large difference is indeed the essence of the QND advantage.",
"Table: NO_CAPTIONFigure: (a) Number of excitation cycles n ¯\\bar{n} vs. photon number p ¯\\overline{p} in the probe pulse.",
"In the non-QND case, n ¯\\bar{n} is limited by spin flipping; In QND, n ¯\\bar{n} is limited by atom loss from the trap due to heating.",
"(b)Optical noise σ 𝐨𝐩 2 \\sigma ^2_{\\textbf {op}} vs. p ¯\\overline{p}.",
"Number of atoms probed is N a =5×10 4 N_{a}=5\\times 10^{4}.",
"The optimum conditions for QND and non-QND cases occur at different p ¯\\overline{p} values.Figure: The variance σ S z 2 \\sigma ^2_{S_{z}} vs. the number of trapped atoms N a N_{a}.",
"Blue data points are for non-QND cases, and the green data points for QND cases.",
"The red dashed line indicates the 1/N a 1/N_{a} atomic quantum projection noise.",
"For QND cases, the green dashed line models the optical noise, and the green solid line is combined variance of both the optical noise and the atomic quantum projection noise.",
"The blue dashed line and blue solid line are overlapped.Fig.",
"REF shows the average number of excitation cycles $\\bar{n}$ and the optical noise $\\sigma ^2_{\\textbf {op}}$ as a function of the photon number ($\\overline{p}$ ) in the probe pulse.",
"When $\\overline{p}$ is small, $\\bar{n}$ is proportional to $\\overline{p}$ , while $\\sigma ^2_{\\textbf {op}}$ is inversely proportional to $\\overline{p}$ .",
"As $\\overline{p}$ increases, loss mechanisms come into effect, $\\bar{n}$ becomes saturated and $\\sigma ^2_{S_{z}}$ increases due to the photon shot noise.",
"The optimum choice for $\\overline{p}$ occurs at the point when $\\bar{n}$ starts to saturate, and $\\bar{n}$ is different between QND and non-QND cases.",
"Fig.",
"shows the variances of $S_{z}$ for both the QND and non-QND cases with the number of trapped atoms varying in the range of $10^{3}-10^{4}$ .",
"The measured results agree well with calculated ones.",
"From non-QND to QND cases, the optical noise is reduced by 19 dB, independent of the number of atoms.",
"The QND optical noise goes below the atomic quantum projection noise when $N_{a} > 3 \\times 10^4$ , and is 2.3 dB below at $N_{a} = 5 \\times 10^{4}$ ."
],
[
"Discussion and Outlook",
"In this work, we have demonstrated a QND phase measurement of the spin precession of atoms in an optical dipole trap.",
"The 19 dB gain in the optical noise can be further improved by reducing heating loss of the atoms due to optical probing and scattering loss due to impure laser polarization of the control beam.",
"In the current setup, the atoms are transferred into the ODT of 200 $\\mu $ K depth at a temperature of 100 $\\mu $ K, and are heated out of the ODT after an average of 80 excitation cycles.",
"By applying laser cooling in the ODT prior to the measurement sequence, the atom temperature could be lowered down toward the Doppler-cooling limit of $4.4\\ \\mathrm {\\mu K}$ , thus increasing the number of excitation cycles $\\bar{n}$ .",
"Laser cooling would also reduce the Doppler-broadened width of 400 kHz, towards the natural linewidth of 182 kHz, and thus increase the photon absorption cross-section.",
"Furthermore, replacing the traveling wave ODT with an optical lattice would increase the trap depth and reduce heating losses due to the probe beam.",
"The impure laser polarization of control beam causes excitation to the $|c\\rangle $ state, followed by decay into the lower-lying P levels.",
"This leakage in the cycling transition can be reduced with better polarization control.",
"All these steps would combine to reduce the loss branching ratio $b_{l}$ , increase $\\bar{n}$ and further suppress the optical noise.",
"The use of the QND method introduced in this work can be expanded to a wide range of applications.",
"For example, QND measurements can help improve the search sensitivity of a permanent electric dipole moment (EDM) of atoms [10], [54], [51].",
"The recently demonstrated tweezer array of Yb atoms, with their spin states acting as qubits, is an emerging platform for quantum computation [16], [17], [20], [21], [23], on which the QND approach would help improve readout fidelity.",
"Suppression of spin-flip shown in this work can also be used to decrease spin noise and increase interrogation time in spin squeezing experiments [32].",
"Moreover, the QND approach can be employed to implement quantum error correction that requires non-destructive detection of error syndromes [55], as well as real-time feedback control on atomic spin states [56], [57].",
"We emphasize that the all-optical control allows quick switching and real-time programming [58].",
"While we have focused on the $(F,F+1,F+1)$ ladder-type system, the approach can be generalized to other configurations, such as a $\\Lambda $ -type systems or $(F,F+1,F)$ systems.",
"Moreover, instead of the dissipative readout demonstrated in this work, the optical dressing effect can also be applied to dispersive atom-photon interaction [35], leading to applications such as measurement-based spin squeezing, or generation of entanglement between distant atomic ensembles for distributed quantum sensing [59].",
"We would like to thank D. Sheng, Z.-S. Yuan, Y.-G. Zheng, and Y.-N. Lv for helpful discussions.",
"This work has been supported by the National Natural Science Foundation of China through Grants No.",
"91636215, No.",
"12174371, No.",
"11704368, the Strategic Priority Research Program of the Chinese Academy of Sciences through Grant No.",
"XDB21010200, and Anhui Initiative in Quantum Information Technologies through Grant No.",
"AHY110000.",
"C.-L.Z.",
"was supported by NSFC through Grant No.",
"11922411."
],
[
"Optical noise in absorption imaging",
"Population is detected by measuring the optical depth ($OD$ ) of the atomic ensemble.",
"$OD$ is derived from the number of detected photons in the reference image without atoms ($p_{1}$ ) and that in the absorption image of the atomic ensemble ($p_{2}$ ), $OD=\\ln ({p_{1}}/{p_{2}}).$ For the number of incident probe photons of $p$ , and take into account the quantum efficiency of the camera $\\epsilon $ , the reference image has $p_{1}=\\sum _{i=1}^{p}d_i,$ where $d_i$ is a binary variable indexing whether the $i$ -th photon is detected: $d_i\\sim B(1,\\epsilon )$ .",
"The number of photons in the absorption image is $p_{2}=\\sum _{i=1}^{p-\\sum _{j=1}^{N_{\\pm }}n_j}d_i,$ where $N_{\\pm }$ is the number of atoms in the probed state and $n_j$ is the number of photons scattered (absorbed) by the $j$ -th atom.",
"The expectation values and variances for $p_{1}$ and $p_{2}$ can be expressed as $\\mathrm {E}(p_{1})&=&\\sigma _{p_{1}}^{2}=\\overline{p}\\epsilon ,\\\\\\mathrm {E}(p_{2})&=&\\overline{p}\\epsilon -\\frac{N_{a}}{2}\\bar{n}\\epsilon ,$ $\\sigma _{p_{2}}^{2}&=&\\mathrm {E}(p-\\sum _{j=1}^{N_{\\pm }}n_j)\\mathrm {Var}(d_i)+\\mathrm {Var}(p-\\sum _{j=1}^{N_{\\pm }}n_j)[\\mathrm {E}(d_i)]^2\\\\&=&\\overline{p}\\epsilon +N_{a}\\overline{n}(2\\epsilon ^{2}-\\epsilon )/2+N_{a}\\overline{n}^2\\epsilon ^2/4.$ When calculating $\\sigma _{p_{2}}^{2}$ , it is assumed that $n_j$ follows the Poisson distribution.",
"In our case, $\\overline{p}\\gg N_{a}\\bar{n}\\gg 1$ .",
"The expectation value and variance for $OD$ are $\\mathrm {E}(OD)&\\simeq &\\frac{N_{a}\\bar{n}}{2\\overline{p}},\\\\\\sigma _{OD}^{2}&\\simeq &\\frac{2}{\\overline{p}\\epsilon }+\\frac{N_{a}\\overline{n}^{2}}{4\\overline{p}^2}.$ These results follow the more detailed derivations found in Ref.",
"[60], [61].",
"It is worth noting that the optical noise in absorption imaging originates from not only the intrinsic photon shot noise discussed above, but also technical noise.",
"In order to reach the fundamental photon-shot-noise level, a total of 30 images without atoms are taken after the detection pulses for a fringe-removal algorithm [62]."
],
[
"Variance of measured $S_{z}$",
"As the populations of $|g;\\frac{1}{2},-\\frac{1}{2}\\rangle $ and $|g;\\frac{1}{2},+\\frac{1}{2}\\rangle $ states are equal, the expectation value and variance for $\\rho _{+}$ are $\\mathrm {E}(\\rho _{+})&=&\\frac{1}{2},\\\\\\sigma _{\\rho _{+}}^{2}&\\simeq &\\frac{5\\overline{p}}{2(N_{a}\\bar{n})^{2}\\epsilon }+\\frac{1}{4N_{a}}.\\\\$ For non-QND measurements, the variance of measured $S_{z}$ is $\\sigma _{S_{z}}^{2}&=&\\mathrm {Var}(2\\rho _{+}-1)\\\\&\\simeq &\\frac{10\\overline{p}}{(N_{a}\\bar{n})^{2}\\epsilon }+\\frac{1}{N_{a}}.\\\\$ For QND measurements, the variance of measured $S_{z}$ is $\\sigma _{S_{z}}^{2}&=&\\mathrm {Var}(\\rho _{+}-\\rho _{-})\\\\&\\simeq &\\frac{4\\overline{p}}{(N_{a}\\bar{n})^{2}\\epsilon }+\\frac{1}{N_{a}}.$ The reduction in variances from non-QND to QND is largely due to the difference in the $\\bar{n}$ values, and is independent of $N_{a}$ .",
"Figure: The measured light shift δ p \\delta _{p} of the probe transition against the control laser detuning.",
"At a control laser beam of 4 mW and 300 μ\\mu m radius, the Rabi frequency Ω c /(2π)=12.5(1) MHz \\Omega _{c}/(2\\pi )=12.5(1)\\,\\mathrm {MHz} is determined."
],
[
"Parameters for the control laser",
"The 423 nm control beam drives the $6s6p\\,{}^3P_1\\leftrightarrow 6s8s\\,{}^3S_1$ transition.",
"The natural linewidth of this transition is estimated to be $2\\pi \\times 1.7$ MHz, based on the known lifetime of the $6s8s\\,{}^3S_1$ level[63] and the estimated branching ratios.",
"The detuning $\\Delta _{c}$ and the Rabi frequency $\\Omega _{c}$ are experimentally determined by measuring the light shift $\\delta _{p}$ of the 556 nm probe transition (Fig.",
"REF ), $\\delta _{p}= \\frac{\\Delta _{c}}{2}\\ln (1+\\frac{2\\Omega _{c}^2}{\\Gamma _{c}^2+4\\Delta _{c}^2}).$"
]
] | 2209.08218 |
[
[
"Adaptation for Validation of a Consolidated Control Barrier Function\n based Control Synthesis"
],
[
"Abstract We develop a novel adaptation-based technique for safe control design in the presence of multiple control barrier function (CBF) constraints.",
"Specifically, we introduce an approach for synthesizing any number of candidate CBFs into one consolidated CBF candidate, and propose a parameter adaptation law for the weights of its constituents such that the controllable dynamics of the consolidated CBF are non-vanishing.",
"We then prove that the use of our adaptation law serves to certify the consolidated CBF candidate as valid for a class of nonlinear, control-affine, multi-agent systems, which permits its use in a quadratic program based control law.",
"We highlight the success of our approach in simulation on a multi-robot goal-reaching problem in a crowded warehouse environment, and further demonstrate its efficacy experimentally in the laboratory via AION ground rovers operating amongst other vehicles behaving both aggressively and conservatively."
],
[
"Introduction",
"Since the arrival of control barrier functions (CBFs) to the field of safety-critical systems [1], much attention has been devoted to the development of their viability for safe control design [2], [3], [4].",
"As a set-theoretic approach founded on the notion of forward invariance, CBFs encode safety in that they ensure that any state beginning in a safe set remains so for all future time.",
"In the context of control design, CBF conditions are often used as constraints in quadratic program (QP) based control laws, either as safety filters [5] or in conjunction with stability constraints (e.g.",
"control Lyapunov functions) [6].",
"Their utility has been successfully demonstrated for a variety of safety-critical applications, including mobile robots [7], [8], unmanned aerial vehicles (UAVs) [9], [10], and autonomous driving [11], [12].",
"But while it is now well-established that CBFs for controlled dynamical systems serve as certificates of safety, the verification of candidate CBFs as valid is in general a challenging problem.",
"Though for a single CBF there exist guarantees of validity under certain conditions for systems with either unbounded [2] or bounded control authority [13], [14], these results do not generally extend to control systems seeking to satisfy multiple candidate CBF constraints.",
"Recent approaches to control design in the presence of multiple CBF constraints have mainly circumvented this challenge by considering only one such constraint at a given time instance, either by assumption [15] or construction in a non-smooth manner [16], [17].",
"In contrast, the authors of [18] and [19] each propose smoothly synthesizing one candidate CBF for the joint satisfaction of multiple constraints, but make no attempt to validate their candidate function.",
"The problem of safe control design under a multitude of constraints is especially relevant in practical applications involving autonomous mobile robots, where the main challenge is in the robot completing its nominal objective while satisfying constraints related to collision avoidance with respect to obstacles both static and dynamic.",
"Figure: Parameter adaptation for our C-CBF leads to a gain-dependent (and time-varying) controlled-invariant set C(k)⊂S=⋂ i=1 c S i C(k) \\subset S = \\bigcap _{i=1}^c S_i.",
"C(k)C(k) is shown here with a dotted white boundary for gains k 0 k_0 at time t 0 t_0 and k 1 k_1 at t 1 t_1.It is with this problem in mind that we propose a consolidated CBF (C-CBF) based approach to control design for multi-agent systems in the presence of both non-communicative and non-responsive (though non-adversarial) agents.",
"Constructed by smoothly synthesizing any arbitrary number of candidate CBFs into one, our C-CBF defines a new super-level set that can under-approximate the intersection of its constituent sets arbitrarily closely (see Figure REF ).",
"We further propose a parameter adaptation law for the weighting of the constituent functions, and prove that its use renders our C-CBF valid and the super-level set controlled invariant for the class of nonlinear, control-affine, multi-agent systems under consideration.",
"And while various works have utilized parameter adaptation in the context of control for safety-critical systems, usually in an attempt to either learn [20], [21] or compensate for [22] unknown parameters in the system dynamics, our proposed adaptation law is the first to our knowledge to be used for the simultaneous verified satisfaction of multiple CBF constraints.",
"To show the effectiveness of our proposed control formulation, we study a decentralized multi-robot goal-reaching problem in a crowded warehouse environment amongst non-responsive agents.",
"As a practical demonstration, we tested our controller experimentally on a collection of ground rovers in the laboratory setting and found that it succeeded in safely driving the rovers to their goal locations amongst non-responsive agents behaving both aggressively and conservatively.",
"The paper is organized as follows.",
"Section introduces some preliminaries, including set invariance, optimization based control, and our first problem statement.",
"In Section , we introduce the form of our C-CBF and propose a parameter adaptation law for rendering it valid.",
"Sections and contain the results of our simulated and experimental case studies respectively, and in Section we conclude with final remarks and directions for future work."
],
[
"Mathematical Preliminaries",
"We use the following notation throughout the paper.",
"$\\mathbb {R}$ denotes the set of real numbers.",
"The set of integers between $i$ and $j$ (inclusive) is $[i..j]$ .",
"$\\Vert \\cdot \\Vert $ represents the Euclidean norm.",
"A function $\\alpha : \\operatorname{\\mathbb {R}}\\rightarrow \\operatorname{\\mathbb {R}}$ is said to belong to class $\\mathcal {K}_\\infty $ if $\\alpha (0)=0$ and $\\alpha $ is increasing on the interval $(-\\infty ,\\infty )$ , A function $\\phi : \\operatorname{\\mathbb {R}}\\times \\operatorname{\\mathbb {R}}\\rightarrow \\operatorname{\\mathbb {R}}$ is said to belong to class $\\mathcal {L}\\mathcal {L}$ if for each fixed $r$ (resp.",
"$s$ ), the function $\\phi (r,s)$ is decreasing with respect to $s$ (resp.",
"$r$ ) and is such that $\\phi (r,s) \\rightarrow 0$ for $s \\rightarrow \\infty $ (resp.",
"$r \\rightarrow \\infty $ ).",
"The Lie derivative of a function $V:\\mathbb {R}^n\\rightarrow \\mathbb {R}$ along a vector field $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ at a point $x\\in \\mathbb {R}^n$ is denoted $L_fV(x) \\triangleq \\frac{\\partial V}{\\partial x} f(x)$ .",
"In this paper we consider a multi-agent system, each of whose $A$ constituent agents may be modelled by the following class of nonlinear, control-affine dynamical systems: $\\dot{x}_i = f_i(x_i(t)) + g_i(x_i(t))u_i(t), \\quad x_i(0) = x_{i0}$ where $x_i \\in \\operatorname{\\mathbb {R}}^n$ and $u_i \\in \\mathcal {U}_i \\subseteq \\operatorname{\\mathbb {R}}^m$ are the state and control input vectors for the ith agent, with $\\mathcal {U}_i$ the input constraint set, and where $f_i: \\operatorname{\\mathbb {R}}^n \\rightarrow \\operatorname{\\mathbb {R}}^n$ and $g_i: \\operatorname{\\mathbb {R}}^{n \\times m} \\rightarrow \\operatorname{\\mathbb {R}}^n$ are known, locally Lipschitz, and not necessarily homogeneous $\\forall i \\in \\mathcal {A} = [1..A]$ .",
"We denote the concatenated state vector as $x = [x_1,\\hdots ,x_A]^T \\in \\operatorname{\\mathbb {R}}^N$ , the concatenated control input vector as $u = [u_1,\\hdots ,u_A]^T \\in \\mathcal {U} \\subseteq \\operatorname{\\mathbb {R}}^M$ , and as such express the full system dynamics as $\\dot{x} = F(x(t)) + G(x(t))u(t), \\quad x(0) = x_0,$ where $F = [f_1,\\hdots ,f_A]^T: \\operatorname{\\mathbb {R}}^N \\rightarrow \\operatorname{\\mathbb {R}}^N$ and $G = \\textrm {diag}([g_1,\\hdots ,g_A]): \\operatorname{\\mathbb {R}}^{M \\times N} \\rightarrow \\operatorname{\\mathbb {R}}^N$ .",
"We assume that a (possibly empty) subset of the agents are communicative, denoted $j \\in \\mathcal {A}_c = [1..A_c]$ , in the sense that they share information (e.g.",
"states, control objectives, etc.)",
"with one another, and that the remaining agents are non-communicative, denoted $k \\in \\mathcal {A}_n = [(A_c+1)..A]$ , in that they do not share information, where $A_c \\ge 0$ and $A_n = A - A_c \\ge 0$ are the number of communicative and non-communicative agents respectively.",
"We further assume that all agents are non-adversarial in that they do not seek to damage or otherwise deceive others, though there may be non-communicative agents which are non-responsive ($l \\in \\mathcal {A}_{n,n} \\subseteq \\mathcal {A}_n$ ) in that they do not actively avoid unsafe situations."
],
[
"Safety and Forward Invariance",
"Consider a set of safe states $S$ defined implicitly by a continuously differentiable function $h: \\operatorname{\\mathbb {R}}^N \\rightarrow \\operatorname{\\mathbb {R}}$ , as follows: $S = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; h(x) \\ge 0\\rbrace ,$ where the boundary and interior of $S$ are denoted as $\\partial S = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; h(x) = 0\\rbrace $ and $\\textrm {int}(S) = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; h(x) > 0\\rbrace $ respectively.",
"In many works (e.g.",
"[23], [24]), the set $S$ defined by (REF ) is referred to as safe if it is forward-invariant, i.e.",
"if $x(0) \\in S \\Rightarrow x(t) \\in S$ , $\\forall t \\ge 0$ .",
"Nagumo's Theorem provides a necessary and sufficient condition for rendering the set $S$ forward-invariant for the system (REF ).",
"Lemma 1 (Nagumo's Theorem[25]) Suppose that there exists $u(t) \\in \\mathcal {U}$ such that (REF ) admits a globally unique solution for each $x_0 \\in S$ .",
"Then, the set $S$ is forward-invariant for the controlled system (REF ) if and only if $L_Fh(x) + L_Gh(x)u \\ge 0, \\; \\forall x \\in \\partial S.$ One way to render a set $S$ forward-invariant is to use CBFs in the control design.",
"Definition 1 [2] Given a set $S \\subset \\operatorname{\\mathbb {R}}^N$ defined by (REF ) for a continuously differentiable function $h: \\operatorname{\\mathbb {R}}^N \\rightarrow \\operatorname{\\mathbb {R}}$ , the function $h$ is a control barrier function (CBF) defined on a set $D \\supseteq S$ if there exists a Lipschitz continuous class $\\mathcal {K}_\\infty $ function $\\alpha : \\operatorname{\\mathbb {R}}\\rightarrow \\operatorname{\\mathbb {R}}$ such that, for all $x \\in D$ , $\\sup _{u \\in \\mathcal {U}}\\left[L_Fh(x) + L_Gh(x)u\\right] \\ge -\\alpha (h(x)).$ In this paper, we assume that $\\frac{\\partial h}{\\partial x}$ is Lipschitz continuous so that $L_Fh(x)$ and $L_Gh(x)$ are likewise.",
"In other works (e.g.",
"[26]), the function $h$ responsible for defining $S$ is a CBF if there exists a class $\\mathcal {K}_\\infty $ function $\\alpha $ satisfying $L_Gh(x) = \\mathbf {0}_{1\\times M} \\Rightarrow L_Fh(x) + \\alpha (h(x)) > 0.$ We note, however, that with unbounded control authority (i.e.",
"$\\mathcal {U} = \\operatorname{\\mathbb {R}}^{M}$ ) a sufficient condition for the existence of some $\\alpha \\in \\mathcal {K}_\\infty $ satisfying (REF ), and thus for $h$ to be a CBF, is $L_Gh(x) \\ne \\mathbf {0}_{1\\times M}$ , $\\forall x \\in S$ , though this does not generally hold for a system with multiple CBF constraints."
],
[
"Control Design using CBFs",
"Decentralized controllers, in which agents compute inputs based on local information, have found empirical success as a control strategy for multi-agent systems of the form (REF ) [27], [28].",
"The following is an example of one such controller for an agent $i\\in \\mathcal {A}$ with safety constraints encoded via $c>1$ candidate CBFs: $u_i^* = \\operatornamewithlimits{arg\\,min}_{u_i \\in \\mathcal {U}_i} &\\frac{1}{2}\\Vert u_i-u_i^0\\Vert ^2 \\\\\\textrm {s.t.}",
"\\quad &\\forall s\\in [1..c] \\nonumber \\\\a_{s,i} + b_{s,i}u_{i} &\\ge 0, $ where (REF ) seeks to produce a control solution $u_i^*$ that deviates minimally from some nominal input $u_i^0$ , and () encodes $c$ safety constraints of the form (REF ) via candidate CBFs $h_s$ , where $a_{s,i} = L_{f_i}h_s + \\alpha _s(h_s)$ and $b_{s,i} = L_{g_i}h_s$ .",
"Notably, for many classes of systems (REF ) is neither guaranteed to be feasible nor to preserve safety between agents [8].",
"When some subset of agents are able to communicate with one another, i.e.",
"agents $j \\in \\mathcal {A}_c$ share information, their control inputs $u_{\\mathcal {A}_c} = [u_1,\\hdots ,u_{A_c}]^T$ may be computed in a centralized fashion as follows: $u_{\\mathcal {A}_c}^* = \\operatornamewithlimits{arg\\,min}_{u_{\\mathcal {A}_c} \\in \\mathcal {U}_{\\mathcal {A}_c}} &\\frac{1}{2}\\Vert u_{\\mathcal {A}_c}-u_{\\mathcal {A}_c}^0\\Vert ^2 \\\\\\textrm {s.t.}",
"\\quad \\forall j,k&\\in \\mathcal {A}_c, \\; k \\ne j \\nonumber \\\\a_{s,j} + b_{s,j}u_{j} &\\ge 0, \\; \\forall s \\in [1..c_I], \\\\a_{s,jk} + b_{s,j}u_{j} + b_{s,k}u_{k} &\\ge 0, \\; \\forall s \\in [c_I + 1..c] $ where $u_{\\mathcal {A}_c}^0 = [u_1^0,\\hdots ,u_{A_c}^0]^T$ is the nominal input vector shared amongst communicative agents, $\\mathcal {U}_{\\mathcal {A}_c} = \\bigoplus _{j=1}^{A_c}\\mathcal {U}_j$ is the Minkowski sum of their input constraint sets, () denotes the $c_I \\ge 0$ individual CBF constraints for agent $j$ (e.g.",
"speed), and () represents combinations of safety constraints between agents (e.g.",
"collision avoidance), where $a_{s,jk} = L_{f_j}h_s + L_{f_k}h_s + \\alpha _s(h_s)$ , $b_{s,j} = L_{g_j}h_s$ , and $b_{s,k} = L_{g_k}h_s$ .",
"When all agents are communicative, (REF ) is guaranteed to be safe provided that it is feasible.",
"A challenge when it comes to both (REF ) and (REF ) is in satisfying all of the safety constraints simultaneously, especially when it comes to the design of $\\alpha _s$ .",
"In some recent works, authors have proposed setting $\\alpha _s(h_s)=p_sh_s$ and including the parameters $p_s$ as decision variables in the QP [29] (and thus an additional term $\\sum _{s=1}^c\\frac{1}{2}q_sp_s^2$ for $q_s>0$ in the objective function), but the performance of these approaches are still heavily dependent on the gains $q_s$ .",
"Other techniques have avoided the issue of multiple candidate CBFs by assuming that only one constraint is in need of satisfaction at once [15] or by synthesizing a single non-smooth candidate CBF [16], [17], both of which may lead to undesirable chattering behavior or the loss of existence and uniqueness of solutions.",
"We seek to address this open problem, and require the following assumption to do so.",
"Assumption 1 The intersection of the safe sets $S_s$ for all $s \\in [1..c]$ is non-empty, i.e.",
"$S = \\bigcap _{s=1}^cS_s \\ne \\emptyset $ .",
"Problem 1 Given that Assumption REF holds for a collection of $c>1$ candidate control barrier functions $h_s$ corresponding to safe sets $S_s$ , design a consolidated control barrier function candidate $H: \\operatorname{\\mathbb {R}}^N \\times \\operatorname{\\mathbb {R}}_+^c \\rightarrow \\operatorname{\\mathbb {R}}$ with constituent gains $k = [k_1,\\hdots ,k_c]^T \\in \\operatorname{\\mathbb {R}}_+^c$ for the zero super-level set $C(k) = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; H(x, k) \\ge 0\\rbrace $ such that $C(k) \\subseteq S$ for all $k$ satisfying $0 < k_s < \\infty $ , $\\forall s \\in [1..c]$ ."
],
[
"Consolidated CBF based Control",
"In this section, we first introduce our proposed solution to Problem REF , a consolidated control barrier function (C-CBF) candidate that smoothly synthesizes multiple candidate CBFs into one, and then design a parameter adaptation law which renders the candidate C-CBF valid for safe control design."
],
[
"Consolidated CBFs",
"Let the vector of $c>1$ candidate CBFs evaluated at a given state $x$ be denoted $h(x) = [h_1(x) \\; \\hdots \\; h_c(x)]^T \\in \\operatorname{\\mathbb {R}}^c$ , and define a gain vector as $k = [k_1 \\; \\hdots \\; k_c]^T \\in \\operatorname{\\mathbb {R}}^c$ , where $0< k_s < \\infty $ for all $s \\in [1..c]$ .",
"Our C-CBF candidate $H: \\operatorname{\\mathbb {R}}^N \\times \\operatorname{\\mathbb {R}}_+^c \\rightarrow \\operatorname{\\mathbb {R}}$ is the following: $H(x, k) = 1 - \\sum _{s=1}^c\\phi \\Big (h_s(x), k_s\\Big ),$ where $\\phi : \\operatorname{\\mathbb {R}}_{\\ge 0} \\times \\operatorname{\\mathbb {R}}_{\\ge 0} \\rightarrow \\operatorname{\\mathbb {R}}_+$ belongs to class $\\mathcal {L}\\mathcal {L}$ , is continuously differentiable, and satisfies $\\phi (h_s,0)=\\phi (0,k_s)=\\phi (0,0)=1$ .",
"For example, the decaying exponential function, i.e.",
"$\\phi (h_s,k_s)=e^{-h_sk_s}$ , satisfies these requirements over the domain $\\operatorname{\\mathbb {R}}_{\\ge 0} \\times \\operatorname{\\mathbb {R}}_{\\ge 0}$ .",
"With $\\phi $ possessing these properties, it follows then that the new zero super level-set $C(k) = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; H(x, k) \\ge 0\\rbrace $ is a subset of $S$ (i.e.",
"$C(k \\subset S$ ), where the level of closeness of $C(k)$ to $S$ depends on the choices of gains $k$ .",
"This may be confirmed by observing that if any $h_s(x) = 0$ then $H(x) \\le 1 - 1 - \\sum _{j=1, j\\ne s}^c \\phi (h_j(x), k_j) < 0$ , and thus for $H(x) \\ge 0$ it must hold that $h_s(x) > 0$ , for all $s \\in [1..c]$ .",
"As such, $H$ defined by (REF ) is a solution to Problem REF , i.e.",
"$H$ is a C-CBF candidate.",
"This implies via Lemma REF that if $H$ is valid over the set $C(k)$ , then $C(k)$ is controlled invariant and thus the trajectories of (REF ) remain safe with respect to each constituent safe set $S_s$ , $\\forall s \\in [1..c]$ .",
"By Definition REF , for a static gain vector (i.e.",
"$\\dot{k} = \\mathbf {0}_{c \\times 1}$ ) the function $H$ is a CBF on the set $S$ if there exists $\\alpha _H \\in \\mathcal {K}_\\infty $ such that the following condition holds for all $x \\in S \\supset C(k)$ : $L_FH(x, k) + L_GH(x, k)u \\ge -\\alpha _H(H(x, k)),$ where from (REF ) it follows that $L_FH(x) &= -\\sum _{s=1}^c\\frac{\\partial \\phi }{\\partial h_s}L_Fh_c(x), \\\\L_GH(x) &= -\\sum _{s=1}^c\\frac{\\partial \\phi }{\\partial h_s}L_Gh_c(x) .$ Again taking $\\phi (h_s, k_s) = e^{-h_sk_s}$ as an example, we obtain that $\\frac{\\partial \\phi }{\\partial h_s} = -k_se^{-h_sk_s}$ , in which case it is evident that the role of the gain vector $k$ is to weight the constituent candidate CBFs $h_s$ and their derivative terms $L_Fh_s$ and $L_Gh_s$ in the CBF condition (REF ).",
"Thus, a higher value $k_s$ indicates a weaker weight in the CBF dynamics, as the exponential decay overpowers the linear growth.",
"Due to the combinatorial nature of these gains, for an arbitrary $k$ there may exist some $x \\in C(k)$ such that $L_GH(x) = \\mathbf {0}_{1\\times M}$ , which may violate (REF ) and lead to the state exiting $C(k)$ (and potentially $S$ as a result).",
"Using online adaptation of $k$ , however, it may be possible to achieve $L_GH(x) \\ne \\mathbf {0}_{1\\times M}$ for all $t \\ge 0$ , which motivates the following problem.",
"Problem 2 Given a C-CBF candidate $H: \\operatorname{\\mathbb {R}}^N \\times \\operatorname{\\mathbb {R}}_+^c \\rightarrow \\operatorname{\\mathbb {R}}$ defined by (REF ) and associated with the set $C(k)$ , design an adaptation law $\\dot{k} = \\kappa (x, k)$ such that $L_GH \\ne \\mathbf {0}_{1\\times M}$ for all $t \\ge 0$ ."
],
[
"Adaptation for Control Synthesis",
"Before proceeding with our main result, we require the following assumption.",
"Assumption 2 The matrix of controlled candidate CBF dynamics $L_g \\in \\operatorname{\\mathbb {R}}^{c \\times M}$ is not all zero, i.e.",
"$L_g = \\begin{bmatrix} L_gh_1 \\\\ \\vdots \\\\ L_gh_c \\end{bmatrix} \\ne \\mathbf {0}_{c \\times M}.$ We now present our main result, an adaptation law that solves Problem REF and thus renders $H$ a valid CBF for the set $C(k(t))$ , for all $t \\ge 0$ .",
"Theorem 1 Suppose that there exist $c>1$ candidate CBFs $h_s: \\operatorname{\\mathbb {R}}^N \\rightarrow \\operatorname{\\mathbb {R}}$ defining sets $S_s = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; h_s(x) \\ge 0\\rbrace $ , $\\forall s \\in [1..c]$ , and that it is known that $\\mathcal {U}=\\operatorname{\\mathbb {R}}^M$ .",
"If $k(0)$ is such that $L_GH \\ne \\mathbf {0}_{1 \\times M}$ at $t=0$ , then, under the ensuing adaptation law, $\\kappa (x, k) = \\operatornamewithlimits{arg\\,min}_{\\mu \\in \\operatorname{\\mathbb {R}}^c} \\; \\frac{1}{2}(\\mu - \\mu _0)^TP&(\\mu - \\mu _0) \\\\\\mathrm {s.t.}",
"\\quad \\quad \\quad &\\nonumber \\\\\\mu + \\alpha _k(k-k_{min}) &\\ge 0, \\\\p^TQ\\dot{p} + p^T\\dot{Q}p + \\alpha _{p}(h_{p}) &\\ge 0, $ the controlled CBF dynamics $L_GH \\ne \\mathbf {0}_{1 \\times M}$ for all $t \\ge 0$ , and thus the function $H$ defined by (REF ) is a valid CBF for the set $C(k(t)) = \\lbrace x \\in \\operatorname{\\mathbb {R}}^N \\; | \\; H(x, k) \\ge 0\\rbrace $ , for all $t \\ge 0$ , where $P \\in \\operatorname{\\mathbb {R}}^{c \\times c}$ is a positive-definite gain matrix, $\\alpha _k, \\alpha _p \\in \\mathcal {K}_\\infty $ , $\\mu _0$ is the nominal $\\dot{k}$ , $k_{min} = [k_{1,min},\\hdots ,k_{c,min}]^T$ is the vector of minimum allowable values $k_{s,min} > 0$ , and $p &= \\left[\\frac{\\partial \\phi }{\\partial h_1} \\; \\hdots \\; \\frac{\\partial \\phi }{\\partial h_c}\\right]^T, \\\\Q &= I - (NN^T)^T - NN^T - (NN^T)^TNN^T $ with $h_p = \\frac{1}{2}p^TQp - \\varepsilon $ , $\\varepsilon > 0$ , and $N = [n_1 \\; \\hdots \\; n_r],$ such that $\\lbrace n_1,\\hdots ,n_r\\rbrace $ constitutes a basis for the null space of $L_g^T$ , i.e.",
"$\\mathcal {N}(L_g^T) = \\mathrm {span}\\lbrace n_1,\\hdots ,n_r\\rbrace $ , where $L_g$ is given by (REF ).",
"First, given (REF ), we have that $\\dot{H} &= -\\sum _{s=1}^c\\left(\\frac{\\partial \\phi }{\\partial h_s}\\dot{h}_s + \\frac{\\partial \\phi }{\\partial k_s}\\dot{k}_c\\right) \\nonumber \\\\&= p^T \\dot{h} + q^T \\dot{k} \\nonumber \\\\&= p^T (L_f + L_gu) + q^T \\dot{k} \\nonumber $ where $p$ is given by (REF ), $L_g$ by (REF ), $L_f = [L_Fh_1 \\; \\hdots \\; L_Fh_c]^T$ , and $q = [\\frac{\\partial \\phi }{\\partial k_1} \\; \\hdots \\; \\frac{\\partial \\phi }{\\partial k_c}]^T$ .",
"As such, $L_FH = p^TL_f + q^T\\dot{k}$ and $L_GH = p^TL_g$ .",
"With $\\mathcal {U} = \\operatorname{\\mathbb {R}}^M$ , it follows that as long as $L_GH \\ne \\mathbf {0}_{1 \\times M}$ it is possible to choose $u$ such that $\\dot{H}(x,u) \\ge -\\alpha _H(H)$ .",
"We will now show that with $\\dot{k} = \\kappa (x, k)$ given by (REF ) it holds that $L_GH \\ne \\mathbf {0}_{1 \\times M}$ and thus $H$ is a CBF for $C(k(t))$ , for all $t \\ge 0$ .",
"Since $L_GH = p^TL_g$ , the problem of showing that $L_GH \\ne \\mathbf {0}_{1\\times M}$ is equivalent to proving that $p \\notin \\mathcal {N}(L_g^T) = \\mathrm {span}\\lbrace n_1,\\hdots ,n_r\\rbrace $ .",
"Since the vector $p$ can be expressed as a sum of vectors perpendicular to and parallel to $\\mathcal {N}(L_g^T)$ (respectively $p^\\perp $ and $p^\\parallel $ ), it follows that $p \\notin \\mathcal {N}(L_g^T)$ as long as $\\Vert p^\\perp \\Vert >0$ , where $p^\\perp = \\left(I - NN^T\\right)p$ by vector projection, and $N$ is given by (REF ).",
"Thus, a sufficient condition for $p \\notin \\mathcal {N}(L_g^T)$ is that $\\frac{1}{2}\\Vert (I-NN^T)p\\Vert ^2 = \\frac{1}{2}p^TQp > \\varepsilon $ for some $\\varepsilon >0$ , where $Q$ is given by ().",
"Then, by defining a function $h_p = \\frac{1}{2}p^TQp - \\varepsilon $ , it follows from (REF ) that when (REF ) is true at $t=0$ , it is true $\\forall t \\ge 0$ as long as () holds.",
"Therefore, gains $k$ adapted according to the law (REF ) are guaranteed to result in $L_GH \\ne \\mathbf {0}_{1 \\times M}$ .",
"Thus, $H$ is a CBF for the set $C(k(t))$ , for all $t \\ge 0$ .",
"This completes the proof.",
"Remark 1 With $Q$ depending on basis vectors spanning $\\mathcal {N}(L_g^T)$ , it is not immediately obvious under what conditions $\\dot{Q}$ is continuous (or even well-defined).",
"Prior results show that if the rank of $\\mathcal {N}(L_g^T)$ is constant then $\\dot{Q}$ varies continuously $\\forall x \\in B_{\\epsilon }(x)$ [30], but analytical derivations of $\\dot{Q}$ are not available to the best of our knowledge.",
"In practice, we observe that the rank of $\\mathcal {N}(L_g^T)$ is indeed constant, and we approximate $\\dot{Q}$ numerically using finite-difference methods.",
"With $H$ consolidating the many constituent constraints into one CBF condition, we can then replace the centralized CBF-QP controller (REF ) with the following: $u_{\\mathcal {A}_c}^* &= \\operatornamewithlimits{arg\\,min}_{u_{\\mathcal {A}_c} \\in \\mathcal {U}_{\\mathcal {A}_c}} \\frac{1}{2}\\Vert u_{\\mathcal {A}_c}-u_{\\mathcal {A}_c}^0\\Vert ^2 \\\\&\\quad \\quad \\textrm {s.t.}",
"\\nonumber \\\\a &+bu_{\\mathcal {A}_c} \\ge 0, $ where $a = L_FH + \\alpha _H(H)$ and $b = L_GH_{[i \\in \\mathcal {A}_c]}$ .",
"If all agents are communicative, i.e.",
"$\\mathcal {A}_c = \\mathcal {A}$ , then since $H$ is a CBF for the set $C(k(t)) \\subset S$ , for all $t \\ge 0$ , the system trajectories are guaranteed to stay within $C(k(t)) \\subset S$ and thus remain safe.",
"In the presence of non-communicative agents, we replace the decentralized CBF-QP controller (REF ) with $u_i^* &= \\operatornamewithlimits{arg\\,min}_{u_i \\in \\mathcal {U}_{i}} \\frac{1}{2}\\Vert u_{i}-u_{i}^0\\Vert ^2 \\\\&\\textrm {s.t.}",
"\\nonumber \\\\a &+ b_iu_i \\ge d, $ where $b_i = L_GH_{[mi:m(i+1)]}$ , i.e.",
"the portion of the dynamics of $H$ that agent $i$ controls, and $d = e^{-rH}\\max _{u \\in \\mathcal {U}}\\sum _{j=1,j \\ne i}^AL_GH_{[jm:j(m+1)]}u_j$ , where $r>0$ .",
"While for the case of unbounded control authority $d$ is similarly unbounded, in practice it is reasonable to assume that agents have limited control authority and thus to use (REF ) assuming some bounded $\\mathcal {U}$ ."
],
[
"Multi-Robot Numerical Study",
"In this section, we demonstrate our C-CBF controller on a decentralized multi-robot goal-reaching problem.",
"Consider a collection of 3 non-communicative, but responsive robots ($i \\in \\mathcal {A}_n \\setminus \\mathcal {A}_{n,n}$ ) in a warehouse environment seeking to traverse a narrow corridor intersected by a passageway occupied with 6 non-responsive agents ($j \\in \\mathcal {A}_{n,n}$ ).",
"The non-responsive agents may be e.g.",
"humans walking or some other dynamic obstacles.",
"Let $\\mathcal {F}$ be an inertial frame with a point $s_0$ denoting its origin, and assume that each robot may be modeled according to a dynamic extension of the kinematic bicycle model described by [31], provided here for completeness: $\\dot{x}_i &= v_{i}\\left(\\cos {\\psi _i} - \\sin {\\psi _i}\\tan {\\beta _i}\\right) \\\\\\dot{y}_i &= v_{i}\\left(\\sin {\\psi _i} + \\cos {\\psi _i}\\tan {\\beta _i}\\right) \\\\\\dot{\\psi }_i &= \\frac{v_{i}}{l_r}\\tan {\\beta _i} \\\\\\dot{\\beta }_i &= \\omega _i \\\\\\dot{v}_{i} &= a_{i},$ where $x_i$ and $y_i$ denote the position (in m) of the center of gravity (c.g.)",
"of the ith robot with respect to $s_0$ , $\\psi _i$ is the orientation (in rad) of its body-fixed frame, $\\mathcal {B}_i$ , with respect to $\\mathcal {F}$ , $\\beta _i$ is the slip angle$\\beta _i$ is related to the steering angle $\\delta _i$ via $\\tan {\\beta _i} = \\frac{l_r}{l_r+l_f}\\tan {\\delta _i}$ , where $l_f+l_r$ is the wheelbase with $l_f$ (resp.",
"$l_r$ ) the distance from the c.g.",
"to the center of the front (resp.",
"rear) wheel.",
"(in rad) of the c.g.",
"of the vehicle relative to $\\mathcal {B}_i$ (assume $|\\beta _i|<\\frac{\\pi }{2}$ ), and $v_{i}$ is the velocity of the rear wheel with respect to $\\mathcal {F}$ .",
"The state of robot $i$ is denoted $z_i = [x_i \\; y_i \\; \\psi _i \\; \\beta _i \\; v_{i}]^T$ , and its control input is $u_i=[a_{i} \\; \\omega _i]^T$ , where $a_{i}$ is the acceleration of the rear wheel (in m/s$^2$ ), and $\\omega _i$ is the angular velocity (in rad/s) of $\\beta _i$ .",
"The challenges of this scenario relate to preserving safety despite multiple non-communicative and non-responsive agents present in a constrained environment.",
"A robot is safe if it 1) obeys the speed restriction, 2) remains inside the corridor area, and 3) avoids collisions with all other robots.",
"Speed is addressed with the following candidate CBF: $h_{v}(z_i) &= s_M - v_i, $ where $s_M > 0$ , while for corridor safety and collision avoidance we used forms of the relaxed future-focused CBF introduced in [11] for roadway intersections, namely $\\begin{split}h_{c}(z_i) &= (m_L(x_i + \\dot{x}_i) + b_L - (y_i + \\dot{y}_i)) \\cdot \\\\ &\\quad \\; (m_R(x_i + \\dot{x}_i) + b_R - (y_i + \\dot{y}_i))\\end{split}\\\\\\begin{split}h_{r}(z_i,z_j) &= D(z_i,z_j,t + \\hat{\\tau })^2 \\\\ & \\quad + \\epsilon D(z_i,z_j,t)^2 - (1 + \\epsilon )(2R)^2,\\end{split}$ where (REF ) prevents collisions with the corridor walls (defined as lines in the $xy$ -plane via $m_L,b_L,m_R,b_R \\in \\operatorname{\\mathbb {R}}$ ), and () prevents inter-robot collisions and is defined $\\forall i \\in \\mathcal {A}_n \\setminus \\mathcal {A}_{n,n}$ , $\\forall j \\in \\mathcal {A}_n$ , where $\\epsilon >0$ , $D(z_i, z_j, t_a)$ is the Euclidean distance between agents $i$ and $j$ at arbitrary time $t_a$ , and $\\hat{\\tau }$ denotes the time in the interval $[0,T]$ at which the minimum inter-agent distance will occur under constant velocity future trajectories.",
"For a more detailed discussion on future-focused CBFs, see [11].",
"As such, (REF ), (REF ), and () define the sets $S_{v,i} &= \\lbrace z_i \\in \\operatorname{\\mathbb {R}}^n \\; | \\; h_{v}(z_i) \\ge 0\\rbrace , \\nonumber \\\\S_{c,i} &= \\lbrace z_i \\in \\operatorname{\\mathbb {R}}^n \\; | \\; h_{c}(z_i) \\ge 0\\rbrace , \\nonumber \\\\S_{r,i} &= \\bigcap \\limits _{j=1,j\\ne i}^A \\lbrace z \\in \\operatorname{\\mathbb {R}}^N \\; | \\; h_{r}(z_i, z_j) \\ge 0\\rbrace , \\nonumber $ the intersection of which constitutes the safe set for agents $i$ , i.e.",
"$S_i(t) = S_{v,i} \\cap S_{c,i} \\cap S_{r,i}$ .",
"We control robots $i \\in \\mathcal {A}_n \\setminus \\mathcal {A}_{n,n}$ using a C-CBF based decentralized controller of the form (REF ) with constituent functions $h_c$ , $h_s$ , $h_r$ , an LQR based nominal control input (see [11]), and initial gains $k(0) = \\mathbf {1}_{10 \\times 1}$ .",
"The non-responsive agents used a similar LQR controller to move through the passageway in pairs of two, with the first two pairs passing through the intersection without stopping and the last pair stopping at the intersection before proceeding.",
"As shown in Figure REF , the non-communicative robots traverse both the narrow corridor and the busy intersection to reach their goal locations safely.",
"The trajectories of the gains $k$ for each warehouse robot are shown in Figure REF , while their control inputs are depicted in Figure REF .",
"The CBF time histories for the constituent and consolidated functions are highlighted in Figures REF and REF respectively, and show that the C-CBF controllers maintained safety at all times.",
"Figure: XY paths for the warehouse robots (blue) and non-responsive agents (red) in the warehouse control problem.Figure: Gains kk for the C-CBF controllers in the warehouse study.",
"Robot 1 denoted with solid lines, dotted for robot 2, dash-dots for robot 3.",
"AgentA and AgentB denote the other two non-communicative robots from the perspective of one (e.g.",
"AgentA=Agent1 and AgentB=Agent3 for robot 2).Figure: Warehouse robot controls: accel.",
"(aa) and slip angle rate (ω\\omega ).Figure: Evolution of warehouse robot constituent CBF candidates, h s h_s ∀s∈[1..c]\\forall s \\in [1..c], synthesized to construct C-CBF.Figure: Evolution of C-CBF HH for warehouse robots 1, 2, and 3."
],
[
"Experimental Case Study",
"For experimental validation of our approach, we used an AION R1 UGV ground rover as an ego vehicle in the laboratory setting and required it to reach a goal location in the presence of two non-responsive rovers: one static and one dynamic.",
"We modeled the rovers as bicycles using (), and sent angular rate $\\omega _i$ and velocity $v_i$ (numerically integrated based on the controller's acceleration output) commands to the rovers' on-board PID controllers.",
"The ego rover used our proposed C-CBF (REF ) with constituent candidate CBFs (REF ) (with $s_M = 1$ m/s) and the rff-CBF defined in () for collision avoidance.",
"The nominal input to the C-CBF controller was the LQR law from the warehouse robot example, as was the controller used by the dynamic non-responsive rover.",
"A Vicon motion capture system was used for position feedback, and the state estimation was performed by extended Kalman filter via the on-board PX4.",
"Figure: A rover avoids a static and dynamic rover using our proposed C-CBF controller en route to a target in the laboratory setting.For the setup, the static rover was placed directly between the ego rover and its goal, while the dynamic rover was stationary until suddenly moving across the ego's path as it approached its target.",
"As highlighted in Figure REF , the ego rover first headed away from the static rover and then decelerated and swerved to avoid a collision with the second rover before correcting course and reaching its goal.",
"Videos and code for both this experiment and the simulation in Section is available on GithubLink to Github repo: github.com/6lackmitchell/CCBF-Control."
],
[
"Conclusion",
"In this paper, we addressed the problem of safe control under multiple state constraints via a C-CBF based control design.",
"To ensure that the synthesized C-CBF is valid, we introduced a parameter adaptation law on the weights of the C-CBF constituent functions and proved that the resulting controller is safe.",
"We then demonstrated the success of our approach on a multi-robot control problem in a crowded warehouse environment, and further validated our work on a ground rover experiment in the lab.",
"In the future, we plan to explore conditions under which the C-CBF approach may preserve guarantees in the presence of input constraints, including whether alternative adaptation laws for the weights assist in guarantees of liveness in addition to safety."
]
] | 2209.08170 |
[
[
"Report of the Topical Group on Cosmic Probes of Dark Matter for Snowmass\n 2021"
],
[
"Abstract Cosmological and astrophysical observations currently provide the only robust, positive evidence for dark matter.",
"Cosmic probes of dark matter, which seek to determine the fundamental properties of dark matter through observations of the cosmos, have emerged as a promising means to reveal the nature of dark matter.",
"This report summarizes the current status and future potential of cosmic probes to inform our understanding of the fundamental nature of dark matter in the coming decade."
],
[
"Cosmic Probes of Dark Matter",
"Conveners: Alex Drlica-Wagner, Chanda Prescod-Weinstein, Hai-Bo Yu Contributors: Andrea Albert, Arka Banerjee, Masha Baryakhtar, Keith Bechtol, Simeon Bird, Simon Birrer, Torsten Bringmann, Regina Caputo, Sukanya Chakrabarti, Thomas Y. Chen, Djuna Croon, Francis-Yan Cyr-Racine, William A. Dawson, Cora Dvorkin, Vera Gluscevic, Daniel Gilman, Daniel Grin, Renée Hložek, Rebecca K. Leane, Ting S. Li, Yao-Yuan Mao, Joel Meyers, Siddharth Mishra-Sharma, Julian B. Muñoz, Ferah Munshi, Ethan O. Nadler, Aditya Parikh, Kerstin Perez, Annika H. G. Peter, Stefano Profumo, Katelin Schutz, Neelima Sehgal, Joshua D. Simon, Kuver Sinha, Monica Valluri, Risa H. Wechsler Abstract Cosmological and astrophysical observations currently provide the the only robust, positive evidence for dark matter.",
"Cosmic probes of dark matter, which seek to determine the fundamental properties of dark matter through observations of the cosmos, have emerged as a promising means to reveal the nature of dark matter.",
"This report summarizes the current status and future potential of cosmic probes to inform our understanding of the fundamental nature of dark matter in the coming decade."
],
[
"Executive Summary",
"The existence of dark matter, which constitutes $ {\\sim }\\, 85\\%$ of the matter density and $ {\\sim }\\, 26\\%$ of the total energy density of the universe, is a clear demonstration that we lack a complete understanding of fundamental physics.",
"The importance, impact, and interdisciplinary nature of the dark matter problem make it one of the most exciting questions in science today.",
"In this report, we explain how cosmic probes of dark matter, which determine microscopic properties of dark matter through cosmic observations on macroscopic scales, have emerged as one of the most promising ways to unveil the nature of dark matter.",
"The HEP community must seize the opportunity to fully realize the potential of cosmic probes of dark matter and broaden its approach to the dark matter problem accordingly in the coming decade.",
"Cosmological and astrophysical observations currently provide the only robust, empirical evidence for dark matter.",
"These observations, together with numerical simulations of cosmic structure formation, have established the cold dark matter (CDM) paradigm and lay the groundwork for all other experimental efforts to characterize the fundamental properties of dark matter.",
"In the coming decade, we will measure the distribution of dark matter in the cosmos with unprecedented precision.",
"In particular, we are at the threshold of definitively testing the predictions of CDM on galactic and sub-galactic scales, and any observed deviation would revolutionize our understanding of the fundamental nature of dark matter.",
"Furthermore, by accessing extreme scales and environments, cosmic probes are sensitive to very rare interactions between dark matter and normal matter that are inaccessible in conventional dark matter searches.",
"Cosmic measurements of dark matter properties are entering the precision era.",
"This report summarizes the ways in which cosmic probes have emerged as a new field in the endeavor to measure the fundamental, microscopic properties of dark matter.",
"We first identify three core HEP community priorities for cosmic probes of dark matter: [nosep] Current/near-future HEP cosmology experiments have direct sensitivity to dark matter particle physics [1], [2], [3].",
"Cosmological studies of dark matter should be supported as a key component of the HEP Cosmic Frontier program due to their unique ability to probe dark matter microphysics and link the results of terrestrial dark matter experiments to cosmological measurements.",
"The construction of future cosmology experiments is critical for expanding our understanding of dark matter physics.",
"Proposed facilities across the electromagnetic spectrum, as well as gravitational waves, can provide sensitivity to dark matter physics [4].",
"The HEP community should make strategic investments in the design, construction, and operation of these facilities in order to maximize their sensitivity to dark matter physics.",
"Cosmic probes provide robust sensitivity to the microphysical properties of dark matter due to enormous progress in theoretical modeling, numerical simulations, and astrophysical observations.",
"Theory, simulation, observation, and experiment must be supported together to maximize the efficacy of cosmic probes of dark matter physics.",
"We have identified the following major scientific opportunities for cosmic probes of dark matter physics in the coming decade.",
"These opportunities are summarized below, presented at length in the subsequent sections of this report, and discussed in detail in a set of community white papers [5], [6], [7], [8], [4], [1], [2], [3], [9], [10], [11].",
"Major Scientific Opportunities The Standard Model of particle physics and cosmology can be tested at unprecedented levels of precision by measuring the cosmic distribution of dark matter.",
"These measurements span an enormous range of scales from the observable universe to sub-stellar-mass systems (e.g., the matter power spectrum, the mass spectrum of dark matter halos, dark matter halo density profiles, and abundances of compact objects) [5], [7], [12].",
"Novel particle properties of dark matter (e.g., self-interactions, quantum wave features, tight couplings with radiation) can lead to observable signatures in the distribution of dark matter that differ from the CDM prediction.",
"The HEP community should support measurements of the dark matter distribution as a key element of its Cosmic Frontier program.",
"The CDM model makes the strong, testable prediction that the mass spectrum of dark matter halos extends below the threshold at which galaxies form, $\\mathcal {O}(10^7)~M_{\\odot }$ [5].",
"Sub-galactic dark matter halos are less influenced by baryonic processes making them especially clean probes of fundamental physics of dark matter.",
"We are on the cusp of detecting dark matter halos that are devoid of visible stars through several cosmic probes (e.g., strong lensing and the dynamics of stars around the Milky Way).",
"The HEP community should pursue the detection of dark matter halos below the threshold of galaxy formation as an exceptional test of fundamental dark matter properties.",
"Extreme astrophysical environments provide unique opportunities to explore dark matter couplings to the Standard Model that are inaccessible with conventional dark matter search experiments and span ${\\sim }50$ orders of magnitude in dark matter particle mass [8].",
"Instruments, observations, and theorists that study extreme astrophysical environments should be supported as an essential way to explore the expanding landscape of dark matter models.",
"Numerical simulations of structure formation and baryonic physics are critical to robustly interpret observational results and disentangle new dark matter physics from astrophysical effects [6].",
"Simulations are thus essential to address particle physics questions about the nature of dark matter.",
"HEP computational expertise and resources should be combined with astrophysical simulation expertise to rapidly advance numerical simulations of dark matter physics.",
"The interdisciplinary nature of cosmic probes of dark matter calls for innovative new ways to support the pursuit of scientific opportunities across traditional disciplinary boundaries.",
"Sustained collaboration between particle theorists, gravitational dynamicists, numerical simulators, observers, and experimentalists is required to fully realize the power of cosmic probes of dark matter.",
"Large experimental collaborations are naturally matched to the HEP community, but new mechanisms are also needed to support these emerging, interdisciplinary efforts.",
"These major opportunities represent pathways for transforming our understanding of dark matter.",
"We observe that the trend in dark matter research has changed profoundly in the last 10 years.",
"In particular, particle physicists and astrophysicists have collaborated to make a series of breakthroughs in studying dark matter theories beyond the CDM paradigm.",
"For example, the idea of dark sectors, i.e., that dark matter may reside in its own sector and carry its own forces, has been firmly established, together with associated theoretical tools.",
"Such a dark force could operate at the most fundamental level with a range of $\\mathcal {O}(10^{-12})~{\\rm cm}$ , but it would change the dark matter distribution within $\\mathcal {O}(10^{22})~{\\rm cm}$ in galactic halos.",
"Simulators can now perform cosmological hydrodynamical simulations of structure formation that account for the particle physics properties of dark matter, such as self-interactions [13], [14], [15], quantum wave features [16], and tight couplings with radiation [17].",
"These studies set the basis for quantifying astrophysical uncertainties and disentangling dark matter physics from baryon physics, a key step for extracting the Lagrangian parameters that describe a particular dark matter model.",
"Advances since the last Snowmass study combined with the scientific outlook assembled in this Snowmass study motivate the possibility that cosmic probes may result in a transformational change in our understanding of dark matter in the coming decade.",
"On the experimental side, new observational facilities that directly measure dark matter in the cosmos will be taking data within the next decade, see [4] for details.",
"For example, the DOE- and NSF-supported Rubin Observatory Legacy Survey of Space and Time (LSST) is scheduled to start full operations in 2024 [18].",
"The primary HEP interest in Rubin LSST is the study of dark energy; however, it has enormous potential to discover new physics beyond CDM [19], [2].",
"Rubin LSST will observe the faintest satellite galaxies of the Milky Way, stellar streams, and strong lens systems to detect the smallest dark matter halos, thereby probing the minimum mass of ultra-light dark matter and thermal warm dark matter.",
"Gravitational lensing will provide precise measurements of the densities and shapes of dark matter halos, which are sensitive probes of interactions in the dark sector.",
"Microlensing measurements will directly probe primordial black holes and the compact object fraction of dark matter at the sub-percent level over a wide range of masses.",
"On a longer time horizon, the CMB Stage-4 (CMB-S4) project [20], [21], another cosmology experiment supported by DOE and NSF, has access to rich dark matter physics as well [3].",
"For instance, detection of additional relativistic degrees of freedom in the early universe would immediately imply the existence of a dark sector.",
"The HEP community strongly supports the inclusion of dark matter physics in the research programs of these experiments alongside studies of dark energy and inflation.",
"Since cosmic probes of dark matter are multidisciplinary and interdisciplinary, we must develop new ways to cultivate and sustain collaboration, as well as new approaches to mentoring the next-generation of scientists.",
"A program like the DOE Dark Matter New Initiatives is well-suited to support a small-scale collaborative effort from particle physicists and astrophysicists with a well-defined scientific goal.",
"Within upcoming HEP cosmology experiments such as Rubin LSST and CMB-S4, new effort is needed to assemble collaborative teams to study cosmic probes of dark matter physics.",
"Additional support must be provided to enable dark matter as a parallel branch of fundamental physics being pursued by these experiments.",
"Such a program will fully leverage the unprecedented capabilities of these facilities [2].",
"On large scales, the construction of future cosmology experiments is critical for expanding our understanding of dark matter physics.",
"HEP involvement will be essential for the design and construction of these facilities [4], and dark matter physics should be a core component of their scientific missions.",
"Identifying the nature of dark matter is one of the most important tasks in science.",
"Cosmic probes are the most “expansive” (and may be the only viable) approach to understanding fundamental properties of dark matter, because they are valid even if the coupling between dark matter and normal matter is extremely weak, even as weak as gravity.",
"With new observational facilities coming online, plans for future facilities emerging, and tremendous progress being made in numerical simulations and semi-analytical modeling, we are entering a transformative decade in the effort to measure the microscopic properties of dark matter from cosmic observations.",
"In this report, we present the possibilities created by a community commitment to a decade of dark matter."
],
[
"Introduction",
"Over the past several decades, experimental searches for non-baryonic dark matter have proceeded along several complementary avenues.",
"Collider experiments attempt to produce and detect the presence of dark matter particles, while direct detection experiments attempt to measure energy deposition from very rare interactions between dark matter and Standard Model particles.",
"In parallel, indirect dark matter searches seek to detect the energetic Standard Model products from the annihilation or decay of dark matter particles in situ in astrophysical systems.",
"Despite these extensive efforts, the only positive measurements of dark matter to date come from astrophysical and cosmological observations.",
"This report summarizes the exciting scientific opportunities presented by cosmic probes of fundamental dark matter physics in the coming decade, as highlighted in Fig.",
"REF schematically.",
"The content of this report has been primarily guided by five solicited white papers [5], [6], [7], [8], [4] and six contributed white papers from the HEP community [1], [2], [3], [9], [10], [11].",
"Astrophysical and cosmological observations are a unique, powerful, and complementary technique to study the fundamental properties of dark matter.",
"They probe dark matter directly through gravity, the only force to which dark matter is known to couple.",
"On large cosmological scales, current observational data can be described by a simple cosmological model containing stable, non-relativistic, collisionless, cold dark matter.",
"However, many viable theoretical models of dark matter predict observable deviations from CDM that are testable with current and future experimental programs.",
"Fundamental physical properties of dark matter—e.g., particle mass, time evolution, self-interaction cross section, and coupling to the Standard Model or other dark sector particles—can imprint themselves on the macroscopic distribution of dark matter in a detectable manner.",
"In addition, astrophysical observations complement terrestrial dark matter searches by providing input to direct and indirect dark matter experiments, and by enabling alternative tests of any non-gravitational coupling(s) between dark matter and the Standard Model.",
"For example, astrophysical observations are required to (i) measure the local density and velocity distribution of dark matter, an important input for designing and interpreting direct dark matter searches, (ii) identify and characterize regions of high dark matter density, an important input for targeting and interpreting indirect searches, and (iii) set strong constraints on the particle properties of dark matter, an important input for designing novel terrestrial dark matter experiments with viable discovery potential.",
"In the event of a terrestrial dark matter detection—e.g., the detection of a weakly interacting massive particle (WIMP) or axion—cosmic observations will be crucial to interpret terrestrial measurements in the context of cosmic dark matter.",
"Furthermore, cosmic probes provide critical information to direct future terrestrial searches for novel dark matter candidates.",
"Finally, in many cases, astrophysical and cosmological observations provide the only robust constraints on the viable range of dark matter models.",
"There is also immense discovery potential at the intersection of particle physics, cosmology, and astrophysics.",
"The detection of dark matter halos that are completely devoid of visible galaxies would provide an extremely sensitive probe of new dark matter physics.",
"Measuring a deviation from the gravitational predictions of CDM in these halos would provide much-needed experimental guidance on dark matter properties that are not easily measured in particle physics experiments (e.g., dark matter self-interaction cross sections).",
"Likewise, results from terrestrial particle physics experiments can suggest specific deviations from the CDM paradigm that can be tested with astrophysical observations.",
"The expanding landscape of theoretical models for dark matter strongly motivates the exploration of dark matter parameter space beyond the current sensitivity of the HEP program.",
"In fact, cosmology has a long history of testing the fundamental properties of dark matter.",
"For instance, neutrinos were long considered a viable candidate to make up all the dark matter [25], [26].",
"The 30 eV neutrino dark matter candidate of the 1980s is an especially interesting case study of the interplay between particle physics experiments and astrophysical observations.",
"In 1980, Lubimov et al.",
"reported the discovery of a non-zero neutrino rest mass in the range $14\\,{\\rm eV} < m_{\\nu } < 46\\,{\\rm eV}$ [27].",
"Neutrinos with this mass would provide a significant fraction of the critical energy density of the universe, but would be relativistic at the time of decoupling, thus manifesting as hot dark matter.",
"Over the next decade, this “discovery” was aggressively tested by several other tritium $\\beta $ -decay experiments.",
"During this same period, the first measurements of the stellar velocity dispersion of dwarf spheroidal galaxies showed that these galaxies are highly dark matter dominated.",
"The inferred dark matter density within the central regions of dwarf galaxies was used to place lower limits on the neutrino rest mass that were incompatible with the 30 eV neutrino dark matter candidate [28], [29].",
"Furthermore, numerical simulations of structure formation demonstrated that large-scale structure observations are incompatible with a universe dominated by hot dark matter in the form of neutrinos [30].",
"Similar stories can be told of stable heavy leptons and other dark matter candidates that have been excluded by cosmological and astrophysical measurements [31], [32], [33].",
"Cosmology has continually shown that the microscopic physics governing the fundamental nature of dark matter and the macroscopic distribution of dark matter are intimately intertwined.",
"The strong connection between cosmology, astrophysics, and particle physics serve as the motivation for the Dark Matter: Cosmic Probes Topical Group (CF3) within the Snowmass Cosmic Frontier.",
"CF3 focuses on the use of cosmological techniques and astrophysical observations to study the fundamental nature of dark matter over the full range of allowed dark matter masses.",
"While many experimental studies of dark matter search for a previously undetected interactions between dark matter and Standard Model particles, CF3 also seeks to measure the behavior of dark matter ever more precisely in order to compare against the predictions of $\\Lambda {\\rm CDM}$ .",
"Thus, some of the scientific approaches and experimental facilities proposed by CF3 overlap significantly with cosmological studies of dark energy and the early universe.",
"CF3 discussions took place between November 2020 and July 2022 through a series of meetings that occurred on a roughly bi-weekly cadence.",
"CF3 received 75 letters of intent,https://snowmass21.org/cosmic/dm_probes which resulted in the coordinated submission of five solicited white papers: [nosep] WP1 - Dark matter physics from dark matter halo measurements [5].",
"WP2 - Cosmological simulations for dark matter physics [6].",
"WP3 - Primordial black hole dark matter [7].",
"WP4 - Dark matter in extreme astrophysical environments [8].",
"WP5 - Observational facilities to study dark matter physics [4].",
"These solicited white papers were complemented by six white papers that were contributed directly to CF3 [1], [2], [3], [9], [10], [11] and numerous related white papers that were contributed to other topical groups by the HEP community [12], [34], [35], [36], [24].",
"Furthermore, we present three cases that highlight how cosmic probes can play a central role in identifying fundamental properties of dark matter and/or providing complementary information for designing search strategies in terrestrial experiments.",
"This report summarizes nearly two years of community input, and its structure largely follows the CF3 solicited community white papers [5], [6], [7], [8], [4]."
],
[
"Dark Matter Halo Measurements",
"In the standard model of cosmic structure formation, dark matter in the late-time universe is clustered into gravitationally bound over-densities called halos.",
"These halos provide sites for baryons to cool, collapse, and form galaxies.",
"Astronomical observations show that dark matter halos are distributed according to a power-law mass spectrum extending from the scales of galaxy clusters ($ {\\sim }\\, 10^{15} M_\\odot $ ) to those of ultra-faint dwarf galaxies ($ {\\sim }\\, 10^8 M_\\odot $ ).",
"In the prevailing CDM theory, dark matter is made of collisionless, cold particles or compact objects.",
"The CDM theory does a good job of explaining the large-scale structure of the universe [46] and overall properties of galaxies [47], [48].",
"However, there are many reasons to believe that CDM is an approximation and that the dark sector is more complex and vibrant.",
"From the theory perspective, CDM provides a parametric description of cosmic structure, but it is far from a complete theory.",
"In CDM, the particle properties of dark matter, such as the mass, spin, interaction(s), and production mechanism(s), remain unspecified.",
"In fact, many theoretical models describing the particle physics of dark matter predict that the simplest CDM model breaks down at small physical scales [5].",
"On the observational side, CDM has faced long-standing challenges in explaining detailed measurements of dark matter distributions on galactic and sub-galactic scales [49], [50], where we are pushing the boundaries of both observations and numerical simulations.",
"In the next decade, observations of dark matter halos over a wide range of mass scales will provide unique opportunities to test the vast landscape of dark matter theories and potentially discover deviations from the predictions of CDM.",
"Using halo measurements to study dark matter physics has several advantages.",
"First, there is a strong connection between dark matter halos and the physics of the early universe.",
"The seeds of cosmological structure formation were established in the earliest moments after the Big Bang.",
"As we measure the distribution of dark matter across a broad range of physical scales, we simultaneously learn about the initial conditions of the universe and probe periods of cosmic history that might be inaccessible by other means.",
"Second, halo measurements are sensitive to a broad range of dark matter models.",
"To date, all positive experimental evidence for the existence and properties of dark matter comes from astrophysical observations.",
"Measurements of the abundance, density profiles, and spatial distribution of dark matter halos offer sensitivity to an enormous range of dark matter models, and are complementary to both terrestrial experiments and indirect searches for dark matter annihilation and decay products.",
"Third, our understanding of how the fundamental properties of dark matter at a microscopic scale impact structure formation throughout cosmic history is rapidly advancing.",
"Recently, there has been tremendous progress in modeling the formation and evolution of dark matter halos in the context of novel dark matter theories beyond CDM.",
"There is enormous potential to further develop detailed phenomenology for a broader range of dark matter models, and to explore new regions of theory space with new and archival data.",
"Thus, halo measurements provide a window into both dark matter physics and early universe cosmology.",
"Fig.",
"REF illustrates these connections by showing the linear matter power spectrum predicted by several theoretical models of dark matter, together with the relevant scales of the halo mass and temperature of the universe when that mode entered the horizon.",
"In Fig.",
"REF , we show the complementarity between measurements of the matter power spectrum and dark matter halos and terrestrial direct detection searches in the context of the spin-independent dark matter–nucleon scattering cross section.",
"To further leverage halo measurements and extract fundamental properties of dark matter, we set the following observational milestones.",
"First, precision measurements of galaxy-scale dark matter halos are critical.",
"Current and near-future facilities will provide a detailed mapping between luminous galaxies and their invisible dark matter halos across 13 billion years of cosmic history and more than 8 orders of magnitude in dark matter halo mass ($10^{15}\\,M_{\\odot } \\gtrsim M_{\\rm halo} \\gtrsim 10^{7} \\,M_{\\odot }$ ).",
"Detailed measurements of halo abundances and density profiles across cosmic time will provide increasingly stringent tests of the CDM paradigm.",
"Second, within the next decade, several observational techniques will become sensitive to dark matter halos at or below the minimum mass required to host stellar populations ($M_{\\rm halo} \\lesssim 10^{7} \\,M_{\\odot }$ ).",
"Population studies of completely dark halos offer unique advantages to study the microphysical properties of dark matter because the evolution of these halos is less affected by baryonic physics.",
"Many theoretical models of dark matter predict conspicuous deviations from CDM in low-mass halos.",
"Third, a suite of innovative and ambitious observational techniques can be used to search for compact stellar- and planetary-mass-scale halos via their subtle gravitational effects ($M_{\\rm halo} \\lesssim 10^{2}\\,M_{\\odot }$ ).",
"The discovery of such low-mass halos would immediately transform our understanding of both dark matter properties and the physics of the early universe.",
"These observational breakthroughs have the potential to revolutionize our understanding the nature of dark matter.",
"For example, if a cored density profile is inferred in ultra-faint dwarf galaxies, where baryonic feedback is minimal, it will indicate that dark matter may have strong self-interactions or quantum wave features.",
"In this case, combining the measurements of dark matter density profiles from ultra-faint dwarfs to clusters of galaxies, we may narrow down the mass of dark matter particle(s), as well as that of the dark mediator(s).",
"Observations of dark matter halos below the galaxy formation threshold will put strong constraints on the “warmth” of dark matter and set upper limits on the interaction strength between dark matter and any warm species, including baryons or dark radiation, in the early universe.",
"On the other hand, if such halos are not detected, it would imply a cutoff in the matter power spectrum and a radical deviation from CDM.",
"Since a cutoff in the matter power spectrum could also reduce the abundance of Milky Way satellite galaxies and affect the Lyman-alpha forest, we could further confirm a departure from CDM with these complementary measurements.",
"Numerical simulations are essential to understand baryonic systematics and to connect the Lagrangian parameters that describe a particle physics model for dark matter to halo observables.",
"Simulations are the topic of the following section.",
"Figure: Cosmic probes of the matter power spectrum and dark matter halos set strong constraints on the minimum thermal dark matter particle mass , and spin-independent dark matter–nucleon scattering cross section , , , , , , (green regions).Projected improvements in sensitivity coming from future facilities and observations are indicated with a dashed green line, based on the estimate with a potential discovery of 10 5 M ⊙ 10^5~M_\\odot subhalos using Rubin LSST stream observations.Constraints from Big Bang nucleosynthesis and and cosmic rays upscattering dark matter in the XENON1T experiment are indicated in light green and are subject to additional model dependence.These constraints are highly complementary to constraints from direct detection experiments , (gray regions).",
"The neutrino fog for xenon direct detection experiments is shown with dashed black line ."
],
[
"Cosmological Simulations of Dark Matter",
"Cosmological N-body simulations are essential to predict and interpret the imprints of fundamental dark matter physics on structure formation in the nonlinear regime.",
"Properly modelling baryon physics associated with galaxy formation in these simulations is often a key step to distinguish effects of baryon physics from those of new dark matter physics.",
"With enormous datasets expected from Rubin LSST and other forthcoming facilities, we are at the critical stage to develop techniques for efficient forward simulations in order to extract information about dark matter from the data.",
"In what follows, we give a brief overview and propose a plan for building simulation program to interpret observations so that we can robustly search for novel signatures of dark matter microphysics across a large dynamic range of length scales and cosmic time [6].",
"Over the last 40 years, cosmological simulations have played a vitally important role in studying dark matter particle properties.",
"They have been essential to the development of the CDM paradigm and to eliminating neutrinos as a dominant component of the dark matter.",
"During the last decade, particle physicists and simulators have come together to generate cosmological predictions for a subset of novel dark matter scenarios beyond CDM (e.g., Fig.",
"REF [65], [66], [67]).",
"The challenge and opportunity for this decade is to develop a robust and vibrant simulation program that connects the ground-breaking capabilities of observational facilities [2], [1], [3], [4], [68], [69] to an expanding landscape of particle models for dark matter and targeted terrestrial experiments [22].",
"Because a well-synthesized program of theory, simulation, observation, and experiment is critical to revealing the nature of dark matter, we identify six areas of focus for simulations that advance along the key opportunities described in Sec. .",
"Figure: An example flowchart for distinguishing cold and warm dark matter models in the context of dark matter halo substructure as observed in strong gravitational lens systems.",
"This example highlights the need for collaborative efforts among particle physicists, simulators and observers, in order to harness the full power of new observational facilities to quantitatively test dark matter models.",
"The two right columns show images of simulations and lensing observables assuming cold and warm dark matter models.",
"From top to bottom: large-box numerical simulations of structure formation, simulated dark matter substructure within a galaxy halo, a possible realization of dark matter structure generated under the model, and a particular realization of dark matter structure generated under the model consistent with observations of the strong lens system WGDJ0405-3308.",
"Figure adapted from .First, increased collaboration between simulators and particle theorists will help identify significant dark matter models and areas of parameter space for further study.",
"Model builders and observers both rely on simulations as a crucial link that draws their ideas and work together.",
"This approach underpins the key opportunity of using cosmic probes to understand fundamental properties of dark matter by mapping dark matter microphysics to astrophysical structure formation and observables associated with it.",
"For example, knowing the scale on which structures are expected to be modified relative to CDM can enable simulators to efficiently target well-motivated regions of parameter space.",
"In turn, targeted parameter space searches can help theorists focus their work on realistic model-building efforts.",
"Guidance from theorists will be particularly valuable to rigorously develop initial conditions for simulations of specific dark matter models.",
"Second, it is important to advance algorithm development and develop code benchmarks to ensure that simulations meet the required precision targets set by the sensitivity of new facilities.",
"Broadly speaking, there are four major classes of dark matter models that currently capture the attention of simulators: CDM, fuzzy dark matter, self-interacting dark matter, and warm dark matter.",
"Each of these presents distinct challenges in numerical implementation, requiring benchmarks for validating simulations and ensuring that they achieve the necessary precision to successfully support dark matter inference.",
"Key predictions include measurements of (sub)halo mass functions; (sub)halo density profiles; and subhalo radial distributions, infall times, and phase space distributions.",
"Third, it is critical to perform simulations with full hydrodynamics using validated subgrid models and numerical resolution at the relevant redshifts and cosmological scales.",
"Understanding the role of baryonic physics at small scales is critically important, since key discrepancies between the predictions of CDM and alternative dark matter models occur at small scales where baryonic physics plays an important role [50].",
"Degeneracies between baryon physics and alternative dark matter models presents a challenge.",
"Breaking these degeneracies requires full inclusion of baryonic physics in simulations and dedicated comparisons between validated simulations.",
"Data from current and near-future facilities will usher in the discovery of many new types of systems with the potential to provide better sensitivity to dark matter physics, should support be provided for that specific scientific goal.",
"Fourth, we will benefit significantly from the analysis of simulation outputs in the realm of observations.",
"Forward modeling simulations into the space of observables enables apples-to-apples comparisons between models and data.",
"Such investigations are necessary to fully prepare for and utilize unprecedented datasets from DESI, Rubin LSST, CMB-S4, and other forthcoming instruments.",
"Rigorous comparisons between theory and observation, as well as tools that translate theoretical predictions to observable parameter spaces will continue to be essential.",
"These comparisons will help us determine when a problem arises from numerical techniques and when it is a true physical problem.",
"As data analysis pipelines and simulations become more elaborate – and datasets become larger – strengthening our capacity to disentangle numerical effects from physical phenomena will be of critical importance.",
"Furthermore, translating simulations into observable parameter spaces will assist in designing and evaluating new facilities.",
"Fifth, we need fast realizations of observables to infer dark matter properties from observation on feasible timescales.",
"Cosmological simulations with full hydrodynamics are a critical tool to reveal how the physical properties of dark matter alter the abundance and internal structure of dark matter halos and subhalos, which can result in observable differences in astronomical objects and systems.",
"These simulations produce “mock universes” that allow us to compare theoretical prediction with observations in the space of observables.",
"As such, running these simulations will become the bottleneck of parameter inference and model comparison, because these tasks typically require the generation of a large sample of simulated datasets covering different input parameters (dark matter properties in this case).",
"Multiple methods have been identified to address these challenges [6].",
"They broadly fall into the categories of (1) reducing the computational cost of individual simulations by swapping some simulation components with models, and (2) reducing the number of simulations needed for analyses.",
"We will likely need to combine these approaches to cover the vast space of untested dark matter theories and the diversity of observational measurements.",
"These efforts will benefit from the introduction of machine learning and artificial intelligence techniques that are described later in this chapter.",
"Finally, numerical simulations and fast realizations should inform observers and experimentalists where to look for new signatures of dark matter physics.",
"Simulations can play a major role in motivating new observational strategies by revealing unforeseen signatures of, and affirming analytic predictions for, dark matter physics.",
"One example of this dynamic in operation is the development of an accurate model for the dark matter distribution in our Milky Way galaxy.",
"Understanding the local density and velocity distribution of dark matter is key to properly design terrestrial direct detection experiments.",
"Furthermore, in the event that a positive signal is detected in a terrestrial experiment, we need to interpret that signal in an astrophysical context and confirm whether it is consistent with what cosmic constraints on dark matter.",
"Another example is the identification of dark matter (sub)halos that do not contain baryons, as predicted by CDM.",
"Simulations can inform observational strategies to characterize phenomena that are potentially entirely sourced by non-luminous objects.",
"In Fig.",
"REF , we show an example flowchart for distinguishing cold and warm dark matter models using strong gravitational lens systems with the collaborative efforts from the six areas discussed above.",
"With upcoming observational and experimental facilitates, the next decade will be transformative in the HEP community's ability to learn about dark matter in the sky and in the lab.",
"Numerical simulations of structure formation are a bridge between theory and observation.",
"A close collaboration among simulators, particle physicists, and observers is essential to interpret observational data, break degeneracies between baryon physics and dark matter physics, and reveal the microscopic nature of dark matter.",
"Only with a vibrant and cohesive simulation program will we be able to leverage the full power and precision of cosmic probes of dark matter."
],
[
"Primordial Black Holes and the Early Universe",
"As potentially the first density perturbations to collapse, primordial black holes may be our earliest window into the birth of the universe and energies between the QCD phase transition and the Planck scale.",
"The corresponding length scales ($k = 10^{7} - 10^{19}$ $h\\,\\mathrm {Mpc}^{-1}$ ) are much smaller than those measured by other current and future cosmological probes, see Fig.",
"REF .",
"While previous estimates suggested that primordial black holes were constrained to be a subdominant component of dark matter over much of the viable mass range, more recent analyses have relaxed many of these constraints, re-opening the possibility that, in certain mass ranges, primordial black holes may comprise a dominant component of dark matter, as shown in Fig.",
"REF .",
"The detection of primordial black holes would change our understanding of the fundamental physics of the early universe and the composition of dark matter [7], [12].",
"Primordial black holes are a probe of primordial density fluctuations in a range that is inaccessible to other techniques.",
"These curvature fluctuations are imprinted on space-time hypersurfaces during inflation, at extremely high energies, beyond those currently accessible by terrestrial and cosmic accelerators.",
"Our understanding of the universe at these high energies ($\\gtrsim 10^{15}$ GeV) comes predominantly from extrapolations of known physics at the electroweak scale.",
"Measurements of the primordial density fluctuations via the abundance of primordial black holes would provide unique insights into physics at these very high energy scales.",
"This significant reward motivates the development of several complementary techniques that are sensitive to primordial black holes and subject to different astrophysical systematics, such as gravitational microlensing, gravitational wave detection, and gamma-ray signatures of black hole evaporation.",
"In many cases, the science of primordial black holes can be performed by facilities that have well-motivated and multi-faceted scientific programs, e.g., optical/near-infrared time-domain imaging surveys, gravitational wave detectors, precision astrometry from radio interferometry, future MeV–TeV energy gamma-ray facilities.",
"That said, realizing primordial black hole science from these facilities often requires specialized observing schemes, dedicated data analysis, and devoted theoretical studies.",
"Therefore, it is important to closely integrate scientific efforts with enabling facilities across the scientific funding agencies (DOE, NASA, NSF).",
"Current and near-future observations can provide unprecedented sensitivity to the search for primordial black holes.",
"However, it is necessary to ensure that these facilities acquire their data with a cadence and sky coverage that enables the searches [19], [90], [91].",
"In addition, the sensitivity of the searches will be maximized by combining datasets from multiple observational facilities.",
"Development of joint processing and analyses of Rubin LSST, the Nancy Grace Roman Space Telescope (Roman), and Euclid will maximize the opportunity to detect primordial black holes.",
"Furthermore, current and future gravitational wave facilities will provide an unparalleled opportunity to detect primordial black holes directly through gravity.",
"These facilities include both ground-based detectors, such as LIGO and Cosmic Explorer, and space-based detectors, such as LISA and AEDGE [92].",
"The scale of current and near-future datasets and the complexity of analyses benefit from collaborative scientific teams.",
"These teams will develop the tools to perform rigorous and sensitive searches for primordial black holes in current and near-future observational data.",
"The computational challenges presented by these searches are well-matched to the capabilities of HEP scientists and facilities.",
"In addition, theoretical studies will help us better understand the production mechanisms, clustering, and spin properties of primordial black holes.",
"These characteristics will inform the expected abundance of black hole microlensing and gravitational-wave events and systematics with cosmic surveys, as well as the connections to primordial physics in the early universe.",
"Furthermore, improved simulations of the merger rate of primordial black holes and of specific accretion rates will help inform observational constraints."
],
[
"Dark Matter in Extreme Astrophysical Environments",
"Astro-particle searches for dark matter have historically focused on measuring cosmic-ray or photon products from the annihilation or decay of dark matter particles.",
"However, dark matter interactions could also alter the physical processes occurring in the interiors of stars or stellar remnants, the dynamics of black holes, or the mergers of compact objects.",
"These alterations would imprint characteristic signals in electromagnetic and gravitational wave observations.",
"Exploring dark matter via observations of these extreme astrophysical environments—defined here as heavy compact objects such as white dwarfs, neutron stars, and black holes, as well as supernovae and compact object merger events—has been a major field of growth since the last Snowmass study.",
"In the coming decade, observations of extreme astrophysical targets have the potential to open sensitivity to novel dark matter parameter space across a broad mass range (Fig.",
"REF ) [8].",
"Exploiting these opportunities relies on both advances in theoretical work and on current and near-future observatories, including both gravitational-wave instruments and instruments spanning the full electromagnetic spectrum, from radio to gamma-rays.",
"To help guide the discussion on the search in extreme astrophysical environments, we organize these searches by the dark matter mass range that is probed: ultralight dark matter ($<1$ keV), light dark matter (keV–MeV), and heavy dark matter ($\\gtrsim $ GeV).",
"Despite this categorization, we emphasize that many of these probes overlap in mass range, as summarized in Fig.",
"REF .",
"In addition, we note that the parameter space of the dark matter that is probed does not always saturate the relic abundance; instead, dark matter is broadly defined as matter that does not interact appreciably with Standard Model matter.",
"Extreme astrophysical environments provide unique opportunities to probe ultralight dark matter ($<1$ keV).",
"Ultralight particles can be produced in the hot, dense cores of stars and stellar remnants and affect their evolution.",
"Ultralight dark matter—either ambient in the environment or produced in a neutron star—can convert in the high magnetic field environment of the neutron star into radio waves or X-rays that could be detected by telescopes.",
"In the last decade, new ideas unique to bosonic dark matter have been developed.",
"Specific models of ultralight dark matter can alter the shape of the gravitational waveforms of merging neutron stars through their coupling to the dense neutron star matter.",
"Black hole superradiance is a process that can extract energy and angular momentum from rotating black holes and place it into bound states of exponentially large numbers of ultralight bosons, as long as the Compton wavelength of the particle is comparable to the size of black holes.",
"These systems yield signals of coherent gravitational waves as well as black hole spin down, which do not depend on particle interactions but only on gravity.",
"Finally, ultralight dark matter can form collapsed structures like compact halos and boson stars that could be detected using gravitational waves, microlensing, or electromagnetic signals.",
"Opportunities to probe light dark matter (keV–MeV) exist from a variety of astrophysical situations: supernova explosions, the properties of neutron stars, binary neutron star mergers, and black hole population statistics (measured with gravitational waves from binary inspirals).",
"Key observational targets for dark matter in this mass range include observation of gamma rays, neutrinos, and the populations of neutron stars and black holes as observed electromagnetically and via gravitational waves.",
"Light dark matter produced in core collapse supernovae can be constrained from limits of their supernova cooling, or lead to visible signals in the X-ray or gamma-ray bands.",
"During a binary neutron star merger, dark matter can lead to a bright transient gamma-ray signal.",
"In the cores of blue supergiants, dark matter can affect stellar evolution, ultimately changing black hole population properties including the location of the black hole mass gap.",
"Dark matter scattering and annihilating in exoplanets, brown dwarfs, Population III stars, and stellar remnants can be probed through infrared and optical radiation, and through gamma rays.",
"Neutron stars can be heated by light dark matter via the Auger effect, which is probed by telescopes in the ultraviolet, optical and infrared ranges of the electromagnetic spectrum.",
"Lastly, accumulation of dark matter (in particular bosonic light dark matter) can lead to the collapse of astrophysical objects.",
"Most of the signals arise from couplings to Standard Model photons and fermions.",
"As an example, in Fig.",
"REF we show the complementarity between cosmic probes of light and ultra-light dark matter and terrestrial helioscope and haloscope in the context of searches for the signatures of axion-like particles.",
"Compact astrophysical objects such as neutron stars and black holes provide unique environments to test heavy dark matter ($>$ GeV).",
"Dark matter captured by neutron stars and their subsequent heating can be observed by upcoming infrared and radio telescopes.",
"Dark matter that is produced in neutron stars may collect in the interior or form neutron star halos, with implications for the equation of state, mass-radius relation, and gravitational wave signals.",
"Moreover, dark matter can collect in high-density spikes around black holes enhancing annihilation rates.",
"A black hole–compact object binary can form a dark matter spike that can be observed by future space-based gravitational wave observatories.",
"Merging compact objects can also give insight into a wide variety of dark sector particles that modify the dynamics of the merger process.",
"This includes fifth forces and modifications to gravity.",
"Finally, sufficient accumulation of dark matter around a compact object can cause the dark matter particles themselves to collapse into a black hole.",
"Upcoming pulsar searches and gravitational wave observatories will be sensitive to this kind of dark matter signature.",
"The coming decade presents a key opportunity to maximize the sensitivity of observations to novel dark matter theory space.",
"Observational and theoretical astrophysicists should collaborate to constrain the standard astrophysical properties of these extreme environments.",
"We need coordination between the experimental collaborations responsible for upcoming major observatories across all astrophysical messengers: gravitational waves, neutrinos and electromagnetic radiation at all wavelengths.",
"Theoretical developments will improve our understanding of the signatures of dark matter in extreme environments, which will in turn help optimize the design and data taking strategies of future instruments.",
"Figure: Cosmic probes of extreme astrophysical environments , , , , combined with measurements of dark matter halos , and other cosmological observations , , , set strong constraints on the parameter space of axion-like particles (green regions).Projected improvements in sensitivity coming from future facilities and observations are indicated with a dashed green line.Cosmic probes are sensitive well-motivated regions for the QCD axion , , , and axion-like particles , and they are highly complementary to other experimental searches with helioscopes and haloscopes (gray regions taken from ).",
"This figure uses limit data available from ."
],
[
"Facilities for Cosmic Probes of Dark Matter",
"Over the next decade, observational facilities spanning the electromagnetic spectrum, as well as gravitational waves, offer the potential to significantly expand our understanding of dark matter physics.",
"In this section, we briefly discuss current and near-future facilities that are aligned with the HEP Cosmic Frontier program and offer the opportunity to greatly enhance our understanding of dark matter physics.",
"In many cases, these facilities have multi-faceted scientific portfolios that include sensitivity to dark energy, inflation, neutrino physics, and modifications to gravity.",
"Furthermore, the technology used in these facilities leverages the core technical and scientific capabilities of the HEP community.",
"Strong involvement from the HEP community will maximize the scientific output of these facilities, and in many cases, HEP involvement is necessary for the construction and/or operation of these facilities.",
"The capability to probe dark matter physics should be considered in the design phase of these new facilities.",
"The discussion in this section focuses on a series of facilities-oriented white papers submitted to CF3 as part of the Snowmass process [4], [1], [2], [3], [11].",
"We note that the facilities described here complement other multi-messenger facilities [110], gamma-ray and X-ray experiments [111], [112], and gravitational wave facilities [92] submitted to other topical groups in the Snowmass process."
],
[
"Current/Near-Future Facilities",
"Dark Energy Spectroscopic Instrument The Dark Energy Spectroscopic Instrument (DESI) began regular operations in 2021 and is currently performing one of the most powerful wide-field spectroscopic surveys [78], [113].",
"The synergy between DESI and other current and near-future observing facilities will yield datasets of unprecedented size and quality, with sensitivity to dark matter physics over a wide range of scales and redshifts.",
"DESI will detect four times as many Lyman-$\\alpha $ quasars as were observed in the largest previous survey, yielding about 1 million medium-resolution spectra.",
"Observations of these spectra will constrain the formation of dark matter halos through measurements of the clustering of low-density intergalactic gas out to $z \\sim 5$ .",
"Measurements of stellar radial velocities from DESI in conjunction with astrometry from $Gaia$ (and eventually Roman) will enable us to constrain the global distribution of dark matter within the Milky Way, its dwarf satellites, and stellar streams.",
"However, suites of numerical simulations of non-CDM cosmologies are needed to interpret observations from DESI in the context of fundamental dark matter physics.",
"Such simulations must be transformed into realistic mock datasets that account for observational selection effects.",
"The creation of these mock datasets is a significant investment that could heavily leverage HEP infrastructure that is already integrated into the DESI Collaboration [1].",
"Vera C. Rubin Observatory The Rubin Observatory Legacy Survey of Space and Time (LSST), which is scheduled to start in 2024, has the potential to become a flagship dark matter experiment [18].",
"LSST will probe dark matter through a wide suite of observations including measurements of Milky Way satellites, stellar streams, galaxy clusters, weak lensing, microlensing searches for primordial black holes, and studies of stellar populations and stellar remnants [19], [114], [2].",
"Due to the size and complexity of the Rubin LSST dataset and the need for devoted, high-resolution numerical simulations, a coordinated effort is required to perform rigorous dark matter analyses.",
"A large collaborative team of scientists with the necessary organizational and funding support is needed to lead this effort.",
"Furthermore, studies of dark matter with Rubin LSST will also guide the design of, and confirm the results from, other dark matter experiments.",
"Transforming Rubin LSST into a dark matter experiment is key to achieving the dark matter science goals that have been identified as a high priority by the HEP community [2].",
"CMB-S4 CMB-S4 is a ground-based experiment that will perform exquisite measurements of the CMB temperature and polarization anisotropies [23], [21].",
"These measurements (on their own and in combination with other surveys) will provide new means to probe the nature of dark matter.",
"These measurements will provide a snapshot of the universe as it was around the time of recombination, and they will also reveal the imprints of structure growth at much later times.",
"Gravitational lensing of the CMB leads to characteristic distortions of the primary CMB maps [115], allowing us to statistically reconstruct maps of the integrated line-of-sight density.",
"Scattering of CMB photons in galaxy clusters (the Sunyaev-Zel’dovich effect) [116], [117] allows for the identification of the most massive bound structures in the universe out to very high redshifts.",
"Cosmological measurements in general, and CMB measurements in particular, provide insights into dark matter physics that are complementary to direct, indirect, and collider searches for dark matter.",
"Cosmological observables are impacted by the influence of dark matter on the entire cosmic history.",
"Dark matter constraints derived from cosmology do not rely on assumptions about the dark matter density in the neighborhood of the Earth or of any specific astrophysical object.",
"Furthermore, CMB observations are sensitive to regions of parameter space that are out of reach of current direct searches.",
"Several aspects of the dark matter program are already included among the CMB-S4 core science cases; however, support must be provided to achieve these science goals [3]."
],
[
"Future Facilities",
"Here we briefly describe the landscape of proposed future facilities, starting with those probing the most energetic photons and moving to those with lower energies, and concluding with gravitational wave detectors.",
"While these facilities are synergistic with a broad range of scientific objectives, strong support from the HEP community is necessary to enable the design, construction, and operation of these facilities.",
"X-ray/Gamma-ray Facilities Instruments operating at X-ray and gamma-ray energies have been indispensable for the indirect dark matter detection and multi-messenger communities [111], [112], [110].",
"While indirect detection is discussed elsewhere, these experiments also provide an important test of PBH evaporation and dark matter in extreme astrophysical environments.",
"In particular MeV-scale $\\gamma $ -ray experiments like AMEGO-X [118] and GECCO [119] are important for probing PBH evaporation, while X-ray experiments like XRISM [120], Athena [121], and proposed facilities such as STROBE-X [122] are important for probing the physics of extreme environments around neutron stars and black holes.",
"Optical/Near-Infrared Facilities Optical/near-infrared telescopes have been the work-horse of dark matter studies on galactic scales.",
"Proposed optical/near-infrared facilities include wide-field multi-object spectroscopic (WFMOS) surveys, such as DESI-II [123], MegaMapper [124], [125], MSE [126], and SpecTel [127], and the US Extremely Large Telescope program [128], including the Giant Magellan Telescope (GMT) [129] and the Thirty Meter Telescope (TMT) [130].",
"Both classes of instruments plan to target stars in the Milky Way halo, stellar streams, and dwarf galaxies, as well as dark matter dominated galaxies in the local universe, and strong lens systems and galaxy clusters at higher redshift.",
"The WFMOS survey instruments will provide medium- to high-resolution spectra of millions of stars, as well as providing information on a large number of higher redshift galaxies discovered by Rubin LSST [131].",
"In contrast, the US ELTs will provide unprecedented sensitivity, image resolution, astrometry, and extreme precision radial velocity observations.",
"In all cases, these facilities seek to extend measurements of the dark matter halo mass function below the threshold of galaxy formation ($10^6$ –$10^8 M_\\odot $ ) and to measure the density profiles of dark matter halos in the local universe (e.g., for our Milky Way, its satellites, and other local galaxies) and at higher redshifts (e.g., strong lens systems and galaxy clusters).",
"Measurements of the dark matter halo mass function and density profiles can be translated into sensitivity to the dark matter particle mass and interaction cross-sections.",
"Furthermore, these facilities provide multi-faceted fundamental physics programs that include measurements of dark energy and inflation [131].",
"They would leverage HEP technology developed for Stage III and IV dark energy experiments (DECam and DESI) and advance technology toward a Stage V dark energy experiment [131].",
"HEP technical expertise is well-matched to the design and construction of fiber positioners, CCD detectors, and spectrographs.",
"Furthermore, HEP computational resources and expertise are well matched to the task of data processing and distribution.",
"These experiments have strong support from the astronomy community; however, HEP support will be critical for their construction and to ensure that they maximize fundamental physics output.",
"Microwave Facilities The proposed millimeter-wavelength facility CMB-HD [132] will extend the resolution of cosmic microwave background surveys by a factor of five and the sensitivity by a factor of three or more.",
"These observations will open a new window of small-scale CMB observations and will uniquely enable measurements of the small-scale matter power spectrum (scales of $k \\sim 10\\,h\\,{\\rm Mpc}^{-1}$ ) from weak gravitational lensing using the CMB as a backlight.",
"These observations will also enable measurements that rule out or detect any new light particle species ($N_{\\rm {eff}}$ ) that were in thermal equilibrium with the Standard Model at any time in the early Universe [36], and enable probes of axion-like particles through CMB photon conversion, time-dependent CMB polarization rotation, cosmic birefringence, and ultra-high-resolution kinetic Sunyaev-Zel'dovich measurements.",
"CMB-HD would leverage and extend the scientific and technical investment of HEP in CMB-S4.",
"Radio Facilities Proposed centimeter-wavelength radio observatories, including the ngVLA [133] and DSA-2000 [134], can employ pulsar timing measurements to map the dark matter halo of the Milky Way and the substructures it contains.",
"These experiments could complement other gravitational wave facilities at higher frequency.",
"Proposed low-frequency radio experiments, such as LuSEE Night [11], [135], PUMA [136], and successors to HERA [137] can use the 21-cm line of hydrogen from the Dark Ages ($z\\sim 50$ ) through cosmic dawn and reionization ($z \\sim 6$ ) to probe dark matter physics via the thermal history of intergalactic gas and the timing of the formation of the first stars and galaxies.",
"These facilities would have complementary programs to probe dark energy (measuring the expansion history and growth of the universe up to $z=6$ ) and the physics of inflation (constraining primordial non-Gaussianity and primordial features).",
"Gravitational Wave Facilities Proposed gravitational wave facilities, such as Cosmic Explorer and LIGO-Voyager, can probe dark matter directly through gravity [92].",
"These experiments are sensitive to channels including the detection of axion-like particles in binary neutron star mergers, ultralight bosons through superradiant instabilities of rotating black holes, the identification of boson stars in compact binaries, dark matter density spikes around black holes, and the existence of sub-solar-mass primordial black holes."
],
[
"Tools for Cosmic Probes of Dark Matter Physics",
"Collaborative Infrastructure Historically, many cosmic probes of dark matter physics have been pursued by small groups of scientists.",
"However, as the scale and complexity of cosmic survey experiments increase, the need for numerical simulations to interpret data grow, and the range of possible dark matter models expands, it becomes difficult to find sufficient expertise within a small group of scientists.",
"Similar challenges have been faced by the dark energy community, which has motivated the formation of large collaborative efforts to build and analyze data from new facilities.",
"These collaborations bring together the efforts of university groups, international collaborators, and scientists at national laboratories to accomplish scientific tasks that are too large for any single investigator.",
"Modern efforts to assemble collaborative teams to study cosmic probes of dark matter physics have already started in the context of the Dark Energy Survey [138] and the Rubin LSST Dark Energy Science Collaboration [2].",
"In many cases, these teams can reside within existing collaborative infrastructure that has been established for other HEP mission goals.",
"However, additional support must be provided to enable dark matter as a parallel branch of fundamental physics being pursued by these experiments.",
"New Support Mechanisms Cosmic probes of dark matter provide a rich, diverse, and interdisciplinary area of research.",
"While this leads to an exciting discovery space, it also leads to logistical difficulties in classifying the research within the existing research support structures (i.e., DOE HEP, NSF-PHY, NSF-AST, and NASA).",
"In particular, support for cosmic probes of fundamental dark matter properties often falls in the cracks between these disciplines and agencies.",
"This has been especially challenging for theoretical research in this domain, which has been increasingly been difficult to fund.",
"We recommend that inter-agency coordination is required to assure that cosmic probes of dark matter physics are firmly supported.",
"Multi-agency support extending across the spectrum of theory, simulation, and experiment will enable large gains in the coming decade.",
"Artificial Intelligence/Machine Learning The interplay between models and observations is a cornerstone of the scientific method, aiming to inform which theoretical models are reflected in the observed data.",
"Within cosmology, as both models and observations have substantially increased in complexity over time, the tools needed to enable a rigorous comparison have required updating as well.",
"In the next decade, vast data volumes will be delivered by ongoing and upcoming cosmology experiments, as well as the ever-expanding theoretical search space.",
"We are now at a crucial juncture where we may be limited by the statistical and data-driven tools themselves rather than the quality or volume of the available data.",
"Methods based on artificial intelligence and machine learning have recently emerged as promising tools for cosmological applications, demonstrating the ability to overcome some of the computational bottlenecks associated with traditional statistical techniques.",
"Machine learning is starting to see increased adoption across different sub-fields of and for various applications within cosmology.",
"At the same time, the nascent and emergent nature of practical artificial intelligence motivates careful continued development and significant care when it comes to their application in the sciences, as well as cognizance of their potential for broader societal impact [9].",
"Cosmology Data Preservation Cosmology datasets and simulations have useful lifetimes that extend long beyond the operational period of individual projects.",
"As datasets from facilities and simulations grow in size, the “take out” model of manual download followed by local computation will become insufficient and unwieldy.",
"Future work needs to focus on co-locating data with computing, and automating the coordination between multiple data/compute centers.",
"Furthermore, special attention should be paid toward facilitating the joint analysis of datasets beyond the lifetime of individual projects.",
"Implementing a comprehensive data preservation system will require support not only for hardware, but also for personnel to develop and maintain the technologies to simplify cross-site data sharing and personnel to curate the relevant datasets.",
"The authors of [34] recommend that the HEP community support a new cosmology data archive center to coordinate this work across multiple HEP computing facilities."
],
[
"Roads to New Physics",
"Cosmic probes of dark matter present exciting opportunities with powerful synergies to other dark matter search techniques, namely ground-based direct and indirect detection facilities.",
"In this section, we consider three exemplar scenarios where astrophysical observations lead to the characterization of fundamental dark matter properties, and discuss their implications for revealing the particle nature of dark matter and understanding early-universe cosmology.",
"In particular, we highlight how astrophysical observations, terrestrial experiments, and theoretical modelling can work together to extract fundamental parameters of dark matter."
],
[
"Warm and Self-Interacting Dark Matter",
"We first consider a scenario where dark matter differs from the standard CDM model by having a warm thermal velocity distribution and an appreciable self-interaction cross section.",
"The matter power spectrum of warm dark matter is suppressed compared to CDM, resulting in a reduction in the number and the central density of low-mass dark matter halos.",
"For thermal warm dark matter, the power spectrum is completely determined by the dark matter particle mass, $m_{\\rm WDM}$ [50].",
"The observed population of satellite dwarf galaxies of the Milky Way can be used to measure $m_{\\rm WDM}$ if such a suppression is observed.",
"Furthermore, dark matter self-interactions can thermalize the inner regions of dark matter halos and change the dark matter density profile.",
"Thus, the self-interaction cross section per unit mass, $\\sigma _{\\rm SIDM}/m_\\chi $ , can be inferred from measurements of stellar velocities of galaxies, which are sensitive to the central dark matter content.",
"Figure: Potential measurements of the self-interacting cross section and warm dark matter mass from upcoming observations by the Rubin Observatory and a future spectroscopic survey (such as MSE or MegaMapper , ).",
"The projection assumes a dark matter model with a cross section of σ SIDM /m χ =2 cm 2 /g\\sigma _{\\rm SIDM}/m_\\chi =2~{\\rm cm^2/g} and a suppressed matter power spectrum corresponding to a warm dark matter mass of m WDM =6 keV m_{\\rm WDM} = 6~{\\rm keV} (red asterisk).",
"The uncertainties contours are created by following a procedure similar to .",
"Figure adapted from .Fig.",
"REF demonstrates the ability of Rubin LSST combined with a future spectroscopic survey to measure the particle properties of dark matter from observations of Milky Way satellite galaxies, adapted from [19].",
"This projection assumes a self-interaction cross section of $\\sigma _{\\rm SIDM}/m_\\chi =2~{\\rm cm^2/g}$ and a matter power spectrum corresponding to thermal warm dark matter with $m_{\\rm WDM}=6~{\\rm keV}$ ; see [19] for details.More recent studies show that a larger cross section on dwarf scales may be needed to fully explain diverse dark matter densities of the Milky Way satellite galaxies in self-interacting dark matter models [140], [141], [142], [143], [144], [145], [146].",
"In this case, dark matter (sub)halos with a high concentration can experience gravothermal collapse, resulting in a high inner density, which can be probed using strong lensing observations with Rubin LSST, see, e.g., [147], [148], [149], [150], [151] for relevant discussion.",
"The projection should be regarded as an illustration of the capability of future facilities to measure fundamental dark matter particle properties using observations of the cosmic distribution of dark matter.",
"This measurement does not assume that dark matter couples to the Standard Model.",
"Furthermore, we can break degeneracies between dark matter particle properties and the physics of galaxy formation (e.g., the long tail towards large dark matter mass) by combining satellite galaxy measurements with a probe that is independent of subhalo luminosity, such as strong lensing and stellar streams, ultimately resulting in closed contours at high statistical significance.",
"Astrophysical observations can provide precision measurements of interactions in the dark sector.",
"To achieve this goal, a collaborative effort is crucial.",
"First, after Rubin LSST discovers new satellite galaxies, spectroscopic followup measurements of their stellar velocity dispersion are needed to constrain the dark matter density.",
"Second, with the population of newly-discovered satellites, it will be possible to update models that capture the connection between invisible subhalos and visible galaxies to better control baryonic uncertainties.",
"Third, dedicated N-body simulations are needed to make concrete predictions of self-interacting and warm dark matter models in terms of the properties of subhalos, such as their mass function, orbital parameters and central density.",
"Fourth, in order to implement the novel dark matter properties in the simulations and interpret the observational results, we need to use the methods of particle physics to calculate the self-interaction cross section and determine how it depends on the velocity and scattering angle, as well as the linear matter power spectrum that encodes the evolution of the dark matter candidate in the early universe.",
"Last but not the least, we will make predictions for observables on different galactic systems, such as density profiles in isolated dwarf galaxies and substructures of galaxy clusters, and search for signatures of new physics beyond CDM in these systems.",
"The possibility that dark matter has strong self-interactions indicates that there is a light mass scale ${\\cal O}(1)~{\\rm MeV}$ in the dark sector, which is much below the weak scale.",
"Such a light dark sector motivates dedicated searches in upcoming and proposed terrestrial experiments in the intensity frontier, such as FASER [152] and LDMX [153]; see [154] for an overview of relevant experiments and facilities.",
"It is also natural to expect that the light dark sector contains relativistic degrees of freedom [155], [156], i.e., dark radiation, which can be probed in CMB-S4 [20].",
"In addition, after combining observations from dwarf to cluster scales, spanning from $\\sim 10^8~M_\\odot $ to $10^{15}~M_\\odot $ , we may determine masses of dark matter and mediator particles [157].",
"The discovery of a large self-interaction cross section and a cut-off in the matter power spectrum will have profound implications for constructing particle theories of dark matter and understanding its evolutionary history in the early universe, see [49], [155], [69], [158] for related discussion."
],
[
"Primordial Black Holes",
"Next, we consider the discovery of primordial black holes, a realization of compact object dark matter.",
"Primordial black holes represent a gravitational dark matter candidate that cannot be produced in terrestrial experiments and can only be detected and studied observationally.",
"Much of the primordial black hole dark matter parameter space has been constrained by existing observations [70], but windows still remain where primordial black holes can make up some or all of the dark matter, see Fig.",
"REF .",
"Even if primordial black holes make up a subdominant component of the dark matter, their existence would revolutionize our understanding of early universe physics at extreme temperatures that are inaccessible to laboratory experiments [7].",
"Rubin LSST provides an exciting opportunity to directly measure the mass function of compact objects through microlensing observations.",
"If scheduled optimally, LSST will provide sensitivity to microlensing event rates corresponding to $10^{-4}$ of the dark matter density in compact objects with masses $\\gtrsim 0.1\\,{\\rm M_\\odot }$ , a factor $10^{2}\\textup {--}10^{3}$ improvement compared to the existing limits.",
"In addition, the Roman Space Telescope also provides microlensing opportunities in the search for primordial black holes in the next decade.",
"As a high resolution space-based imaging system, Roman has the potential to detect or constrain primordial black holes through various types of lensing.",
"Fig.",
"REF shows discovery potential for primordial black holes making up a subdominant component of the dark matter.",
"We follow the prescription of Carr et al.",
"2017 [159] and model the normalized primordial black hole mass function with a log-normal distribution (see their Eq.",
"3).",
"We set the parameters of our mass function to be consistent with existing observational constraints [159], choosing a peak mass of $M_c = 30\\,{\\rm M_\\odot }$ , a width of $\\sigma = 1$ , and an integrated contribution to the dark matter density of $f_{\\rm PBH} = 0.03$ (3%).",
"We bin into logarithmic mass bins with width of $0.33$ dex and calculated the integrated contribution to the dark matter abundance and the expected number of microlensing events that would be observed by LSST using the projected sensitivity [19].This width of our mass bins is roughly comparable to the mass uncertainty reported for the recent detection of a microlensing event [160].",
"We assign uncertainties on the measured compact object fraction from the fractional Poisson uncertainties on the number of observed events.",
"Rubin LSST will be sensitive to the existence of this primordial black hole population with high confidence; however, this analysis does not include contamination from astrophysical black holes, which is expected to be small at these high masses [161], [19].",
"The detection of primordial black holes by Rubin LSST and/or Roman would provide insights into early universe cosmology.",
"Primordial black holes could form at early times from the direct gravitational collapse of large density perturbations that originated during inflation.",
"The same fluctuations that initialize seeds of galaxies, if boosted on small scales, can lead to some small areas having a Schwarzschild mass within the horizon, which collapse to form black holes.",
"Thus the detection of primordial black holes would directly constrain the amplitude of density fluctuations.",
"These constraints probe small scales between $k = 10^7\\textup {--}10^{19}~{\\rm h/Mpc}$ , much smaller than those measured by other current and future probes."
],
[
"Axions and Axion-like Particles",
"Interest in axion and axion-like particles as potential dark matter candidates has increased significantly since the last Snowmass study [162], [108], [24].",
"The diversity of terrestrial axion experiments has increased dramatically during this time period [108].",
"Current experiments now probe the QCD axion model space in the $\\mu $ eV mass range [163], [164], [165], and future experiments are poised to extend sensitivity to higher and lower masses [108].",
"Meanwhile, the expanding theoretical landscape of axion-like particle models makes it increasingly important to understand the cosmological implications of these models and their detectable signatures in the cosmos.",
"Cosmic probes provide important complementary information in the search for QCD axions and more generally, axion-like particles.",
"In particular, a detailed understanding of the local dark matter distribution is important for both designing terrestrial search strategies and interpreting any positive experimental results from those searches [166].",
"For example, many terrestrial cavity experiments search for a distinct, narrow feature coming from axion conversion into photons [108].",
"The width of the axion feature depends on the velocity dispersion of dark matter in the Milky Way, and more precisely, the Solar neighborhood.",
"Thus, precision cosmic measurements of the local dark matter velocity and density distributions are an important guide when planning future searches for axions and axion-like particles.",
"The existence of dynamically cold dark matter substructure within the Solar System can change the signal strength and temporal modulation behavior in dark matter direct detection [167], [168].",
"In addition, gravitational focusing of dark matter streams by the Sun, Moon and Earth can affect the phase-space distribution of local dark matter particles [169], [170], [171] and have an impact on direct detection signals accordingly.",
"Many current studies are based on WIMPs models, and more work is needed to extend them to axion detection.",
"For instance, since axions are much colder than WIMPs in the streams, the axion signal strength can be boosted more significantly due to gravitational focusing [172].",
"In fact, the possible existence of cold dark matter structures has motivated alternative “high-resolution” experimental scan strategies [173].",
"Conversely, the discovery of an axion signal in terrestrial haloscope experiment would immediately provide information about the local velocity dispersion of dark matter and enable improved modeling of the local dark matter distribution [174].",
"Cosmic probes can also help guide future terrestrial searches over the much broader parameter space of axion-like particles.",
"Cosmic observations are currently the most sensitive probes of axion-like particles coupling to photons in the mass range below $ {\\sim }\\, 10^{-6}$ eV, see Fig.",
"REF .",
"In this regime, upcoming probes of extreme astrophysical environments, such as observations of neutron stars or a Galactic supernova, may provide positive signal of axion-like particles [8].",
"Such a signal could help guide the design of future terrestrial searches using lumped element approaches or superconducting radiofrequency cavities [108].",
"While these terrestrial experiments are currently envisioned to cover large regions axion-like particle parameter space [108], their design and operation could be accelerated if cosmic probes identified a specific target mass range and coupling strength.",
"In the regime below $ {\\sim }\\, 10^{-12}$ eV, cosmic probes such as searches for axion-like particle condensates through microlensing [175] and black hole superradiance [97] may yield a positive signal that could motivate the design of novel atomic clock or nuclear magnetic resonance experiments [24].",
"Moreover, below $ {\\sim }\\, 10^{-12}$ eV, the length scales associated with equilibrium states for axion dark matter approach scales of astrophysical significance, making observations of cosmic phenomena an essential probe in this mass regime.",
"For example, an axion-like particle with mass $ {\\sim }\\, 10^{-21}$ eV produces a dwarf galaxy-scale halo with a distinct soliton core [176], [177], [178].",
"Cosmic probes are currently our only option for experimentally testing models in this mass regime."
],
[
"Summary and Outlook",
"More than 80 years after the first astrophysical discovery of dark matter, its fundamental nature remains an open question.",
"Over the last several decades, the HEP community has designed and executed extensive searches for dark matter in a wide variety of terrestrial experiments.",
"Despite these heroic efforts, the only positive measurements of dark matter continue to come from cosmic observations.",
"Scientific advances over the last decade have made it possible to use precision measurements of macroscopic astrophysical systems to probe the microscopic particle properties of dark matter.",
"This Snowmass report presents the critical opportunity for the HEP community to fully realize the potential of cosmic probes of dark matter.",
"In this report, we have described methods of measuring fundamental properties of dark matter that are valid even when the coupling between dark matter and normal matter is extremely weak (e.g., as weak as gravity).",
"Cosmic measurements of the distribution of dark matter, including the matter power spectrum, dark matter halos, and compact objects, can constrain particle properties of dark matter, such as the mass, interaction cross section, and production mechanism.",
"Moreover, if dark matter has feeble non-gravitational interactions with normal matter, extreme astrophysical environments, such as neutron stars and black holes, provide unique opportunities to explore dark matter physics over 50 orders of magnitude in the mass; much of this model space is inaccessible with terrestrial experiments.",
"In addition, precision astrophysical measurements of dark matter with current and near-future observational facilities are critical for interpreting results from conventional dark matter searches.",
"We have further demonstrated that with the unprecedented coverage and sensitivity of current and near-future observational facilities, the rapidly improving scale and accuracy of numerical simulations, and better theoretical modelling, astrophysical uncertainties can be controlled and the fundamental parameters of dark matter can be extracted.",
"This makes it possible to map Lagrangian parameters describing a particular dark matter model to astrophysical observables, and vice versa.",
"Thus cosmic probes can provide precision measurements of particle physics properties of dark matter in a way that is similar to how HEP experiments have enabled the construction of the Standard Model of particle physics.",
"Cosmic probes of particle properties of dark matter have emerged as a new research field since the last Snowmass community study, largely due to tremendous progress in theoretical investigations of novel dark matter scenarios, N-body simulations of structure formation, as well as astrophysical observations of dark matter distributions.",
"There is a new and exciting trend in the HEP community that more and more theoretical particle physicists have begun working on astrophysical aspects of dark matter.",
"At the same time, astrophysicists working on N-body simulations have started to develop simulation algorithms that can model novel dark matter scenarios beyond CDM.",
"We must encourage and support this promising and evolving trend from both communities.",
"Furthermore, we must develop new mechanisms to further support synergistic efforts among theorists, simulators, dynamicists, observers, and experimentalists/instrumentalists, who are traditionally supported by different agencies and/or programs.",
"Cosmic probes of dark matter are fundamentally multidisciplinary and interdisciplinary, and traditional disciplinary divisions limit scientific outcomes.",
"New support mechanisms can be pursued from small to large scales.",
"On small scales, a program like the DOE Dark Matter New Initiatives (DMNI) is well suited to support a small-scale collaborative effort from particle physicists and astrophysicists with a well-defined scientific goal.",
"Cosmic probes of dark matter were not included in the current DMNI program.",
"If the program continues, we strongly urge that DOE integrates cosmic probes into its portfolio.",
"Alternatively, a similar program could be established to make rapid progress in this emerging field.",
"Dark matter physics associated with current and near-future facilities, such as DESI, Rubin LSST, and CMB-S4, is extremely rich.",
"Dark matter science should be supported within these projects on intermediate scales in parallel to studies of dark energy and inflation.",
"Such a program will fully leverage the unprecedented capabilities of these facilities.",
"On large scales, the construction of future cosmology experiments is critical for expanding our understanding of dark matter physics.",
"HEP involvement will be essential for the design and construction of these facilities, and dark matter physics should be a core component of their scientific mission.",
"Cosmic probes of dark matter are unique and important, because they have sensitivity to microscopic physics of dark matter and provide precision measurements, regardless of whether dark matter has sizable interactions with normal matter.",
"The HEP community has recognized the power of this approach, and it is now time to maximize its full potential.",
"The support for comic probes, which may be the only viable way to measure dark matter properties, is essential for the decade of dark matter to come."
]
] | 2209.08215 |
[
[
"Deep Plug-and-Play Prior for Hyperspectral Image Restoration"
],
[
"Abstract Deep-learning-based hyperspectral image (HSI) restoration methods have gained great popularity for their remarkable performance but often demand expensive network retraining whenever the specifics of task changes.",
"In this paper, we propose to restore HSIs in a unified approach with an effective plug-and-play method, which can jointly retain the flexibility of optimization-based methods and utilize the powerful representation capability of deep neural networks.",
"Specifically, we first develop a new deep HSI denoiser leveraging gated recurrent convolution units, short- and long-term skip connections, and an augmented noise level map to better exploit the abundant spatio-spectral information within HSIs.",
"It, therefore, leads to the state-of-the-art performance on HSI denoising under both Gaussian and complex noise settings.",
"Then, the proposed denoiser is inserted into the plug-and-play framework as a powerful implicit HSI prior to tackle various HSI restoration tasks.",
"Through extensive experiments on HSI super-resolution, compressed sensing, and inpainting, we demonstrate that our approach often achieves superior performance, which is competitive with or even better than the state-of-the-art on each task, via a single model without any task-specific training."
],
[
"Introduction",
"Hyperspectral image (HSI) consists of numerous discrete bands from a wide range of continuous spectrums for each spatial location and provides rich spectral information beyond the visible, which makes it exceedingly useful in the applications of face recognition [1], [2], remote sensing [3], [4], [5], classification [6], [7], [8], and more.",
"However, due to the cost and complexity of hyperspectral imaging, HSIs are often limited to low spatial resolution and suffered from different types of degradations, such as noise, missing pixels, and undersampling.",
"As a result, restoring clean, high-resolution, and complete hyperspectral data becomes a crucial initial step for the success of subsequent HSI applications.",
"In general, HSI restoration can be considered as an inverse imaging problem, i.e.",
"reconstructing the original HSI $x$ from the degraded observation $y$ that is obtained via a forward process $y=D(x)+e$ , where $D$ is the degradation operation, and $e$ is assumed to be additive Gaussian white noise (AWGN).",
"Many tasks, such as HSI denoising, super-resolution, inpainting, and compressed sensing, can fit into this framework by specifying $D$ as identity, downsampling, masking, and sensing operation, respectively.",
"However, the problems are that most of these forward processes are ill-posed, which means the corresponding inverse problems are difficult or even impossible to be solved without extra prior knowledge about HSIs.",
"To address the aforementioned problems, classical HSI restoration approaches minimize a cost function that consists of a data-fidelity term, which measures how well the reconstructed HSIs match the observations, and a regularization term, which reflects certain prior knowledge with respect to unknown HSIs.",
"Following this methodology, numerous handcrafted priors are developed, e.g.",
"total variation [9], [10], low-rank tensor modeling [11], [12], and adaptive spatial-spectral dictionary learning [13].",
"These methods are generally flexible in solving multiple restoration tasks with tiny adjustments and able to achieve reasonable performance under certain assumptions about input data.",
"However, they also suffer from two serious drawbacks.",
"First, their performance is inherently restricted by the matching degree of handcrafted priors and intrinsic properties of HSIs.",
"Second, most of these methods are computationally expensive as they require the iterative minimization of the cost function for each input HSI.",
"With the emergence of deep learning, many works [14], [15] have been developed to circumvent the design of handcrafted priors by taking advantage of the powerful representation capability of deep neural networks.",
"Such methods directly learn the mapping from degraded observations to reconstructed HSIs from a large number of training pairs and implicitly embed the learned prior knowledge into the parameters of the neural network.",
"In this way, not only do they achieve the state-of-the-art performance in many ill-posed HSI restoration tasks [16], [17], [18], but they also reduce tremendous inference time in comparison with classical methods.",
"Despite the strong superiority, deep-learning-based methods literally lose the flexibility to tackle different restoration tasks in one model.",
"One has to design and train separate models for different tasks or even identical tasks with slightly different settings, which is cumbersome.",
"As an alternative, the commonly used plug-and-play (PnP) framework [19], [20] in Gray/RGB image restoration seems to be a promising choice for its capability of leveraging the flexibility of optimization-based methods and the data-driven prior by replacing the traditional handcrafted regularizer with a deep-learning one.",
"However, unlike the conventional Gray/RGB images that mainly contain spatial information, HSIs provide richer spectral information that should be considered, which makes the direct use of Gray/RGB denoisers, such as FFDNet [21], unsuitable as they generally ignore the important spatial-spectral correlation and the global correlation along the spectrum in HSIs.",
"Besides, as a general solution to various image restoration tasks, the PnP method often requires different denoising strengths to achieve desirable performance under different settings, so it is better for PnP denoisers to be capable of handling a wide range of continuous noise levels, which is also missing in most existing HSI denoising algorithms.",
"In this paper, we propose to leverage the plug-and-play framework with a novel deep HSI denoiser to solve multiple HSI restoration problems in a unified approach.",
"In detail, we first use the alternating direction method of multipliers (ADMM) [22] to decouple the traditional optimization objective into two independent subproblems where one subproblem can be solved in closed-form and another subproblem related to image prior can be implicitly solved by an off-the-shelf denoiser.",
"Then, a deep HSI denoiser, i.e.",
"a gated recurrent convolutional neural network (GRCNN), is introduced to circumvent the shortcomings of existing commonly used Gray/RGB plug-and-play denoisers.",
"Specifically, a gated recurrent convolution block is adopted as the basic building block to exploit the rich spectrum correlation in HSIs, and its effectiveness is further enhanced by a residual encoder-decoder architecture.",
"Meanwhile, an additional noise level map is employed to help to train a robust denoiser that is able to handle a wide range of continuous noise levels.",
"Over extensive experiments, it is shown that our deep HSI denoiser not only outperforms the existing HSI denoising algorithms, but also supports the proposed plug-and-play method achieving the superior generalizability and performance against the learning-based and optimization-based methods, respectively, in the tasks of HSI super-resolution, compressed sensing, and inpainting.",
"In summary, our main contributions are that: A deep Plug-and-Play ADMM approach is presented to solve the HSI restoration tasks in a unified approach, in which the prior modeling is implicitly handled by a deep HSI denoiser.",
"A new deep HSI denoiser is introduced to better exploit the intrinsic characteristics of HSIs via gated recurrent convolution units, short- and long-term skip connections, and an auxiliary noise level map, meanwhile supporting the proposed plug-and-play HSI restoration framework.",
"Extensive experiments on four typical HSI restoration tasks, including denoising, super-resolution, compressed sensing, and inpainting, demonstrate that our plug-and-play approach is able to effectively and flexibly tackle a variety of HSI restoration problems without any task-specific training.",
"The rest of this paper is organized as follows.",
"Section reviews the related works of HSI restoration and plug-and-play.",
"Section presents the detailed illustration of the proposed plug-and-play HSI restoration method and the proposed deep denoiser.",
"Section provides the experimental results for four HSI restoration tasks and gives a series of discussions.",
"Section concludes the paper with a brief summary."
],
[
"Related Works",
"In this section, we review the relevant studies on HSI restoration and plug-and-play methods."
],
[
"HSI Restoration",
"Traditional optimization-based HSI restoration methods usually solve an inverse imaging problem with extra regularizations by exploiting the underlying characteristics of HSIs.",
"By considering the spectral correlation, many works, such as total variational methods [9], [10], wavelet methods [23], and low-rank models [24], [25], [10], [26], have been developed.",
"The non-local self-similarity is another important property of HSIs and was exploited in works like block-matching and 4-D filtering (BM4D) [27] and the tensor dictionary learning [28].",
"With the development of deep learning, many learning-based methods were proposed, e.g.",
"Wei et al.",
"[14] presented a novel network QRNN3D for HSI denoising, Mei et al.",
"[17] introduced a 3D fully convolution neural network for HSI super-resolution.",
"Although these methods achieved remarkable results in their respective fields, most of them are specifically developed for a single task.",
"Recently, a number of works [11], [13], [29], [30] have been developed to solve multiple HSI restoration problems in a unified approach.",
"For example, Chang et al.",
"[11] utilized the low-rank tensor prior to model the spatial non-local self-similarity and spectral correction simultaneously and applied their method to HSI denoising, deblurring, destriping, and super-resolution.",
"By considering the sparsity on the subspace bases on gradient maps, Peng et al.",
"[30] proposed an enhanced 3D total variation prior for both HSI denoising and compressed sensing.",
"In the work of Fu et al.",
"[13], a novel adaptive spatial-spectral dictionary learning method was introduced for HSI denoising and super-resolution.",
"Despite the superior performance and flexibility, they are time-consuming and heavily rely on how well the handcrafted prior matches with the intrinsic properties of HSIs."
],
[
"Plug-and-Play Methods",
"Since introduced in [31], [32], [33], the idea of plug-and-play has received great attention for its flexibility and effectiveness to solve a wide range of inverse imaging problems.",
"In the area of traditional Gray/RGB image restoration, there have been many attempts [32], [34], [20], [35], [36] for injecting denoising priors into the plug-and-play framework.",
"In [32], Danielyan et al.",
"combined the augmented Lagrangian technique with a BM3D prior [37].",
"By leveraging DnCNN [38] denoiser as the implicit regularizer, Sun et al.",
"[34] developed a block coordinate regularization-by denoising (RED) algorithm.",
"Inspired by the design of FFDNet [21], Zhang et al.",
"[20] proposed a DRUNet that adopts a noise level map as additional input for plug-and-play denoising.",
"The recent work in [36], from a new perspective, proposed to use a deep super-resolver instead of denoiser as the implicit prior for plug-and-play single image super-resolution.",
"Although all the denoisers [38], [35], [21] for plug-and-play Gray/RGB image restoration can be directly extended to the HSI cases, none of them specifically explore the extensive domain knowledge of HSIs.",
"Some recent works [39], [40] also attempt to inject plug-and-play deep denoising prior for HSI restoration.",
"In [39], Liu et al.",
"employed BM3D [37] as an extra plug-and-play regularization for its fibered rank constrained tensor restoration framework.",
"In [40], FFDNet [21] was used for local regularity, where the global structures are constrained by Kronecker-basis-representation-based tensor low-rankness.",
"However, the plug-and-play priors in these works are still Gray/RGB denoisers that cannot regularize the global spatial-spectral correlation, which limits the performance of these methods for extremely ill-posed HSI restoration tasks, such as inpainting and compressed sensing.",
"On the contrary, by considering both local and global information simultaneously, the single 3D denoiser we used can not only accelerate the iterative process but also benefit the ill-posed HSI restoration by rich prior knowledge encoded in the network.",
"We refer interested readers to [41] for a thorough review of the plug-and-play algorithms."
],
[
"Deep Plug-and-Play HSI Restoration",
"We consider the degradation model for HSI restoration as $y=Dx+e,$ where $x \\in \\mathbb {R}^{MNB}$ is the latent clean image, $y\\in \\mathbb {R}^{M_hN_hB}$ is the degraded observation, $D$ is the noise-irrelevant degradation matrix, $e$ is assumed to be additive white Gaussian noise and $M,N,B$ denote the number of columns, rows, and bands of HSIs, respectively.",
"In the following, we first introduce the plug-and-play ADMM algorithm in the proposed deep plug-and-play framework.",
"Then, a deep HSI denoiser acting as the implicit image prior is presented."
],
[
"Plug-and-Play ADMM",
"The main idea of plug-and-play ADMM is to separate the prior term and data-fidelity term of the traditional optimization objective into two subproblems, where the subproblem related to prior can be treated as a denoising problem that can be solved by an off-the-shelf denoiser, e.g.",
"a deep HSI denoiser.",
"According to this, we can not only retain the flexibility to solve multiple HSI restoration problems in a unified model, but also leverage the powerful prior-modeling ability provided by deep neural networks.",
"In detail, the adopted PnP-ADMM solves the following regularized objective: $(\\widehat{{x}}, \\widehat{{v}})=\\underset{{x}}{\\operatorname{argmin}} \\hspace{5.0pt}\\frac{1}{2} || {D}{x}- {y} ||^{2}+\\lambda g({v}), \\quad \\text{\\emph {s.t. }}",
"{x}={v},$ where $g$ denotes the regularization and $\\lambda $ is a non-negative weighting term.",
"By considering its augmented Lagrangian function: $\\begin{aligned}\\mathcal {L}({x}, {v}, {u})= & \\frac{1}{2} || {D}{x} - {y} ||^{2} +\\lambda g({v}) +{u}^{T}({x}-{v}) \\\\&+\\frac{\\rho }{2}\\Vert {x}-{v}\\Vert ^{2}.\\end{aligned}$ Equation (REF ) can be solved by minimizing $\\mathcal {L}$ , which is achieved by alternately solving the following three subproblems: ${x}^{(k+1)} & = \\underset{{x}}{\\operatorname{argmin}} \\hspace{5.0pt}\\frac{1}{2} || {D}{x} - {y} ||^{2}+\\frac{\\rho }{2}\\left\\Vert {x}-\\tilde{{x}}^{(k)}\\right\\Vert ^{2} , \\\\{v}^{(k+1)} & = \\underset{{v} \\in \\mathbb {R}^{n}}{\\operatorname{argmin}} \\hspace{5.0pt}g({v})+\\frac{1}{2 \\sigma ^{2}}\\left\\Vert {v}-\\tilde{{v}}^{(k)}\\right\\Vert ^{2} , \\\\\\bar{{u}}^{(k+1)} & = \\bar{{u}}^{(k)}+\\left({x}^{(k+1)}-{v}^{(k+1)}\\right),$ where $\\bar{{u}}^{(k)} \\equiv (1/\\rho ) {u}^{(k)}$ is the scaled dual variable of ${v}$ and $\\rho $ is a penalty parameter, $\\tilde{{x}}^{(k)} \\equiv {v}^{(k)} - \\tilde{{u}}^{(k)}$ , $\\tilde{{v}}^{(k)} \\equiv {x}^{(k+1)} + \\bar{{u}}^{(k)}$ and $\\sigma \\equiv \\sqrt{\\lambda / \\rho }$ .",
"Equation () can be considered as solving a Gaussian HSI denosing problem, where $\\sigma $ is the noise level, ${v}$ is the \"clean\" image, and $\\tilde{{v}}^{k}$ is the corresponding \"noisy\" image.",
"To explain, we could first consider the MAP estimate of \"clean\" image $v$ : $v = \\underset{{x}}{\\operatorname{argmin}} \\hspace{5.0pt}\\lbrace -\\ln p(\\tilde{{v}}^{k}|v) - \\ln p(v)\\rbrace .$ If we assume the noise is AWGN with the variance of $\\sigma ^2$ , then we have $-\\ln p(\\tilde{{v}}^{k}|v) = \\frac{1}{2 \\sigma ^{2}}\\left\\Vert {v}-\\tilde{{v}}^{(k)}\\right\\Vert ^{2} + \\ln (\\sigma \\sqrt{2\\pi }).$ By substituting Equation (REF ) into Equation (REF ) and considering $-\\ln p(v)$ as $g(v)$ , Equation () is technically equivalent with Equation (REF ), which proves the fact that Equation () can be considered as a Gaussian denoising problem.",
"As a result, we could practically solve Equation () with any off-the-shelf HSI denoising algorithm $\\mathcal {D}_{\\sigma }$ , e.g.",
"a deep HSI denoiser.",
"${v}^{(k+1)}=\\mathcal {D}_{\\sigma }\\left(\\widetilde{{v}}^{(k)}\\right).$"
],
[
"Update $x$",
"As for the subproblem in Equation (REF ), the general solution could be obtained by solving a least square problem in the closed-form, i.e., $x ^{(k+1)} = \\left( D^TD +\\rho I \\right)^{-1}\\left( D^Ty +\\rho \\tilde{x}^{(k)} \\right).$ For the super-resolution task, $D=SH$ where $H$ is a circulant matrix denoting the blur operation.",
"$S$ is a binary matrix denoting the $k$ times downsampling.",
"The fast solution of Equation (REF ) is given by [31], which is expressed as: ${x}=\\rho ^{-1} {b}-\\rho ^{-1} {G}^{T}\\left(\\mathcal {F}^{-1}\\left\\lbrace \\frac{\\mathcal {F}({G} {b})}{\\left|\\mathcal {F}\\left(\\widetilde{h}_{0}\\right)\\right|^{2}+\\rho }\\right\\rbrace \\right),$ where $\\mathcal {F}$ and $\\mathcal {F}^{-1}$ denote Fast Fourier Transform (FFT) and inverse FFT, ${G}={S H}$ , ${b}={G}^{T} {y}+\\rho \\widetilde{{x}}$ , and $\\tilde{h}_0$ is the 0th polyphase component of the filter $HH^T$ .",
"For the compressed sensing task, $D = \\Phi \\in R^{MN\\times MNB}$ is the sensing matrix.",
"The fast solution of Equation (REF ) can be found in [42], which is shown below: $x^{(k+1)} = x^{(k)} +\\Phi ^{\\top } \\left[\\frac{y-\\Phi \\tilde{x}^{(k)}}{\\operatorname{diag}\\left\\lbrace \\rho +\\psi _{1}, \\ldots , \\rho +\\psi _{n}\\right\\rbrace } \\right],$ where $\\Phi \\Phi ^{\\top } \\stackrel{\\text{ def }}{=} \\operatorname{diag}\\left\\lbrace \\psi _{1}, \\ldots , \\psi _{n}\\right\\rbrace $ .",
"For the inpainting task, $D=S$ is a diagonal masking matrix.",
"The solution is exactly the same as the general solution in Equation (REF ), but we could implement matrix inverse as element-wise division.",
"$x ^{(k+1)} = \\left( S^TS +\\rho I \\right)^{-1}\\left( S^Ty +\\rho \\tilde{x}^{(k)} \\right).$ Specifically, since $S$ is diagonal, $S^T = S$ and $S^TS$ are diagonal, which makes $(S^TS +\\rho I)$ diagonal as well.",
"Hence, the matrix inversion of $(S^TS +\\rho I)^{-1}$ can be implemented as element-wise division.",
"Furthermore, $S^T = S$ , $S^Ty=Sy$ can be viewed as a masking process and can be efficiently implemented as element-wise multiplication.",
"Following the similar derivations described above, our PnP framework can also be used for other tasks, e.g., HSI deblurring [43], [44], HSI and MSI fusion [45], [46].",
"In short, solving HSI deblurring is identical with HSI super-resolution with the scale factor set to 1, while HSI-MSI fusion needs some modifications of the regularized objective shown in Equation (REF )."
],
[
"Parameter Setting",
"There are two parameters that must be set for each iteration in the PnP-ADMM algorithm, i.e., the penalty parameter $\\rho $ and the noise level of denoiser $\\sigma $ .",
"We adopt a similar strategy presented in [20] to set these parameters.",
"In detail, $\\sigma $ is uniformly sampled from a large noise level $\\sigma _1$ to a small one $\\sigma _2$ in log space.",
"$\\rho $ is determined by the relation $\\sigma =\\sqrt{\\lambda / \\rho }$ where $\\lambda $ is empirically set to $1.5$ .",
"It is noted that the performance may be further improved by tuning the parameters for each image with recent method [47] based on reinforcement learning.",
"The summary of the proposed deep plug-and-play HSI restoration algorithm can be found in Algorithm REF .",
"Degraded HSI ${y}$ , degradation operator $D$ , parameters $\\rho $ and $\\sigma $ for each iteration, number of iterations $K$ and deep denoiser $\\mathcal {D}$ Restored HSI ${x}$ Perform task-specific initialization of $x$ Initialize $v = x$ , $u = 0$ $k=1,2,...,K$ ${x}^{(k+1)} = \\underset{{x}}{\\operatorname{argmin}} \\frac{1}{2} || {D}\\tilde{{x}}^{(k)} - {y} ||^{2}+\\frac{\\rho }{2}\\left\\Vert {x}-\\tilde{{x}}^{(k)}\\right\\Vert ^{2}$ ${v}^{(k+1)}=\\mathcal {D}_{\\sigma }\\left(\\widetilde{{v}}^{(k)}\\right)$ $\\bar{{u}}^{(k+1)} = \\bar{{u}}^{(k)}+\\left({x}^{(k+1)}-{v}^{(k+1)}\\right)$ Deep Plug-and-Play HSI Restoration with Alternating Direction Method of Multipliers."
],
[
"Deep HSI Denoiser",
"In contrast to traditional Gray/RGB image denoising task, HSI denoising for plug-and-play has its unique requirements and challenges.",
"First, HSI is made up of a massive number of bands in a wide range of spectrum that correlate with each other and should be considered in the process of denoising.",
"As such, direct use of Gray/RGB denoisers, such as FFDNet [21], DnCNN [38], and IRCNN [35], would not achieve decent performance due to the lack of the exploration of spatial-spectral correlation in HSIs.",
"Second, most existing denoisers are only able to handle a or a set of fixed noise levels, which is not suitable for the plug-and-play framework.",
"In the following, we present our Gated Recurrent Convolutional Neural Network (GRCNN) to address the aforementioned issues."
],
[
"Network Architecture",
"Inspired by the commonly used UNet [48] and the recently proposed QRNN3D [14] architectures, we design our gated recurrent convolutional neural network as a deep encoder-decoder to exploit the complex information underlying HSIs.",
"As shown in Figure REF , the encoder consists of repeated application of a downsample gated recurrent convolution (GRConv) unit to decrease the spatial size and a GRConv residual block to increase the number of features.",
"The decoder is symmetrically set up, and every step in the expansive path includes an upsample GRConv unit and a residual block to reconstruct the clean HSIs.",
"At the first and final layers, we use two bidirectional GRConv units to handle the mapping between input/output HSIs and extracted features.",
"Meanwhile, long-term skip connections in the form of concatenation are also used to inject shallow features from the encoder into the decoder to guide the reconstruction of clear images."
],
[
"Gated Recurrent Convolution Unit",
"Intuitively, if we assume the noise is randomly and independently added to each individual pixel at each band, it is highly possible that one noisy pixel in one band remains relatively clean in another band.",
"Such intuition suggests that if we can somehow know the quality of pixels in different bands at the same spatial location, we might be able to utilize the clean pixels to guide the denoising of the noisy ones via the global spectral correlation.",
"Based on the above discussion, we introduce a gated recurrent convolution (GRConv) unit to replace the traditional convolutional layer as the basic building block in our network, and its overall structure is shown in Figure REF .",
"Figure: The overall structure of the gated recurrent convolution unit.",
"II, WW, FF, HH denote the input feature maps, weight maps, candidate feature maps, and merged feature maps, respectively.In detail, the proposed GRConv first separately performs two 3D convolution/deconvolution on input features $I$ to obtain a set of pixel-wise weight maps $W$ and feature maps $F$ band by band, as described in Equation (REF ) where $h_w, h_f$ are two 3D filters, $\\otimes $ represents the convolution, and $\\sigma $ is the sigmoid activation.",
"The weight map ranges from 0 to 1 and acts as an indicator telling what percentage of information in the corresponding feature map we should keep.",
"$\\begin{aligned}W & = \\sigma \\left(h_w \\otimes I \\right) \\\\F & = \\tanh \\left(h_f \\otimes I \\right)\\end{aligned}$ Then, a weighted merging step is applied to fuse the candidate features among the bands recurrently as shown in Equation (REF ): $h_i = (1-w_i) \\odot h_{i-1} + w_i \\odot f_i, \\,\\,\\, \\forall i\\in [1,b],$ where $\\odot $ denotes the element-wise multiplication and $w_i$ , $f_i$ , $h_i$ denote the weight map, the candidate feature map and the fused feature map at the $i^{th}$ band, respectively.",
"It should be noted that Equation (REF ) only merges the features in one direction.",
"In the bidirectional GRConv unit, we perform the same fusion with different weight maps in another direction and stack the fused feature maps in both directions to form the final feature maps.",
"To reduce the computational complexity, all the GRConv units except the first and the last ones are single directional, and we alternatively change the merging directions to make sure each band can receive guidance from both directions."
],
[
"Residual Block",
"With the network goes deeper and deeper, it becomes increasingly difficult to stabilize the training and make the model converge to a desirable local minimum.",
"Inspired by the ResNet [49], we build our basic network component as a residual block which consists of two $3\\times 3\\times 3$ gated recurrent convolution/deconvolution units and one projection shortcut implemented with a $1\\times 1\\times 1$ GRConv unit to keep the shallow information flow.",
"The shortcuts together with the skip connections enable both the short- and long-term information interaction, thereby enlarging the representation capacity of the network.",
"Figure: The illustration of Resblock.",
"It contains two 3×33\\times 3 GRConv unit and one 1×11\\times 1 GRConv unit to increase the number of feature maps, while keeping the spatial resolution unchanged."
],
[
"Noise Level Map",
"As indicated in Equation (), the denoiser for our PnP-ADMM algorithm should be a Gaussian denoiser and be able to handle a relatively wide range of continuous noise levels.",
"Existing Gray/RGB/HSI denoisers are not ideal for this task as most of them are designed for only a small set of noise levels.",
"Although we can directly train the network with a training dataset covering a range of continuous noise levels, such a strategy lacks effective guidance, which might impose an extra burden on the network for inferencing the noise strength.",
"With the aim of improving the generalizability to remove noise at continuous levels, we propose to train the denoiser with noise level map as an additional input, which acts as a teacher that directly tells the network the demanded denoising strength."
],
[
"Experiments",
"In this section, we first evaluate the proposed deep HSI denoiser on Gaussian and complex HSI denoising (Section REF ).",
"Then, without any additional training, the proposed denoiser is plugged into our plug-and-play framework to solve the other three classical HSI restoration tasks.",
"The experimental results in super-resolution, compressed sensing, and inpainting are provided in Section REF , Section REF and Section REF , respectively.",
"The detailed ablation study of our approach is presented in Section REF in the final."
],
[
"Dataset",
"We train the proposed denoiser with the natural hyperspectral image dataset, i.e., ICVL [50], which consists of 201 images with 31 spectral bands in the size of $1392 \\times 1300$ .",
"We randomly select 100 images for training, 5 images for validation, and 50 images for testing.",
"Each image in the training set is cropped into multiple overlapped volumes of size $64\\times 64\\times 31$ to enlarge the dataset.",
"Besides, data augmentation techniques such as rotation and scaling are also adopted, resulting in roughly 50k training samples in total.",
"The main region of each test image with the size of $512\\times 512\\times 31$ is used for testing."
],
[
"Implementation Details",
"We implement the proposed denoising network in PyTorch [51] .",
"The Adam [52] optimizer with default parameters is adopted to minimize the L2 loss between reconstructed HSI and ground truth.",
"In order to obtain a robust denoiser for our plug-and-play framework, the network is first trained on a fixed noise level of 50 for 30 epochs and then fine-tuned on random noise levels ranging from 0 to 50 for another 20 epochs.",
"This version of the trained denoiser is used for the experiments of HSI super-resolution, compressed sensing and inpainting.",
"When compared with other gaussian denoising methods, we further fine-tune our model with noisy images randomly corrupted by four fixed noise levels, i.e.",
"10, 30, 50, 70, to obtain the model for the evaluation on Gaussian denoising task.",
"When compared with other complex denoising methods, our network is trained without the noise level map using the same strategy of Gaussian noise for the first 50 epochs and then fine-tune on complex noise for another 50 epochs.",
"The initial learning rate is set to $10^{-3}$ and gradually decayed to stabilize the training.",
"It takes roughly two days for a complete training with an NVIDIA RTX 3090 GPU.",
"Without any additional training, the pretrained denoiser can be plugged into our plug-and-play framework to solve other HSI restoration tasks, but the best hyperparameters vary for different tasks.",
"In order to achieve the best performance, we empirically choose the best hyperparameters of our PnP-ADMM algorithm for HSI super-resolution, compressed sensing, and inpainting tasks.",
"Besides, the inputs for the PnP-ADMM are initialized with bicubic interpolation and triangulation-based linear interpolation for HSI super-resolution and inpainting, respectively."
],
[
"Quantitative Metrics",
"To evaluate the performance of the proposed method comprehensively, three quantitative quality indices are used, i.e., PSNR, SSIM [53], and SAM [15].",
"PSNR and SSIM measure the spatial quality while SAM focuses on the spectral quality.",
"The larger PSNR and SSIM values the better the restore images are, while the smaller values of SAM imply better performance.",
"PSNR and SSIM are calculated as the average of the bandwise results for each HSI.",
"Table: Quantitative denoising results of different methods under several noise levels on ICVL dataset.",
"[30, 70] suggests each image is corrupted by Gaussian noise with random σ\\sigma (ranged from 30 to 70).Figure: Simulated Gaussian noise removal results at the 20th band of images under noise level σ=50\\sigma =50 on ICVL dataset.",
"The first row shows the denoising results.",
"The second row includes the corresponding error maps.",
"The error maps are the absolution errors between the ground truth and the recovered results.",
"The brighter position indicates the higher error.Table: Quantitative denoising results of different methods under three complex noise cases on ICVL dataset.",
"non-iid denotes Non i.i.d Gaussian noise.",
"g+stripe denotes Non i.i.d Gaussian noise and stripe noise.",
"g+impulse denotes Non i.i.d Gaussian noise and impulse noise.",
"(Please refer to Section for the detailed explanation.",
")Figure: Simulated Complex noise removal results at the 20th band of images on ICVL dataset.",
"The first row shows the denoising results.",
"The second row includes the corresponding error maps."
],
[
"Experiments on Gaussian Noise",
"As shown in Equation (), a plug-and-play denoiser should be able to tackle Gaussian denoising with a relatively wide range of noise strength.",
"Therefore, we first evaluate the performance of our denoiser on simulated Gaussian noise.",
"Under this setting, additive Gaussian white noise is added to each input HSI with different strengths, including 30, 50, 70, and random strengths ranging from 30 to 70.",
"We compare our network with five recently developed state-of-the-art HSI denoising algorithms, including three traditional methods, BM4D [27], weighted low-Rank tensor recovery (WLRTR) [11] and non-local meets global (NGmeet) [29], and two deep learning methods, including HSID-CNN [15] and QRNN3D [14].",
"For traditional methods, the parameters are manually or automatically set up according to the original papers.",
"For deep learning methods, we fine-tune/retrain their pre-trained models using the same dataset as ours.",
"The quantitative results under different noise levels for ICVL dataset are summarized in Table REF .",
"As we can see, the proposed method outperforms most of the competing methods in terms of PSNR and SSIM.",
"In particular, our method achieves the highest PNSR in the experiment of random noise levels ranging from 30 to 70, which makes it more suitable for plug-and-play denoising.",
"NGmeet is the second-best method and achieves similar performance, but it is far more time-consuming than ours ($\\approx $ 100 times).",
"Figure REF shows the visual comparison of different methods under noise level $\\sigma =50$ .",
"As we can see, our method can properly removing the Gaussian noise while preserving the fine-grained details in HSIs.",
"Traditional methods, such as BM4D and WLRTR, introduce evident artifacts in some areas.",
"NGmeet performs better for removing artifacts, but it produces the over-smooth image.",
"The image restored by QRNN3D demonstrates the most similar visual appearance as ours, but it fails to restore a smooth result while facing a large area of white pixels (see the mirror in the upper left corner)."
],
[
"Experiments on Complex Noise",
"To further demonstrate the capability of our denoiser for HSI denoising.",
"We also conduct experiments on simulated complex noise as it is more common in real scenes.",
"Three complex noise cases are tested, including: non-iid: each band in the HSI is corrupted by zero-mean Gaussian noise with different intensities ranging from 0 to 70; g+stripe: apart from non-iid noise, stripe noise (5%-15% percentages of columns) is randomly added to one-third of bands; g+impulse: apart from non-iid noise, impulse noise with intensity ranging from 10% to 70% is randomly added to one-third of bands.",
"As the noise level map is not available for complex noise, the proposed network without the noise level map is used for evaluation under these cases.",
"Since traditional methods work best under specific noise assumptions, we compare four traditional baselines different from those used in Gaussian noise.",
"The complete list of competing methods includes low-rank matrix recovery approaches (LRMR [55], LRTV [56], NMoG [57]), low-rank tensor approach (TDTV [10]), and deep-learning methods (HSID-CNN [15], QRNN3D [14]).",
"Table REF shows the quantitative results under three complex noise cases on ICVL dataset.",
"It can be seen that our method achieves the best performance for all three noise cases.",
"Specifically, our method is significantly better than traditional baselines.",
"QRNN3D achieves the most competitive results as ours, but it is apparently inferior to ours under g+impulse noise case.",
"Figure REF provides the visual comparison of our method and competing ones.",
"It can be seen that our method can properly remove the noise while retaining more details than the others.",
"Traditional methods either fail to remove most of the noise (LRMR, NMoG), or produce over-smooth results (LRTV, TTDTV).",
"QRNN3D produces most similar results as ours under non-iid and g+stripe cases but is worse than ours under g+impulse cases, which is consistent with the quantitative results.",
"Table: Quantitative super-resolution results of different methods under different scale factors on ICVL dataset.",
"Our method is training-free.",
"Clean means the competing methods are trained and tested on clean data.",
"Noisy means the competing methods are trained and tested on noisy data.",
"Clean2Noisy means the competing methods are trained on clean data but tested on noisy one.Figure: The super-resolution results at the 20th band of images on the ICVL dataset for Noisy (row 1-2, scale factor=2) and Clean (row 2-3, scale factor=4) cases.",
"The first row shows the super-resolution results.",
"The second row includes the corresponding error maps."
],
[
"HSI Super-Resolution",
"Without any extra training, our denoiser can be plugged into the PnP-ADMM algorithm to solve the HSI super-resolution problem.",
"To demonstrate the flexibility and powerfulness of our method, we test our method under Clean, Noisy, and Clean2Noisy settings for three scale factors, i.e., 2, 4, 8.",
"Specifically, the low-resolution HSIs are generated by first blurring the original HSIs via a $8\\times 8$ Gaussian kernel with $\\sigma =3$ , and then downsampled by a scaling factor of 2, 4, or 8.",
"Under Noisy and Clean2Noisy settings, additional Gaussian noise with $\\sigma _n=10$ is added.",
"For comparison, we adopt the bicubic interpolation as the baseline and compare our method against four deep-learning-based methods, i.e., IFN [16], 3D-FCNN [17], SSPSR [58], and Bi-3DQRNN [59].",
"Traditional HSI super-resolution methods [60], [61], [62] are not considered as most of them focus on super-resolution with additional inputs, such as multiple frames and RGB images.",
"We directly apply our PnP framework to the aforementioned settings without any extra training, while all the competing methods are trained with the corresponding datasets for Clean and Noisy settings.",
"In the setting of Clean2Noisy, the models from Clean settings are directly applied to evaluate the generalizability of each method.",
"The quantitative results of comparison on ICVL dataset are provided in Table REF .",
"As we can see, our method obtains the best results under Clean and Clean2Noisy settings.",
"Under Noisy setting, our method achieves better results against IFN and 3D-FCNN and achieves the best PSNR at the scale factors of $\\times 2$ and $\\times 8$ .",
"SSPSR is the second-best method and achieves better results than ours when scale factor = 2 under Noisy setting.",
"It can be partially explained that SSPSR is specifically trained on the noisy dataset, while ours is training-free.",
"Nevertheless, when it comes to Clean2Noisy setting, SSPSR trained on the clean dataset completely fails to generalize to the noisy testset, as all the other competing methods do.",
"As for visual comparisons shown in Figure REF , we can observe that the images produced by our method retain most details for Clean case.",
"As for Noisy case, ours is prominently better than IFN and 3DFCNN but a little more blurred than SSPSR.",
"Overall, our method achieves the competitive or even better performance against the deep-learning-based methods without any task-specific training."
],
[
"HSI Compressed Sensing",
"Compressive HSI imaging is a common technique for improving the spatial and temporal resolution of HSI.",
"Such a system captures a single snapshot measurement frame that encodes the information that can be used for recovering the original high-dimensional data with specific algorithms.",
"In this experiment, we extend our method for coded aperture snapshot spectral imagers (CASSI [63]).",
"Three state-of-the-art compressed HSI reconstruction methods for CASSI are chosen for comparison, including GAP-TV [64], DeSCI [42] and NGmeet [65].",
"Following the setting in [42], the CAVE Toy image is used for the experiment and the simulated data is generated by the same compressed operator.",
"Table: Quantitative compressed HSI reconstruction results of different methods on the CAVE Toy image.The reconstruction results of GAP-TV [64] and DeSCI [42] were provided in [42], and the results of NGmeet were provided in [65].",
"Table REF shows the quantitative results of our method and the competing ones.",
"It can be seen that our method is significantly better in the metrics of both PSNR and SSIM."
],
[
"HSI Inpainting",
"We also extend our method for HSI inpainting and compare the performance against two recent proposed HSI inpainting methods, i.e., weighted low-rank tensor recovery (WLRTR) [11], and FastHyIn [66].",
"We generate the simulated data with stripe masks where stripes are randomly distributed in all bands.",
"To simulate the real degraded HSIs, the Gaussian noise of two noise levels 30 and 50 are added.",
"As shown in Figure REF , although WLRTR and FastHyIn can remove the extra Gaussian noise properly, both of them fail to remove the stripes completely.",
"In contrast, our method produces better results with diminished stripes and noise.",
"The qualitative assessment results are listed in Table REF .",
"Compared with all competing methods, our plug-and-play method achieves the best performance in all qualitative/quantitative assessments, further confirming the high fidelity of our method.",
"Table: Quantitative inpainting results of different methods under stripe mask and different noise levels on ICVL dataset."
],
[
"Discussion and Analysis",
"In this section, we further discuss and analyze the proposed method.",
"We first demonstrate the functionality of each network component in our deep denoiser, whose effectiveness is further verified by comparing it with two traditional Gray/RGB PnP denoisers.",
"Then, we present the inpainting results of our method on the other three unseen HSI datasets, i.e., CAVE [67], Harvard [68], and Pavia University [69], to evaluate the generalizability of our method.",
"At the end of this section, we analyze the computational complexity of the proposed method and provide comparisons against other methods."
],
[
"Ablation Study on the Proposed Denoiser",
"To verify the effectiveness of the proposed components in our deep denoiser, i.e.",
"residual block, gated recurrent convolution, and additional noise level map input, we compare the performance of the complete network and the variants that are generated by gradually removing the aforementioned components one by one.",
"All networks are trained with the same dataset and training strategy.",
"Specifically, we denote the complete network as Full, the network without noise level map as -Map, the network without both noise level map and residual connection as -Res, and the network without all the three components as -Gate.",
"The quantitative results of these variants on ICVL dataset are listed in Table REF .",
"Take noise level 30 as an example, it can be easily observed that there is a significant improvement (1.84dB) after adding the gated recurrent convolution, which demonstrates the importance of exploiting the spectral correlation in HSIs.",
"Besides, there are also prominent improvements after adding residual block and noise level map, i.e.",
"0.83dB and 0.52dB, which verifies the corresponding effectiveness.",
"Table: Ablation study of the proposed denoising network on ICVL dataset.Figure: Visual comparison with traditional PnP denoisers on super-resolution task and ICVL dataset.Figure: The computational complexity of different competing methods for denoising, super-resolution, and inpainting tasks.",
"Since the source code of NGmeet for compressed sensing is not available, the results of compressed sensing are not included."
],
[
"Comparison with Traditional PnP Denoisers",
"In order to prove the superiority of our denoiser within the PnP framework against traditional Gray/RGB PnP denoiser, we conduct an experiment on super-resolution for two widely used state-of-the-art denoisers in grayscale image restoration, i.e., IRCNN [38] and DRUnet [20].",
"By denoising each band of HSIs sequentially, these denoisers are plugged into our plug-and-play framework for evaluation.",
"As shown in Table REF , our denoiser is consistently better than the competing grayscale denoisers in terms of all metrics, indicating it serves as a better plug-and-play prior for HSIs.",
"The visual comparison in Figure REF also verifies the superiority of our method.",
"It can be easily observed that the restored images by grayscale denoiser are blurred and introduce evident artifacts, while our method produces clearer results without evident artifacts.",
"Table: Quantitative comparison against other PnP denoisers on super-resolution task and ICVL dataset.",
"All the denoisers are directly plugged into the same PnP framework to evaluate the performance."
],
[
"Comparisons on More Datasets",
"In order to test the generalizability of our plug-and-play method over different scenes, we conduct a series of experiments for HSI inpainting on other three HSI datasets, including two natural HSI datasets, i.e., CAVE [67] and Harvard [68], and one remote-sensed dataset, i.e., Pavia University [69].",
"It should be noted that we directly apply the denoiser trained on the ICVL dataset for these experiments.",
"As shown in Table REF , our method outperforms the competing methods even though these scenes are completely unseen by the denoiser, suggesting the strong robustness of our method.",
"Table: Quantitative inpainting results (PSNR) of different methods on the CAVE, Harvard, and Pavia University datasets.Table: Quantitative evaluation of the proposed PnP approach against competing deep-learning-based methods by the number of parameters (Params) and PSNR.",
"The number of parameters under different settings is the same, and we provide the PSNR results on two settings for comparison, i.e., mixed noise levels of [30,70] for Gaussian denoising, and 4×4\\times scale factor for super-resolution."
],
[
"Computational Efficiency",
"In this section, we analyze and compare the computational efficiency of our method and the competing classical and deep-learning-based methods.",
"As indicated in Algorithm REF , our method is iterative based, and most of the computation lies in the forward computation of deep denoiser since the data-fidelity term can generally be solved in closed-form.",
"Hence, the approximate time complexity of our method is $O(nD)$ where $n$ is the number of iteration and $D$ is the inference time of denoiser.",
"For denoising task, $n=1$ .",
"For super-resolution, compressed sensing, and inpainting, $n$ is set to 25, 50, 100, respectively.",
"Figure REF shows the approximate running time of different methods on denoising, super-resolution, and inpainting tasks.",
"All the tests are run on an i9-10850k CPU and an Nvidia RTX 3090 GPU.",
"It can be seen that our method is significantly faster than most traditional methods while preserving superior performance for denoising tasks.",
"QRNN3D is slightly faster than ours at the cost of slight performance degradation.",
"For super-resolution, our method is training-free and achieves the best performance, but is slower than the competing pure deep-learning-based methods, which are all trained specifically.",
"Table REF also provides the quantitative comparison by the number of parameters on the denoising and super-resolution task.",
"For inpainting tasks, all the methods are training-free.",
"It can be observed that our method is significantly faster than WLRTR and achieve better performance.",
"FastHyIn is faster than ours, but its performance also degrades severely.",
"Overall, when compared with traditional methods, our method is significantly faster than most of them and achieves better performance.",
"Simultaneously, when compared with specifically trained deep-learning-based methods, our method can also achieve better performance without any specific training at the cost of a tiny increase in running time."
],
[
"Conclusion",
"In this paper, we develop a flexible and effective PnP-ADMM approach for HSI restoration, in which a novel deep HSI denoiser is introduced as an implicit PnP image prior.",
"With careful design, our denoiser not only achieves the state-of-the-art performance in HSI denoising task by properly exploiting the intrinsic characteristics underlying HSIs, but also, acting as a powerful HSI prior, finely supports the plug-and-play framework to solve different HSI restoration tasks without any additional training.",
"The extensive experimental results and analysis show that our method is able to achieve superior generalizability and competitive or even better performance against the learning-based methods in a variety of HSI restoration problems."
],
[
"Acknowledgement",
"This work was supported by the National Natural Science Foundation of China under Grants No.",
"62171038, No.",
"61827901, and No.",
"62088101."
]
] | 2209.08240 |
[
[
"Deterministic-Statistical Approach for an Inverse Acoustic Source\n Problem using Multiple Frequency Limited Aperture Data"
],
[
"Abstract We propose a deterministic-statistical method for an inverse source problem using multiple frequency limited aperture far field data.",
"The direct sampling method is used to obtain a disc such that it contains the compact support of the source.",
"The Dirichlet eigenfunctions of the disc are used to expand the source function.",
"Then the inverse problem is recast as a statistical inference problem for the expansion coefficients and the Bayesian inversion is employed to reconstruct the coefficients.",
"The stability of the statistical inverse problem with respect to the measured data is justified in the sense of Hellinger distance.",
"A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm is implemented to explore the posterior density function of the unknowns.",
"Numerical examples show that the proposed method is effective for both smooth and non-smooth sources given limited-aperture data."
],
[
"Introduction",
"In recent years, the inverse problem of determining an unknown source function has attracted significant attention due to its practical importance in many applications such as the biomedical imaging and the identification of pollution sources [19], [11], [12], [9], [15], [14].",
"The reconstruction of the acoustic source using single frequency data is challenging.",
"Inverse source problems at a fixed frequency do not possess a unique solution due to the existence of non-radiating sources [2], [1].",
"For multiple-frequency data, the uniqueness of the inverse source problem is derived in [14] for a chosen unbounded set of the Dirichlet eigenvalues of the Laplacian using near field data (see also [21] for the uniqueness with the measurements taken on a bounded band of frequency).",
"The use of multiple frequency data improves the stability of the inverse source problem [7].",
"Accordingly, many researchers consider the reconstruction of an extended acoustic source problem using multiple frequency data.",
"Various methods have been proposed in the last decade including the continuation methods [8], [5], eigenfunction expansion methods [14], [13], and sampling type methods [9], [3], [16], [4].",
"Bayesian statistics is a classical approach for inverse problems [18].",
"Due to the increase of the computational power, Bayesian inversion has been becoming more popular [27], [28], [6], [32], [33].",
"Recently, focusing on partial data, we combined the deterministic methods and Bayesian inversion to successfully treat several inverse problems including an inverse scattering problem, an inverse source problem, and the reconstruction of moving point sources using limited-aperture data [23], [22], [24].",
"In particular, we use certain deterministic method to obtain qualitative information of the unknowns.",
"Such information is built into the priors for the Bayesian inversion, which is then used to compute more details of the unknowns.",
"Both the deterministic method and the Bayesian inversion use the same measured data.",
"Numerical results show that such a combination can provide better reconstructions.",
"In this paper, we propose a deterministic-statistical approach for an inverse source problem using multiple frequency limited aperture data.",
"The direct sampling method is used to find the support of the source.",
"A disc is identified such that the support of the source is contained in the disc.",
"Using the Dirichlet eigenfunctions of the disc (Bessel's functions) as the basis, we expand the source function.",
"These coefficients are the unknowns for the Bayesian inverse problem, whose posterior density function is explored using an M-H (Metropolis-Hastings) MCMC (Markov chain Monte Carlo) algorithm.",
"The conditional mean (CM) is used to represent the solution.",
"Numerical examples show that the proposed approach is effective for limited-aperture data.",
"The rest of the paper is organized as follows.",
"In Section 2, we introduce the inverse acoustic source problem of interest.",
"Section 3 presents the direct sampling method to reconstruct a disc that contains the support of the source.",
"In Section 4, we first expand the unknown source using the Dirichlet eigenfunctions of the disc and propose a Bayesian approach to reconstruct the expansion coefficients.",
"The proposed method is validated by various numerical examples in Section 5.",
"Finally, we discuss the method and make some conclusions in Section 6."
],
[
"The inverse source problem",
"Let $\\Omega $ be a bounded domain in $\\mathbb {R}^2$ with a Lipschitz boundary $\\partial \\Omega $ .",
"We assume that $\\mathbb {R}^2 \\setminus {\\overline{\\Omega }}$ is connected.",
"Let $u$ be the outgoing solution to the inhomogeneous Helmholtz equation in $\\mathbb {R}^2$ : $\\begin{split}&\\Delta u(x,k)+k^2 u(x,k) =f(x), ~ x=(x_{1},x_{2}) \\in \\mathbb {R}^2, \\\\&\\lim _{r\\rightarrow \\infty } \\sqrt{r} \\left(\\frac{\\partial u}{\\partial r}-iku\\right)=0, \\qquad r=|x|,\\end{split}$ where $k \\in K$ , $K=[k_a, k_b]$ , $0 < k_a < k_b$ , is the wavenumber and $f(x)\\in L^2(\\Omega )$ with $\\text{supp}f\\subset \\Omega $ .",
"Note that $k$ is proportional to the frequency.",
"There exists a unique solution $u$ to (REF ) given by (see [17]) $u({x},k)=\\int _{\\Omega }\\Phi _k(x,y) f(y)dy,$ where $\\Phi _k(x,y)=-\\frac{i}{4}H_{0}^{(1)}(k|x-y|)$ is the fundamental solution to the Helmholtz equation and $H_{0}^{(1)}$ denotes the zeroth-order Hankel function of the first kind.",
"Furthermore, $u(x,k)$ has the asymptotic behavior [17] $u(x,k)=\\frac{e^{i\\frac{\\pi }{4}}}{\\sqrt{8k\\pi }}\\frac{e^{ikr}}{\\sqrt{r}}\\left\\lbrace u^{\\infty }(\\hat{x},k)+\\mathcal {O}\\left(\\frac{1}{r}\\right)\\right\\rbrace \\quad \\text{as}\\; r\\rightarrow \\infty ,$ where $\\hat{x}={x}/{|x|}\\in \\mathbb {S}$ , $\\mathbb {S}:=\\lbrace |\\hat{x}| = 1 : \\hat{x} \\in \\mathbb {R}^2\\rbrace $ .",
"The far field pattern $u^{\\infty }(\\hat{x},k)$ of $u(x,k)$ is given by $u^{\\infty }(\\hat{x},k)=\\int _{\\Omega }\\Phi ^{\\infty }_k(\\hat{x},y) f(y)dy,$ where $\\Phi ^{\\infty }_k(\\hat{x},y)=\\exp {(-ik\\hat{x}\\cdot y)}$ is the far field pattern of the fundamental solution $\\Phi _k(x,y)$ .",
"We are interested in the inverse source problem of determining the unknown source $f(x)$ from the partial measurement of the far field pattern $u^{\\infty }(\\hat{x},k)$ prescribed on the unit circle $\\mathbb {S}$ for multiple $k$ 's, i.e.",
"reconstruct $f(x)$ from $U:=\\lbrace u^{\\infty }(\\hat{x},k)|\\hat{x}\\in \\Gamma ,k\\in K \\rbrace $ , where $\\Gamma \\subset \\mathbb {S}$ .",
"In practice, the measurement data is usually discrete $u^{\\infty }(\\hat{x}_i,k_j)$ for $\\hat{x}_i \\in \\Gamma , i=1, 2, \\ldots , I$ and $k_j \\in K, j=1, 2, \\ldots , J$ .",
"We propose a deterministic-statistical approach to reconstruct the source function in two steps.",
"Firstly, the direct sampling method (DSM) is applied to obtain a disc $\\hat{B}$ which contains the compact support of the source function $f(x)$ .",
"Secondly, we expand $f(x)$ in terms of the Dirichlet eigenfunctions of $\\hat{B}$ and employ the Bayesian statistics to recover the expansion coefficients.",
"Note that, ideally, the disc $\\hat{B}$ should be such that $\\text{supp}f\\subset \\hat{B}$ and $\\hat{B}\\setminus \\text{supp}f$ is not too large."
],
[
"Direct Sampling Method",
"The direct sampling method was proposed in [20] to reconstruct small scattering objects.",
"It is simple and effective to reconstruct the support of the unknown target (obstacle, inhomogeneous medium, source) and can process limited aperture data.",
"Following [23], [16], for multiple frequency far field pattern, we employ the direct sampling method to determine a disc such that it contains the compact support of the source.",
"It turns out that the DSM is effective to obtain a disc, which is important for the success of the Bayesian inversion.",
"Assume that a domain $D$ is known such that $\\Omega \\subset D$ , i.e., the source function $f(x)$ lies inside $D$ .",
"Usually, $D$ is the region of interest and is quite large.",
"Let $D$ be covered by a set of uniformly distributed sampling points $S$ .",
"For each point $x_{p} \\in S$ , we define an indicator function $I(x_{p})=\\frac{ | \\sum _{k_j} \\langle u^{\\infty }(\\hat{x},k_j),\\Phi _{k_j}^{\\infty }(\\hat{x},x_{p}) \\rangle _{L^2(\\Gamma )} | }{ \\sum _{k_j} \\Vert u^{\\infty }(\\hat{x},k_j) \\Vert _{L^2(\\Gamma )} \\Vert \\Phi _{k_j}^{\\infty }(\\hat{x},x_{p}) \\Vert _{L^2(\\Gamma )} },$ where the inner product $ \\langle \\cdot ,\\cdot \\rangle _{L^2(\\Gamma )}$ is defined as $\\langle u^{\\infty }(\\hat{x},k_j),\\Phi _{k_j}^{\\infty }(\\hat{x},x_{p}) \\rangle _{L^2(\\Gamma )}=\\int _{L^2(\\Gamma )} u^{\\infty }(\\hat{x},k_j) \\bar{\\Phi } _{k_j}^{\\infty }(\\hat{x},x_{p}) ds(\\hat{x})$ and $\\bar{\\Phi } _{k_j}^{\\infty }(\\hat{x},x_{p})$ is the conjugate of ${\\Phi } _{k_j}^{\\infty }(\\hat{x},x_{p})$ .",
"In the case of discrete data $u^{\\infty }(\\hat{x}_i,k_j), i=1, \\ldots , I, j=1, \\ldots , J$ , the indicator function becomes $I(x_{p})=\\frac{ \\sum _{j=1}^J |\\sum _{i=1}^I u^{\\infty }(\\hat{x}_i,k_j) \\cdot \\overline{\\Phi _{k_j}^{\\infty }(\\hat{x}_i,x_{p})}| }{ \\sum _{j=1}^J \\sqrt{\\sum _{i=1}^I |u^{\\infty }(\\hat{x}_i,k_j)|^2} \\sqrt{\\sum _{i=1}^I|\\Phi _{k_j}^{\\infty }(\\hat{x}_i,x_{p})|^2}}.$ The DSM uses the indicator function to obtain the support of $f(x)$ approximately.",
"It is clear that $I(x_{p}) \\in [0,1]$ .",
"If $ I(x_{p})$ is small (close to 0), then the point $x_{p}$ is likely to lie outside the source.",
"On the other hand, if $ I(x_{p})$ is large (close to 1), $x_{p}$ is likely to lie inside the source.",
"We refer to [4] for some theoretical justification of the indicator function.",
"Based on the value of the indicator function, we are able to find a subdomain $\\hat{B} \\subset D$ containing the support of the source such that $I(x_{p})$ is larger than a cutoff value $\\gamma $ for $x_p \\in \\hat{B}$ .",
"In particular, we will take $\\hat{B}$ as a disc with radius $R$ .",
"The radius $R$ is given by $R=\\max _{ x_{p}\\in D, I(x_{p}) \\ge \\gamma } \\Vert x_{p}\\Vert ,$ The motivation to use a disc $\\hat{B}$ is two folds.",
"Firstly, a disc can easily cover the compact support of the source.",
"Secondly, the Dirichlet eigenfunctions for a disc are known.",
"Note that a square/rectangle domain also works.",
"The algorithm for multiple frequency limited aperture inverse source problems (MFLAISP) is as follows.",
"DSM for MFLAISP 1.",
"Collect the data $u^{\\infty }(\\hat{x}_i,k_j), i=1, \\ldots , I, j=1, \\ldots , J$ for $x_i \\in \\Gamma $ and $k_j\\in K$ .",
"2.",
"Generate sampling points set $S$ for $D$ .",
"3.",
"For each $x_p \\in S$ , compute ${I}(x_p)$ using (REF ).",
"4.",
"Identify a disc $\\hat{B}$ using $I(x_p)$ with radius $R$ given by (REF ).",
"We remark that other deterministic methods such as the orthogonality sampling method and extended sampling method [30], [31] can also be used as long as such a method can provide a good prediction of a disc (or a square) that contains the compact support of the source."
],
[
"Bayesian Inversion",
"We expand the source using the Dirichlet eigenfunctions of $\\hat{B}$ obtained by the DSM, and use Bayesian inversion to explore the posterior density function of the expansion coefficients.",
"In particular, we shall construct an approximation $f_{BE}$ for the source $f$ in a finite-dimensional subspace spanned by the Dirichlet eigenfunctions of $\\hat{B}$ .",
"Let $\\Vert \\cdot \\Vert $ be the usual $L^2$ -norm.",
"The Dirichlet eigenvalue problem (see, e.g., [25]) is to find $\\lambda $ and nontrivial $w$ such that $ \\begin{split}-\\Delta w &= \\lambda w \\quad {\\text{in}}~ \\hat{B}, \\\\w&=0 \\quad \\text{on} ~\\partial \\hat{B}.\\end{split}$ We call $\\lambda $ the Dirichlet eigenvalue and $w$ the eigenfunction corresponding to $\\lambda $ .",
"All the eigenvalues are positive and have no finite point of accumulation.",
"Since the Dirichlet eigenfunctions are associated to an elliptic self-adjoint compact operator on $L^2(\\hat{B})$ , $\\lbrace w_{n}\\rbrace _{n=1}^\\infty $ forms a complete orthonormal set [14].",
"Consequently, one can expand $f$ as $f(x)= \\sum _{n=1}^\\infty A_{n}w_{n},$ where the Fourier coefficients are given by $A_{n}=\\int _{\\hat{B}} f(x)w_{n}(x)~ dx.$ Using polar coordinate $x=(r \\cos (\\theta ), r\\sin (\\theta ))$ , the Dirichlet eigenfunctions of a disc $\\hat{B}$ centered at the origin with radius $R$ are given by [14] $\\begin{split}& Q_{mn}^{1}(x)=\\frac{1}{\\sqrt{\\pi }RJ_{n+1}(q_{mn})}J_{n}\\left(\\frac{q_{mn}r}{R}\\right)\\cos {(n\\theta )},\\quad m=1,2,3,\\cdots , n=0,1,2,\\cdots , \\\\& Q_{mn}^{2}(x)=\\frac{1}{\\sqrt{\\pi }RJ_{n+1}(q_{mn})}J_{n}\\left(\\frac{q_{mn}r}{R}\\right)\\sin {(n\\theta )},\\quad m=1,2,3,\\cdots , n=1,2,\\cdots ,\\end{split}$ where $J_{n}$ is the Bessel function of order $n$ and $q_{mn}$ is the $m$ th zero of $J_{n}$ .",
"These eigenfunctions satisfy $\\Delta Q_{mn}^{j} + k_{mn}^2 Q_{mn}^{j} =0, \\quad j=1,2,$ with wavenumber $k_{mn}=q_{mn}/R $ .",
"An approximation $f_{BE}$ of $f$ on the disc $\\hat{B}$ is given by $f_{BE}(x)= \\sum _{m=1} ^M \\left( \\sum _{n=0}^N A_{mn}^{1} Q_{mn}^{1}(x)+\\sum _{n=1}^N A_{mn}^{2} Q_{mn}^{2}(x) \\right),$ where $A_{mn}^1 =\\int _{\\hat{B}} f(x) Q_{mn}^1(x) dx , \\quad A_{mn}^2 = \\int _{\\hat{B}} f(x) Q_{mn}^1(x) dx.$ Denote $ H^{s}(\\hat{B})$ the Sobolev space of order $s>0$ equipped with the standard norm $\\Vert \\cdot \\Vert _s$ .",
"Moreover, $ H^{s}_{0}(\\hat{B})$ is defined as the closure of $C_{0}^\\infty (\\hat{B})$ with respect to the norm in $ H^{s}(\\hat{B})$ .",
"The property of the Dirichlet eigenfunction expansion is stated in the following lemma [14].",
"Lemma 4.1 Let $f\\in H^{s}_{0}(\\hat{B})$ with $s>1$ .",
"Furthermore, let $\\lbrace w_{n}\\rbrace _{n=1}^\\infty $ be the set of normalized Dirichlet eigenfunctions of $\\hat{B}$ .",
"There exists a constant $C$ depending only on $\\hat{B}$ such that $\\Vert f- f_{BE}\\Vert \\le C \\Vert f \\Vert _{s} N^{(1-s)/2}.$ Remark 4.1 Here we choose a disc $\\hat{B}$ since the Dirichlet eigenfunctions are known analytically.",
"One can also use rectangular domains containing the support of the source.",
"If a general domain, e.g., a polygon, is used, the Dirichlet eigenfunctions can be computed using numerical methods such as the finite element methods (see, e.g., [25]).",
"Let $A$ be the vector $\\lbrace A_{m0}^{1},A_{mn}^{1},A_{mn}^{2}\\rbrace _{n=1,m=1}^{N,M }$ and $X$ be the vector space $\\mathbb {R}^{(2N+1)M}$ .",
"The inverse problem of the reconstruction of the source function becomes the determination of the coefficients $A \\in X$ given the measurement $U$ .",
"Based on the eigenfunction expansion (REF ), we employ the Bayesian inversion to reconstruct $A$ for the source $f(x)$ from the measurement data [27], [28].",
"The statistical model of the inverse source problem can be written as $ U=\\mathcal {F}(A)+\\eta ,$ where $\\mathcal {F}(A)=\\int _{\\hat{B}}\\Phi ^{\\infty }_k(\\hat{x},y) f_{BE}(y)dy$ and $\\eta \\sim \\mathcal {N}(0,\\sigma ^2 \\mathbb {I})$ is the Gaussian noise.",
"Using Bayes' formula [27], [28], the posterior density of the random variable $A$ satisfies $ \\pi (A|U) \\propto \\pi (U|A)\\pi (A),\\\\$ where $\\propto $ means “proportional to”, $\\pi (A)$ represents the prior density of the unknown $A$ , the conditional distribution $\\pi (U|A)=\\mathcal {N}(U-\\mathcal {F}(A),\\sigma ^2 \\mathbb {I})$ is the likelihood function, and the posterior distribution $\\pi (A|U)$ is solution to the Bayesian inverse problem.",
"To represent the statistical information of the unknown $A$ , point estimators are often used, e.g., the conditional mean (CM) $A_{\\text{CM}}=\\mathbb {E}(\\pi (A|U)).$ We now analyze the stability of the Bayesian inverse problem.",
"Define $G(A;U)=\\frac{1}{2\\sigma ^2}\\Vert U-\\mathcal {F}(A)\\Vert ^2_{L^2(\\Gamma )}.$ The relationship (REF ) in terms of measures $\\mu ^{U}$ and $\\mu _{0}$ corresponding to posterior and prior densities can be written as $\\frac{d \\mu ^{U}}{ d \\mu _{0}}(A)=\\frac{1}{L(U)} \\exp \\left( -G(A;U) \\right),$ where $L(U)=\\int _{X} \\exp \\left(-G(A;U) \\right) \\mathrm {d} \\mu _{0}(A)$ is the normalization constant.",
"Lemma 4.2 For integer values of $n$ , the Bessel function of the first kind $ J_{n}(y)$ can be defined by the Hansen-Bessel Formula [29] $J_{n}(y)=\\frac{1}{\\pi } \\int _{0}^{\\pi } \\cos (y \\sin t -nt) dt.$ We prove a property of the operator $\\mathcal {F}$ following [28].",
"Lemma 4.3 There exists a constant $C$ such that, for all $A\\in X$ , $\\Vert \\mathcal {F}(A)\\Vert _{L^2(\\Gamma )}\\leqslant C\\Vert A\\Vert _{1}.$ From the Fourier-Bessel expansion (REF ) and the definition of $\\mathcal {F}(A)$ , we have $\\begin{split}|\\mathcal {F}(A)| &= \\left| \\int _{\\hat{B}}\\Phi ^{\\infty }_k(\\hat{x},y) \\sum _{n,m} A_{nm} Q_{nm}(y) ~dy \\right|\\\\&\\le \\sum _{n,m} |A_{nm}| \\left| \\int _{\\hat{B}}\\exp (-ik\\hat{x}\\cdot y) Q_{nm}(y) ~dy \\right|.\\\\\\end{split}$ For simplicity, we consider $\\hat{B}=B(0,R)$ , the disc centered at the origin with radius $R$ for the proof.",
"The case for a general $\\hat{B}$ is similar.",
"It is clear that $|\\mathcal {F}(A)|\\le \\sum _{n,m} \\pi R^2 |A_{nm}| |Q_{nm}(y)|.\\\\$ According to (REF ), for $y \\in B(0,R)$ , we have that $|J_{n}(y)|= \\left|\\frac{1}{\\pi } \\int _{0}^{\\pi } \\cos (y\\sin t -nt) dt\\right|\\le 1,$ which implies that $ | Q_{nm}(y)| \\le \\frac{1}{\\sqrt{\\pi }RJ_{n+1}(q_{mn})}.$ Combining (REF ) and (REF ), we obtain $\\begin{split}|\\mathcal {F}(A)| & \\le \\frac{ \\sqrt{\\pi } R}{J_{n+1}(q_{mn})} \\sum _{n,m} |A_{nm}|.\\\\\\end{split}$ Consequently, we have that $\\left\\Vert \\mathcal {F}(A)\\right\\Vert _{L^2(\\Gamma )} \\le \\frac{ \\sqrt{2}\\pi R}{J_{n+1}(q_{mn})} \\sum _{n,m} |A_{nm} |=C \\Vert A\\Vert _{1},$ where $C=\\frac{ \\sqrt{2}\\pi R}{J_{n+1}(q_{mn})}$ .",
"Corollary 4.1 For all $A_{1},A_{2}\\in X$ , there exists a constant $C$ , such that $\\Vert \\mathcal {F}(A_1)-\\mathcal {F}(A_2)\\Vert _{L^2(\\Gamma )} \\leqslant C \\Vert A_1-A_2\\Vert _1.$ Definition 4.1 The Hellinger distance between two probability measures $\\mu _1$ and $\\mu _2$ with common reference measure $\\nu $ is defined as $d_{\\rm Hell}(\\mu _1, \\mu _2)=\\left( \\int \\left(\\sqrt{\\mathrm {d} \\mu _{1} / \\mathrm {d} \\nu }-\\sqrt{\\mathrm {d} \\mu _{2} / \\mathrm {d} \\nu }\\right)^2 ~ \\mathrm {d} \\nu \\right)^{1/2}.$ The following theorem states the well-posedness of the Bayesian inverse problem under investigation.",
"Theorem 4.2 Let $\\mu _{0}$ be a Gaussian measure such that $\\mu _0(X)=1$ and $\\mu ^{U} \\ll \\mu _{0}$ .",
"For $U_1$ and $U_2$ with $\\max \\lbrace \\Vert U_1\\Vert _{L^2(\\Gamma )} , \\Vert U_2\\Vert _{L^2(\\Gamma )} \\rbrace \\le r$ , there exists $M=M(r)>0$ such that $d_{\\rm Hell}(\\mu _{U_1}, \\mu _{U_2}) \\le M \\Vert U_1-U_2\\Vert _{L^2(\\Gamma )} .$ From $L(U)=\\int _{X} \\exp \\left( -\\frac{1}{2\\sigma ^2} \\Vert U-\\mathcal {F}(A)\\Vert ^2_{L^2(\\Gamma )} \\right) \\mathrm {d} \\mu _{0}(A),$ we have that $0\\le L(U)\\le 1.$ Using Lemma REF , we obtain that $\\begin{split}L(U) & \\ge \\int _{{X}} \\exp \\left(- \\frac{1}{2\\sigma ^2}~\\Vert U\\Vert ^2_{L^2(\\Gamma )} -\\frac{1}{2\\sigma ^2}~\\Vert \\mathcal {F}(A)\\Vert ^2_{L^2(\\Gamma )} \\right) \\mathrm {d} \\mu _{0}(A) \\\\& \\ge \\int _{\\Vert A\\Vert _{1}\\le 1} \\exp \\left(- \\frac{1}{2\\sigma ^2}~\\Vert U\\Vert ^2_{L^2(\\Gamma )}-\\frac{C}{2\\sigma ^2}~\\Vert A\\Vert _{1}\\right) \\mathrm {d} \\mu _{0}(A) \\\\& = \\exp (-M)\\mu _{0}\\lbrace \\Vert A\\Vert _{1}\\le 1 \\rbrace \\\\& > 0\\end{split}$ since $\\mu _{0}$ is a Gaussian measure.",
"Using the mean value theorem and Lemma REF , for $\\mu _{0}$ , it holds that $ \\begin{split}& \\,| L(U_1)-L(U_2)| \\\\\\le &\\, \\int _{X} \\left|\\exp \\left( -G(A;U_1) \\right)- \\exp \\left( -G(A;U_2) \\right) \\right| \\mathrm {d} \\mu _{0}(A) \\\\ \\le & \\int _{X} \\left| -G(A;U_1) - ( -G(A;U_2) ) \\right| \\mathrm {d} \\mu _{0}(A) \\\\= &\\,\\int _{X} \\left|-\\frac{1}{2\\sigma ^2} \\Vert U_1-\\mathcal {F}(A)\\Vert ^2_{L^2(\\Gamma )}+\\frac{1}{2\\sigma ^2} \\Vert U_2-\\mathcal {F}(A)\\Vert ^2_{L^2(\\Gamma )}) \\right| \\mathrm {d}\\mu _{0}(A) \\\\\\le & \\, \\int _{X} \\frac{1}{2\\sigma ^2}\\left(\\left| \\Vert U_1\\Vert _{L^2(\\Gamma )} ^2-\\Vert U_2\\Vert _{L^2(\\Gamma )} ^2\\right| + 2 \\Vert \\mathcal {F}(A)\\Vert _{L^2(\\Gamma )} ~ \\Vert U_1- U_2\\Vert _{L^2(\\Gamma )} \\right) \\mathrm {d} \\mu _{0}(A)\\\\\\le &\\, \\int _{X} \\frac{1}{2\\sigma ^2}\\left( \\Vert U_1\\Vert _{L^2(\\Gamma )} +\\Vert U_2\\Vert _{L^2(\\Gamma )} + 2C ||A||_{1} \\right) \\mathrm {d} \\mu _{0}(A) \\Vert U_1- U_2\\Vert _{L^2(\\Gamma )} \\\\\\le &\\, M \\Vert U_1-U_2\\Vert _{L^2(\\Gamma )}.\\end{split}$ From the definition of the Hellinger distance, we have that $ \\begin{split}& \\,d_{\\rm Hell}^2(\\mu _{U_1}, \\mu _{U_2}) \\\\ = & \\frac{1}{2} \\int _{X} \\left\\lbrace \\left(\\frac{\\exp (-G(A;U_{1}))}{L(U_{1})} \\right)^{1/2} -\\left(\\frac{\\exp (-G(A;U_{2}))}{L(U_{2})} \\right)^{1/2}\\right\\rbrace ^2 \\mathrm {d} \\mu _{0}(A) \\\\= &\\, \\frac{1}{2} \\int _{X} \\left\\lbrace \\left(\\frac{\\exp (-G(A;U_{1}))}{L(U_{1})} \\right)^{1/2} -\\left(\\frac{\\exp (-G(A;U_{2}))}{L(U_{1})} \\right)^{1/2} \\right.",
"\\\\&\\, \\left.",
"{} +\\left(\\frac{\\exp (-G(A;U_{2}))}{L(U_{1})} \\right)^{1/2} -\\left(\\frac{\\exp (-G(A;U_{2}))}{L(U_2)} \\right)^{1/2} \\right\\rbrace ^2 \\mathrm {d} \\mu _{0}(A) \\\\\\le &\\, L(U_{1})^{-1} \\int _{X} \\left\\lbrace {\\exp \\left(-\\frac{1}{2}G(A;U_{1}) \\right)} -{\\exp \\left(-\\frac{1}{2}G(A;U_{2}) \\right)} \\right\\rbrace ^2 \\mathrm {d} \\mu _{0}(A) \\\\& \\,+ \\left| L(U_{1})^{-1/2}- L(U_{2})^{-1/2} \\right|^2 \\int _{X} {\\exp (-G(A;U_{2}))} \\mathrm {d} \\mu _{0}(A).\\end{split}$ With the mean value theorem and Lemma REF , it holds that $\\begin{split}& \\,\\int _{X} \\left\\lbrace {\\exp \\Big (-\\frac{1}{2}G(A;U_{1}) \\Big )} -{\\exp \\Big (-\\frac{1}{2}G(A;U_{2}) \\Big )} \\right\\rbrace ^2 \\mathrm {d} \\mu _{0}(A) \\\\\\leqslant & \\, \\int _{X} \\Big |\\frac{1}{2}G(A;U_{1}) -\\frac{1}{2}G(A;U_{2}) \\Big |^2 \\mathrm {d} \\mu _{0}(A) \\\\\\leqslant &\\, \\frac{ 1}{16\\sigma ^4} \\int _{X} \\Big |\\Vert U_{1}-\\mathcal {F}(A)\\Vert _{L^2(\\Gamma )} ^2 -\\Vert U_{2}-\\mathcal {F}(A)\\Vert _{L^2(\\Gamma )} ^2\\Big |^2 \\mathrm {d} \\mu _{0}(A) \\\\\\leqslant &\\, M \\Vert U_{1}-U_{2}\\Vert _{L^2(\\Gamma )} ^2.\\end{split}$ Using the bounds on $L(U_1)$ and $L(U_2)$ , we have that $ \\begin{split}\\left| L(U_1)^{-1/2}- L(U_2)^{-1/2} \\right|^2 &\\leqslant M \\max \\Big (L(U_1)^{-3}, L(U_2)^{-3} \\Big ) | L(U_{1})- L(U_{2}) |^2\\\\&\\leqslant M \\Vert U_{1}-U_{2}\\Vert _{L^2(\\Gamma )} ^2.\\end{split}$ Combining (REF )-(REF ), we conclude that $d_{\\rm Hell}(\\mu _{U_1}, \\mu _{U_2}) \\leqslant M\\Vert U_1-U_2\\Vert _{L^2(\\Gamma )}.$ To explore the posterior probability distribution of the unknown $A$ , we employ the preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm for the Markov chain Monte Carlo (MCMC) method [10].",
"pCN-MH: 1.",
"Set $j \\leftarrow 0$ and choose an initial value ${A}^{(0)}$ .",
"2.",
"Propose a move according to $\\tilde{A}^{(j)}=\\left(1-{\\beta }^2\\right)^{1/2}{A}^{(j)}+\\beta W_n,\\quad W_n\\sim \\mathcal {N}(0, \\mathbb {I}).$ 3.",
"Compute $\\alpha ({A}^{(j)}, \\tilde{A}^{(j)})=\\min \\left\\lbrace 1, {\\exp \\left(-G(\\tilde{A}^{(j)};U)+G({A}^{(j)};U)\\right)}\\right\\rbrace .$ 4.",
"Draw $\\tilde{\\alpha }\\sim \\mathcal {U}(0,1)$ .",
"If $\\alpha ({A}^{(j)}, \\tilde{A}^{(j)})\\ge \\tilde{\\alpha }$ , set ${A}^{(j+1)}=\\tilde{A}^{(j)}$ .",
"Else, ${A}^{(j+1)}={A}^{(j)}$ .",
"5.",
"When $j=\\text{MaxIt}$ , the maximum sample size, stop.",
"Otherwise, set $j \\leftarrow j+1$ and go to Step 2."
],
[
"Numerical Examples",
"In this section, we present some numerical experiments to demonstrate the effectiveness of the proposed deterministic-statistical method.",
"In all examples, the synthetic far field data is generated by decomposing $\\Omega $ into a triangular mesh ${\\mathcal {T}}$ and approximating (REF ) by $u^{\\infty }(\\hat{x},k) \\approx \\sum _{T\\in {\\mathcal {T}}} \\Phi ^{\\infty }_k(\\hat{x},y_{T}) f(y_{T}) |T|, \\quad \\hat{x}=(\\cos \\theta ,\\sin \\theta ),$ where $T \\in {\\mathcal {T}}$ is a triangle, $y_T$ is the center of $T$ , and $|T|$ denotes the area of $T$ .",
"The observation directions $\\theta $ 's are chosen from the following three apertures: $\\Gamma _1 = 0: \\frac{\\pi }{26}:2\\pi - \\frac{\\pi }{26}, \\qquad \\Gamma _2 = 0: \\frac{\\pi }{26}:\\pi - \\frac{\\pi }{26}, \\qquad \\Gamma _3 = 0: \\frac{\\pi }{26}:\\frac{\\pi }{2}- \\frac{\\pi }{26},$ i.e.",
"$\\Gamma _1$ is the full aperture, $\\Gamma _2$ is a half of the full aperture and $\\Gamma _3$ is a quarter of the full aperture.",
"To ensure the accuracy of the far field data, we use fine meshes with the mesh size $h \\approx 0.01$ .",
"The perturbed far field measurement is given by $u^{m}(\\hat{x},k):=u^{\\infty }(\\hat{x},k)+0.03 ( \\max _{\\hat{x}} \\Re ( u^{\\infty }(\\hat{x},k))+ i \\max _{\\hat{x}} \\Im ( u^{\\infty }(\\hat{x},k))),$ where $\\Re $ and $\\Im $ represent the real and imaginary part, respectively.",
"For the DSM, the measurement data is the far field pattern correspond to wavenumbers $K_{1}=1:1:3$ .",
"The domain $D$ is the square $[-4,4]^2$ , which is uniformly covered by $81 \\times 81$ sampling points.",
"The cutoff values for the indicator function of three scenarios $\\Gamma _1$ , $\\Gamma _2$ and $\\Gamma _3$ are $\\gamma =0.41, 0.64$ and $0.70$ , correspondingly.",
"These $\\gamma $ 's are obtained by trial and error.",
"In the contour plots of the indicator $I_{x_{p}}$ over the sampling domain $D$ , the red dashed line represents the exact boundary of source $f(x)$ and the estimation of the support is the black circle.",
"In fact, we will see that the DSM uses a smaller set of the far field data than the Bayesian inversion does.",
"In general, a satisfactory reconstruction of the disc $\\hat{B}$ can be obtained using the data for a few smaller $k$ 's.",
"Once we obtain the approximate disc $\\hat{B}$ , we choose $N=2,M=5$ in (REF ) for the approximation $f_{BE}$ (25 terms in total).",
"The measurement for the Bayesian method is corresponding to the wavenumbers $K=1:1:20$ .",
"In the MCMC we take $\\pi (A)=\\mathcal {N}(0,0.01)$ and $\\sigma =0.04$ in the likelihood.",
"To compute the posterior distribution of $A$ , we apply pCN-MH with $\\beta =0.001$ .",
"A Markov chain of sample size $120,000$ is drawn in the Bayesian inversion, of which the first $20,000$ samples are discarded.",
"The CM is then used as a point estimate for $A$ .",
"To evaluate the performance of the reconstruction, we compute both the absolute error (AE) $\\Vert f-f_{BE}\\Vert _2$ and the relative error (RE) $\\frac{\\Vert f_{BE}-f\\Vert _2}{\\Vert f\\Vert _2}$ .",
"Example 1: Let $f(x)=3Q_{11}(x), \\quad x\\in B(0,0.9),$ i.e., the source function is a constant multiple of an eigenfunction $Q_{11}$ for $B(0,0.9)$ .",
"We first show the performance of the Bayesian inversion when the compact support of $f(x)$ is known exactly, namely, $\\hat{B} = B(0,0.9)$ .",
"Due to (REF ), we expect that the CM of the coefficient for $Q_{11}$ is 3 and the CM's of the other coefficients are zeros in (REF ).",
"Using the Bayesian method for three apertures, $\\Gamma _1$ , $\\Gamma _2$ and $\\Gamma _3$ , the reconstructions $f_{BE}$ are shown in Fig.",
"REF .",
"It can be seen that the samples for the coefficient of $Q_{11}(x)$ accumulate around 3 and the rest accumulate around 0 for all three apertures.",
"Figure: Example 1 (exact support known).",
"Top row: the histograms of the coefficients for f BE f_{BE} when the support is known exactly.",
"Bottom row: the reconstructed f BE f_{BE} and exact ff.",
"Left column: Γ 1 \\Gamma _1.",
"Middle column: Γ 2 \\Gamma _2.",
"Right column: Γ 3 \\Gamma _3.Next we use the proposed deterministic-statistical method to reconstruct $f(x)$ without the knowledge of its support.",
"The DSM is first used to find a disc $\\hat{B}$ containing the support of $f(x)$ .",
"For all three apertures $\\Gamma _1$ , $\\Gamma _2$ and $\\Gamma _3$ , the indicator functions $I(x_{p})$ 's and the discs $\\hat{B}$ 's obtained are shown in the top row of Fig.REF .",
"The associated approximate radii of $\\hat{B}$ 's are $ 1.3601$ , $1.4213 $ and $1.0817$ (see Table REF ).",
"All $\\hat{B}$ 's are close to the exact support, which indicates the effectiveness of the DSM.",
"In the Bayesian inversion stage, based on the reconstructed $\\hat{B}$ , we explore the statistical information of the coefficients for $f_{BE}$ using pCN-MH.",
"The second row of Fig.",
"REF shows the histograms of the coefficients, which tend to converge.",
"Note that the eigenfunctions of $\\hat{B}$ are used and the coefficients for $f_{BE}$ are not zero in general.",
"The exact source function $f$ and the reconstructions $f_{BE}$ are shown in the third row of Fig.",
"REF .",
"The absolute and the relative errors of the reconstructions using the CM's are listed in Table REF (first four columns).",
"It can be seen that all the approximate source functions $f_{BE}$ 's are quite close to the exact sources.",
"For all three apertures, the absolute errors are small and the relative errors are less than 7%.",
"Figure: Example 1 (reconstructed support).",
"First row: contour plots of the indicators for the DSM.",
"Second row: the histograms of the coefficients for f BE f_{BE}.",
"Third row: the reconstructed f BE f_{BE} and exact ff.",
"Left column: Γ 1 \\Gamma _1.",
"Middle column: Γ 2 \\Gamma _2.",
"Right column: Γ 3 \\Gamma _3.Example 2: Let $f(x)=2(0.81-(x_{1}^2+x_{2}^2))\\chi _{x_{1}^2+x_{2}^2 \\le 0.81},$ where $\\chi $ is the characteristic function.",
"The exact support of $f(x)$ is $B(0,0.9)$ .",
"The contour plots of the indicator functions by the DSM are shown in the first row of Fig.",
"REF for $\\Gamma _1$ , $\\Gamma _2$ and $\\Gamma _3$ .",
"The radii of the discs $\\hat{B}$ 's are $0.9055$ , $1.1180$ and $1.0817$ , which are listed in Table REF .",
"The histograms of the coefficients are shown in the second row of Fig.",
"REF .",
"The reconstructed $f_{BE}$ 's and the exact $f$ are shown in the third row of Fig.",
"REF .",
"The errors are listed in Table REF .",
"It can be seen that as the measurement aperture becomes less, the errors increase.",
"Table: Exact support of f(x)f(x) and the radii of the discs by the DSM.Table: Absolute error (AE) ∥f-f BE ∥ 2 \\Vert f-f_{BE}\\Vert _2 and the relative error (RE) ∥f BE -f∥ 2 ∥f∥ 2 \\frac{\\Vert f_{BE}-f\\Vert _2}{\\Vert f\\Vert _2}.Figure: Example 2.",
"First row: contour plots of the indicators for the DSM.",
"Second row: the histograms of the coefficients for f BE f_{BE}.",
"Third row: the reconstructed f BE f_{BE} and exact ff.",
"Left column: Γ 1 \\Gamma _1.",
"Middle column: Γ 2 \\Gamma _2.",
"Right column: Γ 3 \\Gamma _3.Example 3: Let $f(x)=5\\exp (-45x_1^2-30x_2^2)).$ In this case, $f(x) \\ne 0$ for all $x\\in \\mathbb {R}^2$ .",
"However, $f(x)$ is very close to 0 when $|x|$ is large and the approximation (REF ) for $f(x)$ is still valid approximately for $\\hat{B}$ large enough.",
"We consider a rough support of $f(x)$ : $B^*=\\lbrace x\\in \\mathbb {R}^2| |f(x)|\\le 10^{-10}\\rbrace $ .",
"We have $B^* \\approx B(0,0.7471)$ .",
"The contour plots of the indicator functions by the DSM are shown in the first row of Fig.",
"REF for $\\Gamma _1$ , $\\Gamma _2$ and $\\Gamma _3$ .",
"The reconstructed domains $\\hat{B}$ contains $B^*$ and are close to it for all three apertures.",
"The radii of the reconstructed discs $\\hat{B}$ 's are $0.8246$ , $1.0198$ and $1.0630$ , which are listed in Table REF .",
"The histograms of the coefficients are shown in the second row of Fig.",
"REF .",
"The reconstructed $f_{BE}$ 's and the exact $f$ are shown in the third row of Fig.",
"REF .",
"The errors are listed in Table REF .",
"Again when the measurement aperture becomes less the errors increase.",
"Figure: Example 3.",
"First row: contour plots of the indicators for the DSM.",
"Second row: the histograms of the coefficients for f BE f_{BE}.",
"Third row: the reconstructed f BE f_{BE} and exact ff.",
"Left column: Γ 1 \\Gamma _1.",
"Middle column: Γ 2 \\Gamma _2.",
"Right column: Γ 3 \\Gamma _3.Example 4: Let $f(x)=15x_1x_2(0.81-(x_{1}^2+(x_{2}/1.2)^2))\\chi _{\\lbrace (x_{1}^2+(x_{2}/1.2)^2)<=0.81\\rbrace }.$ The compact support of $f(x)$ is an ellipse with minor radius 0.9 and major radius 1.08.",
"The approximate discs by the DSM (first row of Fig.",
"REF ) provide reliable estimates for the support $f(x)$ , which are given in Table REF .",
"The histograms of the coefficients are shown in the second row of Fig.",
"REF .",
"The reconstructed $f_{BE}$ 's and the exact $f$ are shown in the third row of Fig.",
"REF .",
"The errors are listed in Table REF .",
"Again the errors increase as the measurement aperture becomes less.",
"Figure: Example 4.",
"First row: contour plots of the indicators for the DSM.",
"Second row: the histograms of the coefficients for f BE f_{BE}.",
"Third row: the reconstructed f BE f_{BE} and exact ff.",
"Left column: Γ 1 \\Gamma _1.",
"Middle column: Γ 2 \\Gamma _2.",
"Right column: Γ 3 \\Gamma _3.Example 5: The last example is a discontinuous source function.",
"Let $f(x)=\\chi _{(x_1^2+x_2^2<=0.81)}.$ The contour plots of the indicator functions by the DSM are shown in the first row of Fig.",
"REF for $\\Gamma _1$ , $\\Gamma _2$ and $\\Gamma _3$ .",
"The radii of the reconstructed discs $\\hat{B}$ 's are $0.9849$ , $1.2166$ and $1.1705$ listed in Table REF .",
"The histograms of the coefficients are shown in the second row of Fig.",
"REF .",
"The reconstructed $f_{BE}$ 's and the exact $f$ are shown in the third row of Fig.",
"REF .",
"The errors are listed in Table REF .",
"The main features of the discontinuous source $f(x)$ such as the value and discontinuity are reconstructed well.",
"Figure: Example 5.",
"First row: contour plots of the indicators for the DSM.",
"Second row: the histograms of the coefficients for f BE f_{BE}.",
"Third row: the reconstructed f BE f_{BE} and exact ff.",
"Left column: Γ 1 \\Gamma _1.",
"Middle column: Γ 2 \\Gamma _2.",
"Right column: Γ 3 \\Gamma _3."
],
[
"Conclusions",
"In this paper, we combine the DSM and Bayesian approach to reconstruct an extended source using the multiple frequency limited aperture far field data.",
"In the first step, the DSM is used to obtain an approximation of the compact support (a disc) of the source.",
"Using the eigenfunctions of the disc, we expand the source and employ the Bayesian inverse to recover the expansion coefficients.",
"Numerical examples, including a discontinuous source function, show the effectiveness of the proposed method.",
"It is observed that as the aperture becomes smaller the reconstruction error increases.",
"Nonetheless, the results are satisfactory for limited aperture data.",
"The cutoff value for the indicator function of the DSM is chosen by trial and error.",
"We are investigating other methods to avoid choosing ad-hoc cutoff values.",
"Algorithms that can improve the acceptance rate of the samplings in the MCMC method are also worth efforts to improve efficiency.",
"Another interesting topic is the case when the source function is also frequency dependent, i.e, $f$ depends on $k$ as well."
],
[
"Acknowledgements",
"The research of ZZ is supported by Hong Kong RGC grant (project 17307921), National Natural Science Foundation of China (project 12171406), and a seed funding from the HKU-TCL Joint Research Center for Artificial Intelligence."
]
] | 2209.08222 |
[
[
"Measurement of Stimulated Raman Side-Scattering Predominance and\n Energetic Importance in the Compression Stage of the Double-Cone Ignition\n Approach to Inertial Confinement Fusion"
],
[
"Abstract Due to its particular geometry, stimulated Raman side-scattering (SRSS) drives scattered light emission at non-conventional directions, leading to scarce and complex experimental observations.",
"Experimental campaigns at the SG-II UP facility have measured the scattered light driven by SRSS over a wide range of angles, showing an emission at large polar angles, sensitive to the plasma profile and laser polarization.",
"Furthermore, direct comparison with back-scattering measurement has evidenced SRSS as the dominant Raman scattering process in the compression stage, leading to the scattering loss of about 5\\% of the total laser energy.",
"The predominance of SRSS was confirmed by 2D particle-in-cell simulations, and its angular spread has been corroborated by ray-tracing simulations.",
"The main implication is that a complete characterization of the SRS instability and an accurate measurement of the energy losses require the collection of the scattered light in a broad range of directions.",
"Otherwise, spatially limited measurement could lead to an underestimation of the energetic importance of stimulated Raman scattering."
],
[
"Measurement of Stimulated Raman Side-Scattering Predominance and Energetic Importance in the Compression Stage of the Double-Cone Ignition Approach to Inertial Confinement Fusion K. Glize These two authors contributed equally X. Zhao These two authors contributed equally Key Laboratory for Laser Plasmas (MoE) and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China Y. H. Zhang Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China C. W. Lian Key Laboratory of Basic Plasma Physics and Department of Engineering and Applied Physics, University of Science and Technology of China, Hefei 230026, China S. Tan Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, School of Physics and Electronics, Hunan University, Changsha, 410082, China F. Y. Wu Key Laboratory for Laser Plasmas (MoE) and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China C. Z. Xiao Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, School of Physics and Electronics, Hunan University, Changsha, 410082, China R. Yan Key Laboratory of Basic Plasma Physics and Department of Engineering and Applied Physics, University of Science and Technology of China, Hefei 230026, China Z. Zhang Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China X. H. Yuan Authors to whom correspondence should be addressed [email protected] and [email protected] Key Laboratory for Laser Plasmas (MoE) and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China J. Zhang Authors to whom correspondence should be addressed [email protected] and [email protected] Key Laboratory for Laser Plasmas (MoE) and School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Due to its particular geometry, stimulated Raman side-scattering (SRSS) drives scattered light emission at non-conventional directions, leading to scarce and complex experimental observations.",
"Experimental campaigns at the SG-II UP facility have measured the scattered light driven by SRSS over a wide range of angles, showing an emission at large polar angles, sensitive to the plasma profile and laser polarization.",
"Furthermore, direct comparison with back-scattering measurement has evidenced SRSS as the dominant Raman scattering process in the compression stage, leading to the scattering loss of about 5% of the total laser energy.",
"The predominance of SRSS was confirmed by 2D particle-in-cell simulations, and its angular spread has been corroborated by ray-tracing simulations.",
"The main implication is that a complete characterization of the SRS instability and an accurate measurement of the energy losses require the collection of the scattered light in a broad range of directions.",
"Otherwise, spatially limited measurement could lead to an underestimation of the energetic importance of stimulated Raman scattering.",
"In inertial confinement fusion (ICF) experiments [1], one of the main obstacles that prevents the implosion from reaching the required conditions to obtain ignition of the fuel are the laser-plasma instabilities (LPIs) [2], consisting of non-linear couplings of the laser driver beam with plasma modes.",
"Stimulated Raman scattering (SRS) is a three-wave coupling resonantly driving an electron plasma wave (EPW) [3].",
"This process leads to the scattering of a part of the incident laser reducing the energy coupling, and generation of a hot electron population that can preheat the fuel core.",
"This instability is of primary concern in most of the ICF schemes, such as Indirect-Drive [4], [5], [6], Direct-Drive [7], [8] or Shock Ignition [9], [10], [11], [12].",
"Stimulated Raman side-scattering (SRSS) is a particular SRS geometry in which the scattered light is emitted perpendicular to the density gradient, enabling an absolute growth (exponential growth in time at a localised spatial position) at density lower than $n_{c}/ 4$ , where $n_{c}$ is the critical density.",
"Despite extensive theoretical investigations in the late 70s [13], [14], [15], most of the interest has been focused on stimulated Raman back-scattering (SRBS) [5], due to the experimental complexity to measure SRSS and the largest SRS growth rate for the backward geometry.",
"Recently, there has been a renewed interest due to observations of SRSS on several planar experiments [7], [8], [16], [17], [10], either from single beam interaction, or by multiple beams [18].",
"Multi-beam processes happen when the laser beams are sharing a common symmetry axis enabling to drive a shared daughter wave, being either an EPW [19] or a scattered wave [16], [17].",
"These experimental observations have led to the development of a more complete analytical description of the SRSS, accounting for the convective nature (finite spatial amplification while propagating through the resonant region) of the instability near the turning point, in order to explain the SRSS growth in region below the absolute threshold [20].",
"It highlighted that ICF experiments are prone to being SRSS unstable as the instability can extend to lower densities in the convective regime due to the large dimensions of the interaction, namely long density scale-length, large laser focal spot and high temperature.",
"Therefore a complete understanding of this detrimental process is imperative as leading to additional laser energy coupling loss and hot electron generation.",
"However, experimental observations have been restricted to a limited number of directions since ICF laser facilities are usually not designed to measure SRS in directions other than back-scattering.",
"Thus, comprehensive measurements of this mechanism and related losses are still not available.",
"In order to improve the overall characterisation of SRS at non-typical angles, crucial new diagnostics are currently being implemented in order to provide further observations at additional angles of observations [21].",
"In this Letter, we present, up to our knowledge, the first highly-resolved 2D angular measurement of SRS in ICF experiment.",
"These measurements have been obtained in the context of the Double-Cone Ignition (DCI) scheme [22], a newly proposed approach to Direct-Drive (DD) ignition [23], in which the compression and ignition stages are separated.",
"During the preliminary experimental campaigns, focused on the compression stage at low laser intensity, a new diagnostic has been designed and implemented to provide an angularly resolved measurement of the SRS emission spectrum: Angular-Resolved Scattered-light Diagnostic Station (ARSDS) [24].",
"It enabled to observe broadband SRS light emitted over a wide spatial area.",
"Direct comparison with the back-scattered light collected in the aperture of one of the driver beam evidenced SRSS as the dominant SRS process.",
"2D PIC simulations confirmed that in this regime, the interaction was below the threshold for the typical SRBS to grow and only SRSS was responsible for the SRS emission.",
"Due to its broad spatial emission, SRSS was identified to be responsible for the scattering of up to $5\\pm 2\\%$ of the total laser energy and requires to be measured over the whole interaction volume.",
"Figure: (a) Schematic of the irradiation geometry on the conical target.",
"(b) Pulse shape of the overlapped power delivered on target, both design (blue) and experimental (dashed red).",
"(c) Spherical coordinate map depicting the positions of the laser beams (red squares), the ARSDS fibers (black dots) and the additional 50 fibers (blue diamonds).The experiment was performed at the SG-II UP laser facility [25] using the setup displayed on Fig.",
"REF (a).",
"For clarity, we define here that beam #7 azimuthal position is the origin $\\varphi =0 $ of the azimuth axis and that the north pole is the origin $\\theta =0 $ of the polar axis.",
"A cone of four beams were used to irradiate a 45$~$ m-thick CH spherical cap target, with an inner radius of 450$~$ m. The CH cap is contained within a 20$~$ m-thick, open-ended Au cone.",
"The cone of four beams, #1, #3, #5 and #7, were incident on the CH target at a polar angle of $\\theta =50 $ , uniformly distributed in the azimuthal direction.",
"The beams are primarily p-polarised plus an angle of 7 for beams #3, #5 and 23 for beams #1, #7.",
"Each beam delivers 1.5 kJ at 351 nm focused by a $f/7.1$ wedged lens on target within a 525$~$ m ($1/e^{2}$ ), CPP smoothed focal spot, with a pulse shape presented on Fig.",
"REF (b), reaching a peak overlapped intensity of $\\approx 1.8 \\times 10^{15}~$ W.cm$^{-2}$ in vacuum.",
"At peak power, plasma parameters were estimated using the hydro-radiative code MULTI2D [26], predicting a density scale-length of $L_{n_{c}/ 4} \\approx 200 ~$ m at $n_{c}/ 4$ up to $L_{n_{c}/ 10} \\approx 250 ~$ m in the coronal plasma, with an uniform electron temperature from $T_{e} \\approx 2.0$ to $2.5~$ keV.",
"The electron temperature near $n_{c} / 4$ was confirmed to be $\\le 2.2~$ keV from the red shifted-spectral feature related to Two-Plasmon Decay (TPD) instability [27], as shown by the inset in Fig.",
"REF (a).",
"A Full-Aperture Back-scattering Station (FABS) was installed on #7 in order to collect the light scattered in the backward direction $[\\theta , \\varphi ]=[50, 0]$ .",
"ARSDS was used to collect the SRS scattered light in both azimuthal and polar axes, at the coordinates depicted by the black dots in Fig.",
"REF (c), in order to obtain temporally integrated spectra resolved in angle.",
"Figure: Typical ARSDS spectrum of the SRS emission against (a) the polar angle collected at an azimuthal angle of 33.733.7 ; (b) the azimuthal angle collected at polar angles displayed in Fig.",
"(c).",
"The inset in (a) presents the contrast enhanced TPD features.",
"(c) (resp.",
"(d)) Red plain line is the scattered energy against the exit angle, integrated over the spectrum (a) (resp.",
"(b)) and normalized to the collection solid angle.",
"Back-scattered energy measured by FABS is displayed as a black dot on (c).",
"The discontinuity in (d) is due to the different polar coordinate, as shown in Fig.",
"(c).",
"On (c) and (d), blue dotted, green dashed and black dash-dot lines present the scattered light signal in arbitrary units from the ray-tracing simulations with respectively ideal spherical compression, flattened compression and rotated polarization.A typical result is presented on Fig.",
"REF , displaying the scattered light spectrum angularly resolved in the (a) polar and (b) azimuthal directions.",
"The overall signal is dominated by light with a spectrum ranging from 540 nm up to 640 nm emitted within a wide area ranging from 30 up to 90 polar angles, over the whole azimuth, corresponding to a density range of $ \\approx [0.1-0.2]n_{c}$ .",
"The plain red curve on Fig.",
"REF (c) (resp.",
"Fig.",
"REF (d)) presents the signal from Fig.",
"REF (a) (resp.",
"Fig.",
"REF (b)) integrated over the spectrum, calibrated in energy and normalised to the solid angle.",
"It shows that most of the scattered light is emitted at large polar angles, peaked around $\\theta = 70 $ .",
"Such signal corresponds to an SRS emission at shorter wavelength and larger angle than previously reported, where usually the detection of SRS light is limited to angles $\\le 50 $ .",
"Furthermore, Fig.",
"REF (d) shows that the amount of scattered energy is sensitive to the azimuthal position.",
"The maximum emission is not contained in the azimuthal plane of the laser beam but is offset by an angle $\\Delta \\varphi \\approx 10$ .",
"Considering that the broadband signal is emitted at polar angles larger than the incident driver beams, over a large azimuthal section, it evidences that side-scattering is responsible for such emission.",
"The SRSS scattered light is emitted orthogonal to the density gradient and experiences refraction on its way out of the plasma [15], [20].",
"This results in a correlation between the scattered light wavelength and the polar exit angle, as observed on Fig.",
"REF (a).",
"Moreover, SRSS scattered light is also emitted perpendicular to the laser polarization plane, as reported in theoretical [2], [15], [28] and numerical [29], [30], [31] studies.",
"Considering the 2D spherical nature of our target, the inherent sensitivity of SRSS to the plasma profile and laser polarization, numerical simulations are required to confirm the observed angular spread of the scattered light.",
"A new 3D ray-tracing code PHANTAM [32], based on the method published by Kaiser [33], has been developed to simulate the propagation of the SRSS light, also accounting for collisional absorption.",
"The simulation box is 0.4cm x 0.4cm x 0.3cm with a 200x200x300 mesh, containing a plasma obtained from the hydrodynamic simulations.",
"SRSS light is generated when incident rays propagate into the the electron density range $[0.12, 0.20]n_{c}$ , at the associated wavelength, consistent with the experimental data, without gain or threshold consideration.",
"SRSS rays are initialized perpendicular to both the local density gradient and the local polarization of the incident ray with a 9-degree random spreading angle.",
"Upon exiting the plasma, SRSS rays are collected at the same positions than ARSDS detectors.",
"As displayed by the blue dotted curves in Fig.",
"REF (c) and (d), such simulations are confirming an emission at large polar angles, maximized near the beam azimuth, despite several discrepancies.",
"These are likely the result of difference in plasma profiles between the ideal spherical compression from the simulation [34] and the actual experiment.",
"This assumption can be confirmed by reducing the discrepancies when considering a plasma profile flattened along the target normal, as depicted by the green dashed curves in Fig.",
"REF (c) and (d).",
"The importance of the laser polarization is highlighted by the black dash-dot curves, showing a completely different scattering profile when the polarization of each beam is rotated by 90$$ .",
"These simulations show that the scattering is a complex combination of plasma profile, polarization and single-beam contribution from beam #7 and the neighbour beam #5.",
"Further investigations, beyond the scope of this paper, are necessary to improve numerical agreement with the experiment and will be the subject of future work.",
"From these spectra, it appears that the scattering is driven by single beam SRSS, as multi-beam process would drive scattered light either in the bisector plan at $\\varphi \\approx $ 45$$ for a shared scattered electromagnetic wave [16], [17], or constrained to density $n_{e} \\le 0.12n_{c}$ for a shared EPW due to the large angle between two neighbouring beams [19].",
"This was further confirmed by experimentally measuring a two-orders-of-magnitude decrease of the signal in the polar direction when #7 is switched off, and retrieving the overall emission profile in the ray-tracing simulations from independent beam contribution.",
"Thus, single-beam intensity is considered in the following discussion.",
"In ICF experiments, SRSS can experience both absolute or convective growth, depending on the interaction conditions.",
"Figure REF presents SRSS convective gain [20] and absolute threshold [15] against the scattered light wavelength and incident laser intensity, for our conditions considering oblique incidence and damping.",
"It appears that the absolute threshold is overcome for most of the density range.",
"The lower density part can be interpreted considering the convective regime, having a moderate gain in our intensity range, where the white curve highlights the iso-contour corresponding to a gain of 1.",
"The absence of signal above 640 nm in our data is likely due to the significant re-absorption at the associated high density, as previously reported [16], [20].",
"At low density, Michel [20] reported that the finite size of the beam is limiting the convective growth of SRSS, which requires a large transverse amplification length.",
"Considering a 525$~$ m focal spot, this corresponds to a cutoff around 530 nm in our conditions, consistent with our experimental observations.",
"Figure: SRSS convective gain against laser intensity and scattered light wavelength, for our experimental plasma conditions L n c /10 ≈250L_{n_{c}/ 10} \\approx 250 ~m and T e =2.0T_{e} = 2.0~keV.",
"The white curve highlights the iso-contour corresponding to a gain of 1.",
"The white region corresponds to the absolute regime.",
"The dashed blue rectangle highlights our experimental measurements.FABS measurements typically show a slightly higher amount of scattered energy than ARSDS ($\\approx 1.5$ higher), with similar spectrum, for the same polar angle as shown on the Fig.",
"REF (c), despite the difference in azimuth, consistent with Fig.",
"REF (d).",
"Such measurement evidences that there is no stronger emission localised in the backward direction which could be attributed to SRBS.",
"This is expected as the laser intensity is one order of magnitude lower than the predicted SRBS threshold of $\\approx ~4.5\\times 10^{15}~$ W.cm$^{-2}$ for our conditions [35], [2], [5].",
"Such low laser intensity also prevents SRBS to grow from the high intensity speckles [36], [37], which have been inferred to reach up to $\\approx 2.5 \\times 10^{15}~$ W.cm$^{-2}$ .",
"This results in a negligible reflectivity measured by FABS, $\\le 0.15\\pm 0.05$ %, as being only a fraction of the total scattered energy.",
"In order to account for the large scattering angular spread, 50 additional fibers [38] were used along with ARSDS and FABS in order to extend the spatial measurement over $\\frac{4}{3} \\pi $ , as depicted on Fig.",
"REF (c).",
"This diagnostic setup is similar to ARSDS, providing an energy measurement of SRS scattering integrated in time.",
"Despite inherent uncertainties due to the limited number of directions probed, a reflectivity up to $5\\pm 2\\%$ of the total laser energy was estimated, without considering the absorption.",
"This evidences that SRSS is energetically significant and necessitates to be measured over the whole interaction volume.",
"This implies that SRSS can lead to additional losses not usually accounted for and could be one candidate to explain some “missing” energy reported in recent experiments [6], [39].",
"Indeed, it appears that SRSS losses can be easily overlooked as only scattering a low amount of energy locally, which would normally not be detected or considered.",
"Figure: (a) 2D spectrum of the scattered light averaged over 2 ps, (b) and associated spectrum from PIC simulations.To confirm the predominance of SRSS over SRBS, 2D plane-wave PIC simulations have been performed using the code EPOCH [40].",
"An s-polarized plane wave is normally incident into a linear-density-profile plasma slab ranging from $0.1n_c$ to $0.2n_c$ , with a density scale-length of $250 ~$ m. The full simulation box is $200 ~$ m in length (x-axis) and $80 ~$ m in width (y-axis), and there is a $10 ~$ m vacuum at the left boundary.",
"The longitudinal boundary conditions are open boundary for fields and thermal boundary for particles, and the transverse boundary condition is periodic.",
"Interaction parameters are as follow: the laser intensity $I_0=4.5\\times 10^{14}~$ W.cm$^{-2}$ , the electron temperature $T_e=2~$ keV, and fixed ions.",
"Figure REF (a) shows a time-averaged 2D spectrum of the scattered light wave over $2~$ ps.",
"The brightest signals are the near 90 scattered light perpendicular to the density gradient.",
"As expected, there is no measurable SRBS due to its negligible convective gain.",
"Figure REF (b) shows the scattered light spectrum confirming SRSS growth over the whole range of density simulated, consistent with the experiment.",
"In conclusion, it was experimentally evidenced and confirmed by 2D PIC simulations that side-scattering is the dominant stimulated Raman scattering geometry in the compression stage of DCI experiment.",
"Due to the sensitivity to both plasma profile and laser polarization, SRSS scattered light is emitted over a wide volume, as observed experimentally and confirmed by ray-tracing simulations.",
"Such broad angular scattering results in a small amount of energy being scattered locally, while being energetically significant when measured over the whole interaction volume.",
"Thus, relying on limited directions to diagnose SRS, such as back-scattering, could lead to a strong underestimation of the actual energy losses, by a factor of 35 in our conditions.",
"Even though the results discussed here were obtained using conical targets, it was verified experimentally that our conclusions were still valid for planar targets.",
"Thus, the importance and relevance of our observations can be extended beyond the DCI scheme.",
"However, to assess the impact and significance of side-scattering on other ICF schemes, further investigations are required such as: (i) SRSS behavior at higher intensities; (ii) the competition between Raman side- and back-scattering when SRBS threshold is overcome; (iii) the effect of laser smoothing techniques such as smoothing by spectral dispersion (SSD) [41] which could reportedly mitigate SRSS [42].",
"We thank the SG-II UP operating group and target fabrication team for their assistance.",
"This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (grants No.",
"XDA25010100, No.",
"XDA25030200 & No.",
"XDA25050700)."
]
] | 2209.08251 |
[
[
"APPDIA: A Discourse-aware Transformer-based Style Transfer Model for\n Offensive Social Media Conversations"
],
[
"Abstract Using style-transfer models to reduce offensiveness of social media comments can help foster a more inclusive environment.",
"However, there are no sizable datasets that contain offensive texts and their inoffensive counterparts, and fine-tuning pretrained models with limited labeled data can lead to the loss of original meaning in the style-transferred text.",
"To address this issue, we provide two major contributions.",
"First, we release the first publicly-available, parallel corpus of offensive Reddit comments and their style-transferred counterparts annotated by expert sociolinguists.",
"Then, we introduce the first discourse-aware style-transfer models that can effectively reduce offensiveness in Reddit text while preserving the meaning of the original text.",
"These models are the first to examine inferential links between the comment and the text it is replying to when transferring the style of offensive Reddit text.",
"We propose two different methods of integrating discourse relations with pretrained transformer models and evaluate them on our dataset of offensive comments from Reddit and their inoffensive counterparts.",
"Improvements over the baseline with respect to both automatic metrics and human evaluation indicate that our discourse-aware models are better at preserving meaning in style-transferred text when compared to the state-of-the-art discourse-agnostic models."
],
[
"Introduction",
"Disclaimer: Due to the nature of this work, figures and examples may contain offensive phrases.",
"The spread of offensive and hateful content on social media can be detrimental to users' psychological well-being [36], [14].",
"Anonymity on platforms such as Reddit can further embolden users to post such hateful content [1].",
"Further, the sheer volume of content on popular social media platforms can render the human moderation process ineffective [15] or psychologically damaging for moderators [11] and calls for AI systems that can mitigate this problem.",
"AI moderation of social media by simply removing content classified as offensive [45], [16] may reduce diversity in online conversations and deter users from using the platform [19].",
"Our exploration reveals that many comments removed by moderators on Reddit contain contributions to the discourse beyond their offensive content.",
"Rather than simply removing these comments from social media platforms, they can be turned into inoffensive statements by using alternative words, removing profanity, or paraphrasing certain parts, while preserving the overall semantic content.",
"Figure: Example of instances where pretrained language models either fail to remove offensiveness (BART/T5) or drastically alter the intended meaning (DialoGPT) when fine-tuned on our style transfer taskWe approach the problem of eliminating offensiveness from text while preserving original semantic content as a supervised style-transfer task, where offensive text is transferred to inoffensive text.",
"As a first step towards this goal, we create the first publicly-available, expert-annotated style transfer corpus for Reddit data, which contains offensive comments that include certain lexical items and more subtle instances that are implicit and grounded in context.",
"This differentiates our work from unsupervised approaches are mostly good at handling instances with explicit lexical cues [30].",
"Although large pretrained transformer models have been successfully deployed for generation tasks, these models come with the risk of either failing to generate desired output or obfuscating the source passage's meaning while still producing coherent text [4].",
"In our work, we target the issue of content preservation using discourse frameworks, which have been successfully employed for various generation tasks [28], [40], [5], but have not been employed in style transfer models.",
"We hypothesize that integrating discourse coherence frameworks within transformer-based style transfer models can contribute to better preservation of semantic content, specifically for short social media comments.",
"We study our hypothesis with a small pilot annotation of style-transferred text produced by pretrained transformer models.",
"Figure REF shows examples of the issues described above in our style transfer task, where BART [21] and T5 [33] do not remove offensiveness from the original comment, but DialoGPT [50] significantly alters the original semantic content.",
"We observe that coherence relations between a comment and its reply are not preserved under style transfer in cases where offensiveness is removed.",
"For instance, the removed comment refers to \"emotionally manipulative behavior\" in the parent comment with \"This is evil\", exhibiting the behavior of \"Same-Unit\" discourse relation, which is not preserved in the style-transferred text generated by DialoGPT.",
"To test our hypothesis, we provide the following contributions: [leftmargin=*] Collect a datasethttps://github.com/sabithsn/APPDIA-Discourse-Style-Transfer of ~2K offensive comments from Reddit that are annotated by expert sociolinguists with inoffensive counterparts.",
"Our data also contains parent comments/posts and are tagged with discourse relations, making it the first publicly available dataset of its kind.",
"Propose two approaches for integrating discourse relation frameworks with pretrained transformer models: i) using Penn Discourse Treebank [29], [32], [38] relations within a single comment, and ii) parsing a comment and the text it is responding to using the Rhetorical Structure Theory Discourse Treebank [27].",
"The results for both discourse-aware approaches indicate improvement in content preservation over the pretrained baselines, providing support for our hypothesis and for the use of discourse frameworks to preserve meaning in style-transfer tasks."
],
[
"Related Work",
"Paraphrase generation is a well-studied problem that has yielded large datasets such as the PDTB paraphrase database [13], WikiAnswer [12], ParaNMT [39], and the MSCOCO dataset [25].",
"Recent works in the related but relatively new field of style transfer primarily target sentiment transfer [23], [42], formality transfer [7] or expertise transfer [6].",
"Very few works have targeted transferring style from offensive to inoffensive text, with [30] and [8] being notable exceptions.",
"Our dataset differs from the aforementioned works in multiple ways.",
"Ours is the first publicly available dataset that contains offensive Reddit comments that are rewritten by experts, paired with parent comment/post, and automatically tagged with discourse relations.",
"Further, both [30] and [8] derive their datasets from political subreddits [35], while our data encompasses subreddits on personal views, question-answer discussions, and gender rights in addition to political subreddits.",
"Figure: Our data collection pipeline for obtaining removed comments from Reddit that are offensive.Development of pretrained language models (PLM) such as BART [21] has changed the landscape of natural language generation research and we are witnessing a shift toward controllable text generation [48], [47], [34], [51].",
"Discourse relations have been proposed as a possible controlled generation method, shown to aid extractive and abstractive summarization [9], [41], text generation from meaning representations [28], and question answering with logical reasoning or complex answers [17], [40].",
"Discourse-aware models have also been shown to generate more coherent texts [5] within a reinforcement learning setting.",
"Our work integrates both RST-DT and PDTB frameworks with pretrained transformer models and provides a comparison of the relative efficacy of the two frameworks for a generation task.",
"Within the context of style transfer, recent works have focused on classification and reconstruction loss [30], [7] in semi-supervised/unsupervised setting, use of copy mechanism [18], and coherence classifier [8] to guide the style transferred text.",
"To our knowledge, our work is the first to utilize discourse coherence frameworks for style transfer."
],
[
"Data Collection and Annotation",
"In order to reduce offensiveness in text, we create an expert-annotated dataset of offensive comments and their style-transferred counterparts.",
"In this section, we first describe our pipeline for collecting and curating a set of offensive comments from Reddit.",
"Then, we describe our annotation process for reducing offensiveness in these comments."
],
[
"Data Collection Pipeline",
"First, we stream 14 subreddits spanning topics of politics, personal views, question-answer discussions, and gender rights for new comments using PRAWhttps://praw.readthedocs.io/en/stable/.",
"The streamed comments are then tagged for offensiveness using a BERT model [10] fine-tuned on the OLID dataset [43], which consists of 14K tweets annotated for offensiveness and was used for the SemEval 2019 [44] shared task.",
"If a comment is tagged as inoffensive by the classifier, we remove it from our data.",
"As our initial exploration revealed that a large portion of the removed comments on Reddit (> 60%) may not be offensive and may have been removed due to violation of subreddit-specific rules, the exclusion of inoffensive comments from the data is essential for a feasible annotation process.",
"Our manual annotation, as described in the next section, eliminates any false positive bias that the classifier may have.",
"More details about the classifier can be found in the Appendix.",
"After running the classifier, the body and metadata of comments that are tagged as offensive, are stored locally.",
"We then periodically check the accessibility status of these comments on Reddit.",
"If it has been removed by moderators, we query Reddit for the parent comment or post that it is in response to.",
"If the comment is a reply to another comment, then the comment it is replying to is considered the parent, and if the comment is a top-level comment, i.e, a direct reply to a post, then the post is considered the parent.",
"If the parent has been deleted or removed, the comment is discarded.",
"Otherwise, we add the comment, along with the parent, to our dataset.",
"Our data collection pipeline is summarized in Figure REF .",
"After filtering out very long comments, we end up with a pool of 5.8K comments for annotation.",
"Table: Examples of applying local and global changes to the comments for different types of offensive speech, as per our annotation protocol."
],
[
"Data Annotation",
"The 5.8K comments obtained from our data collection pipeline are annotated by three expert sociolinguists.",
"The annotators are paid 30 USD per hour and the Institutional Review Board (IRB) protocol, as defined by our institution, the University of Pittsburgh, was followed for recruitment and annotation.",
"The primary goal of our annotation is to remove offensiveness from a comment while retaining the intent of the comment.",
"Similar to the SemEval 2019 shared task [44], we define offensiveness as consisting of insults, profane words, hate speech, or threats of violence for our purposes.",
"We observe that some comments can be made inoffensive by the removal or substitution of offensive words.",
"We call such changes localized changes.",
"For other comments, however, the text needs to be altered/paraphrased substantially to reduce offensiveness.",
"We refer to this type of change as global change.",
"With these principles in mind, the annotators are provided with an annotation protocol, whose key points are listed below: [leftmargin=*,itemsep=1pt] Each comment has to be manually inspected.",
"If a comment is already inoffensive, or cannot be translated into inoffensive text without altering the original intent, it is discarded.",
"If applying localized changes is not possible or doesn't rid the comment of offensiveness, then global changes are made.",
"Examples of our manual annotation can be found in Table REF .",
"The first three rows of Table REF show examples of localized changes and the last three rows show examples of globalized changes in our data.",
"Further details about the distribution of data and subreddits can be found in Appendix .",
"To assess the meaning preservation of annotation, we compute the BLEU score [31] between the annotated text and the original text.",
"We use the BLEU score to measure similarity due to the open-endedness of the task (the inter-rater agreement, for instance, cannot be calculated here).",
"Since BLEU compares the overlap between reference and candidate sentences, it can serve as a metric for measuring content preservation.",
"Our annotations achieve a BLEU score of 60.06 with the original text as reference.",
"Since a BLEU score of 60 generally indicates a high overlap with the reference sentences, we can deduce that our annotation process successfully preserved the original meaning.",
"Further, the offensiveness classifier is used to tag the annotated text, showing that annotators eliminated offensiveness from 68% of the comments.",
"In reality, however, this number is likely to be higher, as the classifier may tag inoffensive comments about sensitive subjects as offensive.",
"For example, \"a rape victim should not be the one to blame\" is tagged as offensive.",
"This highlights the limitations of existing offensiveness classifiers."
],
[
"Discourse-Aware Models",
"We propose two approaches for integrating the PDTB and RST-DT discourse frameworks into pretrained transformer models, as described below.",
"Figure: PDTB-augmented style transfer model.",
"Special tokens represent the beginning and end of each argument, as well as the relation between each argument pair, are passed to the encoder.Figure: RST-augmented style transfer model.",
"A special token representing the relation at the root of the RST tree is prepended to the tokenized text of the removed comment, which is then passed to the encoder."
],
[
"PDTB Within-Comment Relations",
"To extract PDTB relations at the comment level, we parse the comment text in isolation, first using the [26] end-to-end discourse parser to extract explicit discourse relations (signaled by a discourse connective), then running the [20] XLNet-large model to extract implicit discourse relations (not signaled by a discourse connective) from adjacent sentence pairs.",
"Because there is no PDTB-tagged Reddit corpus available, we run these models trained on the PDTB-2 corpus.",
"For the L2 classification task (the more difficult of the tasks we run), [26] report an F-1 score of 80.61, and [20] report an accuracy of 57.74 (they do not report F-1 for the L2 classification task) on the PDTB-2.",
"We then use the positions of the argument pairs, and their discourse relations, in our input."
],
[
"RST-DT Context-Based Relations",
"To obtain a representation of the RST-DT relation between a comment and its parent, we concatenate the contents of the comment and the parent, separating them out as paragraphs.",
"We then run the [22] EDU segmenter on this text, and run the model in [37] on the resulting EDUs.",
"We train and test this parser on the RST-DT and GUM corpus [46] combined, and report the F-1 scores on the test set in Appendix .",
"We use the relation at the root of the RST tree as input to our style-transfer model."
],
[
"Integration with transformer model",
"To integrate the RST-DT and PDTB relations within pretrained transformer models, we first generate special tokens representing each relation for RST-DT and for the start and end of each relation for PDTB.",
"We update the tokenizer with these additional tokens, insert the tokens in the input text, and pass the modified text to the encoder of the model, as shown in Figures REF (PDTB) and REF (RST-DT).",
"We resize the model embedding to accommodate for this additional vocabulary."
],
[
"Experiments",
"In this section, we first describe the experiments with pretrained transformer models, followed by the experiments with discourse-aware models."
],
[
"Pretrained Transformer Models",
"We experiment with three different pretrained transformer models, namely: i) BART-base [21], ii) T5-base [33], and iii) DialoGPT-medium [50].",
"While BART and T5 are pretrained on formal data such as Wikipedia or web data such as C4https://www.tensorflow.org/datasets/catalog/c4, DialoGPT is pretrained on Reddit data for the response generation task."
],
[
"Discourse-aware Transformer Models",
"Due to its higher potential in removing offensiveness, we integrate our discourse-aware approaches with DialoGPT.",
"To integrate PDTB relations, we experiment with the following variations: i) Level 1 and Level 2 explicit PDTB relations, ii) Level 1 and Level 2 implicit PDTB relations, and iii) combining level 2 explicit and implicit relations.",
"To incorporate RST-DT, we use our proposed approach with the top-level RST-DT classes.",
"We limit our scope to only top-level RST-DT classes because we are unlikely to encounter lower-level classes frequently in our dataset.",
"We also experiment with combining both of our approaches.",
"Under this setting, a comment is prepended with root-level RST-DT relation between itself and its parent, and PDTB relations (both implicit and explicit) are inserted in the body of the text.",
"Since PDTB implicit and RST parsers have low accuracy scores, we propose setting a threshold $\\alpha $ for the inclusion of a discourse relation.",
"If the confidence score for a given relation falls below $\\alpha $ , the relation is discarded.",
"This is done to account for higher likelihood of misclassification on instances the discourse classifiers have low confidence on.",
"We experiment on three different $\\alpha $ values as follows: We set $\\alpha =0$ , and thus all predicted RST-DT and PDTB relations are taken We compute the mean ($\\mu $ ) and standard deviation ($\\sigma $ ) of the classifier score for the predicted class and set $\\alpha = \\mu -\\sigma $ We compute the interquartile range of the classifier scores and set $\\alpha = Q1$ , where $Q1$ is the value of first quartile."
],
[
"Results",
"Below, we describe the results of our experiments.",
"We split our dataset into an 80-10-10 split for training, development, and test sets respectively.",
"We first calculate automatic metrics, reporting the BLEU [31] and rescaled BERTScore [49] on our test set.",
"In addition, we compute the SafeScore —percentage of style transferred comments predicted as inoffensive by the BERT classifier that was initially used to identify potential candidates.",
"Further, we ask a human annotator to compare style-transferred text generated by baseline model and our proposed discourse-aware model."
],
[
"Automatic Evaluation",
"Using our automated metrics, we compare semantic similiarity between: i) the manual annotation and style transferred text, and ii) the original comment and style-transferred text."
],
[
"Pretrained Models",
"While BART and T5 are seen to achieve very high BLEU and BERT scores in Table REF , these numbers hide critical failures of the models: staying too close to the original comment and not reducing offensiveness.",
"The goal of an ideal style transfer model would be to have a good SafeScore, while also achieving a good BLEU and BERTScore.",
"A good point of reference for this ideal scenario would be the BLEU, BERTScore, and SafeScore achieved by human annotators.",
"DialoGPT, in contrast to BART and T5, has a lower BLEU and BERTScore, but is notably better at reducing offensiveness and achieves SafeScore comparable to that of human annotators.",
"This could be attributed to the fact that unlike BART and T5, which are pretrained on out-of-domain web or formal data, DialoGPT is specifically pretrained on Reddit data, making it suitable for our task.",
"For the rest of the paper, we refer to DialoGPT as the baseline model.",
"Table: Results of finetuning pretrained models on our dataset.",
"While BART and T5 outputs have a high similarity to the original and annotated text, they do not drastically reduce offensiveness, while the reverse is true for DialoGPT.Table: Results from running our discourse-aware style transfer models, where the average numbers across three runs are reported and the best numbers for each metric are bolded.",
"Improvement over baseline is shown in parenthesis.",
"As the above tables demonstrate, incorporating discourse relations improves model results by a wide margin, with RST root-level relations yielding the best BERTScore results and the combined RST + PDTB model yielding the best offensiveness score and BLEU score results.Table: Examples of style-transferred text generated by the different models.",
"The discourse-aware model refers to our best-performing discourse-aware model, the RST-PDTB model (α=0)(\\alpha =0).",
"The top three examples are ones in which our model performed better than the baseline, while in the fourth example both performed well and in the bottom example the baseline performed better than the discourse-aware model."
],
[
"Discourse-Aware Models",
"Table REF shows improvement in automated metrics achieved by our discourse aware models in comparison to the baseline DialoGPT, providing strong evidence in favor of our hypothesis.",
"In addition to this overarching takeaway, we make the following observations from our experiments:"
],
[
"The choice of framework impacts performance",
"Although discourse models yield improvements on the baseline for each automatic metric, the extent of improvement over the baseline varies depending on the discourse framework used.",
"Most notably, the RST-DT relation between the comment and its parent has the highest individual impact on BLEU and BERTScore, suggesting that the context of a comment is important for models to retain semantic meaning in generated text.",
"While we do not see any major difference between Level 1 and Level 2 PDTB relations, implicit PDTB relations have a higher impact on the BLEU and BERTScore than explicit PDTB relations.",
"Although implicit relation parsers have a lower accuracy, the improvement can be attributed to the fact that implicit relations occur more frequently in our dataset (41% instances) compared to explicit relations that occur in 25% of the instances.",
"Further, explicit relations are lexically signalled by discourse connectives already present in the text, while implicit relations do not have connectives present in the text.",
"Combining implicit and explicit relations does not change the performance notably."
],
[
"Combining discourse frameworks yields the highest improvement",
"Combining our approaches for PDTB and RST-DT relations has the greatest impact on the BLEU score, with an absolute improvement of 4.3 over the baseline.",
"The BLEU score, in this case, is a measure of overlap with the original content, while the SafeScore measures the efficacy of offensiveness removal.",
"The better BLEU score with the highest SafeScore of 67.7 indicates that incorporating both discourse frameworks enables the model to preserve original content better while effectively removing offensiveness compared to other approaches.",
"Although the BERTScore is slightly lower than that achieved by RST-augmented model, the improvement of 3.2 over baseline supports the use of both frameworks."
],
[
"Low-confidence relations are important",
"Our last observation is that dropping low-confidence relations ($\\alpha =\\mu - \\sigma $ ) can negatively impact SafeScore, while BLEU and BERTscore remains relatively unchanged.",
"We notice that, if value of $\\alpha $ is increased ($\\alpha =Q1$ ), then the BLEU and BERTScore begin to degrade.",
"This suggests, while classifier accuracy is a concern for implicit PDTB and RST-DT relations, the classifiers still capture valuable information that can aid the preservation of semantic content and reduction of offensiveness."
],
[
"Human Evaluation",
"Although automated metrics such as BLEU and BertScore can be good indicators of preservation of original content, they have certain limitations.",
"For example, they do not take into account cases where deviating from the original comment is the correct approach for offensiveness reduction.",
"We also observe that, in certain cases, the models may generate text that has a high overlap in words but their coherence may be affected by out-of-place words.",
"Thus, human evaluation is required for a complete understanding of limitations and strengths of our proposed model.",
"To this end, we presented one of our expert annotators with 100 randomly selected examples and style transferred text generated by both the baseline and our best discourse-aware model.",
"The order of the text generated by the two models was randomly shuffled so that the human evaluation was free from any potential bias.",
"Table REF shows examples of style-transferred text generated by the different models.",
"The expert annotator was asked to judge each pair from three angles: i) which of the generated texts preserves the original semantic content most, ii) which of the generated texts is more coherent, and iii) which of the generated text is preferred overall.",
"We report the results of the human evaluation in two different dimensions.",
"First, we analyze all 100 samples to get an overall picture of improvement.",
"Next, we exclude comments that do not contain any discourse relation.",
"This allows us to understand how much effect discourse relations may have on the overall results.",
"From the evaluation results reported in Table REF , we make the following key observations described below.",
"Table: Results of human evaluation"
],
[
"Discourse improves both coherence and content preservation",
"While we see a large preference for our discourse-aware model overall (40% as opposed to 29%), the difference is more prominent in terms of content preservation (48% vs 36%) compared to coherence (37% vs 32%).",
"This further supports our hypothesis that, while the baseline model can generate coherent texts, a discourse-aware model is necessary for content preservation."
],
[
"Improvements are larger for comments containing discourse relations",
"For the subset of data where discourse relations are present, we see an even larger improvement of our discourse model compared to the baseline.",
"Our model is preferred in 56% of cases for content preservation (compared to 30% for the baseline), 46% for coherence (compared to the baseline's 34%) and 46% overall (compared to 26% for the baseline).",
"This implies that the difference between our model and the baseline becomes more important for comments that have discourse structure within them."
],
[
"Conclusion and Future Work",
"In this paper, we have demonstrated that utilizing discourse frameworks and parsing models can help pretrained transformer models preserve original content when transferring style from offensive to inoffensive.",
"We have shown that combining different discourse frameworks can further improve content preservations.",
"The improvements we observe in this paper are significant; however, we hypothesize that utilizing discourse relations for these tasks can be even more impactful if the performance of existing discourse parsers is improved.",
"Discourse parsing is a very challenging task [2], [3], and the largest and most widely-used corpora are composed of news texts over a short time span.",
"Thus, there is a need for further research (and additional annotated corpora) on discourse relations within the context of social media.",
"We hope our publicly available code and data will motivate other researchers to build on the groundwork laid out in this paper.",
"Further research is also necessary in the context of style-transferring for offensive text.",
"After further improving these language models and evaluating their safety, future systems that are proven to be robust and effective can potentially help social media moderators or be deployed in a human-in-the-loop or assistive technology capacity.",
"We expect these models to have the potential to not only improve the psychological well-being of users but also to motivate healthy engagement on social media."
],
[
"Ethical Considerations",
"We acknowledge that our models can not eliminate offensiveness completely from a given text.",
"Thus, deploying our model to display style-transferred text requires taking future safety measures.",
"We also acknowledge that our use of pretrained models can induce biases in certain scenarios, as pretrained models have been shown to be susceptible to bias in the data used for pretraining [24]."
],
[
"Acknowledgement",
"We would like to thank SRI International for their valuable feedback.",
"This project was supported by DARPA grant prime OTA No.",
"HR00112290024 (subcontract No.",
"AWD00005100 and SRA00002145).",
"We also acknowledge the Center for Research Computing at the University of Pittsburgh for providing computational resources.",
"We would also like to thank the human annotators, the anonymous reviewers, Ilana Torres, and Mert Inan for their valuable feedback.",
"Data Annotation Table: Distribution of annotated dataAnnotation Distribution Following the annotation process, we obtain a labeled set of ~2K comments with their corresponding rewrites.",
"Table REF shows the distribution of the annotated data.",
"From this distribution, we observe that frequency of offensive comments are high in political subreddits such as r/Conservative compared to popular subreddits such as r/AskReddit.",
"Subreddits such as r/MensRights did not yield a substantial number of rewrites.",
"Analyzing our data revealed two reasons for the low frequency: i) the traffic on these subs is low compared to other subreddits, and ii) removed comments from these subreddits frequently contain extremely toxic content that cannot be rewritten into non-offensive versions while preserving original intent.",
"These particular subreddits need to be streamed for a longer period to obtain a substantial number of offensive comments that can be rewritten as non-offensive.",
"Pretrained Model Hyperparameters Offensiveness classifier: We fine-tune bert-base-cased [10] for 3 epochs on the OLID training set [43].",
"We use learning rate of 8e-5, batch size of 8 and maximum length of 100.",
"The model achieved an F1 score of 80.2 on the OLID test set.",
"Style transfer models: For all style transfer models, we use the same set of hyperparameters: block size of 512, batch size of 2, learning rate of 5e-5.",
"All models were fine-tuned for 10 epochs.",
"During generation, we again use set of parameters: maximum length of 200, top_p of 0.7 and temperature of 0.8.",
"Performance of RST parser Table: F-1 scores for RST parser trained on RST and GUM data and tested on an evaluation set from each (details in text)"
],
[
"Annotation Distribution",
"Following the annotation process, we obtain a labeled set of ~2K comments with their corresponding rewrites.",
"Table REF shows the distribution of the annotated data.",
"From this distribution, we observe that frequency of offensive comments are high in political subreddits such as r/Conservative compared to popular subreddits such as r/AskReddit.",
"Subreddits such as r/MensRights did not yield a substantial number of rewrites.",
"Analyzing our data revealed two reasons for the low frequency: i) the traffic on these subs is low compared to other subreddits, and ii) removed comments from these subreddits frequently contain extremely toxic content that cannot be rewritten into non-offensive versions while preserving original intent.",
"These particular subreddits need to be streamed for a longer period to obtain a substantial number of offensive comments that can be rewritten as non-offensive."
],
[
"Offensiveness classifier:",
"We fine-tune bert-base-cased [10] for 3 epochs on the OLID training set [43].",
"We use learning rate of 8e-5, batch size of 8 and maximum length of 100.",
"The model achieved an F1 score of 80.2 on the OLID test set."
],
[
"Style transfer models:",
"For all style transfer models, we use the same set of hyperparameters: block size of 512, batch size of 2, learning rate of 5e-5.",
"All models were fine-tuned for 10 epochs.",
"During generation, we again use set of parameters: maximum length of 200, top_p of 0.7 and temperature of 0.8."
]
] | 2209.08207 |
[
[
"Deep learning for reconstructing protein structures from cryo-EM density\n maps: recent advances and future directions"
],
[
"Abstract Cryo-Electron Microscopy (cryo-EM) has emerged as a key technology to determine the structure of proteins, particularly large protein complexes and assemblies in recent years.",
"A key challenge in cryo-EM data analysis is to automatically reconstruct accurate protein structures from cryo-EM density maps.",
"In this review, we briefly overview various deep learning methods for building protein structures from cryo-EM density maps, analyze their impact, and discuss the challenges of preparing high-quality data sets for training deep learning models.",
"Looking into the future, more advanced deep learning models of effectively integrating cryo-EM data with other sources of complementary data such as protein sequences and AlphaFold-predicted structures need to be developed to further advance the field."
],
[
"Introduction",
"Cryo-EM is revolutionizing structural biology due to its unique capability of determining the structures of large protein complexes and assemblies.",
"The atomic-resolution structure determination for proteins enabled by cryogenic electron microscopy (cryo-EM) [3], allows us to understand the complex biological processes carried out by proteins as well as to identify potential therapeutic protein targets for drug discovery.",
"However, reconstructing $\\textit {de novo}$ protein structures from high-resolution ($\\sim $ 3 - 4 $Å$ ) cryo-EM density maps, which accounts for a large portion of cryo-EM density maps deposited currently in the EMDB [2], is time-consuming and challenging when homologous template structures for target proteins are not available.",
"For instance, as shown in Figure REF, in the current year 2022, only about 12,500 out of 22,300 density maps of high-resolutions deposited to EMDB have a complete atomic structure available in Protein Data Bank (PDB) [39].",
"Figure: The growth of cryo-EM density maps and cryo-EM-derived protein structures and the distribution of the resolution of the density maps.",
"The statistics was obtained from EMDataResource , an unified data resource for 3-Dimension electron microscopy (3DEM) on 2022-09-14.Accurately reconstructing protein structures from cryo-EM maps is a challenging process because the data is often noisy and incomplete and target protein structures can be large and complex.",
"Traditional methods based on energy optimization such as EM-Fold [22], Gorgon [23], Rosetta [24], Pathwalking [25], MAINMAST [26], [27], VESPER [50], and Phenix [28] have made valuable progress in reconstructing protein structures from cryo-EM density maps.",
"These methods rely on extensive physics-based or statistical potential-based optimization algorithms that require high computational resources.",
"These methods often need manual intervention and trials to extract features from the cryo-EM density maps to obtain accurate reconstruction of protein structure.",
"A different strategy to automatically determine protein structures from cryo-EM density maps is to use the data-driven machine learning approach [43], a kind of artificial intelligence (AI) technology, to directly learn a mapping from cryo-EM density maps to protein structures from the large amount of known cryo-EM data and their corresponding protein structures (i.e., labels).",
"Early AI methods in the field are based on shallow machine learning techniques such as k-nearest neighbor, support-vector machines, or k-means clustering techniques.",
"These methods such as RENNSH [29], SSELearner [30], and Pathwalking [25] are able to identify only secondary structures or simplified backbone structures and often are unable to achieve the optimal solution.",
"To overcome the challenges of the traditional optimization methods and early machine learning methods, deep learning methods [44] have been developed to automatically reconstruct three-dimensional (3D) protein structures from cryo-EM density maps with significant success in recent years (see Figure REF for a summary of a general cryo-EM protein structure determination pipeline powered by deep learning).",
"In this article, we review the recent development of deep learning technology in the field, analyze their impacts, investigate the challenging issues in preparing data to train deep learning models, and discuss some new trends to further advance the field.",
"Figure: A summary of a cryo-EM density map generation and protein structure reconstruction pipeline powered by deep learning.",
"The density map (EMD-22898) illustrated in the figure is for SARS-CoV-2 ORF3a .",
"PDB ID: 7KJR."
],
[
"Deep learning reconstruction of protein structures from cryo-EM density maps",
"Deep learning, also called deep neural network, is currently the most powerful machine learning method of predicting the properties of an object from the input data describing the object.",
"It has achieved great success in many fields including a recent major breakthrough in predicting protein structure from sequence by AlphaFold [1].",
"Compared to other machine learning methods, deep learning has a unique capability of extracting informative features for pattern recognition from raw data automatically, making it suitable for reconstructing protein structures from raw density maps in which only a large amount of numbers rather than informative features are available.",
"It is worth noting that deep learning has been applied to almost all the areas of cryo-EM data analysis [34], [31], [18], [19], [20], [21], [37] from sample preparation, particle picking, density map denoising, and to the final step of 3-D structure determination.",
"Due to the space limit, this review is focused on the last step of cryo-EM data analysis - reconstructing protein structures from density maps.",
"The deep learning architectures designed for this task and how to prepare data to train them are discussed in the two subsections below."
],
[
"Deep learning architectures for reconstructing protein structures from cryo-EM density maps",
"Deep learning methods for inferring protein structures from cryo-EM density maps can be classified into different categories based on the neural network architectures (e.g., convolutional neural network (CNN) [32], U-Net [33], [42], graph convolutional network (GCN) [40], and long- and short-term memory network (LSTM) [41] they use and the output (e.g., 3D structure and secondary structure) they generate from density map input.",
"Early deep learning methods aimed to identity secondary structures from low- and medium-resolution density maps [11].",
"As more and more high-resolution density maps became available [3], recent deep learning methods targeted at directly reconstruct 3D backbone structures (i.e., locations of carbon and nitrogen atoms on the protein backbone) and even full-atom 3D structures (i.e., locations of all/most heavy atoms and amino acid identity/type) from density maps [10], [7], [14], [15], [16].",
"An example of deep learning reconstruction of protein structure from cryo-EM density map is showed in Figure REF.",
"Figure: An example of reconstructing a structure from the cryo-EM density map of SARS-CoV spike gycoprotein by deep learning.",
"(a) Density map of SARS-CoV spike glycoprotein (EMD-6732) in resolution of 3.8 ÅÅ at recommended contour level of 0.06 (11.0 σ\\sigma ).",
"(b) The structure reconstructed from EMD-6732 by a deep learning method - DeepTracer.",
"The RMSD is 1.023 ÅÅ with respect to the ground truth structure (PDB ID: 5XLR).",
"(c) The overlay of the density map and reconstructed structure at 0.5 transparency level by UCSF ChimeraX .One of the most widely used deep learning architectures of obtaining protein structural information from density maps is convolution neural network (CNN).",
"CNNs use a mathematical operation known as convolution to extract features from spatially organized data such as a 2D-image or 3D density map to predict the properties of the data (e.g., classifying voxels in a density map into amino acid types).",
"Several CNN methods (mostly 3D-CNN architecture) including Generator [7], Emap2sec [8], AAnchor [9], CNN Based [11], Cascaded-CNN [10], and CR-I-TASSER (mostly 3D CNN) [15] have been developed to determine secondary structures [8], [11], backbone-/full-atom 3D structures [15], [7], [9] or both from cryo-EM density maps [10].",
"Cascaded-CNN is the first deep learning de novo method of directly reconstructing 3D structures of proteins from cryo-EM density maps, even though it focuses on building backbone structures.",
"CR-I-TASSER combines the 3D-CNN prediction from cryo-EM maps and an advanced protein structure prediction method - I-TASSER [45] to build full-atom protein structures.",
"Another widely used deep learning architecture in the field is U-Net [33], originally designed for biomedical image classification and segmentation tasks.",
"U-Net consists of a series of convolution-based down-sampling layers to condense the input images into smaller dimensions and a series of convolution-based up-sampling counterpart layers to reconstruct the data of the same dimension as in the down-sampling process to classify/segment pixels in the input images.",
"Compared to the standard CNN architectures, U-Nets can be more effectively in extracting multi-level abstract representations of the data through the down-sampling and up-sampling processes.",
"The 2D U-Net architecture has been generalized to 3D U-Net architectures in Haruspex [12] and EMNUSS [6] to detect secondary structures from cryo-EM density maps and DeepTracer [13] and EMBuild [16] to reconstruct 3D protein structures from cryo-EM density maps.",
"DeepTracer has been successfully applied to reconstruct the structures of some SARS-CoV proteins from cryo-EM density maps (e.g., Figure REF).",
"In addition to CNN and U-Net, other deep learning architectures such as graph convolutional networks (GCN) and long- and short-term memory network (LSTM) have also been used with CNN to reconstruct protein structures from cryo-EM density maps [7].",
"A summary of different deep learning-based methods, their function (e.g., input and output) and availability is presented in Table 1.",
"|p2.7cm||p2.7cm|p6.5cm|p1cm| Methods Architecture Function Open source Structure Generator[7] 3-D CNN, GCN, Bidirectional LSTM First use 3-D CNN to identify amino acids and their rotameric identities in an EM map and then GCN and LSTM to build protein structures $$ Emap2sec[8] 3-D CNN Take voxel cubes as input to identify secondary structures of protein $$ AAnchor[9] 3-D CNN Take in voxel cubes to identify amino acid types and locations $$ A CNN Based Method[11] 3-D CNN Take in voxel cubes to detect secondary structures of protein from background $\\times $ CascadedCNN [10] Cascaded 3-D CNN Take in voxel cubes to identify C$\\alpha $ atoms of protein backbone and secondary structures to generate 3D protein structures $$ Haruspex[12] 3-D U-Net Take in voxel cubes to predict the probabilities of 4 different classes; $\\alpha $ -helix, $\\beta $ -sheet, nucleotide, or unassigned to assign secondary structures $$ DeepTracer[13] 3-D U-Net Take in voxel cubes to identify the location of backbone atoms, secondary structures and amino acid types simultaneously to build 3D structure $$ DeepTracer ID [14] DeepTracer (3-D U-Net) and pre-calculated AlphaFold2 protein library Use DeepTracer to generate an initial 3D protein structure to search AlphaFold2DB to identify similar structural hits for refinement $$ CR-I-TASSER [15] 3-D CNN, I-TASSER Predict C$\\alpha $ using 3-D CNN for selecting structural templates for I-TASSER to generate 3D protein structure $$ EMBuild [16] 3-D U-Net++, AlphaFold Integrate AlphaFold structure prediction, FFT-based global fitting, domain-based semi-flexible refinement, and graph-based iterative assembling with main-chain probability maps predicted by U-Net++ to 3D build protein structure $$ EMNUSS [6] 3-D U-Net++ Take in voxel cubes to identify secondary structures of protein $$ Summary of deep learning based methods for protein structure reconstruction from cryo-EM density maps.",
"Inspired by the recent breakthrough in developing deep learning methods of predicting protein structures from sequences such as AlphaFold [1] and RoseTTAFold [5], a new trend is to integrate deep learning methods of reconstruct protein structures from cryo-EM density maps with the advanced computational (e.g., deep learning) methods of predicting protein structures from sequences to obtain more accurate structural models.",
"For instance, DeepTracer ID [14] first uses DeepTracer to build an initial structure from cryo-EM density maps and then search the structure against a database of AlphaFold-predicted structures to identify similar structural hits to enhance the reconstructed structure.",
"EMBuild [16] combines the structures reconstructed from cryo-EM maps, AlphaFold-predicted structural models and other protein structural refinement methods to construct accurate structures for protein complexes.",
"DeepProLigand [4] integrates the protein structural models reconstructed from cryo-EM density maps by DeepTracer with the known template structures containing ligands to model protein-ligand interaction, which was ranked first in the ligand prediction in 2021 EMDataResource Ligand Model Challenge.",
"Collecting a sufficient amount of high-quality data to train and test deep learning models is critical for any deep learning task.",
"The common way to acquire the experimental cryo-EM density maps is through the EMDB [2].",
"An alternative approach employed by some methods such as Cascaded-CNN [10] and SSELearner [30] is to simulate the density map from the PDB protein structure.",
"Cascaded-CNN applies pdb2mrc from EMAN2 package [49], and VESPER uses pdb2vol from Situs package [51] to generate the simulated maps.",
"However, simulated maps lack complex noise, missing density values, and experimental artifacts which can arise from particle alignment errors, interaction of electron beam with the atoms, or movement of atoms during image capture.",
"Therefore, the deep learning models trained on simulated maps may not work as expected on very noisy experimental data.",
"To address the problem, CR-I-TASSER, EMNUSS and Emap2sec employs a hybrid training approach that uses both simulated maps and experimental maps in the training and validation process."
],
[
"Training data preprocessing",
"Prior to using the cryo-EM density map to train deep learning models, it is generally necessary to normalize and standardize the data to make them suitable for machine learning as shown by Cascaded-CNN and DeepTracer, which perform data grid resampling, density value normalization, and grid division.",
"These preprocessing steps ensure the uniformity among density maps and help deep learning models to extract features and recognize patterns more easily.",
"During the grid division, the 3D cryo-EM is splitted into the cubes of a specific size (e.g., 64 $\\times $ 64 $\\times $ 64 $Å^{3}$ by Cascaded-CNN and DeepTracer, 50 $\\times $ 50 $\\times $ 50 $Å^{3}$ by CR-I-TASSER, 40 $\\times $ 40 $\\times $ 40 $Å^{3}$ by Haruspex, and 11 $\\times $ 11 $\\times $ 11 $Å^{3}$ by Emap2sec and AAnchor).",
"Each of these cubes is then processed by the deep learning method to classify the voxels into the targeted classes such as amino acid types (identities) and secondary structures."
],
[
"Future directions",
"Deep learning has made a significant impact on protein structure reconstruction from cryo-EM density maps.",
"However, the field is still in the early stage of development.",
"The latest deep learning technology such as graph neural networks [52] and attention mechanisms [46] have not been used in the field.",
"While CNNs and U-Nets based on convolution are currently the most used methods for structure reconstruction, they have some shortcoming for 3D structural modeling.",
"CNNs are translation-equivariant, but not fully rotation invariant that is desirable for 3D structure analysis.",
"Moreover, the convolution mechanism propagates message in the constrained local receptive field, which is not as effective as the attention mechanism [46] that can leverage all the input information by automatically weighting the input features according to their relevance as demonstrated by the remarkable success of AlphaFold2 in protein structure prediction.",
"More sophisticated deep learning models like attention-based Transformer models [35], 3D-equivariant graph neural networks [36], and AlphaFold2-like deep learning models need to be developed to better use cryo-EM data to improve reconstruction accuracy.",
"Another important direction is to use deep learning to integrate cryo-EM data with multiple other sources of complementary data such as protein structural models predicted from sequences, structural templates in the Protein Data Bank, and protein sequences to more accurately reconstruct protein structures from noisy density maps that often miss the density values of some atoms.",
"The current integration process is limited to shallow data combination.",
"For instance, DeepTracer ID uses AlphaFold models to refine the structural models predicted from structural models reconstructed from deep learning.",
"More comprehensive, end-to-end deep learning models to combine multiple sources of data to generate accurate final protein structures can be developed to automatically and accurately reconstruct protein structures from the data.",
"Moreover, it is important to integrate cryo-EM based deep learning methods of reconstructing protein structures with the advanced methods developed in the field of protein structure prediction.",
"The structural models directly reconstructed from cryo-EM data by deep learning generally have correct overall topology, but the reconstructed models may not satisfy physicochemical restraints such as bond length and bond angles and not have all the molecular details (e.g., the precise location of all side chain atoms) [10], [4].",
"Linking the atoms of amino acids identified from the density maps into full peptide chains consistent with protein sequences and physicalchemical restraints is still challenging.",
"However, the modeling techniques such as protein structure refinement and molecular dynamics to fix these problems have been established for protein structure prediction [1].",
"Some methods such as CR-I-TASSER have started to integrate the two kinds of technologies.",
"More synergistic integration of the two are needed to generate high-quality realistic protein structures from cryo-EM data.",
"The development of high-quality deep learning models to reconstruct protein structures from cryo-EM density maps critically depends on the availability of sufficient high-quality training data.",
"Curating a large amount of high-quality training and test data is challenging and time consuming, but often receives little attention.",
"Currently, there are few well-curated data sets available for training and evaluating deep learning models in the field.",
"Therefore, more effort needs to be devoted to creating such data sets and make them to publicly available for the community to use."
],
[
"Conclusion",
"A number of useful deep learning models have been developed to reconstruct protein structures from cryo-EM density maps, demonstrating deep learning is a promising technology to further push the frontier of applying cryo-EM technology to determine protein structures.",
"As the deep learning field is evolving very fast, many more state-of-the-art deep learning architectures (e.g., AlphaFold2-like models and transformers) have yet to be applied to further advance the emerging field.",
"More sophisticated deep learning methods need to be developed to seemlessly integrate cryo-EM data with other complementary data such as predicted protein structures, protein sequences, and template structures to further improve cryo-EM-based structure determination.",
"A synergistic integration of cryo-EM based protein structure determination techniques and latest protein structure prediction techniques is also important for generating highly accurate native-like protein structures.",
"To speed up the development, more effort is need to create a large amount of high-quality cryo-EM training and test data for the community to use."
],
[
"Conflict of interest statement",
"The authors declare that there is no conflict of interest."
],
[
"Acknowledgements",
"This work was supported in part by Department of Energy grants (DE-AR0001213, DE-SC0020400, and DE-SC0021303), two NSF grants (DBI1759934 and IIS1763246), and NIH grants (R01GM093123 and R01GM146340)."
]
] | 2209.08171 |
[
[
"Confidence-Guided Data Augmentation for Deep Semi-Supervised Training"
],
[
"Abstract We propose a new data augmentation technique for semi-supervised learning settings that emphasizes learning from the most challenging regions of the feature space.",
"Starting with a fully supervised reference model, we first identify low confidence predictions.",
"These samples are then used to train a Variational AutoEncoder (VAE) that can generate an infinite number of additional images with similar distribution.",
"Finally, using the originally labeled data and the synthetically generated labeled and unlabeled data, we retrain a new model in a semi-supervised fashion.",
"We perform experiments on two benchmark RGB datasets: CIFAR-100 and STL-10, and show that the proposed scheme improves classification performance in terms of accuracy and robustness, while yielding comparable or superior results with respect to existing fully supervised approaches"
],
[
"Introduction",
"Deep learning models have achieved state of the art performances, especially for computer vision applications.",
"Much of the recent successes, however, can be attributed to the existence of large, high quality, labeled datasets.",
"In many real-world applications, collecting similar datasets is often cumbersome and time consuming.",
"Semi-Supervised Learning (SSL) aims to alleviate heavy labeling needs by leveraging the availability of unlabeled data to learn more robust models[4].",
"Data Augmentation (DA) is another solution to the problem of limited data.",
"It aims to increase the size and variability of training datasets in order to reduce overfitting and improve the model's generalizability [7].",
"Ideally, a good training set should contain enough variations within each class for the model to learn the most optimal decision boundaries.",
"However, when there are under-represented regions in the training feature space, especially in low data regime or in presence of low-quality inputs, the model risks learning sub-optimal decision boundaries, resulting in less accurate predictions (increased misclassifications) [21], [22].",
"One way to address this limitation is to augment the training dataset.",
"A common data augmentation approach is to apply different transformations to the training samples [20], or to generate new data by mixing existing samples and fusing their labels [10], [26].",
"Such data augmentations, however, may not fully cover the underrepresented regions because they only look at the immediate vicinity of input samples.",
"Another approach is to use existing detailed annotations, like part landmarks, to select additional samples from external labeled resources [24] and thus, rely on similar strongly supervised tasks to obtain new data to be added.",
"This approach may hurt the model's performance by introducing out-of-distribution samples.",
"Hence, generating the appropriate samples to include in the training set is a critical task.",
"In this paper, we use the model's classification confidence to guide synthetic data generation in order to augment the training data with samples from the under-performing regions that include the most misclassified samples We investigate the effect of generating synthetic data by a trained Variational Auto-Encoder (VAE [11]), and the use of these as unsupervised information in a deep semi-supervised learning framework (MixMatch [2]).",
"Our contributions can be summarized as follows: [noitemsep] Augment the training dataset by generating synthetic images drawn from the same distribution as those samples that are misclassified by a baseline model Alleviate the need to label augmented data by using an SSL framework Increase the diversity of the available training set, and thus, learn more accurate decision boundaries, without actually collecting new data Our approach can be easily integrated into existing neural networks for SSL with little efforts We perform experiments on two benchmark datasets, and show that, despite its simplicity, the proposed scheme yields promising improvements in terms of accuracy and robustness over fully-supervised settings."
],
[
"Related works",
"To highlight the main motivations behind our proposed approach, we first review VAEs and their main applications for image data augmentation or generative modeling in general.",
"Next, SSL state-of-the art methods are presented along with some of their main challenges."
],
[
"Variational Auto-Encoder (VAE)-Based Methods for Data Augmentation ",
"A Variational Auto-Encoder (VAE) implements a latent variable model using a composition of two neural networks.",
"A neural network decoder maps a latent variable configuration to an observation, and a neural network encoder approximately infers the latent variable configuration given an input observation [11].",
"VAEs are used for learning disentangled representations, generating discrete data, and nonlinear dimensionality reduction [15].",
"The key difference between VAEs and typical autoencoders is that VAEs learn latent variables with continuous distributions, which has shown to be a very helpful trait for tackling generative modeling problems.",
"Given a set of data $x \\in \\chi $ , a VAE seeks to maximize the likelihood of the associated parametric model $\\lbrace \\mathbb {P}(\\theta ), \\theta \\in \\Theta \\rbrace $ .",
"Based on the assumption that there exist latent variables $z$ living in a lower dimensional space $\\mathcal {Z}$ ($z \\in \\mathcal {Z}$ ), the marginal distribution can be expressed as follows: $p_{\\theta }(x)=\\int _{\\mathcal {Z}} p_{\\theta }(x \\mid z) q(z) d z$ where $q$ is a prior distribution over the latent variables acting as a regulation component and $p_{\\theta }(x \\mid z)$ is often a simple distribution and is referred to as the $\\it {decoder}$ .",
"A variational distribution $q_{\\varphi }$ is most of the time taken as a simple parameterized distribution (Gaussian, Bernoulli, ).",
"aiming at approximating the true posterior distribution and referred to as the $\\it {encoder}$ is then introduced.",
"Using Importance Sampling allows to derive an unbiased estimate of $p_{\\theta }(x)$ such that $\\mathbb {E}_{z \\sim q_{\\varphi }}\\left[\\hat{p}_{\\theta }\\right]=p_{\\theta }(x)$ .",
"Therefore, a lower bound on the logarithm of the objective function of eq:vae can be derived using Jensen’s inequality [3]: $\\log p_{\\theta }(x) \\ge \\mathbb {E}_{z \\sim q_{\\varphi }}\\left[\\log p_{\\theta }(x, z)-\\log q_{\\varphi }(z \\mid x)\\right]$ eq:vaejens is usually referred to as the $ELBO$ .",
"The $ELBO$ can be made tractable thanks to the reparametrization trick, allowing optimization with respect to both $\\theta $ and $\\varphi $ .",
"Once trained, the model's decoder functions as a generative model, allowing for the generation of new data by simply drawing a sample using the prior $q$ and passing it to the decoder.",
"In recent years, generative models with improved performance, such as VAEs and Generative Adversarial Networks (GAN) [8], have become increasingly popular models for DA.",
"As opposed to GANS, VAEs have witnessed a limited interest to perform DA and were mostly used for audio applications [9], [14], [23].",
"There have also been some other attempts on medical images for both classification [27], [13] and segmentation tasks [18], [19].",
"The biggest barrier preventing VAEs from being used more widely is the fact that they frequently generate fuzzy and blurry samples.",
"This drawback is further highlighted when VAEs are trained with a low number of samples which makes it hard to use them to perform DA in a low-data regime.",
"To address this issue, we use a VAE with a pretrained encoder which is fine-tuned on the samples of interest.",
"These VAE training samples are selected based mainly on model's confidence as well as additional samples from the training set."
],
[
"Semi-Supervised Learning (SSL)",
"Semi-supervised learning [4] (SSL) is a classical learning paradigm that has gained increasing interests with the rise of deep learning models which rely heavily on labeled data.",
"By leveraging the availability of unlabeled data, SSL methods can infer information about the inherent data structure, and thus, learn a more robust representation.",
"In recent years, SSL algorithms based on deep neural networks (deep-SSL) have proven successful on standard benchmark tasks for various applications including computer vision [16].",
"Figure: Confidence-guided data augmentation for semi-supervised image classification: (i) Fully supervised training, (ii) Softmax filtering, (iii) VAE data augmentation, (iv) Semi-supervised training.Most modern deep semi-supervised learning methods consist of adding a loss term which is computed on unlabeled data to encourage the model to generalize better to unseen data.",
"In much recent works, this loss term falls into one of three classes [16], [17]: entropy minimization, which encourages the model to output confident predictions on unlabeled data; consistency regularization, which encourages the model to produce the same output distribution when its inputs are perturbed; and generic regularization, which encourages the model to generalize well and avoid overfitting the training data.",
"Despite the promising performance achieved by deep SSL, there are still several theoretical challenges regarding the actual benefits of using unlabelled patterns in a supervised learning framework [1].",
"The inclusion of unlabeled data has proven to be very helpful when learning with limited labeled data.",
"However, this only holds true under the appropriate assumptions or conditions.",
"According to recent empirical studies, there are scenarios where the unlabeled data can degrade the model's performance, especially when there is a distribution misalignment between labeled and unlabeled data, or when the unlabeled data contains a lot of outliers and samples from unknown classes [25].",
"Based on these observations, we propose a novel data augmentation approach that aims not only to alleviate the distribution mismatch between the labeled data and the unlabeled data, but also to boost the semi-supervised modeling performance by emphasizing the samples that are most likely to be misclassified."
],
[
"Proposed method",
"Our approach seeks to sequentially generate new synthetic samples and integrate them within the training process in a semi-supervised fashion.",
"The proposed data augmentation is guided by a trained, fully-supervised, baseline model's performance.",
"The generated samples are drawn from the same distribution as the least performing samples based on a fully supervised reference model's outputs.",
"This can be especially beneficial when learning from limited labeled data and/or when collecting additional data is not an option.",
"By training a VAE to learn the latent representation of the baseline model's under-performing regions, we can adaptively provide the model with additional inputs that can bridge the performance gap across the different regions of the training feature space.",
"Adopting a semi-supervised training framework increases the model's robustness and resistance to the noise that might be introduced at the data augmentation stage.",
"In this section, we introduce our proposed data augmentation method for deep semi-supervised learning.",
"fig:approach shows a diagram of the proposed approach.",
"The training pipeline includes four main steps: (i) Fully supervised training, (ii) Softmax filtering, (iii) VAE data augmentation, (iv) Semi-supervised training.",
"We detail these below.",
"Formally, aside from a held out test set ($\\mathcal {D}_{\\mathcal {TEST}}$ ), we randomly split our input training dataset into three different partitions: [noitemsep] $\\mathcal {D}_{\\mathcal {L}}=\\left(\\mathbf {x}_{i}, y_{i}\\right)_{i=1}^{n}$ denotes the labeled training subset which contains $n$ images $x_i$ with respective labels $y_i$ , where $ 1<i<n$ .",
"$\\mathcal {D}_{\\mathcal {V}}=\\left(\\mathbf {x}_{i}, y_{i}\\right)_{i=1}^{n_{v}}$ denotes the validation subset of size $n_{v}$ which is used for model selection and hyper-parameters tuning.",
"$\\mathcal {D}_{\\mathcal {REF}}=\\left(\\mathbf {x}_{i}, y_{i}\\right)_{i=1}^{n_{ref}}$ , is a newly introduced subset, that is used to select the samples which will be used to train a VAE for data augmentation.",
"In all our experiments, $\\mathcal {D}_{\\mathcal {L}}$ (respectively $\\mathcal {D}_{\\mathcal {V}}$ and $\\mathcal {D}_{\\mathcal {REF}}$ ) constitutes 60% (respectively 20% and 20%) of the input data."
],
[
"Fully supervised training",
"We first train and validate a fully supervised model $f_{\\theta }^{FS}$ ($\\theta $ denotes the model's parameter) using the labeled training set $\\mathcal {D}_{\\mathcal {L}}$ and the validation set $\\mathcal {D}_{\\mathcal {V}}$ .",
"As a fully supervised baseline, we train a Wide Residual Networks (WideResNet) model.",
"WideResNets are shallower versions of ResNets where the depth is decreased and the width is increased.",
"WideResNets have achieved state of the art performances in most standard computer vision tasks, and are usually used as backbone models for deep semi-supervised models.",
"Moreover, WideResNet are faster to train since GPUs become more efficient on parallel computing with wider layers."
],
[
"Softmax filtering",
"The obtained model $f_{\\theta }^{FS}$ is afterwards tested on the third partition of the training set: $\\mathcal {D}_{\\mathcal {REF}}$ .",
"$\\mathcal {D}_{\\mathcal {REF}}$ serves as a held-out reference subset that is separate from the validation and testing subsets.",
"Based on the assumption that all three partitions are drawn from the same distribution, we expect that misclassifications from $\\mathcal {D}_{\\mathcal {REF}}$ are likely to be similar to the potential misclassifications from $\\mathcal {D}_{\\mathcal {TEST}}$ .",
"This intermediate evaluation step aims to identify and select the under-performing reference samples based on the model's prediction confidence.",
"$f_{\\theta }^{FS}$ generates a logits vector $z$ for each input sample $x$ .",
"We approximate the model's confidence score on a given prediction using $softmax$ function $S$ (eq:softmax).",
"$S(z)=\\frac{\\exp \\left(z\\right)}{\\sum \\exp \\left(z\\right)}$ Softmax converts the logits vector into a vector of probabilities, where the probabilities of each value are proportional to the relative scale of each value in the model's logits.",
"Hence, Softmax can reflect the prediction likelihood of each class.",
"We define $\\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}}$ as the subset of samples with the VAE reconsence in their true classes from $\\mathcal {D}_{\\mathcal {REF}}$ (eq:Dref), i.e., $\\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}} = \\lbrace x \\in \\mathcal {D}_{\\mathcal {REF}} \\mid S(f_{\\theta }^{FS}(x)) <= \\gamma \\rbrace $ where $\\gamma $ is a user predefined confidence threshold."
],
[
"Data augmentation",
"$\\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}}$ is used to train a VAE in order to learn the latent distribution of the misclassified samples and generate similar synthetic samples.",
"$\\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}}$ tends to have a relatively small size which makes training the VAE from scratch challenging.",
"To address this issue, we (i) use a pretrained encoder network; (ii) increase the size of $\\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}}$ by randomly adding more samples from $\\mathcal {D}_{\\mathcal {L}}$ .",
"The trained VAE is then used to generate two synthetic datasets: [noitemsep] $D_{Rec}$ denotes the set of VAE reconstructions of all images in $\\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}}$ as defined in eq:Drec.",
"These images are similar to their seed images and can be assigned the same labels.",
"$D_{Rec} = \\lbrace VAE(x) \\mid x \\in \\mathcal {D}_{{\\mathcal {REF}}_{\\mathcal {LOW}}} \\rbrace $ $D_{Synth}$ is a subset of $K$ randomly generated images.",
"Since these images are generated from random seeds, they cannot be assigned labels, and thus, are treated as unlabeled during semi-supervised training.",
"This can be advantageous as we can generate as many unlabeled samples as desired by sampling from different random seeds.",
"In our experiments we use $K=5000$ ."
],
[
"Semi-supervised training",
"A semi-supervised model is trained using: [noitemsep] $D_{Rec} \\cup \\mathcal {D}_{\\mathcal {L}}$ as training labeled data.",
"$D_{Synth}$ as unlabeled training samples.",
"$\\mathcal {D}_{\\mathcal {V}}$ is used for model selection and hyperparameters tuning.",
"In this research, we adopt MixMatch [2], an SSL algorithm that proposes a holistic approach which seamlessly unifies ideas and components from the dominant paradigms in SSL, resulting in an algorithm that is greater than the sum of its parts and surpasses the performance of the traditional approaches."
],
[
"Experimental analysis ",
"In this section, we evaluate our proposed data augmentation technique for semi-supervised learning on two relatively challenging benchmark datasets: STL-10, and CIFAR-100."
],
[
"Experimental setup",
"All experiments are implemented on Pytorch and ran on a computer equipped with an Intel Core i7-5930K CPU (12CPUs), an NVIDIA GeForce GTX TITAN X GPU with 12GB of VRAM and 128GB of RAM.",
"STL-10[5]: is a benchmark image classification dataset mainly used for developing unsupervised feature learning, deep learning, self-supervised learning algorithms.",
"It contains 96x96 RGB images from ten different classes.",
"Each class has few labeled training example.",
"STL-10 consists of 5000 training samples and 8000 testing samples split over 10 predefined folds.",
"It also comes with an additional 100000 unlabeled images for unsupervised learning.",
"However, we focus our experiments on the provided labeled partition since we want to simulate a low data training regime where access to additional unlabeled data is not possible.",
"We want to check the model's performance can be improved by only using the available labeled data.",
"CIFAR-100[12]: CIFAR-100 is another computer vision benchmark data that is an extension of CIFAR-10.",
"It contains 100 classes containing 600 32x32 images each.",
"There are 500 training images and 100 testing images per class.",
"The 100 classes in the CIFAR-100 are grouped into 20 superclasses.",
"Each image comes with a fine label (the class to which it belongs) and a coarse label (the superclass to which it belongs).",
"Considering its small size, and the large number of classes in CIFAR-100, and their variable granularity levels, it is also a challenging benchmark dataset.",
"For both datasets, we use the predefined training sets as our input data, which we randomly split into three subsets ($\\mathcal {D}_{\\mathcal {L}}$ , $\\mathcal {D}_{\\mathcal {V}}$ and $\\mathcal {D}_{\\mathcal {REF}}$ ) as shown in fig:approach.",
"We report the average performances over five runs.",
"Each run corresponds to a different random split.",
"We use the WideResNet-50 as the backbone network.",
"For both supervised and semi-supervised training, the network is trained for $10^4$ epochs using a stochastic gradient descent optimizer with a momentum of 0.9.",
"The learning rate starts from $3e^{-2}$ , and automatically decays by a factor of $10^{-2}$ based on the validation ($\\mathcal {D}_{\\mathcal {V}}$ ) loss.",
"For MixMatch algorithm, we use a mixup ratio $\\alpha = 0.5$ , a sharpening temperature $T=0.5$ , and an unsupervised loss factor $\\beta = 100$ ."
],
[
"Results and analysis",
"The core novelty of our approach is data augmentation using a VAE trained on low confidence samples based on the baseline fully supervised predictions.",
"For each dataset, we start by training and tuning the WideResNet-50 model on the labeled partition ($\\mathcal {D}_{\\mathcal {L}}$ ).",
"We evaluate the obtained models on $\\mathcal {D}_{\\mathcal {REF}}$ to identify and select the low confidence predictions: $\\mathcal {D}_{\\mathcal {REF_{LOW}}}$ .",
"Figure: Reconstructions generated by the VAE with and without using a pre-trained encoder - STL10We experiment with difference confidence thresholds $\\gamma $ .",
"The value that yields the best results is the lower outlier boundary of the reference prediction scores as defined in eq:gamma.",
"$\\gamma = Q_1 - 1.5*IQR$ where $Q_1$ is the lower quartile and $IQR$ is the interquartile range.",
"Figure: Synthetic samples generated by the VAE with and without using a pre-trained encoder - STL10Depending on the baseline performance, and on the number of reference samples, the size of $\\mathcal {D}_{\\mathcal {REF_{LOW}}}$ can be relatively small.",
"Hence, training a VAE from scratch on such a small set cannot be reliable to generate synthetic images that can capture the inner distribution of the inputs.",
"Therefore, we opt to using a pretrained encoder network [6].",
"We use a ResNet18 model pretrained on CIFAR-10, which is a similar natural images dataset.",
"fig:drec and fig:dsyn show some image samples generated from both $D_{Rec}$ and $D_{Synth}$ for STL-10 with and without using a pretrained VAE encoder.",
"Even though both VAEs can generate faithful reconstructions of the input images ($D_{Rec}$ ) in either cases (fig:drec), we notice that the synthetic images randomly generated by the pretrained VAE have more realistic patterns (colors, edges and shapes) (fig:dsyn).",
"It is worth noting that, the images generated by VAE are blurry.",
"They lack sharp edges and fine details.",
"This is because training VAEs is based on the assumption that the data follow a single Gaussian distribution whereas, natural images have multi-modal distribution.",
"Despite this limitation, our experiments show that the VAE outputs can still improve the training process.",
"Table: Classification testing accuracy of the first two iterations of the proposed approach on STL-10 and CIFAR-100Our experiment were designed to answer the following three main questions: Does using semi-supervised training with the synthetically generated data improve upon fully supervised training using only the available data?",
"Is there a benefit of generating new data at multiple iterations?",
"What is the impact of training with the raw reference subset $\\mathcal {D_{REF}}$ versus using it to guide augmented data generation?",
"tab:iter summarizes the classification accuracies on STL-10 and CIAR-100 for the first two iterations of our approach.",
"In the first iteration, we use the fully supervised model (first row) as a reference to select the low confidence samples from $\\mathcal {D}_{Ref}$ , and then we use the output to train a MixMatch model (second row).",
"For the second iteration, we use the obtained MixMatch model as the new reference, and we follow the same approach to retrain a second MixMatch model (third row).",
"As shown in tab:iter, our approach drastically improves the classification performance on both datasets compared to the fully supervised reference models.",
"Moreover, iterating further gives a slight improvement which is promising for potential future works.",
"This answers the first two questions as it proves that sequentially training a semi-supervised model using our proposed data augmentation technique improves upon fully supervised training using only the available data.",
"Table: MixMatch testing accuracies for training with the raw 𝒟 Ref \\mathcal {D}_{Ref} vs. training with the datasets generated by the VAE starting from 𝒟 Ref \\mathcal {D}_{Ref} (i.e., 𝒟 Synth \\mathcal {D}_{Synth} and 𝒟 Rec \\mathcal {D}_{Rec})To answer the third question, we train two MixMatch models.",
"In the first model, we use all the images in $\\mathcal {D_{REF}}$ as unlabeled data.",
"In the second model, we only use the synthetic data generated by the VAE starting from $\\mathcal {D_{REF}}$ .",
"We use $D_{Rec}$ as additional unlabeled data, and $D_{Synth}$ as unlabeled data.",
"tab:bmk shows the obtained results.",
"For both datasets, we obtain a slight improvement from using $\\mathcal {D_{REF}}$ to guide data augmentation instead of using it as it is."
],
[
"Conclusions",
"In this work, we introduced a new data augmentation technique for semi-supervised training.",
"By fine-tuning a pretrained VAE based on low confidence samples from a held-out training subset, we can generate both labeled and unlabeled augmentation that we can use to train a deep semi-supervised model.",
"Experiments on the benchmark CIFAR100 and STL10 datasets have validated the effectiveness of our approach.",
"Our experimental analysis has shown that: 1) confidence-guided synthetic image generation can both improve classification accuracy, and alleviate the need to collect additional data; 2) the usage of synthetic data as unsupervised knowledge in a semi-supervised setting helps to improve the model's performance; and 3) the approach can be extended to additional iterations while yielding sequential improvements.",
"Future work comprises experimenting with different deep generative models such as GANs, and designing a learning framework that integrates the four proposed components: (i)Fully supervised training; (ii)Softmax filtering; (iii) Data augmentation; and (iv) Semi-supervised training within a unified pipeline."
]
] | 2209.08174 |
[
[
"A Category of Ordered Algebras Equivalent to the Category of\n Multialgebras"
],
[
"Abstract It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (CABA's) taking a set to its power-set and, reciprocally, a complete, atomic Boolean algebra to its set of atomic elements.",
"Of course, such a correspondence induces an equivalence between the opposite category of $\\textbf{Set}$ and the category of CABA's.",
"We extend this result by taking multialgebras over a signature $\\Sigma$, specifically those whose non-deterministic operations cannot return the empty-set, to CABA's with their zero element removed and a structure of $\\Sigma$-algebra compatible with its order; reciprocally, one of these \"almost Boolean\" $\\Sigma$-algebras is taken to its set of atomic elements equipped with a structure of multialgebra over $\\Sigma$.",
"This leads to an equivalence between the category of $\\Sigma$-multialgebras and a category of ordered $\\Sigma$-algebras.",
"The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures."
],
[
"Introduction",
"It is a seminal result (see [8] for a proof) that a correlation between sets and complete, atomic Boolean algebras (CABA's) exists: a set is taken to its power-set, while a CABA is taken to its set of atomic elements.",
"These two assignments can be made into functors, giving rise to an equivalence of $\\textbf {Set}^{op}$ and $\\textbf {CABA}$ , the category with CABA's as objects.",
"This is part of a broader area of study, known as Stone dualities, which studies relationships between posets and topological spaces and was established by Stone ([6]) and his representation theorem, which states that every Boolean algebra is isomorphic to a field of sets, specifically the algebra of clopen sets of its Stone space (a topological space where points are ultrafilters of the original Boolean algebra).",
"Of course, this corresponds to an equivalence between the category $\\textbf {BA}$ of Boolean algebras and that of Stone spaces.",
"In the search of dualities with categories of posets, we focus on a more concrete equivalence closely related to the one between $\\textbf {Set}^{op}$ and $\\textbf {CABA}$ .",
"We arrive at a category of multialgebras (originally developed in [5]) over a signature $\\Sigma $ by adding further structure to $\\textbf {Set}$ ; correspondingly, we replace $\\textbf {CABA}$ by a category of CABA's equipped with $\\Sigma $ -operations compatible with its order.",
"However, we choose to stress slightly distinct categories, although results as per the aforementioned terms do exist.",
"We are most interested in non-partial multialgebras, where the result of an operation never returns the empty set.",
"Consequently, we exchange CABA's by posets corresponding to power-sets with the empty-set removed (that is, CABA's without minimum elements).",
"This way, a multialgebra, with universe $A$ , is taken to an algebra over the set of non-empty subsets of $A$ , with order given by inclusion and operations given by “accumulating” the operations of the multialgebra, while reciprocally, an “almost Boolean” algebra is taken to its set of atomic elements, transformed into a multialgebra.",
"In the area of research of non-deterministic semantics ([1]), specially paraconsistent logics ([3]), this offers an alternative: many logicians are reluctant to appeal to multialgebras in order to characterze a given logic, and the equivalence we here present shows one can, if one chooses to, replace such non-deterministic structures for more classically-behaved algebras, with an added underlying order.",
"In the first section, we give the definition of multialgebras we will use and introduce a brief characterization of power-sets without the empty-set.",
"In the second section, we introduce a naive approach to what we would like to accomplish, and show why it fails.",
"In the third section, we introduce the categories for which our desired result actually holds and the functors that will establish an equivalence between them, equivalence we detail in section four.",
"The final section is reserved for related results.",
"A preliminary version of this paper can be found in the PhD thesis [7]."
],
[
"Preliminary notions",
"A signature is a collection $\\Sigma =\\lbrace \\Sigma _{n}\\rbrace _{n\\in \\mathbb {N}}$ of possibly empty, disjoint sets indexed by the natural numbers; when there is no risk of confusion, the union $\\bigcup _{n\\in \\mathbb {N}}\\Sigma _{n}$ will also be denoted by $\\Sigma $ .",
"A $\\Sigma -$ multialgebra (also known as multialgebra) is a pair $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma })$ such that: $A$ is a non-empty set and, for $\\sigma \\in \\Sigma _{n}$ , $\\sigma _{\\mathcal {A}}$ is a function of the form $ \\sigma _{\\mathcal {A}}:A^{n}\\rightarrow \\mathcal {P}(A)\\setminus \\lbrace \\emptyset \\rbrace ,$ where $\\mathcal {P}(A)$ denotes the power-set of $A$ .",
"A homomorphism between $\\Sigma -$ multialgebras $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma })$ and $\\mathcal {B}=(B, \\lbrace \\sigma _{\\mathcal {B}}\\rbrace _{\\sigma \\in \\Sigma })$ is a function $h:A\\rightarrow B$ satisfying, for any $n\\in \\mathbb {N}$ , $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}\\in A$ , $ \\lbrace h(a): a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace \\subseteq \\sigma _{\\mathcal {B}}(h(a_{1}), \\ldots , h(a_{n})).$ If the inclusion, in the previous equation, were to be replaced by an equality, the resulting $h$ would be a full homomorphism; and a bijective full homomorphism is called an isomorphism.",
"Whenever $h$ is a homomorphism from $\\mathcal {A}$ to $\\mathcal {B}$ , we write $h:\\mathcal {A}\\rightarrow \\mathcal {B}$ ."
],
[
"Complete, atomic and bottomless Boolean algebras",
"Here, we will understand Boolean algebras mostly as partially-ordered sets (poset).",
"A pair $(A, \\le )$ is a Boolean algebra if: $\\le $ is reflexive, anti-symmetric and transitive; there are a maximum (denoted by 1) and a minimum (0), which we shall assume distinct; for every pair of elements $(a, b)\\in A^{2}$ , the set $\\lbrace a, b\\rbrace $ has a supremum, denoted by $\\sup \\lbrace a, b\\rbrace $ or $a\\vee b$ , and an infimum, denoted by $\\inf \\lbrace a, b\\rbrace $ or $a\\wedge b$ ; and every element $a$ has a complement $b$ which satisfies $ b=\\inf \\lbrace c\\in A: \\sup \\lbrace a, c\\rbrace =1\\rbrace $ and $ b=\\sup \\lbrace c\\in A: \\inf \\lbrace a, c\\rbrace =0\\rbrace .$ A Boolean algebra $(A, \\le )$ is said to be complete if every $S\\subseteq A$ has a supremum.",
"Lemma 1 Every Boolean algebra $(A, \\le )$ is distributive, meaning $a\\vee (b\\wedge c)=(a\\vee b)\\wedge (a\\vee c)$ and $a\\wedge (b\\vee c)=(a\\wedge b)\\vee (a\\wedge c)$ for any $a, b, c\\in A$ ; every complete Boolean algebra $(A, \\le )$ is infinite distributive, meaning that for any $S\\cup \\lbrace a\\rbrace \\subseteq A$ , $\\sup \\lbrace \\inf \\lbrace a, s\\rbrace : s\\in S\\rbrace =\\inf \\lbrace a, \\sup S\\rbrace $ and $\\inf \\lbrace \\sup \\lbrace a, s\\rbrace : s\\in S\\rbrace =\\sup \\lbrace a, \\inf S\\rbrace $ .",
"An element $a$ of a Boolean algebra is said to be an atom if it is minimal in the underlying order of the algebra when restricted to $A\\setminus \\lbrace 0\\rbrace $ , meaning that if $b\\le a$ , then either $b=0$ or $b=a$ .",
"The set of all the atoms $d$ such that $d \\le a$ will be denoted by $A_{a}$ .",
"Finally, a Boolean algebra is said to be atomic if, for every one of its elements $a$ , $a=\\sup A_{a}$ .",
"Notice that complete, atomic Boolean algebras are power-sets.",
"Essentially, if one takes for a Boolean algebra $\\mathcal {A}=(A, \\le )$ the set $A_{1}$ of all of its atoms, one sees $\\mathcal {A}$ is isomorphic to $\\mathcal {P}(A_{1})$ , the power-set of $A_{1}$ , being an arbitrary element $a\\in A\\setminus \\lbrace 0\\rbrace $ taken, by this isomorphism, to $A_{a}$ , while 0 is taken to $\\emptyset $ .",
"Reciprocally, one associates to a set $X$ its obvious corresponding complete, atomic Boolean algebra $\\mathcal {P}(X)$ .",
"For more information, look at Theorem $2.4$ of [8].",
"We would like to work with Boolean algebras that are, simultaneously, complete, atomic and bottomless, meaning it lacks a bottom element: this seems a contradiction, given we assume Boolean algebras to have bottom elements, but this can be adequately formalized.",
"Definition 1 Given a non-empty partially ordered set $\\mathcal {A}=(A, \\le _{\\mathcal {A}})$ , we define $\\mathcal {A}_{0}$ as the partially ordered set $ (A\\cup \\lbrace 0\\rbrace , \\le _{\\mathcal {A}_{0}}),$ where we assume $0\\notin A$ , such that $a\\le _{\\mathcal {A}_{0}} b$ if and only if: either $a\\le _{\\mathcal {A}} b$ ; or $a=0$ .",
"Definition 2 The non-empty partially ordered set $\\mathcal {A}$ is a complete, atomic and bottomless Boolean algebra whenever $\\mathcal {A}_{0}$ is a complete, atomic Boolean algebra.",
"Notice that, since $\\mathcal {P}(\\emptyset )$ only has $\\emptyset $ as element, for any complete, atomic and bottomless Boolean algebra $\\mathcal {A}$ we cannot have $\\mathcal {A}_{0}=\\mathcal {P}(\\emptyset )$ , given $\\mathcal {A}$ has at least one element and therefore $\\mathcal {A}_{0}$ must have at least two.",
"This means complete, atomic and bottomless Boolean algebras correspond to the powerset of non-empty sets with $\\emptyset $ removed.",
"Proposition 1 If $\\mathcal {A}$ is a partially ordered set, so is $\\mathcal {A}_{0}$ .",
"Lemma 2 Given a partially ordered set $(A, \\le )$ , for elements $a, b\\in A$ we have that the supremum of the lower bounds of $\\lbrace a,b\\rbrace $ , if it exists, is itself a lower bound for $\\lbrace a,b\\rbrace $ .",
"Let $s$ be he supremum of the lower bounds of $\\lbrace a, b\\rbrace $ : by definition, this means that for any upper bound $d$ for the set $\\lbrace c\\in A: c\\le a, c\\le b\\rbrace $ of lower bounds, $s\\le d$ ; but, since $a$ and $b$ are such upper bounds, we find that $s\\le a$ and $s\\le b$ .",
"Theorem 1 A partially ordered set $(A, \\le )$ which satisfies all following conditions is a complete, atomic and bottomless Boolean algebra.",
"It has a maximum element 1.",
"All non-empty subsets $S$ of $A$ have a supremum.",
"For every $a\\in A$ different from 1 there exists $b\\in A$ , named the complement of $a$ , such that $ b=\\inf \\lbrace c\\in A: \\sup \\lbrace a, c\\rbrace =1\\rbrace $ and $ b=\\sup \\lbrace c\\in A: \\text{$\\inf \\lbrace a, c\\rbrace $ does not exist}\\rbrace ,$ property we call being semi-complemented.",
"Denoting by $A_{a}$ the set of minimal elements smaller than $a$ , $a=\\sup A_{a}$ .",
"Suppose that $\\mathcal {A}=(A, \\le _{\\mathcal {A}})$ is a partially ordered set satisfying the previous list of conditions.",
"Since $\\mathcal {A}$ is a partially ordered set, so is $\\mathcal {A}_{0}$ from Proposition REF .",
"The maximum 1 of $\\mathcal {A}$ remains a maximum in $\\mathcal {A}_{0}$ , while 0 becomes a minimum.",
"For non-zero elements $a$ and $b$ , the supremum in $\\mathcal {A}_{0}$ remains the same as in $\\mathcal {A}$ , while if $a=0$ or $b=0$ the supremum is simply the largest of the two.",
"If $a$ or $b$ are equal to 0, the infimum is 0, while if $a, b\\in A$ there are two cases to consider: if $\\inf \\lbrace a, b\\rbrace $ was defined in $\\mathcal {A}$ , it remains the same in $\\mathcal {A}_{0}$ ; if the infimum was not defined in $\\mathcal {A}$ , this means that there were no lower bounds for both $a$ and $b$ , since otherwise we would have $ \\inf \\lbrace a,b\\rbrace =\\sup \\lbrace c\\in A: c\\le _{\\mathcal {A}}a\\quad \\text{and}\\quad c\\le _{\\mathcal {A}}b\\rbrace $ by Lemma REF , and therefore the infimum of both in $\\mathcal {A}_{0}$ is 0.",
"Every element $a\\in A\\setminus \\lbrace 1\\rbrace $ already has a complement $b$ in $\\mathcal {A}$ such that $b=\\inf \\lbrace c\\in A: \\sup \\lbrace a, c\\rbrace =1\\rbrace $ and $ b=\\sup \\lbrace c\\in A: \\text{$\\inf \\lbrace a, c\\rbrace $ does not exist}\\rbrace ;$ of course the first equality keeps on holding in $\\mathcal {A}_{0}$ , while the second becomes, remembering that the non-defined infima in $\\mathcal {A}$ become 0 in $\\mathcal {A}_{0}$ , $ b=\\sup \\lbrace c\\in A: \\inf \\lbrace a, c\\rbrace =0\\rbrace ;$ the complement of 1 is clearly 0 and vice-versa.",
"This proves $\\mathcal {A}_{0}$ is a Boolean algebra.",
"Since $\\mathcal {A}$ is closed under suprema of non-empty sets and $\\sup \\emptyset =0$ in $\\mathcal {A}_{0}$ , it is clear that $\\mathcal {A}_{0}$ is closed under any suprema.",
"Clearly $\\mathcal {A}_{0}$ remains atomic, since $\\mathcal {A}$ is atomic, what finishes the proof that the previous list of conditions imply $\\mathcal {A}$ is a complete, atomic and bottomless Boolean algebra.",
"Theorem 2 The reciprocal of Theorem REF holds, meaning that complete, atomic and bottomless Boolean algebras satisfy the list of conditions found in REF .",
"Given a partially ordered set $\\mathcal {A}$ , suppose $\\mathcal {A}_{0}$ is a complete, atomic Boolean algebra.",
"The maximum 1 of $\\mathcal {A}_{0}$ is still a maximum in $\\mathcal {A}$ .",
"The supremum of any non-empty set in $\\mathcal {A}$ is just its supremum in $\\mathcal {A}_{0}$ .",
"Given any element $a\\ne 1$ , its complement $b$ in $\\mathcal {A}_{0}$ ends up being also its complement in $\\mathcal {A}$ .",
"Clearly $ b=\\inf \\lbrace c\\in A: \\sup \\lbrace a,c\\rbrace =1\\rbrace .$ Now, $\\inf \\lbrace a,c\\rbrace $ does not exist in $\\mathcal {A}$ if, and only if, $\\inf \\lbrace a,c\\rbrace =0$ in $\\mathcal {A}_{0}$ : we already proved that if $\\inf \\lbrace a,c\\rbrace $ does not exist in $\\mathcal {A}$ then $\\inf \\lbrace a,c\\rbrace =0$ in $\\mathcal {A}_{0}$ , remaining to show the reciprocal; if the infimum of $a$ and $c$ existed in $\\mathcal {A}$ , it would equal 0 in $\\mathcal {A}_{0}$ given the unicity of the infimum, contradicting that 0 is not in $\\mathcal {A}$ .",
"This way, we find that in $\\mathcal {A}$ $ b=\\sup \\lbrace c\\in A: \\text{$\\inf \\lbrace a,c\\rbrace $ does not exist}\\rbrace ,$ as required.",
"Clearly $\\mathcal {A}_{0}$ being atomic implies $\\mathcal {A}$ being atomic.",
"Proposition 2 If $(A, \\le _{\\mathcal {A}})$ is a complete, atomic and bottomless Boolean algebra, for any $S\\subseteq A$ , if $ S^{a}=\\lbrace s\\in S: \\text{$\\inf \\lbrace a, s\\rbrace $ exists}\\rbrace \\ne \\emptyset ,$ then $ \\sup \\lbrace \\inf \\lbrace a, s\\rbrace : s\\in S^{a}\\rbrace =\\inf \\lbrace a, \\sup S\\rbrace ;$ if $S^{a}=\\emptyset $ , $\\inf \\lbrace a, \\sup S\\rbrace $ also does not exist.",
"If $S^{a}=\\emptyset $ this means that $\\inf \\lbrace a,s\\rbrace =0$ for every $s\\in S$ in $\\mathcal {A}_{0}$ , and therefore $\\inf \\lbrace a, \\sup S\\rbrace =0$ , so that the same infimum no longer exists in $\\mathcal {A}$ .",
"If $S^{a}\\ne \\emptyset $ , all infima and suprema in $\\sup \\lbrace \\inf \\lbrace a,s\\rbrace : s\\in S^{a}\\rbrace $ and $\\inf \\lbrace a, \\sup S\\rbrace $ exist in $\\mathcal {A}$ and are therefore equal to their counterparts in $\\mathcal {A}_{0}$ ; given $\\sup \\lbrace \\inf \\lbrace a, s\\rbrace : s\\in S^{a}\\rbrace =\\sup \\lbrace \\inf \\lbrace a, s\\rbrace : s\\in S\\rbrace $ in $\\mathcal {A}_{0}$ , since $s\\in S\\setminus S^{a}$ implies $\\inf \\lbrace a, s\\rbrace =0$ , by the infinite-distributivity of $\\mathcal {A}_{0}$ one proves the desired result.",
"The lesson to be taken from this short exposition is that a complete, atomic and bottomless Boolean algebra is a power-set (of a non-empty set) with the empty-set removed.",
"This will be important to us given our multialgebras cannot return the empty-set as the result of an operation."
],
[
"A first attempt",
"Consider the categories $\\textbf {Alg}(\\Sigma )$ of $\\Sigma $ -algebras, with homomorphisms between $\\Sigma $ -algebras as morphisms, and $\\textbf {MAlg}(\\Sigma )$ of $\\Sigma $ -multialgebras, with homomorphisms between $\\Sigma $ -multialgebras as morphisms.",
"For simplicity, denote the set of non-empty subsets of $A$ , $\\mathcal {P}(A)\\setminus \\lbrace \\emptyset \\rbrace $ , by $\\mathcal {P}^{*}(A)$ .",
"For a $\\Sigma $ -multialgebra $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma })$ , consider the $\\Sigma $ -algebra $\\mathsf {P}(\\mathcal {A})=(\\mathcal {P}^{*}(A), \\lbrace \\sigma _{\\mathsf {P}(\\mathcal {A})}\\rbrace _{\\sigma \\in \\Sigma })$ where, for a $\\sigma \\in \\Sigma _{n}$ and nonempty $A_{1}, \\ldots , A_{n}\\subseteq A$ , $ \\sigma _{\\mathsf {P}(\\mathcal {A})}(A_{1}, \\ldots , A_{n})=\\bigcup _{(a_{1}, \\ldots , a_{n})\\in A_{1}\\times \\cdots \\times A_{n}}\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}).$ Again, for simplicity, we may write the previous equation as $\\sigma _{\\mathsf {P}(\\mathcal {A})}(A_{1}, \\ldots , A_{n})=\\bigcup \\lbrace \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}) : a_{i}\\in A_{i}\\rbrace $ .",
"We also define, for $\\mathcal {A}$ and $\\mathcal {B}$ two $\\Sigma $ -multialgebras and a homomorphism $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ , the function $\\mathsf {P}( h ):\\mathsf {P}(A)\\rightarrow \\mathsf {P}(B)$ such that, for a $\\emptyset \\ne A^{\\prime }\\subseteq A$ , $ \\mathsf {P}( h )(A^{\\prime })=\\lbrace h (a)\\in B: a\\in A^{\\prime }\\rbrace .$ One could hope that $\\mathsf {P}( h )$ is actually a $\\Sigma $ -homomorphism, perhaps making of $\\mathsf {P}$ a functor from $\\textbf {MAlg}(\\Sigma )$ to $\\textbf {Alg}(\\Sigma )$ , but the following result shows this is usually not to be expected.",
"Lemma 3 For $\\mathcal {A}$ and $\\mathcal {B}$ two $\\Sigma $ -multialgebras and $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ a homomorphism, $\\mathsf {P}( h )$ satisfies $ \\mathsf {P}( h )(\\sigma _{\\mathsf {P}(\\mathsf {A})}(A_{1}, \\ldots , A_{n}))\\subseteq \\sigma _{\\mathsf {P}(\\mathsf {B})}(\\mathsf {P}( h )(A_{1}), \\ldots , \\mathsf {P}( h )(A_{n}))$ for all $\\sigma \\in \\Sigma $ and nonempty $A_{1}, \\ldots , A_{n}\\subseteq A$ .",
"If $ h $ is a full homomorphism, $\\mathsf {P}( h )$ is a homomorphism.",
"Given $\\sigma \\in \\Sigma _{n}$ and nonempty $A_{1}, \\ldots , A_{n}\\subseteq A$ , we have that $ \\sigma _{\\mathsf {P}(\\mathcal {B})}(\\mathsf {P}( h )(A_{1}), \\ldots , \\mathsf {P}( h )(A_{n}))=\\bigcup \\lbrace \\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n}) : b_{i}\\in \\mathsf {P}( h )(A_{i})\\rbrace =$ $\\bigcup \\lbrace \\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n}) : b_{i}\\in \\lbrace h (a) : a\\in A_{i}\\rbrace \\rbrace =\\bigcup \\lbrace \\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n})) : a_{i}\\in A_{i}\\rbrace ,$ which clearly contains $ \\bigcup \\lbrace \\lbrace h (a): a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace : a_{i}\\in A_{i}\\rbrace = \\lbrace h (a): a\\in \\bigcup \\lbrace \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}) : a_{i}\\in A_{i}\\rbrace \\rbrace =$ $ \\lbrace h (a): a\\in \\sigma _{\\mathsf {P}(\\mathcal {A})}(A_{1}, \\ldots , A_{n})\\rbrace =\\mathsf {P}( h )(\\sigma _{\\mathsf {P}(\\mathcal {A})}(A_{1}, \\ldots , A_{n})),$ so that $\\mathsf {P}( h )$ satisfies the required property.",
"If $ h $ is a full homomorphism, $\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n}))=\\lbrace h (a) : a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace $ , and the inclusion in the equations above becomes an equality.",
"So, let us restrict $\\mathsf {P}$ for a moment to the category $\\textbf {MAlg}_{=}(\\Sigma )$ , of $\\Sigma $ -multialgebras with only full homomorphisms between them as morphisms, and let us call this new transformation $\\mathsf {P}_{=}:\\textbf {MAlg}_{=}(\\Sigma )\\rightarrow \\textbf {Alg}(\\Sigma )$ .",
"Proposition 3 $\\mathsf {P}_{=}$ is, in fact, a functor from $\\textbf {MAlg}_{=}(\\Sigma )$ to $\\textbf {Alg}(\\Sigma )$ .",
"Unfortunately, $\\mathsf {P}_{=}$ is not injective on objects: take the signature $\\Sigma _{s}$ with a single unary operator $s$ , and consider the $\\Sigma $ -multialgebras $\\mathcal {A}=(\\lbrace 0,1\\rbrace , \\lbrace s_{\\mathcal {A}}\\rbrace )$ and $\\mathcal {B}=(\\lbrace 0,1\\rbrace , \\lbrace s_{\\mathcal {B}}\\rbrace )$ such that: $s_{\\mathcal {A}}(0)=s_{\\mathcal {A}}(1)=\\lbrace 1\\rbrace $ and $s_{\\mathcal {B}}(0)=s_{\\mathcal {B}}(1)=\\lbrace 0,1\\rbrace $ .",
"Figure: *Clearly the two of then are not isomorphic, given that the result of an operation in $\\mathcal {A}$ always has cardinality 1 and in $\\mathcal {B}$ alway has cardinality 2.",
"However, we have that $s_{\\mathsf {P}_{=}(\\mathcal {A})}(\\lbrace 0\\rbrace )=s_{\\mathsf {P}_{=}(\\mathcal {A})}(\\lbrace 1\\rbrace )=s_{\\mathsf {P}_{=}(\\mathcal {A})}(\\lbrace 0, 1\\rbrace )=\\lbrace 1\\rbrace $ , while $s_{\\mathsf {P}_{=}(\\mathcal {B})}(\\lbrace 0\\rbrace )=s_{\\mathsf {P}_{=}(\\mathcal {A})}(\\lbrace 1\\rbrace )=s_{\\mathsf {P}_{=}(\\mathcal {A})}(\\lbrace 0, 1\\rbrace )=\\lbrace 0,1\\rbrace $ .",
"Figure: *Taking the function $ h :\\mathcal {P}^{*}(A)\\rightarrow \\mathcal {P}^{*}(B)$ such that $ h (\\lbrace 0\\rbrace )=\\lbrace 0\\rbrace $ , $ h (\\lbrace 1\\rbrace )=\\lbrace 0,1\\rbrace $ , and $ h (\\lbrace 0,1\\rbrace )=\\lbrace 1\\rbrace $ , we see that it is a bijection and a homomorphism, and therefore $ h :\\mathsf {P}_{=}(\\mathcal {A})\\rightarrow \\mathsf {P}_{=}(\\mathcal {B})$ is an isomorphism."
],
[
"An improvement",
"The problem with our definition of $\\mathsf {P}_{=}$ is that it disregards the structure of the universe of $\\mathcal {P}(\\mathcal {A})$ .",
"So, we change our target category to reflect this structure.",
"Definition 3 Given a signature $\\Sigma $ , a $(\\Sigma , \\le )$ -algebra $\\mathcal {A}$ is a triple $(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma }, \\le _{\\mathcal {A}})$ such that: $(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma })$ is a $\\Sigma $ -algebra; $(A,\\le _{\\mathcal {A}})$ is a complete, atomic and bottomless Boolean algebra; if $A_{a}$ is the set of minimal elements of $(A, \\le _{\\mathcal {A}})$ (atoms) less than or equal to $a$ , for all $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}$ we have that $ \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})=\\sup \\lbrace \\sigma _{\\mathcal {A}}(b_{1}, \\ldots , b_{n}): (b_{1}, \\ldots , b_{n})\\in A_{a_{1}}\\times \\cdots \\times A_{a_{n}}\\rbrace .$ Proposition 4 For $\\mathcal {A}$ a $(\\Sigma , \\le )$ -algebra, any $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}, b_{1}, \\ldots , b_{n}\\in A$ such that $a_{1}\\le _{\\mathcal {A}} b_{1}$ , ..., $a_{n}\\le _{\\mathcal {A}} b_{n}$ , one has $\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\le _{\\mathcal {A}} \\sigma _{\\mathcal {A}}(b_{1}, \\ldots , b_{n})$ .",
"Since, for every $i\\in \\lbrace 1, \\ldots , n\\rbrace $ , $a_{i}\\le _{\\mathcal {A}}b_{i}$ , we have that $A_{a_{i}}\\subseteq A_{b_{i}}$ , one concludes that $A_{a_{1}}\\times \\cdots \\times A_{a_{n}}\\subseteq A_{b_{1}}\\times \\cdots \\times A_{b_{n}}$ ; this way, $ \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})=\\sup \\lbrace \\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n}): c_{i}\\in A_{a_{i}}\\rbrace \\le _{\\mathcal {A}} \\sup \\lbrace \\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n}): c_{i}\\in A_{b_{i}}\\rbrace =\\sigma _{\\mathcal {A}}(b_{1}, \\ldots , b_{n}).$ For a $\\Sigma $ -multialgebra $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\sigma })$ , we define $\\mathbb {P}(\\mathcal {A})$ as the $(\\Sigma , \\le )$ -algebra $ (\\mathcal {P}^{*}(A), \\lbrace \\sigma _{\\mathbb {P}(\\mathcal {A})}\\rbrace _{\\sigma \\in \\Sigma }, \\le _{\\mathbb {P}(\\mathcal {A})})$ such that $(\\mathcal {P}^{*}(A), \\lbrace \\sigma _{\\mathbb {P}(\\mathcal {A})}\\rbrace _{\\sigma \\in \\Sigma })$ is exactly the $\\Sigma $ -algebra $\\mathsf {P}(\\mathcal {A})$ defined at the beginning of Section and, for nonempty subsets $A_{1}$ and $A_{2}$ of $A$ , $A_{1}\\le _{\\mathbb {P}(\\mathcal {A})} A_{2}$ if and only if $A_{1}\\subseteq A_{2}$ .",
"Since: $\\mathsf {P}(\\mathcal {A})$ is a $\\Sigma $ -algebra; $(\\mathcal {P}^{*}(A), \\le _{\\mathbb {P}(\\mathcal {A})})$ is a complete, atomic and bottomless Boolean algebra, given that $\\mathcal {P}(A)$ is a complete, atomic Boolean algebra with at least two elements; and, for $\\sigma \\in \\Sigma _{n}$ and $\\emptyset \\ne A_{1}, \\ldots , A_{n}\\subseteq A$ , since the atoms of $A_{i}$ are exactly $A_{A_{i}}=\\lbrace \\lbrace a\\rbrace : a\\in A_{i}\\rbrace $ , $ \\sigma _{\\mathbb {P}(\\mathcal {A})}(A_{1}, \\ldots , A_{n})=\\bigcup \\lbrace \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}) : a_{i}\\in A_{i}\\rbrace =\\bigcup \\lbrace \\sigma _{\\mathbb {P}(\\mathcal {A})}(\\lbrace a_{1}\\rbrace , \\ldots , \\lbrace a_{n}\\rbrace ) : \\lbrace a_{i}\\rbrace \\in A_{A_{i}}\\rbrace ;$ we truly have that $\\mathbb {P}(\\mathcal {A})$ is a $(\\Sigma , \\le )$ -algebra.",
"Definition 4 Given $(\\Sigma , \\le )$ -algebras $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma }, \\le _{\\mathcal {A}})$ and $\\mathcal {B}=(B, \\lbrace \\sigma _{\\mathcal {B}}\\rbrace _{\\sigma \\in \\Sigma }, \\le _{\\mathcal {B}})$ , a function $ h :A\\rightarrow B$ is said to be a $(\\Sigma , \\le )$ -homomorphism, in which case we write $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ , when: for all $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}\\in A$ we have that $ h (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}))\\le _{\\mathcal {B}}\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n}));$ $ h $ is continuous, meaning that, for every non-empty subset $A^{\\prime }\\subseteq A$ , $ h (\\sup A^{\\prime })=\\sup \\lbrace h (a): a\\in A^{\\prime }\\rbrace $ ; $ h $ maps minimal elements of $(A, \\le _{\\mathcal {A}})$ to minimal elements of $(B, \\le _{\\mathcal {B}})$ .",
"Notice that a $(\\Sigma , \\le )-$ homomorphism is essentially an “almost $\\Sigma -$ homomorphism” which is also continuous and minimal-elements-preserving.",
"Notice also that a $(\\Sigma , \\le )$ -homomorphism is order preserving: if $a\\le _{\\mathcal {A}}b$ , then $b=\\sup \\lbrace a,b\\rbrace $ , and therefore $ h (b)=\\sup \\lbrace h (a), h (b)\\rbrace $ , meaning that $ h (a)\\le _{\\mathcal {B}} h (b)$ ."
],
[
"$\\mathbb {P}$ is a functor",
"Lemma 4 The composition of $(\\Sigma , \\le )$ -homomorphisms returns a $(\\Sigma , \\le )$ -homomorphism.",
"Take $(\\Sigma , \\le )$ -algebras $\\mathcal {A}$ , $\\mathcal {B}$ and $\\mathcal {C}$ , and $(\\Sigma , \\le )$ -homomorphisms $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ and $ h^{\\prime } :\\mathcal {B}\\rightarrow \\mathcal {C}$ .",
"$ h^{\\prime } \\circ h $ obviously is a function from $A$ to $C$ , so let $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}\\in A$ : we have that, since both $ h^{\\prime } $ and $ h $ are $(\\Sigma , \\le )$ -homomorphisms, $ h^{\\prime } \\circ h (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}))= h^{\\prime } ( h (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})))\\le _{\\mathcal {C}} h^{\\prime } (\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n}))),$ because $ h^{\\prime } $ is order-preserving and $ h (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}))\\le _{\\mathcal {B}}\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n}))$ , and $ h^{\\prime } (\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n})))\\le _{\\mathcal {C}}\\sigma _{\\mathcal {C}}( h^{\\prime } ( h (a_{1})), \\ldots , h^{\\prime } ( h (a_{n})))= \\sigma _{\\mathcal {C}}( h^{\\prime } \\circ h (a_{1}), \\ldots , h^{\\prime } \\circ h (a_{n}))$ since $ h^{\\prime } $ is an “almost homomorphism”.",
"Given a non-empty $A^{\\prime }\\subseteq A$ , we have that $ h (\\sup A^{\\prime })=\\sup \\lbrace h (a): a\\in A^{\\prime }\\rbrace $ and, denoting $\\lbrace h (a): a\\in A^{\\prime }\\rbrace $ as $B^{\\prime }$ , we have that $ h^{\\prime } (\\sup B^{\\prime })=\\sup \\lbrace h^{\\prime } (b): b\\in B^{\\prime }\\rbrace $ ; since $\\sup B^{\\prime }= h (\\sup A^{\\prime })$ , we obtain $ h^{\\prime } \\circ h (\\sup A^{\\prime })=\\sup \\lbrace h^{\\prime } (b): b\\in B^{\\prime }\\rbrace =\\sup \\lbrace h^{\\prime } \\circ h (a): a\\in A^{\\prime }\\rbrace ,$ which means that $ h^{\\prime } \\circ h $ is continuous.",
"Finally, if $a\\in A$ is a minimal element, $ h (a)\\in B$ is a minimal element, since $ h $ preserves minimal elements, and for the same reason $ h^{\\prime } \\circ h (a)= h^{\\prime } ( h (a))\\in C$ remains a minimal element still, and from all of the above $ h^{\\prime } \\circ h $ is a $(\\Sigma , \\le )$ -homomorphism.",
"Proposition 5 When we take as objects all $(\\Sigma , \\le )$ -algebras and as morphisms all the $(\\Sigma , \\le )$ -homomorphisms between them, the resulting object is a category, denoted by $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ .",
"So the transformation taking a $\\Sigma $ -multialgebra $\\mathcal {A}$ to $\\mathbb {P}(\\mathcal {A})$ and a homomorphism $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ to the $(\\Sigma , \\le )$ -homomorphism $\\mathbb {P}( h ):\\mathbb {P}(\\mathcal {A})\\rightarrow \\mathbb {P}(\\mathcal {B})$ such that, for an $\\emptyset \\ne A^{\\prime }\\subseteq A$ , $ \\mathbb {P}( h )(A^{\\prime })=\\lbrace h (a)\\in B: a\\in A^{\\prime }\\rbrace ,$ is a functor, of the form $\\mathbb {P}:\\textbf {MAlg}(\\Sigma )\\rightarrow \\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ .",
"First we must show that $\\mathbb {P}( h )$ is, in fact, an $(\\Sigma , \\le )$ -homomor- phism: given Lemma REF and the fact that $\\mathsf {P}( h )=\\mathbb {P}( h )$ , we have that $\\mathbb {P}( h )$ satisfies the first condition for being a $(\\Sigma , \\le )-$ homomorphism; and, if $\\emptyset \\ne A^{\\prime \\prime }$ is a subset of $\\mathcal {P}(A)$ , we have that $ \\mathbb {P}( h )(\\sup A^{\\prime \\prime })=\\lbrace h (a): a\\in \\sup A^{\\prime \\prime }\\rbrace =\\lbrace h (a): a\\in \\bigcup A^{\\prime \\prime }\\rbrace =\\bigcup \\lbrace \\lbrace h (a): a\\in A^{\\prime }\\rbrace : A^{\\prime }\\in A^{\\prime \\prime }\\rbrace =$ $\\bigcup \\lbrace \\mathbb {P}( h )(A^{\\prime }) : A^{\\prime }\\in A^{\\prime \\prime }\\rbrace =\\sup \\lbrace \\mathbb {P}( h )(A^{\\prime }): A^{\\prime }\\in A^{\\prime \\prime }\\rbrace ,$ what proves the satisfaction of the second condition; for the third condition, we remember that the minimal elements of $(\\mathcal {P}^{*}(A), \\subseteq )$ are the singletons, that is, sets of the form $\\lbrace a\\rbrace $ with $a\\in A$ , and since $\\mathbb {P}( h )(\\lbrace a\\rbrace )=\\lbrace h (a)\\rbrace $ , $\\mathbb {P}( h )$ preserves minimal elements.",
"Theorem 3 $\\mathbb {P}:\\textbf {MAlg}(\\Sigma )\\rightarrow \\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ is a functor."
],
[
"$\\mathbb {P}$ may be seem as part of a monad",
"As is the case with the power-set functor, from $\\textbf {Set}$ to itself, we may see $\\mathbb {P}$ , or even $\\mathsf {P}$ and $\\mathsf {P}_{=}$ , as being part of a monad, although some minor modifications are necessary.",
"So, consider the endofunctor $ \\tilde{\\mathbb {P}}:\\textbf {MAlg}(\\Sigma )\\rightarrow \\textbf {MAlg}(\\Sigma )$ such that, for a $\\Sigma $ -multialgebra $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma })$ , $\\tilde{\\mathbb {P}}\\mathcal {A}$ is the $\\Sigma $ -multialgebra with universe $\\mathcal {P}^{*}(A)$ and operations given by $ \\sigma _{\\tilde{\\mathbb {P}}\\mathcal {A}}(A_{1}, \\ldots , A_{n})=\\lbrace \\lbrace a\\rbrace \\in \\mathcal {P}^{*}(A) : a\\in \\bigcup \\lbrace \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}) : a_{i}\\in A_{i}\\rbrace \\rbrace ,$ for $\\sigma $ an $n$ -ary symbol and $A_{1}$ through $A_{n}$ non-empty subsets of $A$ ; and for $\\Sigma $ -multialgebras $\\mathcal {A}$ and $\\mathcal {B}$ , a homomorphism $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ and a non-empty $A^{\\prime }\\subseteq A$ , $\\tilde{\\mathbb {P}} h :\\tilde{\\mathbb {P}}\\mathcal {A}\\rightarrow \\tilde{\\mathbb {P}}\\mathcal {B}$ satisfies $\\tilde{\\mathbb {P}} h (A^{\\prime })=\\lbrace h (a)\\in B: a \\in A^{\\prime }\\rbrace $ .",
"Notice that $\\tilde{\\mathbb {P}}\\mathcal {A}$ is almost the same as $\\mathsf {P}(\\mathcal {A})$ , with the difference that in the latter, operations return subsets of $A$ , while in the former they return sets of singletons of $A$ , whose union is exactly the result of the operation as performed in $\\mathsf {P}(\\mathcal {A})$ .",
"For the natural transformations to form a monad together with $\\tilde{\\mathbb {P}}$ , we chose the obvious candidates: $\\eta :1_{\\textbf {MAlg}(\\Sigma )}\\rightarrow \\tilde{\\mathbb {P}}$ and $\\epsilon :\\tilde{\\mathbb {P}}\\circ \\tilde{\\mathbb {P}}\\rightarrow \\tilde{\\mathbb {P}}$ given by, for a $\\Sigma $ -multialgebra $\\mathcal {A}$ , an element $a$ of $\\mathcal {A}$ and a non-empty collection $\\lbrace A_{i}\\rbrace _{i\\in I}$ of non-empty subsets of $A$ , $\\eta _{\\mathcal {A}}(a)=\\lbrace a\\rbrace $ and $\\epsilon _{\\mathcal {A}}(\\lbrace A_{i}\\rbrace _{i\\in I})=\\bigcup \\lbrace A_{i} : i\\in I\\rbrace $ .",
"Proposition 6 For any $\\Sigma $ -multialgebra $\\mathcal {A}$ , $\\eta _{\\mathcal {A}}$ and $\\epsilon _{\\mathcal {A}}$ are, indeed, homomorphisms.",
"Proposition 7 For any $\\Sigma $ -multialgebras $\\mathcal {A}$ and $\\mathcal {B}$ , and homomorphism $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ , the identities $\\tilde{\\mathbb {P}} h \\circ \\eta _{\\mathcal {A}}=\\eta _{\\mathcal {B}}\\circ h $ and $\\tilde{\\mathbb {P}} h \\circ \\epsilon _{\\mathcal {A}}=\\epsilon _{\\mathcal {B}}\\circ \\tilde{\\mathbb {P}}\\tilde{\\mathbb {P}} h $ are satisfied, meaning $\\eta $ and $\\epsilon $ are natural transformations.",
"Let $a$ be an element of $\\mathcal {A}$ .",
"We have that $\\tilde{\\mathbb {P}} h \\circ \\eta _{\\mathcal {A}}(a)=\\tilde{\\mathbb {P}} h (\\eta _{\\mathcal {A}}(a))$ , and since $\\eta _{\\mathcal {A}}(a)=\\lbrace a\\rbrace $ , we have that $\\tilde{\\mathbb {P}} h \\circ \\eta _{\\mathcal {A}}(a)=\\lbrace h (a)\\rbrace $ .",
"Meanwhile, $\\eta _{\\mathcal {B}}\\circ h (a)=\\eta _{\\mathcal {A}}( h (a))=\\lbrace h (a)\\rbrace $ , and as stated both expressions coincide.",
"Now, let $\\lbrace A_{i}\\rbrace _{i\\in I}$ be an element of $\\tilde{\\mathbb {P}}\\tilde{\\mathbb {P}}\\mathcal {A}$ , meaning it is a non-empty set of non-empty subsets of $\\mathcal {A}$ : $\\tilde{\\mathbb {P}} h \\circ \\epsilon _{\\mathcal {A}}(\\lbrace A_{i}\\rbrace _{i\\in I})=\\tilde{\\mathbb {P}} h (\\epsilon _{\\mathcal {A}}(\\lbrace A_{i}\\rbrace _{i\\in I}))$ , and since $\\epsilon _{\\mathcal {A}}(\\lbrace A_{i}\\rbrace _{i\\in I})=\\bigcup \\lbrace A_{i} : i\\in I\\rbrace $ , the whole expression simplifies to $\\lbrace h (a)\\ :\\ a\\in \\bigcup \\lbrace A_{i} : i\\in I\\rbrace \\rbrace $ .",
"In turn, $ \\epsilon _{\\mathcal {B}}\\circ \\tilde{\\mathbb {P}}\\tilde{\\mathbb {P}} h (\\lbrace A_{i}\\rbrace _{i\\in I})=\\epsilon _{\\mathcal {B}}(\\lbrace \\lbrace h (a)\\ :\\ a\\in A_{i}\\rbrace \\ :\\ i\\in I\\rbrace ),$ which is equal to $ \\bigcup \\lbrace \\lbrace h (a)\\ :\\ a\\in A_{i}\\rbrace i\\in I\\rbrace =\\lbrace h (a)\\ :\\ a\\in \\bigcup \\lbrace A_{i} : i\\in I\\rbrace \\rbrace ,$ giving us the desired equality.",
"Theorem 4 The triple of $\\tilde{\\mathbb {P}}$ , $\\eta $ and $\\epsilon $ forms a monad.",
"Let $\\mathcal {A}$ be a $\\Sigma $ -multialgebra.",
"We first must prove $\\epsilon \\circ \\tilde{\\mathbb {P}}\\epsilon =\\epsilon \\circ \\epsilon \\tilde{\\mathbb {P}}$ , what amounts to $\\epsilon _{\\mathcal {A}}\\circ \\tilde{\\mathbb {P}}\\epsilon _{\\mathcal {A}}=\\epsilon _{\\mathcal {A}}\\circ \\epsilon _{\\tilde{\\mathbb {P}}\\mathcal {A}}$ , as homomorphisms from $\\tilde{\\mathbb {P}}^{3}\\mathcal {A}$ to $\\tilde{\\mathbb {P}}\\mathcal {A}$ .",
"So, let $\\lbrace \\lbrace A_{i}^{j}\\rbrace _{i\\in I}\\rbrace _{j\\in J}$ be an element of $\\tilde{\\mathbb {P}}^{3}\\mathcal {A}$ , where $I$ and $J$ are non-empty sets of indexes and all $A_{i}^{j}$ are non-empty subsets of $A$ : $ \\epsilon _{\\mathcal {A}}\\circ \\tilde{\\mathbb {P}}\\epsilon _{\\mathcal {A}}(\\lbrace \\lbrace A_{i}^{j}\\rbrace _{i\\in I}\\rbrace _{j\\in J})=\\epsilon _{\\mathcal {A}}(\\lbrace \\epsilon _{\\mathcal {A}}(\\lbrace A_{i}^{j}\\ :\\ i\\in I\\rbrace )\\ :\\ j\\in J\\rbrace )=\\epsilon _{\\mathcal {A}}(\\lbrace \\bigcup \\lbrace A_{i}^{j} : i\\in I\\rbrace \\ :\\ j\\in J\\rbrace ),$ what equals $\\bigcup \\lbrace \\bigcup \\lbrace A_{i}^{j} : i\\in I\\rbrace : j\\in J\\rbrace $ , while $\\epsilon _{\\mathcal {A}}\\circ \\epsilon _{\\tilde{\\mathbb {P}}\\mathcal {A}}(\\lbrace \\lbrace A_{i}^{j}\\rbrace _{i\\in I}\\rbrace _{j\\in J})=\\epsilon _{\\mathcal {A}}(\\bigcup \\lbrace \\lbrace A_{i}^{j} : j\\in J\\rbrace \\rbrace _{i\\in I})=\\bigcup \\lbrace \\bigcup \\lbrace A_{i}^{j} : j\\in J\\rbrace : i\\in I\\rbrace ,$ and it is clear that both sets are the same.",
"It remains to be proven $\\epsilon \\circ \\tilde{\\mathbb {P}}\\eta =\\epsilon \\circ \\eta \\tilde{\\mathbb {P}}=1_{\\tilde{\\mathbb {P}}}$ , meaning that $\\epsilon _{\\mathcal {A}}\\circ \\eta _{\\tilde{\\mathbb {P}}\\mathcal {A}}=\\epsilon _{\\mathcal {A}}\\circ \\tilde{\\mathbb {P}}\\eta _{\\mathcal {A}}$ , as homomorphisms from $\\tilde{\\mathbb {P}}\\mathcal {A}$ to $\\tilde{\\mathbb {P}}\\mathcal {A}$ , and this coincides with the identity homomorphism on this multialgebra as well.",
"So, we take a non-empty subset $A^{\\prime }$ of $A$ , and we have that $\\epsilon _{\\mathcal {A}}\\circ \\eta _{\\tilde{\\mathbb {P}}\\mathcal {A}}(A^{\\prime })=\\epsilon _{\\mathcal {A}}(\\lbrace A^{\\prime }\\rbrace )=A^{\\prime }$ , while for the other expression one derives $ \\epsilon _{\\mathcal {A}}\\circ P\\eta _{\\mathcal {A}}(A^{\\prime })=\\epsilon _{\\mathcal {A}}(\\lbrace \\eta _{\\mathcal {A}}(a)\\ :\\ a\\in A^{\\prime }\\rbrace )=\\epsilon _{\\mathcal {A}}(\\lbrace \\lbrace a\\rbrace \\ :\\ a\\in A^{\\prime }\\rbrace )=\\bigcup \\lbrace \\lbrace a\\rbrace : a\\in A^{\\prime }\\rbrace =A^{\\prime },$ what finishes the proof."
],
[
"Multialgebras of atoms",
"Given a $(\\Sigma , \\le )$ -algebra $\\mathcal {A}$ , take the set $\\mathbb {A}((A, \\le _{\\mathcal {A}}))$ of atoms of $(A,\\le _{\\mathcal {A}})$ , that is, the set of minimal elements of this partially ordered set (equal to $A_{1}$ as well).",
"For a $\\sigma \\in \\Sigma _{n}$ and atoms $a_{1}, \\ldots , a_{n}\\in \\mathbb {A}((A, \\le _{\\mathcal {A}}))$ , we define $ \\sigma _{\\mathbb {A}(\\mathcal {A})}(a_{1}, \\ldots , a_{n})=\\lbrace a\\in \\mathbb {A}((A, \\le _{\\mathcal {A}})): a\\le _{\\mathcal {A}} \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace =A_{\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})}.$ This way, $(\\mathbb {A}((A, \\le _{\\mathcal {A}})), \\lbrace \\sigma _{\\mathbb {A}(\\mathcal {A})}\\rbrace _{\\sigma \\in \\Sigma })$ becomes a $\\Sigma $ -multialgebra, that we will denote by $\\mathbb {A}(\\mathcal {A})$ and call the multialgebra of atoms of $\\mathcal {A}$ .",
"Given $(\\Sigma , \\le )$ -algebras $\\mathcal {A}$ and $\\mathcal {B}$ and a $(\\Sigma , \\le )$ -homomorphism $ h :\\mathcal {A}\\rightarrow \\mathcal {B}$ , we also define $\\mathbb {A}( h ):\\mathbb {A}((A, \\le _{\\mathcal {A}}))\\rightarrow \\mathbb {A}((B, \\le _{\\mathcal {B}}))$ as the restriction of $ h $ to $\\mathbb {A}((A, \\le _{\\mathcal {A}}))\\subseteq A$ .",
"It is well-defined since every $(\\Sigma , \\le )$ -homomorphism preserves minimal elements, that is, atoms.",
"For $\\sigma \\in \\Sigma _{n}$ and atoms $a_{1}, \\ldots , a_{n}\\in \\mathbb {A}((A, \\le _{\\mathcal {A}}))$ we have that $ \\lbrace \\mathbb {A}( h )(a): a\\in \\sigma _{\\mathbb {A}(\\mathcal {A})}(a_{1}, \\ldots , a_{n})\\rbrace =\\lbrace h (a): a\\in \\sigma _{\\mathbb {A}(\\mathcal {A})}(a_{1}, \\ldots , a_{n})\\rbrace =$ $\\lbrace h (a)\\in \\mathbb {A}((B, \\le _{\\mathcal {B}})): a\\le _{\\mathcal {A}} \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace $ and, since $a\\le _{\\mathcal {A}} \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})$ implies $ h (a)\\le _{\\mathcal {B}} h (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}))$ given $ h $ is order preserving, which in turn implies $ h (a)\\le _{\\mathcal {B}}\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n}))$ since $ h $ is an “almost homomorphism”, we get that $ \\lbrace h (a)\\in \\mathbb {A}((B, \\le _{\\mathcal {B}})): a\\le _{\\mathcal {A}} \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace \\subseteq \\lbrace b\\in \\mathbb {A}((B, \\le _{\\mathcal {B}})): b\\le _{\\mathcal {B}}\\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n})\\rbrace =$ $ \\sigma _{\\mathbb {A}(\\mathcal {B})}( h (a_{1}), \\ldots , h (a_{n}))=\\sigma _{\\mathbb {A}(\\mathcal {B})}(\\mathbb {A}( h )(a_{1}), \\ldots , \\mathbb {A}( h )(a_{n})),$ what proves $\\mathbb {A}( h )$ is a homomorphism between $\\Sigma $ -multialgebras, and we may write $\\mathbb {A}( h ):\\mathbb {A}(\\mathcal {A})\\rightarrow \\mathbb {A}(\\mathcal {B})$ .",
"The natural question is if $\\mathbb {A}:\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )\\rightarrow \\textbf {MAlg}(\\Sigma )$ is a functor, to which the answer is yes: it is easy to see that it distributes over the composition of morphisms and preserves the identical ones."
],
[
"$\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ and {{formula:c1c32ac1-0b6d-4025-8f7a-566596a72df2}} are equivalent",
"Now, we aim to prove that $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ and $\\textbf {MAlg}(\\Sigma )$ are actually equivalent categories, the equivalence being given by the functors $\\mathbb {P}$ and $\\mathbb {A}$ .",
"In order to prove that $\\mathbb {P}$ and $\\mathbb {A}$ form an equivalence of categories it is enough proving that both are full and faithful and $\\mathbb {A}$ is a right adjoint of $\\mathbb {P}$ ."
],
[
"$\\mathbb {P}$ and {{formula:48b3abf9-45db-45df-a84a-717f8cd90a9c}} are full and faithful",
"It is easy to see $\\mathbb {P}$ is faithful: given $\\Sigma $ -multialgebras $\\mathcal {A}$ and $\\mathcal {B}$ , and homomorphisms $ h , h^{\\prime } :\\mathcal {A}\\rightarrow \\mathcal {B}$ , if $\\mathbb {P}( h )=\\mathbb {P}( h^{\\prime } )$ , we have that, for every $a\\in A$ , $ \\lbrace h (a)\\rbrace =\\mathbb {P}( h )(\\lbrace a\\rbrace )=\\mathbb {P}( h^{\\prime } )(\\lbrace a\\rbrace )=\\lbrace h^{\\prime } (a)\\rbrace ,$ and therefore $ h = h^{\\prime } $ .",
"Proposition 8 $\\mathbb {A}$ is faithful.",
"Given $(\\Sigma , \\le )$ -algebras $\\mathcal {A}$ and $\\mathcal {B}$ , and $(\\Sigma , \\le )$ -homomorphisms $ h , h^{\\prime } :\\mathcal {A}\\rightarrow \\mathcal {B}$ , suppose that $\\mathbb {A}( h )=\\mathbb {A}( h^{\\prime } )$ .",
"Then, for every $a\\in A$ , we can write $a=\\sup A_{a}$ , since $(A, \\le _{\\mathcal {A}})$ is atomic.",
"Since $ h $ and $ h^{\\prime } $ are continuous, $ h (a)=\\sup \\lbrace h (a^{\\prime }): a^{\\prime }\\in A_{a}\\rbrace $ and $ h^{\\prime } (a)=\\sup \\lbrace h^{\\prime } (a^{\\prime }): a^{\\prime }\\in A_{a}\\rbrace $ .",
"But, since $\\mathbb {A}( h )=\\mathbb {A}( h^{\\prime } )$ , $ h $ and $ h^{\\prime } $ are the same when restricted to atoms, and therefore $\\lbrace h (a^{\\prime }): a^{\\prime }\\in A_{a}\\rbrace =\\lbrace h^{\\prime } (a^{\\prime }): a^{\\prime }\\in A_{a}\\rbrace $ .",
"This means that $ h (a)= h^{\\prime } (a)$ and, since $a$ is arbitrary, $ h = h^{\\prime } $ .",
"Now, given $\\Sigma $ -multialgebras $\\mathcal {A}$ and $\\mathcal {B}$ , and a $(\\Sigma , \\le )$ -homomorphism $ h :\\mathbb {P}(\\mathcal {A})\\rightarrow \\mathbb {P}(\\mathcal {B})$ , to prove that $\\mathbb {P}$ is also full we must find a homomorphism $ h^{\\prime } :\\mathcal {A}\\rightarrow \\mathcal {B}$ such that $\\mathbb {P}( h^{\\prime } )= h $ .",
"For every $a\\in A$ , $\\lbrace a\\rbrace $ is an atom and, since $ h $ preserves atoms, $ h (\\lbrace a\\rbrace )$ is an atom of $\\mathbb {P}(B)$ , and therefore of the form $\\lbrace b_{a}\\rbrace $ for some $b_{a}\\in B$ .",
"We define $ h^{\\prime } :\\mathcal {A}\\rightarrow \\mathcal {B}$ by $ h^{\\prime } (a)=b_{a}$ .",
"First of all, we must show $ h^{\\prime } $ is in fact a homomorphism, which is quite analogous to the proof of the same fact for $\\mathbb {A}( h )$ .",
"Given $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}\\in A$ , $ \\lbrace h^{\\prime } (a): a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace =\\lbrace b_{a}: a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace =\\sup \\lbrace \\lbrace b_{a}\\rbrace : a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace =$ $\\sup \\lbrace h (\\lbrace a\\rbrace ): a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace =h (\\sup \\lbrace \\lbrace a\\rbrace : a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace ),$ given that $ h $ is continuous, and since it is an “almost homomorphism” this equals $ h (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}))= h (\\sigma _{\\mathbb {P}(\\mathcal {A})}(\\lbrace a_{1}\\rbrace , \\ldots , \\lbrace a_{n}\\rbrace ))\\subseteq \\sigma _{\\mathbb {P}(\\mathcal {B})}( h (\\lbrace a_{1}\\rbrace ), \\ldots , h (\\lbrace a_{n}\\rbrace ))=$ $ \\sigma _{\\mathbb {P}(\\mathcal {B})}(\\lbrace b_{a_{1}}\\rbrace , \\ldots , \\lbrace b_{a_{n}}\\rbrace )=\\sigma _{\\mathcal {B}}(b_{a_{1}}, \\ldots , b_{a_{n}})=\\sigma _{\\mathcal {B}}( h^{\\prime } (a_{1}), \\ldots , h^{\\prime } (a_{n})).$ Now, when we consider $\\mathbb {P}( h^{\\prime } )$ , we see that, for every atom $\\lbrace a\\rbrace $ of $\\mathbb {P}(\\mathcal {A})$ , $\\mathbb {P}( h^{\\prime } )(\\lbrace a\\rbrace )=\\lbrace b_{a}\\rbrace = h (\\lbrace a\\rbrace )$ , and so the restrictions of $ h $ and $\\mathbb {P}( h^{\\prime } )$ to atoms are the same, and therefore $\\mathbb {A}( h )=\\mathbb {A}(\\mathbb {P}( h^{\\prime } ))$ .",
"Since $\\mathbb {A}$ is faithful, we discover that $ h =\\mathbb {P}( h^{\\prime } )$ and, as we stated before, $\\mathbb {P}$ is full.",
"Now it remains to be shown that $\\mathbb {A}$ is also full.",
"Given $(\\Sigma , \\le )$ -algebras $\\mathcal {A}$ and $\\mathcal {B}$ , and a homomorphism $ h :\\mathbb {A}(\\mathcal {A})\\rightarrow \\mathbb {A}(\\mathcal {B})$ , we then define $ h^{\\prime } :\\mathcal {A}\\rightarrow \\mathcal {B}$ by $ h^{\\prime } (a)=\\sup \\lbrace h (c): c\\in A_{a}\\rbrace .$ First of all, we must prove that $ h^{\\prime } $ is a $(\\Sigma , \\le )$ -homomorphism, for which we shall need a few lemmas.",
"Lemma 5 In a complete, atomic and bottomless Boolean algebra $\\mathcal {A}$ , take a non-empty family of indexes $I$ and, for every $i\\in I$ , $X_{i}\\subseteq A$ .",
"Suppose we have $x_{i}=\\sup X_{i}$ , for $i\\in I$ , and $X=\\bigcup \\lbrace X_{i} : i\\in I\\rbrace $ .",
"Then, $\\sup \\lbrace x_{i}: i\\in I\\rbrace =\\sup X$ .",
"We define $a=\\sup \\lbrace x_{i}: i\\in I\\rbrace $ and $b=\\sup X$ : first, we show that $a$ is an upper bound for $X$ , so that $a\\ge _{\\mathcal {A}} b$ .",
"For every $x\\in X$ , we have that, since $X=\\bigcup \\lbrace X_{i} : i\\in I\\rbrace $ , there exists $j\\in I$ such that $x\\in X_{j}$ , and therefore $x_{j}\\ge _{\\mathcal {A}} x$ .",
"Since $a=\\sup \\lbrace x_{i}: i\\in I\\rbrace $ , we have that $a\\ge _{\\mathcal {A}} x_{j}$ , and by transitivity $a\\ge _{\\mathcal {A}} x$ , and therefore $a$ is indeed an upper bound for $X$ .",
"Now we show that $b$ is an upper bound for $\\lbrace x_{i}: i\\in I\\rbrace $ , and so $b\\ge _{\\mathcal {A}} a$ (and $a=b$ ).",
"For every $i\\in I$ , we have that $b$ is an upper bound for $X_{i}$ , since $X_{i}\\subseteq X$ and $b$ is an upper bound for $X$ , and therefore $b\\ge _{\\mathcal {A}} x_{i}$ , since $x_{i}$ is the smallest upper bound for $X_{i}$ .",
"It follows that $b$ is indeed an upper bound for $\\lbrace x_{i}: i\\in I\\rbrace $ , what finishes the proof.",
"Lemma 6 In a complete, atomic and bottomless Boolean algebra $\\mathcal {A}$ , for a non-empty $C\\subseteq A$ one has that $\\bigcup \\lbrace A_{c} : c\\in C\\rbrace =A_{\\sup C}$ .",
"If $d\\in A_{c}$ for a $c\\in C$ , $d$ is an atom such that $d \\le _{\\mathcal {A}} c$ .",
"Since $c\\le _{\\mathcal {A}}\\sup C$ , $d\\le _{\\mathcal {A}}\\sup C$ , and therefore $d$ belongs to $A_{\\sup C}$ .",
"Thus $\\bigcup \\lbrace A_{c} : c\\in C\\rbrace \\subseteq A_{\\sup C}$ .",
"Reciprocally, suppose that $d\\in A_{\\sup C}$ .",
"Then, $d$ is an atom such that $d\\le _{\\mathcal {A}}\\sup C$ , and therefore $\\inf \\lbrace d, \\sup C\\rbrace =d$ .",
"It follows that the subset $C^{d}\\subseteq C$ , of $c\\in C$ such that $\\inf \\lbrace d,c\\rbrace $ exists, is not empty, by Proposition REF .",
"But if $c\\in C^{d}$ , $\\inf \\lbrace d,c\\rbrace $ exists, and since $d$ is an atom, we have that $d\\in A_{c}\\subseteq \\bigcup \\lbrace A_{c} : c\\in C\\rbrace $ , and from that $\\bigcup \\lbrace A_{c} : c\\in C\\rbrace =A_{\\sup C}$ .",
"Since $\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})$ is equal to the supremum of $\\lbrace \\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n}): (c_{1}, \\ldots , c_{n})\\in A_{a_{1}}\\times \\cdots \\times A_{a_{n}}\\rbrace $ , from Lemma REF we have that $A_{\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})}$ is equal to $ \\bigcup \\lbrace A_{\\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n})} : c_{1}\\in A_{a_{1}}, \\ldots , c_{n}\\in A_{a_{n}}\\rbrace ,$ that is, we have the following lemma.",
"Lemma 7 For a $\\sigma \\in \\Sigma _{n}$ , and $a_{1}, \\ldots , a_{n}\\in A$ , $ A_{\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})}=\\bigcup \\lbrace A_{\\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n})} : c_{i}\\in A_{a_{i}}\\rbrace .$ Theorem 5 $\\mathbb {A}$ is full.",
"So, first of all, we prove that $ h^{\\prime } $ is a $(\\Sigma , \\le )$ -homomorphism.",
"First, it is clear that $ h^{\\prime } $ maps atoms into atoms: if $a$ is an atom, $A_{a}=\\lbrace a\\rbrace $ and $ h^{\\prime } (a)=\\sup \\lbrace h (c): c\\in A_{a}\\rbrace =\\sup \\lbrace h (a)\\rbrace = h (a),$ which is an atom since $ h $ is a map between $\\mathbb {A}(\\mathcal {A})$ and $\\mathbb {A}(\\mathcal {B})$ .",
"$ h^{\\prime } $ is continuous: for any non-empty set $C\\subseteq A$ , we remember that $ h^{\\prime } (c)=\\sup \\lbrace h (d): d\\in A_{c}\\rbrace $ , and from Lemmas REF and REF we get that $ \\sup \\lbrace h^{\\prime } (c): c\\in C\\rbrace =\\sup \\lbrace \\sup \\lbrace h (d): d\\in A_{c}\\rbrace : c\\in C\\rbrace =$ $ \\sup \\bigcup \\lbrace \\lbrace h (d): d\\in A_{c}\\rbrace : c\\in C\\rbrace =\\sup \\lbrace h (d): d\\in A_{\\sup C}\\rbrace = h^{\\prime } (\\sup C).$ Since $\\lbrace h (a): a\\in \\sigma _{\\mathbb {A}(\\mathcal {A})}(a_{1}, \\ldots , a_{n})\\rbrace \\subseteq \\sigma _{\\mathbb {A}(\\mathcal {B})}( h (a_{1}), \\ldots , h (a_{n}))$ , given that $ h $ is a homomorphism of multialgebras, it follows from Lemma REF that $ h^{\\prime } (\\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}))=\\sup \\lbrace h (c): c\\in \\bigcup \\lbrace A_{\\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n})} : c_{i}\\in A_{a_{i}}\\rbrace \\rbrace =$ $ \\sup \\bigcup \\lbrace \\lbrace h (c): c\\in \\sigma _{\\mathbb {A}(\\mathcal {A})}(c_{1}, \\ldots , c_{n})\\rbrace : c_{i}\\in A_{a_{i}}\\rbrace \\le _{\\mathcal {B}}$ $ \\sup \\bigcup \\lbrace \\sigma _{\\mathbb {A}(\\mathcal {B})}( h (c_{1}), \\ldots , h (c_{n})) : c_{i}\\in A_{a_{i}}\\rbrace ,$ where we have used that, for atoms $c_{1}, \\ldots , c_{n}$ of $\\mathcal {A}$ , $\\sigma _{\\mathbb {A}(\\mathcal {A})}(c_{1}, \\ldots , c_{n})=A_{\\sigma _{\\mathcal {A}}(c_{1}, \\ldots , c_{n})}$ ; since, for atoms $d_{1}, \\ldots , d_{n}$ of $\\mathcal {B}$ , we also have that $\\sigma _{\\mathbb {A}(\\mathcal {B})}(d_{1}, \\ldots , d_{n})=A_{\\sigma _{\\mathcal {B}}(d_{1}, \\ldots , d_{n})}$ , this is equal to $ \\sup \\bigcup \\lbrace A_{\\sigma _{\\mathcal {B}}( h (c_{1}), \\ldots , h (c_{n}))} : c_{i}\\in A_{a_{i}}\\rbrace =\\sup \\bigcup \\lbrace A_{\\sigma _{\\mathcal {B}}( h^{\\prime } (c_{1}), \\ldots , h^{\\prime } (c_{n}))} : c_{i}\\in A_{a_{i}}\\rbrace .$ Since $ h^{\\prime } $ is continuous, $c_{i}\\le _{\\mathcal {A}}a_{i}$ , for every $i\\in \\lbrace 1, \\ldots , n\\rbrace $ , implies $ h^{\\prime } (c_{i})\\le _{\\mathcal {B}} h^{\\prime } (a_{i})$ , and therefore $\\sigma _{\\mathcal {B}}( h^{\\prime } (c_{1}), \\ldots , h^{\\prime } (c_{n}))\\le _{\\mathcal {B}}\\sigma _{\\mathcal {B}}( h^{\\prime } (a_{1}), \\ldots , h^{\\prime } (a_{n}))$ for $(c_{1}, \\ldots , c_{n})\\in A_{a_{1}}\\times \\cdots \\times A_{a_{n}}$ .",
"It follows that the union, for $(c_{1}, \\ldots , c_{n})$ in $A_{a_{1}}\\times \\cdots \\times A_{a_{n}}$ , of $A_{\\sigma _{\\mathcal {B}}( h^{\\prime } (c_{1}), \\ldots , h^{\\prime } (c_{n}))}$ , is contained on $A_{\\sigma _{\\mathcal {B}}( h^{\\prime } (a_{1}), \\ldots , h^{\\prime } (a_{n}))}$ , and therefore $ \\sup \\bigcup \\lbrace A_{\\sigma _{\\mathcal {B}}( h^{\\prime } (c_{1}), \\ldots , h^{\\prime } (c_{n}))} : c_{i}\\in A_{a_{i}}\\rbrace \\le _{\\mathcal {B}}\\sup A_{\\sigma _{\\mathcal {B}}( h^{\\prime } (a_{1}), \\ldots , h^{\\prime } (a_{n}))}=\\sigma _{\\mathcal {B}}( h^{\\prime } (a_{1}), \\ldots , h^{\\prime } (a_{n})).$ Now, for every atom $a$ of $\\mathcal {A}$ , we have that $ h^{\\prime } (a)= h (a)$ , and therefore the restriction of $h^{\\prime }$ to atoms coincides with $h$ , that is, $\\mathbb {A}( h^{\\prime } )= h $ , and since $ h $ was taken arbitrarily, $\\mathbb {A}$ is full."
],
[
"$\\mathbb {P}$ and {{formula:bd7c7d5d-1ed3-4eed-8aeb-f2986e2db562}} are adjoint",
"It remains to be shown that $\\mathbb {P}$ and $\\mathbb {A}$ are adjoint.",
"To this end, consider the bijections $ \\Phi _{\\mathcal {B}, \\mathcal {A}}:Hom_{\\textbf {MAlg}(\\Sigma )}(\\mathbb {A}(\\mathcal {B}), \\mathcal {A})\\rightarrow Hom_{\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )}(\\mathcal {B}, \\mathbb {P}(\\mathcal {A})),$ for $\\mathcal {A}$ a $\\Sigma $ -multialgebra and $\\mathcal {B}$ a $(\\Sigma , \\le )$ -algebra, given by, for $ h :\\mathbb {A}(\\mathcal {B})\\rightarrow \\mathcal {A}$ a homomorphism and $b$ an element of $\\mathcal {B}$ , $ \\Phi _{\\mathcal {B},\\mathcal {A}}( h )(b)=\\lbrace h (c): c\\in A_{b}\\rbrace .$ Proposition 9 $\\Phi _{\\mathcal {B}, \\mathcal {A}}( h )$ is a $(\\Sigma , \\le )$ -homomorphism.",
"If $b$ is an atom, $A_{b}=\\lbrace b\\rbrace $ , and therefore $\\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(b)=\\lbrace h (c): c\\in A_{b}\\rbrace =\\lbrace h (b)\\rbrace $ , which is a singleton and therefore an atom of $\\mathbb {P}(\\mathcal {A})$ .",
"Let $D$ be a non-empty subset of $\\mathcal {B}$ .",
"We have that $ \\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(\\sup D)=\\lbrace h (c): c\\in A_{\\sup D}\\rbrace =\\lbrace h (c): c\\in \\bigcup \\lbrace A_{d} : d\\in D\\rbrace \\rbrace =$ $\\bigcup \\lbrace \\lbrace h (c): c\\in A_{d}\\rbrace : d\\in D\\rbrace =\\bigcup \\lbrace \\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(d) : d\\in D\\rbrace =\\sup \\lbrace \\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(d): d\\in D\\rbrace ,$ since $A_{\\sup D}=\\bigcup _{d\\in D}A_{d}$ and the supremum in $\\mathbb {P}(\\mathcal {A})$ is simply the union.",
"For $\\sigma \\in \\Sigma _{n}$ and $b_{1}, \\ldots , b_{n}$ elements of $\\mathcal {B}$ , we have that $ \\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n}))=\\lbrace h (c): c\\in A_{\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n})}\\rbrace =\\lbrace h (c): c\\in \\bigcup \\lbrace A_{\\sigma _{\\mathcal {B}}(c_{1}, \\ldots , c_{n})} : c_{i}\\in A_{b_{i}}\\rbrace \\rbrace =$ $ \\bigcup \\lbrace \\lbrace h (c): c\\in A_{\\sigma _{\\mathcal {B}}(c_{1}, \\ldots , c_{n})}\\rbrace : c_{i}\\in A_{b_{i}}\\rbrace ,$ and, since $c_{1}, \\ldots , c_{n}$ are atoms, this is equal to $ \\bigcup \\lbrace \\lbrace h (c): c\\in \\sigma _{\\mathbb {A}(\\mathcal {B})}(c_{1}, \\ldots , c_{n})\\rbrace : c_{i}\\in A_{b_{i}}\\rbrace \\subseteq \\bigcup \\lbrace \\sigma _{\\mathcal {A}}( h (c_{1}), \\ldots , h (c_{n})) : c_{i}\\in A_{b_{i}}\\rbrace =$ $ \\bigcup \\lbrace \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}) : a_{i}\\in \\lbrace h : c\\in A_{b_{i}}\\rbrace \\rbrace =\\bigcup \\lbrace \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n}) : a_{i} \\in \\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(b_{i})\\rbrace =$ $ \\sigma _{\\mathbb {P}(\\mathcal {A})}(\\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(b_{1}), \\ldots , \\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(b_{n})).$ Now, the $\\Phi _{\\mathcal {B}, \\mathcal {A}}$ must be bijections between $Hom_{\\textbf {MAlg}(\\Sigma )}(\\mathbb {A}(\\mathcal {B}), \\mathcal {A})$ and $Hom_{\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )}(\\mathcal {B}, \\mathbb {P}(\\mathcal {A}))$ .",
"They are certainly injective: if $\\Phi _{\\mathcal {B}, \\mathcal {A}}( h )=\\Phi _{\\mathcal {B}, \\mathcal {A}}( h^{\\prime } )$ , for every atom $b$ we have that $ \\lbrace h (b)\\rbrace =\\Phi _{\\mathcal {B}, \\mathcal {A}}( h )(b)=\\Phi _{\\mathcal {B}, \\mathcal {A}}( h^{\\prime } )(b)=\\lbrace h^{\\prime } (b)\\rbrace ,$ and therefore $ h = h^{\\prime } $ .",
"For the surjectivity, take a $(\\Sigma , \\le )$ -homomorphism $ h :\\mathcal {B}\\rightarrow \\mathbb {P}(\\mathcal {A})$ .",
"We then define $ h^{\\prime } :\\mathbb {A}(\\mathcal {B})\\rightarrow \\mathcal {A}$ by $ h^{\\prime } (b)=a$ for an atom $b$ in $\\mathcal {B}$ , where $ h (b)=\\lbrace a\\rbrace $ .",
"It is well-defined since a $(\\Sigma , \\le )$ -homomorphism takes atoms to atoms, and the atoms of $\\mathbb {P}(\\mathcal {A})$ are exactly the singletons.",
"We must show that $ h^{\\prime } $ is truly a homomorphism.",
"For $\\sigma \\in \\Sigma _{n}$ and atoms $b_{1}, \\ldots , b_{n}$ in $\\mathbb {A}(\\mathcal {B})$ such that $ h (b_{i})=\\lbrace a_{i}\\rbrace $ for every $i\\in \\lbrace 1, \\ldots , n\\rbrace $ , we have that $ h (\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n}))\\subseteq \\sigma _{\\mathbb {P}(\\mathcal {A})}( h (b_{1}), \\ldots , h (b_{n}))$ , since $ h $ is a $(\\Sigma , \\le )$ -homomorphism, and therefore $ \\lbrace h^{\\prime } (b): b\\in \\sigma _{\\mathbb {A}(\\mathcal {B})}(b_{1}, \\ldots , b_{n})\\rbrace =\\lbrace h^{\\prime } (b): b\\in A_{\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n})}\\rbrace = \\bigcup \\lbrace h (b) : b\\in A_{\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n})}\\rbrace =$ $ h (\\sup A_{\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n})})= h (\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n}))\\subseteq \\sigma _{\\mathbb {P}(\\mathcal {A})}( h (b_{1}), \\ldots , h (b_{n}))=\\sigma _{\\mathbb {P}(\\mathcal {A})}(\\lbrace a_{1}\\rbrace , \\ldots , \\lbrace a_{n}\\rbrace )=$ $ \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})=\\sigma _{\\mathcal {A}}( h^{\\prime } (b_{1}), \\ldots , h^{\\prime } (b_{n})).$ Finally, we state that $\\Phi _{\\mathcal {B}, \\mathcal {A}}( h^{\\prime } )= h $ since, for any element $b$ in $\\mathcal {B}$ , we have that $ \\Phi _{\\mathcal {B}, \\mathcal {A}}( h^{\\prime } )(b)=\\lbrace h^{\\prime } (c): c\\in A_{b}\\rbrace =\\bigcup \\lbrace h (c) : c\\in A_{b}\\rbrace = h (\\sup A_{b})= h (b),$ and therefore the $\\Phi _{\\mathcal {B}, \\mathcal {A}}$ are, indeed, bijective.",
"Given $\\mathcal {A}$ and $\\mathcal {C}$ two $\\Sigma $ -multialgebras, $\\mathcal {B}$ and $\\mathcal {D}$ two $(\\Sigma , \\le )$ -algebras, $ h :\\mathcal {A}\\rightarrow \\mathcal {C}$ a homomorphism and $ h^{\\prime } :\\mathcal {D}\\rightarrow \\mathcal {B}$ a $(\\Sigma , \\le )$ -homomorphism, we must now only prove that the following diagram commutes.",
"$ {Hom_{\\textbf {MAlg}(\\Sigma )}(\\mathbb {A}(\\mathcal {B}), \\mathcal {A}) [rr]^{\\Phi _{\\mathcal {B}, \\mathcal {A}}} [dd]^{Hom(\\mathbb {A}( h^{\\prime } ), h )} & & Hom_{\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )}(\\mathcal {B}, \\mathbb {P}(\\mathcal {A})) [dd]^{Hom( h^{\\prime } , \\mathbb {P}( h ))} \\\\& & \\\\Hom_{\\textbf {MAlg}(\\Sigma )}(\\mathbb {A}(\\mathcal {D}), \\mathcal {C}) [rr]^{\\Phi _{\\mathcal {D}, \\mathcal {C}}} & & Hom_{\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )}(\\mathcal {D}, \\mathbb {P}(\\mathcal {C}))}$ So, we take a homomorphism $g:\\mathbb {A}(\\mathcal {B})\\rightarrow \\mathcal {A}$ and an element $d$ of $\\mathcal {D}$ .",
"We have that $ Hom( h^{\\prime } , \\mathbb {P}( h ))(\\Phi _{\\mathcal {B}, \\mathcal {A}}(g))=\\mathbb {P}( h )\\circ \\Phi _{\\mathcal {B}, \\mathcal {A}}(g)\\circ h^{\\prime } ,$ and therefore the right side of the diagram gives us $ \\mathbb {P}( h )\\circ \\Phi _{\\mathcal {B}, \\mathcal {A}}(g)\\circ h^{\\prime } (d)=\\mathbb {P}( h )(\\lbrace g(b): b\\in A_{ h^{\\prime } (d)}\\rbrace )=\\lbrace h \\circ g(b): b\\in A_{ h^{\\prime } (d)}\\rbrace .$ The left side gives us $ \\Phi _{\\mathcal {D}, \\mathcal {C}}( h \\circ g\\circ \\mathbb {A}( h^{\\prime } ))(d)=\\lbrace h \\circ g\\circ \\mathbb {A}( h^{\\prime } )(e): e\\in A_{d}\\rbrace =\\lbrace h \\circ g\\circ h^{\\prime } (e): e\\in A_{d}\\rbrace .$ If $d$ is an atom, the right side becomes the singleton containing only $ h \\circ g\\circ h^{\\prime } (d)$ , since in this case $A_{d}=\\lbrace d\\rbrace $ and, given that $ h^{\\prime } $ preserves atoms, $A_{ h^{\\prime } (d)}=\\lbrace h^{\\prime } (d)\\rbrace $ .",
"The left side becomes also the singleton formed by $ h \\circ g\\circ h^{\\prime } (d)$ , because again $A_{d}=\\lbrace d\\rbrace $ .",
"As a $(\\Sigma , \\le )$ -homomorphism is determined by its action on atoms, we find that the left and right sides of the diagram are equal, and therefore the diagram commutes.",
"As observed before, this proves $\\mathbb {A}$ and $\\mathbb {P}$ are adjoint and, therefore, that $\\textbf {MAlg}(\\Sigma )$ and $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ are equivalent."
],
[
"Some consequences and related results",
"The result that $\\textbf {MAlg}(\\Sigma )$ and $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ are equivalent has a few consequences, and related results, we would like to stress.",
"First of all, we start by taking the empty signature: in that case, given all multialgebras are non-empty, $\\textbf {MAlg}(\\Sigma )$ becomes the category of non-empty sets $\\textbf {Set}^{*}$ , with functions between them as morphisms.",
"Meanwhile, $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ becomes the category with complete, atomic and bottomless Boolean algebras as objects (given we simply drop the operations from a $(\\Sigma , \\le )$ -algebra), with continuous, atoms-preserving functions between them as morphisms.",
"Notice this is very closely related to the equivalence between $\\textbf {CABA}$ and $\\textbf {Set}^{op}$ : the morphisms on the former are merely continuous functions, so the only extra requirement to the morphisms we are making is that they should preserve atoms.",
"This, of course, allows one to exchange the opposite category of $\\textbf {Set}$ by $\\textbf {Set}$ itself (or rather $\\textbf {Set}^{*}$ ).",
"A generalization of our result is to partial multialgebras.",
"That is, pairs $\\mathcal {A}=(A, \\lbrace \\sigma _{\\mathcal {A}}\\rbrace _{\\sigma \\in \\Sigma })$ such that, if $\\sigma \\in \\Sigma _{n}$ , $\\sigma _{\\mathcal {A}}$ is a function from $A^{n}$ to $\\mathcal {P}(A)$ (no longer $\\mathcal {P}(A)\\setminus \\lbrace \\emptyset \\rbrace $ ).",
"In other words, a partial multialgebra is a multialgebra where operations may return the empty-set.",
"Given partial $\\Sigma $ -multialgebras $\\mathcal {A}$ and $\\mathcal {B}$ , a homomorphism between them is a function $ h :A\\rightarrow B$ such that, as is the case for homomorphisms for multialgebras, $ \\lbrace h (a): a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})\\rbrace \\subseteq \\sigma _{\\mathcal {B}}( h (a_{1}), \\ldots , h (a_{n})),$ for $\\sigma \\in \\Sigma _{n}$ and $a_{1}, \\ldots , a_{n}\\in A$ .",
"The class of all partial $\\Sigma $ -multialgebras, with these homomorphisms between them as morphisms, becomes a category, which we shall denote by $\\textbf {PMAlg}(\\Sigma )$ .",
"It is easy to find an equivalence, much alike the one between $\\textbf {MAlg}(\\Sigma )$ and $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ , between $\\textbf {PMAlg}(\\Sigma )$ and a category related to $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ : it is sufficient to replace the requirement that, in a $(\\Sigma , \\le )$ -algebra, $(A, \\le _{\\mathcal {A}})$ is a complete, atomic and bottomless Boolean algebra, by the requisite that it is actually a complete, atomic Boolean algebra, and accordingly, change the morphisms in the correspondent category by requiring they preserve the supremum of any sets, not necessarily non-empty.",
"Finally, consider a modified notion of homomorphism between $\\Sigma $ -multialgebras $\\mathcal {A}$ and $\\mathcal {B}$ , that of a function $ h :A\\rightarrow \\mathcal {P}(B)\\setminus \\lbrace \\emptyset \\rbrace $ such that $ \\bigcup _{a\\in \\sigma _{\\mathcal {A}}(a_{1}, \\ldots , a_{n})} h (a)\\subseteq \\bigcup _{(b_{1}, \\ldots , b_{n})\\in h (a_{1})\\times \\cdots \\times h (a_{n})}\\sigma _{\\mathcal {B}}(b_{1}, \\ldots , b_{n}).$ The category with $\\Sigma $ -multialgebras as objects and these homomorphisms as morphisms will be denoted by $\\textbf {MMAlg}(\\Sigma )$ .",
"If, in the category $\\textbf {Alg}_{\\mathsf {B}}(\\Sigma )$ , we change morphisms by not longer demanding that they map atoms into atoms, it is easy to adapt the proof given in Section to show that the resulting category is equivalent to $\\textbf {MMAlg}(\\Sigma )$ ."
],
[
"Conclusion and Future Work",
"As we explained before, the main results here presented are a generalization of the equivalence between the categories of complete, atomic Boolean algebras and $\\textbf {Set}^{op}$ .",
"On the one hand, we add operations to Boolean algebras that are compatible with its order, while on the other we allow for non-deterministic operations taking us to a category of multialgebras.",
"Although not specially complicated, this result is useful as it allows to treat non-deterministic matrices (Nmatrices) as, not precisely algebraic semantics, but mixed methods that combine both an algebraic component and one relative to its order.",
"This may seem to increase the complexity of decision methods, but this sacrifice is made precisely to avoid non-determinism and use merely classical concepts.",
"This is made, not because we distrust the use of multialgebras as semantics for non-classical logics, but as an alternative to those logicians that have philosophical objections against that very use.",
"More pragmatically, we are encouraged to further study the categories of multialgebras, now from the viewpoint of categories of partially ordered sets, far better understood than the former ones; moreover, we can now recast several non-deterministic characterizations of logics found in the literature in the terms here presented.",
"Specifically, there are several paraconsistent logics uncharacterizable by finite matrices, but characterized by finite Nmatrices, which can now have semantics presented only in classical terms of algebras and orders."
],
[
"Acknowledgements",
"The first author acknowledges financial support from the National Council for Scientific and Technological Development (CNPq), Brazil, under research grant 306530/2019-8.",
"The second author was initially supported by a doctoral scholarship from the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001 (Brazil), and later on by a postdoctoral fellowship by NSF-BSF, grant number 2020704."
]
] | 2209.08158 |
[
[
"Mitigating Both Covariate and Conditional Shift for Domain\n Generalization"
],
[
"Abstract Domain generalization (DG) aims to learn a model on several source domains, hoping that the model can generalize well to unseen target domains.",
"The distribution shift between domains contains the covariate shift and conditional shift, both of which the model must be able to handle for better generalizability.",
"In this paper, a novel DG method is proposed to deal with the distribution shift via Visual Alignment and Uncertainty-guided belief Ensemble (VAUE).",
"Specifically, for the covariate shift, a visual alignment module is designed to align the distribution of image style to a common empirical Gaussian distribution so that the covariate shift can be eliminated in the visual space.",
"For the conditional shift, we adopt an uncertainty-guided belief ensemble strategy based on the subjective logic and Dempster-Shafer theory.",
"The conditional distribution given a test sample is estimated by the dynamic combination of that of source domains.",
"Comprehensive experiments are conducted to demonstrate the superior performance of the proposed method on four widely used datasets, i.e., Office-Home, VLCS, TerraIncognita, and PACS."
],
[
"Introduction",
"Computer vision has made great progress in recent years with the help of deep learning under the basic assumption that all data are independently and identically distributed.",
"However, in practical applications, images collected by different devices in different environments often follow different distributions.",
"In such an out-of-distribution scenario, existing deep learning models suffer from the distribution shift and fail to generalize well [1].",
"Figure: An overview of the proposed VAUE.",
"A visual alignment module is proposed to align P(X)P(X) in visual space via normalizing the style distributions.",
"Uncertainty-guided belief ensemble is introduced to approximate P(Y|X)P(Y|X) with the combination of that of source domains at test time.To tackle the distribution shift problem, great efforts have been made in Domain Adaptation (DA), which generally aims to transfer knowledge from a labeled source domain to an unlabeled target domain so that the learned model can perform well on the target domain [1].",
"However, DA requires the target domain to be accessible which is hard to meet when the target domain changes dynamically.",
"Furthermore, DA needs to retrain the model when applying it to another target domain, and it is time-consuming.",
"In recent years, Domain Generalization (DG) has attracted much attention, which tries to enable the model to generalize to unseen domains utilizing multiple source domains.",
"Without access to target domains, DG improves the generalizability of models in out-of-distribution scenarios and has broad application prospects.",
"Among existing works, domain-invariant representation learning is a classic approach to DG.",
"Let $X$ denote the input variable, i.e., an image, and $Y$ denote the output variable, i.e., a predicted label.",
"As analysed in previous works, traditional models often suffer from covariate shift [2], i.e., $P^i(X) \\ne P^j(X)$ , and conditional shift [3], i.e., $P^i(Y|X) \\ne P^j(Y|X)$ in out of distribution scenarios.",
"Some works [4], [5] try to learn a representation space $Z = F(X)$ where the marginal distribution $P(Z)$ keeps the same across source domains so that the covariate shift can be eliminated assuming that $P(Y|X)$ keeps stable.",
"Another line of research [6] tries to align the class-conditional distribution $P(Z|Y)$ in the representation space for a fine-grained distribution matching assuming that $P(Y)$ keeps stable.",
"These works often hold impractical assumptions and fail to eliminate both the covariate and conditional shift.",
"Additionally, though distributions of source domains are aligned, there is no guarantee that the distributions of unseen domains would be also aligned with that of source domains.",
"In this paper, we propose a new approach to eliminate both the covariate shift and conditional shift, as shown in Figure REF .",
"For the covariate shift, we propose to align $P(X)$ in visual space.",
"In most scenarios, the domain shift is mainly caused by the image style which can be represented as feature statistics [7].",
"We attempt to model the common real-world style distribution, i.e., the real-world distribution of feature statistics, which has yielded domain-specific style distributions with different selection biases.",
"After that, we normalize all the domain-specific style distributions to the common style distribution so that the covariate shift is eliminated via visual alignment.",
"For the conditional shift, instead of aligning $P(Y|X)$ across source domains, we design a nonlinear ensemble scheme based on uncertainty modeling to dynamically approximate $P(Y | X=\\mathbf {x}^t)$ given a test sample $\\mathbf {x}^t$ .",
"The subject logic and Dempster-Shafer theory of evidence are first introduced to solve DG in our method.",
"Comprehensive experiments have been conducted on four widely used datasets to demonstrate the effectiveness of our method.",
"Let input sample and output label spaces be denoted as $\\mathcal {X}$ and $\\mathcal {Y}$ respectively.",
"A domain is a set of data sampled from a joint distribution, which can be denoted as $\\mathcal {D} = \\lbrace (\\mathbf {x}_i, y_i)\\rbrace _{i=1}^n \\sim P(X, Y)$ , where $\\mathbf {x} \\in \\mathcal {X} \\subset \\mathbb {R}^d, y\\in \\mathcal {Y} \\subset \\mathbb {R}$ , and $P(X,Y)$ denotes the joint distribution of the sample and label.",
"$X$ and $Y$ are the corresponding random variables.",
"$C = |\\mathcal {Y}|$ denoting the number of classes.",
"Given $N$ source domains $\\lbrace \\mathcal {D}^i\\rbrace _{i=1}^N$ which follow different distributions, DG aims to learn a model which can generalize well on unseen target domains with unknown distribution shifts.",
"In this paper, vectors are shown in bold, and the subscript indicates the corresponding dimension of the vector, e.g., $\\mathbf {v}$ and $v_i$ ."
],
[
"Covariate Shift",
"Assumption 1 (Independence Assumption) Let $f_{sem}(X) $ and $f_{sty}(X) $ be the semantic component and the style component extracted from $X$ .",
"$ P(f_{sem}(X))$ keeps stable across domains, while $P(f_{sty}(X))$ changes due to different selection biases on various domains.",
"$f_{sem}(X)$ is independent with $f_{sty}(X)$ so that $P(f_{sem}(X), f_{sty}(X)) = P(f_{sem}(X))\\times P(f_{sty}(X)) $ .",
"Under Assumption REF , we can align $\\lbrace P^i(X)\\rbrace _{i=1}^N$ of source domains by normalizing $\\lbrace P^i(f_{sty}(X))\\rbrace _{i=1}^N$ to the same target distribution.",
"According to the previous work [7], we know that feature statistics of the features in intermediate layers of deep networks, i.e., means and standard deviations computed on each feature channel, have encoded the style information of images.",
"Given an image, the intermediate feature after a certain layer of a network is denoted as $\\mathbf {z} \\in \\mathbb {R}^{K\\times H\\times W}$ .",
"The vector of feature statistics $\\mathbf {\\mu } \\in \\mathbb {R}^{K}$ and $\\mathbf {\\sigma } \\in \\mathbb {R}^{K}$ can be calculated as follows: $\\begin{aligned}{\\mu }_{k}(\\mathbf {z}) & = \\frac{1}{HW} \\sum _{h=1}^H \\sum _{w=1}^W{z}_{khw}\\\\\\sigma _{k}(\\mathbf {z}) & = \\sqrt{\\frac{1}{HW}\\sum _{h=1}^H \\sum _{w=1}^W ({z}_{khw} - \\mu _{k}(\\mathbf {z}))^2}\\end{aligned}$ As analysed in Assumption REF , we can approximately represent $f_{sty}(X)$ with $\\lbrace \\mathbf {\\mu }, \\mathbf {\\sigma }\\rbrace $ .",
"Hence we can normalize the distribution of $\\lbrace \\mathbf {\\mu }, \\mathbf {\\sigma }\\rbrace $ of images to align $P(X)$ in visual space.",
"Assuming that all domain-specific feature statistics are sampled from the real-world style distribution with various selection biases, given a batch of images equally sampled from each domain, we aggregate all samples to approximate the real-world style distribution with an empirical distribution of feature statistics $P^{emp}(f_{sty}(X))$ .",
"$P^{emp}(f_{sty}(X))$ is formalized as a Gaussian distribution.",
"Its parameters are calculated as follows: $\\begin{aligned}\\mathbf {m}_\\mu & = \\frac{1}{B} \\sum _{b=1}^{B} \\mathbf {\\mu }(\\mathbf {z}^{b}), \\ \\mathbf {m}_\\sigma = \\frac{1}{N} \\sum _{b=1}^{B} \\mathbf {\\sigma }(\\mathbf {z}^b)\\\\\\mathbf {\\Sigma }_\\mu & = \\frac{1}{B-1} \\sum _{b=1}^B(\\mathbf {\\mu }(\\mathbf {z}^b) - \\mathbf {m}_\\mu )(\\mathbf {\\mu }(\\mathbf {z}^b) - \\mathbf {m}_\\mu )^T\\\\\\mathbf {\\Sigma }_\\sigma & = \\frac{1}{B-1} \\sum _{b=1}^N(\\mathbf {\\sigma }(\\mathbf {z}^b) - \\mathbf {m}_\\sigma )(\\mathbf {\\sigma }(\\mathbf {z}^b) - \\mathbf {m}_\\sigma )^T\\end{aligned}$ where $B$ denotes the batch size.",
"Then we can get that: $P^{emp}\\left(f_{sty}(X) = \\lbrace \\mathbf {\\mu }_s, \\mathbf {\\sigma }_s\\rbrace \\right) = \\mathcal {N}( \\mathbf {\\mu }_s|\\mathbf {m}_\\mu , \\mathbf {\\Sigma }_\\mu ) \\times \\mathcal {N}( \\mathbf {\\sigma }_s| \\mathbf {m}_\\sigma , \\mathbf {\\Sigma }_\\sigma )$ We normalize $\\lbrace P^i(f_{sty}(X))\\rbrace _{i=1}^N$ to $P^{emp}(f_{sty}(X))$ for the marginal distribution alignment.",
"For each image, we replace the original feature statistics with new values, $\\mathbf {\\sigma }_{s}$ and $\\mathbf {\\mu }_{s}$ , sampled from the empirical distribution: $\\lbrace {\\mathbf {\\mu }}_{s}, \\mathbf {\\sigma }_{s}\\rbrace \\sim P^{emp}(f_{sty}(X)) \\\\\\mathbf {z} = ((\\mathbf {z} - \\mathbf {\\mu }(\\mathbf {z})) / \\mathbf {\\sigma }(\\mathbf {z}) ) \\odot \\mathbf {\\sigma }_{s} + \\mathbf {\\mu }_{s}$ where $/$ and $\\odot $ denote channel-wise division and multiplication respectively.",
"By normalizing the distribution of feature statistics to the empirical distribution $P^{emp}(f_{sty}(X))$ , which is a reasonable substitute for the real-world style distribution, the visual style distributions $P(f_{sty}(X))$ of source domains are aligned.",
"The operation of visual alignment is a plug-and-play module, which can be flexibly inserted into the neural networks at different positions like Batch Normalization layers.",
"Each visual alignment module is randomly enabled according to a probability factor $\\tau $ to control the ratio of samples to be transformed."
],
[
"Conditional Shift",
"To overcome the conditional shift, we also design an uncertainty-guided belief ensemble strategy to dynamically approximate $P(Y|X)$ given a test sample based on the Subjective Logic (SL) and Dempster-Shafer theory of evidence.",
"For more details, we refer readers to published works [10], [12], [13], [11]."
],
[
"Subjective Logic",
"We construct a linear domain-specific classifier for each source domain following a shared feature extractor, which is expected to fit the domain-specific posterior distribution.",
"Instead of the softmax operation, we choose $\\exp (\\cdot )$ to make outputs of linear classifiers non-negative.",
"The unnormalized non-negative outputs of each domain-specific classifier, $\\lbrace \\mathbf {e}^i\\rbrace _{i=1}^N, \\mathbf {e}^i \\in \\mathbb {R}^C$ , can be regarded as the collected evidences in favor of a sample to be classified into a certain category.",
"Following SL, we define the belief masses, $\\lbrace \\mathbf {b}^i\\rbrace _{i=1}^N$ , $\\mathbf {b}^i \\in \\mathbb {R}^{C}$ , and uncertainty masses, $\\lbrace u^i\\rbrace _{i=1}^N$ , $u^i \\in \\mathbb {R}$ , for $i$ -th domain-specific classifier as follows: $\\begin{aligned}\\mathbf {b}^i = \\frac{\\mathbf {e}^i}{S^i},\\ u^i = \\frac{C}{S^i},\\ \\text{and}\\ u^i + \\sum _{c=1}^C b_c^i = 1\\end{aligned}$ where $S^i = \\sum _{c=1}^C(e^i_c + 1)$ .",
"Under this definition, the uncertainty mass is inversely proportional to the total evidence.",
"The output of classical deep learning classifiers is a probability assignment over all classes.",
"SL parameterizes a Dirichlet distribution with the evidence, which is a probability density function for all possible probability assignments over all classes.",
"In other words, Dirichlet distribution is defined on a $C$ -dimensional unit simplex, $\\mathcal {S}_C = \\lbrace \\mathbf {p} | \\sum _{c=1}^C p_c = 1\\ \\text{and}\\ \\forall c,\\ p_c \\ge 0\\rbrace $ , where every point is a $C$ -dimensional probability assignments.",
"Specifically, the parameters of Dirichlet distribution for $i$ -th domain-specific classifier is defined as $\\mathbf {\\alpha }^i =\\mathbf {e}^i +1 $ , and then the $i$ -th Dirichlet distribution can be denoted as: $\\text{Dir}(\\mathbf {p} \\mid \\alpha ^i)= {\\left\\lbrace \\begin{array}{ll}\\frac{1}{B(\\alpha ^i)} \\prod _{c=1}^{C} p_{c}^{\\alpha ^i_{c}-1} & \\text{ for } \\mathbf {p} \\in \\mathcal {S}_{C}, \\\\ 0 & \\text{ otherwise ,}\\end{array}\\right.",
"}$ where $B({\\cdot })$ is a $C$ -dimensional multinomial beta function.",
"Under above definitions, given a sample, all domain-specific classifiers will output their evidence collected from the sample, $\\lbrace \\mathbf {e}^i\\rbrace _{i=1}^N$ .",
"And then belief masses $\\lbrace \\mathbf {b}^i\\rbrace _{i=1}^N$ for each category and uncertainty masses $\\lbrace u^i\\rbrace _{i=1}^N$ can be derived.",
"What is more, Dirichlet distributions for each domain-specific classifier $\\lbrace \\text{Dir}(\\mathbf {p}| \\mathbf {\\alpha }^i)\\rbrace _{i=1}^N$ will be formalized with derived evidences."
],
[
"Reduced Dempster's Combinational Rule",
"We adopt a reduced Dempster's combinational rule [11] to nonlinearly combine the predictions of all domain-specific classifiers.",
"Definition 1 (Reduced Dempster's Combinational Rule) Given two sets of masses $\\mathcal {M}^1=\\lbrace \\mathbf {b}^1, u^1\\rbrace $ and $\\mathcal {M}^2=\\lbrace \\mathbf {b}^2, u^2\\rbrace $ , the combination $\\mathcal {M}=\\lbrace \\mathbf {b}, u\\rbrace $ can be calculated as follows: $b_c = \\frac{1}{1-F}(b^1_c b^2_c + b^1_c u^2 + b^2_c u^1), u = \\frac{1}{1-F} u^1 u^2,$ where $F = \\sum _{i\\ne j}b_i^1b_j^2$ reflecting the conflict between two mass sets.",
"The combination can be denoted as $\\mathcal {M} = \\mathcal {M}^1\\oplus \\mathcal {M}^2$ .",
"All mass sets given by domain-specific classifiers can be combined as $\\mathcal {M} = \\mathcal {M}^1\\oplus \\mathcal {M}^2\\oplus \\cdots \\oplus \\mathcal {M}^N$ .",
"By doing so, we can formalize the overall Dirichlet distribution based on the combined belief masses and uncertainty mass $\\lbrace \\mathbf {b}, u\\rbrace $ .",
"Specifically, parameters of the combined Dirichlet distribution can be derived as follows: $S = \\frac{C}{u}, \\mathbf {e} = \\mathbf {b} \\times S\\ \\text{and}\\ \\mathbf {\\alpha } = \\mathbf {e} + 1$ ."
],
[
"Single-Domain and Cross-Domain Training",
"After illustrating the definitions of evidence, belief, uncertainty and combination rule, we now specifically show the detailed training process.",
"Given an input sample, the $i$ -th domain-specific classifiers will output a mass set $\\mathcal {M}^i$ and a Dirichlet distribution $\\text{Dir}(\\mathbf {p} | \\mathbf {\\alpha }^i) $ .",
"Let $\\mathbf {x}$ denote an input sample, and $\\mathbf {y}$ denote the corresponding one-hot label.",
"To train this classifier with $\\mathbf {x}$ , the loss function is designed as follows [10]: $\\begin{aligned}&\\mathcal {L}_{ece}(\\mathbf {x}, \\mathbf {y},\\mathbf {\\alpha }^i) \\\\= & \\mathbb {E}_{\\text{Dir}(\\mathbf {p} | \\mathbf {\\alpha }^i)}\\left[\\sum _{c=1}^{C}-y_{c} \\log \\left(p_{c}\\right)\\right]+\\lambda \\text{KL}[\\text{Dir}(\\mathbf {p} | \\mathbf {\\tilde{\\alpha }}^i) || \\text{Dir}(\\mathbf {p} | \\mathbf {1})] \\\\= & \\sum _{c = 1}^C y_c \\left(\\psi (S^i) - \\psi (\\alpha ^i_c) \\right) +\\lambda \\text{KL}[\\text{Dir}(\\mathbf {p} | \\mathbf {\\tilde{\\alpha }}^i) || \\text{Dir}(\\mathbf {p} |\\mathbf {1} )] ,\\end{aligned}$ where $\\mathbf {\\tilde{\\alpha }}^i = \\mathbf {y} + (\\mathbf {1}-\\mathbf {y}) \\odot \\mathbf {\\alpha }^i$ , $\\mathbf {1}$ is a vector with all elements equal to 1, and $\\psi (\\cdot )$ is the digamma function.",
"In this paper, $\\lambda $ is set to 0.01.",
"The first term is an expectation of cross entropy computed over the Dirichlet distribution which is essentially a Bayes risk.",
"And the second term is proposed to enforce the evidence for incorrect labels to shrink to 0 [10].",
"To model the uncertainty well, we design a single-domain training part and a cross-domain training part.",
"For the single-domain part, the data of $i$ -th domain are only fed into $i$ -th domain-specific classifier and compute the loss $\\mathcal {L}_{ece}$ .",
"For the cross-domain part, the data of $i$ -th domain are fed into all domain-specific classifiers except the $i$ -th domain.",
"After that, $N-1$ mass sets are combined.",
"And the combined Dirichlet distribution is derived, parameters of which are denoted as $\\mathbf {\\alpha }^{/i}$ .",
"Hence the loss function can be designed as follows: $\\mathcal {L}_{D}= \\frac{1}{N} \\sum _{i=1}^N \\frac{1}{|\\mathcal {D}^i|} \\sum _{j=1}^{|\\mathcal {D}^i|}\\left(\\mathcal {L}_{ece}(\\mathbf {x}^j, \\mathbf {y}^j, \\mathbf {\\alpha }^i) + \\mathcal {L}_{ece}(\\mathbf {x}^j, \\mathbf {y}^j, \\mathbf {\\alpha }^{/i}) \\right)$ Furthermore, we enforce the correlation across different dimensions of the extracted feature to shrink to 0.",
"We found that this design can prevent the numerical computation problem of the proposed method during training.",
"Specifically, for a batch of features $\\mathbf {z} \\in \\mathbb {R}^{B \\times d}$ , the mean of features $m = \\frac{1}{B} \\sum _{b=1}^B \\mathbf {z}^b$ , then the decorrelation loss can be designed as: $\\mathcal {L}_{decor} = \\left|\\frac{1}{B-1} \\left(\\mathbf {z} - m\\right)^T\\left(\\mathbf {z} - m\\right) \\odot (\\mathbf {1}-I) \\right|_1$ where $I$ is a identity matrix.",
"The finally loss function can be designed as : $\\mathcal {L} = \\mathcal {L}_{D} + \\mathcal {L}_{decor}$"
],
[
"Testing",
"At test time, a test image is fed into the feature extractor firstly, and then the extracted feature is fed into all domain-specific classifiers.",
"After that, all mass sets produced by classifiers are combined based on the reduced Dempster's combinational rule.",
"The class which has the highest combined belief mass is the final prediction.",
"The combined uncertainty mass shows the overall confidence of the prediction.",
"Given a test sample, the real working labeling function is the nonlinear combination of that of source domains, which automatically adjusts the weight of each labeling function of source domains according to the uncertainty of the sample at test time.",
"By this way the conditional shift between train data and test data is eliminated."
],
[
"Datasets",
"For demonstrating the effectiveness of the proposed method VAUE, we evaluate it on four widely used DG datasets, namely Office-Home (4 domains, 65 classes, and 75,588 images), VLCS (4 domains, 5 classes, and 10,729 images), TerraIncognita (4 domains, 10 classes, and 24,788 images), PACS (4 domains, 7 classes, and 9,991 images).",
"For all experiments, one domain is selected as the unseen test domain, and the others are treated as training domains."
],
[
"Implementation Details",
"For all experiments, the networks, which are pre-trained on ImageNet, are trained by AdamW with a batch size of 64 for each domain and weight decay of 5e-4.",
"The batch normalization is frozen during the training.",
"The exponential moving average of model parameters with a momentum of 0.999 is conducted to make the training processes more stable.",
"We adopt the standard data augmentations following [14].",
"The visual alignment modules are inserted after 1,2,3-th ConvBlock of ResNet.",
"All results are reported based on the average top-1 classification accuracy over three repetitive runs.",
"For Office-Home, VLCS, and TerraIncognita, following [14], we randomly split each training domain into 8:2 training/validation splits.",
"All validation splits of training domains are aggregated as an overall validation set, which is used for model selection.",
"The probability factor $\\tau $ of visual alignment modules is set to 0.1.",
"For Office-Home and TerraIncognita, the models are trained with a learning rate of 1e-5 for up to 5k iterations.",
"For VLCS, models are trained for up to 2k iterations.",
"We summarize the results of comparison methods reported in [14] in Table REF .",
"For PACS, the original train-validation split provided by [15] is adopted for a fair comparison with more diverse and novel competitors.",
"The probability factor $\\tau $ is set to 1.",
"The models are trained with a learning rate of 1e-4 for up to 4k iterations.",
"We summarize the results of competitors reported in original papers in Table REF .",
"Table: Performance comparison on Office-Home, VLCS, and TerraIncognita dataset with ResNet-50 as backbone.As shown in Table REF , we evaluate the proposed method VAUE on three datasets with ResNet-50 as the feature extractor.",
"We can see that VAUE achieves the best average accuracy on Office-Home and VLCS and the second average accuracy on TerraIncognita.",
"The domain shift on TerraIncognita is less about the image style.",
"So the performance improvement is not as obvious as that on other datasets.",
"We also evaluate VAUE on PACS dataset with ResNet-18 and ResNet-50 as backbone respectively to compare our method with a greater variety of methods, as shown in Table REF .",
"We can see that VAUE achieves the best average accuracy with both ResNet-18 and ResNet-50.",
"We notice that there is an obvious performance drop on the Photo (P) domain compared to DeepAll, which minimizes the empirical risk by aggregating samples from all source domains.",
"This is mainly due to the ImageNet pretraining [29].",
"The images of the Photo (P) domain are highly similar to those of ImageNet.",
"If the training strategy is changed, this benefit of pre-training may be reduced."
],
[
"Ablation Study",
"To better demonstrating the effectiveness of the proposed VAUE, we conduct an ablation study by constructing four variant methods as shown in Table REF .",
"VAUE w/o VA is a variant of VAUE without visual alignment modules.",
"VAUE w/o EC is a variant constructed by replacing the reduced Dempster's combinational rule with a vanilla average combination.",
"VAUE w/o CD is a variant that trains the models without cross-domain training mentioned in Section REF .",
"VAUE w/o UE is a variant constructed by removing the whole part of the uncertainty-guided belief ensemble.",
"We can see that all four designs provide significant performance improvement to the final accuracy.",
"We note that the visual alignment modules produce much performance gain except on Photo domain.",
"Because after being aligned to a common Gaussian distribution, the resulting image style could be more different from that of ImageNet.",
"Table: Performance comparison on PACS dataset with ResNet-18 and ResNet-50 as backbone.Table: Ablation Study on PACS dataset."
],
[
"Conclusion",
"In this paper, we design the visual alignment module for dealing with the covariate shift by aligning the distribution of image style to a common Gaussian distribution.",
"Another uncertainty-guided ensemble strategy is proposed to deal with the conditional shift between training domains and test samples by a dynamic adjustment.",
"Experiment results show the excellent performance of the proposed VAUE."
]
] | 2209.08253 |
[
[
"The no-hair theorem and black hole shadows"
],
[
"Abstract The successful observation of M87 supermassive black hole by the Black Hole Event Horizon Telescope(EHT) provides a very good opportunity to study the theory of gravity.",
"In this work, we obtain the exact solution for the short hair black hole (BH) in the rotation situation, and calculate in detail how hairs affect the BH shadow.",
"For the exact solution part, using the Newman-Janis algorithm, we generalize the spherically symmetric short-hair black hole metric to the rotation case (space-time lie element (2.25)).",
"For the BH shadow part, we study two hairy BH models.",
"In model 1, the properties of scalar hair are determined by the parameters $\\alpha_{0}$ and $L$.",
"In model 2, the scalar hair of the BH is short hair.",
"In this model, the shape of the BH shadow is determined by scalar charge $Q_{m}$ and $k$.",
"In general, various BH hairs have different effects on the shadows, such as non-monotonic properties and intersection phenomena mentioned in this work.",
"Using these characteristics, it is possible to test the no-hair theorem in future EHT observations, so as to have a deeper understanding of the quantum effect of BHs.",
"In future work, we will use numerical simulations to study the effects of various hairs on BHs and their observed properties."
],
[
"Introduction",
"In general relativity (GR), black hole is an exact solution to Einstein's field equation, such as the common Schwarzschild black hole(e.g.",
"see [1], [2]), which describes the vacuum gravitational field near a point mass, while for spherically symmetric internal solutions, which show various forms of solutions due to different equations of state.",
"In the 1960s, Kerr generalized the Schwarzschild black hole to the rotational situation through precise mathematical calculation [1], [3].",
"Kerr solution is extremely important to the study of black hole physics, and the physical object described by it is real in the universe.",
"The introduction of the spin of the black hole leads to many new properties of the solution(e.g.",
"see [1], [4], [5]).",
"For example, Kerr black hole has an ergosphere, which makes it have a negative energy orbit, the negative energy orbits are orbits with negative angular momentum in the ergosphere of the black hole, and their rotation direction is opposite to that of the black hole; the introduction of spin parameter results in two event horizons for Kerr black hole.",
"When the spin approaches 1, Kerr black hole will become an extreme black hole, which is very important for the study of the quantum effect of the black hole.",
"In classical general relativity, the solution of Einstein's field equation describes only the curvature of space-time and does not include any quantum effects.",
"The existence of Schwarzschild and Kerr black holes is believed to be due to the possibility of forming black holes through gravitational collapse during late stellar evolution (e.g.",
"see [6]) (EHT's observations of black holes will be described later).",
"Black holes form through gravitational collapse, which has been extensively discussed in detail(e.g.",
"see [1], [7]), and physicists have found it interesting that the final result of spherically symmetric gravitational collapse is described by the mass of the system, that is, the mass of the system completely determines all the properties of spherically symmetric space-time in a vacuum.",
"If this is generalized to Kerr black hole and Kerr-Newman black hole, then the space-time after gravitational collapse will be completely determined by mass, spin and charge of the black hole, and has nothing to do with the properties of the precursor star and the process of gravitational collapse, which is the so-called no-hair theorem[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"The no-hair theorem is one of the main characteristics of classical black holes.",
"Due to the extreme nature of the black hole event horizon, quantum effects near the horizon have to be considered[20], [21], [22], [23], [24], [25], [26].",
"However, considering the non-trivial matter field in the black hole space-time, the black hole no-hair theorem may be violated.",
"There are many types of black hole hairs, among which the scalar hairs are the main ones.",
"Due to the influence of scalar hairs on black holes, the space-time metric of black holes changes.",
"For example, J.Ovalle et al.",
"studied black hole solutions with scalar hairs using gravitational decoupling approach[27].",
"In their work, scalar hairs can be reflected by mode parameters $\\alpha _{0}$ and $L$ .",
"These spherically symmetric black holes with scalar hairs have been extended to the rotational case by E.Contreras et al., using gravitational decoupling method, and their basic physical properties have been calculated in detail[28].",
"In addition, some black hole solutions with hairs have also been obtained analytically (e.g.,[29], [30], [31]).",
"These approximate and exact solutions provide good conditions for physicists to understand the quantum effects of black holes.",
"In recent years, scientists have made a series of breakthroughs in the observation of black holes, the most important of which are the observation of gravitational waves and the measurement of the shadow of M87 supermassive black hole[32], [33], [34].",
"In fact, the gravitational wave signal generated by the merger of black holes is enough to indicate the existence of black holes, but this is an indirect way.",
"A more straightforward approach is to measure the black hole shadow.",
"In April 2019, the EHT research group published the first image of the shadow of M87 black hole[33], which provided the possibility for physicists to study the physical properties of the strong gravitational field.",
"Using the observation data of EHT, scientists have conducted extensive and in-depth research on black hole physics, general relativity, etc.",
"(e.g.,([35], [36])).",
"Recently, the accuracy of EHT measurement has been further improved, and it is very possible to further test the basic properties of black holes, such as magnetic field[37].",
"Some progress has been made in testing the no-hair theorem by using the observation of the black hole shadow.",
"Mohsen Khodadi et al.",
"studied the effect of hair on black hole shadow by using the rotational black hole with hair[38], [39].",
"Because the metric of the black hole with hair they used only takes into account conventional correction to Kerr black hole, the calculations, while complete, are difficult to relate the details of specific quantum hair, such as scalar hair[20], [21], [22], [23], [24], [25], [26].",
"Therefore, we need to calculate the shadow shape corresponding to the black hole with the detailed scalar hairs.",
"In reference [38], the authors made a few calculations on this, but did not show more details about this.",
"In our work, we will perform detailed calculations of the shadow of the hairy black hole; using the metric of different scalar hairy black holes, we discuss how these scalar hairs change the properties of black hole shadows.",
"The logical structure of this article is as follows.",
"In section 2, we introduce the basic properties of the solutions of the black holes with scalar hairs and derive the rotational case of the metric of black holes with short hairs.",
"In section 3, the geodesic equation of photon in rotational black hole is derived and its analytical solution is obtained.",
"In section 4, we calculate the effect of scalar hairs on the geometric properties of black hole shadows.",
"In section 5, we calculate how scalar hairs affect the rate of energy emission.",
"The sixth section is the summary and discussion of the whole paper.",
"In classical general relativity, black holes carry no charge other than mass, charge and spin, which is known in black hole physics as no-hair theorem[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"However, it is possible that the interaction of black hole space-time with matter introduces other charges, such as internal norm symmetries and certain fields, which could make it possible for black holes to carry hairs.",
"When these corresponding physical effects are introduced into the space-time of black hole, i.e., to make black hole hairy, the presence of these hairs will change the space-time background of black hole, thus bringing the quantum effects of black hole into the classical space-time geometry.",
"Using extended gravitational decoupling (EGD) approach, J.Ovalle et al.",
"obtained a spherically symmetric space-time metric with hair[27], because in the EGD approach, there is no certain matter field, which makes the hairy black holes obtained by them have great generality.",
"In EGD, the corresponding Einstein field equation can be expressed as follows: $R_{\\mu \\nu }-\\dfrac{1}{2}R g_{\\mu \\nu }=8\\pi \\tilde{T}_{\\mu \\nu },$ where $R_{\\mu \\nu }$ is Ricci curvature tensor, $R$ is Ricci curvature scalar, $g_{\\mu \\nu }$ is the space-time metric, $\\tilde{T}_{\\mu \\nu }$ is the total energy-momentum tensor, and can be written as $\\tilde{T}_{\\mu \\nu }=T_{\\mu \\nu }+\\theta _{\\mu \\nu }$ where $T_{\\mu \\nu }$ is the energy-momentum tensor in GR, $\\theta _{\\mu \\nu }$ is the energy-momentum tensor caused by a quantum field or gravitational branch.",
"According to general relativity, $G_{\\mu \\nu }=R_{\\mu \\nu }-\\frac{1}{2}Rg_{\\mu \\nu }$ needs to satisfy the Bianchi identity, namely $\\bigtriangledown _{\\mu }\\tilde{T}^{\\mu \\nu }=0$ .",
"When $\\theta _{\\mu \\nu }=0$ , i.e., without considering hairs, it can be proved that the solution of field equation (REF ) degenerates into Schwarzschild black hole solution.",
"By proper treatment of the tensor $\\theta _{\\mu \\nu }$ , the spherically symmetric solution of Einstein field equation can be obtained [27] $ds^{2}=-f(r)dt^{2}+\\dfrac{1}{g(r)}dr^{2}+r^{2}d\\Omega ^{2}$ where $d\\Omega ^{2}=d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2}$ , $f(r)$ and $g(r)$ are the metric coefficients, and the expressions in this case are $f(r)=g(r)=1-\\dfrac{2M}{r}+\\alpha _{0} \\exp \\left({-\\frac{r}{M-\\frac{\\alpha _{0}L}{2}}}\\right)$ Here, $\\alpha _{0}$ describes the deformation parameters due to the consideration of hairs, so the size of $\\alpha _{0}$ represents the physical meaning related to the strength of hairs.",
"$L$ is a constant with the dimension of distance.",
"In this work, we limit the values of $\\alpha _{0}$ and $L$ as follows [27], [28]: $0 \\le \\alpha _{0} \\le 1$ $0 \\le L \\le 2$ By adjusting the parameters $\\alpha _{0}$ and $L$ , we can analyse how hairs change the properties of the space-time metric of black holes.",
"Meanwhile, the space-time metric provides a basis for testing the no-hair theorem of black holes from an astronomical perspective.",
"Using gravitational decoupling approach, E.Contreras et al.",
"generalized the spherically symmetric black hole solution to the rotational case[28].",
"By applying some restrictions on $S_{\\mu \\nu }$ in the energy-momentum tensor $\\tilde{T}_{\\mu \\nu }=T_{\\mu \\nu }+S_{\\mu \\nu }$ , such as meeting the strong energy condition (SEC), the space-time metric of the black hole with hair was obtained $ds^{2}=-\\left[1-\\dfrac{2rm(r)}{\\rho ^{2}}\\right]dt^{2}-\\dfrac{4arm(r)\\sin ^{2}\\theta }{\\rho ^{2}}dtd\\phi +\\dfrac{\\rho ^{2}}{\\Delta }dr^{2}+\\rho ^{2}d\\theta ^{2}+\\dfrac{\\Sigma \\sin ^{2}\\theta }{\\rho ^{2}}d\\phi ^{2}$ where, the expressions of related symbols are $\\rho ^{2}=r^{2}+a^{2}\\cos ^{2}\\theta $ $\\Delta =r^{2}-2r\\tilde{m}(r)+a^{2}=r^{2}\\left[ 1+\\alpha _{0} \\exp \\left({-\\frac{r}{M-\\frac{\\alpha _{0}L}{2}}}\\right) \\right]-2Mr+a^{2}$ $\\Sigma =(r^{2}+a^{2})^{2}-a^{2}\\Delta \\sin ^{2}\\theta $ $f(r)=g(r)=1-\\dfrac{2\\tilde{m}(r)}{r}$ In these expressions, $a$ is the spin of the black hole, and other parameters have the same physical meaning as in the spherically symmetric case.",
"Since rotational black holes are real in the universe, it is more feasible to test the no-hair theorem of black holes using observational data such as EHT and gravitational waves.",
"In addition, to generalize the rotational hairy black hole to the charged case[27], we only need to replace the metric coefficient in the spherically symmetric case by $f(r)=g(r)=1-\\dfrac{2M}{r}+\\dfrac{Q^{2}}{r^{2}}-\\dfrac{\\alpha _{0}}{r} \\left( M-\\dfrac{\\alpha _{0}L}{2} \\right) \\exp \\left({-\\frac{r}{M-\\frac{\\alpha _{0}L}{2}}}\\right)$ where $Q$ is the charge carried by the black hole.",
"The form of Kerr-Newman black hole with hairs is similar to (REF ) $\\sim $ (REF )."
],
[
"Model 2: Short hair black hole",
"When considering the coupling of gravity with some anisotropic fluids in general relativity, the black hole solution with hair can be obtained.",
"These anisotropic fluids satisfy the following conditions, that is, in some conditions, the gravity will produce de Sitter and Reissner-Nordstron (RN) black holes, and in other conditions, the solution of the black hole with hair can be obtained[31], whose linear element expression is $f(r)=g(r)=1-\\dfrac{2M}{r}+\\dfrac{Q_{m}^{2k}}{r^{2k}},$ $Q_{m}$ is the strength parameter of the hair.",
"When $k=1$ , the space-time metric degrades into an RN black hole; when $k>1$ , the space-time metric is a short-hair black hole, and $Q_{m}$ is the charged value of the short hair.",
"In this work, we discuss the case of $k=\\frac{3}{2}$ in order to understand the properties of short-hair black hole.",
"For the black hole corresponding to the metric (REF ), its energy density and pressure are respectively $\\rho =\\dfrac{Q_{m}^{2k}(2k-1)}{8\\pi r^{2k+2}},$ $p=\\dfrac{Q_{m}^{2k}(2k-1)k}{8\\pi r^{2k+2}}.$ As discussed in the original literature, the black hole solution satisfies three classical conditions, namely the weak energy condition (WEC), the energy density decays faster than $r^{-4}$ , and $T=T_{ab}g^{ab}\\leqslant 0$ .",
"Therefore, this black hole does not violate the no-short-hair theorem.",
"Next, based on the metric of spherically symmetric short-hair black hole, we derive the solution of rotating short-hair black hole using Newman-Janis (NJ) method.",
"In short, the NJ method is to generalize spherically symmetric space-time to rotational space-time by complex transformation.",
"The key is to simplify Einstein field equation into a second-order partial differential equations.",
"If there is an analytic solution to this equations, the rotation form of the corresponding space-time metric can be obtained.",
"The content of the NJ method can be referred to the relevant literature (e.g.,[40], [41], [42]).",
"Follow the idea of this method, we will show the key derivation.",
"Since the Schwarzschild coordinate system is used in the space-time form (REF ), if the space-time considered is a black hole, there is coordinate singularity in this kind of coordinate system.",
"Therefore, in the NJ algorithm, this coordinate system needs to be transformed to the advanced null coordinates (ANC) $(u,r,\\theta , \\varphi )$ , and the coordinate transformation is $du=dt-\\dfrac{dr}{f(r)g(r)}=dt-\\dfrac{dr}{\\left(1-\\dfrac{2M}{r}+\\dfrac{Q_{m}^{2k}}{r^{2k}}\\right)^{2}},$ In ANC, we can choose the advanced null basis vector $(e^{\\mu },n^{\\mu },m^{\\mu }, \\bar{m}^{\\mu })$ to expand the space-time metric, that is, the inverse form of the space-time metric is expressed as a linear combination of the basis vector, namely $g^{\\mu \\nu }=-e^{\\mu }n^{\\nu }-e^{\\nu }n^{\\mu }+m ^{\\mu }\\bar{m}^{\\nu }+m^{\\nu }\\bar{m}^{\\mu }$ .",
"For the metric (REF ) considered here, its basis vector in ANC is $\\begin{aligned}& L^{\\mu }=\\delta _{r}^{\\mu }, \\\\& n^{\\mu }=\\delta _{\\mu }^{\\mu }-\\dfrac{1}{2}\\left(1-\\dfrac{2M}{r}+\\dfrac{Q_{m}^{2k}}{r^{2k}}\\right)^{2}, \\\\& m^{\\mu }=\\dfrac{1}{\\sqrt{2}r}\\delta _{\\theta }^{\\mu }+\\dfrac{i}{\\sqrt{2}r\\sin \\theta }\\delta _{\\phi }^{\\mu }, ~~~\\bar{m}^{\\mu }=\\dfrac{1}{\\sqrt{2}r}\\delta _{\\theta }^{\\mu }-\\dfrac{i}{\\sqrt{2}r\\sin \\theta }\\delta _{\\phi }^{\\mu }.\\end{aligned}$ To generalize spherically symmetric space-time to the rotation case, we can perform the following operations in ANC.",
"In complex space, the coordinate $(u,r)$ is rotated by an angle of $\\theta $ , that is, $u\\rightarrow u-ia\\cos \\theta , r\\rightarrow r+ ia\\cos \\theta $ ; where $a$ is a constant, which can also be interpreted as the spin of the space-time, and $\\theta $ is the rotation angle.",
"At this time, the spherical symmetry metric coefficients $f(r)\\rightarrow F(r,\\theta ,a), g(r)\\rightarrow G(r,\\theta ,a)$ and $h(r)\\rightarrow \\psi (r,\\theta ,a)$ .",
"$h(r)$ is the coefficient of $d\\Omega ^{2}$ in the metric (REF ), and here $h(r)=r^{2}$ .",
"Through these operations, the basis vectors in ANC become functions expressed by $F(r,\\theta ,a)$ , $G(r,\\theta ,a)$ and $\\psi (r,\\theta ,a)$ .",
"After calculation, it is found that the components of the inverse metric $g^{\\mu \\nu }$ are as follows: $\\begin{aligned}& g^{uu}=\\dfrac{a^{2}\\sin ^{2}\\theta }{\\psi }, ~~~ g^{\\theta \\theta }=\\dfrac{1}{\\psi }, ~~~ g^{ur}=g^{ru}=\\sqrt{\\frac{G}{F}}-\\dfrac{a^{2}\\sin \\theta }{\\psi }, \\\\& g^{\\phi \\psi }=\\dfrac{1}{\\psi \\sin ^{2}\\theta }, ~~~ g^{u\\phi }=g^{\\phi u}=\\dfrac{a}{\\psi }, ~~~ g^{r\\phi }=g^{\\phi r}=\\dfrac{a}{\\psi }, ~~~ g^{rr}=G+\\dfrac{a\\sin ^{2}\\theta }{\\psi }.\\end{aligned}$ By processing these inverse metric, we can get the covariance metric tensor, and thus obtain the expression of its line element as $ds^{2}=-Fdu^{2}+2\\sqrt{\\frac{F}{G}}dudr+2a\\sin ^{2}\\theta \\left(\\sqrt{\\frac{F}{G}}+F\\right)dud\\phi -2a\\sin ^{2}\\theta \\sqrt{\\frac{F}{G}}drd\\phi +\\psi d\\theta ^{2}-\\sin ^{2}\\theta \\left(-\\psi +a^{2}\\sin ^{2}\\theta \\left(2\\sqrt{\\frac{F}{G}}+F\\right)\\right)d\\phi ^{2}$ Next, using coordinate transformation, the space-time metric can be transformed from the Eddington-Finkelstin coordinates (EFC) to the Boyer-Lindquist coordinate (BLC)(???",
"), and the final result is $ds^{2}=-\\dfrac{\\psi }{\\rho ^{2}}\\left(1-\\dfrac{2\\bar{f}}{\\rho ^{2}} \\right)dt^{2}+\\dfrac{\\psi }{\\Delta }dr^{2}-\\dfrac{4a\\bar{f}\\psi \\sin ^{2}\\theta }{\\rho ^{4}}dtd\\phi +\\psi d\\theta ^{2}+\\dfrac{\\psi \\Sigma \\sin ^{2}\\theta }{\\rho ^{4}}d\\phi ^{2}$ Here, the relationship of the related symbols is $\\begin{aligned}& k(r)=h(r)\\sqrt{\\dfrac{f(r)}{g(r)}}=r^{2}, ~~~ \\rho ^{2}=k(r)+a^{2}\\cos ^{2}\\theta , \\\\& \\bar{f}=\\dfrac{1}{2}k(r)-\\dfrac{1}{2}h(r)f(r), \\\\& \\Delta (r)=r^{2}f(r)+a^{2}, ~~~ \\Sigma =\\left(k(r)+a^{2}\\right)^{2}-a^{2}\\Delta (r)\\sin ^{2}\\theta .\\end{aligned}$ In this way, we get the general form of the metric of the rotational space-time in the NJ method.",
"However, there is an unknown function $\\psi $ in the metric expression, which needs to be solved by Einstein field equation.",
"Since the space-time metric at this time satisfies rotation symmetry, the component of the Einstein tensor $G_{r\\theta }=0$ .",
"Meanwhile, the metric (REF ) should also satisfy Einstein field equation[41], [42].",
"Through a series of calculations, the equation of the gravitational field can be simplified into the following equations $\\left(k(r)+a^{2}y^{2}\\right)^{2}\\left(3\\dfrac{\\partial \\psi }{\\partial r}\\dfrac{\\partial \\psi }{\\partial y^{2}}-2\\psi \\dfrac{\\partial ^{2}\\psi }{\\partial r \\partial y^{2}} \\right)=3a^{2}\\dfrac{\\partial k}{\\partial r}\\psi ^{2}$ $\\psi \\left( \\left( \\dfrac{\\partial k}{\\partial r} \\right)^{2}+k \\left(2-\\dfrac{\\partial ^{2}k}{\\partial r^{2}}\\right)-a^{2}y^{2}\\left(2+\\dfrac{\\partial ^{2}k}{\\partial r^{2}}\\right)\\right)+\\left(k+a^{2}y^{2}\\right)\\left(4y^{2}\\dfrac{\\partial \\psi }{\\partial y^{2}}-\\dfrac{\\partial k}{\\partial r}\\dfrac{\\partial \\psi }{\\partial r} \\right)=0$ For the short-hair black hole considered in this section, $f(r)=g(r)=1-\\frac{2M}{r}+\\frac{Q_{m}^{2k}}{r^{2k}}, h(r)=r^{2}$ , we can obtain $k(r)=h(r)\\sqrt{\\frac{f(r)}{g(r)}}=r^{2}$ .",
"Substituting $k(r)$ into equations (REF ) and (REF ), by solving the equations, we can get $\\psi (r,\\theta ,a)=r^{2}+a^{2}\\cos ^{2}\\theta ,$ So far, we have the expressions for all the unknown functions, thereby obtaining the rotation form of the short-hair black hole, which is $ds^{2}=-\\left(1-\\dfrac{r^{2}-r^{2}f(r)}{\\rho ^{2}}\\right)dt^{2}+\\dfrac{\\rho ^{2}}{\\Delta }dr^{2}-\\dfrac{2a\\sin ^{2}\\theta \\left(r^{2}-r^{2}f(r)\\right)}{\\rho ^{2}}dtd\\phi +\\rho ^{2}d\\theta ^{2}+\\dfrac{\\Sigma \\sin ^{2}\\theta }{\\rho ^{2}}d\\phi ^{2}=-\\left(1-\\dfrac{2Mr-\\frac{Q_{m}^{2k}}{r^{2k-2}}}{\\rho ^{2}} \\right)dt^{2}+\\dfrac{\\rho ^{2}}{\\Delta }dr^{2}-\\dfrac{2a\\sin ^{2}\\theta \\left(2Mr-\\frac{Q_{m}^{2k}}{r^{2k-2}}\\right)}{\\rho ^{2}}dtd\\phi +\\rho ^{2}d\\theta ^{2}+\\dfrac{\\Sigma \\sin ^{2}\\theta }{\\rho ^{2}}d\\phi ^{2}.$ Where $\\rho ^{2}=r^{2}+a^{2}\\cos ^{2}\\theta $ , $\\Sigma =(r^{2}+a^{2})^{2}-a^{2}\\Delta (r)\\sin ^{2}\\theta $ , $\\Delta =r^{2}-2Mr+\\frac{Q_{m}^{2k}}{r^{2k-2}}+a^{2}$ .",
"If we consider a no-hair black hole, that is, $Q_{m}=0$ , then the metric (2.25) degrades to a Kerr black hole, and $a$ can be interpreted as the black hole spin.",
"When $Q_{m}\\ne 0$ and $k>1$ , it is a short-hair black hole in rotation situation.",
"When $k=\\frac{3}{2}$ , $\\Delta =r^{2}-2Mr+\\frac{Q_{m}^{3}}{r}+a^{2}=0$ will determine the structure of the black hole event horizon.",
"By adjusting the range of values of $M$ , $Q_{m}$ and $a$ , the number of the black hole event horizon may be 1, 2 or 3.",
"This is caused by the introduction of the short hair.",
"Then we analyse the specific properties of each case.",
"From $\\Delta =0$ , it can be found that the event horizon of the black hole should satisfy the following equation $r^{3}-2Mr^{2}+a^{2}r+Q_{m}^{3}=0$ The discriminant of the root is $\\Delta =\\left(\\dfrac{a^{2}}{3}-\\dfrac{4M^{2}}{9}\\right)^{3}+\\left(\\dfrac{Q_{m}^{3}}{2}-\\dfrac{8M^{3}}{27}+\\dfrac{Ma^{2}}{3}\\right)^{2}$ When $\\Delta >0$ , equation (REF ) has only one real root, so the black hole has only one event horizon, the radius of which is $r=\\dfrac{2M}{3}+\\@root 3 \\of {\\dfrac{8M^{3}}{27}-\\dfrac{Q_{m}^{3}}{2}-\\dfrac{Ma^{2}}{3}+\\sqrt{\\Delta }}+\\@root 3 \\of {\\dfrac{8M^{3}}{27}-\\dfrac{Q_{m}^{3}}{2}-\\dfrac{Ma^{2}}{3}-\\sqrt{\\Delta }}$ When $\\Delta =0$ , equation (REF ) has three real roots, at least two of which are equal, so the event horizons of the black hole are $r_{1}=\\dfrac{2M}{3}-2\\@root 3 \\of {\\dfrac{Q_{m}^{3}}{2}-\\dfrac{8M^{3}}{27}+\\dfrac{Ma^{2}}{3}}$ $r_{2}=r_{3}=\\dfrac{2M}{3}+\\@root 3 \\of {\\dfrac{Q_{m}^{3}}{2}-\\dfrac{8M^{3}}{27}+\\dfrac{Ma^{2}}{3}}$ When $\\Delta <0$ , equation (REF ) has three unequal real roots, that is, the black hole has three different event horizons, and their values are $r_{1}=\\dfrac{2M}{3}+2\\sqrt{\\dfrac{4M^{2}}{9}-\\dfrac{a^{2}}{3}}\\cos \\left[\\dfrac{1}{3}\\arccos \\left(\\dfrac{\\left(\\dfrac{8M^{3}}{27}-\\dfrac{Ma^{2}}{3}-\\dfrac{Q_{m}^{3}}{2}\\right)\\sqrt{\\dfrac{4}{9}M^{2}-\\dfrac{1}{3}a^{2}}}{\\left(\\dfrac{1}{3}a^{2}-\\dfrac{4}{9}M^{2}\\right)^{2}}\\right)\\right]$ $r_{2}=\\dfrac{2M}{3}+2\\sqrt{\\dfrac{4M^{2}}{9}-\\dfrac{a^{2}}{3}}\\cos \\left[\\dfrac{1}{3}\\arccos \\left(\\dfrac{\\left(\\dfrac{8M^{3}}{27}-\\dfrac{Ma^{2}}{3}-\\dfrac{Q_{m}^{3}}{2}\\right)\\sqrt{\\dfrac{4}{9}M^{2}-\\dfrac{1}{3}a^{2}}}{\\left(\\dfrac{1}{3}a^{2}-\\dfrac{4}{9}M^{2}\\right)^{2}}\\right)+\\dfrac{2}{3}\\pi \\right]$ $r_{3}=\\dfrac{2M}{3}+2\\sqrt{\\dfrac{4M^{2}}{9}-\\dfrac{a^{2}}{3}}\\cos \\left[\\dfrac{1}{3}\\arccos \\left(\\dfrac{\\left(\\dfrac{8M^{3}}{27}-\\dfrac{Ma^{2}}{3}-\\dfrac{Q_{m}^{3}}{2}\\right)\\sqrt{\\dfrac{4}{9}M^{2}-\\dfrac{1}{3}a^{2}}}{\\left(\\dfrac{1}{3}a^{2}-\\dfrac{4}{9}M^{2}\\right)^{2}}\\right)+\\dfrac{4}{3}\\pi \\right]$ Through calculations, we find that the structure of the event horizon of the short-hair black hole is particularly complicated.",
"$\\Delta $ determines the number of black hole event horizons.",
"Here, the mass of the black hole is taken as the unit, i.e., $M=1$ .",
"When the black hole spin $0\\leqslant a \\leqslant 1$ , the critical value condition of $Q_{m}$ derived from $\\Delta =0$ is $Q_{m}^{3}=2\\left(\\dfrac{4}{9}-\\dfrac{a^{2}}{3}\\right)^{\\frac{3}{2}}+\\dfrac{16}{27}-\\dfrac{2}{3}a^{2}$ When $\\Delta >0$ , i.e., $Q_{m}^{3}<2(\\frac{4}{9}-\\frac{a^{2}}{3})^{\\frac{3}{2}}+\\frac{16}{27}-\\frac{2a^{2}}{3}$ , only $Q_{m}<0$ can meet this situation, which indicates that the short-hair black hole has no event horizon, so it is not the case we consider.",
"When $\\Delta =0$ , the value range of $Q_{m}$ is $0 \\leqslant Q_{m} \\leqslant \\frac{2}{3}\\times 4^{\\frac{1}{3}}$ .",
"When $\\Delta <0$ , the value range is $Q_{m}>\\frac{2}{3}\\times 4^{\\frac{1}{3}}$ .",
"In general, when $0 \\leqslant Q_{m} \\leqslant \\frac{2}{3}\\times 4^{\\frac{1}{3}}$ , the short-hair black hole has three event horizons, two of which are equal.",
"From a physical point of view, there are only two event horizons, which are modifications of Kerr black hole.",
"When $Q_{m}>\\frac{2}{3}\\times 4^{\\frac{1}{3}}$ , the short-hair black hole has three different event horizons, which indicates that the appearance of the short-hair makes the black hole appear a new event horizon, and essentially changes the structure of the black hole event horizon.",
"As we all know, AdS-Kerr black hole has three event horizons $(r_{+},r_{-},r_{\\Lambda })$ .",
"The radius of the event horizon generated by the introduction of cosmological constant $\\Lambda $ is very large, namely $r_{\\Lambda }>>r_{+}$ , while the radius of the event horizon generated by the short hair is close to $r_{+}$ .",
"Since the physical effects of $r_{+}$ are relatively easy to measure, the corresponding effects caused by the short hair are also relatively easy to detect, which makes the third event horizon of the short-hair black hole interesting."
],
[
"Geodesic equations of photons and analytic solution",
"Next, we derive the geodesic equations of photons based on the rotational space-time metrics (REF ) and (2.25), and obtain the analytical solution.",
"We will use the Hamilton-Jacobi equation to calculate.",
"In this method, Carter et al.",
"introduced a new integral constant and obtained analytical solutions of the geodesic equations by using variable separation approach[43], [44].",
"Now, let's introduce the main process.",
"For a rotational black hole, the Hamilton-Jacobi equation for the test particle is of the following form $\\dfrac{\\partial S}{\\partial \\sigma }=-\\dfrac{1}{2}g^{\\mu \\nu }\\dfrac{\\partial S}{\\partial x^{\\mu }}\\dfrac{\\partial S}{\\partial x^{\\nu }},$ where $S$ is the Jacobi action and $\\sigma $ is the affine parameter of a geodesic.",
"In order to be able to separate variables from geodesic equations, the action $S$ should have the form $S=\\dfrac{1}{2}m^{2}\\sigma -Et+\\mathcal {L}\\phi +S_{r}(r)+S_{\\theta }(\\theta ).$ $m$ is the mass of the test particle, and for photons, $m=0$ .",
"$E$ and $\\mathcal {L}$ are the two initial integrals of the motion equation, corresponding to energy and angular momentum respectively.",
"$S_{r}(r)$ and $S_{\\theta }(\\theta )$ are radial and angular functions respectively.",
"Due to the introduction of these conserved quantities and unknown functions, the geodesic equations can be simplified to four component equations.",
"These results can be achieved in the following ways.",
"By substituting the action (REF ) into the Hamilton-Jacobi equation, equation (REF ) can be reduced to such equations $\\begin{aligned}& \\rho ^{2}\\dfrac{dt}{d\\sigma }=\\dfrac{r^{2}+a^{2}}{\\Delta (r)}\\left(E(r^{2}+a^{2})-a\\mathcal {L} \\right)-a\\left(aE\\sin ^{2}\\theta -\\mathcal {L} \\right), \\\\& \\rho ^{2}\\dfrac{dr}{d\\sigma }=\\sqrt{R}, ~~~~~~~~ \\rho ^{2}\\dfrac{d\\theta }{d\\sigma }=\\sqrt{H}, \\\\& \\rho ^{2}\\dfrac{d\\phi }{d\\sigma }=\\dfrac{a}{\\Delta }\\left(E(r^{2}+a^{2})-a\\mathcal {L} \\right)-\\left(aE-\\dfrac{\\mathcal {L}}{\\sin ^{2}\\theta } \\right),\\end{aligned}$ Since unknown functions $S_{r}(r)$ and $S_{\\theta }(\\theta )$ are set in the action $S$ , we need to introduce functions $R$ and $H$ , whose relationship with other initial integrals is $R=\\left(E(r^{2}+a^{2})-a\\mathcal {L} \\right)^{2}-\\Delta \\left(m^{2}r^{2}+(aE-\\mathcal {L})^{2}+K \\right)$ $H=K-\\left(\\dfrac{\\mathcal {L}^{2}}{\\sin ^{2}\\theta }-a^{2}E^{2} \\right)\\cos ^{2}\\theta $ Where $K$ is the constant introduced by Carter et al., which is later called Carter constant.",
"By introducing Carter constant, we can separate variables from the motion equation, and (REF ) $\\sim $ (REF ) are complete geodesic equations.",
"In this work, we calculate the motion of the test particle on the equatorial plane of the black hole, so $\\theta =\\frac{\\pi }{2}$ .",
"In addition, in the process of calculating the black hole shadow, the test particle is generally considered as photon, whose static mass is 0.",
"The motion of photons near black holes is very complex and is discussed in detail in the relevant classical textbooks.",
"The most obvious feature is that at a certain radius close to the event horizon of the black hole, the motion of the photon in this critical radius is very different from that outside.",
"There is no stable motion orbit of the photon within this critical radius (the region satisfies the condition $R^{^{\\prime \\prime }}<0, R^{^{\\prime }}=dR/dr$ ), but there exist stable motion orbit outside this radius(the region satisfies the condition $R^{^{\\prime \\prime }}>0$ ).",
"The critical orbit should satisfy $R=0$ and $\\frac{dR}{dr}=0$ .",
"In order to better introduce the coordinates of the shape of the black hole shadow, we first introduce the parameter pair $(\\xi ,\\eta )$ , so that $\\mathcal {L}, K, E$ can be expressed.",
"By substituting $R$ and considering $\\xi =\\frac{\\mathcal {L}}{E}$ and $\\eta =\\frac{K}{E^{2}}$ , $\\xi $ and $\\eta $ can be reduced to $ $ For Hairy black hole $\\xi ={\\dfrac{1}{a}\\left[\\left(2\\alpha _{0}-\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right)\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right)+2-\\dfrac{2M}{r} \\right]^{-1}}\\left\\lbrace \\left(r^{2}+a^{2}\\right)\\left[\\left(2\\alpha _{0}-\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right) \\right.\\right.",
"\\left.\\left.",
"\\times \\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right)+2-\\dfrac{2M}{r} \\right] -4\\left[a^{2}+r^{2}\\left(1+\\alpha _{0}\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right) \\right)-2Mr \\right] \\right\\rbrace $ $\\eta ={\\dfrac{1}{a^{2}}\\left[\\left(2\\alpha _{0}-\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right)\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right)+2-\\dfrac{2M}{r} \\right]^{-1}}\\left\\lbrace 16Ma^{2}-\\dfrac{8a^{2}\\alpha _{0}r^{2}}{M-\\frac{\\alpha _{0}L}{2}} \\right.",
"\\left.",
"\\times \\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right) -r^{3}\\left[2-\\dfrac{6M}{r}+\\left(2\\alpha _{0}+\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right)\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right) \\right]^{2} \\right\\rbrace $ $ $ For short-hair black hole $\\xi =\\dfrac{\\left(r^{2}+a^{2}\\right) \\left[2-\\dfrac{2M}{r}+\\dfrac{2Q_{m}^{2k}(1-k)}{r^{2k}}\\right] -4\\left[r^{2}-2Mr+\\dfrac{Q_{m}^{2k}}{r^{2k-2}}+a^{2}\\right]}{a\\left[ 2-\\dfrac{2M}{r}+\\dfrac{2Q_{m}^{2k}(1-k)}{r^{2k}} \\right]}$ $\\eta =\\dfrac{16a^{2}\\left(M-\\dfrac{kQ_{m}^{2k}}{r^{2k-1}}\\right) -4r^{3}\\left[1-\\dfrac{3M}{r}+\\dfrac{Q_{m}^{2k}(k+1)}{r^{2k}}\\right]^{2}}{a^{2}\\left[ 2-\\dfrac{2M}{r}+\\dfrac{2Q_{m}^{2k}(1-k)}{r^{2k}} \\right]}.$"
],
[
"The shape of black hole shadows",
"Once the geodesic of the photon is known, we can calculate the motion of the photon as measured by an observer anywhere.",
"In general, we assume that the earth is infinitely far away from the black hole and the observer is a mass point, which makes the calculation very convenient.",
"In the calculation of black hole shadows, physicists introduce the celestial coordinate system, which is a two-dimensional coordinate system[45], and its relationship with the BL coordinates of the black hole is $\\alpha =\\lim \\limits _{r_{0}\\rightarrow \\infty } \\left(-r_{0}^{2}\\sin \\theta _{0}\\dfrac{d\\phi }{dr} \\right)$ $\\beta =\\lim \\limits _{r_{0}\\rightarrow \\infty } \\left(r_{0}^{2}\\dfrac{d\\theta }{dr} \\right)$ where $\\theta _{0}$ is the angle between the line from the earth to the black hole and the axis of rotation of the black hole, and $r_{0}$ is the distance between the earth and the black hole.",
"In a celestial coordinate system, photons emitted from the vicinity of the black hole correspond to the coordinates $(\\alpha , \\beta )$ one by one, so that the geometry of the black hole shadow to be understood by pushing back the edge of the geodesic motion of the photon by $(\\alpha , \\beta )$ .",
"Through calculation, we get the expressions of $\\alpha $ and $\\beta $ as $ $ For Hairy black hole $\\alpha = -\\dfrac{\\xi }{\\sin \\theta }\\mid _{\\theta =\\frac{\\pi }{2}}=-\\xi = -{\\dfrac{1}{a}\\left[\\left(2\\alpha _{0}-\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right)\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right)+2-\\dfrac{2M}{r} \\right]^{-1}}\\left\\lbrace \\left(r^{2}+a^{2}\\right)\\left[\\left(2\\alpha _{0}-\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right) \\right.\\right.",
"\\left.\\left.",
"\\times \\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right)+2-\\dfrac{2M}{r} \\right] -4\\left[a^{2}+r^{2}\\left(1+\\alpha _{0}\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right) \\right)-2Mr \\right] \\right\\rbrace $ $\\beta =\\pm \\sqrt{\\eta +a^{2}\\cos ^{2}\\theta -\\xi ^{2}\\cot ^{2}\\theta }\\mid _{\\theta =\\frac{\\pi }{2}}=\\pm \\sqrt{\\eta }= \\pm {\\dfrac{1}{a^{2}}\\left[\\left(2\\alpha _{0}-\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right)\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right)+2-\\dfrac{2M}{r} \\right]^{-1}}\\left\\lbrace 16Ma^{2}-\\dfrac{8a^{2}\\alpha _{0}r^{2}}{M-\\frac{\\alpha _{0}L}{2}} \\right.",
"\\left.",
"\\times \\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right) -r^{3}\\left[2-\\dfrac{6M}{r}+\\left(2\\alpha _{0}+\\dfrac{\\alpha _{0}r}{M-\\frac{\\alpha _{0}L}{2}} \\right)\\exp \\left(\\dfrac{-r}{M-\\frac{\\alpha _{0}L}{2}} \\right) \\right]^{2} \\right\\rbrace $ $ $ For short-hair black hole $\\alpha =-\\xi =-\\dfrac{\\left(r^{2}+a^{2}\\right) \\left[2-\\dfrac{2M}{r}+\\dfrac{2Q_{m}^{2k}(1-k)}{r^{2k}}\\right] -4\\left[r^{2}-2Mr+\\dfrac{Q_{m}^{2k}}{r^{2k-2}}+a^{2}\\right]}{a\\left[ 2-\\dfrac{2M}{r}+\\dfrac{2Q_{m}^{2k}(1-k)}{r^{2k}} \\right]}$ $\\beta =\\pm \\sqrt{\\eta }=\\pm \\dfrac{16a^{2}\\left(M-\\dfrac{kQ_{m}^{2k}}{r^{2k-1}}\\right) -4r^{3}\\left[1-\\dfrac{3M}{r}+\\dfrac{Q_{m}^{2k}(k+1)}{r^{2k}}\\right]^{2}}{a^{2}\\left[ 2-\\dfrac{2M}{r}+\\dfrac{2Q_{m}^{2k}(1-k)}{r^{2k}} \\right]}$ Figure: The shadow shape of hairy black hole under different model parameters (Model 1).",
"From left to right represents the process of increasing the spin of the black hole, which is a=0,0.5,0.9,0.99a=0, 0.5, 0.9, 0.99 respectively.",
"From top to bottom represents the process of increasing deformation parameter, which is α 0 =0.1,0.5,0.9\\alpha _{0}=0.1, 0.5, 0.9 respectively.",
"Curves of various colors correspond to different LL values.Figure: The shadow shape of hairy black hole under different model parameters (Model 1).",
"From left to right represents the process of increasing the spin of the black hole, which is a=0,0.5,0.9,0.99a=0, 0.5, 0.9, 0.99 respectively.",
"From top to bottom represents the process of increasing parameter LL, which is L=0.01,1,2L=0.01, 1, 2 respectively.",
"Curves of various colors correspond to different α 0 \\alpha _{0} values.Figure: The shadow shape of short-hair black hole under different model parameters (Model 2).",
"From left to right is the process of increasing the black hole spin aa.",
"Curves of various colors correspond to different values of intensity parameter Q m Q_{m}.",
"Here k=1.5k=1.5.Using expressions (REF ) $\\sim $ (REF ), the shape of the black hole shadow can be calculated, so as to make clear the influence of the hairs on the shape of black hole shadow, as shown in figures REF , REF , and REF .",
"The main results are as follows: We obtain the shadow shapes of two kinds of black holes with hairs.",
"In model 1 (corresponding to figures REF and REF ), the values of parameters $\\alpha _{0}$ and $L$ have a significant influence on the shadow shapes (mainly referring to the size of the shadow boundary here).",
"When the parameter $\\alpha _{0}$ , which represents the intensity of the scalar hair, increases gradually from 0, the boundary of the black hole shadow will shrink continuously.",
"When $\\alpha _{0}$ increases to a certain critical value, the black hole shadow changes in reverse.",
"On the other hand, the boundary of the black hole shadow will always increase with the increase of the characteristic length $L$ .",
"Therefore, the change of $\\alpha _{0}$ and $L$ to the black hole shadow is complicated.",
"In Model 2 (corresponding to figure REF ), when $k=1.5$ , namely the short hair case, if the black hole spin $a=0$ , the shape of the shadow boundary of the black hole is a standard circle.",
"It can be found that when the charge $Q_{m}$ of the short hair keeps increasing, the shadow boundary of the black hole will decrease monotonically, but its shape is all circle.",
"As the black hole spin $a$ increases, the shape of the black hole shadow is distorted.",
"We find that the change of the hairy black hole in model 1 to the shadow is basically consistent with the case $\\varepsilon >0$ in reference [38], but the specific changes are different.",
"The change of the short-hair black hole in model 2 to the shadow corresponds to the case $\\varepsilon <0$ in reference [38].",
"If the effects of various hairs carried by black holes on the metric of black holes are known, it is possible to test them by means of EHT observations.",
"However, the parameter values of the real hairy black hole are smaller than those discussed by us (since the shape of the black hole is mainly contributed by the mass, and the quantum effect should not contribute much to it), which requires us to further improve the resolution of EHT before we can test the no-hair theorem of the black hole."
],
[
"The scale and distortion characteristics of shadows",
"From the previous analysis, we can see that the boundary shape of the black hole shadow is a circle only when $a=0$ .",
"When the black hole spin $a\\ne 0$ , the boundary of the black hole shadow will be distorted in different degrees, which makes it necessary to introduce new parameters to accurately describe the shape of the black hole shadow.",
"Generally speaking, scientists introduce the following two parameters to describe the shadow shape.",
"The first parameter is the shadow radius $R_{s}$ , which describes the scale of the shadow shape of the black hole.",
"The second parameter is the degree of distortion of the shadow ($\\delta _{s}$ ), which describes how far the boundary of the black hole shadow deviates from the circle.",
"According to previous discussions, its definition is as follows: $R_{s}=\\dfrac{(\\alpha _{t}-\\alpha _{r})^{2}+\\beta _{t}^{2}}{2|\\alpha _{r}-\\alpha _{t}|},$ $\\delta _{s}=\\dfrac{d_{s}}{R_{s}}=\\dfrac{|\\alpha _{p}-\\tilde{\\alpha }_{p}|}{R_{s}}.$ Figure: The radius (left) and distortion (right) of the black hole shadow vary with parameter LL (Model 1).",
"From top to bottom, they correspond to different deformation parameters α 0 =0,0.5,1\\alpha _{0}=0, 0.5, 1 respectively.",
"The curves of various colors correspond to different black hole spins aa.Figure: The radius (left) and distortion (right) of the black hole shadow vary with deformation parameter α 0 =0\\alpha _{0}=0 (Model 1).",
"From top to bottom, they correspond to different parameters L=0,1,2L=0, 1, 2 respectively.",
"The curves of various colors correspond to different black hole spins aa.Figure: The radius (left) and distortion (right) of the black hole shadow vary with the intensity parameter Q m Q_{m} (Model 2).",
"The curves of various colors correspond to different black hole spins aa.",
"Here k=1.5k=1.5.The physical meanings of $\\alpha _{p}$ and $\\tilde{\\alpha }_{p}$ are shown in reference [45].",
"When the black hole spin $a=0$ (i.e.",
"in the case of a spherically symmetric black hole), $\\alpha _{p}=\\tilde{\\alpha }_{p}$ , at this time, the black hole shadow is not distorted.",
"When extreme black holes are considered (i.e.",
"$a\\rightarrow 1$ ), $\\alpha _{p}-\\tilde{\\alpha }_{p}$ is at its maximum at this point.",
"Through numerical calculation, we find the following results: (1) for model 1, when Kerr black hole is considered, namely $\\alpha _{0}=0$ (no-hair black hole), the change of the scale parameter $L$ hardly affects the value of $R_{s}$ .",
"When $\\alpha _{0}=0.5$ (i.e.",
"black hole carry hairs), $R_{s}$ increases slowly with the increase of parameter $L$ .",
"When $\\alpha _{0}=0.8$ , that is, more hairs are considered, $R_{s}$ rises rapidly and then changes gently with $L$ (see the left part of Figure REF ).",
"When $L=0$ , $R_{s}$ decreases with the increase of $\\alpha _{0}$ .",
"When $L=1$ , $R_{s}$ decreases first and then increases with the increase of $\\alpha _{0}$ .",
"As $L$ increases further, the size of $R_{s}$ varies with $\\alpha _{0}$ to show some interesting phenomenon (see the left part of Figure REF ).",
"(2) For model 1, when Kerr black hole (i.e.",
"no-hair black hole) is considered, the distortion parameter $\\delta _{s}$ is a constant function of the scale parameter $L$ .",
"When hair black holes are considered, i.e.",
"$\\alpha _{0}\\ne 0$ , $\\delta _{s}(L)$ is a decreasing function and decays rapidly at the beginning, while $\\delta _{s}$ changes slowly when $L$ is very large (see the right part of Figure REF ).",
"In the case that $L$ is constant, $\\delta _{s}(\\alpha _{0})$ is an increasing function when $L=0$ , and the first half is an increasing function and the second half is a decreasing function when $L=1$ .",
"Its variation becomes more singular at larger spins of the black hole and $L$ (see the right part of Figure REF ).",
"(3) For model 2, i.e., the short-hair black hole, $R_{s}(Q_{m})$ is a monotone decreasing function, $R_{s}(a)$ , $\\delta _{s}(Q_{m})$ and $\\delta _{s}(a)$ are all increasing functions (see Figure REF ).",
"Since $R_{s}$ and $\\delta _{s}$ vary significantly with scalar, quantum or short hairs, it is more likely to be tested in EHT observations."
],
[
"The rate of energy emission",
"When particles or photons pass near a black hole, they are likely to be absorbed by the black hole, so the effect is similar to that of a black body.",
"If the black hole is regarded as a black body, the size of the black hole shadow will be proportional to the high-energy absorption cross section of the particle.",
"In addition, the theoretical calculation of the high-energy absorption cross section ($\\sigma _{eim}$ ) of black holes shows that $\\sigma _{eim}$ is very close to a constant, which brings great convenience to our calculation.",
"For a spherically symmetric black hole with hairs (scalar, quantum or short hair), the geometric absorption cross section $\\sigma _{eim}$ is approximately equal to the photon sphere of the black hole space-time, i.e., $\\sigma _{eim}\\approx \\pi R_{s}^{2}$[46], [47].",
"The shape of the black hole shadow is approximately a circle except for the near extreme black hole, so the formula $\\sigma _{eim}\\approx \\pi R_{s}^{2}$ is approximately applicable.",
"Knowing the results of $\\sigma _{eim}$ , we can well define the rate of energy emission of a rotational black hole as $\\dfrac{d^{2}E(\\omega )}{d\\omega dt}=\\dfrac{2\\pi ^{2}\\omega ^{3}\\sigma _{eim}}{e^{\\frac{\\omega }{T}}-1}.$ Here, $\\omega $ is the frequency of the particle and $T$ is the Hawking temperature corresponding to the event horizon of the black hole with hairs, whose mathematical form is directly determined by the metric coefficients of the black hole ($T=\\lim \\limits _{\\theta \\rightarrow 0,r\\rightarrow r_{+}}\\frac{1}{2\\pi \\sqrt{g_{rr}}}\\frac{\\partial \\sqrt{g_{tt}}}{\\partial r}$ ).",
"Using Python for numerical calculation, we get the relationship between the rate of energy emission and frequency $\\omega $ , and then get the following results.",
"Figure: The energy emissivity changes with the particle frequency under different model parameters (Model 1).",
"Curves of various colors correspond to different LL values, where the deformation parameter α 0 =0.8\\alpha _{0}=0.8.Figure: The energy emissivity changes with the particle frequency under different model parameters (Model 1).",
"Curves of various colors correspond to different deformation parameters α 0 \\alpha _{0}, where parameter L=0.5L=0.5.Figure: The energy emissivity changes with the particle frequency under different model parameters (Model 1).",
"Curves of various colors correspond to different deformation parameters α 0 \\alpha _{0}, where parameter L=1L=1.Figure: The energy emissivity changes with the particle frequency under different model parameters (Model 1).",
"Curves of various colors correspond to different deformation parameters α 0 \\alpha _{0}, where parameter L=1.5L=1.5.Figure: The energy emissivity changes with the particle frequency under different model parameters (Model 2).",
"Curves of various colors correspond to different values of intensity parameters Q m Q_{m}.",
"Here k=1.5k=1.5.Firstly, it can be seen from the energy emissivity formula (REF ) that its form is very close to Planck black-body radiation formula, so its main numerical results (see Figures REF , REF , REF , REF , REF ) are close to the black-body radiation curve.",
"However, there are some differences, $\\sigma _{eim}$ is different at various Hawking temperatures.",
"Secondly, in model 1, for the case of $\\alpha _{0}$ with a constant value ($\\alpha _{0}=0.8$ ), with the increase of parameter $L$ , the overall trend of its energy emissivity increases, however, a more complicated situation appears on the right side of the emissivity peak, that is, the emissivity curves corresponding to different $L$ values intersect.",
"The intersection of energy emissivity curves may reveal a new physical process.",
"We take Figure REF as an example for analysis.",
"When $\\alpha _{0}=0.8$ , the existence of the intersections indicates that the change of the energy emissivity with parameter $L$ is not monotonic.",
"As the spin of the black hole increases, the position of the intersection shifts to the right (i.e.",
"the greater the particle frequency $\\omega $ ).",
"In Figure REF , when parameter $L=1$ , the energy emissivity curves corresponding to different $\\alpha _{0}$ also show the phenomenon of intersections.",
"We find that with the increase of the spin of the black hole, the position of the intersection moves to the left (that is, the direction in which the particle frequency decreases).",
"The intersection phenomenon of energy emissivity curves is the characteristic of model 1.",
"For model 2, namely the case of the black hole with short hairs, Figure REF shows the results of numerical calculation, including: (1) the energy emissivity decreases with the increase of short-hair parameter $Q_{m}$ , different from model 1, the energy emissivity curves corresponding to these different $Q_{m}$ values do not exist intersection phenomenon; (2) due to the close relationship between short-hair parameter $Q_{m}$ and the black hole spin $a$ , the value of $Q_{m}$ is greatly limited.",
"Finally, comparing model 1 and model 2, the main difference of energy emissivity is whether there is intersection phenomenon, which provides a new possibility for EHT to measure the black hole hairs.",
"That is, whether a black hole has a normal or short hair should show up in EHT measurements."
],
[
"Summary",
"In this work, we obtain the exact solution (space-time linear element (2.25)) of the short-hair black hole in the rotation case.",
"Combined with the space-time metric of the black hole with hairs, we calculate the influence of short hair and hairs on the black hole shadows in detail, and analyse the physical significance of these results.",
"Specifically, our main results are as follows: (1) Using the NJ method, we generalize the spherically symmetric short-hair black hole metric to the rotation case (space-time linear element (2.25)).",
"In the case of spherically symmetric short-hair black hole, the value range of parameter $k$ is $k>1$ .",
"For the rotational short-hair black hole (2.25), the range of short-hair charge value $Q_{m}$ is greatly reduced due to the introduction of the black hole spin $a$ .",
"When $0\\leqslant Q_{m}\\leqslant \\frac{2}{3}\\times 4^{\\frac{1}{3}}$ , the rotational short-hair black hole has three event horizons, two of which are equal, so from physical observation, the black hole has only two event horizons at this time.",
"When $Q_{m}> \\frac{2}{3}\\times 4^{\\frac{1}{3}}$ , the rotational short-hair black hole has three unequal event horizons, so the space-time structure of the black hole is significantly different from that of Kerr black hole.",
"In the space-time metric (2.25), the value range of $k$ is $k>1$ .",
"If the value of $k$ is larger, some interesting phenomena will occur in the rotational short-hair black hole.",
"We take $k=\\frac{5}{2}$ as an example for rough analysis.",
"When $k=\\frac{5}{2}$ , the roots of equations (REF ) and (2.25) will be very complex; As long as the values of $Q_{m}$ and $a$ are appropriate, short-hair black holes may have more event horizons.",
"Then, what is the special physical meaning of the short-hair black holes with multiple event horizons (especially more than 3)?",
"We look forward to studying this issue in future work.",
"(2) We calculate the shadow shapes for two kinds of black holes with hairs in detail, and further study the scale, distortion properties and energy emissivity of black hole shadows.",
"First of all, for model 1, the effect of scalar hair on the black hole shadows corresponds to that of $\\varepsilon >0$ in reference[38], but the specific changes of the shadows in model 1 are different.",
"This is because the black hole hair in reference[38] is considered as a perturbation to the black hole, while the space-time metric of model 1 is accurate and does not have perturbation property.",
"For model 2, that is, the change of the black hole shadow caused by short hairs, the main change trend is consistent with that of $\\varepsilon <0$ in reference[38].",
"Because of the special structure of the short-hair black hole, the specific changes of black hole shadows are different.",
"Secondly, the variation of $R_{s}$ and $\\delta _{s}$ with $L$ and $\\alpha _{0}$ is not a monotone function in model 1, but in model 2, it is.",
"These results show that scalar hairs (model 1) have different effects on Kerr black hole shadows than short hairs (model 2), so it is possible to distinguish the types and properties of these hairs if they are detected by EHT observations.",
"Finally, as for the effects of the hairs on energy emissivity, the main results in model 1, different energy emissivity curves have intersection phenomenon, while in model 2 (short-hair black hole), there is no similar intersection phenomenon.",
"In general, various black hole hairs have different effects on the shadows, such as non-monotonic properties and intersection phenomena mentioned in this paper.",
"Using these characteristics, it is possible to test the no-hair theorem in future EHT observations, so as to have a deeper understanding of the quantum effect of black holes.",
"In future work, we will use numerical simulations to study the effects of various hairs on black holes and their observed properties.",
"We acknowledge the anonymous referee for a constructive report that has significantly improved this paper.",
"We acknowledge the Special Natural Science Fund of Guizhou University (grant No.",
"X2020068) and the financial support from the China Postdoctoral Science Foundation funded project under grants No.",
"2019M650846."
]
] | 2209.08202 |
[
[
"MA2QL: A Minimalist Approach to Fully Decentralized Multi-Agent\n Reinforcement Learning"
],
[
"Abstract Decentralized learning has shown great promise for cooperative multi-agent reinforcement learning (MARL).",
"However, non-stationarity remains a significant challenge in decentralized learning.",
"In the paper, we tackle the non-stationarity problem in the simplest and fundamental way and propose \\textit{multi-agent alternate Q-learning} (MA2QL), where agents take turns to update their Q-functions by Q-learning.",
"MA2QL is a \\textit{minimalist} approach to fully decentralized cooperative MARL but is theoretically grounded.",
"We prove that when each agent guarantees a $\\varepsilon$-convergence at each turn, their joint policy converges to a Nash equilibrium.",
"In practice, MA2QL only requires minimal changes to independent Q-learning (IQL).",
"We empirically evaluate MA2QL on a variety of cooperative multi-agent tasks.",
"Results show MA2QL consistently outperforms IQL, which verifies the effectiveness of MA2QL, despite such minimal changes."
],
[
"Introduction",
"Cooperative multi-agent reinforcement learning (MARL) is a well-abstracted model for a broad range of real applications, including logistics [12], traffic signal control [32], power dispatch [30], and inventory management [7].",
"In cooperative MARL, centralized training with decentralized execution (CTDE) is a popular learning paradigm, where the information of all agents can be gathered and used in training.",
"Many CTDE methods [14], [8], [23], [19], [29], [36], [33] have been proposed and shown great potential to solve cooperative multi-agent tasks.",
"Another paradigm is decentralized learning, where each agent learns its policy based on only local information.",
"Decentralized learning is less investigated but desirable in many scenarios where the information of other agents is not available, and for better robustness, scalability, and security [35].",
"However, fully decentralized learning of agent policies (i.e., without communication) is still an open challenge in cooperative MARL.",
"The most straightforward way for decentralized learning is directly applying independent learning at each agent [26], which however induces the well-known non-stationarity problem for all agents [35] and may lead to the learning instability and a non-convergent joint policy, though the performance varies as shown in empirical studies [19], [4], [17], [33].",
"In the paper, we directly tackle the non-stationarity problem in the simplest and fundamental way, i.e., fixing the policies of other agents while one agent is learning.",
"Following this principle, we propose multi-agent alternate Q-learning (MA2QL), a minimalist approach to fully decentralized cooperative multi-agent reinforcement learning, where agents take turns to update their policies by Q-learning.",
"MA2QL is theoretically grounded and we prove that when each agent guarantees an $\\varepsilon $ -convergence at each turn, their joint policy converges to a Nash equilibrium.",
"In practice, MA2QL only requires the minimal changes to independent Q-learning (IQL) [26], [25] and also independent DDPG [13] for continuous action, i.e., simply swapping the order of two lines of codes as follows.",
"Figure: MA2QLWe evaluate MA2QL on a didactic game to empirically verify its convergence, and multi-agent particle environments [14], multi-agent MuJoCo [18], and StarCraft multi-agent challenge [20] to verify its performance in discrete and continuous action spaces, fully and partially observable environments.",
"We find that MA2QL consistently outperforms IQL, despite such minimal changes.",
"The effectiveness of MA2QL suggests that simpler approaches may have been left underexplored for fully decentralized cooperative multi-agent reinforcement learning."
],
[
"Preliminaries",
"Dec-POMDP.",
"Decentralized partially observable Markov decision process (Dec-POMDP) is a general model for cooperative MARL.",
"A Dec-POMDP is a tuple $M=\\left\\lbrace S,A,P,Y,O,I,n,r,\\gamma \\right\\rbrace $ .",
"$S$ is the state space, $n$ is the number of agents, $\\gamma \\in [0,1)$ is the discount factor, and $I = \\lbrace 1,2\\cdots n\\rbrace $ is the set of all agents.",
"$A = A_1 \\times A_2 \\times \\cdots \\times A_n$ represents the joint action space where $A_i$ is the individual action space for agent $i$ .",
"$P(s^{\\prime } |s,\\mathbf {a} ): S \\times A \\times S \\rightarrow [0,1]$ is the transition function, and $r(s,\\mathbf {a} ): S \\times A \\rightarrow \\mathbb {R}$ is the reward function of state $s$ and joint action $\\mathbf {a}$ .",
"$Y$ is the observation space, and $O(s,i):S \\times I \\rightarrow Y $ is a mapping from state to observation for each agent.",
"The objective of Dec-POMDP is to maximize $J({\\mathbf {\\pi }}) = \\mathbb {E}_{\\mathbf {\\pi }}\\left[ \\sum _{t = 0}^\\infty \\gamma ^t r(s_t,\\mathbf {a}_t ) \\right],$ and thus we need to find the optimal joint policy ${\\mathbf {\\pi }}^{*} = \\arg \\max _{{\\mathbf {\\pi }}} J({\\mathbf {\\pi }})$ .",
"To settle the partial observable problem, history $\\tau _i \\in \\mathcal {T}_i: (Y \\times A_i)^*$ is often used to replace observation $o_i \\in Y$ .",
"Each agent $i$ has an individual policy $\\pi _i(a_i|\\tau _i)$ and the joint policy $\\mathbf {\\pi }$ is the product of each $\\pi _i$ .",
"Though individual policy is learned as $\\pi _i(a_i|\\tau _i)$ in practice, we will use $\\pi _i(a_i|s)$ in analysis and proofs for simplicity.",
"Dec-MARL.",
"Although decentralized cooperative multi-agent reinforcement learning (Dec-MARL) has been previously investigated [34], [4], the setting varies across these studies.",
"In this paper, we consider Dec-MARL as a fully decentralized solution to Dec-POMDP, where each agent learns its policy/Q-function from its own action individually without communication or parameter-sharing.",
"Therefore, in Dec-MARL, each agent $i$ actually learns in the environment with transition function $P_i(s^{\\prime }|s,a_i) = \\mathbb {E}_{a_{-i} \\sim \\pi _{-i}}[P(s^{\\prime }|s,a_i,a_{-i})]$ and reward function $r_i(s,a_i) = \\mathbb {E}_{a_{-i} \\sim \\pi _{-i}} [r(s,a_i,a_{-i})]$ , where $\\pi _{-i}$ and $a_{-i}$ respectively denote the joint policy and joint action of all agents expect $i$ .",
"As other agents are also learning (i.e., $\\pi _{-i}$ is changing), from the perspective of each individual agent, the environment is non-stationary.",
"This is the non-stationarity problem, the main challenge in Dec-MARL.",
"IQL.",
"Independent Q-learning (IQL) is a straightforward method for Dec-MARL, where each agent $i$ learns a Q-function $Q(s,a_i)$ by Q-learning.",
"However, as all agents learn simultaneously, there is no theoretical guarantee on the convergence due to non-stationarity, to the best of our knowledge.",
"In practice, IQL is often taken as a simple baseline in favor of more elaborate MARL approaches, such as value-based CTDE methods [19], [21].",
"However, much less attention has been paid to IQL itself for Dec-MARL."
],
[
"Multi-Agent Alternate Policy Iteration",
"To address the non-stationarity problem in Dec-MARL, a fundamental way is simply to make the environment stationary during the learning of each agent.",
"Following this principle, we let agents learn by turns; in each turn, one agent performs policy iteration while fixing the policies of other agents.",
"This procedure is referred to as multi-agent alternate policy iteration.",
"As illustrated in Figure REF , multi-agent alternate policy iteration differs from policy iteration in single-agent RL.",
"In single-agent RL, policy iteration is performed on the same MDP.",
"However, here, for each agent, policy iteration at a different round is performed on a different MDP.",
"As $\\pi _{-i}$ is fixed at each turn, $P_i(s^{\\prime }|s,a_i)$ and $r_i(s,a_i)$ are stationary and we can easily have the following lemma.",
"Lemma 1 (multi-agent alternate policy iteration) If all agents take turns to perform policy iteration, their joint policy sequence $\\lbrace \\mathbf {\\pi }\\rbrace $ monotonically improves and converges to a Nash equilibrium.",
"In each turn, as the policies of other agents are fixed, the agent $i$ has the following update rule for policy evaluation, $Q_{\\pi _i}(s,a_i) \\leftarrow r_i(s,a_i) + \\gamma \\mathbb {E}_{s^{\\prime }\\sim P_i,a_i^{\\prime } \\sim \\pi _i}[Q_{\\pi _i}(s^{\\prime },a_i^{\\prime })].$ We can have the convergence of policy evaluation in each turn by the standard results [24].",
"Moreover, as $\\pi _{-i}$ is fixed, it is straightforward to have $Q_{\\pi _i}(s,a_i) = \\mathbb {E}_{a_{-i} \\sim \\pi _{-i}}[Q_{\\mathbf {\\pi }}(s,a,a_i)].$ Then, the agent $i$ performs policy improvement by $ \\pi ^{\\operatorname{new}}_i(s) = \\arg \\max _{a_i} \\mathbb {E}_{\\pi ^{\\operatorname{old}}_{-i} }\\left[ Q_{\\mathbf {\\pi }^{\\operatorname{old}}}(s,a_i,a_{-i} ) \\right].$ As the policies of other agents are fixed (i.e., $\\pi ^{\\operatorname{new}}_{-i}=\\pi ^{\\operatorname{old}}_{-i}$ ), we have $\\begin{aligned}V_{\\mathbf {\\pi }^{\\operatorname{old}}}(s) & = \\mathbb {E}_{\\mathbf {\\pi }^{\\operatorname{old}}}[Q_{\\mathbf {\\pi }^{\\operatorname{old}}}(s,a_i,a_{-i})] = \\mathbb {E}_{\\pi _i^{\\operatorname{old}}}\\mathbb {E}_{\\pi _{-i}^{\\operatorname{old}}}[Q_{\\mathbf {\\pi }^{\\operatorname{old}}}(s,a_i,a_{-i})] \\\\& \\le \\mathbb {E}_{\\pi _i^{\\operatorname{new}}}\\mathbb {E}_{\\pi _{-i}^{\\operatorname{old}}}[Q_{\\mathbf {\\pi }^{\\operatorname{old}}}(s,a_i,a_{-i}) ] = \\mathbb {E}_{\\pi _i^{\\operatorname{new}}}\\mathbb {E}_{\\pi _{-i}^{\\operatorname{new}}}[Q_{\\mathbf {\\pi }^{\\operatorname{old}}}(s,a_i,a_{-i}) ] \\\\& = \\mathbb {E}_{\\mathbf {\\pi }^{\\operatorname{new}}}[Q_{\\mathbf {\\pi }^{\\operatorname{old}}}(s,a_i,a_{-i}) ] = \\mathbb {E}_{\\mathbf {\\pi }^{\\operatorname{new}}}[r(s,a_i,a_{-i}) + \\gamma V_{\\mathbf {\\pi }^{\\operatorname{old}}}(s^\\prime ) ]\\\\& \\le \\cdots \\le V_{\\mathbf {\\pi }^{\\operatorname{new}}}(s),\\end{aligned}$ where the first inequality is from (REF ).",
"This proves that the policy improvement of agent $i$ in each turn also improves the joint policy.",
"Thus, as agents perform policy iteration by turn, the joint policy sequence $\\lbrace \\mathbf {\\pi }\\rbrace $ improves monotonically, and $\\lbrace \\mathbf {\\pi }\\rbrace $ will converge to a Nash equilibrium since no agents can improve the joint policy unilaterally at convergence.",
"Figure: Illustration of multi-agent alternate policy iteration (upper panel) and multi-agent alternate Q-iteration (lower panel) of three agents.",
"As essentially the MDP differs at the different turn of each agent, policy iteration/Q-iteration of each agent iterates over different MDPs.Lemma REF immediately indicates an approach for Dec-MARL with convergence guarantee and also tells us that if we find the optimal policy for agent $i$ in each round $k$ given the other agents' policies $\\pi _{-i}^k$ , then the joint policy will obtain the largest improvement.",
"This result can be formulated as following, $\\begin{aligned}&\\pi ^{*,k}_i = \\arg \\max _{\\pi _i} \\mathbb {E}_{\\pi ^{k}_{-i}}\\left[ Q_{\\pi _i,\\pi _{-i}^{k}}(s,a_i,a_{-i} ) \\right] \\\\&V_{\\pi _i,{\\pi }_{-i}^{k}}(s) \\le V_{\\pi ^{*,k}_i,{\\pi }_{-i}^k}(s) \\quad \\forall \\pi _i, \\forall s.\\end{aligned}$ We could obtain this $\\pi ^{*,k}_i$ by policy iteration with many on-policy iterations.",
"However, such a method will face the issue of sample inefficiency which may be amplified in MARL settings.",
"We will use Q-iteration to settle this problem in the next section."
],
[
"Method",
"To address the problem of multi-agent alternate policy iteration, we propose multi-agent alternate Q-iteration, which is sufficiently truncated for fast learning but still has the same theoretical guarantee.",
"Further, based on multi-agent alternate Q-iteration, we derive multi-agent alternate Q-learning, which makes the minimal change to IQL to form a simple yet effective value-based decentralized learning method for cooperative MARL."
],
[
"Multi-Agent Alternate Q-Iteration",
"Instead of policy iteration, we let agents perform Q-iteration by turns as depicted in Figure REF .",
"Let $\\mathcal {M}^k_i=\\lbrace P_i^k,r^k_i\\rbrace $ denote the MDP of agent $i$ in round $k$ , where we have $\\mathcal {M}^{k}_i \\ne \\mathcal {M}^{k-1}_i$ unless $\\pi _{-i}$ has converged, and $Q^{t,k}_{i}(s,a_i)$ denote the Q-function of agent $i$ with $t$ updates in the round $k$ .",
"We define the Q-iteration as follows, $Q^{t+1,k}_{i}(s,a_i) \\leftarrow r^k_i(s,a_i) + \\gamma \\mathbb {E}_{s^\\prime \\sim P_i^k} \\left[ \\max _{a_i^\\prime } Q^{t,k}_{i}(s^\\prime ,a^\\prime _i) \\right].$ Then, the sequence $\\lbrace Q^{t,k}_{i}\\rbrace $ converges to $Q^{*,k}_{i}$ with respect to the MDP $\\mathcal {M}_i^k=\\lbrace P_i^k,r_i^k\\rbrace $ , and we have the following lemma.",
"Lemma 2 ($\\varepsilon $ -convergent Q-iteration) By iteratively applying Q-iteration (REF ) at each agent $i$ for each turn, for any $\\varepsilon >0$ , we have $\\begin{aligned}\\big \\Vert {Q^{t,k}_{i} - Q^{*,k}_{i}}\\big \\Vert _\\infty \\le \\varepsilon , \\quad \\text{when \\,\\,} t \\ge \\frac{\\log \\left( (1- \\gamma ) \\varepsilon \\right) - \\log (2R + 2\\varepsilon )}{\\log \\gamma },\\end{aligned}$ where $R = \\frac{r_{\\max }}{1-\\gamma }$ and $r_{\\max } = \\max _{s,\\mathbf {a}}r(s,\\mathbf {a})$ .",
"From the definition of $Q^{t,k}_{i}$ (REF ), we have $\\begin{aligned}\\big \\Vert Q^{t+1,k}_{i} - Q^{t,k}_{i}\\big \\Vert _\\infty & = \\big \\Vert \\gamma \\mathbb {E}_{s^{\\prime }\\sim P_i^k}[\\operatorname{max}_{a_i^\\prime } Q^{t,k}_i(s^\\prime ,a_i^\\prime ) - \\operatorname{max}_{a_i^\\prime } Q^{t-1,k}_i(s^\\prime ,a_i^\\prime )] \\big \\Vert _\\infty \\\\& \\le \\gamma \\big \\Vert Q^{t,k}_i - Q^{t-1,k}_i \\big \\Vert _\\infty \\le \\gamma ^t \\big \\Vert Q^{1,k}_i - Q^{0,k}_i \\big \\Vert _\\infty .\\end{aligned}$ Then for any integer $m \\ge 1$ , we have $\\begin{aligned}\\big \\Vert Q^{t+m,k}_{i} - Q^{t,k}_{i}\\big \\Vert _\\infty & \\le \\big \\Vert Q^{t+m,k}_{i} - Q^{t+m-1,k}_{i}\\big \\Vert _\\infty + \\cdots + \\big \\Vert Q^{t+1,k}_{i} - Q^{t,k}_{i}\\big \\Vert _\\infty \\\\& \\le \\gamma ^t \\frac{1 - \\gamma ^m}{1-\\gamma } \\big \\Vert Q^{1,k}_i - Q^{0,k}_i\\big \\Vert _\\infty .\\end{aligned}$ Let $m \\rightarrow \\infty $ , and we have $\\begin{aligned}\\big \\Vert Q^{*,k}_{i} - Q^{t,k}_{i} \\big \\Vert _\\infty & \\le \\frac{\\gamma ^t}{1-\\gamma }\\big \\Vert Q^{1,k}_i - Q^{0,k}_i \\big \\Vert _\\infty \\\\& \\le \\frac{\\gamma ^t}{1-\\gamma } \\max _{s,a_i} \\big | r^k_i(s,a_i) + \\gamma \\mathbb {E}_{s^{\\prime }\\sim P_i^k}[\\operatorname{max}_{a_i^\\prime } Q^{0,k}_i(s^\\prime ,a_i^\\prime )] - Q^{0,k}_i(s,a_i) \\big |.\\end{aligned}$ As all agents update by turns, we have $Q^{0,k}_i = Q^{t^{k-1}_{i},k-1}_{i}$ , where $t^{k-1}_{i}$ is the number of Q-iteration for agent $i$ in the $k-1$ round.",
"Therefore, we have $\\big \\Vert {Q^{0,k}_i - Q^{*,k-1}_{i}\\big \\Vert _\\infty = \\big \\Vert Q^{t^{k-1}_{i},k-1}_{i-1} - Q^{*,k-1}_{i}} \\big \\Vert _\\infty \\le \\varepsilon .$ With this property, we have $& \\big | r^k_i(s,a_i) + \\gamma \\mathbb {E}_{s^{\\prime } \\sim P_i^k}[\\operatorname{max}_{a_i^\\prime } Q^{0,k}_i(s^\\prime ,a_i^\\prime )] - Q^{0,k}_i(s,a_i) \\big | \\nonumber \\\\& = \\big | r^k_i(s,a_i) + \\gamma \\mathbb {E}_{s^{\\prime } \\sim P_i^k}[\\operatorname{max}_{a_i^\\prime } Q^{0,k}_i(s^\\prime ,a_i^\\prime )] - Q^{*,k-1}_{i}(s,a_i) + Q^{*,k-1}_{i}(s,a_i) - Q^{0,k}_i(s,a_i) \\big | \\nonumber \\\\& \\le \\big | r^k_i - r^{k-1}_{i} \\big | + \\gamma \\big | \\mathbb {E}_{s^{\\prime } \\sim P_i^k}[\\operatorname{max}_{a_i^\\prime } Q^{0,k}_i(s^\\prime ,a_i^\\prime )] - \\mathbb {E}_{s^{\\prime } \\sim P_i^{k-1}}[\\operatorname{max}_{a_i^\\prime } Q^{*,k-1}_{i}(s^\\prime ,a_i^\\prime )] \\big | \\nonumber \\\\& \\quad + \\big |Q^{*,k-1}_{i}(s,a_i) - Q^{0,k}_i(s,a_i) \\big | \\le 2r_{\\max } + (\\frac{2\\gamma r_{\\max }}{1-\\gamma } + \\varepsilon ) +\\varepsilon = 2R + 2\\varepsilon ,$ where the second term in the last inequality is from $\\Vert Q^{*,k-1}_{i}\\Vert _\\infty \\le \\frac{r_{\\max }}{1-\\gamma }$ , $\\Vert Q^{0,k}_{i}\\Vert _\\infty \\le \\Vert Q^{*,k-1}_{i}\\Vert _\\infty + \\varepsilon $ , and (REF ).",
"Finally, by combining (REF ) and (REF ), we have $\\big \\Vert Q^{*,k}_{i} - Q^{t,k}_{i}\\big \\Vert _\\infty \\le \\frac{\\gamma ^t}{1-\\gamma }(2R+2\\varepsilon ).$ We need $\\Vert Q^{*,k}_{i} - Q^{t,k}_{i}\\Vert _\\infty \\le \\varepsilon $ , which can be guaranteed by $t \\ge \\frac{\\log \\left( (1- \\gamma ) \\varepsilon \\right) - \\log (2R +2\\varepsilon )}{\\log \\gamma }$ .",
"Corollary 1 For any $\\varepsilon > 0$ , if we take sufficient Q-iteration $t^k_i$ , i.e., $Q^k_i = Q^{t_i^k,k}_i$ , then we have $\\big \\Vert Q^k_i - Q^{*,k}_i\\big \\Vert _\\infty \\le \\varepsilon \\quad \\forall k, i.$ With Lemma REF , Lemma REF , and Corollary REF , we have the following theorem.",
"Theorem 1 (multi-agent alternate Q-iteration) Suppose that $Q^*_i(s,\\cdot )$ has the unique maximum for all states and all agents.",
"If all agents in turn take Q-iteration to $\\Vert Q^k_i - Q^{*,k}_i\\Vert _\\infty \\le \\varepsilon $ , then their joint policy sequence $\\lbrace \\pi ^k\\rbrace $ converges to a Nash equilibrium, where $\\pi _i^k(s) = \\arg \\max _{a_i}Q_i^k(s,a_i)$ .",
"First, from Lemma REF , we know $Q^{*,k}_i$ also induces a joint policy improvement, thus $Q^{*,k}_i$ converges to $Q^{*}_i$ .",
"Let $\\pi _i^*(s)=\\arg \\max _{a_i}Q^*_i(s,a_i)$ , then $\\mathbf {\\pi }^*$ is the joint policy of a Nash equilibrium.",
"Then, we define $\\Delta $ as $\\Delta = \\min _{s,i} \\max _{a_i \\ne \\pi ^*_i(s)} |Q^*_i(s,\\pi ^*_i(s)) - Q^*_i(s,a_i) |.$ From the assumption we know that $\\Delta > 0$ .",
"We take $\\varepsilon = \\frac{\\Delta }{6}$ , and from Lemma REF , we know there exists $k_0$ such that $\\big \\Vert Q^*_i - Q^{*,k}_i\\big \\Vert _\\infty \\le \\varepsilon \\quad \\forall k \\ge k_0.$ For $k\\ge k_0$ and any action $a_i \\ne \\pi ^*_i(s)$ , we have $\\begin{aligned}Q^k_i(s,& \\pi ^*_i(s)) - Q^k_i(s,a_i) \\\\& = Q^k_i(s,\\pi ^*_i(s)) - Q^{*,k}_i(s,\\pi ^*_i(s)) + Q^{*,k}_i(s,\\pi ^*_i(s)) - Q^{*}_i(s,\\pi ^*_i(s)) \\\\& \\quad + Q^{*}_i(s,\\pi ^*_i(s)) - Q^{*}_i(s,a_i) + Q^{*}_i(s,a_i) - Q^{*,k}_i(s,a_i) + Q^{*,k}_i(s,a_i) - Q^k_i(s,a_i) \\\\& \\ge Q^{*}_i(s,\\pi ^*_i(s)) - Q^{*}_i(s,a_i) - |Q^k_i(s,a_i) - Q^{*,k}_i(s,a_i)| - |Q^{*,k}_i(s,a_i) - Q^{*}_i(s,a_i)|\\\\& \\quad - |Q^{*}_i(s,\\pi ^*_i(s)) - Q^{*,k}_i(s,\\pi ^*_i(s))| - |Q^{*,k}_i(s,\\pi ^*_i(s)) -Q^k_i(s,\\pi ^*_i(s))| \\\\& = \\Delta - 4\\varepsilon = \\Delta /3 > 0,\\end{aligned}$ which means $\\pi ^k_i(s) = \\arg \\max _{a_i} Q^k_i(s,a_i) = \\arg \\max _{a_i} Q^*_i(s,a_i) = \\pi ^*_i(s)$ .",
"Thus, $Q_i^{k}$ of each agent $i$ induces $\\pi _i^*$ and all together induce $\\mathbf {\\pi }^*$ , which the joint policy of a Nash equilibrium.",
"Theorem REF assumes that for each agent, $Q_i^*$ has the unique maximum for all states.",
"Although this may not hold in general, in practice we can easily settle this by introducing a positive random noise to the reward function.",
"Suppose the random noise is bounded by $\\delta $ , then we can easily derive that the performance drop of optimizing environmental reward plus noise is bounded by $\\delta /(1-\\gamma )$ .",
"As we can make $\\delta $ arbitrarily small, the bound is tight."
],
[
"Multi-Agent Alternate Q-Learning",
"From Theorem REF , we know that if each agent $i$ guarantees $\\varepsilon $ -convergence to $Q_i^{*,k}$ in each round $k$ , multi-agent alternate Q-iteration also guarantees a Nash equilibrium of the joint policy.",
"This immediately suggests a simple, practical decentralized learning method, namely multi-agent alternate Q-learning (MA2QL).",
"For learning Q-table or Q-network, MA2QL makes the minimal changes to IQL.",
"For learning Q-tables, all agents in turn update their Q-tables.",
"At a round $k$ of an agent $i$ , all agents interact in the environment, and the agent $i$ updates its Q-table a few times using the collected transitions $\\left<s,a_i,r,s^{\\prime } \\right>$ .",
"For learning Q-networks, all agents in turn update their Q-networks.",
"At a round of an agent $i$ , all agents interact in the environment and each agent $j$ stores the collected transitions $\\left<s,a_j,r,s^{\\prime } \\right>$ into its replay buffer, and the agent $i$ updates its Q-network using sampled mini-batches from its replay buffer.",
"There is a slight difference between learning Q-table and Q-network.",
"Strictly following multi-agent alternate Q-iteration, Q-table is updated by transitions sampled from the current MDP.",
"On the other hand, Q-network is updated by minibatches sampled from the replay buffer.",
"If the replay buffer only contains the experiences sampled from current MDP, learning Q-network also strictly follows multi-agent alternate Q-iteration.",
"However, in practice, we slightly deviate from that and allow the replay buffer to contain transitions of past MDPs, following IQL [23], [19], [17] for sample efficiency, the convergence may not be theoretically guaranteed though.",
"MA2QL and IQL can be simply summarized and highlighted as MA2QL agents take turns to update Q-functions by Q-learning, whereas IQL agents simultaneously update Q-functions by Q-learning."
],
[
"Related Work",
"CTDE.",
"The most popular learning paradigm in cooperative MARL is centralized training with decentralized execution (CTDE), including value decomposition and multi-agent actor-critic.",
"For value decomposition [23], [19], [21], [29], a joint Q-function is learned in a centralized manner and factorized into local Q-functions to enable decentralized execution.",
"For multi-agent actor-critic, a centralized critic, Q-function or V-function, is learned to provide gradients for local policies [14], [8], [33].",
"Moreover, some studies [31], [18], [22] combine value decomposition and multi-agent actor-critic to take advantage of both, while others rely on maximum-entropy RL to naturally bridge the joint Q-function and local policies [10], [36].",
"Decentralized learning.",
"Another learning paradigm in cooperative MARL is decentralized learning, where the simplest way is for each agent to learn independently, e.g., independent Q-learning (IQL) or independent actor-critic (IAC).",
"These methods are usually taken as simple baselines for CTDE methods.",
"For example, IQL is taken as a baseline in value decomposition methods [23], [19], while IAC is taken as a baseline in multi-agent actor-critic [8], [33].",
"Some study further considers decentralized learning with communication for parameter-sharing [34], [11].",
"However, parameter-sharing should be considered as centralized training [27].",
"More recently, IAC (i.e., independent PPO) has been empirically investigated and found remarkably effective in several cooperative MARL tasks [4], [33], including multi-agent particle environments (MPE) [14] and StarCraft multi-agent challenge (SMAC).",
"On the other hand, IQL has also been thoroughly benchmarked and its performance is close to CTDE methods in a few tasks [17].",
"This sheds some light on the potential of value-based decentralized cooperative MARL.",
"We go one step further and propose MA2QL, a value-based fully decentralized learning method, which is theoretically grounded and in practice makes the minimal changes to IQL.",
"In the next section, we provide the empirical comparison between MA2QL and IQL."
],
[
"Experiments",
"In this section, we empirically study MA2QL on a set of cooperative multi-agent tasks, including a didactic game, multi-agent particle environments (MPE) [14], multi-agent MuJoCo [18], and StarCraft multi-agent challenge (SMAC) [20], to investigate the following questions.",
"1.",
"Does MA2QL converge and what does it converge to empirically, compared with the optimal solution and IQL?",
"Does the number of Q-function updates affect the convergence?",
"2.",
"As MA2QL only makes the minimal changes to IQL, is MA2QL indeed better than IQL in both discrete and continuous action spaces, and in more complex tasks?",
"In all the experiments, the training of MA2QL and IQL is based on the same number of environmental steps (i.e., the same number of samples).",
"Moreover, as the essential difference between MA2QL and IQL is that MA2QL agents take turns to update Q-function while IQL agents update Q-function simultaneously, for a fair comparison, the total number of Q-function updates for each agent in MA2QL is set to be the same with that in IQL.",
"For example, in a setting of $n$ agents, if IQL agents update Q-function one step (e.g., one gradient step) every environmental step, MA2QL agents update Q-function $n$ steps every $n$ environmental steps.",
"For learning Q-networks, the size of replay buffer is also the same for IQL and MA2QL.",
"Again, we do not use parameter-sharing, which should not be allowed in decentralized settings [27].",
"More detailed experimental settings, hyperparameters, and additional results are available in Appendix , , and REF , respectively.",
"All results are presented using the mean and standard deviation of five random seeds."
],
[
"A Didactic Game",
"The didactic game is a cooperative stochastic game, which is randomly generated for the reward function and transition probabilities with 30 states, 3 agents, and 5 actions for each agent.",
"Each episode in the game contains 30 timesteps.",
"For comparison, we use dynamic programming to find the global optimal solution, denoted as OPTIMAL.",
"For MA2QL and IQL, each agent independently learns a $30 \\times 5$ Q-table.",
"First, we investigate how the number of Q-iterations empirically affects the convergence of multi-agent alternate Q-iteration, where Q-iteration is performed by dynamic programming and denoted as MA2QL-DP.",
"As shown in Figure REF , we can see that different numbers of Q-iterations (i.e., $t=1,5,10,50$ ) that each agent takes at each turn do not affect the convergence in the didactic game, even when $t=1$ .",
"This indicates $\\varepsilon $ -convergence of Q-iteration can be easily satisfied with as few as one iteration.",
"Next, we compare the performance of MA2QL and IQL.",
"As illustrated in Figure REF , IQL converges slowly (about 2000 steps), while MA2QL converges much faster (less than 100 steps) to a better return and also approximates OPTIMAL.",
"Once again, MA2QL and IQL use the same number of samples and Q-table updates for each agent.",
"One may notice that the performance of MA2QL is better than MA2QL-DP.",
"This may be attributed to sampling and exploration of MA2QL, which induces a better Nash equilibrium.",
"Then, we investigate MA2QL in terms of the number of Q-table updates at each turn, which resembles the number of Q-iterations by learning on samples.",
"Specifically, denoting $K$ as the number of Q-table updates, to update an agent, we repeat $K$ times of the process of sampling experiences and updating Q-table.",
"This mean with a larger $K$ , agents take turns less frequently.",
"As shown in Figure REF , with larger $K$ , the learning curve is more stair-like, which means in this game a small number of Q-table updates is enough for convergence at each turn.",
"Thus, with larger $K$ , the learning curve converges more slowly.",
"Last, we investigate how the number of collected transitions at each turn impacts the performance of MA2QL.",
"As depicted in Figure REF , the performance of MA2QL is better with more samples.",
"This is because, with more samples, the update of Q-learning using these samples is more like to induce a full iteration of Q-table.",
"In summary, as illustrated in Figure REF , the number of Q-function updates or samples may influence the performance of MA2QL.",
"However, in the following experiments, we tune the performance of IQL as best as we can and correspondingly change the configuration of MA2QL as stated above for a fair comparison."
],
[
"MPE",
"MPE is a popular environment in cooperative MARL.",
"We consider three partially observable tasks: 5-agent simple spread, 5-agent line control, and 7-agent circle control [2], where the action space is set to discrete.",
"Moreover, we use the sparse reward setting of these tasks, thus they are more difficult than the original ones.",
"More details are available in Appendix .",
"For both IQL and MA2QL, Q-network is learned by DQN [16].",
"Figure REF shows the learning curve of MA2QL compared with IQL in these three MPE tasks.",
"In simple spread and circle control, at the early training stage, IQL learns faster and better than MA2QL, but eventually MA2QL converges to a better joint policy than IQL.",
"IQL always converges to a worse sub-optimum than MA2QL, similar to that in the didactic game.",
"Moreover, unlike the didactic game, simultaneous learning of IQL may also make the learning unstable even at the late training stage as in line control and circle control, where the episode rewards may decrease.",
"On the other hand, learning by turns gradually improves the performance and converges to a better joint policy than IQL.",
"As MA2QL and IQL both use replay buffer that contains old experiences, why does MA2QL outperform IQL?",
"The reason is that their experiences are generated in different manners.",
"In the Q-learning procedure for each agent $i$ , the ideal target is $y_i = r^{\\pi }_i(s,a_i) + \\mathbb {E}_{s^\\prime \\sim P^{\\pi }_i(\\cdot |s,a_i)}[ \\max _{a^\\prime _i} Q_i(s^\\prime ,a^\\prime _i)]$ and the practical target is $\\tilde{y}_i = r^{\\pi _{\\operatorname{D}}}_i(s,a_i) + \\mathbb {E}_{s^\\prime \\sim P^{\\pi _{\\operatorname{D}}}_i(\\cdot |s,a_i)}[ \\max _{a^\\prime _i} Q_i(s^\\prime ,a^\\prime _i)]$ , where $\\pi _{\\operatorname{D}}$ is the average joint policy for the experiences in the replay buffer.",
"We then can easily obtain a bound for the target that $\\left| y_i - \\tilde{y}_i \\right| \\le \\frac{2-\\gamma }{1-\\gamma }r_{\\max } D_{\\operatorname{TV}}\\left(\\pi ^{-i}(\\cdot |s)\\Vert \\pi _{\\operatorname{D}}^{-i}(\\cdot |s)\\right)$ where $r_{\\max } = \\max _{s,a}r(s,a)$ .",
"We can then give an analysis from the aspect of the divergence between $\\pi $ and $\\pi _{\\operatorname{D}}$ .",
"MA2QL obtains experiences with only one agent learning, so the variation for the joint policy is smaller than that of IQL.",
"Thus, in general, the divergence between $\\pi $ and $\\pi _{\\operatorname{D}}$ is smaller for MA2QL, which is beneficial to the learning.",
"Figure: Learning curve of MA2QL compared with IQL in 5-agent simple spread, 5-agent line control, and 7-agent circle control in MPE, where x-axis is environment steps.Figure: Learning curve of MA2QL compared with IQL in 2-agent HalfCheetah, 3-agent Hopper and 3-agent Walker2d in multi-agent MuJoCo, where x-axis is environment steps."
],
[
"Multi-Agent MuJoCo",
"Multi-agent MuJoCo has become a popular environment in cooperative MARL for continuous action space.",
"We choose three robotic control tasks: 2-agent HalfCheetah, 3-agent Hopper, and 3-agent Walker2d.",
"To investigate continuous action space in both partially and fully observable environments, we configure 2-agent HalfCheetah and 3-agent Hopper as fully observable, and 3-agent Walker2d as partially observable.",
"For both IQL and MA2QL, we use DDPG [13] as the alternative of DQN to learn a Q-network and a deterministic policy for each agent to handle continuous action space.",
"In comparison to discrete action space, training multiple cooperative agents in continuous action space still remains challenging due to the difficulty of exploration and coordination in continuous action space.",
"Thus, the evaluation on these multi-agent MuJoCo tasks can better demonstrate the effectiveness of decentralized cooperative MARL methods.",
"As illustrated in Figure REF , in all the tasks, we find that MA2QL consistently and significantly outperforms IQL while IQL struggles.",
"We believe the reason is that the robotic control tasks are much more dynamic than MPE and the non-stationarity induced by simultaneous learning of IQL may be amplified, which makes it hard for agents to learn effective and cooperative policies.",
"On the other hand, alternate learning of MA2QL can deal with the non-stationarity and sufficiently stabilize the environment during the learning process, especially in HalfCheetah and Hopper, where MA2QL stably converges to much better performance than IQL.",
"According to these experiments, we can verify the superiority of MA2QL over IQL in the continuous action space."
],
[
"SMAC",
"SMAC is a popular partially observable environment for benchmarking cooperative MARL algorithms.",
"SMAC has a much larger exploration space, where agents are much easy to get stuck in sub-optimal policies especially in the decentralized setting.",
"We test our method on three representative maps for three difficulties: $\\mathtt {3s\\_vs\\_4z}$ (easy), $\\mathtt {5m\\_vs\\_6m}$ (hard), and $\\mathtt {corridor}$ (super hard), where harder map has more agents.",
"It is worth noting that we do not use any global state in the decentralized training and each agent learns on its own trajectory.",
"The results are shown in Figure REF .",
"On the map $\\mathtt {3s\\_vs\\_4z}$ , IQL and MA2QL both converge to the winning rate of 100%.",
"However, on the hard and super hard map $\\mathtt {5m\\_vs\\_6m}$ and $\\mathtt {corridor}$ , MA2QL achieves stronger than IQL.",
"Moreover, in these maps there are different numbers of agents, 3 agents in $\\mathtt {3s\\_vs\\_4z}$ , 5 agents in $\\mathtt {5m\\_vs\\_6m}$ , and 6 agents in $\\mathtt {corridor}$ , thus the results may also indicate the good scalability of MA2QL, though the difficulty of the map varies.",
"It is worth noting that recent study [17] shows that IQL performs well in SMAC, even close to CTDE methods like QMIX [19].",
"Here, we show that MA2QL can still outperform IQL in three maps with various difficulties, which indicates that MA2QL can also tackle the non-stationarity problem and bring the performance gain in more complex tasks."
],
[
"Conclusion and Discussion",
"In the paper, we propose MA2QL, a simple yet effective value-based fully decentralized cooperative MARL algorithm.",
"MA2QL is theoretically grounded and requires the minimal changes to independent Q-learning.",
"Empirically, we verify MA2QL in a variety of cooperative multi-agent tasks, including a cooperative stochastic game, MPE, multi-agent MuJoco, and SMAC.",
"The results show that, in spite of such minimal changes, MA2QL outperforms IQL in both discrete and continuous action spaces, fully and partially observable environments.",
"MA2QL makes the minimal changes IQL and improves IQL.",
"MA2QL also has the convergence guarantee, yet is limited to Nash equilibrium in tabular cases.",
"As a Dec-POMDP usually has many Nash equilibria, the converged performance of MA2QL may not be the optimal as shown in the stochastic game.",
"Nevertheless, learning the optimal joint policy in fully decentralized setting is still an open problem.",
"In the stochastic game, we see that IQL also converges, though much slower than MA2QL.",
"This indicates that IQL may also have the convergence guarantee under some conditions, which however is not well understood.",
"We believe decentralized learning for cooperative MARL is an important and open research area.",
"However, much less attention has been paid to decentralized learning than centralized training with decentralized execution.",
"This work may provide some insights to further studies of decentralized learning."
],
[
"MPE",
"The three tasks are built upon the origin MPE [14] (MIT license) and [2] (MIT license) with sparse reward, more agents, and larger space.",
"These tasks are more difficult than the original ones.",
"In the original tasks, we found IQL and MA2QL perform equally well, which corroborates the good performance of IQL on simple MPE tasks [17].",
"Specifically, we change the dense reward setting to the sparse reward setting that will be described below.",
"We increase the number of agents from 3 to 5 in the task of simple spread and enlarge the range of initial positions from [-1, 1] to [-2, 2].",
"A larger space would require the agents to explore more and the agents would be easier to get stuck in sub-optimum.",
"As there is no distance signal in the sparse reward setting, we add a boundary penalty into the environment.",
"If the agents move outside the boundaries, the environment would return the negative reward of -1.",
"The boundaries in our experiment are [-3, 3].",
"Simple spread.",
"In this task, the agents need to find some landmarks in a 2D continuous space.",
"In the sparse reward setting, only if the agents cover the landmarks, the environment would return the positive reward.",
"There is no reward after the landmarks are covered.",
"Whether the landmark is covered by the agent is based on the distance between the landmark and the agent.",
"We set a threshold of 0.3 in our experiment.",
"If the distance is smaller than the threshold, we consider the landmark is covered by the agent.",
"We set 5 agents and 4 landmarks in this task.",
"Fewer landmarks require the agents to do more exploration in the space to find the landmarks.",
"Line control.",
"In this task, there are 5 agents in the space.",
"The goal of the task is that the agents need to arrange themselves in a straight line.",
"The reward depends on the number of agents in the straight line.",
"If the straight line is formed by the agents of 5, the environment would return the reward of 5.",
"Since the action space is discrete, we set a threshold of 15$^\\circ $ to judge whether it is a line.",
"Circle control.",
"There are 7 agents and 1 landmark in the space.",
"This task is similar to the task of line control, while the agents are asked to form a circle where the landmark is the center.",
"We also set a threshold of 0.15."
],
[
"Multi-Agent MuJoCo",
"Multi-agent MuJoCo [18] (Apache-2.0 license) is built upon the single-agent MuJoCo [28].",
"A robot can be considered as the combination of joints and body segments.",
"These joints and body segments in a robot can be regarded as the vertices and connected edges in a graph.",
"We can divide a graph into several disjoint sub-graphs and a sub-graph can be considered as an agent.",
"We can design new observation and action spaces for each agent based on the division.",
"For example, the robot of Walker2d has 6 joints in total.",
"We can divide joints and body segments into left parts and right parts as two agents.",
"The details about our experiment settings in multi-agent Mujoco are listed in Table REF .",
"The configuration defines the number of agents and the joints of each agent.",
"The “agent obsk” defines the number of nearest agents an agent can observe.",
"Thus, HalfCheetah and Hopper are fully observable, while Walker2D is partially observable.",
"Table: The environment settings of multi-agent MuJoCo"
],
[
"MPE",
"We use DRQN with an RNN layer and MLP as the architecture of Q-network.",
"The RNN layer is composed of a GRU with the 64-dimension hidden state.",
"We utilize ReLU non-linearities.",
"All networks are trained with a batch size of 128 and Adam optimizer with the learning rate of 0.0005.",
"There are 16 parallel environments that collect transitions and the replay buffer with the size of 5000 contains trajectories.",
"For exploration, we use $\\epsilon $ -greedy policy and $\\epsilon $ starts from 1.0 and decreases linearly to 0.1 in the first 50000 timesteps.",
"For MA2QL, we define $K$ is the training steps for updating an agent at each turn.",
"For instance, it is the turn to update the agent $i$ .",
"After training $K$ steps, it would change the turn to update the agent $i+1$ .",
"We set $K$ as 30 in this experiment."
],
[
"Multi-Agent MuJoCo",
"We use DDPG for continuous action space and the setting is similar to the one used in OpenAI Spinning Up [1] (MIT license).",
"The architecture of both actor network and critic network is MLP with two hidden layers with 256-dimension hidden state.",
"We utilize ReLU non-linearities except the final output layer of the actor network.",
"The activation function of the final output layer of the actor network is tanh function that can bound the actions.",
"All networks are trained with a batch size of 100 and Adam optimizer with the learning rate of 0.001.",
"There is only one environment that collects transitions and the replay buffer contains the most recent $10^6$ transitions.",
"For exploration, the uncorrelated, mean-zero Gaussian noise with $\\sigma =0.1$ is used.",
"For MA2QL, we set $K$ as 8000.",
"Unlike other two environments where agents learn on trajectories, in multi-agent MuJoCo, agents learns on experiences.",
"Thus, in each turn, the agent can be updated every environment step and hence $K$ is set to be large."
],
[
"SMAC",
"Our code is based on the [9] (Apache-2.0 license) and the version of SMAC is SC2.4.10.",
"The architecture of Q-network is DRQN with an RNN layer and MLP.",
"The RNN layer is composed of a GRU with the 64-dimension hidden state.",
"All networks are trained with a batch size of 32 and Adam optimizer with the learning rate of 0.0005.",
"All target networks are updated every 200 training steps.",
"There are 2 parallel environments that collect transitions and the replay buffer with the size of 5000 contains trajectories.",
"For exploration, we use $\\epsilon $ -greedy policy and $\\epsilon $ starts from 1.0 and decreases linearly to 0.05 in the first 100000 timesteps.",
"We add the action of last timestep into the observation as the input to the network.",
"For MA2QL, we set $K$ as 50.",
"Figure: Learning curves of MA2QL with different KK in 3-agent Hopper, compared with IQL, where x-axis is environment steps."
],
[
"Resources",
"We perform the whole experiment with a total of ten Tesla V100 GPUs."
],
[
"Ablation Study",
"In this section, we study the effect of $K$ in the robotic control task: 3-agent Hopper.",
"We consider $K=[4000, 8000, 40000]$ .",
"As shown in Figure REF , when $K$ is small, it outperforms IQL but still gets stuck in sub-optimal policies.",
"On the contrary, if $K$ is large, different $K$ affects the efficiency of the learning, but not the final performance."
],
[
"Additional Comparison with IPPO in Multi-Agent MuJoCo",
"As MA2QL is derived from Q-learning and thus a value-based method, we focus on the comparison with IQL.",
"Here we provide the results additionally compared with an actor-critic method, i.e., independent PPO (IPPO) [4], in the experiment of multi-agent MuJoCo.",
"The setting and the architecture of IPPO are also similar to the one used in OpenAI Spinning Up [1] (MIT license).",
"As illustrated in Figure REF , MA2QL also consistently outperforms IPPO."
],
[
"Additional SMAC Results",
"We also demonstrate the results on more maps in SMAC.",
"As shown in Figure REF , IQL shows strong performance in these maps, which corroborates the good performance of IQL in SMAC [17].",
"MA2QL performs similarly to IQL on the maps of $\\mathtt {3s5v}$ and $\\mathtt {8m}$ , but better on the map $\\mathtt {2c\\_vs\\_64zg}$ .",
"Together with the results in Section REF , we can see that although IQL performs well in SMAC, MA2QL can still bring performance gain in more difficult maps, which again verifies the superiority of MA2QL over IQL.",
"Figure: Learning curve of MA2QL compared with IQL in 15-agent simple spread in MPE."
],
[
"Scalability of MA2QL",
"We demonstrate the result of 15-agent simple spread in MPE.",
"As illustrated in Figure REF , MA2QL brings large performance gains over IQL.",
"More agents mean the environments become more complex and unstable.",
"IQL is easy to get stuck by the non-stationarity problem while MA2QL can handle it well.",
"The results show again that alternate learning of MA2QL can sufficiently stabilize the environment during the learning process.",
"It also indicates the good scalability of MA2QL.",
"Figure: Learning curve of MA2QL compared with IQL with different learning rates in simple spread in MPE and different batch sizes in HalfCheetah in multi-agent MuJoCo, where the default learning rate in MPE is 0.0005 and the default batch size in multi-agent MuJoCo is 100."
],
[
"Study on Hyperparameters of IQL",
"As MA2QL is essentially an add-on for IQL, the hyperparameters are the same for IQL and MA2QL.",
"In the experiments, the hyperparameters of IQL are taken from previous studies as aforementioned for well-tuned performance, and we do not tune the hyperparameters of IQL for MA2QL.",
"To further study the effect of hyperparameters of IQL on MA2QL, we conduct additional experiments in simple spread with different learning rates and HalfCheetah with different batch sizes.",
"As shown in Figure REF , under these different hyperparameters, the performance of IQL and MA2QL varies, but MA2QL consistently outperforms IQL, which can be evidence of the gain of MA2QL over IQL is insensitive to the hyperparameters of IQL."
],
[
"Alternate Update of IPPO",
"The principle of alternate update can also be applied to independent actor-critic methods.",
"Here we additionally provide the empirical results of alternate update of IPPO (termed as MA2PPO) in the cooperative stochastic game and multi-agent MujoCo.",
"For the cooperative stochastic game, as illustrated in Figure REF , MA2PPO substantially outperforms IPPO.",
"For multi-agent MujoCo, as shown in Figure REF , MA2PPO brings performance gain in HalfCheetah and performs comparably with IPPO in Hopper and Walker2D.",
"These results may suggest that such a simple principle can also be applied to independent actor-critic methods for better performance.",
"A more thorough investigation on independent actor-critic methods is beyond the scope of this paper and left as future work.",
"Figure: Learning curves of MA2PPO compared with IPPO in the cooperative stochastic game.Figure: Learning curves of MA2PPO compared with IPPO in 2-agent HalfCheetah, 3-agent Hopper and 3-agent Walker2d in multi-agent MuJoCo, where x-axis is environment steps.Figure: Learning curves of MA2PPO-CTDE compared with MAPPO on the map 3m in SMAC."
],
[
"Alternate Update in CTDE",
"Though the principle of alternate update can still be applied to CTDE methods, there may not be many benefits.",
"In the CTDE setting, there are actually a number of algorithms that have convergence guarantee.",
"Without the advantage of convergence, alternate update will actually make the learning procedure slower.",
"We modified the CTDE algorithm MAPPO [33] with alternate update (termed as MA2PPO-CTDE) and compare it with MAPPO in the simple map 3m in SMAC.",
"The empirical result is illustrated in Figure REF .",
"We can find that MA2PPO-CTDE obtains low performance while MAPPO has the performance which is close to the optimum.",
"Though alternate update may become better after a longer training, alternate update is not quite meaningful in CTDE."
]
] | 2209.08244 |
[
[
"The Interstellar Interlopers"
],
[
"Abstract Interstellar interlopers are bodies formed outside of the solar system but observed passing through it.",
"The first two identified interlopers, 1I/`Oumuamua and 2I/Borisov, exhibited unexpectedly different physical properties.",
"1I/`Oumuamua appeared unresolved and asteroid-like whereas 2I/Borisov was a more comet-like source of both gas and dust.",
"Both objects moved under the action of non-gravitational acceleration.",
"These interlopers and their divergent properties provide our only window so far onto an enormous and previously unknown galactic population.",
"The number density of such objects is $\\sim$ 0.1 AU$^{-3}$ which, if uniform across the galactic disk, would imply 10$^{25}$ to 10$^{26}$ similar objects in the Milky Way.",
"The interlopers likely formed in, and were ejected from, the protoplanetary disks of young stars.",
"However, we currently possess too little data to firmly reject other explanations."
],
[
"Background",
"Tiny solid particles of dust carry about 1% of the mass of the interstellar medium.",
"Formed in the expanding, unstable atmospheres of evolved stars, these particles have characteristic sizes $\\sim $ 0.2 to 0.3 $\\mu $ m or less.",
"Until recently, the existence, abundance, and nature of possible macroscopic (meter-sized and larger) interstellar bodies has been a matter for speculation only.",
"With the discovery of 1I/`Oumuamua and 2I/Borisov, two sub-kilometer bodies passing through the solar system but formed elsewhere, the new population of macroscopic interstellar interlopers has finally been revealed.",
"The large sizes of the interlopers imply formation in dense environments, presumably the protoplanetary disks where interstellar dust hierarchically accumulates to form planets.",
"Gravitational scattering by growing planets can excite the velocity dispersion of planetesimals, launching some beyond the control of the parent star.",
"Judging from our own solar system, most of the mass of the protoplanetary disk, and most of the material ejected to the interstellar medium by planetary scattering, originated in the cold regions beyond the H$_2$ O snowline.",
"This freezeout line is currently close to the orbit of Jupiter in our solar system.",
"The ice-rich material draws an immediate parallel between the expected properties of the interstellar bodies and those of the comets native to the Solar System, which we briefly discuss here.",
"Solar system comets are divided into two distinct dynamical types corresponding to two distinct long-term storage reservoirs.",
"The short-period comets (SPCs), with modest eccentricities ($e <$ 1) and inclinations, are derived from the trans-Neptunian Kuiper belt (specifically, from the scattered disk component of the Kuiper belt) by long-term chaotic instabilities [185].",
"The mass of the present-day Kuiper belt is $\\sim $ 0.1 M$_{\\oplus }$ (reduced from an estimated starting mass of $\\sim $ 20 M$_{\\oplus }$ or 30 M$_{\\oplus }$ ).",
"The belt contains $\\sim 10^5$ bodies larger than 100 km and perhaps $\\sim 10^{10}$ of kilometer-size.",
"Some Kuiper belt objects (the so-called “Cold Classical” objects) likely formed in-situ [152], [145], while other components of the belt are suspected to have formed at smaller distances and were subsequently scattered out gravitationally.",
"Figure: The distribution of osculating orbital eccentricities of LPCs (blue), SPCs (pink) and ISOs (purple).",
"The dashed grey line indicates e=1e=1, separating bound from unbound orbits.",
"The LPCs cluster at e≃1e\\simeq 1.",
"The eccentricities of 1I and 2I are indicated with black and purple dotted lines, respectively.In contrast, the long-period comets (LPCs) have an isotropic distribution of inclinations, large semimajor axes, and eccentricities, $e \\sim $ 1 (Figure REF ).",
"Their source is the Oort cloud, a 50,000 AU scale gravitationally bound swarm estimated to contain $N_{OC} \\sim 10^{11}$ [54] to 10$^{12}$ [24], [40] kilometer-sized comets, as first recognized by [147].",
"Although values for the total mass of the Oort cloud $\\sim $ 1 M$_{\\oplus }$ are widely quoted, masses up to $\\sim $ 20 M$_{\\oplus }$ [54] are allowed by the data.",
"This is because the sizes and size distribution of LPC nuclei are poorly known, and because the inner Oort cloud (semimajor axes $\\lesssim $ few $\\times 10^3$ AU) might contain substantial mass and yet go undetected, because it is relatively immune to external perturbations (but see [99]).",
"Even during the earliest stages of planetary formation, typical densities at Oort cloud distances were too low for comets to have formed in situ there.",
"Instead, the LPCs likely formed in the region now occupied by the giant planets, whose growth lead to the gravitational scattering of many comets into highly eccentric and unbound ($e >$ 1) orbits.",
"A fraction of those ejected comets, 0.01 $\\lesssim f \\lesssim $ 0.1 [68], [23], became captured (by external perturbations from nearby stars, perhaps in the Sun's birth cluster [40], and by the galactic tide) into circularized orbits with raised perihelia.",
"Once removed from the gravitational control of the planets, the inclinations of these captured orbits were randomized over the course of several Gyr, transforming the initial disk-like distribution into the spherical, modern-day Oort cloud [78].",
"Continuing external perturbations on the Oort cloud supply LPCs back to the planetary region, while dispersing others to interstellar space.",
"The remaining $(1-f)\\simeq $ 90% - 99% of the ejected objects were lost by the Sun to the interstellar medium.",
"For example, taking $f$ = 0.01, as many as $N_{OC}/f \\sim $ 10$^{13}$ to 10$^{14}$ comets were ejected from the solar system alone.",
"Given $N_{\\star } \\sim 10^{11}$ stars in the galaxy, and assuming that all stars have Sun-like comet populations, a first order guess as to the number of ejected comets in the galaxy is $N_{ii} = N_{\\star } N_{OC}/f \\sim $ 10$^{24}$ to 10$^{25}$ .",
"The combined mass of this population is a large but very uncertain $\\sim $10$^6$ M$_{\\odot }$ to $\\sim $10$^7$ M$_{\\odot }$ (assuming a nominal 1 km radius, as is typical of the measured comets).",
"Depending on the nucleus size distribution, the total ejected mass could be much larger, as could the numbers of ejected sub-kilometer comets.",
"These simplified estimates are necessarily extrapolations based on limited knowledge.",
"As discussed previously, the fraction of scattered comets that remained bound to the Solar System is uncertain by an order of magnitude.",
"Moreover, the analogous fraction for other stars depends sensitively on the architecture of their planetary systems and on the star cluster environment when they formed.",
"The ejection and capture efficiencies must therefore differ from star to star, perhaps by large factors.",
"Nevertheless, the serendipitous discoveries of 1I/`Oumuamua and 2I/Borisov prove the existence of a vast galactic population of macroscopic interlopers beyond reasonable doubt.",
"There is explosive interest in both the physical nature and the possible origins of the galactic reservoir of interstellar interlopers.",
"Figure: (left:) 1I/`Oumuamua showing point-like appearance on UT 2017 October 26 at the 2.5 m Nordic Optical Telescope (right:) 2I/Borisov showing cometary activity on UT 2019 October 12 from the 2.4 m Hubble Space Telescope."
],
[
"Dynamical Properties",
"The basic dynamical properties of the first two interstellar interlopers are summarized in Table REF while their appearances at optical wavelengths are shown in Figure REF .",
"Table: Dynamical Properties of the Interstellar Interlopers a,b ^{a,b}"
],
[
"Discovery and Orbits",
"1I/`Oumuamua The first confirmed interstellar object (formerly C/2017 U1) was discovered by Robert Weryk on 2017 October 19 from the summit of Haleakalā on Maui, Hawaii (announced as 2017 U1 in [190]).",
"Observations soon revealed that the orbit was hyperbolic.",
"Specifically, the object had an osculating eccentricity $e =$ 1.201, retrograde inclination, $i$ = 122.8 deg, and a perihelion distance $q =$ 0.256 AU.",
"(Removal of planetary perturbations gives only slightly different pre-entry barycentric orbital elements, as listed in Table REF ).",
"Figure REF , shows the trajectory of `Oumuamua and its position on 2017 October 17.",
"`Oumuamua was discovered close to the Earth, passing within only about 0.16 AU three days prior to discovery.",
"This serendipitous detection near the peak of ground-based visibility implies the existence of an unnoticed population of similar objects with less favorable observing geometries.",
"1I/`Oumuamua was point-like in all observations (Figure REF ).",
"Figure: The trajectory of 1I/`Oumuamua near perihelion, from 2017 August to December.",
"The positions of the Earth and `Oumuamua at discovery on October 17 are indicated.",
"Even black points indicate the positions of the Earth and `Oumuamua at time intervals of one month.",
"Arrows indicate the directions of motion of both objects through their orbits.2I/Borisov: The second known interstellar interloper, 2I/Borisov (also known as C/2019 Q4), was discovered on UT 2019 August 30 when only 38$^{\\circ }$ from the Sun.",
"This remarkable observation was made by Gennadiy Borisov who used a 0.65 m self-built telescope to target an area of the sky barely examined by other survey telescopes.",
"Its orbit is robustly hyperbolic with eccentricity $e = 3.358$ , and consequently the interstellar origin of 2I is not in doubt.",
"`Oumuamua was discovered outbound from perihelion only by virtue of its close approach to Earth, and it was only observable for a short time-span.",
"In sharp contrast – and fortunately – 2I/Borisov was discovered about three months before perihelion (UT 2019 December 8 at $q$ = 2.00 AU) because of its bright coma of ejected dust (Figure REF ).",
"The early discovery and intrinsic brightness of 2I/Borisov enabled physical and astrometric observations to be obtained for about 1 year, whereas `Oumuamua faded beyond detection in only 2.5 months.",
"The trajectory of 2I/Borisov is shown schematically in Figure REF .",
"Figure: The trajectory of 2I/Borisov.",
"The positions of the Earth and 2I/Borisov at discovery are marked, with month-separated points as in Figure ."
],
[
"Non-Gravitational Acceleration",
"1I/`Oumuamua: Observations over a 2.5 month arc allowed an accurate determination of the orbit of `Oumuamua and unexpectedly revealed the existence of non-gravitational acceleration.",
"In active comets, non-gravitational acceleration is caused by recoil from the asymmetric ejection of mass.",
"Its detection in 1I/`Oumuamua was especially puzzling because observations provided no evidence for mass loss at any level.",
"By convention, such accelerations are described by three orthogonal components, $A_1, A_2, A_3$ , each expressed as AU day$^{-2}$ .",
"Here, $A_1$ is the acceleration away from the Sun, $A_3$ is perpendicular to the orbit plane and $A_2$ completes the right-handed triple.",
"The total acceleration at $r_H$ = 1 AU is computed from $\\alpha _{ng}(1) = g(r_H) (A_1^2 + A_2^2 + A_3^3)^{1/2}$ and is expressed in m s$^{-2}$ (see Table REF ).",
"The quantity $g(r_H)$ is a function to represent the radial dependence of the mass loss, normalized such that $g(1$ AU) = 1 [122].",
"Table REF lists the accelerations provided by JPL Horizonshttps://ssd.jpl.nasa.gov/horizons/app.html and shows that $A_1$ is the dominant component.",
"This is consistent with a force acting radially away from the Sun, as expected of outgassing recoil (see Section REF ) and radiation pressure (Section REF ).",
"The magnitude of the acceleration, $\\alpha _{ng}(1)$ , corresponds to about $10^{-3}$ times the local solar gravitational acceleration at 1 AU, and is three to four orders of magnitude larger than typical accelerations in solar system comets (Figure REF ).",
"However, when considering the relatively small size of `Oumuamua, the non-gravitational force is lower than those observed in typical comets.",
"2I/Borisov: A 10 month astrometric series also revealed non-gravitational acceleration in 2I/Borisov, with $A_1$ again being the dominant component and with a smaller magnitude (Table REF ).",
"Unlike the case of 1I/`Oumuamua, the non-gravitational acceleration of 2I/Borisov was matched by obvious, on-going mass loss and the cause is not mysterious.",
"Figure: (left panel) Dashed lines show the measured non-gravitational accelerations of 1I/`Oumuamua and 2I/Borisov compared to values for short-period (pink) and long-period (blue) comets.",
"(right panel) Non-gravitational force computed assuming spherical nuclei and a bulk density ρ=500\\rho =500 kg/m 3 ^3.",
"For the interlopers, we use object radii as listed in Table (); 2I is represented by a range to reflect the uncertainty of the radius.",
"Non-gravitational accelerations and nucleus diameters are taken from the JPL Small Body Database.",
"The sizes (hence, masses) of long-period comets are largely unknown.Figure REF compares the non-gravitational accelerations and forces acting on the comets and interlopers.",
"Comparison of the left and right panels shows that, while the acceleration of `Oumuamua is larger than in most comets, this is simply because it is very small (acceleration is $\\propto r_n^{-1}$ ).",
"When plotted as force, 1I is on the weaker side of the comet distribution, consistent with a low rate of mass loss.",
"The larger nucleus of 2I/Borisov shows a high but less extreme value of $\\alpha _{ng}(1)$ , but is unremarkable on the histogram of forces in the figure.",
"In both interlopers, $\\alpha _{ng}$ varies with heliocentric distance approximately as $r_H^{-2}$ , although the heliocentric dependence is not well measured because of the limited range of distances over which astrometric observations were secured."
],
[
"Physical Properties",
"Key physical properties of `Oumuamua and 2I/Borisov are summarized in Table REF and more fully discussed in the following text.",
"Table: Physical Properties of the Interstellar Interlopers a ^a"
],
[
"Nucleus Sizes",
"The apparent magnitude of an object viewed in reflected sunlight, $V$ , is related to the albedo, $p$ , and effective radius, $r_n$ by [163] $p \\,\\Phi (\\alpha ) \\, r_n^2 = \\,\\bigg (\\,2.25\\times 10^{22} \\,\\bigg )\\,r_H^2\\, \\Delta ^2 \\,10^{0.4(V_{\\odot } - V)}\\,,$ where $r_H$ and $\\Delta $ are the heliocentric and geocentric distances in AU, $\\alpha $ is the phase (Sun-object-observer) angle, and $V_{\\odot }$ is the apparent magnitude of the Sun.",
"The phase function, $\\Phi (\\alpha )$ , represents the dimming of the object observed at phase angle $\\alpha $ relative to $\\alpha $ = 0.",
"Given only optical observations, the number of unknowns ($p$ , $r_n$ and $\\Phi (\\alpha )$ ) in Equation REF exceeds the number of constraints.",
"Therefore, there is ambiguity in estimates of $r_n$ depending on the assumed values of $p$ and $\\Phi (\\alpha )$ .",
"Additional complications arise because published observations use different filters, and assume different phase functions and geometric albedos.",
"Albedos of fresh icy surfaces tend to be large, $p \\sim $ 0.8 - 0.9, while the carbonaceous surfaces of primitive asteroids tend to be small, $p \\sim $ 0.04.",
"To proceed, we scale relevant properties assuming that $p$ = 0.1.",
"With this albedo, the effective nuclear radius of `Oumuamua lies in the range 55 m [94], 70$\\pm $ 3 m [131], 80 m [42] to 114 m [104].",
"With no objective way to decide amongst these estimates, we take a middle value, $r_n =$ 80 m, as our best estimate of the effective radius, included in Table REF , and note that this value is uncertain by a factor of order two.",
"The bright coma prevented direct detection of the nucleus of 2I/Borisov.",
"However, [91] constrained the nuclear radius to lie within the range 0.2 $\\le r_n \\le $ 0.5 km.",
"The upper limit is set by the non-detection of the nucleus in high resolution surface brightness data.",
"The lower limit is derived from the non-gravitational acceleration assuming a comet-like bulk density $\\rho _n$ = 500 kg m$^{-3}$ .",
"We adopt $r_n$ = 400 m as the nominal value."
],
[
"Colors",
"Broadband colors are not compositionally diagnostic, but they do provide another metric with which to compare the interlopers with other solar system bodies.",
"The conventional reference point is provided by the color of the Sun, which has optical color index B-R = 0.99$\\pm $ 0.02.",
"For objects in which the albedo varies linearly with wavelength, as is commonly the case in distant solar system bodies, it is useful to use the reflectivity gradient, defined as the fractional change in the brightness per unit wavelength, $S^{\\prime }$ [% (1000Å)$^{-1}$ ], relative to that of the Sun.",
"The Sun, by definition, has $S^{\\prime }$ = 0 % (1000Å)$^{-1}$ .",
"Objects with B-R $>$ 1.6, corresponding to $S^{\\prime } \\sim $ 25% (1000Å)$^{-1}$ , are described as “ultrared”, and widely suspected to consist of irradiated, carbon-rich material [96].",
"Figure: Color distributions of minor bodies in the solar system compared to those of `Oumuamua and 2I/Borisov (shown as vertical dotted lines).Slope S ' S^{\\prime } = 0 % (1000Å) -1 ^{-1} corresponds to the solar color, B-R = 0.99, while S ' ∼S^{\\prime } \\sim 25 % (1000Å) -1 ^{-1} corresponds to B-R = 1.6.",
"Colors are from and .1I/`Oumuamua: Published optical measurements of the color of `Oumuamua consistently show a surface reddened with respect to the Sun, but with wide scatter.",
"This scatter reflects the difficulty of obtaining measurements for such a highly variable, rapidly moving target.",
"The reflectivity increases linearly with wavelength from at least 4500Å to 10,500Å[52], [131].",
"Values of the spectral slope range from $S^{\\prime } =$ 7$\\pm $ 3%/1000Å [94] to 23$\\pm $ 3%/1000Å [131]; we adopt a middle value from independent measurements near $S^{\\prime } =$ 15$\\pm $ 5%/1000Å.",
"2I/Borisov: Whereas the colors of 1I/`Oumuamua refer to the bare nucleus, those of 2I/Borisov measure the color of coma dust, with negligible contribution from the nucleus.",
"As with the case of `Oumuamua, reported measurements of the color of 2I show a reddened optical continuum with a scatter larger than the formal measurement errors [93], [67], [21], [84], [125].",
"Some of these differences in measurements may be attributed to the usage of a variety of filters.",
"Specifically, it is known that the U and B filters may be contaminated by resonance fluorescence bands from gas.",
"The continuum color of 2I/Borisov might also have varied with time and distance from the Sun [125].",
"We adopt a nominal spectral slope $S^{\\prime } \\sim $ 12$\\pm $ 1% (1000Å)$^{-1}$ .",
"In Figure REF , we show the color distributions (expressed as $S^{\\prime }$ ) of different populations of minor bodies in the solar system.",
"These data are drawn from the online updated databaseMBOSS Database: https://www.eso.org/~ohainaut/MBOSS/ originally described in [69].",
"We also indicate the positions of `Oumuamua and Borisov with respect to these distributions.",
"In the inner solar system, the reddish colors of asteroids are due to an abundance of nano-phase (1 to 100 nm) iron produced in surface materials by energetic particle bombardment (“space weathering”).",
"Optical colors in the outer solar system can be much redder.",
"Ultrared colors probably reflect the presence of irradiated, macro-molecular carbon compounds that are unstable or otherwise depleted in the warm, inner solar system.",
"The Cold Classical KBOs are the least dynamically evolved objects in the belt and located at about 43 AU.",
"They also exhibit a larger fraction of ultra-red surface colors compared to most other small bodies in the solar system, with the exception of the Centaurs (recently escaped Kuiper belt objects).",
"This difference is not understood, and may be attributed to longer exposure to space weathering if the Cold Classicals formed in situ.",
"The colors of 1I and 2I are similar to each other and to most inner solar system populations (see the vertical dotted lines in Figure REF ).",
"The lack of distinctive evidence for ultra-red matter, expected from long-term exposure to the interstellar environment, is consistent with the colors of LPCs.",
"Figure: Images (upper row) and dust dynamics models (lower row) of 2I/Borisov as a function of date.",
"The models indicate a coma consisting of large (effective size ∼\\sim 1 mm), slowly ejected (V∼V \\sim 2 m s -1 ^{-1}) in steady-state.",
"Small particles are more strongly swept by radiation pressure, creating a coma and tail unlike those observed.",
"Adapted from ."
],
[
"Activity",
"1I/`Oumuamua: The morphology of `Oumuamua was at all times point-like, even in the highest resolution data from the Hubble Space Telescope and in the deepest data from large ground-based telescopes.",
"The optical data provide no evidence for outgassing activity.",
"The optical scattering cross-sections of typical cometary comae are dominated by dust, with minor contributions from resonance fluorescence from gas molecules.",
"Accordingly, measurements of the surface brightness profile of `Oumuamua were used to place model-dependent limits on the mass loss in micron-sized dust particles; $\\dot{M} \\le 2\\times 10^{-4}$ kg s$^{-1}$ [94], $\\dot{M} \\le 2\\times 10^{-3}$ kg s$^{-1}$ [131] (Table REF ).",
"Both upper limits are orders of magnitude smaller than the 10$^2$ kg s$^{-1}$ to 10$^3$ kg s$^{-1}$ mass loss rates from typical near-Sun comets estimated in the same way.",
"Limits on the production of water, the dominant cometary volatile, are $\\dot{M} \\le $ 30 kg s$^{-1}$ , as determined from radio lines of the dissociation product, OH (see Section REF ).",
"2I/Borisov: In sharp contrast to `Oumuamua, 2I displayed (and, indeed, was discovered because of) obvious cometary activity in the form of an extended optical dust coma (Figure (REF ) [93], [67].",
"Dust dynamics models reproduce the slowly changing morphology of the comet (Figure REF ) and show that the coma is dominated by sub-millimeter and larger particles [103], [193].",
"Total dust production rates estimated from imaging data alone range from $\\dot{M} \\sim $ 2 kg s$^{-1}$ [91] to $\\dot{M} \\sim $ 35 kg s$^{-1}$ [31], [103].",
"Less model-dependent spectroscopic production rates gave gas production rates up to $\\dot{M}$ = 20 to 40 kg s$^{-1}$ .",
"Continued activity in 2I/Borisov included a photometric outburst [41] and subsequent break-up, illustrated in Figure REF [92].",
"The post-perihelion breakup might be related to a seasonal response, the likes of which are commonly observed in solar system comets [103].",
"Figure: Spatially filtered (coma suppressed) Hubble Space Telescope images of 2I/Borisov showing the appearance of a split nucleus on UT 2019 March 30.",
"Each panel shows a region 0.44\" wide corresponding to about 800 km at the comet.",
"Note that the resolution of the telescope projects to ∼150\\sim 150km at the 2.65 AU geocentric distance of 2I/Borisov on March 30.",
"From ."
],
[
"Lightcurves",
"1I/`Oumuamua: Photometric observations immediately showed that `Oumuamua had an extremely large lightcurve peak-to-peak amplitude of $\\Delta V \\sim $ 2.5 magnitudes, corresponding to a factor of $\\sim $ 10 in brightness and a stable period close to 8 hours [131], [94], [104], [9], [22], [42].",
"The lightcurve is shown in Figure REF , adapted from the data in Figure 1 of [11].",
"While the period of `Oumuamua is unremarkable compared with the distribution of rotational periods of the asteroids, the brightness variations are extreme (Figure REF ), as will be discussed below.",
"Lightcurves of atmosphereless solar system bodies are caused by azimuthal shapes or surface albedo variations modulated by rotation.",
"To first order, the shape of an irregular body projected into the sky-plane can be approximated as an ellipsoid with semi-axes $a \\times b$ .",
"The “effective radius”, equal to the radius of a circle having equal area, is simply $r_n = (ab)^{1/2}$ (Table REF ).",
"At opposition (Sun-target-Earth angle = 0o), the axis ratio $b/a$ is related to the apparent lightcurve amplitude, $\\Delta V$ in magnitudes, by $b/a = 10^{0.4\\Delta V}$ .",
"Substitution gives $b/a$ = 10:1 for `Oumuamua [131].",
"However, most observations of `Oumuamua were obtained at phase angles $\\alpha \\sim $ 20o, where illumination and self-shadowing effects act to magnify the lightcurve amplitude relative to that at zero phase [98], [106], [115].",
"Taking these effects into account, the projected axis ratio of `Oumuamua corresponding to $\\Delta V$ = 2.5 magnitudes is reduced to $b/a$ = 6:1 or 7:1 [94], [42], [128].",
"This is still an extreme shape compared to known solar system bodies (c.f.",
"Figure REF ).",
"The shapes of the asteroids are controlled by repeated energetic and sometimes disruptive collisions.",
"Even for these violent cases, the average projected axis ratios are $b/a \\sim $ 1.4:1 and values $b/a \\ge $ 2 are rare.",
"With $b/a$ = 6:1 and effective radius $r_n$ = 80 m, the projected sky-plane shape of `Oumuamua is an $a \\times b$ = 32$\\times $ 196 m ellipse.",
"It is worth noting that the NEO 2016 AK$_{193}$ exhibited brightness variations of $\\sim 2.5-3$ magnitudes during its discovery apparition and subsequent followup observations [77].",
"Elongated solar system bodies are typically prolate in shape and rotating around a minor axis.",
"A large lightcurve amplitude can also be produced by an oblate body rotating around a long axis.",
"Detailed numerical and analytic calculations performed by [123] demonstrated that the most likely shape of `Oumuamua was a 6:6:1 oblate ellipsoid, as opposed to the 6:1:1 prolate geometry popularized in several solicited artist impressions.",
"Statistically, a randomly oriented oblate body with a given axis ratio is more likely than its prolate counterpart to yield a large rotational amplitude consistently during each revolution.",
"It is apparent in Figure REF that the object exhibited consistently deep brightness minima.",
"In this sense, we should assume that `Oumuamua is a flattened, disk-like body, not an elongated, cigar-shaped one.",
"Energy dissipation favors relaxation to the minimum energy (maximum moment of inertia) rotational state, which would work in favor of a prolate body shape, but the timescale for this relaxation depends on unknown physical parameters of the body and is suspected to be very long [42].",
"Figure: The photometric lightcurve of `Oumuamua, similar to that computed by .",
"This figure uses the photometric data presented by , , , , , and .",
"The points are color coded based on the observational facility that obtained the data.Other Interpretations It is possible that the lightcurve of `Oumuamua could result from azimuthal albedo variations instead of from projected shape variations.",
"In solar system bodies, however, azimuthal albedo variations are almost always so small as to be immeasurable.",
"The physical reason is that the surface material is homogenized, both by space weathering and by gardening (the churning of exposed surface materials by micrometeorite bombardment).",
"These processes, acting together or alone, render the surfaces of almost all asteroids and comets uniform in their scattering properties.",
"The one notable exception to this is provided by Saturn's 1500 km diameter satellite Iapetus, which sports hemispheric $\\sim $ 10:1 albedo variations.",
"Iapetus is a special case, however, because the satellite is in a spin orbit resonance, exposing one hemisphere to the impact of debris from another Saturnian satellite while the other hemisphere survives unscathed [178].",
"It is difficult to imagine how any comparable asymmetry could arise over the surface a body when free-floating in interstellar space.",
"Therefore, although the possibility that the lightcurve of `Oumuamua is caused by extreme azimuthal albedo variations cannot be formally rejected, it seems less plausible than an origin in the shape of the body.",
"2I/Borisov: The optical cross-section of 2I/Borisov was dominated by dust in the coma, preventing the separation of the signal from the underlying nucleus.",
"For this reason, the rotational lightcurve could not be measured.",
"Photometric variations in the coma instead provide a measure of the dust production rates in the object, subject to corrections for the changing viewing geometry.",
"2I/Borisov was observed over a wide range of phase angles, 40$^o \\lesssim \\alpha \\lesssim $ 90o, necessitating a large and correspondingly uncertain phase correction.",
"Depending on the details of this correction, some observers [93] reported a net increase in dust cross-section on the approach to perihelion (commensurate with increasing gas production rates; Section REF ) while others inferred steady fading [84].",
"Figure: The distribution of asteroid lightcurve amplitudes (left) and rotational periods (right).",
"The measured values for `Oumuamua are shown as dotted black lines (these are unconstrained for Borisov).",
"The top and bottom panels show the distribution for all objects greater than and less than 1km in diameter, respectively.",
"Data from ."
],
[
"Gas Production",
"1I/`Oumuamua: No spectroscopic evidence for volatile outgassing was found in `Oumuamua ([194], [52] and [181]).",
"We summarize the reported upper limits to production rates, $Q$ , in Table REF .",
"The corresponding mass production rates are $\\dot{M} = \\mu m_H Q$ , where $\\mu $ is the molecular weight of the species in question and $m_H$ is the mass of the hydrogen atom.",
"From the Table we compute limits for water, $\\dot{M} \\le $ 30 kg s$^{-1}$ , and for CO, $\\dot{M} <$ 0.04 kg s$^{-1}$ .",
"Table: Upper limits to gas production rates in `Oumuamua.2I/Borisov: In contrast, the spectrum of 2I/Borisov showed distinct gas emission bands rising above its dusty continuum, corresponding to the classical resonance fluorescence features observed in short- and long-period comets.",
"Tables REF and REF summarize the spectroscopic results.",
"The production rate data are also plotted in Figure REF as a function of the time of observation.",
"The figure shows a relatively small variation in the production rates, reflecting the small range of heliocentric distances over which data were obtained.",
"Figure: Time dependence of 2I/Borisov production rates.",
"The top panel shows production rates of H 2 _2O, CO and OH (Table ), and the bottom panel shows production rates of CN, C 2 _2, C 3 _3, NH 2 _2 and HCN (Table ).",
"Adapted from .Table: Production rates of CO, H 2 _2O and OH measured for 2I/Borisov.",
"Adapted from .Table: Production rates of CN, C 2 _2 and C 3 _3 measured for 2I/Borisov.",
"Adapted from .The mass-dominant cometary volatile in long and short period comets, H$_2$ O, was measured systematically in 2I/Borisov by [192].",
"They obtained observations at six epochs before and after perihelion with the Neil Gehrels Swift Observatory's Ultraviolet/Optical Telescope.",
"These data revealed a water production rate peaking near $Q({\\rm H_2O}) = (10.7\\pm 1.2)\\times 10^{26}$ s$^{-1}$ at perihelion, corresponding to mass production rate $\\dot{M}({\\rm H_2O}) =$ 32 kg s$^{-1}$ .",
"Adopting a nominal equilibrium sublimation rate of water ice at 2 AU equal to $f_s = 5\\times 10^{-5}$ kg m$^{-2}$ s$^{-1}$ (obtained from solution of the energy balance Equation REF ), the required area of exposed and sublimating water ice is $C = \\dot{M}({\\rm H_2O})/f_s \\sim 0.64$ km$^2$ .",
"This surface area corresponds to that of a circle with radius $r_n = (C/\\pi )^{1/2} \\sim $ 0.45 km.",
"This “sublimation radius” lies close to the nominal 0.4 km nucleus radius and to the upper limit (0.5 km) obtained by [92] in an independent calculation.",
"These two independent estimates both corroborate each other and imply that the surface of 2I/Borisov had a large active fraction, $f_A \\sim $ 1.",
"This, in turn, matches measurements showing that short-period comet nuclei of size comparable to 2I/Borisov typically have $f_A \\sim 1$ [88].",
"2I/Borisov was also a productive source of carbon monoxide, CO, with a production rate ratio measured on multiple dates to be $Q({\\rm CO}) / Q({\\rm H_2O}) =$ 0.7$\\pm $ 0.3 [30], and 1.3 to 1.6 [20].",
"This is substantially higher than mean values $Q({\\rm CO}) / Q({\\rm H_2O}) =$ 0.04 in solar system comets at distances $\\lesssim $ 2.5 AU [18], [127].",
"The high relative abundance of CO implies low temperature formation of 2I/Borisov in order to trap the CO, presumably in the outer regions of a protoplanetary disk.",
"It also implies low temperature storage since formation in order to retain the CO against sublimation (see section REF )."
],
[
"Physical Models",
"A central puzzle is that 1I/`Oumuamua showed no visible coma in deep composite images, and yet had a non-gravitational acceleration of 30$\\sigma $ significance.",
"These two observations have prompted theories, from the mundane to the fantastical, regarding the provenance of the object.",
"In comparison, the properties of 2I/Borisov were closer to those observed in numerous solar system comets."
],
[
"Recoil from outgassing",
"Non-gravitational accelerations in solar system comets are caused by recoil in response to the sublimation of surface ice in the heat of the Sun, and the subsequent anisotropic mass ejection.",
"The recoil force may be written as $k_R \\dot{M} V_{s}$ , where $\\dot{M}$ is the sublimation rate, $V_{s}$ is the speed of the ejected material, and $0 \\le k_R \\le 1$ is a dimensionless constant representing the degree of anisotropy of the outflow.",
"For perfectly isotropic flow, $k_R$ = 0, while for perfectly collimated outflow, $k_R$ = 1.",
"Newton's law applied to a body of spherical-equivalent radius $r_n$ and bulk density $\\rho _n$ then gives, $\\dot{M} = \\,\\bigg (\\,\\frac{4\\pi \\rho _n r_n^3}{3}\\,\\bigg )\\,\\bigg (\\,\\frac{ \\alpha _{ng}(1)}{k_R V_s}\\,\\bigg )\\,,$ for the mass loss rate needed to generate the acceleration, $\\alpha _{ng}(1)$ .",
"Measurements of comets show that sublimation proceeds mostly from the hot dayside of the nucleus.",
"Correspondingly, the recoil force acts primarily in the radial direction away from the Sun (i.e.",
"along $A_1$ ), as can be seen in Table REF .",
"The magnitude of $k_R$ has only been determined with confidence for the short-period comet 67P/Churyumov-Gerasimenko at $k_R$ = 0.5, and is otherwise not well known.",
"We adopt this value here.",
"The speed of sublimated gas is close to the thermal velocity of gas molecules at the sublimation temperature of ice.",
"For water ice at $r_H \\lesssim $ 2 AU, this is $T \\sim $ 200 K, giving $V_s \\sim $ 500 m s$^{-1}$ , a value we adopt throughout.",
"The average density of solar system cometary nuclei is $\\rho _n \\sim $ 500 kg m$^{-3}$ [62], which we also adopt here.",
"Substitution into Equation REF , with our nominal spherical-equivalent radius estimate for `Oumuamua $r_n$ = 80 m, gives $\\dot{M} \\sim $ 24 kg s$^{-1}$ .",
"For the larger nucleus of 2I/Borisov, with 200 $\\le r_n \\le $ 500 m, mass loss rates $\\dot{M}$ = 400 to 6000 kg s$^{-1}$ (scaled to 1 AU) are needed to account for $\\alpha _{ng}(1)$ .",
"Can these rates be supplied by sublimation of cometary volatiles?",
"The equilibrium rate of sublimation of a volatile surface exposed to the Sun is given by solutions to the following equation, $\\bigg (\\,\\frac{ (1-A)\\,L_{\\odot }}{4\\pi r_H^2}\\,\\bigg )\\, \\cos (\\theta ) =\\, \\varepsilon \\sigma T^4 + H f_s(T) + C(T)\\,.$ Here, $L_{\\odot }$ (W) is the luminosity of the Sun, $r_H$ (m) the heliocentric distance, $A$ and $\\varepsilon $ are the Bond albedo and thermal emissivity of the surface, $\\sigma $ (W m$^{-2}$ K$^{-4}$ ) is the Stefan-Boltzmann constant, $T$ (K) is the surface temperature, $H$ (J kg$^{-1}$ ) is the latent heat of sublimation, and $f_s(T)$ (kg m$^{-2}$ s$^{-1}$ ) is the specific sublimation rate.",
"The angle $\\theta $ is the angle between the surface normal and the direction to the Sun.",
"The term on the left hand side represents the absorbed solar power.",
"The terms on the right hand side account for the thermal radiation, bond breaking in sublimation and conduction into the interior, respectively.",
"The surface materials on solar system bodies tend to be porous and have low thermal conductivity, justifying the neglect of the conduction term $C(T)$ in most cases.",
"The latent heats of water and CO ice are $H = 2.8\\times 10^6$ J kg$^{-1}$ and $H = 2\\times 10^5$ J kg$^{-1}$ , respectively.",
"Equation REF cannot be solved alone, and the Clausius-Clepeyron equation is commonly used in addition to represent the temperature dependence of the sublimation phase boundary in $f_s(T)$ .",
"Moreover, the equation must be solved for a surface divided into elements, each with its own angle, $\\theta $ , to the solar direction.",
"The averaged value of $\\cos (\\theta )$ is as follows: $\\overline{\\cos (\\theta )}$ = 1 for a flat surface oriented normal to the Sun, $\\overline{\\cos (\\theta )}$ = 1/2 for a spherical body sublimating only from the sun-facing hemisphere, and $\\overline{\\cos (\\theta )}$ = 1/4 for a uniformly sublimating sphere.",
"$\\overline{\\cos (\\theta )}$ = 1 represents the highest possible temperature, corresponding to the noon-day equatorial Sun, while $\\overline{\\cos (\\theta )}$ = 1/4 corresponds to the lowest possible temperature on a sublimating, isothermal sphere.",
"The optical properties $A$ and $\\varepsilon $ are generally not well constrained, but Equation REF is insensitive to both provided $A \\ll $ 1 and $\\varepsilon \\gg $ 0.",
"Here, we assume $A$ = 0 and $\\varepsilon $ = 0.9.",
"We adopt a nominal reference distance of $r_H$ = 1 AU and assume that sublimation occurs from the sun-facing hemisphere of a spherical body (i.e.",
"$\\overline{\\cos (\\theta )}$ = 1/2).",
"Under these assumptions, Equation REF gives $f_s$ (H$_2$ O) = 2.2$\\times 10^{-4}$ kg m$^{-2}$ s$^{-1}$ and $f_s$ (CO) = 2.3$\\times 10^{-3}$ kg m$^{-2}$ s$^{-1}$ for the more volatile substance.",
"We emphasize that these values are strictly valid only for sublimation at the surface.",
"A volatile that is protected from direct heat by an overlying layer of less volatile material (as is likely to be the case for highly volatile CO, for example) will sublimate at a smaller but much more model-dependent rate.",
"Production rates of water in solar system comets varies approximately as $r_H^{-2}$ out to $r_H \\sim $ 2 AU.",
"Production rates of CO vary similarly, but out to $r_H \\sim $ 50 AU.",
"This is due to the exponential rise in the sublimation term in Equation REF which is only in competition with the radiation term with $\\sim T^4$ dependence.",
"The model mass loss rate, $\\dot{M}$ , is given by $\\dot{M} = 2 \\pi r_n^2 f_s f_A$ , where $f_A$ is the fraction of the nucleus surface from which ice sublimates.",
"By substituting this into Equation REF , the expected non-gravitational acceleration, $\\alpha _{ng}$ for each volatile may be calculated.",
"Figure: Active sublimation surface fractions measured in solar system comets.",
"Adapted from Figure 2 in .As shown in Figure REF , the active fraction on short-period comets shows a clear trend with nucleus radius [88], represented by a best-fit power law $f_A \\simeq 0.1 \\, r_N^{-2}\\,,$ for $r_n \\gtrsim $ 0.3 km (and $f_A$ = 1 otherwise).",
"This trend is likely produced, at least in part, by observational bias, stemming from the fact that small nuclei with small active fractions are intrinsically faint and less likely to be discovered in flux-limited surveys.",
"We plot and label the estimated radii of 1I and 2I in Figure REF .",
"Both objects are small enough to suggest large active fractions, $f_A \\sim $ 1, by analogy with the SPCs, and we adopt $f_A$ = 1 here.",
"In the case of 1I/`Oumuamua, there are two immediate problems.",
"The first concerns the identity of the sublimating volatile.",
"Water ice sublimates too slowly at 1 AU to supply the $\\dot{M} =$ 24 kg s$^{-1}$ needed to provide $\\alpha _{ng}(1)$ as originally pointed out in an unrefereed preprint by [166].",
"Instead, super-volatile ices (H$_2$ , N$_2$ , CO, Ne, Ar) sublimating in equilibrium with sunlight are required to generate sufficient recoil, as a result of their relatively small latent heats of sublimation.",
"The noble gases have low abundance and are unlikely sources of activity.",
"[168] proposed that 1I/`Oumuamua was composed of solid hydrogen, H$_2$ , presumably formed in the failed prestellar core of a giant molecular cloud (c.f.",
"[111], [80]).",
"However, uncommonly frigid temperatures ($T <$ 6 K) are needed to accrete and retain solid H$_2$ and the survival of such a volatile object in the open interstellar medium is in doubt.",
"Next, [37] and [86] proposed that sublimating N$_2$ could be the cause of the acceleration.",
"Nitrogen has the advantage of being spectroscopically inert, consistent with its non-detection in `Oumuamua.",
"However, in the solar system, large nitrogen ice reservoirs are known to exist only on the surfaces of large, thermally differentiated Kuiper belt objects, with Pluto providing the premier example.",
"It is unlikely that sufficient exposed solid nitrogen exists on differentiated extrasolar Kuiper belt objects to act as a galaxy-wide supply of 1I/`Oumuamua-like bodies [110].",
"Finally, [171] suggested that carbon monoxide (CO) sublimation could supply the recoil acceleration.",
"Equilibrium hemispheric sublimation of CO from an 80 m radius body at 1 AU is indeed sufficient to provide $\\dot{M}$ = 24 kg s$^{-1}$ needed to accelerate `Oumuamua.",
"Furthermore, CO is an abundant and observationally well-established volatile in solar system comets; its existence in `Oumuamua would not be surprising.",
"Unfortunately, empirical limits to CO production in `Oumuamua [181] (even when corrected for a numerical error to $Q_{CO} = 9\\times 10^{23}$ s$^{-1}$ ($\\sim $ 0.04 kg s$^{-1}$ ) by [171]; Table REF ), are three orders of magnitude too small for CO to account for the non-gravitational acceleration.",
"The identity of a possible gaseous driver of the non-gravitational acceleration thus remains unresolved.",
"The second problem with outgassing as an explanation of the non-gravitational acceleration of `Oumuamua is that no dust coma was detected.",
"Gas and dust mass production rates are comparable in typical comets, but the ejected dust typically dominates the optical appearance.",
"This is because the dust has a much larger scattering cross-section per unit mass of material than does the gas, which is only rendered visible through resonance fluorescence.",
"Therefore, if sublimation caused the non-gravitational acceleration of 1I/`Oumuamua, it is particularly puzzling that no comet-like, dusty coma was evident even in the deepest images.",
"The empirical limits on the production rate of micron-sized particles ($<$ (0.2 to 2)$\\times 10^{-3}$ kg s$^{-1}$ ([94], [131]) are orders of magnitude smaller than required to supply the non-gravitational acceleration.",
"A possible solution to this problem is that dust could be hidden from view given a sufficiently large effective particle radius [134].",
"The cross-section per unit mass of a collection of spheres with radius, $a$ , varies as $\\propto 1/a$ .",
"Millimeter-sized particles in `Oumuamua would present 10$^{-3}$ of the cross-section of an equal mass of micron-sized particles.",
"Consequently, they would be 10$^3$ times fainter in scattered light, perhaps allowing them to have escaped detection.",
"This suggestion is ad-hoc in the specific case of `Oumuamua, but measurements of some weakly active comets (e.g.",
"[85]) and active asteroids ([90]) indeed indicate mean particle radii of 10$^2$ $\\mu $ m and greater.",
"The physical explanation for this is not well understood, but a possible cause is that inter-particle cohesion (which itself varies as $1/a$ ) prevents small particles from escaping into the coma under the action of weak gas drag, leaving only the large particles to be ejected [174].",
"Another possibility is that 1I/'Oumuamua outgassed through a porous mantle having enough strength to resist the expulsion of dust particles.",
"Again, this explanation is both ad-hoc and untestable.",
"As a result, a consistent explanation of the origin of `Oumuamua's non-gravitational acceleration by outgassing has not been reached, and alternative explanations resting on the action of radiation pressure have been proposed (Section REF ).",
"The case of 2I/Borisov is more clear-cut.",
"We possess independent estimates of the nucleus radius (200 $\\le r_n \\le $ 500 m), the non-gravitational acceleration (Table REF ), and the mass loss rate in gas (Table REF ).",
"The latter is $\\sim $ 80 kg s$^{-1}$ at 2 AU or $\\dot{M} \\sim $ 320 kg s$^{-1}$ when scaled to 1 AU by the inverse square law.",
"Using these values to solve Equation REF for the nucleus density then gives 100 $\\le \\rho _n \\le $ 1600 kg m$^{-3}$ , a plausible range that brackets the nominal $\\rho _n$ = 500 kg m$^{-3}$ density of solar system comets ([62]).",
"Ultra-low densities like that posited for `Oumuamua (Section REF ) are specifically excluded from the allowable range of solutions for 2I/Borisov.",
"The coma morphology was consistent with a dust differential power law size distribution index -3.5 and the absence of small particles.",
"The effective mean particle size was $a \\gtrsim $ 100 $\\mu $ m [93], [103], [84], suggesting the role of small particle sticking as posited for 1I/`Oumuamua."
],
[
"Radiation pressure: Fractal Bodies",
"Radiation pressure offers an entirely different interpretation of the non-gravitational acceleration of 1I/`Oumuamua.",
"At 1 AU, the radiative pressure is $F_{\\odot }/c$ , where $F_{\\odot }$ = 1360 W m$^{-2}$ is the solar constant and $c = 3\\times 10^8$ m s$^{-1}$ is the speed of light.",
"The force on a spherical body of radius $r_n$ is then $\\pi F_{\\odot } r_n^2/c$ and, by Newton's law, the density required to account for the measured non-gravitational acceleration may be written as $\\rho _n = \\,\\bigg (\\,\\frac{3}{4 r_n}\\,\\bigg )\\,\\bigg (\\,\\frac{ F_{\\odot }}{ c \\alpha _{ng}(1)}\\,\\bigg )\\,.$ By substitution of $\\alpha _{ng}(1)$ for `Oumuamua, we find that $\\rho _n \\sim 0.01(100/r_n)$ kg m$^{-3}$ .",
"This is two orders of magnitude less dense than air and implies a highly porous structure.",
"For comparison, the least dense artificial solid is Aerographite with $\\rho $ = 0.2 kg m$^{-3}$ [130].",
"Built of a complex assemblage of thin carbon sheets and tubes, Aerographite is still an order of magnitude denser than required of `Oumuamua by Equation REF .",
"Could such a low density material form naturally?",
"One suggestion is that `Oumuamua could have a fractal structure produced by Ballistic Cluster-Cluster Aggregation (BCCA) in an ultra-low energy protoplanetary disk environment [139].",
"The mass of a fractal, $m_f$ , varies with its size, $a_f$ , as $m_f \\propto a_f^D$ , and the density scales as $\\rho _f \\propto a_f^{D-3}$ , where $D$ is the fractal dimension.",
"Therefore, smaller values of $D$ correspond to less dense, more open and even “stringy” structures enveloping large void spaces.",
"Fractal aggregates have been investigated in the context of planetary accretion, where they offer several attractive features.",
"These include tight dynamical coupling with the gas, providing a way to overcome the radial drift, fragmentation and bouncing barriers to the growth of large bodies [129], [177], [58].",
"Tiny (0.1 $\\mu $ m?)",
"monomers would be well-coupled to the disk gas and therefore would collide gently, at speeds set initially by Brownian motion.",
"In BCCA, comparably sized clusters collide and stick, producing larger clusters of lower density.",
"The fractal dimension in BCCA is $D \\sim $ 2, meaning that the density varies as $\\rho _f \\propto a_f^{-1}$ .",
"As the aggregate sizes grow to $a_f \\sim $ 1 to 10 mm, numerical models show that the density reaches a minimum value $\\rho _f \\sim 10^{-2}$ kg m$^{-3}$ (i.e.",
"comparable to that required of 1I/`Oumuamua by Equation REF ) [101].",
"At larger sizes, the particle clusters are compressed, first by about an order of magnitude due to ram pressure from the surrounding gas and then to comet-like densities at size scales $a_f \\gtrsim 10^2$ m, by gravitational self-compression.",
"The particles in a fractal structure are held together by incredibly weak van der Waals forces but, counter-intuitively, even with densities as small as $\\sim 10^{-2}$ kg m$^{-3}$ , fractal structures could survive the stresses induced by rotation and Solar tides [53].",
"Indeed, Aerographite has a tensile strength $\\sim 10^3$ N m$^{-2}$ [130].",
"There is some evidence for fractal structure in comets, but only on very small spatial scales.",
"Sub-millimeter aggregates gently collected from the coma of 67P/Churyumov-Gerasimenko had densities $<$ 1 kg m$^{-3}$ [56] and fractal dimensions $D$ = 1.7$\\pm $ 0.1 [120].",
"However, this density is still larger, by a factor of 10$^2$ , than the 0.01 kg m$^{-3}$ density implied by Equation REF for `Oumuamua.",
"Moreover, although some individual submillimeter particles in the nucleus of 67P show low densities and fractal structure, a majority do not.",
"The bulk density of the nucleus is a much more compact $\\rho _n \\sim $ 500 kg m$^{-3}$ [62].",
"There are no known ultra-low density bodies of significant size in the solar system, but this could be a selection effect; even if the solar system formed with an abundance of gently-agglomerated, ultra low density bodies, it is unlikely that any would survive today.",
"Compression or destruction by impact in later, more energetic phases of planetary accretion would erase any evidence of their existence.",
"Only fractal bodies ejected early to the collisionless environment of interstellar space would have had a chance to survive.",
"As a novel caveat to this idea, [116] suggested that fractal bodies might actively form in the comae of other comets, as particles lifted from the surface collide and stick to form “dust bunny” aggregates.",
"While most cometary fragments appear short-lived, in this scenario we should be able to observe cometary fragments that are progressively accelerated into hyperbolic orbits.",
"Table: Empirical spin-changes of short-period comet nuclei"
],
[
"Radiation pressure: Membrane",
"The measured acceleration of `Oumuamua, if due to radiation pressure, could instead imply a thin sheet geometry with a low column density, $\\Sigma $ (kg m$^{-2}$ ), given by $\\Sigma =\\,\\bigg (\\, \\frac{F_{\\odot }}{c \\,\\alpha _{ng}(1)}\\,\\bigg )\\,.$ Substitution for the case of `Oumuamua gives $\\Sigma \\sim $ 0.8 kg m$^{-2}$ , a value that is more typical of a thin sheet of cardboard (density $\\sim 10^3$ kg m$^{-3}$ , thickness $\\sim 10^{-3}$ m) than of any natural, macroscopic object.",
"[13] suggested that `Oumuamua could be a razor-thin sheet or membrane, perhaps akin to a light-sail.",
"Such a structure would have a column density small enough to be accelerated by radiation pressure.",
"With a dimension $r_n \\sim $ 100 m, the corresponding sheet mass would be $r_n^2 \\Sigma \\, \\sim $ 10$^4$ kg, or smaller if the albedo is higher.",
"Because the orbit of `Oumuamua is gravitationally unbound, they inferred that `Oumuamua could be the manufactured product of an alien civilization.",
"Active radio signals transmitted from `Oumuamua were not detected [46], [180], [75].",
"The alien membrane hypothesis is consistent with the existence of non-gravitational acceleration without detectable mass loss, provided the membrane maintains an orientation nearly perpendicular to sunlight.",
"It is also qualitatively consistent with `Oumuamua's extreme lightcurve, albeit with a low ($\\sim $ 1%) probability for having the orientation needed to generate the observed large amplitude [197].",
"The alien membrane hypothesis is highly questionable on other grounds, however.",
"For example, `Oumuamua cannot be a probe targeted at Earth because it missed the Earth by $\\sim $ 40 million km; intelligent aliens could surely do better.",
"Could `Oumuamua instead be a piece of alien space trash?",
"If so, it is difficult to see why an intelligent civilization would flood the galaxy with $10^{25}$ or $10^{26}$ (Section REF ) pieces of 100 meter scale, Mylar-like debris."
],
[
"Stability against spin-up destruction",
"Models of the unexpected properties of 1I/'Oumuamua thus require either strained explanations in terms of the outgassing of unseen volatiles, or explanations invoking material of such low column density that radiation pressure has a strong effect.",
"Whichever model applies, the strong non-gravitational acceleration of this object implies that substantial torques should have modified the spin of the nucleus, potentially driving it to rotational instability [157].",
"Figure: Spin-up timescales measured in solar system comets.Equation is shown as a dashed line.",
"Adapted from Figure 1 in .The nuclei of short-period comets show clear evidence for the action of outgassing torques, in the form of small changes in the rotation period per orbit, $\\Delta P$ [105].",
"When sustained over sufficiently long times, these outgassing torques can drive comets to rotational instability, in which centripetal forces exceed those of gravity and material cohesion acting to bind the nucleus.",
"The result is nucleus breakup or disintegration, widely observed in both short-period [88] and long-period [89] comets.",
"Empirically, the timescale for changing the nucleus spin can be computed from $\\tau _S=\\,\\bigg (\\,\\frac{P}{\\Delta P}\\, P_K\\bigg )\\,,$ where, $P$ is the measured rotational period, $\\Delta P$ is the change in rotational period, and $P_K$ is the Keplerian orbital period.",
"Table REF lists measurements of $\\Delta P$ while Figure REF shows $\\tau _S$ as a function of nucleus radius for the nuclei of short-period comets having perihelion distance 1 $\\le q \\le $ 2 AU.",
"To a good level of approximation, the timescale is represented by $\\tau _S \\simeq 100\\, r_N^{\\,2} \\textrm {~~(years)}\\,,$ where $r_N$ is the nucleus radius expressed in km.",
"Equation REF is shown in the figure as a dashed line.",
"The small nuclei of 1I/`Oumuamua and 2I/Borisov, both indicated in Figure REF , have very short spin-up times, with that for `Oumuamua being $<$ 1 year.",
"In fact, 1I/`Oumuamua should spin-up even more quickly than indicated by Equation REF , both because its small perihelion distance would induce larger outgassing rates than in the comparison comet population, and because of its extreme aspect ratio.",
"The latter would give a longer lever arm than for a more nearly spherical nucleus.",
"Based on these considerations, [158] argued that the lack of steady spin-up and constant periodicity in the lightcurve implied that outgassing could not explain the anomalous acceleration.",
"Figure REF shows the phased lightcurve of `Oumuamua with photometric data obtained from only one telescope and presented by [42].",
"The lightcurve is highly – but not completely – repetitive.",
"For example, differences at the $\\sim $ 10% level are evident near phase = 0.01 and 0.35.",
"If these differences are not due to measurement error, they could indicate a slightly excited or “tumbling” rotational state, which [42] interpreted as the result of an ancient collision in the protoplanetary disk of another star.",
"Rotation can also be excited contemporaneously by mass loss and other torques.",
"Figure: (left panel:) Phased lightcurve data from 1I/`Oumuamua showing small differences between measurements three rotations apart, from UT 2017 October 27 (pink cicles) and October 28 (blue circles).",
"The data are phase folded at a period of 7.5483 hr.",
"(right panel:) Zoom view of the region near phase 0.3 to show differences in the brightness.",
"Adapted from .",
"[11], [42], and [55] searched for periodicities in the composite lightcurve, after correcting the data for the changing phase angle and distance to `Oumuamua and for the use of different filters by different observers.",
"An analysis by [11] identified dominant periodic signals at $8.67\\pm 0.34$ and $3.74 \\pm 0.11$ hrs, consistent with a single rigidly rotating and precessing nucleus and these authors found no evidence for the action of torques on the rotation of 1I/`Oumuamua.",
"However, the [11] analysis assumes a simple decomposition into discrete frequencies, potentially concealing evidence for the action of torques.",
"[53] noticed that reported rotation periods of 1I/`Oumuamua increased linearly from 7.3 hour to 8.2 hour over the three day period UT 2017 October 26-29, corresponding to a spin-change timescale $\\tau _s = P/\\Delta P \\sim $ 26 days.",
"This is broadly consistent both with the YORP spin-up timescale for an ultra-low density body only 80 m in radius and with the outgassing torque timescale computed from Equation REF .",
"An independent analysis revealed that the addition of a torque significantly improves the fit to the lightcurve [123].",
"Moreover, the best-fit moment arm for the torque ($k_T$ = 0.0046) is similar to the median moment arm measured in short-period comets ($k_T$ = 0.007; [88].",
"(The same study also revealed that an oblate body shape is more likely than a prolate one).",
"Although different in detail, the [53] and [123] studies still reveal fractional changes in the rotation period of order unity on timescales of $\\sim $ 1 month, leaving it unclear how to reconcile the survival of the nucleus against rotational disruption with the large non-gravitational acceleration.",
"Point-source outgassing from the subsolar point would eliminate secular spin-up [169] but this pathological geometry does not occur on the known comets.",
"Likewise, a symmetric elongated body with jets uniformly covering the illuminated surface can produce stable spin dynamics [171] but, again, this would be unlike outgassing from any of the known comets.",
"One intriguing possibility is that `Oumuamua did rotationally disrupt from outgassing torques near perihelion, and that the body observed in 2017 is merely the remnant of a rotationally disrupted precursor nucleus that did not survive perihelion.",
"The drifting spin period reported in Figure 1 of [53], when extrapolated backwards, reaches zero in early October, a few weeks after perihelion.",
"Moreover, this possibility is consistent with the observed disintegration of the nuclei of long-period comets, a likely result of rotational instability when near the Sun [89].",
"Such rotational disintegration occurs preferentially in sub-kilometer nuclei with perihelia $<$ 1 AU; both conditions (radius $\\sim $ 0.08 km, perihelion 0.25 AU) apply to `Oumuamua.",
"Residual outgassing from volatiles exposed by the breakup could further modify the rotation, and provide the non-gravitational acceleration.",
"2I/Borisov, with its larger nucleus and perihelion distance, is less susceptible to rotational breakup, for which the timescale (Equation REF ) is 4 $\\le \\tau _s \\le $ 25 years.",
"It did, however, release several fragments (Figure REF ), possibly under the influence of nucleus rotation."
],
[
"Tidal Remnant",
"Comets in the solar system are occasionally tidally disrupted when passing within the Roche lobe of the Sun or a giant planet.",
"The most famous and best-studied example is that of the kilometer-sized, periodic comet P/Shoemaker-Levy 9 (1993 F2, “SL9”), which disrupted upon passing Jupiter at 1.6 times the planetary radius in 1992 [187].",
"The example provided by SL9 motivated [159] to consider `Oumuamua as a tidally shredded fragment of a precursor cometary body from the protoplanetary disk of another star.",
"They obtained an estimate of the fractional disruption rate of ejected bodies in the range 10$^{-3}$ to 10$^{-2}$ , although this number sensitively depends on the model assumptions.",
"They also pointed out that a majority of disrupted fragments would be de-volatilized by subsequent close approaches to the host star prior to escape (further investigated in [160]).",
"The fragment shapes produced by tidal disruption, subject to assumptions about the stellar impact parameter, rubble-pile structure, internal friction and more, are prolate and can be as elongated as suggested by the lightcurve of `Oumuamua [196].",
"However, the galactic rate at which cometary disruptions and ejections occur, given the many unknowns, is highly uncertain.",
"Other potential sources of tidal fragments exist including post main-sequence stars [74], [157], [102], close stellar flybys in clusters [154] and circumbinary systems [32], [87].",
"[26] argued that misaligned circumbinary disks are particularly efficient progenitors of interstellar asteroids, specifically of objects that are close enough to have lost their volatiles.",
"In addition to being tidally shredded, an object must be ejected from the gravitational control of its host star if it is to join the interstellar rank.",
"In a given planetary system, the ejection efficiency depends on the stellar mass, $M_{\\star }$ , and on the distance, $a_P$ , size, $R_P$ , and mass, $M_P$ , of the scattering planet.",
"The distance and mass of the star set the Kepler velocity, $V_{K} = (GM_{\\star }/a_P)^{1/2}$ .",
"The size and mass of the planet set the escape velocity from the surface of planet, $V_e = (2GM_P/R_P)^{1/2}$ .",
"The escape velocity is a reasonable approximation for the maximum velocity that can be imparted in a scattering event.",
"Figure: Safronov number (color coded) for the known exoplanets showing that those with Θ>\\Theta > 1 are preferentially located in the outer regions of their systems (c.f. ).",
"Large pentagonal symbols show solar system planets.",
"The illustrative dashed line shows Θ=1\\Theta =1, for a Jupiter radius planet orbiting a solar mass star.",
"For the most part, planets above the line are capable of ejecting comets to the interstellar medium while those below it are not.A useful parameter to quantify the efficiency of ejection for any given perturber is the Safronov number, $\\Theta = V_e^2/(2V_K^2)$ or, equivalently, $\\Theta = \\left(\\frac{a_{\\rm P}}{R_{\\rm P}}\\right)\\left(\\frac{M_{\\rm P}}{M_*}\\right) \\,.$ To a very good approximation, only planets with $\\Theta >1$ can eject objects via scattering.",
"In the modern-day solar system, the terrestrial planets have $\\Theta <$ 1 but all four giant planets satisfy $\\Theta >$ 1 and are capable of ejecting comets into the interstellar medium.",
"The mass of material that can be ejected depends on the details of the planet-disk interaction.",
"Importantly, for all planets where $\\Theta >1$ , larger values do not necessarily imply that the planet ejected more debris.",
"For instance, Jupiter has a much higher mass than Neptune, and naively one would imagine that it dominated the cometary ejection in the Solar System.",
"However, Neptune migrated over a larger range of distances than did Jupiter, providing it with access to a larger mass of nearby cometesimals.",
"To complicate matters further, the density of the disk near Neptune was smaller than at Jupiter.",
"[68] modeled this process and found that the Oort cloud emplacement efficiency is broadly distributed across the Jupiter to Neptune region.",
"In Figure REF , we show the Safronov number for currently confirmed extrasolar planets (c.f.",
"Figure 1 in [109]).",
"The figure shows that ejection through planetary scattering is unlikely to occur from the inner regions of the known planetary systems, consistent with the idea that most interstellar interlopers are ice-rich bodies formed beyond their snow-lines (see the discussion in Section REF ).",
"Unfortunately, given the uncertainties in the architectures and evolution of other planetary system, it is not possible to meaningfully estimate the galactic rate of ejection to the interstellar medium other than by measurements of the interloper population.",
"Whatever the mechanism, objects that are ejected from close to the host star must acquire higher velocities than those launched from more distant locations in order to overcome the gravity of the star.",
"These higher velocities require scattering from more massive planets and, in planetary systems like our own, high velocity ejections should be rare.",
"Therefore, unless the progenitor system is a special case like a circumbinary system where it is easier to eject from closer in, ejection at high velocity should be less common.",
"A first order corollary of this is that the distribution of velocities of interstellar objects should closely resemble that of the stars upon ejectionThese dispersions would be modified by subsequent dynamical heating.."
],
[
"2I as a more \"normal\" comet",
"The $Q({\\rm CO}) / Q({\\rm H_2O}) \\sim $ 1 production rate ratio in 2I/Borisov [20], [30] distinguishes this object from most solar system comets, in which H$_2$ O is the dominant molecule (Figure REF ).",
"The average cometary ratio is $Q({\\rm CO}) / Q({\\rm H_2O}) \\sim $ 4$\\%$ , albeit with a wide range from 0.5% to 20% [19].",
"However, a few exceptionally CO-rich comets exist.",
"For example, C/1995 O1 (Hale-Bopp) had $Q({\\rm CO}) / Q({\\rm H_2O}) >$ 12 at $r_H$ = 6 AU [15], while short-period comet/Centaur 29P/Schwassmann-Wachmann 1 had $Q({\\rm CO}) / Q({\\rm H_2O}) =$ 10$\\pm $ 1, also at $r_H \\sim $ 6 AU [15], [19]).",
"The outstanding example is C/2016 R2, in which $Q({\\rm CO}) / Q({\\rm H_2O}) = $ 308$\\pm $ 35 at $r_H \\sim $ 2.8 AU [127] (c.f.",
"$Q({\\rm CO}) / Q({\\rm H_2O}) > $ 10, [17]).",
"C/1995 O1 (Hale-Bopp) and C/2016 R2 have barycentric orbital eccentricities $e <$ 1 and are from the Oort cloud, while 29P is a recent arrival from the Kuiper belt.",
"Figure: The composition of the LPC C/2016 R2, 2I/Borisov, and typical carbon enriched and depleted solar system comets.",
"This is a generalized version of an analogous figure in , and adapted from .The carbon depleted comet is representative of many of the solar system comets for which production rate measurements of CO 2 _2, CO and H 2 _2O exist (see Table 1 in ).",
"The carbon enriched comet is W3 Christensen .",
"The composition for 2I/Borisov is derived from Table and the references therein, and R2 is from .",
"The lack of CO 2 _2 for 2I is only because no measurement of CO 2 _2 was reportedHow can these large $Q({\\rm CO}) / Q({\\rm H_2O})$ ratios be understood?",
"One effect is distance; the $Q({\\rm CO}) / Q({\\rm H_2O})$ in a given object should naturally grow with $r_H$ because the volatility of water ice falls faster with $r_H$ than that of CO.",
"This contributes to the high ratios in 29P and C/1995 O1 at 6 AU, where water is largely frozen out.",
"However, 2I/Borisov and C/2016 R2 were observed at much more modest heliocentric distances ($\\sim $ 2 to 2.5 AU) and the distance effect can be ignored.",
"A second effect is that the outgassed species might not represent the bulk composition of the nucleus.",
"For example, highly volatile CO molecules can be mobilized at much lower temperatures than H$_2$ O molecules, providing a source zone extending more deeply into the nucleus.",
"This effect must be transient, because CO could be depleted from the thermally heated skin of the nucleus long before the water ice is sublimated away.",
"But the effect is difficult to model, because cometary volatiles exist within a complex, porous regolith with a large and varying temperature profile and unmeasured permeability to deep gas flow.",
"The most interesting interpretation of these observations is that super-volatile enriched objects like 2I/Borisov and C/2016 R2 were accreted at very large stellocentric distances compared to most of the comets remaining in the short- and long-period populations.",
"Comets formed at large distances beyond the CO snow-line would be the least strongly bound to their parent star and perhaps the most likely to be lost to interstellar space.",
"If so, we should expect to find higher average $Q({\\rm CO}) / Q({\\rm H_2O})$ in interstellar interlopers than in the bound comets."
],
[
"Evidence from Protoplanetary Disks",
"The study of interstellar interlopers is closely related to the field of protostellar disk evolution.",
"Recent observations that resolve protostellar disk substructure are beginning to revolutionize our understanding of the earliest stages of planet formation [119], [135].",
"Observations of face-on disks have revealed the ubiquity of remarkable structures including gaps, spirals and rings [114], [2], [1], [146], [12].",
"Protostellar gas disks are typically a factor of two larger than disks of millimeter sized dust grains, possibly a result of radial migration of the dust under gas drag [3].",
"Disk observations from near the projected mid-plane allow for the measurement of the vertical scale heights of both gas and dust.",
"Gas is vertically supported by a pressure gradient while, in the absence of substantial turbulence, dust settles towards the mid-plane under the action of stellar gravity.",
"As a result, the dust mass density in the mid-plane grows towards a critical value.",
"In the modern view, a “streaming instability” is triggered if the dust to gas ratio of $\\sim 1$ is reached [195] and the subsequent agglomeration of macroscopic bodies is rapid.",
"While volatile molecules can be trapped in water either as clathrates or in the amorphous form, bulk ice can only freeze past the snowline.",
"Therefore, we are especially interested in measurements of disk structure, including the disk scale-heights in gas and dust, at the largest distances from the parent star.",
"[184] presented ALMA images of the protoplanetary disk SSTC2D J163131.2-242627 (Oph 163131) whose disk is inclined to the line of sight by only $i\\sim 6^\\circ $ .",
"The scale height in millimeter-sized dust is $\\sim $ 0.5 AU at 100 AU from this Sun-like star (the mass is 1.2$\\pm $ 0.2 M$_{\\odot }$ ), compared with a scale height in $^{12}$ CO gas of $\\sim $ 10 AU at the same distance (Figure REF ).",
"Enhanced mid-plane densities may allow macroscopic bodies to grow rapidly in this disk, even at 50 AU to 100 AU from the star.",
"SSTC2D thus constitutes a possible analog for the formation site of CO-rich 2I/Borisov [20], [30].",
"Other works also find a dust scale height of $\\sim 1$ AU at 100AU from the central star [155], [183], [39].",
"Figure: ALMA continuum and CO observations of the edge on protoplanetary disk SSTC2D J163131.2-242627 (Oph163131, i∼84 ∘ i\\sim 84^\\circ ).",
"The color scale shows the 12 ^{12}COintensity map from ALMA observations.",
"The blue solid and dashed lines show 2.52.5 and 5σ5\\sigma contours of the dust continuum from ALMA.",
"The gas is significantly more extended in the vertical direction than the dust (with scale heights different by more than an order of magnitude), implying conditions amenable for supervolatile enriched planetesimal formation at large stellocentric distances.",
"Adapted from .Given the greater abundance of H$_2$ O relative to CO in molecular clouds and protoplanetary disk gas, some mechanism of concentration is needed to account for comets in which CO/H$_2$ O $\\gtrsim $ 1.",
"Radial transport of solids under the action of viscous forces may play a role.",
"New models including both the migration of solids and the diffusion and freezing of CO gas in a protoplanetary disk show the formation of an extensive region in which grains grow with CO/H$_2$ O $>$ 1 [156], [47].",
"These models are highly idealised, involving many assumptions and poorly characterized physical processes.",
"However, the growth of a CO-enriched region, spanning from about 10 AU to 100 AU after 10$^6$ years of disk evolution, appears to be a consistent result.",
"Such regions could be the source of CO-enhanced comets and interstellar objects.",
"Perversely, the above models predict CO enrichment over such a large fraction of the protoplanetary disk that one wonders why CO-enriched comets are observationally rare.",
"A possible answer is given by [113], who argued that the final super-volatile abundances of comets should also depend on their disk residence time prior to ejection from the parent star.",
"For example, [175] argued that the solar radiation received by Arrokoth and other Cold Classical KBOs, when integrated over $>10-100$ million year timescales, would be sufficient to deplete all subsurface super-volatiles.",
"Comets ejected to interstellar space soon after their growth can preserve a high CO fraction, while those lingering longer would lose the bulk of their super-volatiles by sublimation.",
"In this view, the low $Q({\\rm CO}) / Q({\\rm H_2 O})$ in most measured comets reflects a long interval between the growth of the nucleus and the formation of the Oort cloud.",
"CO-enriched objects like C/2016 R2 and 2I/Borisov would then have preserved their CO by virtue of unusually early ejection from the parent disks.",
"The apparent rarity of CO-enriched comets may simply reflect the time profile of cometary ejection.",
"Unfortunately, even in the solar system, the details of cometary accretion are subjects of uncertainty and contention [33], [34] and the timing of Oort cloud formation is essentially unknown, making these ideas difficult to test.",
"Still, it is evident that future measurement of the distribution of compositions of interstellar interlopers will provide insights into both the timing and the structure and evolution of the disks from which they are likely ejected.",
"Prior to the discovery of 1I and 2I, only upper limits to the galactic number density of interstellar bodies could be estimated.",
"These estimates are scattered over a wide range, depending on assumptions made about the observational parameters of different surveys, and about the nature and distribution of the sources of interlopers.",
"The most “sophisticated” estimates were not necessarily the most accurate.",
"The detection of `Oumuamua in the relatively well-characterized Pan STARRS sky survey allows for more confident estimates of the galactic number density of similar bodies.",
"The value is now estimated to be between $n_{o} \\sim $ 0.1 AU$^{-3}$ [94], [182] to 0.2 AU$^{-3}$ [38].",
"Since 5 years have elapsed since the discovery of `Oumuamua without another detection of a similar object, we adopt the lower estimate, $n_{o}\\sim 0.1$ AU$^{-3}$ .",
"This corresponds to $\\sim $ 10$^4$ similar objects closer to the Sun than Neptune (i.e.",
"distance $\\le $ 30 AU) at any instant.",
"With a solar system crossing time $\\sim $ 10 years, the flux of interlopers into the planetary region is an incredible $\\sim $ 10$^3$ year$^{-1}$ (3 day$^{-1}$ ).",
"Considering the galaxy as a disk of radius 10 kpc and thickness 1 kpc, this density implies a population $\\sim $ 10$^{26}$ objects of 100 m scale, with a combined mass $\\sim $ 6$\\times 10^{35}$ kg ($10^{11}$ M$_{\\oplus }$ , or about 1 M$_{\\oplus }$ per star).",
"This is comparable to the canonical $\\sim $ 1 M$_{\\oplus }$ estimated mass of the Oort cloud, but refers to objects 10 times smaller.",
"However, the interloper mass is very uncertain because it is based on a single object and because interstellar objects presumably occupy a size distribution in which the mass is dominated by objects larger than `Oumuamua.",
"Given these uncertainties, it is not yet clear if the inferred interloper population contradicts the hypothesis that these objects are ejected comets.",
"Conversely, 2I/Borisov was discovered as part of a near-Sun survey whose depth and areal coverage have not been published, making it impossible to estimate a useful number density.",
"Furthermore, 2I was discovered because of its bright coma, the uncertain optical properties of which would undercut any attempt to derive a meaningful object number density.",
"For these reasons, it is not possible to use the detection of the second interloper to strengthen the density estimate obtained from `Oumuamua."
],
[
"Dynamics",
"Gravitational interactions with giant molecular clouds (GMCs) and other sub-structures in the disk of the galaxy cause a progressive excitation of the velocities of passing stars and tracer particles.",
"This process of “disk heating” should similarly excite the motions of interstellar bodies, providing a method to estimate the length of time they have spent in the interstellar environment.",
"In fact, since interstellar objects outnumber stars by many orders of magnitude, the population is in principle a much better realization of the fine-grain assumption in the collisionless Boltzmann equation.",
"Figure REF shows an empirical stellar age vs. velocity dispersion relation together with the excess velocities relative to the local standard of rest of both interlopers [81].",
"`Oumuamua's low velocity (26 km s$^{-1}$ , compared to 15$\\pm $ 2 km s$^{-1}$ for the velocity of the Sun relative to LSR [161]) implies an age $\\tau _s \\sim $ 100 Myr, originally noted by [118], [57] and [73].",
"The larger velocity of 2I/Borisov (32 km s$^{-1}$ ), indicates greater disk heating and therefore a greater age since ejection, probably $\\tau \\sim 10^9$ yr [73].",
"These estimates are statistical in nature and also subject to surprisingly large systematic uncertainties in the velocity of the Sun ([165] vs. [161]); they cannot be used to infer highly accurate ages.",
"Still, the strong likelihood is that 1I/`Oumuamua has spent less time in interstellar space than 2I/Borisov.",
"Figure: Age kinematics of stars and interstellar interlopers.",
"Blue pentagonal points show average measured stellar velocity dispersions (Figure 7 in ), with a best fit σ∼τ 2/5 \\sigma \\sim \\tau ^{2/5} as a blue solid line.",
"The black and purple dotted lines indicate the velocity for 1I/`Oumuamua and 2I/Borisov (with only statistical uncertainties shaded).",
"Median measured dispersions of stars, White Dwarfs and GMCs are indicated with solid lines.With an age $\\sim $ 100 Myr, `Oumuamua has probably not travelled far from its origin.",
"[57] identified a match with the $45\\pm 10$ Myr old Carina and Columba moving groups (co-moving but unbound aggregates of recently formed stars) [10].",
"He suggested that `Oumuamua formed in a protostellar disk around a star there and was ejected with low peculiar velocity to explain the kinematics.",
"Independent dynamical integrations of the galactic trajectory confirm that `Oumuamua was very likely associated with the local Orion Arm, consistent with the Carina or Columbia stellar associations [73].",
"[83] argued that this was evidence that the object was produced in a giant molecular cloud core, instead of a protostellar disk - because these cloud products would have significantly lower velocity dispersions due to the transient nature of star forming regions in the galaxy.",
"While linkage to a general region of the galaxy is possible, attempts to identify the particular star from which `Oumuamua was ejected are futile, given the many observational and dynamical uncertainties.",
"There is even less hope of identifying a home system for the much older 2I/Borisov [73].",
"One consequence of a local origin for `Oumuamua is that the inferred density of similar objects, $n_0 \\sim $ 0.1 AU$^{-3}$ , may not apply uniformly to the whole galaxy.",
"[138] argued that `Oumuamua was ejected from the planetesimal disk of a young nearby star, and that this ejection was highly anisotropic.",
"[83] traced the motions of test particles ejected from stars in the Carina and Columba stellar associations and found them to be statistically consistent with the orbit of 'Oumuamua.",
"They inferred ejection speeds $\\sim $ 1 km s$^{-1}$ ."
],
[
"Effects of the interstellar environment on the interlopers",
"Direct interactions between stars and interlopers are extraordinarily rare.",
"The timescale for a single object to pass within a distance, $d$ , of a star is just $t \\sim (\\pi N_{\\star } \\Delta V d^2)^{-1}$ , where $N_{\\star } \\sim $ 0.1 pc$^{-3}$ is the number density of stars, and $\\Delta V$ = 50 km s$^{-1}$ is the nominal velocity dispersion.",
"The paths of interlopers will be affected by gravitational focusing, which increases the effective cross-section by a factor $ \\sim (V/\\Delta V)^2$ , where $V$ is the local escape speed at the minimum distance from the star.",
"For example, a Sun impact would have $d \\sim 10^9$ m and $V/\\Delta V \\sim (600/50)^2 \\sim $ 140.",
"Substituting, we find $t \\sim 10^{17}$ years, showing that star-interloper collisions can be ignored.",
"Even interactions as close as those of `Oumuamua (minimum distance $d$ = 0.25 AU) and 2I/Borisov ($d$ = 2.0 AU) are incredibly unlikely ($t \\sim 10^7$ Gyr), for a given object.",
"As a result, the pre-entry thermal evolution of interstellar interlopers should be minimal.",
"Their surface temperatures, set by equilibrium with the interstellar UV flux, will be just a few degrees above the microwave background temperature.",
"On the other hand, the interlopers travel through and interact with the gas and dust of the interstellar medium.",
"[176] showed that while some mass is added by the implantation of interstellar gas into the surfaces of Oort cloud comets (which are as fully exposed to the ISM as are the interlopers), much more material is eroded by impact with interstellar dust grains.",
"The net effect is the loss of the upper $\\Delta L \\sim 0.1$ m of surface for every billion years of exposure.",
"Compared to the size scales of `Oumuamua and 2I/Borisov, the lost material is mass-wise unimportant.",
"Only decimeter scale and smaller interstellar debris should be substantially depleted by impact erosion on billion year timescales.",
"Interstellar space is also pervaded by cosmic rays, with energies that are orders of magnitude larger than the few eV binding energies of common molecular bonds.",
"Cosmic rays severely damage the molecular structure of materials with which they interact.",
"In the open interstellar medium, the cosmic ray energy spectrum resembles a broken power law, having the largest fluences at the smallest energies.",
"(The energy spectrum in the Kuiper belt is different, owing to shielding of low energy particles by the Sun's magnetic field).",
"The cosmic ray penetration depths into solid matter, $d_{CR}$ depend on the particle energies, $E$ .",
"For $E \\lesssim $ 0.1 GeV the penetration depths, $d_{CR} <$ 0.1 m [29], [61], are smaller than the impact-eroded layer thickness.",
"With $d_{CR} < \\Delta L$ , the degree of surface processing by low energy particles is limited by the steady loss of surface material to impact and the continual exposure of fresh material from beneath.",
"However, the energetic $E \\gtrsim $ 1 GeV protons and alpha particles have much larger penetration depths in ice, $d_{CR} \\sim $ 1 m to 10 m. Since $d_{CR} > \\Delta L$ , comets and interstellar interlopers should develop a processed surface layer or “crust” that is too thick to be eroded away by impact with interstellar dust.",
"Laboratory experiments show that the principal effect of energetic particle bombardment is to sever molecular bonds, resulting in the formation of new bonds (and radicals).",
"Being light, hydrogen atoms easily escape, leading to a progressive build-up of macro-molecular, carbon-rich solids.",
"This irradiated material can have low volatility and very low albedo, quite different from the initial material.",
"An unfortunate consequence of the destruction of bonds is that irradiated materials lack the characteristic vibrational spectral features on which spectroscopic chemical identifications are based (e.g. [49]).",
"Probably for this reason, the reflection spectra of most comets, like those of `Oumuamua and 2I/Borisov, tend to be linear, featureless and difficult or impossible to compositionally diagnose.",
"The similarity between the redder-than-sunlight colors of the interlopers and those of solar system comets (Section REF ) is broadly consistent with cosmic ray processing of both."
],
[
"Capture of Interstellar Objects ",
"Galactic tides, although very weak, can temporarily trap slowly passing interstellar objects on loosely bound orbits, building a swarm of such bodies estimated to number $\\sim 10^7$ [153].",
"Likewise, a small fraction of the interstellar objects passing through the planetary region of the solar system can be trapped by gravitational interactions with the planets, albeit with very low and strongly encounter-velocity dependent efficiency [143], [35].",
"The action of non-gravitational acceleration could also trap smaller objects.",
"[144] calculated that the total mass of interstellar objects trapped in the Solar System was $\\simeq 10^{-9}M_\\oplus $ , by estimating their typical dynamical lifetimes.",
"Most of this interstellar material in the present day Solar System was captured during the Sun's cluster phase.",
"Under a different set of assumptions, [36] argued that there were only $\\sim 8$ interstellar objects captured within 5AU at any given time.",
"While the statistics of capture are forbidding, it has nevertheless been suggested that some Centaurs and Trojans with extreme orbits could be captured interstellar bodies [142].",
"However, it is more likely that these objects are transient captures from distant solar system reservoirs, such as Halley type comets or the Oort cloud [137].",
"In this regard, just as there is a small probability for unbound objects to become trapped into bound orbits, it is possible for comets from our own Oort cloud to be scattered onto hyperbolic orbits through interactions with passing stars and sub-stellar objects.",
"On rare occasions, ejected Oort cloud comets passing through the planetary region might be mistaken for interstellar interlopers arriving from afar.",
"[79] estimated that about 0.1% of objects with orbits similar to `Oumuamua, and 0.01% of those similar to the more eccentric orbit of 2I/Borisov, could originate as comets deflected from our own Oort cloud.",
"While these probabilities depend on poorly constrained estimates of the number density of nearby sub-stellar and even sub-Jovian perturbers, it is clear that almost all unbound objects entering the planetary region originate from elsewhere in the galaxy."
],
[
"Interstellar Meteors and Impactors",
"Existing constraints on the flux of interstellar particles across a wide range of sizes are shown in Figure REF .",
"The observations consist of (pink pentagons) in-situ dust detections from the Ulyssess and Galileo spacecraft [66], [65], [108], [64], [107], (blue hexagons) radar measurements from Arecibo [124], [132], [133], radar lower limits with the (dark blue X) Advanced Meteor Orbit Radar (AMOR) [7], [5], [179] and (red X) Canadian Meteor Orbit Radar (CMOR)[188], (purple triangle) optical data from [76] and from the Canadian Automated Meteor Observatory (CAMO) [141], and (orange X) upper limits from optical images of meteoroids from the photographic database of the IAU Meteor Data Center [70], [71].",
"The flux inferred from the 0.1 AU$^{-3}$ density of 1I/`Oumuamua-like objects is shown as a red filled circle.",
"Figure: Observational constraints on the fluxes of interstellar bodies.",
"The line is added to guide the eye and is not a fit to the data.",
"Figure modified from .The data in Figure REF are broadly compatible with a $r_n^{-3}$ radius distribution, subject to a roll-over at sizes $r_n \\lesssim $ 0.1 $\\mu $ m. The latter occurs because dust particles smaller than $\\sim $ 0.1 $\\mu $ m are largely deflected from the inner solar system by a combination of radiation pressure and Lorenz forces.",
"Slightly larger particles (radius $\\gtrsim $ 0.4 $\\mu $ m) can penetrate and have been recorded from impact detectors on spacecraft, with a flux $\\sim 10^{-4}$ m$^{-2}$ s$^{-1}$ [63] (see Figure REF ).",
"These interstellar dust particles can be reliably identified because their velocities are accurately measured using time-of-flight detectors and found to be higher than the local solar system escape speed.",
"Millimeter-sized interstellar particles should, in principle, give rise to meteors, which should also be readily identifiable by their velocitiesWhile $e >$ 1 is a necessary condition for the identification of interstellar material, it is not definitive, because planetary perturbations can generate hyperbolic trajectories, albeit with excess velocities of only $\\sim $ 0.1 km s$^{-1}$ [189], [79].",
"However, rather stringent measurements of speed (accurate to $\\pm $ 1 km s$^{-1}$ ) and of direction (to $\\pm $ 1o or 2o) are needed to distinguish some hyperbolic from bound orbits (c.f. [72].",
"Unfortunately, measurements of sufficient accuracy are difficult given the short flight lengths (a few $\\times $ 10 km) and times ($\\sim $ 1 s) of meteors in the atmosphere.",
"As a result, there is a long history of false detections of interstellar meteors.",
"Famously, [149] used optical data to infer high speeds requiring a substantial fraction of terrestrial meteors to be of interstellar origin, a result that we now know to be wholly incorrect.",
"Similarly, [173] claimed that a previously reported meteor was interstellar but relied, in part, on secret (US Dept.",
"Defense) data concerning the accuracy of the trajectory.",
"Interferometric radar determinations of meteor velocities [6] show that most meteors are unambiguously bound and thus have a solar system origin, with only a small high velocity tail, again likely due to measurement errors [141].",
"Large interstellar objects must occasionally strike the Earth but, because of their high impact velocities, they are less likely than asteroids to deposit meteorites onto the surface.",
"The rate of impact of 100 m scale interstellar bodies into Earth is $\\sim $ (5 to 10)$\\times 10^{-9}$ year$^{-1}$ [91], giving only 25 to 50 such events over the age of the Earth.",
"It is possible that a majority of these would have exploded above the ground as airbursts.",
"This estimated rate is $\\sim 10^{4}$ times less than the rate of impact by solar system material (mostly asteroids) of comparable size.",
"Therefore, we are unlikely to find evidence for craters formed by interstellar projectiles or for interstellar meteorites in our collections.",
"Even if the flux were higher, it is unclear how to distinguish a crater formed by an interstellar impact from one formed by a solar system projectile [25].",
"The mass distribution of free-floating planets resembles a differential power law having an index $p$ = 0.92$\\pm $ 0.06 [60].",
"This corresponds to a size distribution index 3.8$\\pm $ 0.2, which is comparable to but slightly steeper than suggested by Figure REF .",
"While this similarity is likely a coincidence, as noted by [60], it is also an interesting reminder that young planetary systems can eject objects with size scales ranging from dust to planets."
],
[
"Ground-Based",
"It is possible that interstellar objects in the solar system have been recorded but went unnoticed in existing survey data.",
"A systematic search for such objects in archival data would be a valuable first step towards improving our population estimates, even before the advent of powerful, new sky survey telescopes.",
"Many new interstellar interlopers are expected to be found as products of all-sky surveys both planned and already operational.",
"In particular, the Rubin Observatory Legacy Survey of Space and Time (LSST) should offer a substantial increase in sensitivity to transient objects, while surveying the entire night sky in the southern hemisphere with a close to nightly cadence [140], [45], [28], [167].",
"[82] used an elaborate simulation to estimate not only the LSST detection rates as a function of size and absolute magnitude, but also the distribution of orbital elements and trajectories of detectable interstellar objects.",
"They predicted that the LSST will detect between 1-2 `Oumuamua-like interstellar objects every year (for details see Table REF ).",
"Table: Predicted detection rates of interstellar objects a ^a adapted from .The flux curve in Figure REF indicates that meter-sized interstellar objects, with a flux $f \\sim 10^{-24}$ m$^{-2}$ s$^{-1}$ , should strike the Earth's atmosphere on a $t \\sim (4\\pi R_{\\oplus }^2 f)^{-1} \\sim 10^2$ year timescale.",
"It is therefore not surprising that convincing examples of hyperbolic bolides have yet to be reported.",
"However, the cumulative fluxes of 1 cm and 10 cm scale interstellar meteors should be higher by 4 and 2 orders of magnitude, respectively, corresponding to timescales from a few days to a year.",
"These timescales approach the duration of long-term meteor surveys, such that we can reasonably hope for convincing detections of small scale interstellar meteors in the next few years, provided that adequate velocity resolution can be attained.",
"Firm detections of material in the centimeter to decimeter size range would help bridge the gap between the spacecraft detected micron-sized interstellar dust particles and the macroscopic objects discussed here."
],
[
"Space Based",
"In space, the forthcoming NEO Surveyor (consisting of a 50 cm diameter telescope located interior to Earth's orbit at L1 ([117]) is expected to provide thermal infrared (10 $\\mu $ m) detections and orbits of small bodies out to the orbit of Jupiter, with a sensitivity to near-Earth objects rivalling that of the LSST.",
"Located interior to Earth's orbit, it will also provide greater coverage of the sky at small elongation (Sun-telescope-object) angles than is possible with a large telescope from the ground.",
"This allows for the detection of objects at heliocentric distances $r_H <$ 1 AU.",
"Combined with optical data, NEO Surveyor thermal flux density measurements will break the degeneracy between the sizes and albedos that afflicts optical data alone.",
"Thermal data from JWST (James Webb Space Telescope) can also fulfill this role, albeit with much more restricted telescope pointing constraints.",
"Interstellar comet analogues might also provide a source of volatile enrichment to short period planets.",
"The discovery of the first interstellar object implies that, on average, every star contributes $\\sim 1 M_\\oplus $ worth of cometary material to the galactic population.",
"If every star ejects $\\sim 1 M_\\oplus $ of material and because ejection is a chaotic process, a comparable amount of material should be injected into the interior of every contributing system and potentially accreted by exoplanets [170].",
"This enrichment could constitute a non-negligible fraction of the atmospheric metal content of exoplanetary atmospheres.",
"Recently, the European Space Agency selected the Comet Interceptor [97] to launch in 2029.",
"The Interceptor has a low $\\Delta V \\sim $ 1 km s$^{-1}$ budget and can only reach objects whose orbits bring them fortuitously close to its loitering location at L2 within the 3 year duration of its mission [164].",
"With these mission parameters, the likelihood that Interceptor will find an accessible interstellar target is negligible (c.f.",
"Table REF , [82]).",
"More optimistically, an impactor mission to `Oumuamua, sent from the Earth, would have been achievable with a modest impulse ($\\Delta V \\sim $ 4 km s$^{-1}$ ) given sufficient forewarning of the approach [167].",
"[82] estimated that 10% to 30% of the interstellar interlopers to be detected by the LSST will be reachable by a mission with $\\Delta $ V $<$ 15 km s$^{-1}$ [82].",
"Optimistically, a handful of rendezvous-suitable targets will be detected each decade.",
"[SUMMARY POINTS] The two known interstellar interlopers are both sub-kilometer bodies exhibiting non-gravitational acceleration but otherwise are surprisingly physically different.",
"1I/`Oumuamua appears asteroidal while 2I/Borisov outgasses strongly.",
"It is not clear whether these differences reflect two different populations of interstellar body or different evolutionary stages of the same type of object.",
"Dynamical considerations suggest that 1I/`Oumuamua and 2I/Borisov have different ages, likely $\\sim 10^8$ years and $\\sim 10^9$ years, respectively.",
"Interlopers formed by accretion in protoplanetary disks are ejected to interstellar space by strong gravitational scattering from planets.",
"`Oumuamua is tentatively associated with the Carina or Columbia stellar associations.",
"The origin of 2I/Borisov is unknown.",
"The number density of `Oumuamua-like (100 m scale) bodies is $\\sim $ 0.1 AU$^{-3}$ , and the implied galactic population is $\\sim $ 10$^{14}$ to 10$^{15}$ per star.",
"[FUTURE ISSUES] Current interloper population estimates are extremely uncertain but of great importance, in relation to the likely protoplanetary disk origins of these bodies and their galactic total mass.",
"Physical measurements of interstellar interlopers are needed to understand the reason for the divergent appearances of the first two examples, and to better relate these bodies to solar system comets.",
"Planned deep, all-sky surveys (in particular, by NASA's NEO Surveyor and by the LSST) are expected to reveal $\\sim $ 1 new interstellar interloper per year.",
"Spacecraft intercepts of interlopers will be possible but difficult, given the high average encounter velocities and limited forewarning of arrival."
],
[
"DISCLOSURE STATEMENT",
"The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.",
"We thank Marion Villenave, Karen Meech and Olivier Hainaut for providing us with data and Robert Jedicke, Davide Farnocchia, Yoonyoung Kim, Marco Micheli, Greg Laughlin, Aster Taylor, Jane Luu, Pedro Lacerda, Amy Mainzer, Adina Feinstein, Andrew Youdin, Jing Li, Benjamin Donitz and Alan Stern for useful conversations and suggestions."
]
] | 2209.08182 |
[
[
"Covariance regression with random forests"
],
[
"Abstract Capturing the conditional covariances or correlations among the elements of a multivariate response vector based on covariates is important to various fields including neuroscience, epidemiology and biomedicine.",
"We propose a new method called Covariance Regression with Random Forests (CovRegRF) to estimate the covariance matrix of a multivariate response given a set of covariates, using a random forest framework.",
"Random forest trees are built with a splitting rule specially designed to maximize the difference between the sample covariance matrix estimates of the child nodes.",
"We also propose a significance test for the partial effect of a subset of covariates.",
"We evaluate the performance of the proposed method and significance test through a simulation study which shows that the proposed method provides accurate covariance matrix estimates and that the Type-1 error is well controlled.",
"We also demonstrate an application of the proposed method with a thyroid disease data set."
],
[
"Introduction",
"Most existing multivariate regression analyses focus on estimating the conditional mean of the response variable given its covariates.",
"For example, in traditional regression analysis, the expectation of the response variables is related to linear combinations of covariates.",
"While estimating the conditional covariances or correlations among multiple responses based on covariates is also important, it is a less studied problem.",
"For example, functional brain connectivity focuses on the exploration of the co-occurrence of brain activity in different brain regions, and this co-variability can be explained as a function of covariates [33].",
"As another example, human biomarkers such as glucose, cholesterol, iron, albumin, and so on, are important for biomedical research and the covariance of these biomarkers is influenced by age [19].",
"In microbiome studies, the changes in the co-occurrence patterns among taxa with respect to the covariates have been studied [21], [24].",
"In general terms, let $\\mathbf {Y}_{n \\times q}$ be a matrix of $q$ response variables measured on $n$ observations, where $\\mathbf {y}_i$ represents the $i$ th row of $\\mathbf {Y}$ .",
"Similarly, let $\\mathbf {X}_{n \\times p}$ be a matrix of $p$ covariates available for all $n$ observations, where $\\mathbf {x}_i$ represents the $i$ th row of $\\mathbf {X}$ .",
"For an observation with covariates $\\mathbf {x}_i$ and responses $\\mathbf {y}_i$ , the goal is to estimate the covariance of the response variables based on the covariates $\\Sigma _{\\mathbf {x}_i}$ and to analyze how this conditional covariance matrix varies with respect to the covariates.",
"For this problem, [41] use a kernel estimator to estimate the conditional covariance matrix for a single continuous covariate.",
"However, it is not clear how to extend this approach to situations with multiple covariates.",
"[14] propose a linear covariance regression model $\\mathbf {y}_i = (\\mathbf {A} + \\gamma _i \\mathbf {B}) \\mathbf {x}_i + \\epsilon _i,$ where the mean and covariance of the multivariate response is parameterized as functions of covariates.",
"This model can also be interpreted as a special random-effects model where $\\mathbf {A}_{q\\times (p+1)}$ and $\\mathbf {B}_{q\\times (p+1)}$ characterize the fixed and random parts of the model, respectively.",
"The scalar $\\gamma _i$ can be interpreted as an individual-level variability in addition to the random error $\\epsilon _i$ .",
"The rows of $\\mathbf {B}$ indicate how much this additional variability affects $\\mathbf {y}_i$ .",
"The vector $\\epsilon _i$ is of dimension $q \\times 1$ and is assumed to be normally distributed.",
"In this framework, they assume that $E[\\gamma _i]=0$ , $E[\\epsilon _i]=0$ , $E[\\gamma _i \\epsilon _i]=0$ , $Var[\\gamma _i]=1$ , $Var[\\epsilon _i]=\\Psi $ , leading to the following covariance matrix $\\Sigma _{\\mathbf {x}_i} = \\Psi + \\mathbf {B} \\mathbf {x}_i \\mathbf {x}_i^T \\mathbf {B}^T.$ [29] illustrate an application of this model with a four-dimensional health outcome.",
"[11] propose a Bayesian nonparametric model for covariance regression within a high-dimensional response context.",
"Their approach relates the high-dimensional multivariate response set to a lower-dimensional subspace through covariate-dependent factor loadings obtained with a latent factor model.",
"The conditional covariance matrix is a quadratic function of these factor loadings.",
"The method is limited to data sets with smaller sample sizes.",
"[12] proposes a parametric Bayesian model for high-dimensional responses.",
"In this model, the conditional covariance matrices vary with continuous covariates.",
"[42] propose another covariance regression model where the covariance matrix is linked to the linear combination of similarity matrices of covariates.",
"In this study, we propose a nonparametric covariance regression method for estimating the covariance matrix of a multivariate response given a set of covariates, using a random forest framework.",
"The above-mentioned methods are very useful in modelling covariance matrix but compared to them the proposed method offers higher flexibility in estimating the covariance matrix given the set of covariates.",
"For example, with the proposed method, we can estimate the conditional covariance matrix for a set of covariates including multiple continuous and categorical variables, and the proposed method can be used to capture complex interaction patterns with the set of covariates.",
"Moreover, the proposed method is nonparametric and needs less computational time compared to the parametric models, and can be applied to data sets with larger sample sizes.",
"Random forest [8] is an ensemble tree-based algorithm involving many decision trees, and can also be seen as an adaptive nearest neighbour predictor [15], [22], [26], [27], [32], [38], [2].",
"In the proposed random forest framework, we grow each tree with a splitting rule specially designed to maximize the difference in the sample covariance of $\\mathbf {Y}$ between child nodes.",
"For a new observation $\\mathbf {y}^*$ with covariates $\\mathbf {x}^*$ , the proposed random forest finds the set of nearest neighbour observations among the out-of-bag (OOB) observations that are not used in the tree growing process.",
"This set of nearest neighbour observations is then used to estimate the conditional covariance matrix of $\\mathbf {y}^*$ given $\\mathbf {x}^*$ .",
"In each tree built in the proposed random forest framework, the set of covariates is used to find subgroups of observations with similar conditional covariance matrices, assuming that they are related to conditional covariance matrices.",
"We propose a hypothesis test to evaluate the effect of a subset of covariates on the estimated covariance matrices while controlling for the others.",
"We investigate two particular cases, the global effect of the covariates and the partial effect of a single covariate.",
"This paper is organized as follows.",
"In Section , we give the details of the proposed method, significance test and variable importance measure.",
"The simulation study results for accuracy evaluation, global and partial effects of covariates, and variable importance are presented in Section .",
"We provide a real data example in Section , and conclude with some remarks in Section ."
],
[
"Proposed method",
"Let $\\Sigma _{\\mathbf {x}_i}$ be the true conditional covariance matrix of $\\mathbf {y}_i$ based on covariates $\\mathbf {x}_i$ , and $\\Sigma _{\\mathbf {X}}$ be the collection of all conditional covariance matrices for $n$ observations.",
"Similarly, let $\\hat{\\Sigma }_{\\mathbf {x}_i}$ be the estimated conditional covariance matrix of $\\mathbf {y}_i$ based on covariates $\\mathbf {x}_i$ , and $\\hat{\\Sigma }_{\\mathbf {X}}$ be the collection of all estimated conditional covariance matrices for $n$ observations.",
"In this section, we describe the proposed method in detail."
],
[
"Tree growing process and estimation of covariance matrices for new observations with random forests",
"We aim to train a random forest with the set of covariates $\\mathbf {X}$ to find subgroups of observations with similar covariance matrices of $\\mathbf {Y}$ , based on many unsupervised decision trees built with a specialized splitting criterion.",
"The tree growing process follows the CART approach [9].",
"The basic idea of the CART algorithm is to select the best split at each parent node among all possible splits, all evaluated with a selected splitting criterion, to obtain the purest child nodes.",
"The algorithm evaluates all possible splits to determine the split variable and split point.",
"Instead of considering all possible splits at each parent node, the best split search in random forests is confined to a randomly chosen subset of covariates that varies from node to node.",
"The splitting process continues until all nodes are terminal.",
"Our goal is to obtain subgroups of observations with distinct covariance matrices.",
"Hence, we propose a customized splitting rule that will seek to increase the difference in covariance matrices between two child nodes in the tree [4], [26], [38], [2].",
"We define $\\Sigma ^L$ as the sample covariance matrix estimate of the left node as follows: $\\Sigma ^L = \\frac{1}{n_L-1}\\sum _{i \\in t_L} (\\mathbf {y}_i - \\mathbf {\\bar{Y}}_L)(\\mathbf {y}_i - \\mathbf {\\bar{Y}}_L)^T,$ where $t_L$ is the set of indices of the observations in the left node, $n_L$ is the left node size and $\\mathbf {\\bar{Y}}_L = \\frac{1}{n_L} \\sum _{i \\in t_L} \\mathbf {y}_i$ .",
"The sample covariance matrix estimate of the right node, $\\Sigma ^R$ , is computed in the same way, where $n_R$ is the right node size.",
"The proposed splitting criterion is $ \\sqrt{n_Ln_R}*d(\\Sigma ^L, \\Sigma ^R),$ where $d(\\Sigma ^L, \\Sigma ^R)$ is the Euclidean distance between the upper triangular part of the two matrices and computed as follows: $ d(A, B) = \\sqrt{\\sum _{i=1}^{q}\\sum _{j=i}^{q} (\\mathbf {A}_{ij} - \\mathbf {B}_{ij})^2},$ where $\\mathbf {A}_{q \\times q}$ and $\\mathbf {B}_{q \\times q}$ are symmetric matrices.",
"The best split among all possible splits is the one that maximizes (REF ).",
"The final covariance matrices are estimated based on the random forest.",
"For a new observation, we use the nearest neighbour observations to estimate the final covariance matrix.",
"The idea of finding the nearest neighbour observations, a concept very similar to the ‘nearest neighbour forest weights’ [15], [22], was introduced in [26] and later used in [27], [32], [38], [2].",
"[32] called this set of observations the Bag of Observations for Prediction (BOP).",
"For a new observation $\\mathbf {x}^{*}$ , we form the set of nearest neighbour observations with the out-of-bag (OOB) observations [23], [3].",
"We can define the $BOP_{oob}$ for a new observation as $BOP_{oob}(\\mathbf {x}^{*}) = \\bigcup \\limits _{b=1}^{B} O_b(\\mathbf {x}^{*}),$ where $B$ is the number of trees and $O_b(\\mathbf {x}^{*})$ is the set of OOB observations in the same terminal node as $\\mathbf {x}^{*}$ in the $b$ th tree.",
"Each tree is built with a selected random sub-sample, i.e.",
"in-bag observations ($I_b$ ), which has about 63 percent distinct observations from the original sample.",
"The remaining training observations, namely $O_b$ , are OOB observations for that tree and are not used to build the $b$ th tree.",
"$BOP_{oob}$ is slightly different than the nearest neighbour sets in the previous papers who use in-bag observations to form BOP.",
"Since the OOB observations are not used in the tree building process, for the trees where they are OOB, they act as new observations.",
"Therefore, OOB observations represent a new observation better than in-bag observations.",
"Using OOB observations for neighbourhood construction is similar to the idea of honesty in the context of forests.",
"An honest double-sample tree splits the training subsample into two parts: one part for tree growing and another part for estimating the desired response [39].",
"We use the nearest neighbour construction idea to estimate the covariance matrices for the new observations.",
"Algorithm REF describes how to estimate the covariance matrix with OOB observations for a new or training observation.",
"After training the random forest with the specialized splitting criterion, for a new observation $\\mathbf {x}^{*}$ , we form $BOP_{oob}(\\mathbf {x}^{*})$ and then we estimate the covariance matrix by computing the sample covariance matrix of the observations in $BOP_{oob}(\\mathbf {x}^{*})$ ."
],
[
"The number of observations in the nodes decreases as we progress down the tree during the tree-building process.",
"The nodesize parameter is the target average size for the terminal nodes.",
"Lowering this parameter results in deeper trees, which means more splits until the terminal nodes.",
"Tuning the nodesize parameter can potentially improve the prediction performance [22].",
"In typical supervised problems where the target is the observed true response, random forests search for the optimal level of the nodesize parameter by using out-of-bag (OOB) prediction errors computed using the true responses and OOB predictions.",
"The nodesize value with the smallest OOB error is chosen.",
"However, in our problem, the target is the conditional covariance matrix which is unknown.",
"Therefore, we propose a heuristic method for tuning the nodesize parameter.",
"For nodesize tuning, we use the OOB covariance matrix estimates, as described in Algorithm REF .",
"The general idea of the nodesize tuning method is to find the nodesize level where the OOB covariance matrix predictions at two consecutive nodesize levels become similar.",
"We first train separate random forests for a set of nodesize values (see the Parameter settings section in simulation study).",
"Then, we compute the OOB covariance matrix estimates as described in Algorithm REF for each random forest.",
"Define $MAD(\\mathbf {A},\\mathbf {B})= \\frac{2}{q(q+1)} \\sum _{i=1}^{q}\\sum _{j=i}^{q} | \\mathbf {A}_{ij} - \\mathbf {B}_{ij} |$ .",
"Let $\\hat{\\Sigma }^s_{\\mathbf {x}_{i}}$ be the estimated covariance matrix for observation $i$ when nodesize$=s$ .",
"Let $s(1) < \\ldots < s(M)$ be a set of increasing node sizes.",
"For $j=\\lbrace 1,\\ldots M-1\\rbrace $ , let $MAD_j=\\frac{1}{n}\\sum _{i=1}^{n} MAD \\left( \\hat{\\Sigma }_{\\mathbf {x}_{i}}^{s(j)},\\hat{\\Sigma }_{\\mathbf {x}_{i}}^{s(j+1)} \\right).$ Then we select $s(j)$ that corresponds to the value $j$ for which $MAD_j$ is the minimum among $\\lbrace MAD_1,\\ldots , MAD_M\\rbrace $ .",
"See Section of the Supplementary Material for the results of a nodesize tuning experiment.",
"When a node sample size $n_d$ is smaller than the number of responses $q$ , the sample covariance matrix becomes highly variable.",
"In fact, if $n_d - 1 < q$ , the estimate is singular and hence non-invertible.",
"Therefore, the tuning set of nodesize levels should be larger than $q$ .",
"In fact, we need more than $q$ distinct values, so we use sub-sampling instead of bootstrap resampling for tree building to guarantee distinctness, assuming the observations in the original sample are distinct.",
"[htbp] Estimation of covariance matrix for a new or training observation [1] A forest built with the proposed method $BOP_{oob}(\\mathbf {x}_i) = \\emptyset $ b=1,...,B $\\mathbf {x}_i$ is a new observation ($\\mathbf {x}_i$ is a training observation $\\mathbf {x}_i \\in O_b$ ) Find the terminal node of $\\mathbf {x}_i$ at tree $b$ , say $d$ $BOP_{oob}(\\mathbf {x}_i) = BOP_{oob}(\\mathbf {x}_i) \\cup O_b^d(\\mathbf {x}_i)$ (where $O_b^d(\\mathbf {x}_i)$ is the set of OOB observations in the same terminal node $d$ as $\\mathbf {x}_i$ , excluding $\\mathbf {x}_i$ itself when $\\mathbf {x}_i$ is a training observation) Compute sample covariance matrix with the observations in $BOP_{oob}(\\mathbf {x}_i)$"
],
[
"Significance test",
"The proposed method uses covariates to find groups of observations with similar covariance matrices with the assumption that the set of covariates is important to distinguish between these covariance matrices.",
"However, some (or all) covariates might not be relevant.",
"In this paper, we propose a hypothesis test to evaluate the effect of a subset of covariates on the covariance matrix estimates, while controlling for the other covariates.",
"If a subset of covariates has an effect on the covariance matrix estimates obtained with the proposed method, then the conditional covariance matrix estimates given all covariates should be significantly different from the conditional covariance matrix estimates given the controlling set of covariates.",
"We propose a hypothesis test to evaluate the effect of a subset of covariates on the covariance matrix estimates for the null hypothesis $ H_0 : \\Sigma _\\mathbf {X} = \\Sigma _{\\mathbf {X}^c},$ where $\\Sigma _\\mathbf {X}$ is the conditional covariance matrix of $\\mathbf {Y}$ given all $X$ variables, and $\\Sigma _{\\mathbf {X}^c}$ is the conditional covariance matrix of $\\mathbf {Y}$ given only the set of controlling $X$ variables.",
"The proposed significance test is described in Algorithm REF .",
"After computing the covariance matrix estimates for all covariates and control variables only, we compute the test statistic with $ T = \\frac{1}{n} \\sum _{i=1}^{n}{d \\big (\\hat{\\Sigma }_{\\mathbf {x}_i}, \\hat{\\Sigma }_{\\mathbf {x}^c_i}\\big )},$ where $d(.,.",
")$ is computed as (REF ).",
"The test statistic specifies how much the covariance matrix estimates given all covariates differ from the estimates given only the controlling set of covariates.",
"As $T$ becomes larger, we have more evidence against $H_0$ .",
"We conduct a permutation test under the null hypothesis (REF ) by randomly permuting rows of $\\mathbf {X}$ .",
"Let $R$ be the total number of permutations and $T_r$ be the global test statistic (REF ) computed for the $r$ th permuted $\\mathbf {X}$ .",
"We estimate the test $p$ -value with $ p = \\frac{1}{R} \\sum _{r=1}^{R}{I(T_r > T)},$ and we reject the null hypothesis (REF ) at a pre-specified level $\\alpha $ if the $p$ -value is less than $\\alpha $ .",
"[] Permutation test for a subset of covariates effect [1] Train RF with $\\mathbf {X}$ and $\\mathbf {Y}$ , estimate covariance matrices as described in Algorithm REF , say $\\hat{\\Sigma }_{\\mathbf {x}_i}$ $\\forall i=\\lbrace 1,\\ldots ,n\\rbrace $ Train RF with $\\mathbf {X}^{c}$ and $\\mathbf {Y}$ , and estimate covariance matrices as described in Algorithm REF , say $\\hat{\\Sigma }^{c}_{\\mathbf {x}_i}$ $\\forall i=\\lbrace 1,\\ldots ,n\\rbrace $ Compute test statistic with $T = \\frac{1}{n} \\sum _{i=1}^{n}{d \\big (\\hat{\\Sigma }_{\\mathbf {x}_i}, \\hat{\\Sigma }^{c}_{\\mathbf {x}_i}\\big )}$ , where $d(.,.",
")$ is computed as (REF ) $r = 1:R$ Permute rows of $\\mathbf {X}$ , say $\\mathbf {X}_{r}$ Train RF with $\\mathbf {X}_{r}$ and $\\mathbf {Y}$ Estimate covariance matrices as described in Algorithm REF , say $\\hat{\\Sigma }^{^{\\prime }}_{\\mathbf {x}_i}$ $\\forall i=\\lbrace 1,\\ldots ,n\\rbrace $ Train RF with $\\mathbf {X}^{c}_{r}$ and $\\mathbf {Y}$ Estimate covariance matrices as described in Algorithm REF , say $\\hat{\\Sigma }^{c^{\\prime }}_{\\mathbf {x}_i}$ $\\forall i=\\lbrace 1,\\ldots ,n\\rbrace $ Compute test statistic with $T_r = \\frac{1}{n} \\sum _{i=1}^{n}{d \\big (\\hat{\\Sigma }^{^{\\prime }}_{\\mathbf {x}_i}, \\hat{\\Sigma }^{c^{\\prime }}_{\\mathbf {x}_i}\\big )}$ Approximate the permutation $p$ -value with $p = \\frac{1}{R} \\sum _{r=1}^{R}{I(T_r > T)}$ Reject the null hypothesis when $p < \\alpha $ .",
"Otherwise, do not reject the null hypothesis.",
"In the significance test described above, we need to apply the proposed method many times: for the original data with (i) all covariates and (ii) the set of control covariates, and at each permutation for the permuted data with (iii) all covariates and (iv) the set of control covariates.",
"The proposed method applies a nodesize tuning as described in the previous section.",
"Since tuning the nodesize parameter can be computationally demanding, we tune the nodesize for the original data with all covariates and with the set of control covariates only and use those tuned values for their corresponding permutation steps.",
"The proposed significance test has two particular cases of interest.",
"The first is to evaluate the global effect of the covariates on the conditional covariance estimates.",
"If $\\mathbf {X}$ has a global effect on the covariance matrix estimates obtained with the proposed method, then the conditional estimates $\\Sigma _\\mathbf {X}$ should be significantly different from the unconditional covariance matrix estimate $\\Sigma _{root}$ which is computed as the sample covariance matrix of $\\mathbf {Y}$ .",
"The null hypothesis (REF ) becomes $ H_0 : \\Sigma _\\mathbf {X} = \\Sigma _{root}.$ See Section of the Supplementary Material for the details of the global significance test.",
"The second case is to evaluate the effect of a single covariate when the other covariates are in the model.",
"In that particular case, the null hypothesis (REF ) remains.",
"The only difference between the global and partial significance tests is the number of forests we need to train.",
"In the partial significance test, we need to train two random forests per sample, one for all covariates and one for the controlling variables, which makes a total $2R+2$ random forests.",
"However, when we test for the global effect, we need to train only one random forest per sample (in total $R+1$ random forests) since we do not need to build a random forest for the root node."
],
[
"Variable importance",
"For traditional regression tree problems, we can get the variable importance (VIMP) measures by computing the average change in prediction accuracy using the OOB samples.",
"However, the covariance regression problem does not have an observed target.",
"We can compute the VIMP measures by using the fit-the-fit approach which has been applied to enhance interpretability of the covariates on the response [20], [2], [5], [35], [36], [25].",
"In the univariate response case, we get the importance measures by fitting a regression forest to re-predict the predicted values.",
"However, in covariance regression, we have a predicted covariance matrix for each observation and not a single value.",
"Therefore, we use a multivariate splitting rule based on the Mahalanobis distance [17] to re-predict the predicted covariance matrices.",
"We begin by applying the proposed method using the original covariates and responses and estimate the covariance matrices as described in Algorithm REF .",
"Next, we train a random forest with the original covariates and the vector of upper-triangular estimated covariance matrix elements as a multivariate response.",
"VIMP measures are obtained from this random forest.",
"Covariates with higher VIMP measures indicate higher importance for the estimation of covariance matrices."
],
[
"Implementation",
"We have developed an R package called CovRegRF.",
"We used the custom splitting feature of the randomForestSRC package [16] to implement our specially designed splitting criterion in the tree building process.",
"The package is available on CRAN, https://CRAN.R-project.org/package=CovRegRF."
],
[
"Simulations",
"In this section, we perform a simulation study to demonstrate the performance of the proposed method, validate the proposed significance test with two particular cases—global and partial significance tests—and evaluate the variable importance estimations of the covariates."
],
[
"Data generating process",
"We carry out a simulation study using four Data Generating Processes (DGPs).",
"The details of the DGPs are given in Section of the Supplementary Material.",
"The first two DGPs are variations of the first simulated data set used in [14].",
"Both DGPs include one covariate and two response variables.",
"The covariate $x$ is generated uniformly on $[-1, 1]$ .",
"In DGP1, the covariance matrix for the observation $x_i$ is $\\Sigma _{\\mathbf {x_i}} = \\mathbf {\\Psi } + \\mathbf {B}\\mathbf {x}_i\\mathbf {x}_i^T\\mathbf {B}^T$ where $\\mathbf {x}_i^T=(1, x_i)^T$ .",
"DGP2 is similar to DGP1, except that we add a quadratic term to the covariance matrix equation such as $\\Sigma _{\\mathbf {x_i}} = \\mathbf {\\Psi } + \\mathbf {B} \\mathbf {\\dot{x}}_i\\mathbf {\\dot{x}}_i^T \\mathbf {B}^T$ where $\\mathbf {\\dot{x}}_i^T=(1, (x_i + x_i^2))^T$ .",
"In DGP3, the vector of covariates includes seven independent variables generated from the standard normal distribution.",
"For the covariance structure, we use an AR(1) structure with heterogeneous variances.",
"The correlations are generated with all seven covariates according to a tree model with a depth of three and eight terminal nodes.",
"The variances are functions of the generated correlations.",
"In DGP4, the covariance matrix has a compound symmetry structure with heterogeneous variances.",
"Both variances and correlations are functions of covariates.",
"The covariates are generated from the standard normal distribution.",
"The correlations are generated with a logit model and the variances are functions of these generated correlations.",
"The number of covariates and response variables varies depending on the simulation settings.",
"For all DGPs, after generating $\\Sigma _{\\mathbf {x}_i}$ , $\\mathbf {y}_i$ is generated from a multivariate normal distribution $N(\\mathbf {0},\\Sigma _{\\mathbf {x}_i})$ ."
],
[
"Accuracy evaluation",
"We perform a simulation study based on the four DGPs described above to evaluate the accuracy of the proposed method for estimating the covariance matrices.",
"For DGP3 and DGP4, we consider five response variables.",
"For each DGP, we use several values of the training sample size $n_{train}=\\lbrace 50,100,200,500,1000\\rbrace $ , which generates a total of 20 settings (4 DGPs $\\times $ 5 training sample sizes).",
"We repeat each setting 100 times.",
"In each run of the simulations, we generate an independent test set of new observations with $n_{test} = 1000$ .",
"We evaluate the performance of the covariance matrix estimates using the mean absolute errors (MAE) computed for both the estimated correlations and standard deviations separately.",
"For the estimated correlations, we compute the MAE between the upper triangular (off-diagonal) matrices of the true and estimated correlations over all observations as follows: $MAE^{cor}(\\mathbf {\\hat{C}}_\\mathbf {X},\\mathbf {C}_\\mathbf {X}) = \\frac{2}{q(q-1)n_{test}}\\sum _{i=1}^{n_{test}} \\sum _{j=1}^{q} \\sum _{k=j+1}^q |\\hat{\\rho }_{ijk} - \\rho _{ijk}|,$ where $\\mathbf {C}_\\mathbf {X}$ and $\\mathbf {\\hat{C}}_\\mathbf {X}$ are the collection of all correlation matrices corresponding to $\\Sigma _\\mathbf {X}$ and $\\hat{\\Sigma }_\\mathbf {X}$ , respectively.",
"The values $\\rho _{ijk}$ and $\\hat{\\rho }_{ijk}$ represent the correlations in row $j$ and column $k$ of $\\mathbf {C}_{\\mathbf {x}_i}$ and $\\mathbf {\\hat{C}}_{\\mathbf {x}_i}$ , respectively.",
"For the estimated standard deviations, we compute the normalized MAE between the true and estimated standard deviations over all observations as follows: $MAE^{sd}(\\hat{\\Sigma }_{\\mathbf {X}},\\Sigma _{\\mathbf {X}}) = \\frac{1}{qn_{test}}\\sum _{i=1}^{n_{test}} \\sum _{j=1}^{q} \\Bigg |\\frac{\\hat{\\sigma }_{ij} - \\sigma _{ij}}{\\sigma _{ij}}\\Bigg |.$ The values $\\sigma ^2_{ij}$ and $\\hat{\\sigma }^2_{ij}$ represent the $j$ th diagonal element of $\\Sigma _{\\mathbf {x}_i}$ and $\\hat{\\Sigma }_{\\mathbf {x}_i}$ , respectively.",
"Smaller values of $MAE^{cor}$ and $MAE^{sd}$ indicate better performance.",
"We compare our proposed method with the original Gaussian-based covariance regression model covreg which was presented in the Introduction.",
"This method is currently available in the covreg R package [28].",
"Moreover, as a simple benchmark method, we compute the sample covariance matrix without covariates, which is then used as the covariance matrix estimate for all new observations from the test set."
],
[
"Variable importance",
"For the variable importance evaluation simulations, we use DGP3 and DGP4 in which we add five noise variables $X$ to the covariates set.",
"As above, we consider several values for the training sample sizes $n_{train}=\\lbrace 50, 100, 200, 500, 1000\\rbrace $ , for a total of 10 scenarios studied.",
"We examine whether the estimated VIMP measures tend to rank the important variables first.",
"The variable with the highest VIMP measure has a rank of 1.",
"For each scenario, we compute the average rank for the important variables group and for the noise variables group."
],
[
"Evaluating the power of the global significance test",
"We studied four scenarios to evaluate the global effect of the covariates, two of which are under the null hypothesis (REF ) and the other two under the alternative hypothesis.",
"We generate the data sets for these scenarios as follows: $H_0$ (case 1): we generate 5 $Y$ with a constant population covariance matrix and 10 $X$ variables which are all independent following a standard normal distribution.",
"In this case, the covariance of $Y$ is independent of $X$ and we are therefore under the null hypothesis.",
"$H_0$ (case 2): we first generate 7 $X$ and 5 $Y$ under DGP3.",
"Then, we replace the $\\mathbf {X}$ matrix with 10 independent $X$ variables generated from a standard normal distribution.",
"In this case, the covariance of $\\mathbf {Y}$ varies with some of the $X$ variables but those $X$ variables are not available in the training set.",
"Therefore, we are again under the null hypothesis.",
"$H_1$ (without noise): we generate 7 $X$ and 5 $Y$ under DGP3, and the covariates are available in the training set.",
"In this case, the covariance of $\\mathbf {Y}$ varies with all $X$ variables.",
"$H_1$ (with noise): we generate 7 $X$ and 5 $Y$ under DGP3 and we add 3 independent $X$ variables to the covariates' training set.",
"In this case, the covariance of $\\mathbf {Y}$ varies with some of the $X$ variables but not all."
],
[
"Evaluating the power of the partial significance test",
"We can consider three scenarios to evaluate the effect of a single covariate, where one is under the null hypothesis (REF ) and the other two under the alternative hypothesis.",
"We generate the data sets for these scenarios as follows: $H_0$ : We first generate 2 $X$ and 5 $Y$ with DGP4 and we add 1 independent $X$ variable to the covariates' training set.",
"In this case, the covariance of $\\mathbf {Y}$ varies only with the first two $X$ variables.",
"The control set of variables is $\\lbrace X_1, X_2\\rbrace $ and we evaluate the effect of the $X_3$ variable.",
"Therefore, we are under the null hypothesis.",
"$H_1 (weakest)$ : We generate 3 $X$ and 5 $Y$ with DGP4.",
"In this case, the covariance of $\\mathbf {Y}$ varies with all $X$ variables.",
"The control set of variables is $\\lbrace X_1, X_2\\rbrace $ and we evaluate the effect of $X_3$ , which has the weakest effect on the covariance matrix.",
"$H_1 (strongest)$ : We generate 3 $X$ and 5 $Y$ with DGP4.",
"In this case, the covariance of $\\mathbf {Y}$ again varies with all $X$ variables.",
"But now the control set of variables is $\\lbrace X_2, X_3\\rbrace $ and we evaluate the effect of $X_1$ , which has the strongest effect on the covariance matrix.",
"For both the global and partial significance test simulations, we use training sample sizes of $n_{train}=\\lbrace 50,100,200,300,500\\rbrace $ .",
"The number of permutations and the number of replications for each scenario are set to 500.",
"We estimate the type-1 error as the proportion of rejection in the scenarios simulated under $H_0$ and the power as the proportion of rejection in the scenarios simulated under $H_1$ .",
"We estimate a $p$ -value for each replication and we reject the null hypothesis if the $p$ -value is less than the significance level $\\alpha =0.05$ .",
"Finally, we compute the proportion of rejection over 500 replications."
],
[
"Parameter settings",
"For the simulations, we use the following parameters for the proposed method.",
"We set the number of trees to 1000.",
"Letting $p$ be the number of covariates, then the number of covariates to randomly split at each node, mtry, is set to $\\lceil p/3 \\rceil $ .",
"The number of random splits for splitting a covariate at each node, nsplit, is set to $\\max \\lbrace n_{train}/50, 10\\rbrace $ .",
"We tune the nodesize parameter with the set of nodesize$=\\lbrace [sampsize \\times (2^{-1},2^{-2},2^{-3},\\ldots )]>q\\rbrace $ where $q$ is the number of responses and sampsize$=0.632n_{train}$ .",
"In each replication, covreg is run in four independent chains for 8000 iterations, with the first half taken as burn-in."
],
[
"Accuracy evaluation",
"Figures REF and REF present the accuracy results for 100 repetitions.",
"For each method, we can see the change in $MAE^{cor}$ and $MAE^{sd}$ computed for 100 repetitions with an increasing training sample size.",
"As demonstrated in Figure REF , for DGP1 and DGP2 when $n_{train}=50$ , the proposed method and covreg both have a similar performance with respect to the correlation estimation, with a slight advantage for covreg.",
"For DGP1, covreg performs better for both the correlation and standard deviation compared to the proposed method as the sample size increases.",
"This is expected since DGP1 is generated exactly under the covreg model.",
"However, the proposed method still remains competitive.",
"For DGP2, in which a quadratic term is added, the proposed method performs better for the correlation than covreg with increasing sample size.",
"covreg shows better standard deviation estimation performance for smaller sample sizes, but after $n_{train}=500$ the proposed method performs slightly better.",
"As demonstrated in Figure REF , for DGP3, the proposed method shows a significantly smaller $MAE^{cor}$ and $MAE^{sd}$ than covreg for all sample sizes.",
"Moreover, for the smaller sample sizes, the proposed method has considerably lower variance in MAE.",
"For DGP4, both methods improve with increasing sample size, but the proposed method shows smaller or equal MAEs for both correlation and standard deviation estimations.",
"For DGP3 and DGP4, these results are expected, since the proposed method can capture a nonlinear effect.",
"For the nodesize tuning, we compare the accuracy results for different levels of nodesize along with the proposed tuning method.",
"Figures REF and REF in the Supplementary Material present the MAE results for all DGPs which show that the tuning method works well.",
"Figure: Accuracy evaluation results for DGP1 and DGP2.",
"Smaller values of MAE cor MAE^{cor} and MAE sd MAE^{sd} are better.Figure: Accuracy evaluation results for DGP3 and DGP4.",
"Smaller values of MAE cor MAE^{cor} and MAE sd MAE^{sd} are better."
],
[
"Variable importance",
"Figure REF in the Supplementary Material presents the average ranks of the VIMP measures for both the important and noise sets of variables for DGP3 and DGP4.",
"In all scenarios, the important variables have smaller average ranks than noise variables.",
"As the sample size increases, the difference between the average ranks of important and noise variables increases, as expected."
],
[
"Global significance test",
"The left plot in Figure REF presents the estimated type-1 error and power for different training sample sizes for the two $H_0$ scenarios and two $H_1$ scenarios, respectively.",
"We expect the type-1 error to be close to the significance level ($\\alpha =0.05$ ) and we can see that it is well controlled in both cases studied.",
"In both $H_1$ scenarios, the power increases with the sample size.",
"When the sample size is small, adding noise covariates slightly decreases the power, but this effect disappears as the sample size increases.",
"Figure: Significance test results.",
"The left and right plots present the results for global and partial significance tests, respectively.",
"The proportion of rejection corresponds to the type-1 error for H 0 H_0 scenarios, and power for H 1 H_1 scenarios.",
"The dotted line represents the significance level of α=0.05\\alpha =0.05."
],
[
"Partial significance test",
"The right plot in Figure REF presents the estimated type-1 error and power for different training sample sizes for the $H_0$ scenario and two $H_1$ scenarios, respectively.",
"As can be seen from the $H_0$ line, the type-1 error is close to the significance level ($\\alpha =0.05$ ).",
"In both $H_1$ scenarios, the power increases with the sample size as expected.",
"However, the power is much smaller when one tests the weakest covariate compared to the strongest covariate."
],
[
"Real data example",
"Thyroid hormone, the collective name for two hormones, is widely known for regulating several body processes, including growth and metabolism [40], [34].",
"The main hormones produced by the thyroid gland are triiodothyronine (T3) and thyroxine (T4).",
"The synthesis and secretion of these hormones are primarily regulated by thyroid stimulating hormone (TSH), which is produced by the pituitary gland.",
"Primary hypothyroidism is a condition that occurs when the thyroid gland is underactive and the thyroid hormone produced is insufficient to meet the body's requirements, which leads to an increase of TSH.",
"Contrarily, when the thyroid gland produces levels of thyroid hormones that are too high, leading to decreased levels of TSH, the resulting condition is hyperthyroidism.",
"Serum levels of the thyroid hormones and TSH are used to evaluate subjects' thyroid function status and to identify subjects with a thyroid dysfunction.",
"Therefore, establishing reference intervals for these hormones is critical in the diagnosis of thyroid dysfunction.",
"However, reference ranges are affected by age and sex [18], [1], [7], [37], [30].",
"Furthermore, there is a relationship between TSH and thyroid hormone, and the effects of age and sex on this relationship have not been well described [13], [20].",
"Serum levels of these hormones are also affected by the subject's diagnosis, i.e.",
"hormone levels would be within the reference ranges for normal subjects and out of range for subjects with thyroid dysfunction.",
"The conditional mean of these hormones based on the covariates is studied in the literature, but to our knowledge, no study has yet explicitly investigated the effect of covariates on the conditional covariance matrix of these hormones.",
"Hence, our contribution is to study the effect of age, sex and diagnosis on the covariance matrix of the thyroid hormones and TSH.",
"In this study, we investigate the thyroid disease data set from the UCI machine learning repository [10].",
"This data set originally included 9172 subjects and 30 variables including age, sex, hormone levels and diagnosis.",
"Following the exclusion criteria applied in [13] and [37], we exclude pregnant women, subjects who have euthyroid sick syndrome (ESS), goitre, hypopituitarism or tumour, subjects who use antithyroid medication, thyroxine or lithium, who receive I131 treatment, or who have had thyroid surgery.",
"The subjects have different diagnoses including hypothyroidism and hyperthyroidism, as well as normal subjects.",
"Since the sample size of hyperthyroidism subjects is small, we exclude them from the analysis.",
"We also exclude the very young and very old subjects, since there are only a few subjects on the extremes.",
"The remaining data set consists of 324 hypothyroidism and 2951 normal subjects ($n=3275$ ) between 20 and 80 years of age (2021 females/1254 males).",
"We want to estimate the covariance matrix of four thyroid-related hormones—TSH, T3, TT4 (total T4) and FTI (free thyroxine index/free T4)—based on covariates and investigate how the relationship between these hormones varies with the covariates.",
"We apply the proposed method with the covariates age, sex and diagnosis to estimate the covariance matrix of the four hormones.",
"We first perform the significance test with 500 permutations to evaluate the global effect of the three covariates.",
"The estimated p-value with (REF ) is 0 and we reject the null hypothesis (REF ), which indicates that the conditional covariance matrices vary significantly with the set of covariates.",
"Next, we apply the proposed method and obtain the covariance matrix estimates.",
"We analyze the correlations between hormones as a function of covariates, and as shown in Figure REF , age seems not to have much effect on the estimated correlations.",
"We also compute the variable importance measures, and age (0.001) is found to be the least important variable where diagnosis (1.000) is the most important variable, followed by sex (0.011).",
"Therefore, we apply the significance test to evaluate the effect of age on covariance matrices while controlling for sex and diagnosis.",
"Using 500 permutations, the estimated p-value with (REF ) is 0.42 and we fail to reject the null hypothesis (REF ), indicating that we have insufficient evidence to prove that age has an effect on the estimated covariance matrices while sex and diagnosis are in the model.",
"Although the mean levels of TSH and thyroid hormones differ with age [18], [1], [7], [30], the correlation between these hormones may not be affected by aging.",
"Similarly, we apply the significance test for diagnosis and sex while controlling for the remaining two covariates, and the estimated p-values for both tests are 0, which indicates that both diagnosis and sex, taken individually, have an effect on the covariance matrix of the four hormones.",
"We compare the estimated correlations using the proposed method to the sample correlations computed using the whole sample, which are represented with the black dashed lines in Figure REF .",
"For example, the sample correlation between TSH and T3 over all samples is -0.28 which is not close to the estimated correlation of either hypothyroidism or normal subjects.",
"Furthermore, the estimated variances of the four hormones as a function of age, sex and diagnosis are presented in Figure REF of the Supplementary Material.",
"We can see that the variances also differ with covariates.",
"For a mean regression analysis for any of these hormones, assuming a constant variance could yield misleading results.",
"Figure: Estimated correlations between the four hormones as a function of age, sex and diagnosis.",
"Dashed lines represent the sample correlations computed using the whole sample."
],
[
"Concluding remarks",
"In this study, we propose a nonparametric covariance regression method, using a random forest framework, for estimating the covariance matrix of a multivariate response given a set of covariates.",
"Random forest trees are built with a new splitting rule designed to maximize the distance between the sample covariance matrix estimates of the child nodes.",
"For a new observation, the random forest provides the set of nearest neighbour out-of-bag (OOB) observations which is used to estimate the conditional covariance matrix for that observation.",
"We perform a simulation study to test the performance of the proposed method and compare it to the original Gaussian-based covariance regression model covreg.",
"The average computational times of both methods for the simulations are presented in Table REF of the Supplementary Material.",
"We can see from the table that the proposed method is significantly faster than covreg.",
"For the real data analysis, the computational time was 200.14 seconds.",
"Furthermore, we propose a significance test to evaluate the effect of a subset of covariates while the other covariates are in the model.",
"We investigate two particular cases: the global effect of covariates and the effect of a single covariate.",
"We also propose a way to compute variable importance measures.",
"The proposed method can be extended in a variety of ways.",
"One of them is to compute the weighted Euclidean distance between covariance matrices of the child nodes as $d(\\mathbf {A}, \\mathbf {B}) = \\sqrt{\\sum _{i=1}^{q}\\sum _{j=i}^{q} w_{ij} (\\mathbf {A}_{ij} - \\mathbf {B}_{ij})^2} $ for the splitting rule.",
"The weights are like the measures of importance for the elements of the covariance matrices.",
"Another one is that for the final covariance matrix estimation for a new observation, we can use sparse or robust covariance matrix estimations [31], [6] using the nearest neighbour observations.",
"Similarly, it is theoretically possible to use the sparse or robust covariance matrix estimations instead of the sample covariance matrix for the tree building process.",
"However, the computational time could be a limiting factor.",
"The proposed method can be applied to larger $\\mathbf {X}$ dimensions.",
"The computational time increases linearly with $\\texttt {mtry}$ which is the number of covariates to randomly split at each node.",
"It can also be adapted to larger $\\mathbf {Y}$ dimensions, but the computational time could be a limitation for very large $\\mathbf {Y}$ dimensions.",
"Computing the sample covariance matrix has a time complexity $\\mathcal {O}(nq^2)$ for $q$ response variables and we compute covariance matrix for each node split in each tree of the forest which necessitates many covariance matrix computations."
],
[
"Funding",
"This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by Fondation HEC Montréal.",
"Supplementary Material for Covariance regression with random forests 0"
],
[
"Figures REF and REF present the accuracy results for different levels of nodesize along with the proposed method which applies a nodesize tuning as described in Section REF of the main paper.",
"In the figures, the red boxplots illustrate the MAE results for the proposed nodesize tuning heuristic, and the remaining boxplots show the accuracy obtained when we set the nodesize to a specific value.",
"For the set of nodesize levels to be searched in the proposed method, we have nodesize$=\\lbrace [(2^{-1},2^{-2},2^{-3},\\ldots )s]>q\\rbrace $ where $q$ is the number of outcomes and $s$ is the sub-sample size computed as $s=0.632n_{train}$ .",
"As can be seen from the results in Figure REF , as nodesize decreases, first $MAE^{cor}$ and $MAE^{sd}$ decrease and after a point increase for both DGP1 and DGP2.",
"Since we have more levels of nodesize in the larger sample scenarios, it is easier to see this behaviour.",
"For these two DGPs, smaller nodesize values do not mean better performance.",
"As can be seen from Figure REF , for DGP3 and DGP4, contrary to results of DGP1 and DGP2, $MAE^{cor}$ and $MAE^{sd}$ decrease as the nodesize increases.",
"Hence, the best performing nodesize is mostly the smallest.",
"Overall, when we compare the accuracy of the proposed nodesize tuning heuristic to the individual results of different nodesize values, we can see that it mostly performs well, especially for the larger sample sizes.",
"Figure: MAE results for different nodesize values for DGP1 and DGP2.",
"Smaller values of MAE cor MAE^{cor} and MAE sd MAE^{sd} are better.",
"ss is the sub-sample size, i.e.",
"s=.632n train s=.632 n_{train}.",
"Red boxplots illustrate the accuracy for the proposed nodesize tuning, and the rest of the boxplots show the accuracy obtained when we set the nodesize to a specific value.Figure: MAE results for different nodesize values for DGP3 and DGP4.",
"Smaller values of MAE cor MAE^{cor} and MAE sd MAE^{sd} are better.",
"ss is the sub-sample size, i.e.",
"s=.632n train s=.632 n_{train}.",
"Red boxplots illustrate the accuracy for the proposed nodesize tuning, and the rest of the boxplots show the accuracy obtained when we set the nodesize to a specific value."
],
[
"Global significance test",
"The proposed global significance test is described in Algorithm .",
"After computing the unconditional and conditional covariance matrices, $\\Sigma _{root}$ and $\\Sigma _{\\mathbf {x}_i}$ , respectively, we compute the global test statistic with $ T = \\frac{1}{n} \\sum _{i=1}^{n}{d \\big (\\hat{\\Sigma }_{\\mathbf {x}_i}, \\Sigma _{root}\\big )}$ where $d(.,.",
")$ is computed as (REF ) in the main paper.",
"[] Global permutation test for covariates' effects [1] Compute sample covariance matrix of $\\mathbf {Y}$ in the root node, say $\\Sigma _{root}$ Train a RF with $\\mathbf {X}$ and $\\mathbf {Y}$ Estimate covariance matrices as described in Algorithm REF of the main paper, say $\\hat{\\Sigma }_{\\mathbf {x}_i}$ $\\forall i=\\lbrace 1,\\ldots ,n\\rbrace $ Compute test statistic $T$ as in (REF ) $r = 1:R$ Permute rows of $\\mathbf {X}$ to obtain $\\mathbf {X}_{r}$ Train a RF with $\\mathbf {X}_{r}$ and $\\mathbf {Y}$ Estimate covariance matrices as described in Algorithm REF of the main paper, say $\\hat{\\Sigma }^{^{\\prime }}_{\\mathbf {x}_i}$ $\\forall i=\\lbrace 1,\\ldots ,n\\rbrace $ Compute test statistic with $T^{\\prime }_r = \\frac{1}{n} \\sum _{i=1}^{n}{d \\big (\\hat{\\Sigma }^{^{\\prime }}_{\\mathbf {x}_i}, \\Sigma _{root}\\big )}$ Approximate the permutation $p$ -value with $p = \\frac{1}{R} \\sum _{r=1}^{R}{I(T^{\\prime }_r > T)}$ Reject the null hypothesis at level $\\alpha $ when $p < \\alpha $ .",
"Otherwise, do not reject the null hypothesis."
],
[
"Data generating process",
"In DGP1, the covariance matrix for the observation $x_i$ is $\\Sigma _{\\mathbf {x_i}} = \\mathbf {\\Psi } + \\mathbf {B}\\mathbf {x}_i\\mathbf {x}_i^T\\mathbf {B}^T$ where $\\mathbf {x}_i^T=(1, x_i)^T$ , $\\mathbf {B}_0 = [(1,-1)^T,(1, 1)^T]$ , $\\mathbf {B} = \\frac{w}{w+1}\\mathbf {B}_0 $ , $\\mathbf {\\Psi }_0 = \\mathbf {B}_0 [(1,0)^T,(0, 1/3)^T] \\mathbf {B}^T_0$ , $\\mathbf {\\Psi } = \\frac{1}{w+1}\\mathbf {\\Psi }_0 $ and $w=1$ .",
"In DGP3, the correlations are generated with all seven covariates according to a tree model with a depth of three and eight terminal nodes: $\\rho (\\mathbf {x}_i) & = u_1 I\\left(x_{i1}<0, x_{i2}<0, x_{i4}<0\\right)\\\\& + u_2 I\\left(x_{i1}<0, x_{i2}<0, x_{i4} \\ge 0\\right)\\\\& + u_3 I\\left(x_{i1}<0, x_{i2} \\ge 0, x_{i5}<0\\right)\\\\& + u_4 I\\left(x_{i1}<0, x_{i2} \\ge 0, x_{i5} \\ge 0\\right)\\\\& + u_5 I\\left(x_{i1} \\ge 0, x_{i3}<0, x_{i6}<0\\right)\\\\& + u_6 I\\left(x_{i1} \\ge 0, x_{i3}<0, x_{i6} \\ge 0\\right)\\\\& + u_7 I\\left(x_{i1} \\ge 0, x_{i3} \\ge 0, x_{i7}<0\\right)\\\\& + u_8 I\\left(x_{i1} \\ge 0, x_{i3} \\ge 0, x_{i7} \\ge 0\\right)$ where the terminal node values are $u=\\left(0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\\right)$ and $I$ is the indicator function.",
"The variances are functions of $\\rho $ and computed as $Var(y_j|\\mathbf {x}_i) = (1+\\rho (\\mathbf {x}_i))^j$ , $j=\\lbrace 1,\\ldots ,q\\rbrace $ .",
"In DGP4, for an observation $\\mathbf {x}_i$ , we can generate the correlation with the logit model, $\\rho (\\mathbf {x}_i) = \\frac{1}{1 + \\exp {\\big (-(\\beta _0 + \\sum _{j=1}^{p} \\beta _j x_{ij} + x_{i1}^2)}\\big )}$ where $\\beta _0^{}$ is the intercept parameter fixed to $\\beta _0 = -1$ and $\\beta _j$ are the weights for the covariates, fixed to $(1, 1-\\frac{1}{p}, 1-\\frac{2}{p}, \\ldots , 1-\\frac{(p-1)}{p})$ .",
"For an observation $\\mathbf {x}_i$ , the variance of each response is generated as $Var(y_j|\\mathbf {x}_i) = (1+\\rho (\\mathbf {x}_i))^j$ , $j=\\lbrace 1,\\ldots ,q\\rbrace $ ."
],
[
"Variable importance measures",
"As stated in Section REF of the main paper, Figure REF presents the average rank, from the estimated VIMP measures, for the important and noise variables groups for DGP3 and DGP4.",
"The variable with the highest VIMP measure has rank 1.",
"As rank increases, variable importance decreases.",
"In all scenarios, the important variables have smaller average ranks than noise variables.",
"As expected, the difference between the average ranks of important and noise variables increases with increasing sample size.",
"Figure: Average ranks from estimated VIMP measures for DGP3 and DGP4.",
"Smaller rank values indicate a more important variable (the most important variable has rank 1)."
],
[
"Real data example",
"Figure REF presents the estimated variances of the four hormones (TSH, T3, TT4 and FTI) as a function of age, sex, and diagnosis.",
"As we can see from the plots, the variances differ with diagnosis and sex, whereas age does not seem to have much effect on the estimated variances.",
"Figure: Estimated variances for the four hormones as a function of age, sex and diagnosis.",
"Dashed lines represent the sample variances computed using the whole sample."
],
[
"Comparison of computational times",
"All simulations were run in R version 3.6.0 on a Linux machine with Intel(R) Xeon(R) E5-2667 v3 @ 3.20GHz with 396 GB of memory.",
"The average computational time of each method for the four DGPs is presented in Table REF .",
"For both methods, the time for a setting consists of the time for training and the time for prediction for a new data set.",
"We can see that the proposed method is significantly faster than covreg.",
"Table: Average computational time (in seconds) of both methods over 100 replications for each simulated data set."
]
] | 2209.08173 |
[
[
"Delving Globally into Texture and Structure for Image Inpainting"
],
[
"Abstract Image inpainting has achieved remarkable progress and inspired abundant methods, where the critical bottleneck is identified as how to fulfill the high-frequency structure and low-frequency texture information on the masked regions with semantics.",
"To this end, deep models exhibit powerful superiority to capture them, yet constrained on the local spatial regions.",
"In this paper, we delve globally into texture and structure information to well capture the semantics for image inpainting.",
"As opposed to the existing arts trapped on the independent local patches, the texture information of each patch is reconstructed from all other patches across the whole image, to match the coarsely filled information, specially the structure information over the masked regions.",
"Unlike the current decoder-only transformer within the pixel level for image inpainting, our model adopts the transformer pipeline paired with both encoder and decoder.",
"On one hand, the encoder captures the texture semantic correlations of all patches across image via self-attention module.",
"On the other hand, an adaptive patch vocabulary is dynamically established in the decoder for the filled patches over the masked regions.",
"Building on this, a structure-texture matching attention module anchored on the known regions comes up to marry the best of these two worlds for progressive inpainting via a probabilistic diffusion process.",
"Our model is orthogonal to the fashionable arts, such as Convolutional Neural Networks (CNNs), Attention and Transformer model, from the perspective of texture and structure information for image inpainting.",
"The extensive experiments over the benchmarks validate its superiority.",
"Our code is available at https://github.com/htyjers/DGTS-Inpainting."
],
[
"Introduction",
"Image inpainting, which aims at inpainting the masked regions of the image, supports a vast range of applications, such as image editing and restoration.",
"The pioneering diffusion-based [2], [8] and patch-based [1], [5], [11] methods can only be applied to inpaint masked regions with small size by simple color information from pixel level, as they failed to capture the high-level semantics for inpainting.",
"To resolve it, substantial attention has shifted to deep models, where the convolutional neural networks (CNNs) based models [26], [15], [21], [39], [34] follow the encoder-decoder architecture to learn high level semantic information.",
"However, the local inductive priors for CNNs only received the filled information from the bounded known regions within the local spatial ranges of masked regions.",
"To remedy such issue, attention-based mechanisms [38], [41], [20], [42], [32] have been proposed.",
"In particular, the masked regions represented as patches are initially filled with coarse content, which serves as the query to attend to all known patches of the image, and subsequently selects the candidate with large score for substitution.",
"Notably, [41] proposes a cross-layer attention module to calculate the attention score over the feature map in deep layer, and perform the patch substitution over the low layer as per the attention score, the inpainting output is finally obtained via upsampling.",
"Although it considers all known patches across image, each known patches is independently considered upon masked region, such strategy will mislead the masked patch to be inpainted by only one dominant known region with the largest attention score, which may suffer from non-ideal inpainting output.",
"Akin to attention-based methods, transformer based models [29], [40], [4] also consider the information from all known regions.",
"Instead of focusing on the patch pool, it stands upon the pixel level, where each pixel within the masked regions serves as the query to attend to all other pixels from the known regions, so as to be reconstructed, which is further projected into the color vocabulary to select the most relevant color for inpainting.",
"The inpainting pixel then joins into the known pixel pool, such process repeats until all pixels are inpainted as per a predefined order.",
"Technically, [29], [40] propose the decoder-only transformer to capture the structure priors of pixel level and project into the visual color vocabulary for selecting the corresponding color via the dense attention module [28].",
"On one hand, it delves into all known regions rather than confirming only limited known regions for inpainting, hence it is superior to attention models.",
"On the other hand, the pixel level cannot well capture the semantics as patch level, and therefore inferior to attention models.",
"Besides, only the positional information is utilized to yield attention score, which is far from the texture semantic level.",
"Furthermore, the computational burden for large number of pixels enforces transformer model to avoid quadratical complexity raised by possible self-attention module.",
"Going one step further to the viewpoint of texture and structure, all the above methods are basically classified as either texture-only [41], [39] or structure-texture based methods [29], [40], [42], [35], [24], [18], [27], [19], [14], [36], [10], [30], [31].",
"As the texture-only methods, such as CNNs-based [39] and attention-based model [41] that heavily rely on the known texture information to restore the masked regions, but ignoring the structure may prohibit the reasonable texture to be recovered; worse still, the texture information exploited for inpainting only comes from the bounded known regions rather than the whole image, hence the semantic correlations among texture from the global image may not be well captured.",
"Different from that, the structure-texture based methods aim to generate better texture semantics for the masked regions guided by the structure constraints.",
"Following that, the texture is recovered via varied upsampling network.",
"To sum up, their core problem lies in how to fill the structure information into masked regions.",
"Figure: Intuition of the Coarse Filled Attention module.EdgeConnect [24] restores the edge information as structure information based on edge map and masked grayscale via CNNs.",
"The repaired edge map combines with the ground-truth masked image containing texture information to restore the masked regions via encoder-decoder model.",
"Wang et al.",
"[30] adopts the CNNs model to reconstruct the masked regions of monochrome images as structure constraint, upon which, the color information as texture flow is propagated within the image via multi-scale learning.",
"[14] follows an encoder-decoder architecture, where the encoder aims to equalize the structure feature from deep layers and texture feature from shallow layers via channel and spatial equalization, then combined as input feed to decoder to generate inpainting image.",
"Despite the intuition of capturing the structure information, it failed to exploit the information from all known patches, we hence mildly refer it as \"pseudo global structure\", and possibly misguided the non-ideal texture recovery compared with the transformer model [29], [40].",
"[10] recently proposed that structure and texture can mutually guide each other via a two-stream architecture based on U-Net variant.",
"However, it suffers from utilizing the local texture to guide the global structure, leading to the blurry artifacts as illustrated in Fig.REF (d).",
"Based on the above, it is quite beneficial as how to yield global texture and structure information to well exploit the semantics across the whole image, and how to match these two types of global information for image inpainting.",
"As motivated above, we propose to delve globally into texture and structure for image inpainting.",
"To well capture the global semantic correlations from the texture for all patches across the whole image, we adopt the transformer pipeline paired with both encoder and decoder, unlike the decoder-only transformer, the encoder encodes the correlations of the texture information regarding all the patches throughout the self-attention module, with each patch summarized as an entry over the feature map by CNNs, so as to characterize semantics.",
"By this means, the texture information of each entry, represented as the texture reference vector, named texture reference for short, serves as the query to attend to all other texture references for reconstruction.",
"In other words, each texture references encode varied attention scores for semantic correlation degree with all the others across the global image, yielding to a global texture reference, yet a local structure information caused by CNNs.",
"The transformer decoder aims to inpaint all the masked patches by all texture references.",
"To this end, a Coarse Filled Attention module is developed to initially fill in all the masked patches by exploiting all the known patches.",
"Rather than its inaccurate coarse texture information, we prefer its global structure information benefited from exploiting all known patches of the image.",
"Together with all the texture references enjoying global semantic correlations, we propose a novel structure-texture matching attention module bridged by all the known patches, where the each masked patch encoding structure information serves as the query to attend to all the known patches, while each known patch serves as the query to attend to all texture references.",
"By such means, the best matching of these two worlds can be exploited with such transition manner, together with the adaptive patch vocabulary consisting of restored patches, to progressively inpaint all masked patches via a probabilistic diffusion process.",
"The overall pipeline is shown in Fig.REF .",
"Figure: (a) are the input masked image and ground truth; (b) attention methods independently calculate attention score between coarse information for M b M_b and all known regions, to be restored by only one dominant known region; (c-i) ours constructs global semantic correlations via texture references.",
"(c-ii) and (c-iii) exhibit the attention map of texture references over M c M_c.Figure: Overall of the Transformer structure-texture Decoder.",
"(a) The Structure-Texture Matching Attention outputs the attention score map between each coarsely filled patch m ˜ t \\widetilde{m}_{t} and whole texture references R ˜ T \\widetilde{R}_{T}, where they cannot directly match up well due to coarsely filled information.",
"So we propose the bridge attention module over R ˜ T \\widetilde{R}_{T} based on the known patches with precise global texture and structure information, via an attention transition manner, to reconstruct m ˜ t \\widetilde{m}_{t}; (b) The candidate inpainting patches over coarsely filled patches form an adaptive patch vocabulary to progressively select the most relevant candidate with the largest value of Eq., to expand the unmasked patch set via a probabilistic diffusion process.In summary, our contributions are summarized below: [topsep = 5pt, leftmargin = 15pt] We propose a transformer pipeline paired with both encoder and decoder, where the encoder module aims at capturing the semantic correlations of the whole images within texture references, leading to a global texture reference set; we design a coarse filled attention module to exploit all the known image patches to fill in the masked regions, yielding a global structure information.",
"To endow the decoder with the capacity of marring the best of the two worlds, i.e., global texture reference and structure information.",
"we equip the decoder with a structure-texture matching attention module via an intuitive attention transition manner, where an adaptive patch vocabulary is dynamically established for the filled patches over the masked regions via a probabilistic diffusion process.",
"To ease the computational burden, we disclose several training tricks to overcome memory overhead for GPUs while achieve the state-of-the-art performance on typical benchmarks."
],
[
"Overall Framework",
"Image inpainting aims to transform the input image $I_{gt} \\in \\mathbb {R}^{3\\times H \\times W}$ with mask $M$ sharing the same size, where each entry values either 1 or 0, and performing element wise multiplication over $I_{gt}$ to yield a masked image, i.e., $I_{m} = I_{gt} \\odot M$ into a complete image $I_{out}$ .",
"In this section, we technically explain our pipeline shown in Fig.REF with more details.",
"Sec.",
"REF discusses how to encode the global semantic correlations over $I_{m}$ .",
"Based on that, Sec.",
"REF discusses how to fill into the masked patches for image inpainting.",
"We finally offer the overall loss functions in Sec.",
"REF ."
],
[
"Transformer Texture Encoder",
"To capture the textural semantic correlations across the whole image, we need to learn the explicit texture representation for each patch.",
"In particular, we learn that via the typical CNNs backbone, i.e., ResNet50 [12], to yield the high level semantic feature map $f \\in \\mathbb {R}^{C \\times H_{0} \\times W_{0}}$ , where each entry of the feature map encodes the texture information for one specific patch of $I_{m}$ .",
"It is apparent that each entry of the feature map with large size, e.g., $32\\times 32$ w.r.t.",
"shallow layer fails to capture high-level semantics; when going too deep, e.g., $8 \\times 8$ , each entry of the feature map bears too much semantics, resulting the texture information from one patch into mixed texture with other patches.",
"To balance, we set: $H_{0}=\\frac{H}{16},W_{0}=\\frac{W}{16}$ as [3], where we confirm the dimension for each entry as per the channel number regarding $C$ , i.e., 2048 for ResNet50.",
"Then we project each entry into a low dimension vector representation $r_{T}$ with dimension $d_{E} = 256$ , which is performed by merging all 2048 channels first, followed by 256 filters with $1 \\times 1$ , to then reshape the size of feature map to be $\\mathbb {R}^{d_{E} \\times H_{0} \\times W_{0}}$ .",
"To preserve spatial order information, a positional embedding $E_{p_{e}} \\in \\mathbb {R}^{d_{E} \\times H_{0}W_{0}}$ , similar as [25], regarding all entries are added to form the final representation $E_{T}$ for encoder.",
"Now we are ready to calculate the texture correlations across global image by performing the self-attention over $E_{T}$ .",
"Following transformer encoder architecture, including $N$ layers paired each layer with a multi-head self-attention (MSA) and feed forward network (FFN).",
"For the lth layer, it yields: $\\begin{aligned}& H_{T}^{l} = {\\rm LN}\\left({\\rm MSA}\\left(E_{T}^{l}\\right)\\right)+E_{T}^{l}\\\\& E_{T}^{l+1} = {\\rm LN}\\left({\\rm FFN}\\left(H_{T}^{l}\\right)\\right)+H_{T}^{l},\\end{aligned}$ where LN($\\cdot $ ) denotes layer normalization and FFN($\\cdot $ ) consists of two fully connected (FC) layers with each comprised by two sub-layers.",
"MSA($\\cdot $ ) reconstructs each $r_{T}$ via multi-head self-attention module over all the others to capture the global semantic correlations, the two fully connected layer subsequently converting it to be the input of the (l+1)th layer till the last layer, here residual connections around each of the sub-layers is employed.",
"We formulate the Multi-head self-attention, MSA($\\cdot $ ), over $l$ th layer is computed as $\\begin{aligned}& head^l_{j} = {\\rm softmax}\\left(\\frac{{\\rm W}^{j_{l}}_{Q}E_{T}^{l}\\left({\\rm W}^{j_{l}}_{K}E_{T}^{l}\\right)^T}{\\sqrt{d_{l}}}\\right){\\rm W}^{j_{l}}_{V}E_{T}^{l}\\\\& {\\rm MSA} = \\left[head^l_{1};\\cdots ;head^l_{h}\\right]{\\rm W}^l,\\end{aligned}$ where $h$ is the number of head, $d_{l}$ is the embedding dimension, ${\\rm W}^{j_{l}}_{Q}$ , ${\\rm W}^{j_{l}}_{K}$ and ${\\rm W}^{j_{l}}_{V}$ are three learnable linear projection matrices, $1 \\le j \\le h$ .",
"${\\rm W}^l$ represents a learnable FC layer to fuse the outputs from different heads.",
"After the encoder layers, we can reconstruct each texture feature vector $r_{T}$ to be the reference vector $\\widetilde{r}_{T}$ , with all forming the reference set $\\widetilde{R}_{T}$ .",
"It is easily seen that each $\\widetilde{r}_{T}$ encodes the global texture correlations among all other, where the texture correlations varied across different locations."
],
[
"Transformer Structure-Texture Decoder",
"Besides texture reference set $\\widetilde{R}_{T}$ from encoder, it is crucial to model the representation regarding the masked patches.",
"Unlike the existing decoder-only transformer over pixel level, we need to consider the patch size for the semantics matching with $\\widetilde{R}_{T}$ .",
"We simply reshape $I_{m}$ to $I_{m}^{\\prime }$ to get low resolution image such that $I_{m}^{\\prime } \\in \\mathbb {R}^{3\\times \\frac{H}{4} \\times \\frac{W}{4}}$ to strength the global structure information, while achieving a medium patch size.",
"Inspired by [7], [17], we convert $I_{m}^{\\prime }$ to a sequence of flattened 2D patches $I_{p}^{\\prime } \\in \\mathbb {R}^{N_{0} \\times \\left(3 \\times P \\times P\\right)}$ , with $P \\times P$ resolution for each patch, and $N_{0}$ is the patch number of patches, then we flatten the patches and map to $d_{D}$ dimensions with a trainable linear matrix as the patch embedding.",
"For both known patches and unknown patches, the extra spatial position embedding $E_{p_{d}}\\in \\mathbb {R}^{d_{D} \\times N_{0}}$ is added into the flattened patches for spatial order preservation.",
"Before discussing the correlations between $\\widetilde{R}_{T}$ and masked regions, we need to fill the masked regions with coarse information based on the known regions.",
"Unlike [38] relying on CNNs with local inductive prior to exploit the known patches to fill in the coarse content , we propose a globally filled attention mechanism to exploit all the known regions of the image.",
"To ease the understanding, we illustrate it in Fig.REF .",
"To coarsely fill in the $k^{th}$ masked patch $m_{k}$ , we leverage the known patch and previous k-1 coarsely filled patches.",
"Specifically, all masked patches are sorted by ascending order by the mask ratio to be filled with coarse content, and reconstruct each of them throughout the attention over the known patches and other coarsely filled patches $\\widetilde{m}_{i}, 1 \\le i \\le k-1$ , both of them form the set $P_{k-1}$ , together with the filled $\\widetilde{m}_{k}$ to further become $P_{k}$ .",
"The coarsely filled $\\widetilde{m}_{k}$ w.r.t.",
"${m}_ {k}$ is reconstructed via attention score over $P_{k-1}$ as: $\\begin{aligned}& \\widetilde{m}_{k} = {\\rm softmax}\\left(\\frac{{\\rm W}^C_{Q}m_{k}\\left({\\rm W}^C_{K}P_{k-1}\\right)^T}{\\sqrt{d_{m}}}\\right){\\rm W}^C_{V}P_{k-1}\\\\& P_{k} = P_{k-1} \\cup \\widetilde{m}_{k},\\end{aligned}$ where $d_{m}$ is the embedding dimension, ${\\rm W}^C_{Q}$ , ${\\rm W}^C_{K}$ and ${\\rm W}^C_{V}$ are three learnable linear projection matrices.",
"Figure: Intuition of the bridge module.",
"The masked patches filled with coarse texture but fine global structure information serve as the query to attend to all known patches, while each known patch servers as the query to attend to all texture references via an attention transition manner.Now we discuss how to inpaint each $\\widetilde{m}_{t}$ by selecting the desirable $\\widetilde{r}_{T}$ from $\\widetilde{R}_{T}$ .",
"As previously discussed, the conventional attention mechanism shown in Fig.REF (b) independently calculates the attention score between masked and each individual known regions, misleading the masked patch to be inpainted by only one dominant known region.",
"Unlike it, our proposed method as shown in Fig.REF (c) exploits the global texture semantic correlations over $\\widetilde{R}_{T}$ for inpainting.",
"We observe that both $\\widetilde{r}_{T}$ and $\\widetilde{m}_{t}$ are formed by exploiting unmasked regions across the whole image.",
"It is apparent that $\\widetilde{r}_{T}$ well captures global texture information, which is much more precise than the coarsely filled texture information for $\\widetilde{m}_{k}$ .",
"However, the structure information by exploiting all the unmasked patches for each $\\widetilde{m}_{t}$ is much better, and strengthened via downsampling.",
"Motivated by this, beyond reconstructing $\\widetilde{m}_{t}$ via attention module directly from $\\widetilde{R}_{T}$ .",
"We also propose to serve the unmasked regions, which enjoys both the ideal texture and structure information, as a bridge module to match $\\widetilde{r}_{T}$ and $\\widetilde{m}_{k}$ , as depicted in Fig.REF .",
"For $M$ layers of the decoder, each layer consists of two sub-layers: a structure-texture matching attention (STMA) module and a FFN function with two fully connected layers, which subsequently converts it to be the input of the (l+1)th layer till the $M$ th layer as similar as encoder.",
"We equally employ residual connections around each of them.",
"For $\\widetilde{m}_{t}, t \\le N_{0}$ over the $l$ th layer, it yields: $\\begin{aligned}& \\widetilde{h}^{l}_{t} = {\\rm LN}\\left({\\rm STMA}\\left(\\widetilde{m}^{l}_{t}\\right)\\right)+\\widetilde{m}^{l}_{t}\\\\& \\widetilde{m}^{l+1}_{t} = {\\rm LN}\\left({\\rm FFN}\\left(\\widetilde{h}^{l}_{t}\\right)\\right)+\\widetilde{h}^{l}_{t},\\end{aligned}$ where STMA($\\cdot $ ) denotes the structure-texture matching attention, to obtain the attention score map between $\\widetilde{m}_{t}$ and $\\widetilde{R}_{T}$ based on known regions including inpainting patches $m^K_{j}, 1 \\le j \\le t-1$ , denoted as $O_{t-1}$ .",
"The STMA($\\cdot $ ) performs over the $l$ th layer as: $\\begin{aligned}& S^{d_{l}}_{t} = \\left(\\frac{{\\rm W}^{d_{l}}_{Q}\\widetilde{m}^{l}_{t}\\left({\\rm W}^{d_{l}}_{K}\\widetilde{R}_{T}\\right)^T}{\\sqrt{d_{i}}}\\right),\\\\\\end{aligned}$ where $S^{d_{l}}_{t}$ directly computes the attention scores between $\\widetilde{m}_{t}$ and $\\widetilde{R}_{T}$ .",
"${\\rm W}^{d_{l}}_{Q}$ and ${\\rm W}^{d_{l}}_{K}$ are learnable linear projection matrices and $d_{i}$ is the embedding dimension.",
"As aforementioned, the coarse texture information from $\\widetilde{m}_{t}$ may not directly match $\\widetilde{R}_{T}$ well, so we propose bridging attention module over $\\widetilde{R}_{T}$ based on the unmasked patches to reconstruct $\\widetilde{m}_{t}$ over the $l$ th layer below: $\\begin{aligned}&S^{b_{l}}_{t} = \\left(\\frac{{\\rm W}^{b_{l}}_{Q_{c}}\\widetilde{m}^{l}_{t}\\left({\\rm W}^{b_{l}}_{K_{c}}O^{l}_{t-1}\\right)^T}{\\sqrt{d_{c}}}\\right) \\left(\\frac{{\\rm W}^{b_{l}}_{Q_{r}}O^{l}_{t-1}\\left({\\rm W}^{b_{l}}_{K_{r}}\\widetilde{R}_{T}\\right)^T}{\\sqrt{d_{r}}}\\right),\\\\\\end{aligned}$ where $S^{b_{l}}_{t}$ denotes the bridge attention module.",
"${\\rm W}^{b_{l}}_{Q_{c}}$ , ${\\rm W}^{b_{l}}_{K_{c}}$ , ${\\rm W}^{b_{l}}_{Q_{r}}$ and ${\\rm W}^{b_{l}}_{K_{r}}$ are learnable linear projection matrices, $d_{c}$ and $d_{r}$ are the embedding dimension.",
"Figure: Intuition of the training tricks for computational efficiency.We remark that Eq.REF implies an attention transition manner by serving $\\widetilde{m}^{l}_{t}$ as the query to attend to $O^{l}_{t-1}$ , each of $O^{l}_{t-1}$ further serves as the query to attend to $\\widetilde{R}_{T}$ , to finally reconstruct $\\widetilde{m}^{l}_{t}$ ; note that we only achieve the attention score with the known patches as query to attend to $\\widetilde{R}_{T}$ to reconstruct $\\widetilde{m}^{l}_{t}$ , rather than $O^{l}_{t-1}$ to further reconstruct $\\widetilde{m}^{l}_{t}$ , as it is apparent that the known patches are ideal ground truth, and should not be reconstructed.",
"Combining Eq.REF with REF , it has: $\\begin{aligned}& m^{C_{l}}_{t} = \\lambda {\\rm softmax}\\left( S^{d_{l}}_{t}\\right){\\rm W}^{d_{l}}_{V}\\widetilde{R}_{T}+ \\left(1- \\lambda \\right){\\rm softmax}\\left(S^{b_{l}}_{t}\\right){\\rm W}^{b_{l}}_{V}\\widetilde{R}_{T},\\end{aligned}$ where ${\\rm W}^{d_{l}}_{V}$ and ${\\rm W}^{b_{l}}_{V}$ are learnable linear projection matrices.",
"Following this, we require a decoder vocabulary over the patch level, from which we generate each inpainting patch.",
"Particularly, each coarsely filled patch $\\widetilde{m}^{M}_{z}$ is reconstructed via Eq.REF over the final $M$ th layer to become $m^{C}_{z}$ , while form vocabulary set $m^C$ , from which we select the most relevant candidate $m^K_{t}$ to become the final inpainting patch with the largest probability via Eq.REF .",
"$\\begin{aligned}m^K_{t} &= \\mathop {{\\rm arg\\,max}}_{m^{C}_{z}} p\\left(m^{C}_{z} | O_{t-1}, \\widetilde{R}_{T}\\right)\\\\&= \\mathop {{\\rm arg\\,max}}_{m_{z}^C} \\frac{\\lambda \\left\\Vert {\\rm softmax}\\left( S^{d_{M}}_{z}\\right)\\right\\Vert _{1} +\\left(1- \\lambda \\right)\\left\\Vert {\\rm softmax}\\left(S^{b_{M}}_{z}\\right) \\right\\Vert _{1}}{ \\sum \\limits _{t\\le j \\le N_{c}}\\left(\\lambda \\left\\Vert {\\rm softmax}\\left( S^{d_{M}}_{j}\\right)\\right\\Vert _{1} +\\left(1- \\lambda \\right)\\left\\Vert {\\rm softmax}\\left(S^{b_{M}}_{j}\\right) \\right\\Vert _{1}\\right)},\\\\&O_{t} = O_{t-1} \\cup m_{t}^K,\\end{aligned}$ where softmax$(S^{d_{M}}_{z})$ , softmax$(S^{b_{M}}_{z})$ are the output vector representation with 256 dimensions via the last $M$ th layer, the $i$ th entry softmax$(S^{d_{M}}_{z})$ (i) and softmax$(S^{b_{M}}_{z})$ (i) denote the attention score between the $z$ th coarsely filled patch $\\widetilde{m}^{M}_{z}$ and the $i$ th texture reference $\\widetilde{r}_{T}^{i}$ via Eq.REF and REF , $\\left\\Vert \\cdot \\right\\Vert _{1}$ sums up all entries over $\\widetilde{R}_{T}$ , which help reconstruct all $\\widetilde{m}^{M}_{z}$ to form $m^{C}_{z}$ in $m^C$ ; $N_{c}$ is the number of elements for $m^C$ .",
"The most relevant candidate $m_{z}^C$ is selected to be the winner $m_t^K$ in the $t$ th iteration with the largest attention score summation, via $\\left\\Vert \\cdot \\right\\Vert _{1}$ norm, over $\\widetilde{R}_{T}$ , i.e., the summation of attention score over $\\widetilde{R}_{T}$ for $\\widetilde{m}^{M}_{z}$ , to join the known patch set to expand $O_{t-1}$ to $O_{t}$ via a probabilistic diffusion process throughout Eq.REF , while further help select the others candidate to be inpainted in the $(t+1)$ th iteration, and end up with all patches to be inpainted.",
"We remark that our decoder patch vocabulary $m^C$ is adaptively constructed based on coarsely filled patches, and dynamically updated.",
"The overall architecture for our transformer decoder is shown in Fig.REF .",
"Computational efficiency.",
"One may wonder the computational complexity caused by the attention module for each iteration.",
"We clarify it to be efficient as shown in Fig.REF , the attention map among coarsely filled patch set, known patch set and texture reference set can be computed off the shelf.",
"When restoring a coarsely filled patch into inpainting patch, there is no need to recalculate whole the attention map between different sets, but only remove the corresponding row of attention map meanwhile make up the column for the known patch w.r.t.",
"the coarsely filled patch as per Fig.REF (a) and REF (b).",
"We also supplement the new row to the attention map between newly inpainting patch and texture reference set in REF (c).",
"After inpainting most of candidate patches from vocabulary $m^{C}$ , there is no need to repeat the probabilistic diffusion process via Eq.REF , especially for the candidate with quite small ratio for coarse content, we simply average such content with the neighborhood patches to further reduce the attention complexity to $\\widetilde{R}_{T}$ and $m^K$ .",
"Table: Quantitative results with varied mask ratios under ℓ 1 {\\ell _{1}}, PSNR, SSIM, and FID on PSV , CelebA-HQ and Places2 with irregular mask dataset (↑\\uparrow : Higher is better; ↓\\downarrow : Lower is better; - : no reported results from the methods).",
"The two best scores are colored byredred and blueblue."
],
[
"Loss Functions",
"After all masked patches are inpainted, we obtain the reconstructed vector set with each locating in 384 dimensional feature space, which are further restored to RGB image following [16], denoted by $I_{out}^{\\prime } \\in \\mathbb {R}^{3\\times \\frac{H}{4} \\times \\frac{W}{4}}$ .",
"We basically follow [14], [24] over typical losses to measure reconstruction error between inpainting images $I_{out}^{\\prime }$ and downsampling ground truth $I_{gt}^{\\prime }$ , such as reconstruction loss $\\mathcal {L}_{rec}$ , perceptual loss $\\mathcal {L}_{prec}$ , style loss $\\mathcal {L}_{style}$ and adversarial loss $\\mathcal {L}_{adv}$ .",
"Afterwards, we perform the upsampling over the $I_{out}^{\\prime }$ to be $I_{out} \\in \\mathbb {R}^{3\\times H \\times W}$ through a adversarial network [40] guided by $\\mathcal {L}_{adv}$ , which will not added into our final loss function.",
"Specifically, we aim to learn desirable $I_{out}^{\\prime }$ via the followings: Reconstruction Loss.",
"We employ the ${\\ell _{1}}$ loss to measure the pixel wise difference between the ground truth $I_{gt}^{\\prime }$ after downsampling and our inpainting output $I_{out}^{\\prime }$ , then yields the following: $\\begin{aligned}& \\mathcal {L}_{rec} = \\left\\Vert I_{out}^{\\prime } - I_{gt}^{\\prime } \\right\\Vert _{1}.\\end{aligned}$ Perceptual Loss.",
"To simulate human perception of images quality, we utilize the perceptual loss by defining a distance measure between activation feature maps of a pre-trained network over our inpainting output and ground truth, we have the following: $\\begin{aligned}& \\mathcal {L}_{prec} = \\mathbb {E}\\left[ \\sum \\limits _{i}\\frac{1}{N_{i}}\\left\\Vert \\phi _{i}\\left(I_{out}^{\\prime }\\right) - \\phi _{i}\\left(I_{gt}^{\\prime }\\right) \\right\\Vert _{1}\\right],\\end{aligned}$ where $\\phi _{i}$ is the activation feature map $N_i$ with the size of $C_{i} \\times H_{i} \\times W_{i}$ of the $i$ th layer of VGG backbone.",
"In our work, $\\phi _{i}$ represents the activation maps from layers ReLu1_1, ReLu2_1, ReLu3_1, ReLu4_1 and ReLu5_1.",
"Style Loss.",
"The above activation maps from Eq.REF are further utilized to compute style loss to measure the differences between covariances of the activation maps to mitigate \"checkerboard\" artifacts.",
"Given the $j$ th layer activation feature map of VGG backbone , the style loss is formulated below: $\\begin{aligned}& \\mathcal {L}_{style} = \\mathbb {E}_{j}\\left[ \\left\\Vert G_{j}^\\phi \\left(I_{out}^{\\prime }\\right) - G_{j}^\\phi \\left(I_{gt}^{\\prime }\\right) \\right\\Vert _{1}\\right],\\end{aligned}$ where $G_{j}^\\phi \\in R^{C_{j} \\times C_{j}}$ is a Gram matrix as the selected activation maps.",
"Overall Loss.",
"Based on the above, we finally offer the overall loss as Eq.REF that is minimized to train the overall transformer pipeline: $\\begin{aligned}& \\mathcal {L}_{tran} = \\lambda _{r}\\mathcal {L}_{rec} + \\lambda _{p}\\mathcal {L}_{prec} +\\lambda _{s}\\mathcal {L}_{style}.\\end{aligned}$ For our experiments, we set $\\lambda _{r} = 10$ ; following [14], we set $\\lambda _{p}=0.1$ and $\\lambda _{s}=250$ .",
"Finally, the upsampling operation is performed over the $I_{out}^{\\prime }$ to be final inpainting out $I_{out}$ .",
"Table: Ablation study over PSV (best results are boldface).Figure: Visual comparison between our method and the competitors (zoom-in to see more details).",
"Our method generates more realistic image inpainting over PSV , CelebA-HQ and Places2 with irregular masks .Figure: (a) show the attention maps of exemplar texture references for coarsely filled information over masked region; (b) exhibit the exemplar texture reference vectors to inpaint the masked patches; (c) and (d) visualize the inpainting output together with corresponding structure information.Figure: (a) are the input masked images; (b) are the baseline model; (c) and (d) are the varied models based on (b), Compared with (c) and (d), our method (e) achieves the better results by addressing the artifacts and synthesize plausible semantics on the patches marked by boxes."
],
[
"Implementation Details",
"Our proposed method is implemented in Python and Pytorch.",
"Following [3], we train architecture with AdamW [23] optimizer; the learning rate of transformer and backbone set to $10^{-4}$ and $10^{-5}$ respectively, weight decay to $10^{-4}$ .",
"All transformer weights are initialized with Xavier init [9], and we adopt the ImageNet-pretrained ResNet50 [12] from TORCHVISION with frozen batchnorm layers.",
"We also increase the feature resolution via a dilation to the last stage of the backbone and removing a stride from the first convolution of this stage.",
"Both the transformer encoder and decoder include four layers.",
"The network is trained using $256 \\times 256$ images with irregular masks [21], we conduct experiments on three public datasets that have different characteristics: Paris StreetView (PSV) [6], CelebA-HQ [22] and Places2 [44].",
"We use 2 NVIDIA 2080TI GPU with batch size 32 for PSV, 4 NVIDIA 2080TI GPU for CelebA-HQ and Places2 with batch size 64 to train the transformers."
],
[
"Comparison with state-of-the art methods",
"We quantitatively evaluate our proposed method and state-of-the-arts as per four evaluation metrics: 1) ${\\ell _{1}}$ error; 2) peak signal-to-noise ratio (PSNR); 3) structural similarity index (SSIM) [33]; and 4) Fréchet Inception Score (FID) [13].",
"The ${\\ell _{1}}$ , PSNR and SSIM are used to compare the low-level differences over pixel level between the generated image and ground truth.",
"FID evaluates the perceptual quality by measuring the feature distribution distance between the synthesized and real images.",
"The irregular masked regions of the image are testified with varied ratios over the whole image size.",
"Quantitative Comparisons.",
"We compare our method with the latest competitorsWe only choose the competitors that report the quantitative results over the above three data sets with varied mask ratio.",
": 1) CNNs texture-only methods: GC [39] and PIC [43]; 2) Attention-based method: HiFill [37]; 3) Structure-Texture based methods: MEDFE [14], EC [24], CTSDG [10], EII [30] and 4) decoder-only transformer based methods: ICT [29], BAT [40].",
"For fairness, we directly report the comparable results from [40], [30], [10].",
"As shown in Table.",
"REF , we can see that our method enjoys a smaller ${\\ell _{1}}$ error and FID score, together with larger PSNR and SSIM than the competitors.",
"Particularly, small FID score validates the pros for the global texture references and structure feature representation.",
"As the texture-only method GC and PIC only fill the masked patches via the bounded known regions.",
"While HiFill independently calculates similarities between each coarsely filled patch and all known patches, which misleads the masked patch to be inpainted by only one dominant known region.",
"Despite the intuition of MEDFE to capture the structure information, it failed to exploit the information from all known patches.",
"The similar limitation also holds for EC, CTSDG and EII, which keep consistent with our analysis in Sec.",
"REF .",
"BAT and ICT restore the masked regions upon the pixel level, which cannot well capture the global texture semantics, in line with the principle in Sec.",
"REF .",
"Our method outperforms the others.",
"Qualitative Comparisons.",
"To shed more light on the observation, Fig.REF showcases the visualization results over all methods over three datasets.",
"It can be seen that the inpainting output by our method is more semantically coherent based on surrounding known regions.",
"We validate such intuition by Fig.REF , where REF (a) show the examples of the texture reference vectors encoding global semantic correlations to be reconstructed via other texture references beyond the local regions, to yield a better inpainting output in REF (c).",
"We further visualize the structure information of inpainting output in REF (d), which validate the intuition of our structure-texture matching attention in Sec.",
"REF .",
"User Study.",
"We further perform subjective user study over dataset PSV, CelebA-HQ and Places2.",
"Concretely, we randomly sampled 20 test images from each dataset, Total 10 volunteers are invited to choose the most realistic images from inpainting results generated by the proposed method and some state-of-the-art approaches.",
"As shown in the last column of Table.",
"REF , the result of our method outperforms state-of-the-arts by large margins."
],
[
"Ablation Study",
"To further validate the advantage of each component for our pipeline, we conduct the ablation studies over the following varied models: Baseline model.",
"The baseline model contains neither transformer texture encoder nor the bridge module formulated as Eq.REF , which is equivalent to transformer decoder only model.",
"Unlike the existing models over pixel level, we perform such baseline model over patch level and reconstruct the masked regions.",
"Comparing Fig.REF (b) and (e), the baseline model inattentively produces the blurry noise in the restoring images, especially around complex boundaries such as the green architecture in the red box.",
"Table.REF reports the quantitative results, which further discloses the contribution of global texture reference and bridge module to the performance gain.",
"The importance of bridge module.",
"We abandon Eq.REF for bridge module, and leave Eq.REF only to validate the importance of our bridge module.",
"Our proposed attention transition manner aims at well matching the global texture and structure for inpainting.",
"Comparing Fig.REF (c) with (e), the bridge module can obviously benefit the restoration of structure guided by the consistent global textures.",
"Table.",
"REF also validates such observation.",
"The importance of global texture reference.",
"We testify the model that can only restore the masked regions with local texture representation, i.e., obtaining texture representation via ResNet50 yet without self-attention to yield texture references.",
"Fig.REF (d) offers the visualized results of such model, comparing with our method in REF (e) together with the results in Table.",
"REF , which confirm the benefits of our global texture reference to foster the inpainting performance."
],
[
"Conclusion",
"In this paper, we delve globally into texture and structure information for image inpainting.",
"Technically, the transformer pipeline paired with both encoder and decoder is proposed.",
"The encoder aims at capturing the global texture semantic correlations across the whole image, while the decoder module restores the masked patches, featured with a structure-texture matching attention module to well match the global texture and structure information.",
"An adaptive patch vocabulary is established, to progressively inpaint all the coarsely filled patches via a probabilistic diffusion process.",
"Experimental results over benchmarks validate the advantages of our model over state-of-the-art counterparts.",
"Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No U21A20470, 62172136, 61725203, U1936217.",
"Key Research and Technology Development Projects of Anhui Province (no.",
"202004a5020043)."
]
] | 2209.08217 |
[
[
"Fast, Accurate and Object Boundary-Aware Surface Normal Estimation from\n Depth Maps"
],
[
"Abstract This paper proposes a fast and accurate surface normal estimation method which can be directly used on depth maps (organized point clouds).",
"The surface normal estimation process is formulated as a closed-form expression.",
"In order to reduce the effect of measurement noise, the averaging operation is utilized in multi-direction manner.",
"The multi-direction normal estimation process is reformulated in the next step to be implemented efficiently.",
"Finally, a simple yet effective method is proposed to remove erroneous normal estimation at depth discontinuities.",
"The proposed method is compared to well-known surface normal estimation algorithms.",
"The results show that the proposed algorithm not only outperforms the baseline algorithms in term of accuracy, but also is fast enough to be used in real-time applications."
],
[
"Introduction",
"Surface normal vectors estimation [7], [3], [14] is the common process in different 3D vision and 3D processing task such as 3D surface reconstruction [10], scene segmentation [13], object recognition [16], and others.",
"A complete 3D processing pipeline may fail due to lack of effective surface normal estimation process.",
"Thus, fast and accurate normal estimation is of great importance in practical application.",
"There are several approaches dedicated to normal estimation in the literature depending on the type of input 3D data.",
"For an unorganized point clouds (an unordered set of 3D points), at the first step, a graph should be constructed to identify the neighboring points for each query points.",
"Then, the vector normal to the query point is estimated.",
"The most common approach for surface normal estimation is called plane-PCA ([1]).",
"In this method, a plane is fitted to the all neighboring points (including the query point).",
"Then, the vector normal to the fitted plane is determined by eigen decomposition of the scattering matrix.",
"The eigen vector correspond to the smallest eigen value is considered as the surface normal vector.",
"The plane-PCA method is a robust and accurate method.",
"However, it is not suitable for large-scale point clouds due to its high computational complexity.",
"Beside the plane-PCA, too many research have been done to improve the accuracy of the surface normal estimation process.",
"In [8], in order to improve the normal estimation accuracy for points belonging to high curvature surfaces, an optimum tangent plane is fitted using robust statistics.",
"Randomized Hough Transform (RHT) along with statistical exploration bounds is used to preserve sharp features in [2].",
"To reduce the computational complexity, the authors used a fixed-size accumulator.",
"A GPU-based implementation is proposed in [9] to speed-up a computationally intensive tensor voting algorithm.",
"In order to exclude the outliers, the Deterministic MM-estimator (DetMM) is used in [6].",
"In addition to classical data processing techniques, deep learning-based methods have recently attracted the attention of the research community for surface normal vector estimation [7], [17].",
"However, these methods usually require richly labeled datasets.",
"Comparing to unorganized point clouds, surface normal estimation directly from a depth map (organized point cloud) has received less attention in the literature.",
"However, estimating surface normal from depth maps has the following advantages: No extra processing step is needed to determine the points belonging to the neighborhood of a query point.",
"The complete normal estimation process can be implemented through 2D image processing operators which are much faster than 3D ones.",
"Some research in the literature focuses on the estimation of normal vectors directly from input depth images.",
"In [15] a closed-form expression is proposed for estimating normal vectors at each point.",
"However, improper tangent vector selection led to inaccurate normals map.",
"Holzer et al.",
"[5] proposed a surface normal estimation method based on adaptive neighboring size selection and integral images.",
"However, the accuracy of their method depends on hyperparameters values which are chosen empirically.",
"Also, the performance of their method is degraded facing small objects with high surface curvature.",
"One of the most accurate and fast surface normal estimation methods is presented in [12].",
"While the surface tangent vectors are constructed perfectly, the approximation error of first-order partial derivatives decreases the accuracy of the estimation process.",
"Authors in [4] proposed a fast and accurate method for surface normal estimation.",
"The method is also implemented using GPU to achieve higher performance.",
"Despite the superiority of the method, the method is unable to estimate normal vectors for uniform areas.",
"In our previous work [11], a fast and accurate surface normal estimation method is proposed.",
"In that work, a closed-form expression is proposed for each component of the surface normal vectors.",
"Also, the method is capable of multi-scale implementation which in turn decreases the effect of the measurement noise.",
"However, using multi-scale approach increases the execution time of the algorithm.",
"To address this issue, a fast and accurate surface normal estimation method is presented in this paper.",
"The contributions of this work are as follows: The averaging process in multi-scale approach is used in multi-direction manner to suppress the effect of measurement noise.",
"The multi-direction method is implemented in an efficient manner.",
"The erroneous estimated normal vectors are excluded from final normal map using a simple yet effective method.",
"The rest of this paper is organized as follows: In section 2, our previous work is reviewed briefly as motivation to current work.",
"In section 3 the proposed method is explained in details.",
"Section 4 is dedicated to experiments and results.",
"Finally, the paper is concluded in section 5."
],
[
"Motivation and Background",
"In our previous work [11], a fast method was proposed to estimate surface normal vectors directly from depth maps.",
"In this work, first, the projection of two surface tangent vectors in the depth map is constructed (fig:surfacetangentvectors).",
"Then, the cross product of these two tangent vectors is considered as the surface normal vector at the query point.",
"The closed-form solution for the normal vectors were derived as follows: Figure: 2D projection of surface tangent vectors on the depth map .$n_x=-\\frac{\\alpha }{f_y}d_3 \\left( d_2 -d_1\\right)$ $n_y=-\\frac{\\alpha }{f_x} d_2 \\left( d_3 -d_1\\right)$ $\\begin{split}n_z=&\\frac{\\alpha }{f_x} v_1d_2 \\left( d_3 -d_1 \\right)+\\frac{\\alpha }{f_y} u_1d_3 \\left( d_2 -d_1 \\right)\\\\&+\\frac{\\alpha ^2}{f_xf_y}d_2d_3\\end{split}$ where, $f_x$ and $f_y$ denote the focal lengths.",
"Also, $d_i=g(r_i,c_i)$ , $u_i=\\frac{r_i-o_x}{f_x}$ , $v_i=\\frac{c_i-o_y}{f_y}$ .",
"$o_x$ and $o_y$ are the coordinates of the optical center.",
"In case of noisy input, the averaging process on multi-scale results will reduce the effect of noise.",
"Therefore: $n_x=\\frac{-1}{K}\\sum _{i=1}^{K}\\frac{\\alpha _i}{f_y}d_3 \\left( d_2 -d_1\\right) \\quad i=1,2,\\cdots ,K$ $n_y=\\frac{-1}{K}\\sum _{i=1}^{K}\\frac{\\alpha _i}{f_x} d_2 \\left( d_3 -d_1\\right) \\quad i=1,2,\\cdots ,K$ $\\begin{split}n_z=&\\frac{1}{K}\\sum _{i=1}^{K}\\frac{\\alpha _i}{f_x} v_1d_2 \\left( d_3 -d_1 \\right)+\\frac{\\alpha _i}{f_y} u_1d_3 \\left( d_2 -d_1 \\right)\\\\ &+\\frac{\\alpha _i^2}{f_xf_y}d_2d_3 \\quad i=1,2,\\cdots ,K\\end{split}$ While the single-scale version of this method is fast and gives an accurate estimation of normal vectors for smooth surfaces, the multi-scale normal estimation is slower by a factor equal to the number of scales.",
"Moreover, the effect of depth discontinuity at object boundaries is not considered.",
"In the next section, both shortcomings of our previous work are addressed and a fast, accurate, and object boundary-aware surface normal vector estimation method is presented."
],
[
"Fast normal estimation",
"In our previous work [11], a multi-scale approach is used to reduce the effect of measurement noise on the final estimated surface normal vectors.",
"The averaging operation among different scales can effectively reduce the noise effect.",
"However, using $K$ scales for final normal construction increases the execution time by a factor of $K$ compared to the single-scale approach.",
"In order to benefit from averaging operation in noise effect reduction, here, we use multiple pairs of different tangent vectors to estimate the surface normal at a query point.",
"Then, the final normal vector at each point is obtained by taking the average value of the resulting normal vectors.",
"fig:surfacetangentvectorsmemar shows the projection of four different surface tangent vectors on depth maps.",
"While, the cross product of every vectors pair can be used to estimate the normal vectors, only perpendicular tangent vectors pairs are considered, here.",
"The average value of all resulting normal vectors is the final normal vector for each query point.",
"The averaging process makes this estimation robust against measurement noise.",
"Figure: Projection of surface tangent vectors construction in all four main directionsConsidering all four tangent vector pairs, the normal vector can be estimated as: $\\begin{split}n&=0.25(n_{23}+n_{34}+n_{45}+n_{52}) \\\\&=0.25(s_{12}\\times s_{13}+s_{13}\\times s_{14}+s_{14}\\times s_{15}+s_{15}\\times s_{12}) \\\\&=0.25(s_{12}\\times s_{13}-s_{14}\\times s_{13}+s_{14}\\times s_{15}-s_{12}\\times s_{15}) \\\\&=0.25(s_{12}-s_{14})\\times s_{13}+(s_{14}-s_{12})\\times s_{15}) \\\\&=0.25(s_{12}-s_{14})\\times s_{13}-(s_{12}-s_{14})\\times s_{15}) \\\\&=0.25((s_{12}-s_{14})\\times (s_{13}-s_{15}))\\\\&=0.25(s_{24}\\times s_{35})\\end{split}$ where, $\\mathbf {s_{24}}$ and $ \\mathbf {s_{35}}$ are two tangent vectors which are depicted in fig:surfacetangentvectorsfinal.",
"eq:proofofefficiency proves that the result of the summation of four different cross products can be achieved using a single cross product.",
"This means that using this approach can accelerate the normal estimation process by a factor of 4.",
"Finally, the closed-form solution for the normal vectors can be determined as: $\\mathbf {s_{24}}=\\begin{bmatrix}x_4-x_2\\\\y_4-y_2\\\\z_4-z_2\\end{bmatrix}=\\begin{bmatrix}u_4d_4-u_2d_2\\\\v_4d_4-v_2d_2\\\\d_4-d_2\\end{bmatrix}$ $\\mathbf {s_{35}}=\\begin{bmatrix}x_5-x_3\\\\y_5-y_3\\\\z_5-z_3\\end{bmatrix}=\\begin{bmatrix}u_5d_5-u_3d_3\\\\v_5d_5-v_3d_3\\\\d_5-d_3\\end{bmatrix}$ $n_x=-\\frac{\\alpha }{4f_y}\\left( d_3 + d_5\\right) \\left( d_2 -d_4\\right)$ $n_y=-\\frac{\\alpha }{4f_x} \\left( d_2 + d_4\\right) \\left( d_3 -d_5\\right)$ $n_z=-u_1n_x-v_1n_y+ \\frac{\\alpha ^2}{4f_xf_y}\\left( d_2 + d_4\\right) \\left( d_3 +d_5\\right)$ Figure: Projection of final surface tangent vectors"
],
[
"Considering object boundaries",
"The proposed method as well as our previous work [11] works well when facing the points belonging to surfaces without depth discontinuances.",
"However, at the object boundaries, at least one of the surface tangent vectors is not valid.",
"Therefore, the orientation of the estimated normal vector may be erroneous.",
"To tackle this problem, a new approach is presented here.",
"Surface normals are unit vectors and their orientations are the only important parameters in 3D processing.",
"This is why only $\\phi $ and $\\theta $ components are taken into account when a normal vector is converted from Cartesian coordinates into spherical one.",
"However, the length of estimated normal vector depends on the length of tangent vectors as fallows: $\\parallel n\\parallel _2 ~\\propto ~ \\parallel s_{24}\\times s_{35}\\parallel _2 ~\\propto ~ \\parallel s_{24}\\parallel _2 .\\parallel s_{35}\\parallel _2$ Since at least one of the tangent vectors has a large length in object boundaries, eq:veclength indicates that the length of the normal also should be large in those areas.",
"Thus, the length of the vector ($r$ component in spherical coordinates) can be used as a mask to find and exclude erroneous normal estimation.",
"A simple thresholding on $r$ values gives the location of outliers.",
"The final normal estimation is performed by applying the invalid points mask to the estimated normal map."
],
[
"Experimental results",
"In order to evaluate the normal estimation performance of the proposed method, experiments were carried out on real data captured by a Microsoft Kinect Azure RGB-D camera as well as synthetic depth from 3F2N [4] dataset.",
"All the algorithms are implemented in MATLAB.",
"The full specifications of implementation environment are reported in tab:specruntime.",
"There are two ways to demonstrate a surface normal vector as an image.",
"In the first one, each component of the normal vector is considered as a color channel and the resulting vector from depth map can be treated as a color image.",
"While this way is straightforward, it can not highlight small errors in the estimated normal vectors.",
"The second way is to convert the normal vector to spherical coordinates and investigate $I_{\\phi }$ and $I_{\\theta }$ components.",
"$I_\\phi $ and $I_\\theta $ can be calculated as: $I_\\phi =\\tan ^{-1}\\left(\\frac{n_y}{n_x}\\right)$ $I_\\theta =\\tan ^{-1} \\left( \\frac{\\sqrt{n_x^2+n_y^2}}{n_z} \\right)$ Figure: 𝐈 θ \\mathbf {I_\\theta } images of the estimation results of different algorithms (Real images).",
"From left: the ground truth, Fan's method , Holzer's method , Nakagawa's method , Moradi's method , the proposed method.Figure: 𝐈 φ \\mathbf {I_\\phi } images of the estimation results of different algorithms (Real images).",
"From left: the ground truth, Fan's method , Holzer's method , Nakagawa's method , Moradi's method , the proposed method.Figure: 𝐈 θ \\mathbf {I_\\theta } images of the estimation results of different algorithms (Synthetic images from 3F2N dataset ).",
"From left: the depth image, ground truth, Fan's method , Nakagawa's method , Moradi's method , the proposed method.Figure: 𝐈 φ \\mathbf {I_\\phi } images of the estimation results of different algorithms (Synthetic images from 3F2N dataset ).",
"From left: the depth image, ground truth, Fan's method , Nakagawa's method , Moradi's method , the proposed method.The results of $I_{\\theta }$ and $I_{\\phi }$ for different real depth maps are depicted in fig:normal-estimationtheta and fig:normal-estimationphi, respectively.",
"As shown in these figures, the proposed method outperforms all the baseline algorithms in term of similarity to the ground truth images.",
"The second place belongs to our previous work [11].",
"In order to have a fair comparison, the $\\alpha $ value for both algorithms is set to 2.",
"Also, the results of the plane fitting-based method with 25 neighboring points are considered as ground truths.",
"fig:normal-estimationthetafan and fig:normal-estimationphifan show the result of normal estimation on 3F2N dataset.",
"Again, the proposed method and our previous work outperform the other works.",
"Since, the synthetic images are smooth enough, the Nakagawa's method does not affected by sensitivity of partial derivatives to noise.",
"This is why this method performs well on synthetic data compared to real data.",
"Table: The full specifications of the implementation environmentFor quantitative comparison, the Mean Squared error (MSE) is used as the performance metric.",
"tab:tabp1, tab:tabp2, tab:tabp1fan, and tab:tabp2fan show the MSE values of both $\\theta $ and $\\phi $ image components.",
"As reported in the tables, the proposed method shows the best performance among all the baseline algorithms except for the $\\phi $ component of the synthetic depth images which there is not a significant difference among all the baseline algorithms.",
"Since the synthetic data are relatively smooth, the $\\alpha $ value for the proposed method as well as our previous work [11] is set to 1.",
"In this situation, Nakagawa's method shows same behavior as our previous work (tab:tabp1fan and and tab:tabp2fan).",
"Note that all images are normalized in $[0-2\\pi ]$ range.",
"Table: Mean Squared error (MSE) of I θ I_\\theta imagesTable: Mean Squared error (MSE) of I φ I_\\phi imagesTable: Mean Squared error (MSE) of I θ I_\\theta images (3F2N dataset )Table: Mean Squared error (MSE) of I φ I_\\phi images (3F2N dataset )The color image representation of normal vectors can be used to demonstrate the erroneous surface normal estimation at object boundaries.",
"fig:refinednormals shows the invalid normal vectors removal operation, step by step.",
"Fig.",
"fig:initnormals shows the estimated surface normal vectors using eq:proofofefficiency.",
"As shown in this figure, there are some erroneous estimation at the locations with the depth discontinuity.",
"As mentioned in sec:obc, the large values of $r$ component (Fig.",
"fig:rcomponent) in spherical coordinates have a high potential to be the erroneous estimation.",
"A simple thresholding on $r$ values gives the location of outliers (Fig.",
"fig:rmask).",
"The final normal estimation is performed by applying the mask in Fig.",
"fig:rmask to Fig.",
"fig:initnormals.",
"Fig.",
"fig:refindnormalssub shows the final result of surface normal estimation.",
"Figure: Surface normal vectors refinement at object boundaries.",
"a) initial surface normal estimation using eq:proofofefficiency, b) value of rr component for each point, c) outlier removal mask, and d) final result of surface normal estimationFinally, in order to evaluate the running time of the baseline algorithms as well as the proposed one, all the algorithms are implemented in the MATLAB environment (The MATLAB implementation of the 3F2N method is available in their github repository [4]).",
"Each algorithm is executed 200 times and the average running times are reported in tab:msexetime.",
"As reported in the table, the proposed method ranked in second place after Moradi's method (our previous work) [11].",
"The proposed method is slightly slower than our previous work.",
"Note that, a single scale implementation of the proposed method is equivalent to four scales implementation of our previous work.",
"Thus, this work is almost four times faster than our previous work.",
"The normal estimation process can be performed in 135 fps using the proposed method.",
"Fan's method has a straightforward implementation.",
"However, searching for invalid points ($\\Delta Z=0$ ) in the normal map increases the execution time of this method.",
"Table: The average execution time for different normal estimation methods for a 576×640576\\times 640 depth image"
],
[
"Conclusion",
"In this paper an improved version of our previous work is presented.",
"In our previous work, the normal estimation process is formulated as a closed-form solution which was efficiently implemented.",
"However, multi-scale implementation is necessary for tackling measurement noise which in turn leads to lower computational efficiency.",
"In this work, instead of changing the pixel distance value parameter $\\alpha $ the averaging process is adopted for different directions to reduce the effects of the measurement noise.",
"Next, the multi-direction approach is reformulated so that it can be efficiently implemented.",
"So far, the proposed normal estimation method is fast and accurate for smooth surfaces.",
"However, it produces erroneous results when facing object boundaries and depth discontinuities.",
"To address this issues, a simple yest effective mask is constructed based on the length of the estimated normal vectors.",
"This mask is used to exclude erroneous normal vectors from final normal map.",
"The proposed method is compared to some well-known surface normal estimation algorithms.",
"Qualitative and quantitative comparisons on real as well as synthetic depth images show that the proposed surface normal estimation method outperforms the baseline algorithms in both terms of accuracy and computational complexity."
]
] | 2209.08241 |
[
[
"Geometric Eisenstein Series, Intertwining Operators, and Shin's\n Averaging Formula"
],
[
"Abstract In the geometric Langlands program over function fields, Braverman-Gaitsgory and Laumon constructed geometric Eisenstein functors which geometrize the classical construction of Eisenstein series.",
"Fargues and Scholze very recently constructed a general candidate for the local Langlands correspondence, via a geometric Langlands correspondence occurring over the Fargues-Fontaine curve.",
"We carry some of the theory of geometric Eisenstein series over to the Fargues-Fontaine setting.",
"Namely, given a quasi-split connected reductive group $G/\\mathbb{Q}_{p}$ with simply connected derived group and maximal torus $T$, we construct an Eisenstein functor $\\mathrm{nEis}(-)$, which takes sheaves on $Bun_{T}$ to sheaves on $Bun_{G}$.",
"We show that, given a sufficiently nice $L$-parameter $\\phi_{T}: W_{\\mathbb{Q}_{p}} \\rightarrow \\phantom{}^{L}T$, there is a Hecke eigensheaf on $Bun_{G}$ with eigenvalue $\\phi$, given by applying $\\mathrm{nEis}(-)$ to the Hecke eigensheaf $\\mathcal{S}_{\\phi_{T}}$ on $Bun_{T}$ attached to $\\phi_{T}$ by Fargues and Zou.",
"We show that $\\mathrm{nEis}(\\mathcal{S}_{\\phi_{T}})$ interacts well with Verdier duality, and, assuming compatibility of the Fargues-Scholze correspondence with a suitably nice form of the local Langlands correspondence, provide an explicit formula for the stalks of the eigensheaf in terms of parabolic inductions of the character $\\chi$ attached to $\\phi_{T}$.",
"This has several surprising consequences.",
"First, it recovers special cases of an averaging formula of Shin for the cohomology of local Shimura varieties with rational coefficients, and generalizes it to the non-minuscule case.",
"Second, it refines the averaging formula in the cases where the parameter $\\phi$ is sufficiently nice, giving an explicit formula for the degrees of cohomology that certain parabolic inductions sit in, and this refined formula holds even with torsion coefficients."
],
[
"Geometric Eisenstein Series over Function Fields",
"In the Langlands program, Eisenstein series are a way of describing the non-cuspidal automorphic spectrum of a group in terms of the cuspidal automorphic spectrum of its proper Levi subgroups.",
"Over function fields these objects have several geometric incarnations, as first studied extensively by Laumon in the case of $\\mathrm {GL}_{n}$ and later refined by Braverman-Gaitsgory for general reductive groups.",
"In particular, if one is interested in the function field of a curve $Y$ over a finite field $k$ then one replaces the functions defining Eisenstein series by certain automorphic sheaves.",
"Namely, let $G/k$ be a split connected reductive group; then one wishes to construct \"Eisenstein sheaves\" on $\\mathrm {Bun}_{G}$ the moduli stack of $G$ -bundles on $Y$ .",
"To do this, for $P \\subset G$ a proper parabolic subgroup with Levi factor $M$ one considers the following diagram of moduli stacks of bundles $\\begin{tikzcd}& \\mathrm {Bun}_{P} [dr,\"\\mathfrak {p}_{P}\"] [d,\"\\mathfrak {q}_{P}\"] & \\\\& \\mathrm {Bun}_{M} & \\mathrm {Bun}_{G}\\end{tikzcd}$ where $\\mathrm {Bun}_{P}$ is the moduli stack of $P$ -bundles $\\mathcal {G}_{P}$ on $Y$ and the maps $\\mathfrak {p}_{P}$ (resp.",
"$\\mathfrak {q}_{P}$ ) send $\\mathcal {G}_{P}$ to the $G$ -bundle (resp.",
"$M$ -bundle) $\\mathcal {G}_{P} \\times ^{P} G$ (resp.",
"$\\mathcal {G}_{P} \\times ^{P} M$ ).",
"Using this, one can define an Eisenstein functor which (up to shifts and twists) is given by $ \\mathrm {Eis}_{P}(-) := \\mathfrak {p}_{P!",
"}\\mathfrak {q}_{P}^{*}(-) $ taking $\\ell $ -adic sheaves on $\\mathrm {Bun}_{M}$ to $\\ell $ -adic sheaves on $\\mathrm {Bun}_{G}$ .",
"Under the function-sheaf dictionary, the values of this functor give rise to the classical Eisenstein series.",
"In this geometric context, one can ask for even more.",
"Namely, in the geometric Langlands correspondence one is interested in constructing Hecke eigensheaves on $\\mathrm {Bun}_{G}$ , and it is natural to ask whether one could upgrade this construction to a functor that is well-behaved with respect to the eigensheaf property.",
"In particular, if $\\hat{M}$ (resp.",
"$\\hat{G}$ ) denotes the Langlands dual group of $M$ (resp.",
"$G$ ) one can consider a $\\hat{M}$ -local system $E_{\\hat{M}}$ and a Hecke eigensheaf $\\mathcal {S}_{E_{\\hat{M}}}$ with eigenvalue $E_{\\hat{M}}$ .",
"One then considers the induced $\\hat{G}$ local system $E_{\\hat{G}}$ given by the natural embedding $\\hat{M} \\hookrightarrow \\hat{G}$ , and one would like to construct a functor that produces a eigensheaf with eigenvalue $E_{\\hat{G}}$ from the Hecke eigensheaf $\\mathcal {S}_{E_{\\hat{M}}}$ .",
"One might hope that $\\mathrm {Eis}_{P}(\\mathcal {S}_{E_{\\hat{M}}})$ works; however, this is too naive.",
"Namely, one expects such sheaves to be well-behaved under Verdier duality, and one can easily check that $\\mathrm {Eis}_{P}(-)$ will not commute with Verdier duality, since the map $\\mathfrak {p}_{P}$ is not proper.",
"To remedy this, one considers relative Drinfeld compactifications of the map $\\mathfrak {p}_{P}$ , denoted $\\widetilde{\\mathrm {Bun}}_{P}$ and $\\overline{\\mathrm {Bun}}_{P}$ , respectively.",
"These compactifications are equipped with open immersions $\\widetilde{j}: \\mathrm {Bun}_{P} \\hookrightarrow \\widetilde{\\mathrm {Bun}}_{P}$ and $j: \\mathrm {Bun}_{P} \\hookrightarrow \\overline{\\mathrm {Bun}}_{P}$ , which realize $\\mathrm {Bun}_{P}$ as an open and dense subspace, and are defined by considering parabolic structures with torsion at finitely many Cartier divisors.",
"Moreover, they both have maps $ \\overline{\\mathfrak {p}}_{P}: \\overline{\\mathrm {Bun}}_{P} \\rightarrow \\mathrm {Bun}_{G} $ and $ \\widetilde{\\mathfrak {p}}_{P}: \\widetilde{\\mathrm {Bun}}_{P} \\rightarrow \\mathrm {Bun}_{G} $ which are proper after restricting to a connected component and extend $\\mathfrak {p}_{P}$ , as well as maps $\\widetilde{\\mathfrak {q}}_{P}: \\widetilde{\\mathrm {Bun}}_{P} \\rightarrow \\mathrm {Bun}_{M}$ and $\\overline{\\mathfrak {q}}_{P}: \\overline{\\mathrm {Bun}}_{P} \\rightarrow \\mathrm {Bun}_{M^{ab}}$ extending the natural maps $\\mathfrak {q}_{P}: \\mathrm {Bun}_{P} \\rightarrow \\mathrm {Bun}_{M}$ and $\\mathfrak {q}_{P}^{\\dagger }: \\mathrm {Bun}_{P} \\xrightarrow{} \\mathrm {Bun}_{M} \\rightarrow \\mathrm {Bun}_{M^{ab}}$ , respectively.",
"To obtain a sheaf that interacts well with Verdier duality, one needs to take account for the singularities of the compactification.",
"Namely, if one considers the intersection cohomology sheaf $\\mathrm {IC}_{\\widetilde{\\mathrm {Bun}}_{P}}$ of $\\widetilde{\\mathrm {Bun}}_{P}$ then the desired functor is given by $ \\widetilde{\\mathrm {Eis}}_{P}(-) := \\widetilde{\\mathfrak {p}}_{P*}(\\widetilde{\\mathfrak {q}}_{P}^{*}(-) \\otimes \\mathrm {IC}_{\\widetilde{\\mathrm {Bun}}_{P}})$ One of the main results of Braverman-Gaitsgory is that this satisfies the desired Hecke eigensheaf property when applied to $\\mathcal {S}_{E_{\\hat{M}}}$ .",
"One may also wonder what this corresponds to at the level of functions.",
"We recall that the classical Eisenstein series is known to satisfy a functional equation after multiplying by an appropriate ratio of $L$ -values.",
"The compactified Eisenstein series corresponds to this completed Eisenstein series under the function-sheaf dictionary.",
"In fact, in certain cases one can see the usual functional equation at the sheaf-theoretic level.",
"If $P = B$ is the Borel and we consider the appropriately normalized Hecke eigensheaf $\\mathcal {S}_{E_{\\hat{T}}}$ associated to $E_{\\hat{T}}$ via geometric class field theory then $\\widetilde{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ satisfies a functional equation under a regularity hypothesis on the local system $E_{\\hat{T}}$ .",
"Now let $w \\in W_{G}$ be an element of the Weyl group of $G$ with $\\widetilde{w} \\in N(T)$ a choice of representative.",
"Then $\\widetilde{w}$ acts on $\\mathrm {Bun}_{T}$ , and, if we write $\\mathcal {S}_{E_{\\hat{T}}}^{w}$ for the pullback of $\\mathcal {S}_{E_{\\hat{T}}}$ along this automorphism, we have the following result.",
"Theorem 1.1 For $E_{\\hat{T}}$ a regular $\\hat{T}$ -local system on $Y$ and a choice of representative $\\widetilde{w} \\in N(\\hat{T})$ of $w \\in W_{G}$ , we have an isomorphism $ \\widetilde{\\mathrm {Eis}}_{B}(\\mathcal {S}_{E_{\\hat{T}}}) \\simeq \\widetilde{\\mathrm {Eis}}_{B}(\\mathcal {S}^{w}_{E_{\\hat{T}}}) $ of $\\ell $ -adic sheaves on $\\mathrm {Bun}_{G}$ .",
"As alluded to above, one can see that, after passing to functions, this gives precisely the well-known functional equation for the Eisenstein series multiplied by the appropriate ratio of $L$ -values ().",
"Moreover, by , under the regularity assumption the sheaf $\\widetilde{\\mathrm {Eis}}(\\mathcal {S}_{E_{\\hat{T}}})$ is perverse.",
"The main goal of this note is to explore what this theory of geometric Eisenstein series has to tell us in the context of the recent geometric Langlands correspondence constructed by Fargues and Scholze."
],
[
"Hecke Eigensheaves over the Fargues-Fontaine Curve",
"Consider two distinct primes $\\ell \\ne p$ .",
"Let $G/\\mathbb {Q}_{p}$ be a quasi-split connected reductive group with simply connected derived group over the $p$ -adic numbers, and set $\\breve{\\mathbb {Q}}_{p}$ to be the completion of the maximal unramified extension of $\\mathbb {Q}_{p}$ with Frobenius $\\sigma $ .",
"Let $W_{\\mathbb {Q}_{p}} \\subset \\Gamma := \\mathrm {Gal}(\\overline{\\mathbb {Q}}_{p}/\\mathbb {Q}_{p})$ denote the Weil group.",
"In , Fargues and Scholze developed the geometric framework required to make sense of objects like $\\mathrm {Bun}_{G}$ , the moduli stack of $G$ -bundles on the Fargues-Fontaine curve $X$ , and show that the local Langlands correspondence for $G$ can be viewed as a geometric Langlands correspondence over $X$ .",
"Namely, they prove a version of Geometric Satake in this setup, allowing them to construct Hecke operators and in turn excursion operators.",
"Their Hecke operators take sheaves on $\\mathrm {Bun}_{G}$ to sheaves on $\\mathrm {Bun}_{G}$ with a continuous $W_{\\mathbb {Q}_{p}}$ -action, via some version of Drinfeld's Lemma.",
"As a consequence, they were able to construct semi-simple Langlands parameters $\\phi ^{\\mathrm {FS}}_{\\pi }: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ attached to any smooth irreducible representation $\\pi $ of $G(\\mathbb {Q}_{p})$ .",
"Predating the construction of the Fargues-Scholze local Langlands correspondence was Fargues' conjecture as formulated in .",
"This asserted the existence of Hecke Eigensheaves attached to supercuspidal $L$ -parameters $\\phi : W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ , where $\\phantom{}^{L}G = \\hat{G} \\ltimes W_{\\mathbb {Q}_{p}}$ is the $L$ -group of $G$ .",
"More precisely, given such a $\\phi $ , Fargues conjectured the existence of a $\\overline{\\mathbb {Q}}_{\\ell }$ -sheaf $\\mathcal {S}_{\\phi }$ on $\\mathrm {Bun}_{G}$ such that, if one acts via a Hecke operator $T_{V}$ corresponding to a representation $V \\in \\mathrm {Rep}_{\\overline{\\mathbb {Q}}_{\\ell }}(\\phantom{}^{L}G)$ then there is an isomorphism $ T_{V}(\\mathcal {S}_{\\phi }) \\simeq \\mathcal {S}_{\\phi } \\boxtimes r_{V} \\circ \\phi $ of sheaves on $\\mathrm {Bun}_{G}$ with continuous $W_{\\mathbb {Q}_{p}}$ -action satisfying natural compatiblities.",
"The moduli stack $\\mathrm {Bun}_{G}$ is stratified by elements of the Kottwitz set $B(G) := G(\\breve{\\mathbb {Q}}_{p})/(b \\sim gb\\sigma (g)^{-1})$ , giving rise to Harder-Narasimhan (= HN)-strata $\\mathrm {Bun}_{G}^{b}$ for all $b \\in B(G)$ .",
"It was conjectured that the sheaf $\\mathcal {S}_{\\phi }$ must be supported on the basic locus $\\bigsqcup _{b \\in B(G)_{basic}} \\mathrm {Bun}_{G}^{b}$ or, in bundle-theoretic terms, the locus defined by semistable bundles.",
"Each of the basic strata $\\mathrm {Bun}_{G}^{b}$ are isomorphic to the classifying stack $[\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}]$ , where $J_{b}$ is the $\\sigma $ -centralizer attached to $b \\in B(G)$ .",
"The $\\sigma $ -centralizers of the basic elements parameterize extended pure inner forms of $G$ in the sense of Kottwitz , and the restrictions of the sheaf to these classifying stacks can be interpreted as smooth representations of $J_{b}(\\mathbb {Q}_{p})$ .",
"Using this, Fargues also gave a conjectural description for what the restrictions of the sheaf should be given by.",
"In particular, for $b \\in B(G)_{basic}$ the restriction to $\\mathrm {Bun}_{G}^{b}$ should be a direct sum (up to multiplicities)There are no higher multiplicities if the centralizer of $\\phi $ is abelian.$\\bigoplus _{\\pi \\in \\Pi _{\\phi }(J_{b})} \\pi $ , where $\\Pi _{\\phi }(J_{b})$ is the $L$ -packet over $\\phi $ as conjectured by Kaletha's refined local Langlands correspondence for $G$ .",
"Assuming the refined local Langlands, the verification of the Hecke eigensheaf property ultimately reduces to a strong form of the Kottwitz conjecture for the cohomology of a space of shtukas parameterizing modifications $ \\mathcal {F}_{b} \\rightarrow \\mathcal {F}_{b^{\\prime }} $ on $X$ bounded by a geometric dominant cocharacter $\\mu $ , where $b,b^{\\prime } \\in B(G)_{basic}$ are appropriately chosen basic elements with respect to $\\mu $ , and $\\mathcal {F}_{b}$ and $\\mathcal {F}_{b^{\\prime }}$ denote the bundles on $X$ corresponding to $b,b^{\\prime } \\in B(G)$ .",
"Since the work of Fargues-Scholze, the construction of this eigensheaf has been carried out in several cases.",
"For tori, it follows from the work of Fargues , , and Zou .",
"For $G = \\mathrm {GL}_{n}$ , this is a result of Anschütz and Le-Bras .",
"For general reductive groups, a somewhat general strategy for constructing this eigensheaf in the particular case of the group $\\mathrm {GSp}_{4}$ is layed out in , by showing compatibility of the Fargues-Scholze correspondence with the refined local Langlands correspondence of Kaletha , and then using this to deduce the non-minuscule cases of the Kottwitz conjecture required for the verification of the Hecke eigensheaf property via the spectral action .",
"This strategy is carried out for odd unitary groups in the paper .",
"We recall that the fibers of the local Langlands correspondence over supercuspidal parameters should solely consist of supercuspidal representations.",
"Therefore, the above eigensheaves can be thought of as analogous to supercuspidal representations in the classification of smooth irreducible representations.",
"To have a more definitive connection between the theory of smooth representations and Fargues' eigensheaves, it becomes desirable to construct eigensheaves that serve as the analogues of parabolic inductions of supercuspidals, which will analogously be \"parabolically induced\" from the eigensheaves attached to supercuspidal parameters.",
"This will be precisely what carrying over the theory of the previous section to the Fargues-Fontaine setting gives us."
],
[
"Geometric Eisenstein Series over the Fargues-Fontaine Curve",
"Let $A \\subset T \\subset B \\subset G$ be a choice of maximal split torus, maximal torus, and Borel, respectively.",
"In this paper, we will be concerned with studying the geometric Eisenstein functor over the Fargues-Fontaine curve in the principal case (i.e where the parabolic $P \\subset G$ is the Borel).",
"Restricting to the principal case has several advantages.",
"For one, one can unconditionally speak about the Hecke eigensheaf attached to a parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T$ valued in the maximal torus.",
"Additionally, in this case there exists only one Drinfeld compactification $\\overline{\\mathrm {Bun}}_{B}$ of the moduli space of $B$ -structures $\\mathrm {Bun}_{B}$ , which has a fairly manageable geometry.",
"As mentioned in §1.1, there are in general two compactifications $\\overline{\\mathrm {Bun}}_{P}$ and $\\widetilde{\\mathrm {Bun}}_{P}$ .",
"The compactification $\\overline{\\mathrm {Bun}}_{P}$ has a relatively simple geometry and can be understood more or less the same as $\\overline{\\mathrm {Bun}}_{B}$ .",
"The problem is that $\\overline{\\mathrm {Bun}}_{P}$ admits only a map to $\\mathrm {Bun}_{M^{ab}}$ and not to $\\mathrm {Bun}_{M}$ .",
"This means we can prove analogous results for the Hecke eigensheaves attached to characters induced from $\\phi : W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}M^{ab}$ which correspond to generalized principal series representations induced from $M$ .",
"We have chosen not to do this in this note for simplicity.",
"However, if one wants to consider inductions of Hecke eigensheaves attached to a general supercuspidal parameter $\\phi _{M}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}M$ then one is forced to understand the much more complicated geometry of the space $\\widetilde{\\mathrm {Bun}}_{P}$ .",
"Certainly, some analogues of the results proven in this paper should be possible, but there are many technical hurdles that need to be overcome."
],
[
"Geometric Eisenstein Series",
"Throughout, we will let $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ be a continuous parameter valued in the $L$ -group of $T$ , where $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ has the discrete topology.",
"Our aim is to construct an eigensheaf (Definition REF ) with eigenvalue $\\phi $ , the composition of $\\phi _{T}$ with the natural embedding $\\phantom{}^{L}T(\\Lambda ) \\rightarrow \\phantom{}^{L}G(\\Lambda )$ .",
"This will be an object in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ the category of lisse-étale solid $\\Lambda $ -sheaves, as defined in .",
"We do not work directly with this category of solid sheaves in our argument, as the usual six functors are not as well behaved in this case.",
"Instead, we will first restrict to the case where $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ and one has an isomorphism $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {F}}_{\\ell }) \\simeq \\mathrm {D}_{\\text{ét}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {F}}_{\\ell })$ with the usual category of étale $\\overline{\\mathbb {F}}_{\\ell }$ sheaves as defined in .",
"We then construct the lisse-étale sheaves with $\\overline{\\mathbb {Z}}_{\\ell }$ and $\\overline{\\mathbb {Q}}_{\\ell }$ coefficients from this sheaf.",
"For the first part of this section, we will always assume that $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ unless otherwise stated, and to simplify the notation, we adopt the convention that, when working with $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ , we will denote the derived category of étale sheaves on a $v$ -stack or diamond $Z$ by simply writing $\\mathrm {D}(Z)$ .",
"We will assume that the prime $\\ell $ is very good with respect to the group $G$ () throughout, to avoid complications in this $\\ell $ -modular setting.",
"We let $\\mathcal {S}_{\\phi _{T}}$ be the eigensheaf on the moduli stack $\\mathrm {Bun}_{T}$ parameterizing $T$ -bundles on $X$ attached to $\\phi _{T}$ by Fargues , and Zou .",
"Our aim is to construct an eigensheaf with respect to the parameter $\\phi $ by applying a geometric Eisenstein functor to $\\mathcal {S}_{\\phi _{T}}$ .",
"To do this, one needs to show that the relevant geometric objects used in defining geometric Eisenstein series are well-behaved in this framework.",
"Namely, one can show that the moduli stack of $B$ -bundles on $X$ , denoted $\\mathrm {Bun}_{B}$ , gives rise to an Artin $v$ -stack and the maps in the natural diagram ${}\\begin{tikzcd}& \\mathrm {Bun}_{B} [dr,\"\\mathfrak {p}\"] [d,\"\\mathfrak {q}\"] & \\\\& \\mathrm {Bun}_{T} & \\mathrm {Bun}_{G}\\end{tikzcd}$ have good geometric properties; $\\mathfrak {q}$ is a cohomologically smooth (non-representable) map of Artin $v$ -stacks, and $\\mathfrak {p}$ is compactifiable and representable in locally spatial diamonds (See §REF ).",
"It follows that one has a well-defined functor given by $\\mathfrak {p}_{!",
"}\\mathfrak {q}^{*}:\\mathrm {D}(\\mathrm {Bun}_{T}) \\rightarrow \\mathrm {D}(\\mathrm {Bun}_{G})$ using the functors constructed in .",
"In order to make the functor $\\mathfrak {p}_{!",
"}\\mathfrak {q}^{*}$ behave well with respect to Verdier duality, one needs to take into account the appropriate shifts and twists coming from the dualizing object on $\\mathrm {Bun}_{B}$ .",
"Namely, using the cohomological smoothness of $\\mathfrak {q}$ and $\\mathrm {Bun}_{T}$ , it is easy to see that the moduli stack $\\mathrm {Bun}_{B}$ is cohomologically smooth of some $\\ell $ -dimension given by a locally constant function $\\mathrm {dim}(\\mathrm {Bun}_{B}): |\\mathrm {Bun}_{B}| \\rightarrow \\mathbb {Z}$ , where $|\\mathrm {Bun}_{B}|$ is the underlying topological space of $\\mathrm {Bun}_{B}$ .",
"It follows that, $v$ -locally on $\\mathrm {Bun}_{B}$ , the dualizing object is given by $\\Lambda [2\\mathrm {dim}(\\mathrm {Bun}_{B})]$ .",
"This leads us to our first attempt to define the Eisenstein functor as $\\mathrm {Eis}(-) := \\mathfrak {p}_{!",
"}(\\mathfrak {q}^{*}(-)[\\mathrm {dim}(\\mathrm {Bun}_{B})])$ .",
"While this definition is closer to what we want, it has one key flaw; even though the dualizing object on $\\mathrm {Bun}_{B}^{\\nu }$ is $v$ -locally isomorphic to $\\Lambda [2\\mathrm {dim}(\\mathrm {Bun}_{B})]$ it is not equal to this sheaf on the nose.",
"In particular, usually one would need to include some Tate twists.",
"However, all these spaces are defined over the base $\\ast = \\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}}$ , so one cannot naively make sense of this.",
"Typically, this would be encoded via some kind of Frobenius descent datum, but here the answer is even more interesting.",
"Understanding this requires one to explicitly compute the dualizing object on $\\mathrm {Bun}_{B}$ .",
"This can be accomplished by noting that each of the connected components of $\\mathrm {Bun}_{B}$ are related to Banach-Colmez spaces, the diamonds parameterizing global sections $H^{0}(X,\\mathcal {E})$ and $H^{1}(X,\\mathcal {E})$ for $\\mathcal {E}$ a bundle on the Fargues-Fontaine curve $X$ , as studied in and .",
"These objects have a relatively explicit description in terms of pro-étale quotients of perfectoid open unit discs and also have absolute versions defined over $\\mathop {\\rm Spd}{\\mathbb {F}_{p^{s}}}$ , for some $s \\ge 1$ .",
"Using this explicit description, one can compute that the dualizing object on these absolute spaces is given by the constant sheaf with the appropriate shift and Tate twist by the dimension.",
"However, these Tate twists do not disappear over the algebraic closure $\\overline{\\mathbb {F}}_{p}$ ; namely, the action of geometric Frobenius is manifestly related to the action of $p^{\\mathbb {Z}} \\in H^{0}(X,\\mathcal {O}_{X})^{*} = \\mathbb {Q}_{p}^{*}$ by scaling of global sections, essentially by definition of the Fargues-Fontaine curve.",
"This allows one to see that the dualizing object on the Banach-Colmez space over $\\overline{\\mathbb {F}}_{p}$ is isomorphic to a shift of the constant sheaf together with a descent datum with respect to the scaling action by $\\mathbb {Q}_{p}^{*}$ given by the norm character $|\\cdot |: \\mathbb {Q}_{p}^{*} \\rightarrow \\Lambda ^{*}$ , which serves the role of a Frobenius descent datum.",
"Unravelling this all, this will ultimately tell us that the dualizing object on $\\mathrm {Bun}_{B}$ is related to the modulus character $\\delta _{B}$ .",
"One way of nicely encoding these modulus character twists is to consider the sheaf $\\Delta _{B}^{1/2}$ on $\\mathrm {Bun}_{T}$ , which will be the Hecke eigensheaf attached to the $L$ -parameter $ \\hat{\\rho } \\circ |\\cdot |: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda ) $ where $\\hat{\\rho }$ denotes the half sum of all positive roots of $G$ and we abusively write $|\\cdot |: W_{\\mathbb {Q}_{p}} \\rightarrow \\Lambda ^{*}$ for the norm character of $W_{\\mathbb {Q}_{p}}$ .",
"As a sheaf on $\\mathrm {Bun}_{T}$ , the stalks on the connected components $\\mathrm {Bun}_{T}^{\\nu } \\simeq [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ will just be given by the character $\\delta _{B}^{1/2}:T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ .",
"Similarly, we write $\\Delta _{B}$ for the sheaf given by the $L$ -parameter $2\\hat{\\rho } \\circ |\\cdot |$ .",
"We can now state our first Theorem.",
"Theorem 1.2 (Theorem REF ) The dualizing object on $\\mathrm {Bun}_{B}$ is isomorphic to $ \\mathfrak {q}^{*}(\\Delta _{B})[2\\mathrm {dim}(\\mathrm {Bun}_{B})] $ In particular, the sheaf $ \\mathrm {IC}_{\\mathrm {Bun}_{B}} := \\mathfrak {q}^{*}(\\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})] $ is Verdier self-dual on $\\mathrm {Bun}_{B}$ .",
"With this in hand, we can refine the previous definition of the Eisenstein functor.",
"We define the normalized Eisenstein functor to be $ \\mathrm {nEis}(-) := \\mathfrak {p}_{!",
"}(\\mathfrak {q}^{*}(-) \\otimes \\mathrm {IC}_{\\mathrm {Bun}_{B}}) $ This is already very suggestive.",
"Indeed, the (unnormalized) Eisenstein functor will have stalks related to the unnormalized parabolic induction of the characters $\\chi $ , and the normalized Eisenstein series will have stalks related to the normalized parabolic induction.",
"Moreover, just as smooth duality interacts nicely with normalized parabolic induction so too does Verdier duality with the normalized Eisenstein functor.",
"In order to study how the normalized Eisenstein functor interacts with Verdier duality, it becomes very natural to want a nice compactification of the morphism $\\mathfrak {p}$ , as $\\mathrm {Eis}$ involves the functor $\\mathfrak {p}_{!",
"}$ .",
"As seen in §$1.1$ , this can be accomplished by considering an analogue of the Drinfeld compactification $\\overline{\\mathrm {Bun}}_{B}$ .",
"We show that such a compactification exists and has the right properties.",
"Theorem 1.3 (Proposition REF , Proposition REF ) There exists an Artin $v$ -stack $\\overline{\\mathrm {Bun}}_{B}$ together with an inclusion $ j: \\mathrm {Bun}_{B} \\hookrightarrow \\overline{\\mathrm {Bun}}_{B} $ which realizes $\\mathrm {Bun}_{B}$ as an open and dense substack.",
"Moreover, $\\overline{\\mathrm {Bun}}_{B}$ has natural maps $\\overline{\\mathfrak {q}}: \\overline{\\mathrm {Bun}}_{B} \\rightarrow \\mathrm {Bun}_{T}$ (resp.",
"$\\overline{\\mathfrak {p}}: \\overline{\\mathrm {Bun}}_{B} \\rightarrow \\mathrm {Bun}_{G}$ ) extending the map $\\mathfrak {q}$ (resp.",
"$\\mathfrak {p})$ along $j$ .",
"The map $\\overline{\\mathfrak {p}}$ is representable in nice diamonds and proper after restricting to connected components.",
"Now, we would like to claim that the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a Hecke eigensheaf with respect to the parameter $\\phi $ given by composing $\\phi _{T}$ with the natural embedding $\\phantom{}^{L}T \\rightarrow \\phantom{}^{L}G$ ; however, in analogy with §$1.1$ , the right object to consider in this case is not $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ , but rather a compactified version $\\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ .",
"Unfortunately, there is currently no well-behaved formalism for intersection cohomology in the context of diamonds and $v$ -stacks.",
"This prevents us from even defining the kernel sheaf $\\mathrm {IC}_{\\overline{\\mathrm {Bun}}_{B}}$ typically used in the definition of $\\overline{\\mathrm {Eis}}$ in any naive way.",
"There is however a way out if we impose some conditions on our parameter $\\phi _{T}$ .",
"We write $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ for the set of $\\Gamma $ -orbits of geometric cocharacters.",
"Given an element $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ , we can attach to it a representation of $\\phantom{}^{L}T$ by inducing the representation of $\\hat{T}$ defined by a representative of the orbit $\\nu $ in $\\mathbb {X}_*(T_{\\overline{\\mathbb {Q}}_{p}})$ .",
"We consider the composition $\\nu \\circ \\phi _{T}$ , and we will say that $\\phi _{T}$ is generic if the Galois cohomology complexes $ R\\Gamma (W_{\\mathbb {Q}_{p}}, \\alpha \\circ \\phi _{T}) $ are trivial for all $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ defined by the $\\Gamma $ -orbits of roots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ .",
"This condition may appear mysterious; but there are several ways to see why it is a morally right condition.",
"Perhaps the most compelling comes from local representation theory.",
"As mentioned in §$1.1$ , the compactified Eisenstein series in the function field setting corresponds to the completed Eisenstein series under the function sheaf dictionary, while the non-compactified Eisenstein series corresponds to just the usual Eisenstein series.",
"In particular, the sheaf $\\mathrm {IC}_{\\overline{\\mathrm {Bun}}_{B}}$ encodes the zeros and poles of the meromorphic continuation of the Eisenstein series.",
"If $\\chi $ denotes the character of $T(\\mathbb {Q}_{p})$ attached to $\\phi _{T}$ via local class field theory then we recall that, if $w \\in W_{G}$ is an element of the relative Weyl group, the local analogue of the meromorphic continuation of Eisenstein series is the theory of intertwining operators.",
"In particular, suppose that $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ , then we have maps $ i_{\\chi ,w}: i_{B}^{G}(\\chi ) \\rightarrow i_{B}^{G}(\\chi ^{w}) $ of smooth $G(\\mathbb {Q}_{p})$ -representations, which can be viewed as meromorphic functions on the set of complex unramified characters.",
"Now, using local Tate-duality, it is easy to see that the vanishing of the above complexes will imply that $\\chi $ pre-composed with coroots is not trivial or isomorphic to a power of the norm character.",
"These are precisely the type of conditions one expects to guarantee that the intertwining operators are holomorphic and give rise to an isomorphism.",
"In fact, we show that, for $\\chi $ attached to a generic parameter $\\phi _{T}$ , we always have an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w})$ for any $w \\in W_{G}$ (Proposition REF ).",
"This suggests that, at least heuristically, we should always have an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ , for $\\phi _{T}$ satisfying some version of genericity and any reasonable definition of $\\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ .",
"Indeed, this makes sense when we look at the geometry of $\\overline{\\mathrm {Bun}}_{B}$ ; in particular, we will show that the closed complement of $\\mathrm {Bun}_{B}$ in $\\overline{\\mathrm {Bun}}_{B}$ admits a locally closed stratification given by $\\mathrm {Div}^{(\\overline{\\nu })} \\times \\mathrm {Bun}_{B}$ , where $\\mathrm {Div}^{(\\overline{\\nu })}$ is a certain partially symmetrized version of the mirror curve $\\mathrm {Div}^{1}$ parameterizing effective Cartier divisors in $X$ attached to $\\overline{\\nu }$ inside the coinvariant lattice $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ .",
"We recall that there is a natural map $(-)_{\\Gamma }: \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , from $\\Gamma $ -orbits to coinvariants.",
"This map defines an injection on the $\\Gamma $ -orbits of the simple positive coroots.",
"In particular, for each vertex $i \\in \\mathcal {J}$ of the relative Dynkin diagram of $G$ , we get an element $\\alpha _{i}$ in the coinvariant lattice, which corresponds to a $\\Gamma $ -orbit of positive simple roots.",
"The natural strata of $\\overline{\\mathrm {Bun}}_{B}$ are specified by elements $\\overline{\\nu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ lying in the positive span of these $\\alpha _{i}$ , and each strata corresponds to the locus of $B$ -bundles with torsion specified by $\\overline{\\nu }$ .",
"Now, the restriction of $\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}})$ to this strata is given by pulling back a Hecke operator on $\\mathrm {Bun}_{T}$ applied to $\\mathcal {S}_{\\phi _{T}}$ along $\\mathfrak {q}$ .",
"One can deduce that the factor appearing on the divisor curve $\\mathrm {Div}^{(\\overline{\\nu })}$ will be related to $\\alpha _{i} \\circ \\phi _{T}$ , via the Hecke eigensheaf property $T_{\\alpha _{i}}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\alpha _{i} \\circ \\phi _{T} \\boxtimes \\mathcal {S}_{\\phi _{T}}$ for $\\mathcal {S}_{\\phi _{T}}$ , where we have identified $\\alpha _{i}$ with its associated $\\Gamma $ -orbit via $(-)_{\\Gamma }$ .",
"This implies that the complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha _{i} \\circ \\phi _{T})$ appears in $\\overline{\\mathfrak {p}}_{!",
"}$ applied to this restriction as a tensor factor, and will in turn vanish for $\\phi _{T}$ generic, suggesting an isomorphism of the form $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ in this case.",
"We can turn these heuristics into actual math.",
"In particular, since we expect an isomorphism of the form $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ under some verison of genericity, we should expect for such parameters that $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ behaves well under Verdier duality, is a perverse Hecke eigensheaf with eigenvalue $\\phi $ , and satisfies the analogue of the functional equation seen in Theorem REF in this case.",
"The precise conditions on $\\phi _{T}$ that we will need to prove our results will depend on the result and the particular group $G$ .",
"For this reason, we break up our condition into several parts.",
"Condition/Definition 1.4 (Condition/Definition REF ) Given a parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ , we impose the following conditions on $\\phi _{T}$ in what follows.",
"For all $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ defined by the $\\Gamma $ -orbits of simple coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the Galois cohomology complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ is trivial.",
"For all $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ defined by the $\\Gamma $ -orbits of coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the Galois cohomology complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ is trivial.",
"For all $w \\in W_{G}$ in the relative Weyl group, if $\\chi $ denotes the character attached to $\\phi _{T}$ by local class field theory then we have $ \\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq (\\chi \\otimes \\delta _{B}^{-1/2})^{w} $ For all $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ defined by the $\\Gamma $ -orbits of coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the Galois cohomology complex $R\\Gamma (W_{\\mathbb {Q}_{p}},(\\alpha \\circ \\phi _{T})^{2})$ is trivial.",
"If $\\phi _{T}$ satisfies (1) we say that it is weakly generic, and if it satisfies (2) then we say it is generic.",
"If it satisfies (2)-(3) we say that it is weakly normalized regular, and if it satisfies (2)-(4) we say that it is normalized regular.",
"Remark 1.5 The relationship between these various conditions appears to be complicated in general, and is related to the behavior of the principal series representations $i_{B}^{G}(\\chi )$ of $G$ .",
"Roughly speaking, Condition (2) guarantees the irreducibility of non-unitary principal series representations and that the intertwining operators for $i_{B}^{G}(\\chi )$ are isomorphisms, while Condition (3) guarantees the irreducibility of certain unitary principal series representations.",
"As we will see in Appendix , genericity is enough to guarantee that one has an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w})$ for all $w \\in W_{G}$ .",
"However, one could still have an isomorphism of this form if $i_{B}^{G}(\\chi )$ is the reducible induction of a unitary character $\\chi $ .",
"This can happen if the character $\\chi $ is not regular (i.e it is fixed by some $w \\in W_{G}$ ).",
"Conditions (3) and (4) imply such regularity conditions (Lemma REF ), so that when $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ , the representation $i_{B}^{G}(\\chi )$ satisfies the stronger condition of being irreducible (Corollary REF ).",
"To illustrate this, note that, for $G = \\mathrm {GL}_{n}$ , if we write $\\phi _{T} = \\bigoplus _{i = 1}^n \\phi _{i}$ as a sum of characters then genericity is equivalent to supposing that $ R\\Gamma (W_{\\mathbb {Q}_{p}},\\phi _{i}^{\\vee } \\otimes \\phi _{j}) $ is trivial for all $i \\ne j$ .",
"We note that, by local Tate-duality and using that the Euler-Poincaré characteristic of this complex is 0, genericity is equivalent to assuming that $\\phi _{i}$ is not isomorphic to $\\phi _{j}$ or $\\phi _{j}(1)$ .",
"If we write $\\chi = \\chi _{1} \\otimes \\ldots \\otimes \\chi _{n}$ as a product of characters of $\\mathbb {Q}_{p}^{*}$ then this implies that $\\chi _{i}^{-1}\\chi _{j} \\lnot \\simeq |\\cdot |^{\\pm 1}$ for all $i > j$ , which is precisely the condition guaranteeing irreducibility.",
"Moreover, in this case we will see that Condition (2) implies Condition (3) (Lemma REF ) and Condition (4) will be unnecessary for the desired applications.",
"On the other hand, if $G = \\mathrm {SL}_{2}$ , and we write $\\chi $ for the character of $\\mathbb {Q}_{p}^{*}$ attached to $\\phi _{T}$ via local class field theory, we need that $\\chi \\lnot \\simeq |\\cdot |^{\\pm 1}$ and $\\chi ^{2} \\lnot \\simeq \\mathbf {1}$ to guarantee irreducibility of $i_{B}^{G}(\\chi )$ .",
"The condition $\\chi \\lnot \\simeq |\\cdot |^{\\pm 1}$ is guaranteed by Condition (2), and Condition (3) is equivalent to $\\chi ^{2} \\lnot \\simeq \\mathbf {1}$ .",
"Remark 1.6 The choice of calling a parameter satisfying Condition (2) generic is motivated by the analogous notion of decomposed generic considered by Caraiani and Scholze .",
"In particular, we note that if $\\phi _{T}$ is unramified and $G = \\mathrm {GL}_{n}$ then, by the previous remark, we see that Condition (2) is precisely equivalent to $\\phi _{T}$ being decomposed generic.",
"Weak genericity will be required to show our Eisenstein functor commutes with Verdier duality, while weak normalized regularity will be needed to compute the stalks of our Eisenstein series and show that it satisfies the functional equation.",
"Unfortunately, in general to get the Hecke eigensheaf property for $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ we still need more, which is why we have Condition (4).",
"Write $(-)^{\\Gamma }: \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})\\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ for the natural map from geometric cocharacters to their Galois orbits.",
"Given a geometric dominant cocharacter $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ with Galois orbit $\\mu ^{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}/\\Gamma $ and reflex field $E_{\\mu }$ , we have an associated representation $V_{\\mu ^{\\Gamma }} \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ .",
"The weights of $V_{\\mu }|_{\\phantom{}^{L}T}$ can be interpreted in terms of the representations corresponding to the Galois orbits of weights in the usual highest weight representation.",
"The following condition will guarantee the Hecke eigensheaf property for the Hecke operator defined by $V_{\\mu ^{\\Gamma }}$ .",
"Definition 1.7 (Definition REF ) For a toral parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ and a geometric dominant cocharacter $\\mu $ , we say $\\phi _{T}$ is strongly $\\mu $ -regular if the Galois cohomology complexes $ R\\Gamma (W_{\\mathbb {Q}_{p}},(\\nu - \\nu ^{\\prime })^{\\Gamma } \\circ \\phi _{T}) $ are trivial for $\\nu $ ,$\\nu ^{\\prime }$ distinct weights of the highest weight representation of $\\hat{G}$ of highest weight $\\mu _{k}$ for all $k = 1,\\ldots ,n$ .",
"We say that $\\phi _{T}$ is $\\mu $ -regular if there exists a set of geometric dominant cocharacters $\\mu _{k} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ for $k = 1,\\ldots ,m$ such that the following hold: $\\mu _{k} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ is quasi-minuscule or minuscule (as in or ).",
"Up to central elements, we have an equality $\\mu = \\sum _{k = 1}^{m} n_{k}\\mu _{k}$ as elements in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ for $n_{k} \\in \\mathbb {N}_{> 0}$ .",
"The parameter $\\phi _{T}$ is strongly $\\mu _{k}$ -regular for all $k = 1,\\ldots ,m$ .",
"Remark 1.8 Using the highest weight theory for minuscules (resp.",
"quasi-minuscules), one can also show (Lemma REF ) that $\\mu $ -regularity holds for any $\\mu $ generated by minuscules (resp.",
"minuscules/quasi-minuscules) if $\\phi _{T}$ satisfies Condition (2) (resp.",
"Conditions (2) and (4)).",
"In particular, normalized regularity will always imply the strongest form of our results.",
"However, the precise condition can be relaxed depending on the weights appearing in the highest weight representations corresponding to the choice of generators $\\mu _{i}$ .",
"For example, for $G = \\mathrm {GL}_{n}$ we can always choose the $\\mu _{i}$ to be the minuscule fundamental weights and then $\\phi _{T}$ being $\\mu $ -regular for all $\\mu $ will be guaranteed by Condition (2).",
"Let's now see how these conditions manifest in our results on Eisenstein series.",
"We begin with studying how Verdier duality interacts with Eisenstein series.",
"To do this, we need to make the following assumption.",
"Assumption 1.9 (Assumption REF ) If $j: \\mathrm {Bun}_{B} \\hookrightarrow \\overline{\\mathrm {Bun}}_{B}$ is the open inclusion into the Drinfeld compactification the sheaf $j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ is ULA with respect to $\\overline{\\mathfrak {q}}$ .",
"Remark 1.10 This is a precise analogue of .",
"The proof in this case seems a bit subtle, since being ULA over a point in this case is not a trivial condition, but should appear in upcoming work of Hansen-Scholze on Geometric Eisenstein series and the Harris-Viehmann conjecture.",
"The relevance for studying how Verdier duality interacts with $\\mathrm {nEis}$ is as follows.",
"It follows that $j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ is reflexive with respect to Verdier duality on $\\mathrm {Bun}_{B}$ .",
"In particular, using the Verdier self-duality of $\\mathrm {IC}_{\\mathrm {Bun}_{B}}$ , this allows us to see that the Verdier dual of $j_*(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ is isomorphic to $j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ .",
"This reduces the problem of how Eisenstein series interact with Verdier duality to the problem of describing the cone of the map $ j_{!",
"}(\\mathfrak {q}^{*}(\\mathcal {S}_{\\phi _{T}})) \\rightarrow \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}}) $ after applying $\\overline{\\mathfrak {p}}_{!",
"}$ .",
"As already mentioned above, if we look at the restriction of $\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}})$ to the strata $\\mathrm {Div}^{(\\overline{\\nu })} \\times \\mathrm {Bun}_{B}$ described above, this vanishes for weakly generic $\\phi _{T}$ after applying $\\overline{\\mathfrak {p}}_{!",
"}$ , as the Galois cohomology complexes $R\\Gamma (W_{\\mathbb {Q}_{p}}, \\alpha \\circ \\phi _{T})$ will appear for $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ corresponding to a $\\Gamma $ -orbit of simple coroots.",
"In particular, if $\\mathbb {D}_{Z}$ denotes Verdier duality on a $v$ -stack or diamond $Z$ , we can show the following.",
"Theorem 1.11 (Theorem REF ) For $\\phi _{T}$ a weakly generic toral parameter there is an isomorphism of objects in $\\mathrm {D}(\\mathrm {Bun}_{G})$ $ \\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}})) $ where we note that $\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\mathcal {S}_{\\phi _{T}^{\\vee }}$ , if $\\phi _{T}^{\\vee }$ is the parameter dual to $\\phi _{T}$ .",
"We will assume the validity of the ULA Theorem and thereby the validity of this theorem for the rest of the section.",
"We now turn our attention to the Hecke eigensheaf property.",
"In particular, consider a finite index set $I$ and a representation $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ .",
"Given $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ , we write $V((\\nu _{i})_{i \\in I})$ for the multiplicity of the weight space of the corresponding representation of $\\phantom{}^{L}T^{I}$ in $V|_{\\phantom{}^{L}T^{I}}$ , and write $T_{(\\nu _{i})_{i \\in I}}$ for the associated Hecke operator.",
"By applying excision to the aforementioned locally closed stratification of $\\overline{\\mathrm {Bun}}_{B}$ and combining it with the geometric Satake correspondence of Fargues-Scholze , we can show the following result.",
"Theorem 1.12 (Theorem REF ) For $\\mathcal {F} \\in \\mathrm {D}(\\mathrm {Bun}_{T})$ , $I$ a finite index set, and $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ with associated Hecke operator $T_{V}$ , the sheaf $T_{V}(\\mathrm {nEis}(\\mathcal {F}))$ on $\\mathrm {Bun}_{G}$ with continuous $W_{\\mathbb {Q}_{p}}^{I}$ -action has a $W_{\\mathbb {Q}_{p}}^{I}$ -equivariant filtration indexed by $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ .",
"The filtration's graded pieces are isomorphic to $ \\bigoplus _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}} \\mathrm {nEis}(T_{(\\nu _{i})_{i \\in I}}(\\mathcal {F})) \\otimes V((\\nu _{i})_{i \\in I}) $ The filtration is natural in $I$ and $V$ , as well as compatible with compositions and exterior tensor products.",
"The argument for proving this \"filtered eigensheaf property\" is very similar to that given by in their proof of the Hecke eigensheaf property for the compactified geometric Eisenstein functor in the function field setting.",
"However, there Braverman and Gaitsgory rely on the decomposition theorem applied to the perverse sheaf $\\mathrm {IC}_{\\overline{\\mathrm {Bun}}_{B}}$ , which would not make sense in this context.",
"Nevertheless, we still have access to the excision spectral sequence one usually uses in proving the decomposition theorem.",
"In particular, in the proof of the decomposition theorem one uses the excision spectral sequence and then invokes the theory of weights to show that it degenerates.",
"Something similar happens here.",
"Namely, if we apply this to the sheaf $\\mathcal {F} = \\mathcal {S}_{\\phi _{T}}$ then we see that $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a filtered eigensheaf in the sense that, up to passing to the graded pieces of this filtration, it is an eigensheaf with eigenvalue $\\phi $ , and later we will see, by looking at the Weil group action, this filtration must always split under the normalized regularity hypothesis on $\\phi _{T}$ .",
"Even without knowing that the filtration on $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ splits, the filtered eigensheaf property can already be used to tell us a lot about the structure of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ .",
"In particular, given $b \\in B(G)$ , the restriction $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ to the HN-strata $\\mathrm {Bun}_{G}^{b}$ defines a complex of smooth $J_{b}(\\mathbb {Q}_{p})$ -representations, and we are interested in describing this restriction.",
"Here $J_{b}$ is the $\\sigma $ -centralizer of $b$ and it is an extended pure inner form of a Levi subgroup $M_{b}$ of $G$ .",
"The fact that $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a filtered Hecke eigensheaf with eigenvalue $\\phi $ implies that, if $\\rho $ is an irreducible constituent of this restriction, then the Fargues-Scholze parameter $\\phi _{\\rho }^{\\mathrm {FS}}$ must be equal to $\\phi $ under the twisted embedding $\\phantom{}^{L}J_{b} \\rightarrow \\phantom{}^{L}G$ .",
"If we believe that the Fargues-Scholze local Langlands correspondence is the true local Langlands correspondence then this seems to suggest that $\\rho $ should be given by a normalized parabolic induction of the character $\\chi $ attached to $\\phi _{T}$ via local class field theory.",
"In particular, we note, by deformation theory, that a toral parameter being generic implies that every lift of $\\phi _{T}$ to $\\overline{\\mathbb {Z}}_{\\ell }$ factors through $\\phantom{}^{L}T$ and that the induced $\\overline{\\mathbb {Q}}_{\\ell }$ -parameter cannot come from the semi-simplification of a parameter with monodromy (LemmaREF ).",
"This imposes a very rigid constraint on $J_{b}$ .",
"In particular, the Borel $B \\cap M_{b}$ of $M_{b}$ should transfer to a Borel $B_{b}$ of $J_{b}$ .",
"The elements where this occurs are the elements in the image of the map $B(T) \\rightarrow B(G)$ , called the unramified elements $B(G)_{\\mathrm {un}}$ , as studied in the work of Xiao-Zhu .",
"Corollary 1.13 (Corollary REF ) For $\\phi _{T}$ a generic parameter and $b \\in B(G)$ , assuming compatibility of the Fargues-Scholze and the conjectural local Langlands correspondence (Assumption REF ), the restriction $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ vanishes unless $b \\in B(G)_{\\mathrm {un}}$ .",
"We now assume compatibility (Assumption REF ) in addition to validity of the ULA theorem (Assumption REF ) for the rest of the section.",
"The previous corollary tells us that if we are interested in understanding the complex of $J_{b}(\\mathbb {Q}_{p})$ -representations defined by the stalks $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ then we can restrict to the case where $b \\in B(G)_{\\mathrm {un}}$ , and here we expect the stalks to be given by the inductions $i_{B_{b}}^{J_{b}}(\\chi ) \\otimes \\delta _{P_{b}}^{1/2}$ , where $\\delta _{P_{b}}$ is the modulus character of the standard parabolc $P_{b}$ with Levi factor $M_{b}$ transferred to $J_{b}$These twists by the modulus character come from the fact that the excursion algebra on $\\mathrm {Bun}_{G}$ acts on a smooth irreducible representation $\\rho \\in \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda ) \\simeq \\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\subset \\mathrm {D}(\\mathrm {Bun}_{G})$ via the Fargues-Scholze parameter $\\phi _{\\rho }^{FS}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}J_{b}(\\Lambda )$ of a smooth irreducible representation $\\rho $ of $J_{b}$ , composed with the twisted embedding $\\phantom{}^{L}J_{b}(\\Lambda ) \\rightarrow \\phantom{}^{L}G(\\Lambda )$ , as in ..",
"This is indeed the case.",
"To understand this, we use that each element $b \\in B(G)_{\\mathrm {un}}$ has a unique element $b_{T}$ with anti-dominant slope homomorphism mapping to it.",
"We call such an element HN-dominant, where we recall that the Harder-Narasimhan slopes are the negatives of the isocrystal slopes (See ).",
"The set of elements in $B(T)$ mapping to $b$ can be described as $w(b_{T})$ , where we identify $w \\in W_{b} := W_{G}/W_{M_{b}}$ with a set of representatives of minimal length in $W_{G}$ .",
"The connected components of $\\mathrm {Bun}_{B}$ and $\\mathrm {Bun}_{T}$ are indexed by elements in $B(T) \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , giving a direct sum decomposition: $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) = \\bigoplus _{\\overline{\\nu } \\in B(T)} \\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}}) $ After restricting to $\\mathrm {Bun}_{G}^{b}$ , the point is that only the summands $\\overline{\\nu } = w(b_{T})$ for $w \\in W_{b}$ survive.",
"It is fairly easy to see this when $\\overline{\\nu } = b_{T}$ .",
"In particular, the connected component $\\mathrm {Bun}_{B}^{b_{T}}$ will parametrize split $B$ -structures since the HN-slopes are dominant, and the diagram (REF ) (essentially) becomes $ \\begin{tikzcd}& \\left[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}\\right] [dr,\"\\mathfrak {p}\"] [d,\"\\mathfrak {q}\"] & \\\\& \\left[\\ast /\\underline{T(\\mathbb {Q}_{p})}\\right] & \\left[\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}\\right]\\end{tikzcd} $ Using this, it is easy to see that $\\mathfrak {p}_{!",
"}\\mathfrak {q}^{*}(\\chi )$ will be given by compactly supported functions of $J_{b}/B_{b}(\\mathbb {Q}_{p})$ which transform under $B_{b}(\\mathbb {Q}_{p})$ via $\\chi $ .",
"In other words, the representation $\\mathrm {Ind}_{B_{b}}^{J_{b}}(\\chi )$ .",
"When one accounts for the twists coming from the dualizing object, one finds that the exact formula becomes $ \\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}}) \\simeq j_{b!",
"}(i_{B_{b}}^{J_{b}}(\\chi ) \\otimes \\delta _{P_{b}}^{1/2})[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] $ where $j_{b}: \\mathrm {Bun}_{G}^{b} \\rightarrow \\mathrm {Bun}_{G}$ is the inclusion of the HN-strata corresponding to $b$ .",
"Now, what about the connected components $\\overline{\\nu } = w(b_{T})$ with $w$ non-trivial?",
"Here the HN-slopes of $\\overline{\\nu }$ is at least partially anti-dominant, and therefore $\\mathrm {Bun}_{B}^{w(b_{T})}$ will parameterize some non-split extensions.",
"Nonetheless, one finds that $\\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}})$ behaves similarly to the contribution of the connected component given by the HN-dominant reduction.",
"Note that, a priori, the complex $\\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}})$ could be supported on all $b^{\\prime } \\in B(G)$ with $b \\succeq b^{\\prime }$ in the natural partial ordering on $B(G)$ .",
"If one imposes the previous compatibility assumption and assumes $\\phi _{T}$ is generic then one can use the previous corollary to assume that $b^{\\prime } \\in B(G)_{\\mathrm {un}}$ .",
"In this case, the complex $\\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ can be computed in terms of the cohomology of the space of simulatenous reductions of a $G$ -bundle $\\mathcal {F}_{G}$ to two $B$ -bundles with underlying $T$ -bundles given by $w(b_{T})$ and a Weyl group translate of $b^{\\prime }_{T}$ , where $b^{\\prime }_{T}$ is the HN-dominant reduction of $b^{\\prime }$ .",
"This space admits a locally closed stratification by the generic relative position of these two reductions coming from the Bruhat decomposition of $B\\backslash G/B$ .",
"If $b^{\\prime } \\ne b$ then each of the non-empty strata admit a map to a positive symmetric power of the mirror curve $\\mathrm {Div}^{1}$ , and are locally modelled by a semi-infinite flag space called a Zastava space, as studied in the function field setting by Feign, Finkelberg, Kuznetsov, and Mirković .",
"By combining a study of these Zastava spaces with Condition (3) on $\\phi _{T}$ and induction on $b$ , we can show that the restriction of $\\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ to each of these locally closed strata vanishes, from which we can conclude that the restriction $\\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ vanishes unless $b^{\\prime } = b$ , where again only the contribution of the split $B$ -structure matters.",
"All in all, we conclude an isomorphism: $ \\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}}) \\simeq j_{b!",
"}(i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2})[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] $ This parallel behavior between the HN-dominant connected component and the connected components in its Weyl group orbit is no accident.",
"In analogy with §$1.1$ , we expect, for a choice of representative $\\tilde{w} \\in N(T)$ of $w \\in W_{G}$ in the relative Weyl group, to have an isomorphism $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}^{\\tilde{w}}) $ where $\\mathcal {S}_{\\phi _{T}}^{\\tilde{w}}$ is the pullback of $\\mathcal {S}_{\\phi _{T}}$ along the map $\\mathrm {Bun}_{T} \\rightarrow \\mathrm {Bun}_{T}$ induced by $\\tilde{w}$ .",
"This involution sends the connected component $\\mathrm {Bun}_{T}^{b_{T}}$ to $\\mathrm {Bun}_{T}^{w(b_{T})}$ and sends the character $\\chi $ to $\\chi ^{w}$ .",
"In particular, we see that, by our previous description of stalks, this gives the precise analogue of Theorem REF .",
"We summarize the above discussion as follows.",
"Theorem 1.14 (Corollary REF ) Consider $\\phi _{T}$ a weakly normalized regular parameter with associated character $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ .",
"Given $b \\in B(G)_{\\mathrm {un}}$ , we consider $J_{b}$ , $M_{b}$ , $B_{b}$ , and $W_{b}$ as defined above.",
"For $b \\in B(G)$ , the stalk $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\in \\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ is given by an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq \\bigoplus _{w \\in W_{b}} i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ if $b \\in B(G)_{\\mathrm {un}}$ , an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq 0$ if $b \\notin B(G)_{\\mathrm {un}}$ .",
"In particular, $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a perverse sheaf on $\\mathrm {Bun}_{G}$ with respect to the standard $t$ -structure defined by the HN-strata.",
"We note that the previous Corollaries imply that the stalks of the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ are valued in smooth admissible representation when $\\phi _{T}$ is weakly normalized regular.",
"This implies that the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is ULA with respect to the structure map $\\mathrm {Bun}_{G} \\rightarrow \\ast $ , using the characterization given in .",
"This ULA property allows us to extend the construction of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ to $\\overline{\\mathbb {Z}}_{\\ell }$ and $\\overline{\\mathbb {Q}}_{\\ell }$ -coefficients, where passing to the world of lisse-étale solid sheaves is a non-trivial matter because of the difference between $\\ell $ -adic and discrete topologies.",
"We will need to work with $\\phi _{T}$ that is integral in the sense that, if $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ , it is of the form $\\overline{\\phi }_{T} \\otimes _{\\overline{\\mathbb {Z}}_{\\ell }} \\overline{\\mathbb {Q}}_{\\ell }$ for a $\\overline{\\mathbb {Z}}_{\\ell }$ -valued parameter $\\overline{\\phi }_{T}$ .",
"Given an integral parameter $\\phi _{T}$ such that the mod $\\ell $ -reduction is weakly normalized regular, we get a sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ for all $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ .",
"The description of the stalks, the filtered eigensheaf property with eigenvalue $\\phi $ , and the commutation with Verdier duality all extend in a natural way to these coefficient systems, and we would now like to say that the filtered eigensheaf property implies it is a genuine eigensheaf under the conditions on $\\phi _{T}$ .",
"For a representation $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ , the filtered eigensheaf property tells us that $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ has a filtration whose graded pieces have Weil group action given by $\\nu ^{\\Gamma } \\circ \\phi _{T}$ , for $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ a non-zero weight of $V$ in $\\hat{T}$ .",
"In order to see this splits, it suffices to show for $\\nu , \\nu ^{\\prime }$ distinct non-zero weights of $V$ in $\\hat{T}$ that the extension group $H^{1}(W_{\\mathbb {Q}_{p}}, (\\nu - \\nu ^{\\prime })^{\\Gamma } \\circ \\phi _{T})$ vanishes for the $\\Gamma $ -orbit $(\\nu - \\nu ^{\\prime })^{\\Gamma }$ defined by $\\nu - \\nu ^{\\prime }$ .",
"However, this is equivalent to saying that the entire complex $ R\\Gamma (W_{\\mathbb {Q}_{p}},(\\nu - \\nu ^{\\prime })^{\\Gamma } \\circ \\phi _{T}) $ is trivial, and this was precisely the kind of vanishing result that $\\mu $ -regularity guaranteed.",
"Moreover, by the vanishing of the $H^{0}$ the splitting will be unique.",
"In particular, if $V = V_{\\mu ^{\\Gamma }}$ for $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ such that $\\phi _{T}$ is strongly $\\mu $ -regular then this allows us to see that we get a unique splitting for $V = V_{\\mu ^{\\Gamma }}$ .",
"Moreover, if we assume the weaker condition that $\\phi _{T}$ is $\\mu $ -regular then we can write $\\mu = \\sum _{k = 1}^{m} n_{k}\\mu $ for some minuscule/quasi-minuscule cocharacters $\\mu _{k}$ , and we know that we have a unique splitting for the Hecke operators defined by the $\\mu _{k}$ , by definition of $\\mu $ -regularity.",
"However, using the compatibilities of the filtration, we can reduce to showing the splitting for the Hecke operators defined by $\\mu _{k}$ via realizing $V_{\\mu }$ as a direct summand of $\\bigotimes _{k = 1}^{m} V_{\\mu _{k}}^{\\otimes n_{i}}$ , but in this case we cannot guarantee that this splitting is unique.",
"We need to be a bit careful when running the argument sketched above.",
"In particular, if $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ , then the category $\\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ is semi-simple with irreducible objects parametrized by $\\Gamma $ -orbits of dominant cocharacters $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}/\\Gamma $ and the above argument goes through.",
"If $\\Lambda \\in \\lbrace \\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {F}}_{\\ell }\\rbrace $ this is no longer true.",
"However, in these cases, we can replace $\\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ by a sub-category of tilting modules $\\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G)$ (,), which will be semi-simple with indecomposable objects given by $\\mu ^{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}/\\Gamma $ , denoted $\\mathcal {T}_{\\mu ^{\\Gamma }} \\in \\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G)$ .",
"Extending the theory of tilting modules to the full $L$ -group $\\phantom{}^{L}G$ is a bit subtle.",
"However, this is precisely what our assumption that $\\ell $ is very good with respect to $G$ will allow us to do.",
"This category is preserved under taking tensor products, and therefore we can define the notion of a \"tilting eigensheaf\" (Definition REF ) by replacing $\\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ with $\\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G)$ in the usual definition.",
"For $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ , we write $r_{V}: \\phantom{}^{L}G \\rightarrow \\mathrm {GL}(V)$ for the associated map.",
"Our main theorem is then as follows.",
"Theorem 1.15 (Theorem REF ) For $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ , we consider $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ an integral parameter such that its mod $\\ell $ -reduction is weakly normalized regular.",
"There then exists a perverse sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ which is a filtered eigensheaf with eigenvalue $\\phi $ .",
"If $V \\in \\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G)$ is a direct sum of tilting modules $\\boxtimes _{i \\in I} \\mathcal {T}_{\\mu _{i}^{\\Gamma }}$ for geometric dominant cocharacters $\\mu _{i}$ , and $\\phi _{T}$ is $\\mu _{i}$ -regular (resp.",
"strongly $\\mu _{i}$ -regular), the filtration on $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ splits (resp.",
"splits uniquely), and we have a natural isomorphism $ T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\boxtimes r_{V} \\circ \\phi $ of sheaves in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G})^{BW_{\\mathbb {Q}_{p}}^{I}}$ .",
"In particular, if $\\phi _{T}$ is $\\mu $ -regular (resp.",
"strongly $\\mu $ -regular) for all geometric dominant cocharacters $\\mu $ (e.g if it is normalized regular) then $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a weak tilting eigensheaf (resp.",
"tilting eigensheaf).",
"For $b \\in B(G)$ , the stalk $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\in \\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ is given by an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq \\bigoplus _{w \\in W_{b}} i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ if $b \\in B(G)_{\\mathrm {un}}$ , an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq 0$ if $b \\notin B(G)_{\\mathrm {un}}$ .",
"Moreover, if $\\mathbb {D}_{\\mathrm {Bun}_{G}}$ denotes Verdier duality on $\\mathrm {Bun}_{G}$ , we have an isomorphism $ \\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) $ of sheaves in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ .",
"Remark 1.16 The notion of a weak tilting eigensheaf means that we always have isomorphisms $ \\eta _{V,I}: T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\boxtimes r_{V} \\circ \\phi $ for $V \\in \\mathrm {Tilt}(\\phantom{}^{L}G^{I})$ and a finite index set $I$ , but do not necessarily know that the desired compatibilities with respect to $I$ and $V$ .",
"Even though we know these compatibilities for the filtration, it is not necessarily clear that the splitting we produce through our argument respects these compatibilities without assuming strong $\\mu $ -regularity.",
"Only knowing the compatibilities of the splittings under such restrictive conditions is a bit unfortunate; fortunately, for most of the applications to local Shtuka spaces with one leg it suffices to only know a splitting exists.",
"This eigensheaf has several suprising applications to the cohomology of local Shimura varieties and shtuka spaces.",
"To formalize this, we define, for $b \\in B(G)$ , a complex of $J_{b}(\\mathbb {Q}_{p})$ -representations denoted $\\mathrm {Red}_{b,\\phi }$ .",
"If $b \\notin B(G)_{\\mathrm {un}}$ we set this to be equal to 0 and if $b \\in B(G)_{\\mathrm {un}}$ to be equal to $\\bigoplus _{w \\in W_{b}} i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho },\\nu _{b}\\rangle ]$ .",
"Now let's consider $\\mu $ a geometric dominant cocharacter of $G$ with reflex field $E$ and set $B(G,\\mu ) \\subset B(G)$ to be the subset of $\\mu $ -admissible elements (Definition REF ).",
"We let $\\mathcal {T}_{\\mu }$ be the associated highest weight tilting module of $\\hat{G}$ .",
"This defines a representation of $W_{E} \\ltimes \\hat{G}$ with associated Hecke operator $T_{\\mu }$ .",
"We write $r_{\\mu }: W_{E} \\ltimes \\hat{G} \\rightarrow \\mathrm {GL}(\\mathcal {T}_{\\mu })$ for the associated map.",
"We consider the cohomology of the local shtuka spaces $\\mathrm {Sht}(G,b,\\mu )_{\\infty }$ , as defined in .",
"In particular, the representation $\\mathcal {T}_{\\mu }$ defines a sheaf $\\mathcal {S}_{\\mu }$ on $\\mathrm {Sht}(G,b,\\mu )_{\\infty }$ via geometric Satake, and we can consider the complex $R\\Gamma _{c}(G,b,\\mu )$ of $J_{b}(\\mathbb {Q}_{p}) \\times G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules attached to cohomology valued in this sheaf.",
"Remark 1.17 We note that, since we have used the tilting module $\\mathcal {T}_{\\mu }$ in the definition of $R\\Gamma _{c}(G,b,\\mu )$ instead of the usual highest weight representation $V_{\\mu }$ this is slightly different then the usual definition appearing in the literature.",
"The two definitions will coincide when the representation $V_{\\mu }$ defines a tilting module, which is equivalent to $V_{\\mu }$ being irreducible with coefficients in $\\Lambda $ .",
"We say such a $\\mu $ is tilting if this holds.",
"This will always hold if $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ or if $\\mu $ is minuscule, and we study this notion more carefully in Appendix .",
"We can use $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ to describe the cohomology of $R\\Gamma _{c}(G,b,\\mu )$ .",
"Assume that $\\phi _{T}$ is $\\mu $ -regular, the Hecke eigensheaf property then tells us that we have an isomorphism $ T_{\\mu }(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\boxtimes r_{\\mu } \\circ \\phi |_{W_{E}} $ of sheaves with continuous $W_{E}$ -action.",
"If we restrict to the open HN-strata $j_{\\mathbf {1}}: \\mathrm {Bun}_{G}^{\\mathbf {1}} \\rightarrow \\mathrm {Bun}_{G}$ defined by the trivial element $\\mathbf {1} \\in B(G)$ then this gives an isomorphism $ j_{\\mathbf {1}}^{*}T_{\\mu }(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq i_{B}^{G}(\\chi ) \\boxtimes r_{\\mu } \\circ \\phi |_{W_{E}} $ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules.",
"Now the point is that only the elements $b \\in B(G,\\mu )$ occur as a modifications $\\mathcal {F}_{b} \\rightarrow \\mathcal {F}_{G}^{0}$ of type $\\mu $ , where $\\mathcal {F}_{G}^{0}$ is the trivial $G$ -bundle.",
"Therefore, only these stalks contribute to the LHS.",
"By applying excision to the locally closed stratification given by $\\mathrm {Bun}_{G}^{b}$ for $b \\in B(G,\\mu )$ , we find that the LHS has a filtration with graded pieces isomorphic to $j_{\\mathbf {1}}^{*}T_{\\mu }(j_{b!",
"}(\\mathrm {Red}_{b,\\phi }))$ , but these are isomorphic to $R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]$ .",
"From the above analysis, we deduce the following.",
"Theorem 1.18 (Theorem REF ) For $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ an integral toral parameter such that its mod $\\ell $ -reduction is weakly normalized regular and any geometric dominant cocharacter $\\mu $ such that $\\phi _{T}$ is $\\mu $ -regular, we have an equality $ \\sum _{b \\in B(G,\\mu )} [R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes i_{B}^{G}(\\chi )] $ in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\Lambda )$ .",
"If we now consider the case where $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ then it follows that the averaging formula is valid for all normalized regular parameters $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell })$ , which admit a $\\overline{\\mathbb {Z}}_{\\ell }$ -lattice and all $\\mu $ .",
"Moreover, with $\\overline{\\mathbb {Q}}_{\\ell }$ -coefficients, we can interpret both sides as trace forms on $K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ , and use that the set of characters obtained from such parameters is Zariski dense in the variety of unramified characters to conclude the following more general claim.",
"Theorem 1.19 (Theorem REF ) For $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell })$ any toral parameter and $\\mu $ any geometric dominant cocharacter of $G$ , we have an equality $ \\sum _{b \\in B(G,\\mu )} [R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes i_{B}^{G}(\\chi )] $ in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\overline{\\mathbb {Q}}_{\\ell })$ .",
"If $\\mu $ is minuscule and $G = \\mathrm {GL}_{n}$ then this recovers special cases of an averaging formula of Shin , which was formalized for more general reductive groups by Alexander Bertoloni-Meli .",
"In particular, for all $\\chi $ the induction $i_{B}^{G}(\\chi )$ defines a class $[i_{B}^{G}(\\chi )] \\in K_{0}^{st}(G(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ in the subgroup of the Grothendieck group with stable character sum.",
"To such a class, the averaging formula gives a description of the RHS in terms of an average over $B(G,\\mu )$ of the isotypic parts of $R\\Gamma _{c}(G,b,\\mu )$ with respect to $\\mathrm {Red}_{b}^{\\mathfrak {c}}(i_{B}^{G}(\\chi ))$ , where $\\mathfrak {c}$ is a refined endoscopic datum (Definition REF ).",
"In Appendix REF , we verify that this indeed agrees with the conjectured averaging formula when $\\mathfrak {c}$ is the trivial endoscopic datum.",
"This is rather remarkable.",
"Such formulae are typically proven in the minuscule case by stabilizing the trace formula on the Igusa varieties indexed by $b \\in B(G,\\mu )$ , and our analysis gives a more conceptual explanation for them.",
"By combining our work here with the compatibility results proven in and , this should give a proof of this averaging formula in cases where the non-basic Igusa varieties haven't even been properly studied yet!However, in the case where $G$ is split and $\\mu $ is minuscule this formula can actually be checked by hand (See Proposition REF ), but in the case of unitary groups and $\\mu $ minuscule this already gives new information.",
"We recall that, in the proof of the averaging formula, we used excision to produce a filtration whose graded pieces were isomorphic to $ j_{\\mathbf {1}}^{*}T_{\\mu }(j_{b!",
"}(\\mathrm {Red}_{b,\\phi })) \\simeq j_{\\mathbf {1}}^{*}T_{\\mu }(j_{b!",
"}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}})) $ By using the isomorphism $\\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}^{\\vee }})$ , we can show (See Corollary REF ) that we have an isomorphism: $j_{\\mathbf {1}}^{*}T_{\\mu }(j_{b!",
"}(\\mathrm {Red}_{b,\\phi })) \\simeq j_{\\mathbf {1}}^{*}T_{\\mu }(j_{b*}(\\mathrm {Red}_{b,\\phi }))$ of objects in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ .",
"This implies that the excision spectral sequence degenerates, allowing us to conclude the following refined averaging formula.",
"Theorem 1.20 (Theorem REF ) For $\\phi _{T}$ an integral parameter with weakly normalized regular mod $\\ell $ -reduction, and $\\mu $ any geometric dominant cocharacter such that $\\phi _{T}$ is $\\mu $ -regular, we have an isomorphism $ \\bigoplus _{b \\in B(G,\\mu )_{\\mathrm {un}}} \\bigoplus _{w \\in W_{b}} R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}][\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ] = \\bigoplus _{b \\in B(G,\\mu )} R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }] \\simeq i_{B}^{G}(\\chi ) \\boxtimes r_{\\mu } \\circ \\phi |_{W_{E}} $ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules, where $\\rho _{b,w} := i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}$ .",
"We now assume that $\\phi _{T}$ is an integral parameter with weakly normalized regular mod $\\ell $ reduction in all that follows.",
"The previous Theorem leads to a very explicit descriptions of the complexes $R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}]$ and the degrees of cohomology they sit in.",
"Corollary 1.21 (Corollary REF ) For $\\mu $ a geometric dominant cocharacter with reflex field $E$ such that $\\phi _{T}$ is $\\mu $ -regular, fixed $b \\in B(G,\\mu )_{\\mathrm {un}}$ , and varying $w \\in W_{b}$ , the complex $R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}]$ is isomorphic to $\\phi _{b,w}^{\\mu } \\boxtimes i_{B}^{G}(\\chi )[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ , for $\\phi _{b,w}^{\\mu }$ a representation of $W_{E}$ .",
"Moreover, we have an isomorphism $ \\bigoplus _{b \\in B(G,\\mu )_{\\mathrm {un}}} \\bigoplus _{w \\in W_{b}} \\phi _{b,w}^{\\mu } \\simeq r_{\\mu } \\circ \\phi |_{W_{E}} $ of $W_{E}$ -representations.",
"Remark 1.22 When $G = \\mathrm {GL}_{n}$ , we can deduce these consequences for all $\\mu $ under the assumption that $\\phi _{T}$ is generic (cf.",
"Remark REF ).",
"We anticipate that by combining this statement with the approach to torsion vanishing taken by Koshikawa via using compatibility of the Fargues-Scholze and the usual local Langlands correspondence it should lead to generalizations of Caraiani and Scholze's results.",
"It is now natural to wonder what the representations $\\phi _{b,w}^{\\mu }$ exactly are.",
"It was already observed by Xiao-Zhu that the elements of the set $B(G,\\mu )_{\\mathrm {un}}$ correspond to Weyl group orbits of weights of the highest weight module $\\mathcal {T}_{\\mu }$ or rather its restriction $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ (Corollary REF ).",
"If we let $b_{T} \\in B(T)$ be the HN-dominant reduction of an element $b \\in B(G,\\mu )_{\\mathrm {un}}$ then the orbit of the character $b_{T}$ under the Weyl group $W_{G}$ can be described as $w(b_{T})$ for $w \\in W_{b}$ varying.",
"For varying $b \\in B(G,\\mu )_{\\mathrm {un}}$ , this describes the set of non-zero weights which can occur in the representation $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ .",
"In particular, given such a $\\overline{\\nu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , we can look at the direct sum of weight spaces $ \\bigoplus _{\\begin{array}{c}\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\\\ \\nu _{\\Gamma } = \\overline{\\nu }\\end{array}} \\mathcal {T}_{\\mu }(\\nu ) $ and this coincides with the weight space of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}(\\nu )$ via the isomorphism $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma } \\simeq \\mathbb {X}^{*}(\\hat{T}^{\\Gamma })$ .",
"The refined averaging formula suggests the following relationship.",
"Conjecture 1.23 (Conjecture REF ) For all geometric dominant cocharacters $\\mu $ such that $\\phi _{T}$ is $\\mu $ -regular, an unramified element $b \\in B(G,\\mu )_{\\mathrm {un}}$ , and a Weyl group element $w \\in W_{b}$ , we have an isomorphism $ \\bigoplus _{\\begin{array}{c}\\widetilde{w(b_{T})} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\\\\\widetilde{w(b_{T})}_{\\Gamma } = w(b_{T})\\end{array}} \\widetilde{w(b_{T})} \\circ \\phi _{T}|_{W_{E}} \\otimes \\mathcal {T}_{\\mu }(\\widetilde{w(b_{T})}) \\simeq \\phi _{b,w}^{\\mu } $ of $W_{E}$ -representations.",
"We verify this conjecture in some particular cases, by noting that the contribution from the $\\mu $ -ordinary locus can be explicitly computed using a shtuka analogue of Boyer's trick , as studied by Gaisin-Imai .",
"To do this, we note that we have a distinguished element in $B(G,\\mu )_{\\mathrm {un}}$ called the $\\mu $ -ordinary element, which we denote by $b_{\\mu }$ .",
"It is the maximal element in the partial ordering on $B(G,\\mu )$ , and we let $b_{\\mu _{T}}$ be its HN-dominant reduction.",
"The conjecture suggests that this should correspond to the Weyl group orbit of the highest weight of $\\mathcal {T}_{\\mu }$ .",
"In this case, the space $\\mathrm {Sht}(G,b_{\\mu },\\mu )_{\\infty }$ with its $G(\\mathbb {Q}_{p}) \\times J_{b}(\\mathbb {Q}_{p})$ -action is parabolically induced from the space $\\mathrm {Sht}(T,b_{\\mu _{T}},\\mu )_{\\infty }$ with its $T(\\mathbb {Q}_{p}) \\times T(\\mathbb {Q}_{p})$ -action.",
"In particular, using this we can deduce the following isomorphism (See Proposition REF ): $ R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\rho _{b_{\\mu },w}] \\simeq i_{B}^{G}(\\chi ^{w}) \\boxtimes w(\\mu ) \\circ \\phi _{T}|_{W_{E}}[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] $ This calculation has a very interesting consequence.",
"In particular, when combined with the refined averaging formula, we see that we must have an isomorphism $i_{B}^{G}(\\chi ^{w}) \\simeq i_{B}^{G}(\\chi )$ .",
"So, by choosing $\\mu $ to be sufficiently regular so that $W_{b_{\\mu }} = W_{G}$ , we can deduce the following.",
"Theorem 1.24 (Corollary REF ) For $\\phi _{T}$ an integral parameter with weakly normalized regular mod $\\ell $ -reduction such that there exists a $\\mu $ which is not fixed under $W_{G}$ and $\\phi _{T}$ is $\\mu $ -regular, we have an isomorphism $ i_{\\chi ,w}: i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w}) $ of smooth $G(\\mathbb {Q}_{p})$ -representations for all $w \\in W_{G}$ .",
"This showcases the strong connection between the theory of geometric Eisenstein series and the theory of intertwining operators and the Langlands quotient that has been our philosophical guide throughout.",
"A relation that holds even with mod $\\ell $ -coefficients!",
"With mod $\\ell $ coefficients, there is no good theory of intertwining operators or the Langlands quotient (See however , for the current state of the art), and we suspect that further developing the theory of geometric Eisenstein series should provide some insights into these notions in the $\\ell $ -modular setting.",
"We saw above that our previous conjecture on $\\phi _{b,w}^{\\mu }$ can be completely verified using Boyer's trick in the case that the only weights of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ are orbits of the highest weight.",
"This will be the case when the image $\\mu _{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+} \\simeq \\mathbb {X}^{*}(\\hat{T}^{\\Gamma })^{+}$ of $\\mu $ is minuscule with respect to the pairing with $\\mathbb {X}_{*}(\\hat{T}^{\\Gamma })$ .",
"If we combine this with the refined averaging formula, then we can also deduce the claim when $B(G,\\mu )$ has two elements.",
"I.e the case where $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ has two weight spaces; one corresponding to the $\\mu $ -ordinary element and the other corresponding to the basic element.",
"This will prove the previous conjecture in all cases where $\\mu _{\\Gamma } \\in \\mathbb {X}^{*}(\\hat{T}^{\\Gamma })^{+}$ is minuscule or quasi-minuscule with respect to the pairing with characters $\\mathbb {X}^{*}(\\hat{T}^{\\Gamma })$ .",
"Theorem 1.25 (Corollary REF ) For $\\mu $ a geometric dominant cocharacter such that $\\mu _{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+}$ is minuscule or quasi-minuscule with respect to the pairing with $\\mathbb {X}_{*}(\\hat{T}^{\\Gamma })$ , the previous conjecture is true.",
"Remark 1.26 Even for $\\mu $ minuscule it can be the case that the image $\\mu _{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}_{\\Gamma }$ is no longer minuscule with respect to the above pairing, as this corresponds to restricting the highest weight representation of $\\hat{G}$ defined by $\\mu $ to $\\hat{G}^{\\Gamma }$ .",
"Therefore, even for $\\mu $ minuscule, we can still have that the basic element $b \\in B(G,\\mu )$ is unramified (See ) In these cases, a very analogous result was proven by , where they describe the irreducible components of affine Deligne-Lusztig varieties in terms of the weight space defined by the basic element.",
"These affine Deligne-Lusztig varieties describe the special fibers of the local shtuka spaces $\\mathrm {Sht}(G,b,\\mu )_{\\infty }/\\underline{K}$ in the case that $G$ is unramified, and $K$ is a hyperspecial level.",
"Moreover, we suspect that, by using nearby cycles, one could deduce some special cases of our result from theirs.",
"Throughout our results, we have introduced various technical conditions on $\\phi _{T}$ .",
"We suspect that some of these conditions are artifacts of the proofs we have used to overcome the technical geometry of $\\mathrm {Bun}_{B}$ and its compacitifications in this diamond world.",
"While the conditions are manageable for specific applications to specific groups it leaves one wanting for a more conceptually clear picture.",
"In particular, we conjecture that the following is true, which (modulo checking the compatibilities of the isomorphisms in eigensheaf property) our methods show for $\\mathrm {GL}_{n}$ and integral parameters.",
"Conjecture 1.27 For $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ and $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ a generic toral $L$ -parameter, there exists a sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ which is a perverse Hecke eigensheaf with eigenvalue $\\phi $ such that one has an isomorphism $\\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}^{\\vee }})$ , and its stalk at all $b \\in B(G)$ is isomorphic to $\\mathrm {Red}_{b,\\phi }$ .",
"This conjecture would follow from knowing the existence of $\\mathrm {IC}_{\\overline{\\mathrm {Bun}}_{B}}$ and in turn the compactified Eisenstein functor $\\overline{\\mathrm {Eis}}$ with all the various desiderata proven by Braverman-Gaitsgory in the function field setting.",
"In particular, we expect that $\\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}})$ should always commute with Verdier duality, and satisfy the functional equation if $\\alpha \\circ \\phi _{T}$ is non-trivial for all $\\Gamma $ -orbits of roots.",
"Moreover, by the analogue of the results of , there should be a natural map $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\rightarrow \\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}}) $ whose cone should be given by Eisenstein functors tensored with complexes admitting a filtration isomorphic to $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ for $\\alpha $ a $\\Gamma $ -orbit of roots of $G$ .",
"In particular, we should have an isomorphism $\\overline{\\mathrm {Eis}}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ precisely when $\\phi _{T}$ is generic.",
"It follows by our above analysis that this would imply an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w})$ for all generic $\\chi $ , which is precisely what we show in the appendix (Proposition REF ).",
"In §2, we start by defining the set of unramified elements in $B(G)$ and discussing their relationship with highest weight theory, as in .",
"In §3, we review the construction of eigensheaves on $\\mathrm {Bun}_{T}$ attached to parameters $\\phi _{T}$ , introducing the conditions on our parameter $\\phi _{T}$ and working through some useful lemmas and examples related to them.",
"In §4, we review the geometric Satake correspondence of Fargues-Scholze, recalling the key results and relating the highest weight theory of $\\phantom{}^{L}G$ to the cohomology of semi-infinite Schubert cells.",
"In §5, we introduce Drinfeld's compactifications over the Fargues-Fontaine curve and establish Theorem REF .",
"We also introduce a locally closed stratification of $\\overline{\\mathrm {Bun}}_{B}$ and show it is well-behaved.",
"In §6, we move into the sheaf theory introducing the normalized Eisenstein functor and establishing Theorem 1.2.",
"In §7, we will study how the Eisenstein functor interacts with Hecke operators, establishing Theorem 1.12.",
"This will ultimately be done via a key diagram relating the action of Hecke operators of $\\mathrm {Bun}_{G}$ base-changed along the map $\\mathfrak {p}:\\mathrm {Bun}_{B} \\rightarrow \\mathrm {Bun}_{G}$ to semi-infinite Schubert cells, where it reduces to the results in §4.",
"In §8, we study Verdier duality and show Theorem 1.11.",
"In §9, we will carry out the computation of the stalks of the Eisenstein series $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ establishing Theorem 1.14.",
"In §10, we describe the theory of tilting modules for the $L$ -group $\\phantom{}^{L}G$ , constructing $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ with $\\overline{\\mathbb {Z}}_{\\ell }$ and $\\overline{\\mathbb {Q}}_{\\ell }$ -coefficients and showing Theorem 1.15.",
"Finally, in §11, we deduce the applications to the cohomology of local shtuka spaces showing Theorems 1.18, 1.20, 1.24, and 1.25."
],
[
"Acknowledgments",
"I would like to thank my advisor David Hansen for his continual support throughout this project, as well as many useful comments and suggestions, and for sharing developments in the parallel project also on Geometric Eisenstein series.",
"Special thanks also go to Peter Scholze for very helpful feedback and corrections on an earlier draft.",
"I would also like to thank Alexander Bertoloni-Meli, Arthur-César Le-Bras, Eric Chen, Dennis Gaitsgory, Thomas Haines, Tasho Kaletha, Robert Kurinczuk, Sam Mundy, Marko Tadić, Eva Viehmann, Geordie Williamson, Liang Xiao, Xinwen Zhu, and Konrad Zou, for helpful exchanges which provided insights that lead to the development and completion of this paper.",
"Lastly, I would like to thank the MPIM for their hospitality throughout part of the completion of this project."
],
[
"Notation",
" Let $\\ell \\ne p$ be distinct primes.",
"Let $G$ be a quasi-split connected reductive group over the $p$ -adic numbers $\\mathbb {Q}_{p}$ with simply connected derived group.",
"We let $\\overline{\\mathbb {Q}}_{\\ell }$ denote the algebraic closure of the $\\ell $ -adic numbers, with residue field $\\overline{\\mathbb {F}}_{\\ell }$ and ring of integers $\\overline{\\mathbb {Z}}_{\\ell }$ , endowed with the discrete topology.",
"Throughout, we will assume that, for our fixed $G$ , $\\ell $ is very good in the sense of .",
"Let $\\Gamma $ be the absolute Galois group of $\\mathbb {Q}_{p}$ , and let $W_{\\mathbb {Q}_{p}} \\subset \\Gamma $ be the Weil group of $\\mathbb {Q}_{p}$ .",
"We set $\\mathcal {L}_{\\mathbb {Q}_{p}} := W_{\\mathbb {Q}_{p}} \\times \\mathrm {SL}_{2}$ to be the Weil-Deligne group.",
"Fix choices $A \\subset T \\subset B \\subset G$ of maximal split torus, maximal non-split torus, and Borel.",
"We use $U$ to denote the unipotent radical of $B$ .",
"We let $W_{G}$ be the relative Weyl group of $G$ and $w_{0}$ be the element of longest length.",
"We write $\\mathrm {Ind}_{B}^{G}(-)$ for the unnormalized parabolic induction functor from $B$ to $G$ .",
"We let $\\delta _{B}^{1/2}$ be the modulus character defined by $B$ and set $i_{B}^{G}(-) := \\mathrm {Ind}_{B}^{G}(- \\otimes \\delta _{B}^{1/2})$ to be the normalized induction.",
"Let $\\breve{\\mathbb {Q}}_{p}$ be the completion of the maximal unramified extension of $\\mathbb {Q}_{p}$ with Frobenius $\\sigma $ .",
"For $E/\\mathbb {Q}_{p}$ a finite extension, we set $\\breve{E}$ to be the compositum $E\\breve{\\mathbb {Q}}_{p}$ .",
"Set $\\mathbb {C}_{p}$ to be the completion of the algebraic closure of $\\mathbb {Q}_{p}$ .",
"Let $B(G) = G(\\breve{\\mathbb {Q}}_{p})/(g \\sim hg\\sigma (h)^{-1})$ denote the Kottwitz set of $G$ .",
"For $b \\in B(G)$ , we write $J_{b}$ for the $\\sigma $ -centralizer of $b$ .",
"We will always work over the fixed base $\\ast := \\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}}$ , unless otherwise stated.",
"Let $\\mathrm {Perf}$ denote the category of (affinoid) perfectoid spaces in characteristic $p$ over $\\ast $ .",
"For $S \\in \\mathrm {Perf}$ , let $\\mathrm {Perf}_{S}$ denote the category of affinoid perfectoid spaces over it.",
"For $S \\in \\mathrm {Perf}$ , let $X_{S}$ denote the relative (schematic) Fargues-Fontaine curve over $S$ .",
"For $\\mathop {\\rm Spa}{(F,\\mathcal {O}_{F})} \\in \\mathrm {Perf}$ a geometric point, we will often drop the subscript on $X_{F}$ and just write $X$ for the associated Fargues-Fontaine curve.",
"For $b \\in B(G)$ , we write $\\mathcal {F}_{b}$ for the associated $G$ -bundle on $X$ .",
"For $S \\in \\mathrm {Perf}$ , we let $\\mathcal {F}_{G}^{0}$ denote the trivial $G$ -bundle on $X_{S}$ .",
"We consider coefficient systems $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ , with a fixed choice of square root of $p \\in \\Lambda $ .",
"We define all half Tate twists with respect to this choice.",
"For an Artin $v$ -stack $X$ , we write $\\mathrm {D}_{\\blacksquare }(X,\\Lambda )$ for the condensed $\\infty $ -category of solid $\\Lambda $ -valued sheaves on $X$ , and write $\\mathrm {D}_{\\mathrm {lis}}(X,\\Lambda ) \\subset \\mathrm {D}_{\\blacksquare }(X,\\Lambda )$ for the full sub-category of $\\Lambda $ -valued lisse-étale sheaves, as defined in .",
"For a $v$ -stack or diamond $X$ , when working with torsion coefficients, we will indicate this by just writing $\\mathrm {D}(X) := \\mathrm {D}_{\\text{ét}}(X,\\Lambda )$ for the category of étale $\\Lambda $ -sheaves on $X$ , as defined .",
"If $X$ is an Artin $v$ -stack () then we will regard it as a condensed $\\infty $ -category via the identification $\\mathrm {D}_{\\mathrm {lis}}(X,\\Lambda ) \\simeq \\mathrm {D}(X)$ when viewed as objects in $\\mathrm {D}_{\\blacksquare }(X,\\Lambda )$ , as described in .",
"We let $\\hat{G}$ denote the Langlands dual group of $G$ with fixed splitting $(\\hat{T},\\hat{B},\\lbrace X_{\\alpha }\\rbrace )$ .",
"If $E$ denotes the splitting field of $G$ then the action of $W_{\\mathbb {Q}_{p}}$ factors through $Q:= W_{E}/W_{\\mathbb {Q}_{p}}$ .",
"We let $\\phantom{}^{L}G := \\hat{G} \\rtimes Q$ denote the $L$ -group.",
"For $I$ a finite index set, we let $\\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ denote the category of finite-dimensional algebraic representations of $\\phantom{}^{L}G^{I}$ .",
"To any condensed $\\infty $ -category $\\mathcal {C}$ , we write $\\mathcal {C}^{BW^{I}_{\\mathbb {Q}_{p}}}$ for the category of objects with continuous $W_{\\mathbb {Q}_{p}}^{I}$ -action, as defined in .",
"We will let $\\mathrm {Div}^{1} := \\mathop {\\rm Spd}{\\breve{\\mathbb {Q}}_{p}}/\\mathrm {Frob}^{\\mathbb {Z}}$ denote the mirror curve, and, for a finite extension $E/\\mathbb {Q}_{p}$ , we write $\\mathrm {Div}^{1}_{E}$ for the base-change to $E$ .",
"For $I$ a finite index set, we let $\\mathrm {Div}^{I}$ denote $|I|$ -copies of the mirror curve.",
"For $n \\in \\mathbb {Z}$ , we let $\\mathrm {Div}^{(n)} = (\\mathrm {Div}^{1})^{n}/S_{n}$ , denote the $n$ th symmetric power of the mirror curve, where $S_{n}$ is the symmetric group on $n$ letters.",
"For a reductive group $H/\\mathbb {Q}_{p}$ , we write $\\mathrm {D}(H(\\mathbb {Q}_{p}),\\Lambda )$ for the unbounded derived category of smooth $\\Lambda $ -representations.",
"We say a map of $v$ -stacks $f: X \\rightarrow Y$ is representable in nice diamonds if it is representable in locally spatial diamonds, is compactifiable, and (locally) $\\mathrm {tr.deg}(f) < \\infty $ .",
"All 6-functors will be implicitly derived unless otherwise stated.",
"For a locally pro-$p$ group $H$ , we write $\\underline{H}$ for the functor sending $S \\in \\mathrm {Perf}$ to $\\mathrm {Cont}(|S|,H)$ , the set of continuous maps from the underlying topological space of $S$ to $H$ .",
"Remark 2.1 At various points, we will need to consider the functors $f_{!",
"}: \\mathrm {D}(X) \\rightarrow \\mathrm {D}(Y)$ and $f^{!",
"}: \\mathrm {D}(Y) \\rightarrow \\mathrm {D}(X)$ for certain \"stacky\" morphisms of Artin $v$ -stacks $f: X \\rightarrow Y$ .",
"The correct definitions of these functors in this case are given in the work of .",
"In particular, they extend the 6-functors studied in , to fine maps of decent $v$ -stacks .",
"In general being a decent $v$ -stack is stronger than being Artin.",
"However, it is easy to check that all the stacks (resp.",
"morphisms) we consider these functors for will be decent (resp.",
"fine).",
"To see this, one can use which states that if $f: X \\rightarrow Y$ is a map of $v$ -stacks which is representable in nice diamonds and $Y$ is decent then $X$ is also decent and $f$ is fine.",
"In the cases we consider, one can apply this if one takes $Y = \\mathrm {Bun}_{G}$ .",
"To see that $\\mathrm {Bun}_{G}$ is decent, one can use the charts studied in , and take advantage of the fact that the maps defining the charts are formally smooth by .",
"This in particular allows one to see that these charts map strictly surjectively to $\\mathrm {Bun}_{G}$ .",
"It remains to explain why the maps appearing in our context fine, to do this one can combine the previous analysis with , which says that fine morphisms satisfy the 2 out of 3 property.",
"When speaking about such fine maps of decent $v$ -stacks we will often just cite theorems that only apply to the setting where $f$ is representable in nice diamonds, and leave it to the reader to check that one can deduce the analogous results from the cited result and the formal properties of the 6-functors defined in .",
"Given a decent $v$ -stack $X \\rightarrow \\ast $ such that $X$ is fine over $\\ast $ , we let $K_{X} := f^{!",
"}(\\Lambda ) \\in \\mathrm {D}(X)$ denote the dualizing object of $X$ .",
"Similarly, for $\\mathcal {F} \\in \\mathrm {D}(X)$ , we will write $R\\Gamma _{c}(X,\\mathcal {F}) := f_{!",
"}(\\mathcal {F}) \\in D(\\Lambda )$ .",
"We write $\\mathbb {D}_{X}(-) := R\\mathcal {H}om(-,K_{X})$ for the Verdier duality functor.",
"For a fine map $f: X \\rightarrow S$ of decent $v$ -stacks, we write $\\mathbb {D}_{X/S} := R\\mathcal {H}om(-,f^{!",
"}(\\Lambda ))$ for relative Verdier duality.",
"We will use the geometric normalization of local class field theory.",
"For $n \\in \\mathbb {Z}$ , we write $(n)$ for the $n$ th power of the cyclotomic character of $W_{\\mathbb {Q}_{p}}$ .",
"We note that, under this normalization, $(1)$ is sent to the norm character $|\\cdot |: \\mathbb {Q}_{p}^{*} \\rightarrow \\Lambda ^{*}$ , which acts trivially on $\\mathbb {Z}_{p}^{*} $ and sends $p$ to $p^{-1} \\in \\Lambda ^{*}$ .",
"Before introducing the rest of the notation, we discuss the relationship between unramified elements in $B(G)$ and the representation theory of the $L$ -group."
],
[
"Unramified Elements in $B(G)$ and Highest Weight Theory",
"In this section, we will study the set of unramified elements in the Kottwitz set of $G$ .",
"As we will show, these elements are connected to the highest weight theory of the Langlands dual group $\\hat{G}$ , as discussed in .",
"First, we recall that the Kottwitz set $B(G)$ of a connected reductive group $G/\\mathbb {Q}_{p}$ is equipped with two maps: The slope homomorphism $ \\nu : B(G) \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+,\\Gamma }_{\\mathbb {Q}} $ $ b \\mapsto \\nu _{b} $ where $\\Gamma := \\mathrm {Gal}(\\overline{\\mathbb {Q}}_{p}/\\mathbb {Q}_{p})$ and $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\mathbb {Q}}^{+}$ is the set of rational dominant cocharacters of $G$ .",
"The Kottwitz invariant $ \\kappa : B(G) \\rightarrow \\pi _{1}(G)_{\\Gamma } $ where $\\pi _1(G)$ denotes the algebraic fundamental group of Borovoi.",
"Now, given a geometric cocharacter $\\mu $ of $G$ with reflex field $E$ , we can define the element: $ \\tilde{\\mu } := \\frac{1}{[E:\\mathbb {Q}_{p}]} \\sum _{\\gamma \\in \\mathrm {Gal}(E/\\mathbb {Q}_{p})} \\gamma (\\mu ) \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+,\\Gamma }_{\\mathbb {Q}} $ We let $\\mu ^{\\flat }$ be the image of $\\mu $ in $\\pi _1(G)_{\\Gamma } \\simeq X^*(Z(\\hat{G})^{\\Gamma })$ .",
"Via the isomorphim $B(G)_{\\mathrm {basic}} \\simeq \\pi _{1}(G)_{\\Gamma }$ , we regard it as a basic element of $B(G)$ , which are the minimal elements in the natural partial ordering on $B(G)$ .",
"Now we recall that, for a torus $T$ , we have an isomorphism $B(T) \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ .",
"We can use this isomorphism to give a nice description of a certain piece of $B(G)$ .",
"Definition 2.2 We let $B(G)_{\\mathrm {un}} \\subset B(G)$ denote the image of the natural map $B(T) \\rightarrow B(G)$ .",
"We refer to this as the set of unramified elements.",
"We now have the following Lemma.",
"We write $(-)_{\\Gamma }$ for the natural quotient map $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ .",
"Lemma 2.3 Let $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma } \\simeq B(T) \\rightarrow B(G)$ be the natural map.",
"Then this induces an isomorphism: $ \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }/W_{G} \\simeq B(G)_{\\mathrm {un}} $ Strictly speaking, the proof given by Xiao-Zhu is only in the case that $G$ is unramified.",
"We remedy this now.",
"Note that it is clear that this map is surjective, so it suffices to check injectivity.",
"Let $\\mu _{1},\\mu _{2} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ be two elements with $b_{1},b_{2}$ their images in $B(G)$ under the natural composite $ \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma } \\simeq B(T) \\rightarrow B(G) $ , and suppose that $b_{1} = b_{2}$ .",
"Since $\\kappa _{G}(b_{1}) = \\kappa _{G}(b_{2})$ , it follows that we have $\\mu _{1} - \\mu _{2} = (\\gamma - 1)\\nu + \\alpha $ for some coroot $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ and $\\gamma \\in \\Gamma $ .",
"We may, without loss of generality, replace $\\mu _{1}$ by $\\mu _{1} + (\\gamma - 1)\\nu $ , and therefore assume that $\\mu _{1} - \\mu _{2} = \\alpha $ .",
"Since the slope homomorphisms of $\\nu _{b_{1}}$ and $\\nu _{b_{2}}$ are equal by assumption, we can assume, after conjugating by an element of $W_{G}$ , that $\\tilde{\\mu }_{1} = \\tilde{\\mu }_{2}$ .",
"Therefore, it follows that, if $E_{\\alpha }$ denotes the reflex field of $\\alpha $ , we have an equality $ \\sum _{g \\in \\mathrm {Gal}(E_{\\alpha }/\\mathbb {Q}_{p})} g(\\alpha ) = 0 $ which in turn implies that $ \\sum _{g \\in \\mathrm {Gal}(E_{\\alpha }/\\mathbb {Q}_{p})} (1 - g)(\\alpha ) = |\\mathrm {Gal}(E_{\\alpha }/\\mathbb {Q}_{p})|\\alpha $ This would imply that $\\alpha _{\\Gamma }$ vanishes in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ assuming that $\\alpha _{\\Gamma }$ isn't torsion.",
"However, $\\Gamma $ permutes the simple coroots, which form a basis of all coroots.",
"Therefore, it follows that $\\alpha _{\\Gamma }$ is not torsion.",
"Now we would like to describe $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }/W_{G}$ slightly differently.",
"To do this, we consider the natural pairing $ \\langle -,- \\rangle : \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma } \\times \\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma } \\rightarrow \\mathbb {Z} $ induced by the usual pairing between cocharacters and characters.",
"We let $\\hat{\\Delta } \\subset \\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ (resp.",
"$\\Delta \\subset \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ ) be the set of (absolute) simple roots (resp.",
"coroots) of $G$ .",
"Then we define $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+}$ to be the set of elements in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ whose inner product with $\\text{Im}{(\\hat{\\Delta } \\rightarrow \\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma })}$ under the natural averaging map is positive.",
"The natural map $ \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+} \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }/W_{G} $ is an isomorphism.",
"We also note that we have a natural partial ordering on $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ .",
"In particular, given $\\overline{\\nu }, \\overline{\\nu }^{\\prime } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ we say that $\\overline{\\nu } \\ge \\overline{\\nu }^{\\prime }$ if $\\overline{\\nu } - \\overline{\\nu }^{\\prime }$ is a positive integral combination of $\\alpha _{\\Gamma }$ for $\\alpha \\in \\Delta $ .",
"We note that we have a natural injective order preserving map: $ \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}_{\\Gamma } \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\mathbb {Q}}^{\\Gamma ,+} \\times \\pi _{1}(G)_{\\Gamma } $ With this, we can reformulate the previous lemma as follows.",
"Lemma 2.4 The following diagram is commutative and respects the partial ordering $ \\begin{tikzcd}& B(G)_{\\mathrm {un}} [r,\"\\simeq \"] [d] & \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}_{\\Gamma } [d] \\\\& B(G) [r,\"\\nu \\times \\kappa \"] & \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\mathbb {Q}}^{\\Gamma ,+} \\times \\pi _{1}(G)_{\\Gamma }\\end{tikzcd} $ Now recall that, for $\\mu $ a geometric dominant cocharacter of $G$ , we have the following.",
"Definition 2.5 We define $B(G,\\mu ) \\subset B(G)$ to be subset of $b \\in B(G)$ for which $\\nu _{b} \\le \\tilde{\\mu }$ with respect to the Bruhat ordering and $\\kappa (b) = \\mu ^{\\flat }$ .",
"The previous lemma allows us to interpret the unramified elements in this set as follows.",
"Corollary 2.6 Under the identifications $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+} \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }/W_{0} \\simeq B(G)_{\\mathrm {un}}$ , we have an equality: $ B(G,\\mu )_{\\mathrm {un}} := B(G)_{\\mathrm {un}} \\cap B(G,\\mu ) = \\lbrace \\lambda _{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+}\\text{ } | \\text{ } \\lambda _{\\Gamma } \\le \\mu _{\\Gamma } \\rbrace $ We now would like to connect this set with the highest weight theory for $\\hat{G}$ .",
"If the group is not split then the unramified elements are naturally connected with the highest weight theory of the subgroup $\\hat{G}^{\\Gamma }$ .",
"Even though $\\hat{G}^{\\Gamma }$ is possibly disconnected its representation theory behaves like a connected reductive group.",
"To see this, first we note that the subgroup $\\hat{T}^{\\Gamma }$ defined by the maximal torus has character group isomorphic to $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , and the partial order described above allows one to talk about the highest weight of a representation.",
"In particular, if we let $\\hat{T}^{\\Gamma ,\\circ }$ (resp.",
"$\\hat{G}^{\\Gamma ,\\circ }$ ) denote the neutral component of $\\hat{T}^{\\Gamma }$ (resp.",
"$\\hat{G}^{\\Gamma }$ ).",
"Then one can use that the natural map $\\hat{T}^{\\Gamma }/\\hat{T}^{\\Gamma ,\\circ } \\rightarrow \\hat{G}^{\\Gamma }/\\hat{G}^{\\Gamma ,\\circ }$ is an isomorphism () to see that usual highest weight theory extends to $\\hat{G}^{\\Gamma }$ .",
"In particular, we have the following.",
"Lemma 2.7 For $\\overline{\\mu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}_{\\Gamma }$ , there is a unique up to isomorphism irreducible representation of $V_{\\overline{\\mu }} \\in \\mathrm {Rep}_{\\overline{\\mathbb {Q}}_{\\ell }}(\\hat{G}^{\\Gamma })$ of highest weight $\\overline{\\mu }$ , which give rise to all the irreducible representations in $\\mathrm {Rep}_{\\overline{\\mathbb {Q}}_{\\ell }}(\\hat{G}^{\\Gamma })$ for varying $\\overline{\\mu }$ .",
"Moreover, the multiplicity of the $\\overline{\\mu }$ weight space in $V_{\\overline{\\mu }}$ is 1, and the non-zero weights $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ of $V_{\\overline{\\mu }}$ lie in the convex hull of the $W_{G}$ -orbit of $\\overline{\\mu }$ .",
"To a geometric dominant cocharacter $\\mu $ , we can attach an irreducible representation $V_{\\mu } \\in \\mathrm {Rep}_{\\overline{\\mathbb {Q}}_{\\ell }}(\\hat{G})$ .",
"This defines a natural representation of $\\hat{G} \\rtimes W_{E_{\\mu }}$ as in , where $E_{\\mu }$ is the reflex field of $\\mu $ .",
"An element $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ defines a representation of $\\hat{T}$ , and we write $V_{\\mu }(\\nu )$ for the corresponding weight space of $V_{\\mu }$ .",
"If we consider the restriction $V_{\\mu }|_{\\hat{G}^{\\Gamma }}$ then the weight space $V_{\\mu }(\\nu )$ gives rise to a $\\nu _{\\Gamma }$ weight space.",
"Using this, it is easy to see we have the following relationship.",
"Lemma 2.8 For $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ and $\\overline{\\nu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , we have the following equality: $ \\mathrm {dim}(V_{\\mu }(\\overline{\\nu })) = \\sum _{\\begin{array}{c}\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})\\\\ \\nu _{\\Gamma } = \\overline{\\nu }\\end{array}} \\mathrm {dim}(V_{\\mu }(\\nu )) $ We will often combine this lemma with the following, which follows from the above discussion.",
"Corollary 2.9 () For $\\mu $ a geometric dominant cocharacter, under the identification $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+} \\simeq B(G)_{\\mathrm {un}}$ the elements $\\overline{\\nu } \\in B(G,\\mu )_{\\mathrm {un}}$ correspond to $W_{G}$ -orbits of the possible non-zero weights in $V_{\\mu }|_{\\hat{G}^{\\Gamma }}$ .",
"Let's study this now more carefully.",
"For $b \\in B(G)$ , we want to use the above discussion to understand the fiber of the map: $ i: B(T) \\rightarrow B(G)_{\\mathrm {un}} \\subset B(G) $ Recall that, given $b \\in B(G)$ , since $G$ is quasi-split the $\\sigma $ -centralizer $J_{b}$ is an extended pure inner form (in the sense of ) of a Levi subgroup $M_{b}$ of $G$ , which is the centralizer of the slope homomorphism $\\nu _{b}$ of $b$ .",
"We make the following definition.",
"Definition 2.10 For $b \\in B(G)$ , we let $W_{M_{b}}$ denote the relative Weyl group of $M_{b}$ and set $W_{b} := W_{G}/W_{M_{b}}$ .",
"We will fix a set of representatives $w \\in W_{G}$ of minimal length, as in , and abuse notation by writing $w$ for both the representative and the corresponding element.",
"When combining this discussion with Lemma REF , we can deduce the following Corollary.",
"Corollary 2.11 For fixed $b \\in B(G)_{\\mathrm {un}}$ , the fiber $i^{-1}(b)$ has a unique element, denoted $b_{T}$ , whose $\\kappa $ -invariant lies in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{-}$ the negative Weyl chamber defined by the choice of Borel.",
"Moreover, we have an equality $i^{-1}(b) = \\lbrace w(b_{T})\\text{ } |\\text{ } w \\in W_{b} \\rbrace $ .",
"Now, given a parabolic $P$ with Levi factor $M$ , the element $b \\in B(G)$ admits a reduction to a Levi subgroup $M$ if and only if the parabolic $P \\cap M$ of $M$ transfers to a parabolic of $J_{b}$ under the inner twisting (apply to the basic reduction of $b$ to $M_{b}$ ).",
"We record this specialized to the case of the Borel for future use.",
"Lemma 2.12 An element $b \\in B(G)$ lies in $B(G)_{\\mathrm {un}}$ if and only if $B \\cap M_{b}$ defines, via the inner twisting, a Borel subgroup of $J_{b}$ .",
"Remark 2.13 We note that, when $b \\in B(G)_{\\mathrm {un}}$ , we have an isomorphism $M_{b} \\simeq J_{b}$ because $J_{b}$ will be a quasi-split inner form of $M_{b}$ .",
"We leave off with an important observation about the map $(-)_{\\Gamma }: \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ from orbits to coinvariants.",
"Let $\\tilde{\\mathcal {J}}$ (resp.",
"$\\mathcal {J}$ ) denote the vertices of the absolute (resp.",
"relative) Dynkin diagram of $G$ .",
"For $i \\in \\tilde{\\mathcal {J}}$ , we write $\\tilde{\\alpha }_{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ for the corresponding simple absolute coroot.",
"We recall that $\\Gamma $ permutes the $\\tilde{\\alpha }_{i}$ , and the orbits under $\\Gamma $ are in bijection with elements of $\\mathcal {J}$ ; namely, the average over the orbit is the (reduced) positive coroot corresponding to $i \\in \\mathcal {J}$ .",
"Therefore, for each $i \\in \\mathcal {J}$ , we obtain an element $\\alpha _{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ given by the common image of the elements in the orbit corresponding to $i \\in \\mathcal {J}$ under the map $(-)_{\\Gamma }$ .",
"This allows us to make the following definition.",
"Definition 2.14 We denote the group of coinvariants by $\\Lambda _{G,B} := \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , and set $\\Lambda _{G,B}^{pos}$ to be the semi-group spanned by the elements $\\alpha _{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ corresponding to the $\\Gamma $ -orbit indexed by $i \\in \\mathcal {J}$ .",
"Now we introduce the rest of the notation motivated by the discussion above.",
"We let $\\Lambda _G^+ := \\mathbb {X}_{*}(A)^{+}$ (resp.",
"$\\hat{\\Lambda }^+_G := \\mathbb {X}^{*}(A)^{+}$ ) be the semi-group of dominant cocharacters (resp.",
"characters), viewed as elements of the positive Weyl chamber in $\\Lambda _{G} := \\mathbb {X}_{*}(A) = \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma }$ (resp.",
"$\\hat{\\Lambda }_{G} = \\mathbb {X}^{*}(A) = \\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma }$ ) defined by the choice of Borel.",
"To a dominant character $\\hat{\\lambda } \\in \\hat{\\Lambda }^+_G$ , we get an associated highest weight Weyl $G$ -module, denoted $\\mathcal {V}^{\\hat{\\lambda }}$ .",
"It has a fixed highest weight vector $v^{\\hat{\\lambda }} \\in \\mathcal {V}^{\\hat{\\lambda }}$ , and, given a pair of such weights $\\hat{\\lambda }_1, \\hat{\\lambda }_2$ , there is a canonical map $ \\mathcal {V}^{\\hat{\\lambda }_1 + \\hat{\\lambda }_2} \\rightarrow \\mathcal {V}^{\\hat{\\lambda }_1} \\otimes \\mathcal {V}^{\\hat{\\lambda }} $ which takes $v^{\\hat{\\lambda }_1 + \\hat{\\lambda }_2}$ to $v^{\\hat{\\lambda }_1} \\otimes v^{\\hat{\\lambda }_2}$ .",
"Let $\\mathcal {J}$ be the set of vertices of the relative Dynkin diagram of $G/\\mathbb {Q}_{p}$ .",
"For each $i \\in \\mathcal {J}$ , we denote the corresponding element by $\\alpha _i \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , as in Definition REF .",
"We warn the reader that this is different then the $\\Gamma $ -orbits defined by the (reduced) simple roots corresponding to $i \\in \\mathcal {J}$ , these we will denote by $\\alpha _{i,A} \\in X_{*}(A) \\subset X_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma }$ , and will be sum over the elements in the Galois orbit associated to $\\alpha _{i}$ .",
"On the other hand, in the root lattice $\\hat{\\Lambda }_{G}$ we will only be interested in the (reduced) simple positive roots corresponding to $i \\in \\mathcal {J}$ , and so we just write this as $\\hat{\\alpha }_{i} \\in \\hat{\\Lambda }_{G}$ .",
"We consider the natural pairing $ \\langle -,- \\rangle : \\hat{\\Lambda }_{G} \\times \\Lambda _{G,B} \\rightarrow \\mathbb {Z} $ given by the identifications $\\hat{\\Lambda }_{G} = \\mathbb {X}^{*}(A) \\simeq \\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma }$ and $\\Lambda _{G,B} = \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ .",
"Using the assumption on the derived group of $G$ being simply connected, we define a set of fundamental weights $\\hat{\\varpi }_i \\in \\hat{\\Lambda }_G$ , non-uniquely characterized by the property that $\\langle \\hat{\\varpi }_{i}, \\alpha _{j} \\rangle = \\delta _{ij}$ .",
"Namely, this can be defined by taking the sum over the Galois orbits of fundamental weights in $\\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ .",
"We will regularly use the natural quotient map $ (-)_{\\Gamma }: \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})\\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ as well as the map $ (-)^{\\Gamma }: \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})\\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ from cocharacters to their Galois orbits.",
"We note that $(-)_{\\Gamma }$ factorizes over $(-)^{\\Gamma }$ .",
"For a geometric dominant cocharacter $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ with reflex field $E$ , we write $V_{\\mu }$ for the natural representation of $W_{E} \\ltimes \\hat{G}$ of highest weight $\\mu $ , as in .",
"We also write $V_{\\mu ^{\\Gamma }} \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ for the induction of this representation to $\\phantom{}^{L}G$ , which only depends on the associated $\\Gamma $ -orbit $\\mu ^{\\Gamma }$ of $\\mu $ .",
"For $b \\in B(G)$ , we let $J_{b}$ be the extended pure inner form of $M_{b}$ considered above.",
"If $b \\in B(G)_{\\mathrm {un}}$ , we write $B_{b}$ for the Borel defined by $B \\cap M_{b}$ under the inner twisting, as in Lemma REF .",
"Given $b \\in B(G)_{\\mathrm {un}}$ , we note that, by Lemma REF , there exists a unique element $b_{T} \\in B(T)$ with anti-dominant slope homomorphism with respect to the choice of Borel.",
"We will refer to this as the HN-dominant reduction of $b$ .",
"Recalling that there is a minus sign when passing between isocrystals and bundles (cf.",
"), this will correspond to the case where the Harder-Narasimhan slopes are dominant.",
"We set $W_{b} := W_{G}/W_{M_{b}}$ , where $W_{M_{b}}$ (resp.",
"$W_{G}$ ) is the relative Weyl group of $M_{b}$ (resp.",
"$G$ ).",
"We will identify $w$ with a representative of minimal length in $W_{G}$ throghout.",
"We will write $\\hat{\\rho }$ for the half sum of all positive roots."
],
[
"Hecke Eigensheaves for Tori",
"In this section, we want to talk about geometric local class field theory.",
"Namely, given a torus $T$ with $L$ -parameter $ \\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda ) $ where $\\phantom{}^{L}T$ denotes the Langlands dual group of $T$ , we want to construct a Hecke eigensheaf, denoted $\\mathcal {S}_{\\phi _{T}}$ , on the moduli stack $\\mathrm {Bun}_{T}$ .",
"We recall that, for a representation $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ of $I$ -copies of the $L$ -group of $G$ for some finite index set $I$ , a Hecke operator is a map defined by the correspondence $ \\begin{tikzcd}& & [dl,\"h^{\\leftarrow } \\times \\pi \"] \\mathrm {Hck}_{G}^{I} [dr,\"h^{\\rightarrow }\"] & & \\\\& \\mathrm {Bun}_{G} \\times \\mathrm {Div}^{I} & & \\mathrm {Bun}_{G} &\\end{tikzcd} $ where $\\mathrm {Hck}_{G}^{I}$ is the functor that parametrizes, for $S \\rightarrow \\mathrm {Div}^{I}$ defining a tuple of Cartier divisors in the relative Fargues-Fontaine $X_{S}$ over $S$ , corresponding to characteristic 0 untilts $S_{i}^{\\sharp }$ for $i \\in I$ of $S$ , a pair of $G$ -torsors $\\mathcal {E}_{1}$ , $\\mathcal {E}_{2}$ together with an isomorphism $ \\beta :\\mathcal {E}_{1}|_{X_{S} \\setminus \\bigcup _{i \\in I} S_{i}^{\\sharp }} \\xrightarrow{} \\mathcal {E}_{2}|_{X_{S} \\setminus \\bigcup _{i \\in I} S_{i}^{\\sharp }}$ where $h^{\\rightarrow }((\\mathcal {E}_{1},\\mathcal {E}_{2},(S_{i}^{\\sharp })_{i \\in I})) = \\mathcal {E}_{2}$ and $(h^{\\leftarrow } \\times \\pi )((\\mathcal {E}_{1},\\mathcal {E}_{2},\\beta ,(S_{i}^{\\sharp })_{i \\in I})) = (\\mathcal {E}_{1},(S_{i}^{\\sharp })_{i \\in I})$ .",
"For an algebraic representation $V \\in \\mathrm {Rep}_{\\Lambda }(^{L}G^{I})$ , the geometric Satake correspondence of Fargues-Scholze furnishes a sheaf $\\mathcal {S}_{V} \\in \\mathrm {D}_{\\blacksquare }(\\mathrm {Hck},\\Lambda )$ .",
"Using this, we can define the Hecke operator as $ T_{V}: \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda ) \\rightarrow \\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G} \\times \\mathrm {Div}^{I},\\Lambda ) $ $ \\mathcal {F} \\mapsto (h^{\\leftarrow } \\times \\pi )_{\\natural }((h^{\\rightarrow })^{*}(\\mathcal {F}) \\otimes \\mathcal {S}_{V})$ where $(h^{\\leftarrow } \\times \\pi )_{\\natural }$ is the natural push-forward.",
"I.e the left adjoint to the restriction functor in the category of solid $\\Lambda $ -sheaves .",
"This induces a functor $ T_{V}: \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda ) \\rightarrow \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{\\mathbb {Q}_{p}}^{I}} $ via a version of Drinfeld's Lemma .",
"When $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ , this is essentially the statement that $\\Lambda $ -valued local systems on $\\mathrm {Div}^{I}$ are equivalent to continuous representations of $W_{\\mathbb {Q}_{p}}^{I}$ on finite projective $\\Lambda $ -modules .",
"In this case, we will freely pass between this perspective of local systems and $W_{\\mathbb {Q}_{p}}^{I}$ -representations.",
"With this in hand, we can define what it means for a sheaf on $\\mathrm {Bun}_{G}$ to be a Hecke eigensheaf.",
"Definition 3.1 Given a continuous $L$ -parameter $\\phi : W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}G(\\Lambda )$ , we say a sheaf $\\mathcal {S}_{\\phi } \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ is a Hecke eigensheaf with eigenvalue $\\phi $ if, for all $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ with associated map $r_{V}: \\phantom{}^{L}G^{I} \\rightarrow \\mathrm {GL}(V)$ , we are given isomorphisms $ \\eta _{V,I}: T_{V}(\\mathcal {S}_{\\phi }) \\simeq \\mathcal {S}_{\\phi } \\boxtimes r_{V} \\circ \\phi $ of sheaves in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{\\mathbb {Q}_{p}}^{I}}$ , that are natural in $I$ and $V$ , and compatible with compositions and exterior tensor products in $V$ .",
"We will similarly say that $\\mathcal {S}_{\\phi }$ is a weak eigensheaf with eigenvalue $\\phi $ if we only know the existence of these isomorphisms.",
"Remark 3.2 We recall that Hecke operators are monoidal and functorial in $(V,I)$ .",
"In particular, given two representations $V,W \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ , we have a natural isomorphism $ (T_{V} \\times \\mathrm {id})(T_{W})(\\cdot )|_{\\Delta } \\simeq T_{V \\otimes W}(\\cdot ) $ where $\\Delta : \\mathrm {Div}^{1} \\rightarrow (\\mathrm {Div}^{1})^{2}$ is the diagonal map.",
"The compatibilities for the isomorphisms $\\eta _{V,I}$ are defined with respect to such isomorphisms.",
"Now let's elucidate what this means for tori.",
"Recall that an irreducible representation of $\\phantom{}^{L}T^{I}$ is parametrized by a tuple of Galois orbits $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ .",
"Similarly, one has a decomposition $ \\mathrm {Hck}_{T}^{I} = \\bigsqcup _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}} \\mathrm {Hck}^{I}_{T,(\\nu _{i})_{i \\in I}} $ of $\\mathrm {Hck}_{T}^{I}$ into open and closed substacks, where $\\mathrm {Hck}^{I}_{T,(\\nu _{i})_{i \\in I}}$ parametrizes a modification $\\mathcal {E}_{1} \\dashrightarrow \\mathcal {E}_{2}$ of meromorphy given by the Galois orbit $\\nu _{i}$ over the $I$ Cartier divisors in $\\mathrm {Div}^{I}$ .",
"If one lets $\\mathcal {S}_{(\\nu _{i})_{i \\in I}}$ be the sheaf defined by the representation of $\\phantom{}^{L}T^{I}$ corresponding to $(\\nu _{i})_{i \\in I}$ , then this sheaf is simply the constant sheaf supported on the component $\\mathrm {Hck}^{I}_{T,(\\nu _{i})_{i \\in I}}$ .",
"Therefore, for studying the Hecke operator $T_{(\\nu _{i})_{i \\in I}}$ , we can restrict the Hecke correspondence to the diagram: $ \\begin{tikzcd}& & [dl,\"h_{(\\nu _{i})_{i \\in I}}^{\\leftarrow } \\times \\pi \"] \\mathrm {Hck}^{I}_{T,(\\nu _{i})_{i \\in I}} [dr,\"h_{(\\nu _{i})_{i \\in I}}^{\\rightarrow }\"] & & \\\\& \\mathrm {Bun}_{T} \\times \\mathrm {Div}^{I} & & \\mathrm {Bun}_{T} &\\end{tikzcd} $ Let $E_{\\nu _{i}}$ denote the reflex field of the $\\Gamma $ -orbit $\\nu _{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ .",
"We can consider the following base-change of $\\mathrm {Div}^{I}$ : $ \\mathrm {Div}^{I}_{E_{(\\nu _{i})_{i \\in I}}} := \\prod _{i \\in I} \\mathrm {Div}^{1}_{E_{\\nu _{i}}} $ We note that, since a modification of $T$ -bundles is uniquely determine by the locus of meromorphy, we have an isomorphism $\\mathrm {Hck}^{I}_{T,(\\nu _{i})_{i \\in I}} \\simeq \\mathrm {Bun}_{T} \\times \\mathrm {Div}^{I}_{E_{(\\nu _{i})_{i \\in I}}}$ .",
"Under this identification, we have a map $ h_{(\\nu _{i})_{i \\in I}}^{\\rightarrow }: \\mathrm {Bun}_{T} \\times \\mathrm {Div}^{I}_{E_{(\\nu _{i})_{i \\in I}}} \\rightarrow \\mathrm {Bun}_{T} $ where, given $(\\mathcal {F}_{T},(D_{i})_{i \\in I})$ , we denote the resulting $T$ -bundle under applying this map as $\\mathcal {F}_{T}(\\sum _{i \\in I} -\\nu _{i}D_{i})$ .",
"We note that the isomorphism class of this bundle is only determined by the image of $\\nu _{i\\Gamma }$ in the coinvariant lattice, and this will be important in the next section.",
"The map $h_{(\\nu _{i})_{i \\in I}}^{\\leftarrow } \\times \\pi $ is defined by the natural finite étale morphism $q_{(\\nu _{i})_{i \\in I}}: \\mathrm {Div}^{I}_{E_{(\\nu _{i})_{i \\in I}}} \\rightarrow \\mathrm {Div}^{I}$ , and this corresponds to inducing the $\\prod _{i \\in I} W_{E_{\\nu _{i}}}$ action to $\\prod _{i \\in I} W_{\\mathbb {Q}_{p}}$ .",
"Now, via local class field theory, there is a character $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ attached to $\\phi _{T}$ .",
"Moreover, each connected component $\\mathrm {Bun}_{T}^{\\nu }$ for varying $\\nu \\in B(T) \\simeq \\Lambda _{G,B}$ is isomorphic to the classifying stack $[\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ .",
"As a consequence, we may interpret $\\chi $ as a sheaf on the connected components $j_{\\overline{\\nu }}: \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{T}$ for $\\overline{\\nu } \\in B(T)$ .",
"One might hope that considering $ \\mathcal {S}_{\\phi _{T}} := \\bigoplus _{\\overline{\\nu } \\in B(T)} j_{\\overline{\\nu }!",
"}(\\chi ) $ the sheaf on $\\mathrm {Bun}_{T}$ whose restriction to each connected component is equal to $\\chi $ gives rise to the desired Hecke eigensheaf.",
"This is indeed the case.",
"In particular, via the realization of local class field theory in the torsion of Lubin-Tate formal groups, we have the following proposition.",
"Proposition 3.3 , The sheaf $\\mathcal {S}_{\\phi _{T}}$ is an eigensheaf with eigenvalue $\\phi _{T}$ .",
"In particular, for all $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ , we have an isomorphism $ T_{(\\nu _{i})_{i \\in I}}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\boxtimes _{i \\in I} \\nu _{i} \\circ \\phi _{T} \\otimes \\mathcal {S}_{\\phi _{T}} $ of objects in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{T},\\Lambda )^{BW_{\\mathbb {Q}_{p}}^{I}}$ .",
"More precisely, if $\\tilde{\\nu }_{i}$ is a representative of the $\\Gamma $ -orbit of $\\nu _{i}$ for all $i \\in I$ , we have an isomorphism $ (h_{(\\nu _{i})_{i \\in I}}^{\\rightarrow })^{*}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\boxtimes _{i \\in I} \\tilde{\\nu }_{i} \\circ \\phi _{T}|_{W_{E_{\\nu _{i}}}} \\otimes \\mathcal {S}_{\\phi _{T}} $ which after applying $q_{(\\nu _{i})_{i \\in I}*}$ gives rise to the previous identification of induced representations.",
"A special role will be played by the eigensheaf attached to the parameter $ \\hat{\\rho }\\circ |\\cdot |: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda ) $ where we recall that $\\hat{\\rho }$ denotes the half sum of all positive roots with respect to the choice of Borel and $|\\cdot |: W_{\\mathbb {Q}_{p}} \\rightarrow \\Lambda ^{*}$ is the norm character.",
"We note that the value of this sheaf on each connected component is given by the representation $\\delta _{B}^{1/2}$ , where $\\delta _{B}$ denotes the modulus character defined by $B$ .",
"This leads to the following definition.",
"Definition 3.4 We let $\\Delta _{B}^{1/2}$ be the eigensheaf on $\\mathrm {Bun}_{T}$ attached to the parameter $ \\hat{\\rho }\\circ |\\cdot |: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda ) $ via Proposition REF .",
"Similarly, we write $\\Delta _{B}$ for the eigensheaf attached to $2\\hat{\\rho }\\circ |\\cdot |$ , where the stalks of this sheaf are given by $\\delta _{B}$ .",
"The key point is that (up to shifts) the pullback of this eigensheaf to the moduli stack $\\mathrm {Bun}_{B}$ gives rise to a sheaf which we will denote by $\\mathrm {IC}_{\\mathrm {Bun}_{B}}$ .",
"We will see later that this sheaf is Verdier self-dual on $\\mathrm {Bun}_{B}$ and therefore tensoring by it will give rise to the morally correct definition of the Eisenstein functor.",
"We note that, given a parameter $\\phi _{T}$ with associated eigensheaf $\\mathcal {S}_{\\phi _{T}}$ , the tensor product $\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}$ will be the eigensheaf attached to the tensor product $\\phi _{T} \\otimes \\hat{\\rho } \\circ |\\cdot |$ of $L$ -parameters.",
"It therefore follows from Proposition REF that the following is true.",
"Corollary 3.5 For all $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ , we have an isomorphism $ T_{(\\nu _{i})_{i \\in I}}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\simeq \\boxtimes _{i \\in I} (\\nu _{i} \\circ \\phi _{T})(\\langle \\hat{\\rho }, \\nu _{i\\Gamma } \\rangle ) \\otimes (\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) $ of objects in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{T},\\Lambda )^{BW^{I}_{\\mathbb {Q}_{p}}}$ .",
"Remark 3.6 Note that, for any representative $\\tilde{\\nu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ of the orbit $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ , we have an equality: $\\langle \\hat{\\rho }, \\tilde{\\nu } \\rangle = \\langle \\hat{\\rho }, \\nu _{\\Gamma } \\rangle $ .",
"Now we discuss the various conditions that we will impose on our parameter $\\phi _{T}$ , as well as discuss their relationship with the irreducibility of principal series through various examples."
],
[
"Genericity, Normalized Regularity, and the Irreducibility of Principal Series",
"Consider the functor $ R\\Gamma (W_{\\mathbb {Q}_{p}},-): \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{T},\\Lambda )^{BW_{\\mathbb {Q}_{p}}} \\rightarrow \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{T},\\Lambda ) $ given by taking continuous cohomology with respect to $W_{\\mathbb {Q}_{p}}$ .",
"As we will see later, computing the Eisenstein functor applied to the eigensheaf $\\mathcal {S}_{\\phi _{T}}$ will reduce to computing the values of $R\\Gamma (W_{\\mathbb {Q}_{p}},(h_{\\nu }^{\\leftarrow })^{*}(\\mathcal {S}_{\\phi _{T}}))$ and $R\\Gamma (W_{\\mathbb {Q}_{p}},(h_{\\nu }^{\\leftarrow })^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}))$ for $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ .",
"In this note, we will want to restrict to the simplest case where these contributions all vanish.",
"The exact conditions we will need are as follows.",
"Condition/Definition 3.7 Given a parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ , we impose the following conditions on $\\phi _{T}$ in what follows.",
"For all $\\Gamma $ -orbits $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ of simple coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the Galois cohomology complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ is trivial.",
"For all $\\Gamma $ -orbits $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ of coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the Galois cohomology complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ is trivial.",
"For all $w \\in W_{G}$ , if $\\chi $ denotes the character attached to $\\phi _{T}$ by local class field theory then we have $ \\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq (\\chi \\otimes \\delta _{B}^{-1/2})^{w} $ For all $\\Gamma $ -orbits $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ of coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the Galois cohomology complex $R\\Gamma (W_{\\mathbb {Q}_{p}},(\\alpha \\circ \\phi _{T})^{2})$ is trivial.",
"If $\\phi _{T}$ satisfies (1) we say that it is weakly generic, and if it satisfies (2) then we say it is generic.",
"If it satisfies (2)-(3) we say that it is weakly normalized regular, and if it satisfies (2)-(4) we say that it is normalized regular.",
"These conditions are related to the irreducibility of the induction $i_{B}^{G}(\\chi )$ .",
"To explain this, let's translate all these conditions to the character $\\chi $ .",
"Let $E/\\mathbb {Q}_{p}$ denote the splitting field of $G$ .",
"Then the action of $W_{\\mathbb {Q}_{p}}$ on $\\hat{G}$ factors through $W_{\\mathbb {Q}_{p}}/W_{E}$ .",
"Given $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ a $\\Gamma $ -orbit of coroots with reflex field $E_{\\alpha }$ , local class field theory gives us a map $ E_{\\alpha }^{*} \\rightarrow T(E_{\\alpha }) $ attached to $\\tilde{\\alpha } \\circ \\phi _{T}|_{W_{E_{\\alpha }}}$ , for a representative $\\tilde{\\alpha }$ in the $\\Gamma $ -orbit of $\\alpha $ .",
"If we post-compose with the norm map $\\mathrm {Nm}_{E_{\\alpha }/\\mathbb {Q}_{p}}$ then we get a map $ E_{\\alpha }^{*} \\rightarrow T(\\mathbb {Q}_{p}) $ which only depends on the Galois orbit $\\alpha $ .",
"We further precompose with the norm map $\\mathrm {Nm}_{E/E_{\\alpha }}: E^{*} \\rightarrow E_{\\alpha }^{*}$ , giving a character: $ E^{*} \\rightarrow T(\\mathbb {Q}_{p}) $ We write $\\chi _{\\alpha }: E^{*} \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }^{*}$ for the precomposition of $\\chi $ with this map.",
"Now, consider the complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ , where $\\phi _{T}$ is the parameter attached to $\\chi $ .",
"It follows by Schapiro's lemma that we have an isomorphism: $ R\\Gamma (W_{E_{\\alpha }},\\tilde{\\alpha } \\circ \\phi _{T}|_{W_{E_{\\alpha }}}) \\simeq R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T}) $ Applying local Tate duality and using that the Euler-Poincaré characteristic of $R\\Gamma (W_{E_{\\alpha }},\\tilde{\\alpha } \\circ \\phi _{T}|_{W_{E_{\\alpha }}})$ is 0, we see that this is equivalent to insisting that $\\alpha \\circ \\phi _{T}|_{W_{E_{\\alpha }}}$ is not the trivial representation or the cyclotomic character $(1)$ .",
"From here, by using compatibility of local class field theory with restriction, we can see that the above conditions on $\\phi _{T}$ translate to the following condition on $\\chi $ .",
"Condition/Definition 3.8 Given a smooth character $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ consider the following conditions on $\\chi $ .",
"For all $\\Gamma $ -orbits $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ of positive roots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the character $\\chi _{\\alpha }$ is not isomorphic to the trivial representation $\\mathbf {1}$ or $|\\cdot |^{\\pm 1}_{E}$ , where $|\\cdot |_{E}$ is the norm character on $E$ the splitting field of $G$ .",
"For all $w \\in W_{G}$ , we have that $ \\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq \\chi ^{w} \\otimes (\\delta _{B}^{1/2})^{w} $ For all $\\Gamma $ -orbits $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ of positive roots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ , the character $\\chi _{\\alpha }^{2}$ is not isomorphic to the trivial representation $\\mathbf {1}$ or $|\\cdot |^{\\pm 1}_{E}$ , where $|\\cdot |_{E}$ is the norm character on $E$ the splitting field of $G$ .",
"We say that $\\chi $ is generic if (1) holds, that it is weakly normalized regular if (1)-(2) hold, and that it is normalized regular if (1)-(3) hold.",
"We now illustrate how this condition is related to irreducibility of $i_{B}^{G}(\\chi )$ in some examples.",
"We will assume that $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ in all of the examples for simplicity.",
"Example 3.9 ($G = \\mathrm {GL}_{2}$ ) We can write $\\chi := \\chi _{1} \\otimes \\chi _{2}$ for $\\chi _{i}: \\mathbb {Q}_{p}^{*} \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }^{*}$ smooth characters and $i = 1,2$ .",
"We see that $\\chi $ being generic implies that $\\chi _{1}\\chi _{2}^{-1} \\lnot \\simeq \\mathbf {1}$ and $\\chi _{1}\\chi _{2}^{-1} \\lnot \\simeq |\\cdot |^{\\pm 1}$ , and this latter condition guarantees that the normalized parabolic induction $i_{B}^{\\mathrm {GL}_{2}}(\\chi _{1} \\otimes \\chi _{2})$ is irreducible.",
"Let's also look at Condition (2) in this case.",
"Suppose it fails, then we have an isomorphism: $ \\chi _{1}(t_{1})\\chi _{2}(t_{2})|t_{1}t_{2}^{-1}|^{1/2} \\simeq \\chi _{1}(t_{2})\\chi _{2}(t_{1})|t_{1}t_{2}^{-1}|^{1/2} $ Evaluating at $(t_{1},t_{2}) = (t,1)$ this would imply that $\\chi _{1}\\chi _{2}^{-1} \\simeq \\mathbf {1}$ , which would contradict $\\chi $ being generic.",
"In particular, we see that $\\chi $ being generic is enough to guarantee irreducibility and weak normalized regularity for $\\mathrm {GL}_{2}$ .",
"In fact, this is a more general phenomenon.",
"Lemma 3.10 If $G = \\mathrm {GL}_{n}$ then $\\chi $ being generic implies that it is weakly normalized regular.",
"(Sketch) We write the character as a product $\\chi := \\chi _{1} \\otimes \\chi _{2} \\otimes \\cdots \\otimes \\cdots \\otimes \\chi _{n}$ of characters $\\chi _{i}: \\mathbb {Q}_{p}^{*} \\rightarrow \\Lambda ^{*}$ , and write $(t_{1},\\ldots ,t_{n})$ for the natural coordinates on $T(\\mathbb {Q}_{p}) \\simeq (\\mathbb {Q}_{p}^{*})^{n}$ .",
"We have an equality: $ \\chi \\otimes \\delta _{B}^{1/2} = \\prod _{i = 1}^{n} \\chi _{i}(t_{i}) \\otimes |t_{i}|^{\\frac{n - 1}{2} - (i - 1)} $ We visualize the coordinates as a set of vertices of a graph: $ (t_{1}) \\leftrightarrow (t_{2}) \\leftrightarrow \\cdots \\leftrightarrow (t_{n - 1}) \\leftrightarrow (t_{n}) $ Then this has an axis of symmetry between $t_{n/2}$ and $t_{n/2 + 1}$ if $n$ is even, and an axis of symmetry going through $t_{(n + 1)/2}$ if $n$ is odd.",
"The element $w$ corresponds to a permutation $\\sigma $ of the vertices of the graph.",
"Now suppose that $\\sigma $ crosses the line of symmetry.",
"If $n$ is odd we assume that it sends $t_{(n - 1)/2}$ to $t_{(n + 1)/2}$ then by evaluating the equality $ \\chi \\otimes \\delta _{B}^{1/2} \\simeq \\chi \\otimes (\\delta _{B}^{-1/2})^{w}$ at $(1,\\ldots ,1,t_{(n - 1)/2} = t,1,\\ldots ,1)$ it reduces to $ \\chi _{t_{(n - 1)/2}}(t)|t| \\simeq \\chi _{t_{(n + 1)/2}}(t) $ which implies $ \\chi _{t_{(n - 1)/2}}(t)^{-1}\\chi _{t_{(n + 1)/2}}(t) \\simeq |t| $ contradicting $\\chi $ being generic.",
"Similarly, if $n$ is even and the permutation sends $t_{n/2}$ to $t_{n/2 + 1}$ then evaluating at $(1,\\ldots ,1,t_{n/2} = t,1,\\ldots ,1)$ the equality becomes $ \\chi _{t_{n/2}}(t)\\chi _{t_{n/2 + 1}}^{-1} \\simeq \\mathbf {1} $ which again contradicts $\\chi $ being generic.",
"Here the point is that the power of the norm character appearing when evaluating at $(1,\\ldots ,1,t = t_{i},1,\\ldots ,1)$ is given by the distance of $t_{i}$ from the reflection of $t_{i + 1}$ across the line of symmetry.",
"In general, we can consider the cycle which is closest to the line of symmetry, suppose it ends in a permutation $t_{i} \\rightarrow t_{i + 1}$ .",
"If we just evaluated at $(1,\\ldots ,1,t = t_{i},1,\\ldots ,1)$ then we would get that $\\chi _{i}(t)\\chi _{i + 1}(t)^{-1}$ is equal to a very large power of the norm character.",
"However, since we choose the permutation to be as close as possible to the axis of symmetry, the permutation will fix all the vertices from the axis of symmetry to $t_{i + 1}$ reflected across it.",
"Therefore, we are free to also evaluate at a power of $t$ on these invariant coordinates, since the characters $\\chi _{j}$ corresponding to an invariant coordinate $t_{j}$ on either side of the equation will cancel.",
"This allows us to reduce the power of the norm character appearing to 1 or 0, and then we see that this contradicts $\\chi $ being generic again.",
"In general, genericity does not always imply weak normalized regularity.",
"In particular, Condition REF (3) seems to be related to the irreducibility of some unitary principal series representations.",
"Example 3.11 ($G = \\mathrm {SL}_{2}$ ) In this case, $\\chi : \\mathbb {Q}_{p}^{*} \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }^{*}$ is a character of $\\mathbb {Q}_{p}^{*}$ .",
"The induction $i_{B}^{G}(\\chi )$ will be irreducible if and only if $\\chi \\lnot \\simeq |\\cdot |^{\\pm 1}$ and $\\chi ^{2} \\lnot \\simeq \\mathbf {1}$ .",
"The condition that $\\chi \\lnot \\simeq |\\cdot |^{\\pm 1}$ is guaranteed by $\\chi $ being generic but the condition $\\chi ^{2} \\lnot \\simeq \\mathbf {1}$ is not.",
"However, we note that $\\chi $ being weakly normalized regular enforces the additional condition that $\\chi ^{2} \\lnot \\simeq \\mathbf {1}$ guaranteeing irreducibility in this case.",
"We notice in the previous example that the Condition that $ \\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq \\chi \\otimes (\\delta _{B}^{-1/2})^{w} $ for $w \\in W_{G}$ non-trivial is guaranteed already by the Condition that $\\chi _{\\alpha }^{2} \\lnot \\simeq \\mathbf {1}$ , for the unique positive coroot $\\alpha $ of $\\mathrm {SL}_{2}$ .",
"This is also implied by the vanishing of the $H^{0}$ s in Condition REF (4).",
"In particular, it is easy to check for $\\mathrm {SL}_{n}$ that, for normalized regular $\\chi $ , the condition $\\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq \\chi \\otimes (\\delta _{B}^{-1/2})^{w}$ is superfluous.",
"This seems to always be the case, but we were unable to verify it.",
"However, it is easy to show that it is superfluous for the reflections.",
"Lemma 3.12 Suppose that, for all (reduced) coroots $\\alpha \\in \\mathbb {X}_{*}(A) \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma } \\subset \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ , we have that $\\chi _{\\alpha }^{2} \\lnot \\simeq \\mathbf {1}$ then we also have that $\\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq \\chi ^{w} \\otimes (\\delta _{B}^{-1/2})^{w}$ for all reflections $w \\in W_{G}$ .",
"We show the contrapositive.",
"Since $w$ is a reflection, there exists a positive (reduced) coroot $\\alpha \\in \\mathbb {X}_{*}(A) \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma }$ such that $w(\\alpha ) = -\\alpha $ .",
"If we have an isomorphism $\\chi \\otimes \\delta _{B} \\simeq \\chi ^{w} \\otimes (\\delta _{B}^{-1/2})^{w}$ then precomposing both sides with the composition $ E^{*} \\xrightarrow{} \\mathbb {Q}_{p}^{*} \\xrightarrow{} A(\\mathbb {Q}_{p}) \\subset T(\\mathbb {Q}_{p}) $ where $E$ is the splitting field of $G$ gives an isomorphism $\\chi _{\\alpha }^{2} \\simeq \\mathbf {1}$ .",
"We recall that a character $\\chi $ is regular if $\\chi \\lnot \\simeq \\chi ^{w}$ for any non-trivial $w \\in W_{G}$ .",
"This is indeed guaranteed by the condition that $\\chi _{\\alpha }^{2} \\lnot \\simeq \\mathbf {1}$ , for all $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ corresponding to a $\\Gamma $ -orbit of positive roots.",
"Lemma 3.13 Suppose that, for all positive (reduced) coroots $\\alpha \\in \\mathbb {X}_{*}(A) \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma }$ , we have $\\chi _{\\alpha }^{2} \\lnot \\simeq \\mathbf {1}$ then $\\chi \\lnot \\simeq \\chi ^{w}$ for any $w \\in W_{G}$ .",
"In particular, if $\\chi $ is normalized regular then $\\chi $ is regular.",
"Since $\\chi \\mapsto \\chi ^{w}$ defines a group action of $W_{G}$ on the set of characters it suffices to show that $\\chi \\lnot \\simeq \\chi ^{w}$ for $w$ a reflection, since reflections generate $W_{G}$ .",
"If we have that $\\chi \\simeq \\chi ^{w}$ for a reflection then, arguing as in the previous lemma, we deduce that $ \\chi _{\\alpha }^{2} \\simeq \\mathbf {1} $ for the positive (reduced) coroot such that $w(\\alpha ) = -\\alpha $ , as desired.",
"The additional condition of normalized regularity will be necessary to verify the weak Hecke eigensheaf property for the highest weight representations defined by all geometric cocharacters $\\mu $ .",
"In particular, the exact condition we will need is as follows.",
"Definition 3.14 For a toral parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ and a geometric dominant cocharacter $\\mu $ , we say $\\phi _{T}$ is strongly $\\mu $ -regular if the Galois cohomology complexes $ R\\Gamma (W_{\\mathbb {Q}_{p}},(\\nu - \\nu ^{\\prime })^{\\Gamma } \\circ \\phi _{T}) $ are trivial for $\\nu $ ,$\\nu ^{\\prime }$ distinct weights of the highest weight representation of $\\hat{G}$ of highest weight $\\mu _{k}$ for all $k = 1,\\ldots ,n$ .",
"We say that $\\phi _{T}$ is $\\mu $ -regular if there exists a set of geometric dominant cocharacters $\\mu _{k} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ for $k = 1,\\ldots ,m$ such that the following hold: $\\mu _{k} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ is quasi-minuscule or minuscule (as in or ).",
"Up to central elements, we have an equality $\\mu = \\sum _{k = 1}^{m} n_{k}\\mu _{k}$ as elements in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ for $n_{k} \\in \\mathbb {N}_{> 0}$ .",
"The parameter $\\phi _{T}$ is strongly $\\mu _{k}$ -regular for all $k = 1,\\ldots ,m$ .",
"We have the following.",
"Lemma 3.15 Let $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ be a toral parameter.",
"The following is true.",
"If $\\phi _{T}$ is generic then, for any minuscule $\\mu $ , $\\phi _{T}$ is strongly $\\mu $ -regular.",
"If $\\phi _{T}$ is generic and satisfies Condition REF (4) then, for any quasi-minuscule $\\mu $ , $\\phi _{T}$ is strongly $\\mu $ -regular.",
"This in particular implies the following.",
"If $\\phi _{T}$ is generic then, for any $\\mu $ a geometric dominant cocharacter such that the $\\mu _{k}$ in Definition REF can be chosen to be minuscule, $\\phi _{T}$ is $\\mu $ -regular.",
"If $\\phi _{T}$ is generic and satisfies Condition REF (4) (e.g $\\phi _{T}$ is normalized regular) then, for all geometric dominant cocharacters $\\mu $ , the parameter $\\phi _{T}$ is $\\mu $ -regular.",
"Since $\\mu $ -regularity is a condition on $\\Gamma $ -orbits of weights of the representation $V_{\\mu }$ , we may without loss of generality assume $G$ is split.",
"Recalling that we can always write a geometric dominant cocharacter as a linear combination of minuscule/quasi-minuscule ones, the claim reduces to the following standard lemma.",
"Lemma 3.16 Let $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ be a minuscule (resp.",
"quasi-minuscule) geometric dominant cocharacter.",
"Given $\\nu ,\\nu ^{\\prime } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ distinct elements lying in the convex hull of the Weyl orbit of $\\mu $ the difference $\\nu - \\nu ^{\\prime }$ is equal to $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ (resp.",
"$\\alpha $ or $2\\alpha $ ) for $\\alpha $ a root of $G$ .",
"If $\\mu $ is quasi-minuscule the Weyl group orbits in the convex hull of $\\mu $ are of size 2.",
"One orbit will be given by the minimal central element $\\mu _{\\mathrm {min}}$ lying under $\\mu $ in the Bruhat ordering, and the other is the orbit of the highest weight $\\mu $ .",
"We also know that $\\langle \\hat{\\alpha }, \\mu \\rangle \\in \\lbrace 0,\\pm 1, \\pm 2\\rbrace $ for all roots $\\hat{\\alpha }$ and that there exists a unique root $\\hat{\\gamma } \\in \\hat{\\Lambda }_{G}$ such that $\\langle \\hat{\\gamma }, \\mu \\rangle = 2$ and $\\mu = \\hat{\\gamma }^{\\vee } + \\mu _{\\mathrm {min}}$ , where $\\hat{\\gamma }^{\\vee }$ is the coroot dual to $\\gamma $ (cf.",
").",
"The claim immediately follows from these observations in this case.",
"If $\\mu $ is minuscule then there is a unique Weyl group orbit in the convex hull of the Weyl orbit of $\\mu $ , and we have that $\\langle \\hat{\\alpha }, \\nu \\rangle \\in \\lbrace 0,\\pm 1\\rbrace $ for all roots $\\hat{\\alpha } \\in \\hat{\\Lambda }_{G}^{+}$ .",
"It therefore follows that any distinct $\\nu ,\\nu ^{\\prime }$ in the convex hull are Weyl orbits of the highest weight and must differ by a root (cf.",
").",
"As mentioned in the introduction, weak normalized regularity and $\\mu $ -regularity for a geometric dominant cocharacter which isn't fixed under any element of $W_{G}$ , will imply the existence of isomorphisms $i_{\\chi ,w}: i_{B}^{G}(\\chi ) \\xrightarrow{} i_{B}^{G}(\\chi ^{w})$ for all $w \\in W_{G}$ once the theory of Eisenstein series has been developed.",
"By Lemma REF , this will always hold for $\\chi $ normalized regular.",
"Similarly, we will show the following.",
"Theorem 3.17 (Corollary REF ) If $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }^{*}$ is a normalized regular character then the smooth $G(\\mathbb {Q}_{p})$ -representation $i_{B}^{G}(\\chi )$ is irreducible.",
"In particular, this guarantees that the $G(\\mathbb {Q}_{p})$ -representations $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w})$ for all $w \\in W_{G}$ .",
"In fact, for regular characters $\\chi $ , the existence of such isomorphisms is equivalent to irreducibility, and, as we saw in Lemma REF , regularity is implied by normalized regularity.",
"More generally, we show that such isomorphisms exist assuming $\\chi $ is generic (Proposition REF ), but this does not guarantee irreducibility of certain unitary principal series as seen when $G = \\mathrm {SL}_{2}$ .",
"Using the Langlands classification, the proof of this theorem will essentially reduce to a calculation of reducibility points in rank 1, where it reduces to Example REF and the following example, which illustrates the behavior of our conditions in the non-split case.",
"Example 3.18 ($G = \\mathrm {U}_{3}/E$ ) Let $E/\\mathbb {Q}_{p}$ be a quadratic extension.",
"We write $\\overline{(-)}$ for the non-trivial automorphism of $E$ over $\\mathbb {Q}_{p}$ .",
"If $e_{1},e_{2},e_{3}$ is the standard basis for the cocharacter lattice $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ then $\\overline{(-)}$ acts by $ e_{1} \\longleftrightarrow -e_{3} $ $ e_{2} \\longleftrightarrow -e_{2} $ It follows that the simple coroot $\\alpha _{1} := e_{1} - e_{2}$ is sent to the simple coroot $\\alpha _{2} := e_{2} - e_{3}$ under $\\overline{(-)}$ .",
"Thus, the $\\Gamma $ -orbits of positive coroots in $\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ are given by $\\lbrace \\alpha _{1},\\alpha _{2}\\rbrace $ with reflex field $E$ and $\\alpha _{1} + \\alpha _{2}$ with reflex field $\\mathbb {Q}_{p}$ .",
"Now recall that the maximal torus $T(\\mathbb {Q}_{p}) \\subset \\mathrm {U}_{3}(\\mathbb {Q}_{p})$ is isomorphic to $E^{*} \\times E^{1}$ , via the embedding $ E^{*} \\times E^{1} \\rightarrow \\mathrm {U}_{3}(\\mathbb {Q}_{p}) $ $ t \\mapsto \\begin{pmatrix} t & 0 & 0 \\\\0 & s & 0 \\\\0 & 0 & \\overline{t}^{-1} \\end{pmatrix} $ where $E^{1}$ denotes the set of norm 1 elements.",
"Then if we write the character $\\chi (t,s): E^{*} \\times E^{1} \\rightarrow \\Lambda ^{*}$ as $\\chi (t,s) = \\chi _{1}(t)\\chi _{2}(ts\\overline{t}^{-1})$ the reducibility of $i_{B}^{G}(\\chi )$ depends solely on $\\chi _{1}$ , as in , where here it reduces to the analogous question for $\\mathrm {SU}_{3}$ , and there the reducibility points were studied in .",
"The induction $i_{B}^{G}(\\chi )$ is reducible if and only if one of the following hold: $\\chi _{1} = \\eta |\\cdot |_{E}^{\\pm 1/2}$ , where $\\eta |_{\\mathbb {Q}_{p}^{*}} = \\eta _{E/\\mathbb {Q}_{p}}$ , $\\chi _{1} = |\\cdot |_{E}^{\\pm 1}$ , $\\chi _{1}|_{\\mathbb {Q}_{p}^{*}}$ is trivial, but $\\chi $ is not.",
"Here $|\\cdot |_{E}$ is the norm character of $E$ which is in particular the splitting field of $G$ , and $\\eta _{E/\\mathbb {Q}_{p}}: \\mathbb {Q}_{p}^{*} \\rightarrow \\Lambda ^{*}$ is the unique quadratic character with kernel given by $\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(E^{*})$ .",
"We note that that the cocharacters of $T(\\mathbb {Q}_{p})$ given by the $\\Gamma $ -orbits of positive roots are $ \\lbrace \\alpha _{1},\\alpha _{2}\\rbrace : E^{*} \\rightarrow T(\\mathbb {Q}_{p}) = E^{*} \\times E^{1} $ $ t \\mapsto (t,t^{-1}\\overline{t}) $ and $ \\lbrace \\alpha _{1} + \\alpha _{2}\\rbrace : E^{*} \\rightarrow E^{*} \\times E^{1} $ $ t \\mapsto (\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t),1) $ By precomposing $\\chi $ with the first character, we see that $\\chi $ being generic implies $\\chi _{1} \\lnot \\simeq \\mathbf {1}$ and $\\chi _{1} \\lnot \\simeq |\\cdot |^{\\pm 1}_{E}$ , which implies reducibility point $(2)$ cannot occur.",
"By precomposing $\\chi $ with the second character, we see that $\\chi $ being generic implies that $\\chi (\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t)) \\simeq \\chi _{1}(\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t)) \\lnot \\simeq \\mathbf {1}$ and $\\chi _{1}(\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t)) \\lnot \\simeq |\\cdot |^{\\pm 1}_{E}$ .",
"Note that if $\\chi \\simeq \\eta |\\cdot |_{E}^{\\pm 1/2}$ then we have, for all $t \\in E^{*}$ , an isomorphism: $ \\chi (\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t)) = \\chi _{1}(t\\overline{t}) \\simeq \\eta (\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t))|\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t)|_{E}^{\\pm 1/2} \\simeq |t|_{E}^{\\pm 1} $ Thus, we see again that $\\chi $ being generic guarantees irreducibility of the two non-unitary inductions.",
"Now, if $\\chi |_{\\mathbb {Q}_{p}^{*}}$ is trivial then we have that $ \\chi (\\mathrm {Nm}_{E/\\mathbb {Q}_{p}}(t)) = \\chi _{1}(t\\overline{t}) \\simeq \\mathbf {1} $ Thus, we see $\\chi $ being generic guarantees the irreducibility of all principal series.",
"Moreover, $\\chi $ being weakly normalized regular enforces the additional constraint that $ \\chi (t\\overline{t}) \\lnot \\simeq \\mathbf {1} $ which we just saw follows from $\\chi $ being generic, so weak normalized regularity follows from generic in this case.",
"Moreover, we also see that the $\\mu _{k}$ in the Definition of $\\mu $ -regularity can always be chosen to be minuscule fundamental weights.",
"By Lemma REF (1), this implies that $\\mu $ -regularity is guaranteed by genericity for all $\\mu $ .",
"The connection between genericity and irreducibility of non-unitary principal series fits in nicely with the general Langlands philosophy.",
"In particular, since we are inducing from a Borel, we expect that a parameter should have monodromy if it arises as a constituent of the reducible induction of a non-unitary character.",
"We saw in the above examples that this shouldn't occur when the parameter $\\phi _{T}$ attached to $\\chi $ is generic.",
"We can analyze when a parameter comes from the semi-simplification of a parameter with non-trivial monodromy and relate this to genericity.",
"Lemma 3.19 We let $\\phi : \\mathcal {L}_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ be an $L$ -parameter.",
"Suppose that $\\phi : \\mathcal {L}_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ has non-trivial monodromy, and that the semi-simplification $\\phi ^{\\mathrm {ss}}$ (See Assumption REF ) factors through $\\phantom{}^{L}T \\rightarrow \\phantom{}^{L}G$ via the natural embedding.",
"If we write $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell })$ for the parameter induced by $\\phi $ then $\\phi _{T}$ is not generic.",
"If $\\phi $ has non-trivial monodromy and the semi-simplification factors through $\\phantom{}^{L}T$ , there exists a lift $ \\begin{tikzcd}& \\phantom{}^{L}B(\\overline{\\mathbb {Q}}_{\\ell }) [d] & \\\\W_{\\mathbb {Q}_{p}} [ur,\"\\tilde{\\phi }\",dashed] [r,\"\\phi _{T}\"] & \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell }) & \\\\\\end{tikzcd}$ of $\\phi _{T}$ , which is not given by the inclusion $\\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\phantom{}^{L}B(\\overline{\\mathbb {Q}}_{\\ell })$ .",
"Therefore, there exists some Galois orbit of positive coroots $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ such that $\\alpha \\circ \\tilde{\\phi } = \\alpha \\circ \\phi _{T}$ is non-trivial.",
"This implies that there exists a non-trivial class in the $H^{0}$ of $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T})$ , which would in turn imply $\\phi _{T}$ is not generic.",
"We finish this section by deducing a geometric consequence of weak genericity that will be useful for studying how Eisenstein series interact with Verdier duality."
],
[
"A Geometric Consequence of Weak Genericity",
"We work with torsion coefficeints $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ .",
"Consider the following easy lemma.",
"Lemma 3.20 Assume $\\phi _{T}$ is weakly generic then it follows that, for all the $\\Gamma $ -orbits of simple positive coroots $\\alpha $ , the complex $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T}(\\langle \\hat{\\rho },\\alpha _{\\Gamma } \\rangle ))$ is trivial.",
"We note that, since $\\alpha $ is a $\\Gamma $ -orbit of simple positive coroots, we have that $\\langle \\hat{\\rho },\\alpha _{\\Gamma } \\rangle = 1$ ; therefore, we want to check that $ R\\Gamma (W_{\\mathbb {Q}_{p}},(\\alpha \\circ \\phi _{T})(1))$ is trivial.",
"By using local Tate-duality and that the Euler-Poincaré characteristic of this complex is 0, this is equivalent to checking that $(\\alpha \\circ \\phi _{T})(1)$ is not the trivial representation or the cyclotomic twist by $(-1)$ , which is equivalent to insisting that $(\\alpha \\circ \\phi _{T})^{-1}$ is not the trivial representation or the cyclotomic twist by $(1)$ .",
"This is the same condition guaranteeing that the complex $ R\\Gamma (W_{\\mathbb {Q}_{p}},(\\alpha \\circ \\phi _{T})^{-1}) $ is trivial, which follows from weak genericity applied to the $\\Gamma $ -orbit of negative simple coroots $-\\alpha $ .",
"This has an important geometric consequence, related to the vanishing of certain Galois cohomology groups appearing in the moduli space of $B$ -bundles and its compactifications.",
"To explain this, let $\\overline{\\nu }$ be an element of $\\Lambda _{G,B}^{pos} \\setminus \\lbrace 0\\rbrace $ .",
"We can write this as a linear combination $\\sum _{i \\in \\mathcal {J}} n_{i}\\alpha _{i}$ for positive integers $n_{i}$ , where the $\\alpha _{i}$ correspond to the Galois orbits of simple absolute roots as in Definition REF .",
"Given such an $\\alpha _{i}$ , we can consider the reflex field $E_{i}$ of the associated Galois orbit, and define the following partially symmetrized curve: $ \\mathrm {Div}^{(\\overline{\\nu })} := \\prod _{i \\in \\mathcal {J}} \\mathrm {Div}_{E_{i}}^{(n_{i})} $ Points of this curve correspond to tuples of Cartier divisors $D_{i}$ over $E_{i}$ of degree $n_{i}$ for all $i \\in \\mathcal {J}$ .",
"We can consider the map $ h_{(\\overline{\\nu })}^{\\rightarrow }: \\mathrm {Bun}_{T} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\rightarrow \\mathrm {Bun}_{T} $ given by sending $(\\mathcal {F}_{T},(D_{i})_{i \\in \\mathcal {J}})$ to $\\mathcal {F}_{T}(\\sum -\\alpha _{i} \\cdot D_{i})$ , where we are identifying $\\alpha _{i}$ with its $\\Gamma $ -orbit.",
"This partially symmetrized mirror curve $\\mathrm {Div}^{(\\overline{\\nu })}$ behaves a bit strangely if $G$ is not split.",
"To illustrate this, consider the following example.",
"Example 3.21 Let $G = \\mathrm {U}_{3}$ be a unitary group in 3 variables attached to a quadratic extension $E/\\mathbb {Q}_{p}$ and write $\\alpha _{1}$ and $\\alpha _{2}$ for the two absolute positive simple roots.",
"We recall, as in REF , that the Galois group exchanges $\\alpha _{1}$ and $\\alpha _{2}$ .",
"Therefore, they both map to a unique element $\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ which spans the lattice $\\Lambda _{G,B}^{pos}$ by Definition.",
"Consider the element $\\overline{\\nu } := 2\\alpha \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ .",
"We note that we have an equality $\\mathrm {Div}^{(\\overline{\\nu })} = \\mathrm {Div}^{(2)}_{E}$ in this case.",
"The pre-image of $2\\alpha $ under the natural map $(-)_{\\Gamma }: \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma \\rightarrow \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ consists of two elements: the $\\Gamma $ -orbit $\\lbrace 2\\alpha _{1},2\\alpha _{2}\\rbrace $ with reflex field $E$ and the $\\Gamma $ -orbit of $\\lbrace \\alpha _{1} + \\alpha _{2}\\rbrace $ with reflex field $\\mathbb {Q}_{p}$ .",
"We saw in the previous section that the space of modifications defined by the $\\Gamma $ -orbit $\\lbrace 2\\alpha _{1},2\\alpha _{2}\\rbrace $ is given by $\\mathrm {Bun}_{T} \\times \\mathrm {Div}^{1}_{E}$ , correspondingly we have a natural map $ \\triangle _{\\lbrace 2\\alpha _{1},2\\alpha _{2}\\rbrace }: \\mathrm {Div}^{1}_{E} \\xrightarrow{} \\mathrm {Div}^{2}_{E} \\rightarrow \\mathrm {Div}^{(2)}_{E} $ given by the diagonal embedding composed with the quotient map.",
"It is easy to check we have an equality $h_{\\lbrace 2\\alpha _{1},2\\alpha _{2}\\rbrace }^{\\rightarrow }(-) := (\\mathrm {id} \\times \\triangle _{\\lbrace 2\\alpha _{1},2\\alpha _{2}\\rbrace }) \\circ h_{\\overline{\\nu }}^{\\rightarrow }$ .",
"Perhaps more interestingly, attached to the $\\Gamma $ -orbit $\\lbrace \\alpha _{1} + \\alpha _{2}\\rbrace $ , we have a twisted diagonal map $ \\triangle _{\\lbrace \\alpha _{1} + \\alpha _{2}\\rbrace }: \\mathrm {Div}^{1}_{\\mathbb {Q}_{p}} \\rightarrow \\mathrm {Div}^{(2)}_{E} $ given by sending a Cartier divisor $D$ to its pre-image under the natural finite-étale covering $X_{S,E} \\rightarrow X_{S}$ of Fargues-Fontaine curves induced by the extension $E/\\mathbb {Q}_{p}$ .",
"By , this map defines a closed embedding whose image lies in the complement of the image of $\\triangle _{\\lbrace 2\\alpha _{1},2\\alpha _{2}\\rbrace }$ in $\\mathrm {Div}^{(2)}_{E}$ , and we similarly see that we have a relationship $h_{(\\overline{\\nu })}^{\\rightarrow } \\circ (\\mathrm {id} \\times \\triangle _{\\lbrace \\alpha _{1} + \\alpha _{2}\\rbrace })(-) = h_{\\lbrace \\alpha _{1} + \\alpha _{2}\\rbrace }^{\\rightarrow }(-)$ .",
"The previous example illustrates that we can can understand $\\mathrm {Bun}_{T} \\times \\mathrm {Div}^{(\\overline{\\nu })}$ as realizing the set of all modifications specified by Galois orbits $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma $ such that $\\nu _{\\Gamma } = \\overline{\\nu }$ .",
"This is indeed a general phenomenon.",
"In particular, using that $\\Gamma $ permutes the simple absolute coroots of $G$ , given any such $\\nu $ with reflex field $E_{\\nu }$ , we can define a twisted diagonal embedding $ \\Delta _{\\nu }: \\mathrm {Div}^{1}_{E_{\\nu }} \\rightarrow \\mathrm {Div}^{(\\overline{\\nu })} $ such that we have a relationship $h_{\\nu }^{\\rightarrow }(-) := (\\mathrm {id} \\times \\triangle _{\\nu }) \\circ h_{\\overline{\\nu }}^{\\rightarrow }(-)$ .",
"If $\\overline{\\nu } = \\alpha _{i}$ for $i \\in \\mathcal {J}$ this map is an isomorphism for the unique $\\Gamma $ -orbit of simple coroots corresponding to $\\alpha _{i}$ .",
"Therefore, the pullback of $\\mathcal {S}_{\\phi _{T}}$ along $h_{(\\overline{\\nu })}^{\\leftarrow }$ is isomorphic to $\\mathcal {S}_{\\phi _{T}} \\boxtimes \\tilde{\\alpha }_{i} \\circ \\phi _{T}$ for a choice of representative $\\tilde{\\alpha }_{i}$ of the $\\Gamma $ -orbit corresponding to $\\alpha _{i}$ .",
"In general, recall that given a local system $\\mathbb {L}$ and $n$ a positive integer, we can consider the symmetric powers $ \\mathbb {L}^{(n)} := \\pi _{*}(\\boxtimes _{i = 1}^{n} \\mathbb {L})^{S_{n}} $ where $\\pi $ denotes the push-forward along the $S_{n}$ -torsor: $ \\pi : (\\mathrm {Div}^1)^{n} \\rightarrow \\mathrm {Div}^{(n)} $ Using this, we can define a local system on $\\mathrm {Div}^{(\\overline{\\nu })}$ given by $ E_{\\phi _{T}}^{(\\overline{\\nu })} := \\boxtimes _{i \\in \\mathcal {J}} E_{\\phi _{i}}^{(n_{i})} $ where $\\phi _{i}$ is the local system on $\\mathrm {Div}^{1}_{E_{\\alpha _{i}}}$ corresponding to the character $\\tilde{\\alpha }_{i} \\circ \\phi _{T}|_{W_{E_{\\alpha _{i}}}}$ of $W_{E_{\\alpha _{i}}}$ .",
"With this in hand, we can describe the pullback of $\\mathcal {S}_{\\phi _{T}}$ along $h_{(\\overline{\\nu })}^{\\leftarrow }$ as the sheaf $E_{\\phi _{T}}^{(\\overline{\\nu })} \\boxtimes \\mathcal {S}_{\\phi _{T}}$ by using that Hecke operators are monoidal, and the natural compatibilities of the eigensheaf.",
"We now state a key vanishing result that will be important for studying how geometric Eisenstein series interact with Verdier duality.",
"Lemma 3.22 If $\\phi _{T}$ is weakly generic then, for all $\\overline{\\nu } \\in \\Lambda _{G,B}^{pos} \\setminus \\lbrace 0\\rbrace $ , the complexes $ R\\Gamma _{c}(\\mathrm {Div}^{(\\overline{\\nu })},E_{\\phi _{T}}^{(\\overline{\\nu })}(\\langle \\hat{\\rho }, \\overline{\\nu } \\rangle )) $ and $ R\\Gamma _{c}(\\mathrm {Div}^{(\\overline{\\nu })},E_{\\phi _{T}}^{(\\overline{\\nu })}) $ are trivial.",
"By Künneth formula, this easily reduces to showing that $R\\Gamma _{c}(\\mathrm {Div}^{(n_{i})},E^{(n_{i})}_{\\phi _{i}}(n_{i}))$ is trivial for all $i \\in \\mathcal {J}$ and $n_{i} \\in \\mathbb {N}_{> 0}$ .",
"However, $E^{(n_{i})}_{\\phi _{i}}(n_{i})$ is given by taking the $n_{i}$ th symmetric power of $E_{\\phi _{i}}(1)$ .",
"Therefore, by Künneth formula again, this reduces us to showing that $R\\Gamma (W_{\\mathbb {Q}_{p}},\\alpha _{i} \\circ \\phi _{T}(1)) \\simeq R\\Gamma _{c}(W_{E_{\\alpha _{i}}},\\tilde{\\alpha }_{i} \\circ \\phi _{T}|_{W_{E_{\\alpha _{i}}}}(1)) \\simeq R\\Gamma _{c}(\\mathrm {Div}^{1}_{E_{\\alpha _{i}}},E_{\\phi _{i}}(1)) $ vanishes, where the first isomorphism follows from Schapiro's lemma and the second isomorphism follows from the correspondence between $\\Lambda $ -valued local systems on $\\mathrm {Div}^{1}_{E_{\\alpha _{i}}}$ and representations of $W_{E_{\\alpha _{i}}}$ .",
"The vanishing of the LHS follows from weak genericity and Lemma REF .",
"The second vanishing statement follows from the same argument and weak genericity.",
"We will now review the next ingredient in our calculations of geometric Eisenstein series, the Geometric Satake correspondence."
],
[
"The Geometric Satake Correspondence",
"We will now recall some facts about the geometric Satake correspondence for the $B_{dR}^{+}$ Grassmannian, as proven in .",
"For any finite set $I$ , we consider the Hecke stack $ \\pi _{G}: \\mathrm {Hck}_{G}^{I} \\rightarrow \\mathrm {Div}^{I} $ as in §.",
"For a point $S \\rightarrow \\mathrm {Div}^{I}$ , we can consider the completion of the structure sheaf $\\mathcal {O}_{X_{S}}$ at the union of the $I$ Cartier divisors in $X_{S}$ defined by $S$ .",
"This defines a ring which we denote by $B^{+}$ , and inverting $D$ , we get a ring which we denote by $B$ .",
"The mapping sending $S \\in \\mathrm {Div}^{I}$ to $G(B^{+})$ and $G(B)$ defines étale sheaves on $\\mathrm {Perf}$ , which we denote by $L^{+}_{\\mathrm {Div}^{I}}G$ and $L_{\\mathrm {Div}^{I}}G$ , respectively.",
"We note that, for $I = \\lbrace \\ast \\rbrace $ and $S = \\mathop {\\rm Spa}(F,\\mathcal {O}_{F}) \\rightarrow \\mathrm {Div}^{I}$ a geometric point with associated untilt $(C,\\mathcal {O}_{C})$ , we have $B^{+} = B_{dR}^{+}(C,\\mathcal {O}_{C})$ and $B = B_{dR}(C,\\mathcal {O}_{C})$ , the usual de-Rham period rings attached to the untilt.",
"For simplicity, we will often just drop the subscript $\\mathrm {Div}^{I}$ and just write $L^{+}G$ and $LG$ for these étale sheaves.",
"By , the Hecke stack can be described as the quotient: $ [L^{+}G\\backslash LG/L^{+}G] \\rightarrow \\mathrm {Div}^{I} $ To study this, we can uniformize this by the quotient $ \\mathrm {Gr}_{G}^{I} := LG/L^{+}G \\rightarrow \\mathrm {Div}^{I} $ which is the Fargues-Fontaine analogue of the Beilinson-Drinfeld Grassmannian.",
"It follows by the results of that the Beilinson-Drinfeld Grassmannian is a well-behaved geometric object; it can be written as a closed union of subsheaves that are proper and representable in spatial diamonds over $\\mathrm {Div}^{I}$ , given by bounding the meromorphy of the modifications.",
"We can consider the category $ \\mathrm {D}(\\mathrm {Hck}_{G}^{I})^{bd} $ of objects with support in one of the aforementioned quasi-compact subsets.",
"This carries a monoidal structure coming from the diagram: $ \\mathrm {Hck}_{G}^{I} \\times _{\\mathrm {Div}^{I}} \\mathrm {Hck}_{G}^{I} \\xleftarrow{} L^{+}G\\backslash LG \\times ^{L^{+}G} LG/L^{+}G \\xrightarrow{} \\mathrm {Hck}_{G}^{I} $ Here the middle space can be interpreted as parameterizing a pair of $G$ -bundles $\\mathcal {E}_{1},\\mathcal {E}_{2}$ together with a pair of modifications $\\beta _{1}: \\mathcal {E}_{1} \\dashrightarrow \\mathcal {E}_{0}$ and $\\beta _{2}: \\mathcal {E}_{0} \\dashrightarrow \\mathcal {E}_{2}$ to/from the trivial bundle with the same locus of meromorphy.",
"The maps $p_{i}$ are the natural projections remembering only the data $(\\mathcal {E}_{i},\\beta _{i})$ for $i = 1,2$ , and $m$ is defined by sending $(\\mathcal {E}_{i},\\beta _{i})_{i = 1,2}$ to $\\beta _{2} \\circ \\beta _{1}: \\mathcal {E}_{1} \\dashrightarrow \\mathcal {E}_{2}$ .",
"Given $A,B \\in \\mathrm {D}(\\mathrm {Hck}_{G}^{I})^{bd}$ , we define $ A \\star B := Rm_{*}(p_{1}^{*}(A) \\otimes p_{2}^{*}(B)) \\in \\mathrm {D}(\\mathrm {Hck}_{G}^{I})^{bd}$ the convolution of $A$ and $B$ .",
"Since the map $m$ is a fibration in $\\mathrm {Gr}_{G}^{I}$ it is proper over any quasi-compact subset, so by proper base-change it gives a well-defined associative monoidal structure.",
"One can further refine our category of sheaves as follows.",
"In particular, we note that the locus of $\\mathrm {Hck}_{G}^{I}$ where the meromorphy is equal to the Galois orbit of $\\mu _{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ at the $i$ th Cartier divisor for $i \\in I$ is uniformized by a cohomologically smooth diamond of relative dimension $\\sum _{i \\in I} \\langle 2\\hat{\\rho }, \\mu _{i} \\rangle $ over $\\mathrm {Div}^{I}$ by .",
"One can check that this gives rise to a well-defined perverse $t$ -structure on $\\mathrm {D}(\\mathrm {Hck}_{G}^{I})^{bd}$ over $\\mathrm {Div}^{I}$ given by insisting that $!$ -restriction (resp.",
"$*$ -restriction) to these strata sit in cohomological degrees $\\ge $ (resp.",
"$\\le $ ) $\\sum _{i \\in I} \\langle 2\\hat{\\rho }, \\mu _{i} \\rangle $ , and that convolution preserves perversity .",
"With this in hand, we arrive at the key definition.",
"Definition 4.1 We define the Satake category $ \\mathrm {Sat}_{G}^{I} \\subset \\mathrm {D}(\\mathrm {Hck}_{G}^{I})^{bd} $ as the category of all $A \\in \\mathrm {D}(\\mathrm {Hck}_{G}^{I})^{bd}$ which are perverse, flat (i.e for all $\\Lambda $ -modules $M$ $A \\otimes M$ is also perverse), and ULA over $\\mathrm {Div}^{I}$ , as defined in .",
"The ULA and flatness property in the above definition has the key consequence that the pullback of $A \\in \\mathrm {Sat}_{G}^{I}$ to $\\mathrm {Gr}_{G}^{I}$ composed with the push-forward to $\\mathrm {Div}^{I}$ has cohomology sheaves valued in $\\Lambda $ -valued local systems on $\\mathrm {Div}^{I}$ .",
"In particular, using an analogue of Drinfeld's lemma , this gives rise to a fiber functor $ F_{G}^{I}: \\mathrm {Sat}_{G}^{I} \\rightarrow \\mathrm {Rep}_{W_{\\mathbb {Q}_{p}}^{I}}(\\Lambda ) $ $ A \\mapsto \\bigoplus _{i} \\mathcal {H}^{i}(R\\pi _{G*}(A)) $ where $\\mathrm {Rep}_{W_{\\mathbb {Q}_{p}}^{I}}(\\Lambda )$ denotes the category of continuous representations of $W_{\\mathbb {Q}_{p}}^{I}$ on finite projective $\\Lambda $ -modules, and $R\\pi _{G*}$ is the functor given by pulling back to $\\mathrm {Gr}_{G}^{I}$ and taking the push-forward to $\\mathrm {Div}^{I}$ , as in .",
"Now, by using the factorization structure on these Grassmannians, one can also construct an analogue of the fusion product .",
"Let us recall its construction.",
"We consider a partition $I = I_{1} \\sqcup \\ldots \\sqcup I_{k}$ of this index set.",
"We consider the open subspace $ \\mathrm {Div}^{I;I_{1},\\ldots ,I_{k}} \\subset \\mathrm {Div}^{I} $ parameterizing a tuple of Cartier divisors $(D_{i})_{i \\in I}$ such that $D_{i}$ and $D_{i^{\\prime }}$ are disjoint whenever $i,i^{\\prime } \\in I = I_{1} \\sqcup \\ldots \\sqcup I_{k}$ lie in different $I_{j}$ s, and let $\\mathrm {Hck}_{G}^{I;I_{1},\\ldots ,I_{k}}$ be the base-change of $\\mathrm {Hck}_{G}^{I}$ to this open subspace.",
"We can define the subcategory $\\mathrm {Sat}_{G}^{I;I_{1},\\ldots ,I_{k}} \\subset \\mathrm {D}(\\mathrm {Hck}_{G}^{I;I_{1},\\ldots ,I_{k}})^{\\mathrm {bd}}$ in an analogous manner to Definition REF .",
"We have a natural restriction map $ \\mathrm {Sat}_{G}^{I} \\rightarrow \\mathrm {Sat}_{G}^{I;I_{1},\\ldots ,I_{k}} $ which, by , defines a fully faithful embedding.",
"There is also an identification $ \\mathrm {Hck}^{I}_{G} \\times _{\\mathrm {Div}^{I}} \\mathrm {Div}^{I;I_{1},\\ldots ,I_{k}} \\simeq \\prod _{j = 1}^{k} \\mathrm {Hck}_{G}^{I_{j}} \\times _{\\prod _{j} \\mathrm {Div}^{I_{j}}} \\mathrm {Div}^{I;I_{1},\\ldots ,I_{k}} $ giving a natural map $ \\mathrm {Sat}_{G}^{I_{1}} \\times \\ldots \\times \\mathrm {Sat}_{G}^{I_{k}} \\rightarrow \\mathrm {Sat}_{G}^{I;I_{1},\\ldots ,I_{k}} $ via taking exterior products.",
"Then, by , this lies in the full subcategory $\\mathrm {Sat}_{G}^{I}$ , and the resulting map $ \\mathrm {Sat}_{G}^{I_{1}} \\times \\ldots \\times \\mathrm {Sat}_{G}^{I_{k}} \\rightarrow \\mathrm {Sat}_{G}^{I} $ is called the fusion product.",
"Now consider the composite $ \\mathrm {Sat}_{G}^{I} \\times \\mathrm {Sat}_{G}^{I} \\rightarrow \\mathrm {Sat}_{G}^{I \\sqcup I} \\rightarrow \\mathrm {Sat}_{G}^{I} $ where the first map is the fusion product, and the last map is given by restricting along the diagonal embedding $\\mathrm {Hck}^{I} \\rightarrow \\mathrm {Hck}^{I \\sqcup I}$ .",
"Then this is naturally isomorphic to the convolution product.",
"By this comparison between fusion and convolution, one can deduce that the convolution product is in fact symmetric monoidal, and that the functor $F_{G}^{I}$ takes this monoidal structure to the usual tensor product on $\\mathrm {Rep}_{W_{\\mathbb {Q}_{p}}^{I}}(\\Lambda )$ .",
"Now, using Tannakian duality, one deduces the following.",
"Theorem 4.2 For a finite index set $I$ , the category $\\mathrm {Sat}_{G}^{I}$ is naturally in $I$ identified with $\\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ the category of representations of $\\phantom{}^{L}G^{I}$ on finite projective $\\Lambda $ -modules.",
"Remark 4.3 One needs to be a bit careful here.",
"In particular, $\\phantom{}^{L}G$ as defined here differs from the usual definition of the $L$ -group.",
"In particular, here the usual action of $W_{\\mathbb {Q}_{p}}$ on $\\hat{G}$ is twisted by a cyclotomic character (See for more details).",
"One of the key points that also plays an important role in the proof of this theorem is that this construction respects the natural map $ \\mathrm {res}_{T}^{I,G}: \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I}) \\rightarrow \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}T^{I}) $ given by restricting a representation of $\\phantom{}^{L}G^{I}$ to $\\phantom{}^{L}T^{I}$ along the natural embedding: $ \\phantom{}^{L}T^{I} \\rightarrow \\phantom{}^{L}G^{I} $ To explain this, we need to explain how this functor is realized in the geometry of Beilinson-Drinfeld Grassmannians.",
"First note that, as in §, we have an identification $ \\mathrm {Gr}_{T}^{I} = \\bigsqcup _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}} \\mathrm {Gr}_{T,(\\nu _{i})_{i \\in I}}^{I} \\simeq \\bigsqcup _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}} \\mathrm {Div}^{I}_{E_{(\\nu _{i})_{i \\in I}}} $ where $\\mathrm {Gr}_{T,(\\nu _{i})_{i \\in I}}$ parametrizes modifications of $T$ -bundles with meromorphy equal to the Galois orbit defined by $\\nu _{i}$ at a Cartier divisor $D_{i}$ for all $i \\in I$ , and $\\mathrm {Div}^{I} := \\prod _{i \\in I} \\mathrm {Div}^{1}_{E_{\\nu _{i}}}$ , where $E_{\\nu _{i}}$ denotes the reflex field of $\\nu _{i}$ .",
"In particular, we see that Theorem REF is trivial in this case, as $F_{T}^{I}$ will induce an equivalence between $\\mathrm {Sat}_{T}^{I}$ and $(\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ -graded objects in $\\mathrm {Rep}_{\\Lambda }(W_{\\mathbb {Q}_{p}}^{I})$ under this isomorphism.",
"We consider the diagram $ \\begin{tikzcd}& \\mathrm {Gr}_{B}^{I} [dl,\"p\"] [dr,\"q\"] & \\\\\\mathrm {Gr}_{T}^{I}& & \\mathrm {Gr}_{G}^{I} \\\\\\end{tikzcd}$ and define the functor: $ p_{!",
"}q^{*}: \\mathrm {D}(\\mathrm {Gr}_{G}^{I})^{bd} \\rightarrow \\mathrm {D}(\\mathrm {Gr}_{T}^{I})^{bd} $ As we will discuss further in the next section, the fibers of the morphism $q$ over the connected components $\\mathrm {Gr}_{T,(\\nu _{i})_{i \\in I}}^{I}$ give rise to a locally closed stratification of $\\mathrm {Gr}_{G}^{I}$ which embed via the morphism $p$ (cf.",
").",
"These are the so called semi-infinite Schubert cells.",
"If one considers $\\mathbb {G}_{m}$ acting on $\\mathrm {Gr}_{G}^{I}$ via a suitably chosen cocharacter $\\mu $ composed with the $L^{+}G$ action on $\\mathrm {Gr}_{G}^{I}$ then one can identify these semi-infinite cells with a union $\\mathbb {G}_{m}$ -orbits (the attractor of the $\\mathbb {G}_{m}$ -action), and the fixed points will be precisely the connected components: $ \\mathrm {Gr}_{T}^{I} = \\bigsqcup _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}} \\mathrm {Gr}_{T,(\\nu _{i})_{i \\in I}}^{I} $ This allows one to apply a diamond analogue of Braden's hyperbolic localization theorem .",
"In particular, since sheaves in $\\mathrm {Sat}_{G}^{I}$ pullback to $L^{+}G$ -equivariant sheaves on $\\mathrm {Gr}_{G}^{I}$ , they will be $\\mathbb {G}_{m}$ -equivariant.",
"From this, one can deduce that $p_{!",
"}q^{*}$ is a hyperbolic localization with respect to this $\\mathbb {G}_{m}$ -action and will therefore preserve perverse, flat, and ULA objects over $\\mathrm {Div}^{I}$ .",
"We therefore get an induced functor $ \\mathrm {CT}: \\mathrm {Sat}_{G}^{I} \\rightarrow \\mathrm {Sat}_{T}^{I} $ called the constant term functor, as in .",
"We now consider the following function $ \\mathrm {deg}: |\\mathrm {Gr}_{T}^{I}| \\rightarrow \\mathbb {Z} $ which has value $\\langle 2\\hat{\\rho }, |(\\nu _{i})_{i \\in I}| \\rangle $ on the connected component indexed by $(\\nu _{i})_{i \\in I}$ , where $|(\\nu _{i})_{i \\in I}| := \\sum _{i \\in I} \\overline{\\nu }_{i\\Gamma } \\in \\Lambda _{G,B}$ .",
"Now, by applying excision with respect to the stratatification by semi-infinite cells one can show that, for all $A \\in \\mathrm {Sat}_{G}$ , one has an isomorphism $\\bigoplus _{i} \\mathcal {H}^{i}(\\pi _{G*}(A)) \\simeq \\mathcal {H}^{0}(\\pi _{T*}(\\mathrm {CT}(A)[\\mathrm {deg}]))$ (See the proof of ).",
"This in particular gives us the following Proposition.",
"Proposition 4.4 For all finite index sets $I$ , there exists a commutative diagram $\\begin{tikzcd}& \\mathrm {Sat}_{G}^{I} [d,\"F_{G}^{I}\"] [r,\"\\mathrm {CT}{[\\mathrm {deg}]}\"] & \\mathrm {Sat}_{T}^{I} [d,\"F_{T}^{I}\"] & \\\\& \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I}) [r,\"\\mathrm {res}^{I,G}_{T}\"] & \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}T^{I}) &\\end{tikzcd} $ where here $F_{G}^{I}$ (resp.",
"$F_{T}^{I}$ ) is the equivalence given by Theorem REF for $G$ (resp.",
"$T$ ).",
"Moreover, it follows by , that the fusion product respects the constant term functors.",
"Proposition 4.5 For all finite index sets $I$ , with a partition $I_{1} \\sqcup \\ldots \\sqcup I_{k}$ , we have a commutative diagram $\\begin{tikzcd}& [d,\"\\mathrm {CT}^{I_{1}}{[\\deg ]} \\times \\ldots \\times \\mathrm {CT}^{I_{k}}{[\\deg ]}\"] \\mathrm {Sat}_{G}^{I_{1}} \\times \\ldots \\times \\mathrm {Sat}_{G}^{I_{k}} [rr] & & \\mathrm {Sat}_{G}^{I} [d,\"\\mathrm {CT}^{I}{[\\deg ]}\"] \\\\& \\mathrm {Sat}_{T}^{I_{1}} \\times \\ldots \\times \\mathrm {Sat}_{T}^{I_{k}} [rr] & & \\mathrm {Sat}_{T}^{I} &\\end{tikzcd} $ which commutes functorially in $I$ , where the top (resp.",
"bottom) vertical arrow is given by the fusion product for $G$ (resp.",
"$T$ ).",
"We now turn our attention to the semi-infinite cells."
],
[
"The Cohomology of Semi-Infinite Cells",
"Theorem REF , Proposition REF , and Proposition REF have implications for the cohomology of spaces related to moduli spaces of $B$ -structures.",
"We will record this now.",
"We let $E/\\mathbb {Q}_{p}$ be the splitting field of $G$ .",
"Then we have an identification $\\mathrm {Gr}_{G,E}^{I} \\simeq \\mathrm {Gr}_{G_{E}}^{I}$ over the base change $\\mathrm {Div}^{I}_{E}$ of $\\mathrm {Div}^{I}$ to $E$ .",
"Sheaves on this space will be equivalent algebraic representations of $I$ copies of the dual group $\\hat{G}$ .",
"First, we recall that we have the following natural stratification of $\\mathrm {Gr}_{G,E}^{I}$ .",
"Definition 4.6 For $(\\lambda _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+})^{I}$ , we let $\\mathrm {Gr}_{G,(\\lambda _{i})_{i \\in I},E}^{I}$ (resp.",
"$\\mathrm {Gr}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ ) be the locally closed (resp.",
"closed) subfunctor of $\\mathrm {Gr}_{G}$ parameterizing modifications $ \\mathcal {F}_{G}^{0} \\rightarrow \\mathcal {F}_{G} $ between the trivial $G$ -bundle $\\mathcal {F}_{G}^{0}$ and a $G$ -bundle $\\mathcal {F}_{G}$ of meromorphy equal to (resp.",
"less than) then $\\sum _{D_{i} = D_{j}} \\lambda _{i}$ at a Cartier divisor $D_{j}$ , for some fixed $j \\in I$ .",
"As mentioned in the previous section, $\\mathrm {Gr}^{I}_{G,(\\lambda _{i})_{i \\in I},E}$ is representable in nice diamonds and is cohomologically smooth of dimension equal to $\\sum _{i \\in I} \\langle 2\\hat{\\rho }, \\lambda _{i} \\rangle $ over $\\mathrm {Div}^{I}_{E}$ , by .",
"Similarly, by , $\\mathrm {Gr}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ is representable in nice diamonds and proper over $\\mathrm {Div}^{I}_{E}$ .",
"Fix $\\boxtimes _{i \\in I} V_{i} = V \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G}^{I})$ with fixed central character, and suppose the highest weight of $V_{i}$ is given by $\\lambda _{i}$ for $\\lambda _{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ .",
"Attached to this, we get a $\\Lambda $ -valued perverse sheaf $\\mathcal {S}_{V}$ supported on $\\mathrm {Gr}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ by Theorem REF and .",
"We now fix a geometric point $x \\rightarrow \\mathrm {Div}^{I}_{E}$ .",
"In what follows, for a space $?$ over $\\mathrm {Div}^{I}_{E}$ we write $_{x}?$ for the base-change to $x$ .",
"Since a local system on $\\mathrm {Div}^{I}_{E}$ will be determined by the $W_{E}^{I}$ -representation given by its pullback to this geometric point, looking at this pullback will be sufficient for most calculations.",
"We now consider another stratification of $\\mathrm {Gr}_{G,E}^{I}$ .",
"Definition 4.7 Consider the natural map: $ p: \\mathrm {Gr}_{B,E}^{I} \\rightarrow \\mathrm {Gr}_{T,E}^{I} $ For $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , we set $\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I},E}$ to be the fiber of $\\mathrm {Gr}_{B,E}^{I}$ over the connected component $\\mathrm {Gr}_{T,(\\nu _{i})_{i \\in I},E}^{I}$ in $\\mathrm {Gr}_{T,E}^{I}$ parameterizing modifications of type $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ .",
"We note, by , that the natural map $ q: \\mathrm {Gr}_{B,E}^{I} \\rightarrow \\mathrm {Gr}_{G,E}^{I} $ is a bijection on geometric points and it defines a locally closed embedding on the $\\mathrm {S}_{G,(\\nu _{i})_{i \\in I},E}^{I}$ .",
"Therefore, the spaces $\\mathrm {S}_{G,(\\nu _{i})_{i \\in I},E}^{I}$ for varying $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , form a locally-closed stratification of $\\mathrm {Gr}_{G,E}^{I}$ .",
"Remark 4.8 In particular, given a modification $\\beta : \\mathcal {F}_{G}^{0} \\dashrightarrow \\mathcal {F}_{G}$ of $G$ -bundles, by the Tannakian formalism this is equivalent to specifying a set of meromorphic maps $ \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}^{0}} \\dashrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}} $ for all dominant characters $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ satisfying the Plücker relations (cf.",
"Definition REF ).",
"The trivial $G$ -bundle $\\mathcal {F}_{G}^{0}$ admits a natural split $B$ -structure whose associated $T$ -bundle is the trivial $T$ -bundle $\\mathcal {F}_{T}^{0}$ , and this defines a set of maps $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{0}} \\hookrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}^{0}_{G}} $ where $\\mathcal {L}^{\\hat{\\lambda }} := (\\mathcal {V}^{\\hat{\\lambda }})^{U}$ and $U$ is the unipotent radical of $B$ .",
"For a set of divisors $(D_{i})_{i \\in I} \\in \\mathrm {Div}_{E}^{I}$ , we can also consider the meromorphic map $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{0}}(\\sum _{i \\in I} -\\langle \\hat{\\lambda },\\nu _{i\\Gamma } \\rangle \\cdot D_{i}) \\dashrightarrow \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{0}} \\rightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}^{0}} \\dashrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}} $ defined by modifying the $T$ -bundles by $\\nu _{i}$ at $D_{i}$ for all $i \\in I$ .",
"We claim that $\\beta $ defines a point in $\\mathrm {S}_{G,(\\nu _{i})_{i \\in I},E}^{I}$ if and only if this map does not have a zero or pole for all $\\hat{\\lambda }$ .",
"This is easy to see.",
"In particular, if this map does not have a zero or a pole then the maps $\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{0}}(-\\sum _{i \\in I} \\langle \\hat{\\lambda },\\nu _{i\\Gamma } \\rangle \\cdot D_{i}) \\rightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}} $ define a $B$ -structure $\\mathcal {F}_{B}$ on $\\mathcal {F}_{G}$ whose underlying $T$ -bundle is given by $\\mathcal {F}_{T}^{0}(-\\sum _{i \\in I} \\langle \\hat{\\lambda },\\nu _{i\\Gamma } \\rangle \\cdot D_{i})$ .",
"Moreover, the construction induces a map of $B$ -bundles $ \\mathcal {F}_{B}^{0} \\dashrightarrow \\mathcal {F}_{B} $ which when applying $\\times ^{B} G$ gives the modification $\\beta $ and when applying $\\times ^{B} T$ gives rise to a modification defining a point in the union of connected components $\\mathrm {Gr}_{T,(\\nu _{i})_{i \\in I},E}^{I}$ over the point attached to $(D_{i})_{i \\in I}$ in $\\mathrm {Div}^{I}_{E}$ .",
"In other words, $\\mathcal {F}_{B}^{0} \\dashrightarrow \\mathcal {F}_{B}$ defines an element of the locally closed stratum $\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I},E}$ .",
"For $V \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G}^{I})$ , we again consider the sheaf $\\mathcal {S}_{V}$ on $\\mathrm {Gr}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ , and pullback to a fixed geometric point $x \\rightarrow \\mathrm {Div}^{I}_{E}$ defined by $\\mathop {\\rm Spa}(C)$ for $C$ an algebraically closed perfectoid field.",
"If we write $p_{(\\nu _{i})_{i \\in I}}: \\phantom{}_{x}\\mathrm {S}_{G,(\\nu _{i})_{i \\in I},E}^{I} \\rightarrow \\phantom{}_{x}\\mathrm {Gr}^{I}_{T,(\\nu _{i})_{i \\in I},E} \\simeq \\mathop {\\rm Spa}(C)$ for the induced map on connected components indexed by $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , we note that we have an isomorphism $ p_{!",
"}q^{*}(\\mathcal {S}_{V}) \\simeq \\bigoplus _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}} p_{(\\nu _{i})_{i \\in I}!",
"}(\\mathcal {S}_{V}|_{\\phantom{}_{x}\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I},E}}) = \\bigoplus _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}} \\mathbb {H}^{*}_{c}(\\phantom{}_{x}\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I}},\\mathcal {S}_{V}|_{\\phantom{}_{x}\\mathrm {S}_{G,(\\nu _{i})_{i \\in I}}}) $ (cf.",
").",
"However, this is simply the constant term functor in the previous section.",
"In particular, by Proposition REF , we deduce the following.",
"Corollary 4.9 For $V = \\boxtimes _{i \\in I} V_{i} \\in \\mathrm {Rep}(\\hat{G}^{I})$ , a geometric point $x \\rightarrow \\mathrm {Div}^{I}_{E}$ , and all tuples $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , we have an isomorphism $ \\mathbb {H}^{-\\langle 2\\hat{\\rho }, |(\\nu _{i})_{i \\in I}| \\rangle }_{c}(_{x}\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I},E},\\mathcal {S}_{V}|_{_{x}\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I},E}}) \\simeq \\boxtimes _{i \\in I} V_{i}(\\nu _{i})(-\\langle \\hat{\\rho }, \\nu _{i}) \\rangle ) $ of $W_{E}^{I}$ -modules.",
"Remark 4.10 The Tate twists appearing here are due to the difference between the standard definition of $\\phantom{}^{L}G$ and the one used in the geometric Satake equivalence, as in the remark proceeding Theorem REF .",
"This will be the key proposition required for the proof of the filtered Hecke eigensheaf property.",
"More specifically, to show the compatibilities of the filtered eigensheaf property, we need to show that this isomorphism is functorial in $I$ .",
"In particular, consider a map of finite index sets $\\pi : I \\rightarrow J$ .",
"For $j \\in J$ , we set $I_{j} := \\pi ^{-1}(j)$ and consider the natural map $\\Delta _{IJ}: \\mathrm {Div}^{J}_{E} \\rightarrow \\mathrm {Div}^{I}_{E}$ , which diagonally embeds the $j$ th copy of $\\mathrm {Div}^{1}_{E}$ in $\\mathrm {Div}^{J}_{E}$ into $\\mathrm {Div}^{I_{j}}_{E}$ .",
"Then, by the relationship between fusion product and tensor product under Theorem REF , we have a identification $ \\Delta _{IJ}^{*}(\\mathcal {S}_{V}) \\simeq \\mathcal {S}_{\\Delta ^{*}_{IJ}(V)} $ of sheaves on $\\mathrm {Gr}_{G,E}^{J}$ , where $\\Delta ^{*}_{IJ}(V)$ is given by restriction along the corresponding map $\\hat{G}^{J} \\rightarrow \\hat{G}^{I}$ .",
"Now, by Proposition REF , we have the following.",
"Corollary 4.11 For all finite index sets $I,J$ with a map $f: I \\rightarrow J$ , a tuple $(\\nu _{j})_{j \\in J} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{J}$ , a representation $V \\in \\mathrm {Rep}(\\hat{G}^{I})$ , and a geometric point $x \\rightarrow \\mathrm {Div}_{E}^{J}$ , the identification $\\Delta _{IJ}^{*}(\\mathcal {S}_{V}) \\simeq \\mathcal {S}_{\\Delta ^{*}_{IJ}(V)}$ induces an isomorphism $ \\mathbb {H}^{*}_{c}(_{x}\\mathrm {S}^{J}_{G,(\\nu _{j})_{j \\in J},E},\\mathcal {S}_{\\Delta ^{*}_{IJ}(V)}|_{\\phantom{}_{x}\\mathrm {S}^{J}_{G,(\\nu _{j})_{j \\in J},E}}) \\simeq \\bigoplus _{\\begin{array}{c}(\\nu _{i})_{i \\in I} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})\\\\ \\sum _{i \\in I_{j}} \\nu _{i} = \\nu _{j}\\end{array}} \\mathbb {H}^{*}_{c}(_{x}\\mathrm {S}_{G,(\\nu _{i})_{i \\in I},E},\\mathcal {S}_{V}|_{_{x}\\mathrm {S}^{I}_{G,(\\nu _{i})_{i \\in I},E}}) $ of $W_{E}^{J}$ -modules, where the action on the RHS is via the natural map $\\Delta _{IJ}: W_{E}^{J} \\rightarrow W_{E}^{I}$ .",
"This is compatible with the identification $ \\Delta _{IJ}^{*}(V((\\nu _{j})_{j \\in J})) \\simeq \\bigoplus _{\\begin{array}{c}(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I} \\\\ \\sum _{i \\in I_{j}} \\nu _{i} = \\nu _{j}\\end{array}} V((\\nu _{i})_{i \\in I}) $ under the isomorphisms of Corollary REF .",
"We note that the previous result has some very useful geometric consequences.",
"Let's explain this in the case that $I = \\lbrace \\ast \\rbrace $ is a singleton for a fixed geometric point $x \\rightarrow \\mathrm {Div}^{1}_{E}$ .",
"Fix $\\lambda \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ , and consider the highest weight representation $V_{\\lambda } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ defined by $\\lambda $ .",
"Since the sheaf $\\mathcal {S}_{\\lambda }$ is supported on $_{x}\\mathrm {Gr}_{G,\\le \\lambda }$ , we can deduce the following.",
"Corollary 4.12 For $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ and $\\lambda \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ with associated highest weight representation $V_{\\lambda } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ , if the weight space $V_{\\lambda }(\\nu )$ is non-trivial then the intersection $_{x}\\mathrm {S}_{G,\\nu ,E} \\cap \\phantom{}_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E}$ is non-empty.",
"Remark 4.13 One can also see this by using the Iwasawa decomposition of $G$ and working explicitly with the loop group of $G$ (See the analysis proceeding and ).",
"For example, one can show that the intersection $_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E} \\cap \\phantom{}_{x}\\mathrm {S}_{G,w_{0}(\\lambda ),E}$ is simply the point given by $\\xi ^{\\lambda }$ , where $\\xi \\in B_{dR}^{+}(C,\\mathcal {O}_{C})$ is the uniformizing parameter defined by the geometric point $x$ .",
"This corresponds to the lowest weight space $V_{\\lambda }(w_{0}(\\lambda ))$ .",
"We will now finish our analysis by recording some facts about the closure relationships for these strata.",
"Proposition 4.14 For $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ and $\\lambda \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ , the closure of the intersection $ _{x}\\mathrm {S}_{G,\\nu ,E} \\cap \\phantom{}_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E} $ in $\\mathrm {Gr}_{G,\\le \\lambda ,E}$ is equal to the disjoint union $ \\bigsqcup _{\\nu ^{\\prime } \\le \\nu } \\phantom{}_{x}\\mathrm {S}_{G,\\nu ^{\\prime },E} \\cap \\phantom{}_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E} $ where this defines a closed subspace in $_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E}$ by .",
"In particular, using the previous Corollary, we deduce that $_{x}\\mathrm {S}_{G,w_{0}(\\lambda ),E} \\cap \\phantom{}_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E}$ is a closed subspace and $_{x}\\mathrm {S}_{G,\\lambda ,E} \\cap \\phantom{}_{x}\\mathrm {Gr}_{G, \\le \\lambda ,E}$ is an open subspace."
],
[
"Moduli stacks of $B$ -structures",
"In this section, we will study the moduli stack of $B$ -structures $\\mathrm {Bun}_{B}$ and its basic geometric properties.",
"This will allow us to define the geometric Eisenstein functor.",
"For understanding many of the finer properties of this functor, it is important to consider a compactification of the natural morphism $\\mathfrak {p}: \\mathrm {Bun}_{B} \\rightarrow \\mathrm {Bun}_{G}$ taking $B$ -bundles to their induced $G$ -bundles.",
"This compactification will be an analogue of Drinfeld's compactification in the function-field setting, denoted $\\overline{\\mathrm {Bun}}_{B}$ .",
"We will show that this gives rise to an Artin $v$ -stack, which admits $\\mathrm {Bun}_{B}$ as an open and dense substack, and that the natural map $\\overline{\\mathfrak {p}}: \\overline{\\mathrm {Bun}}_{B} \\rightarrow \\mathrm {Bun}_{G}$ extending $\\mathfrak {p}$ is indeed proper after restricting to connected components.",
"We will also define a locally closed stratification of $\\overline{\\mathrm {Bun}}_{B}$ .",
"These strata will play an important role in the proof of the Hecke eigensheaf property and understanding how the Eisenstein functor interacts with Verdier duality."
],
[
"The Geometry of $\\mathrm {Bun}_{B}$",
"We will start by collecting some basic facts about the moduli stack $\\mathrm {Bun}_{B}$ parameterizing, for $S \\in \\mathrm {Perf}$ , the groupoid of $B$ -bundles on $X_{S}$ .",
"Given a $B$ -bundle $\\mathcal {G}_{B}$ , we can send it to the induced $T$ -bundle and $G$ -bundle via the natural maps $\\begin{tikzcd}& B [r] [d] & G \\\\& T &\\end{tikzcd}$ which induces a diagram of $v$ -stacks: $\\begin{tikzcd}& \\mathrm {Bun}_{B} [r,\"\\mathfrak {p}\"] [d,\"\\mathfrak {q}\"] & \\mathrm {Bun}_{G} \\\\& \\mathrm {Bun}_{T} &\\end{tikzcd}$ Let's first start by breaking this up into connected components.",
"As seen in §3, the connected components of $\\mathrm {Bun}_{T}$ are indexed by elements $\\overline{\\nu } \\in B(T) \\simeq \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma } = \\Lambda _{G,B}$ .",
"This allows us to define the following.",
"Definition 5.1 For $\\overline{\\nu } \\in B(T)$ , we write $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ for the pre-image of the connected component $\\mathrm {Bun}_{T}^{\\overline{\\nu }}$ defined by $\\overline{\\nu }$ under the map $\\mathfrak {q}^{\\overline{\\nu }}$ .",
"We write $\\mathfrak {p}^{\\overline{\\nu }}: \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{G}$ and $\\mathfrak {q}^{\\overline{\\nu }}: \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}^{\\overline{\\nu }}_{T}$ for the restriction of $\\mathfrak {p}$ and $\\mathfrak {q}$ to $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ , respectively.",
"We claim that this induces a decomposition of the moduli stack $\\mathrm {Bun}_{B}$ into connected components.",
"To each element $\\overline{\\nu }$ , we define the integer $d_{\\overline{\\nu }} := \\langle 2\\hat{\\rho }, \\overline{\\nu } \\rangle $ , where $2\\hat{\\rho }$ is the sum of all positive roots with respect to the choice of Borel.",
"We note that, if $\\overline{\\nu }$ is anti-dominant with respect to the choice of Borel, $d_{\\overline{\\nu }}$ is negative.",
"This will be the case where the HN-slopes are dominant so the $B$ -bundles will split, and the negative dimension comes from quotienting out by the torsor of splittings.",
"On the other hand, if $\\overline{\\nu }$ is dominant then the connected component will parametrize non-split $B$ -structures and we analogously see that the dimension will be positive.",
"We have the following claim.",
"Proposition 5.2 The map $\\mathfrak {q}$ is a cohomologically smooth (non-representable) morphism of Artin $v$ -stacks in the sense of .",
"In particular, for $\\overline{\\nu } \\in \\Lambda _{G,B}$ , the map $\\mathfrak {q}^{\\overline{\\nu }}$ is pure of $\\ell $ -dimension equal to $d_{\\overline{\\nu }}$ , in the sense of .",
"This follows from , where we note that $\\mathrm {Bun}_{T}$ is an Artin $v$ -stack that is cohomologically smooth of $\\ell $ -dimension 0 (See also ).",
"In particular, this implies using , that $\\mathfrak {q}$ is a universally open morphism of Artin $v$ -stacks.",
"Moreover, one can check that the fibers of this morphism are connected (See the proof of ).",
"As a consequence, we can deduce that, since $\\bigsqcup _{\\overline{\\nu } \\in \\Lambda _{G,B}} \\mathrm {Bun}_{T}^{\\overline{\\nu }}$ is a decomposition of $\\mathrm {Bun}_{T}$ into connected components, the following is true.",
"Corollary 5.3 The connected components of $\\mathrm {Bun}_{B}$ are given by $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ , for varying $\\overline{\\nu } \\in B(T)$ .",
"We now comment briefly on the geometry of the map $\\mathfrak {p}$ .",
"In particular, we have the following.",
"Lemma 5.4 The map $\\mathfrak {p}$ is representable in nice diamonds.",
"These properties allow us to define the Eisenstein functor using the derived functors defined in .",
"Definition 5.5 We define a locally constant function $ \\mathrm {dim}(\\mathrm {Bun}_{B}): |\\mathrm {Bun}_{B}| \\rightarrow \\mathbb {Z} $ $ x \\in |\\mathrm {Bun}_{B}^{\\overline{\\nu }}| \\mapsto d_{\\overline{\\nu }} $ and with it the unnormalized Eisenstein functor $ \\mathrm {Eis}: \\mathrm {D}(\\mathrm {Bun}_{T}) \\rightarrow \\mathrm {D}(\\mathrm {Bun}_{G}) $ $ \\mathcal {F} \\mapsto \\mathfrak {p}_{!",
"}(\\mathfrak {q}^{*}(\\mathcal {F})[\\mathrm {dim}(\\mathrm {Bun}_{B})]) $ In particular, since this definition involves the functor $\\mathfrak {p}_{!",
"}$ it is natural to consider a compactification of the morphism $\\mathfrak {p}$ to understand the finer properties of $\\mathfrak {p}_{!",
"}$ .",
"This leads us to our study of the Drinfeld compactification."
],
[
"The Definition and Basic Properties",
"We recall that classically (curve over a finite or complex field) there is a rather straight-forward way of compactifying the map: $ \\mathfrak {p}: \\mathrm {Bun}_B \\rightarrow \\mathrm {Bun}_G $ This is called a Drinfeld Compactification of $\\mathfrak {p}$ , denoted $\\overline{\\mathrm {Bun}}_B$ .",
"Its main property is that there exists an open immersion $\\mathrm {Bun}_{B} \\rightarrow \\overline{\\mathrm {Bun}}_{B}$ , with topologically dense image, and it has a map $\\overline{\\mathfrak {p}}: \\overline{\\mathrm {Bun}}_B \\rightarrow \\mathrm {Bun}_G$ extending $\\mathfrak {p}$ , which is proper after restricting to a connected component.",
"First, as a warm up, let us explain the construction when $G = \\mathrm {GL}_2$ .",
"For this, we recall that $\\mathrm {Bun}_B$ can be viewed as parameterizing tuples $(\\mathcal {M},\\mathcal {L},\\kappa : \\mathcal {L} \\hookrightarrow \\mathcal {M})$ , where $\\mathcal {E}$ is a rank 2 vector bundle, $\\mathcal {L}$ is a rank 1 vector bundle, and $\\kappa $ is an injective bundle map.",
"To compactify this space, we will allow $\\kappa $ to be a map of $\\mathcal {O}_{X_S}$ -modules whose pullback to each geometric point is an injective map of coherent sheaves on $X$ .",
"In other words, we allow $\\mathcal {M}/\\mathcal {L}$ to have torsion.",
"For a general $G$ , the idea is to apply the Tannakian formalism.",
"In particular, given a $G$ -bundle $\\mathcal {F}_G$ we get, for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_G^+$ , an induced highest weight bundle, denoted $\\mathcal {V}_{\\mathcal {F}_G}^{\\hat{\\lambda }}$ .",
"A point of $\\mathrm {Bun}_B$ mapping to $\\mathcal {F}_G$ via $\\mathfrak {p}$ then defines a set of line subbundles $\\kappa ^{\\hat{\\lambda }}: \\mathcal {L}^{\\hat{\\lambda }} \\hookrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_G}$ which satisfy some Plücker relations.",
"Using this interpretation, $\\overline{\\mathrm {Bun}}_B$ can then be defined as classifying $G$ -bundles $\\mathcal {F}_G$ together with a system of maps $\\overline{\\kappa }^{\\hat{\\lambda }}: \\mathcal {L}^{\\hat{\\lambda }} \\rightarrow \\mathcal {V}_{\\mathcal {F}_G}^{\\hat{\\lambda }}$ for all $\\hat{\\lambda } \\in \\Lambda _G^+$ , which are injective after pulling back to a geometric point and satisfy the same Plücker relations.",
"We now explain how to construct the aforementioned compactification $\\overline{\\mathrm {Bun}}_B$ of $\\mathrm {Bun}_B$ over $\\mathrm {Bun}_G$ in the Fargues-Fontaine setting.",
"In order to describe the Drinfeld compactification, we note that, for $S \\in \\mathrm {Perf}$ , $\\mathrm {Bun}_B$ can be viewed as a stack parameterizing triples: A $G$ -bundle $\\mathcal {F}_G$ on $X_{S}$ .",
"A $T$ -bundle $\\mathcal {F}_T$ on $X_{S}$ .",
"A $G$ -equivariant map $\\kappa : \\mathcal {F}_G \\rightarrow G/U \\times ^T \\mathcal {F}_T$ .",
"By the Tannakian formalism, (3) can be described as a collection of injective bundle maps on $X_S$ , $\\kappa ^{\\mathcal {V}}: (\\mathcal {V}^{U})_{\\mathcal {F}_{T}} \\rightarrow \\mathcal {V}_{\\mathcal {F}_{G}}$ for every $G$ -module $\\mathcal {V}$ satisfying the following Plücker relations: For the trivial representation $\\mathcal {V}$ , $\\kappa ^{\\mathcal {V}}$ must be the identity map $\\mathcal {O}_{X_S} \\rightarrow \\mathcal {O}_{X_S}$ .",
"For a $G$ -module map $\\mathcal {V}^1 \\rightarrow \\mathcal {V}^2$ , the induced square $\\begin{tikzcd}& ((\\mathcal {V}^1)^U) _{\\mathcal {F}_T}[d] [r, \"\\kappa ^{\\mathcal {V}^1}\"] & \\mathcal {V}^1_{\\mathcal {F}_G} [d] & \\\\& ((\\mathcal {V}^2)^U)_{\\mathcal {F}_T} [r, \"\\kappa ^{\\mathcal {V}^2}\"] & \\mathcal {V}^2_{\\mathcal {F}_G} &\\end{tikzcd}$ commutes.",
"For two $G$ -modules $\\mathcal {V}^1$ and $\\mathcal {V}^2$ , we have that the diagram $\\begin{tikzcd}& ((\\mathcal {V}^1)^U \\otimes (\\mathcal {V}^2)^U)_{\\mathcal {F}_T} [d] [r,\"\\kappa ^{\\mathcal {V}^1} \\otimes \\kappa ^{\\mathcal {V}^2}\"] & \\mathcal {V}^1_{\\mathcal {F}_G} \\otimes \\mathcal {V}^2_{\\mathcal {F}_G} [d,\"id\"] & \\\\& ((\\mathcal {V}^1 \\otimes \\mathcal {V}^2)^U)_{\\mathcal {F}_T} [r, \"\\kappa ^{\\mathcal {V}^1 \\otimes \\mathcal {V}^2}\"] & \\mathcal {V}^1_{\\mathcal {F}_G} \\otimes \\mathcal {V}^2_{\\mathcal {F}_G} &\\end{tikzcd}$ commutes.",
"As mentioned above, the idea will now be to introduce torsion in the above definition.",
"In particular, we have the following definition for $\\overline{\\mathrm {Bun}}_{B}$ .",
"Definition 5.6 We define $\\overline{\\mathrm {Bun}}_{B}$ to be the $v$ -stack parameterizing, for $S \\in \\mathrm {Perf}$ , triples $(\\mathcal {F}_G,\\mathcal {F}_T, \\overline{\\kappa }^{\\mathcal {V}})$ , where $\\overline{\\kappa }^{\\mathcal {V}}$ is a map of $\\mathcal {O}_{X_S}$ -modules defined for every $G$ -module $\\mathcal {V}$ $ (\\mathcal {V}^U)_{\\mathcal {F}_T} \\rightarrow \\mathcal {V}_{\\mathcal {F}_G} $ satisfying the following conditions: For every geometric point $s \\rightarrow S$ , the pullback of $\\overline{\\kappa }^{\\mathcal {V}}$ to the Fargues-Fontaine curve over $s$ is an injection of coherent sheaves.",
"The Plücker relations hold in the following sense: For the trivial representation $\\mathcal {V}$ , $\\overline{\\kappa }^{\\mathcal {V}}$ is the identity map $\\mathcal {O} \\rightarrow \\mathcal {O}$ .",
"For a $G$ -module map $\\mathcal {V}^1 \\rightarrow \\mathcal {V}^2$ , the induced square $\\begin{tikzcd}& ((\\mathcal {V}^1)^U)_{\\mathcal {F}_T} [d] [r, \"\\overline{\\kappa }^{\\mathcal {V}^1}\"] & \\mathcal {V}^1_{\\mathcal {F}_G} [d] & \\\\& ((\\mathcal {V}^2)^U)_{\\mathcal {F}_T} [r, \"\\overline{\\kappa }^{\\mathcal {V}^2}\"] & \\mathcal {V}^2_{\\mathcal {F}_G} &\\end{tikzcd}$ commutes.",
"For two $G$ -modules $\\mathcal {V}^1$ and $\\mathcal {V}^2$ , we have that the diagram $\\begin{tikzcd}& ((\\mathcal {V}^1)^U \\otimes (\\mathcal {V}^2)^U)_{\\mathcal {F}_T} [d] [r,\"\\overline{\\kappa }^{\\mathcal {V}^1} \\otimes \\overline{\\kappa }^{\\mathcal {V}^2}\"] & \\mathcal {V}^1_{\\mathcal {F}_G} \\otimes \\mathcal {V}^2_{\\mathcal {F}_G} [d,\"id\"] & \\\\& ((\\mathcal {V}^1 \\otimes \\mathcal {V}^2)^U)_{\\mathcal {F}_T} [r, \"\\overline{\\kappa }^{\\mathcal {V}^1 \\otimes \\mathcal {V}^2}\"] & \\mathcal {V}^1_{\\mathcal {F}_G} \\otimes \\mathcal {V}^2_{\\mathcal {F}_G} &\\end{tikzcd}$ commutes.",
"Remark 5.7 To simplify the notation, we will write $\\mathcal {L}^{\\hat{\\lambda }} := (\\mathcal {V}^{\\hat{\\lambda }})^{U}$ and $\\overline{\\kappa }^{\\hat{\\lambda }}$ for the embedding attached to the highest weight module of $G$ of highest weight $\\hat{\\lambda } \\in \\Lambda _{G}^{+}$ .",
"It follows by construction that it suffices to consider only the embeddings induced by the highest weight Weyl $G$ -modules attached to the fundamental weights $\\hat{\\varpi }_{i} \\in \\hat{\\Lambda }_{G}^{+}$ for $i \\in \\mathcal {J}$ .",
"This gives rise to a well-defined $v$ -stack and, using this description, we get well-defined morphisms $\\overline{\\mathfrak {p}}: \\overline{\\mathrm {Bun}}_B \\rightarrow \\mathrm {Bun}_G$ and $\\overline{\\mathfrak {q}}: \\overline{\\mathrm {Bun}}_B \\rightarrow \\mathrm {Bun}_T$ via projecting the data $(\\mathcal {F}_{G},\\mathcal {F}_{T},\\overline{\\kappa })$ to the first and second factor, respectively.",
"We also get a natural map $j: \\mathrm {Bun}_{B} \\rightarrow \\overline{\\mathrm {Bun}}_{B}$ .",
"Now, to conclude this section, we prove some basic things about its geometry.",
"Proposition 5.8 The $v$ -stack $\\overline{\\mathrm {Bun}}_{B}$ is an Artin $v$ -stack, and the map $j: \\mathrm {Bun}_{B} \\rightarrow \\overline{\\mathrm {Bun}}_{B}$ is an open immersion.",
"It suffices to show the claim after base-change to an algebraically closed perfectoid field $\\mathop {\\rm Spa}(F,\\mathcal {O}_{F})$ .",
"We write $X$ for the associated Fargues-Fontaine curve.",
"Recall that, given a scheme $Y$ , one defines the affine closure to be $\\overline{Y} = \\mathop {\\rm Spec}{\\Gamma (Y,\\mathcal {O}_{Y})}$ .",
"We let $\\overline{G/U}$ be the affine closure of $G/U$ .",
"Viewing this as a constant scheme over $X$ , we consider the stack: $ Z := [(\\overline{G/U})/(T \\times G)] \\rightarrow X $ Now, it follows by , that, for $S \\in \\mathrm {Perf}$ , a section $\\begin{tikzcd}& & Z [d] \\\\& X_{S} [ur,\"s\",dotted] [r] & X\\end{tikzcd}$ is equivalent to the datum of a $T$ -bundle (resp.",
"$G$ -bundle) $\\mathcal {F}_{T}$ (resp.",
"$\\mathcal {F}_{G}$ ) on $X_{S}$ together with a family of maps $\\overline{\\kappa }^{\\mathcal {V}}$ of $\\mathcal {O}_{X_{S}}$ -modules, satisfying the Plücker conditions of Definition REF .",
"Therefore, if we consider $\\mathcal {M}_{Z}$ , the moduli stack parameterizing such sections, then $\\overline{\\mathrm {Bun}}_{B}$ is the sub-functor corresponding to the locus where these maps are injective after pulling back to a geometric point.",
"By , this is an open subfunctor.",
"By , $\\mathcal {M}_{Z}$ is an Artin $v$ -stack; therefore, the same is true for $\\overline{\\mathrm {Bun}}_{B}$ .",
"It remains to see that $\\mathrm {Bun}_{B}$ is an open sub-functor.",
"Now, by the work of , it follows that $\\overline{G/U}$ is strongly quasi-affine in the sense that $G/U \\hookrightarrow \\overline{G/U}$ is an open immersion.",
"This induces an open immersion of stacks $ [(G/U)/(T \\times G)] = [X/B] \\hookrightarrow Z $ which, after passing to moduli stacks of sections, gives a natural map $\\mathrm {Bun}_{B} \\rightarrow \\mathcal {M}_{Z}$ factoring through the open immersion $\\overline{\\mathrm {Bun}}_{B} \\hookrightarrow \\mathcal {M}_{Z}$ .",
"It now suffices to show that $\\mathrm {Bun}_{B} \\rightarrow \\mathcal {M}_{Z}$ is an open immersion, but this follows from the fact that $[X/B] \\hookrightarrow Z$ is an open immersion, by arguing as in ."
],
[
"Properness of Compactifications",
"We now seek to show that the morphism $\\overline{\\mathfrak {p}}: \\overline{\\mathrm {Bun}}_{B} \\rightarrow \\mathrm {Bun}_{G}$ is indeed a compactification of the map $\\mathfrak {p}$ .",
"In particular, we write $\\overline{\\mathrm {Bun}}^{\\overline{\\nu }}_{B}$ for the pre-image of the connected component $\\mathrm {Bun}_{T}^{\\overline{\\nu }} \\subset \\mathrm {Bun}_{T}$ .",
"Later we will see that $\\overline{\\mathrm {Bun}}^{\\overline{\\nu }}_{B}$ defines the connected components of $\\overline{\\mathrm {Bun}}_{B}$ .",
"This will follow from Corollary REF and showing that $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ is dense inside $\\overline{\\mathrm {Bun}}_{B}^{\\overline{\\nu }}$ (Proposition REF ).",
"We write $\\overline{\\mathfrak {q}}^{\\overline{\\nu }}: \\overline{\\mathrm {Bun}}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{T}^{\\overline{\\nu }}$ and $\\overline{\\mathfrak {p}}^{\\overline{\\nu }}: \\overline{\\mathrm {Bun}}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{G}$ for the maps induced by $\\overline{\\mathfrak {q}}$ and $\\overline{\\mathfrak {p}}$ , respectively.",
"This section will be dedicated to proving the following.",
"Proposition 5.9 For all $\\overline{\\nu } \\in \\Lambda _{G,B}$ , the map $\\overline{\\mathfrak {p}}^{\\overline{\\nu }}$ is representable in nice diamonds and proper.",
"Consider $S \\in \\mathrm {Perf}$ and an $S$ -point of $\\mathrm {Bun}_{G}$ corresponding to a $G$ -bundle $\\mathcal {F}_{G}$ .",
"We let $Z$ denote the fiber of $\\overline{\\mathfrak {p}}$ over the $S$ -point defined by $\\mathcal {F}_{G}$ .",
"We need to show $Z \\rightarrow S$ is representable in nice diamonds and proper.",
"To this end, let $\\hat{\\varpi }_{i}$ for $i \\in \\mathcal {J}$ be the set of fundamental weights.",
"The $G$ -bundle $\\mathcal {F}_{G}$ then defines a finite set of vector bundles $(\\mathcal {V}^{\\hat{\\varpi }_{i}})_{\\mathcal {F}_{G}} =: \\mathcal {V}_{i}$ for $i \\in \\mathcal {J}$ .",
"For $\\overline{\\nu } \\in \\Lambda _{G,B}$ , we set $d_{i} := \\langle \\hat{\\varpi }_{i}, \\overline{\\nu } \\rangle $ .",
"We consider the space $ P_{i} := (\\mathcal {H}^{0}(\\mathcal {V}_{i}(-d_{i})) \\setminus \\lbrace 0\\rbrace )/\\underline{\\mathbb {Q}}^{*}_{p} \\rightarrow S $ where $\\mathcal {H}^{0}(\\mathcal {V}_{i}(-d_{i}))$ is the Banach-Colmez space parameterizing global sections of the vector bundle $\\mathcal {V}_{i}$ twisted by $-d_{i}$ and $\\lbrace 0\\rbrace $ denotes the 0-section.",
"It follows, by , that $P_{i} \\rightarrow S$ is representable in nice diamonds and proper.",
"Therefore, the same is true for the product: $ \\prod _{i \\in \\mathcal {J}} P_{i} \\rightarrow S $ This will parametrize line bundles $\\mathcal {L}_{i}$ of degree $d_{i}$ together with a map $ \\mathcal {L}_{i} \\rightarrow \\mathcal {V}_{i} $ of $\\mathcal {O}_{X_{S}}$ -modules whose pullback to each geometric point is an injection of coherent sheaves.",
"Therefore, we get a natural map $ X \\rightarrow \\prod _{i = 1}^{r} P_{i} $ remembering only the embeddings defined by the fundamental weights $\\hat{\\varpi }_{i}$ .",
"Now, by the Remark proceeding REF , it follows that this is an injective map mapping to the subspace where the Plücker relations are satisfied.",
"The desired claim would therefore follow from showing that this locus is closed.",
"To see this, note that the commutativity conditions describing the Plücker relations can be expressed in terms of the vanishing of some linear combination of morphisms of $\\mathcal {O}_{X_S}$ -modules.",
"Using this, one reduces to checking that, given a vector bundle $\\mathcal {T}$ on $X_{S}$ , the point in $\\mathcal {H}^{0}(\\mathcal {T})$ defined by the zero section of $\\mathcal {T}$ is closed.",
"By choosing an injection $\\mathcal {T} \\hookrightarrow \\mathcal {O}(m)^{N}$ for sufficiently large $m$ and $N$ , one obtains an injective map $\\mathcal {H}^{0}(\\mathcal {T}) \\rightarrow \\mathcal {H}^{0}(\\mathcal {O}(m)^{N})$ of diamonds compatible with the zero section.",
"Thus, one reduces to checking the claim for $\\mathcal {T} = \\mathcal {O}(m)^{N}$ .",
"In this case, it is nothing more than .",
"Now, we seek to describe the finer structure of these compactifications.",
"In particular, a key role will be played by their stratifications.",
"To do this, we will need to take a brief detour to discuss some properties of some $\\mathcal {O}_{X_{S}}$ -modules on the Fargues-Fontaine that occur as cokernels of fiberwise injective maps, as considered in the definition of the Drinfeld compactification."
],
[
"$\\mathcal {O}_S$ -flat coherent sheaves on the Fargues-Fontaine Curve",
"Throughout this section, we fix $S \\in \\mathrm {Perf}$ and note that, in Definition REF , we imposed the condition that the intervening maps of vector bundles $\\mathcal {F} \\rightarrow \\mathcal {F}^{\\prime }$ on $X_{S}$ satisfy the property that their pullback to each geometric point of $S$ is an injection of coherent sheaves on $X$ .",
"The stratification on the Drinfeld compactification will be given by fixing the length of the torsion of the cokernel of such morphisms.",
"Normally, one could appeal to classical results on flat coherent sheaves in families to get a handle on the structure of such strata; however, as the Fargues-Fontaine curve $X_{S}$ is non-Noetherian unless $S$ is a field, one cannot naively define a category of coherent sheaves on it.",
"Nonetheless, we can still define the following.",
"Definition 5.10 For $S \\in \\mathrm {Perf}$ , a flat coherent $\\mathcal {O}_{X_{S}}$ -module on $X_{S}$ is an $\\mathcal {O}_{X_{S}}$ -module $\\mathcal {F}$ which can, locally for the analytic topology on $X_{S}$ , be written as the cokernel of a fiberwise injective map of bundles on $X_{S}$ .",
"Equivalently, the map remains an injection after pulling back to any $T \\in \\mathrm {Perf}_{S}$ .",
"Remark 5.11 By , it follows that we can always find a global presentation of an $S$ -flat coherent sheaf $\\mathcal {F}$ as a two term complex of vector bundles on $X_{S}$ .",
"Remark 5.12 We note that in the classical context of a relative projective curve $X \\rightarrow S$ over a reduced Noetherian scheme $S$ the analogue of this condition for a coherent sheaf $\\mathcal {F}$ is equivalent to insisting that $\\mathcal {F}$ is $\\mathcal {O}_{S}$ -flat (See ).",
"However, as discussed above, the notion of coherent sheaves make no sense in this context, and the notion of flatness also does not make sense as $\\mathcal {O}_{X_{S}}$ is not a module over $\\mathcal {O}_{S}$ .",
"We have the following easy lemma, which gives a homological criterion characterizing flat coherent $\\mathcal {O}_{X_{S}}$ -modules.",
"Lemma 5.13 Let $\\mathcal {F}$ be an $\\mathcal {O}_{X_{S}}$ -module.",
"For integers $a \\le b$ write $\\mathrm {Perf}_{[a,b]}(X_{S})$ for the derived category of perfect complexes of Tor-amplitude $[a,b]$ , as in .",
"The following conditions are equivalent.",
"$\\mathcal {F}$ is a flat coherent $\\mathcal {O}_{X_{S}}$ -module.",
"$\\mathcal {F}$ is represented by an object in $\\mathrm {Perf}_{[-1,0]}(X_{S})$ and $\\mathrm {Tor}_{1,\\mathcal {O}_{X_{S}}}(\\mathcal {O}_{X_{T}},\\mathcal {F})$ is trivial for all $T \\in \\mathrm {Perf}_{S}$ .",
"For the forward direction, by definition $\\mathcal {F}$ is represented by an object in $\\mathrm {Perf}_{[-1,0]}(X_{S})$ .",
"For the other condition, we choose a presentation $ 0 \\rightarrow \\mathcal {F}_{-1} \\rightarrow \\mathcal {F}_{0} \\rightarrow \\mathcal {F} \\rightarrow 0 $ of $\\mathcal {F}$ as the cokernel of a fiberwise injective map of vector bundles.",
"Now, since the first map is injective after tensoring by $- \\otimes _{\\mathcal {O}_{X_{S}}} \\mathcal {O}_{X_{T}}$ for any $T \\in \\mathrm {Perf}_{S}$ , it follows easily by the associated long exact sequence that $\\mathrm {Tor}_{1,\\mathcal {O}_{X_{S}}}(\\mathcal {O}_{X_{T}},\\mathcal {F}) = 0$ .",
"For the converse direction, it follows from the proof of that if $\\mathcal {F}$ is represented by an element in $\\mathrm {Perf}_{[-1,0]}(X_{S})$ then it can be globally represented by a two term complex of vector bundles on $X_{S}$ .",
"Choosing such a presentation $\\mathcal {F}_{-1} \\rightarrow \\mathcal {F}_{0}$ , it follows that the defining map must be fiberwise injective by the vanishing of $\\mathrm {Tor}_{1,\\mathcal {O}_{X_{S}}}(\\mathcal {O}_{X_{T}},\\mathcal {F})$ for all $T \\in \\mathrm {Perf}_{S}$ .",
"This simple lemma allows us to see the following.",
"Lemma 5.14 Flat coherent sheaves on $X_{S}$ are stable under taking kernels of surjections and cokernels of fiberwise injective $\\mathcal {O}_{X_{S}}$ -module maps.",
"We explain the case of taking cokernels of injections with the case of surjections being similar.",
"We consider two flat coherent sheaves $\\mathcal {F}$ and $\\mathcal {F}^{\\prime }$ on $X_{S}$ and a short exact sequence $ 0 \\rightarrow \\mathcal {F} \\rightarrow \\mathcal {F}^{\\prime } \\rightarrow \\mathcal {F}^{\\prime }/\\mathcal {F} \\rightarrow 0 $ of $\\mathcal {O}_{X_{S}}$ -modules where the first map is fiberwise injective.",
"It easily follows from the corresponding long exact sequences of $\\mathrm {Tor}$ s, the previous lemma applied to $\\mathcal {F}$ and $\\mathcal {F}^{\\prime }$ , and the fiberwise injectivity of the first map that $\\mathrm {Tor}_{i,\\mathcal {O}_{X_{S}}}(\\mathcal {O}_{X_{T}},\\mathcal {F}^{\\prime }/\\mathcal {F})$ is trivial for all $T \\in \\mathrm {Perf}_{S}$ and $i \\ge 1$ .",
"Therefore, by the previous lemma, it suffices to show that $\\mathcal {F}^{\\prime }/\\mathcal {F} \\in \\mathrm {Perf}_{[-1,0]}(X_{S})$ .",
"Since $\\mathcal {F},\\mathcal {F}^{\\prime } \\in \\mathrm {Perf}_{[-1,0]}(X_{S})$ it follows that $\\mathcal {F}^{\\prime }/\\mathcal {F} \\in \\mathrm {Perf}_{[-2,0]}(X_{S})$ , and, by the vanishing of $\\mathrm {Tor}_{i,\\mathcal {O}_{X_{S}}}(\\mathcal {O}_{X_{T}},\\mathcal {F}^{\\prime }/\\mathcal {F})$ for $i \\ge 2$ , it must lie in $\\mathrm {Perf}_{[-1,0]}(X_{S})$ .",
"Now, consider the underlying topological space of $S$ , denoted $|S|$ .",
"We can consider a geometric point $s \\in |S|$ and the pullback of such a flat coherent $\\mathcal {O}_{X_{S}}$ -module $\\mathcal {F}$ to $X_{s}$ .",
"The scheme $X_{s}$ will just be the usual Fargues-Fontaine curve over a geometric point so it is in particular a Dedekind scheme, and $\\mathcal {F}|_{X_{s}}$ will just be a coherent sheaf.",
"Therefore, we have a decomposition $ \\mathcal {F}|_{X_{s}} \\simeq \\mathcal {F}^{\\mathrm {tors}}|_{X_{s}} \\oplus \\mathcal {F}^{\\mathrm {vb}}|_{X_{s}} $ where $\\mathcal {F}^{\\mathrm {tors}}|_{X_{s}}$ (resp.",
"$\\mathcal {F}^{\\mathrm {vb}}|_{X_{s}}$ ) is a torsion sheaf (resp.",
"vector bundle) on $X_{s}$ .",
"Given such a torsion sheaf, we write $\\lambda (\\mathcal {F}|_{X_{s}}) := \\ell (\\mathcal {F}^{\\mathrm {tors}}|_{X_{s}})$ for the length of this torsion sheaf.",
"Our main aim is to prove the following proposition.",
"We would like to thank David Hansen for supplying the idea behind its proof.",
"Proposition 5.15 If $\\mathcal {F}$ is a flat coherent sheaf on $X_{S}$ then the function $ |S| \\rightarrow \\mathbb {N}_{\\ge 0} $ $ s \\mapsto \\lambda (\\mathcal {F}|_{X_{s}}) $ is upper semi-continuous.",
"Moreover, if this function is locally constant on $S$ , then there exists a unique short exact sequence of flat coherent sheaves on $X_{S}$ $ 0 \\rightarrow \\mathcal {F}^{\\mathrm {tors}} \\rightarrow \\mathcal {F} \\rightarrow \\mathcal {F}^{\\mathrm {vb}} \\rightarrow 0 $ where $\\mathcal {F}^{\\mathrm {vb}}$ is a vector bundle on $X_{S}$ , and $\\mathcal {F}^{\\mathrm {tors}}$ is a torsion sheaf in the sense that its pullback to each geometric point is a torsion sheaf.",
"To do this, we will need to prove the following lemma.",
"Lemma 5.16 Let $\\mathcal {F}$ be a flat coherent sheaf on $X_{S}$ .",
"We consider the $v$ -sheaf on $\\mathrm {Perf}_{S}$ , denoted $\\mathcal {H}^{0}(\\mathcal {F})$ , which sends $T \\in \\mathrm {Perf}_{S}$ to the set of global sections of $\\mathcal {F}$ .",
"The following is true.",
"$\\mathcal {H}^{0}(\\mathcal {F}) \\rightarrow S$ is separated and representable in nice diamonds.",
"The $v$ -sheaf $(\\mathcal {H}^{0}(\\mathcal {F}) \\setminus \\lbrace 0\\rbrace )/\\underline{\\mathbb {Q}^{*}_{p}} \\rightarrow S$ given by deleting the 0-section and quotienting out by the scaling action by $\\underline{\\mathbb {Q}_{p}}^{*}$ is proper and representable in nice diamonds over $S$ .",
"We can check all claims analytically locally on $S$ .",
"First note, by , that analytically locally on $S$ we can find an exact sequence $ 0 \\rightarrow \\mathcal {E} \\rightarrow \\mathcal {O}_{X_{S}}(-n)^{m} \\rightarrow \\mathcal {F} \\rightarrow 0 $ where $\\mathcal {E}$ is a vector bundle, for all $n$ sufficiently large and fixed $m$ .",
"Passing to cohomology, this gives us an exact sequence of $v$ -sheaves: $ 0 \\rightarrow \\mathcal {H}^0(\\mathcal {F}) \\hookrightarrow \\mathcal {H}^{1}(\\mathcal {E}) \\rightarrow \\mathcal {H}^{1}(\\mathcal {O}(-n)^{m}) $ Now note that the slopes of $\\mathcal {E}$ are necessarily negative after pulling back to a geometric point for $n$ sufficiently large.",
"Therefore, we can assume that $\\mathcal {H}^{1}(\\mathcal {E})$ is a nice diamond by .",
"It follows by that $\\mathcal {H}^{1}(\\mathcal {O}_{X_{S}}(-n)^{m})$ is separated which implies that the injective map $\\mathcal {H}^{0}(\\mathcal {F}) \\rightarrow \\mathcal {H}^{1}(\\mathcal {E})$ is a closed embedding.",
"Therefore, $\\mathcal {H}^{0}(\\mathcal {F})$ is also nice.",
"For the second part, we note that the closed embedding $ \\mathcal {H}^{0}(\\mathcal {F}) \\hookrightarrow \\mathcal {H}^{1}(\\mathcal {E}) $ is compatible with the 0-section and the scaling action.",
"This reduces us to checking that $(\\mathcal {H}^{1}(\\mathcal {E}) \\setminus \\lbrace 0\\rbrace )/\\underline{\\mathbb {Q}}_{p}^{*} \\rightarrow S$ is proper, which follows from .",
"In particular, we will need the following Corollary.",
"Corollary 5.17 Let $\\mathcal {F}$ and $\\mathcal {F}^{\\prime }$ be two flat coherent sheaves on $X_{S}$ .",
"We consider the $v$ -sheaf on $\\mathrm {Perf}_{S}$ , denoted $\\mathcal {H}om(\\mathcal {F},\\mathcal {F}^{\\prime })$ , which sends $T \\in \\mathrm {Perf}_{S}$ to the set of $\\mathcal {O}_{X_{S}}$ -module homomorphisms $\\mathcal {F} \\rightarrow \\mathcal {F}^{\\prime }$ .",
"The following is true.",
"$\\mathcal {H}om(\\mathcal {F},\\mathcal {F}^{\\prime }) \\rightarrow S$ is separated and representable in nice diamonds.",
"The diamond $(\\mathcal {H}om(\\mathcal {F},\\mathcal {F}^{\\prime }) \\setminus \\lbrace 0\\rbrace )/\\underline{\\mathbb {Q}^{*}_{p}} \\rightarrow S$ given by deleting the 0-map and quotienting out by the scaling action by $\\underline{\\mathbb {Q}}_{p}^{*}$ is proper over $S$ .",
"By applying , we have analytically locally on $S$ a short exact sequence $ 0 \\rightarrow \\mathcal {E} \\rightarrow \\mathcal {O}_{X_{S}}(-n)^{m} \\rightarrow \\mathcal {F} \\rightarrow 0 $ for all $n$ sufficiently large.",
"Applying $\\mathrm {Hom}(-,\\mathcal {F}^{\\prime })$ , we get an exact sequence of $v$ -sheaves: $ 0 \\rightarrow \\mathcal {H}om(\\mathcal {F},\\mathcal {F}^{\\prime }) \\rightarrow \\mathcal {H}om(\\mathcal {O}_{X_{S}}(-n)^{m},\\mathcal {F}^{\\prime }) \\rightarrow \\mathcal {H}om(\\mathcal {E},\\mathcal {F}) $ In other words, $\\mathcal {H}om(\\mathcal {F},\\mathcal {F}^{\\prime })$ is the fiber of the last map over the 0-section, and in turn the first map is a closed immersion by Lemma REF (1) applied to $\\mathcal {H}^{0}(\\mathcal {E}^{\\vee } \\otimes \\mathcal {F}^{\\prime })$ .",
"This allows us to replace $\\mathcal {F}^{\\prime }$ with a vector bundle, and the claim then follows by Lemma REF applied to $\\mathcal {F}^{\\vee } \\otimes \\mathcal {F}^{\\prime }$ .",
"Now that we have gotten this out of the way we can finally turn to the proof of our claim.",
"We consider the $v$ -sheaf $ \\mathcal {S}_{\\mathcal {F}} \\rightarrow S $ of fiberwise non-zero global sections $s$ of $\\mathcal {F}$ which are annihilated by $\\mathcal {I}_{D} \\subset \\mathcal {O}_{X_{S}}$ for $D$ a degree 1 relative Cartier divisor $D \\subset X_{S}$ .",
"Alternatively, we can view this as parameterizing pairs of $(D,f)$ of a degree 1 Cartier divisor in $X_{S}$ and a point of $\\mathcal {H}om(\\mathcal {O}_{X_{S}}/\\mathcal {I}_{D},\\mathcal {F}) \\setminus \\lbrace 0\\rbrace $ .",
"Using this description, we can factorize the map $\\mathcal {S}_{\\mathcal {F}} \\rightarrow S$ as $\\mathcal {S}_{\\mathcal {F}} \\rightarrow \\mathrm {Div}^{1}_{S} \\rightarrow S$ .",
"We let $\\mathcal {S}_{\\mathcal {F}}/\\underline{\\mathbb {Q}}_{p}^{*}$ be the quotient of this space by the scaling action on the section $f$ .",
"Now the projection $\\mathrm {Div}^{1}_{S} \\rightarrow S$ is proper and representable in nice diamonds by combining and .",
"Moreover, by the previous Corollary, we know that $\\mathcal {S}_{\\mathcal {F}}/\\underline{\\mathbb {Q}}_{p}^{*} \\rightarrow \\mathrm {Div}^{1}_{S}$ is proper and representable in nice diamonds because it is a fibration in the spaces $\\mathcal {H}om(\\mathcal {O}_{X_{T}}/\\mathcal {I}_{D},\\mathcal {F}_{T}) \\setminus \\lbrace 0\\rbrace /\\underline{\\mathbb {Q}_{p}}^{*}$ for $T \\in \\mathrm {Perf}_{S}$ and $D$ a degree 1 relative Cartier divisor in $X_{T}$ .",
"It follows that the image of $\\mathcal {S}_{\\mathcal {F}} \\rightarrow S$ is closed.",
"We note that this coincides with the locus where $\\lambda (\\mathcal {F}) > 0$ , so we have deduced a special case of the upper semi-continuity resultWe could have also deduced this special case by arguing as in the proof of Proposition REF .. Now we argue by induction.",
"In particular, by replacing $S$ by its image, it follows by properness that $\\mathcal {S}_{\\mathcal {F}} \\rightarrow S$ is a $v$ -cover.",
"Therefore, after $v$ -localization, we can assume that $\\mathcal {S}_{\\mathcal {F}} \\rightarrow S$ admits a section, which implies that there exists a degree 1 Cartier divisor $D \\subset X_{S}$ and a fiberwise non-zero section $s: \\mathcal {O}_{X_{S}}/\\mathcal {I}_{D} \\rightarrow \\mathcal {F}$ .",
"Using Lemma REF , we can replace $\\mathcal {F}$ with $\\mathcal {F}^{\\prime } = \\mathcal {F}/s(\\mathcal {O}_{X_{S}})$ , noting that the locus where $\\lambda (\\mathcal {F}^{\\prime }) > 0$ coincides with the locus where $\\lambda (\\mathcal {F}) > 1$ .",
"Therefore, the claimed upper semi-continuity result follows.",
"For the second claim, uniqueness is clear, and, by , the category of flat coherent sheaves on $X_{S}$ satisfies $v$ -descent, so it suffices to show the claim up to a $v$ -localization.",
"However, now via induction we can just argue as above, using the sections $s$ produced above and Lemma REF to give rise to the map $\\mathcal {F}^{\\mathrm {tors}} \\rightarrow \\mathcal {F}$ .",
"Now we can reap the fruit of our labor in this section, and use it to show that the compactification $\\overline{\\mathrm {Bun}}_{B}$ has a well-behaved stratification."
],
[
"Stratifications",
"We now turn our attention to stratifying $\\overline{\\mathrm {Bun}}_B$ .",
"In particular, for an element $\\overline{\\nu } \\in \\Lambda _{G,B}^{pos} \\setminus \\lbrace 0\\rbrace $ , we write $\\overline{\\nu } = \\sum _{i \\in \\mathcal {J}} n_{i}\\alpha _{i}$ as a positive linear combination of the elements corresponding to $\\Gamma $ -orbits of simple positive coroots $\\alpha _{i}$ .",
"We let $\\mathrm {Div}^{(\\overline{\\nu })}$ be the partially symmetrized power of the mirror curve attached to it, as in §REF .",
"We have a map of Artin $v$ -stacks $ j_{\\overline{\\nu }}: \\mathrm {Div}^{(\\overline{\\nu })} \\times \\mathrm {Bun}_{B} \\rightarrow \\overline{\\mathrm {Bun}}_{B} $ sending a tuple $(\\lbrace (D_{i})_{i \\in \\mathcal {J}}\\rbrace , \\mathcal {F}_{G},\\mathcal {F}_{T},\\kappa ^{\\hat{\\lambda }})$ , to the tuple $ (\\mathcal {F}_{G},\\mathcal {F}_{T}(-\\sum _{i \\in J}\\alpha _{i} \\cdot D_{i}), \\overline{\\kappa }^{\\hat{\\lambda }}) $ where $\\overline{\\kappa }^{\\hat{\\lambda }}$ is the natural composition $ (\\mathcal {L}^{\\hat{\\lambda }})_{\\mathcal {F}_{T}}(-\\sum _{i \\in J}\\langle \\alpha _{i},\\hat{\\lambda } \\rangle \\cdot D_{i}) \\rightarrow \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}} \\xrightarrow{} (\\mathcal {V}^{\\hat{\\lambda }})_{\\mathcal {F}_{G}} $ defined by the unique effective modification of $T$ -bundles of the specified meremorphy and support.",
"We now make the following definition.",
"Definition 5.18 For $\\overline{\\nu } \\in \\Lambda _{G,B} \\setminus \\lbrace 0\\rbrace $ , we define the $v$ -stack $\\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ (resp.",
"$_{\\ge \\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ ) to be the locus where, for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ , the cokernels $\\mathcal {V}^{\\hat{\\lambda }}/\\text{Im}{(\\overline{\\kappa }^{\\hat{\\lambda }})}$ have torsion of length equal to (resp.",
"greater than) $\\langle \\hat{\\lambda }, \\overline{\\nu } \\rangle $ after pulling back to any geometric point.",
"By Proposition REF (i), $_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ is a locally closed substack of $\\overline{\\mathrm {Bun}}_{B}$ , and the closure of $_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ in $\\overline{\\mathrm {Bun}}_{B}$ is contained in $_{\\ge \\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ (In fact, is equal to it, as will follow from Proposition REF ).",
"To work with these strata, we will need the following.",
"Proposition 5.19 For $\\overline{\\nu } \\in \\Lambda _{G,B}^{pos} \\setminus \\lbrace 0\\rbrace $ , the map $j_{\\overline{\\nu }}$ induces an isomorphism $\\mathrm {Bun}_{B} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\simeq \\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ In particular, $j_{\\overline{\\nu }}$ is a locally closed embedding.",
"It is clear that $j_{\\overline{\\nu }}$ induces a map into the locally closed stratum $_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ .",
"We need to exhibit an inverse of this map.",
"We first begin with the following lemma.",
"Lemma 5.20 Let $\\mathrm {Coh}$ be the $v$ -stack parameterizing, for $S \\in \\mathrm {Perf}$ , flat coherent sheaves on $X_{S}$ , as in .",
"For $k,n \\in \\mathbb {N}_{\\ge 0}$ , we set $\\mathrm {Coh}^{k}_{n}$ to be the locally closed (by Proposition REF (i)) substack parameterizing flat coherent sheaves whose torsion length (resp.",
"vector bundle rank) is equal to $k$ (resp.",
"$n$ ) after pulling back to a geometric point.",
"There is a well-defined map $ \\mathrm {Coh}_{n}^{k} \\rightarrow \\mathrm {Div}^{(k)} $ of $v$ -stacks, sending a $S$ -flat coherent sheaf $\\mathcal {F}$ with attached short exact sequence $0 \\rightarrow \\mathcal {F}^{\\mathrm {tor}} \\rightarrow \\mathcal {F} \\rightarrow \\mathcal {F}^{\\mathrm {vb}} \\rightarrow 0$ , as in Proposition REF (ii), to the support of $\\mathcal {F}^{\\mathrm {tor}}$ .",
"We note that, given $\\mathcal {F}$ a $S$ -flat coherent sheaf in $\\mathrm {Coh}_{n}^{k}$ , we have, by Proposition REF (ii), a unique short exact sequence $ 0 \\rightarrow \\mathcal {F}^{\\mathrm {tor}} \\rightarrow \\mathcal {F} \\rightarrow \\mathcal {F}^{\\mathrm {vb}} \\rightarrow 0 $ where we note that $\\mathcal {F}^{\\mathrm {tor}}$ will be a $S$ -flat coherent sheaf of generic rank 0, using Lemma REF .",
"Now we can choose a presentation of this $ 0 \\rightarrow \\mathcal {F}_{-1} \\rightarrow \\mathcal {F}_{0} \\rightarrow \\mathcal {F}^{\\mathrm {tor}} \\rightarrow 0 $ for two vector bundles $\\mathcal {F}_{-1}$ and $\\mathcal {F}_{0}$ .",
"Since $\\mathcal {F}^{\\mathrm {tor}}$ is of constant rank 0 by construction, the first map must be a fiberwise injective map of vector bundles locally of the same rank.",
"Therefore, analytically locally on $S$ , we can then take the top exterior power of the map $\\mathcal {F}_{-1} \\rightarrow \\mathcal {F}_{0}$ , and this will give rise to a fiberwise injective map of line bundles, which in turn gives rise to a relative Cartier divisor $D$ in $X_{S}$ (cf.",
"to see that this is independent of the choice of presentation).",
"If $\\mathcal {F}$ defines a point in $\\mathrm {Coh}_{n}^{k}$ then $D$ must be of degree $k$ , and we get the desired map.",
"We need to exhibit an inverse to the natural map $\\mathrm {Bun}_{B} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\rightarrow \\phantom{}_{\\overline{\\nu }}\\mathrm {Bun}_{B}$ .",
"To do this, for $S \\in \\mathrm {Perf}$ , consider a short exact sequence of $ 0 \\rightarrow \\mathcal {V}_{1} \\rightarrow \\mathcal {V}_{n + 1} \\rightarrow \\mathcal {V}_{n} \\rightarrow 0 $ of $\\mathcal {O}_{X_{S}}$ -modules, where $\\mathcal {V}_{1}$ (resp.",
"$\\mathcal {V}_{n + 1}$ ) is a line bundle (resp.",
"rank $n + 1$ vector bundle), and the first map is a fiberwise-injective map, so that $\\mathcal {V}_{n}$ is $S$ -flat.",
"Assume that $\\mathcal {V}_{n}$ defines a point in $\\mathrm {Coh}_{n}^{k}$ for some $k$ , and let $D$ be the degree $k$ Cartier divisor in $X_{S}$ defined by the previous Lemma.",
"It follows by an application of Proposition REF (ii) that we have a short exact sequence $ 0 \\rightarrow \\mathcal {V}_{n}^{\\mathrm {tors}} \\rightarrow \\mathcal {V}_{n} \\rightarrow \\mathcal {V}_{n}^{\\mathrm {vb}} \\rightarrow 0 $ where $\\mathcal {V}_{n}^{\\mathrm {tors}}$ will define a point in $\\mathrm {Coh}^{k}_{0}$ and $\\mathcal {V}_{n}^{\\mathrm {vb}}$ is a rank $n$ vector bundle.",
"Let $\\widetilde{\\mathcal {V}}_{1}$ denote the preimage of $\\mathcal {V}_{n}^{\\mathrm {tors}}$ in $\\mathcal {V}_{n + 1}$ .",
"It is then easy to see that $\\mathcal {V}_{1} \\rightarrow \\mathcal {V}_{n + 1}$ gives rise to an isomorphism $\\mathcal {V}_{1}(D) \\simeq \\widetilde{\\mathcal {V}}_{1}$ .",
"Now, given a $S$ -point of $_{\\overline{\\nu }}\\mathrm {Bun}_{B}$ , we can construct the desired inverse by applying the above argument to the short exact sequences coming from the embeddings $\\overline{\\kappa }^{\\hat{\\omega }_{i}}$ , for the fundamental weights $\\hat{\\omega }_{i} \\in \\hat{\\Lambda }_{G}^{+}$ .",
"With this locally closed stratification in hand, we can now study how Hecke correspondences base-changed to $\\overline{\\mathrm {Bun}}_{B}$ interact with it.",
"This will play a key role in the proof of the Hecke eigensheaf property, and reduce showing the density of $\\mathrm {Bun}_{B} \\subset \\overline{\\mathrm {Bun}}_{B}$ to studying the usual closure relationships for semi-infinite Schubert cells."
],
[
"The Key diagram and Density of the Compactification",
"We would now like to describe how Hecke correspondences on $\\mathrm {Bun}_{G}$ interact with pullback along the map $\\overline{\\mathfrak {p}}: \\overline{\\mathrm {Bun}}_{B} \\rightarrow \\mathrm {Bun}_{G}$ .",
"This will be used to show the filtered Hecke eigensheaf property for the geometric Eisenstein series, analogous to the analysis carried out in .",
"In this section, we will just study the geometry of the relevant diagram and use it to deduce that the open inclusion $\\mathrm {Bun}_{B} \\subset \\overline{\\mathrm {Bun}}_{B}$ defines a dense subset in the underlying topological space of $\\overline{\\mathrm {Bun}}_{B}$ .",
"We fix a finite index set $I$ , and consider the Hecke stacks $\\mathrm {Hck}^{I}_{G,E}$ base-changed to the field $E$ over which $G$ splits.",
"We consider the usual diagram $ \\mathrm {Bun}_{G} \\times \\mathrm {Div}^{I}_{E} \\xleftarrow{} \\mathrm {Hck}_{G,E}^{I} \\xrightarrow{} \\mathrm {Bun}_{G} $ as in § and §.",
"We now fix a tuple of geometric dominant characters $(\\lambda _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+})^{I}$ , and restrict to the locus $\\mathrm {Hck}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ where the meromorphy of this modification is bounded by $(\\lambda _{i})_{i \\in I}$ .",
"We define the $v$ -stack $\\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}}$ by the Cartesian diagram: $\\begin{tikzcd}& \\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}} [r,\"^{\\prime }h_{G}^{\\rightarrow }\"] [d,\"^{\\prime }\\overline{\\mathfrak {p}}\"] & \\overline{\\mathrm {Bun}}_{B} [d,\"\\overline{\\mathfrak {p}}\"] \\\\& \\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E} [r,\"h_{G}^{\\rightarrow }\"] & \\mathrm {Bun}_{G}\\end{tikzcd}$ By definition, $\\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}}$ parametrizes pairs of $G$ -bundles $(\\mathcal {F}_{G},\\mathcal {F}_{G}^{\\prime })$ together with a modification $\\mathcal {F}_{G} \\dashrightarrow \\mathcal {F}_{G}^{\\prime }$ with meromorphy bounded by $\\lambda _{i}$ at Cartier divisors $D_{i}$ for $i \\in I$ and an enhanced $B$ -structure on $\\mathcal {F}_{G}^{\\prime }$ specified by maps $\\overline{\\kappa }^{\\prime \\hat{\\lambda }}$ for $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ .",
"The fact that the modification $\\mathcal {F}_{G} \\dashrightarrow \\mathcal {F}_{G}^{\\prime }$ has meromorphy bounded by $(\\lambda _{i})_{i \\in I}$ implies that, for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ , we have an inclusion: $ \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}^{\\prime }} \\subset \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}}(\\sum _{i \\in I} \\langle \\hat{\\lambda }, -w_{0}(\\lambda _{i\\Gamma }) \\rangle \\cdot D_{i}) $ Therefore, the embeddings $ \\overline{\\kappa }^{\\prime \\hat{\\lambda }}: \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{\\prime }} \\hookrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}^{\\prime }} $ give rise to a map: $ \\overline{\\kappa }^{\\hat{\\lambda }}: \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{\\prime }}(\\sum _{i \\in I} \\langle \\hat{\\lambda },w_{0}(\\lambda _{i\\Gamma }) \\rangle \\cdot D_{i}) \\hookrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}} $ This defines for us a morphism $ \\phi _{(\\lambda _{i})_{i \\in I}}: \\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}} \\rightarrow \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} $ which records the point in $\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ defined by the pair $(\\overline{\\kappa }^{\\hat{\\lambda }},(D_{i})_{i \\in I})$ .",
"This sits in a commutative diagram $\\begin{tikzcd}& \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} [d,\"\\overline{\\mathfrak {p}} \\times \\mathrm {id}\"] & [l,\"\\phi _{(\\lambda _{i})_{i \\in I}}\"] \\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}} [r,\"^{\\prime }h_{G}^{\\rightarrow }\"] [d,\"\\phantom{}^{^{\\prime }}\\mathfrak {p}\"] & \\overline{\\mathrm {Bun}}_{B} [d,\"\\overline{\\mathfrak {p}}\"] \\\\& \\mathrm {Bun}_{G} \\times \\mathrm {Div}^{I}_{E} & [l,\"h_{G}^{\\leftarrow } \\times \\pi \"] \\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E} [r,\"h_{G}^{\\rightarrow }\"] & \\mathrm {Bun}_{G}\\end{tikzcd}$ where we note that left square is not Cartesian.",
"Similarly, we will write $Z^{I}_{(\\lambda _{i})_{i \\in I}}$ for the space obtained by replacing $\\overline{\\mathrm {Bun}}_{B}$ with $\\mathrm {Bun}_{B}$ on the right hand side of the diagram.",
"This sits in an analogous diagram ${}\\begin{tikzcd}& \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} [d,\"\\overline{\\mathfrak {p}} \\times \\mathrm {id}\"] & [l,\"\\phi _{(\\lambda _{i})_{i \\in I}}\"] Z^{I}_{(\\lambda _{i})_{i \\in I}} [r,\"^{\\prime }h_{G}^{\\rightarrow }\"] [d,\"\\phantom{}^{^{\\prime }}\\mathfrak {p}\"] & \\mathrm {Bun}_{B} [d,\"\\mathfrak {p}\"] \\\\& \\mathrm {Bun}_{G} \\times \\mathrm {Div}^{I}_{E} & [l,\"h_{G}^{\\leftarrow } \\times \\pi \"] \\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E} [r,\"h_{G}^{\\rightarrow }\"] & \\mathrm {Bun}_{G}\\end{tikzcd}$ As we will see, the proof of the filtered Hecke eigensheaf Property will ultimately reduce to contemplating the fibers of the morphism $\\phi _{(\\lambda _{i})_{i \\in I}}$ .",
"For our purposes, it will suffice to consider the pullback of this diagram to a geometric point $\\mathop {\\rm Spa}(F,\\mathcal {O}_{F}) \\rightarrow \\mathrm {Div}^{I}_{E}$ .",
"We denote the resulting space by $_{x}\\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}}$ .",
"It sits in a diagram of the form $\\begin{tikzcd}& \\phantom{}_{x}\\overline{\\mathrm {Bun}}_{B} [d,\"\\overline{\\mathfrak {p}}\"] & [l,\"\\phi _{(\\lambda _{i})_{i \\in I}}\"] _{x}\\overline{Z}^{I}_{(\\lambda _{i})_{i \\in I}} [r,\"^{\\prime }h_{G}^{\\rightarrow }\"] [d,\"^{\\prime }\\overline{\\mathfrak {p}}\"] & \\overline{\\mathrm {Bun}}_{B} [d,\"\\overline{\\mathfrak {p}}\"] \\\\& \\phantom{}_{x}\\mathrm {Bun}_{G} & [l,\"h_{G}^{\\leftarrow } \\times \\pi \"] _{x}\\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E} [r,\"h_{G}^{\\rightarrow }\"] & \\mathrm {Bun}_{G}\\end{tikzcd}$ where $_{x}\\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ is the Hecke stack parameterizing modifications at the tuple of Cartier divisors $(D_{i})_{i \\in I}$ corresponding to $x$ and $\\phantom{}_{x}\\overline{\\mathrm {Bun}}_{B}$ (resp.",
"$\\phantom{}_{x}\\mathrm {Bun}_{G}$ ) denotes the base change of $\\overline{\\mathrm {Bun}}_{B}$ (resp.",
"$\\mathrm {Bun}_{G}$ ) to $\\mathop {\\rm Spa}(F,\\mathcal {O}_{F})$ .",
"Consider a tuple $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ and write $\\overline{\\nu } := \\sum _{i \\in I} \\nu _{i\\Gamma }$ .",
"Let $E_{\\nu _{i}}$ denote the reflex field of $\\nu _{i}$ .",
"We view the geometric point $x \\rightarrow \\mathrm {Div}^{I}$ as a geometric point of $\\mathrm {Div}^{(\\overline{\\nu })}$ via composing with the map $ \\Delta _{(\\nu _{i})_{i \\in I}}: \\mathrm {Div}^{I}_{E} \\rightarrow \\prod _{i \\in I} \\mathrm {Div}^{1}_{E_{\\nu _{i}}} \\xrightarrow{} \\prod _{i \\in I} \\mathrm {Div}^{(\\nu _{i\\Gamma })} \\rightarrow \\mathrm {Div}^{(\\overline{\\nu })} $ where the last map is given by taking the union of Cartier divisors and $\\Delta _{\\nu _{i}}$ is the twisted diagonal embedding described in §REF .",
"We set $_{x,(\\nu _{i})_{i \\in I}}\\overline{\\mathrm {Bun}}_{B}$ to be the pullback of the locally closed stratum $\\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B} \\simeq \\mathrm {Div}^{(\\overline{\\nu })} \\times \\mathrm {Bun}_{B}$ to this geometric point.",
"The substack $_{x,(\\nu _{i})_{i \\in I}}\\overline{\\mathrm {Bun}}_{B}$ corresponds to the locus where the embeddings $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}} \\rightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}} $ have a zero of order given by $\\nu _{i}$ at $D_{i}$ for all dominant characters $\\hat{\\lambda }$ of $G$ and all $i \\in I$ , and a zero nowhere else.",
"We now consider the open substack $_{x,0}\\overline{\\mathrm {Bun}}_{B} = \\phantom{}_{x}\\mathrm {Bun}_{B}$ .",
"Since the maps $\\kappa $ have no zero at the Cartier divisors corresponding to $x$ , it follows that they define an $L^{+}B$ torsor over $_{x,0}\\overline{\\mathrm {Bun}}_{B}$ , which we denote by $_{x}\\mathcal {B}$ .",
"We then consider the map $ i_{(\\nu _{i})_{i \\in I}}: \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} \\rightarrow \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} $ given by sending $(\\mathcal {F}_{G},\\mathcal {F}_{T},\\overline{\\kappa }^{\\hat{\\lambda }},(D_{i})_{i \\in I})$ to the object $(\\mathcal {F}_{G},\\mathcal {F}_{T}(-\\sum _{i \\in I} \\overline{\\nu }_{i} \\cdot D_{i}),\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}}(- \\sum _{i \\in I} \\langle \\overline{\\nu }_{i}, \\hat{\\lambda } \\rangle \\cdot D_{i}) \\hookrightarrow \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}} \\hookrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}},(D_{i})_{i \\in I})$ , and note that the map $i_{(\\nu _{i})_{i \\in I}}$ defines an isomorphism between the pullbacks $_{x,0}\\overline{\\mathrm {Bun}}_{B}$ and $_{x,(\\nu _{i})_{i \\in I}}\\overline{\\mathrm {Bun}}_{B}$ .",
"Therefore, by transport of structure, we get a $L^{+}B$ -torsor, denoted $_{x}\\mathcal {B}^{(\\nu _{i})_{i \\in I}}$ , over $_{x,(\\nu _{i})_{i \\in I}}\\overline{\\mathrm {Bun}}_{B}$ .",
"For $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , we let $\\phantom{}_{x}Z^{I,\\ast ,(\\nu _{i})_{i \\in I}}_{(\\lambda _{i})_{i \\in I}}$ (resp.",
"$\\phantom{}_{x}Z^{I,(\\nu _{i})_{i \\in I},\\ast }_{(\\lambda _{i})_{i \\in I}}$ ) be the fibers of $^{\\prime }h_{G}^{\\rightarrow }$ (resp.",
"$\\phi _{(\\lambda _{i})_{i \\in I})}$ ) over $_{x,(\\nu _{i})_{i \\in I}}\\overline{\\mathrm {Bun}}_{B}$ .",
"We now have the following Lemma describing these subspaces, which is an analogue of .",
"Lemma 5.21 For tuples $(\\nu _{i})_{i \\in I},(\\nu ^{\\prime }_{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , geometric dominant cocharacters $(\\lambda _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+})^{I}$ , and a geometric point $x \\rightarrow \\mathrm {Div}^{I}_{E}$ , the following is true.",
"There is an isomorphism $ _{x}Z^{I,\\ast ,(\\nu ^{\\prime }_{i})_{i \\in I}}_{(\\lambda _{i})_{i \\in I}} \\simeq \\phantom{}_{x}\\mathrm {Gr}^{I}_{G,\\le (-w_{0}(\\lambda _{i}))_{i \\in I}} \\times ^{L^{+}B} \\phantom{}_{x}\\mathcal {B}^{(\\nu ^{\\prime }_{i})_{i \\in I}} $ where the $L^{+}B$ action on $_{x}\\mathrm {Gr}^{I}_{G,\\le (-w_{0}(\\lambda _{i}))_{i \\in I}}$ is given by the inclusion $L^{+}B \\hookrightarrow L^{+}G$ .",
"Under the identification in (1), the substack $\\phantom{}_{x}Z^{I,(\\nu _{i})_{i \\in I},(\\nu ^{\\prime }_{i})_{i \\in I}}_{(\\lambda _{i})_{i \\in I}} \\hookrightarrow \\phantom{}_{x}Z^{I,\\ast ,(\\nu ^{\\prime }_{i})_{i \\in I}}_{(\\lambda _{i})_{i \\in I}}$ identifies with the substack $ \\phantom{}_{x}\\mathrm {Gr}^{I}_{G,\\le (-w_{0}(\\lambda _{i}))_{i \\in I},E} \\cap \\phantom{}_{x}\\mathrm {S}^{I}_{G,(-w_{0}(\\lambda _{i}) - \\nu _{i} + \\nu ^{\\prime }_{i})_{i \\in I},E} \\times ^{L^{+}B} \\phantom{}_{x}\\mathcal {B}^{(\\nu ^{\\prime }_{i})_{i \\in I}} \\subset \\phantom{}_{x}\\mathrm {Gr}^{I}_{G,\\le (-w_{0}(\\lambda _{i}))_{i \\in I},E} \\times ^{L^{+}B} \\phantom{}_{x}\\mathcal {B}^{(\\nu ^{\\prime }_{i})_{i \\in I}} $ When viewed as a stack projecting to $_{x,(\\nu _{i})_{i \\in I}}\\overline{\\mathrm {Bun}}_{B}$ , the stack $\\phantom{}_{x}Z_{(\\lambda _{i})_{i \\in I}}^{I,(\\nu _{i})_{i \\in I},(\\nu ^{\\prime }_{i})_{i \\in I}}$ identifies with $ \\phantom{}_{x}\\mathrm {Gr}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E} \\cap \\phantom{}_{x}\\mathrm {S}^{I}_{G,(\\nu _{i} - \\nu ^{\\prime }_{i} + w_{0}(\\lambda _{i}))_{i \\in I},E} \\times ^{L^{+}B} \\phantom{}_{x}\\mathcal {B}^{(\\nu _{i})_{i \\in I}} $ Follows from the definitions and the description of the semi-infinite cells mentioned in the remark proceeding Definition REF .",
"As mentioned earlier, this description of the fibers will serve a key role in the proof of the Hecke eigensheaf property.",
"For now, we content ourselves by using it to prove density.",
"Proposition 5.22 $\\mathrm {Bun}_{B}$ defines a substack of $\\overline{\\mathrm {Bun}}_{B}$ which is topologically dense.",
"Consider a geometric point of $s \\rightarrow \\overline{\\mathrm {Bun}}_{B}$ defined by a triple $(\\mathcal {F}_{G},\\mathcal {F}_{T},\\overline{\\kappa })$ , and an open substack of $U \\subset \\overline{\\mathrm {Bun}}_{B}$ containing $s$ .",
"It suffices to show that $U$ contains a point in $\\mathrm {Bun}_{B}$ .",
"We assume that $\\overline{\\kappa }$ has a singularity $\\nu _{i}$ defined at distinct Cartier divisors $D_{i}$ for $i \\in I$ corresponding to geometric points $x_{i} \\rightarrow \\mathrm {Div}^{1}_{E}$ .",
"We write $x \\rightarrow \\mathrm {Div}^{I}_{E}$ for the associated geometric point given by the product of the $x_{i}$ , and apply the previous Lemma in the situation that $\\nu ^{\\prime }_{i} = 0$ for all $i \\in I$ .",
"By Lemma REF (1), we have that $\\phantom{}_{x}\\mathrm {Gr}^{I}_{G,\\le (-w_{0}(\\lambda _{i}))_{i \\in I},E} \\times ^{L^{+}B} \\phantom{}_{x}\\mathcal {B} \\simeq \\prod _{i \\in I} \\phantom{}_{x_{i}}\\mathrm {Gr}_{G,\\le -w_{0}(\\lambda _{i}),E} \\times ^{L^{+}B} \\phantom{}_{x_{i}}\\mathcal {B}$ maps to $\\overline{\\mathrm {Bun}}_{B}$ via the morphism $\\phi _{(\\lambda _{i})_{i \\in I}}$ .",
"Now, for the given $(\\nu _{i})_{i \\in I}$ , we can choose $\\lambda _{i}$ for all $i \\in I$ such that the weight space $V_{\\lambda _{i}}(w_{0}(\\lambda _{i}) + \\nu _{i})$ is non-zero.",
"Then, by Lemma REF (2) and Corollary REF , the fiber of $\\phi _{(\\lambda _{i})_{i \\in I}}$ over $s$ is non-empty.",
"Therefore, pulling back $U$ , we get a non-empty open subset of $\\prod _{i \\in I} \\phantom{}_{x_{i}}\\mathrm {Gr}_{G,\\le -w_{0}(\\lambda _{i}),E} \\times ^{L^{+}B} \\phantom{}_{x_{i}}\\mathcal {B}$ .",
"By the closure relations of Proposition REF , we get that this open subset must have non-empty intersection with the open subspace $\\prod _{i \\in I} \\phantom{}_{x_{i}}\\mathrm {Gr}_{G,\\le -w_{0}(\\lambda _{i}),E} \\cap \\phantom{}_{x_{i}}\\mathrm {S}_{G,-w_{0}(\\lambda _{i}),E} \\times ^{L^{+}B} \\phantom{}_{x_{i}}\\mathcal {B}$ , but, by another application of Lemma REF (2), this tells us that $U$ must have non-trivial intersection with $\\mathrm {Bun}_{B}$ ."
],
[
"The Normalized Eisenstein Functor and Verdier Duality on $\\mathrm {Bun}_{B}$",
"Now that we have finished our geometric preparations, we can start to understand the sheaf theory on the moduli stack of $B$ -structures.",
"Our first order of business is to refine our definition of the Eisenstein functor given in the previous section to better respect Verdier duality on the moduli stack $\\mathrm {Bun}_{B}$ ."
],
[
"The Normalized Eisenstein Functor",
"Before proceeding with our analysis of Eisenstein series, we refine the definition of the Eisenstein functor.",
"There is one key problem with our definition, the sheaf $\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]$ is not Verdier self-dual.",
"In particular, the dualizing object $K_{\\mathrm {Bun}_{B}}$ on $\\mathrm {Bun}_{B}$ is not isomorphic to $\\Lambda [2\\mathrm {dim}(\\mathrm {Bun}_{B})]$ ; this is only $v$ -locally true on $\\mathrm {Bun}_{B}$ .",
"To elucidate the problem, note that, given $\\overline{\\nu } \\in \\Lambda _{G,B}$ , we can consider the natural map $\\mathfrak {q}^{\\overline{\\nu }}: \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{T}^{\\overline{\\nu }} = [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ .",
"Given a character $\\kappa _{\\overline{\\nu }}$ of $T(\\mathbb {Q}_{p})$ , we can pull this character back along $\\mathfrak {q}^{\\overline{\\nu }}$ to get a sheaf on $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ .",
"These characters give us sheaves on $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ , which $v$ -locally will be constant, but are not constant on the nose.",
"Our main Theorem is as follows.",
"Theorem 6.1 The dualizing object on $\\mathrm {Bun}_{B}$ is isomorphic to $\\mathfrak {q}^{*}(\\Delta _{B})[2\\mathrm {dim}(\\mathrm {Bun}_{B})]$ , with $\\Delta _{B} \\in \\mathrm {D}(\\mathrm {Bun}_{T})$ as in Definition REF .",
"Before tackling the proof, we record the key consequence of this theorem for us.",
"Corollary 6.2 The sheaf $\\mathfrak {q}^{*}(\\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})]$ on $\\mathrm {Bun}_{B}$ is Verdier self-dual.",
"This motivates the definition of the normalized Eisenstein functor.",
"Definition 6.3 We let $\\mathrm {IC}_{\\mathrm {Bun}_{B}} := \\mathfrak {q}^{*}(\\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})]$ .",
"We define the normalized Eisenstein functor: $ \\mathrm {nEis}: \\mathrm {D}(\\mathrm {Bun}_{T}) \\rightarrow \\mathrm {D}(\\mathrm {Bun}_{G}) $ $ \\mathcal {F} \\mapsto \\mathfrak {p}_{!",
"}(\\mathfrak {q}^{*}(\\mathcal {F}) \\otimes \\mathrm {IC}_{\\mathrm {Bun}_{B}})$ In particular, we note that we have a natural isomorphism: $ \\mathrm {nEis}(-) \\simeq \\mathrm {Eis}(- \\otimes \\Delta _{B}^{1/2})$ We now tackle the proof of Theorem REF ."
],
[
"The Proof of Theorem ",
"Definition 6.4 Let $b \\in B(G)_{\\mathrm {un}}$ be an unramified element.",
"We recall, by Corollary REF , this admits a unique HN-dominant reduction $b_{T}$ with anti-dominant slope homomorphism, and we can describe the set of all elements mapping to $b \\in B(G)$ as $w(b_{T})$ , for varying $w \\in W_{b}$ .",
"Given $\\overline{\\nu } \\in B(T) \\simeq \\Lambda _{G,B}$ mapping to $b$ , we write $w_{\\overline{\\nu }} \\in W_{b}$ for the unique element satisfying the relation $\\overline{\\nu } = w_{\\overline{\\nu }}(b_{T})$ .",
"As a warm up, we explain the proof in the case that $G = \\mathrm {GL}_{2}$ and $B$ is the upper triangular Borel, proving some key lemmas along the way.",
"We recall that in this case $B(T) \\simeq \\Lambda _{G,B} \\simeq \\mathbb {Z}^{2}$ via the Kottwitz invariant, and so we can index the connected components of $\\mathrm {Bun}_{B}$ by a pair of integers.",
"We write $d_{\\overline{\\nu }}$ for the $\\ell $ -cohomological dimension of the connected component $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ .",
"Example 6.5 We first consider the case that $\\overline{\\nu } = (d,d)$ for $d \\in \\mathbb {Z}$ , so $\\overline{\\nu }$ is already HN-dominant and $w_{\\overline{\\nu }} = 1$ .",
"Note that the connected component $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ parametrizes split reductions, and is isomorphic to $[\\ast /\\underline{B(\\mathbb {Q}_{p})}]$ .",
"Now we have the following lemma, which is .",
"Lemma 6.6 Let $H$ be a locally pro-$p$ group then $K_{[\\ast /H]} \\simeq Haar(H,\\Lambda )$ , where $Haar(H,\\Lambda )$ is the space of $\\Lambda $ -valued Haar measures on $H$ .",
"Therefore, the dualizing object on $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ is canonically isomorphic to $Haar(B(\\mathbb {Q}_{p}),\\Lambda )$ , the space of Haar measures on $B(\\mathbb {Q}_{p})$ .",
"We note that, as a $B(\\mathbb {Q}_{p})$ -representation, this is isomorphic to $\\delta _{B}$ by definition of the modulus character.",
"Thus, Theorem REF is true in this case.",
"Given a diamond or $v$ -stack $X \\rightarrow \\mathop {\\rm Spd}{\\mathbb {F}_{p}}$ , we define the local systems $\\Lambda (d)$ by pulling back the local system on $\\mathop {\\rm Spd}{\\mathbb {F}_{p}}$ given by the representation of $\\mathrm {Gal}(\\overline{\\mathbb {F}}_{p}/\\mathbb {F}_{p})$ for which the geometric Frobenius acts via scaling by $p^{-d} \\in \\Lambda ^{*}$ .",
"We consider the following key example.",
"Example 6.7 We let $Z_{\\mathbb {F}_{p}} := \\mathcal {H}^{0}(\\mathcal {O}(1))_{\\mathbb {F}_{p}} \\rightarrow \\mathop {\\rm Spd}{\\mathbb {F}_{p}}$ be the absolute positive Banach-Colmez space parmeterizing sections of the line bundle $\\mathcal {O}(1)$ on $X_{S}$ for $S \\in \\mathrm {Perf}_{\\mathbb {F}_{p}}$ , as in .",
"This is isomorphic to $\\mathop {\\rm Spd}{\\mathbb {F}_{p}[|x^{1/p^{\\infty }}|]}$ .",
"We let $\\mathrm {Frob}_{Z_{\\mathbb {F}_{p}}}$ denote the geometric Frobenius sending $x \\mapsto x^{p}$ .",
"Then, by , we have a natural isomorphism $ K_{Z_{\\mathbb {F}_{p}}} \\simeq \\Lambda [2](1) $ of étale sheaves on $Z_{\\mathbb {F}_{p}}$ .",
"Now, $K_{Z_{\\mathbb {F}_{p}}}$ has an action of $\\mathbb {Q}_{p}^{*}$ coming from the action of $\\mathbb {Q}_{p}^{*}$ on $Z_{\\mathbb {F}_{p}} = \\mathcal {H}^{0}(\\mathcal {O}(1))_{\\mathbb {F}_{p}}$ by scaling of global sections.",
"This defines for us a character $\\kappa : \\mathbb {Q}_{p}^{*} \\rightarrow \\Lambda ^{*}$ .",
"We claim that $\\kappa (\\mathbb {Z}_{p}^{*})$ is trivial.",
"To see this, consider the $v$ -stack quotient by the $\\mathbb {Z}_{p}^{*}$ action: $ q: Z_{\\mathbb {F}_{p}} \\rightarrow [Z_{\\mathbb {F}_{p}}/\\underline{\\mathbb {Z}_{p}^{*}}] $ Now, by Strictly speaking, this Proposition only works for the maximal pro-$p$ subgroup $\\mathbb {Z}_{p} \\subset \\mathbb {Z}_{p} \\times \\mathbb {Z}/(p - 1)\\mathbb {Z} \\simeq \\mathbb {Z}_{p}^{*}$ , but it is easy to see the proof extends to further quotienting out by a finite group., fixing a Haar measure on $\\mathbb {Z}_{p}^{*}$ determines a unique isomorphism: $ (q_{*}(K_{Z_{\\mathbb {F}_{p}}}))^{\\mathbb {Z}_{p}^{*}} \\simeq K_{[K_{Z_{\\mathbb {F}_{p}}}/\\underline{\\mathbb {Z}_{p}^{*}}]} $ The only way $\\mathbb {Z}_{p}^{*}$ could act non-trivially on $\\Lambda (1)$ is if it acted on $Z_{\\mathbb {F}_{p}}$ by a power of geometric Frobenius.",
"It follows that the action is trivial, since there are no non-trivial homomorphisms from $\\mathbb {Z}_{p}^{*}$ to $\\mathbb {Z}$ .",
"Therefore, we conclude that $ K_{[K_{Z_{\\mathbb {F}_{p}}}/\\underline{\\mathbb {Z}_{p}^{*}}]} \\simeq \\Lambda [2](1) $ as étale sheaves on $[Z_{\\mathbb {F}_{p}}/\\underline{\\mathbb {Z}_{p}^{*}}]$ , which in particular implies that the action of $\\mathbb {Z}_{p}^{*}$ is trivial on $q^{*}(K_{[Z_{\\mathbb {F}_{p}}/\\underline{\\mathbb {Z}_{p}^{*}}]}) \\simeq K_{Z_{\\mathbb {F}_{p}}}$ , where this isomorphism follows from .",
"Now, it remains to determine the value of $\\kappa (p)$ .",
"To elucidate this, we note that, under the identification $Z_{\\mathbb {F}_{p}} \\simeq \\mathop {\\rm Spd}{\\mathbb {F}_{p}[|x^{1/p^{\\infty }}|]}$ , the element $p \\in \\mathbb {Q}_{p}^{*}$ acts via the geometric Frobenius $\\mathrm {Frob}_{Z_{\\mathbb {F}_{p}}}$ on $Z_{\\mathbb {F}_{p}}$ .",
"It follows by the previous isomorphism that the value of the character $\\kappa (p)$ is determined by the action of $\\mathrm {Frob}_{Z_{\\mathbb {F}_{p}}}$ on $\\Lambda (1)$ , which is just the multiplication by $p^{-1}$ map on $\\Lambda $ by Definition.",
"In summary, we have concluded an identification $K_{Z_{\\mathbb {F}_{p}}} \\simeq |\\cdot |[2]$ of sheaves with $\\mathbb {Q}_{p}^{*}$ -action, where $|\\cdot |$ is the rank 1 local system on $Z_{\\mathbb {F}_{p}}$ with $\\mathbb {Q}_{p}^{*}$ -action given by the norm character.",
"From here, we conclude the analogous isomorphism $ K_{Z_{\\overline{\\mathbb {F}}_{p}}} \\simeq |\\cdot |[2] $ of sheaves with $\\mathbb {Q}_{p}^{*}$ -action over the algebraic closure.",
"Let's now push this a bit further and consider the case of a general positive absolute Banach-Colmez space $Z_{\\mathbb {F}_{p}} := \\mathcal {H}^{0}(\\mathcal {O}(d))_{\\mathbb {F}_{p}} \\rightarrow \\mathop {\\rm Spd}{\\mathbb {F}_{p}}$ for $d \\ge 1$ .",
"Now we don't have such a simple presentation as in the case that $d = 1$ ; however, we claim that we still have a similar relationship between the geometric Frobenius and the scaling action by $p^{\\mathbb {Z}} \\in \\mathbb {Q}_{p}^{*}$ on $Z_{\\mathbb {F}_{p}}$ .",
"To understand this, we invoke the following explicit description.",
"Recall that, for $S \\in \\mathrm {Perf}$ , we can view the adic Fargues-Fontaine curve $\\mathcal {X}_{S}$ as given by gluing the open Fargues-Fontaine curve $\\mathcal {Y}_{S,[1,p]}$ along the map $\\phi : \\mathcal {Y}_{S,[1,1]} \\simeq \\mathcal {Y}_{S,[p,p]}$ induced by the geometric Frobenius, with notation as in .",
"We write $B_{S,[1,p]}$ (resp.",
"$B_{S,[1,1]}$ ) for the ring of functions of $\\mathcal {Y}_{S,[1,p]}$ (resp.",
"$\\mathcal {Y}_{S,[1,1]}$ ).",
"We make use of the following lemma, which follows from the analysis in .",
"Lemma 6.8 For $d \\in \\mathbb {Z}$ , let $\\mathcal {O}(d)_{X_{S}}$ be the natural line bundle of degree $d$ on $X_{S}$ .",
"We have an isomorphism: $ R\\Gamma (X_{S},\\mathcal {O}(d)) \\simeq \\lbrace B_{S,[1,p]} \\xrightarrow{} B_{S,[1,1]} \\rbrace $ Therefore, if we write $\\mathbf {B}_{[1,p]}$ for the sheaf of rings on $\\mathrm {Perf}_{\\mathbb {F}_{p}}$ defined by sending $S \\in \\mathrm {Perf}_{\\mathbb {F}_{p}}$ to the global sections of $Y_{S,[1,p]}$ then we have an isomorphism $ \\mathcal {H}^{0}(\\mathcal {O}(d)) \\simeq \\mathbf {B}_{[1,p]}^{\\phi = p^{d}} $ of diamonds over $\\mathop {\\rm Spd}(\\mathbb {F}_{p})$ .",
"As in the above example, this identification tells us that action of the geometric Frobenius $\\mathrm {Frob}_{Z_{\\mathbb {F}_{p}}}$ on $Z_{\\mathbb {F}_{p}}$ is the same as the scaling action by $p^{d} \\in \\mathbb {Q}_{p}^{*}$ on $Z_{\\mathbb {F}_{p}}$ .",
"Again, by , we have an isomorphism $ K_{Z_{\\mathbb {F}_{p}}} \\simeq \\Lambda [2d](d) $ and, it follows that $p^{d} \\in \\mathbb {Q}_{p}^{*}$ acts on the sheaf $\\Lambda $ by $p^{-d} \\in \\Lambda ^{*}$ , which, arguing as above, allows us to conclude an isomorphism $ K_{Z_{\\mathbb {F}_{p}}} \\simeq |\\cdot |[2d] $ of sheaves with $\\mathbb {Q}_{p}^{*}$ -action.",
"We record the content of the above example as a Lemma for future use.",
"Lemma 6.9 Let $d \\in \\mathbb {N}_{\\ge 1}$ be a positive integer and consider the absolute positive Banach-Colmez space $\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\mathbb {F}_{p}} \\rightarrow \\mathop {\\rm Spd}{\\mathbb {F}_{p}}$ .",
"Then we have an isomorphism $K_{\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\mathbb {F}_{p}}} \\simeq |\\cdot |[2d]$ (resp.",
"$K_{\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\overline{\\mathbb {F}}_{p}}} \\simeq |\\cdot |[2d]$ ) as sheaves with $\\mathbb {Q}_{p}^{*}$ -action, where $|\\cdot |$ denotes the rank 1 sheaf on $\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\mathbb {F}_{p}}$ (resp.",
"$\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\overline{\\mathbb {F}}_{p}}$ ) that transforms under the scaling action by $\\mathbb {Q}_{p}^{*}$ via the norm character.",
"Remark 6.10 In what follows, it will be important to formalize this in terms of $v$ -stacks.",
"If we consider the $v$ -stack quotient $ [\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\overline{\\mathbb {F}}_{p}}/\\underline{\\mathbb {Q}}_{p}^{*}] \\rightarrow \\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}} $ then this admits a natural map $q: [\\mathcal {H}^{0}(\\mathcal {O}(d))/\\underline{\\mathbb {Q}}_{p}^{*}] \\rightarrow [\\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}}/\\underline{\\mathbb {Q}}_{p}^{*}]$ to a classifying stack.",
"Then the above isomorphism descends to an identification $K_{[\\mathcal {H}^{0}(\\mathcal {O}(d))_{\\overline{\\mathbb {F}}_{p}}/\\underline{\\mathbb {Q}}_{p}^{*}]} \\simeq q^{*}(|\\cdot |)[2d]$ , where we recall that there is an identification $\\mathrm {D}([\\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}}/\\underline{\\mathbb {Q}}_{p}^{*}]) \\simeq \\mathrm {D}(\\mathbb {Q}_{p}^{*},\\Lambda )$ of the derived category of sheaves on the classifying stack $[\\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}}/\\underline{\\mathbb {Q}}_{p}^{*}]$ with the derived category of smooth $\\mathbb {Q}_{p}^{*}$ -representations on $\\Lambda $ -modules.",
"Let's now push this a bit further and prove an analogous claim for negative Banach-Colmez spaces.",
"Lemma 6.11 Let $d \\in \\mathbb {N}_{\\ge 1}$ and consider the negative absolute Banach-Colmez space $\\mathcal {H}^{1}(\\mathcal {O}(-d)) \\rightarrow \\mathop {\\rm Spd}{\\mathbb {F}_{p}}$ .",
"Then we have isomorphisms $K_{\\mathcal {H}^{1}(\\mathcal {O}(d))_{\\mathbb {F}_{p}}} \\simeq |\\cdot |^{-1}[2d]$ (resp.",
"$K_{\\mathcal {H}^{1}(\\mathcal {O}(d))_{\\overline{\\mathbb {F}}_{p}}} \\simeq |\\cdot |^{-1}[2d]$ ) as sheaves with $\\mathbb {Q}_{p}^{*}$ -action.",
"Using the explicit description above, we know that, if we consider the sheaf of rings $\\mathbf {B}_{[1,p]}$ and $\\mathbf {B}_{[1,1]}$ on $\\mathrm {Perf}_{\\mathbb {F}_{p}}$ given by taking global sections of $Y_{S,[1,p]}$ (resp.",
"$Y_{S,[1,1]}$ ), there is an isomorphism $ \\mathcal {H}^{1}(\\mathcal {O}(-d))_{\\mathbb {F}_{p}} \\simeq \\text{Coker}{(\\mathbf {B}_{[1,p]} \\xrightarrow{} \\mathbf {B}_{[1,1]})} $ of $v$ -sheaves on $\\mathrm {Perf}_{\\mathbb {F}_{p}}$ .",
"As a consequence of this, we can deduce that the geometric Frobenius $\\mathrm {Frob}_{Z_{\\mathbb {F}_{p}}}$ on $Z_{\\mathbb {F}_{p}} = \\mathcal {H}^{0}(\\mathcal {O}(-d))_{\\mathbb {F}_{p}}$ agrees with the scaling action by $p^{-d} \\in \\mathbb {Q}_{p}^{*}$ .",
"Moreover, as in the positive case, we have an isomorphism $ K_{\\mathcal {H}^{1}(\\mathcal {O}(d))_{\\mathbb {F}_{p}}} \\simeq \\Lambda [2d](d) $ of étale sheaves.",
"One can deduce this from the positive case and the proof of .",
"Now, arguing as in the case of positive Banach-Colmez spaces, we deduce that $ K_{\\mathcal {H}^{1}(\\mathcal {O}(d))_{\\mathbb {F}_{p}}} \\simeq |\\cdot |^{-1}[2d] $ as sheaves with $\\mathbb {Q}_{p}^{*}$ -action.",
"The claim over $\\overline{\\mathbb {F}}_{p}$ follows.",
"With these two lemmas in hand, let's continue our calculation of the dualizing object on $\\mathrm {Bun}_{B}$ in the case that $G = \\mathrm {GL}_{2}$ .",
"We will return to working over the base $\\ast = \\mathop {\\rm Spd}{\\overline{\\mathbb {F}}_{p}}$ in all that follows.",
"Example 6.12 Consider the case where $\\overline{\\nu } = (-d,-e)$ is HN-dominant, so that $d > e$ .",
"In this case, $d_{\\overline{\\nu }} = e - d$ and $w_{\\overline{\\nu }} = 1$ .",
"We note that $d_{\\overline{\\nu }}$ is negative, so to keep track of the sign change, we consider the absolute value $|d_{\\overline{\\nu }}|$ .",
"The connected component $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ just parametrizes the split extensions of $\\mathcal {O}(d)$ by $\\mathcal {O}(e)$ .",
"Therefore, its topological space is just a point.",
"More precisely, if $b$ corresponds to the bundle $\\mathcal {O}(d) \\oplus \\mathcal {O}(e)$ then it is isomorphic $[\\ast /\\mathcal {J}_{b}]$ , where $\\mathcal {J}_{b}$ parametrizes automorphisms of $\\mathcal {O}(d) \\oplus \\mathcal {O}(e)$ .",
"The group diamond $\\mathcal {J}_{b}$ is isomorphic to $ \\begin{pmatrix} \\mathrm {Aut}(\\mathcal {O}(d)) & \\mathcal {H}om(\\mathcal {O}(e),\\mathcal {O}(d)) & \\\\ 0 & \\mathrm {Aut}(\\mathcal {O}(e)) & \\end{pmatrix} \\simeq \\begin{pmatrix} \\underline{\\mathbb {Q}}_{p}^{*} & \\mathcal {H}^{0}(\\mathcal {O}(|d_{\\overline{\\nu }}|)) & \\\\ 0 & \\underline{\\mathbb {Q}}_{p}^{*} & \\end{pmatrix} $ In particular, note that in this case $J_{b}(\\mathbb {Q}_{p}) = T(\\mathbb {Q}_{p}) \\simeq \\mathbb {Q}_{p}^{*} \\times \\mathbb {Q}_{p}^{*}$ .",
"We need to compute the dualizing object of $[\\ast /\\mathcal {J}_{b}]$ .",
"To do this, we consider the natural map $ p: [\\ast /\\mathcal {J}_{b}] \\rightarrow [\\ast /\\underline{T(\\mathbb {Q}_{p})}] $ induced via the map $\\mathcal {J}_{b} \\rightarrow \\underline{T(\\mathbb {Q}_{p})}$ given by quotienting out by the unipotent part.",
"This map has a section given by the inclusion $\\underline{T(\\mathbb {Q}_{p})} \\subset \\mathcal {J}_{b}$ , and we denote this by $s: [\\ast /\\underline{T(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\mathcal {J}_{b}]$ .",
"First off note, by Lemma REF , that the dualizing object on $[\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ can be identified with the set of Haar measures on $T(\\mathbb {Q}_{p})$ .",
"Now, since $T(\\mathbb {Q}_{p})$ is unimodular, this implies that, as a $T(\\mathbb {Q}_{p})$ -representation, the sheaf is trivial.",
"Therefore, we are reduced to computing $p^{!",
"}(\\Lambda )$ .",
"To do this, note that the section $s$ is a fibration in the positive Banach-Colmez space $\\mathcal {H}^{0}(\\mathcal {O}(|d_{\\overline{\\nu }}|))$ .",
"It therefore follows by the proof of that adjunction induces an isomorphism: $ s^{!}s_{!",
"}(\\Lambda ) \\simeq \\Lambda $ However, since $s$ is a section of $p$ , that gives us a natural isomorphism: $ s_{!",
"}(\\Lambda ) \\simeq p^{!",
"}(\\Lambda ) $ Therefore, we are reduced to computing $s_{!",
"}(\\Lambda )$ .",
"We now have the following.",
"Lemma 6.13 There is a natural isomorphism $ s^{!",
"}(\\Lambda ) \\simeq \\delta _{B}[2|d_{\\overline{\\nu }}|] $ of sheaves on $[\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}] \\simeq [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ , where $s: [\\ast /\\underline{T(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\mathcal {J}_{b}]$ is the natural map.",
"We consider the Cartesian diagram: $\\begin{tikzcd}& \\mathcal {H}^{0}(|d_{\\overline{\\nu }}|) [r,\"\\tilde{s}\"] [d,\"\\tilde{q}\"]& \\ast [d,\"q\"] & \\\\& \\left[ \\ast /\\underline{T(\\mathbb {Q}_{p})} \\right] [r,\"s\"] & \\left[ \\ast /\\mathcal {J}_{b} \\right]&\\end{tikzcd} $ By base-change, we have a natural isomorphism $\\tilde{q}^{*}s^{!",
"}(\\Lambda ) \\simeq \\tilde{s}^{!",
"}(\\Lambda )$ and, by Lemma REF , the RHS is isomorphic to $|\\cdot |[2|d_{\\overline{\\nu }}|]$ as a sheaf with $\\mathbb {Q}_{p}^{*}$ -action.",
"However, we want to understand how this transforms as a $T(\\mathbb {Q}_{p})$ representation.",
"To do this, we note that the $T(\\mathbb {Q}_{p})$ action on $\\mathcal {H}^{0}(\\mathcal {O}(|d_{\\overline{\\nu }}|)) \\simeq \\mathcal {H}^{0}(\\mathcal {O}(e)^{\\vee } \\otimes \\mathcal {O}(d))$ comes from the semi-direct product structure on $\\mathcal {J}_{b}$ $ \\mathcal {J}_{b} = \\begin{pmatrix} \\mathrm {Aut}(\\mathcal {O}(d)) & \\mathcal {H}om(\\mathcal {O}(e),\\mathcal {O}(d)) & \\\\ 0 & \\mathrm {Aut}(\\mathcal {O}(e)) & \\end{pmatrix} = \\begin{pmatrix} \\mathbb {Q}_{p}^{*} & \\mathcal {H}^{0}(\\mathcal {O}(e)^{\\vee } \\otimes \\mathcal {O}(d)) & \\\\ 0 & \\mathbb {Q}_{p}^{*} & \\end{pmatrix} \\simeq T(\\mathbb {Q}_{p}) \\ltimes \\mathcal {H}^{0}(\\mathcal {O}(d - e)) $ It follows that if we consider the map $ T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*} $ $ \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} \\mapsto ad^{-1} $ then $T(\\mathbb {Q}_{p})$ acts via composing this map with the scaling action on $\\mathcal {H}^{0}(\\mathcal {O}(|d_{\\overline{\\nu }}|))$ .",
"This implies that $\\tilde{q}^{*}s^{!",
"}(\\Lambda ) \\simeq \\tilde{s}^{!",
"}(\\Lambda ) \\simeq \\delta _{B}^{-1}[2|d_{\\overline{\\nu }}|]$ as a sheaf with $T(\\mathbb {Q}_{p})$ -action.",
"This was the desired claim.",
"Using the previous adjunction, the lemma tells us that $p^{!",
"}(\\Lambda ) \\simeq s_{!",
"}(\\Lambda ) \\simeq \\delta _{B}^{-1}[-2|d_{\\overline{\\nu }}|]$ as a sheaf with $T(\\mathbb {Q}_{p})$ action.",
"This is seemingly the inverse of the character we want.",
"We need to be a bit careful.",
"As we will see below, due to the minus sign when passing between isocrystals and bundles, the natural map $ \\mathfrak {q}^{\\overline{\\nu }}: \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{T}^{\\overline{\\nu }} $ identifies with the projection $p: [\\ast /\\mathcal {J}_{b}] \\rightarrow [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ composed with the involution of $T(\\mathbb {Q}_{p})$ given by conjugation by the element of longest length.",
"Therefore, the pullback $\\mathfrak {q}^{\\overline{\\nu }*}(\\delta _{B})$ naturally identifies with $\\delta _{B}^{w_{0}} = \\delta _{B}^{-1}$ under the $T(\\mathbb {Q}_{p})$ action described above.",
"Now let's consider the anti-dominant case.",
"Example 6.14 Let $\\overline{\\nu } = (-d,-e)$ be anti HN-dominant so that $d < e$ .",
"Then $d_{\\overline{\\nu }} = e - d = |d_{\\overline{\\nu }}|$ is positive, and $w_{\\overline{\\nu }} = w_{0}$ is the element of longest length.",
"The connected component $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ is isomorphic to $[\\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }}))/\\underline{T(\\mathbb {Q}_{p})}]$ , where $T(\\mathbb {Q}_{p})$ acts via interpreting $\\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }}))$ as a space parameterizing extensions $ 0 \\rightarrow \\mathcal {O}(d) \\rightarrow \\mathcal {E} \\rightarrow \\mathcal {O}(e) \\rightarrow 0 $ on $X_{S}$ , and $T(\\mathbb {Q}_{p})$ acts via the identification $T(\\mathbb {Q}_{p}) \\simeq (\\mathrm {Aut}(\\mathcal {O}(d)),\\mathrm {Aut}(\\mathcal {O}(e)))$ .",
"Now we can factor the structure map as $ [\\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }}))/\\underline{T(\\mathbb {Q}_{p})}] \\xrightarrow{} [\\ast /\\underline{T(\\mathbb {Q}_{p})}] \\rightarrow \\ast $ and, using that $T(\\mathbb {Q}_{p})$ is unimodular, we reduce to computing $f^{!",
"}(\\Lambda )$ as before.",
"We consider the Cartesian diagram $\\begin{tikzcd}& \\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }})) [r,\"\\tilde{f}\"] [d,\"\\tilde{q}\"] & \\ast [d,\"q\"] & \\\\&\\left[\\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }}))/\\underline{T(\\mathbb {Q}_{p})}\\right] [r,\"f\"] & \\left[\\ast /\\underline{T(\\mathbb {Q}_{p})}\\right]&\\end{tikzcd}$ and again, by base-change, this gives us an isomorphism $\\tilde{q}^{*}f^{!",
"}(\\Lambda ) \\simeq \\tilde{f}^{!",
"}(\\Lambda )$ .",
"Now, applying Lemma REF , we deduce that $\\tilde{q}^{*}f^{!",
"}(\\Lambda )$ is isomorphic to $|\\cdot |^{-1}[2d_{\\overline{\\nu }}]$ as a sheaf with $\\mathbb {Q}_{p}^{*}$ action.",
"We note that the $T(\\mathbb {Q}_{p})$ -action comes from the identification $\\mathcal {H}^{1}(\\mathcal {O}(d) \\otimes \\mathcal {O}(e)^{\\vee }) \\simeq \\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }}))$ .",
"Therefore, $T(\\mathbb {Q}_{p})$ acts via the map $ T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*} $ $ \\begin{pmatrix} a & 0 & \\\\ 0 & d & \\end{pmatrix} \\rightarrow ad^{-1} $ composed with the action of $\\mathbb {Q}_{p}^{*}$ on $\\mathcal {H}^{1}(\\mathcal {O}(-d_{\\overline{\\nu }}))$ by scaling of global sections.",
"As a consequence, we deduce that, as a $T(\\mathbb {Q}_{p})$ -representation, $\\Lambda [2d_{\\overline{\\nu }}](d_{\\overline{\\nu }})$ is isomorphic to $\\delta _{B}^{-1}[2d_{\\overline{\\nu }}]$ .",
"However, as seen in the previous example, the pullback $\\mathfrak {q}^{\\overline{\\nu }}$ twists the above $T(\\mathbb {Q}_{p})$ action by the element of longest length, so this gives the desired claim.",
"We will need some formalism to move beyond the case of $\\mathrm {GL}_{2}$ .",
"Let $G$ now be a general quasi-split connected reductive group.",
"We let $b \\in B(G)_{\\mathrm {un}}$ be an unramified element, and set $b_{T}$ to be the HN-dominant reduction as before.",
"We let $\\mathcal {J}_{b}$ be the group diamond parameterizing automorphisms of the bundle $\\mathcal {F}_{b}$ associated to $b$ .",
"Now, we can write $\\mathcal {J}_{b}^{\\ge \\lambda }$ for the sub-diamond of automorphisms $\\gamma $ of $\\mathcal {F}_{b}$ such that $ (\\gamma - 1)(\\rho _{*}(\\mathcal {F}_{b}))^{\\ge \\lambda ^{\\prime }} \\subset (\\rho _{*}\\mathcal {F}_{b})^{\\ge \\lambda ^{\\prime } + \\lambda } $ for all representations $\\rho $ of $G$ .",
"We set $\\mathcal {J}_{b}^{> \\lambda } = \\cup _{\\lambda ^{\\prime } > \\lambda } \\mathcal {J}_{b}^{\\ge \\lambda ^{\\prime }}$ .",
"We note that, since $J_{b}(\\mathbb {Q}_{p})$ is the automorphism group of the isocrystal defined by $b$ , and, for $S \\in \\mathrm {Perf}$ , there is an identification: $H^{0}(X_{S},\\mathcal {O}_{X_{S}}) = \\underline{\\mathbb {Q}_{p}}(S)$ , we have a natural injection $ \\underline{J_{b}(\\mathbb {Q}_{p})} \\hookrightarrow \\mathcal {J}_{b} $ which has a natural section given by letting $\\mathcal {J}_{b}$ act on the graded-pieces of the HN-filtration.",
"This gives us a semi-direct product decomposition $ \\mathcal {J}_{b} \\simeq \\underline{J_{b}(\\mathbb {Q}_{p})} \\ltimes \\mathcal {J}_{b}^{> 0}$ of group diamonds (See ).",
"We can relate this to $B$ -bundles as follows.",
"We consider the $B$ -torsor $Q := \\mathcal {F}_{b_{T}} \\times ^{T} B$ over $X_{S}$ for $S \\in \\mathrm {Perf}$ .",
"We suppose now that $b_{T}$ is the element corresponding to Levi factor of the HN-reduction of $\\mathcal {F}_{b}$ .",
"Then, as in Here Fargues-Scholze consider instead the opposite Borel $B^{-}$ ; however, since we have defined the HN-dominant reduction $b_{T}$ to have anti-dominant slope homorphism this agrees with their definition., we have equalities $ \\mathcal {J}_{b}(S) = Q(X_{S}) $ $ \\mathcal {J}_{b}^{U}(S) = (\\mathcal {E}_{b_{T}} \\times ^{T} U)(X_{S}) = R_{u}Q(X_{S}) $ where $U$ is the unipotent radical of $B$ .",
"Remark 6.15 This identification is a manifestation of the following easy fact, that we will use implicitly throughout.",
"Given a bundle $\\mathcal {E}$ on $X_{S}$ , the set of $\\mathcal {E}$ -torsors on $X_{S}$ is parametrized by $H^{1}(X_{S},\\mathcal {E})$ , and the automorphisms of such $\\mathcal {E}$ -torsors are parametrized by $H^{0}(X_{S},\\mathcal {E})$ .",
"In particular, since $Q$ defines a $B$ -structure on $\\mathcal {F}_{b}$ , every such global section induces an automorphism of $\\mathcal {F}_{b}$ .",
"Moreover, since $b_{T}$ corresponds to the Levi factor of an HN-reduction of $\\mathcal {F}_{b}$ this will imply that every element of $\\mathcal {J}_{b}(S)$ can be obtained in this way.",
"Let's now consider the case where $b_{T}$ is HN-dominant, but does not correspond to an HN-reduction of $\\mathcal {F}_{b}$ .",
"In other words, there exist some positive roots $\\hat{\\alpha }$ such that $\\langle \\hat{\\alpha }, \\nu _{b_{T}} \\rangle = 0$ (cf.",
"Example REF ).",
"In this case, all we can conclude is a strict containment $Q(X_{S}) \\subset \\mathcal {J}_{b}(S)$ .",
"In particular, if we write $Q^{>0}(X_{S})$ for the subset mapping to $\\mathcal {J}_{b}^{> 0}$ then we still have an isomorphism $ \\mathcal {J}_{b}^{> 0}(S) \\simeq Q^{> 0}(X_{S}) $ but, on the slope 0 part, $Q^{= 0}(X_{S})$ is only properly contained in $\\underline{J_{b}(\\mathbb {Q}_{p})}(S)$ .",
"In particular, $Q^{= 0}(S)$ will be identified with $\\underline{B_{b}(\\mathbb {Q}_{p})}(S)$ , where $B_{b} \\subset J_{b}$ is the Borel subgroup defined in Lemma REF , and $R_{u}Q^{= 0}$ is identified with the unipotent radical of this Borel.",
"More precisely, if $J_{b}$ is an inner twisting of $M_{b} \\subset G$ then $M_{b}$ is the Levi subgroup in $G$ corresponding to the positive simple roots $\\hat{\\alpha }_{i}$ such that $\\langle \\hat{\\alpha }_{i}, \\nu _{b_{T}} \\rangle = 0$ , by definition of $M_{b}$ as the maximal Levi admitting a basic reduction of $b$ .",
"Now let's refine this further, recall that $U$ has a filtration by commutator subgroups $ (1) \\subset \\cdots \\subset U_{i + 1} \\subset U_{i} \\subset \\cdots \\subset U_{1} \\subset U_{0} = U $ where $U_{i}$ is generated by the root subgroups $U_{\\hat{\\alpha }}$ for $\\hat{\\alpha } = \\sum _{i \\in \\mathcal {J}} n_{i}\\hat{\\alpha }_{i}$ , where $\\sum _{i \\in \\mathcal {J}} n_{i} > i$ .",
"As a consequence, we note that $ U_{i}/U_{i - 1} = \\bigoplus _{\\hat{\\alpha }} U_{\\hat{\\alpha }} $ where the direct sum runs over all roots $\\hat{\\alpha } = \\sum _{i \\in \\mathcal {J}} n_{i}\\hat{\\alpha }_{i}$ such that $\\sum _{i \\in \\mathcal {J}} n_{i} = i$ .",
"This gives rise to a filtration on $R_{u}Q$ whose graded pieces will be a direct sum of the line bundles $ \\hat{\\alpha }_{*}(Q) $ of degree equal to $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle $ .",
"This allows us to write $\\mathcal {J}_{b}^{> 0}$ as an iterated fibration of positive Banach-Colmez spaces $ \\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q)) $ for $\\hat{\\alpha }$ such that $\\mathrm {deg}(\\hat{\\alpha }_{*}(Q)) = \\langle \\hat{\\alpha }, \\nu _{b} \\rangle > 0$ .",
"The subgroup $T(\\mathbb {Q}_{p}) \\subset J_{b}(\\mathbb {Q}_{p})$ will act on this positive Banach-Colmez space via the previous semi-direct product structure.",
"The action will be the composite of the character $ T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*} $ $ t \\mapsto \\mathrm {Ad}(t|\\mathfrak {g}_{w_{0}(\\hat{\\alpha })}) = \\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})^{-1} $ and the scaling action on $\\mathbb {Q}_{p}^{*}$ , where $\\mathfrak {g}_{\\hat{\\alpha }}$ is the $\\hat{\\alpha }$ -isotypic root space of $\\mathrm {Lie}(G)$ .",
"This action will respect the projection $\\mathrm {Bun}_{B}^{b_{T}} \\rightarrow \\mathrm {Bun}_{T}^{b_{T}} = [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ , and the minus sign occurs because of the passage between isocrystals and bundles.",
"These observations combined with the above Lemmas relating the dualizing objects on positive Banach-Colmez to norm characters give us everything we need to pin down the dualizing object on $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ in the case that $\\overline{\\nu }$ is HN-dominant.",
"Proposition 6.16 Let $b_{T} \\in B(T)$ be an element with anti-dominant slope homomorphism with respect to $B$ .",
"The dualizing object $K_{\\mathrm {Bun}_{B}^{b_{T}}}$ on $\\mathrm {Bun}_{B}^{b_{T}}$ is isomorphic to $(\\mathfrak {q}^{b_{T}})^{*}(\\delta _{B})[2d_{b_{T}}]$ .",
"Since $b_{T}$ has anti-dominant slope homomorphism its HN-slopes are dominant, and therefore $\\mathrm {Bun}_{B}^{b_{T}}$ parametrizes split reductions.",
"In particular, its underlying topological space is just a point.",
"More specifically, it is isomorphic to $[\\ast /\\mathcal {G}_{b_{T}}]$ , where $\\mathcal {G}_{b_{T}}(S) := Q(X_{S})$ , and $Q$ is the torsor defined above.",
"The semi-direct product structure on $\\mathcal {J}_{b}$ and the above discussion imply that we have a semi-direct product structure $ \\mathcal {G}_{b_{T}} \\simeq \\underline{B_{b}(\\mathbb {Q}_{p})} \\ltimes \\mathcal {J}_{b}^{> 0} $ on $\\mathcal {G}_{b_{T}}$ .",
"Therefore, we have a natural map $ [\\ast /\\mathcal {G}_{b_{T}}] \\rightarrow [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] $ which when composed with $[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ identifies with the map $\\mathrm {Bun}_{B}^{b_{T}} \\rightarrow \\mathrm {Bun}_{T}^{b_{T}}$ .",
"Now, by Lemma REF , the dualizing object on $[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}]$ is defined by the modulus character $\\delta ^{= 0} := \\delta _{B_{b}}: B_{b}(\\mathbb {Q}_{p}) \\rightarrow T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ where the first map is the natural projection and the last map, by definition of $B_{b}$ , will be given by $ t \\mapsto \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ \\langle \\hat{\\alpha },\\nu _{b} \\rangle = 0\\end{array}} |\\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})| $ Therefore, we reduce to computing the dualizing object with respect to the map: $ p: [\\ast /\\mathcal {G}_{b_{T}}] \\rightarrow [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] $ This map has a natural section $ s: [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\mathcal {G}_{b_{T}}] $ the fibers of which are given by $\\mathcal {J}_{b}^{> 0}$ , which, as seen above, is an iterated fibration of positive Banach-Colmez spaces.",
"Therefore, by , we deduce that the adjunction $ s^{!}s_{!",
"}(\\Lambda ) \\simeq \\Lambda $ is an isomorphism.",
"Arguing as in Example REF , this reduces us to showing that $ s^{!",
"}(\\Lambda ) \\simeq (\\delta ^{> 0})^{-1}[-2d_{b_{T}}] $ where $\\delta ^{> 0}: T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ is the character $ t \\mapsto \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ \\langle \\hat{\\alpha },\\nu _{b} \\rangle \\ne 0\\end{array}} |\\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})|$ so that we have $\\delta _{B} = \\delta ^{= 0}\\delta ^{>0}$ .",
"To see this, we consider the Cartesian diagram $\\begin{tikzcd}& \\mathcal {J}_{b}^{> 0} [r,\"\\tilde{s}\"] [d,\"\\tilde{q}\"]& \\ast [d,\"q\"] & \\\\& \\left[\\ast /\\mathcal {G}_{b_{T}}\\right] [r,\"s\"] & \\left[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}\\right] &\\end{tikzcd} $ which, by base-change, gives us an isomorphism $\\tilde{q}^{*}s^{!",
"}(\\Lambda ) \\simeq \\tilde{s}^{!",
"}(\\Lambda )$ .",
"As above, we can write $\\mathcal {J}_{b}^{> 0}$ as an iterated fibration of the positive Banach-Colmez spaces $\\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q))$ for $\\hat{\\alpha } > 0$ such that $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle > 0$ .",
"By Lemma REF , the dualizing object on $\\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q))$ is isomorphic to $|\\cdot |[2\\langle \\hat{\\alpha }, \\nu _{b} \\rangle ]$ as a $\\mathbb {Q}_{p}^{*}$ representation.",
"However, as a $B_{b}(\\mathbb {Q}_{p})$ representation, this is acted on by the natural surjection $B_{b}(\\mathbb {Q}_{p}) \\rightarrow T(\\mathbb {Q}_{p})$ composed with the character $ T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*} $ $ t \\mapsto \\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})^{-1} $ and the $\\mathbb {Q}_{p}^{*}$ action via scaling the global sections of $\\hat{\\alpha }_{*}(Q)$ .",
"Hence, we obtain that the dualizing object on $\\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q))$ as a $B_{b}(\\mathbb {Q}_{p})$ -representation is given by $|\\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})|^{-1}$ .",
"Therefore, by using the formula $ (g\\circ f)^{!",
"}(\\Lambda ) = g^{!",
"}(\\Lambda ) \\otimes g^{*}(f^{!",
"}(\\Lambda )) $ for cohomologically smooth morphisms $f$ and $g$ , we deduce that the dualizing object on $\\mathcal {J}_{b}^{> 0}$ is isomorphic to $ \\Lambda [\\sum _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ \\langle \\hat{\\alpha },\\nu _{b} \\rangle \\ne 0\\end{array}} \\langle \\hat{\\alpha }, \\nu _{b} \\rangle ](\\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ \\langle \\hat{\\alpha },\\nu _{b} \\rangle \\ne 0\\end{array}} |\\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})|^{-1}) = \\Lambda [2\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ](\\delta ^{> 0})^{-1} = \\Lambda [-2d_{b_{T}}](\\delta ^{> 0})^{-1}$ as a sheaf with $B_{b}(\\mathbb {Q}_{p})$ -action.",
"We now push this further.",
"Let $\\overline{\\nu } \\in B(T)$ be an element mapping to $b \\in B(G)_{\\mathrm {un}}$ with HN-dominant reduction $b_{T}$ .",
"We write $\\overline{\\nu } = w(b_{T})$ for $w := w_{\\overline{\\nu }} \\in W_{b}$ identified with a representative of minimal length in $W_{G}$ .",
"We consider the bundle $ Q_{\\overline{\\nu }} = \\mathcal {F}_{b_{T}} \\times ^{T} B^{w} = \\mathcal {F}_{w(b_{T})} \\times ^{T} B $ and $ R_{u}Q_{\\overline{\\nu }} = \\mathcal {F}_{b_{T}} \\times ^{T} U^{w} $ Now, since $Q_{\\overline{\\nu }}$ defines a reduction of $\\mathcal {F}_{b}$ , we have as before an inclusion $ Q_{\\overline{\\nu }}(X_{S}) \\hookrightarrow \\mathcal {J}_{b}(S) $ giving rise to a slope filtration on $Q_{\\overline{\\nu }}$ .",
"This gives the following definition.",
"Definition 6.17 For $\\overline{\\nu } = w(b_{T}) \\in B(T)$ mapping to $b \\in B(G)_{\\mathrm {un}}$ as above, we let $\\mathcal {G}_{\\overline{\\nu }}$ be the group diamond on $\\mathrm {Perf}$ whose $S$ -valued points are equal to $Q_{\\overline{\\nu }}(X_{S})$ .",
"For any $\\lambda \\in \\mathbb {Q}$ , we define subsets $\\mathcal {G}_{\\overline{\\nu }}^{\\ge \\lambda }$ and $\\mathcal {G}_{\\overline{\\nu }}^{> \\lambda }$ , as above.",
"We get a semi-direct product structure $ \\mathcal {G}_{\\overline{\\nu }}^{> 0} \\rtimes \\mathcal {G}_{\\overline{\\nu }}^{= 0} $ where $\\mathcal {G}_{\\overline{\\nu }}^{> 0} \\subset \\mathcal {J}_{b}^{> 0}$ and $\\mathcal {G}_{\\overline{\\nu }}^{= 0} \\subset \\underline{J_{b}(\\mathbb {Q}_{p})}$ .",
"In particular, $\\mathcal {G}_{\\overline{\\nu }}^{= 0}$ will be identified with the Borel subgroup $\\underline{B_{b}^{w}(\\mathbb {Q}_{p})} = \\underline{B_{b}(\\mathbb {Q}_{p})}$ , the constant group diamond defined by the standard Borel $B_{b}$ conjugated by $w$ .",
"Here the equality follows since we are identifying $w \\in W_{b}$ with a representative of minimal length.",
"Now, as above, $R_{u}Q_{w}$ has a filtration given by the natural filtration on $U_{w}$ $ (1) \\subset \\cdots \\subset U_{i + 1} \\subset U_{i} \\subset \\cdots \\subset U_{1} \\subset U_{0} = U_{w} $ where $U_{i - 1}/U_{i} \\simeq \\bigoplus _{\\hat{\\alpha } > 0} U_{w(\\hat{\\alpha })}$ , and $\\hat{\\alpha }$ runs over positive roots $\\hat{\\alpha } = \\sum _{i \\in \\mathcal {J}} n_{i}\\hat{\\alpha }_{i}$ satisfying $\\sum _{i \\in \\mathcal {J}} n_{i} = i$ .",
"We have a containment $\\mathcal {G}_{\\overline{\\nu }}^{> 0}(S) \\subset \\mathcal {J}_{b}^{> 0}(X_{S})$ , which allows us to write $\\mathcal {G}^{> 0}$ as an iterated fibration of positive Banach-Colmez spaces $\\mathcal {H}^{0}((w(\\hat{\\alpha }))_{*}(Q))$ for $\\hat{\\alpha }$ such that $\\langle ww_{0}(\\hat{\\alpha }), \\nu _{b} \\rangle > 0$ , which is equivalent to insisting that $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle \\ne 0$ and $w(\\hat{\\alpha }) > 0$ .",
"Now, with this in hand, let's prove Theorem REF .",
"(Theorem REF ) We begin with the following Lemma.",
"Lemma 6.18 If $\\overline{\\nu } = w(b_{T}) \\in B(T)$ is an element mapping to $b \\in B(G)_{\\mathrm {un}}$ with HN-dominant reduction $b_{T}$ then the $\\ell $ -dimension $d_{\\overline{\\nu }}$ of the connected component $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ is equal to $ \\langle 2\\hat{\\rho }, \\nu _{b} \\rangle - 2\\langle 2\\hat{\\rho }^{w}, \\nu _{b_{T}} \\rangle $ where $2\\hat{\\rho }^{w}$ is the sum of the positive roots $\\hat{\\alpha }$ such that $w(\\hat{\\alpha }) > 0$ .",
"By definition $d_{\\overline{\\nu }} = \\langle 2\\hat{\\rho },w(b_{T}) \\rangle $ , and this is easily identified with the above quantity, recalling that $w_{0}(\\nu _{b_{T}}) = \\nu _{b}$ by construction.",
"Write $\\overline{\\nu } = w(b_{T})$ as in the lemma.",
"The key point is that we can factor the structure morphism $\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\ast $ as $ \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\xrightarrow{} [\\ast /\\mathcal {G}_{\\overline{\\nu }}] \\rightarrow [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] \\rightarrow \\ast $ We can fit the projection $\\mathfrak {q}^{\\overline{\\nu }}: \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{T}^{\\overline{\\nu }} = [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ into this picture as follows.",
"In particular, this is equal to the map $\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\underline{T(\\mathbb {Q}_{p})}] \\simeq [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ , where the second to last map is the natural projection and the isomorphism is given by conjugating by $w$ .",
"It follows that we want to show that $K_{\\mathrm {Bun}_{B}} \\simeq \\delta _{B}^{w}[2(\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle - 2\\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b_{T}} \\rangle )]$ with respect to the $T(\\mathbb {Q}_{p})$ -action coming from the natural projection.",
"Let us look at the modulus character $\\delta _{B}^{w}: T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*}$ , this is given by $ t \\mapsto |\\mathrm {det}(\\mathrm {Ad}(t|\\mathrm {Lie}(U^{w}))| = \\prod _{\\hat{\\alpha } > 0} |\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })})| = \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) < 0\\end{array}} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{ww_{0}(\\hat{\\alpha }))})|^{-1} \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) > 0\\end{array}} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })}))| $ where the product runs over all positive roots $\\hat{\\alpha } > 0$ .",
"The dualizing object on $[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}]$ is given by the character defined by composing the surjection $B_{b}(\\mathbb {Q}_{p}) \\rightarrow T(\\mathbb {Q}_{p})$ with the map $ t\\mapsto \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ \\langle \\hat{\\alpha },\\nu _{b} \\rangle = 0\\end{array}} |\\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }})| = \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ \\langle \\hat{\\alpha },\\nu _{b} \\rangle = 0\\end{array}} |\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })})| $ which we denote by $\\delta ^{= 0}$ .",
"Here equality follows since we are identifying $w \\in W_{G}/W_{M_{b}}$ with a representative $w \\in W_{G}$ of minimal length.",
"Analogously, by arguing as in the proof of Proposition REF , we can see that the dualizing complex with respect to the map $[\\ast /\\mathcal {G}_{\\overline{\\nu }}] \\rightarrow [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}]$ is given by $\\delta ^{>0}[-2\\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle ]$ .",
"Here $\\delta ^{> 0}$ is defined by $ t \\mapsto \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) > 0 \\\\ \\langle \\hat{\\alpha }, \\nu _{b} \\rangle \\ne 0\\end{array}} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })}))| $ Thus, it suffices to show that $ f^{!",
"}(\\Lambda ) \\simeq \\Lambda [2(\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle - \\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b_{T}} \\rangle )]\\delta ^{< 0} $ where $\\delta ^{< 0}$ is the character defined by $ t \\mapsto \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) < 0 \\end{array}} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{ww_{0}(\\hat{\\alpha })}))|^{-1} $ so that $\\delta _{B}^{w} = \\delta ^{> 0}\\delta ^{= 0}\\delta ^{< 0}$ , where we note that if $w(\\hat{\\alpha }) < 0$ it is automatic that $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle \\ne 0$ since $w$ is a representative of minimal length.",
"To do this, we should proceed analogously to Example REF .",
"Namely, define the space $X^{\\overline{\\nu }}$ by the Cartesian diagram $\\begin{tikzcd}& X^{\\overline{\\nu }} [r,\"\\tilde{f}\"] [d,\"\\tilde{q}\"] & \\ast [d,\"q\"] & \\\\& \\mathrm {Bun}_{B}^{\\overline{\\nu }} [r,\"f\"] & \\left[\\ast /\\mathcal {G}_{\\overline{\\nu }}\\right]&\\end{tikzcd}$ and we need to elucidate the space $X^{\\overline{\\nu }}$ .",
"We claim it is an iterated fibration of negative Banach-Colmez spaces.",
"To see this, we consider $Q_{\\overline{\\nu }}$ as above, and look at the negative slope part $Q_{\\overline{\\nu }}^{< 0}$ .",
"We can identify $X^{\\overline{\\nu }}(S)$ with the set of $Q_{\\overline{\\nu }}^{< 0}$ -torsors over $X_{S}$ (cf.",
"the proof of ).",
"By considering the filtration of $U^{w}$ by commutator subgroups, we can write $X^{\\overline{\\nu }}$ as an iterated fibration of the negative Banach-Colmez spaces $ \\mathcal {H}^{1}((w(\\hat{\\alpha }))_{*}(Q)) $ for positive roots $\\hat{\\alpha } > 0$ such that $\\langle w(\\hat{\\alpha }), \\nu _{b} \\rangle < 0$ , which since $\\nu _{b}$ is dominant and $w$ is a representative of minimal length is equivalent to insisting that $w(\\hat{\\alpha }) < 0$ .",
"Then, using Lemma REF , we deduce that the dualizing object on $\\mathcal {H}^{1}((w(\\hat{\\alpha }))_{*}(Q))$ is equal to $|\\cdot |^{-1}[\\langle \\hat{\\alpha }, \\nu _{b} \\rangle ]$ as a sheaf with $\\mathbb {Q}_{p}^{*}$ -action.",
"Now $B_{b}(\\mathbb {Q}_{p})$ acts on $\\mathcal {H}^{1}((w(\\hat{\\alpha }))_{*}(Q))$ via the scaling action on $\\mathcal {H}^{1}((w(\\hat{\\alpha }))_{*}(Q))$ pre-composed with the natural projection $B_{b}(\\mathbb {Q}_{p}) \\rightarrow T(\\mathbb {Q}_{p})$ and the character $ T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*} $ $ t \\mapsto \\mathrm {Ad}(t|\\mathfrak {g}_{ww_{0}(\\hat{\\alpha })}) = \\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })})^{-1} $ for varying $\\hat{\\alpha } > 0$ such that $w(\\hat{\\alpha }) < 0$ , where twist by $w_{0}$ comes from the minus sign when passing between isocrystals and bundles as above.",
"This tells us that we have an isomorphism $ \\tilde{f}^{!",
"}(\\Lambda ) \\simeq \\Lambda [2(\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle - \\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b_{T}} \\rangle )]\\delta ^{< 0} $ as a sheaf with $B_{b}(\\mathbb {Q}_{p})$ action, but, by base change, we have an isomorphism $q^{*}f^{!",
"}(\\Lambda ) \\simeq \\tilde{f}^{!",
"}(\\Lambda )$ , and the result follows."
],
[
"Proof of the Filtered Eigensheaf Property",
"We would now like the describe how Hecke correspondences on $\\mathrm {Bun}_{G}$ interact with the Eisenstein functor.",
"This will be used to show the Hecke eigensheaf property.",
"Our analysis is heavily inspired by , where an analogous claim is proven in the classical case.",
"However, unlike the arguments there, we cannot appeal to the decomposition theorem, as the usual formalism of weights doesn't exist in this context.",
"Nonetheless, we still have the excision spectral sequence, which will give us a filtration on the Eisenstein series.",
"In §, we will show that, under the condition of $\\mu $ -regularity (Definition REF ), this filtration splits for the Hecke operator defined by $V_{\\mu ^{\\Gamma }} \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ when applied to $\\mathcal {F} = \\mathcal {S}_{\\phi _{T}}$ .",
"Our goal is the following Theorem.",
"Theorem 7.1 For $\\mathcal {F} \\in \\mathrm {D}(\\mathrm {Bun}_{T})$ , $I$ a finite index set, and $V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ , the sheaf $T_{V}(\\mathrm {Eis}(\\mathcal {F}))$ has a $W_{\\mathbb {Q}_{p}}^{I}$ -equivariant filtration indexed by $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ .",
"The filtration's graded pieces are isomorphic to $ \\mathrm {Eis}(T_{(\\nu _{i})_{i \\in I}}(\\mathcal {F})) \\otimes V((\\nu _{i})_{i \\in I})(-\\langle \\hat{\\rho }, \\sum _{i \\in I} \\nu _{i\\Gamma } \\rangle ) $ as sheaves in $\\mathrm {D}(\\mathrm {Bun}_{G})^{BW_{\\mathbb {Q}_{p}}^{I}}$ .",
"The filtration is natural in $I$ and $V$ , as well as compatible with compositions and exterior tensor products in $V$ .",
"Let $E/\\mathbb {Q}_{p}$ be an extension over which $G$ splits.",
"The value of $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ is determined by the value of the Hecke correspondence base-changed to $E$ , using (See ).",
"Here the correspondence is determined by a representation $V = \\boxtimes _{i \\in I} V_{i} \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G}^{I})$ of $I$ -copies of the dual group.",
"We let $\\lambda _{i} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ be the highest weight of $V_{i}$ .",
"By taking direct sums, we can assume WLOG that $V$ has a fixed central character.",
"We let $\\mathcal {S}_{V}$ be the $\\Lambda $ -valued sheaf on $\\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E}$ defined via Theorem REF .",
"Our aim is to construct a $W_{E}^{I}$ -equivariant filtration on the sheaf: $ T_{V}(\\mathrm {Eis}(\\mathcal {F})) = (h^{\\rightarrow }_{G} \\times \\pi )_{!",
"}(h_{G}^{\\rightarrow *}(\\mathfrak {p}_{!",
"}\\mathfrak {q}^{*}(\\mathcal {F})[\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\mathcal {S}_{V}) \\in \\mathrm {D}(\\mathrm {Bun}_{G})^{BW_{E}^{I}}$ This will be accomplished by contemplating the diagram $\\begin{tikzcd}& \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} [d,\"\\overline{\\mathfrak {p}} \\times \\mathrm {id}\"] & [l,\"\\phi _{(\\lambda _{i})_{i \\in I}}\"] Z^{I}_{(\\lambda _{i})_{i \\in I}} [r,\"^{\\prime }h_{G}^{\\rightarrow }\"] [d,\"^{\\prime }\\mathfrak {p}\"] & \\mathrm {Bun}_{B} [d,\"\\mathfrak {p}\"] \\\\& \\mathrm {Bun}_{G} \\times \\mathrm {Div}^{I}_{E} & [l,\"h_{G}^{\\leftarrow } \\times \\pi \"] \\mathrm {Hck}^{I}_{G,\\le (\\lambda _{i})_{i \\in I},E} [r,\"h_{G}^{\\rightarrow }\"] & \\mathrm {Bun}_{G}\\end{tikzcd}$ as defined in §REF (REF ).",
"Using base-change on the right Cartesian square, we get an isomorphism $ T_{V}(\\mathrm {Eis}(\\mathcal {F})) \\simeq (h^{\\rightarrow }_{G} \\times \\pi )_{!",
"}(\\phantom{}^{^{\\prime }}\\mathfrak {p}_{!",
"}\\phantom{}^{^{\\prime }}h_{G}^{\\rightarrow *}\\mathfrak {q}^{*}(\\mathcal {F})[\\mathrm {dim}(\\mathrm {Bun}_{B})] \\otimes \\mathcal {S}_{V}) $ but, applying the projection formula with respect to $\\mathfrak {p}^{\\prime }$ , this becomes $ T_{V}(\\mathrm {Eis}(\\mathcal {F})) \\simeq (h^{\\rightarrow }_{G} \\times \\pi )_{!",
"}\\phantom{}^{^{\\prime }}\\mathfrak {p}_{!",
"}(^{\\prime }h_{G}^{\\rightarrow *}(\\mathfrak {q}^{*}(\\mathcal {F})[\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\phantom{}^{^{\\prime }}\\mathfrak {p}^{*}(\\mathcal {S}_{V})) $ We define $ K_{V} := \\phantom{}^{^{\\prime }}h_{G}^{\\rightarrow *}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\phantom{}^{^{\\prime }}\\mathfrak {p}^{*}(\\mathcal {S}_{V}) $ allowing us to rewrite our formula nicely as $T_{V}(\\mathrm {Eis}(\\mathcal {F})) \\simeq (h^{\\rightarrow }_{G} \\times \\pi )_{!}",
"\\phantom{}^{^{\\prime }}\\mathfrak {p}_{!",
"}(^{\\prime }h_{G}^{\\rightarrow *}\\mathfrak {q}^{*}(\\mathcal {F}) \\otimes K_{V}) = (\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!",
"}\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(^{\\prime }h_{G}^{\\rightarrow *}\\mathfrak {q}^{*}(\\mathcal {F}) \\otimes K_{V})$ Now we would like to reduce the claim to applying excision to $\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{V})$ with respect to a locally closed stratification of $\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ .",
"The claim should then follow from Corollary REF and Lemma REF .",
"In order to do this, let's further rewrite the formula.",
"We recall that, for $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , we have a map $ i_{(\\nu _{i})_{i \\in I}}: \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} \\rightarrow \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} $ sending the tuple $(\\mathcal {F}_{G},\\mathcal {F}_{T},\\overline{\\kappa }^{\\hat{\\lambda }},(D_{i})_{i \\in I})$ to the tuple $(\\mathcal {F}_{G},\\mathcal {F}_{T}(-\\sum _{i \\in I} \\nu _{i} \\cdot D_{i}),\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}}(- \\sum _{i \\in I} \\langle \\nu _{i}, \\hat{\\lambda } \\rangle \\cdot D_{i}) \\hookrightarrow \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}} \\hookrightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}},(D_{i})_{i \\in I})$ .",
"We also have the map $ h_{(\\nu _{i})_{i \\in I}}^{\\rightarrow }: \\mathrm {Bun}_{T} \\times \\mathrm {Div}^{I}_{E} \\simeq \\mathrm {Hck}^{I}_{T,(\\nu _{i})_{i \\in I}} \\rightarrow \\mathrm {Bun}_{T} $ as in § which is given by modifying a $T$ -bundle by $(\\nu _{i})_{i \\in I}$ at a tuple of divisors $(D_{i})_{i \\in I}$ defining a point in $\\mathrm {Div}^{I}_{E}$ .",
"Then, for $\\mathcal {F} \\in \\mathrm {D}(\\mathrm {Bun}_{T})$ , we recall that we have an identification $(h_{(\\nu _{i})_{i \\in I}}^{\\rightarrow })^{*}(\\mathcal {F}) = T_{(\\nu _{i})_{i \\in I}}(\\mathcal {F})$ of sheaves in $\\mathrm {D}(\\mathrm {Bun}_{T})^{BW_{E}^{I}}$ .",
"Now, we can verify the following easy Lemma, which follows from the definition of $\\phi _{(\\lambda _{i})_{i \\in I}}$ .",
"Lemma 7.2 The following is true.",
"The maps $h_{(w_{0}(\\lambda _{i}))_{i \\in I}}^{\\rightarrow } \\circ (\\overline{\\mathfrak {q}} \\times \\mathrm {id}) \\circ \\phi _{(\\lambda _{i})_{i \\in I}}$ and $\\mathfrak {q} \\circ \\phantom{}^{^{\\prime }}h_{G}^{\\rightarrow }$ from $Z^{I}_{(\\lambda _{i})_{i \\in I}}$ to $\\mathrm {Bun}_{T}$ coincide.",
"For every $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , the maps $(\\overline{\\mathfrak {q}} \\times \\mathrm {id}) \\circ i_{(\\nu _{i})_{i \\in I}}$ and $(h_{(\\nu _{i})_{i \\in I}}^{\\rightarrow } \\times \\mathrm {id}) \\circ (\\overline{\\mathfrak {q}} \\times \\mathrm {id})$ from $\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ to $\\mathrm {Bun}_{T} \\times \\mathrm {Div}^{I}_{E}$ coincide.",
"Now, with this in hand, let's revisit equation (4): $ T_{V}(\\mathrm {Eis}(\\mathcal {F})) \\simeq (\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!}",
"\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(^{\\prime }h_{G}^{\\rightarrow *}\\mathfrak {q}^{*}(\\mathcal {F}) \\otimes K_{V}) $ Using Lemma REF (1), we have that $ ^{\\prime }h_{G}^{\\rightarrow *}\\mathfrak {q}^{*}(\\mathcal {F}) \\simeq \\phi _{(\\lambda _{i})_{i \\in I}}^{*}(\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}(h_{(w_{0}(\\lambda _{i}))_{i \\in I}}^{\\rightarrow })^{*}(\\mathcal {F}) $ substituting this in and applying projection formula with respect to $\\phi _{(\\lambda _{i})_{i \\in I}}$ , we can rewrite the RHS as $(\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!",
"}((\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}(h_{(w_{0}(\\lambda _{i}))_{i \\in I}}^{\\rightarrow })^{*}(\\mathcal {F}) \\otimes \\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{V})) $ Now we claim that we have the following description of $\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{V})$ .",
"Theorem 7.3 The sheaf $\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{V}) \\in \\mathrm {D}(\\overline{\\mathrm {Bun}}_{B})^{BW_{E}^{I}}$ has a $W_{E}^{I}$ -equivariant filtration indexed by $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ .",
"The graded pieces of this filtration are given by $ \\boxtimes _{i \\in I} (i_{\\nu _{i}!",
"}(j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})])) \\otimes V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ Assuming this for now, we get that $T_{V}(\\mathrm {Eis}(\\mathcal {F}))$ has a $W_{E}^{I}$ -equivariant filtration with graded pieces given by $ \\boxtimes _{i \\in I} (\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!",
"}((\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}h_{w_{0}(\\lambda _{i})}^{\\rightarrow *}(\\mathcal {F}) \\otimes (i_{\\nu _{i}!",
"}(j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})])) \\otimes V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ))) $ Applying projection formula with respect to $i_{\\nu _{i}}$ , we obtain $ \\boxtimes _{i \\in I} (\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!",
"}i_{\\nu _{i}!",
"}(i_{\\nu _{i}}^{*}(\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}h_{w_{0}(\\lambda _{i})}^{\\rightarrow *}(\\mathcal {F}) \\otimes (j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})])) \\otimes V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ but now, by Lemma REF (2), we have $ i_{\\nu _{i}}^{*}(\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}h_{w_{0}(\\lambda _{i})}^{\\rightarrow *}(\\mathcal {F}) \\simeq (\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}h^{\\rightarrow *}_{w_{0}(\\lambda _{i}) + \\nu _{i}}(\\mathcal {F}) \\simeq (\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}(T_{w_{0}(\\lambda _{i}) + \\nu _{i}}(\\mathcal {F})) $ so substituting this into the previous formula we get $ \\boxtimes _{i \\in I} (\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!",
"}i_{\\nu _{i}!",
"}((\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}(T_{w_{0}(\\lambda _{i}) + \\nu _{i}}(\\mathcal {F}) \\otimes (j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]))) \\otimes V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ Now $i_{\\nu _{i}}$ does nothing to the $G$ -bundle $\\mathcal {F}_{G}$ and the copy of $\\mathrm {Div}^{1}_{E}$ .",
"Therefore, this becomes $ \\boxtimes _{i \\in I} (\\overline{\\mathfrak {p}} \\times \\mathrm {id})_{!",
"}((\\overline{\\mathfrak {q}} \\times \\mathrm {id})^{*}(T_{w_{0}(\\lambda _{i}) + \\nu _{i}}(\\mathcal {F})) \\ \\otimes (j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})])) \\otimes V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ which is just $ \\boxtimes _{i \\in I} (\\mathrm {Eis}\\boxtimes \\mathrm {id})(T_{w_{0}(\\lambda _{i}) + \\nu _{i}}(\\mathcal {F})) \\otimes V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ by an application of projection formula to $j \\times \\mathrm {id}$ .",
"Since $w_{0}(\\lambda _{i})$ is the lowest weight of $V_{i}$ this implies the desired result.",
"Thus, to construct the filtration all we have to do is prove Theorem REF .",
"(Theorem REF ) For $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ , let $\\overline{\\nu } := \\sum _{i \\in I} \\nu _{i\\Gamma }$ .",
"Consider the locally closed stratum $_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} \\simeq \\mathrm {Bun}_{B} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\times \\mathrm {Div}^{I}_{E} \\subset \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ .",
"We have the natural projection $ p_{1}: \\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} \\simeq \\mathrm {Bun}_{B} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\times \\mathrm {Div}^{I}_{E} \\rightarrow \\mathrm {Div}^{(\\overline{\\nu })} $ as well as the map $ p_{2}: \\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} \\rightarrow \\mathrm {Div}^{I}_{E} \\xrightarrow{} \\mathrm {Div}^{(\\overline{\\nu })} $ where the first map is the natural projection and $\\Delta _{(\\overline{\\nu }_{i})_{i \\in I}}$ is as defined in §REF .",
"Since $\\mathrm {Div}^{(\\overline{\\nu })}$ is proper using and in particular separated, it follows that if we let $\\phantom{}_{(\\nu _{i})_{i \\in I}}(\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}) \\rightarrow \\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ be the pullback of the diagonal morphism $\\Delta _{\\mathrm {Div}^{(\\overline{\\nu })}}$ along $(p_{1},p_{2})$ that this is a closed immersion.",
"Therefore, we see that the composite $\\phantom{}_{(\\nu _{i})_{i \\in I}}(\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}) \\rightarrow \\phantom{}_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B} \\rightarrow \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ is a locally closed immersion parameterizing $(\\overline{\\kappa }^{\\hat{\\lambda }}: (\\mathcal {L}^{\\hat{\\lambda }})_{\\mathcal {F}_{T}} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }})_{\\mathcal {F}_{G}}, \\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+},(D_{i})_{i \\in I})$ such that $\\text{Coker}(\\overline{\\kappa }^{\\hat{\\lambda }})$ has torsion of length $\\langle \\hat{\\lambda }, \\nu _{i\\Gamma } \\rangle $ supported at $D_{i}$ , in the sense of Lemma REF , and a 0 nowhere else for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ and $i \\in I$ .",
"If we let $j_{(\\nu _{i})_{i \\in I}} := i_{(\\nu _{i})_{i \\in I}} \\circ (j \\times \\mathrm {id})$ then we see that this maps isomorphically onto $\\phantom{}_{(\\nu _{i})_{i \\in I}}(\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E})$ , and for varying $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}))^{I}$ these form a locally closed stratification of $\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}$ .",
"Now, by applying the excision with respect to this locally closed stratification, we obtain a filtration on $\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{Z})$ whose graded pieces are isomorphic to: $ j_{(\\nu _{i})_{i \\in I}!",
"}j_{(\\nu _{i})_{i \\in I}}^{*}\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{V}) $ Moreover, since the maps $j_{(\\nu _{i})_{i \\in I}}$ are defined over the projection to $\\mathrm {Div}^{I}_{E}$ , it follows that this filtration is $W_{E}^{I}$ -equivariant.",
"It remains to determine the $W_{E}^{I}$ -action on the graded pieces.",
"To do this, we can consider the pullback to a geometric point $x = \\mathop {\\rm Spa}(C,\\mathcal {O}_{C}) \\rightarrow \\mathrm {Div}^{I}_{E}$ , where we can regard $x$ as a point of $\\mathrm {Div}^{(\\overline{\\nu })}$ , as in §REF .",
"We recall that by definition $ K_{V} := \\phantom{}^{^{\\prime }}\\mathfrak {p}^{*}(\\mathcal {S}_{V}) \\otimes (^{\\prime }h_{G}^{\\rightarrow })^{*}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]) $ so, by Lemma REF (3), this identifies with $ R\\Gamma _{c}(\\phantom{}_{x}\\mathrm {Gr}_{G,\\le (\\lambda _{i})_{i \\in I},E} \\cap \\phantom{}_{x}\\mathrm {S}_{G,(\\nu _{i} + w_{0}(\\lambda _{i}))_{i \\in I},E}, \\mathcal {S}_{V}|_{\\phantom{}_{x}\\mathrm {Gr}_{G,\\le (\\lambda _{i})_{i \\in I},E} \\cap \\phantom{}_{x}\\mathrm {S}_{G,(\\nu _{i} + w_{0}(\\lambda _{i}))_{i \\in I},E}})[-\\langle 2\\hat{\\rho }, \\sum _{i \\in I} (\\nu _{i} + w_{0}(\\lambda _{i})) \\rangle + \\mathrm {dim}(\\mathrm {Bun}_{B})] $ We need to justify that the contribution of the $L^{+}B$ torsor $\\phantom{}_{x}\\mathcal {B}$ in Lemma REF (3) is isomorphic to $\\Lambda [-\\langle 2\\hat{\\rho }, \\sum _{i \\in I} (\\nu _{i} + w_{0}(\\lambda _{i})) \\rangle ]$ .",
"To see this, consider the case where $I = \\lbrace \\ast \\rbrace $ is a singleton, and we have elements $\\lambda \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ and $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ .",
"We consider the $L^{+}B$ -action on $\\phantom{}_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E} \\cap \\phantom{}_{x}\\mathrm {S}_{G,\\nu + w_{0}(\\lambda ),E}$ .",
"We note that $L^{+}T$ will act trivially on this space, as can be seen from the definition of the semi-infinite cells.",
"Consider the remaining unipotent part $L^{+}U$ .",
"Using the filtration on $U$ by commutator subgroups, we can write $L^{+}U$ as an iterated fibration of $L^{+}\\mathbb {G}_{a,\\hat{\\alpha }}$ indexed by the positive roots $\\hat{\\alpha }$ of $G$ .",
"One can check that the $L^{+}\\mathbb {G}_{a,\\hat{\\alpha }}$ action on $\\phantom{}_{x}\\mathrm {Gr}_{G,\\le \\lambda ,E} \\cap \\phantom{}_{x}\\mathrm {S}_{G,\\nu + w_{0}(\\lambda ),E}$ factors through the truncated loop group $L^{+}_{n_{\\hat{\\alpha }}}\\mathbb {G}_{a,\\hat{\\alpha }}$ , where $n_{\\hat{\\alpha }} = \\langle \\hat{\\alpha }, \\nu + w_{0}(\\lambda ) \\rangle $ (See for example the proof of ).",
"If we let $\\mathcal {O}_{X,x}$ denote the completed local ring at the fixed untilt $x$ , with uniformizing parameter $t_{x}$ then, by writing $\\mathcal {O}_{X,x}/t_{x}^{n_{\\hat{\\alpha }}}$ as an iterated extension of $\\mathcal {O}_{X,x}/t_{x} \\simeq C$ , we can describe $L^{+}_{n_{\\hat{\\alpha }}}\\mathbb {G}_{a,\\hat{\\alpha }}$ as a fibration of $(\\mathbb {A}^{1}_{C})^{\\diamond }$ iterated $n_{\\hat{\\alpha }}$ times.",
"This tells us that the compactly supported cohomology of $L^{+}_{n_{\\hat{\\alpha }}}\\mathbb {G}_{a,\\hat{\\alpha }}$ over $\\mathop {\\rm Spa}(C,\\mathcal {O}_{C})$ is isomorphic to $\\Lambda [-2n_{\\hat{\\alpha }}]$ .",
"As a consequence, we deduce that the contribution of $\\phantom{}_{x}\\mathcal {B}$ to the above formula is $\\Lambda [-\\sum _{\\hat{\\alpha } > 0} n_{\\hat{\\alpha }}] = \\Lambda [-\\langle 2\\hat{\\rho }, \\nu + w_{0}(\\lambda ) \\rangle ]$ , by Künneth.",
"Now, by Corollary REF , $j_{(\\nu _{i})_{i \\in I}}^{*}\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{V})$ identifies with $ \\boxtimes _{i \\in I} V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, (w_{0}(\\lambda _{i}) + \\nu _{i}) \\rangle )[-\\langle 2\\hat{\\rho }, \\sum _{i \\in I} (w_{0}(\\lambda _{i}) + \\nu _{i}) \\rangle + \\langle 2\\hat{\\rho }, \\sum _{i \\in I} (w_{0}(\\lambda _{i}) + \\nu _{i}) \\rangle + \\mathrm {dim}(\\mathrm {Bun}_{B})] $ or rather $ \\boxtimes _{i \\in I} V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i}) \\rangle )[\\mathrm {dim}(\\mathrm {Bun}_{B})] $ and so we get the desired result.",
"It remains to see that this filtration satisfies the desired compatibilities.",
"Consider a map of finite index sets $\\pi : I \\rightarrow J$ .",
"For $j \\in J$ , we set $I_{j} := \\pi ^{-1}(j)$ and consider the natural map $\\Delta _{IJ}: \\mathrm {Div}^{J}_{E} \\rightarrow \\mathrm {Div}^{I}_{E}$ , which diagonally embeds the $j$ th copy of $\\mathrm {Div}^{1}_{E}$ in $\\mathrm {Div}^{J}_{E}$ into $\\mathrm {Div}^{I_{j}}_{E}$ .",
"Attached to this, we have a Cartesian diagram $ \\begin{tikzcd}\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{J}_{E} [d,\"\\mathrm {id} \\times \\Delta _{IJ}\"] & Z^{J}_{(\\lambda _{j})_{j \\in J}} [l,\"\\phi _{(\\lambda _{j})_{j \\in J}}\"] [d,\"\\tilde{\\Delta }_{IJ}\"]\\\\\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} & Z^{I}_{(\\lambda _{i})_{i \\in I}} [l,\"\\phi _{(\\lambda _{i})_{i \\in I}}\"]\\end{tikzcd}$ where $\\lambda _{j} := \\sum _{i \\in I_{j}} \\lambda _{i}$ for all $j \\in J$ .",
"Base change gives us a natural isomorphism: $(\\mathrm {id} \\times \\Delta _{IJ})^{*}\\phi _{(\\lambda _{i})_{i \\in I}!",
"}(K_{\\boxtimes _{i \\in I} V_{i}}) \\simeq \\phi _{(\\lambda _{j})_{j \\in J}!",
"}\\tilde{\\Delta }_{IJ}^{*}(K_{\\boxtimes _{i \\in I} V_{i}})$ However, by the relationship between fusion product and tensor product under Theorem REF , we deduce that $\\tilde{\\Delta }_{IJ}^{*}(K_{\\boxtimes _{i \\in I} V_{i}}) \\simeq K_{\\boxtimes _{j \\in J} V_{j}}$ , where $V_{j} := \\otimes _{i \\in I_{j}} V_{i}$ .",
"We now compare the two filtrations on the LHS and the RHS of this isomorphism.",
"To do this, we define $(\\nu _{j})_{j \\in J}$ by $\\nu _{j} := \\sum _{i \\in I_{j}} \\nu _{i}$ .",
"We note that we have a natural Cartesian diagram $ \\begin{tikzcd}\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{J}_{E} [r,\"i_{(\\nu _{j})_{j \\in J}}\"] [d,\"\\mathrm {id} \\times \\Delta _{IJ}\"] & \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{J}_{E} [d,\"\\mathrm {id} \\times \\Delta _{IJ}\"] \\\\\\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E} [r,\"i_{(\\nu _{i})_{i \\in I}}\"]& \\overline{\\mathrm {Bun}}_{B} \\times \\mathrm {Div}^{I}_{E}\\end{tikzcd}$ On the LHS of (5), we have a filtration with graded pieces isomorphic to $ (\\mathrm {id} \\times \\Delta _{IJ})^{*}i_{(\\nu _{i})_{i \\in I}!",
"}(j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\boxtimes _{i \\in I} V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ which is naturally isomorphic to $ i_{(\\nu _{j})_{j \\in J}!",
"}(\\mathrm {id} \\times \\Delta _{IJ})^{*}(j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\boxtimes _{i \\in I} V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) $ by base-change applied to the previous Cartesian square.",
"We can further rewrite this as $ i_{(\\nu _{j})_{j \\in J}!",
"}\\circ (j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\boxtimes _{j \\in J} \\otimes _{i \\in I_{j}} V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle ) ) $ On the other hand, for such a $(\\nu _{j})_{j \\in J}$ , the RHS of (5) has a filtration with graded pieces isomorphic to $ i_{(\\nu _{j})_{j \\in J}!",
"}(j \\times \\mathrm {id})_{!",
"}(\\Lambda [\\mathrm {dim}(\\mathrm {Bun}_{B})]) \\otimes \\boxtimes _{j \\in J} V_{j}(w_{0}(\\lambda _{j}) + \\nu _{j})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{j}) + \\nu _{j} \\rangle )) $ but now note that $ V_{j}(w_{0}(\\lambda _{j}) + \\nu _{j})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{j}) + \\nu _{j} \\rangle )) = \\bigoplus _{\\begin{array}{c} (\\nu _{i})_{i \\in I} \\in \\Lambda _{G,B}^{I} \\\\ \\sum _{i \\in I_{j}} \\nu _{i} = \\nu _{j}\\end{array}} \\bigotimes _{i \\in I_{j}} V_{i}(w_{0}(\\lambda _{i}) + \\nu _{i})(-\\langle \\hat{\\rho }, w_{0}(\\lambda _{i}) + \\nu _{i} \\rangle )) $ for all $j \\in J$ .",
"Therefore, the graded piece indexed by $(\\nu _{j})_{j \\in J}$ on the RHS have a split filtration with graded pieces isomorphic to the graded pieces coming from the filtration on the LHS.",
"The comaptibility of these two filtrations now follows from Corollary REF , and the fact that the filtration came from restricting the sheaf $\\mathcal {S}_{V}$ to semi-infinite cells.",
"Now, we can reap the fruit of this section using the filtered eigensheaf property to get some control on the stalks of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ ."
],
[
"Consequences of the Filtered Eigensheaf Property",
"First, we note, by applying Theorem REF when $\\mathcal {F} = \\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}$ together with Corollary REF , we obtain the following.",
"Corollary 7.4 For all finite index sets $I$ and $V = \\boxtimes _{i \\in I} V_{i} \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G^{I})$ , the sheaf $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ admits a $W_{\\mathbb {Q}_{p}}^{I}$ -equivariant filtration indexed by $(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}$ .",
"The filtration's graded pieces are isomorphic to $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\otimes \\boxtimes _{i \\in I} (\\nu _{i} \\circ \\phi _{T}) \\otimes V_{i}(\\nu _{i})$ .",
"The filtration is natural in $I$ and $V$ , as well as compatible with compositions and exterior tensor products in $V$ .",
"In particular, we note that the direct sum of the graded pieces of the filtration on $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ is isomorphic to $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\otimes \\bigoplus _{(\\nu _{i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})/\\Gamma )^{I}} \\boxtimes _{i \\in I} \\nu _{i} \\circ \\phi _{T} \\otimes V_{i}(\\nu _{i}) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\boxtimes r_{V} \\circ \\phi $ as sheaves in $\\mathrm {D}(\\mathrm {Bun}_{G})^{BW^{I}_{\\mathbb {Q}_{p}}}$ , where $\\phi $ is the parameter $\\phi _{T}$ composed with the natural embedding $\\phantom{}^{L}T \\rightarrow \\phantom{}^{L}G$ .",
"In other words, $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ is a filtered eigensheaf with eigenvalue $\\phi $ .",
"We now would like to use this to deduce some consequences about the stalks of the Eisenstein series $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ .",
"In particular, let's consider some Schur irreducible constituent $A$ of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ .",
"We will now need our assumption that the prime $\\ell $ is very good.",
"Under this assumption, the excursion algebra will define endomorphims of $A$ which determine and are determined by the parameter $\\phi _{A}^{\\mathrm {FS}}$ , as in .",
"Since the excursion algebra is determined by natural transformations of Hecke operators, the filtered Hecke eigensheaf property tells us that these scalars must be specified by $\\phi $ , so that we have an equality: $\\phi = \\phi _{A}^{\\mathrm {FS}}$ , as conjugacy classes of semi-simple parameters.",
"In particular, if we consider $b \\in B(G)$ and look at the restriction $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ to the locally closed HN-strata $\\mathrm {Bun}_{G}^{b} \\subset \\mathrm {Bun}_{G}$ indexed by $b$ .",
"Then, by , we have a canonical isomorphism $\\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ .",
"By the previous discussion and compatibility of the Fargues-Scholze correspondence with restriction to $J_{b}$ , we deduce that any irreducible constituent $\\rho $ of the restriction $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ has Fargues-Scholze parameter $\\phi _{\\rho }^{\\mathrm {FS}}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}J_{b}(\\Lambda )$ equal to $\\phi $ under the appropriately Tate twisted embedding: $ \\phantom{}^{L}J_{b}(\\Lambda ) \\rightarrow \\phantom{}^{L}G(\\Lambda ) $ Now, since our parameter $\\phi $ is induced from the maximal torus, we would like to say that this is impossible unless $J_{b}$ itself admits a maximal torus, which is in turn equivalent to assuming that $b$ is unramified.",
"Here we need to be a bit careful.",
"In particular, if we consider $G = \\mathrm {GL}_{2}$ and $b$ the element of slope $\\frac{1}{2}$ then $J_{b} = D^{*}_{\\frac{1}{2}}$ the units in the quaternion division algebra.",
"The trivial representation $\\mathbf {1}$ of $D^{*}_{\\frac{1}{2}}$ has Fargues-Scholze parameter given by $ W_{\\mathbb {Q}_{p}} \\rightarrow \\mathrm {GL}_{2}(\\Lambda ) $ $ g \\mapsto \\begin{pmatrix} |g|^{1/2} & 0 \\\\ 0 & |g|^{-1/2} \\end{pmatrix} $ This parameter is induced from a maximal torus of $\\mathrm {GL}_{2}$ ; however, it is not generic.",
"In particular, the composite of this parameter with the unique simple root defined by the upper triangular Borel gives a Galois representation isomorphic to the norm character $|\\cdot |$ .",
"Therefore, one might hope that assuming compatibility of some suitably nice form of the local Langlands correspondence for $G$ with the Fargues-Scholze correspondence together with genericity of $\\phi _{T}$ is enough to give us the desired description of the stalks.",
"This is indeed the case.",
"The assumption we need is as follows.",
"Assumption 7.5 For a connected reductive group $H/\\mathbb {Q}_{p}$ , we have: $\\Pi (H)$ the set of smooth irreducible $\\overline{\\mathbb {Q}}_{\\ell }$ -representations of $H(\\mathbb {Q}_{p})$ , $\\Phi (H)$ the set of conjugacy classes of continuous semisimple maps $ W_{\\mathbb {Q}_{p}} \\times \\mathrm {SL}(2,\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\phantom{}^{L}H(\\overline{\\mathbb {Q}}_{\\ell }) $ where $\\overline{\\mathbb {Q}}_{\\ell }$ has the discrete topology and $\\mathrm {SL}(2,\\overline{\\mathbb {Q}}_{\\ell })$ acts via an algebraic representation and the map respects the action of $W_{\\mathbb {Q}_{p}}$ on $\\phantom{}^{L}H(\\overline{\\mathbb {Q}}_{\\ell })$ , the $L$ -group of $H$ .",
"$\\Phi ^{\\mathrm {ss}}(H)$ the set of continuous semi-simple homomorphisms $ W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}H(\\overline{\\mathbb {Q}}_{\\ell }) $ $(-)^{\\mathrm {ss}}: \\Phi (G) \\rightarrow \\Phi ^{\\mathrm {ss}}(G)$ the map defined by precomposition with $ W_{\\mathbb {Q}_{p}} \\rightarrow W_{\\mathbb {Q}_{p}} \\times \\mathrm {SL}(2,\\overline{\\mathbb {Q}}_{\\ell }) $ $ g \\mapsto (g,\\begin{pmatrix} |g|^{1/2} & 0 \\\\ 0 & |g|^{-1/2} \\end{pmatrix}) $ Then, we assume, for all $b \\in B(G)$ , that there exists a map $ \\mathrm {LLC}_{b}: \\Pi (J_{b}) \\rightarrow \\Phi (J_{b}) $ $ \\rho \\mapsto \\phi _{\\rho } $ satisfying the following properties: The diagram $ \\begin{tikzcd}[ampersand replacement=\\&]\\Pi (J_{b}) [rr, \"\\mathrm {LLC}_{b}\"] [drr,\"\\mathrm {LLC}^{\\mathrm {FS}}_{b}\"] \\& \\& \\Phi (J_{b}) [d,\"(-)^{\\mathrm {ss}}\"] \\\\\\& \\& \\Phi ^{\\mathrm {ss}}(J_{b})\\end{tikzcd}$ commutes, where $\\mathrm {LLC}_{b}^{\\mathrm {FS}}$ is the Fargues-Scholze local Langlands correspondence for $J_{b}$ .",
"Consider $\\phi _{\\rho }$ as an element of $\\Phi (G)$ given by composing with the twisted embedding $\\phantom{}^{L}J_{b}(\\overline{\\mathbb {Q}}_{\\ell }) \\simeq \\phantom{}^{L}M_{b}(\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ (as defined in ).",
"Then $\\phi _{\\rho }$ factors through the natural embedding $\\phantom{}^{L}T \\rightarrow \\phantom{}^{L}G$ if and only if $b \\in B(G)_{\\mathrm {un}}$ .",
"If $\\rho $ is a representation such that $\\mathcal {L}_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}J_{b}(\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ factors through $\\phantom{}^{L}T$ , where the last map is the twisted embedding then, by (2), the element $b$ is unramified, and we require that $\\rho $ is isomorphic to an irreducible constituent of $i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}$ for $w \\in W_{b}$ and $\\chi $ the character attached to the induced toral parameter $\\phi _{T}$ , where $\\delta _{P_{b}}^{1/2}$ is the modulus character of $M_{b} \\simeq J_{b}$ .",
"Remark 7.6 This assumption might seem a bit daunting, but is verifiable in many cases.",
"In particular, the first assumption follows from the compatibility of the Fargues-Scholze correspondence with the Harris-Taylor correspondence for groups of type $A_{n}$ and its inner forms ().",
"Similarly, for groups of type $C_{2}$ and their inner forms over a unramified extension $L$ with $p > 2$ , this follows from the main theorem of , and, for odd unramified unitary groups over $\\mathbb {Q}_{p}$ this follows from the main theorem of .",
"The methods employed in these two papers should generalize to at least a few other cases.",
"Assumption (2) is also a standard and verifiable conjecture in the cases where the local Langlands correspondence is known to exist.",
"If $\\rho $ is a representation such that $\\phi _{\\rho }$ factors through $\\phantom{}^{L}T$ then we are claiming that $J_{b}$ has a Borel subgroup.",
"For non quasi-split groups, it is conjectured that one should only consider $L$ -parameters coming from the $L$ -groups of the Levi subgroups of the non quasi-split group.",
"These are referred to as relevant $L$ -parameters.",
"In particular, one expects the $L$ -packets of $\\mathrm {LLC}_{b}$ over irrelevant $\\phi $ to be empty (See for example ), so if $\\phi _{\\rho }$ factors through $\\phantom{}^{L}T$ for some $\\rho $ under $\\mathrm {LLC}_{b}$ it should imply that the group $J_{b}$ has a Borel $B_{b}$ .",
"Assumption (3) is just the expectation that, when $\\phi _{\\rho }$ factors through $\\phantom{}^{L}T$ , the members of the $L$ -packet should be given by the irreducible constituents of the parabolic inductions from $T$ to $J_{b}$ , where the Weyl group twists appear since the Weyl group conjugates of $\\phi _{\\rho }: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}J_{b}(\\overline{\\mathbb {Q}}_{\\ell }) \\simeq \\phantom{}^{L}M_{b}(\\overline{\\mathbb {Q}}_{\\ell })$ all map to the same parameter when viewed as a parameter valued in $\\phantom{}^{L}G$ , and the modulus twist by $\\delta _{P_{b}}^{1/2}$ appears since we are comparing this to an $L$ -parameter of $G$ via the twisted embedding $\\phantom{}^{L}J_{b}(\\overline{\\mathbb {Q}}_{\\ell }) \\simeq \\phantom{}^{L}M_{b}(\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {Q}}_{\\ell })$ .",
"Under this assumption, we will deduce our main Corollary of the filtered eigensheaf property.",
"Corollary 7.7 Under Assumption REF , consider $b \\in B(G)$ with corresponding locally closed HN-strata $\\mathrm {Bun}_{G}^{b} \\subset \\mathrm {Bun}_{G}$ .",
"For $\\phi $ a generic parameter, the following is true.",
"If $b \\notin B(G)_{\\mathrm {un}}$ the restriction $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} $ vanishes.",
"If $b \\in B(G)_{\\mathrm {un}}$ is an unramified element and $\\rho $ is a smooth irreducible $\\overline{\\mathbb {F}}_{\\ell }$ -representation occurring as a constituent of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ then $\\rho $ is an irreducible constituent of $i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}$ for some $w \\in W_{b}$ .",
"As noted above, $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ will be valued in an unbounded complex of smooth $\\overline{\\mathbb {F}}_{\\ell }$ -representations of the $\\sigma $ -centralizer $J_{b}$ of $b$ .",
"If we consider a smooth irreducible constituent of this restriction $\\rho $ then, as already discussed above, it follows that the Fargues-Scholze parameter $\\phi _{\\rho }^{\\mathrm {FS}}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}J_{b}(\\overline{\\mathbb {F}}_{\\ell })$ under the twisted embedding $ \\phantom{}^{L}J_{b}(\\overline{\\mathbb {F}}_{\\ell }) \\rightarrow \\phantom{}^{L}G(\\overline{\\mathbb {F}}_{\\ell }) $ agrees with $\\phi $ .",
"Using , we can choose $\\tilde{\\rho }$ a lift of $\\rho $ to a smooth irreducible $\\overline{\\mathbb {Q}}_{\\ell }$ -representation admitting a $J_{b}(\\mathbb {Q}_{p})$ -stable $\\overline{\\mathbb {Z}}_{\\ell }$ -lattice such that $\\rho $ occurs as a subquotient of $\\tilde{\\rho }$ mod $\\ell $ .",
"Since the Fargues-Scholze correspondence is compatible with reduction mod $\\ell $ , it follows that the Fargues-Scholze parameter $\\phi _{\\tilde{\\rho }}^{\\mathrm {FS}}$ factors through $\\phantom{}^{L}G(\\overline{\\mathbb {Z}}_{\\ell })$ and that it equals $\\phi _{\\rho }^{\\mathrm {FS}}$ mod $\\ell $ .",
"Now, since $\\phi _{\\rho }^{\\mathrm {FS}}$ factors through $\\phantom{}^{L}T$ and induces a generic parameter, we claim that the same is true for $\\phi _{\\tilde{\\rho }}^{\\mathrm {FS}}$ .",
"This follows through standard deformation theory.",
"In particular, if $ H^{1}(W_{\\mathbb {Q}_{p}},\\alpha \\circ \\phi _{T}) $ vanishes for all $\\Gamma $ -orbits $\\alpha $ of roots, then any lift of $\\phi _{\\rho }^{\\mathrm {FS}}$ will factor through $\\phantom{}^{L}T$ , but this vanishing is guaranteed by $\\phi _{T}$ being generic (cf.",
").",
"It is also easy to see that $\\phi _{\\tilde{\\rho }}^{\\mathrm {FS}}$ must be generic since its mod $\\ell $ reduction is.",
"Now, by Assumption REF , we note that $\\phi _{\\tilde{\\rho }}^{\\mathrm {FS}}$ is the semi-simplification of $\\phi _{\\tilde{\\rho }}$ , the $L$ -parameter attached to $\\tilde{\\rho }$ , but by Lemma REF and genericity that implies that $\\phi _{\\tilde{\\rho }}|_{W_{\\mathbb {Q}_{p}}} = \\phi _{\\tilde{\\rho }}^{\\mathrm {FS}}$ .",
"The two claims now follow from Assumptions REF (2) and (3), respectively.",
"This statement will allow us to give a complete description of the eigensheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ for $\\phi _{T}$ satisfying the slightly stronger condition of weak normalized regularity.",
"In particular, as we will see in §9, for the restrictions of the sheaf to $\\mathrm {Bun}_{G}^{b}$ for $b \\in B(G)_{\\mathrm {un}}$ , we will always be able to evaluate the stalks in terms of normalized parabolic inductions of Weyl group translates of the character $\\chi $ ,and, by the previous Lemma, we know these are the only possible non-zero stalks.",
"Now we turn our attention to studying how the geometric Eisenstein functor interacts with Verdier duality."
],
[
"Eisenstein Series and Verdier Duality",
"We would like to study how the normalized Eisenstein functor interacts with Verdier duality.",
"This will be done assuming the following claim.",
"Assumption 8.1 We assume that the sheaf $j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ is ULA with respect to the morphism defined by $\\overline{\\mathfrak {q}}$ , in the sense of .",
"Remark 8.2 A proof of this claim should appear in upcoming work of Hansen-Scholze , where this statement is used in their proof of the Harris-Viehmann conjecture.",
"We assume this claim, and use it to show that this implies that our Eisenstein functor commutes with Verdier duality when $\\phi _{T}$ is weakly generic.",
"In particular, if $\\mathbb {D}_{\\mathrm {Bun}_{G}}$ (resp.",
"$\\mathbb {D}_{\\mathrm {Bun}_{T}}$ ) denotes Verdier duality on $\\mathrm {Bun}_{G}$ (resp.",
"$\\mathrm {Bun}_{T}$ ), our main goal is to prove the following.",
"Theorem 8.3 Assuming REF then, for $\\phi _{T}$ a weakly generic toral parameter, there is an isomorphism of objects in $\\mathrm {D}(\\mathrm {Bun}_{G})$ $ \\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}})) $ where we note that $\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\mathcal {S}_{\\phi _{T}^{\\vee }}$ , if $\\phi _{T}^{\\vee }$ denotes the parameter dual to $\\phi _{T}$ .",
"First, let's record some implications of Assumption REF .",
"Lemma 8.4 The sheaf $j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ is also ULA with respect to $\\overline{\\mathfrak {q}}$ , and we have isomorphisms $ \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}) $ and $ \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}) $ of objects in $\\mathrm {D}(\\overline{\\mathrm {Bun}}_{B})$ .",
"The first claimed isomorphism just follows from Corollary REF , and the fact that we always have an isomorphism of derived functors $\\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}} \\circ j_{!}",
"\\simeq j_{*} \\circ \\mathbb {D}_{\\mathrm {Bun}_{B}}$ , by projection formula.",
"Now, since the dualizing object on $\\mathrm {Bun}_{T}$ is just the constant sheaf, by Lemma REF and the fact that $T$ is unimodular, it follows that we have a unique (up to fixing Haar measures on $T(\\mathbb {Q}_{p})$ ) isomorphism $\\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}} \\simeq \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}/\\mathrm {Bun}_{T}}$ .",
"Therefore, by , we can show that $j_{*}(\\mathbb {D}_{\\mathrm {Bun}_{B}}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}) \\simeq \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}/\\mathrm {Bun}_{T}}(j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}))$ is also ULA with respect to $\\overline{\\mathfrak {q}}$ , where the first isomorphism follows from Corollary REF .",
"The second claimed isomorphism now follows from the first and the fact that ULA objects are reflexive with respect to Verdier duality, again by .",
"We will now combine this with the following lemma.",
"Lemma 8.5 Let $f: X \\rightarrow S$ be a map of decent $v$ -stacks which are fine over a base $\\ast $ .",
"Suppose that $A$ is ULA with respect to $f$ and $B \\in \\mathrm {D}(S)$ , then we have a natural isomorphism $ \\mathbb {D}_{X}(A \\otimes f^{*}(B)) \\simeq \\mathbb {D}_{X}(A) \\otimes f^{*}(\\mathbb {D}_{S}(B)) $ in $\\mathrm {D}(X)$ .",
"Let $g: S \\rightarrow \\ast $ denote the structure morphism.",
"It follows, by and $A$ being ULA, that we can rewrite the RHS of the above isomorphism as $ \\mathbb {D}_{X}(A) \\otimes f^{*}(\\mathbb {D}_{S}(B)) \\simeq R\\mathcal {H}om(A,f^{!",
"}(\\mathbb {D}_{S}(B))) $ which in turn is equal to $ R\\mathcal {H}om(A,f^{!",
"}(R\\mathcal {H}om(B,g^{!",
"}(\\Lambda )))) $ by definition.",
"Now, by projection formula, we can further rewrite this as $ R\\mathcal {H}om(A,R\\mathcal {H}om(f^{*}(B),f^{!}g^{!",
"}(\\Lambda ))) $ but, by Hom-Tensor duality, this is just $ R\\mathcal {H}om(A \\otimes f^{*}(B),f^{!}g^{!",
"}(\\Lambda )) = \\mathbb {D}_{X}(A \\otimes f^{*}(B)) $ as desired.",
"Combining the previous two Lemmas, we deduce the following.",
"Corollary 8.6 There is an isomorphism $ \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}}) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq \\overline{\\mathfrak {q}}^{*}(\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}})) \\otimes \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\otimes j_*(\\mathrm {IC}_{\\mathrm {Bun}_{B}})$ of objects in $\\mathrm {D}(\\overline{\\mathrm {Bun}}_{B})$ .",
"Similarly, using Lemma REF , we also have an isomorphism $ \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}}) \\otimes j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}) $ of objects in $\\mathrm {D}(\\overline{\\mathrm {Bun}}_{B})$ .",
"Now let's apply these results to prove Theorem REF .",
"(Theorem REF ) First note that, using projection formula with respect to $j$ , we have an isomorphism: $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) = \\mathfrak {p}_{!",
"}(\\mathfrak {q}^{*}(\\mathcal {S}_{\\phi _{T}}) \\otimes \\mathrm {IC}_{\\mathrm {Bun}_{B}}) = \\overline{\\mathfrak {p}}_{!}j_{!",
"}(j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}}) \\otimes \\mathrm {IC}_{\\mathrm {Bun}_{B}}) \\simeq \\overline{\\mathfrak {p}}_{!",
"}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}}) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) $ Now, by Proposition REF , we have that $\\overline{\\mathfrak {p}}_{!",
"}$ is equivalent to $\\overline{\\mathfrak {p}}_{*}$ , this means that we have an isomorphism $ \\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\overline{\\mathfrak {p}}_{!",
"}(\\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}}) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}))) $ but now, by Corollary REF , this is isomorphic to $\\overline{\\mathfrak {p}}_{!",
"}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\otimes j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}))$ Therefore, we need to exhibit an isomorphism between this sheaf and $\\mathrm {nEis}(\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\overline{\\mathfrak {p}}_{!",
"}(\\overline{\\mathfrak {q}}^{*}(\\mathbb {D}_{\\mathrm {Bun}_{T}}(\\mathcal {S}_{\\phi _{T}})) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\simeq \\overline{\\mathfrak {p}}_{!",
"}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}))$ In other words, we need to show that the cone of the natural map $ \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\otimes j_{!",
"}(\\mathrm {IC}_{\\mathrm {Bun}_{B}})) \\rightarrow \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\otimes j_{*}(\\mathrm {IC}_{\\mathrm {Bun}_{B}}))$ is trivial after applying $\\overline{\\mathfrak {p}}_{!",
"}$ .",
"We will do this by factorizing (8).",
"In particular, note, by applying projection formula to $j$ as above and rewriting $\\mathrm {IC}_{\\mathrm {Bun}_{B}} \\simeq \\mathfrak {q}^{*}(\\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})] \\simeq j^{*}\\overline{\\mathfrak {q}}^{*}(\\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})]$ , we see that $(7)$ is isomorphic to $ \\overline{\\mathfrak {p}}_{!}(j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }} \\otimes \\Delta _{B}^{1/2}))[\\mathrm {dim}(\\mathrm {Bun}_{B})] $ and therefore applying Verdier duality and Theorem REF it follows that $(6)$ is isomorphic to $ \\overline{\\mathfrak {p}}_{!",
"}(j_{*}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }} \\otimes \\Delta _{B}^{1/2}))[\\mathrm {dim}(\\mathrm {Bun}_{B})] $ Therefore, we can rewrite the map (8) as $ j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\rightarrow j_{*}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2})) $ Now note that we can factorize this morphism via the adjunction maps as $ j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }} \\otimes \\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})] \\xrightarrow{} \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }} \\otimes \\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})] \\xrightarrow{} j_{*}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }} \\otimes \\Delta _{B}^{1/2}))[\\mathrm {dim}(\\mathrm {Bun}_{B})] $ By the Octahdral axiom, it suffices to show the cone of $(1)$ and $(2)$ are trivial after applying $\\overline{\\mathfrak {p}}_{!",
"}$ .",
"The cone of (1) is relatively easy to get a handle on, but the cone of (2) is more tricky.",
"To do this, we note it suffices to show the claim after applying Verdier duality on $\\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}$ .",
"This follows because $\\overline{\\mathfrak {p}}_{!}",
"\\simeq \\overline{\\mathfrak {p}}_{*}$ and Verdier duality on $\\mathrm {Bun}_{G}$ can be checked to be a conservative functor.",
"In particular, one can use the semi-orthogonal decomposition of $\\mathrm {D}(\\mathrm {Bun}_{G})$ into the HN-strata $\\mathrm {D}(\\mathrm {Bun}_{G}^{b})$ to reduce it to the claim that smooth duality on the unbounded derived category of smooth $\\overline{\\mathbb {F}}_{\\ell }$ -representations of a $p$ -adic reductive group is conservative.",
"Now, by the above discussion, the Verdier Dual of the target of $(2)$ is equal to $j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})]$ .",
"Hence, we deduce that the Verdier dual of (2) is equal to a map $ j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2})) \\rightarrow \\mathbb {D}_{\\overline{\\mathrm {Bun}}_{B}}(\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}^{\\vee }} \\otimes \\Delta _{B}^{1/2})) \\simeq \\overline{\\mathfrak {q}}^{!",
"}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{-1/2}) $ where $\\Delta _{B}^{-1/2}$ is the sheaf on $\\mathrm {Bun}_{T}$ whose restriction to each connected component is given by the character $\\delta _{B}^{-1/2}$ .",
"As a quick sanity check, note that, by Theorem REF , we have an isomorphism $j^{*}\\overline{\\mathfrak {q}}^{!",
"}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{-1/2}) \\simeq j^{!",
"}\\overline{\\mathfrak {q}}^{!",
"}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{-1/2}) \\simeq \\mathfrak {q}^{!",
"}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{-1/2} \\otimes \\Delta _{B})[\\mathrm {dim}(\\mathrm {Bun}_{B})] \\simeq \\mathfrak {q}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2})[\\mathrm {dim}(\\mathrm {Bun}_{B})]$ , and so we can see that this is the natural map coming from adjunction.",
"Now, observe that $\\phi _{T}$ is weakly generic if and only if $\\phi _{T}^{\\vee }$ is weakly generic (since taking duals just exchanges the role of positive and negative roots).",
"Therefore, it suffices to show the following.",
"Lemma 8.7 Assume that $\\phi _{T}$ is weakly generic toral parameter, then the cone of the morphism $ j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\simeq j_{!",
"}\\mathfrak {q}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\rightarrow \\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) $ vanishes after applying $\\overline{\\mathfrak {p}}_{!",
"}$ .",
"Similarly, the cone of the natural map $ j_{!",
"}j^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\simeq j_{!",
"}\\mathfrak {q}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\rightarrow \\overline{\\mathfrak {q}}^{!",
"}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{-1/2}) $ vanishes after applying $\\overline{\\mathfrak {p}}_{!",
"}$ .",
"We start with the first map.",
"By excision, it suffices to show that the restriction to the locally closed stratum $_{\\overline{\\nu }}\\overline{\\mathrm {Bun}}_{B}$ vanishes for all $\\overline{\\nu } \\in \\Lambda _{G,B}^{\\text{pos}} \\setminus \\lbrace 0\\rbrace $ after applying $\\overline{\\mathfrak {p}}_{!",
"}$ .",
"In particular, we are tasked with computing $ \\overline{\\mathfrak {p}}_{!",
"}j_{\\overline{\\nu }!",
"}j_{\\overline{\\nu }}^{*}\\overline{\\mathfrak {q}}^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) $ for all such $\\overline{\\nu }$ .",
"Note that the composite map $ \\mathrm {Bun}_{B} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\xrightarrow{} \\overline{\\mathrm {Bun}}_{B} \\xrightarrow{} \\mathrm {Bun}_{T} $ can be identified with $ \\mathrm {Bun}_{B} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\xrightarrow{} \\mathrm {Bun}_{T} \\times \\mathrm {Div}^{(\\overline{\\nu })} \\xrightarrow{} \\mathrm {Bun}_{T} $ where the last map is the Hecke operator defined in §REF .",
"Therefore, we are reduced to computing $ \\overline{\\mathfrak {p}}_{!",
"}j_{\\overline{\\nu }!",
"}(\\mathfrak {q} \\times \\mathrm {id})^{*}(h_{(\\overline{\\nu })}^{\\rightarrow })^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) \\simeq \\overline{\\mathfrak {p}}_{!",
"}j_{\\overline{\\nu }!",
"}(\\mathfrak {q} \\times \\mathrm {id})^{*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2} \\boxtimes E_{\\phi _{T}}^{(\\overline{\\nu })}(\\langle \\hat{\\rho }, \\overline{\\nu } \\rangle )) $ However, note that we have an equality: $\\overline{\\mathfrak {p}} \\circ j_{\\overline{\\nu }} = \\mathfrak {p} \\times g$ , where $g: \\mathrm {Div}^{(\\overline{\\nu })} \\rightarrow \\ast $ is the structure map.",
"Therefore, by Künneth formula, we obtain that the RHS can be identified with $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\otimes R\\Gamma _{c}(\\mathrm {Div}^{(\\overline{\\nu })}, E_{\\phi _{T}}^{(\\overline{\\nu })}(\\langle \\hat{\\rho }, \\overline{\\nu } \\rangle )) $ but now it follows, by Corollary REF and the weak genericity assumption on $\\phi _{T}$ , that the complex $R\\Gamma _{c}(\\mathrm {Div}^{(\\overline{\\nu })},E_{\\phi _{T}}^{(\\overline{\\nu })}(\\langle \\hat{\\rho }, \\overline{\\nu } \\rangle ))$ is trivial.",
"Now for the second map we argue similarly.",
"In particular, it suffices to show that $\\overline{\\mathfrak {q}}^{!",
"}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{-1/2})$ vanishes after applying $\\overline{\\mathfrak {p}}_{!",
"}j_{\\overline{\\nu }*}j_{\\overline{\\nu }}^{!",
"}$ , but then, as before, we observe that $j_{\\overline{\\nu }}^{!}",
"\\circ \\overline{\\mathfrak {q}}^{!}",
"= (\\mathfrak {q} \\times \\mathrm {id})^{!}",
"\\circ (h_{\\overline{\\nu }}^{\\rightarrow })^{!",
"}$ .",
"By uniformizing $\\mathrm {Div}^{1}$ by punctured positive Banach-Colmez spaces, the dualizing object on $\\mathrm {Div}^{(\\overline{\\nu })}$ can be identified up to a shift with $\\Lambda (\\langle \\hat{\\rho }, \\overline{\\nu } \\rangle )$ .",
"It follows that $h_{\\overline{\\nu }}^{!",
"}(\\Delta _{B}^{-1/2} \\otimes \\mathcal {S}_{\\phi _{T}})$ can be identified up to a shift with $E_{\\phi _{T}}^{(\\overline{\\nu })}(-\\langle \\hat{\\rho }, \\overline{\\nu } \\rangle + \\langle \\hat{\\rho }, \\overline{\\nu } \\rangle ) = E_{\\phi _{T}}^{(\\overline{\\nu })}$ .",
"Similarly, we can compute $(\\mathfrak {q} \\times \\mathrm {id})^{!",
"}$ using Theorem REF , and this will not effect the $\\mathrm {Div}^{(\\overline{\\nu })}$ -factor.",
"Using that $\\overline{\\mathfrak {p}}_{!}",
"\\circ j_{\\overline{\\nu }*} \\simeq \\overline{\\mathfrak {p}}_{*} \\circ j_{\\overline{\\nu }*} = (\\mathfrak {p} \\times g)_{*}$ together with the properness of $g$ , we can apply Künneth to see the desired vanishing follows from the vanishing of the complex $ R\\Gamma _{c}(\\mathrm {Div}^{(\\overline{\\nu })},E_{\\phi _{T}}^{(\\overline{\\nu })}) $ which again follows from Corollary REF .",
"For the rest of the paper, we will assume REF and thereby the validity of Theorem REF .",
"In addition, we will assume compatibility with a suitably nice form of the local Langlands correspondence (Assumption REF )."
],
[
"Stalks of Geometric Eisenstein Series",
"Now our aim is to explicitly determine the stalks of the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ , when $\\phi _{T}$ is weakly normalized regular.",
"Recall this means that $\\phi _{T}$ is generic, and, for all $w \\in W_{G}$ , $\\chi \\otimes \\delta _{B}^{1/2} \\lnot \\simeq \\chi ^{w} \\otimes (\\delta _{B}^{-1/2})^{w}$ .",
"Genericity will allow us to apply the results of Corollary REF , and the second condition will appear naturally in the computation of the stalks.",
"It is helpful to treat each connected component separately.",
"In particular, using Corollary REF , we consider the decomposition: $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) = \\bigoplus _{\\overline{\\nu } \\in B(T)} \\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}}) $ The main result of this section is as follows.",
"Theorem 9.1 Consider $\\phi _{T}$ a weakly normalized regular parameter with associated character $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ .",
"We fix $\\overline{\\nu } \\in B(T)$ with image $b \\in B(G)_{\\mathrm {un}}$ , HN-dominant reduction $b_{T}$ , and associated Borel $B_{b}$ .",
"Using Corollary REF , we can write $\\overline{\\nu } = w(b_{T})$ for a unique $w \\in W_{b}$ .",
"Then we have an isomorphism $ \\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}}) \\simeq j_{b!",
"}(i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ] $ under the identification $\\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda ) \\simeq \\mathrm {D}(\\mathrm {Bun}_{G}^{b})$ , where $j_{b}: \\mathrm {Bun}_{G}^{b} \\rightarrow \\mathrm {Bun}_{G}$ is the locally closed immersion defined by the HN-stratum corresponding to $b$ , and $P_{b}$ is the standard parabolic with Levi factor $M_{b} \\simeq J_{b}$ .",
"By varying $\\overline{\\nu }$ over all connected components, we obtain the following.",
"Corollary 9.2 Consider $\\phi _{T}$ a weakly normalized regular parameter, with associated character $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ .",
"For $b \\in B(G)$ , the stalk $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\in \\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ is given by an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq \\bigoplus _{w \\in W_{b}} i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ if $b \\in B(G)_{\\mathrm {un}}$ , an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq 0$ if $b \\notin B(G)_{\\mathrm {un}}$ .",
"In particular, $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a perverse sheaf on $\\mathrm {Bun}_{G}$ with respect to the standard $t$ -structure defined by the HN-strata.",
"First, consider the following easy Lemma.",
"Lemma 9.3 For $\\overline{\\nu } \\in B(T)$ with image $b$ in $B(G)$ , the restriction $ \\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}} $ for $b^{\\prime } \\in B(G)$ vanishes unless $b \\succeq b^{\\prime }$ in the natural partial ordering on $B(G)$ .",
"This follows from the observation that the image of $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ under $\\mathfrak {p}^{\\overline{\\nu }}$ is contained in the open substack $\\mathrm {Bun}_{B}^{\\le b}$ parametrizing bundles with associated Kottwitz element less than $b$ .",
"In particular, using the Tannakian formalism , this reduces to the observation that, for $\\mathrm {GL}_{n}$ , a bundle $\\mathcal {E}$ with a filtration by vector subbundles has Harder-Narasimhan polygon less than or equal to Harder-Narasimhan polygon of the direct sum of the graded pieces of the filtration, which is an easy consequence of the formalism of Harder-Narasimhan reductions (See for example ).",
"Thus, for a fixed $b^{\\prime } \\in B(G)$ , Lemma REF tells us that $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ is a direct sum of $\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ for $\\overline{\\nu }$ whose image $b \\in B(G)$ satisfies $b \\succeq b^{\\prime }$ .",
"Now the key point is that, under the weak normalized regularity assumption, all the contributions will vanish except when $b^{\\prime } = b$ .",
"This is one of the many reasons that weak normalized regularity (or at least genericity) is absolutely necessary to get a reasonable eigensheaf.",
"In general, all possible $\\overline{\\nu }$ contribute to $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ , and therefore $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ will be equal to an infinite direct sum of smooth irreducible representations sitting in infinitely many degrees.",
"We now reduce Theorem REF to two propositions.",
"We first have the following proposition describing the contribution of the split reduction in the connected components $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ .",
"Proposition 9.4 Let $\\overline{\\nu } \\in B(T)$ be an element mapping to $b \\in B(G)_{\\mathrm {un}}$ .",
"We write $\\overline{\\nu } = w(b_{T})$ as above.",
"Then, if $\\xi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ is any smooth character, we have an isomorphism $ \\mathrm {Eis}^{\\overline{\\nu }}(\\xi )|_{\\mathrm {Bun}_{G}^{b}} \\simeq \\mathrm {Ind}_{B_{b}}^{J_{b}}(\\xi ^{w} \\otimes \\delta _{B}^{1/2} \\otimes (\\delta _{B}^{w})^{-1/2})[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] $ of complexes of smooth $J_{b}(\\mathbb {Q}_{p})$ -modules, under the identification $\\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ .",
"In particular, using the isomorphism $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\simeq \\mathrm {Eis}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}) $ we have, for all $\\overline{\\nu } \\in B(T)$ mapping to $b \\in B(G)_{\\mathrm {un}}$ and $\\chi $ the character attached to $\\phi _{T}$ , an isomorphism $\\begin{split}\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} = \\mathrm {Eis}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2})|_{\\mathrm {Bun}_{G}^{b}} & \\simeq \\mathrm {Ind}_{B_{b}}^{J_{b}}(\\chi ^{w} \\otimes (\\delta _{B}^{w})^{1/2} \\otimes \\delta _{B}^{1/2} \\otimes (\\delta _{B}^{w})^{-1/2})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ] \\\\ & \\simeq \\mathrm {Ind}_{B_{b}}^{J_{b}}(\\chi ^{w} \\otimes \\delta _{B}^{1/2})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]\\\\ & \\simeq i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]\\end{split}$ This tells us that all the claimed contributions to the restriction $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}}$ appear.",
"All that remains is to show is that there are no additional contributions, and this is precisely what weak normalized regularity will allow us to do.",
"Proposition 9.5 Assume $\\phi _{T}$ is weakly normalized regular, then, for all $\\overline{\\nu } \\in B(T)$ mapping to $b \\in B(G)_{\\mathrm {un}}$ , the sheaf $\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})$ is only supported on the HN-strata $\\mathrm {Bun}_{G}^{b}$ .",
"We dedicate the remainder of this section to the proof of these two propositions."
],
[
"The Proof of Proposition ",
"We let $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ be the connected component defined by $\\overline{\\nu } \\in B(T)$ mapping to $b \\in B(G)_{\\mathrm {un}}$ , and let $\\mathrm {Bun}_{B}^{\\overline{\\nu },b}$ be the preimage of $\\mathrm {Bun}_{G}^{b}$ along $\\mathfrak {p}^{\\overline{\\nu }}:\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{G}$ .",
"Topologically, the stack $\\mathrm {Bun}_{B}^{\\overline{\\nu },b}$ is just a point defined by the split reduction $\\mathcal {F}_{\\overline{\\nu }} \\times ^{T} B$ .",
"We already saw in the proof of Theorem REF what the automorphisms of this $B$ -bundle are.",
"In particular, they are given by the group diamond $\\mathcal {G}_{\\overline{\\nu }}$ , where $\\mathcal {G}_{\\overline{\\nu }}(S) = Q_{\\overline{\\nu }}(X_{S})$ for $S \\in \\mathrm {Perf}$ , and $Q_{\\overline{\\nu }}$ is defined as $\\mathcal {F}_{w(b_{T})} \\times ^{T} B = \\mathcal {F}_{b_{T}} \\times ^{T} B^{w}$ , as in Definition REF .",
"In other words, we have an isomorphism $ \\mathrm {Bun}_{B}^{\\overline{\\nu },b} \\simeq [\\ast /\\mathcal {G}_{\\overline{\\nu }}] $ and the map $\\mathfrak {p}^{\\overline{\\nu }}:\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{G}$ induces a map of the form $ [\\ast /\\mathcal {G}_{\\overline{\\nu }}] \\rightarrow [\\ast /\\mathcal {J}_{b}] \\simeq \\mathrm {Bun}_{G}^{b} $ which we will abusively denote by $\\mathfrak {p}$ .",
"This map is given by the natural inclusion $ \\mathcal {G}_{\\overline{\\nu }} \\hookrightarrow \\mathcal {J}_{b} $ of group diamonds, coming from the fact that $Q_{\\overline{\\nu }}$ defines a reduction of $\\mathcal {F}_{b}$ , as in §REF .",
"Moreover, the map $\\mathfrak {q}^{\\overline{\\nu }}: \\mathrm {Bun}_{B}^{\\overline{\\nu },b} \\rightarrow \\mathrm {Bun}_{T}$ is identified with a map of the form $ [\\ast /\\mathcal {G}_{\\overline{\\nu }}] \\rightarrow [\\ast /\\underline{T(\\mathbb {Q}_{p})}] $ which we will abusively denote by $\\mathfrak {q}$ .",
"This map factors as $ \\begin{tikzcd}\\left[\\ast /\\mathcal {G}_{\\overline{\\nu }}\\right] [r] & \\left[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}\\right] [r,\"q^{\\natural }\"] & \\left[\\ast /\\underline{T(\\mathbb {Q}_{p})}\\right] \\simeq ^{w} \\left[\\ast /\\underline{T(\\mathbb {Q}_{p})}\\right] \\\\\\end{tikzcd} $ where the first map is given by the semi-direct decomposition $ \\mathcal {G}_{\\overline{\\nu }}^{> 0} \\ltimes \\mathcal {G}_{\\overline{\\nu }}^{= 0} $ and the identification $\\mathcal {G}_{\\overline{\\nu }}^{= 0} \\simeq B_{b}(\\mathbb {Q}_{p})$ , the second map $q^{\\natural }$ is the natural projection, and the last isomorphism is given by conjugating by $w$ .",
"It follows by base change that we have an isomorphism $ \\mathrm {Eis}^{\\overline{\\nu }}(\\xi )|_{\\mathrm {Bun}_{G}^{b}} \\simeq \\mathfrak {p}_{!",
"}\\mathfrak {q}^{*}(\\xi )[d_{\\overline{\\nu }}] $ where $d_{\\overline{\\nu }} = \\mathrm {dim}(\\mathrm {Bun}_{B}^{\\overline{\\nu }})$ .",
"We let $s: [\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\mathcal {J}_{b}]$ be the natural map, as in §REF .",
"Our goal is to compute $ s^{*}\\mathfrak {p}_{!",
"}\\mathfrak {q}^{*}(\\xi )[d_{\\overline{\\nu }}] $ as a complex of $J_{b}(\\mathbb {Q}_{p})$ -representations.",
"To do this, we consider the stack $Y^{\\overline{\\nu }}$ defined by the Cartesian diagram $\\begin{tikzcd}& Y^{\\overline{\\nu }} [r,\"\\tilde{\\mathfrak {p}}\"] [d,\"\\tilde{s}\"] & \\left[\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}\\right] [d,\"s\"] \\\\& \\left[\\ast /\\mathcal {G}_{\\overline{\\nu }}\\right] [d,\"\\mathfrak {q}\"] [r,\"\\mathfrak {p}\"] & \\left[\\ast /\\mathcal {J}_{b}\\right] \\\\& \\left[\\ast /\\underline{T(\\mathbb {Q}_{p})}\\right] &\\end{tikzcd}$ By base-change, it suffices to compute: $\\tilde{\\mathfrak {p}}_{!",
"}\\tilde{s}^{*}\\mathfrak {q}^{*}(\\xi )[d_{\\overline{\\nu }}] $ Let's now describe the stack $Y^{\\overline{\\nu }}$ .",
"We define $\\mathcal {G}^{\\overline{\\nu }}$ to be the cokernel of the natural map of group diamonds: $ 0 \\rightarrow \\mathcal {G}_{\\overline{\\nu }}^{> 0} \\rightarrow \\mathcal {J}^{> 0}_{b} \\rightarrow \\mathcal {G}^{\\overline{\\nu }} \\rightarrow 0 $ We recall that $\\mathcal {G}_{\\overline{\\nu }}^{= 0} \\simeq \\underline{B_{b}(\\mathbb {Q}_{p})} \\subset \\underline{J_{b}(\\mathbb {Q}_{p})} \\simeq \\mathcal {J}_{b}^{= 0}$ acts on the first two terms by the semi-direct product structure.",
"It follows that this is inclusion is equivariant with respect to this action and hence $\\mathcal {G}^{\\overline{\\nu }}$ also acquires an action of $\\underline{B_{b}(\\mathbb {Q}_{p})}$ .",
"We deduce that the space $Y^{\\overline{\\nu }}$ is isomorphic to the $v$ -stack quotient: $ [\\mathcal {G}^{\\overline{\\nu }}/\\underline{B_{b}(\\mathbb {Q}_{p})}] $ With this in hand, we further refine diagram (9) $\\begin{tikzcd}& \\left[\\mathcal {G}^{\\overline{\\nu }}/\\underline{B_{b}(\\mathbb {Q}_{p})}\\right] [rr,\"\\tilde{\\mathfrak {p}}\"] [dd,\"\\tilde{s}\"] [dr,\"j\"]& & \\left[\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}\\right] [dd,\"s\"] \\\\& & \\left[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}\\right] [ur,\"i\"] [ddl,\"q\"] & \\\\& \\left[\\ast /\\mathcal {G}_{\\overline{\\nu }}\\right] [d,\"\\mathfrak {q}\"] [rr,\"\\mathfrak {p}\"] & & \\left[\\ast /\\mathcal {J}_{b}\\right] \\\\& \\left[\\ast /\\underline{T(\\mathbb {Q}_{p})}\\right] & &\\end{tikzcd}$ where $i$ and $j$ are the natural maps, and $q$ is the composition of the natural projection $q^{\\natural }:[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\underline{T(\\mathbb {Q}_{p})}]$ followed by conjugation by $w$ .",
"This allows us to further reduce to computing $\\tilde{\\mathfrak {p}}_{!",
"}\\tilde{s}^{*}\\mathfrak {q}^{*}(\\xi )[d_{\\overline{\\nu }}] \\simeq i_{!}j_{!",
"}\\tilde{s}^{*}\\mathfrak {q}^{*}(\\xi )[d_{\\overline{\\nu }}] \\simeq i_{!}j_{!",
"}j^{*}q^{*}(\\xi )[d_{\\overline{\\nu }}]$ but now, by projection formula, this is isomorphic to: $ i_{!",
"}(q^{*}(\\xi ) \\otimes j_{!",
"}(\\Lambda ))[d_{\\overline{\\nu }}] $ Now recall by Lemma REF we have an equality $ d_{\\overline{\\nu }} = \\langle 2\\hat{\\rho }, \\nu _{b} \\rangle - 2\\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle $ where $\\hat{\\rho }^{w}$ is the sum of the positive roots $\\hat{\\alpha } > 0$ such that $w(\\hat{\\alpha }) > 0$ .",
"To proceed further, we consider the character $ \\delta (t) := (\\delta _{B}^{w})^{-1/2} \\otimes \\delta _{B}^{1/2}(t) = \\prod _{\\hat{\\alpha } > 0} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })}))|^{-1/2} \\prod _{\\hat{\\alpha } > 0} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{\\hat{\\alpha }}))|^{1/2} = \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) < 0\\end{array}} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })}))| $ of $T(\\mathbb {Q}_{p})$ .",
"We have the following lemma.",
"Lemma 9.6 $j_{!",
"}(\\Lambda )$ is isomorphic to $(q^{\\natural })^{*}(\\delta )[2(\\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle - \\langle 2\\hat{\\rho },\\nu _{b} \\rangle )]$ .",
"Let's see why the result follows from this.",
"In particular, using the formula $d_{\\overline{\\nu }} = \\langle 2\\hat{\\rho }, \\nu _{b} \\rangle - 2\\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle $ , this gives us an isomorphism: $ \\mathrm {Eis}^{\\overline{\\nu }}(\\xi ) \\simeq i_{!",
"}(q^{*}(\\xi ) \\otimes (q^{\\natural })^{*}(\\delta ))[-2\\langle 2\\hat{\\rho },\\nu _{b} \\rangle + 2\\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle + \\langle 2\\hat{\\rho },\\nu _{b} \\rangle - 2\\langle 2\\hat{\\rho }^{w}_{G},\\nu _{b} \\rangle ] $ $ \\simeq i_{!",
"}((q^{\\natural })^{*}(\\xi ^{w} \\otimes \\delta ))[\\langle -2\\hat{\\rho }, \\nu _{b} \\rangle ] $ However, now $i_{!",
"}((q^{\\natural })^{*}(\\xi \\otimes \\delta )))$ will be identified with compactly supported functions on $J_{b}(\\mathbb {Q}_{p})$ which transform under the action of $B_{b}(\\mathbb {Q}_{p})$ by the character $\\xi ^{w} \\otimes \\delta $ via the natural projection $B_{b}(\\mathbb {Q}_{p}) \\rightarrow T(\\mathbb {Q}_{p})$ .",
"However, this is precisely the parabolic induction $\\mathrm {Ind}_{B_{b}}^{J_{b}}(\\xi ^{w} \\otimes \\delta )$ , and the claim follows.",
"We now prove the lemma.",
"(Lemma REF ) We need to determine $j_{!",
"}(\\Lambda )$ , where $j$ is the map: $ [\\mathcal {G}^{\\overline{\\nu }}/\\underline{B_{b}(\\mathbb {Q}_{p})}] \\rightarrow [\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}] $ Recall that $\\mathcal {G}^{\\overline{\\nu }}$ is defined via the short exact sequence of group diamonds: $ 0 \\rightarrow \\mathcal {G}_{\\overline{\\nu }}^{> 0} \\rightarrow \\mathcal {J}_{b}^{> 0} \\rightarrow \\mathcal {G}^{\\overline{\\nu }} \\rightarrow 0 $ However, as in §REF , the map $\\mathcal {G}_{\\overline{\\nu }}^{> 0} \\rightarrow \\mathcal {J}_{b}^{> 0}$ respects the filtration by commutator subgroup, and we similarly see the cokernel $\\mathcal {G}^{\\overline{\\nu }}$ has an induced filtration by commutator subgroups.",
"Namely, since $\\mathcal {G}_{\\overline{\\nu }}^{> 0}$ breaks up in terms of the positive Banach-Colmez spaces $\\mathcal {H}^{0}((w(\\hat{\\alpha }))_{*}(Q))$ , for $\\hat{\\alpha }$ a positive root such that $w(\\hat{\\alpha }) > 0$ and $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle \\ne 0$ , and $\\mathcal {J}_{b}^{>0}$ breaks up as $\\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q))$ , for $\\hat{\\alpha }$ a positive root such that $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle \\ne 0$ , we can write $\\mathcal {G}^{\\overline{\\nu }}$ as an iterated fibration of the positive Banach-Colmez spaces $ \\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q)) $ where $\\hat{\\alpha }$ is a positive root such that $w(\\hat{\\alpha }) < 0$ .",
"Here we note that it automatically follows that $\\langle \\hat{\\alpha }, \\nu _{b} \\rangle \\ne 0$ , since $w \\in W_{G}/W_{M_{b}}$ is identified with a representative of minimal length in $W_{G}$ , by definition of $M_{b}$ .",
"Therefore, we deduce that $j$ is an iterated fibration of positive Banach-Colmez spaces, and we can apply the proof of to deduce that the adjunction $ j^{!}j_{!",
"}(\\Lambda ) \\simeq \\Lambda $ is an isomorphism.",
"The claim is therefore reduced to showing that: $ j^{!",
"}(\\Lambda ) \\simeq (\\delta )^{-1}[2(\\langle 2\\hat{\\rho },\\nu _{b} \\rangle - \\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle )] $ As per usual, we consider the Cartesian diagram $\\begin{tikzcd}& \\mathcal {G}^{\\overline{\\nu }} [r,\"\\tilde{j}\"] [d,\"\\tilde{\\pi }\"] & \\ast [d,\"\\pi \"] & \\\\&\\left[\\mathcal {G}^{\\overline{\\nu }}/\\underline{B_{b}(\\mathbb {Q}_{p})} \\right] [r,\"j\"] & \\left[\\ast /\\underline{B_{b}(\\mathbb {Q}_{p})}\\right]&\\end{tikzcd}$ and, by base-change, obtain an isomorphism: $ \\tilde{j}^{!",
"}(\\Lambda ) \\simeq \\tilde{\\pi }^{*}j^{!",
"}(\\Lambda ) $ By Lemma REF , the dualizing object on the Banach-Colmez spaces $\\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q))$ is isomorphic to $|\\cdot |[2\\langle \\hat{\\alpha },\\nu _{b} \\rangle ]$ as a sheaf with $\\mathbb {Q}_{p}^{*}$ -action.",
"However, $B_{b}(\\mathbb {Q}_{p})$ acts on this positive Banach-Colmez spaces via the natural surjection $ B_{b}(\\mathbb {Q}_{p}) \\rightarrow T(\\mathbb {Q}_{p}) $ composed with the character given by $ T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {Q}_{p}^{*} $ $ t \\mapsto \\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })})^{-1} $ and the scaling action of $\\mathbb {Q}_{p}^{*}$ on $\\mathcal {H}^{0}(\\hat{\\alpha }_{*}(Q))$ , where the inverse comes from the minus sign when passing between bundles and isocrystals.",
"This tells us that the dualizing object on $\\mathcal {G}^{\\overline{\\nu }}$ as a sheaf with $B_{b}(\\mathbb {Q}_{p})$ -action is isomorphic to $ \\prod _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) < 0\\end{array}} |\\mathrm {det}(\\mathrm {Ad}(t|\\mathfrak {g}_{w(\\hat{\\alpha })})|^{-1}[2\\sum _{\\begin{array}{c}\\hat{\\alpha } > 0 \\\\ w(\\hat{\\alpha }) < 0\\end{array}} \\langle \\hat{\\alpha }, \\nu _{b} \\rangle ] $ but this is isomorphic to $ \\delta ^{-1}[2(\\langle 2\\hat{\\rho },\\nu _{b} \\rangle - \\langle 2\\hat{\\rho }^{w}_{G}, \\nu _{b} \\rangle )] $ as desired."
],
[
"The Proof of Proposition ",
"We argue by induction on $b \\in B(G)_{\\mathrm {un}}$ with respect to the partial ordering on $B(G)$ and the following stronger statement.",
"\"For $b \\in B(G)_{\\mathrm {un}}$ with HN-dominant reduction $b_{T}$ , and $\\overline{\\nu } = w(b_{T}) \\in B(T)$ mapping to $b$ for varying $w \\in W_{b}$ , we have an isomorphism $ \\mathrm {nEis}^{w(b_{T})}(\\mathcal {S}_{\\phi _{T}}) \\simeq j_{b!",
"}(i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2})[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] $ of sheaves in $\\mathrm {D}(\\mathrm {Bun}_{G})$ .\"",
"The base case will be when $b$ is such that any $b^{\\prime } \\in B(G)_{\\mathrm {un}}$ satisfying $b \\succeq b^{\\prime }$ is equal to $b$ .",
"Since the stalk of $\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}}$ will only be non-trivial for $b^{\\prime } \\in B(G)$ such that $b \\succeq _{\\ne } b^{\\prime }$ by Lemma REF , the result in this case follows follows from Proposition REF and Corollary REF (1), where we note that $\\phi _{T}$ is weakly normalized regular and therefore generic.",
"For the inductive step, assume the claim is true for all $b^{\\prime } \\in B(G)_{\\mathrm {un}}$ such that $b \\succeq _{\\ne } b^{\\prime }$ .",
"Let $\\overline{\\nu } \\in B(T)$ be an element mapping to $b$ .",
"By Proposition REF and Corollary REF (1) again, it suffices to show that the restriction of $\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})$ to $\\mathrm {Bun}_{G}^{b^{\\prime }}$ vanishes for all such $b^{\\prime }$ .",
"By Corollary 7.7 (2), it suffices to show, for all $w \\in W_{b^{\\prime }}$ , that the complex $ R\\mathcal {H}om(\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b^{\\prime }}},i_{B_{b^{\\prime }}}^{J_{b^{\\prime }}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}) = R\\mathcal {H}om(\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}}),j_{b*}(i_{B_{b^{\\prime }}}^{J_{b^{\\prime }}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2})) $ is trivial.",
"To do this, let $\\overline{\\nu }^{\\prime } = w(b^{\\prime }_{T})$ be the element mapping to $b^{\\prime } \\in B(G)$ defined by $w \\in W_{b^{\\prime }}$ .",
"Our inductive hypothesis tells us that we have an isomorphism $ j_{b^{\\prime }!",
"}(i_{B_{b^{\\prime }}}^{J_{b^{\\prime }}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2})[-\\langle 2\\hat{\\rho },\\nu _{b^{\\prime }} \\rangle ] \\simeq \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}(\\mathcal {S}_{\\phi _{T}}) $ varying over $\\overline{\\nu }^{\\prime }$ mapping to $b^{\\prime } \\in B(G)$ .",
"If we write $\\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}})$ for the sheaf defined by replacing $\\mathfrak {p}_{!",
"}$ with $\\mathfrak {p}_{*}$ in the definition of $\\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}})$ .",
"It follows, using Theorem REF and Theorem REF , that this is isomorphic to $j_{b^{\\prime }*}(i_{B_{b^{\\prime }}}^{J_{b^{\\prime }}}(\\chi ^{w}))[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ (See the discussion preceding Proposition REF for details).",
"Therefore, it suffices to show, for all $\\overline{\\nu }^{\\prime }$ mapping to $b^{\\prime } \\in B(G)_{\\mathrm {un}}$ and $\\overline{\\nu }$ mapping to $b \\in B(G)_{\\mathrm {un}}$ , that $ R\\mathcal {H}om(\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}}),\\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}})) $ is trivial.",
"To aid our analysis, we consider the following functor $ \\mathrm {CT}^{\\overline{\\nu }}(-) := \\mathfrak {q}^{\\overline{\\nu }}_{*} \\circ \\mathfrak {p}^{\\overline{\\nu }!",
"}(-): \\mathrm {D}(\\mathrm {Bun}_{G}) \\rightarrow \\mathrm {D}(\\mathrm {Bun}_{T}^{\\overline{\\nu }})[-\\mathrm {dim}(\\mathrm {Bun}_{B}^{\\overline{\\nu }})] $ which is in particular the right adjoint of the unnormalized Eisenstein functor $\\mathrm {Eis}^{\\overline{\\nu }}(-) := \\mathfrak {p}_{!",
"}(\\mathfrak {q}^{*}(-)[\\mathrm {dim}(\\mathrm {Bun}_{B}^{\\overline{\\nu }})])$ .",
"Writing $\\mathrm {nEis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}})$ as $\\mathrm {Eis}^{\\overline{\\nu }}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2})$ and using adjunction, it suffices to show that the complex $ R\\mathcal {H}om_{T(\\mathbb {Q}_{p})}(\\chi \\otimes \\delta _{B}^{1/2},\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}})) $ is trivial in $\\mathrm {D}(\\mathrm {Bun}_{T}^{\\overline{\\nu }}) \\simeq \\mathrm {D}(T(\\mathbb {Q}_{p}),\\Lambda )$ .",
"To show this, we first look at the diagram $\\begin{tikzcd}& \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{G}} \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }} [r,\"\\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }}\"] [d,\"\\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }^{\\prime }}\"] & \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }} [d,\"\\mathfrak {p}^{\\overline{\\nu }^{\\prime }}\"] [r,\"\\mathfrak {q}^{\\overline{\\nu }^{\\prime }}\"] & \\mathrm {Bun}_{T}^{\\overline{\\nu }^{\\prime }} \\\\& \\mathrm {Bun}_{B}^{\\overline{\\nu }} [r,\"\\mathfrak {p}^{\\overline{\\nu }}\"] [d,\"\\mathfrak {q}^{\\overline{\\nu }}\"] & \\mathrm {Bun}_{G} & \\\\& \\mathrm {Bun}_{T}^{\\overline{\\nu }} & &\\end{tikzcd}$ and note, by base-change, that we have a natural isomorphism $ \\mathfrak {q}^{\\overline{\\nu }}_{*} \\circ \\mathfrak {p}^{\\overline{\\nu }!}",
"\\circ \\mathfrak {p}^{\\overline{\\nu }^{\\prime }}_{*} \\circ \\mathfrak {q}^{\\overline{\\nu }^{\\prime }*}(-) \\simeq \\mathfrak {q}_{*}^{\\overline{\\nu }} \\circ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }^{\\prime }}_{*} \\circ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }!}",
"\\circ \\mathfrak {q}^{\\overline{\\nu }^{\\prime }*}(-) $ of derived functors $\\mathrm {D}(\\mathrm {Bun}_{T}^{\\overline{\\nu }^{\\prime }}) \\rightarrow \\mathrm {D}(\\mathrm {Bun}_{T}^{\\overline{\\nu }})$ .",
"This tells us that $\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}})$ is the direct image of the complex $ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }!",
"}(\\mathfrak {q}^{\\overline{\\nu }^{\\prime }*}(\\mathcal {S}_{\\phi _{T}} \\otimes \\Delta _{B}^{1/2}))[\\mathrm {dim}(\\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }}) -\\mathrm {dim}(\\mathrm {Bun}_{B}^{\\overline{\\nu }})] $ on $\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{G}} \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }}$ onto $\\mathrm {Bun}_{T}^{\\overline{\\nu }}$ .",
"By , the space $\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{G}} \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }}$ has a locally closed stratification given by the generic relative position of the two bundles $ \\bigsqcup _{w \\in W_{G}} (\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{G}} \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }})_{w}$ which we denote by $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ for varying $w \\in W_{G}$ .",
"Using the excision spectral sequence, this implies that $\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}})$ also admits a filtration whose graded pieces we write as $(\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}}))_{w}$ .",
"Consider the following claim.",
"Proposition 9.7 Let $\\overline{\\nu }$ and $\\overline{\\nu }^{\\prime }$ be two elements mapping to $b$ and $b^{\\prime }$ in $B(G)_{\\mathrm {un}}$ , respectively.",
"Suppose that $b \\ne b^{\\prime }$ then the stack $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ is empty if $w = 1$ .",
"If $w \\ne 1$ then $(\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}}))_{w}$ is an extension of complexes in $\\mathrm {D}(T(\\mathbb {Q}_{p}),\\Lambda )$ isomorphic to $(\\delta _{B}^{-1/2})^{w} \\otimes \\chi ^{w}$ .",
"First, let's finish the proof of Proposition REF assuming this.",
"By the above discussion, it suffices to show that the complex $ R\\mathcal {H}om_{T(\\mathbb {Q}_{p})}(\\chi \\otimes \\delta _{B}^{1/2},(\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}}))_{w}) $ is trivial for all $w \\in W_{G}$ .",
"This is trivial if $w = 1$ by Point (1).",
"If $w \\ne 1$ , then it follows from Point (2) and the fact that the existence of a map $\\chi \\otimes \\delta _{B}^{1/2} \\rightarrow \\chi ^{w} \\otimes (\\delta _{B}^{-1/2})^{w}$ would contradict Condition REF (3) in the definition of weak normalized regularity by Schur's Lemma.",
"Therefore, since $\\phi _{T}$ is weak normalized regular by assumption, the claim follows.",
"Let's now finish up by reducing this Proposition to a simpler claim, which we will prove in the next section.",
"Proposition REF is an analogue of and the idea behind its proof is the same.",
"(Proposition REF ) We first begin by elucidating the geometry of the spaces $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ a bit more.",
"For $S \\in \\mathrm {Perf}$ , note that a $S$ -point of $\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{G}} \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }}$ corresponds to a pair of $B$ -structures on a $G$ -bundle $\\mathcal {F}_{G}$ on $X_{S}$ .",
"Namely, it parametrizes a pair $\\mathcal {F}_{B}^{1}$ (resp.",
"$\\mathcal {F}_{B}^{2}$ ) of two $B$ -structures on a $G$ -bundle $\\mathcal {F}_{G}$ whose reduction to $T$ , denoted $\\mathcal {F}_{T}^{1}$ (resp.",
"$\\mathcal {F}_{T}^{2}$ ) is isomorphic to $\\mathcal {F}_{\\overline{\\nu }}$ (resp.",
"$\\mathcal {F}_{\\overline{\\nu }^{\\prime }}$ ) after pulling back to any geometric point of $S$ .",
"We can think of it as parameterizing sections $ X_{S} \\rightarrow B\\backslash G/B$ such that the degree is of the specified form.",
"More transparently, we can think of a point of $\\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{G}} \\mathrm {Bun}_{B}^{\\overline{\\nu }^{\\prime }}$ as the $B$ -bundle $\\mathcal {F}_{B}^{2}$ together with a section $ s: X_{S} \\rightarrow \\mathcal {F}_{B}^{2} \\times ^{B} G/B $ where $B$ acts via conjugation.",
"For $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ , we recall that by interpreting $\\mathcal {V}^{\\hat{\\lambda }}$ as global sections of the appropriately twisted bundle on $G/B$ corresponding to $\\hat{\\lambda }$ under Borel-Weil-Bott, every point in $G/B$ gives rise to a line $\\ell ^{\\hat{\\lambda }} \\subset \\mathcal {V}^{\\hat{\\lambda }}$ .",
"We consider the $B$ -stable subspace $\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w} \\subset \\mathcal {V}^{\\hat{\\lambda }}$ consisting of weights greater than or equal to $w(\\hat{\\lambda })$ and $\\mathcal {V}^{\\hat{\\lambda }}_{> w} \\subset \\mathcal {V}^{\\hat{\\lambda }}_{\\ge w}$ the codimension 1 subspace consisting of weights strictly greater than $w(\\hat{\\lambda })$ .",
"We let $(G/B)_{w} := BwB/B$ be the locally closed Schubert cell attached to $w \\in W_{G}$ .",
"We write $(G/B)_{\\ge w}$ for its closure.",
"The closure is stratified by the Schubert cells indexed by elements $w^{\\prime } \\in W_{G}$ with length less than or equal to $w$ .",
"Then the line $\\ell ^{\\hat{\\lambda }} \\subset \\mathcal {V}^{\\hat{\\lambda }}$ will correspond to a point in $(G/B)_{\\ge w}$ if and only if it belongs to $\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w}$ for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ .",
"Moreover, the point belongs to the stratum $(G/B)_{w}$ if and only the projection to $\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w}/\\mathcal {V}^{\\hat{\\lambda }}_{> w}$ is non-zero.",
"This allows us to explain what it means to lie in the locally closed stratum $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ .",
"In particular, by definition , lying in this stratum is equivalent to the condition that $s$ factors through $\\mathcal {F}_{B}^{2} \\times ^{B} (G/B)_{\\succeq w}$ and is not contained in any closed strata defined by $\\mathcal {F}_{B}^{2} \\times ^{B} (G/B)_{\\succeq w^{\\prime }}$ for any $w^{\\prime } > w$ in the Bruhat order.",
"This implies that the section $s$ determines a set of line subbundles $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w})_{\\mathcal {F}_{B}^{2}} $ for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ .",
"Moreover, via the inclusion $\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w} \\subset \\mathcal {V}^{\\hat{\\lambda }}$ , these give the Plücker description of the $B$ -structure $\\mathcal {F}_{B}^{1}$ such that $\\mathcal {F}_{G} \\simeq \\mathcal {F}_{B}^{1} \\times ^{B} G \\simeq \\mathcal {F}_{B}^{2} \\times ^{B} G$ .",
"For this to define a point in $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ , these need to satisfy the condition that the induced map $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w})_{\\mathcal {F}_{B}^{2}} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w}/\\mathcal {V}^{\\hat{\\lambda }}_{> w})_{\\mathcal {F}_{B}^{2}} = (\\mathcal {L}^{\\hat{\\lambda }})_{(\\mathcal {F}_{T}^{2})^{w}} $ is non-zero map of $\\mathcal {O}_{X_{S}}$ -modules (cf.",
",,).",
"In particular, since $\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}}$ is a line bundle, the map being non-zero implies it is a fiberwise injective map of line bundles.",
"Now, recalling the minus sign when passing between isocrystals and bundles, if we define $\\theta := \\overline{\\nu } - w(\\overline{\\nu }^{\\prime })$ then the support of the torsion of the cokernel of this map of line bundles determines a point in $\\mathrm {Div}^{(\\theta )}$ , by the assumptions on the degrees and Lemma REF .",
"For this strata to be non-empty, we must have that $\\theta \\in \\Lambda _{G,B}^{pos}$ .",
"Therefore, for all non-empty strata, we have a map: $ \\pi _{w}: Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w} \\rightarrow \\mathrm {Div}^{(\\theta )} $ Now, with these preparations out of the way, let's start with the proof.",
"For Point (1), note that if $w = 1$ then we have an injective map of line bundles $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}} \\rightarrow \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{2}} $ for all $\\hat{\\lambda }$ , which give rise to the embeddings defined by $\\mathcal {F}_{B}^{1}$ when composed with the injections of bundles $\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{2}} \\rightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}}$ defined by $\\mathcal {F}_{B}^{2}$ , by construction.",
"However, since the composition $\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}} \\rightarrow \\mathcal {V}^{\\hat{\\lambda }}_{\\mathcal {F}_{G}}$ is also a map of vector bundles, this is impossible unless $\\mathcal {F}_{T}^{1} \\simeq \\mathcal {F}_{T}^{2}$ , which would contradict our assumption that $\\overline{\\nu }, \\overline{\\nu }^{\\prime } \\in B(T)$ map to $b \\ne b^{\\prime }$ in $B(G)$ .",
"Therefore, we have established point (1).",
"For point (2), we write $\\mathfrak {q}_{1}$ (resp.",
"$\\mathfrak {q}_{2}$ ) for the natural projections of $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ to $\\mathrm {Bun}_{T}^{\\overline{\\nu }}$ (resp.",
"$\\mathrm {Bun}_{T}^{\\overline{\\nu }^{\\prime }}$ ).",
"We note that $\\mathfrak {q}_{2}$ is equal to the composition $ Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} \\xrightarrow{} \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )} \\xrightarrow{} \\mathrm {Bun}_{T}^{w(\\overline{\\nu }^{\\prime })} \\xrightarrow{} \\mathrm {Bun}_{T}^{\\overline{\\nu }^{\\prime }} $ where $\\phantom{}^{\\mathrm {op}}h^{\\rightarrow }_{(\\theta )}$ is the map sending $(\\mathcal {F}_{T},(D_{i})_{i \\in \\mathcal {J}})$ to the bundle $\\mathcal {F}_{T}(\\sum _{i \\in \\mathcal {J}} \\alpha _{i} \\cdot D_{i})$ and the last map is given by conjugation by $w$ .",
"Recall, $(\\mathrm {CT}^{\\overline{\\nu }} \\circ \\mathrm {nEis}^{\\overline{\\nu }^{\\prime }}_{*}(\\mathcal {S}_{\\phi _{T}}))_{w}$ is given (up to a shift) by the sheaf $ \\mathfrak {q}_{1*} \\circ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }!}",
"\\circ \\mathfrak {q}^{\\overline{\\nu }*}(\\chi \\otimes \\delta _{B}^{1/2}) $ We can write this as the Verdier dual of $\\mathfrak {q}_{1!}",
"\\circ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }*} \\circ \\mathfrak {q}^{\\overline{\\nu }!",
"}(\\chi ^{-1} \\otimes (\\delta _{B}^{1/2})^{-1}) \\simeq \\mathfrak {q}_{1!}",
"\\circ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }*} \\circ \\mathfrak {q}^{\\overline{\\nu }*}(\\chi ^{-1} \\otimes \\delta _{B}^{1/2})$ , where the last isomorphism is Theorem REF .",
"Replacing $\\chi $ by $\\chi ^{-1}$ , this reduces us to showing that $\\mathfrak {q}_{1!}",
"\\circ \\phantom{}^{^{\\prime }}\\mathfrak {p}^{\\overline{\\nu }*} \\circ \\mathfrak {q}^{\\overline{\\nu }*}(\\chi \\otimes \\delta _{B}^{1/2}) \\simeq \\mathfrak {q}_{1!}",
"\\circ \\mathfrak {q}_{2}^{*}(\\chi \\otimes \\delta _{B}^{1/2})$ is an extension of complexes which are isomorphic to $(\\delta _{B}^{1/2})^{w} \\otimes \\chi ^{w}$ .",
"Using the above factorization of $\\mathfrak {q}_{2}$ , we rewrite this as $\\mathfrak {q}_{1!}",
"\\circ (\\mathfrak {q}_{1} \\times \\pi _{w})^{*} \\circ (\\phantom{}^{\\mathrm {op}}h^{\\rightarrow }_{(\\theta )})^{*} \\circ (w^{-1})^{*}(\\chi \\otimes \\delta _{B}^{1/2}) \\simeq \\mathfrak {q}_{1!}",
"\\circ (\\mathfrak {q}_{1} \\times \\pi _{w})^{*}((\\chi ^{w} \\otimes (\\delta _{B}^{1/2})^{w}) \\boxtimes (E^{(\\theta )}_{\\phi _{T}^{w} \\otimes (\\hat{\\rho }^{w}\\circ |\\cdot |)})^{\\vee })$ but now, by projection formula, we are reduced to the following, which is an analogue of Proposition 9.8 The direct image $(\\mathfrak {q}_{1} \\times \\pi _{w})_{!",
"}(\\Lambda )$ is an extension of complexes which are pullbacks of complexes on $\\mathrm {Div}^{(\\theta )}$ .",
"We will prove Proposition REF by relating the spaces $Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w}$ to some variants of what are called Zastava or semi-infinite flag spaces in the classical literature, as first studied over function fields by Feign, Finkelberg, Kuznetsov, and Mirkovic , ."
],
[
"Zastava Spaces",
"We let $U^{\\prime } \\subset U$ be the subgroup defined by the positive root spaces $\\hat{\\alpha } > 0$ such that $w(\\hat{\\alpha }) < 0$ .",
"We set $B^{\\prime } := TU^{\\prime } \\subset B$ to be the subgroup of the Borel defined by these root spaces.",
"We recall that $U^{\\prime }$ acts simply transitively on the closed Schubert cell $(G/B)_{\\ge w}$ and use this to define the $w$ -twisted version of the Zastava space.",
"Definition 9.9 For $\\theta \\in \\Lambda _{G,B}^{pos}$ , we let $W_{w}^{\\theta } \\rightarrow \\mathrm {Div}^{(\\theta )}$ be the $v$ -sheaf parameterizing, for $S \\in \\mathrm {Perf}$ , a triple $ (\\mathcal {F}_{U^{\\prime }},s,D) $ of the datum: A $U^{\\prime }$ -bundle $\\mathcal {F}_{U^{\\prime }}$ on $X_{S}$ .",
"A section $s: X_{S} \\rightarrow \\mathcal {F}_{U^{\\prime }} \\times ^{U^{\\prime }} (G/B)_{\\ge w}$ that does not lie in $(G/B)_{\\ge w^{\\prime }}$ for any $w^{\\prime } > w$ in the Bruhat order.",
"A divisor $D \\in \\mathrm {Div}^{(\\theta )}$ such that the induced non-zero ($\\Rightarrow $ fiberwise injective) maps of line bundles $ \\mathcal {L}^{\\hat{\\lambda }} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w})_{\\mathcal {F}_{U^{\\prime }}} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w}/\\mathcal {V}^{\\hat{\\lambda }}_{> w})_{\\mathcal {F}_{U^{\\prime }}} = \\mathcal {O}_{X_{S}} $ for all $\\hat{\\lambda } \\in \\hat{\\Lambda }_{G}^{+}$ have cokernel with torsion supported on $D$ .",
"Classically, the usual Zastava space in the literature is the same datum as above in the case that $w = w_{0}$ together with a level structure on the bundle $\\mathcal {L}^{\\hat{\\lambda }}$ so that it encodes information about enhanced $B$ -structures on one of the factors.",
"It's importance is that it provides a local model for the singularities of the Drinfeld compactification $\\overline{\\mathrm {Bun}}_{B}$ .",
"The space we have defined above in the case that $w = w_{0}$ is the open part of the Zastava space which models just the space $\\mathrm {Bun}_{B}$ .",
"As seen in our description of $Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w}$ in the previous section, this will clearly have a relationship to the spaces we are interested in describing.",
"Let us first just consider the case of the element of longest length.",
"We claim that the following is true.",
"Lemma 9.10 For $w = w_{0}$ the element of longest length and $\\theta = \\overline{\\nu } - w(\\overline{\\nu }^{\\prime }) \\in \\Lambda _{G,B}^{pos}$ for $\\overline{\\nu }^{\\prime }$ and $\\overline{\\nu }$ as above, there exists a commutative diagram $ \\begin{tikzcd}& Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w} [d,\"\\mathfrak {q}_{1} \\times \\pi _{w}\"] [r] & W^{\\theta }_{w} [d] \\\\& \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )} [r,\"p_{2}\"] & \\mathrm {Div}^{(\\theta )}\\end{tikzcd} $ which is a Cartesian square.",
"When $w = w_{0}$ , we have that $U^{\\prime } = U$ and $B^{\\prime } = B$ .",
"Given an $S$ -point of $W_{w}^{\\theta }$ , the maps $ \\mathcal {L}^{\\hat{\\lambda }} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w})_{\\mathcal {F}_{U}} \\hookrightarrow (\\mathcal {V}^{\\hat{\\lambda }})_{\\mathcal {F}_{G}} $ of vector bundles on $X_{S}$ define a $B$ -structure $\\mathcal {F}_{B}$ on the $G$ -bundle $\\mathcal {F}_{U} \\times ^{U} G$ .",
"We note that the induced map $ \\mathcal {L}^{\\hat{\\lambda }} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w})_{\\mathcal {F}_{U}} \\rightarrow (\\mathcal {V}^{\\hat{\\lambda }}_{\\ge w}/\\mathcal {V}^{\\hat{\\lambda }}_{> w})_{\\mathcal {F}_{U}} = \\mathcal {O}_{X_{S}} $ with torsion cokernel of support given by $D \\in \\mathrm {Div}^{(\\theta )}$ induces an identification $\\mathcal {L}^{\\hat{\\lambda }} \\simeq \\mathcal {O}_{X_{S}}(-\\langle \\hat{\\lambda },\\theta \\rangle \\cdot D)$ implying that $\\mathcal {F}_{B} \\times ^{B} T$ has Kottwitz invariant given by $\\theta = \\overline{\\nu } - w(\\overline{\\nu }^{\\prime })$ after pulling back to a geometric point.",
"Given a bundle $\\mathcal {F}_{T}^{1} \\in \\mathrm {Bun}_{T}^{\\overline{\\nu }}$ of degree $\\overline{\\nu }$ , we can dualize the above maps to get a fiberwise injection $ \\mathcal {O}_{X_{S}} \\rightarrow (\\mathcal {L}^{\\hat{\\lambda }})^{\\vee } \\simeq \\mathcal {O}_{X_{S}}(\\langle \\hat{\\lambda }, \\theta \\rangle \\cdot D) $ of line bundles, and then tensor by $\\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}}$ to get an injection $ \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}} \\rightarrow \\mathcal {L}^{\\hat{\\lambda }}_{\\mathcal {F}_{T}^{1}}(\\langle \\hat{\\lambda },\\theta \\rangle \\cdot D) $ Similarly, by taking duals and twisting the $U$ -torsor $\\mathcal {F}_{U}$ by $\\mathcal {F}_{T}^{1}$ , we obtain $B = T \\ltimes U$ -bundles $\\mathcal {F}_{B}^{1}$ and $\\mathcal {F}_{B}^{2}$ defining points in $Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w}$ , giving rise to a map $ (\\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )}) \\times _{\\mathrm {Div}^{(\\theta )}} W_{w}^{\\theta } \\rightarrow Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} $ which we can see is an isomorphism.",
"In particular, given a point in $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ corresponding to $B$ -bundles $\\mathcal {F}_{B}^{1}$ and $\\mathcal {F}_{B}^{2}$ then we can define a $U$ -bundle $\\mathcal {F}_{B}^{2} \\times ^{B} U$ , and, as already seen in the previous section, we get a section $s: X_{S} \\rightarrow \\mathcal {F}_{U} \\times ^{U} (G/B)_{\\ge w}$ , $D \\in \\mathrm {Div}^{(\\theta )}$ , and a $T$ -bundle $\\mathcal {F}_{T}^{1}$ of the desired form.",
"Now this lemma implies Proposition REF in the case that $w = w_{0}$ .",
"In particular, under the isomorphism $ (\\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )}) \\times _{\\mathrm {Div}^{(\\theta )}} W_{w}^{\\theta } \\simeq Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} $ $\\mathrm {Bun}_{T}^{\\overline{\\nu }}$ splits off as direct factor, and so, by Künneth, we deduce the claim.",
"Now we would like to apply a similar argument using the spaces $W_{w}^{\\theta }$ in the case that $w$ is a general element.",
"However, we run into a problem that, in general, all we get is a map $ (\\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )}) \\times _{\\mathrm {Div}^{(\\theta )}} W_{w}^{\\theta } \\rightarrow Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} $ where attached to a point in $(\\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )}) \\times _{\\mathrm {Div}^{(\\theta )}} W_{w}^{\\theta }$ we only get a $B^{\\prime }$ -torsor $\\mathcal {F}_{B^{\\prime }}^{1}$ with $T$ -factor of degree $\\overline{\\nu }$ .",
"The above map is then given by a base-change of the natural map $f_{\\overline{\\nu }}: \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }}$ .",
"In particular, if we let $\\tilde{Z}_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} \\rightarrow Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ be the base-change of $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ along the map $f_{\\overline{\\nu }}$ then the analogue of this Lemma holds with $\\tilde{Z}_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ in place of $Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ by the same argument.",
"Now let's study the map $f_{\\overline{\\nu }}: \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }}$ in a particular example and see how to prove Proposition REF in this case.",
"Example 9.11 Suppose that $G = \\mathrm {GL}_{3}$ and let $\\overline{\\nu }$ correspond to a tuple of integers $(-e,-f,-g) \\in \\mathbb {Z}^{3} \\simeq B(T)$ via the Kottwitz invariant.",
"We suppose $w$ corresponds to the simple reflection exchanging the first and second basis vectors.",
"After rigidifying the $T$ -bundle $\\mathcal {F}_{T}$ to be isomorphic to $(\\mathcal {O}(e),\\mathcal {O}(f),\\mathcal {O}(g))$ , we can view $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ as the moduli space of torsors under the $U$ -torsor $ \\begin{pmatrix} 1 & \\mathcal {O}(e) \\otimes \\mathcal {O}(f)^{\\vee } & \\mathcal {O}(e) \\otimes \\mathcal {O}(g)^{\\vee } \\\\ 0 & 1 & \\mathcal {O}(f) \\otimes \\mathcal {O}(g)^{\\vee } \\\\ 0 & 0 & 1 \\end{pmatrix} $ over $X$ , where the automorphisms (up to rigidification) of a point in $\\mathrm {Bun}_{B}^{\\overline{\\nu }}$ are given by considering the $\\mathcal {H}^{0}$ Banach-Colmez spaces attached to these bundles.",
"Similarly, after rigidification, we can view $\\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }}$ as the moduli space of torsors under the $U^{\\prime }$ -torsor $ \\begin{pmatrix} 1 & \\mathcal {O}(e) \\otimes \\mathcal {O}(f)^{\\vee } & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} $ over $X$ , and the map $f_{\\overline{\\nu }}$ is given by taking direct sums of the extension of $\\mathcal {O}(e)$ by $\\mathcal {O}(f)$ defined by the point in $\\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }}$ with $\\mathcal {O}(g)$ .",
"In particular, we can see that the fibers of the map $f_{\\overline{\\nu }}: \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }}$ are an iterated fibration in the Banach-Colmez spaces $\\mathcal {H}^{0}(\\mathcal {O}(e) \\otimes \\mathcal {O}(g)^{\\vee })$ and $\\mathcal {H}^{0}(\\mathcal {O}(f) \\otimes \\mathcal {O}(g)^{\\vee })$ .",
"If we assume that these Banach-Colmez spaces are positive it follows from the proof of that the adjunction $ f_{\\overline{\\nu }!",
"}f_{\\overline{\\nu }}^{!}",
"\\rightarrow \\mathrm {id} $ is an equivalence.",
"In particular, combining this with the above discussion would give us the proof of Proposition REF in this case.",
"We now consider $d \\in \\mathbb {N}_{> 0}$ and fix a closed point $\\infty \\in X$ in the Fargues-Fontaine curve over an algebraically closed complete field $F$ in characteristic $p$ .",
"We look at the short-exact sequence of $\\mathcal {O}_{X}$ -modules $ 0 \\rightarrow \\mathcal {O}_{X}(-d) \\rightarrow \\mathcal {O}_{X} \\rightarrow \\mathcal {O}_{X,\\infty }/t_{\\infty }^{d} \\rightarrow 0 $ where $\\mathcal {O}_{X,\\infty }$ is the completed local ring and $t_{\\infty }$ is the uniformizing parameter corresponding to an untilt $C$ of $F$ .",
"Tensoring by $\\mathcal {O}(g)$ , we get a short exact sequence: $ 0 \\rightarrow \\mathcal {O}_{X}(g - d) \\rightarrow \\mathcal {O}_{X}(g) \\rightarrow \\mathcal {O}_{X,\\infty }/t_{\\infty }^{d} \\rightarrow 0 $ Let $\\overline{\\nu }_{d}$ correspond to the tuple of integers $(e,f,g - d) \\in \\mathbb {Z}^{3}$ .",
"Then we consider the natural map $f_{\\overline{\\nu }_{d}}: \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }_{d}} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }_{d}}$ .",
"If we choose $d$ sufficiently large such that the spaces $\\mathcal {H}^{0}(\\mathcal {O}(e) \\otimes \\mathcal {O}_{X}(g - d)^{\\vee })$ and $\\mathcal {H}^{0}(\\mathcal {O}(f) \\otimes \\mathcal {O}_{X}(g - d)^{\\vee })$ are positive Banach-Colmez spaces then the fibers of $f_{\\overline{\\nu }_{d}}$ will be an iterated fibration in these positive Banach-Colmez spaces, and therefore we can again conclude that the adjunction $ f_{\\overline{\\nu }_{d}}f_{\\overline{\\nu }_{d}}^{!}",
"\\rightarrow \\mathrm {id} $ is an isomorphism.",
"Now we claim that we have a map: $ \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }_{d}} $ Explicitly, given a point $\\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }}$ , we have an exact sequence $0 \\rightarrow \\mathcal {E}_{2} \\rightarrow \\mathcal {E} \\rightarrow \\mathcal {O}(g) \\rightarrow 0 $ of bundles, where $\\mathcal {E}_{2}$ is an extension of $\\mathcal {O}(e)$ and $\\mathcal {O}(f)$ .",
"We can then consider the pullback of this exact sequence with respect to the map $\\mathcal {O}(g - d) \\rightarrow \\mathcal {O}(g)$ given by the modification, which will give us a point in $\\mathrm {Bun}_{B}^{\\overline{\\nu }_{d}}$ .",
"We write $f_{\\overline{\\nu },d\\infty }: \\mathrm {Bun}_{B}^{\\overline{\\nu },d\\infty } := \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\times _{\\mathrm {Bun}_{B}^{\\overline{\\nu }_{d}}} \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }_{d}} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }}$ for the pullback of $f_{\\overline{\\nu }_{d}}$ along this map.",
"We again conclude that the adjunction $ f_{\\overline{\\nu },d\\infty !",
"}f_{\\overline{\\nu },d\\infty }^{!}",
"\\rightarrow \\mathrm {id} $ is an isomorphism.",
"If we could use this map instead of $f_{\\overline{\\nu }}$ , we could prove the claim by arguing as above.",
"Indeed, consider $\\mathfrak {q}_{1} \\times \\pi _{w}: Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w} \\rightarrow \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times \\mathrm {Div}^{(\\theta )}$ .",
"If we base-change all the above spaces to $F$ , and let $(\\mathrm {Div}\\setminus \\infty )^{(\\theta )}$ be the partially symmetrized power defined by $\\mathrm {Div}^{1} \\setminus \\infty $ , the open complement of the closed point $\\infty \\rightarrow \\mathrm {Div}^{1}$ defined by the fixed untilt, then, if we consider the map $\\mathfrak {q}_{1} \\times \\pi _{w}: Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w} \\rightarrow \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times (\\mathrm {Div}\\setminus \\infty )^{(\\theta )}$ restricted to this locus, we can show that the base-change $Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w}$ along $f_{\\overline{\\nu },d\\infty }$ sits in a analogous Cartesian square to Lemma REF , and deduce Proposition REF for the restriction of $(\\mathfrak {q}_{1} \\times \\pi _{w})_{!",
"}f_{\\overline{\\nu },d\\infty !",
"}f_{\\overline{\\nu },d\\infty }^{!",
"}(\\Lambda ) \\simeq (\\mathfrak {q}_{1} \\times \\pi _{w})_{!",
"}(\\Lambda )$ to the open strata $(\\mathrm {Div}\\setminus \\infty )^{(\\theta )} \\subset \\mathrm {Div}^{(\\theta )}$ .",
"However, by excision, we can reduce Proposition REF to studying this restriction with the claim over the closed complement being trivial.",
"The claim in this case follows.",
"With this motivating example, all that remains is to formalize the above argument.",
"In particular, first off note, by , that for the proof of Proposition REF it suffices to consider the base-change of all the above spaces to the base $\\ast = \\mathop {\\rm Spd}(F)$ for $F$ an algebraically closed perfectoid field in characteristic $p$ .",
"We consider such a field with fixed characteristic 0 untilt $C$ , and let $\\infty \\rightarrow \\mathrm {Div}^{1}$ be the closed $F$ -point defined by this untilt.",
"We consider the open complement $\\mathrm {Div}^{1} \\setminus \\infty $ , and the partially symmetrized powers $(\\mathrm {Div}^{1} \\setminus \\infty )^{(\\theta )}$ defined by the open subset.",
"By applying excision, it suffices to verify Proposition REF over this open subset with the claim over the closed complement being trivial.",
"We abuse notation and write $Z^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}_{w}$ and $W_{w}^{\\theta }$ for the base-change to this open subspace for the rest of the section.",
"Now let's consider the spaces defined by the locally pro-finite sets $\\underline{B(\\mathbb {Q}_{p})}$ (resp.",
"$\\underline{B^{\\prime }(\\mathbb {Q}_{p})}$ ), and the natural map $ f_{0}: \\mathrm {Bun}_{B^{\\prime }}^{0} \\simeq [\\ast /\\underline{B^{\\prime }(\\mathbb {Q}_{p})}] \\rightarrow \\mathrm {Bun}_{B}^{0} \\simeq [\\ast /\\underline{B(\\mathbb {Q}_{p})}] $ of $v$ -stacks.",
"The fibers of this map are an iterated fibration in $\\mathcal {H}^{0}(\\mathcal {O}_{X}) = \\underline{\\mathbb {Q}_{p}}$ indexed by the positive roots $\\hat{\\alpha } > 0$ such that $w(\\hat{\\alpha }) > 0$ .",
"We choose $\\overline{\\nu }_{\\infty } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ to be an element such that $\\langle \\overline{\\nu }_{\\infty }, \\hat{\\alpha } \\rangle < 0$ for all $\\hat{\\alpha } > 0$ such that $w(\\hat{\\alpha }) > 0$ .",
"Recalling the minus sign when passing between isocrystals and bundles, this implies that the map $ f_{\\overline{\\nu }_{\\infty }}: \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }_{\\infty }} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }_{\\infty }} $ is a fibration in iterated positive Banach-Colmez spaces.",
"We consider a modification $\\mathcal {F}_{T}^{0} \\dashrightarrow \\mathcal {F}_{\\overline{\\nu }_{\\infty }}$ at $\\infty $ of meromorphy $\\overline{\\nu }_{\\infty }$ .",
"This modification induces a map $ \\mathrm {Bun}_{B}^{0} \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }_{\\infty }} $ which we precompose with the map $ \\mathrm {Bun}_{B}^{\\overline{\\nu }} \\rightarrow \\mathrm {Bun}_{B}^{0} $ also given by an appropriate modification.",
"This allows us to define $ \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu },\\infty } := \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu }_{\\infty }} \\times _{\\mathrm {Bun}_{B}^{\\overline{\\nu }_{\\infty }}} \\mathrm {Bun}_{B}^{\\overline{\\nu }} $ by base-changing $f_{\\overline{\\nu }_{\\infty }}$ .",
"We write $ f_{\\overline{\\nu },\\infty }: \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu },\\infty } \\rightarrow \\mathrm {Bun}_{B}^{\\overline{\\nu }} $ for the base-change of $f_{\\overline{\\nu }_{\\infty }}$ .",
"By the proof of , we have that the adjunction $ f_{\\overline{\\nu },\\infty !",
"}f_{\\overline{\\nu },\\infty }^{!}",
"$ is an isomorphism, since $f_{\\overline{\\nu }_{\\infty }}$ and in turn $f_{\\overline{\\nu },\\infty }$ is an iterated fibration of positive Banach-Colmez spaces.",
"Now we define $ \\tilde{Z}_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} := Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} \\times _{\\mathrm {Bun}_{B}^{\\overline{\\nu }}} \\mathrm {Bun}_{B^{\\prime }}^{\\overline{\\nu },\\infty } $ By the previous adjunction, it suffices to show the analogue of Proposition REF for the composition $ \\tilde{Z}_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} \\rightarrow Z_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }} \\xrightarrow{} \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times (\\mathrm {Div}\\setminus \\infty )^{(\\theta )} $ and the shriek pullback of the constant sheaf along the first map.",
"However, by now arguing exactly as in the proof of Lemma REF , with $B^{\\prime }$ and its unipotent radical $U^{\\prime }$ replacing $B$ and $U$ , we can deduce that the base-change of $W_{w}^{\\theta } \\rightarrow (\\mathrm {Div}\\setminus \\infty )^{(\\theta )}$ along $p_{2}: \\mathrm {Bun}_{T}^{\\overline{\\nu }} \\times (\\mathrm {Div}\\setminus \\infty )^{(\\theta )} \\rightarrow (\\mathrm {Div}\\setminus \\infty )^{(\\theta )}$ is precisely the space $\\tilde{Z}_{w}^{\\overline{\\nu },\\overline{\\nu }^{\\prime }}$ .",
"This concludes the proof of Proposition REF by Künneth."
],
[
"Tilting Eigensheaves",
"We would now like to combine our work in the previous sections and use it to construct eigensheaves.",
"We would like to do this in a uniform way for coefficient systems $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ , where $\\Lambda $ has the discrete topology unless otherwise stated.",
"One of the issues is that the representation theory of $\\phantom{}^{L}G/\\Lambda $ is substantially different in each of these three cases.",
"The structure of the representation theory of $\\mathrm {Rep}_{\\overline{\\mathbb {Q}}_{\\ell }}(\\phantom{}^{L}G)$ was described in §2.",
"With $\\overline{\\mathbb {Q}}_{\\ell }$ -coefficients, the category is semisimple with simple objects given by $V_{\\mu ^{\\Gamma }}$ for $\\mu ^{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}/\\Gamma $ a $\\Gamma $ -orbit of a geometric dominant cocharacter $\\mu $ .",
"If $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell }\\rbrace $ this is no longer so straightforward; the representation $V_{\\mu ^{\\Gamma }}$ can fail to be irreducible.",
"To develop a good theory of algebraic representations with these coefficients, we will need to invoke our assumption that $\\ell $ is very good with respect to $G$ and use the theory of tilting modules.",
"We first discuss the general notion of a tilting module.",
"Fix $H$ a split connected reductive group over $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ .",
"We consider the involution $ \\mathbb {D}: \\mathrm {Rep}_{\\Lambda }(H) \\rightarrow \\mathrm {Rep}_{\\Lambda }(H) $ $ V \\mapsto (V^{*})^{\\sigma } $ where $V^{*}$ is the dual representation and $\\sigma $ is the Chevalley involution.",
"For $\\lambda \\in \\mathbb {X}^{*}(H)^{+}$ a dominant character, we let $V^{\\lambda }$ denote the highest weight representation attached to $\\lambda $ by Borel-Weil-Bott, and we write $V_{\\lambda } := \\mathbb {D}(V^{\\lambda })$ for the dual highest weight representation.",
"We now come to our key definition.",
"Definition 10.1 Given $V \\in \\mathrm {Rep}_{\\Lambda }(H)$ , we say that $V$ has a Weyl (resp.",
"good) filtration if it admits a filtration whose successive quotients are isomorphic to $V^{\\lambda }$ (resp.",
"$V_{\\lambda }$ ).",
"We say $V$ is tilting if it admits both a good and a Weyl filtration.",
"We write $\\mathrm {Tilt}_{\\Lambda }(H) \\subset \\mathrm {Rep}_{\\Lambda }(H)$ for the full sub-category of tilting modules.",
"The category of tilting modules is additive, but usually not abelian.",
"If $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ the highest weight modules are simple, and the category is semi-simple.",
"Therefore, this is only an interesting notion if $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell }\\rbrace $ .",
"The key point of moving to this sub-category is that we have the following generalization of usual highest weight theory due Ringel and Donkin , .",
"Theorem 10.2 For each $\\lambda \\in \\mathbb {X}^{*}(H)^{+}$ , there exists a unique indecomposable tilting module $\\mathcal {T}_{\\lambda } \\in \\mathrm {Tilt}_{\\Lambda }(H)$ with highest weight $\\lambda $ .",
"We have that $\\mathrm {dim}(\\mathcal {T}_{\\lambda }(\\lambda )) = 1$ , and, for varying $\\lambda $ , this parametrizes all indecomposable tilting modules.",
"We also get the usual classification of all tilting modules in terms of highest weight tilting modules.",
"Proposition 10.3 , For all $V \\in \\mathrm {Tilt}_{\\Lambda }(H)$ , there exists unique integers $n(\\lambda ) \\in \\mathbb {N}_{\\ge 0}$ for all $\\lambda \\in \\mathbb {X}^{*}(H)^{+}$ and an isomorphism $ V \\simeq \\bigoplus _{\\lambda \\in \\mathbb {X}^{*}(H)^{+}} (\\mathcal {T}_{\\lambda })^{n(\\lambda )} $ of tilting modules.",
"Now we come to a difficult result which was proven by for groups of type $A_{n}$ , for almost all groups, and in general.",
"Theorem 10.4 If we have two tilting modules $V,V^{\\prime } \\in \\mathrm {Tilt}_{\\Lambda }(H)$ then the tensor product $V \\otimes V^{\\prime }$ is tilting.",
"We can now extend this to the $L$ -group using our assumption that $\\ell $ is very good with respect to $G$ .",
"By Theorem REF applied to $H = \\hat{G}$ , we deduce that we have a well-defined category $\\mathrm {Tilt}_{\\Lambda }(\\hat{G})$ of tilting modules, where each object can be written as a direct sum of highest weight tilting modules $\\mathcal {T}_{\\mu }$ for $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ using Proposition REF .",
"Now, given such a $\\mu $ , we consider the reflex field $E_{\\mu }$ , and extend this to a representation of $W_{E_{\\mu }} \\ltimes \\hat{G}$ , as in .",
"We define the tilting module $\\mathcal {T}_{\\mu ^{\\Gamma }}$ as the induction of this representation from $W_{E_{\\mu }} \\ltimes \\hat{G}$ to $W_{\\mathbb {Q}_{p}} \\ltimes \\hat{G}$ , and let $\\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G) \\subset \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ be the full sub-category given by direct sums of such modules, where we note that $\\mathcal {T}_{\\mu ^{\\Gamma }}$ only depends on the $\\Gamma $ -orbit of $\\mu ^{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}/\\Gamma $ of $\\mu $ .",
"Now, since $\\ell $ is very good, it follows that $W_{\\mathbb {Q}_{p}}$ acts on $\\hat{G}$ via a quotient $Q$ that is prime to $\\ell $ by definition , and therefore $W_{\\mathbb {Q}_{p}}/W_{E_{\\mu }}$ is also of order prime to $\\ell $ .",
"Combing this observation, Frobenius Reciprocity/Mackey theory, and Theorem REF , we conclude that $\\mathcal {T}_{\\mu }$ is indeed an irreducible representation of $\\phantom{}^{L}G$ , and that $\\mathrm {Tilt}(\\phantom{}^{L}G)$ is preserved under tensor products.",
"This allows us to define the following.",
"Definition 10.5 Given a continuous $L$ -parameter $\\phi : W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}G(\\Lambda )$ , we say a sheaf $\\mathcal {S}_{\\phi } \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ is a tilting eigensheaf with eigenvalue $\\phi $ if, for all $V \\in \\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G^{I})$ , we are given isomorphisms $ \\eta _{V,I}: T_{V}(\\mathcal {S}_{\\phi }) \\simeq \\mathcal {S}_{\\phi } \\boxtimes r_{V} \\circ \\phi $ of sheaves in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{\\mathbb {Q}_{p}}^{I}}$ , which are natural in $I$ and $V$ , and compatible with compositions and exterior tensor products in $V$ .",
"If $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ this recovers Definition REF .",
"We similarly say $\\mathcal {S}_{\\phi }$ is a weak tilting eigensheaf if only the isomorphisms $\\eta _{V,I}$ exist.",
"Remark 10.6 Our discussion of highest weight theory in §REF for $\\hat{G}^{\\Gamma }$ also extends to the tilting modules $\\mathcal {T}_{\\mu }$ under our assumption that $\\ell $ is very good.",
"In particular, using Proposition REF , we can understand the possible weights occurring in $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ in terms of $\\overline{\\nu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , which lie in the convex hull of the $W_{G}$ orbit of $\\mu _{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+}$ , and the $\\overline{\\nu }$ weight space of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ will be a direct sum over the weight spaces $\\mathcal {T}_{\\mu }(\\nu )$ for $\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})$ mapping to $\\overline{\\nu }$ , as in Lemma REF .",
"To see this, we note, by , $\\ell $ being very good implies that we have the following: $\\hat{G}^{\\Gamma }$ is a smooth linear algebraic group, and $\\hat{G}^{\\Gamma ,\\circ }$ is reductive.",
"$\\hat{G}^{\\Gamma }/\\hat{G}^{\\Gamma ,\\circ } \\simeq \\hat{T}^{\\Gamma }/\\hat{T}^{\\Gamma ,\\circ }$ is of order prime to $\\ell $ , where $\\hat{G}^{\\Gamma ,\\circ }$ (resp.",
"$\\hat{T}^{\\Gamma ,\\circ }$ ) denotes the neutral component of $\\hat{G}^{\\Gamma }$ (resp.",
"$\\hat{T}^{\\Gamma }$ ), and the isomorphism follows as in .",
"These observations allows us the see the highest weight theory of the (possibly disconnected) group $\\hat{G}^{\\Gamma }$ behaves as expected with modular coefficients by mimicking the proof of Lemma REF .",
"We now define our candidate tilting eigensheaf for each of the possible coefficient systems $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ ."
],
[
"The Construction of the Eigensheaf",
"We fix a toral parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ , with induced parameter $\\phi : W_{\\mathbb {Q}_{p}} \\xrightarrow{} \\phantom{}^{L}T(\\Lambda ) \\rightarrow \\phantom{}^{L}G(\\Lambda )$ .",
"Our goal is to construct a candidate tilting eigensheaf with respect to the parameter $\\phi $ .",
"If $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ , we have already carried this out.",
"It is simply the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}(\\mathrm {Bun}_{G})$ viewed as a sheaf in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {F}}_{\\ell })$ via the identification $\\mathrm {D}(\\mathrm {Bun}_{G}) \\simeq \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {F}}_{\\ell })$ , obtained by embedding both categories into $\\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G},\\overline{\\mathbb {F}}_{\\ell })$ .",
"To move beyond this case, we need to invoke the following Lemma.",
"Lemma 10.7 For $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ and $\\phi _{T}$ weakly normalized regular, the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}(\\mathrm {Bun}_{G})$ is ULA with respect to the structure map $\\mathrm {Bun}_{G} \\rightarrow \\ast $ .",
"Given $A \\in \\mathrm {D}(\\mathrm {Bun}_{G})$ , we recall that $A$ being ULA with respect to the map $\\mathrm {Bun}_{G} \\rightarrow \\ast $ is equivalent to saying that its stalks $A|_{\\mathrm {Bun}_{G}^{b}}$ are valued in a complex of smooth representations such that $A^{K}|_{\\mathrm {Bun}_{G}^{b}}$ is a perfect complex of $\\Lambda $ -modules for all open pro-$p$ subgroups $K \\subset J_{b}(\\mathbb {Q}_{p})$ .",
"In particular, the result follows from Corollary REF .",
"Now, if $\\Lambda = \\overline{\\mathbb {Z}}_{\\ell }$ then, by taking inverse limits with respect to the mod $\\ell ^{n}$ reductions of $\\phi _{T}$ , and considering the systems of sheaves given by applying the Eisenstein functor to the eigensheaf attached to these reductions, we obtain a sheaf $ \\widehat{\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})} \\in \\mathrm {D}_{\\text{ét}}^{\\mathrm {ULA}}(X,\\overline{\\mathbb {Z}}_{\\ell }) $ Now, we have a fully faithful embedding $ \\mathrm {D}^{\\mathrm {ULA}}_{\\text{ét}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell }) \\hookrightarrow \\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell }) $ given as in .",
"This embedding is used to define the Hecke operators in the setting of solid sheaves (See ), utilizing that sheaves in the Satake category are ULA over $\\mathrm {Div}^{I}$ .",
"In particular, this embedding is Hecke equivariant in the appropriate sense, and so the filtered eigensheaf property transfers to the image of $\\widehat{\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})}$ in $\\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ .",
"We also have a natural embedding $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell }) \\hookrightarrow \\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ , and we can analogously define the set of ULA objects in it , denoted $\\mathrm {D}_{\\mathrm {lis}}^{\\mathrm {ULA}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ .",
"We have the following claim.",
"Lemma 10.8 Under the embeddings of $\\mathrm {D}_{\\mathrm {lis}}^{\\mathrm {ULA}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ and $\\mathrm {D}^{\\mathrm {ULA}}_{\\text{ét}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ into $\\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ described above, these two full subcategories are isomorphic.",
"We have a semi-orthogonal decomposition of $\\mathrm {D}_{\\mathrm {lis}}^{\\mathrm {ULA}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ () and $\\mathrm {D}_{\\text{ét}}^{\\mathrm {ULA}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ into $\\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Z}}_{\\ell })_{\\mathrm {adm}}$ and $\\hat{\\mathrm {D}}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Z}}_{\\ell })_{\\mathrm {adm}}$ by excision, respectively.",
"Here $\\hat{\\mathrm {D}}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Z}}_{\\ell })$ denotes the derived category of $\\ell $ -complete smooth representations of $J_{b}(\\mathbb {Q}_{p})$ , and the subscript $\\mathrm {adm}$ is used to denote the full subcategory of objects with admissible cohomology.",
"Since the semi-orthogonal decompositions are compatible with the two embeddings into $\\mathrm {D}_{\\blacksquare }(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ it suffices to show that we have an identification $ \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Z}}_{\\ell })_{\\mathrm {adm}} \\simeq \\hat{\\mathrm {D}}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Z}}_{\\ell })_{\\mathrm {adm}} $ ,but this follows from .",
"Using the isomorphism supplied by the previous Lemma, we can regard $\\widehat{\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})} \\in \\mathrm {D}_{\\text{ét}}^{\\mathrm {ULA}}(X,\\overline{\\mathbb {Z}}_{\\ell })$ as an object in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ , which we denote by $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ .",
"Since these isomorphisms are compatible with Hecke operators it follows that Corollary REF transfers to this sheaf.",
"It now remains to describe the desired sheaf when $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ .",
"In this case, we need to assume the parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell })$ is of the form $\\overline{\\phi }_{T} \\otimes \\overline{\\mathbb {Q}}_{\\ell }$ , where $\\overline{\\phi }_{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Z}}_{\\ell })$ is a parameter with weakly normalized regular mod $\\ell $ -reduction.",
"Then we consider the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\overline{\\phi }_{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Z}}_{\\ell })$ constructed above, and define $ \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) := \\mathrm {nEis}(\\mathcal {S}_{\\overline{\\phi _{T}}})[\\frac{1}{\\ell }] \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\overline{\\mathbb {Q}}_{\\ell }) $ by taking the colimit over the multiplication by $\\ell $ maps.",
"Now with the candidate eigensheaf defined, we begin the proof of the eigensheaf property.",
"To capture the necessary integrality conditions, we define the following.",
"Definition 10.9 For $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ with the discrete topology and a continuous toral parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ we say that $\\phi _{T}$ is integral if it admits a mod $\\ell $ -reduction.",
"In particular, if $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ , we assume it is of the form $\\overline{\\phi }_{T} \\otimes _{\\overline{\\mathbb {Z}}_{\\ell }} \\overline{\\mathbb {Q}}_{\\ell }$ for some continuous parameter $\\overline{\\phi }_{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Z}}_{\\ell })$ ."
],
[
"The Hecke Eigensheaf Property",
"Our goal is to prove the following Theorem.",
"Theorem 10.10 For $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ with the discrete topology, we consider $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ an integral parameter with weakly normalized regular mod $\\ell $ reduction.",
"There then exists a perverse sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ which is a filtered eigensheaf with eigenvalue $\\phi $ as in Corollary REF .",
"If $V$ is a direct sum of $\\boxtimes _{i \\in I} \\mathcal {T}_{\\mu _{i}^{\\Gamma }}$ for geometric dominant cocharacters $\\mu _{i}$ , and $\\phi _{T}$ is $\\mu _{i}$ -regular (resp.",
"strongly $\\mu _{i}$ -regular), the filtration on $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ splits (resp.",
"splits uniquely), and we have a natural isomorphism $ T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\boxtimes r_{V} \\circ \\phi $ of sheaves in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G})^{BW_{\\mathbb {Q}_{p}}^{I}}$ .",
"In particular, if $\\phi _{T}$ is $\\mu $ -regular (resp.",
"strongly $\\mu $ -regular) for all geometric dominant cocharacters $\\mu $ then $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ is a weak tilting eigensheaf (resp.",
"tilting eigensheaf).",
"For $b \\in B(G)$ , the stalk $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\in \\mathrm {D}(\\mathrm {Bun}_{G}^{b}) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ is given by an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq \\bigoplus _{w \\in W_{b}} i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ if $b \\in B(G)_{\\mathrm {un}}$ , an isomorphism $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{\\mathrm {Bun}_{G}^{b}} \\simeq 0$ if $b \\notin B(G)_{\\mathrm {un}}$ .",
"Moreover, if $\\mathbb {D}_{\\mathrm {Bun}_{G}}$ denotes Verdier duality on $\\mathrm {Bun}_{G}$ , we have an isomorphism $ \\mathbb {D}_{\\mathrm {Bun}_{G}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) $ of sheaves in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ .",
"The existence of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ and the transfer of the filtered Hecke eigensheaf property to this sheaf was already discussed above.",
"The claim on Verdier duality follows from Theorem REF , and the discussion in .",
"The description of the stalks follows from the construction and Corollary REF .",
"It remains to show that the filtration on $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ splits (resp.",
"splits uniquely) for $\\boxtimes _{i \\in I} \\mathcal {T}_{\\mu _{i}^{\\Gamma }} = V \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ such that $\\phi _{T}$ is $\\mu _{i}$ -regular (resp.",
"strongly $\\mu _{i}$ -regular) for all $i \\in I$ .",
"To do this, we note that an extension between the graded pieces of the filtration on $T_{V}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ is specified by a cohomology class in the $H^{1}$ of $R\\Gamma (W_{\\mathbb {Q}_{p}}^{I}, \\boxtimes _{i \\in I} (\\nu _{i} - \\nu ^{\\prime }_{i})^{\\Gamma } \\circ \\phi _{T}) \\simeq \\bigotimes _{i \\in I} R\\Gamma (W_{\\mathbb {Q}_{p}}, (\\nu _{i} - \\nu ^{\\prime }_{i}) \\circ \\phi _{T})$ for $\\nu _{i}$ and $\\nu ^{\\prime }_{i}$ two distinct non-zero weights of the representation $\\mathcal {T}_{\\mu }$ in $\\hat{T}$ for all $i \\in I$ .",
"In particular, we note if $\\phi _{T}$ is strongly $\\mu _{i}$ -regular then the $H^{1}$ vanishes and the filtration splits, and similarly the $H^{0}$ vanishes so the splitting is unique.",
"Now assume that $\\phi _{T}$ is just $\\mu _{i}$ -regular for all $i \\in I$ .",
"If the $\\mu _{i}$ are replaced with the quasi-minuscule or minuscule generators appearing in the definition of $\\mu _{i}$ -regularity then, by the above discussion we have a unique splitting.",
"Now, by induction, it suffices to show that, given $(\\mu _{1i})_{i \\in I},(\\mu _{2i})_{i \\in I} \\in (\\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+})^{I}$ with $V_{1} := \\boxtimes _{i \\in I} \\mathcal {T}_{\\mu _{1i}^{\\Gamma }}$ and $V_{2} := \\boxtimes _{i \\in I} \\mathcal {T}_{\\mu _{2i}^{\\Gamma }}$ such that we know the filtration on $T_{V_{1}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ and $T_{V_{2}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ splits, the same is true for the filtration on $T_{\\boxtimes _{i \\in I} \\mathcal {T}_{(\\mu _{1i} + \\mu _{2i})^{\\Gamma }}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ .",
"To do this, we can use the isomorphism $ T_{V_{1}}T_{V_{2}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))|_{\\triangle } \\simeq T_{V_{1} \\otimes V_{2}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) $ coming from the fusion product, where $\\triangle : \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{\\mathbb {Q}_{p}}^{I \\sqcup I}} \\rightarrow \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{\\mathbb {Q}_{p}}^{I}}$ is the natural map given by diagonal restriction.",
"By our inductive hypothesis, the diagonal restriction of the filtration on the LHS splits uniquely.",
"Moreover, by the compatibilities of the filtration, we know that the filtration on the LHS refines the filtration on the RHS.",
"In particular, we deduce that the filtration on $T_{V_{1} \\otimes V_{2}}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ has a splitting.",
"However, by considerations of highest weight, the irreducible representation $\\boxtimes _{i \\in I} \\mathcal {T}_{(\\mu _{1i} + \\mu _{2i})^{\\Gamma }}$ is an indecomposable summand of $V_{1} \\otimes V_{2}$ , and this gives the desired claim."
],
[
"Applications",
"Now we will deduce some applications to the cohomology of local shtuka spaces.",
"In §9.1, we will use our eigensheaf to derive an analogue of an averaging formula of Shin for the cohomology of local shtuka spaces.",
"In §9.2 and §9.3, we will discuss a refined version of this averaging formula, and use it to derive a very explicit formula for the isotypic parts of local shtuka spaces with respect to parabolic inductions of characters coming from normalized regular parameters.",
"By combining this with a shtuka analogue of Boyer's trick, we will show that this gives rise to a geometric construction of intertwining operators, and recovers a result analogous to a result of Xiao and Zhu on the irreducible components of affine Deligne-Lusztig varieties, but on the generic fiber.",
"Let's first recall the key definitions.",
"We say a local shtuka datum is a triple $(G,b,\\mu )$ for $\\mu $ a geometric dominant cocharacter of $G$ and $b \\in B(G,\\mu )$ an element of the $\\mu $ -admissible locus of the Kottwtiz set of $G$ (Definition REF ).",
"We let $E$ be the reflex field of $\\mu $ .",
"The triple $(G,b,\\mu )$ defines a diamond $ \\mathrm {Sht}(G,b,\\mu )_{\\infty } \\rightarrow \\mathop {\\rm Spd}(\\breve{E}) $ parameterizing modifications $\\mathcal {F}_{b} \\rightarrow \\mathcal {F}_{G}^{0}$ with meromorphy bounded by $\\mu $ on $X$Note that this is the space denoted $\\mathrm {Sht}(G,b,\\mu ^{-1})$ in .",
"We find that our convention simplifies certain formulae..",
"It carries an action of $G(\\mathbb {Q}_{p}) \\times J_{b}(\\mathbb {Q}_{p})$ and a (non-effective) descent datum from $\\breve{E}$ down to $E$ .",
"This allows us to consider the tower of quotients $ \\mathrm {Sht}(G,b,\\mu )_{\\infty }/\\underline{K} =: \\mathrm {Sht}(G,b,\\mu )_{K} $ for varying open compact subgroups $K \\subset G(\\mathbb {Q}_{p})$ .",
"We write $\\mathcal {S}_{\\mu }$ for the $\\Lambda $ -valued sheaf attached to the highest weight tilting module $\\mathcal {T}_{\\mu }$ of $W_{E} \\ltimes \\hat{G}$ as in the previous section.",
"This is given by pulling back the sheaf on $\\mathrm {Hck}_{G,E}$ defined by $\\mathcal {T}_{\\mu }$ and Theorem REF along the natural map $\\mathrm {Sht}(G,b,\\mu ) \\rightarrow \\mathrm {Hck}_{G,E}$ .",
"In particular, the sheaf $\\mathcal {S}_{\\mu }$ is equivariant with respect to the actions of $G(\\mathbb {Q}_{p})$ and $J_{b}(\\mathbb {Q}_{p})$ by construction.",
"Letting $\\mathrm {Sht}(G,b,\\mu )_{K,\\mathbb {C}_{p}}$ denote the base-change of these spaces to $\\mathbb {C}_{p}$ , we can now define the complex $ R\\Gamma _{c}(G,b,\\mu ) := \\mathop {\\rm colim}\\nolimits _{K \\rightarrow \\lbrace 1\\rbrace } R\\Gamma _{c}(\\mathrm {Sht}(G,b,\\mu )_{K,\\mathbb {C}_{p}},\\mathcal {S}_{\\mu }) $ of $G(\\mathbb {Q}_{p}) \\times J_{b}(\\mathbb {Q}_{p}) \\times W_{E}$ -modules.",
"We now want to disentangle the $G(\\mathbb {Q}_{p})$ and $J_{b}(\\mathbb {Q}_{p})$ action.",
"To do this, for $\\pi $ (resp.",
"$\\rho $ ) a smooth irreducible representation of $G(\\mathbb {Q}_{p})$ (resp.",
"$J_{b}(\\mathbb {Q}_{p})$ ) on $\\Lambda $ -modules, we define the $\\pi $ (resp.",
"$\\rho $ )-isotypic part.",
"I.e the complexes $ R\\Gamma _{c}(G,b,\\mu )[\\pi ] := R\\Gamma _{c}(G,b,\\mu ) \\otimes _{\\mathcal {H}(G)} \\pi $ and $ R\\Gamma _{c}(G,b,\\mu )[\\rho ] := R\\Gamma _{c}(G,b,\\mu ) \\otimes _{\\mathcal {H}(J_{b})} \\rho $ where $\\mathcal {H}(G) := C^{\\infty }_{c}(G(\\mathbb {Q}_{p}),\\Lambda )$ (resp.",
"$\\mathcal {H}(J_{b})$ ) is the usual smooth Hecke algebra of $G$ (resp.",
"$J_{b}$ ).",
"We recall that $\\mathrm {Sht}(G,b,\\mu )_{\\infty }$ has dimension equal to $\\langle 2\\hat{\\rho }, \\mu \\rangle =: d$ .",
"The complexes $R\\Gamma _{c}(G,b,\\mu )[\\pi ]$ (resp.",
"$R\\Gamma _{c}(G,b,\\mu )[\\rho ]$ ) are concentrated in degrees $-d \\le i \\le d$ and are valued in admissible $J_{b}(\\mathbb {Q}_{p})$ (resp.",
"$G(\\mathbb {Q}_{p})$ ) representations, which are moreover of finite length with $\\overline{\\mathbb {Q}}_{\\ell }$ -coefficients .",
"Similarly, we define the complexes $ R\\Gamma _{c}^{\\flat }(G,b,\\mu )[\\pi ] := R\\mathcal {H}om(R\\Gamma _{c}(G,b,\\mu ),\\pi ) $ and $ R\\Gamma _{c}^{\\flat }(G,b,\\mu )[\\rho ] := R\\mathcal {H}om(R\\Gamma _{c}(G,b,\\mu ),\\rho ) $ We will make regular use of the relationship between these complexes and Hecke operators.",
"We write $ T_{\\mu }: \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda ) \\rightarrow \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{E}} $ for the Hecke operator defined by the representation $\\mathcal {T}_{\\mu }$ .",
"We then have the following lemma.",
"Lemma 11.1 Given a local shtuka datum $(G,b,\\mu )$ as above and $\\pi $ (resp.",
"$\\rho $ ) a smooth representation of $G(\\mathbb {Q}_{p})$ (resp.",
"$J_{b}(\\mathbb {Q}_{p})$ ) on $\\Lambda $ -modules, we can consider the associated sheaves $\\rho \\in \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda ) \\simeq \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G}^{b},\\Lambda )$ and $\\pi \\in \\mathrm {D}(G(\\mathbb {Q}_{p}),\\Lambda ) \\simeq \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G}^{\\mathbf {1}},\\Lambda )$ on the HN-strata $j_{b}: \\mathrm {Bun}_{G}^{b} \\hookrightarrow \\mathrm {Bun}_{G}$ and $j_{\\mathbf {1}}: \\mathrm {Bun}_{G}^{\\mathbf {1}} \\hookrightarrow \\mathrm {Bun}_{G}$ , respectively.",
"There then exists an isomorphism $ R\\Gamma _{c}(G,b,\\mu )[\\rho ] \\simeq j_{\\mathbf {1}}^{*}T_{\\mu }j_{b!",
"}(\\rho ) $ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules and an isomorphism $ R\\Gamma _{c}(G,b,\\mu )[\\pi ] \\simeq j_{b}^{*}T_{\\mu ^{-1}}j_{\\mathbf {1}!",
"}(\\pi ) $ of complexes of $J_{b}(\\mathbb {Q}_{p}) \\times W_{E}$ -modules, where $\\mu ^{-1} := -w_{0}(\\mu )$ is a dominant inverse of $\\mu $ .",
"Similarly, we have an isomorphism $ R\\Gamma _{c}^{\\flat }(G,b,\\mu )[\\rho ] \\simeq j_{\\mathbf {1}}^{*}T_{\\mu }j_{b*}(\\rho ) $ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules, and an isomorphism $ R\\Gamma _{c}^{\\flat }(G,b,\\mu )[\\pi ] \\simeq j_{b}^{*}T_{\\mu ^{-1}}j_{\\mathbf {1}*}(\\pi ) $ of complexes of $J_{b}(\\mathbb {Q}_{p}) \\times W_{E}$ -modules.",
"We note that, since we used the tilting module $\\mathcal {T}_{\\mu }$ in the definition of $R\\Gamma _{c}(G,b,\\mu )$ and the Hecke operator $T_{\\mu }$ , the complex $R\\Gamma _{c}(G,b,\\mu )$ will in general be different from the usual complex defined with respect to the highest weight module $V_{\\mu }$ .",
"However, they will agree when we impose the following condition on $\\mu $ with respect to our coefficient system $\\Lambda $ .",
"Definition 11.2 For $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ , we will say $\\mu $ is tilting if the representation $V_{\\mu } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ lies in the full subcategory $\\mathrm {Tilt}_{\\Lambda }(\\hat{G})$ of tilting modules or equivalently if it is irreducible with coefficients in $\\Lambda $ .",
"Remark 11.3 If $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ this condition always holds.",
"Moreover, by Lemma REF , this always holds if $\\mu $ is minuscule.",
"In Appendix , we give more insight into this notion.",
"We now consider a toral parameter $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ with associated smooth character $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ .",
"Unless otherwise stated, we will assume that $\\phi _{T}$ is integral with weakly normalized regular mod $\\ell $ -reduction.",
"Given such a $\\phi _{T}$ , we set $\\phi $ to be the $L$ -parameter of $G$ induced via the natural embedding $\\phantom{}^{L}T(\\Lambda ) \\rightarrow \\phantom{}^{L}G(\\Lambda )$ and consider the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ given by Theorem REF .",
"We begin our analysis by relating our eigensheaves to the averaging formula of Shin."
],
[
"The Averaging Formula",
"For $\\mu $ a geometric dominant cocharacter with reflex field $E$ , we write $r_{\\mu }: W_{E} \\ltimes \\hat{G} \\rightarrow \\mathrm {GL}(\\mathcal {T}_{\\mu })$ for the map defined by $\\mathcal {T}_{\\mu }$ .",
"Since the Hecke operator $T_{\\mu ^{\\Gamma }}$ attached to $\\mathcal {T}_{\\mu ^{\\Gamma }} \\in \\mathrm {Rep}_{\\Lambda }(\\phantom{}^{L}G)$ factors through the Hecke operator $T_{\\mu }$ attached to $\\mathcal {T}_{\\mu }$ (cf. )",
"Theorem REF guarantees that, if $\\phi _{T}$ is $\\mu $ -regular, we have an isomorphism $ T_{\\mu }(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})) \\simeq r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) $ of objects in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )^{BW_{E}}$ .",
"Now let's apply the restriction functor $j_{\\mathbf {1}}^{*}(-)$ to both sides of this isomorphism.",
"By the description of the stalks, we know that the RHS is equal to $r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\pi $ , where $\\pi := i_{B}^{G}(\\chi )$ is the normalized parabolic induction of the character $\\chi $ attached to $\\phi _{T}$ by class field theory.",
"We can also simplify the RHS.",
"In particular, first off note that, since any $G$ -bundle $\\mathcal {F}_{G}$ on $X$ that occurs as a modification $\\mathcal {F}_{G} \\dashrightarrow \\mathcal {F}_{G}^{0}$ of type $\\mu $ lies in the set $B(G,\\mu )$ by , we have an isomorphism $ j_{\\mathbf {1}}^{*}(T_{\\mu }(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))) \\simeq j_{\\mathbf {1}}^{*}T_{\\mu }(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{B(G,\\mu )}) $ where here we view $B(G,\\mu )$ as the open subset of $\\mathrm {Bun}_{G}$ defined by the identification $B(G) \\simeq |\\mathrm {Bun}_{G}|$ describing the underlying topological space $|\\mathrm {Bun}_{G}|$ of $\\mathrm {Bun}_{G}$ , where $B(G)$ has the natural topology given by the partial ordering.",
"We can further refine this by applying excision with respect to the locally closed stratification by the Harder-Narasimhan strata $\\mathrm {Bun}_{G}^{b} \\subset \\mathrm {Bun}_{G}$ for $b \\in B(G,\\mu )$ , using .",
"The excision spectral sequence then tells us that the LHS has a filtration whose graded pieces are isomorphic to: $ j_{\\mathbf {1}}^{*}(T_{\\mu }(j_{b!",
"}j_{b}^{*}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})))) $ In particular, in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\Lambda )$ , the Grothendieck group of $\\Lambda $ -valued admissible $G(\\mathbb {Q}_{p})$ -representations with a smooth action of $W_{E}$ , this tells us that we have an equality: $ \\sum _{b \\in B(G,\\mu )} [j_{\\mathbf {1}}^{*}(T_{\\mu }(j_{b!",
"}j_{b}^{*}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})))] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\pi ]$ Now, using the description of the stalks of $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})$ and Lemma REF , we can spell out the LHS more clearly.",
"In particular, we define the following.",
"Definition 11.4 For $\\phi _{T}$ an arbitrary toral parameter with induced parameter $\\phi $ and $b \\in B(G)$ , we define the complex of smooth admissible $J_{b}(\\mathbb {Q}_{p})$ -representations $\\mathrm {Red}_{b,\\phi }$ as follows.",
"If $b \\notin B(G)_{\\mathrm {un}}$ , we set $\\mathrm {Red}_{b,\\phi }$ to be equal to 0, and, if $b \\in B(G)_{\\mathrm {un}}$ , we set $\\mathrm {Red}_{b,\\phi }$ to be equal to $ \\bigoplus _{w \\in W_{b}} i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] \\in \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda ) $ where $\\delta _{P_{b}}^{1/2}$ is the modulus character of $J_{b}$ defined by the standard parabolic $P_{b} \\subset G$ with Levi factor $M_{b} \\simeq J_{b}$ .",
"This allows us to deduce the following from equation (11).",
"Theorem 11.5 For $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ an integral parameter with weakly normalized regular mod $\\ell $ reduction, if $\\pi := i_{B}^{G}(\\chi )$ is the normalized parabolic induction of the smooth character $\\chi $ attached to $\\phi _{T}$ then, for any geometric dominant cocharacter $\\mu $ such that $\\phi _{T}$ is $\\mu $ -regular, we have an equality $ \\sum _{b \\in B(G,\\mu )} [R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\pi ] $ in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\Lambda )$ .",
"Similarly, one also has an equalityTo deduce this other claim, one can simply act by Verdier duality on the filtration giving rise to (11) and then relax the contragradients, or apply Corollary REF .",
"$ \\sum _{b \\in B(G,\\mu )} [R\\Gamma ^{\\flat }_{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\pi ] $ in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\Lambda )$ .",
"Remark 11.6 We note that, in the above analysis, we didn't necessarily have to restrict to the HN-strata $\\mathrm {Bun}_{G}^{\\mathbf {1}}$ or even a single cocharacter.",
"In particular, by considering Hecke operators defined by representations in $\\mathrm {Tilt}_{\\Lambda }(\\phantom{}^{L}G^{I})$ for a finite index set $I$ , we could have deduced an analogous formula for shtuka spaces with $I$ legs for an arbitrary finite index set $I$ .",
"We could have also restricted to any HN-stratum; however, if the HN-stratum is not defined by a basic element, the answer is not as clean as above.",
"In particular, given a $G$ -bundle $\\mathcal {F}_{b}$ on $X$ corresponding to a general element $b \\in B(G)$ , it is to the best of our knowledge completely unknown exactly which $G$ -bundles $\\mathcal {F}_{G}$ occur as modifications $\\mathcal {F}_{G} \\dashrightarrow \\mathcal {F}_{b}$ of type $\\mu $ .",
"It would be interesting to understand this question better.",
"We leave it to the reader to work out the precise statements of these more general implications.",
"We can use this claim to deduce the averaging formula for an arbitrary toral parameter $\\phi _{T}$ when $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ , by viewing both sides as trace forms on $K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ and using a continuity argument.",
"We note that, in this case, $\\mu $ is always tilting so we have that $R\\Gamma _{c}(G,b,\\mu )$ and $R\\Gamma ^{\\flat }_{c}(G,b,\\mu )$ are just the usual complexes.",
"We recall that $f: K_{0}(G(\\mathbb {Q}_{p})) \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }$ is a trace form if it can be written as $\\mathrm {tr}(\\delta |-)$ for $\\delta \\in C^{\\infty }_{c}(G(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ .",
"We now fix a $\\delta \\in C^{\\infty }_{c}(G(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ and $\\gamma \\in W_{E}$ , we define the following functions attached to this datum $ f^{\\delta ,\\gamma }_{L}: K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\overline{\\mathbb {Q}}_{\\ell } $ $ \\chi \\mapsto \\mathrm {tr}(\\delta \\times \\gamma |i_{B}^{G}(\\chi ) \\boxtimes r_{\\mu } \\circ \\iota (\\chi )|_{W_{E}}) $ $ f^{\\delta ,\\gamma }_{R}: K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow \\overline{\\mathbb {Q}}_{\\ell } $ $ \\chi \\mapsto \\sum _{b \\in B(G,\\mu )_{\\mathrm {un}}} \\sum _{w \\in W_{b}} \\mathrm {tr}(\\delta \\times \\gamma | R\\Gamma _{c}(G,b,\\mu )[i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}]) $ where $\\iota (\\chi ) \\simeq \\phi _{T}$ is the isomorphism given by local class field theory.",
"We have the following lemma.",
"Lemma 11.7 The functions $f^{\\delta ,\\gamma }_{L}$ and $f^{\\delta ,\\gamma }_{R}$ define trace forms on $K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ .",
"This follows from the fact that normalized parabolic induction takes trace forms to trace forms as can be checked from the characterization of trace forms in the trace Paley-Wiener theorem , and the fact that $\\mathrm {tr}(\\delta \\times \\gamma |R\\Gamma _{c}(G,b,\\mu )[-])$ defines a trace form on $K_{0}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ by .",
"This gives the following.",
"Theorem 11.8 For $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell })$ an arbitrary toral parameter with associated character $\\chi $ , and $i_{B}^{G}(\\chi ) =: \\pi $ , we have equalities $ \\sum _{b \\in B(G,\\mu )} [R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\pi ] $ and $ \\sum _{b \\in B(G,\\mu )} [R\\Gamma ^{\\flat }_{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]] = [r_{\\mu } \\circ \\phi |_{W_{E}} \\boxtimes \\pi ] $ in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\overline{\\mathbb {Q}}_{\\ell })$ .",
"It suffices to show that the trace forms $f^{\\delta ,\\gamma }_{L}(\\chi )$ and $f^{\\delta ,\\gamma }_{R}(\\chi )$ agree for varying $\\delta $ and $\\gamma $ and all $\\chi \\in K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ .",
"We define the difference $\\Delta _{\\delta ,\\gamma } := f^{\\delta ,\\gamma }_{L}(\\chi ) - f^{\\delta ,\\gamma }_{R}(\\chi )$ .",
"We say that a subset $S \\in K_{0}(T(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ is dense if $\\Delta _{\\delta ,\\gamma }(x) = 0$ for all $x \\in S$ implies that $\\Delta _{\\delta ,\\gamma } = 0$ .",
"Using Theorem REF and Lemma REF , we can reduce to showing that the subset $S$ of all characters $\\chi $ which are normalized regular and admit a $\\overline{\\mathbb {Z}}_{\\ell }$ -lattice is dense.",
"This is relatively easy to show.",
"In particular, if we view $\\Delta _{\\delta ,\\gamma }$ as a regular function on the variety of unramified characters then the set of characters admitting a $\\overline{\\mathbb {Z}}_{\\ell }$ lattice is Zariski-dense in this variety, and the locus where $\\chi $ is normalized regular is also clearly Zariski-dense.",
"Therefore, the claim follows.",
"This theorem is compatible with existing results.",
"We recall that Shin and Bertoloni-Meli , have described similar averaging formulas.",
"In particular, given a refined endoscopic datum $\\mathfrak {e} = (H,\\mathcal {H},s,\\eta )$ (Definition REF ), Shin and Bertoloni-Meli define maps $ \\mathrm {Red}_{b}^{\\mathfrak {e}}(-): K_{0}^{st}(H(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell }) \\rightarrow K_{0}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell }) $ where $K_{0}^{st}(H(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ denotes the Grothendieck group of stable virtual $\\overline{\\mathbb {Q}}_{\\ell }$ -representations of $H(\\mathbb {Q}_{p})$ .",
"If we are given an $L$ -parameter $\\phi $ which factors as $\\mathcal {L}_{\\mathbb {Q}_{p}} \\xrightarrow{} \\mathcal {H} \\xrightarrow{} \\phantom{}^{L}G$ then, using the local Langlands correspondence for $G$ , we are able to attach a stable distribution $S\\Theta _{\\phi } \\in K_{0}^{st}(H(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ which should satisfy the endoscopic character identities as in .",
"The averaging formula (Conjecture REF ) is a conjectural formula for $ \\sum _{b \\in B(G,\\mu )} R\\Gamma ^{\\flat }_{c}(G,b,\\mu )[\\mathrm {Red}_{b}^{\\mathfrak {c}}(S\\Theta _{\\phi })] $ in $K_{0}(G(\\mathbb {Q}_{p}) \\times W_{E},\\overline{\\mathbb {Q}}_{\\ell })$ .",
"Our averaging formula is related to the case when $\\mathfrak {c}_{\\mathrm {triv}} = (G,1,\\phantom{}^{L}G,\\mathrm {id})$ is the trivial endoscopic datum.",
"In particular, if $\\phi _{T}$ is a generic toral parameter then, by Lemma REF , $\\phi $ should define an actual $L$ -parameter with trivial monodromy, and we can consider the $L$ -packet $\\Pi _{\\phi }(G)$ under the local Langlands correspondence for $G$ appearing in Assumption REF .",
"By Assumption REF (3), the members of the $L$ -packet will be given by the irreducible constituents of $i_{B}^{G}(\\chi )$ .",
"Therefore, we have that $S\\Theta _{\\phi } = [\\pi ]$ in $K_{0}^{st}(G(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ , and in the appendix we verify that the following is true.",
"Proposition 11.9 Let $\\chi : W_{\\mathbb {Q}_{p}} \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }^{*}$ be a smooth generic character, so that, using Lemma REF , we have an equality $ S\\Theta _{\\phi } = [\\pi ] $ in $K_{0}(G(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })^{\\mathrm {st}}$ under the local Langlands correspondence appearing in Assumption REF .",
"Then we always have $ [\\mathrm {Red}_{b,\\phi }] = \\mathrm {Red}^{\\mathfrak {e}_{\\mathrm {triv}}}([\\pi ]) $ in the Grothendieck group $K_{0}(J_{b}(\\mathbb {Q}_{p}),\\overline{\\mathbb {Q}}_{\\ell })$ , and Conjecture REF holds true for the $L$ -parameter $\\phi $ attached to $\\chi $ .",
"We would now like to refine our averaging formula further.",
"In particular, using Theorem REF , we can upgrade this equality in the Grothendieck group to a genuine isomorphism of complexes."
],
[
"The Refined Averaging Formula and Intertwining Operators",
"Consider an element $b \\in B(G)_{\\mathrm {un}}$ with HN-dominant reduction $b_{T} \\in B(T)$ .",
"For $w \\in W_{b}$ , we set $\\rho _{b,w} := i_{B_{b}}^{J_{b}}(\\chi ^{w}) \\otimes \\delta _{P_{b}}^{1/2}$ to be the twisted normalized parabolic induction, and consider $j_{b}: \\mathrm {Bun}_{G}^{b} \\hookrightarrow \\mathrm {Bun}_{G}$ the inclusion of the locally closed HN-strata corresponding to $b$ .",
"Temporarily, we will work with a more general integral toral parameter.",
"As seen in §, the sheaf $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ with its desired properties might not be well-defined for $\\Lambda \\in \\lbrace \\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {Q}}_{\\ell }\\rbrace $ if $\\phi _{T}$ isn't integral with weakly normalized regular mod $\\ell $ -reduction; however, we note that, since $b_{T}$ is HN-dominant, when $\\Lambda = \\overline{\\mathbb {F}}_{\\ell }$ , we always have that $\\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}})$ is only supported on $\\mathrm {Bun}_{G}^{b}$ , since $\\mathrm {Bun}_{B}^{b_{T}}$ will only parametrize split reductions.",
"Moreover, $\\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}})$ will be isomorphic to $j_{b!",
"}(\\rho _{b,1})$ by Proposition REF .",
"Therefore, by the same procedure carried out in §, we always get a well-defined sheaf $\\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}}) \\in \\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ defined by the HN-dominant connected component for any integral toral parameter $\\phi _{T}$ such that we have an isomorphism $j_{b!",
"}(\\rho _{b,1})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ] \\simeq \\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}})$ and, if $\\phi _{T}$ is an integral parameter with weakly generic mod $\\ell $ -reduction, then, as in §, Theorem REF extends in a natural way to $\\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}})$ .",
"We can act by $\\mathbb {D}_{\\mathrm {Bun}_{G}}(-)$ on both sides of (12).",
"By the commutation of Eisenstein series with Verdier duality, the RHS of (12) becomes $ \\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}^{\\vee }}) \\simeq j_{b!",
"}(\\rho ^{*}_{b,1} \\otimes \\delta _{P_{b}})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ] $ where $\\rho ^{*}_{b,1} = (i_{B_{b}}^{J_{b}}(\\chi ) \\otimes \\delta _{P_{b}}^{1/2})^{*} = i_{B_{b}}^{J_{b}}(\\chi ^{-1}) \\otimes \\delta _{P_{b}}^{-1/2}$ denotes the contragradient.",
"On the other hand, the LHS of (12) becomes $ j_{b*}(\\mathbb {D}_{\\mathrm {Bun}_{G}^{b}}(\\rho _{b,1}))[\\langle 2\\hat{\\rho },\\nu _{b}\\rangle ] $ We now need to be a bit careful.",
"In particular, we recall that $\\mathrm {Bun}_{G}^{b} \\simeq [\\ast /\\mathcal {J}_{b}]$ , where $\\mathcal {J}_{b}$ is the group diamond parameterizing automorphisms of $\\mathcal {F}_{b}$ , and we are implicitly using the identification $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G}^{b},\\Lambda ) \\simeq \\mathrm {D}(J_{b}(\\mathbb {Q}_{p}),\\Lambda )$ given by pullback along the natural map $p: [\\ast /\\mathcal {J}_{b}] \\rightarrow [\\ast /\\underline{J_{b}(\\mathbb {Q}_{p})}]$ , as in .",
"Therefore, we need to account for the shifts and twists given by $p^{!",
"}$ .",
"As in the proof of Proposition REF , we can use that the natural section $s$ of $p$ is an iterated fibration of positive Banach-Colmez space to show that $p^{!",
"}(-) \\simeq p^{*}(- \\otimes \\delta _{P_{b}})[-2\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ , and therefore the LHS of (12) becomes $j_{b*}(\\rho _{b,1}^{*} \\otimes \\delta _{P_{b}})[\\langle 2\\hat{\\rho },\\nu _{b}\\rangle - 2\\langle 2\\hat{\\rho },\\nu _{b}\\rangle ] = j_{b*}(\\rho _{b,1}^{*} \\otimes \\delta _{P_{b}})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$ .",
"In conclusion, we have an isomorphism: $ j_{b*}(\\rho _{b,1}^{*} \\otimes \\delta _{P_{b}})[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] \\simeq j_{b!",
"}(\\rho _{b,1}^{*} \\otimes \\delta _{P_{b}})[-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ] $ Relaxing the contragradients and cancelling the shifts and twists, we deduce an isomorphism: $ j_{b*}(\\rho _{b,1}) \\simeq j_{b!",
"}(\\rho _{b,1}) $ Now given $w \\in W_{b}$ , we can replace $\\mathcal {S}_{\\phi _{T}}$ by $\\mathcal {S}_{\\phi _{T}^{w}}$ in the above argument, where $\\phi _{T}^{w}$ is the conjugate of $\\phi _{T}$ by $w$ .",
"This tells us that, if $\\phi _{T}^{w}$ is integral with weakly generic mod $\\ell $ reduction, we have an isomorphism: $ j_{b*}(\\rho _{b,w}) \\simeq j_{b!",
"}(\\rho _{b,w}) $ In conclusion, we deduce the following.",
"Proposition 11.10 For $b \\in B(G)_{\\mathrm {un}}$ , $w \\in W_{b}$ , and $\\rho _{b,w} = i_{B_{b}}^{J_{b}}(\\chi ^{w})$ as defined above, where $\\chi $ is the character attached to an integral toral parameter $\\phi _{T}$ such that its conjugate $\\phi _{T}^{w}$ has weakly generic mod $\\ell $ reduction, we have an isomorphism $ j_{b!",
"}(\\rho _{b,w}) \\simeq j_{b*}(\\rho _{b,w}) $ of objects in $\\mathrm {D}_{\\mathrm {lis}}(\\mathrm {Bun}_{G},\\Lambda )$ .",
"Remark 11.11 Note that if we want this to hold for all $b \\in B(G)_{\\mathrm {un}}$ and $w \\in W_{b}$ this is equivalent to assuming that the mod $\\ell $ -reduction of $\\phi _{T}$ is generic, since $W_{G}$ acts transitively on the $\\Gamma $ -orbits of roots.",
"Remark 11.12 If $b$ is basic then this precisely says that the sheaf defined by $\\rho _{b,w}$ is inert in the sense of .",
"In particular, this Proposition, in conjunction with and Lemma REF , seems to suggest that inert sheaves should correspond precisely to the representations whose (non semi-simplified) $L$ -parameter should have non-trivial monodromy.",
"For example, if one takes the constant sheaf on $\\mathrm {Bun}_{G}^{\\mathbf {1}}$ and considers $j_{\\mathbf {1}!",
"}(\\Lambda )$ then we have that $\\mathbb {D}_{\\mathrm {Bun}_{G}}(j_{\\mathbf {1}!",
"}(\\Lambda )) \\simeq j_{\\mathbf {1}*}(\\Lambda ) \\simeq \\Lambda $ which is not isomorphic to $j_{!",
"}(\\Lambda )$ .",
"Similarly, we see that the $L$ -parameter attached to the trivial representation has monodromy.",
"Now consider $\\mu $ a geometric dominant cocharacter of $G$ with reflex field $E$ and an element $b \\in B(G,\\mu )$ .",
"Applying $j_{\\mathbf {1}}^{*}T_{\\mu }(-)$ to both sides of the previous isomorphism, we conclude, using Lemma REF , an isomorphism: $ R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}] \\simeq R\\Gamma ^{\\flat }_{c}(G,b,\\mu )[\\rho _{b,w}]$ We record this now.",
"Corollary 11.13 Let $(G,b,\\mu )$ be a local shtuka datum.",
"For $b \\in B(G)_{\\mathrm {un}}$ , $w \\in W_{b}$ , and $\\phi _{T}$ an integral toral parameter such that $\\phi _{T}^{w}$ has weakly generic mod $\\ell $ reduction, there is an isomorphism $ R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}] \\simeq R\\Gamma ^{\\flat }_{c}(G,b,\\mu )[\\rho _{b,w}] $ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules.",
"In particular, by Remark REF , if the mod $\\ell $ -reduction of $\\phi _{T}$ is generic, then this is true for all $w \\in W_{b}$ .",
"We now claim that the cohomology of $R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}]$ should be concentrated in degree $\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle $ , for $\\rho _{b,w}$ as above.",
"To do this, let's put ourselves back in the position of an integral $\\phi _{T}$ with weakly normalized regular mod $\\ell $ reduction.",
"We saw that in the previous section that the excision spectral sequence applied to $\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{B(G,\\mu )}$ gives rise to a filtration whose graded pieces are isomorphic to $j_{b!",
"}j_{b}^{*}(\\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}}))$ , but Lemma REF implies that these graded pieces are also isomorphic to $j_{b*}j_{b}^{*}(\\mathrm {Eis}(\\mathcal {S}_{\\phi _{T}}))$ .",
"In particular, this allows us to deduce that the edge maps in the spectral sequence split, and therefore the sequence degenerates, giving an isomorphism: $ \\bigoplus _{b \\in B(G,\\mu )} j_{b!",
"}j_{b}^{*}(\\mathrm {nEis}^{b_{T}}(\\mathcal {S}_{\\phi _{T}})) \\simeq \\mathrm {nEis}(\\mathcal {S}_{\\phi _{T}})|_{B(G,\\mu )} $ We now would like to apply the eigensheaf property.",
"So fix a geometric dominant cocharacter, and assume that $\\phi _{T}$ is $\\mu $ -regular.",
"If $\\pi = i_{B}^{G}(\\chi )$ is the normalized parabolic induction of $\\chi $ as above then, using our description of the stalks, we deduce the following \"refined averaging formula\".",
"Theorem 11.14 For $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\Lambda )$ an integral toral parameter with weakly normalized regular mod $\\ell $ -reduction and $\\mu $ a geometric dominant cocharacter such that $\\phi _{T}$ is $\\mu $ -regular, we have an isomorphism $ \\bigoplus _{b \\in B(G,\\mu )_{\\mathrm {un}}} \\bigoplus _{w \\in W_{b}} R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}] = \\bigoplus _{b \\in B(G,\\mu )} R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }] \\simeq (i_{B}^{G}(\\chi ) \\boxtimes r_{\\mu } \\circ \\phi |_{W_{E}})[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] $ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -modules.",
"Unless otherwise stated, we will from now on assume that $\\phi _{T}$ is integral with weakly normalized regular mod $\\ell $ reduction.",
"Using the definition of the functors $\\mathrm {Red}_{b,\\phi }$ , we can give a very explicit description of the complexes $R\\Gamma _{c}(G,b,\\mu )[\\mathrm {Red}_{b,\\phi }]$ , for $b \\in B(G,\\mu )_{\\mathrm {un}}$ .",
"Corollary 11.15 For $\\mu $ a geometric dominant cocharacter with reflex field $E$ such that $\\phi _{T}$ is $\\mu $ -regular, fixed $b \\in B(G,\\mu )_{\\mathrm {un}}$ , and varying $w \\in W_{b}$ , the complex $R\\Gamma _{c}(G,b,\\mu )[\\rho _{b,w}]$ is isomorphic to $\\phi _{b,w}^{\\mu } \\boxtimes \\pi [-\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$We note that there is a minus sign appearing because we took the derived tensor product of $R\\Gamma _{c}(G,b,\\mu )$ with $\\rho _{b,w}[\\langle 2\\hat{\\rho },\\nu _{b} \\rangle ]$, for $\\phi _{b,w}^{\\mu }$ a representation of $W_{E}$ .",
"Moreover, we have an isomorphism $ \\bigoplus _{b \\in B(G,\\mu )_{\\mathrm {un}}} \\bigoplus _{w \\in W_{b}} \\phi _{b,w}^{\\mu } \\simeq r_{\\mu } \\circ \\phi |_{W_{E}} $ of $W_{E}$ -representations.",
"This leads to a natural question.",
"How can we describe the $W_{E}$ -representations $\\phi _{b,w}^{\\mu }$ in terms of the weights appearing in $r_{\\mu } \\circ \\phi |_{W_{E}}$ .",
"We recall, by Corollary REF , that the orbit of $b_{T}$ under the Weyl group $W_{G}$ can be described as $w(b_{T})$ for $w \\in W_{b}$ varying; moreover, using Corollary REF and Remark REF , we see that we have a correspondence between $B(G,\\mu )_{\\mathrm {un}}$ and the set of Weyl orbits of weights which can occur in the representation $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ .",
"In particular, given $\\overline{\\nu } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }$ , we consider the subspace $ \\bigoplus _{\\begin{array}{c}\\nu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\\\\\tilde{\\nu }_{\\Gamma } = \\overline{\\nu }\\end{array}} \\tilde{\\nu } \\circ \\phi _{T}|_{W_{E}} \\otimes \\mathcal {T}_{\\mu }(\\nu ) $ of $(r_{\\mu } \\circ \\phi )|_{W_{E}}$ , where we note that if we forget the Galois action then this identifies with the $\\nu $ weight space of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ by Lemma REF .",
"Now the refined averaging formula suggests the following.",
"Conjecture 11.16 For all geometric dominant cocharacters $\\mu $ such that $\\phi _{T}$ is $\\mu $ -regular, an unramified element $b \\in B(G,\\mu )_{\\mathrm {un}}$ , and a Weyl group element $w \\in W_{b}$ , we have an isomorphism $ \\bigoplus _{\\begin{array}{c}\\widetilde{w(b_{T})} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\\\\\widetilde{w(b_{T})}_{\\Gamma } = w(b_{T})\\end{array}} \\widetilde{w(b_{T})} \\circ \\phi _{T}|_{W_{E}} \\otimes \\mathcal {T}_{\\mu }(\\widetilde{w(b_{T})}) \\simeq \\phi _{b,w}^{\\mu } $ of $W_{E}$ -representations, where $b_{T}$ is a HN-dominant reduction of $b$ .",
"For the rest of this section, let us look at some cases where this can be shown explicitly, using a shtuka analogue of Boyer's trick.",
"To illustrate the idea, we begin with a particularly nice example, where Theorem REF and Conjecture REF can be checked by hand.",
"Example 11.17 Let $G = \\mathrm {GL}_{2}$ and $\\mu = (1,0)$ .",
"Write $\\phi _{T} = \\phi _{1} \\oplus \\phi _{2}$ , and consider the set $B(G,\\mu )$ .",
"It consists of two elements: the $\\mu $ -ordinary element and the basic element.",
"Only the $\\mu $ -ordinary element lies in $B(G,\\mu )_{\\mathrm {un}}$ ; therefore, only this element contributes to the expression in Theorem REF .",
"Namely, if $b_{\\mu }$ denotes the $\\mu $ -ordinary element, we note that $\\langle 2\\hat{\\rho }, \\nu _{b_{\\mu }} \\rangle = \\langle 2\\hat{\\rho }, \\mu \\rangle = 1$ .",
"We conclude that Theorem REF is an isomorphism $ R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\chi \\otimes \\delta _{B}^{1/2}] \\oplus R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\chi ^{w_{0}} \\otimes \\delta _{B}^{1/2}] \\simeq i_{B}^{GL_{2}}(\\chi ) \\boxtimes \\phi [-1] $ of $G(\\mathbb {Q}_{p}) \\times W_{\\mathbb {Q}_{p}}$ -representations.",
"This can be seen through direct computation.",
"In particular, we have an isomorphism $J_{b_{\\mu }} \\simeq T$ , and, since $\\mu $ is minuscule, we have that $\\mathcal {S}_{\\mu } \\simeq \\Lambda [1](\\frac{1}{2})$ .",
"The space $\\mathrm {Sht}(G,b_{\\mu },\\mu )_{\\infty ,\\mathbb {C}_{p}}$ is the moduli space parameterizing modifications $\\mathcal {O}(-1) \\oplus \\mathcal {O} \\dashrightarrow \\mathcal {O}^{2}$ of type $(1,0)$ .",
"Every such modification is determined by an injection $\\mathcal {O}(-1) \\hookrightarrow \\mathcal {O}$ of line bundles.",
"Formally, this implies that the space $\\mathrm {Sht}(G,b,\\mu )_{\\infty ,\\mathbb {C}_{p}}$ is parabolically induced from the space parameterizing such injections as a space with $T(\\mathbb {Q}_{p})$ action.",
"Here $T(\\mathbb {Q}_{p})$ acts on the space of injections $\\mathcal {O}(-1) \\hookrightarrow \\mathcal {O}$ via the scaling action precomposed with projection to the first factor of $T(\\mathbb {Q}_{p})$ .",
"This is a manifestation of the fact that $\\mathrm {Sht}(G,b_{\\mu },\\mu )_{\\infty ,\\mathbb {C}_{p}}$ is a $\\mathcal {J}_{b}$ -torsor over the flag variety $\\underline{(G/B)(\\mathbb {Q}_{p})} \\simeq \\mathbb {P}^{1}(\\mathbb {Q}_{p}) \\subset \\mathbb {P}^{1}_{\\mathbb {C}_{p}} \\simeq \\mathrm {Gr}_{G,\\le \\mu ,\\mathbb {C}_{p}}$ , where the last isomorphism is the Bialynicki-Birula map.",
"In particular, note that the compactly supported cohomology of $\\underline{G/B(\\mathbb {Q}_{p})}$ is precisely the space of compactly supported functions on $(G/B)(\\mathbb {Q}_{p})$ .",
"All in all, this allows us to conclude isomorphisms $ R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\chi \\otimes \\delta _{B}^{1/2}] = \\mathrm {Ind}_{B}^{G}(\\chi \\otimes \\delta _{B}^{1/2}) \\boxtimes \\phi _{1}[-1] \\simeq i_{B}^{G}(\\chi ) \\boxtimes \\phi _{1}[-1] $ $ R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\chi ^{w_{0}} \\otimes \\delta _{B}^{1/2}] = \\mathrm {Ind}_{B}^{G}(\\chi ^{w_{0}} \\otimes \\delta _{B}^{1/2}) \\boxtimes \\phi _{2}[-1] \\simeq i_{B}^{G}(\\chi ^{w_{0}}) \\boxtimes \\phi _{2}[-1] $ where there is a cancellation of the $\\frac{1}{2}$ Tate twist in $\\mathcal {S}_{\\mu }$ and the Tate twist coming from $\\delta _{B}^{1/2}$ , as in §, with the Tate twists coming from the compactly supported cohomology of $\\mathcal {J}_{b}$ .",
"However, if $\\chi $ is attached to a generic parameter $\\phi _{T}$ , this implies that $i_{B}^{G}(\\chi )$ is irreducible as in Example REF , and it follows that we have an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w_{0}})$ , which allows us to conclude the result.",
"Now let's generalize this example.",
"In particular, recall that $B(G,\\mu )$ has a distinguished $\\mu $ -ordinary element, denoted $b_{\\mu }$ , which is the maximal element with respect to the partial ordering on $B(G,\\mu )$ , and has the property that $\\tilde{\\mu } = \\nu _{b_{\\mu }}$ , where $\\tilde{\\mu }$ is the weighted average over the Galois orbit of $\\mu $ as in §REF .",
"If we write $\\mu _{T}$ for $\\mu $ viewed as a geometric cocharacter of $T$ in the negative Weyl chamber defined by the choice of Borel, we can see that $b_{\\mu }$ admits a HN-dominant reduction to the unique element $b_{\\mu _{T}} \\in B(T,\\mu _{T})$ .",
"In other words, the element $b_{\\mu }$ always lies in $B(G,\\mu )_{\\mathrm {un}} := B(G,\\mu ) \\cap B(G)_{\\mathrm {un}}$ .",
"Conjecture REF suggests to us that this should give rise to the contribution given by the highest weight of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ , which will have multiplicity one.",
"We now prove the following result using a shtuka analogue of Boyer's trick proven by Gaisin-Imai .",
"Proposition 11.18 For $\\mu $ any geometric dominant cocharacter with reflex field $E$ , $b_{\\mu } \\in B(G,\\mu )_{\\mathrm {un}}$ the $\\mu $ -ordinary element with HN-dominant reduction $b_{\\mu _{T}}$ , $w \\in W_{b}$ varying, and $\\phi _{T}$ any toral parameter, we have an isomorphism $ R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\rho _{b_{\\mu },w}] \\simeq w(\\mu _{T}) \\circ \\phi _{T}|_{W_{E}} \\boxtimes i_{B}^{G}(\\chi ^{w})[-\\langle 2\\hat{\\rho }, \\nu _{b_{\\mu }} \\rangle ] $ of $W_{E} \\times G(\\mathbb {Q}_{p})$ -representations.",
"We note that the element $b_{\\mu } \\in B(G,\\mu )$ is Hodge-Newton reducible in the sense of .",
"In particular, $b_{\\mu }$ is induced from the unique element $b_{\\mu _{T}} \\in B(T,\\mu _{T})$ via the natural map $B(T) \\rightarrow B(G)$ .",
"Consider a rank $k$ vector bundle of the form $\\bigoplus _{i = 1}^{k} \\mathcal {O}(n_{i})$ for $n_{i} \\in \\mathbb {Z}$ and suppose we have a modification: $ \\bigoplus _{i = 1}^{k} \\mathcal {O}(n_{i}) \\dashrightarrow \\mathcal {O}^{n} $ Then it is easy to see that such a modification will be determined by a tuple of modifications $ \\mathcal {O}(n_{i}) \\dashrightarrow \\mathcal {O} $ for all $i = 1,\\ldots ,k$ .",
"If we apply the Tannakian formalism (See ), this tells us that the space $\\mathrm {Sht}(G,b_{\\mu },\\mu )_{\\infty ,\\mathbb {C}_{p}}$ parameterizing modifications of the form $ \\mathcal {F}_{b_{\\mu }} \\dashrightarrow \\mathcal {F}_{G}^{0} $ will be determined by the spaces $\\mathrm {Sht}(T,w(b_{\\mu _{T}}),w(\\mu _{T}))_{\\infty ,\\mathbb {C}_{p}}$ parametrizing modifications of the form $ \\mathcal {F}_{w(b_{\\mu _{T}})} \\dashrightarrow \\mathcal {F}_{T}^{0}$ with meromorphy equal to $w(\\mu _{T})$ for varying $w \\in W_{b_{\\mu }}$ .",
"More specifically, $\\mathrm {Sht}(G,b,\\mu )_{\\infty ,\\mathbb {C}_{p}}$ will be parabolically induced from $\\mathrm {Sht}(T,w(b_{T}),w(\\mu _{T}))_{\\infty ,\\mathbb {C}_{p}}$ for any $w \\in W_{b_{\\mu }}$ (See for details) as spaces with $T(\\mathbb {Q}_{p})$ action.",
"In particular, this tells us that the moduli space $\\mathrm {Sht}(G,b_{\\mu },\\mu )_{\\infty ,\\mathbb {C}_{p}}$ parameterizing modifications of meromorphy $\\le \\mu $ is actually equal to the open subspace $\\mathrm {Sht}(G,b,\\mu )_{\\infty }^{\\mu }$ parameterizing modifications of meromorphy equal to $\\mu $ .",
"This is because any modification induced from a modification $ \\mathcal {F}_{b_{\\mu _{T}}} \\dashrightarrow \\mathcal {F}_{T}^{0}$ of type $\\mu _{T}$ will be of type $\\mu $ , which implies that we have an isomorphism $\\mathcal {S}_{\\mu } \\simeq \\Lambda [d](\\frac{d}{2})$ , where $d = \\langle 2\\hat{\\rho },\\nu _{b_{\\mu }} \\rangle = \\langle 2\\hat{\\rho },\\mu \\rangle $ using .",
"Here we need to be a bit careful since $\\mathcal {S}_{\\mu }$ is the pullback of the sheaf associated to the tilting module $\\mathcal {T}_{\\mu }$ not $V_{\\mu }$ as per usual.",
"However, we note that the above discussion tells us that the Newton strata in the Schubert cell/variety $\\mathrm {Gr}_{G,\\le \\mu ,\\mathbb {C}_{p}}^{b_{\\mu }} = \\mathrm {Gr}_{G,\\mu ,\\mathbb {C}_{p}}^{b_{\\mu }}$ has only non-empty intersection with the semi-infinite cells $\\mathrm {S}_{G,w(\\mu _{T}),\\mathbb {C}_{p}}$ indexed by the Weyl group orbits of the highest weight, using the Remark proceeding REF , and since both $\\mathcal {T}_{\\mu }$ and $V_{\\mu }$ have highest weight with multiplicity one the discrepancy doesn't matter via Corollary REF .",
"It remains to describe the complex $R\\Gamma _{c}(G,b_{\\mu },\\mu )$ .",
"Using our above observations and in particular Strictly speaking, Imai-Gaisin work with $\\overline{\\mathbb {Q}}_{\\ell }$ coefficients, but it is easy to see the proof extends to other coefficient systems., it follows that we have an isomorphism $ R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\rho _{b,w}] \\simeq \\mathrm {Ind}_{B}^{G}(R\\Gamma _{c}(T,b_{T},\\mu _{T})[\\chi ^{w} \\otimes \\delta _{B}^{1/2}])[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ](-\\hat{\\rho }, \\mu \\rangle ) $ of complexes of $W_{E} \\times T(\\mathbb {Q}_{p})$ -representations.",
"Moreover, by our analysis in §, we know that $ R\\Gamma _{c}(T,b_{T},\\mu _{T})[\\chi ^{w} \\otimes \\delta _{B}^{1/2}] \\simeq \\chi ^{w} \\otimes \\delta _{B}^{1/2} \\boxtimes w(\\mu _{T}) \\circ \\phi _{T}|_{W_{E}}(\\langle \\hat{\\rho }, \\mu \\rangle ) $ and in turn we have a chain of isomorphisms $\\begin{split}R\\Gamma _{c}(G,b_{\\mu },\\mu )[\\rho _{b,w}] \\simeq \\mathrm {Ind}_{B}^{G}(R\\Gamma _{c}(T,b_{T},\\mu _{T})[\\chi ^{w} \\otimes \\delta _{B}^{1/2}])[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ](-\\hat{\\rho }, \\mu \\rangle ) & \\simeq \\\\ \\mathrm {Ind}_{B}^{G}(\\chi ^{w} \\otimes \\delta _{B}^{1/2}) \\boxtimes w(\\mu _{T}) \\circ \\phi _{T}|_{W_{E}}(\\langle \\hat{\\rho },\\mu \\rangle - \\langle \\hat{\\rho },\\mu \\rangle )[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ] & \\simeq \\\\ i_{B}^{G}(\\chi ^{w}) \\boxtimes w(\\mu _{T}) \\circ \\phi _{T}|_{W_{E}}[-\\langle 2\\hat{\\rho }, \\nu _{b} \\rangle ]\\end{split}$ of complexes of $G(\\mathbb {Q}_{p}) \\times W_{E}$ -representations which gives the desired result.",
"Remark 11.19 We note that in the proof we did not use any of our results on geometric Eisenstein series.",
"It would be interesting to generalize some of these computations to some non-principal situations.",
"In particular, if one works with a general parabolic $P$ with Levi factor $M$ and a supercuspidal parameter $\\phi _{M}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}M$ and assumes that Fargues' conjecture holds on the Levi subgroup $M$ , then the results of guarantee that one has similar formulas relating the cohomology of the shtuka spaces of $G$ to basic local shtuka spaces of $M$ for Hodge-Newton reducible $b \\in B(G,\\mu )$ admitting a reduction to $M$ .",
"The description of the eigensheaf in Fargues' conjecture, as described for odd unramified unitary groups in and partially for $\\mathrm {GSp}_{4}$ in , gives one a very explicit description of these basic local shtuka spaces, which would in turn give a computational approach to understanding and generalizing these formulas beyond the principal case.",
"We fully anticipate that some of the methods we use in the principal case generalize to the non-principal case; however, the results seem much more technical, and these computations would give a nice foothold into the problem.",
"This explicit calculation has some interesting consequences.",
"In particular, we already saw in Example REF that relating Proposition REF to Theorem REF required using the existence of an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w})$ ; i.e.",
"intertwining operators.",
"This phenomenon actually persists.",
"In particular, we deduce the following.",
"Corollary 11.20 Let $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\Lambda ^{*}$ be a character obtained from an integral toral parameter $\\phi _{T}$ whose mod $\\ell $ -reduction is weakly normalized regular.",
"Suppose that there exists a geometric dominant cocharacter $\\mu $ which is not fixed by any element of $W_{G}$ and such that $\\phi _{T}$ is $\\mu $ -regular.",
"Then, for all $w \\in W_{G}$ , there is an isomorphism $ i_{\\chi ,w}: i_{B}^{G}(\\chi ) \\xrightarrow{} i_{B}^{G}(\\chi ^{w}) $ of smooth $G(\\mathbb {Q}_{p})$ -representations.",
"Since $\\mu $ is not fixed by any element of $W_{G}$ , this implies that $M_{b_{\\mu }} = J_{b_{\\mu }} = T$ and $W_{b_{\\mu }} = W_{G}$ .",
"The previous Proposition then tells us that we have an isomorphism $ R\\Gamma _{c}(G,b,\\mu )[\\chi ^{w}] \\simeq w(\\mu _{T}) \\circ \\phi _{T}|_{W_{E}} \\boxtimes i_{B}^{G}(\\chi ^{w}) $ for all $w \\in W_{G}$ .",
"On the other hand, since $\\phi _{T}$ is $\\mu $ -regular, Corollary REF tells us that the LHS must be isomorphic as a $G(\\mathbb {Q}_{p})$ representation to $i_{B}^{G}(\\chi )$ .",
"The claim follows.",
"In particular, the refined averaging formula, together with the direct computation of $R\\Gamma _{c}(G,b,\\mu )[\\chi ^{w}]$ provided above, gives rise to an isomorphism: $i_{\\chi ,w}: i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w})$ .",
"If we combine this with Proposition REF , Lemma REF , and Lemma REF (2) then this recovers the following special case of Corollary REF .",
"Corollary 11.21 Suppose that $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\overline{\\mathbb {Q}}_{\\ell }^{*}$ is a normalized regular character admitting a $\\overline{\\mathbb {Z}}_{\\ell }$ -lattice then the representation $i_{B}^{G}(\\chi )$ is irreducible.",
"This suggests an interesting relationship between the theory of geometric Eisenstein series over the Fargues-Fontaine curve and the classical theory of intertwining operators and the Langlands quotient, which we hope is explored more in the future.",
"This prospect becomes even more exciting in the $\\ell $ -modular situation.",
"The theory of intertwining operators in this context has been partially explored by Dat , and the Langlands quotient theorem does not naively hold in this context, as the following example illustrates.",
"Example 11.22 We thank Robert Kurinczuk for bringing this example to our attention.",
"Suppose that $\\ell \\ne 2$ , $\\ell \\mid p - 1$ , and $G = \\mathrm {GL}_{2}$ .",
"We let $\\chi = \\mathbf {1}_{T}$ be the trivial character.",
"We note that, by our assumption that $\\ell \\mid p - 1$ , we have an isomorphism $|\\cdot | \\simeq \\mathbf {1}_{T}$ .",
"We consider the usual short exact sequence $ 0 \\rightarrow \\mathbf {1}_{G} \\rightarrow i_{B}^{G}(|\\cdot |) \\rightarrow \\mathrm {St}_{G} \\rightarrow 0 $ where $\\mathrm {St}_{G}$ denotes the Steinberg representation.",
"However, under our assumption on $\\ell $ , we have isomorphisms $|\\cdot |^{1} \\simeq \\mathbf {1}_{T} \\simeq |\\cdot |^{-1}$ , and therefore acting by smooth duality actually gives a splitting of the short exact sequence and in turn a chain of isomorphisms $ i_{B}^{G}(\\chi ^{w_{0}}) \\simeq \\mathrm {St}_{G} \\oplus \\mathbf {1}_{G} \\simeq i_{B}^{G}(\\chi ) $ of smooth $G(\\mathbb {Q}_{p})$ -representations.",
"In particular, we see that $i_{B}^{G}(\\mathbf {1}_{T})$ does not have a unique irreducible quotient in this case, so that the Langlands quotient theorem cannot naively hold.",
"Let's now explore a case in which Proposition REF can be used to verify Conjecture REF .",
"Suppose that $\\mu $ is a geometric dominant cocharacter such that the image $\\mu _{\\Gamma } \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})_{\\Gamma }^{+}$ is quasi-minuscule or minuscule with respect to the pairing with $\\mathbb {X}_{*}(\\hat{T}^{\\Gamma })$ .",
"In this case, we recall that the orbit of the highest weight space $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}(b_{\\mu })$ forms a closed orbit under the relative Weyl group $W_{G}$ , where $b_{\\mu } \\in B(G,\\mu )_{\\mathrm {un}}$ is the $\\mu $ -ordinary element.",
"It then follows that all the weight spaces of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ in this orbit will be given by the $\\kappa $ -invariants of $w(b_{\\mu _{T}}) \\in B(T) \\simeq \\mathbb {X}^{*}(\\hat{T}^{\\Gamma })$ , for $w \\in W_{G}$ varying.",
"However, by Proposition REF , we see that all the weight spaces of this form come from the contribution of the $\\mu $ -ordinary element to the refined averaging formula.",
"If $\\mu _{\\Gamma }$ is minuscule with respect to the above pairing this is the only weight space in $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ , and we see that Conjecture REF is true.",
"If $\\mu _{\\Gamma }$ is quasi-minuscule, the only other element in $B(G,\\mu )_{\\mathrm {un}}$ is the basic element, denoted $\\mu ^{\\flat }$ .",
"By Corollary REF , the highest weight representation $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ admits a central weight space $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}(\\mu _{T}^{\\flat })$ in this case, where $\\mu _{T}^{\\flat } \\in B(T)$ is the (unique) reduction to $T$ of $\\mu ^{\\flat } \\in B(G,\\mu )_{\\mathrm {un}}$ .",
"We deduce the following from the refined averaging formula.",
"Corollary 11.23 Let $\\mu $ be a geometric dominant cocharacter such that $\\phi _{T}$ is $\\mu $ -regular with reflex field $E$ .",
"Assume that $\\mu _{\\Gamma }$ is quasi-minuscule with respect to the pairing with $\\mathbb {X}_{*}(\\hat{T}^{\\Gamma })$ .",
"Let $\\mu ^{\\flat } \\in B(G,\\mu )$ be the unique basic element.",
"It follows by Corollary REF that $\\mu ^{\\flat }$ is unramified in this case.",
"We let $\\mu ^{\\flat }_{T}$ be its unique reduction to $B(T)$ .",
"There is an isomorphism $ R\\Gamma _{c}(G,b,\\mu ^{\\flat })[i_{B_{b}}^{J_{b}}(\\chi )] \\simeq \\bigoplus _{\\begin{array}{c}\\tilde{\\mu }^{\\flat }_{T} \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}}) \\\\\\tilde{\\mu }^{\\flat }_{T\\Gamma } = \\mu ^{\\flat }_{T}\\end{array}} \\tilde{\\mu }^{\\flat }_{T} \\circ \\phi _{T}|_{W_{E}} \\otimes \\mathcal {T}_{\\mu }(\\tilde{\\mu }^{\\flat }_{T}) \\boxtimes i_{B}^{G}(\\chi ) $ of complexes of $W_{E} \\times G(\\mathbb {Q}_{p})$ -modules.",
"This follows from combining Theorem REF and Proposition REF .",
"Noting that, by the $\\mu $ -regularity assumption on $\\phi _{T}$ , the Weil group action of the contribution of the highest weight to the refined averaging formula must be distinct from the Weil group action on the contribution of the central weight space.",
"Remark 11.24 If $G$ is split then the condition that the central weight space of $\\mathcal {T}_{\\mu }|_{\\hat{G}^{\\Gamma }}$ is non-zero cannot occur if $\\mu $ is minuscule.",
"However, if $G$ is not split then it can occur that the central weight space is non-trivial even if $\\mu $ is minuscule.",
"For example, if one considers an odd quasi-split unitary group $\\mathrm {U}_n$ and the cocharacter $(1,0,\\ldots ,0,0)$ then the $\\sigma $ -centralizer will be isomorphic to $\\mathrm {U}_n$ , and therefore the basic element $b \\in B(G,\\mu )$ lies in $B(G)_{\\mathrm {un}}$ .",
"A more in depth characterization of when this can occur is given in .",
"This result is in some sense a generic fiber manifestation of some of the results in .",
"Here, in the case that $G$ is unramified, Xiao and Zhu relate the irreducible components of affine Deligne-Luztig varieties to the central weight spaces appearing above, and use these irreducible components to construct cohomological correspondences on the special fibers of certain Shimura varieties using uniformization.",
"These affine Deligne-Luztig varieties are precisely the special fibers of a natural integral model of $\\mathrm {Sht}(G,\\mu ^{\\flat },\\mu )_{\\infty }/\\underline{K}$ for a choice of hyperspecial subgroup $K \\subset G(\\mathbb {Q}_{p})$ ."
],
[
"Intertwining Operators and the Irreducibility of Principal Series",
"We want to show the irreducibility of principal series representations obtained from the characters $\\chi $ attached to parameters $\\phi _{T}: W_{\\mathbb {Q}_{p}} \\rightarrow \\phantom{}^{L}T(\\overline{\\mathbb {Q}}_{\\ell })$ satisfying the Conditions in REF .",
"We let $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {C}^{*}$ be the character attached to $\\phi _{T}$ via local class field theory and a fixed isomorphism $\\overline{\\mathbb {Q}}_{\\ell } \\simeq \\mathbb {C}$ sending $p^{1/2}$ to the fixed choice of square root in $\\overline{\\mathbb {Q}}_{\\ell }$ .",
"Our goal will be to show the following two facts.",
"Proposition 1.1 Suppose that $\\chi $ is a regular character in the sense that it is not fixed under any $w \\in W_{G}$ then if we have an isomorphism $ i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w_{0}}) $ the representation $i_{B}^{G}(\\chi )$ is irreducible.",
"Proposition 1.2 If $\\chi : T(\\mathbb {Q}_{p}) \\rightarrow \\mathbb {C}^{*}$ is a generic character then, for all $w \\in W_{G}$ , we have an isomorphism $ i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w}) $ of smooth $G(\\mathbb {Q}_{p})$ -representations.",
"By combining these two Propositions, we deduce the following.",
"Corollary 1.3 For $\\chi $ a generic regular character, the induction $i_{B}^{G}(\\chi )$ is always irreducible.",
"As a consequence, by Lemma REF we deduce the following.",
"Corollary 1.4 For $\\chi $ a normalized regular character the induction $i_{B}^{G}(\\chi )$ is always irreducible.",
"The idea behind proving such results is due to Speh and Vogan in the archimedean case.",
"They study the reducibility of principal series using the Langlands classification .",
"Strictly speaking, their analysis is for the archimedean place, but it is easy to see that it extends to the case of a $p$ -adic group using the analogous Langlands classification there .",
"They (roughly) break the problem of understanding the reducibility points of principal series representations into two parts: () Understanding the reducibility points of non-unitary principal series with respect to the parabolic inductions from $T$ to $M_{i}$ for $i \\in \\mathcal {J}$ , where $M_{i}$ is the rank 1 Levi subgroup of $G$ whose relative Dynkin diagram is given by $\\lbrace i\\rbrace \\subset \\mathcal {J}$ .",
"() Understanding the reducibility points of unitary principal series representations with respect to induction from $T$ to a (not necessarily rank 1) proper Levi subgroup of $G$ .",
"We will now explain this heuristic in our case.",
"To see analogous analysis worked out more explicitly for specific $p$ -adic reductive groups, we point the reader to , , for a small sample.",
"For $(2)$ , a very definitive answer to such questions can be found in the paper of Keys .",
"In particular, by the number of irreducible components of such a unitary parabolic induction can be computed in terms of the Knapp-Stein R-Group , , , which Keys determines for all split groups.",
"While this is very interesting, we will not address this here.",
"In particular, we have the following.",
"Corollary 1.5 If $\\chi $ is a regular unitary character then the normalized parabolic induction $i_{B}^{G}(\\chi )$ is irreducible.",
"This follows from the Bruhat decomposition and the fact that $i_{B}^{G}(\\chi )$ is unitary and therefore fully decomposable (See or ).",
"Now consider $\\mathbb {X}^{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{\\Gamma } \\otimes \\mathbb {R} \\simeq \\mathbb {X}_*(A) \\otimes \\mathbb {R} \\simeq \\mathbb {R}^{d}$ the set of unramified characters.",
"For $s \\in \\mathbb {R}^{d}$ , we write $\\nu ^{s}$ for the associated unramified character.",
"We will say that $\\nu ^{s}$ is positive (resp.",
"strictly positive) if, for all simple (reduced) positive coroots $\\alpha _{i,A}$ , the precomposition of $\\nu ^{s}$ with $\\alpha _{i,A}$ is positive (resp.",
"strictly positive), or in other words that $s$ lies in the positive Weyl chamber of $\\mathbb {X}_*(A) \\otimes \\mathbb {R}$ defined by the Borel.",
"We write $\\chi = \\mu _{\\chi }\\nu ^{s_{\\chi }}$ , where $\\mu _{\\chi }$ is a unitary character and $\\nu ^{s_{\\chi }}$ is an unramified character of $T$ , for some $s_{\\chi } \\in \\mathbb {R}^{d}$ .",
"We now consider intertwining operators.",
"Recall that, for $w \\in W_{G}$ and $s \\in \\mathbb {R}^{d}$ , we have the intertwining operator $ I_{w}(\\mu _{\\chi },s): i_{B}^{G}(\\mu _{\\chi }\\nu ^{s}) \\rightarrow i_{B}^{G}((\\mu _{\\chi }\\nu ^{s})^{w}) $ $ f(g) \\mapsto \\int _{U_{w}} f(w^{-1}ug)du $ where $U_{w} := U \\cap wU^{-}w^{-1}$ and $U^{-}$ is the unipotent radical of the opposite Borel.",
"This integral will converge if $\\nu ^{s}$ lies sufficiently deep in the dominant Weyl chamber, and admits a meromorphic continuation as a function of $s$ (where one allows $s$ to be complex and imposes this constraint on the real part).",
"Away from these poles, it gives rise to an intertwining operator between $i_{B}^{G}(\\mu _{\\chi }\\nu ^{s})$ and $i_{B}^{G}((\\mu _{\\chi }\\nu ^{s})^{w})$ in the usual representation theory sense.",
"For our purposes, we will be interested in the intertwining operator $I_{w_{0}}(\\mu _{\\chi },s)$ for the element of longest length $w_{0}$ .",
"In this case, one can see that the operator is convergent for all $s$ which are strictly positive, and the image of the operator is the unique irreducible Langlands quotient of $i_{B}^{G}(\\mu _{\\chi }\\nu ^{s})$ (See ).",
"This quotient has multiplcity one and therefore the intertwining space between $i_{B}^{G}(\\chi )$ and $i_{B}^{G}(\\chi ^{w_{0}})$ is one-dimensional.",
"It follows that, if $\\nu ^{s_{\\chi }}$ is strictly positive, it suffices to exhibit an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w_{0}})$ to show irreducibility.",
"We will now use this kind of analysis to prove Proposition REF .",
"(Proposition REF ) First off note that, for all $\\chi $ , we have an equality $ [i_{B}^{G}(\\chi )] = [i_{B}^{G}(\\chi ^{w_{0}})] $ in $K_{0}(G(\\mathbb {Q}_{p}))$ (See for example ).",
"This allows us to, without loss of generality, assume that $\\nu ^{s_{\\chi }}$ is positive.",
"Now, consider the set of $i \\in \\mathcal {J}$ such that the precomposition of $\\nu ^{s_{\\chi }}$ with $\\alpha _{i,A}$ is equal to 0.",
"This defines a parabolic $P_{\\chi }$ of $G$ which we decompose as $P_{\\chi } = M_{\\chi }A_{\\chi }N_{\\chi }$ , where $A_{\\chi }$ is the maximal split torus in the center of $M_{\\chi }$ .",
"If $\\mathcal {J}_{M_{\\chi }}$ denotes the vertices of the relative Dynkin diagram of $M_{\\chi }$ then $\\mathcal {J}_{M_{\\chi }} \\subset \\mathcal {J}$ corresponds to the set of simple positive coroots where this precomposition vanishes.",
"Now, set $\\nu _{1} := \\nu ^{s_{\\chi }}|_{A \\cap M}$ and $\\nu _{2} := \\nu ^{s_{\\chi }}|_{A_{\\chi }}$ .",
"We consider the parabolic induction $ i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1}) $ where we now note that $\\mu _{\\chi } \\otimes \\nu _{1}$ is unitary by construction.",
"Therefore, $i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1})$ is unitary and thereby fully decomposable.",
"It follows that, since $i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1})$ and $i_{B \\cap M_{\\chi }}^{M_{\\chi }}((\\mu _{\\chi } \\otimes \\nu _{1})^{w_{0}^{M_{\\chi }}})$ are equal in the Grothendieck group, we have an isomorphism $i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1}) \\simeq i_{B \\cap M_{\\chi }}^{M_{\\chi }}((\\mu _{\\chi } \\otimes \\nu _{1})^{w_{0}^{M_{\\chi }}})$ , where $w_{0}^{M_{\\chi }}$ is the element of longest length of the Weyl group of $M_{\\chi }$ .",
"Now, we have an isomorphism: $ i_{B}^{G}(\\chi ) \\simeq i_{P_{\\chi }}^{G}((i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1})) \\otimes \\nu _{2}) $ Since $\\chi $ is regular then, it follows by Lemma REF , that the unitary induction $i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1})$ is irreducible.",
"Therefore, the RHS is the induction of an irreducible tempered representation times an unramified character $\\nu _{2}$ satisfying the property that $\\langle \\alpha _{i,A}, \\nu _{2} \\rangle > 0$ for all $i \\in \\mathcal {J} \\setminus \\mathcal {J}_{M_{\\chi }}$ by construction.",
"Again applying the Langlands classification , the intertwining operator attached to the parabolic $P_{\\chi }$ and the element $w_{0}w_{0}^{M_{\\chi }}$ converges for $s = s_{\\chi }$ , and since it maps to a unique quotient of multiplicity one it suffices to exhibit an isomorphism between $i_{P_{\\chi }}^{G}((i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1})) \\otimes \\nu _{2})$ and the induction twisted by $w_{0}w_{0}^{M_{\\chi }}$ .",
"However, since we just saw that $i_{B \\cap M_{\\chi }}^{M_{\\chi }}((\\mu _{\\chi } \\otimes \\nu _{1})^{w_{0}^{M_{\\chi }}}) \\simeq i_{B \\cap M_{\\chi }}^{M_{\\chi }}(\\mu _{\\chi } \\otimes \\nu _{1})$ , it suffices to show we have an isomorphism $i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w_{0}})$ .",
"This establishes the claim.",
"Now we just need to show Proposition REF .",
"We claim that this reduces to the analogous question for $G$ a group of rank 1.",
"In particular, let's consider for $i \\in \\mathcal {J}$ the simple positive (reduced) coroot $\\alpha := \\alpha _{i,A}$ and the rank 1 parabolic $P_{\\alpha } = M_{\\alpha }N_{\\alpha }A_{\\alpha }$ attached to it.",
"As before, we write $\\nu ^{\\alpha }_{1} := \\nu ^{s_{\\chi }}|_{A \\cap M_{\\alpha }}$ and $\\nu ^{\\alpha }_{2} := \\nu ^{s^{\\chi }}|_{A_{\\alpha }}$ .",
"For all simple positive coroots $\\alpha $ , we have an isomorphism $ i_{B}^{G}(\\chi ) \\simeq i_{P_{\\alpha }}^{G}((i_{B \\cap M_{\\alpha }}^{M_{\\alpha }}(\\mu _{\\chi } \\otimes \\nu _{1}^{\\alpha })) \\otimes \\nu _{2}^{\\alpha }) $ However, if $w_{\\alpha }$ is the simple reflection corresponding to $\\alpha $ , we have that $ i_{B}^{G}(\\chi ^{w_{\\alpha }}) \\simeq i_{P_{\\alpha }}^{G}((i_{B \\cap M_{\\alpha }}^{M_{\\alpha }}((\\mu _{\\chi } \\otimes \\nu _{1}^{\\alpha })^{w_{\\alpha }}) \\otimes \\nu _{2}^{\\alpha }) $ Therefore, if we can show the existence of an isomorphism: $ i_{B \\cap M_{\\alpha }}^{M_{\\alpha }}(\\mu _{\\chi } \\otimes \\nu _{1}^{\\alpha }) \\simeq i_{B \\cap M_{\\alpha }}^{M_{\\alpha }}((\\mu _{\\chi } \\otimes \\nu _{1}^{\\alpha })^{w_{\\alpha }}) $ It will imply that we have an isomorphism: $ i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w_{\\alpha }}) $ Now we can proceed by induction on the length of Weyl group elements.",
"We just described the base case, but then, by replacing $\\chi $ with $\\chi ^{w_{\\alpha }}$ we can proceed by considering another simple reflection attached to another simple positive coroot distinct from $\\alpha $ .",
"We note that, since $\\chi $ being generic is a condition for all coroots (not just simple), at each step of the induction we are tasked with showing the following.",
"Proposition 1.6 Let $G$ be a absolutely simple, simply connected, quasi-split connected reductive group of split rank 1.",
"Let $\\alpha $ be the unique simple (reduced) positive coroot of $G$ and $w_{\\alpha }$ the corresponding simple reflection.",
"Then, for all $\\chi $ a generic character of $T(\\mathbb {Q}_{p})$ , we have an isomorphism $ i_{B}^{G}(\\chi ) \\simeq i_{B}^{G}(\\chi ^{w_{\\alpha }}) $ of smooth $G(\\mathbb {Q}_{p})$ -representations.",
"It remains to justify the absolutely simple simply connected assumption.",
"To do this, note that given a $G$ not satisfying these conditions, we can find a central isogeny $f: \\tilde{G} \\rightarrow G$ , where $\\tilde{G}$ is a product of torii and absolutely simple simply connected groups.",
"If we let $\\tilde{B}$ be the preimage of the Borel $B$ with maximal torus given by $\\tilde{T}$ the preimage of $T$ then, since $\\text{Ker}(f)$ is contained in the center, we have an isomorphism $\\tilde{G}/\\tilde{B} \\simeq G/B$ .",
"This implies that we have an isomorphism: $ i_{B}^{G}(\\chi )|_{\\tilde{G}(\\mathbb {Q}_{p})} \\simeq i_{\\tilde{B}}^{\\tilde{G}}(\\chi |_{\\tilde{T}}) $ Moreover, since $f$ will induce an isomorphism on the root spaces, it follows that if $\\chi $ is generic with respect to $G$ then $\\chi |_{\\tilde{T}}$ is generic with respect to $\\tilde{G}$ .",
"This reduces us to exhibiting the desired isomorphism for $\\tilde{G}$ .",
"Now, we prove the Proposition REF through brute force.",
"In particular, we will use Tits' classification theorem (See also for the classification in rank 1).",
"We adopt the same notation as in .",
"Since we are assuming the group to be quasi-split, there are two cases.",
"$\\phantom{}^{1}A^{1}_{1,1}$ In this case, we have that $G = \\mathrm {SL}_{2}$ .",
"We saw in Example REF that genericity guaranteed irreducibility aside from the case where $\\chi ^{2} \\simeq \\mathbf {1}$ , but since this is a unitary character it still follows that we have the desired isomorphism.",
"$\\phantom{}^{2}A^{1}_{2,1}$ In this case, the group cannot be split; in particular, we have that $G = \\mathrm {SU}_{3}$ is a quasi-split special unitary group attached to a quadratic extension $E/\\mathbb {Q}_{p}$ .",
"We saw in Example REF that $\\chi $ being generic guaranteed irreducibility and hence the desired isomorphism.",
"Tilting Cocharacters We consider a general quasi-split connected reductive group $G/\\mathbb {Q}_{p}$ , and a geometric dominant cocharacter $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ .",
"For $\\Lambda \\in \\lbrace \\overline{\\mathbb {Q}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {F}}_{\\ell }\\rbrace $ , we are interested in understanding the condition of $\\mu $ being tilting (Definition REF ).",
"Recall that this means that the representation $V_{\\mu } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ attached to $\\mu $ lies in the subcategory $\\mathrm {Tilt}_{\\Lambda }(\\hat{G})$ .",
"If $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ this is always true, and so we fix $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell }\\rbrace $ in what follows.",
"Since this only involves the representation theory $\\hat{G}$ we may, without loss of generality, assume $G$ is split in what follows.",
"This is simply the question of when the Weyl module $V_{\\mu }$ of $\\hat{G}$ is irreducible with $\\Lambda $ -coefficients.",
"This question has been studied extensively (See for example for a comprehensive overview).",
"In the first two sections, we discuss some general theory to determine when $\\mu $ is tilting in this split case, and then provide a table summarizing when $\\mu $ is tilting in the case that $\\mu $ is a fundamental coweight.",
"General Theory We assume $G$ is a split connected reductive group throughout this section, with Langlands dual group $\\hat{G}$ .",
"For $\\mu $ a dominant cocharacter, the condition that $\\mu $ is tilting is equivalent to showing that the highest weight $G$ -module $V_{\\mu } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ is simple.",
"We begin with the following lemma.",
"Lemma 2.1 If $\\mu $ is minuscule then it is tilting.",
"In this case, the weights of $V_{\\mu }$ form a closed Weyl group orbit.",
"It follows that $V_{\\mu }$ is always irreducible and therefore tilting.",
"We would now like to provide a finer criterion for irreducibility.",
"To do this, we will introduce some notation.",
"Given a coroot $\\nu $ and $r \\in \\mathbb {Z}$ , we consider the affine reflection of $\\mathbb {X}_{*}(T) \\otimes \\mathbb {R}$ given as $ s_{\\nu ,r}(\\mu ) := s_{\\nu }(\\mu ) + r\\nu $ where $s_{\\nu } \\in W_{G}$ is the reflection attached to $\\nu $ .",
"We set $W_{\\ell }$ to be the subgroup generated by reflections $s_{\\nu ,n\\ell }$ , where $\\nu $ is a coroot and $n \\in \\mathbb {Z}$ , and write $\\rho $ for the sum of all coroots of $G$ .",
"Elements $w \\in W_{\\ell }$ act on $\\mu \\in \\mathbb {X}_{*}(T) \\otimes \\mathbb {R}$ , via the standard dot action $w\\cdot \\mu := w(\\mu + \\rho ) - \\rho $ .",
"In other words, we regard $s_{\\nu ,n\\ell }$ as a reflection around the hyperplane: $ \\lbrace \\mu \\in \\mathbb {X}_{*}(T) \\otimes \\mathbb {R} \\text{ | } \\langle \\mu + \\rho , \\nu ^{\\vee } \\rangle = n\\ell \\rbrace $ It follows that the standard alcove for this action is given by $ C = \\lbrace \\mu \\in \\mathbb {X}_{*}(T) \\otimes \\mathbb {R} \\text{ | } 0 < \\langle \\mu + \\rho , \\nu ^{\\vee } \\rangle < \\ell \\rbrace $ and we denote the closure by $\\overline{C}$ .",
"We now have the following slightly more general criterion for the irreducibility of $V_{\\mu }$ .",
"Proposition 2.2 Suppose that $\\mu \\in \\overline{C} \\cap \\mathbb {X}_{*}(T)^{+}$ then $\\mu $ is tilting.",
"For a given $\\mu $ , this will give us a lower bound on the $\\ell $ for which $\\mu $ is tilting.",
"However, it is only a sufficient condition and not necessary.",
"In particular, note that we have the following.",
"Theorem 2.3 If $\\mu = (\\ell -1)(\\rho )$ then $\\mu $ is tilting.",
"So $V_{(\\ell - 1)(\\rho )}$ will always be simple, but $(\\ell - 1)(\\rho )$ will not usually lie in $\\overline{C} \\cap \\mathbb {X}_{*}(T)^{+}$ .",
"Moreover, if we define the Coxeter number $h = \\max _{\\nu }\\lbrace \\langle \\rho , \\nu ^{\\vee } \\rangle + 1 \\rbrace $ ranging over all coroots $\\nu $ then it is easy to see that $C \\cap \\mathbb {X}_{*}(T) \\ne \\emptyset $ is equivalent to $\\ell \\ge h$ , and so, for small $\\ell $ , Proposition REF tells us nothing.",
"To tackle these more general cases, we introduce a sum formula for the characters of the representations.",
"Namely, for $V \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ , we write $\\mathrm {ch}(V) := \\sum _{\\nu \\in \\mathbb {X}_{*}(T)} \\mathrm {dim}(V(\\nu ))e^{\\nu }$ for the character of $V$ .",
"For $\\mu $ a dominant cocharacter, we write $\\chi (\\mu ) := \\mathrm {ch}(V_{\\mu })$ for the character of $V_{\\mu }$ .",
"Then we have the following.",
"Proposition 2.4 For each $\\mu \\in \\mathbb {X}_{*}(T)^{+}$ , there is a filtration of $\\hat{G}$ -modules $ \\cdots \\subset V_{\\mu }^{2} \\subset V_{\\mu }^{1} \\subset V_{\\mu }^{0} = V_{\\mu } $ such that $ \\sum _{i > 0} \\mathrm {ch}(V_{\\mu }^{i}) = \\sum _{\\nu } \\sum _{0 < m\\ell < \\langle \\mu + \\rho ,\\nu ^{\\vee } \\rangle } \\nu _{\\ell }(m\\ell )\\chi (s_{\\nu ,m\\ell } \\cdot \\mu ) $ where $\\nu _{\\ell }(-)$ is the $\\ell $ -adic valuation and $\\nu $ ranges over all coroots.",
"Moreover, we have that $V_{\\mu }/V_{\\mu }^{1}$ is isomorphic to the irreducible socle of $V_{\\mu }$ .",
"In particular, we see that $\\mu $ is tilting if and only if $ \\sum _{\\nu } \\sum _{0 < m\\ell < \\langle \\mu + \\rho ,\\nu ^{\\vee } \\rangle } \\nu _{\\ell }(m\\ell )\\chi (s_{\\nu ,m\\ell } \\cdot \\mu ) = 0 $ This generalizes Proposition REF and gives a computational method for verifying when $\\mu $ is tilting.",
"See for example for this worked out for $G = \\mathrm {SL}_{4}$ and $\\ell > 3$ , for a table answering this question for $G$ an exceptional group and certain $\\mu $ , and for an analogous table for certain exceptional groups and low rank classical groups.",
"In general, a precise classification of when $\\mu $ is tilting for all $G$ seems to be quite complicated, and to the best of our knowledge is unknown.",
"However, when $G$ is of type $A_{n - 1}$ , there exists a complete classification.",
"Proposition 2.5 For $G$ of type $A_{n - 1}$ , $\\mu $ is tilting if and only if for each coroot $\\nu $ of $G$ the following is satisfied.",
"Write $\\langle \\mu + \\rho , \\nu ^{\\vee } \\rangle = a\\ell ^{s} + b\\ell ^{s + 1}$ , with $a,b,s \\in \\mathbb {N}$ and $0 < a < \\ell $ .",
"Then there have to be positive coroots $\\beta _{0},\\beta _{1},\\ldots ,\\beta _{b}$ such that $\\langle \\mu + \\rho , \\beta _{i}^{\\vee } \\rangle = \\ell ^{s + 1}$ for $1 \\le i \\le b$ , $\\langle \\mu + \\rho , \\beta _{0}^{\\vee } \\rangle = ap^{s}$ , $\\nu = \\sum _{i = 0}^{b} \\beta _{i}$ , and $\\sum _{i = 1}^{b} \\beta _{i}$ is a coroot.",
"For general types, we will content ourselves with describing the fundamental weights, where we can give a full description of the tilting condition.",
"The Tilting Condition for Fundamental Coweights We assume that $G$ is a split adjoint group, and let $\\hat{\\alpha }_{j}$ denote the simple roots, where we use the enumeration as in .",
"We choose fundamental coweights characterized by $\\langle \\varpi _{i}, \\hat{\\alpha }_{j} \\rangle = \\delta _{ij}$ .",
"We will be interested in the question of when the representation $V_{\\varpi _{i}}$ is irreducible.",
"In this case, we have a complete classification ,.",
"We note that, if $\\varpi _{i}$ is minuscule, this is automatic by Lemma REF , so in what follows we simply provide a list of $\\ell $ for the non-minuscule $\\varpi _{i}$ (See for a classification).",
"This namely implies the case of $A_{n}$ is trivial, since all fundamental weights are minuscule.",
"Additionally, we recall that our results will only apply if $\\ell $ is very good in the sense of , so we have also enumerated the condition that $\\ell $ is very good for the different typesWe warn the reader if $G$ is non-split there can also be additional constraints on $\\ell $ .. Table: NO_CAPTION By comparing the fourth and fifth columns, we deduce the following.",
"Proposition 2.6 For $G$ a split adjoint connected reductive group over $\\mathbb {Q}_{p}$ , if $\\ell $ is very good then $\\varpi _{i}$ for any $i \\in \\mathcal {J}$ is tilting for all $G$ of type $A_{n},B_{n},C_{n},D_{n},E_{6},F_{4}$ , and $G_{2}$ .",
"Relationship to the Classical Averaging Formula, by Alexander Bertoloni-Meli In this appendix, we show that the averaging formula proven in Theorem REF is compatible with existing formulas and conjectures in the literature.",
"Averaging Formulas To begin, we recall the general statement of these averaging formulas.",
"Such formulas first appeared in the book of Harris–Taylor () and are classically deduced by studying the geometry of the mod-$p$ fibers of Shimura varieties.",
"These fibers admit a Newton stratification in terms of the set $B(G,\\mu )$ and the strata are uniformized by Rapoport–Zink spaces and Igusa varieties.",
"The cohomological consequence of this is the formula of Mantovan () $\\sum \\limits _i \\varinjlim \\limits _{K \\subset G(\\mathbb {A}_f)} (-1)^i H^i_c(\\mathrm {Sh}(G,X)_K, \\overline{\\mathbb {Q}}_{\\ell }) = \\sum \\limits _{b \\in B(G,\\mu )} R\\Gamma ^{\\flat }_c(G,b,\\mu )[\\sum \\limits _j (-1)^j \\varinjlim \\limits _{K^p \\subset G(\\mathbb {A}^p_f)} H^j_c(\\mathrm {Ig}^b_{K^p}, \\overline{\\mathbb {Q}}_{\\ell })],$ in $K_0(G(\\mathbb {A}_f) \\times W_{E_{\\mu }})$ , where $\\mathrm {Sh}(G,X)_K$ (resp.",
"$\\mathrm {Ig}^b_{K^p}$ ) is the Shimura variety (resp.",
"Igusa variety) determined by the associated data.",
"We note that the degree shift built into the Satake sheaf $\\mathcal {S}_{\\mu }$ causes a sign in the above formula that does not appear in much of the literature on this topic.",
"This is explained in Remark 2.4.4 of .",
"Averaging formulas can then be deduced by studying isotypic pieces of the above formula.",
"In order to precisely state these averaging formulas, we first recall some facts about stable characters following .",
"Let $G$ be a connected reductive group with Levi subgroup $M$ and parabolic $P$ .",
"Let $K_0(G(\\mathbb {Q}_p), ^{st} \\subset K_0(G(\\mathbb {Q}_p), $ denote the subgroup of virtual representations with stable character.",
"Then the normalized Jacquet module and parabolic induction functors induce morphisms $i^G_P: K_0(M(\\mathbb {Q}_p), \\rightarrow K_0(G(\\mathbb {Q}_p), , \\quad \\quad r^G_P: K_0(G(\\mathbb {Q}_p), \\rightarrow K_0(M(\\mathbb {Q}_p), .$ Moreover, one can show these operations preserve stability so that we get homomorphisms $i^G_P: K_0(M(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(G(\\mathbb {Q}_p), ^{st}, \\quad \\quad r^G_P: K_0(G(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(M(\\mathbb {Q}_p), ^{st}.$ Now let $G^*$ denote the unique quasi-split group that is an inner form of $G$ .",
"We assume that $G$ arises as an extended pure inner twist of $G^*$ and fix this extra structure $(G, \\varrho , z)$ .",
"One can work with more general $G$ using Kaletha's theory of rigid inner twists, but this is not necessary to explore the connections to this paper where $G$ is always quasi-split.",
"We also need to introduce endoscopy for $G$ .",
"For convenience, we recall: Definition 3.1 A refined endoscopic datum for a connected reductive group $G$ over a local field $F$ is a tuple $(H, \\mathcal {H}, s,\\eta )$ which consists of a quasi-split group $H$ over $F$ , an extension $\\mathcal {H}$ of $W_F$ by $\\widehat{H}$ such that the map $W_F \\rightarrow \\mathrm {Out}(\\widehat{H})$ coincides with the map $\\rho _H: W_F \\rightarrow \\mathrm {Out}(\\widehat{H})$ induced by the action of $W_F$ on $\\widehat{H} \\subset {}^LH$ , an element $s \\in Z(\\widehat{H})^{\\Gamma }$ , an $L$ -homomorphism $\\eta : \\mathcal {H} \\rightarrow {}^LG$ , satisfying the condition: we have $\\eta (\\widehat{H}) = Z_{\\widehat{G}}(s)^{\\circ }$ .",
"Now suppose that $(H, \\mathcal {H}, s,\\eta )$ is a refined endoscopic datum for $G$ .",
"Then after fixing splittings of $G, H, \\widehat{G}, \\widehat{H}$ , there is a canonical endoscopic transfer of distributions inducing a morphism $\\mathrm {Trans}^G_H: K_0(H(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(G(\\mathbb {Q}_p), .$ Furthermore, suppose we have a refined endoscopic datum $(H_M, \\mathcal {H}_M, s, \\eta _M)$ of $M$ such that $\\mathcal {H}_M$ is a Levi subgroup of $\\mathcal {H}$ and the following diagram commutes: $\\begin{tikzcd}\\mathcal {H} [r, \"{\\eta }\"] & {}^LG\\\\\\mathcal {H}_M [u, hook] [r, \"{\\eta _M}\"] & {}^LM [u, hook].\\end{tikzcd}$ The datum $(H_M, \\mathcal {H}_M, H, M, s, \\eta )$ along with these compatibilities is called an embedded endoscopic datum in , .",
"Our fixed choice of splittings of $G$ , $H$ and their duals determines from $P$ a parabolic subgroup $P_{H_M}$ of $H$ with Levi subgroup $H_M$ .",
"We then have an equality ${}\\mathrm {Trans}^G_H \\circ i^H_{P_{H_M}} = i^G_P \\circ \\mathrm {Trans}^M_{H_M}.$ There is also a compatibility of $\\mathrm {Trans}$ and $r$ which we now recall.",
"A refined endoscopic datum $\\mathfrak {e}=(H, \\mathcal {H}, s, \\eta )$ of $G$ and a Levi subgroup $M \\subset G$ can be upgraded to the structure of an embedded endoscopic datum in potentially many non-equivalent ways and these are parametrized by a set $D(M, \\mathfrak {e}) \\cong W(\\widehat{H}) \\setminus W(M, H) / W(\\widehat{M})$ , where $W(M,H)$ is defined to be the subset of the Weyl group $W(\\widehat{G})$ of $\\widehat{G}$ such that $Z_{{}^LH}((w \\circ \\eta )^{-1}(Z(\\widehat{M})^{\\Gamma }))$ surjects onto $W_F$ , and where $W(\\widehat{H})$ is identified with a subgroup of $W(\\widehat{G})$ via $\\eta $ .",
"Then we have $r^G_P \\circ \\mathrm {Trans}^G_H = \\sum \\limits _{ D(M, \\mathfrak {e})} \\mathrm {Trans}^{M}_{H_M} \\circ r^H_{P_{H_M}}.$ We now define the map $\\mathrm {Red}^{\\mathfrak {e}}_b$ which plays a crucial role in the statement of the averaging formula.",
"We define $\\mathrm {Red}^{\\mathfrak {e}}_b: K_0(H(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(J_b(\\mathbb {Q}_p), $ by $\\mathrm {Red}^{\\mathfrak {e}}_b=(\\sum \\limits _{D(M,\\mathcal {H})} \\mathrm {Trans}^{H_M}_{J_b} \\circ r^H_{P^{op}_H}) \\otimes \\overline{\\delta }^{\\frac{1}{2}}_P,$ where $\\overline{\\delta }_{P}$ is the transport of the modulus character for $P$ to $J_b$ .",
"Then the (still largely conjectural) averaging formula gives a relation satisfied by $R\\Gamma ^{\\flat }_c(G,b,\\mu )$ at Langlands (or Arthur) parameters $\\phi $ for which there is an associated stable distribution $S\\Theta _{\\phi ,G}$ on $G$ satisfying endoscopic character identities as in .",
"In particular, this is the case for all tempered $L$ -parameters.",
"To describe the expected formula, fix such a parameter $\\phi $ and suppose that $(H, \\mathcal {H}, s, \\eta )$ is an endoscopic datum such that $\\phi $ factors as $\\mathcal {L}_F \\xrightarrow{} \\mathcal {H} \\rightarrow {}^LG$ .",
"Then we expect Conjecture 3.2 (Averaging Formula) We expect the following equality in $K_0(G(\\mathbb {Q}_p) \\times W_{E_{\\mu }})$ : $\\sum \\limits _{b \\in B(G, \\mu )} [R\\Gamma ^{\\flat }_c(G,b,\\mu )[\\mathrm {Red}^{\\mathfrak {e}}_b(S\\Theta _{\\phi ^H, H})]] = \\mathrm {Trans}^G_H(S\\Theta _{\\phi ^H, H}) \\boxtimes \\mathrm {tr}(r_{\\mu } \\circ \\phi |_{W_{E_{\\mu }}} \\mid s).$ In particular, when $\\mathfrak {e}$ is the trivial endoscopic datum given by $\\mathfrak {e}_{\\mathrm {triv}} = (G^*, {}^LG^*, 1, \\mathrm {id})$ then we expect $\\sum \\limits _{b \\in B(G, \\mu )} [R\\Gamma ^{\\flat }_c(G,b,\\mu )[\\mathrm {Red}^{\\mathfrak {e}_{\\mathrm {triv}}}_b(S\\Theta _{\\phi , G^*})]] = S\\Theta _{\\phi ,G} \\boxtimes r_{\\mu } \\circ \\phi |_{W_{E_{\\mu }}}.$ For $\\mathrm {GL}_n$ , this is known in the trivial endoscopic case for all representations by .",
"Because $L$ -packets are singletons for $\\mathrm {GL}_n$ , the trivial endoscopic case implies the endoscopic versions of the formula.",
"In , the formula is proven for discrete parameters and elliptic endoscopy of unramified $\\mathrm {GU}_n$ , for $n$ odd.",
"In , a strategy is outlined to prove this formula in the elliptic endoscopic cases using the cohomology of Igusa and Shimura varieties.",
"This strategy should (eventually) yield results comparable to whenever adequate global results are known about the Langlands correspondence and the cohomology of Shimura and Igusa varieties.",
"The averaging formulas imply strong results about $R\\Gamma ^{\\flat }_c(G,b,\\mu )$ .",
"For instance, in it is shown that the averaging formula for each elliptic endoscopic group and for $\\phi $ a supercuspidal parameter implies the Kottwitz conjecture as in .",
"Proof of Proposition REF The averaging formula in §REF corresponds to the case of the trivial endoscopic triple $\\mathfrak {e}_{\\mathrm {triv}} =(G^*,{}^LG^*,1, \\mathrm {id})$ .",
"Hence to check that Theorem REF agrees with REF , we just need to check that $\\mathrm {Red}^{\\mathfrak {e}{\\mathrm {triv}}}_b(S\\Theta _{\\phi ,G^*})$ coincides with $\\mathrm {Red}_{b, \\phi }$ for $\\phi $ induced from $\\phi _{T}$ generic.",
"Since $\\phi _{T}$ is generic, by Lemma REF $\\phi $ should give rise to a well-defined $L$ -parameter with trivial monodromy.",
"Therefore, under the $\\mathrm {LLC}_{G}$ appearing in Assumption REF , we are assuming the parameter $\\phi $ has an $L$ -packet given by the irreducible constituents of $i^G_B(\\chi )$ , by Assumption REF (3).",
"Suppose first that $b \\in B(G)_{\\mathrm {un}}$ .",
"Then we have $\\mathrm {Red}_{b, \\phi } = \\sum \\limits _{w \\in W_G/W_{M_b}} i^{J_b}_{B_b}( \\chi ^w) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho }_G, \\nu _b \\rangle ].$ The set $D(M, \\mathfrak {e})$ is a singleton and corresponds to the trivial embedded datum where $H_M=M$ .",
"Note that $r^G_{P^{op}_b}(i^G_B(\\chi )) = r^G_{P_b}(i^G_B(\\chi ))$ and that the latter term can be simplified by the geometric lemma of .",
"$\\mathrm {Red}^{\\mathfrak {e}{\\mathrm {triv}}}_b(S\\Theta _{\\phi , G^*}) & = (\\mathrm {Trans}^{J_b}_{M_b} \\circ r^G_{P^{op}_b})(i^G_B(\\chi ))\\otimes \\overline{\\delta }^{\\frac{1}{2}}_P\\\\& = \\mathrm {Trans}^{J_b}_{M_b} \\circ (\\sum \\limits _{w \\in W_G/W_{M_b}} i^{M_b}_{B \\cap M_b} \\chi ^w)\\otimes \\overline{\\delta }^{\\frac{1}{2}}_P\\\\& = (\\sum \\limits _{W_G/W_{M_b}} i^{J_b}_{B_b}) \\circ \\mathrm {Trans}^{T_b}_T \\chi ^w\\otimes \\overline{\\delta }^{\\frac{1}{2}}_P\\\\&= \\mathrm {Red}_{b, \\phi } .$ where the third equality is (REF ).",
"Now consider the case where $b \\notin B(G)_{\\mathrm {un}}$ .",
"We must show that $\\mathrm {Red}^{\\mathfrak {e}_{\\mathrm {triv}}}_b(i^G_B(\\chi )) = 0$ , for which it suffices to show that $\\mathrm {Trans}^{J_b}_{M_b}(i^{M_b}_{B \\cap M_b}(\\chi ^w))=0$ for each $w \\in W_G/W_{M_b}$ .",
"This follows from the fact that $T$ does not transfer to $J_b$ by assumption and the character of $i^{M_b}_{B \\cap M_b}(\\chi ^w)$ is supported on the conjugates of $T$ as per ."
],
[
"Tilting Cocharacters",
"We consider a general quasi-split connected reductive group $G/\\mathbb {Q}_{p}$ , and a geometric dominant cocharacter $\\mu \\in \\mathbb {X}_{*}(T_{\\overline{\\mathbb {Q}}_{p}})^{+}$ .",
"For $\\Lambda \\in \\lbrace \\overline{\\mathbb {Q}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell },\\overline{\\mathbb {F}}_{\\ell }\\rbrace $ , we are interested in understanding the condition of $\\mu $ being tilting (Definition REF ).",
"Recall that this means that the representation $V_{\\mu } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ attached to $\\mu $ lies in the subcategory $\\mathrm {Tilt}_{\\Lambda }(\\hat{G})$ .",
"If $\\Lambda = \\overline{\\mathbb {Q}}_{\\ell }$ this is always true, and so we fix $\\Lambda \\in \\lbrace \\overline{\\mathbb {F}}_{\\ell },\\overline{\\mathbb {Z}}_{\\ell }\\rbrace $ in what follows.",
"Since this only involves the representation theory $\\hat{G}$ we may, without loss of generality, assume $G$ is split in what follows.",
"This is simply the question of when the Weyl module $V_{\\mu }$ of $\\hat{G}$ is irreducible with $\\Lambda $ -coefficients.",
"This question has been studied extensively (See for example for a comprehensive overview).",
"In the first two sections, we discuss some general theory to determine when $\\mu $ is tilting in this split case, and then provide a table summarizing when $\\mu $ is tilting in the case that $\\mu $ is a fundamental coweight."
],
[
"General Theory",
"We assume $G$ is a split connected reductive group throughout this section, with Langlands dual group $\\hat{G}$ .",
"For $\\mu $ a dominant cocharacter, the condition that $\\mu $ is tilting is equivalent to showing that the highest weight $G$ -module $V_{\\mu } \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ is simple.",
"We begin with the following lemma.",
"Lemma 2.1 If $\\mu $ is minuscule then it is tilting.",
"In this case, the weights of $V_{\\mu }$ form a closed Weyl group orbit.",
"It follows that $V_{\\mu }$ is always irreducible and therefore tilting.",
"We would now like to provide a finer criterion for irreducibility.",
"To do this, we will introduce some notation.",
"Given a coroot $\\nu $ and $r \\in \\mathbb {Z}$ , we consider the affine reflection of $\\mathbb {X}_{*}(T) \\otimes \\mathbb {R}$ given as $ s_{\\nu ,r}(\\mu ) := s_{\\nu }(\\mu ) + r\\nu $ where $s_{\\nu } \\in W_{G}$ is the reflection attached to $\\nu $ .",
"We set $W_{\\ell }$ to be the subgroup generated by reflections $s_{\\nu ,n\\ell }$ , where $\\nu $ is a coroot and $n \\in \\mathbb {Z}$ , and write $\\rho $ for the sum of all coroots of $G$ .",
"Elements $w \\in W_{\\ell }$ act on $\\mu \\in \\mathbb {X}_{*}(T) \\otimes \\mathbb {R}$ , via the standard dot action $w\\cdot \\mu := w(\\mu + \\rho ) - \\rho $ .",
"In other words, we regard $s_{\\nu ,n\\ell }$ as a reflection around the hyperplane: $ \\lbrace \\mu \\in \\mathbb {X}_{*}(T) \\otimes \\mathbb {R} \\text{ | } \\langle \\mu + \\rho , \\nu ^{\\vee } \\rangle = n\\ell \\rbrace $ It follows that the standard alcove for this action is given by $ C = \\lbrace \\mu \\in \\mathbb {X}_{*}(T) \\otimes \\mathbb {R} \\text{ | } 0 < \\langle \\mu + \\rho , \\nu ^{\\vee } \\rangle < \\ell \\rbrace $ and we denote the closure by $\\overline{C}$ .",
"We now have the following slightly more general criterion for the irreducibility of $V_{\\mu }$ .",
"Proposition 2.2 Suppose that $\\mu \\in \\overline{C} \\cap \\mathbb {X}_{*}(T)^{+}$ then $\\mu $ is tilting.",
"For a given $\\mu $ , this will give us a lower bound on the $\\ell $ for which $\\mu $ is tilting.",
"However, it is only a sufficient condition and not necessary.",
"In particular, note that we have the following.",
"Theorem 2.3 If $\\mu = (\\ell -1)(\\rho )$ then $\\mu $ is tilting.",
"So $V_{(\\ell - 1)(\\rho )}$ will always be simple, but $(\\ell - 1)(\\rho )$ will not usually lie in $\\overline{C} \\cap \\mathbb {X}_{*}(T)^{+}$ .",
"Moreover, if we define the Coxeter number $h = \\max _{\\nu }\\lbrace \\langle \\rho , \\nu ^{\\vee } \\rangle + 1 \\rbrace $ ranging over all coroots $\\nu $ then it is easy to see that $C \\cap \\mathbb {X}_{*}(T) \\ne \\emptyset $ is equivalent to $\\ell \\ge h$ , and so, for small $\\ell $ , Proposition REF tells us nothing.",
"To tackle these more general cases, we introduce a sum formula for the characters of the representations.",
"Namely, for $V \\in \\mathrm {Rep}_{\\Lambda }(\\hat{G})$ , we write $\\mathrm {ch}(V) := \\sum _{\\nu \\in \\mathbb {X}_{*}(T)} \\mathrm {dim}(V(\\nu ))e^{\\nu }$ for the character of $V$ .",
"For $\\mu $ a dominant cocharacter, we write $\\chi (\\mu ) := \\mathrm {ch}(V_{\\mu })$ for the character of $V_{\\mu }$ .",
"Then we have the following.",
"Proposition 2.4 For each $\\mu \\in \\mathbb {X}_{*}(T)^{+}$ , there is a filtration of $\\hat{G}$ -modules $ \\cdots \\subset V_{\\mu }^{2} \\subset V_{\\mu }^{1} \\subset V_{\\mu }^{0} = V_{\\mu } $ such that $ \\sum _{i > 0} \\mathrm {ch}(V_{\\mu }^{i}) = \\sum _{\\nu } \\sum _{0 < m\\ell < \\langle \\mu + \\rho ,\\nu ^{\\vee } \\rangle } \\nu _{\\ell }(m\\ell )\\chi (s_{\\nu ,m\\ell } \\cdot \\mu ) $ where $\\nu _{\\ell }(-)$ is the $\\ell $ -adic valuation and $\\nu $ ranges over all coroots.",
"Moreover, we have that $V_{\\mu }/V_{\\mu }^{1}$ is isomorphic to the irreducible socle of $V_{\\mu }$ .",
"In particular, we see that $\\mu $ is tilting if and only if $ \\sum _{\\nu } \\sum _{0 < m\\ell < \\langle \\mu + \\rho ,\\nu ^{\\vee } \\rangle } \\nu _{\\ell }(m\\ell )\\chi (s_{\\nu ,m\\ell } \\cdot \\mu ) = 0 $ This generalizes Proposition REF and gives a computational method for verifying when $\\mu $ is tilting.",
"See for example for this worked out for $G = \\mathrm {SL}_{4}$ and $\\ell > 3$ , for a table answering this question for $G$ an exceptional group and certain $\\mu $ , and for an analogous table for certain exceptional groups and low rank classical groups.",
"In general, a precise classification of when $\\mu $ is tilting for all $G$ seems to be quite complicated, and to the best of our knowledge is unknown.",
"However, when $G$ is of type $A_{n - 1}$ , there exists a complete classification.",
"Proposition 2.5 For $G$ of type $A_{n - 1}$ , $\\mu $ is tilting if and only if for each coroot $\\nu $ of $G$ the following is satisfied.",
"Write $\\langle \\mu + \\rho , \\nu ^{\\vee } \\rangle = a\\ell ^{s} + b\\ell ^{s + 1}$ , with $a,b,s \\in \\mathbb {N}$ and $0 < a < \\ell $ .",
"Then there have to be positive coroots $\\beta _{0},\\beta _{1},\\ldots ,\\beta _{b}$ such that $\\langle \\mu + \\rho , \\beta _{i}^{\\vee } \\rangle = \\ell ^{s + 1}$ for $1 \\le i \\le b$ , $\\langle \\mu + \\rho , \\beta _{0}^{\\vee } \\rangle = ap^{s}$ , $\\nu = \\sum _{i = 0}^{b} \\beta _{i}$ , and $\\sum _{i = 1}^{b} \\beta _{i}$ is a coroot.",
"For general types, we will content ourselves with describing the fundamental weights, where we can give a full description of the tilting condition."
],
[
"The Tilting Condition for Fundamental Coweights",
"We assume that $G$ is a split adjoint group, and let $\\hat{\\alpha }_{j}$ denote the simple roots, where we use the enumeration as in .",
"We choose fundamental coweights characterized by $\\langle \\varpi _{i}, \\hat{\\alpha }_{j} \\rangle = \\delta _{ij}$ .",
"We will be interested in the question of when the representation $V_{\\varpi _{i}}$ is irreducible.",
"In this case, we have a complete classification ,.",
"We note that, if $\\varpi _{i}$ is minuscule, this is automatic by Lemma REF , so in what follows we simply provide a list of $\\ell $ for the non-minuscule $\\varpi _{i}$ (See for a classification).",
"This namely implies the case of $A_{n}$ is trivial, since all fundamental weights are minuscule.",
"Additionally, we recall that our results will only apply if $\\ell $ is very good in the sense of , so we have also enumerated the condition that $\\ell $ is very good for the different typesWe warn the reader if $G$ is non-split there can also be additional constraints on $\\ell $ .. Table: NO_CAPTIONBy comparing the fourth and fifth columns, we deduce the following.",
"Proposition 2.6 For $G$ a split adjoint connected reductive group over $\\mathbb {Q}_{p}$ , if $\\ell $ is very good then $\\varpi _{i}$ for any $i \\in \\mathcal {J}$ is tilting for all $G$ of type $A_{n},B_{n},C_{n},D_{n},E_{6},F_{4}$ , and $G_{2}$ ."
],
[
"Relationship to the Classical Averaging Formula, by Alexander Bertoloni-Meli",
"In this appendix, we show that the averaging formula proven in Theorem REF is compatible with existing formulas and conjectures in the literature."
],
[
"Averaging Formulas",
"To begin, we recall the general statement of these averaging formulas.",
"Such formulas first appeared in the book of Harris–Taylor () and are classically deduced by studying the geometry of the mod-$p$ fibers of Shimura varieties.",
"These fibers admit a Newton stratification in terms of the set $B(G,\\mu )$ and the strata are uniformized by Rapoport–Zink spaces and Igusa varieties.",
"The cohomological consequence of this is the formula of Mantovan () $\\sum \\limits _i \\varinjlim \\limits _{K \\subset G(\\mathbb {A}_f)} (-1)^i H^i_c(\\mathrm {Sh}(G,X)_K, \\overline{\\mathbb {Q}}_{\\ell }) = \\sum \\limits _{b \\in B(G,\\mu )} R\\Gamma ^{\\flat }_c(G,b,\\mu )[\\sum \\limits _j (-1)^j \\varinjlim \\limits _{K^p \\subset G(\\mathbb {A}^p_f)} H^j_c(\\mathrm {Ig}^b_{K^p}, \\overline{\\mathbb {Q}}_{\\ell })],$ in $K_0(G(\\mathbb {A}_f) \\times W_{E_{\\mu }})$ , where $\\mathrm {Sh}(G,X)_K$ (resp.",
"$\\mathrm {Ig}^b_{K^p}$ ) is the Shimura variety (resp.",
"Igusa variety) determined by the associated data.",
"We note that the degree shift built into the Satake sheaf $\\mathcal {S}_{\\mu }$ causes a sign in the above formula that does not appear in much of the literature on this topic.",
"This is explained in Remark 2.4.4 of .",
"Averaging formulas can then be deduced by studying isotypic pieces of the above formula.",
"In order to precisely state these averaging formulas, we first recall some facts about stable characters following .",
"Let $G$ be a connected reductive group with Levi subgroup $M$ and parabolic $P$ .",
"Let $K_0(G(\\mathbb {Q}_p), ^{st} \\subset K_0(G(\\mathbb {Q}_p), $ denote the subgroup of virtual representations with stable character.",
"Then the normalized Jacquet module and parabolic induction functors induce morphisms $i^G_P: K_0(M(\\mathbb {Q}_p), \\rightarrow K_0(G(\\mathbb {Q}_p), , \\quad \\quad r^G_P: K_0(G(\\mathbb {Q}_p), \\rightarrow K_0(M(\\mathbb {Q}_p), .$ Moreover, one can show these operations preserve stability so that we get homomorphisms $i^G_P: K_0(M(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(G(\\mathbb {Q}_p), ^{st}, \\quad \\quad r^G_P: K_0(G(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(M(\\mathbb {Q}_p), ^{st}.$ Now let $G^*$ denote the unique quasi-split group that is an inner form of $G$ .",
"We assume that $G$ arises as an extended pure inner twist of $G^*$ and fix this extra structure $(G, \\varrho , z)$ .",
"One can work with more general $G$ using Kaletha's theory of rigid inner twists, but this is not necessary to explore the connections to this paper where $G$ is always quasi-split.",
"We also need to introduce endoscopy for $G$ .",
"For convenience, we recall: Definition 3.1 A refined endoscopic datum for a connected reductive group $G$ over a local field $F$ is a tuple $(H, \\mathcal {H}, s,\\eta )$ which consists of a quasi-split group $H$ over $F$ , an extension $\\mathcal {H}$ of $W_F$ by $\\widehat{H}$ such that the map $W_F \\rightarrow \\mathrm {Out}(\\widehat{H})$ coincides with the map $\\rho _H: W_F \\rightarrow \\mathrm {Out}(\\widehat{H})$ induced by the action of $W_F$ on $\\widehat{H} \\subset {}^LH$ , an element $s \\in Z(\\widehat{H})^{\\Gamma }$ , an $L$ -homomorphism $\\eta : \\mathcal {H} \\rightarrow {}^LG$ , satisfying the condition: we have $\\eta (\\widehat{H}) = Z_{\\widehat{G}}(s)^{\\circ }$ .",
"Now suppose that $(H, \\mathcal {H}, s,\\eta )$ is a refined endoscopic datum for $G$ .",
"Then after fixing splittings of $G, H, \\widehat{G}, \\widehat{H}$ , there is a canonical endoscopic transfer of distributions inducing a morphism $\\mathrm {Trans}^G_H: K_0(H(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(G(\\mathbb {Q}_p), .$ Furthermore, suppose we have a refined endoscopic datum $(H_M, \\mathcal {H}_M, s, \\eta _M)$ of $M$ such that $\\mathcal {H}_M$ is a Levi subgroup of $\\mathcal {H}$ and the following diagram commutes: $\\begin{tikzcd}\\mathcal {H} [r, \"{\\eta }\"] & {}^LG\\\\\\mathcal {H}_M [u, hook] [r, \"{\\eta _M}\"] & {}^LM [u, hook].\\end{tikzcd}$ The datum $(H_M, \\mathcal {H}_M, H, M, s, \\eta )$ along with these compatibilities is called an embedded endoscopic datum in , .",
"Our fixed choice of splittings of $G$ , $H$ and their duals determines from $P$ a parabolic subgroup $P_{H_M}$ of $H$ with Levi subgroup $H_M$ .",
"We then have an equality ${}\\mathrm {Trans}^G_H \\circ i^H_{P_{H_M}} = i^G_P \\circ \\mathrm {Trans}^M_{H_M}.$ There is also a compatibility of $\\mathrm {Trans}$ and $r$ which we now recall.",
"A refined endoscopic datum $\\mathfrak {e}=(H, \\mathcal {H}, s, \\eta )$ of $G$ and a Levi subgroup $M \\subset G$ can be upgraded to the structure of an embedded endoscopic datum in potentially many non-equivalent ways and these are parametrized by a set $D(M, \\mathfrak {e}) \\cong W(\\widehat{H}) \\setminus W(M, H) / W(\\widehat{M})$ , where $W(M,H)$ is defined to be the subset of the Weyl group $W(\\widehat{G})$ of $\\widehat{G}$ such that $Z_{{}^LH}((w \\circ \\eta )^{-1}(Z(\\widehat{M})^{\\Gamma }))$ surjects onto $W_F$ , and where $W(\\widehat{H})$ is identified with a subgroup of $W(\\widehat{G})$ via $\\eta $ .",
"Then we have $r^G_P \\circ \\mathrm {Trans}^G_H = \\sum \\limits _{ D(M, \\mathfrak {e})} \\mathrm {Trans}^{M}_{H_M} \\circ r^H_{P_{H_M}}.$ We now define the map $\\mathrm {Red}^{\\mathfrak {e}}_b$ which plays a crucial role in the statement of the averaging formula.",
"We define $\\mathrm {Red}^{\\mathfrak {e}}_b: K_0(H(\\mathbb {Q}_p), ^{st} \\rightarrow K_0(J_b(\\mathbb {Q}_p), $ by $\\mathrm {Red}^{\\mathfrak {e}}_b=(\\sum \\limits _{D(M,\\mathcal {H})} \\mathrm {Trans}^{H_M}_{J_b} \\circ r^H_{P^{op}_H}) \\otimes \\overline{\\delta }^{\\frac{1}{2}}_P,$ where $\\overline{\\delta }_{P}$ is the transport of the modulus character for $P$ to $J_b$ .",
"Then the (still largely conjectural) averaging formula gives a relation satisfied by $R\\Gamma ^{\\flat }_c(G,b,\\mu )$ at Langlands (or Arthur) parameters $\\phi $ for which there is an associated stable distribution $S\\Theta _{\\phi ,G}$ on $G$ satisfying endoscopic character identities as in .",
"In particular, this is the case for all tempered $L$ -parameters.",
"To describe the expected formula, fix such a parameter $\\phi $ and suppose that $(H, \\mathcal {H}, s, \\eta )$ is an endoscopic datum such that $\\phi $ factors as $\\mathcal {L}_F \\xrightarrow{} \\mathcal {H} \\rightarrow {}^LG$ .",
"Then we expect Conjecture 3.2 (Averaging Formula) We expect the following equality in $K_0(G(\\mathbb {Q}_p) \\times W_{E_{\\mu }})$ : $\\sum \\limits _{b \\in B(G, \\mu )} [R\\Gamma ^{\\flat }_c(G,b,\\mu )[\\mathrm {Red}^{\\mathfrak {e}}_b(S\\Theta _{\\phi ^H, H})]] = \\mathrm {Trans}^G_H(S\\Theta _{\\phi ^H, H}) \\boxtimes \\mathrm {tr}(r_{\\mu } \\circ \\phi |_{W_{E_{\\mu }}} \\mid s).$ In particular, when $\\mathfrak {e}$ is the trivial endoscopic datum given by $\\mathfrak {e}_{\\mathrm {triv}} = (G^*, {}^LG^*, 1, \\mathrm {id})$ then we expect $\\sum \\limits _{b \\in B(G, \\mu )} [R\\Gamma ^{\\flat }_c(G,b,\\mu )[\\mathrm {Red}^{\\mathfrak {e}_{\\mathrm {triv}}}_b(S\\Theta _{\\phi , G^*})]] = S\\Theta _{\\phi ,G} \\boxtimes r_{\\mu } \\circ \\phi |_{W_{E_{\\mu }}}.$ For $\\mathrm {GL}_n$ , this is known in the trivial endoscopic case for all representations by .",
"Because $L$ -packets are singletons for $\\mathrm {GL}_n$ , the trivial endoscopic case implies the endoscopic versions of the formula.",
"In , the formula is proven for discrete parameters and elliptic endoscopy of unramified $\\mathrm {GU}_n$ , for $n$ odd.",
"In , a strategy is outlined to prove this formula in the elliptic endoscopic cases using the cohomology of Igusa and Shimura varieties.",
"This strategy should (eventually) yield results comparable to whenever adequate global results are known about the Langlands correspondence and the cohomology of Shimura and Igusa varieties.",
"The averaging formulas imply strong results about $R\\Gamma ^{\\flat }_c(G,b,\\mu )$ .",
"For instance, in it is shown that the averaging formula for each elliptic endoscopic group and for $\\phi $ a supercuspidal parameter implies the Kottwitz conjecture as in ."
],
[
"Proof of Proposition ",
"The averaging formula in §REF corresponds to the case of the trivial endoscopic triple $\\mathfrak {e}_{\\mathrm {triv}} =(G^*,{}^LG^*,1, \\mathrm {id})$ .",
"Hence to check that Theorem REF agrees with REF , we just need to check that $\\mathrm {Red}^{\\mathfrak {e}{\\mathrm {triv}}}_b(S\\Theta _{\\phi ,G^*})$ coincides with $\\mathrm {Red}_{b, \\phi }$ for $\\phi $ induced from $\\phi _{T}$ generic.",
"Since $\\phi _{T}$ is generic, by Lemma REF $\\phi $ should give rise to a well-defined $L$ -parameter with trivial monodromy.",
"Therefore, under the $\\mathrm {LLC}_{G}$ appearing in Assumption REF , we are assuming the parameter $\\phi $ has an $L$ -packet given by the irreducible constituents of $i^G_B(\\chi )$ , by Assumption REF (3).",
"Suppose first that $b \\in B(G)_{\\mathrm {un}}$ .",
"Then we have $\\mathrm {Red}_{b, \\phi } = \\sum \\limits _{w \\in W_G/W_{M_b}} i^{J_b}_{B_b}( \\chi ^w) \\otimes \\delta _{P_{b}}^{1/2}[-\\langle 2\\hat{\\rho }_G, \\nu _b \\rangle ].$ The set $D(M, \\mathfrak {e})$ is a singleton and corresponds to the trivial embedded datum where $H_M=M$ .",
"Note that $r^G_{P^{op}_b}(i^G_B(\\chi )) = r^G_{P_b}(i^G_B(\\chi ))$ and that the latter term can be simplified by the geometric lemma of .",
"$\\mathrm {Red}^{\\mathfrak {e}{\\mathrm {triv}}}_b(S\\Theta _{\\phi , G^*}) & = (\\mathrm {Trans}^{J_b}_{M_b} \\circ r^G_{P^{op}_b})(i^G_B(\\chi ))\\otimes \\overline{\\delta }^{\\frac{1}{2}}_P\\\\& = \\mathrm {Trans}^{J_b}_{M_b} \\circ (\\sum \\limits _{w \\in W_G/W_{M_b}} i^{M_b}_{B \\cap M_b} \\chi ^w)\\otimes \\overline{\\delta }^{\\frac{1}{2}}_P\\\\& = (\\sum \\limits _{W_G/W_{M_b}} i^{J_b}_{B_b}) \\circ \\mathrm {Trans}^{T_b}_T \\chi ^w\\otimes \\overline{\\delta }^{\\frac{1}{2}}_P\\\\&= \\mathrm {Red}_{b, \\phi } .$ where the third equality is (REF ).",
"Now consider the case where $b \\notin B(G)_{\\mathrm {un}}$ .",
"We must show that $\\mathrm {Red}^{\\mathfrak {e}_{\\mathrm {triv}}}_b(i^G_B(\\chi )) = 0$ , for which it suffices to show that $\\mathrm {Trans}^{J_b}_{M_b}(i^{M_b}_{B \\cap M_b}(\\chi ^w))=0$ for each $w \\in W_G/W_{M_b}$ .",
"This follows from the fact that $T$ does not transfer to $J_b$ by assumption and the character of $i^{M_b}_{B \\cap M_b}(\\chi ^w)$ is supported on the conjugates of $T$ as per ."
]
] | 2209.08175 |
[
[
"Technical Report for Trend Prediction Based Intelligent UAV Trajectory\n Planning for Large-scale Dynamic Scenarios"
],
[
"Abstract The unmanned aerial vehicle (UAV)-enabled communication technology is regarded as an efficient and effective solution for some special application scenarios where existing terrestrial infrastructures are overloaded to provide reliable services.",
"To maximize the utility of the UAV-enabled system while meeting the QoS and energy constraints, the UAV needs to plan its trajectory considering the dynamic characteristics of scenarios, which is formulated as the Markov Decision Process (MDP).",
"To solve the above problem, a deep reinforcement learning (DRL)-based scheme is proposed here, which predicts the trend of the dynamic scenarios to provide a long-term view for the UAV trajectory planning.",
"Simulation results validate that our proposed scheme converges more quickly and achieves the better performance in dynamic scenarios."
],
[
"Introduction",
"Unmanned Aerial Vehicles (UAVs) have been used in providing emergency communication services for some special scenarios that cannot be served satisfactorily with existing terrestrial infrastructures, such as some mass gathering events, like new year events, large conferences, etc.",
"In these situations, the trajectory planning of the UAV involved can seriously affect system performances, e.g., data throughput[1], transmission delay[2], and user fairness[3].",
"For the practical applications, the classical optimization-based UAV trajectory planning strategies [8] may no longer be feasible, as most of them are designed to be operated in an iterative fashion with high computational complexity.",
"In the light of the aforementioned, Reinforcement Learning (RL) [6] based strategies have been regarded as promising solutions for the UAV-enabled system owing to the great self-learning capability, which exhibits significant efficiency gains in static scenarios by optimizing the interaction process [4] and experience selection [5].",
"Furthermore, faced with realistic application scenarios with highly dynamic characteristics, more and more researches have studied on how to extract scene features by adding Deep Neural Networks to further enhance the capability of RL, so as to construct the Deep RL framework (DRL).",
"To achieve above goals, [7], [10], [11], [9] construct the Artificial Neural Network (ANN) to mine the correlations between scene information (e.g.",
"location [7], [10], energy state [10] and throughput [11], [9]).",
"In order to avoid the uncertainty caused by manual feature extraction in ANN-based schemes, [6], [12], [13] construct the Convolutional Neural Network (CNN) and model the scene information as a tensor composed of multiple channels (e.g.",
"communication range [6], location [12] and object tracking information [13]), which makes scene features more hierarchical.",
"However, considering the dynamic characteristics of real scenarios, there are still two major research gaps in existing researches: 1) The number of GUs is related to the structure of the DRL frameworkAssuming that each GU has $n$ features, then the input of the ANN-based model is a vector of dimension $nN\\times 1$ where $N$ denotes the number of GUs in the scenario.",
"Apparently, a change of $N$ can lead to changes in the network structure.",
"designed in existing studies [7], [10], [11], [9], the variation in the number of GUs in dynamic scenarios leads to the need for re-tuning and re-training for the proposed DRL models.",
"2) The scene information of GUs is not fully utilized in existing studies [6], [12], [13] where the whole dynamic process is usually processed as multiple independent frames, which lacks further exploration of their associations.",
"Therefore, the critical issue is how to bridge the above gaps by designing a DRL framework that is flexible enough and can fully exploit the dynamic characteristics of the scene, which motivates this work with the objective to maximizes the long-term performance of the UAV-enabled large-scale dynamic scenarios by optimizing the UAV's movement action, subjected to the constraints of the communication QoS and energy.",
"Therefore, we design a moving Trend Prediction (TP) based DRL framework for the UAV (served as an agent) to perceive the state of the current environment and predict the trend of the future state.",
"Through continuous interaction with the environment, the agent optimizes its actions according to the received feedback (known as reward in DRL).",
"Simulations have been used to verify and validate the performance of the proposed scheme.",
"Consider an area of interest (AoI) where the explosive growth of access requirements have already far outstripped the capacity of existing terrestrial base stations.",
"One UAV with velocity $v_{uav}$ is dispatched to provide the extra communication capacity for the GUs inside the AoI (denoted as a GU set $\\Omega _{all}^t$ ) with fixed flight altitude $H$ .",
"In particular, the entire mission period of the UAV is discretized into multiple individual time slots and each with equal duration $\\tau $ .",
"Similarly, referring to [9], the AoI has been divided into $K\\times K$ equal grids, of which the centers are used as way-points of the UAV in each time slot.",
"Thus, the location of the UAV (projected on the ground) and the $i$ -th mobile GU in time slot $t$ can be formulated as $\\mathbf {L_u^t}=[x_u^t,y_u^t]$ and $\\mathbf {L_i^t}=[x_i^t,y_i^t]$ , respectively.",
"Following the model proposed in [9], the velocity and moving direction of the GU $i$ will be updated as $v_i^t&=k_1 v_i^{t-1}+(1-k_1)\\overline{v},\\\\\\theta _i^t&=\\theta _i^{t-1}+\\widetilde{\\theta }k_2.$ where $\\overline{v}$ and $\\widetilde{\\theta }$ denote average velocity and steering angle, respectively, and $k_2$ follows $\\epsilon $ -greedy model$\\endcsname $In each time slot, the GU will keep the same moving direction with probability $\\epsilon $ , otherwise the GU chooses one of the remaining directions randomly.",
".",
"Since the terrestrial infrastructures cannot provide communication services, the GUs' data will be temporarily stored in their on-board buffer and wait for the UAV to start the data upload process.",
"Here, the GU-to-UAV channel is modeled as Rician models [1] that can capture the shadowing and small-scale fading effects due to multi-path propagation, in which the channel coefficient from the GU $i$ to the UAV in time slot $t$ can be expressed as $h_i^t=\\sqrt{\\beta _i^t}\\hat{h_i^t},$ where $\\beta _i^t=\\frac{\\alpha }{(H^2+||\\mathbf {L_u^t}-\\mathbf {L_i^t}||^2)^{k_{ps}/2}}$ and $\\hat{h_i^t}=\\sqrt{\\frac{k_s}{k_s+1}}\\tilde{h_i^t}+\\sqrt{\\frac{k_s}{k_s+1}}\\tilde{\\tilde{h_i^t}}$ , $|\\tilde{h_i^t}|=1$ , $|\\tilde{\\tilde{h_i^t}}|\\sim CN(0,1)$ , and $K_{ps}$ and $K_s$ denote the GU-to-UAV path loss component and Rician factor, respectively.",
"$\\alpha $ denotes the channel power gain at the reference distance $||\\mathbf {L_u^t}-\\mathbf {L_i^t}||=1$ m. By adopting OFDMA technology, the UAV can pre-divide communication resources into multiple equal and orthogonal resource blocks in advance, of which the bandwidth allocated to each communication GU is $W$ .",
"In this way, the UAV using OFDMA technology can support concurrent data transmissions with surrounding GUs.",
"Thus, the communication rate with respect to each GU is given by $r_i^t=W\\log _2(1+\\frac{|h_i^t|^2p}{\\sigma ^{2}})$ , where $p$ , $W$ , and $\\sigma ^2$ denote the transmission power, bandwidth and noise power, respectively.",
"In this way, the data uploaded throughput of the $i$ -th GU is obtained by $r_i^t\\tau _c$ , in which $\\tau _c$ denotes the duration of the hovering time used for communications within one time slot, which is assumed as $\\tau \\gg \\tau _c$ .",
"Here we denote the data queue length of the GU $i$ in time slot $t$ as $B_i^t$ , which depends on both the newly generated data (i.e.",
"$I_i^t$ ) and the data uploaded (i.e.",
"$r_i^t\\tau _c$ ) to the UAV, which is given by $B_i^t={B_i^{t-1}+I_i^t-r_i^t\\tau _c}$ .",
"In addition, the energy consumption of the UAV in time slot $t$ is associated with the flying distance of the UAV in the past time slot [9], which is formulated as $E_f^t=p_f\\frac{||\\mathbf {L_u^t}-\\mathbf {L_u^{t-1}}||}{v_{uav}},$ where $p_f$ denotes the UAV flying power."
],
[
"Problem Formulation",
"Regarding the dynamic scenarios considered in this work, the location, number, and data queue of GUs that change in real time greatly increases the feature dimension of the UAV-Ground communication scenario, which poses a severe challenge to build the efficient and reliable data links between GUs and the UAV.",
"Therefore, our objective is to find a policy that helps the UAV make the optimal decision on trajectory planning, so that the utility of the UAV-enabled communication system over all the time slots is maximized, subjected to the coverage range, energy consumption, and the communication QoS.",
"Thus, the UAV trajectory planning issue$\\endcsname $We summarize the notations used in this paper in section II of our technical report .",
"is formulated as $\\textup {P0}:&\\max \\ \\sum \\nolimits _{t\\in \\mathcal {T}}\\mathcal {U}^t\\nonumber \\\\\\ \\textup {s.t.",
"}\\ &{\\text{C1}}:\\ h_i^t\\ge \\underline{h},\\ \\forall \\ t,\\nonumber \\\\&{\\text{C2}}:\\ \\sum _{t}E_f^t\\le \\overline{E},\\nonumber $ where the system utility function $\\mathcal {U}^t$ depends on both the throughput and fairness during the data upload processes.",
"Furthermore, C1 ensures that the communication link should meet the required QoS, namely the channel coefficient of each communication GU $i$ (see (REF )) should not be smaller than the threshold $\\underline{h}$ .",
"C2 limits the total energy consumption during the UAV flight process $\\sum _{t}E_f^t$ (see (REF )) to within the portable energy $\\overline{E}$ under the premise that the communication energy consumption is negligible.",
"Furthermore, the utility function $\\mathcal {U}^t$ is defined here, $\\mathcal {U}^t=f_G^t\\sum _{i\\in \\Omega _{all}^t}r_i^t\\tau _c$ , in which $f_G^t$ called as the Jain's fairness index [3] is defined as $f_G^t=\\frac{(\\sum _{i\\in \\Omega _{all}^t} c_i)^2}{|\\Omega _{all}^t|\\sum _{i\\in \\Omega _{all}^t} c_i^2},$ in which $c_i$ denotes the number of UAV-enabled communication services that GU $i$ has participated in.",
"As a widely-used metric for fairness, the value of $f_G^t$ approaches 1 when the total number of time slots that each GU is served are very close, which can be regarded as a measure of the fairness of UAV communication services in existing scenarios."
],
[
"MDP Model",
"In general, conventional methods normally would narrow down P0 for an individual time slot, namely $\\max U^t,\\forall t\\in \\mathcal {T}$ , and solve it by classical convex optimization methods or heuristic algorithms, which may obtain the greedy-like performance due to lacking of a long-term target.",
"To tackle above issue, we apply Markov Decision Process (MDP), defined as a tuple ($\\mathcal {S}$ , $\\mathcal {A}$ , $\\mathcal {R}$ , $\\mathcal {P}$ ), to model the UAV trajectory planning problem in large-scale dynamic scenarios, which are detailed as follows.",
"1) The $state$ $\\mathcal {S}$ contains the locations of the UAV and GUs as well as their real-time status, which is formulated as $\\mathcal {S}\\triangleq \\lbrace s_t=\\lbrace \\mathbf {L_u^t},\\mathbf {L_i^t},B_i^t,h_i^t|i\\in \\Omega _{all}^t, \\forall t\\rbrace \\rbrace .$ 2) The $action$ $\\mathcal {A}$ contains available actions of the UAV in each time slot, which is formulated as $\\mathcal {A}\\triangleq &\\lbrace a_t=\\lbrace Up, Down, Left, Right,Right\\ upper,\\\\\\nonumber & Right\\ lower, Left\\ upper, Left\\ lower|\\forall t\\rbrace \\rbrace .$ 3) The $reward$ $\\mathcal {V}_{\\pi }(s_t)$ represents the discounted accumulated reward from the state $s_t$ to the end of the task with the policy $\\pi $ , which is formulated as $\\mathcal {V}_{\\pi }(s_t)=\\mathbb {E}_{\\pi }[\\sum _{j=0}^{\\mathcal {J}}\\gamma ^j r(s_{t+j},a_{t+j})],$ where $r(s_t,a_t)$ denotes the immediate reward through executing action $a_t$ at the state $s_t$ , and $\\mathcal {J}$ denotes the end of the task.",
"In particularly, the action $a_t$ adopted here is selected following the policy $\\pi $ , i.e., $\\pi (s_t)=a_t$ .",
"According to problem P0, $r(s_t,a_t)$ is defined as $r(s_t,a_t)=\\left\\lbrace \\begin{aligned}\\mathcal {U}^t&,&\\ {\\text{if C1 and C2 are satisfied,}}\\\\0&,&\\ {\\text{otherwise.",
"}}\\end{aligned}\\right.$ 4) The $transition$ $probability$ $\\mathcal {P}\\triangleq \\lbrace p(s_{t+1}|s_t)\\rbrace $ represents the probability that the UAV reaches the next state $s_{t+1}$ while in the state $s_t$ , which is formulated as $p(s_{t+1}|s_t)=\\left\\lbrace \\begin{aligned}&\\eta \\ \\ \\ \\ , & \\text{if}\\ s_{t+1}\\triangleq \\arg \\max _{a_t\\in \\mathcal {A}} \\mathcal {V}_{\\pi }(s_{t+1}|s_t,a_t), \\\\&1-\\eta , & \\text{otherwise},\\end{aligned}\\right.$ where $\\eta $ corresponds to the greedy coefficient during the action selection process.",
"Based on the formulated MDP model ($\\mathcal {S}$ , $\\mathcal {A}$ , $\\mathcal {R}$ , $\\mathcal {P}$ ), the UAV first observes the state $s_t$ in each time slot $t$ .",
"Then, it takes an action $a_t$ following the policy $\\pi $ , i.e., $a_t=\\pi (s_t)$ , and thus obtains the corresponding immediate reward $r(s_t,a_t)$ .",
"Then, the UAV moves to the next way-point and the $state$ updates to $s_{t+1}$ .",
"Therefore, the problem P0 can be transformed into maximizing the discounted accumulated reward by optimizing the policy $\\pi $ .",
"A table that summarizes all notations in this paper is given in Table.",
"1."
],
[
"DRL Framework",
"It is well-known that the DRL algorithm is efficient and effective in solving MDP for the uncertain (i.e.",
"$\\mathcal {V}_{\\pi }(s_t)$ ) and complex (i.e.",
"$\\mathcal {S}$ ) system, in which the critical issue here is how to obtain the optimal policy $\\pi ^{*}$ for the action selection process.",
"To achieve that, we first rephrase $\\mathcal {V}^{\\pi }(s_t)$ into the form of state-action pairs as $\\mathcal {V}^{\\pi }(s_t)=Q^{\\pi }(s_t,a_t)$ where $a_t=\\pi (s_t)$ means the action $a_t$ is selected at the state $s_t$ according to the policy $\\pi $ .",
"Here we call $Q^{\\pi }(s_t,a_t)$ as the Q-function.",
"Referring to [9], it is easy to draw the conclusion that the optimal policy can be derived by $\\pi ^{*}(s_t)=\\mathop {\\arg \\max }\\limits _{a\\in \\mathcal {A}} Q(s_t,a)$ .",
"In other words, once the UAV selects the action that can maximize the corresponding Q-function at each $state$ $s_t\\in \\mathcal {S}$ , the optimal policy can be realized.",
"Then, the remaining issue is how to obtain $Q(s_t,a_t)$ with the given environment $s_t$ and action $a_t$ .",
"Based on above conclusions and (REF ), we have $Q^{\\pi ^{*}}(s_t,a_t)=r(s_t,a_t)+\\gamma Q^{\\pi ^{*}}(s_{t+1},\\pi ^{*}(s_{t+1})),$ where $Q^{\\pi ^{*}}(s_{t+1},\\pi ^{*}(s_{t+1}))=\\mathop {\\max }\\limits _{a\\in \\mathcal {A}} Q^{\\pi ^{*}}(s_{t+1},a)$ , thus the issue about searching the Q-function $Q^{\\pi ^{*}}(s_t,a_t)$ can be formulated as a regression problem, that is, through iteratively optimizing the parameters of the Q-function so that the left-hand side of the (REF ) is infinitely close to the right-hand side.",
"Here, considering the high dimension of $\\mathcal {S}$ and $\\mathcal {A}$ in the scenario we studied, it is common to apply the neural networks (NN) to fit the Q-function mentioned above.",
"To be specific, the NN can address the sophisticate mapping between the $state$ $s_t$ , $action$ $a_t$ and their corresponding Q-function value $Q(s_t,a_t)$ based on a large training data set.",
"Accordingly, two individual deep neural networks are established here.",
"One with parameters $\\theta _P$ is utilized to construct the Evaluation network $Q_{\\theta _P}(s_t,a_t)$ that models the function $ Q^{\\pi ^{*}}(s_t,a_t)$ , and another one with parameters $\\theta _T$ is used to construct the Target network $Q_{\\theta _T}(s_t,a_t)$ for obtaining the target value (i.e., $r(s_t,a_t)+\\gamma Q^{\\pi ^{*}}(s_{t+1},\\pi ^{*}(s_{t+1}))$ ) in the training process.",
"Finally, the parameters $\\theta _P$ of $Q_{\\theta _P}(s_t,a_t)$ are updated by minimizing the loss function $L$ , which is formulated as $L=\\mathbb {E}_{s_t\\in \\mathcal {S}}[(\\underbrace{Q_{\\theta _p}(s_t,a_t)}_{{\\text{Evaluation}}} -\\underbrace{(r(s_t,a_t)+\\gamma \\max Q_{\\theta _t}(s_{t+1},a))}_{{\\text{Target}}})^2].$ Finally, after the NN has been well trained, the UAV can make a decision at the state $s_t$ according to the obtained Q-function that is modeled as the NN, and then the UAV takes the action $a_t=\\arg \\max _{a\\in \\mathcal {A}} Q_{a}(s_t,a)$"
],
[
"Design the Input Layer of Neural Network",
"In order to extract spatial features of the scenario, here we adopt Convolutional Neural Networks (CNNs) to achieve the Evaluation network $Q_{\\theta _P}(s_t,a_t)$ , which sets a three-channel tensor with size $\\mathbb {R}^{K\\times K\\times 3}$ as the input.",
"Meanwhile, each channel of the tensor is modeled as a matrix with the size $\\mathbb {R}^{K\\times K}$ , corresponding to the scenario model that is divided into $K\\times K$ equal grids.",
"(see the system model in Section. II-A).",
"Here, two convolution layers are constructed to extract the features from the input, and two full connected layers are constructed to establish associations between them.",
"The design of the three-channel tensor are given as follows."
],
[
"Channel 1",
"The Channel 1 describes the effective communication range of the UAV with the location $\\mathbf {L_u(t)}$ , which is formulated as a matrix $\\mathbf {T_1}\\in \\mathbb {R}^{K\\times K}$ .",
"To meet the QoS during the Ground-to-UAV communication process, we substitute (3) into the constraint C1, namely $||\\mathbf {L_u(t)}-\\mathbf {L_i(t)}||^2\\le (\\frac{\\alpha \\hat{h_i(t)}^2}{\\underline{h}^2})^{2/K_{ps}}-H^2.\\nonumber $ which indicates that the location of GUs that can establish reliable communication with the UAV must meet above condition.",
"In this way, we can calculate whether the distance from GU $i\\in \\Omega _{all}$ to the location of the UAV $\\mathbf {L_u(t)}$ satisfies the above condition.",
"If so, we assign the corresponding element in matrix $\\mathbf {T_1}$ as the real-time channel coefficient obtained, otherwise set it as 0."
],
[
"Channel 2",
"The Channel 2 describes the interaction information between GUs and the UAV, which is formulated as a matrix $\\mathbf {T_2}\\in \\mathbb {R}^{K\\times K}$ .",
"In $\\mathbf {T_2}$ , we record the communication times between each GU $i$ and the UAV (i.e., $c_i$ ), and set the element of $\\mathbf {T_2}$ corresponding to the location of GU $i$ as $c_i,i\\in \\Omega _{all}$ , otherwise is 0."
],
[
"Channel 3",
"The Channel 3 describes the predicted movement trend of GUs in the future $N$ time slots, which is formulated as a matrix $\\mathbf {T_3}\\in \\mathbb {R}^{K\\times K}$ .",
"The design principle of $\\mathbf {T_3}$ is predicting the future movement situation of GUs by extracting their mobile features from historical records, which enables the UAV a future-oriented view during the trajectory planning to achieve the maximum accumulated reward from the current state until the end of the task.",
"The specific steps for constructing the channel 3 are as follows.",
"$\\bullet $ Step 1: The matrix $\\textbf {G}^t\\in \\mathbb {R}^{K\\times K}$ is produced to record the buffer state of GUs in the scenario, in which the element is set as $\\textbf {G}^t[m,n]=B_i^t$ if GU $i\\in \\Omega _{all}$ is located in the corresponding grid (i.e.",
"the $m$ -th row and the $n$ -th column) of the scenario, otherwise $\\textbf {G}^t[m,n]=0$ .",
"$\\bullet $ Step 2: The difference matrix $\\Delta \\textbf {G}^t\\in \\mathbb {R}^{K\\times K}$ is produced as $\\Delta \\textbf {G}^t=\\textbf {G}^t-\\textbf {G}^{t-1}$ here, showing the changes of $\\textbf {G}^t$ between adjacent time slots.",
"$\\bullet $ Step 3: Four direction kernels $\\mathcal {U}\\in \\mathbb {R}^{2\\times 1}$ , $\\mathcal {D}\\in \\mathbb {R}^{2\\times 1}$ , $\\mathcal {L}\\in \\mathbb {R}^{1\\times 2}$ , and $\\mathcal {R}\\in \\mathbb {R}^{1\\times 2}$ are utilized here to detect the moving direction of GUs (corresponding to up, down, left and right respectively), which are given by $\\mathcal {U}=[1,-1]^T,\\mathcal {D}=[-1,1]^T,\\mathcal {L}=[1,-1],\\mathcal {R}=[-1,1].$ $\\bullet $ Step 4: The SAME convolution operation is performed on the difference matrix $\\Delta \\textbf {G}^t$ using the above four kernels, respectively.",
"The corresponding outputs are four matrices $\\textbf {G}_{\\mathcal {U}}^t$ , $\\textbf {G}_{\\mathcal {D}}^t$ , $\\textbf {G}_{\\mathcal {L}}^t$ and $\\textbf {G}_{\\mathcal {R}}^t$ with the same size $\\mathbb {R}^{K\\times K}$ , which denotes the detection result of the movement direction of GUs in the past two time slots, respectively.",
"Specifically, elements of above four matrices are obtained by $\\left\\lbrace \\begin{aligned}\\textbf {G}^t_{\\mathcal {U}}[m,n]=\\sum \\nolimits _{i=0}^{1} \\Delta \\textbf {G}^t[m+i,n]\\times \\mathcal {U}[i], \\\\\\textbf {G}^t_{\\mathcal {D}}[m,n]=\\sum \\nolimits _{i=0}^{1} \\Delta \\textbf {G}^t[m+i,n]\\times \\mathcal {D}[i], \\\\\\textbf {G}^t_{\\mathcal {L}}[m,n]=\\sum \\nolimits _{i=0}^{1} \\Delta \\textbf {G}^t[m,n+i]\\times \\mathcal {L}[i], \\\\\\textbf {G}^t_{\\mathcal {R}}[m,n]=\\sum \\nolimits _{i=0}^{1} \\Delta \\textbf {G}^t[m,n+i]\\times \\mathcal {R}[i].\\end{aligned}\\ \\forall m,n.\\right.$ $\\bullet $ Step 5: Based on step 4, the trend prediction matrices of up, down, left and right directions are constructed respectively, which are denoted as $\\textbf {T}^N_{\\mathcal {U}}$ , $\\textbf {T}^N_{\\mathcal {D}}$ , $\\textbf {T}^N_{\\mathcal {L}}$ and $\\textbf {T}^N_{\\mathcal {R}}$ .",
"Taking $\\textbf {T}^N_{\\mathcal {R}}$ as an example, the specific steps are given in Algorithm. 1.",
"Generate the matrix $\\textbf {T}^N_{\\mathcal {R}}$ with $N$ steps [1] Initialize $i=1$ , generate $\\textbf {T}^N_{\\mathcal {R}}[m,n]$ as $\\textbf {T}^N_{\\mathcal {R}}[m,n]=\\left\\lbrace \\begin{aligned}\\textbf {G}_{\\mathcal {R}}^t[m,n]&,&\\ {\\text{if}}\\ \\textbf {G}_{\\mathcal {R}}^t[m,n]>0, \\\\0&,&{\\text{otherwise}}.\\end{aligned}\\right.$ each $\\textbf {T}^N_{\\mathcal {R}}[m,n]\\ne 0$ , set $x\\leftarrow m$ , $y\\leftarrow n$ $i\\le N$ With probability $\\epsilon $ , set $\\textbf {T}^N_{\\mathcal {R}}[x+1,y]=\\textbf {T}^N_{\\mathcal {R}}[x+1,y]+\\gamma \\textbf {T}^N_{\\mathcal {R}}[x,y]$ , and $x\\leftarrow x+1$ .",
"(Right) Otherwise, select another direction randomly, and update the corresponding element as $\\textbf {T}^N_{\\mathcal {R}}[x-1,y]=\\textbf {T}^N_{\\mathcal {R}}[x-1,y]+\\gamma \\textbf {T}^N_{\\mathcal {R}}[x,y], x\\leftarrow x-1, (\\text{Left})\\nonumber $ or $\\textbf {T}^N_{\\mathcal {R}}[x,y+1]=\\textbf {T}^N_{\\mathcal {R}}[x,y+1]+\\gamma \\textbf {T}^N_{\\mathcal {R}}[x,y], y\\leftarrow y+1,(\\text{Up})\\nonumber $ or $\\textbf {T}^N_{\\mathcal {R}}[x,y-1]=\\textbf {T}^N_{\\mathcal {R}}[x,y-1]+\\gamma \\textbf {T}^N_{\\mathcal {R}}[x,y], y\\leftarrow y-1.",
"(\\text{Down})\\nonumber $ i=i+1.",
"Normalize $\\textbf {T}^N_{\\mathcal {R}}$ .",
"For $\\textbf {T}^N_{\\mathcal {L}}$ , $\\textbf {T}^N_{\\mathcal {U}}$ , $\\textbf {T}^N_{\\mathcal {D}}$ , their elements are updated with the similar principle as in step 5 of the Algorithm.",
"1, namely $\\left\\lbrace \\begin{aligned}\\textbf {T}^N_{\\mathcal {L}}[x-1,y]=\\textbf {T}^N_{\\mathcal {L}}[x-1,y]+\\gamma \\textbf {T}^N_{\\mathcal {L}}[x,y], x\\leftarrow x-1,\\\\\\textbf {T}^N_{\\mathcal {U}}[x,y+1]=\\textbf {T}^N_{\\mathcal {U}}[x,y+1]+\\gamma \\textbf {T}^N_{\\mathcal {U}}[x,y], y\\leftarrow y+1,\\\\\\textbf {T}^N_{\\mathcal {D}}[x,y-1]=\\textbf {T}^N_{\\mathcal {D}}[x,y-1]+\\gamma \\textbf {T}^N_{\\mathcal {D}}[x,y], y\\leftarrow y-1.\\end{aligned}\\right.$ $\\bullet $ Step 6: output the $N$ -step moving trend prediction matrix $\\mathbf {T_3}=\\textbf {T}^N_{\\mathcal {U}}+\\textbf {T}^N_{\\mathcal {D}}+\\textbf {T}^N_{\\mathcal {L}}+ \\textbf {T}^N_{\\mathcal {R}}$ , and set it as the Channel 3 of the CNN model.",
"In order to enable the UAV to update the DRL model at the same time during executing the communication task, a training framework that combines online and offline manner is proposed, in which the UAV performs the task following the policy obtained at the offline stage, and then updates the its policy online in current episode.",
"The offline learning process is designed to learn the most valuable experiences from the past.",
"Therefore, a large \"Replay Memory\" (denoted as $R_l$ ) is constructed to store the experiences including $state$ $s_t$ , $action$ $a_t$ , and reward $\\mathcal {U}^t$ during the past episodes, of which seventy percent are with the largest reward and the remaining thirty percent are chosen randomly from the rest.",
"In contrast, the online learning process aims to learn from the current experiences, resulting in the current best behavioral decisions.",
"Therefore, it will generate a relative small \"Replay Memory\" (denoted as $R_s$ ) to store the experiences sampled during the current communication task, of which eighty percent are with the largest reward from the current episode and the remaining twenty percent are picked randomly.",
"The combination of the above two training processes not only allows the UAV to take full advantage of past experiences, but also adjust its policy according to the current actual situation, and sum up all past experiences after the end."
],
[
"Computational Complexity",
"The computational complexity of our proposed moving Trend Prediction (TP) based DRL framework is mainly involved in two parts, namely the building process of the channel 3 and the CNN calculation process.",
"The former can be obtained as follows: step 1, 2 and 3 only involve simple calculations, step 4 requires $O(4\\times |\\Omega _{all}^t|)$ and step 5 requires $O(8\\times N\\times |\\Omega _{all}^t|)$ .",
"Then, the computational complexity of the CNN calculation process is given by $O(N_c\\times M_c^2\\times K_c^2\\times C_{in}\\times C_{out})$ [14] in which $N_c$ =2 $M_c$ =32, $K_c$ =3, $C_{in}$ =2 and $C_{out}$ =32 are the parameters of the model in our scheme.",
"The simulation parameters of the scenario are set as $|\\Omega _{all}^t|=50$ , $K=30$ , $k_1=0.9$ , $\\overline{v}=1$ m/s, $v_{uav}=30m/s$ , $p_f=110$ W, $\\widetilde{\\theta }=\\frac{\\pi }{2}$ , $H=40$ m, $K_{ps}=2$ , $K_s=1$ and $k_2$ follows $\\epsilon $ -greedy model with $\\epsilon =0.9$ .",
"The simulation parameters of the communication process are set as $W=2$ Mhz, $\\tau =1$ s, $\\tau _c=0.1$ s, $\\underline{h}=2.5\\times 10^{-9}$ , $\\overline{E}=10^4$ kJ, $p=0.1$ W $\\alpha =10^{-5}$ , $\\sigma =10^{-9}$ and $I_i^t=5\\times 10^{-3}$ bits/s.",
"The Trend Prediction based model proposed in this work is called as TP here, in which the greedy coefficient $\\eta =0.9$ .",
"The results are shown in Fig.",
"1 and Fig. 2.",
"Figure: Overall performance in large-scale scenarios with fixed 50 GUs, in which the maximum number of training steps per episode is set as (a) 3000 and (b) 6000.Figure: Overall performance in large-scale scenarios with fixed 50 GUs, in which the maximum number of training steps per episode is set as (a) 8000 and (b) 10000.We tested the total reward and number of steps available under multiple training parameters in a scenario with 50 GUs, where the total reward can be seen as the overall performance that combines fairness and throughput, while the number of steps reflects the efficiency of the UAV trajectory.",
"From the above results, even if the number of GUs is increased from 20 to 50, the proposed solution can still obtain good performance.",
"We didn't compare the other two schemes because even in a 20-GU scenario, the other two were not only unable to converge but were also significantly at a performance disadvantage.",
"Figure: Overall performance in large-scale scenarios with increasing GUsWe further tested the performance of the proposed strategy when the number of GUs in the scene dynamically increased, in which only 50 GUs are deployed in the base scene, and one new GU is deployed at the end of each training, the results are given in Fig. 3.",
"Figure: The trajectory of UAV in the scenario with 50 GUs.We further give the trajectory of the UAV in Fig. 4.",
"The UAV starts at the green dot and ends at the blue pentagram.",
"From the perspective of the entire trajectory, the UAV has a good effect on the coverage of the entire area, which indirectly shows that the UAV with the trend prediction function has a broader and long-term vision."
],
[
"Notation Table",
"A table that summarizes all notations in this paper is given in Table.",
"1."
]
] | 2209.08235 |
[
[
"LEARNEST: LEARNing Enhanced Model-based State ESTimation for Robots\n using Knowledge-based Neural Ordinary Differential Equations"
],
[
"Abstract State estimation is an important aspect in many robotics applications.",
"In this work, we consider the task of obtaining accurate state estimates for robotic systems by enhancing the dynamics model used in state estimation algorithms.",
"Existing frameworks such as moving horizon estimation (MHE) and the unscented Kalman filter (UKF) provide the flexibility to incorporate nonlinear dynamics and measurement models.",
"However, this implies that the dynamics model within these algorithms has to be sufficiently accurate in order to warrant the accuracy of the state estimates.",
"To enhance the dynamics models and improve the estimation accuracy, we utilize a deep learning framework known as knowledge-based neural ordinary differential equations (KNODEs).",
"The KNODE framework embeds prior knowledge into the training procedure and synthesizes an accurate hybrid model by fusing a prior first-principles model with a neural ordinary differential equation (NODE) model.",
"In our proposed LEARNEST framework, we integrate the data-driven model into two novel model-based state estimation algorithms, which are denoted as KNODE-MHE and KNODE-UKF.",
"These two algorithms are compared against their conventional counterparts across a number of robotic applications; state estimation for a cartpole system using partial measurements, localization for a ground robot, as well as state estimation for a quadrotor.",
"Through simulations and tests using real-world experimental data, we demonstrate the versatility and efficacy of the proposed learning-enhanced state estimation framework."
],
[
"INTRODUCTION",
"In many practical applications, it is paramount for robots to acquire accurate information about their translational and rotational dynamics.",
"This information, collectively known as the state, can be obtained through a wide variety of state estimation algorithms [1].",
"The choice of algorithms often depends on the dynamics of the robot and the sensors used to collect measurements.",
"In applications where the robot dynamics and sensor measurements can be modelled in a linear or quasi-linear manner, algorithms such as the Kalman filter and Extended Kalman filter can be applied.",
"On the other hand, estimation algorithms such as the moving horizon estimator (MHE) and unscented Kalman filter (UKF) allow the integration of nonlinear dynamics and measurement models.",
"Advancements in optimization algorithms and increases in hardware computational power have led to a surge in the use of deep learning tools for robotics applications [2].",
"These learning tools provide a framework in which data can be leveraged to construct models.",
"With these models, robots are able to obtain more or better information about their states and the environment they are operating in.",
"In this work, we use a deep learning tool, knowledge-based neural ordinary differential equations (KNODEs), to derive accurate dynamics models from data.",
"The KNODE model consists of two components: a prior model derived from first principles and a neural ordinary differential equation that accounts for residual dynamics.",
"In our proposed learning-enhanced state estimation framework, LEARNEST, we apply these accurate knowledge-based, data-driven models into two state estimation algorithms, which in turn improve the accuracy of the state estimates.",
"A schematic of LEARNEST is depicted in Fig.",
"REF .",
"Figure: Schematic of our proposed framework, LEARNEST, applied to a ground robot and a quadrotor system.",
"A prior first-principles model is combined with a neural network to form a hybrid KNODE model.",
"This model is integrated into KNODE-MHE or KNODE-UKF to derive accurate state estimates 𝐱 ^\\hat{\\mathbf {x}}, based on measurements 𝐲\\mathbf {y} and control inputs 𝐮\\mathbf {u}.",
"The feedback path indicates that the state estimates are utilized in other tasks such as motion control or path planning.",
"Image sources for robots: , ."
],
[
"RELATED WORK",
"There are a number of works in the literature that use deep learning tools to learn the dynamics of robotic systems.",
"In [5], the authors use a neural network to learn the residual aerodynamic forces of a quadrotor.",
"The authors in [6] model the residual dynamics of an quadrotor system with a KNODE model and use the model for predictive control.",
"O’Connell et al.",
"[7] propose an algorithm to learn basis functions that represent the aerodynamics of a quadrotor and use adaptive control to mitigate wind disturbances for a quadcopter.",
"Gaussian processes (GP) regression is another popular tool used to model residual dynamics in robotic applications [8], [9].",
"However, it is well known that GPs suffer from the curse of dimensionality and are unable to fully utilize large amounts of data for both training and inference.",
"In many of these above-mentioned applications, the robots have access to full state measurements and the state estimation problem is not addressed.",
"In terms of state estimation, there are a few papers in the literature that leverage the power of deep learning to enhance state estimation algorithms.",
"In [10], the authors use a neural network to optimize the cost matrices of the MHE optimization problem.",
"This approach, however, does not address the issue of an inaccurate dynamics model.",
"Muntwiler et.",
"al [11] formulate the MHE problem using a convex optimization layer with trainable parameters in the dynamics and measurement models.",
"However, this is only applicable for linear systems and has not been shown to be applicable in physical experiments.",
"The authors in [12] use GPs to learn the parameters in the unscented transformation within the UKF.",
"In [13], the authors use a variant of recurrent neural networks (RNN) to learn the full dynamics of an electric vehicle.",
"The RNN is applied to a UKF framework to predict the vehicle states.",
"In a similar vein, the authors in [14] propose the combination of a set of LSTM networks into the UKF to filter sensor measurements and estimate the sideslip angle, which is part of the state of a vehicle.",
"In contrast to these works, our framework makes use of a combination of first-principles models and neural ODEs to learn hybrid dynamics models.",
"This integration of first-principles models allows training to be more sample-efficient as the neural network is only required to learn the residual dynamics.",
"Different from RNNs, KNODEs, being lightweight neural networks, do not suffer from exploding and diminishing gradients during training [15], [16].",
"To the best of the authors' knowledge, this is the first work that proposes the integration of a learning-enhanced hybrid model, that combines first principles knowledge with a neural ODE, into both the MHE and UKF algorithms.",
"Our contributions in this work are three-fold.",
"First, we propose a procedure to train a hybrid knowledge-based data-driven model, known as the KNODE model, using partial or indirect measurements that are possibly nonlinear.",
"This is practical because in many applications, robots do not have access to the full state.",
"Second, we propose a general framework, LEARNEST, in which we integrate the KNODE model into two novel learning-enhanced state estimation algorithms, which we denote as KNODE-MHE and KNODE-UKF.",
"These algorithms are more accurate than their non-learning counterparts in terms of estimation accuracy, as data is assimilated into the estimation process in a systematic and amenable manner, through the synthesis of a KNODE model.",
"Third, we demonstrate the versatility of our framework by considering a range of robotic applications.",
"We provide analysis of the LEARNEST framework through simulations, and verify its capability by applying it onto data collected in physical experiments."
],
[
"PROBLEM FORMULATION",
"We consider robot systems with dynamics that are described in the following form, $ \\begin{split}\\mathbf {x}^+ = \\mathbf {f}(\\mathbf {x},\\,\\mathbf {u},\\,\\mathbf {w}),\\quad \\mathbf {y}=\\mathbf {h}(\\mathbf {x},\\,\\mathbf {u},\\,\\mathbf {v}),\\end{split}$ where $\\mathbf {x} \\in \\mathbb {R}^n$ is the state vector, $\\mathbf {u} \\in \\mathbb {R}^p$ is the control input, $\\mathbf {y} \\in \\mathbb {R}^m$ is the measurement vector, and $\\mathbf {x}^+ \\in \\mathbb {R}^n$ is the state at the next time step.",
"The function $\\mathbf {f}:\\mathbb {R}^n \\times \\mathbb {R}^p \\times \\mathbb {R}^q\\rightarrow \\mathbb {R}^n$ describes the true robot dynamics and $\\mathbf {h}:\\mathbb {R}^n \\times \\mathbb {R}^p \\times \\mathbb {R}^r \\rightarrow \\mathbb {R}^m$ is a function that maps the state and control inputs to the measurements.",
"The system dynamics and measurements are subjected to process and measurement noises, which are denoted by $\\mathbf {w} \\in \\mathbb {R}^q$ and $\\mathbf {v} \\in \\mathbb {R}^r$ .",
"In this setup, the objective is to obtain accurate estimates of the state vector $\\mathbf {x}$ , given sequences of the measurement vector $\\mathbf {y}$ and the control input $\\mathbf {u}$ .",
"In a standard state estimation algorithm, a first-principles model $\\mathbf {f}_{\\text{prior}}$ is used to propagate the dynamics of the system.",
"However, due to uncertainty and model errors, it is unlikely that this model matches the true dynamics perfectly.",
"This results in a degradation of the estimation accuracy.",
"In this work, to improve the accuracy of the dynamics model and state estimates, we propose LEARNEST in which we apply accurate data-driven dynamics models into two learning-enhanced state estimation algorithms, KNODE-MHE and KNODE-UKF.",
"These algorithms provide accurate state estimates by minimizing the discrepancy between the dynamics model and the true system."
],
[
"Moving Horizon Estimation",
"The MHE framework provides an optimization-based solution to the state estimation problem.",
"At each time step $k$ , given the past measurements $\\lbrace \\mathbf {y}_i\\rbrace ^k_{i=k-N}$ and control inputs $\\lbrace \\mathbf {u}_i\\rbrace ^{k-1}_{i=k-N}$ , the following nonlinear optimization problem is formulated and solved over a moving horizon of length $N$ [17], $ \\begin{split}\\underset{\\lbrace \\hat{\\mathbf {x}}_{i|k}\\rbrace ,\\lbrace \\hat{\\mathbf {w}}_{i|k}\\rbrace ,\\lbrace \\hat{\\mathbf {v}}_{i|k}\\rbrace }{\\textnormal {minimize}}\\;\\;\\; & ||\\hat{\\mathbf {x}}_{k-N|k} -\\bar{\\mathbf {x}}_{k-N|k} ||^2_{\\mathbf {P}_k} + \\\\&\\sum _{i=k-N}^{k-1} ||\\hat{\\mathbf {w}}_{i|k}||^2_\\mathbf {Q} +\\sum _{i=k-N}^{k} ||\\hat{\\mathbf {v}}_{i|k}||^2_\\mathbf {R}\\\\\\text{subject to}\\;\\;\\; \\hat{\\mathbf {x}}_{i+1|k} &= \\mathbf {f}(\\hat{\\mathbf {x}}_{i|k}, \\mathbf {u}_{i}, \\hat{\\mathbf {w}}_{i|k}), \\\\&\\qquad \\qquad \\forall \\, i=k-N,\\cdots ,k-1,\\\\\\; \\mathbf {y}_{i} &= \\mathbf {h}(\\hat{\\mathbf {x}}_{i|k}, \\mathbf {u}_{i}, \\hat{\\mathbf {v}}_{i|k}),\\\\&\\qquad \\qquad \\forall \\, i=k-N,\\cdots ,k,\\end{split}$ where $\\lbrace \\hat{\\mathbf {x}}_{i|k}\\rbrace ^k_{i=k-N},\\lbrace \\hat{\\mathbf {w}}_{i|k}\\rbrace ^{k-1}_{i=k-N}$ and $\\lbrace \\hat{\\mathbf {v}}_{i|k}\\rbrace ^k_{i=k-N}$ denote the sequences of estimated states, process, and measurement noises respectively.",
"In addition, $||\\mathbf {s}||^2_A$ denotes $\\mathbf {s}^T A \\mathbf {s}$ , $\\lbrace \\mathbf {s}_i\\rbrace ^u_{i=l}$ represents the sequence $\\lbrace \\mathbf {s}_l,\\cdots ,\\mathbf {s}_u\\rbrace $ , and $\\mathbf {Q}$ and $\\mathbf {R}$ are the cost matrices that penalize the influence of the process and measurement noises on the objective function.",
"The cost matrix $\\mathbf {P}_k$ accounts for past state information.",
"The term $\\bar{\\mathbf {x}}_{k-N|k}$ is the a priori estimate for the initial state $\\hat{\\mathbf {x}}_{k-N|k}$ .",
"One way to update $\\bar{x}_{k-N|k}$ is to use the optimal state estimate obtained from the previous time step, i.e., $\\bar{\\mathbf {x}}_{k-N|k} \\leftarrow \\hat{\\mathbf {x}}^{\\star }_{k-N|k-1}$ [18].",
"More details on this formulation can be found in [17], [19].",
"At each time step $k$ , upon solving the optimization problem (REF ), the last element of the sequence of optimal estimated states, $\\hat{\\mathbf {x}}^{\\star }_{k|k} \\in \\lbrace \\hat{\\mathbf {x}}^{\\star }_{i|k}\\rbrace _{i=k-N}^k$ is applied as the current state estimate.",
"Notice that the constraints in (REF ) consist of a dynamics model $\\mathbf {f}$ .",
"In a standard MHE formulation, a prior model $\\mathbf {f}_{\\text{prior}}(\\mathbf {x},\\mathbf {u})$ is derived from first-principles knowledge of the system and used as the dynamics model within (REF ).",
"In the case where the model is accurate, this procedure naturally gives accurate state estimates.",
"However, it is likely that there are residual dynamics or uncertainty that are not accounted for in these models.",
"Therefore, in this work, we apply the KNODE framework to account for these uncertainties.",
"Instead of only using a prior model, we use a hybrid dynamics model, denoted by $\\mathbf {f}_{h}(\\mathbf {x},\\mathbf {u})$ , obtained through the KNODE framework (see Sections REF and REF ).",
"The objective is to learn $\\mathbf {f}_{h}(\\mathbf {x},\\mathbf {u})$ such that it is a more accurate representation of the true system dynamics (REF ).",
"This in turn allows us to integrate it into the KNODE-MHE framework in LEARNEST to provide more accurate state estimates than a conventional MHE formulation."
],
[
"Unscented Kalman Filter",
"The UKF is a variant of the Kalman filter [20], [21].",
"As an extension of the extended Kalman filter, it allows the direct integration of nonlinear dynamics models without linearization.",
"This ability to integrate nonlinear dynamics models enables us to assimilate the hybrid model learned from the KNODE framework into the UKF.",
"At each time step $k$ , the UKF algorithm consists of three main steps; computing the sigma vectors based on the previous estimate, performing a time update of the sigma vectors using the dynamics and measurement models, and lastly, updating the state estimate using the collected measurements [21].",
"By defining an augmented vector $\\mathbf {z} :=[\\mathbf {x}^{\\top }\\; \\mathbf {w}^{\\top }\\; \\mathbf {v}^{\\top }]^{\\top }$ and by combining the sigma vectors computed through the unscented transformation into a matrix $\\mathbf {Z} := [\\mathbf {Z}^{\\mathbf {x}\\top }\\; \\mathbf {Z}^{\\mathbf {w}\\top }\\; \\mathbf {Z}^{\\mathbf {v}\\top }]^{\\top } := \\left[\\hat{\\mathbf {z}} \\;\\,\\hat{\\mathbf {z}} \\pm \\sqrt{(l+\\lambda )\\mathbf {P}^{\\mathbf {z}}}\\right]$ , the equations for the time update are given as [21], $ \\begin{split}\\mathbf {Z}^{\\mathbf {x}}_{+} &= \\mathbf {f}\\left(\\mathbf {Z}^{\\mathbf {x}},\\, \\mathbf {u},\\, \\mathbf {Z}^{\\mathbf {w}}\\right),\\quad \\hat{\\mathbf {z}}^{-} = \\sum _{i=0}^{2l} {W}_{m,i} \\mathbf {Z}^{\\mathbf {x}}_{+,i}, \\\\\\mathbf {Y} &= \\mathbf {h}\\left(\\mathbf {Z}^{\\mathbf {x}},\\, \\mathbf {u},\\, \\mathbf {Z}^{\\mathbf {v}}\\right),\\quad \\hat{\\mathbf {y}}^{-} = \\sum _{i=0}^{2l} {W}_{m,i} \\mathbf {Y}_{i}, \\\\\\mathbf {P}^{-} &= \\sum _{i=0}^{2l} {W}_{c,i} \\left(\\mathbf {Z}^{\\mathbf {x}}_{i,+} - \\hat{\\mathbf {z}}^{-}\\right)\\left(\\mathbf {Z}^{\\mathbf {x}}_{i,+} - \\hat{\\mathbf {z}}^{-}\\right)^{\\top },\\\\\\end{split}$ where $l=2n+m$ is the dimension of the augmented vector, $\\lambda $ is a scaling parameter and $\\mathbf {P}^{\\mathbf {z}}$ is the block diagonal covariance matrix of the augmented vector.",
"$\\lbrace {W}_{m,i}\\rbrace _{i=0}^{2l}$ and $\\lbrace {W}_{c,i}\\rbrace _{i=0}^{2l}$ are weights computed by the unscented transformation.",
"Details on the computation of the sigma vectors and the measurement update can be found in [21].",
"In (REF ), a prediction model $\\mathbf {f}$ is used to propagate the sigma vectors ahead in time.",
"This model takes on a similar role as the dynamics model within the MHE framework described in Section REF .",
"If this dynamics model represents the true system sufficiently well, then the state estimates generated by the UKF will be accurate.",
"In practice, due to uncertainty and unmodelled dynamics, it is unlikely for the model to match the true dynamics exactly.",
"Hence, in the KNODE-UKF algorithm within the LEARNEST framework, we introduce an enhancement, where we replace the prediction model $\\mathbf {f}$ with a learned hybrid KNODE model $\\mathbf {f}_h$ that represents the true system dynamics with a higher accuracy."
],
[
"Knowledge-based Neural ODEs",
"While the dynamics model derived from prior knowledge of the system achieves a certain level of fidelity, it is generally not sufficient for high-performance model-based applications, where a more accurate model is desired.",
"In robotic systems, uncertainty and residual dynamics manifest in different forms [22], [23].",
"There could be uncertainty in the system properties, or additional perturbations due to aerodynamics, friction, and interactions with the environment.",
"Furthermore, it is often difficult to pinpoint the exact sources of these uncertainties and this makes it challenging to account for their effects in the prior dynamics model.",
"The KNODE framework [24] alleviates this issue by learning these uncertainties and residual dynamics through a data-driven procedure.",
"The KNODE framework first considers a continuous-time representation of the true system dynamics, $ \\dot{\\mathbf {x}} = \\mathbf {f}_c(\\mathbf {x},\\mathbf {u}) := \\mathbf {f}_{\\text{prior}}(\\mathbf {x},\\mathbf {u}) + \\Delta (\\mathbf {x},\\mathbf {u}),$ where $\\mathbf {f}_c(\\mathbf {x},\\mathbf {u})$ represents the true dynamics of the system and $\\Delta (\\mathbf {x},\\mathbf {u})$ represents residual dynamics that are not captured by the prior dynamics model $\\mathbf {f}_{\\text{prior}}(\\mathbf {x},\\mathbf {u})$ .",
"The subscript $c$ is used to denote the continuous-time nature of the model.",
"The residual dynamics $\\Delta (\\mathbf {x},\\mathbf {u})$ is parameterized as a neural ODE with parameters $\\theta $ , denoted as $\\Delta _{\\theta }(\\mathbf {x},\\mathbf {u})$ .",
"In other words, the neural ODE is a neural network that approximates the vector field characterizing the residual dynamics.",
"After training, we obtain a hybrid model, $ \\mathbf {f}_{h,c}(\\mathbf {x},\\mathbf {u}) := \\mathbf {f}_{\\text{prior}}(\\mathbf {x},\\mathbf {u}) + \\Delta _{\\theta ^{\\star }}(\\mathbf {x},\\mathbf {u}),$ where we use $^{\\star }$ to denote the optimal parameters.",
"We then discretize this model using standard numerical solvers such as the explicit Runge-Kutta 4th order method (RK4) to obtain the hybrid model, $\\mathbf {f}_{h}(\\mathbf {x},\\mathbf {u}) := RK4\\left(\\mathbf {f}_{h,c}\\left(\\mathbf {x},\\mathbf {u}\\right)\\right)$ .",
"This model is then incorporated into the MHE and UKF frameworks as described in Sections REF and REF .",
"By incorporating prior models, the learned neural networks are typically lightweight feed-forward networks, which significantly reduce computational time, both during training and inference.",
"More details on the network architectures are given in Sections REF and ."
],
[
"KNODE Training with Partial or Indirect Observations",
"The procedure to train a hybrid KNODE model using partial or indirect observations of the state is described in Phase 1 of Algorithm REF .",
"Conceptually, the procedure is similar to the training procedure described in [6], with the exception of Step 1 in Algorithm REF , where the pre-training observer is applied.",
"In [6], measurements of the full state are assumed to be available.",
"In this work, we relax this assumption and consider the case where only partial or indirect measurements are given.",
"To get access to state observations required for training, we propose the addition of a pre-training observer.",
"The pre-training observer $\\mathbf {g}$ is a function that maps input and output measurements, sampled at times $\\lbrace t_i\\rbrace _{i=1}^M$ , to state observations and is expressed as $ \\zeta _{1},\\cdots ,\\zeta _{M} = \\mathbf {g}(\\mathbf {y}_{1},\\cdots ,\\mathbf {y}_{M},\\mathbf {u}_{1},\\cdots ,\\mathbf {u}_{M}),$ where $\\left\\lbrace \\zeta _{i}\\right\\rbrace ^M_{i=1}$ are state observations used for training.",
"After applying the pre-training observer, we collect the state observations, together with the control inputs into a dataset $\\mathcal {O}$ and compute one-step state predictions, $ \\hat{\\zeta }_{{i+1}} = \\zeta _{i} + \\int ^{t_{i+1}}_{t_i} \\mathbf {f}_{\\text{prior}}(\\zeta _{i},\\mathbf {u}_{i}) + \\Delta _{\\theta }(\\zeta _{i}, \\mathbf {u}_{i})\\, dt.$ With these one-step predictions $\\lbrace \\hat{\\zeta }_{{i}}\\rbrace ^M_{i=2}$ , we define a mean-squared error loss by considering the deviation between the state observations and one-step predictions, $ \\mathcal {L}(\\theta ) := \\frac{1}{M-1}\\sum ^{M}_{i=2}\\left\\Vert \\hat{\\zeta }_{i} - {\\zeta }_{i}\\right\\Vert ^2_2$ From (REF ) and (REF ), notice that the prior model $f_{\\text{prior}}\\left(\\zeta _{i},\\mathbf {u}_{i}\\right)$ is embedded into the loss function through an integral.",
"This embedding allows the KNODE framework to achieve sufficiently high accuracy, without the need to access to the true underlying vector field, which is often noisy or inaccessible, especially in robotics applications.",
"By computing the gradients of the loss function with respect to the parameters $\\theta $ , we update the parameters in an iterative manner through a standard back-propagation procedure [25].",
"With the optimal set of parameters $\\theta ^{\\star }$ , we construct the discrete-time hybrid model as described in Section REF and apply it to the proposed KNODE-MHE and KNODE-UKF algorithms.",
"13pt Measurements $\\lbrace \\mathbf {y}_{i}\\rbrace _{i=1}^M$ and control inputs $\\lbrace \\mathbf {u}_{i}\\rbrace _{i=1}^M$ sampled at times $\\lbrace t_i\\rbrace _{i=1}^M$ , prior dynamics model $\\mathbf {f}_{\\text{prior}}$ , pre-training observer $\\mathbf {g}$ State estimates $\\hat{\\mathbf {x}}$ , discrete-time KNODE model $\\mathbf {f}_h$ Phase 1 (offline): Apply $\\mathbf {g}$ in (REF ) to get state observations $\\lbrace \\zeta _{i}\\rbrace _{i=1}^M$ Collect dataset $\\mathcal {O} := \\lbrace {\\zeta }_{i}, \\mathbf {u}_{i}\\rbrace _{i=1}^M$ Compute one-step predictions $\\hat{\\zeta }_{{i}}$ using (REF ) Compute loss $\\mathcal {L}(\\theta )$ with (REF ) Train to get optimal set of parameters $\\theta ^{\\star }$ Construct hybrid model $\\mathbf {f}_{h,c}$ with (REF ) Discretize to get $\\mathbf {f}_h \\leftarrow RK4(\\mathbf {f}_{h,c})$ Set $\\mathbf {f} \\leftarrow \\mathbf {f}_h$ in (REF ) or (REF ) for KNODE-MHE or KNODE-UKF Phase 2 (online), at each time step $k$: Collect measurements $\\mathbf {y}(k)$ and controls $\\mathbf {u}(k)$ Apply KNODE-MHE or KNODE-UKF to get state estimates $\\hat{\\mathbf {x}}(k)$ Learning Enhanced Model-based State Estimation Framework, LEARNEST"
],
[
"Implementation Details",
"The nonlinear optimization problem within the MHE and KNODE-MHE frameworks is formulated in CasADi [26].",
"An interior-point method solver IPOPT [27] is used to solve the optimization problem.",
"To improve efficiency, the solver is warm-started at each time step by providing it with an initial guess of the solution, based on the optimal solution obtained from the previous time step.",
"For the implementation of the Kalman filters, we adopt tools in the FilterPy library [28] and make modifications to accommodate for the hybrid KNODE model.",
"The KNODE training procedure is implemented using the Python-based torchdiffeq library [29].",
"The explicit fourth order Runge-Kutta method is used for training.",
"We use Adam [30] as the optimizer to find the optimal set of parameters $\\theta ^{\\star }$ during the training procedure.",
"The explicit RK4 method is used to discretize $\\mathbf {f}_{h,c}$ to get the final hybrid model, $\\mathbf {f}_h$ , which is integrated into KNODE-MHE and KNODE-UKF.",
"In our experiments and evaluation procedure, we seek to answer these questions pertinent to the LEARNEST framework: (a) How accurate are the learned hybrid models, compared to the models learned with full state information, and against the ground truth, when applied to state estimation algorithms?",
"(b) How much improvement do the KNODE-MHE and KNODE-UKF frameworks provide as compared to the standard MHE and UKF frameworks?",
"(c) Is the framework applicable to real-world data collected from physical experiments of robotic systems?",
"To address the first two questions, we consider two estimation tasks; state estimation for a cartpole given partial state measurements, as well as localization of a ground robot using indirect measurements from both external and onboard sensors.",
"For the third question, we apply the LEARNEST framework on flight data collected from a quadrotor system to estimate its states."
],
[
"State Estimation for a Cartpole System",
"Consider a cartpole system with the following dynamics [31], [32], $ \\begin{split}\\ddot{\\alpha } &= \\frac{g\\sin \\alpha -\\cos \\alpha \\left(F+m_p l \\dot{\\alpha }^2 \\sin \\alpha \\right)}{l\\left(\\frac{4}{3}-\\frac{m_p\\cos ^2\\alpha }{m_c+m_p}\\right)},\\\\\\ddot{p} &= \\frac{F+m_p l \\left(\\dot{\\alpha }^2 \\sin \\alpha - \\ddot{\\alpha }\\cos \\alpha \\right)}{m_c+m_p},\\end{split}$ where $p$ is the position of the cart and $\\alpha $ is the angle between the pole and the vertical.",
"The cart and the pole have masses $m_c$ and $m_p$ and the pole has a length of $2l$ .",
"$g$ is the gravitational force and $F$ is the force acting on the cart, which is the control input acting on the system.",
"A schematic diagram of a cartpole system is shown in Fig.",
"REF .",
"By defining the state vector $\\mathbf {x}:=[p \\;\\dot{p}\\; \\alpha \\; \\dot{\\alpha }]^{\\top }$ , these dynamics are discretized and written in the form of (REF ).",
"To demonstrate the efficacy of the framework, it is assumed that only the position of the cart $p$ and the angle $\\alpha $ are measured.",
"In other words, we obtain partial measurements of the state, and the measurement model is given by $\\mathbf {y} = C\\mathbf {x}$ , where $C:=[1\\; 0\\; 1\\; 0]$ .",
"In Section REF , we consider a more challenging, yet practical application, where nonlinear and indirect measurements of the state are provided.",
"The true cartpole system is numerically simulated using an explicit 5th order Runge-Kutta method (RK45).",
"In terms of uncertainty, it is assumed that the cart has a true mass of 1.5kg, while the mass of the cart considered in the prior dynamics model $\\mathbf {f}_{\\text{prior}}$ within the KNODE model is 1kg.",
"Since the mass of the cart $m_c$ is tightly coupled in the equations of motion as shown in (REF ), this allows us to ascertain the effectiveness of the KNODE framework in terms of modeling residual dynamics.",
"Additionally, we inject zero-mean Gaussian noise with standard deviation of 0.01 across all states, and into the partial state measurements.",
"For training, we collect $M=5000$ data points $\\lbrace \\mathbf {y}_{i},\\,\\mathbf {u}_{i}\\rbrace $ in a single trajectory over a time span of 10 seconds.",
"We implement a KF as the pre-training observer to account for the translational and rotational dynamics and this provides the state observations required for training.",
"The neural ODE architecture consists of 2 layers with 8 hidden neurons in between and uses the hyperbolic tangent as the activation function.",
"The network is trained over 250 epochs and uses the explicit RK4 solver during the training procedure."
],
[
"Localization for a Ground Robot",
"Next, we consider a task of localizing a ground robot using onboard measurements, combined with measurements from multiple external static sensors.",
"This task is more challenging than state estimation for the cartpole system.",
"First, the measurements from the external sensors are nonlinear functions of the state.",
"Furthermore, this application consists of a sensor fusion sub-task, where measurements from multiple sensors are fused together with onboard measurements.",
"Adapted from [33], we model the dynamics of the ground robot as $ \\begin{split}\\dot{x} &= v\\cos (\\psi + \\beta ),\\;\\;\\;\\;\\dot{y} = v\\sin (\\psi + \\beta ),\\\\\\dot{v} &= a,\\quad \\qquad \\qquad \\;\\;\\;\\dot{\\psi } = \\frac{v\\cos \\beta \\tan \\delta }{l_f + l_r},\\end{split}$ where $\\beta = \\tan ^{-1}\\left((l_r\\tan \\delta ) / (l_f+l_r) \\right)$ is the slip angle between the robot velocity vector and its center line.",
"We denote $(x,\\,y)$ as the position of the robot in the world frame, $v$ is the speed, and $\\psi $ is the heading angle of the robot with respect to the world frame.",
"Additionally, $a$ and $\\delta $ are the acceleration and front steering angle of the robot, which act as the control inputs $\\mathbf {u}$ to the system.",
"The variables $l_f$ and $l_r$ are the distances from the front and rear wheels to the centre of gravity.",
"The state vector $\\mathbf {x}$ is defined as $[x\\;y\\;v\\;\\psi ]^{\\top }$ .",
"Each of the external sensors, denoted with index $i$ , with position $(p_{i,x},\\,p_{i,y})$ , provides range measurements $r_i := \\left((x-p_{i,x})^2 + (y-p_{i,y})^2\\right)^{1/2}$ to the ground robot.",
"The ground robot is assumed to have an onboard sensor that measures its orientation, which in this case is the heading angle $\\psi $ .",
"This system configuration is typical in robotic systems.",
"In many applications, the robot has onboard sensors that provide partial measurements of its state and simultaneously, has access to some data or information from some external source such as the Global Positioning System (GPS) or other sensors [34], [35].",
"In our simulations, we consider four sensors providing range measurements to the ground robot.",
"A schematic of the location of the sensors (black circles) and the trajectory of the ground robot (in blue) is shown on the right of Fig.",
"REF .",
"We consider residual dynamics in the form of aerodynamic drag acting on the robot.",
"It is modelled as $a_{\\text{drag}} = 0.5SC_Dv^2/m$ , where $S$ is the reference area, $C_D$ is the drag coefficient and $m$ is the mass of the robot.",
"In practice, the robot may also experience other forces such as friction or forces depending on the environment it is operating in.",
"In addition to the residual dynamics, zero-mean Gaussian noise with standard deviation of $[0.005\\text{m}, 0.005\\text{m}, 0.005\\text{m/s}, 0.1\\text{deg}]$ are added to the states.",
"Gaussian noise with standard deviation of $0.05\\text{m}$ and $1.0\\text{deg}$ are added to the range and heading measurements respectively.",
"For training, we collect $M=2500$ data points in a single trajectory over a time span of 25 seconds.",
"For the pre-training observer, we first apply a precursory UKF to extract the position and heading angle.",
"We then use a linear KF to extract the robot speed from the position observations.",
"The prior model $\\mathbf {f}_{\\text{prior}}$ used in the KNODE model is obtained from (REF ).",
"The neural ODE consists of 2 layers with 8 hidden neurons in between and uses the hyperbolic tangent as the activation function.",
"The network is trained over 10000 epochs and the explicit RK4 solver is used during training.",
"Figure: Left: Schematic diagram for the cartpole system.",
"Right: Schematic diagram of the ground robot, with its trajectory plotted in blue.",
"The sensors providing measurements to the robot are plotted with black circles.",
"The starting and ending positions of the robot are denoted by a green circle and a red triangle respectively."
],
[
"State Estimation for the Crazyflie Quadrotor",
"In this set of experiments, we use the LEARNEST framework to estimate the translational states of the Crazyflie quadrotor system.",
"Measurements are collected when the open-source Crazyflie 2.1 [4] is flown in an indoor environment.",
"A schematic of the experimental setup is depicted on the left of Fig.",
"REF .",
"The Crazyflie is commanded to follow a circular trajectory with a radius of 0.5m and at a speed of 0.5m/s.",
"The 3-dimensional position measurements are obtained from a VICON motion capture system, while the accelerations are measured from a 3-axis accelerometer onboard the Crazyflie.",
"Given these position and acceleration measurements, the objectives of the state estimation algorithms are to generate accurate estimates of the velocities and to provide noise attenuation for the measurements.",
"The state is defined as $\\mathbf {x}:=[\\mathbf {p}^{\\top } \\dot{\\mathbf {p}}^{\\top } \\ddot{\\mathbf {p}}^{\\top }]^{\\top }$ , where $\\mathbf {p}$ is the position vector of the quadrotor.",
"The prior dynamics model in the KNODE model is represented as $\\mathbf {f}_{\\text{prior}}(\\mathbf {x}) = \\mathbf {A}\\mathbf {x}$ , where $\\mathbf {A} \\in \\mathbb {R}^{9\\times 9}$ maps the translational states to their derivatives.",
"Even though this is a linear map, the KNODE model also consists of a neural network, which makes the overall dynamics model nonlinear.",
"Ten sets of flight data are collected, with an approximate sampling rate of 100Hz.",
"Each of them has a duration of about 14 seconds.",
"The neural ODE used in this setup consists of 2 layers with 16 neurons in between and uses the hyperbolic tangent as the activation function.",
"A KF is implemented as the pre-training observer and only one out of the 10 datasets is used for training.",
"The performance of the state estimation algorithms on the remaining datasets ascertains the generalization ability of the learned model.",
"Training of the KNODE model is done over 150 epochs."
],
[
"RESULTS AND DISCUSSION",
"For both the cartpole and ground robot tasks described in Sections REF and REF , we ran 15 simulations for each task and method across different random seeds, with a total of 240 runs.",
"Statistics of the overall mean-squared errors (MSE) between the estimated and true states under KNODE-MHE and KNODE-UKF, against various baselines are summarized in Table REF .",
"The mean and standard deviation of the MSEs are denoted by $\\mu $ and $\\sigma $ respectively.",
"MHE and UKF are the nominal state estimation algorithms with no learning enhancements.",
"In other words, only the prior model $\\mathbf {f}_{\\text{prior}}$ is used as the dynamics model in these two frameworks.",
"To understand the effect of the residual dynamics, another set of experiments is conducted in which the dynamics of the true system $\\mathbf {f}$ is used as the dynamics model.",
"These results are shown under the rows, MHE (true dynamics) and UKF (true dynamics).",
"These can be interpreted as ideal baselines that are free from any errors induced by the learning procedure, and without any unknown dynamics.",
"The results under KNODE-MHE (full state) and KNODE-UKF (full state) are for the case in which full state measurements are available and used for training.",
"These results, in contrast with those under KNODE-MHE and KNODE-UKF, would ascertain the accuracy of the KNODE model learned under partial or indirect measurements.",
"As observed in Table REF , having a more accurate dynamics model significantly improves the accuracy of the state estimates, in both the mean and standard deviation.",
"This can be seen from the comparison between KNODE-MHE, KNODE-UKF and their non-learning counterparts, MHE and UKF.",
"The results imply that learning a dynamics model will improve accuracy of the state estimates, even with partial or indirect measurements.",
"Also shown in the table, in the case where full state measurements are available, a better model can be obtained and we can expect improvements over the case where partial or indirect measurements are used for learning.",
"It is also observed that the results for KNODE-MHE (full state) and MHE (true dynamics) in the cartpole system are comparable, within $0.3\\%$ .",
"This implies that in this setting, the accuracy of the KNODE model learned under full state measurements is close to that of the true dynamics model.",
"Another observation is that for both estimation tasks, MHE performs better than UKF, when the residual dynamics are present, but not accounted for.",
"Table: MSE Statistics under 2 State Estimation Tasks.Figure: Left: Experimental setup for the Crazyflie system.",
"Partial measurements of the state are obtained from the motion capture system and onboard sensors.",
"Right: Statistics of the overall MSE computed over the translational states of the Crazyflie, and across 10 sets of flight data.",
"The orange lines indicate the medians for each method.As described in Section REF , to ascertain the practicality and efficacy of LEARNEST on real-world data, we apply the framework to 10 sets of flight data collected with the Crazyflie quadrotor.",
"A Savitzky–Golay filter [36] with a window length of 150 and a polynomial of order 2 is applied on the accurate position measurements from VICON to compute velocity and acceleration baselines to compare against the state estimates.",
"The statistics obtained under KNODE-MHE and KNODE-UKF and comparisons with MHE and UKF are depicted on the right of Fig.",
"REF .",
"By computing the median of the overall MSE, it is observed that KNODE-UKF and KNODE-MHE outperforms their non-learning counterparts by 26.6% and 11.9% respectively.",
"It is also observed that MHE generally outperforms UKF, and this trend is similar to that from the results of the other state estimation tasks."
],
[
"CONCLUSION AND FUTURE WORK",
"In this work, we present LEARNEST, a versatile framework in which we learn and integrate a knowledge-based, data-driven KNODE model into two state estimation algorithms, MHE and UKF.",
"Simulation results and experiments with real-world data across various applications show that the KNODE-MHE and KNODE-UKF algorithms outperform their non-learning counterparts.",
"This demonstrates the effectiveness of LEARNEST and validates the accuracy of the learning-enhanced state estimation algorithms.",
"In future work, we plan to apply this framework on other model-based state estimation algorithms such as the particle filter."
]
] | 2209.08185 |
[
[
"Learning Distinct and Representative Modes for Image Captioning"
],
[
"Abstract Over the years, state-of-the-art (SoTA) image captioning methods have achieved promising results on some evaluation metrics (e.g., CIDEr).",
"However, recent findings show that the captions generated by these methods tend to be biased toward the \"average\" caption that only captures the most general mode (a.k.a, language pattern) in the training corpus, i.e., the so-called mode collapse problem.",
"Affected by it, the generated captions are limited in diversity and usually less informative than natural image descriptions made by humans.",
"In this paper, we seek to avoid this problem by proposing a Discrete Mode Learning (DML) paradigm for image captioning.",
"Our innovative idea is to explore the rich modes in the training caption corpus to learn a set of \"mode embeddings\", and further use them to control the mode of the generated captions for existing image captioning models.",
"Specifically, the proposed DML optimizes a dual architecture that consists of an image-conditioned discrete variational autoencoder (CdVAE) branch and a mode-conditioned image captioning (MIC) branch.",
"The CdVAE branch maps each image caption to one of the mode embeddings stored in a learned codebook, and is trained with a pure non-autoregressive generation objective to make the modes distinct and representative.",
"The MIC branch can be simply modified from an existing image captioning model, where the mode embedding is added to the original word embeddings as the control signal.",
"In the experiments, we apply the proposed DML to two widely used image captioning models, Transformer and AoANet.",
"The results show that the learned mode embedding successfully facilitates these models to generate high-quality image captions with different modes, further leading to better performance for both diversity and quality on the MSCOCO dataset."
],
[
"Introduction",
"Image captioning aims to generate natural descriptions for a given image.",
"It is widely used in many real-world applications such as human-computer interaction, multi-modal recommendation, and hence has attracted lots of research attention.",
"Recently, many state-of-the-art (SoTA) methods [19], [24], [54] have achieved promising results w.r.t.",
"evaluation metrics like CIDEr [44], BLEU [36], and SPICE [1].",
"However, as discussed in [49], focusing on achieving higher scores on these metrics usually biases the image captioning models towards using only the common words, phrases and language patterns in the training corpus when describing the images (see Figure REF for an example).",
"In other words, the model automatically finds the most general mode to perform captioning.",
"As a result, the generated captions are limited in diversity both semantically and syntactically.",
"This is far away from the ability of human beings as humans are able to describe the image in various ways.",
"The cause of this phenomenon is widely known as the mode collapse problem and has been discussed in many prior works on generative modeling [16], [53].",
"Formally, given an input ${x}$ and a generative model $\\mathcal {G}$ , mode collapse appears when the estimated output distribution $P_{\\mathcal {G}}({x})$ assigns most of its probability mass to a small region of the output space, despite that the real data distribution $P_{\\text{data}}({y})$ has a much larger variance.",
"As a consequence, the randomly sampled outputs $\\lbrace {y}_i \\sim P_{\\mathcal {G}}({x})~|~i \\in \\mathbb {N}\\rbrace $ tend to be very similar.",
"While mode collapse is typically a side effect for generative modeling, it is somewhat “welcomed” in SoTA image captioning models as it usually facilitates a higher evaluation performance on reference-based metrics like CIDEr, BLEU and SPICE.",
"For example, CIDEr optimization [39] based methods [11], [19] have significantly pushed the performance of image captioning to a new level on mainstream reference-based evaluation metrics.",
"However, as shown in [29], [49], the success of CIDEr optimization could be largely attributed to its ability of reducing the modes in the generated captions.",
"Some recent researches in image captioning have attempted to tackle the mode collapse problem so as to improve the diversity of the generated captions.",
"Specifically, [3], [31], [47] adopt conditional variational autoencoders (CVAE) to encode the conditional distribution of the image captions into a low-dimensional continuous latent space $\\mathcal {Z}$ .",
"When performing inference, these methods randomly sample latent variables from $\\mathcal {Z}$ and input them into the caption decoder to drive it towards using different language patterns to describe the image.",
"In this sense, the sampled latent variables can be considered as the continuous representations of modes interpolated from all occurred modes during training, thus alleviating the mode collapse phenomena to some extent.",
"Nevertheless, it is still difficult to interpret the underlying conditional distribution, i.e., which part of $\\mathcal {Z}$ corresponds to which kind of language patterns in the real world.",
"Thus, for the captions derived from two sampled mode representations, it is uncertain how they differ from each other during the sampling process, as well as their qualities.",
"To tackle this uncertainty, another line of works seeks to learn controllable image captioning models.",
"For example, [9], [41] control the style of the image captions with learned sentiment or personality representations like “factual” or “humorous”, “positive” or “negative”, etc.",
"[14] controls the syntactic structure of the image captions through part-of-speech tags.",
"[10] focuses on different image regions to generate region-specific image captions.",
"However, these methods rely on additional tools or annotations to supervise the learning of modes.",
"More critically, this also restricts their modes within a pre-defined domain.",
"Figure: Captions written by humans vs. captions generated by existing models (Transformer and AoANet ), and by our DML.The model-generated captions prefer to use common words or phrases while captions derived from humans are more informative and diverse.The proposed DML generates diverse image captions based on different mode embeddings, and some of the modes tend to yield certain language patterns.E.g., mode-7 is likely to generate captions with the pattern of “There is ...”, mode-3 tends to produce complex sentences while mode-58 is prone to brief sentences.In this paper, we tackle the above problems by learning distinct and representative modes for image captioning through a new Discrete Mode Learning (DML) paradigm.",
"Different from CVAE-based diverse image captioning methods, DML learns a codebook, which is an embedding matrix consisting of a set of mode embeddings that spans a discrete latent space, thus the language patterns encoded in different modes can be easily evaluated and are more perceptible.",
"Different from previous controllable image captioning methods, DML requires no additional supervision for mode and hence is more convenient to use and is not limited to the pre-defined set of control signals.",
"Specifically, we optimize a dual architecture in the proposed DML: Firstly, an image-conditioned discrete variational autoencoder (CdVAE) branch, where an encoder extracts the hidden states for all the reference captions paired with an image, and further quantizes them using their matched mode embeddings in the codebook according to Euclidean distance.",
"For the matching algorithm, we choose Hungarian algorithm instead of the naive nearest neighbor look-up, which we find to be critical for increasing the number of effective modes.",
"Afterward, a decoder is used to reconstruct the reference captions in a fully non-autoregressive manner, which breaks the sequential dependencies on previous tokens and enforces the decoder to rely purely on the mode embeddings to generate different captions based on the same image feature, leading to more distinct and representative mode embeddings.",
"Secondly, a mode-conditioned image captioning (MIC) branch, which can be simply modified from an existing image captioning model by adding the mode embedding to the original word embeddings as a control signal.",
"During inference, the CdVAE branch is dropped, and the MIC branch is used to generate image captions with various language patterns according to the mode embeddings in the codebook.",
"In the experiments, we evaluate the effectiveness of our proposed DML paradigm by applying it to the widely-used Transformer [43] and the state-of-the-art AoANet [19], denoted by Transformer-DML and AoANet-DML, respectively.",
"We observe that some of the modes tend to yield clear language patterns in the generated captions, as shown in Figure REF , demonstrating that DML has successfully learned perceptible modes without explicit supervision.",
"We also find that our models perform surprisingly well under diversity evaluation (using metrics like SelfCIDEr [49]) and oracle performance evaluation (on mainstream reference-based metrics like CIDEr [44]), achieving new state-of-the-art results.",
"This shows that our learned modes are not only distinct but also effectively cover the rich modes that appeared within the dataset, which suggests that they are very representative.",
"Moreover, we find that the models trained with DML outperform their original counterparts in terms of quality on some of the modes, meaning that DML can serve as a cost-free plugin for existing image captioning models."
],
[
"Related Works",
"Image captioning.",
"Image captioning [2], [19], [46], [50] seeks to generate descriptions based on the given images, which has received lots of attention from the researchers [22].",
"The conventional paradigm of image captioning models [15], [20], [46] mainly consists of two parts: a CNN-based image encoder and an RNN-based decoder for caption generation.",
"Based on this diagram, many works [2], [19], [54] introduce the attention mechanism [43], which enforces models to consider more about the highlighted regions.",
"Besides, to improve the performance, [51], [52] explore the visual relationships by constructing a semantic or scene graph, while [17], [39] optimize their models by Reinforcement Learning (RL) and directly use CIDEr [44] to compute the reward.",
"However, these image captioning models mainly consider how to achieve higher evaluation scores, which usually bias the generated captions to an “average” version that contains the common words and phrases in the training corpus only.",
"Diverse and controllable image captioning.",
"Diverse image captioning aims at learning a model that can generate various captions based on the same image.",
"To this end, CVAE-based models [6], [31], [47] learn a latent space during training and then generate diverse captions by sampling different priors from the latent space.",
"GAN-based models [12], [23], [40] predict diverse captions by using different random noises as inputs accompanied with the given images.",
"Although the diverse image captioning models are able to produce different captions during inference, it is still non-trivial to interpret the underlying conditional distribution, leading to an uncertainty of the model behavior.",
"To make the generated captions controllable, some works [7], [8], [10], [13], [33], [34] introduce an additional control signal.",
"For example, Mathews et al.",
"[33] provide a sentiment signal for each caption, and seek to control the sentiment of the generated captions.",
"Deng & Ding [13] take the length of the caption as a control signal.",
"Conditioned on different length level embeddings, the model is able to generate length-controllable descriptions for the input image.",
"However, most of these methods have to rely on additional tools or annotations to supervise the learning of the control signal, which is usually inconvenient to collect and further limits the language pattern within a pre-defined domain."
],
[
"Method",
"The overall architecture of the proposed Discrete Mode Learning (DML) paradigm is illustrated in Figure REF .",
"It consists of two branches: an image-conditioned discrete variational autoencoder (CdVAE) branch, which contains a mode encoder $\\mathcal {E}_m$ and a masked decoder $\\mathcal {D}_m$ ; and a mode-conditioned image captioning (MIC) branch consists of an image encoder $\\mathcal {E}_c$ and a caption decoder $\\mathcal {D}_c$ .",
"The two branches are connected through a codebook $\\Omega $ that contains a set of mode embeddings learned through the DML paradigm.",
"During training, the model takes as inputs an image $x$ and its paired reference captions $\\lbrace y_i\\rbrace _{i=1}^n$ , and no additional supervision is required.",
"During inference, the CdVAE branch is dropped, and the MIC branch is used to generate image captions with various language patterns according to the mode embeddings in the codebook.",
"More details are as follows."
],
[
"Discrete Mode Learning",
"In a typical image captioning model, the training objective is to maximize $\\log p(y_i|x)$ for each $y_i$ in $\\lbrace y_i\\rbrace _{i=1}^n$ , which is ill-posed since the model is optimized to approach multiple different targets conditioned on the same input $x$ , and usually leads to a mode collapse problem.",
"Previous works alleviate this problem by introducing a latent variable $z_i$ to the objective function, i.e., $\\mathbb {E}_{z_i \\sim p(z|x)}[\\log p(y_i|x, z_i)]$ , which serves as an explicit indication of mode and drives the model to generate different targets conditioning on different $z_i$ .",
"E.g., in CVAE-based diverse image captioning models [31], [47], $z_i$ is randomly sampled from a continuous latent space, which leads to an uncertainty of the model behavior.",
"While in some controllable image captioning models [10], [14], $z_i$ needs to be pre-defined with the help of additional tools or annotations, which is usually restricted to a small set.",
"Unlike these methods, our DML samples latent variables from an embedding matrix $\\Omega \\in \\mathbb {R}^{k\\times d}$ which we call codebook.",
"It defines a discrete latent space, where each entry in $\\Omega $ corresponds to a potential mode embedding, and $k$ is a hyper-parameter representing the total number of modes in the codebook.",
"During training, the mode encoder $\\mathcal {E}_m$ is used to extract the representation for each caption $y_i$ , denoted by $e(y_i)$ , and match the representation with one of the entries in $\\Omega $ .",
"The matched entry is then adopted as the mode embedding of $y_i$ , denoted by $q(y_i)$ , and the objective becomes: $\\max _{\\mathcal {E}_m,\\mathcal {D}_m,\\Omega }\\log p(y_i|q(y_i), x), ~~\\text{where}~q(y_i) = \\text{\\texttt {Match}}(e(y_i), \\Omega ).$ Here, $p(y_i|q(y_i), x)$ is the conditional distribution of the caption $y_i$ given its mode embedding $q(y_i)$ and the paired image $x$ , which is approximated with the masked decoder $\\mathcal {D}_m$ .",
"$\\texttt {Match}(\\cdot ,\\cdot )$ indicates the matching operation.",
"Note that, there is usually no real gradient defined for typical matching operators like the nearest neighbor look-up.",
"Therefore, we follow [42] to first use the straight-through estimator to directly copy the gradients from $q(y_i)$ to $e(y_i)$ , and then apply Vector Quantization algorithm to move $q(y_i)$ towards $e(y_i)$ through mean square loss so as to train the matched codebook entries.",
"The new objective function is $-\\log p(y_i|q(y_i), x) + \\Vert \\mathrm {sg}[e(y_i)]-q(y_i)\\Vert _2^2 + \\beta \\Vert e(y_i)-\\mathrm {sg}[q(y_i)]\\Vert _2^2,$ where the first term inherited from Eq.",
"(REF ) is used to optimize $\\mathcal {E}_m$ and $\\mathcal {D}_m$ .",
"The second term is used to train the matched entries in $\\Omega $ .",
"The last term is a commitment loss that is responsible for bounding the embedding space of $\\Omega $ .",
"$\\mathrm {sg}[\\cdot ]$ refers to “stop gradient” and $\\beta $ is a hyper-parameter."
],
[
"Mode Encoding and Assignment",
"We adopt a stack of $N_{e}$ transformer encoder layers as our mode encoder $\\mathcal {E}_m$ .",
"The input of $\\mathcal {E}_m$ is $y_i$ appended with a special [MODE] token to the start.",
"The output of $\\mathcal {E}_m$ , $e(y_i) \\in \\mathbb {R}^{d}$ , is the hidden state of the [MODE] token from the last layer.",
"When performing the matching process in Eq.",
"(REF ), a naive approach is to use the nearest neighbor look-up algorithm, i.e., $q(y_i) = \\Omega _\\iota , ~~\\text{where}~\\iota = \\arg \\min _j \\Vert e(y_i) - \\Omega _j\\Vert _2.$ However, we find in the experiments that in the models trained with this assignment strategy, the output of $\\mathcal {E}_m$ quickly converges to the nearby of two or three mode embeddings in $\\Omega $ , which is a clear sign of mode collapse (see Figure REF ).",
"To avoid this, inspired by the object query assignment in [5], we treat the mode assignment of the captions as a Bipartite Graph Matching problem and solve it using Hungarian algorithm.",
"Specifically, for all the reference captions $\\lbrace y_i\\rbrace _{i=1}^n$ paired with an image, we construct a bipartite graph between the output hidden states $\\lbrace e(y_i)\\rbrace _{i=1}^n$ from $\\mathcal {E}_m$ and the mode embeddings in $\\Omega $ .",
"We first pad $\\lbrace e_i\\rbrace _{i=1}^n$ to the size of the codebook, $k$ , with $\\varnothing $ (assume $n < k$ ).",
"Then, we search for a permutation of $k$ elements $\\tau \\in \\mathfrak {S}_k$ with the lowest assignment cost: $\\hat{\\tau } = \\operatornamewithlimits{arg\\,min}_{\\tau \\in \\mathfrak {S}_k}\\sum _i^{k}\\Vert e(y_i) - \\Omega _{\\tau (i)}\\Vert _2,$ and the assigned mode embedding for each $y_i$ is $q(y_i) = \\Omega _{\\hat{\\tau }(i)}$ .",
"We find this assignment strategy greatly increases the number of effective mode embeddings in $\\Omega $ (see Figure REF )."
],
[
"Fully Non-autoregressive Decoder",
"After getting the mode embeddings $q(y_i)$ for each caption $y_i$ , we feed it together with the image features from $x$ into a decoder to generate $y_i$ and estimate the conditional distribution $p(y_i|q(y_i), x)$ in Eq.",
"(REF ).",
"In general, sequence generation models are often trained through the autoregressive Teacher Forcing scheme, which aims to maximize the likelihood of the ground-truth token $w_t$ given all preceding ground-truth tokens $w_{j<t}$ .",
"In our setting, the objective function would be: $ -\\log p(y_i|q(y_i), x) = \\sum ^{T}_{t=1} -\\log p(w_t|w_{j<t}, q(y_i), x)$ where $w_t$ is the $t$ -th token in $y_i$ and $T$ is the total number of tokens in $y_i$ .",
"However, being able to access the previous tokens causes the decoder to ignore the mode embedding when reconstructing $y_i$ , since the previous tokens already provide enough information for the mode of $y_i$ .",
"As a result, the training signals for the mode embeddings could be useless.",
"Therefore, we propose to use a fully non-autoregressive (NAT) objective for the CdVAE branch of DML to facilitate the training of the mode embeddings.",
"Specifically, the masked decoder $\\mathcal {D}_m$ is a stack of $N_d$ transformer decoder layers that takes $q(y_i)$ , $x$ , and a sequence of $T$ [MASK] tokens as input, and predicts each target token in $y_i$ in a conditionally-independent manner, i.e., $ -\\log p(y_i|q(y_i), x) = \\sum ^{T}_{t=1} -\\log p(w_t|\\texttt {[MASK]}, q(y_i), x).$ Apply this equation to Eq.",
"(REF ) will result in our final objective function for the CdVAE branch.",
"We find this leads to more distinct and representative mode embeddings (see Figure REF )."
],
[
"Learning MIC Models with DML",
"So far we have introduced how DML learns the mode embeddings with the CdVAE branch.",
"Here we show how to attach the CdVAE branch to an existing image captioning model to make it aware of the mode of the generated captions.",
"Specifically, in a standard encoder-decoder-based image captioning model, the encoder $\\mathcal {E}_c$ encodes the information from the image $x$ into a sequence of hidden states, then the decoder $\\mathcal {D}_c$ attends to the encoder hidden states and predicts the ground-truth tokens in the reference caption $y_i$ following the Teach Forcing scheme similar to Eq.",
"(REF ).",
"To adapt it into our DML framework, we only need to make a minor modification to the token embedding layer of $\\mathcal {D}_c$ by adding the mode embedding $q(y_i)$ to the word embeddings of each token in $y_i$ (i.e., $w_{j<t}$ in Eq.",
"(REF )) element-wisely, resulting in our MIC branch.",
"The training of MIC and CdVAE is performed jointly, using the objective functions in Eq.",
"(REF ) and Eq.",
"(REF ), respectively.",
"In practice, we find the CdVAE branch is much harder to train than the MIC branch, due to its non-autoregressive prediction manner.",
"To make the convergence speed of the two branches compatible, we let them use different batch sizes, i.e., for each image, the CdVAE branch takes as input all its paired captions $\\lbrace y_i\\rbrace _{i=1}^n$ , while the MIC branch randomly samples just one caption from $\\lbrace y_i\\rbrace _{i=1}^n$ .",
"Moreover, the gradient from the CdVAE branch will not be back-propagated to the image encoder.",
"Inference.",
"During inference, the CdVAE branch is dropped, and the inference of MIC follows a very similar procedure as the original image captioning model.",
"The only difference is that it requires selecting a mode embedding from the codebook and adding it to the input token embeddings."
],
[
"Dataset and Evaluation Metrics",
"Dataset.",
"We train and evaluate our method on MSCOCO dataset [26] that contains $123,287$ images and each image is corresponding to at least 5 captions.",
"blackFor a fair comparison, we follow the previous works [30], [31] in the area of diverse and controllable image captioning to use the m-RNN split [32] of the COCO dataset, which divides the data into $118,287$ , $4,000$ and $1,000$ for training, validation and testing, respectively.",
"Quality evaluation.",
"To assess the quality of the generated captions, we use five widely used evaluation metrics, i.e., BLEU [36], ROUGE [25], METEOR [4], CIDEr [44], and SPICE [1].",
"Moreover, to further evaluate the performance of the generated captions, we employ another CLIP-based [37] metric namely ClipScore [18], which can assess whether the generated captions are semantically aligned with given images, even when they are totally different from the reference captions.",
"Diversity evaluation.",
"To investigate the diversity of the generated captions, we use SelfCIDEr [48], mBLEU, and n-gram diversity (i.e., Div-$n$ [3]).",
"All of these metrics evaluate the diversity by comparing the $n$ -gram differences among the generated captions that belong to the same image."
],
[
"Implementation Details",
"Choice of base models.",
"blackThe proposed DML is a general learning paradigm and we expect it can be easily applied in many existing image captioning models and improve their controllability and diversity.",
"Thus, for the MIC branch, we choose two widely-used and representative architectures as our base models, i.e., Transformer [43] and AoANet [19], denoted by Transformer-DML and AoANet-DML separately, to show the generalization ability of DML.",
"Most current SoTA image captioning models, like M2Transformer [11], XLAN [35] and many vision-language pre-training models, are based on the Transformer architecture.",
"Moreover, the performance of Transformer and AoANet is also competitive with the SoTA models [28] when vision-language pre-training is not performed.",
"Therefore, they are good baseline models to illustrate the performance of our DML paradigm.",
"Detailed settings.",
"In the CdVAE branch, the number of transformer layers for $\\mathcal {E}_m$ and $\\mathcal {D}_m$ is set to 6 and 2, respectively.",
"We use 12 attention heads and a hidden size of 768 for all transformer layers.",
"$\\beta $ in Eq.",
"(REF ) is set to 0.25, following [42].",
"The number of mode embeddings in $\\Omega $ is set to 64 by default.",
"We convert each image to 100 object proposals by Faster RCNN [38] pre-trained on Visual Genome [21].",
"We train the models for 100,000 iterations with a batch of 64 images and all the paired captions.",
"We use AdamW [27] optimizer with a learning rate of 2e-4, and cosine decay it to 0.",
"We use the label smoothing of 0.1 and the gradient clipping threshold of 1.0.",
"We train Transformer-DML and AoANet-DML on one NVIDIA 3090 GPU with about 10 and 13 GPU hours, respectively."
],
[
"Evaluation on Diversity",
"Quantitative results.",
"In this part, we perform a diversity analysis for the proposed DML paradigm based on Transformer-DML.",
"We compare our model with previous SoTA methods as well as Beam Search (BS) and show the results in Table REF .",
"From the table, our DML-based model achieves higher performance on most of the diversity evaluation metrics, which demonstrates its effectiveness in learning distinct modes.",
"Table: Diversity evaluation on the best-5 sentences obtained from consensus re-ranking.Qualitative results.",
"We further provide several samples of the generated captions of our Transformer-DML model in Figure REF .",
"We find that the learned modes exhibit clear and distinct language patterns.",
"More specifically, mode-7, mode-32 and mode-43 tend to follow the patterns “There is ...”, “Man ...”, and “A close up of ...”, respectively.",
"Mode-3 and mode-58 focus more on the semantic complexity of the caption, i.e., mode-58 is likely to generate a brief caption while mode-3 is prone to the complicated one.",
"These results show that for the same image, the mode embeddings learned through DML facilitate the model to generate captions in various and comprehensible ways.",
"Figure: Samples of captions that are generated based on different modes.",
"Mode-7, mode-32 and mode-43 tend to generate the captions with a certain pattern, i.e., “There is ...”, “Man ...”, and “A close up of ...”.",
"Mode-3 is apt to use long sentences while mode-58 is prone to brief sentences."
],
[
"Evaluation on Quality",
"Oracle results.",
"To investigate whether the embeddings in the codebook are able to cover the rich modes that appeared within the dataset, we calculate the caption evaluation metrics in the oracle setting consistent with prior works [3], [30], [31], i.e., taking the maximum score for each quality metric over all the candidate captions for each image.",
"Specifically, we train Transformer-DML and AoANet-DML with codebook sizes $k=20$ and $k=100$ , and evaluate the oracle results of the captions generated by all modes.",
"In Table REF , the models with DML obtain the best and second best results on all the evaluation metrics w.r.t.",
"both 20 and 100 samples.",
"Moreover, we compare our DML paradigm with the SoTA baseline COS-CVAE by calculating the oracle scores of CIDEr, SPICE and METEOR with different numbers of samples.",
"In Figure REF , the proposed DML consistently outperforms COS-CVAE on all settings.",
"The high gain in quality metrics demonstrates that the proposed DML successfully captures the rich and representative modes in the training corpus.",
"Table: Comparison with baselines w.r.t.",
"oracle performance (i.e., best-1 quality) on COCO dataset.",
"“#Sample” refers to the number of generated captions for each image.",
"The best and the second best results are highlighted with bold and underline, respectively.Figure: METEORResults of individual mode embedding.",
"To assess the quality of captions generated using different modes, we calculate the evaluation scores of captions generated from some representative modes that are manually selected with distinct patterns.",
"In Table REF , on mode-58, Transformer-DML outperforms the original Transformer on all reference-based metrics.",
"Moreover, the Transformer-DML with different mode embeddings can achieve better or at least competitive performance compared with the original Transformer in terms of ClipScore.",
"We further report the best-performed modes w.r.t.",
"CIDEr, SPICE and ClipScore for AoANet-DML.",
"In Table REF , the results show that the proposed DML can gain higher scores of CIDEr, SPICE and ClipScore by choosing the suitable mode embedding.",
"These results suggest that our DML not only generates diverse results, but the results of each individual mode also yield high quality.",
"Table: The performance of image captions generated by different modes.",
"For Transformer-DML, we select some representative modes that exhibit clear language patterns or semantic complexities as shown in Figure .For AoANet-DML, we show the best-performed modes in terms of CIDEr, SPICE and ClipScore, respectively (highlighted with underline).The original results of Transformer and AoANet are obtained using the code and settings in .The best results of the two captioning methods on each metric are highlighted with bold.Further comparison with COS-CVAE.",
"blackTo further evaluate the effectiveness of our DML paradigm, we run experiments using the UpDown [2] model (a two-layer LSTM with a visual attention module) for our MIC branch, which is also the same language generation model used by COS-CVAE.",
"The oracle performance of this model is 1.688 and 1.942 in terms of CIDEr for 20 and 100 samples, respectively, which still outperforms the COS-CVAE by a large margin.",
"In fact, UpDown is a strong model that achieves compatible performance with a 6-layer Transformer model in a general image captioning setting (1.099 CIDEr vs. 1.114 CIDEr on Karpathy’s test split [20]), which means that two-layer LSTMs may already have enough capacity for the COCO dataset.",
"Moreover, considering that COS-CVAE requires a pre-processing step to construct pseudo supervisions with the help of a pretrained joint vision-language embedding model, the proposed end-to-end learning method could be more convenient to use than COS-CVAE."
],
[
"Performance Analysis of the CdVAE branch",
"In this part, we investigate the effect of our mode assignment strategy (Section REF ) and the fully non-autoregressive (NAT) objective (Section REF ) used by the CdVAE branch of DML.",
"Specifically, we train Transformer models with a codebook size of $k=64$ on three different settings: 1) DML w/o NAT objective; 2) DML w/o Hungarian assignment; 3) the proposed DML, and visualize their mode embeddings in the codebook and caption embeddings from the mode encoder output using t-SNE.",
"As shown in Figure REF (a), DML w/o NAT only activates 5 of 64 mode embeddings, i.e., all caption embeddings are assigned to the 5 modes.",
"Moreover, the caption embeddings of different modes are mixed together (zoom in to see the details) and are distributed among the twisted spaces nearby the mode embeddings, which means that the mode embeddings and the caption embeddings may not contain useful information.",
"The low oracle CIDEr score (1.207) also verifies this hypothesis.",
"In Figure REF (b), we show that DML w/o Hungarian assignment activates even fewer modes (i.e., only three), but the caption embeddings now have a more distinct distribution compared with the result in Figure REF (a).",
"This shows the importance of the fully NAT objective in CdVAE.",
"Still, due to the small number of modes, the oracle CIDEr score in this setting is only 1.211, indicating a limited diversity.",
"Lastly, in Figure REF (c), our DML obtains 29 effective mode embeddings, and the caption embeddings are tightly distributed around their corresponding mode embeddings.",
"Moreover, our model achieves an oracle CIDEr of 1.871.",
"These results demonstrate the effectiveness of the Hungarian mode assignment and the fully non-autoregressive objective for learning distinct and representative mode embeddings.",
"Figure: DML"
],
[
"Ablation Studies on the CdVAE branch",
"In this section, we provide more analysis on the CdVAE branch of the proposed DML method.",
"Our experiments are based on the Transformer-DML model with a codebook size of 64."
],
[
"Asymmetric batch size",
"As we mentioned in Section REF , the training of the CdVAE branch is much harder than the training of the MIC branch.",
"Thus, we adopt asymmetric batch sizes when training them jointly.",
"Moreover, the gradient from the CdVAE branch will not be back-propagated to the image encoder, so that the whole MIC branch is trained with a batch size $n$ times smaller than that of the CdVAE branch.",
"Here, we show how this strategy benefits the training of our Transformer-DML model.",
"Specifically, during training, we change the number of sampled captions for the MIC branch as well as the total batch size, and the rest settings are the same as in Section REF .",
"The results are shown in Table REF .",
"Table: Oracle results of different batch sizes for CdVAE branch and MIC branch.",
"“#effective modes” indicates the total number of modes that have ever been used during the whole training process.",
"“#sampled caps” refers to how many captions per image are sampled to optimize the MIC branch.The first row in Table REF is the performance of our default setting, which yields the best results among all metrics, and also obtains a larger number of effective modes than other settings.",
"When increasing the number of sampled captions from 1 to 5 for the MIC branch, the performance of Transformer-DML drops clearly.",
"We believe this is due to the over-fitting of the MIC branch caused by the large batch size.",
"When decreasing the total batch size from 64 to 16, we observe a clear reduction in the number of effective modes, which means the CdVAE branch is not sufficiently trained.",
"Moreover, with such a small batch size, the MIC branch will also suffer from under-fitting if the number of sampled captions is small, i.e., oracle CIDEr drops from 1.799 to 1.760 when reducing the number of sampled captions from 5 to 2.",
"These results show the importance of balancing the training of the two branches in the proposed DML."
],
[
"Masking strategy of the masked decoder $\\mathcal {D}_m$",
"When training the CdVAE branch, we use a fully non-autoregressive objective, where the input of the masked decoder $\\mathcal {D}_m$ are all [MASK] tokens.",
"This prevents the model from using the mode information leaked from the ground-truth tokens, and we find it greatly benefits the learning of modes.",
"Here we evaluate the performance of several different masking strategies, including random masking the input caption using a fixed probability, and random masking with a dynamically changed probability.",
"The results are shown in Table REF .",
"Table: Oracle results of different masking strategies for CdVAE branch.",
"“#effective modes” is the total number of modes that have ever been used during the whole training process.",
"“1.0 →\\rightarrow 0.0” means we gradually reduce the masking probability from 1 to 0 throughout the training process.From the table, the full masking strategy, i.e., a mask probability of 1.0, leads to the best performance, which is also the default setting in DML.",
"Moreover, introducing any additional information to the CdVAE branch, i.e., a mask probability less than 1.0, will severely hamper the learning of modes, where the number of effective modes drops clearly from 29 to less than 10, and the oracle performances also drop significantly."
],
[
"Conclusion",
"In this paper, we study a problem in image captioning that models tend to be biased to generate an “average” caption, which contains the common words or phases only.",
"To tackle this problem, we propose a Discrete Mode Learning (DML) paradigm for image captioning.",
"The idea is to explore multiple rich modes in the training caption corpus to learn a codebook that contains a set of “mode embeddings”, which enables the image captioning models to generate different captions based on various modes.",
"Moreover, the proposed DML paradigm can be easily plugged into the existing image captioning models (e.g., Transformer and AoANet) to generate high-quality and diverse captions."
],
[
"Limitation and Impacts",
"Limitation.",
"Although our method can generate diverse captions based on different mode embeddings in the codebook, the more specific meanings for some of the modes may still be unclear and require more professional analyses.",
"Potential negative societal impacts.",
"We were not able to find some potential negative impact for our image captioning model, but perhaps, the Discrete Mode Learning paradigm may be applied in pre-trained generative models using large-scale web data to learn specific modes for special communities, which may further cause gender/racial bias problems."
],
[
"Captions Generated by All Modes",
"We provide more samples of the generated captions of our method in Figures REF -REF .",
"The results are obtained from the Transformer-DML model trained on MSCOCO dataset.",
"The codebook size is 64.",
"The number of effective modes in this model, i.e., the total number of modes that have ever been used during the whole training process, is 29.",
"From the figures, the learned modes exhibit clear and distinct language patterns.",
"For example, almost all captions in mode-2 use the present progressive tense or passive voice; mode-6 tends to use exaggerated words when describing the image, e.g., “very cute ”, “pretty”; mode-7 tends to follow the pattern “There is ...” while mode-43 would like the pattern “An image of/A close up of ...”; mode-52 tends to describe the colors in the image; mode-57 uses “and” to combine multiple short phrases for most of the cases; the captions generated by mode-3 are usually very long and semantically rich, while on the contrary, the captions generated by mode-58 are always short and simple.",
"For convenience, we highlight these patterns in the figures.",
"Figure: Sample-1.",
"Modes marked in red mean the corresponding generated captions have certain and distinct language patterns.Figure: Sample-2.",
"Modes marked in red mean the corresponding generated captions have certain and distinct language patterns.Figure: Sample-3.",
"Modes marked in red mean the corresponding generated captions have certain and distinct language patterns."
]
] | 2209.08231 |
[
[
"Weakly Supervised Medical Image Segmentation With Soft Labels and Noise\n Robust Loss"
],
[
"Abstract Recent advances in deep learning algorithms have led to significant benefits for solving many medical image analysis problems.",
"Training deep learning models commonly requires large datasets with expert-labeled annotations.",
"However, acquiring expert-labeled annotation is not only expensive but also is subjective, error-prone, and inter-/intra- observer variability introduces noise to labels.",
"This is particularly a problem when using deep learning models for segmenting medical images due to the ambiguous anatomical boundaries.",
"Image-based medical diagnosis tools using deep learning models trained with incorrect segmentation labels can lead to false diagnoses and treatment suggestions.",
"Multi-rater annotations might be better suited to train deep learning models with small training sets compared to single-rater annotations.",
"The aim of this paper was to develop and evaluate a method to generate probabilistic labels based on multi-rater annotations and anatomical knowledge of the lesion features in MRI and a method to train segmentation models using probabilistic labels using normalized active-passive loss as a \"noise-tolerant loss\" function.",
"The model was evaluated by comparing it to binary ground truth for 17 knees MRI scans for clinical segmentation and detection of bone marrow lesions (BML).",
"The proposed method successfully improved precision 14, recall 22, and Dice score 8 percent compared to a binary cross-entropy loss function.",
"Overall, the results of this work suggest that the proposed normalized active-passive loss using soft labels successfully mitigated the effects of noisy labels."
],
[
"Introduction",
"Automatic or semi-automatic clinical features and biomarker measurements based on deep learning (DL) are useful for longitudinal assessment of medical images.",
"DL-based techniques using Convolutional Neural Networks (CNN) have shown great success in tissue and pathology detection and segmentation [1], [2].",
"The accuracy of deep learning methods is highly dependent on the quality of the training data and corresponding ground truth labels.",
"For sensitive applications like medical image segmentation, it is particularly important to use anatomically accurate labels for training of DL models.",
"However, acquiring “ground truth” (GT) labels in the medical domain can be challenging because pixel-wise labeling is expensive, subjective, and error-prone.",
"Inter-reader variability combined with the presence of ambiguous anatomical boundaries between tissues (e.g., due to partial volume effects) makes labels uncertain [3].",
"Furthermore, the presence of lesions and pathologies in unexpected locations increases the chance of inattentional blindness [4].",
"Consequently, medical image datasets are likely to contain sub-optimal, inaccurate and noisy labels [5].",
"A DL model trained on weak/noisy annotations may have biases and overfit to incorrect annotations [6].",
"Therefore, different approaches have been investigated to reduce the effects of noise and imperfect labels in medical image analysis, including label smoothing and label correction [7], [8].",
"Despite the efforts to develop noise-resistant learning approaches, many aspects have remained unexplored, particularly for medical image segmentation tasks.",
"Therefore, for medical image segmentation task, there is a high demand for robust and reliable methods for training DL models based on noisy and suboptimal labels (e.g., annotations that are not pixel-wise or partial labels).",
"In this work, we proposed to train an instance segmentation model with soft labels obtained from noisy/partial region of interest (ROI) labels using a noise resistance loss function.",
"We assume that partial ROI labels as highly noisy labels and attempt to perform weakly supervised instance segmentation under this assumption.",
"Our key contributions are: Exploration of training of an improved version of MaskRCNN [1] using probabilistic partial labels.",
"Creation of probabilistic labels based on multi-rater scoring, MRI spatial redundancy, and tissue/lesion characteristics, as well as considering image contextual information.",
"Proposal of combining a noise-tolerant loss function (active passive loss) and soft ground truth for training DL models.",
"As a practical application, we focus on bone marrow lesion (BML) detection and segmentation, which is one of the inflammatory components of osteoarthritis (OA).Accurate quantification of features related to OA inflammation can provide a basis for effective clinical management and a target for therapy [9].",
"This task is challenging as BMLs do not have distinctive edges and it is challenging to create binary labels.",
"Furthermore, it is hard to generate precise clean annotations for BMLs, since they may appear in multiple locations, and humans are susceptible to inattentional blindness (missing BMLs in plain sight) [4].",
"To mitigate the effect of noisy labels, different approaches like label correction, noise-robust loss, robust regularization, and loss correction have been deployed, which are briefly described in the following.",
"Label correction aims to improve the quality of raw labels.",
"Different methods have been proposed to estimate label noise distribution, correct corrupted labels [10], or separate the noise from the data using properties of learned representations [11], [12].",
"Even though the label correction methods are effective, they usually require additional precise annotated data or an expensive process of noise detection and correction [13], [14]."
],
[
"Noise-robust loss",
"One approach for noise-robust learning is using loss functions that are inherently noise-tolerant or losses that created by using regularization terms or modifying well-known loss functions to make them noise-tolerant.",
"Ghosh et al.",
"theoretically proved that symmetric losses perform significantly better in case of learning with noisy labels [15].",
"The Mean Absolute Error (MAE) loss is symmetric and noise-tolerant for uniform noise and class-conditional label noise and satisfies the symmetry condition.",
"In contrast to that, cross-entropy (CE) is not symmetric and does not perform well in the presence of noise [15].",
"Therefore, training with CE in addition to complementary losses robust to noise to achieve learning sufficiency and robustness has been suggested [16], [17].",
"Wang et al.",
"demonstrated that the Reverse Cross-Entropy (RCE) [15] loss, which is robust to label noise, can be used as the complementary robust loss term [16].",
"They proposed the so-called Symmetric Cross-Entropy (SCE) loss, which is defined as $l_{SCE}=\\alpha l_{ce} +\\beta l_{Rrce}$ , where $l_{ce}$ is the CE loss and $l_{Rrce}$ is the RCE loss.",
"Using the SCE idea, Ma et al.",
"recently addressed the learning under label noise by characterizing existing loss functions into two types Active vs.",
"Passive [13].",
"Based on their characterization the active loss is a loss that maximizes $p(k=y|x)$ and the passive loss is a loss that minimizes $p(k\\ne y|x)$ .",
"Then they proposed the Active and Passive Loss (APL), which combines an active loss and a passive loss in order to maximize the probability of belonging to a given class and minimize the likelihood of belonging to another class [13].",
"The APL loss was shown to be noise-tolerant if both active loss and passive loss have been chosen from noise-tolerant losses [13].",
"In the same work, Ma et al.",
"proved that any loss can be noise-tolerant if normalization is applied to it [13].",
"Their results showed that APL addresses the underfitting problem and can leverage both robustness and convergence advantages."
],
[
"Robust regularization",
"Regularization methods have been widely used to increase robustness of deep learning models against label noise by preventing overfitting.",
"These methods perform well in the presence of moderate noise, and are mostly used in combination with other techniques [18].",
"Different regularization techniques include explicit regularization like weight decay and dropout or implicit regularization like label smoothing."
],
[
"Loss correction",
"Most of loss correction methods estimate noise-transition matrices to adjust the labels during training.",
"These techniques aim to minimize global risk with respect to the probability distribution.",
"The backward and forward correction involves modifying two losses based on the noise transition matrix [19].",
"Some loss correction methods assume the availability of trusted data with clean labels for validation and to estimate the matrix of corruption probabilities [20] or they may require complex modifications to the training process.",
"Recently, Lukasik et al.",
"[21] have shown that label smoothing is related to loss-correction techniques and is effective in coping with label noise by assuming smoothing as a form of shrinkage regularization.",
"To mitigate noise, Label smoothing may require a simple modification to the training process and does not require additional labels and offers loss correction benefits."
],
[
"Label smoothing and soft labels",
"Traditionally, the label ($y$ ) of each pixel is encoded binary (or as a one-hot vector).",
"In binary labeling, one instance belongs to either one or the other class.",
"This hard labeling assigns all probability mass to one class, resulting in large differences between the largest logit and the rest of the logits in networks using the sigmoid (or soft-max) activation function in the output layer [22].",
"Consequently, in applications such as medical imaging, in case of partial volume effects or disagreement between readers, hard labels may cause overfitting and reduce the adaptability of the network in these situations [22].",
"In contrast, soft labels encode the label of each instance as a real value probability, whose k-th entry represents $p(Y=k|X=x)\\in [0,1]$ .",
"For example, the soft label $x1=[0.3,0.7]$ indicates that $p(Y=1|X=x_i)=0.7$ , whereas hard labels only have 0/1 values.",
"Soft labels can provide additional information to the learning algorithm, which was shown to reduce the number of instances required to train a model [22].",
"Label smoothing techniques can be considered as utilization of probabilistic labels (soft labels).",
"These approaches have been shown to give a better generalization, faster learning speed, better calibration, and mitigation of network over-confidence [23].",
"Label smoothing has been deployed and examined in different applications including image classification [22], model network uncertainty [24], and recognition [25].",
"Probabilistic labels have been used to mitigate reader variability and human assessment noise for classification problems [26].",
"While methods based on uniform label smoothing techniques may improve the network calibration, they do not necessarily improve classification accuracy.",
"As a solution, Vega et al.",
"proposed to compute probabilistic labels from relevant features$(Z(X))$ in the raw images $(X)$ for a classification task [22].",
"In that work, a method is trained to estimate $p(Y|Z(X))$ , which is used as a probabilistic labels’ classification task.",
"This probabilistic labeling approach was shown to provide better calibration and improved classification accuracy than uniform label smoothing."
],
[
"Soft labels for medical image segmentation",
"Smoothing labels and probabilistic labels have only been investigated recently in segmentation tasks.",
"This is because earlier developed methods of label smoothing were originally proposed for image classification, in which the hard labels were flattened by assigning a uniform distribution overall to all classes to prevent model overconfidence.",
"However, this is likely problematic in segmentation tasks, since this approach assigns a probability greater than zero to pixels, even those that one can be confident about not belonging to a certain class (e.g., background outside the body).",
"Islam et al.",
"addressed this problem by proposing a Spatially Varying Label Smoothing (SVLS) to capture expert annotation uncertainty [7].",
"SVLS considers the likelihood with neighboring pixels for determining the probability of each class.",
"Li et al.",
"proposed super-pixel label softening to encounter descriptive contextual information for soft labeling.",
"Using super-pixel soft labels and KL (Kullback-Leibler divergence) loss improved Dice coefficient and volumetric similarity [27].",
"The high uncertainty in defining lesion borders was investigated by Kats et al.",
"who developed a model that uses soft labels (generated by morphological dilation of binary labels) with the soft-Dice loss to segment multiple-sclerosis lesions in MRI data to account for the high uncertainty in defining lesion borders [28].",
"However, further analysis revealed that the soft-Dice may introduce a volumetric bias for tasks with high inherent uncertainty [29].",
"Gross et al.",
"used soft labels obtained through data augmentation for the segmentation task [3].",
"They assumed that using soft labels is analogous to a regression problem, and they used the NormReLU activation and a regression loss function for medical image segmentation [3].",
"However, using NormReLU as the last layer has two main drawbacks.",
"First, it is not highly effective when the maximum is an outlier.",
"Second, it only uses one image to normalize ReLU, which may cause the algorithm to not converge to a good solution since a single image may not be a good representation of the entire data set distributions.",
"In some studies, soft labels were generated by fusing multiple manual annotations.",
"One of the best methods to obtain a consensus mask from multi-reader annotation is the Therefore, Kats et al.",
"proposed a soft version of the STAPLE algorithm [30], which showed superior results compared to morphological dilation used in the aforementioned soft Dice loss approach [28]."
],
[
"Method",
"Ma et al.",
"[13] and Lukasik et al.",
"[21] showed that a normalized loss function and label smoothing are both effective ways to mitigate noise effects.",
"In this paper, we are investigating whether the combination of label smoothing, and a normalized loss function leads to quantitative benefits."
],
[
"Problem Formulation",
"The aim of supervised learning for classification problems is to learn the function $f(x;\\theta )$ .",
"This function maps the input $x$ to the output $y$ using a deep neural network parametrized by $\\theta $ .",
"$f$ approximates the underlying conditional distribution $p(y|x;\\theta )$ to minimize the loss function.",
"We can define a strongly labeled dataset with correct annotation as $D_{S}=\\lbrace (x,y_S)_{m}\\rbrace _{1\\le m\\le |D|}$ , where $y_S \\in 0,1$ .",
"In addition, the weakly soft labeled set (noisy labeled) can be defined as $D_{W}=\\lbrace (x,\\bar{y})_{n}\\rbrace _{1\\le n\\le |D|}$ , $\\bar{y}\\in [0,1]$ , where $x^{(i)}\\in \\Re ^{n_{x}}$ denotes input MRI image (feature space), $y_{s}^{(i)}\\in \\Re ^{m_{y}}$ is the distribution correct strong labels, and $\\bar{y}^{(i)}\\in \\Re ^{n_{y}}$ is the distribution of observed labels (weak soft labels), $K$ the number of segmentation classes, and $\\hat{y}_{i}=f(x_i)$ the output of the model given pixel $i$ .",
"Consequently, $p(\\hat{y}_{i,k}=k|x_i;\\theta )$ is the probability that pixel $i$ is assigned to class $k\\in K$ (denoted as $\\hat{p}_{i,k}$ ).",
"The problem with weak labels (poor, noisy, and partial) is that the probability distribution of the observed label is not equal to the correct labels $P(\\bar{Y}_{W}|X)\\ne P(Y_{S}|X)$ , which is typically caused by (a) partial instance segmentations or (b) missing object instances in the observed labels.",
"In classification problems, the aim is to minimize the risk of $f$ , defined as $R(f)=E_{p(x,y)}[l(Y,f(X))]$ where $l(Y,f(X))$ is the loss function.",
"The goal of classification with weak labels continues to minimize the classification risk, defined as $\\bar{R}(f)=E_{p(X,\\bar{Y})}[\\bar{l}(\\bar{Y},f(X))]$ , where $\\bar{l}(\\bar{Y},f(X))$ is a proper loss function for learning noisy labels.",
"Using a noise-tolerant loss, assuming that $f^{*}$ that minimizes $\\hat{R}(f)$ can be determined, would be a global minimizer of $R(f)$ as well.",
"In contrast, DL models using loss functions without robustness to noisy labels tend to memorize the noisy samples to minimize the $\\hat{R}(f)$ risk.",
"Therefore, in this paper, we are using soft labels and a noise-robust loss function to combat weak label problem and to determine $f^{*}$ to minimize $\\hat{R}(f)$ , which is also a global minimizer of $R(f)$ ."
],
[
"Model",
"We used the IMaskRCNN [1] as the baseline deep learning model for training and evaluation of the proposed extensions.",
"The IMaskRCNN model is an improved version of the well-known instance segmentation model Mask RCNN [31] that improves the segmentation accuracy around object boundaries by adding a skip connection and an extra encoder layer to the mask segmentation head (inspired by U-net architecture) [31].",
"Similar to the original Mask RCNN, the IMaskRCNN is constructed from a backbone (ResNet), which is responsible for feature extraction, a region proposal network (RPN) for extracting the ROI bounding box, and two heads: one for mask segmentation (Mask Head), and the other for classifying the extracted bounding boxes (Classification Head)."
],
[
"Loss",
"Similar to the Mask RCNN, the IMaskRCNN has a multi-task learning loss ($L$ ) for each sampled ROI, which is the result of the classification loss, the bounding-box loss, and the mask-loss accumulation $L=L_{cls}+L_{bbox}+L_{mask}$ ."
],
[
"Mask-Loss for pixel-level noise",
"In Mask RCNN, the mask-head is performing binary segmentation of the detected ROI and $L_{mask}$ is only defined on the $k_{th}$ mask (other mask outputs do not contribute to the loss).",
"Therefore, binary CE (BCE) has been chosen as $L_{mask}$ in Mask R-CNN [31].",
"Assuming the segmentation error to be the major source of error in comparison with the classification error, we aimed to modify $L_{mask}$ to mitigate the effect of pixel-level noise.",
"Therefore, we propose to replace the BCE loss used as Mask-loss with the APL loss [13] and adapt the APL loss for soft labels.",
"APL loss is constructed from an active loss term and a passive loss term as: $ L_{APL}=\\alpha .L_{active}+\\beta .L_{passive}$ where $\\alpha $ and $\\beta $ are parameters to balance two terms.",
"Ma et al.",
"[13] have shown that normalized losses guarantee robustness against noise.",
"Thus, we use normalized losses $(L_{norm})$ : $L_{norm}=\\frac{L(f(x),y)}{\\sum ^{K}_{j=1}(f(x),j)}$ In this paper, we considered Normalized BCE (NBCE) and Normalized RCE (NRCE) for the combination of active and passive losses.",
"The cross-entropy loss for two distributions, $q$ (GT distribution) and $p$ (predicted distribution), is defined as $H(q,p)$ (eq.",
"REF ) and RCE is defined as $H(p,q)$ (eq.",
"REF ).",
"By applying the CE and RCE losses to soft labels $(\\bar{y})$ , it follows that: $l_{sce}=H(q,p)=-[\\bar{y}\\log \\hat{y}-(1-\\bar{y})\\log (1-\\hat{y})]$ $l_{srce}=H(p,q)=-[\\hat{y}\\log \\bar{y}-(1-\\hat{y})\\log (1-\\bar{y})]$ and then the normalized soft CE and normalized soft RCE can be defined using eq.",
"REF .",
"As $log0$ is undefined, the following constraint is applied to probabilities.",
"For $H(a,b)=H(a,b^{*})$ , where $b^{*}$ is the probability clipped between two values $b^{*}\\in [P_{min},1-P_{min}]$ and $P_{min}=1\\mathrm {e}{-20}$ : In this study, we used the publicly available multicenter Osteoarthritis Initiative (OAI, https://nda.nih.gov/oai/) dataset.",
"OAI contains the demographic and imaging information from 4796 subjects aged 45-79 years who underwent annual knee assessments, including MRIs.",
"A total of 126 knee MRI scans (sagittal intermediate-weighted fat suppressed (IWFS) 444$\\times $ 448 imaging matrix, slice thickness 3 mm, field-of-view 159$\\times $ 160 mm) were selected and scored by experts (2 to 7 readers musculoskeletal radiologists and rheumatologists) for the presence of BML (BML; bright spots in bone) at the tibia, femur, and patella using the Inflammation MRI Scoring System (KIMRISS) [32].",
"In this work, we used the BML annotations obtained from KIMRISS along with proxy labels of bones (femur and tibia) obtained from automatic segmentation using the IMaskRCNN trained on registered data from our previous work [1].",
"In the following, we explain how we prepared BML annotations for training.",
"KIMRISS is a granular semi-quantitative scoring system, which measures inflammation in patients with knee osteoarthritis (OA) [32].",
"In the KIMRISS [32], by overlaying a transparent grid template on top of the bones (tibia, femur, and patella), the reader specifies regions in slices that contain BML to determine an estimate for BML volume by multiplying granular regions identified as BML [32].",
"Using this scoring system and regions specified to have BML it is possible to obtain rectangular ROIs from specified areas on the KIMRISS granular template.",
"In the following, we attempt to first provide cleaner labels and produce soft labels by using scoring results from different raters, spatial redundancy in MRI scans, and BML characteristics in IWFS scans.",
"The following steps were taken to create soft labels from multiple annotations after data cleaning and fixing major errors (Fig.",
"REF ).",
"Figure: Generating soft pseudo-GT for BML pipeline Scoring results of raters were aggregated and normalized based on the number of raters who rated the scan.",
"Bone proxy mask areas of the overlayed grid template outside bone were excluded.",
"BML is described as the presence of ill-defined hyperintense areas within trabecular bone on IWFS images.",
"Given this knowledge, we consider areas with greater intensity than other areas of the bone in the trabecular bone more likely to be BML.",
"Therefore, if areas are selected in step 3, we would consider their probability equal to one.",
"We improved labels by adding labeling information from the previous and next MRI slices, based on the likelihood with neighboring voxels as shown in Fig.",
"REF (i.e., if the previous/next slices of the slice A contains BML, the slice A is more likely to contain BML as well).",
"We divided the dataset into 108 scans (from 54 subjects, 1280 slices contain BML) for training, 4 scans (from 2 subjects, 66 slices) for validation, and 17 scans (from 9 subjects, 650 slices) for testing.",
"Furthermore, the validation and test data were segmented manually pixel by pixel by two experts using ITK-snap [33].",
"The input images were cropped to 320$\\times $ 320 pixels.",
"To receive more spatial information, we used 2.5D MRI slices (three sequential slices as RGB channels).",
"Furthermore, we have used mirroring as data augmentation to increase number of training data."
],
[
"Implementation",
"All models were trained on one NVIDIA V100 GPU for 200 epochs and 200 iterations using an Adam optimizer and a learning rate of 0.001.",
"The complete DL model was implemented in Keras using the TensorFlow 2 backbone."
],
[
"Evaluation Criteria",
"For evaluation, the metrics used included precision $(=TP/(TP+FP))$ , recall $(=TP/(TP+FN))$ , Intersection of over Union (IoU) $(=(GT\\cap Pred)/(GT\\cup Pred))$ , and average precision (AP), as well as the Dice similarity score $(= 2\\times TP / (2\\times TP + FP + FN))$ for segmentation.",
"For all of these metrics, higher values indicate better performance."
],
[
"Ablation Study",
"Multiple experiments were performed to evaluate the effect of using soft labels and noise resistance loss on BML detection and segmentation with hard and soft labels.",
"Quantitative results of the training with different configurations are summarized in Table REF .",
"Table: RESULTS OF DIFFERENT CONFIGURATIONS (FOR BML)"
],
[
"Soft Labels vs. Hard Labels",
"We investigated two types of labeling methods, binary labels and soft labels.",
"In conventional label smoothing, the labels are smeared based on a $\\alpha $ where $p(Y=k|X=x)\\in (\\alpha ,1-\\alpha )$ .",
"Conventional label smoothing takes binary $(0,1)$ labels and changes its value uniformly.",
"While in our soft labeling approach the $p(Y=k|X=x)$ can have any value between 0 to 1, and label confidence is adjusted using other information like the intensity of pathology, its location and neighboring voxels.",
"The results are shown in TABLE REF demonstrate that using soft labels and a noise resistance loss separately has a positive effect on precision and recall for detecting BML.",
"Comparing the results from BCE (baseline) and soft BCE (soft baseline), shows $4\\%$ improvement in recall and $2\\%$ improvement in precision and Dice similarity metric.",
"Based on Fig.",
"REF (a and d), it can be observed that using soft labels is effective in preventing overconfidence (as we expected under noisy labels [21]).",
"Figure: Results of different configurations for BML detection"
],
[
"APL Loss vs. BCE loss",
"We tested different combinations of active loss $(\\alpha _{1}.NBCE+\\alpha _{2}.NSCE)$ and passive loss $(\\beta .NRCE)$ and compared it to the baseline using the BCE loss.",
"For the active loss, we considered using a combination of the Binary CE and soft CE together.",
"Using APL for binary labels does not seem to be effective.",
"Comparing results from the baseline and APL binary in TABLE REF shows using APL for binary labels is not improving the metric.",
"Further, as shown in Fig.",
"REF (c, d) binary APL also failed to detect patella's BML in the patella."
],
[
"APL + softLabels",
"The combination of APL + softLabels has improved the recall and precision $10\\%$ in comparison to the baseline method.",
"As mentioned above, soft labels decrease results confidence.",
"Adding the BCE term $(\\alpha _{1}.NBCE)$ to the active loss increases confidence mostly on the true positive labels (Fig.",
"REF e, g) and increased recall 22%, precision 14%, and dice 8%.",
"Further, the probability distribution of active loss with BCE term is visually closer to the ground truth."
],
[
"Discussion",
"In this paper, we proposed a noise-resistance deep learning pipeline using soft labels and an active-passive learning loss.",
"The proposed method primarily addresses the problem of training a CNN using partially annotated data.",
"This simple yet effective method has improved the detection rate of BML.",
"Furthermore, we have generated soft labels that conformed to anatomic structures by combining features directly obtained from the images (pixel intensity and anatomy) with the ROI highlighting general areas of pathology, by using medical scores provided by human readers that can be used for semi-supervised learning.",
"The results using the OAI dataset for BML segmentation show that using a noise-resistant loss in combination with soft labels improves performance in both detection and segmentation tasks, compared to using noise-sensitive losses like CE or noise-robust losses such as MAE, SCE, or APL.",
"Moreover, the proposed combination of active and passive losses for APL improved sensitivity on labeled areas without additional punishment for missed-labeled regions."
],
[
"Effect of segmentation loss on detection task",
"Changes in segmentation loss function improved the result of the segmentation task (Dice score) and detection results (recall, precision, and IoU).",
"This improvement can be attributed to the multi-tasking nature of the Mask RCNN approach.",
"Mask RCNN learning tasks aim to concurrently identify instances, classify instances, and segment instances.",
"Multi-task learning is proven to improve learning efficiency and prediction accuracy compared with training separate models for each task [34], [35].",
"In multi-task learning, generic parameters are softly constrained.",
"Moreover, during back-propagation, the loss of one task has an impact on the concurrent tasks as well.",
"Therefore, in IMaskRCNN, adding noise resistance loss to the mask-head contributes to regularization in other tasks as well."
],
[
"Multi-label Data With Label Noise",
"Most of the previous proposed methods are applicable only for a single-label multi-class segmentation problem, where each data example is assumed to have only one true label.",
"However, most medical image applications require the segmentation of multiple labels, for example, some pixels could be associated with multiple true class labels.",
"However, methods that are developed based on MaskRCNN are suitable for multi-label applications since in MaskRCNN the $L_{mask}$ is defined only on positive ROIs and other mask outputs do not contribute to the loss.",
"This constraint leads to no competition among classes for generating masks, which is suitable for identifying lesions in tissues multi-label data with weak labels."
],
[
"Challenges and limitations of the data",
"Using KIMRISS annotation for training a network is challenging since we can only obtain weak labels.",
"The reason is that KIMRISS objectives has not been providing pixelwise BML annotation and exact BML volume.",
"Thus, some discrepancy exists between raters in the exact location and size of the templates.",
"Furthermore, ROIs obtained from KIMRISS are rectangular shapes and do not provide pixel or shape information.",
"In addition, labeling uncertainty (uncertainty of disagreement and uncertainty of single-target label [36]), which is common in medical images annotations, introduces another noise to these partial labels.",
"Due to varying thresholds between readers for defining positive lesions, disagreement uncertainty exists for the KIMRISS annotation, which is measured as inter-observer reliability for the BML volume.",
"Although the measured reliability suggests an acceptable confidence interval for clinical decisions [32], these kinds of discrepancy affect model training.",
"Other sources of noise labeling were investigated through close observation and follow-up reading.",
"The follow-up reading suggests that approximately $50\\%$ of areas with BML were missed or underestimated in the consensus labels and more than $60\\%$ of areas with BML had been underestimated (or missed) by single readers (mainly due to inattentional error and tunnel vision).",
"It is possible to consider a part of these annotation errors as a random variable, since the same annotator may not make the same errors when annotating the same scan, a second time (after a period of time).",
"Furthermore, our visual observation shows smaller or dimmer BML, BMLs in starting slices, and difficult slices were more likely to be missed by readers.",
"Moreover, we had much less data (only 1280 slices for training) compared to similar works who aimed to solve segmentation using weak label problem.",
"To mitigate the effect of low number of annotations, we can use self-supervised or knowledge distillation training.",
"However, in this paper, our focus was only to investigate the effect of using soft-labeling and normalized APL loss."
],
[
"Conclusion and Future Work",
"Combination of soft labeling and noise tolerant loss is an effective method for weakly supervised segmentation of medical images.",
"It provides a convenient approach for improving the performance of DL models with minimal intervention to the existing methods, while revealing a novel way to design noise-robust loss functions for segmentation.",
"The proposed method has the flexibility to quickly adapt to the state-of-the-art architectures and learning algorithms, unlike most of the current approaches that require changing the learning process to estimate correct labels of the training examples and learn under label noise.",
"This method is suitable for knowledge distillation as it gives more information when compared to hard labels and is effective for one stage training.",
"It does not suffer from class imbalance and, unlike U-net based architectures developed to be robust against noise, it is able to perform multi-class segmentation problem.",
"The research result is designed to integrate with the KIMRISS scoring online platform [32] for identifying BMLs and its volume in the clinical domain.",
"Therefore, we need to investigate the effect of our method on network calibration and confidence in future work, and also measure the reader’s uncertainty."
],
[
"Acknowledgment",
"Academic time for JJ, is made available by Medical Imaging Consultants (MIC), Edmonton, Canada.",
"We thank the members of the OMERACT MRI in Arthritis Working Group for their participation and support in this project."
]
] | 2209.08172 |
[
[
"Gaussian dynamics equation in normal product form"
],
[
"Abstract In this paper, we discuss the normal product form of the density operator of multimode Gaussian states, and obtain the correlation equation between the kernel matrix R of the Gaussian density operator in the normal product form and its kernel matrix G in the standard quadratic form.",
"Further, we explore the time evolution mechanism of R and obtain the Gaussian dynamical equation under the normal product R=i(RJH-HJR).",
"Our work is devoted to searching for another mechanism for Gaussian dynamics.",
"By exploring the description of the normal ordered density matrix under the coherent state representation, we find that our mechanism is feasible and easy to operate."
],
[
"Introduction",
"Quantum information science with continuous variable systems is developing rapidly, presenting many exciting prospects in both its experimental realization and theoretical research.",
"Concepts and protocols, such as entanglement and teleportation, initially intended only for discrete quantum systems, have been extended to continuous variable systems, allowing more efficient implementation and measurements.",
"In this context, Gaussian states, as continuous variable quantum states, play an important role in both the experimental and theoretical fields.",
"Gaussian states are defined as quantum states that have Gaussian Wigner functions, while Gaussian dynamics studies the time evolution mechanism of Gaussian state under Gaussian unitary transformation.",
"Two points should be paid special attention to here, one is that the Gaussian state itself must be of Gaussian type, and the other is that the Hamiltonian of the dynamical system in which the Gaussian state evolves is of standard quadratic form.",
"There are many works on the dynamics mechanism of Gaussian state evolution in quadratic systems [1]-[5].",
"However, many studies focused on the evolution mechanism of the covariance matrix of the Gaussian state, which almost became the paradigm of Gaussian dynamics, and most of the research was done in this way.",
"Here, let us make a brief introduction to this mechanism.",
"For a standard quadratic system, its Hamiltonian can be written as follows $\\widehat{H}=\\frac{1}{2}\\widehat{A}^{T}\\mathbf {H}\\widehat{A}, $ where $\\emph {T}$ represents the transpose of the matrix and $\\mathbf {H}$ is a positive definite, Hermitian and symmetric $2n\\times 2n$ matrix, while $\\widehat{A}=(\\widehat{a_{1}},...,\\widehat{a_{n}},\\widehat{a_{1}}^{\\dag },...,\\widehat{a_{n}}^{\\dag })^{T}$ , in which $\\widehat{a_{i}}$ and $\\widehat{a_{i}}^{\\dag }$ represents the creation and annihilation operators for $n$ -mode Gaussian bosonic systems, satisfying the usual bosonic commutation relations $[\\widehat{a_{i}},\\widehat{a_{j}}]=[\\widehat{a_{i}}^{\\dag },\\widehat{a_{j}}^{\\dag }]=0$ and $[\\widehat{a_{i}},\\widehat{a_{j}}^{\\dag }]=\\delta _{ij}$ .",
"Then, for a Gaussian state, its time-evolution covariance matrix $\\mathbf {\\sigma }(t)$ is according to the following rules [6] $\\overset{\\cdot }{\\mathbf {\\sigma }}(t)=\\frac{d\\sigma (t)}{dt}=(\\mathbf {JH})\\mathbf {\\sigma +\\sigma }(\\mathbf {JH})^{T}, $ where $\\mathbf {J=}\\left(\\begin{array}{cc}\\mathbf {0} & \\mathbf {I}_{n} \\\\-\\mathbf {I}_{n} & \\mathbf {0}\\end{array}\\right) $ , $\\mathbf {I}_{n}$ is $n\\times n$ identity matrix.",
"Thus, by solving Eq.",
"(2), the time evolution of the Gaussian state can be mapped as $\\mathbf {\\sigma }(t)\\rightarrow \\mathbf {S}(t)\\mathbf {\\sigma }(0)\\mathbf {S}^{T}(t).",
"$ Note that $\\mathbf {S}(t)\\equiv \\exp (\\mathbf {JH}t)$ , which is a symplectic matrix and satifies with $\\mathbf {S}^{T}\\mathbf {JS=SJS}^{T}=\\mathbf {J.}",
"$ However, can we directly give the law of the time evolution of the Gaussian state $\\rho _{G}(t)$ itself?",
"This is the main topic to be studied in the present paper.",
"In short, we give the law of the time evolution of the kernel $\\mathbf {R}$ of the Gaussian density matrix in the normal product form through effective theoretical derivation, which is an important development of the Gaussian dynamics mechanism.",
"Compared with the previous work, our work is dedicated to directly giving the time evolution of the Gaussian density matrix, breaking the previous theoretical paradigm with the covariance matrix as a bridge.",
"Moreover, due to the operational simplicity of the normal ordered operator in the coherent state representation, we can in principle solve analytically many problems related to the evolution of density matrices, such as the evolution of von Neumann entropy.",
"Our work is arranged as follows: In Sec.",
"2, we first give a brief review of the Gaussian state and its covariance matrix.",
"Then, we use the covariance matrix of the Gaussian state $\\rho _{G}(t)$ as a bridge to obtain the algebraic relationship between the kernel $\\mathbf {G}$ of the Gaussian state density matrix and the kernel $\\mathbf {R}$ of the normal form of the density matrix, so that once we get $\\mathbf {R}$ , we can give $\\mathbf {G}$ , vice versa.",
"In Sec.",
"3, we introduce the coherent state representation description of the Gaussian state, which is the basis for our follow-up work.",
"In Sec.",
"4, we will show the time evolution law of the kernel matrix $\\mathbf {R}$ of the normal product of $\\rho _{G}(t)$ $\\overset{\\cdot }{\\mathbf {R}}=i(\\mathbf {RJH-HJR}).",
"$"
],
[
"Gaussian state and its covariance matrix",
"The density of a Gaussian state can generally be written as [7] $\\rho _{G}=\\frac{e^{-\\widehat{G}}}{Tr(e^{-\\widehat{G}})}.",
"$ Note that $\\widehat{G}=\\frac{1}{2}\\widehat{A}^{T}\\mathbf {G}\\widehat{A}$ .",
"By Williamson's theorem [8], for a positive definite, Hermitian and symmetric $2n\\times 2n$ matrix $\\mathbf {G}$ , it can be decomposed into the following form $\\mathbf {G}=\\mathbf {S}^{T}\\widetilde{\\mathbf {K}}\\mathbf {S}, $ where, $\\mathbf {S}$ denotes a symplectic matrix, $\\widetilde{\\mathbf {K}}=\\left(\\begin{array}{cc}\\mathbf {K} & \\mathbf {0} \\\\\\mathbf {0} & \\mathbf {K}\\end{array}\\right) $ and $\\mathbf {K}=diag(\\omega _{1},\\ldots ,\\omega _{n})$ .",
"According to [7], for the Gaussian state given by Eq.",
"(6), its covariance matrix can be written as $\\mathbf {\\sigma }=\\mathbf {S}^{-1}\\widetilde{\\mathbf {\\nu }}\\mathbf {S}^{-T},$ in which, $\\widetilde{\\mathbf {\\nu }}=\\left(\\begin{array}{cc}\\mathbf {\\nu } & \\mathbf {0} \\\\\\mathbf {0} & \\mathbf {\\nu }\\end{array}\\right) $ , $\\mathbf {\\nu }=diag(\\nu _{1},\\ldots ,\\nu _{n})$ , and $\\nu _{i}=\\frac{1+e^{-\\omega _{i}}}{1-e^{-\\omega _{i}}}$ .",
"Then $\\mathbf {\\sigma }=\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {\\Omega =\\coth (}\\frac{\\mathbf {\\Omega G}}{2})\\mathbf {\\Omega ,} $ where, $\\mathbf {\\Omega }=\\left(\\begin{array}{cc}\\mathbf {I}_{n} & \\mathbf {0} \\\\\\mathbf {0} & \\mathbf {-I}_{n}\\end{array}\\right) $ .",
"We also know that the characteristic function of any Gaussian state can be written as [9] $C(\\mathbf {Z})=e^{-\\frac{1}{2}\\mathbf {Z}^{\\dag }\\mathbf {CZ}}.",
"$ Note that $\\mathbf {Z=(}z_{1},\\ldots ,z_{n},z_{1}^{\\ast },\\ldots ,z_{n}^{\\ast })^{T}$ .",
"By using $e^{\\mathbf {Z}^{\\dag }\\mathbf {\\Omega }\\widehat{A}}=\\colon e^{\\mathbf {Z}^{\\dag }\\mathbf {\\Omega }\\widehat{A}-\\frac{1}{4}\\mathbf {Z}^{\\dag }\\mathbf {Z}}\\colon ,$ where, $\\colon \\cdots \\colon $ represents normal ordering.",
"Then, $\\rho _{G} &=&(d\\mathbf {Z})e^{\\mathbf {Z}^{\\dag }\\mathbf {\\Omega }\\widehat{A}}C(\\mathbf {Z}) \\\\&=&(d\\mathbf {Z})\\colon e^{\\mathbf {Z}^{\\dag }\\mathbf {\\Omega }\\widehat{A}-\\frac{1}{4}\\mathbf {Z}^{\\dag }\\mathbf {Z}}\\colon e^{-\\frac{1}{2}\\mathbf {Z}^{\\dag }\\mathbf {CZ}} \\\\&=&(d\\mathbf {Z})\\colon e^{-\\frac{1}{2}\\mathbf {Z}^{\\dag }\\mathbf {(C+}\\frac{1}{2}\\mathbf {I)Z}}e^{\\mathbf {Z}^{\\dag }\\mathbf {\\Omega }\\widehat{A}}\\colon .",
"$ By using the technique of integration within ordered product (IWOP) [10] and the integeral fomula $(d\\mathbf {Z})e^{-\\frac{1}{2}\\mathbf {Z}^{\\dag }\\mathbf {VZ}}e^{\\mathbf {Z}^{\\dag }\\mathbf {X}}=\\frac{1}{\\sqrt{\\det \\mathbf {V}}}e^{-\\frac{1}{2}\\mathbf {X}^{T}\\mathbf {EV}^{-1}\\mathbf {X}}, $ where, $\\mathbf {E=}\\left(\\begin{array}{cc}\\mathbf {0} & \\mathbf {I}_{n} \\\\\\mathbf {I}_{n} & \\mathbf {0}\\end{array}\\right) ,$ let us continue our derivation $\\rho _{G} &=&\\frac{1}{\\sqrt{\\det \\mathbf {(C+}\\frac{1}{2}\\mathbf {I)}}}\\colon \\exp [-\\frac{1}{2}\\mathbf {(\\Omega }\\widehat{A}\\mathbf {)}^{T}\\mathbf {E(C+}\\frac{1}{2}\\mathbf {I)}^{-1}(\\mathbf {\\Omega }\\widehat{A})]\\colon \\\\&=&\\frac{1}{\\sqrt{\\det \\mathbf {(C+}\\frac{1}{2}\\mathbf {I)}}}\\colon \\exp [-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {\\Omega E(C+}\\frac{1}{2}\\mathbf {I)}^{-1}\\mathbf {\\Omega }\\widehat{A}]\\colon .",
"$ Here, we can set $\\mathbf {R\\equiv \\Omega E(C+}\\frac{1}{2}\\mathbf {I)}^{-1}\\mathbf {\\Omega }$, then $\\rho _{G}=\\sqrt{\\det \\mathbf {R}}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon .",
"$ Since the Wigner function of the Gaussian state $\\rho _{G}$ can be written as $W(\\mathbf {Z})=\\frac{1}{\\sqrt{\\det \\mathbf {\\sigma }}}\\exp (-\\mathbf {Z}^{\\dag }\\mathbf {\\sigma }^{-1}\\mathbf {Z).}",
"$ Note that $\\mathbf {\\sigma }$ here is the covariance matrix in Eq.",
"(2).",
"According to the Fourier transform relationship between $C(\\mathbf {Z})$ and $W(\\mathbf {Z})$ , we can get $\\frac{\\mathbf {\\sigma }^{-1}}{2}=\\mathbf {\\Omega C}^{-1}\\mathbf {\\Omega }$ or $\\mathbf {C}=\\frac{1}{2}\\mathbf {\\Omega \\sigma \\Omega .}",
"$ Substituting Eq.",
"(9) into Eq.",
"(18), we have $\\mathbf {C}=\\frac{\\mathbf {\\Omega }}{2}\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}.",
"$ Then, taking Eq.",
"(19) into Eq.",
"(14), we can get $\\mathbf {R}\\mathbf {=\\Omega E(} &&\\frac{\\mathbf {I}}{2}\\mathbf {+}\\frac{\\mathbf {\\Omega }}{2}\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {)}^{-1}\\mathbf {\\Omega } \\\\&=&-2\\mathbf {E\\Omega (I+\\Omega }\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {)}^{-1}\\mathbf {\\Omega } \\\\&=&-2\\mathbf {E(I+}\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {\\Omega )}^{-1} \\\\&=&-2\\mathbf {(E+}\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {\\Omega E)}^{-1} \\\\&=&-2\\mathbf {(E+}\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {J)}^{-1} \\\\&=&-2\\mathbf {(E+JJ}^{-1}\\frac{\\mathbf {I}+e^{-\\mathbf {\\Omega G}}}{\\mathbf {I}-e^{-\\mathbf {\\Omega G}}}\\mathbf {J)}^{-1} \\\\&=&-2\\mathbf {(E+J}\\frac{\\mathbf {I}+e^{-\\mathbf {J}^{-1}\\mathbf {\\Omega GJ}}}{\\mathbf {I}-e^{-\\mathbf {J}^{-1}\\mathbf {\\Omega GJ}}}\\mathbf {)}^{-1} \\\\&=&-2\\mathbf {(E+J}\\frac{\\mathbf {I}+e^{-\\mathbf {EGJ}}}{\\mathbf {I}-e^{-\\mathbf {EGJ}}}\\mathbf {)}^{-1}.",
"$ In this way, we obtain the relationship of the kernel matrix $\\mathbf {R}$ of the normal product of $\\rho _{G}(t)$ and $\\mathbf {G}$ , which is exactly the same results as in [11].",
"In Gaussian dynamics, as long as we know the time evolution of $\\mathbf {R}$ , we can infer the evolution of $\\mathbf {G}$ from Eq.",
"(20).",
"That is to say, we can directly calculate the time evolution of the density matrix of the Gaussian state by using this method.",
"Moreover, according to the above calculation, we can also deduce the relationship between $\\mathbf {R}$ and $\\mathbf {\\sigma }$ $\\mathbf {R}=-2\\mathbf {E}(\\mathbf {\\sigma +I})^{-1}.",
"$"
],
[
"Coherent state representation of Gaussian state",
"Now we introduce $n$ -mode coherent states $|\\mathbf {Z}\\rangle \\equiv |z_{1},...,z_{n}\\rangle $ and suppose that $\\rho (\\mathbf {Z})=\\langle \\mathbf {Z}|\\rho _{G}|\\mathbf {Z}\\rangle $ .",
"In normal product form, bosonic creation and annihilation operators could be replaced by the complex parameter of the coherent state, thus, we have $\\rho (\\mathbf {Z}) &=&\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon |\\mathbf {Z}\\rangle \\\\&=&\\sqrt{\\det \\mathbf {R}}e^{-\\frac{1}{2}\\mathbf {Z}^{T}\\mathbf {RZ}}.",
"$ For a single-mode coherent state $|z\\rangle $ , we have $|z\\rangle \\langle z|\\widehat{a}=(z+\\frac{\\partial }{\\partial z^{\\ast }})|z\\rangle \\langle z|, $ $\\widehat{a}^{\\dag }|z\\rangle \\langle z|=(z^{\\ast }+\\frac{\\partial }{\\partial z})|z\\rangle \\langle z|.",
"$ We can generalize the relationship given by the above two equations to the multimode case and have $\\widehat{A}|\\mathbf {Z}\\rangle \\langle \\mathbf {Z}| &=&\\left(\\begin{array}{c}\\widehat{a}_{1} \\\\\\vdots \\\\\\widehat{a}_{n} \\\\\\widehat{a}_{1}^{\\dag } \\\\\\vdots \\\\\\widehat{a}_{n}^{\\dag }\\end{array}\\right) |\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|=\\left[ \\left(\\begin{array}{c}z_{1} \\\\\\vdots \\\\z_{n} \\\\z_{1}^{\\ast } \\\\\\vdots \\\\z_{n}^{\\ast }\\end{array}\\right) +\\left(\\begin{array}{c}0 \\\\\\vdots \\\\0 \\\\\\frac{\\partial }{\\partial z_{1}} \\\\\\vdots \\\\\\frac{\\partial }{\\partial z_{n}}\\end{array}\\right) \\right] |\\mathbf {Z}\\rangle \\langle \\mathbf {Z}| \\\\&=&(\\mathbf {Z}+\\frac{\\mathbf {E-J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}^{T}})|\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|.",
"$ Similarly, the following formula can be derived $|\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|\\widehat{A}^{T}=(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})|\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|.",
"$ Taking into account Eqs.",
"(25) and (26), in the coherent state representation, we obtain $\\langle \\mathbf {Z}|\\rho _{G}\\widehat{A}|\\mathbf {Z}\\rangle &=&\\langle \\mathbf {Z}|\\rho _{G}|\\mathbf {Z}\\rangle (\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) \\\\&=&\\rho (\\mathbf {Z})(\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) $ and $\\langle \\mathbf {Z}|\\widehat{A}^{T}\\rho _{G}|\\mathbf {Z}\\rangle &=&(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})\\langle \\mathbf {Z}|\\rho _{G}|\\mathbf {Z}\\rangle \\\\&=&(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})\\rho (\\mathbf {Z}), $ where, we have set $\\rho (\\mathbf {Z})\\equiv \\langle \\mathbf {Z}|\\rho _{G}|\\mathbf {Z}\\rangle $ , which is actually a Husimi-Q function in the phase space representation."
],
[
"Gaussian dynamics equation in normal product form",
"For an open dynamic system, the time evolution mechanism of the system is determined by the following Lindblad equation [12] $\\overset{\\cdot }{\\rho }(t)=-i[\\widehat{H},\\rho (t)]+\\underset{i}{}[\\widehat{c_{i}}\\rho (t)\\widehat{c_{i}}^{\\dag }-\\frac{1}{2}\\widehat{c_{i}}^{\\dag }\\widehat{c_{i}}\\rho (t)-\\frac{1}{2}\\rho (t)\\widehat{c_{i}}^{\\dag }\\widehat{c_{i}}], $ where $\\widehat{H}$ is quadratic, $\\widehat{c_{i}}$ and $\\widehat{c_{i}}^{\\dag }$ are the linear forms of the creation and annihilation operators.",
"Although the content discussed in this paper can be fully extended to the case where the quantum system is affected by the coherent environment, that is, considering the second term on the right side of Eq.",
"(29), for the sake of brevity and beauty of the text, we only analyze the time evolution mechanism of Gaussian states in quadratic Hamiltonian systems independent of the environment.",
"That is to say, we only discuss the quantum Liouville equation $\\overset{\\cdot }{\\rho _{G}}(t)=i[\\rho _{G}(t),\\widehat{H}].",
"$ Note that here $\\widehat{H}=\\frac{1}{2}\\widehat{A}^{T}\\mathbf {H}\\widehat{A}$ and $\\rho _{G}=\\frac{e^{-\\widehat{G}}}{Tr(e^{-\\widehat{G}})}=\\sqrt{\\det \\mathbf {R}}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon $ .",
"Substituting $\\widehat{H}$ and $\\rho _{G}$ into Eq.",
"(30), we get $\\frac{d[\\sqrt{\\det \\mathbf {R}}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]}{dt}=-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}[\\widehat{A}^{T}\\mathbf {H}\\widehat{A},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ].",
"$ By using the commutation formula $[AB,C]=A[B,C]+[A,C]B$ , we obtain $&&\\frac{d[\\sqrt{\\det \\mathbf {R}}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]}{dt} \\\\&=&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\widehat{A}^{T}\\mathbf {H}[\\widehat{A},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\widehat{A}^{T}[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\widehat{A} \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}[\\widehat{A}^{T},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\mathbf {H}\\widehat{A}.",
"$ Considering the following normal product properties [13] $\\colon \\frac{\\partial }{\\partial \\widehat{a}}f(\\widehat{a},\\widehat{a}^{\\dag })\\colon =[\\colon f(\\widehat{a},\\widehat{a}^{\\dag })\\colon ,\\widehat{a}^{\\dag }], $ $\\colon \\frac{\\partial }{\\partial \\widehat{a}^{\\dag }}f(\\widehat{a},\\widehat{a}^{\\dag })\\colon =[\\widehat{a},\\colon f(\\widehat{a},\\widehat{a}^{\\dag })\\colon ], $ and the derivation rule of quadratic matrix $\\frac{d(X^{T}AX)}{dX}=2X^{T}A, $ $\\frac{d(X^{T}AX)}{dX^{T}}=2AX, $ under the condition $A=A^{T}$ ($A$ is a symmetric matrix), we can simplify Eq.",
"(32) into the following form $&&\\frac{d[\\sqrt{\\det \\mathbf {R}}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]}{dt} \\\\&=&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\widehat{A}^{T}\\mathbf {H}\\colon \\mathbf {J}\\frac{\\partial }{\\partial \\widehat{A}^{T}}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon +\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\colon \\frac{\\partial }{\\partial \\widehat{A}}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\mathbf {J}\\colon \\mathbf {H}\\widehat{A} \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\widehat{A}^{T}[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\widehat{A} \\\\&=&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\widehat{A}^{T}\\mathbf {H}\\colon \\mathbf {JR}\\widehat{A}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon -\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\colon \\widehat{A}^{T}\\mathbf {R}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\mathbf {J}\\colon \\mathbf {H}\\widehat{A} \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\widehat{A}^{T}[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\widehat{A}.",
"$ In the dynamics of phase space, the time evolution formula of Husimi-Q function $\\rho (\\mathbf {Z})$ can be derived as follow $\\frac{d\\rho (\\mathbf {Z})}{dt} &=&Tr(\\overset{\\cdot }{\\rho }|\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|) \\\\&=&-iTr(\\rho \\widehat{H}|\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|-\\widehat{H}\\rho |\\mathbf {Z}\\rangle \\langle \\mathbf {Z}|) \\\\&=&-i\\langle \\mathbf {Z}|\\rho \\widehat{H}|\\mathbf {Z}\\rangle +i\\langle \\mathbf {Z}|\\widehat{H}\\rho |\\mathbf {Z}\\rangle , $ In fact, we just need to average the coherent states on both sides of the Liouville equation.",
"By calculating the average value of the coherent states on both sides of Eq.",
"(37), we have $&&\\frac{d[\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon |\\mathbf {Z}\\rangle ]}{dt} \\\\&=&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\widehat{A}^{T}\\mathbf {HJR}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\widehat{A}\\colon |\\mathbf {Z}\\rangle \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\widehat{A}^{T}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon \\mathbf {RJH}\\widehat{A}|\\mathbf {Z}\\rangle \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\widehat{A}^{T}[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\widehat{A}|\\mathbf {Z}\\rangle .",
"$ We first calculate the third part of the right-hand side of Eq.",
"(38) and have $&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\widehat{A}^{T}[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\widehat{A}|\\mathbf {Z}\\rangle \\\\&=&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})\\langle \\mathbf {Z}|[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]|\\mathbf {Z}\\rangle (\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) \\\\&=&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})(\\langle \\mathbf {Z}|\\mathbf {H}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon |\\mathbf {Z}\\rangle \\\\&&-\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon \\mathbf {H}|\\mathbf {Z}\\rangle )(\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) \\\\&=&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})(\\mathbf {H}\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon |\\mathbf {Z}\\rangle \\\\&&-\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon |\\mathbf {Z}\\rangle \\mathbf {H})(\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) \\\\&=&-\\frac{i}{2}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})(\\mathbf {H}\\rho (\\mathbf {Z})-\\rho (\\mathbf {Z})\\mathbf {H)}(\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}).",
"$ Since $\\rho (\\mathbf {Z})$ is a number, $\\mathbf {H}\\rho (\\mathbf {Z})-\\rho (\\mathbf {Z})\\mathbf {H=0}$ .",
"So we show $-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\widehat{A}^{T}[\\mathbf {H},\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon ]\\widehat{A}|\\mathbf {Z}\\rangle =0 $ .",
"We continue to calculate the first two terms on the right-hand side of Eq.",
"(39), $&&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\widehat{A}^{T}\\mathbf {HJR}\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\widehat{A}\\colon |\\mathbf {Z}\\rangle \\\\&=&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})[\\mathbf {HJR}\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\widehat{A}\\colon |\\mathbf {Z}\\rangle \\mathbf {]} $ and $&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\widehat{A}^{T}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon \\mathbf {RJH}\\widehat{A}|\\mathbf {Z}\\rangle \\\\&=&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\widehat{A}^{T}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon \\mathbf {RJH}|\\mathbf {Z}\\rangle (\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}).",
"$ Then, $\\frac{d\\mathbf {\\rho (\\mathbf {Z})}}{dt} &=&\\frac{d[\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon |\\mathbf {Z}\\rangle ]}{dt} \\\\&=&-\\frac{1}{2}\\sqrt{\\det \\mathbf {R}}\\mathbf {Z}^{T}\\overset{\\cdot }{\\mathbf {R}}\\mathbf {Z}\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})+}\\frac{d\\sqrt{\\det \\mathbf {R}}}{dt}\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})} \\\\&=&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})[\\mathbf {HJR}\\langle \\mathbf {Z}|\\colon \\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\widehat{A}\\colon |\\mathbf {Z}\\rangle \\mathbf {]} \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\langle \\mathbf {Z}|\\colon \\widehat{A}^{T}\\exp (-\\frac{1}{2}\\widehat{A}^{T}\\mathbf {R}\\widehat{A})\\colon \\mathbf {RJH}|\\mathbf {Z}\\rangle (\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) \\\\&=&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}(\\mathbf {Z}^{T}+\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}})[\\mathbf {HJR}\\overset{~}{\\mathbf {\\rho }}(\\mathbf {Z})\\mathbf {Z]} \\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}[\\mathbf {Z}^{T}\\overset{~}{\\mathbf {\\rho }}(\\mathbf {Z})\\mathbf {RJH}](\\mathbf {Z}+\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}) \\\\&=&\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\mathbf {Z}^{T}(\\mathbf {HJR-RJH)Z}\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})}+\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}}[\\mathbf {HJR\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})}Z]}\\\\&&-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}[\\mathbf {Z}^{T}\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})RJH]}\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2}).",
"$ Note that here we have set $\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})=\\rho (\\mathbf {Z})/}\\sqrt{\\det \\mathbf {R}}$ .",
"Multipling $\\mathbf {E+J}$ on the left-hand side of Eq.",
"(43) and $\\mathbf {E-J}$ on its right-hand side and noting that $\\left( \\mathbf {E+J}\\right) ^{2}=0$ and $\\left( \\mathbf {E-J}\\right) ^{2}=0$ , we obtain $&&\\left( \\mathbf {E+J}\\right) \\mathbf {Z}^{T}\\overset{\\cdot }{\\mathbf {R}}\\mathbf {Z\\left( \\mathbf {E-J}\\right) -2\\left( \\mathbf {E+J}\\right) }\\frac{1}{\\sqrt{\\det \\mathbf {R}}}\\mathbf {\\frac{d\\sqrt{\\det \\mathbf {R}}}{dt}\\mathbf {\\left( \\mathbf {E-J}\\right) }} \\\\&\\mathbf {=}&-i\\left( \\mathbf {E+J}\\right) \\mathbf {Z}^{T}\\mathbf {(HJR-RJH)Z}\\left( \\mathbf {E-J}\\right) .",
"$ Because $\\mathbf {Z}^{T}\\overset{\\cdot }{\\mathbf {R}}\\mathbf {Z}$ , $\\frac{d\\sqrt{\\det \\mathbf {R}}}{dt}$ and $\\mathbf {Z}^{T}\\mathbf {(HJR-RJH)Z}$ are all numbers, Eq.",
"(44) can be written as $&&\\left( \\mathbf {E+J}\\right) \\mathbf {\\left( \\mathbf {E-J}\\right) Z}^{T}\\overset{\\cdot }{\\mathbf {R}}\\mathbf {Z-2\\left( \\mathbf {E+J}\\right) \\mathbf {\\left( \\mathbf {E-J}\\right) }\\frac{d\\ln \\sqrt{\\det \\mathbf {R}}}{dt}} \\\\&\\mathbf {=}&-i\\left( \\mathbf {E+J}\\right) \\left( \\mathbf {E-J}\\right) \\mathbf {Z}^{T}\\mathbf {(HJR-RJH)Z.}",
"$ Obviously, we have $\\mathbf {Z}^{T}[\\overset{\\cdot }{\\mathbf {R}}-i\\mathbf {(RJH-HJR)]Z=\\frac{d\\ln \\det \\mathbf {R}}{dt}.}",
"$ For any $\\mathbf {R}$, $\\mathbf {H}$ and $\\mathbf {Z}$ , Eq.",
"(46) always holds, then we get Eq.",
"(5) given in the introduction and $\\frac{d\\ln \\det \\mathbf {R}}{dt}=0$ .",
"In this way, we derive the Gaussian dynamics equation in the normal product form.",
"At the same time, there is reason to believe that $\\ln \\det \\mathbf {R}$ is a constant that does not change with time.",
"According to the fomula $\\det e^{A}=e^{Tr(A)}$ , we can obtain $\\ln \\det \\mathbf {R}=Tr(\\ln \\mathbf {R})$ .",
"Actually, in Eq.",
"(43), as long as we know that $\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}}[\\mathbf {HJR\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})}Z]}$ and $-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}[\\mathbf {Z}^{T}\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})RJH]}\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2})$ are all numbers, then, because of the existence of $\\mathbf {E+J}$ and $\\mathbf {E-J}$ , we can conclude that $\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}\\frac{\\mathbf {E+J}}{2}\\frac{\\partial }{\\partial \\mathbf {Z}}[\\mathbf {HJR\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})}Z]}$ and $-\\frac{i}{2}\\sqrt{\\det \\mathbf {R}}[\\mathbf {Z}^{T}\\overset{~}{\\mathbf {\\rho }}\\mathbf {(\\mathbf {Z})RJH]}\\overleftarrow{\\frac{\\partial }{\\partial \\mathbf {Z}^{T}}}\\frac{\\mathbf {E-J}}{2})$ are both equal to 0.",
"In addition, since $\\ln \\det \\mathbf {R}=Tr(\\ln \\mathbf {R})$ , then $\\mathbf {\\frac{d\\ln \\det \\mathbf {R}}{dt}} &=&\\frac{dTr(\\ln \\mathbf {R})}{dt} \\\\&=&Tr(\\overset{\\cdot }{\\mathbf {R}}\\mathbf {R}^{-1}) \\\\&=&Tr[i\\mathbf {(RJH-HJR)R}^{-1}] \\\\&=&iTr(\\mathbf {RJH\\mathbf {R}^{-1}-HJ}) \\\\&=&i[Tr(\\mathbf {RJH\\mathbf {R}^{-1})-}Tr\\mathbf {(HJ})] \\\\&=&i[Tr(\\mathbf {JH)-}Tr\\mathbf {(HJ})] \\\\&=&0.",
"$ So, we show that if $\\overset{\\cdot }{\\mathbf {R}}=i\\mathbf {(RJH-HJR)}$ , then $\\mathbf {\\frac{d\\ln \\det \\mathbf {R}}{dt}}=0$ naturally satisfies.",
"Compared with Eq.",
"(2) and Eq.",
"(5), it is not difficult to draw $\\mathbf {R(t)=U(t)R(0)U}^{T}\\mathbf {(t),} $ where $\\mathbf {U(t)\\equiv \\exp }(-i\\mathbf {JH}t)$ .",
"In this way, we get the solution of Eq.",
"(5) smoothly."
],
[
"Conclusion",
"The time evolution mechanism of Gaussian states is a long-standing and ever-new topic.",
"This paper mainly provides another mechanism for dealing with the dynamics of Gaussian states.",
"Different from the previous covariance mechanism, our work gives the equation for the time evolution of the kernel matrix $\\mathbf {R}$ of Gaussian states in the normal product form, which provides a new perspective for Gaussian quantum information processing.",
"The advantage of writing the density matrix of the Gaussian state in the normal product form is that the specific functional form of the density matrix under the coherent state representation can be directly given, which can be done simply by replacing Bosonic operators in the density matrix with the complex parameters of the coherent state.",
"This processing method will bring us convenience to solve some problems.",
"For example, for the operator matrix trace problem, the product of matrices, such as $Tr(\\mathbf {AB)}$ , is often encountered.",
"For such problems, we can solve them analytically by writing $\\mathbf {A}$ and $\\mathbf {B}$ in the normal product form ($\\colon \\overset{~}{\\mathbf {A}}\\colon $ and $\\colon \\overset{~}{\\mathbf {B}}\\colon $ ) and then inserting the completeness of the coherent state representation ($Tr(\\mathbf {AB)=}(d\\mathbf {Z}d\\mathbf {Z}^{\\prime })\\langle \\mathbf {Z}|\\colon \\overset{~}{\\mathbf {A}}\\colon |\\mathbf {Z}^{\\prime }\\rangle \\langle \\mathbf {Z}^{\\prime }|\\colon \\overset{~}{\\mathbf {B}}\\colon |\\mathbf {Z}\\rangle $ ).",
"It is difficult to solve such problems in a conventional way, especially in the multi-mode case, and may also have to use numerical methods, while our method can be solved analytically in principle.",
"Moreover, in the normal product, we regard Bosonic operators as numbers, so we can perform integration and differentiation operations without any obstacles, which cannot be replaced by conventional methods.",
"This processing method undoubtedly has great potential and has the value of further research and promotion.",
"Following the theoretical ideas proposed in this paper, in principle, the incoherent evolution of the Gaussian state that does not interact with the environment can be extended to the case in which the system is coherent with the environment, that is, the Lindblad equation can be solved smoothly, which will be our follow-up work.",
"ACKNOWLEGEMENT: The work is supported by the School-level teaching and research project of West Anhui University (Grant wxxy2020047) and Provincial Teaching and Research Projects of Higher Education Institutions in Anhui Province (Grant 2021jyxm1666)."
]
] | 2209.08250 |
[
[
"A Sharp Rate of Convergence in the Functional Central Limit Theorem with\n Gaussian Input"
],
[
"Abstract When the underlying random variables are Gaussian, the classical Central Limit Theorem (CLT) is trivial, but the functional CLT is not.",
"The objective of the paper is to investigate the functional CLT for stationary Gaussian processes in the Wasserstein-1 metric on the space of continuous functions.",
"Matching upper and lower bounds are established, indicating that the convergence rate is slightly faster than in the L\\'{e}vy-Prokhorov metric."
],
[
"Introduction",
"By the Central Limit Theorem, given a collection $\\lbrace \\xi _k,\\ k\\ge 1\\rbrace $ of independent and identically distributed random variables, each with mean zero and variance one, the sequence $S_n=n^{-1/2}\\sum _{k=1}^n \\xi _k$ converges in distribution, as $n\\rightarrow \\infty $ , to the standard Gaussian random variable.",
"The Berry-Esseen bound [16] gives the rate of convergence in the Kolmogorov metric: $\\sup _{x\\in {\\mathbb {R}}} |F_n(x)-\\Phi (x)|\\le \\frac{{\\mathbb {E}}|\\xi _1|^3}{\\sqrt{n}},$ where $F_n$ is the cumulative distribution function of $S_n$ and $\\Phi $ is the cumulative distribution function of the standard Gaussian random variable.",
"While the rate $1/\\sqrt{n}$ is sharp in general, it can be improved by imposing additional conditions on the random variables $\\xi _k$ .",
"For example, if ${\\mathbb {E}}\\xi _k^3=0$ and ${\\mathbb {E}}\\xi _k^4<\\infty $ , then the left-hand side of (REF ) is of order $1/n$ ; cf [21].",
"Of course, if each $\\xi _k$ is standard normal, then the left-hand side of (REF ) is zero.",
"The functional version of the Central Limit Theorem, also known as the Donsker invariance principle [16], establishes weak convergence of the sequence of processes $S_n(t)=n^{-1/2}\\sum _{k=1}^{\\lfloor n t \\rfloor } \\xi _k+ \\frac{nt-\\lfloor n t \\rfloor }{\\sqrt{n}} \\xi _{\\lfloor n t \\rfloor +1},\\ t\\ge 0,\\ n\\ge 1,$ to the standard Brownian motion $W$ .",
"An analog of (REF ) becomes a bound on the distance between the distributions of $S_n$ and $W$ on the space of continuous functions ${\\mathcal {C}}(0,T)$ in the Lévy-Prokhorov metric.",
"Compared to (REF ), the corresponding rate of convergence depends on integrability properties of $\\xi _k$ in a more complicated way: if ${\\mathbb {E}}|\\xi _1|^p<\\infty $ , $p>2$ , then the rate $n^{-(p-2)/(2(p+1))}$ is sharp; if ${\\mathbb {E}}e^{t\\xi _1}<\\infty ,\\ |t|<\\delta , \\ \\delta >0,$ then the rate $\\ln n/\\sqrt{n}$ is sharp.",
"For details, see [7]; earlier works on the subject include [17], [18], [25].",
"A more general approach to investigating the rate of convergence is to find a bound on $\\sup _{\\varphi \\in \\mathcal {G}} |{\\mathbb {E}}\\varphi (S_n)-{\\mathbb {E}}\\varphi (W)|$ for a suitable class $\\mathcal {G}$ of functions $\\varphi : {\\mathcal {C}}(0,T)\\rightarrow {\\mathbb {R}}$ .",
"Barbour [3] used an infinite-dimensional version of Stein's method to establish the benchmark result $|{\\mathbb {E}}\\varphi (S_n)-{\\mathbb {E}}\\varphi (W)| \\le \\frac{C}{\\sqrt{n}}\\Big ( \\sqrt{\\ln n} + {\\mathbb {E}}|\\xi _1|^3\\Big )$ for a certain (rather restrictive) class $\\mathcal {G}$ ; the restrictive nature of this class ensures that there is no contradiction with [7] or [25].",
"For various other $\\mathcal {G}$ , there are bounds of the form $|{\\mathbb {E}}\\varphi (S_n)-{\\mathbb {E}}\\varphi (W)| \\le \\frac{C}{{n}^r},\\ 0<r<\\frac{1}{2};$ cf.",
"[6] and references therein.",
"Unlike (REF ), the left-hand side of (REF ) will not be zero even if the random variables $\\xi _k$ are Gaussian, as long as $\\mathcal {G}$ is rich enough to capture the infinite-dimensional nature of the problem.",
"In fact, for certain $\\mathcal {G}$ , one can use (REF ) to define a metric on the space of distributions.",
"For example, if $\\mathcal {G}$ is the collection of bounded Lipschitz continuous functions, then convergence in the corresponding bounded Lipschitz metric is equivalent to weak convergence, that is, convergence in the Lévy-Prokhorov metric; cf.",
"[16] or [9].",
"Removing the boundedness condition (for example, to include linear functionals) leads to the Wasserstein-1 metric, which is the subject of this paper.",
"Recall that, for two probability measures $\\mu $ , $\\nu $ on a complete separable metric space $E$ with distance function $\\rho $ and the corresponding Borel sigma-algebra $\\mathcal {B}(E)$ , the bounded Lipschitz metric is $\\mathrm {d}_{_{BL}}(\\mu ,\\nu )=\\sup _{\\varphi } \\left| \\int _E \\varphi d\\mu - \\int _E \\varphi d\\nu \\right|,$ with supremum over functions $\\varphi :E \\rightarrow \\mathbb {R}$ such that, for all $x,y\\in E$ , $|\\varphi (x)-\\varphi (y)|\\le \\rho (x,y)$ and $|\\varphi (x)|\\le 1$ ; the Wasserstein-1 metric $\\mathrm {d}_{{w}}(\\mu ,\\nu )$ , also known as the Kantorovich-Rubinstein metric, is $\\mathrm {d}_{{w}}(\\mu ,\\nu )=\\sup _{\\varphi } \\left| \\int _E \\varphi d\\mu - \\int _E \\varphi d\\nu \\right|,$ with supremum over functions $\\varphi :E \\rightarrow \\mathbb {R}$ such that, for all $x,y\\in E$ , $|\\varphi (x)-\\varphi (y)|\\le \\rho (x,y)$ ; the Lévy-Prokhorov metric is $\\mathrm {d}_{_{LP}}(\\mu ,\\nu )=\\inf \\lbrace \\varepsilon >0: \\mu (A)\\le \\nu (A^{\\varepsilon })+\\varepsilon ,\\ A\\in \\mathcal {B}(E)\\rbrace ,$ where $A^{\\varepsilon }=\\lbrace x\\in E: \\inf \\limits _{y\\in A} \\rho (x,y)\\le \\varepsilon \\rbrace .$ We have $\\mathrm {d}_{_{BL}}(\\mu ,\\nu )\\le \\mathrm {d}_{{w}}(\\mu ,\\nu )$ (by definition), $\\mathrm {d}_{_{LP}}(\\mu ,\\nu )\\le \\sqrt{\\mathrm {d}_{_{BL}}(\\mu ,\\nu )}$ ([9]), and $\\mathrm {d}_{_{BL}}(\\mu ,\\nu )\\le 4 \\mathrm {d}_{_{LP}}(\\mu ,\\nu )$ ([9]).",
"In particular, convergence in the Wasserstein-1 metric implies weak convergence, that is, convergence in either Lévy-Prokhorov or bounded Lipschitz metric; the converse is not always true [9]; in fact, the diagram in [11] suggests that the Wasserstein-1 metric $\\mathrm {d}_{{w}}$ is the strongest possible for the CLT-type problems in function spaces.",
"Still, a sharp rate of convergence in one metric does not directly lead to a sharp rate in any other metric.",
"The invariance principle can hold if independence requirement for the random variables $\\xi _k$ is relaxed, for example, to a strictly stationary and ergodic martingale difference [20], or a stationary Markov process satisfying Doebling's condition [13] [where a bound of the type (REF ) is also established].",
"In continuous time, if $X=X(t),\\ t\\in {\\mathbb {R}},$ is a strictly stationary process with mean zero and covariance function $R(t)={\\mathbb {E}}\\big (X(t)X(0)\\big )$ satisfying $\\int _{-\\infty }^{+\\infty } R(t)\\, dt =1$ , then, under some additional conditions of weak dependence, the sequence of processes $S_n(t)=\\frac{1}{\\sqrt{n}}\\int _0^{nt} X(s)\\, ds,\\ n=1,2,\\ldots , t\\in [0,1],$ converges weakly to the standard Brownian motion; cf.",
"[20] or [14].",
"The objective of this paper is to show that if $X$ is a stationary Gauss-Markov process, in either discrete or continuous time, then the Wasserstein-1 distance between $S_n$ and $W$ in the space of continuous functions is of order $\\big (n^{-1}\\ln n\\big )^{1/2}$ .",
"In other words, if $S_n$ is Gaussian, then the convergence rate in Wasserstein-1 metric is slightly faster than the Lévy-Prokhorov rate $\\ln n/\\sqrt{n}$ .",
"This difference does not contradict the results from [7] and [25], and the discrepancy by a $\\sqrt{\\ln n}$ factor can be explained as follows: in the limit $\\sigma \\rightarrow 0+$ , the distance between a Gaussian distribution with mean zero and variance $\\sigma ^2$ and a point mass at zero is of order $\\sigma $ in the Wasserstein-1 metric, but it is of order $\\sigma \\sqrt{|\\ln \\sigma |}$ in the Lévy-Prokhorov metric.",
"Section discusses (the easier) continuous-time case (REF ).",
"Discrete-time case, a generalization of (REF ) for a stationary Gaussian sequence $\\lbrace \\xi _k,\\ k\\ge 0\\rbrace $ , is in Section .",
"In Section , the results are applied to weak approximation for some ordinary differential equations with additive noise.",
"Section is a summary.",
"Traditionally, models (REF ) and (REF ) are studied on a bounded time interval $[0,T]$ , and the index parameter $n=1,2,\\ldots $ is discrete.",
"In this paper, the time interval is $(0,+\\infty )$ and the index parameter $\\kappa \\ge 1$ is not necessarily an integer.",
"The following two properties of Gaussian processes will be used on several occasions: 1.",
"The Borell-TIS inequality [1]: If $X=X(t), \\ t\\in \\mathcal {T},$ is a zero-mean Gaussian process indexed by the set $\\mathcal {T}$ , and $\\mathbb {P}\\left( \\sup \\limits _{t\\in \\mathcal {T}} X(t) < \\infty \\right) =1$ , then ${\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} |X(t)| < \\infty $ and, with $X^*={\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} X(t),\\ \\ \\sigma _X^2=\\sup \\limits _{t\\in \\mathcal {T}}{\\mathbb {E}}X^2(t)$ , $\\mathbb {P}\\Big (\\sup \\limits _{t\\in \\mathcal {T}} X(t) - X^*>x\\Big )\\le e^{-x^2/(2\\sigma _X^2)},\\ x>0.$ 2.",
"The Fernique-Sudakov inequality [1]: If $X=X(t), \\ Y=Y(t)\\ t\\in \\mathcal {T},$ are zero-mean Gaussian processes indexed by the set $\\mathcal {T}$ , and, for all $t,s\\in \\mathcal {T}$ , ${\\mathbb {E}}|X(t)-X(s)|^2 \\le {\\mathbb {E}}|Y(t)-Y(s)|^2$ , then ${\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} X(t) \\le {\\mathbb {E}}\\sup \\limits _{t\\in \\mathcal {T}} Y(t).$"
],
[
"Continuous Time",
"Let $X=X(t),\\ t\\in {\\mathbb {R}},$ be a (continuous version of a) stationary Gaussian process with mean zero and covariance ${\\mathbb {E}}X(t)X(s)=e^{-2|t-s|}.$ In particular, $X(t)$ is a standard Gaussian random variable for every $t$ .",
"Equivalent characterizations of $X$ are as follows: $X(t)&=e^{-t}W(e^{2t});\\\\X(t)&=2\\int _{-\\infty }^t e^{-2(t-s)}dW(s);\\\\dX(t)&=-2X(t)dt+2\\,dW(t),\\ t\\ge 0.$ In (REF ), (), and (), $W=W(t), \\ t\\ge 0,$ is a standard Brownian motion; in (), when $t<0$ , $W(t)=V(-t)$ for an independent copy $V$ of $W$ .",
"The initial condition $X(0)$ in () is a standard Gaussian random variable independent of $W$ .",
"Given a real number $\\kappa \\ge 1$ , we define $W^{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\int _0^{\\kappa t} X(s)\\, ds,\\ \\ t\\ge 0.$ Denote by ${\\mathcal {C}}_{(0)}$ the collection of continuous functions $f=f(t)$ on $[0,+\\infty )$ such that $f(0)=0,\\ \\lim _{t\\rightarrow +\\infty } \\frac{|f(t)|}{t}=0.$ Endowed with the norm $\\Vert f\\Vert _{(0)}=\\sup _{t>0}\\frac{|f(t)|}{1+t},$ ${\\mathcal {C}}_{(0)}$ becomes a separable Banach space; cf.",
"[8].",
"We have $\\mathbb {P}(W\\in {\\mathcal {C}}_{(0)})=1$ (either by the law of large numbers for square integrable martingales or using that $t\\mapsto tW(1/t)$ is a standard Brownian motion), and also $\\mathbb {P}(W^{\\kappa }\\in {\\mathcal {C}}_{(0)})=1$ , $\\kappa \\ge 1$ , because the ergodic theorem implies $\\lim _{t\\rightarrow \\infty } \\frac{1}{t} \\int _0^t X(s)\\, ds = {\\mathbb {E}}X(0) = 0$ with probability one.",
"Proposition 2.1 There exists a constant $C_X$ such that, for every $\\kappa \\ge 1$ and every function $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfying $|\\varphi (f)-\\varphi (g)|\\le \\Vert f-g\\Vert _{(0)},$ we have $\\Big |{\\mathbb {E}}\\varphi \\big (W^{\\kappa }\\big )-{\\mathbb {E}}\\varphi (W)\\Big |\\le C_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Using (), $X(t)=X(0)e^{-2t}+2\\int _0^t e^{-2(t-s)}\\, dW(s).$ Changing the order of integration (stochastic Fubini theorem [23]), $W^{\\kappa }(t)=\\kappa ^{-1/2}W(\\kappa t)+\\frac{X(0)-X(\\kappa t)}{2\\sqrt{\\kappa }}.$ Define $V^{\\kappa }(t)=\\kappa ^{-1/2}W(\\kappa t),\\ \\ \\ X^{\\kappa }(t)=\\frac{X(\\kappa t)-X(0)}{2}.$ Because $t\\mapsto V^{\\kappa }(t)$ , $t\\ge 0$ , is standard Brownian motion for every $\\kappa >0$ , we have ${\\mathbb {E}}\\varphi \\big (W\\big )={\\mathbb {E}}\\varphi \\big (V^{\\kappa }\\big )$ and then, using (REF ), $|{\\mathbb {E}}\\varphi \\big (W^{\\kappa }\\big )-{\\mathbb {E}}\\big (\\varphi (W)\\big )\\Big |\\le \\frac{{\\mathbb {E}}\\Vert X^{\\kappa }\\Vert _{(0)}}{\\sqrt{\\kappa }}.$ Next, by [5], $\\lim _{T\\rightarrow +\\infty } \\frac{\\max _{0\\le t\\le T} X(t)}{\\sqrt{2\\ln T}}=1,$ with probability one.",
"Using the same arguments as in [19], we conclude from (REF ) that $\\limsup _{T\\rightarrow +\\infty } \\frac{\\max _{0\\le t\\le T} |X(t)|}{\\sqrt{2\\ln T}}\\le 2,$ and then continuity of $X$ implies that the random variable $\\zeta =\\sup _{t>0} \\frac{|X(t)-X(0)|}{2\\sqrt{\\ln (2+t)}}$ is finite with probability one.",
"Indeed, if $T^*=\\sup \\left\\lbrace t\\ge 0: \\frac{\\max _{0\\le t\\le T} |X(t)-X(0)|}{2\\sqrt{2\\ln (2+ T)}}>5\\right\\rbrace $ then, by (REF ), $\\mathbb {P}(T^*<\\infty )=1$ , so that $\\zeta \\le \\max _{0\\le t\\le T^*} \\frac{|X(t)-X(0)|}{2\\sqrt{\\ln (2+t)}} + 5 <\\infty .$ Moreover, because $t\\mapsto X(t)-X(0)$ is a Gaussian process with mean zero, the Borell-TIS inequality (REF ) implies ${\\mathbb {E}}\\zeta ^p <\\infty $ for all $p>0$ .",
"If $t>0$ and $\\kappa \\ge 1$ , then, by direct computation, $(2+\\kappa t)\\le (1+\\kappa )(1+t),\\ \\frac{1}{2}\\le \\sqrt{\\ln (1+\\kappa )}, \\ \\frac{\\sqrt{\\ln (1+t)}}{1+t}\\le \\frac{1}{2}.$ As a result, $\\frac{\\sqrt{\\ln (2+\\kappa t)}}{1+t}&\\le \\frac{\\sqrt{\\ln (1+t)}}{1+t} + \\sqrt{\\ln (1+\\kappa )}\\le 2\\sqrt{\\ln (1+\\kappa )},\\\\\\Vert X^{\\kappa }\\Vert _{(0)} &=\\sup _{t>0} \\frac{|X(\\kappa t)-X(0)|}{2(1+t)} \\\\&\\le \\left(\\sup _{t,\\kappa >0}\\frac{|X(\\kappa t)-X(0)|}{2\\sqrt{\\ln (2+\\kappa t)}}\\right)\\left( \\sup _{t>0} \\frac{\\sqrt{\\ln (2+\\kappa t)}}{1+t}\\right)\\le 2\\zeta \\sqrt{\\ln (1+\\kappa )},$ and (REF ) follows with $C_X={\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|X(t)-X(0)|}{\\sqrt{\\ln (2+t)}}\\right].$ $\\Box $ Denote by $\\mu _0$ and $\\mu _{\\kappa }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\kappa }$ .",
"The following is the main result of this section, showing that the convergence rate $\\kappa ^{-1/2}\\sqrt{\\ln \\kappa }$ is sharp for the Wasserstein-1 metric.",
"Theorem 2.2 There exist positive constants $C_X$ and $c_X$ such that, for every $\\kappa \\ge 1$ , $c_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ The upper bound in (REF ) follows from (REF ) and Proposition REF .",
"To establish the lower bound, we use (REF ) with a particular $\\varphi $ .",
"If $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ is a bounded linear functional, then (REF ) and (REF ) imply ${\\mathbb {E}}\\varphi (W^{\\kappa }) - {\\mathbb {E}}\\varphi (W)=\\kappa ^{-1/2}{\\mathbb {E}}\\varphi (X^{\\kappa }).$ For $f\\in {\\mathcal {C}}_{(0)},$ define $\\varphi _{\\kappa }: f \\mapsto \\frac{f(t^*_{\\kappa })}{2},$ where $t^*_{\\kappa } = \\arg \\max _{0\\le t \\le 1} X^{\\kappa }(t).$ In particular, $t^*_{\\kappa }\\in [0,1]$ .",
"Then $\\varphi _{\\kappa }$ is a bounded linear functional on ${\\mathcal {C}}_{(0)}$ : $|\\varphi _{\\kappa }(f)|\\le \\frac{\\max _{0\\le t\\le 1} |f(t)|}{2}\\le \\max _{0\\le t\\le 1}\\frac{|f(t)|}{1+t}\\le \\Vert f\\Vert _{(0)}.$ Therefore, by (REF ) and (REF ), $\\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0) \\ge \\kappa ^{-1/2}|{\\mathbb {E}}\\varphi _{\\kappa }(X^{\\kappa })| =\\frac{{\\mathbb {E}}\\max _{0\\le t \\le 1} X^{\\kappa }(t)}{2\\sqrt{\\kappa }}.$ Next, define $\\zeta _{\\kappa }=\\frac{\\max \\limits _{0\\le t \\le 1} X^{\\kappa }(t)}{\\sqrt{\\ln (1+\\kappa )}}=\\frac{\\max \\limits _{0\\le t \\le \\kappa }X(t)-X(0)}{2\\sqrt{\\ln (1+\\kappa )}}, \\ \\ \\kappa \\ge 1;$ the second equality follows from (REF ).",
"By (REF ), the family $\\lbrace \\zeta _{\\kappa },\\ \\kappa \\ge 1\\rbrace $ is uniformly integrable, so that (REF ) implies $\\lim _{\\kappa \\rightarrow \\infty } {\\mathbb {E}}\\zeta _{\\kappa } = \\frac{1}{\\sqrt{2}},$ which, in turn, means $\\inf _{\\kappa \\ge 1} {\\mathbb {E}}\\zeta _{\\kappa }>0.$ The lower bound in (REF ), with $c_X=\\frac{1}{2}\\inf _{\\kappa \\ge 1} \\frac{{\\mathbb {E}}\\left[ \\max \\limits _{0\\le t \\le \\kappa } X(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}},$ now follows from (REF ) and (REF ), because ${\\mathbb {E}}X(0)=0$ .",
"$\\Box $ To get a better idea about numerical values of $C_X$ and $c_X$ , we need Proposition 2.3 Let $X=X(t),\\ t\\in {\\mathbb {R}},$ be a stationary Gaussian process with mean zero and covariance $e^{-2|t-s|}$ .",
"Define the random variable $\\bar{\\eta }=\\max _{0\\le t\\le 1} X(t).$ Then $1.2& < {\\mathbb {E}}\\bar{\\eta } < 3.2;\\\\\\mathbb {P}(\\bar{\\eta }>x)&\\le e^{-(x-3.2)^2/2},\\ x\\ge 3.2.$ Let $B=B(t)$ , $0\\le t\\le 1$ , be the standard Brownian bridge and $W=W(t)$ , $0\\le t\\le 1$ , a standard Brownian motion.",
"Then ${\\mathbb {E}}|B(t)-B(s)|^2 &= |t-s|-|t-s|^2,\\ {\\mathbb {E}}|X(t)-X(s)|^2 = 2(1-e^{-2|t-s|}),\\\\& {\\mathbb {E}}|W(t)-W(s)|^2=|t-s|,$ and, because $x-x^2\\le 1-e^{-2x}\\le 2x,\\ x\\ge 0,$ inequality (REF ) implies $2\\,{\\mathbb {E}}\\max _{0\\le t\\le 1} B(t) \\le {\\mathbb {E}}\\bar{\\eta } \\le 4\\, {\\mathbb {E}}\\max _{0\\le t\\le 1} W(t).$ It is well known (e.g.",
"[9]) that $\\mathbb {P}\\left(\\max _{0\\le t\\le 1} B(t) >x\\right) = e^{-2x^2}, \\ \\ \\mathbb {P}\\left(\\max _{0\\le t\\le 1} W(t) >x\\right) = \\frac{2}{\\sqrt{2\\pi }}\\int _{x}^{+\\infty }e^{-t^2/2}dt.$ Then ${\\mathbb {E}}\\max _{0\\le t\\le 1} B(t) &= \\int _0^{+\\infty } e^{-2x^2}\\, dx = \\frac{\\sqrt{2\\pi }}{4}>0.6,\\\\{\\mathbb {E}}\\max _{0\\le t\\le 1} W(t) & =\\frac{2}{\\sqrt{2\\pi }}\\int _0^{+\\infty }xe^{-x^2/2}\\, dx = \\sqrt{\\frac{2}{\\pi }}<0.8,$ and (REF ) follows.",
"After that, () is a re-statement of the Borell-TIS inequality (REF ).",
"$\\Box $ We can now show that the number $C_X$ defined in (REF ) satisfies $1.4 < C_X < 14.$ For the lower bound, note that $C_X\\ge {\\mathbb {E}}\\left[\\sup _{t>0} \\frac{X(t)-X(0)}{\\sqrt{\\ln (2+t)}}\\right],$ whereas $\\sup _{t>0} \\frac{X(t)-X(0)}{\\sqrt{\\ln (2+t)}}\\ge \\limsup _{T\\rightarrow \\infty } \\frac{\\max \\limits _{0\\le t\\le T} \\big (X(t)-X(0)\\big )}{\\sqrt{\\ln (2+T)}},$ and it remains to apply (REF ).",
"For the upper bound in (REF ), start by writing $C_X\\le {\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|X(t)|}{\\sqrt{\\ln (2+t)}}\\right] + \\frac{{\\mathbb {E}}|X(0)|}{\\sqrt{\\ln 2}}; \\ \\ \\ \\frac{{\\mathbb {E}}|X(0)|}{\\sqrt{\\ln 2}}=\\sqrt{\\frac{2}{\\pi \\ln 2}}\\approx 0.96.$ Next, let $\\bar{X}$ be the process consisting of iid copies of $X(t),\\ t\\in [0,1),$ on each of the intervals $[k-1,k)$ , $k=1,2,\\ldots $ .",
"Then ${\\mathbb {E}}X^2(t)={\\mathbb {E}}\\bar{X}^2(t)$ , ${\\mathbb {E}}X(t)X(s)\\ge {\\mathbb {E}}\\bar{X}(t)\\bar{X}(s)$ , and so ${\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|X(t)|}{\\sqrt{\\ln (2+t)}}\\right] \\le 2{\\mathbb {E}}\\left[\\sup _{t>0} \\frac{X(t)}{\\sqrt{\\ln (2+t)}}\\right] \\le 2{\\mathbb {E}}\\left[\\sup _{t>0} \\frac{\\bar{X}(t)}{\\sqrt{\\ln (2+t)}}\\right],$ where the first inequality follows from [19], and the second, from (REF ).",
"On the other hand, if $\\bar{\\eta }_k$ , $k\\ge 1$ , are iid copies of the random variable $\\bar{\\eta }$ from (REF ), then ${\\mathbb {E}}\\left[\\sup _{t>0} \\frac{\\bar{X}(t)}{\\sqrt{\\ln (2+t)}}\\right]\\le {\\mathbb {E}}\\left[\\sup _{k\\ge 1} \\frac{\\bar{\\eta }_k}{\\sqrt{\\ln (1+k)}}\\right].$ Using () with $x>5$ , $\\mathbb {P}\\Big (\\sup _{k\\ge 1} \\frac{\\bar{\\eta }_k}{\\sqrt{\\ln (1+k)}} > x\\Big ) &\\le \\sum _{k\\ge 1} \\mathbb {P}\\Big (\\bar{\\eta }_k>x\\sqrt{\\ln (1+k)}\\Big )\\\\&\\le \\sum _{k\\ge 1}\\frac{1}{(1+k)^{-(x-3.2)^2/2}}\\le \\frac{2}{(x-3.2)^2-2},$ and therefore ${\\mathbb {E}}\\left[\\sup _{k\\ge 1} \\frac{\\bar{\\eta }_k}{\\sqrt{\\ln (1+k)}}\\right]\\le 5+2\\int _5^{+\\infty } \\frac{dx}{(x-3.2)^2-2} <6.5.$ Then (REF ) follows from (REF ) – (REF ).",
"Next, we will show that the number $c_X$ defined in (REF ) satisfies $0.2 < c_X < 0.8.$ Indeed, the upper bound follows immediately from (REF ).",
"For the lower bound, start by noting that, for $N\\le \\kappa < N+1$ , ${\\mathbb {E}}\\left[ \\max \\limits _{0\\le t \\le \\kappa } X(t)\\right] \\ge {\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} X(k)\\right]$ and ${\\mathbb {E}}|X(k)-X(m)|^2 = 2(1-e^{-2})> 1$ .",
"Now take iid Gaussian $Y_k$ , $k=1,\\ldots , N$ with mean zero and variance $1/2$ .",
"Then ${\\mathbb {E}}|Y(k)-Y(m)|^2=1$ and so ${\\mathbb {E}}|X(k)-X(m)|^2\\ge {\\mathbb {E}}|Y(k)-Y(m)|^2.$ By (REF ), ${\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} X(k)\\right] \\ge {\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} Y(k)\\right],$ and, by [19], ${\\mathbb {E}}\\left[ \\max \\limits _{k=1,\\ldots , N} Y(k)\\right] \\ge 0.4\\sqrt{\\ln N},$ leading to the lower bound in (REF ).",
"To summarize, we can write (REF ) in a more explicit form $0.2\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le 14\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1.$ Theorem REF can be used to study convergence on a bounded interval.",
"For $T>0$ , let ${\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\mu _{0}^T$ and $\\mu _{\\kappa }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\kappa }$ .",
"Theorem 2.4 There exist positive constants $C_{X,T}$ and $c_{_{X,T}}$ such that, for every $\\kappa \\ge 1$ , $c_{_{X,T}}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T)\\le C_{X,T}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ If $f\\in {\\mathcal {C}}_{(0)}$ , then, for $0<t<T$ , $|f(t)|\\le (1+T) |f(t)|/(1+t)$ , so that $\\Vert f\\Vert _{{\\mathcal {C}}(0,T)}=\\max _{0\\le t\\le T} |f(t)|\\le (1+T)\\Vert f\\Vert _{(0)}$ and the upper bound in (REF ) follows from the upper bound in (REF ), with $C_{X,T}=(1+T)C_X$ .",
"The lower bound, with $c_{_{X,T}}=\\frac{1}{2} \\inf _{\\kappa \\ge 1}\\frac{{\\mathbb {E}}\\left[\\max _{0\\le t \\le T} X^{\\kappa }(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}},$ follows after repeating the corresponding steps in the proof of Theorem REF .",
"$\\Box $ Discrete Time Consider a stationary Gaussian sequence $X=\\lbrace X_n,\\ n\\ge 0\\rbrace ,$ with ${\\mathbb {E}}X_n=0$ and ${\\mathbb {E}}X_{k+n} X_k=(1-a)a^n/(1+a),\\ n,k\\ge 0$ , where $a\\in (-1,1)$ and $a=0$ corresponds to the sequence of iid standard Gaussian random variables.",
"The variance of $X_n$ is chosen so that, for all $a\\in (-1,1)$ , the covariance function $R(n)={\\mathbb {E}}X_{k+n} X_n$ of $X$ satisfies $R(0)+2\\sum _{n=1}^{\\infty } R(n)=1.$ Using a collection $\\xi _k,\\ k=0,\\pm 1,\\pm 2,\\ldots $ of iid standard normal random variables, we get the discrete-time analogs of () and (): $X_n&=(1-a)\\sum _{k=-\\infty }^n a^{n-k}\\xi _k,\\\\X_{n+1}&=aX_n+\\xi _{n+1};$ in (), the initial condition $X_0$ is independent of $\\xi _k,\\ k\\ge 1,$ and is a normal random variable with mean 0 and variance $(1-a)/(1+a)$ .",
"For $x>0$ , let $\\lfloor x \\rfloor $ denote the largest integer that is less than or equal to $x$ .",
"Define the processes $W^{\\kappa }$ by $W^{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} X_{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ The second term on the right-hand side of (REF ) ensures that $W^{\\kappa }$ is a continuous function of $t$ .",
"The case $a=0$ , that is, the Gaussian version of the original Donsker theorem, is of special interest; the corresponding process $W^{\\kappa }$ will be denoted by $S_{\\kappa }$ : $S_{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } \\xi _n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} \\xi _{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ We have $\\mathbb {P}(W^{\\kappa }\\in {\\mathcal {C}}_{(0)})=1$ for every $\\kappa \\ge 1$ , and $a\\in (-1,1)$ , because the ergodic theorem implies $\\lim _{t\\rightarrow \\infty } \\frac{1}{\\lfloor \\kappa t \\rfloor }\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n = {\\mathbb {E}}X_0 = 0$ with probability one.",
"Let $W=W(t),\\ t\\ge 0$ , be a standard Brownian motion, and denote by $\\mu _0$ and $\\mu _{\\kappa }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\kappa }$ .",
"The following is the discrete-time analog of Theorem REF .",
"Theorem 3.1 There exist positive constants $C_a$ and $c_a$ such that, for every $\\kappa \\ge 1$ , $c_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2} \\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ The steps are the same as in the proof of Theorem REF .",
"Substituting (REF ) in (REF ) and changing the order of summation, $W^{\\kappa }(t)=S_{\\kappa }(t)+\\frac{X^{\\kappa }(t)}{\\sqrt{\\kappa }},$ where $S_{\\kappa }$ is from (REF ) and $X^{\\kappa }(t) = \\left( (a-a^{\\lfloor \\kappa t \\rfloor })\\sum _{n=-\\infty }^{0}a^{-n}\\xi _n -\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor }a^{\\lfloor \\kappa t \\rfloor -n}\\xi _n\\right).$ As a result, it is enough to establish (REF ) when $a=0$ : $c_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Then, similar to the continuous time case, we see that ${\\mathbb {E}}\\Vert X^{\\kappa }\\Vert _{(0)} \\le \\bar{C}_a \\sqrt{\\ln (1+\\kappa )},$ with a suitable constant $\\bar{C}_a$ , and then (REF ) follows from (REF ) with $C_a=C_0+\\bar{C}_a$ and $c_a=c_0$ .",
"To prove (REF ), we choose the random variables $\\xi _k$ in (REF ) as the increments of the Brownian motion $W$ : $\\frac{\\xi _n}{\\sqrt{\\kappa }} = W(n/\\kappa )- W((n-1)/\\kappa ).$ Then, for $(n-1)/\\kappa \\le t \\le n/\\kappa $ , the process $t\\mapsto S_{\\kappa }- W $ is a Brownian bridge, and, for every function $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfying (REF ), $|{\\mathbb {E}}\\varphi (S_{\\kappa })-{\\mathbb {E}}\\varphi (W)|\\le {\\mathbb {E}}\\Vert B^{\\kappa }\\Vert _{(0)},$ where $B^{\\kappa }$ is a collection of independent Brownian bridges on $[(n-1)/\\kappa , n/\\kappa ]$ , $n=1,2,\\ldots $ .",
"Direct computations show that, for $N=1,2,\\ldots $ , $& \\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)|\\le 8\\sqrt{{\\ln (N+1)}},\\\\&\\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } B^{\\kappa }(t) \\ge 0.3\\sqrt{{\\ln (N+1)}};$ the numbers $0.3$ and 8 do not necessarily provide optimal bounds.",
"Indeed, let $B=B(t),\\ t\\in [0,1],$ be the standard Brownian bridge, and let $\\eta =\\max _{0\\le t\\le 1} B(t).$ Then $\\sqrt{\\kappa } \\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)| = {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k|,$ where $\\eta _k,\\ k=1,\\ldots , N$ , are iid copies of $\\eta $ .",
"Also, ${\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k\\le {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k| \\le 2 {\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k.$ To derive (REF ), we repeat the arguments from the proof of Lemma 10.1 in [19] using (REF ) and conclude that ${\\mathbb {E}}\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\le 4 \\sqrt{\\ln (N+1)}$ .",
"Similarly, for (), we repeat the proof of Lemma 10.2 in [19].",
"Next, denote by $\\bar{B}$ the process $B^{\\kappa }$ corresponding to $\\kappa =1$ , that is, the collection of independent standard Brownian bridges on $[n-1,n]$ , $n\\ge 1$ .",
"Then we get the upper bound in (REF ), with $C_0={\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|\\bar{B}(t)|}{\\sqrt{\\ln (2+t)}}\\right]< 14,$ by combining (REF ) and (REF ); the upper bound on $C_0$ is from (REF ), because, by (REF ), $C_0\\le C_X$ .",
"The lower bound in (REF ), with $c_0=\\frac{1}{2}\\inf _{\\kappa \\ge 1} \\frac{{\\mathbb {E}}\\left[ \\max _{0\\le t \\le \\kappa } \\bar{B}(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}}> 0.2,$ follows from () after the same arguments as in the proof of Theorem REF ; the lower bound on $c_0$ follows from ().",
"$\\Box $ For $T>0$ , let ${\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\mu _{0}^T$ and $\\mu _{\\kappa }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\kappa }$ .",
"The discrete-time version of Theorem REF is obvious.",
"When $a=0$ , and there is no continuous-time analog, we also have the following result (cf.",
"[2]).",
"Proposition 3.2 If $\\mu _{\\kappa }^T$ is the measure on ${\\mathcal {C}}(0,T)$ generated by the process $S_{\\kappa }$ from (REF ), then $\\lim _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{\\kappa }{\\ln \\kappa }} \\,\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) = {\\sqrt{2}}.$ Using the random variables $\\eta _k$ from the proof of Theorem REF , $\\limsup _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }}\\; \\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\le \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|\\right]}{\\sqrt{\\ln N}},$ and $\\liminf _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }} \\;\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\ge \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\right]}{\\sqrt{\\ln N}}.$ By (REF ) and [24], $\\lim _{N\\rightarrow \\infty }\\frac{\\max \\limits _{k=1,\\ldots ,N}\\eta _k}{\\sqrt{\\ln N}}=\\lim _{N\\rightarrow \\infty } \\frac{\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|}{\\sqrt{\\ln N}}={\\sqrt{2}}$ with probability 1; then uniform integrability [22] implies (REF ).",
"$\\Box $ Applications Let $W$ be a standard Brownian motion and let $W^{\\kappa }$ be the process from (REF ) or (REF ).",
"Consider a continuous mapping $\\Psi : {\\mathcal {C}}_{(0)} \\rightarrow {\\mathcal {C}}_{(0)}$ .",
"Denote by $\\mu _{0,\\psi }$ and $\\mu _{\\kappa ,\\psi }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $\\Psi (W)$ and $\\Psi (W^{\\kappa })$ .",
"Proposition 4.1 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi } = \\mu _{0,\\psi }$ .",
"Moreover, if there exists a number $C_{\\psi }$ such that, for all $f,g\\in {\\mathcal {C}}_{(0)}$ , $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le C_{\\psi }\\Vert f-g\\Vert _{(0)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi },\\mu _{0,\\psi })\\le \\bar{C}_XC_{\\psi }\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.",
"Weak convergence follows by the continuous mapping theorem (e.g.",
"[4]).",
"To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfies (REF ), then $|\\varphi (\\Psi (f))-\\varphi (\\Psi (g))|\\le C_{\\psi }\\Vert f-g\\Vert _{(0)}.$ $\\Box $ Example 1.",
"Given ${\\alpha }>0$ , let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=-{\\alpha }\\int _0^t Y(s)\\, ds + W(t),\\ \\ t\\ge 0.$ Then $Y^{\\kappa }, Y\\in {\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\nu _{\\kappa }, \\nu $ on ${\\mathcal {C}}_{(0)}$ satisfy $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le 2\\bar{C}_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1;$ the constant $\\bar{C}_X$ is from Proposition REF .",
"Indeed, by direct computation, $Y^{\\kappa }(t)=\\Psi _{\\alpha }(W^{\\kappa })(t), \\ \\ Y(t)=\\Psi _{\\alpha }(W)(t),$ where $\\Psi _{\\alpha } : f(t)\\mapsto f(t)-{\\alpha }\\int _0^t e^{-{\\alpha }(t-s)} f(s) \\, ds,\\ \\ f \\in \\mathcal {C}_{(0)},$ is a linear operator.",
"To see that $\\Psi _{\\alpha }$ maps ${\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t}&\\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^t e^{-{\\alpha }(t-s)}|f(s)|\\, ds\\\\& \\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^T e^{-{\\alpha }(t-s)}|f(s)|\\, ds \\\\&+ \\frac{{\\alpha }e^{-{\\alpha }t}}{1+t}\\int _T^t (1+s)e^{{\\alpha }s}\\, \\frac{|f(s)|}{1+s}\\, ds.$ If $\\lim _{t\\rightarrow \\infty }|f(t)|/(1+t) = 0$ , then, for every $\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\varepsilon $ , $s>T$ .",
"As a result, keeping in mind that $\\frac{{\\alpha }}{1+t}\\int _T^t (1+s) e^{{\\alpha }s}\\, ds \\le {\\alpha }\\int _0^t e^{{\\alpha }s}\\, ds \\le e^{{\\alpha }t},$ we compute $\\limsup _{t\\rightarrow \\infty } \\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t} \\le \\varepsilon $ and conclude that $\\lim _{t\\rightarrow \\infty } |\\Psi _{\\alpha }(f)(t)|/(1+t)=0$ .",
"Similarly, $\\Vert \\Psi _{\\alpha }(f)\\Vert _{(0)} \\le \\Vert f\\Vert _{(0)} \\left(1+\\alpha \\int _0^{+\\infty }e^{-\\alpha t}\\, dt \\right) =2\\Vert f\\Vert _{(0)},$ so that (REF ) holds with $C_{\\psi }=2$ .",
"$\\Box $ The analog of Proposition REF on a bounded interval is as follows.",
"Let $\\Psi ^T$ be a continuous mapping of ${\\mathcal {C}}(0,T)$ to itself.",
"Denote by $\\mu _{0,\\psi }^T$ and $\\mu _{\\kappa ,\\psi }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $\\Psi ^T(W)$ and $\\Psi ^T(W^{\\kappa })$ .",
"Proposition 4.2 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi }^T = \\mu _{0,\\psi }^T$ .",
"Moreover, if there exists a number $C_{\\psi }^T$ such that, for all $f,g\\in {\\mathcal {C}}(0,T)$ , $\\Vert \\Psi ^T(f)-\\Psi ^T(g)\\Vert _{{\\mathcal {C}}(0,T)}\\le C_{\\psi }^T\\Vert f-g\\Vert _{{\\mathcal {C}}(0,T)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi }^T,\\mu _{0,\\psi }^T)\\le (1+T)\\bar{C}_{X}C_{\\psi }^T\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.",
"Example 2.",
"Let the function $b=b(x), \\ x\\in {\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\le K|x-y|,\\ x,y\\in {\\mathbb {R}},$ and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = \\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ 0\\le t\\le T.$ If $\\nu _{\\kappa }^T, \\nu ^T$ are the corresponding measures on ${\\mathcal {C}}(0,T)$ , then $\\mathrm {d}_{{w}}(\\nu _{\\kappa }^T,\\nu ^T)\\le (1+T)\\bar{C}_Xe^{KT}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}(0,T)$ , define $\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ 0\\le t\\le T.$ By direct computation (e.g.",
"[10]), we have (REF ) with $C_{\\psi }^T = e^{KT}$ .",
"$ $ $\\Box $ Example 3.",
"Let us combine Examples 1 and 2.",
"Take a positive number ${\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t)& = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds+\\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\\\Y(t)&= -{\\alpha }\\int _0^t Y(s)\\, ds+\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ t\\ge 0.$ If ${\\alpha }>K$ , then $Y^{\\kappa }, Y \\in {\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\nu _{\\kappa }, \\nu $ , $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le \\frac{2{\\alpha }\\bar{C}_X}{{\\alpha }-K}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}_{(0)}$ , define $\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\alpha }\\int _0^t y(s)\\, ds + \\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ t\\ge 0.$ Using variation of parameters formula and (REF ), $\\Psi (f)(t)=\\int _0^t e^{-{\\alpha }(t-s)} b\\big (\\Psi (f)(s)\\big )\\, ds + \\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\Psi $ maps ${\\mathcal {C}}_{(0)}$ to itself.",
"In particular, using (REF ) and (REF ), $\\Psi (f)(t)-\\Psi (g)(t) =\\int _0^t e^{-{\\alpha }(t-s)} \\Big ( b\\big (\\Psi (f)(s)\\big )-b\\big (\\Psi (g)(s)\\big )\\Big )\\, ds + \\Psi _{\\alpha }(f-g)(t)$ so that $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\le K \\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\int _0^{\\infty } e^{-as}\\, ds + 2\\Vert f-g\\Vert _{(0)}.$ As a result, if $\\alpha >K$ , then $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le \\frac{2{\\alpha }}{{\\alpha }-K}\\Vert f-g\\Vert _{(0)},$ and (REF ) follows from (REF ).",
"$\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .",
"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.",
"When the underlying processes are Gaussian, some of the approximation errors are not present.",
"In continuous time, the first two steps are the equality (REF ).",
"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.",
"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.",
"For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.",
"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.",
"For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].",
"Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.",
"For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."
],
[
"Discrete Time",
"Consider a stationary Gaussian sequence $X=\\lbrace X_n,\\ n\\ge 0\\rbrace ,$ with ${\\mathbb {E}}X_n=0$ and ${\\mathbb {E}}X_{k+n} X_k=(1-a)a^n/(1+a),\\ n,k\\ge 0$ , where $a\\in (-1,1)$ and $a=0$ corresponds to the sequence of iid standard Gaussian random variables.",
"The variance of $X_n$ is chosen so that, for all $a\\in (-1,1)$ , the covariance function $R(n)={\\mathbb {E}}X_{k+n} X_n$ of $X$ satisfies $R(0)+2\\sum _{n=1}^{\\infty } R(n)=1.$ Using a collection $\\xi _k,\\ k=0,\\pm 1,\\pm 2,\\ldots $ of iid standard normal random variables, we get the discrete-time analogs of () and (): $X_n&=(1-a)\\sum _{k=-\\infty }^n a^{n-k}\\xi _k,\\\\X_{n+1}&=aX_n+\\xi _{n+1};$ in (), the initial condition $X_0$ is independent of $\\xi _k,\\ k\\ge 1,$ and is a normal random variable with mean 0 and variance $(1-a)/(1+a)$ .",
"For $x>0$ , let $\\lfloor x \\rfloor $ denote the largest integer that is less than or equal to $x$ .",
"Define the processes $W^{\\kappa }$ by $W^{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} X_{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ The second term on the right-hand side of (REF ) ensures that $W^{\\kappa }$ is a continuous function of $t$ .",
"The case $a=0$ , that is, the Gaussian version of the original Donsker theorem, is of special interest; the corresponding process $W^{\\kappa }$ will be denoted by $S_{\\kappa }$ : $S_{\\kappa }(t)=\\frac{1}{\\sqrt{\\kappa }}\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } \\xi _n+\\frac{\\kappa t-\\lfloor \\kappa t \\rfloor }{\\sqrt{\\kappa }} \\xi _{\\lfloor \\kappa t \\rfloor +1},\\ t\\ge 0,\\ \\kappa \\ge 1.$ We have $\\mathbb {P}(W^{\\kappa }\\in {\\mathcal {C}}_{(0)})=1$ for every $\\kappa \\ge 1$ , and $a\\in (-1,1)$ , because the ergodic theorem implies $\\lim _{t\\rightarrow \\infty } \\frac{1}{\\lfloor \\kappa t \\rfloor }\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor } X_n = {\\mathbb {E}}X_0 = 0$ with probability one.",
"Let $W=W(t),\\ t\\ge 0$ , be a standard Brownian motion, and denote by $\\mu _0$ and $\\mu _{\\kappa }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $W$ and $W^{\\kappa }$ .",
"The following is the discrete-time analog of Theorem REF .",
"Theorem 3.1 There exist positive constants $C_a$ and $c_a$ such that, for every $\\kappa \\ge 1$ , $c_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2} \\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_a\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ The steps are the same as in the proof of Theorem REF .",
"Substituting (REF ) in (REF ) and changing the order of summation, $W^{\\kappa }(t)=S_{\\kappa }(t)+\\frac{X^{\\kappa }(t)}{\\sqrt{\\kappa }},$ where $S_{\\kappa }$ is from (REF ) and $X^{\\kappa }(t) = \\left( (a-a^{\\lfloor \\kappa t \\rfloor })\\sum _{n=-\\infty }^{0}a^{-n}\\xi _n -\\sum _{n=1}^{\\lfloor \\kappa t \\rfloor }a^{\\lfloor \\kappa t \\rfloor -n}\\xi _n\\right).$ As a result, it is enough to establish (REF ) when $a=0$ : $c_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}\\le \\mathrm {d}_{{w}}(\\mu _{\\kappa },\\mu _0)\\le C_0\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Then, similar to the continuous time case, we see that ${\\mathbb {E}}\\Vert X^{\\kappa }\\Vert _{(0)} \\le \\bar{C}_a \\sqrt{\\ln (1+\\kappa )},$ with a suitable constant $\\bar{C}_a$ , and then (REF ) follows from (REF ) with $C_a=C_0+\\bar{C}_a$ and $c_a=c_0$ .",
"To prove (REF ), we choose the random variables $\\xi _k$ in (REF ) as the increments of the Brownian motion $W$ : $\\frac{\\xi _n}{\\sqrt{\\kappa }} = W(n/\\kappa )- W((n-1)/\\kappa ).$ Then, for $(n-1)/\\kappa \\le t \\le n/\\kappa $ , the process $t\\mapsto S_{\\kappa }- W $ is a Brownian bridge, and, for every function $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfying (REF ), $|{\\mathbb {E}}\\varphi (S_{\\kappa })-{\\mathbb {E}}\\varphi (W)|\\le {\\mathbb {E}}\\Vert B^{\\kappa }\\Vert _{(0)},$ where $B^{\\kappa }$ is a collection of independent Brownian bridges on $[(n-1)/\\kappa , n/\\kappa ]$ , $n=1,2,\\ldots $ .",
"Direct computations show that, for $N=1,2,\\ldots $ , $& \\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)|\\le 8\\sqrt{{\\ln (N+1)}},\\\\&\\sqrt{\\kappa }\\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } B^{\\kappa }(t) \\ge 0.3\\sqrt{{\\ln (N+1)}};$ the numbers $0.3$ and 8 do not necessarily provide optimal bounds.",
"Indeed, let $B=B(t),\\ t\\in [0,1],$ be the standard Brownian bridge, and let $\\eta =\\max _{0\\le t\\le 1} B(t).$ Then $\\sqrt{\\kappa } \\;{\\mathbb {E}}\\max _{0\\le t\\le N/\\kappa } |B^{\\kappa }(t)| = {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k|,$ where $\\eta _k,\\ k=1,\\ldots , N$ , are iid copies of $\\eta $ .",
"Also, ${\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k\\le {\\mathbb {E}}\\max _{k=1,\\ldots ,N}|\\eta _k| \\le 2 {\\mathbb {E}}\\max _{k=1,\\ldots ,N}\\eta _k.$ To derive (REF ), we repeat the arguments from the proof of Lemma 10.1 in [19] using (REF ) and conclude that ${\\mathbb {E}}\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\le 4 \\sqrt{\\ln (N+1)}$ .",
"Similarly, for (), we repeat the proof of Lemma 10.2 in [19].",
"Next, denote by $\\bar{B}$ the process $B^{\\kappa }$ corresponding to $\\kappa =1$ , that is, the collection of independent standard Brownian bridges on $[n-1,n]$ , $n\\ge 1$ .",
"Then we get the upper bound in (REF ), with $C_0={\\mathbb {E}}\\left[\\sup _{t>0} \\frac{|\\bar{B}(t)|}{\\sqrt{\\ln (2+t)}}\\right]< 14,$ by combining (REF ) and (REF ); the upper bound on $C_0$ is from (REF ), because, by (REF ), $C_0\\le C_X$ .",
"The lower bound in (REF ), with $c_0=\\frac{1}{2}\\inf _{\\kappa \\ge 1} \\frac{{\\mathbb {E}}\\left[ \\max _{0\\le t \\le \\kappa } \\bar{B}(t)\\right]}{\\sqrt{\\ln (1+\\kappa )}}> 0.2,$ follows from () after the same arguments as in the proof of Theorem REF ; the lower bound on $c_0$ follows from ().",
"$\\Box $ For $T>0$ , let ${\\mathcal {C}}(0,T)$ be the space of continuous functions on $[0,T]$ with the sup norm, and denote by $\\mu _{0}^T$ and $\\mu _{\\kappa }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $W$ and $W^{\\kappa }$ .",
"The discrete-time version of Theorem REF is obvious.",
"When $a=0$ , and there is no continuous-time analog, we also have the following result (cf.",
"[2]).",
"Proposition 3.2 If $\\mu _{\\kappa }^T$ is the measure on ${\\mathcal {C}}(0,T)$ generated by the process $S_{\\kappa }$ from (REF ), then $\\lim _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{\\kappa }{\\ln \\kappa }} \\,\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) = {\\sqrt{2}}.$ Using the random variables $\\eta _k$ from the proof of Theorem REF , $\\limsup _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }}\\; \\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\le \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|\\right]}{\\sqrt{\\ln N}},$ and $\\liminf _{\\kappa \\rightarrow \\infty } \\sqrt{\\frac{ \\kappa }{\\ln \\kappa }} \\;\\mathrm {d}_{{w}}(\\mu _{\\kappa }^T,\\mu _0^T) \\ge \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}\\left[\\max \\limits _{k=1,\\ldots ,N}\\eta _k\\right]}{\\sqrt{\\ln N}}.$ By (REF ) and [24], $\\lim _{N\\rightarrow \\infty }\\frac{\\max \\limits _{k=1,\\ldots ,N}\\eta _k}{\\sqrt{\\ln N}}=\\lim _{N\\rightarrow \\infty } \\frac{\\max \\limits _{k=1,\\ldots ,N}|\\eta _k|}{\\sqrt{\\ln N}}={\\sqrt{2}}$ with probability 1; then uniform integrability [22] implies (REF ).",
"$\\Box $ Applications Let $W$ be a standard Brownian motion and let $W^{\\kappa }$ be the process from (REF ) or (REF ).",
"Consider a continuous mapping $\\Psi : {\\mathcal {C}}_{(0)} \\rightarrow {\\mathcal {C}}_{(0)}$ .",
"Denote by $\\mu _{0,\\psi }$ and $\\mu _{\\kappa ,\\psi }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $\\Psi (W)$ and $\\Psi (W^{\\kappa })$ .",
"Proposition 4.1 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi } = \\mu _{0,\\psi }$ .",
"Moreover, if there exists a number $C_{\\psi }$ such that, for all $f,g\\in {\\mathcal {C}}_{(0)}$ , $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le C_{\\psi }\\Vert f-g\\Vert _{(0)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi },\\mu _{0,\\psi })\\le \\bar{C}_XC_{\\psi }\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.",
"Weak convergence follows by the continuous mapping theorem (e.g.",
"[4]).",
"To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfies (REF ), then $|\\varphi (\\Psi (f))-\\varphi (\\Psi (g))|\\le C_{\\psi }\\Vert f-g\\Vert _{(0)}.$ $\\Box $ Example 1.",
"Given ${\\alpha }>0$ , let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=-{\\alpha }\\int _0^t Y(s)\\, ds + W(t),\\ \\ t\\ge 0.$ Then $Y^{\\kappa }, Y\\in {\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\nu _{\\kappa }, \\nu $ on ${\\mathcal {C}}_{(0)}$ satisfy $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le 2\\bar{C}_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1;$ the constant $\\bar{C}_X$ is from Proposition REF .",
"Indeed, by direct computation, $Y^{\\kappa }(t)=\\Psi _{\\alpha }(W^{\\kappa })(t), \\ \\ Y(t)=\\Psi _{\\alpha }(W)(t),$ where $\\Psi _{\\alpha } : f(t)\\mapsto f(t)-{\\alpha }\\int _0^t e^{-{\\alpha }(t-s)} f(s) \\, ds,\\ \\ f \\in \\mathcal {C}_{(0)},$ is a linear operator.",
"To see that $\\Psi _{\\alpha }$ maps ${\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t}&\\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^t e^{-{\\alpha }(t-s)}|f(s)|\\, ds\\\\& \\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^T e^{-{\\alpha }(t-s)}|f(s)|\\, ds \\\\&+ \\frac{{\\alpha }e^{-{\\alpha }t}}{1+t}\\int _T^t (1+s)e^{{\\alpha }s}\\, \\frac{|f(s)|}{1+s}\\, ds.$ If $\\lim _{t\\rightarrow \\infty }|f(t)|/(1+t) = 0$ , then, for every $\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\varepsilon $ , $s>T$ .",
"As a result, keeping in mind that $\\frac{{\\alpha }}{1+t}\\int _T^t (1+s) e^{{\\alpha }s}\\, ds \\le {\\alpha }\\int _0^t e^{{\\alpha }s}\\, ds \\le e^{{\\alpha }t},$ we compute $\\limsup _{t\\rightarrow \\infty } \\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t} \\le \\varepsilon $ and conclude that $\\lim _{t\\rightarrow \\infty } |\\Psi _{\\alpha }(f)(t)|/(1+t)=0$ .",
"Similarly, $\\Vert \\Psi _{\\alpha }(f)\\Vert _{(0)} \\le \\Vert f\\Vert _{(0)} \\left(1+\\alpha \\int _0^{+\\infty }e^{-\\alpha t}\\, dt \\right) =2\\Vert f\\Vert _{(0)},$ so that (REF ) holds with $C_{\\psi }=2$ .",
"$\\Box $ The analog of Proposition REF on a bounded interval is as follows.",
"Let $\\Psi ^T$ be a continuous mapping of ${\\mathcal {C}}(0,T)$ to itself.",
"Denote by $\\mu _{0,\\psi }^T$ and $\\mu _{\\kappa ,\\psi }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $\\Psi ^T(W)$ and $\\Psi ^T(W^{\\kappa })$ .",
"Proposition 4.2 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi }^T = \\mu _{0,\\psi }^T$ .",
"Moreover, if there exists a number $C_{\\psi }^T$ such that, for all $f,g\\in {\\mathcal {C}}(0,T)$ , $\\Vert \\Psi ^T(f)-\\Psi ^T(g)\\Vert _{{\\mathcal {C}}(0,T)}\\le C_{\\psi }^T\\Vert f-g\\Vert _{{\\mathcal {C}}(0,T)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi }^T,\\mu _{0,\\psi }^T)\\le (1+T)\\bar{C}_{X}C_{\\psi }^T\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.",
"Example 2.",
"Let the function $b=b(x), \\ x\\in {\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\le K|x-y|,\\ x,y\\in {\\mathbb {R}},$ and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = \\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ 0\\le t\\le T.$ If $\\nu _{\\kappa }^T, \\nu ^T$ are the corresponding measures on ${\\mathcal {C}}(0,T)$ , then $\\mathrm {d}_{{w}}(\\nu _{\\kappa }^T,\\nu ^T)\\le (1+T)\\bar{C}_Xe^{KT}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}(0,T)$ , define $\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ 0\\le t\\le T.$ By direct computation (e.g.",
"[10]), we have (REF ) with $C_{\\psi }^T = e^{KT}$ .",
"$ $ $\\Box $ Example 3.",
"Let us combine Examples 1 and 2.",
"Take a positive number ${\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t)& = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds+\\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\\\Y(t)&= -{\\alpha }\\int _0^t Y(s)\\, ds+\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ t\\ge 0.$ If ${\\alpha }>K$ , then $Y^{\\kappa }, Y \\in {\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\nu _{\\kappa }, \\nu $ , $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le \\frac{2{\\alpha }\\bar{C}_X}{{\\alpha }-K}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}_{(0)}$ , define $\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\alpha }\\int _0^t y(s)\\, ds + \\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ t\\ge 0.$ Using variation of parameters formula and (REF ), $\\Psi (f)(t)=\\int _0^t e^{-{\\alpha }(t-s)} b\\big (\\Psi (f)(s)\\big )\\, ds + \\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\Psi $ maps ${\\mathcal {C}}_{(0)}$ to itself.",
"In particular, using (REF ) and (REF ), $\\Psi (f)(t)-\\Psi (g)(t) =\\int _0^t e^{-{\\alpha }(t-s)} \\Big ( b\\big (\\Psi (f)(s)\\big )-b\\big (\\Psi (g)(s)\\big )\\Big )\\, ds + \\Psi _{\\alpha }(f-g)(t)$ so that $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\le K \\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\int _0^{\\infty } e^{-as}\\, ds + 2\\Vert f-g\\Vert _{(0)}.$ As a result, if $\\alpha >K$ , then $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le \\frac{2{\\alpha }}{{\\alpha }-K}\\Vert f-g\\Vert _{(0)},$ and (REF ) follows from (REF ).",
"$\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .",
"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.",
"When the underlying processes are Gaussian, some of the approximation errors are not present.",
"In continuous time, the first two steps are the equality (REF ).",
"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.",
"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.",
"For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.",
"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.",
"For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].",
"Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.",
"For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."
],
[
"Applications",
"Let $W$ be a standard Brownian motion and let $W^{\\kappa }$ be the process from (REF ) or (REF ).",
"Consider a continuous mapping $\\Psi : {\\mathcal {C}}_{(0)} \\rightarrow {\\mathcal {C}}_{(0)}$ .",
"Denote by $\\mu _{0,\\psi }$ and $\\mu _{\\kappa ,\\psi }$ the measures on ${\\mathcal {C}}_{(0)}$ generated by the processes $\\Psi (W)$ and $\\Psi (W^{\\kappa })$ .",
"Proposition 4.1 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi } = \\mu _{0,\\psi }$ .",
"Moreover, if there exists a number $C_{\\psi }$ such that, for all $f,g\\in {\\mathcal {C}}_{(0)}$ , $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le C_{\\psi }\\Vert f-g\\Vert _{(0)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi },\\mu _{0,\\psi })\\le \\bar{C}_XC_{\\psi }\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.",
"Weak convergence follows by the continuous mapping theorem (e.g.",
"[4]).",
"To establish (REF ), we use either (REF ) or the upper bound in (REF ) and note that if $\\varphi : {\\mathcal {C}}_{(0)}\\rightarrow {\\mathbb {R}}$ satisfies (REF ), then $|\\varphi (\\Psi (f))-\\varphi (\\Psi (g))|\\le C_{\\psi }\\Vert f-g\\Vert _{(0)}.$ $\\Box $ Example 1.",
"Given ${\\alpha }>0$ , let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=-{\\alpha }\\int _0^t Y(s)\\, ds + W(t),\\ \\ t\\ge 0.$ Then $Y^{\\kappa }, Y\\in {\\mathcal {C}}_{(0)}$ , and the corresponding measures $\\nu _{\\kappa }, \\nu $ on ${\\mathcal {C}}_{(0)}$ satisfy $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le 2\\bar{C}_X\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},\\ \\kappa \\ge 1;$ the constant $\\bar{C}_X$ is from Proposition REF .",
"Indeed, by direct computation, $Y^{\\kappa }(t)=\\Psi _{\\alpha }(W^{\\kappa })(t), \\ \\ Y(t)=\\Psi _{\\alpha }(W)(t),$ where $\\Psi _{\\alpha } : f(t)\\mapsto f(t)-{\\alpha }\\int _0^t e^{-{\\alpha }(t-s)} f(s) \\, ds,\\ \\ f \\in \\mathcal {C}_{(0)},$ is a linear operator.",
"To see that $\\Psi _{\\alpha }$ maps ${\\mathcal {C}}_{(0)}$ to itself, note that, for every $t>T>0$ , $\\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t}&\\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^t e^{-{\\alpha }(t-s)}|f(s)|\\, ds\\\\& \\le \\frac{|f(t)|}{1+t}+\\frac{{\\alpha }}{1+t}\\int _0^T e^{-{\\alpha }(t-s)}|f(s)|\\, ds \\\\&+ \\frac{{\\alpha }e^{-{\\alpha }t}}{1+t}\\int _T^t (1+s)e^{{\\alpha }s}\\, \\frac{|f(s)|}{1+s}\\, ds.$ If $\\lim _{t\\rightarrow \\infty }|f(t)|/(1+t) = 0$ , then, for every $\\varepsilon >0$ , we can find $T$ so that $|f(s)|/(1+s)<\\varepsilon $ , $s>T$ .",
"As a result, keeping in mind that $\\frac{{\\alpha }}{1+t}\\int _T^t (1+s) e^{{\\alpha }s}\\, ds \\le {\\alpha }\\int _0^t e^{{\\alpha }s}\\, ds \\le e^{{\\alpha }t},$ we compute $\\limsup _{t\\rightarrow \\infty } \\frac{|\\Psi _{\\alpha }(f)(t)|}{1+t} \\le \\varepsilon $ and conclude that $\\lim _{t\\rightarrow \\infty } |\\Psi _{\\alpha }(f)(t)|/(1+t)=0$ .",
"Similarly, $\\Vert \\Psi _{\\alpha }(f)\\Vert _{(0)} \\le \\Vert f\\Vert _{(0)} \\left(1+\\alpha \\int _0^{+\\infty }e^{-\\alpha t}\\, dt \\right) =2\\Vert f\\Vert _{(0)},$ so that (REF ) holds with $C_{\\psi }=2$ .",
"$\\Box $ The analog of Proposition REF on a bounded interval is as follows.",
"Let $\\Psi ^T$ be a continuous mapping of ${\\mathcal {C}}(0,T)$ to itself.",
"Denote by $\\mu _{0,\\psi }^T$ and $\\mu _{\\kappa ,\\psi }^T$ the measures on ${\\mathcal {C}}(0,T)$ generated by the processes $\\Psi ^T(W)$ and $\\Psi ^T(W^{\\kappa })$ .",
"Proposition 4.2 We have weak convergence $\\lim _{\\kappa \\rightarrow \\infty } \\mu _{\\kappa ,\\psi }^T = \\mu _{0,\\psi }^T$ .",
"Moreover, if there exists a number $C_{\\psi }^T$ such that, for all $f,g\\in {\\mathcal {C}}(0,T)$ , $\\Vert \\Psi ^T(f)-\\Psi ^T(g)\\Vert _{{\\mathcal {C}}(0,T)}\\le C_{\\psi }^T\\Vert f-g\\Vert _{{\\mathcal {C}}(0,T)},$ then $\\mathrm {d}_{{w}}(\\mu _{\\kappa ,\\psi }^T,\\mu _{0,\\psi }^T)\\le (1+T)\\bar{C}_{X}C_{\\psi }^T\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2},$ with $\\bar{C}_X=C_X$ from (REF ) in continuous time and $\\bar{C}_X=C_a$ from (REF ) in discrete time.",
"Example 2.",
"Let the function $b=b(x), \\ x\\in {\\mathbb {R}},$ satisfy $|b(x)-b(y)|\\le K|x-y|,\\ x,y\\in {\\mathbb {R}},$ and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t) = \\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\ \\ Y(t)=\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ 0\\le t\\le T.$ If $\\nu _{\\kappa }^T, \\nu ^T$ are the corresponding measures on ${\\mathcal {C}}(0,T)$ , then $\\mathrm {d}_{{w}}(\\nu _{\\kappa }^T,\\nu ^T)\\le (1+T)\\bar{C}_Xe^{KT}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}(0,T)$ , define $\\Psi ^T(f)(t)=y(t)$ as the solution of $y(t)=\\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ 0\\le t\\le T.$ By direct computation (e.g.",
"[10]), we have (REF ) with $C_{\\psi }^T = e^{KT}$ .",
"$ $ $\\Box $ Example 3.",
"Let us combine Examples 1 and 2.",
"Take a positive number ${\\alpha }$ and a function $b=b(x)$ satisfying (REF ), and let $Y^{\\kappa }, Y$ be the solutions of $Y^{\\kappa }(t)& = -{\\alpha }\\int _0^t Y^{\\kappa }(s)\\, ds+\\int _0^t b\\big (Y^{\\kappa }(s)\\big )\\, ds + W^{\\kappa }(t),\\\\Y(t)&= -{\\alpha }\\int _0^t Y(s)\\, ds+\\int _0^t b\\big (Y(s)\\big )\\, ds + W(t),\\ \\ t\\ge 0.$ If ${\\alpha }>K$ , then $Y^{\\kappa }, Y \\in {\\mathcal {C}}_{(0)}$ and, for the corresponding measures $\\nu _{\\kappa }, \\nu $ , $\\mathrm {d}_{{w}}(\\nu _{\\kappa },\\nu )\\le \\frac{2{\\alpha }\\bar{C}_X}{{\\alpha }-K}\\left(\\frac{\\ln (1+\\kappa )}{\\kappa }\\right)^{1/2}.$ Indeed, for $f\\in {\\mathcal {C}}_{(0)}$ , define $\\Psi (f)(t)=y(t)$ as the solution of $y(t)=-{\\alpha }\\int _0^t y(s)\\, ds + \\int _0^t b\\big (y(s)\\big )\\, ds + f(t),\\ t\\ge 0.$ Using variation of parameters formula and (REF ), $\\Psi (f)(t)=\\int _0^t e^{-{\\alpha }(t-s)} b\\big (\\Psi (f)(s)\\big )\\, ds + \\Psi _a(f)(t).$ Then, similar to Example 1, we conclude that $\\Psi $ maps ${\\mathcal {C}}_{(0)}$ to itself.",
"In particular, using (REF ) and (REF ), $\\Psi (f)(t)-\\Psi (g)(t) =\\int _0^t e^{-{\\alpha }(t-s)} \\Big ( b\\big (\\Psi (f)(s)\\big )-b\\big (\\Psi (g)(s)\\big )\\Big )\\, ds + \\Psi _{\\alpha }(f-g)(t)$ so that $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\le K \\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)} \\int _0^{\\infty } e^{-as}\\, ds + 2\\Vert f-g\\Vert _{(0)}.$ As a result, if $\\alpha >K$ , then $\\Vert \\Psi (f)-\\Psi (g)\\Vert _{(0)}\\le \\frac{2{\\alpha }}{{\\alpha }-K}\\Vert f-g\\Vert _{(0)},$ and (REF ) follows from (REF ).",
"$\\Box $ Concluding Remarks A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .",
"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.",
"When the underlying processes are Gaussian, some of the approximation errors are not present.",
"In continuous time, the first two steps are the equality (REF ).",
"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.",
"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.",
"For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.",
"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.",
"For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].",
"Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.",
"For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."
],
[
"Concluding Remarks",
"A proof of the functional Central Limit Theorem for processes of the type (REF ) or (REF ) usually includes the following steps: A Gordin-type decomposition [12], when $W^{\\kappa }$ is written as a sum of a martingale and an a “small” correction; A coupling argument, when $W^{\\kappa }$ is constructed on the same probability space as $W$ ; A Skorokhod embedding for the martingale component of $W^{\\kappa }$ .",
"Each step leads to an approximation error; in particular, [17], [18] developed a systematic procedure, now known as the KMT approximation, to minimize the error due to the Skorokhod embedding.",
"When the underlying processes are Gaussian, some of the approximation errors are not present.",
"In continuous time, the first two steps are the equality (REF ).",
"There is no need for Skorokhod embedding because the martingale component is the Brownian motion.",
"In discrete time, the first step is the equality (REF ), whereas (REF ) represents coupling and the Skorokhod embedding.",
"For convergence in the space of continuous functions, the $\\sqrt{\\ln \\kappa }$ correction to the classical rate $1/\\sqrt{\\kappa }$ comes from the growth of the maximum of iid standard Gaussian random variables.",
"Keeping in mind that rate of convergence in the functional CLT can depend both on the underlying functional space and on the distance between the measures on that space, the rate $1/\\sqrt{\\kappa }$ is possible to achieve.",
"For example, by considering $W$ and $S_{\\kappa }$ [from (REF )] as processes in $L_1(0,T)$ , as opposed to ${\\mathcal {C}}(0,T)$ , direct computations [2] yield ${\\mathbb {E}}\\int _0^1 |S_{\\kappa }(t)-W(t)|\\, dt=\\frac{1}{\\sqrt{\\kappa }}\\int _0^1 {\\mathbb {E}}|B(t)|\\, dt = \\sqrt{\\frac{2}{\\pi \\,\\kappa }} \\int _0^1\\sqrt{t(1-t)}\\, dt =\\sqrt{\\frac{\\pi }{32\\,\\kappa }},$ that is, the Wasserstein-1 distance between $S_{\\kappa }$ and $W$ in $L_1(0,T)$ is of order $1/\\sqrt{\\kappa }$ ; see also [3].",
"Given the variety of function spaces that can support $W$ and $W^{\\kappa }$ , as well as the variety of ways to measure the distance between the corresponding probability distributions [11], identifying all situations with a sharp $1/\\sqrt{\\kappa }$ bound becomes an interesting challenge.",
"For $\\mathrm {d}_{{w}}(W,W^{\\kappa })$ in the space of continuous functions with the sup norm, there is strong evidence that convergence cannot be faster than $\\sqrt{\\ln \\kappa /\\kappa }$ : the results of this paper demonstrate it in the Gaussian case, and, by [15], the simple symmetric random walk cannot beat this rate either."
]
] | 2209.08249 |
[
[
"General History of X-Ray Polarimetry in Astrophysics"
],
[
"Abstract Soon after the discovery of the first extrasolar X-Ray sources it was suggested that polarimetry could play a major role as a diagnostic tool.",
"Attempts to measure polarization of X-Ray sources was performed by the team of Columbia University lead by Robert Novick.",
"The technique of Bragg diffraction at 45{\\deg} was successful to detect the polarization of the Crab with rockets and with OSO-8 satellite.",
"In the following evolution of X-Ray Astronomy, Polarimetry was too mismatched with the improved sensitivity of imaging and spectroscopy, based on the use of optics.",
"As a consequence no polarimeter was flown any more.",
"At the beginning of the century a new class of instruments based on the photoelectric effect were developed.",
"In the focus of an X-Ray telescope they can perform angular, energy and time resolved polarimetry and benefit of the large increase of sensitivity due to the optics.",
"The Imaging X-Ray Polarimetry Explorer, exploiting this technique, was launched at the end of 2021."
],
[
"The very early stage",
"Only 9 years after the discovery of X-Rays by Roentgen in 1904 Charles Glover Barkla, a student of Stokes, made experiments of scattering of these newly discovered particles and found that they follow the same rules of polarization of optical light and this demonstrated that these mysterious X-rays were electromagnetic radiation.",
"Physics and Polarimetry of X-Rays are born almost together.",
"When in 1962 Giacconi and Rossi found the first evidence for extrasolar X-ray sources theoreticians predicted that polarimetry would have in this wavelength domain a major role than in other domains.",
"The Soviet Physicist Vitali Ginzburg was very active in this promotion also toward the western community.",
"The easiest implementation of this concept would be an instrument based on the angular distribution of photons scattered from a target.",
"The difficulty is that the scattering prevails over the absorption at energies of few keV, where the majority of photons is, only with Hydrogen and Helium, that have a negligible stopping power for X-Ray photons.",
"The best dense scattering material is Lithium (that in any case must be encased with beryllium) for which the scattering prevails only above 10 keV.",
"In 1969 Herbert Schnopper suggested the use of Bragg crystals at incidence angles of $45^{0}$ as a good analyzer of linear polarization[1].",
"The technique would be very robust and simple.",
"The limit is the narrow band of the diffracted radiation that would result in a low effective area.",
"Moreover the detector would be of the same surface of the entrance window, and the poor resolution of the detectors would not allow to separate effectively by the pulse height the diffracted photons, so that a large background could be expected.",
"The team of Columbia University lead by Robert Novick carried on a systematic study to arrive to perform an astronomical experiment.",
"His co-workers Roger Angel and Martin Weisskopf found that the efficiency could be improved by using mosaic crystals of pyrolytic graphite (i.e.",
"a crystal with the spread of orientation of micro-domains artificially enhanced).",
"Moreover they found that a moderate bending of the crystal would allow a concentration of diffracted photons on a small detector, with a much lower background rate, while substantially preserving the modulated response to polarization[2],[3].",
"The Team of Columbia launched the first rocket searching for polarization of ScoX-1 on 1968[4].",
"The rocket included scattering targets of lithium encased in beryllium surrounded by proportional counters.",
"The polarization angle was derived from the angle between the scatterer and the detector and the spinning phase of the rocket.",
"The observation of ScoX1 (performed only 6 years after the discovery of the source!)",
"was unsuccessful in terms of polarimetry.",
"But the same rocket payload, was launched again to point the Crab Nebula.",
"The result was inconclusive but still compatible with the high polarization found in optical and radio bands.",
"When the technique of Bragg diffractors wih mosaic graphite was mature an improved version of the rocket was built.",
"Out of the atmosphere four panels mounting graphite crystals were deployed on the side at an angle of $45^{0}$ to the pointing/spinning axis of the rocket so that the photons parallel to the axis were diffracted toward proportional counters hosted in the rocket below the scattering stage of the payload that was similar to that of the first rockets.",
"Also this result was inconclusive.",
"But by overimposing the data of all flights Novick found at last a statistically significant evidence of polarization[5].",
"This was a paramount result in terms of physics of X-ray sources also confirming the diagnostic relevance of this subtopics, notwithstanding the experimental difficulties.",
"Another achievement for the future was that, in the few keV range, the Bragg approach, notwithstanding the very low efficiency, is better than Thomson in terms of both sensitivity and reliability While performing this first set of experiments the Novick Team also fixed the statistic frame that would be used for all the planned and the (few) performed experiments.",
"Both scattering and Bragg polarimeters would produce a histogram of phase of emerging photons.",
"The histogram can be fitted with a constant term plus a cos$^{2}$ term.",
"The ratio between the two terms carries the information about polarization degree while the phase of the second term carries the information n the polarization angle.",
"$M=\\dfrac{S_{max}-S_{min}}{S_{max}+S_{min}}$ The M is named modulation.",
"The modulation for a beam 100$ \\% $ polarized, conventionally named modulation factor ($\\mu $ ) is a feature of the detector and usually depends on the energy.",
"If counts follow Poisson Statistics, as usually do in X-ray detectors, the modulation, which is a positively defined quantity, follows a distribution of $ \\chi ^{2} $ for two degrees of freedom.",
"This is the basis to define the significance of a possible detection.",
"The Minimum Detectable Modulation can be computed starting from the statistics as the modulation that in only 1$\\%$ of occurrences can be arrived or passed by statistical fluctuation.",
"By dividing for the modulation factor we can derive the Minimum Detectable Polarization (MDP) that is the usual parameter to describe the sensitivity of an experiment of X-ray polarimetry when studying a source of a certain flux and spectrum.",
"$MDP=\\dfrac{4.29}{\\mu \\varepsilon S}\\times \\sqrt{\\dfrac{\\varepsilon S+B}{T}}$ The fact that a 99$ \\% $ confidence was chosen, instead than the most usual 3$ \\sigma $ (or 5$ \\sigma $ ), witnesses the awareness that polarimetry would be a tough challenge.",
"The Columbia team was also active in optical polarimetry of sources of potential interest for X-ray astronomy, so that it was familiar with the Stokes parameters formalism.",
"Actually the results of the first rockets were presented and discussed in this formal frame, although both Bragg and Thomson stages can only detect linear polarization.",
"The V parameter is unmeasurable so that the use of the Stokes is, in the frame of X-ray polarimetry only, of moderate usefulness.",
"To compare results in a sky map a display of the polarization degree and angle could be the most informative.",
"But in order to insert the X-ray result in a general frame of measurements in other wavelength or compare with theories the Stokes parameter representation would be the most suited.",
"In the topical paper where polarization is detected the discussion is concentrated on the technique to combine data of different experiments and flights and the related systematics, so that, in this case, the Stokes Parameters were ignored."
],
[
"Ariel-5 and OSO-8",
"The first two satellites with an X-ray polarimeter aboard, Ariel-5 and OSO-8, and the only ones for the following 45 years, were launched on 1974 and 1975 respectively, only 5 and 6 years after the first X-Ray survey mission UHURU.",
"The ARIEL-5 spectro-polarimeter designed and built by the Team of Leicester University lead by Richard Griffiths[6] was also based on Bragg Diffraction but was somehow different.",
"The direction of the radiation impinging on the instrument and that of the diffracted photons impinging on the detectors were defined by mechanical collimators as well.",
"But the diffracting crystals were of two kinds, one of mosaic graphite, like OSO-8, the second one of LiF.",
"Crystals were flat and their inclination, with respect to the incoming radiation could be adjusted in order to tune the mounting angle and the related diffraction energy.",
"The spinning satellite would be pointed not to the source but to a direction slightly off-set.",
"By this trick the photons from the source would impinge on the crystal with an angle slightly different at each phase of spin.",
"This would allow to perform high resolution spectroscopy of expected Si and S lines.",
"The LiF was used to make spectra of Fe lines.",
"At an angle of $45^{0}$ the diffractometer could be used as a polarimeter around an energy of 2.5 keV.",
"In OSO-8 experiment, lead by Novick and Weisskopf, both functions of spectroscopy and polarimetry were present but devoted to two dedicated instruments, both designed and built from the Team of Columbia University.",
"The spectrometer was based on a flat crystal of pyrolytic graphite with the unavoidably large detectors.",
"The polarimeter was based on graphite mosaic crystals, bent to concentrate photons on proportional counters.",
"The diffraction angles were confined in a range around $45^{0}$ from mechanical collimators.",
"The whole was in continuous rotation around the pointing axis through the satellite spinning.",
"The eventual yield of both spectrometers was very poor.",
"Only in one case a line was detected.",
"Upper limits on narrow lines were fixed of doubtful significance given that the observed sources were likely thick.",
"The real effect was to discourage the future instruments of high resolution spectroscopy.",
"The two polarimeter had different results.",
"Both OSO-8 and Ariel-5 used proportional counters with pulse shape and pulse height discrimination of background.",
"The areas of the two instruments were comparable but OSO-8 would be much more sensitive.",
"Thanks to the bent geometry the detectors of OSO-8 had a surface around 20 times smaller than the crystal, with an almost proportional reduction of the background.",
"In the Leicester experiment the main use for spectroscopy forced the flat geometry and as a consequence a much higher background.",
"Moreover the mosaic angular spread of the graphite was smaller than that of OSO-8, in order to preserve the spectral resolution and this means a lower efficiency for polarimetry.",
"Last but not least an important fraction of the surface was used for the LiF crystal, which had an efficiency much lower than that of graphite.",
"OSO-8 made a long pointing of the Crab.",
"Confirmed the results of the rockets measurement but with a very high significance (19$ \\sigma $ )[7].",
"The measured polarization was significantly above the so called Chandrasekhar limit (15.7$\\%$ vs 12$\\%$ ), namely the maximm polarization deriving from scattering on an asymmetric geometry.",
"This was the final evidence of a not thermal component of the X-ray emission, likely synchrotron.",
"A strong limit to ScoX1 polarization was also found[8].",
"But other results were of much lower impact.",
"These results were in general considered disappointing.",
"While the data on Crab rose a high interest, data on other sources were still consistent with theories but demonstrated that performing polarimetry was difficult, complex and technically cumbersome.",
"But the main trouble with polarimetry arrived from the great transformation that was occurring in X-Ray Astronomy in those years.",
"Soon after the first discovery of extrasolar sources, Giacconi outlined a long term planning that foresaw, after the first explorer mission (that would be UHURU), a first medium size mission based on an X-ray telescope with, in the focal plane a revolver capable to alternate different instruments in the focus.",
"This mission would become the Einstein Mission[9].",
"The High Resolution Imager produced impressive images of a limited sample of extended sources.",
"But in practice the Imaging Proportional Counter performed the large majority of the work for point-like sources.",
"The telescope collected the photons impinging on an area of 400 $cm^{2}$ and concentrated on a focal spot of the order of one $mm^{2}$ with the consequence that the background count was equivalent to that of a very weak source.",
"The increase in sensitivity for the detection of a source was impressive.",
"With an observation of few hours Einstein/IPC could detect a source at the limit of UHURU sensitivity.",
"Einstein disclosed the world of extragalactic sources that had been only touched with collimated missions.",
"The mismatching in sensitivity between imaging and polarimetry increased enormously.",
"Moreover the viable polarimeters implied a rotation of the instrument, easily achieved with spinning satellites.",
"On the contrary the new imaging/spectra instruments in the focus of the optics did not need any more rotation and a three axis pointing was much more useful for the measurement itself, for the use of power resources and for the pointing flexibility.",
"Therefore a polarimeter would be a rotating equipment harbored in the focal plane.",
"In other terms the addition of a polarimeter would represent a serious increase of complexity and a heavy share of observing time to make polarimetry of a small number of brighter sources belonging to classes that were no more the cutting edge of X-Ray astronomy.",
"Not surprisingly the originally foreseen polarimeters were excluded from both Einstein and Chandra.",
"Polarimetry was committed to hypothetic dedicated missions.",
"Some were proposed but none advanced in the way to approval."
],
[
"The Stellar X-Ray Polarimeter",
"The Stellar X-Ray Polarimeter (SXRP) was an important exception.",
"The Spectrum Roentgen Gamma (SRG) mission was planned by the Space Agency and Academy of Sciences of Soviet Union, under the scientific leadership of Rashid Sunyaev, one of the scientists who contributed to the highest level to predict the relevance of X-Ray Polarimetry.",
"SRG included many instruments in the band of soft and medium hard energy X-rays.",
"The largest instrument was the Soviet Danish Roentgen Telescope (SODART), a pair of large area, medium optical quality, optics, with a focal length of 10m and with an effective area of the order of 1200 cm$^{2}$ each.",
"Each telescope had a sliding platform in the focal plane capable to position different instruments.",
"SODART by itself and, a fortiori, the whole satellite were so complex and massive that could host a polarimeter of around 50 kg mass, included rotation, without a major perturbation.",
"SXRP was based on an optimal use of both Bragg and Thomson technique.",
"The difference in band allowed to stack the two stages.",
"Around 10 centimeters above the focal plane a flat graphite crystal at $45^{o}$ from the optical axis was diffracting 2.6 and 5.2 keV photons to a secondary focal plane.",
"Since the diffraction follow the same laws of optical reflection the whole was still preserving the imaging properties of the telescope in a kind of a Newtonian mounting[10].",
"The crystal was highly transparent to photons of higher energies that would impinge on a stick of lithium within a thin beryllium case.",
"The two analyzers, namely the diffractor and the scatterer, were hosted in the center of a well made with four detectors[11] .",
"The detectors were proportional counters filled with a mixture of Xe, A and $CO_{2}$ .",
"The window was 150$ \\mu $ m thick except for a circular sub-window in the positon of the secondary focus for the diffracted photons, only 50$ \\mu $ m thick to leave a reasonable transparency for 2.6 keV photons.",
"The detectors were position sensitive by exploiting the signal induced on a cathode plane subdivided according to a wedge and strip code.",
"The background was minimized by an anticoincidence plane and by pulse shape discrimination.",
"The whole instrument including the analyzers and the detectors was rotated around the optical axis.",
"The rotation would perform the measurement with the modulation of the diffracted image in the auxiliary detector and to compensate systematics in the scattering stage.",
"The Principal Investigator was Novic, replaced in a late stage in a late stage by Philip Kaaret.",
"An Italian team was part of the project and contributed the detectors.",
"This was likely the best way to use the two conventional techniques in combination with a grazing incidence telescope.",
"The telescope had a very large area and a band pass extended to 15 keV, namely with a reasonable overlap with the scattering cross section, so it was the best possible instrument with conventional (one-layer) optics.",
"The main difficulty was the large tolerances in the alignment of the instrument with the focus of the telescope.",
"This drove some design decisions with negative consequences.",
"A diameter of the scatterer larger than required from the convergence of the X-ray beam and from the tolerance on positioning increased the self-absorption in the scatterer with a loss of signal and, most relevant, an increase of the systematics in case of offset from the axis.",
"Also the distance of the detectors from the axis was larger than needed, to minimize the impact of possible offset of the lithium stick in the focus.",
"But in a scattering polarimeter the background is proportional to the surface of the detectors.",
"The predicted background rate would range from 200 to 400 mCrab[12].",
"This means that the benefit of having an instrument in the focus of a telescope would only apply for the observation of a limited subset of brighter sources.",
"The Bragg stage, thanks to the technique that preserves the imaging (although with a poor quality of the optics) would have a low background, virtually negligible.",
"But taking into account all the losses of signal the effective area at 2.6 keV was of the order of 10 $cm^{2}$ on a bandwidth of the order of 100 eV.",
"Polarimetry is a discipline in starvation of photons.",
"A reasonable MDP could be achieved only for sources of tens of mCrab.",
"In the baseline program for SRG the share of observing time with SXRP as a prime was of 5$ \\% $ .",
"So in practice a few brighter sources would be probed in the first years and the extragalactic sky was out of reach unless in the case of an extreme flare.",
"In any case SXRP would perform polarimetry of a certain number of bright galactic sources down to the level of a few $\\%$ , and this would be a serious step forward with respect to OSO-8.",
"The experiment was completed, integrated and fully calibrated.",
"It passed all the acceptance tests and was packed waiting to the delivery to be integrated in the SRG payload.",
"Unfortunately the making of SRG was slowed and eventually stopped by the general sinking of the Soviet System.",
"The fact that the end of the program was never declared had the effect that SXRP in a perpetual floating situation acted as a stopper for any other proposed X-ray polarimeter, as, for instance, for the one proposed for XMM.",
"One interesting heritage of SXRP was the calibration.",
"It was performed at Lawrence Radiation Laboratory in Berkley, by means of X-ray generators and crystal diffractors.",
"A particular attention was devoted to the calibration of the systematics foreseen for SXRP, such as the off-axis or a slight inclination.",
"A special attention was spent, with a large investment of time, to disentangle the intrinsic polarization of the calibrator from the measurement of the small effects of spurious modulation intrinsic to the instrument.",
"From the point of view of data analysis the technique of Fourier Analysis was widely used, with very promising results taking into account a starting point with a high level of systematics[13]."
],
[
"The quest for photoelectric polarimeter",
"The photoelectric effect had been studied in laboratory in the $^{\\prime }$ 20s essentially with two methods.",
"The first consisted in sending X-rays on a thin solid target and collect the emitted electrons with a detector on a goniometer geometry.",
"The method has a fundamental difficulty for astrophysical applications.",
"The modulation strongly depends on the penetration of the photon and this depends on the energy.",
"In a laboratory set up the measurement can be performed with photons of known energy.",
"In an astrophysical measurement the energy is unknown and the result is of difficult interpretation.",
"The other approach is to have a detector where the whole energy is lost in active regions.",
"In the laboratory this was performed with excellent imaging capability with cloud chambers.",
"The photographic read out was providing a very good image of the tracks, with all the details.",
"The quest for a photoelectric polarimeter for space is the search for a device with quality as similar as possible to a cloud chamber but self-triggered and suited to be hosted in a focal plane.",
"It is worth to mention the fact that for a short period the SXRP Team explored the possibility to introduce a new device based on the first of the two concepts described above.",
"Notwithstanding a relatively poor modulation factor of the order of 25$\\%$ , the full exploitation of the optics at low energies and the condition to be background-less down to very weak sources, promised an estimated sensitivity that would enable polarimetry of extragalactic sources.",
"Further measurements showed that the actual modulation was much lower[14] and this photoelectric polarimeter was no more adopted.",
"But this showed to the small community interested on this topic how relevant would be the implementation of a photoelectric imaging device for the focal plane.",
"SXRP was a big hope for X-ray polarimetry but, as the mission was continuously delayed and never stopped, it became a stopper for any new proposal.",
"Every proposal, such as that submitted for XMM, would face the statement that we should wait for the results of SXRP first.",
"In this phase of suspense, the idea that the next step would be a dedicated mission based on photoelectric effect started to spread.",
"In the very beginning of X-Ray Astronomy some laboratories tried to identify a method to perform polarimetry based on the photoelectric effect.",
"Given the limited imaging capability of typical wire chambers, these were based on the search of modulation of some macroscopic parameter deriving from the asymmetric angular distribution of the photoemission.",
"Riegler at GSFC[15] searched for a modulation with phase of coincident signals in nearby anodes for photons entering parallel to the wires.",
"Sanford at MSSL[16] searched for a difference of the rise-time of tracks parallel or perpendicular to the anode, in a proportional counter.",
"Both results were not conclusive and no further efforts were done after Novick results indicated the Bragg as the most effective technique.",
"With ASCA, Chandra and XMM-Newton the CCD took the place of Gas Proportional Counter as tow horse of X-Ray Astronomy.",
"Measurements showed a large majority of single hits but also a certain number of double hits, especially at higher energies.",
"The fluorescence yield for Si is only 3$\\%$ .",
"These hits were interpreted as a transfer of charge to the nearby pixel from photoelectrons created at a distance from the edge lower than the range.",
"Given the angular distribution of the photoelectrons the double hits along rows and those along columns have memory of the polarization of the X-rays.",
"This was verified by measurements and simulations by Osaka University[17], Max Planck[18] and Leicester[19] teams.",
"The method was obviously attractive because it does not require a dedicated instrument.",
"The measurement of polarization is the byproduct of the measurement of position and energy.",
"In truth this is only apparently true.",
"The range of the electrons is much shorter than the pixel size, so that the efficiency is very poor except at higher energies (>10 keV) where the efficiency of the optics is typically low and the chip is transparent to X-Rays.",
"Moreover the square pattern of the sensor is a problem itself.",
"A beam polarized at $45^{o}$ from the array and a beam unpolarized give exactly the same pattern and the system to work would need a rotation of the satellite or more detectors angularly misaligned.",
"But the most serious difficulty is that in these conditions the probability to have a double hit depends more on the distance of the absorption point from the edge then on polarization and with a point spread function that varies very sharply on the detector pattern the result would change completely for aspect change of seconds of arc.",
"For basic reasons the implementation of the photoelectric method was only possible with gas.",
"The team of Marshall Space Science Laboratory lead by Brian Ramsay made an important step forward with a gas detector with an absorption region with an electric field to drift the electrons of the track to a grid acting as a multiplication plane.",
"The gas mixture was based on Argon but included the TAE, a luminescent component.",
"During the multiplication optical photons were emitted so that the track became somehow luminous.",
"An optics would focus an image of the track on a CCD.",
"70 years after the images from cloud chambers these were the first images of photoelectron tracks.",
"They were acquired by optical methods, just like the cloud chamber experiments, with a CCD, that was taking the place of photographic films.",
"The method was very promising but the minimum energy was of 20 keV[20].",
"At the time the multi-layer optics were in a pioneering phase so that the path toward a realistic experiment was still to be defined.",
"But the technology of gas detectors was passing a season of great improvements.",
"The impetuous development of microelectronics also made feasible detection patterns with fine subdivision and all the simulations showed that a 2-dimension pixel imager would be the full exploitation of the photoelectric effect."
],
[
"The first gas pixel detectors",
"The team of IAPS (institutionally evolved from CNR to INAF), lead by the author and by Paolo Soffitta, another component of the SXRP collaboration, started a collaboration with the team of INFN-Pisa lead by Ronaldo Bellazzini, an outstanding personality in the field of gas detectors for particle physcs.",
"This INAF/INFN partnership could benefit of the convergence of two traditions and eventually succeeded to arrive to the first detector suited for the purpose.",
"A first testing on one dimension was performed with a microgap detector with a pitch of 200$ \\mu $ m filled with a Ne-DME mixture at 1 atmosphere pressure.",
"Data showed that the track length of electrons from photons of 5.4 polarized perpendicular to the strips were definitely longer than tracks from photons parallel[21].",
"To my knowledge this is the first unambiguous detection of an effect in the classical range 2-8 keV, viable for polarimetry.",
"The key was the use of a gas of low atomic number and the high spatial resolution, although in one dimension only.",
"Moreover the data were consistent with simulations software performed starting with a program for micro-analysis, given that at the epoch, general purpose software of the GEANT family were not yet adequate below 10 keV.",
"Encouraged from this result the INFN/INAF team made a first prototype that was the real turning point of photoelectric Polarimetry from a wishful thinking to a practical implementation.",
"The development of the GPD was articulated into various steps with some substantial commonality: A gas cell with a conductive window.",
"An electric field parallel to the optical axis to drift the electrons of the track to a multiplication stage.",
"A Gas Electron Multiplier (GEM) to perform a proportional multiplication of the electrons to amplify the track while preserving the shape.",
"In a first generation of devices the signal from the GEM gave the trigger for the data acquisition.",
"A sense plane of metal pads distributed on a honeycomb pattern to act as anodes to collect the charge multiplied in the gas above the pad itself.",
"A dedicated front end electronic chain independent for each pad.",
"An important driver of this design was the search for maximum axial symmetry, aimed to prevent any possible systematics and hopefully to avoid the need for rotation that till then was a mantra of X-ray polarimetry and one of the source of complications that made of polarimetry an odd companion in focal planes.",
"The first prototype based on trigger by GEM and collection plane built with multi-layer printed circuit technology.",
"The pad pitch was 150$ \\mu $ m. The GEM pitch 60$ \\mu $ m. Signals of each pad were routed horizontally to the input of an ASIC chips, relatively far from the gas cell.",
"The detector was filled with Ne and DME as a quencher.",
"The signal from the GEM was used to trigger the ASIC data acquisition.",
"Eventually the charge collected from each pixel was fetched to the output and A/D converted.",
"Photons of 5.4 and 5.9 keV generated ionization tracks in the gas.",
"From the analysis of the track images the direction of the emission was derived.",
"The histogram of angles was flat for unpolarized 5.9 keV photons and modulated for 5.4 keV photons, which had been polarized by scattering at 90$ ^{o} $ on a lithium target[22].",
"Although the statistics was not very high the papers presenting these results was the re-start of activities on X-ray polarimetry that can be classified into three groups.",
"1) An activity to improve and optimize the performances of prototypes and to build a real detector that could be the core of a space experiment 2) A rejuvenation of theoretical analysis to predict the potentiality of this new observable 3) The proposition of missions of polarimetry at every announcement of opportunity or in the context of large multi-instrument missions.",
"The results from the prototype demonstrated that the basic physics was correct but, for a realistic proposal, some problems had to be solved.",
"The major limits of the prototype were: 1) The multi-layer technique could allow for a limited number of pixels, of the order of one thousand.",
"2) The pads could not be smaller than 100$ \\mu $ m 3) The routing of the charge to chips far from the gas cell on an horizontal lay-out were the source of a noise much larger than that intrinsic to the anode read-out.",
"The total encumber of the detector and electronics, although already much better than that of the conventional experiments, was still relatively large compared with the small active sensing surface.",
"Notwithstanding these limitations at the Announcement of Opportunity from NASA for a Medium Size Explorer a Mission named INSPIRE was proposed including two telescopes one with a a Microcalorimeter.",
"The second step was the inclusion of the whole electronics in an ASIC chip.",
"The upper layer of the chip was the array of pads in hexagonal pattern.",
"The analog electronics was in the lower layers under the projection of the pad.",
"The signals were hold on the trigger from the GEM and routed to the output.",
"Two generations of these ASIC were made in sequence.",
"The first one had 2100 pixels, the second one 22000[23], in an hexagonal pattern.",
"The pixel size arrived to 80$ \\mu $ m.The most impressive evolution was to collapse the whole ingredients of the detecting system into a small volume around the active cell of gas.",
"This was basically different from other micro-pattern prototypes ad was named the Gas Pixel Detector (GPD).",
"The following evolution was a further decrease of the pixel size down to 50$ \\mu $ m and the increase of the number of pixels to 104000.",
"With the previous philosophy to trigger on the GEM and A/D convert the charge of each pixel dead time would have diverged.",
"The last step was the addition of self-triggering capability[24].",
"The ASIC was therefore playing different roles: the bottom of the detector, the sensing multi-pad plane, the front end electronics, the triggering electronics.",
"The chip was mounted on the bottom of a sealed ceramic case with a beryllium window and the whole weighted 80g.",
"In practice it was very close to what is needed for a space mission.",
"The chip could be used in principle with different gas mixtures, pressures and different thicknesses of the drift/absorption gap.",
"To operate in the 2-8 keV band Ne with Dymethylether (DME) quenching was used.",
"Following computation and tests a filling with DME and a 20$ \\% $ of He was found slightly more performing.",
"Eventually pure DME was used, although some problems could be expected with this chemically aggressive substance.",
"As a quencher and even more as the main detecting gas DME is outstanding from the point of view of diffusion.",
"Beside the choice of the gas mixture the pressure and the thickness of the absorbing cell was object of optimization.",
"A pressure around one atmosphere allowed for easier operation both in air and in vacuum and could be contained with a 50$ \\mu $ m thick Beryllium window.",
"The ingredients for the trade-off are: The thickness determines the efficiency.",
"The thickness determines the blurring of the track due to lateral diffusin in the drift.",
"The thickness determines a blurring of the image in the focal plane of the optics.",
"Point 1) is relevant because Polarimetry is a photons starving technique.",
"Point 2) impacts on the modulation factor, especially at lower energies.",
"Point 3) is relevant only for optics of good quality of the order of 10 arcseconds or better.",
"In fact the inclination angles are fixed by the band of the instrument and for medium quality telescopes (from 0.5 to 2 arcminutes) the defocusing effect is significantly minor in comparison with the telescope p.s.f..",
"The optimization of course depends on the band of optics and on the spectrum of a particular source.",
"The optimization for a typical optics of 4 m focal length and a spectrum E$ ^{-2} $ was a mixture of 80$\\%$ DME and 20$\\%$ He at one atmosphere.",
"Various sealed GPD with this filling and a window of 50 micron of Beryllium were built, studied, and tested to evaluate the performances included the resistance in a space environment.",
"This instrument was already mature for an actual mission and was proposed for various announcements to various Agencies (NASA, ESA, ASI, CAS)."
],
[
"The Time Projection Chamber ",
"At the NASA Announcement of Opportunity for a SMEX in 2003 a Team of Goddard Space Flight center lead by Jean Swank and based on the Laboratory for High Energy Astrophysics lead by Keith Jahoda and the Italian Teams that had developed the GPD lead by Ronaldo Bellazzini and Enrico Costa submitted a proposal for an Advanced X-Ray Polarimeter.",
"AXP was based on three conical optics like those for ASTRO-E2 three GPDs in the focus.",
"The proposed mission was not selected but NASA granted the LHEA with funds for the development of their own detectors for photoelectric polarimetry.",
"This was the Time Projecton Chamber (TPC)[25].",
"The process to optimize the performance of the GPD had shown the limitation in the axis-symmetric design.",
"The trade-off between efficiency and modulation factor even in mixtures with minimum diffusion was achieved at values around 10-15$\\%$ of efficiency at a peak of 2-3 keV.",
"On the other side in the GPD the ASIC must be orthogonal to the axis.",
"This forbids to recover the efficiency by stacking more GPDs.",
"The physics base of LHEA detector was the same of GPD and of any other modern photoelectric polarimeter, namely to image the ionization track produced by a photoelectron in a gas detector.",
"But in order to achieve a higher efficiency they decided to drop the axial symmetry.",
"In the TPC the electrons of the track are drifted on one side so that the drift path and the associated blurring is kept at an acceptable level.",
"The thickness of the absorption gap can be made very long in order to to increase the efficiency, potentially to values close to 1.",
"At the end of the drift a GEM, just like in the GPD, amplifies the track.",
"The multiplied electrons are collected by a set of vertical strips parallel to the optical axis.",
"The other dimension, namely that of the drift field, the track is measured by the Time Projection method.",
"Given that the the two coordinates of the images are derived with different methods and the drift method scales with the knowledge of the drift velocity an additional care is needed.",
"The track is imaged but, as usual in drift detectors, when a photon is absorbed no signal gives the start of the drift time.",
"The track is imaged, and the angular direction of the photoelectron can be measured, but the absorption point cannot be derived.",
"This impacts negatively in two ways: Image resolved polarimetry cannot be performed The reduction of background for the use of the optics is limited to one dimension only.",
".",
"These limitations are intrinsic to the method.",
"Some other problems were associated to the specific implementation.",
"These include a large dimension that, combined with the need of rotation, made it more cumbersome and a relatively limited band width.",
"Conversely the TPC could be proposed from early times as a medium energy polarimeter.",
"In fact the lateral read-out was compatible with both a thick absorption gap or a thin absorption gap with a back window that would allow the higher energy photons to reach another polarimeter mounted in series (e.g.",
"a scatterig polarimeter).",
"To summarize the TPC is more efficient of the GPD but is not imaging.",
"The GPD should have a lower background.",
"Moreover the TPC needs rotation to compensate for systematics while the GPD, with hexagonal pads, by construction, at the first order could be free from systematic effects.",
"Actually in the implementation for IXPE sysiematics of the order of 1$\\%$ at lower energies were deteced and calibrated.",
"The constructive details and the performances of the GPD and the TPC are discussed in another part of this Handbook.",
"Here I give some hint on the following history."
],
[
"Toward a mission ",
"With the implementation of GPD and the alternative development of TPC X-Ray Polarimetry had reached the maturity to compete in the selections by the space agencies.",
"In theory, beside this baseline range of 2 - 8 keV, other ranges could be covered, such as the soft band or the hard X-Rays.",
"But in this classic band a large set of results could be expected for most classes of X-ray sources, on the basis of theoretical analysis and achievable sensitivity.",
"No comparable wealth of results was to be expected for other band, that would also require more challenging instrumental efforts.",
"This limited the path toward a mission to a comfrontation of GPD and TPC both filled with low Z gas.",
"This band was well matched with that of XEUS, the Large X-Ray mission studed for years by ESA.",
"A GPD polarimeter was added and was part of the baseline focal plane payload in all the various configurations of XEUS and IXO[26].",
"But the first implementation of this new technique could not be committed to such a remote horizon.",
"An Announcement Of Opportunity for Two Small Scientific Satellites was issued by the Italian Space Agency (ASI) on 2007.",
"A team lead by Enrico Costa, Ronaldo Bellazzini and Gianpiero Tagliaferri proposed POLARIX a mission with three telescopes, residual of another experiment foreseen for the SRG mission and three GPD[27].",
"The mission was selected for a phase A study and was ranked second in the following selection.",
"But eventually all the program of Small Scientific Missions was discontinued by ASI.",
"After the development of TPC the AXP collaboration splinted toward competing proposals.",
"At the further SMEX Announcement of Opportunity on 2007 the GSFC team, lead by Jean Swank proposed GEMS[28], based on the TPC, while the MSFC team lead by Martin Weisskopf with the Italian Team proposed the Imaging X-Ray Polarimetry Explorer[29] based on the GPD.",
"Both missions were based on three telescopes and a deployable boom.",
"NASA selected GEMS for an advanced study and eventually for flight.",
"But in 2012 GEMS (that meanwhile had been descooped to two telescopes) was discontinued by NASA for programmatic reasons.",
"For the second time a mission approved and eventually not launched acted as a stopper to other proposals.",
"After the suppression of GEMS instruments aimed to X-ray polarimetry were proposed again.",
"A descoped version of POLARIX was proposed at the first ESA Anouncement for a small mission[30].",
"The most advanced proposal was the X-ray Imaging Polarimetry Explorer proposed by an European team lead by Paolo Soffitta to the M4 AOO of ESA[30].",
"XIPE was selected for a phase A study together with two other missions.",
"XIPE was based on three telescopes with GPDs in the focus.",
"XIPE was similar to IXPE with a collecting area of the 3 mirrors around twice that of IXPE.",
"On the other side of the Ocean at the following AOO for a SMEX in 2014 many projects of X-Ray polarimetry were submitted and two of them were selected for an A phase study.",
"IXPE was a rejuvenated version of the homonym precursor with a major role of the Italian collaboration that would provide the whole instrument[31] in the focal plane of three telescopes built by the MSFC team.",
"Martin Weisskopf was the PI and Paolo Soffitta the Italian PI.",
"Praxys was based on the GEMS design, with two telescopes (following the GEMS design) and TPCs in the focus.",
"Keth Jahoda was the PI of this other proposal.",
"At the end of the phase A study in 2017 NASA selected IXPE for flight.",
"As a consequence XIPE was no more considered for the ESA M4 selection.",
"In parallel to the development of the hardware the analysis tools and the statistical context were better defined by Martin Weisskopf[32] and Stroheymer and Kalmann[33].",
"The use of Stokes Parameters for the analysis was better defined.",
"Interestingly it was proposed to apply the Stokes formalism also to single photons[34].",
"With management by NASA and ASI, IXPE passed all the program stages and was launched on December 9 2021[35]."
],
[
"Not only IXPE",
"IXPE, launched on 2021 is performing as hoped and planned.",
"During the years needed to have this mainstream experiment in orbit a certain level of activity was performed.",
"The activity of polarimetry in the hard X-Rays was performed with stratospheric balloons and was somehow decoupled from the activity at lower energies onboard satellites.",
"The two main experiments have been POGO and X-Calibur.",
"Both are scattering polarimeters but conceptually quite different.",
"POGO is a collimated detector.",
"The scattering element is plastic scintillator.",
"The background is reduced with an heavy passive shielding and with an active anti-coincidence including active collimator.",
"POGO evolved from a first design (POGOLite) to a larger version POGO+[36].",
"Scientific results include an observation of the Crab[37] and of CygX-1[38].",
"X-CALIBUR is based on an optics with 12m focal length and, in the focus, a scattering stick of Beryllium surrounded with an absorption well made of four strips of CZT detectors.",
"X-Calibur was launched from Antarctica.",
"It observed GX301-2 in a flare status and found an upper limit to polarization[39].",
"Following the results, the project evolved to an advanced version named XL-Calibur, with special attention is paid to improve the background shielding[40].",
"In general scattering polarimeters have larger background partially compensated with a higher modulation factor on a wider energy band.",
"This will be greatly improved when these instrumets will be hosted on satellites, allowing for a broader band and suitable observing time.",
"Also in the domain of Hard X-Rays and at the edge of the range of interest of this chapter a small satellite devoted to polarimetry is expected to be launched from ISRO in the next years.",
"POLIX is constituted of collimators, a Beryllium scatterer and proportional counters[41]l, heritage of the ASTROSAT design.",
"Three years before the launch of IXPE Polarlight, a cubesat with a GPD based on the same ASIC of IXPE and a collimator was launched on a sun synchronous orbit by Hua Feng of Tsing Hua University[42].",
"Polarlight by long integrations measured again after 40 years the polarization of the Crab[43], with a possible change of angle after a glitch, and of Sco X-1[44].",
"Interestlngly PolarLight measured a background rate at a level of one 20th of Crab.",
"This shows that a large number of galactic sources could be observed with collimated photoelectric detectors without the programmatic and financial loads of optics.",
"On the opposite approach the enhanced X-ray Timing and Polarimetry mission(eXTP), in an advanced state of approval, includes 4 telescopes devoted to polarimetry[45].",
"The general feature are similar to those of IXPE but the collecting area is 4 timeslarger and the observations will benefit of the simultaneous measurement with high throughput spectroscopy and timig instruments.",
"While the path toward new data was so slow and painful, some progress was done in the development of methods of analysis.",
"These include a better understanding of the parameters defining the sensitivity, more clear use of the Stokes Parameters, and the development of Bayesian algorithms.",
"POGO and PolarLight used for the first time the Bayesian analysis.",
"The methods of analysis are the subject of next chapters."
]
] | 2209.08181 |
[
[
"CompF2: Theoretical Calculations and Simulation Topical Group Report"
],
[
"Abstract This report summarizes the work of the Computational Frontier topical group on theoretical calculations and simulation for Snowmass 2021.",
"We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics).",
"The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements.",
"Calculations and simulations will need to keep up with these increased requirements.",
"The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs.",
"These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns."
],
[
"Executive Summary",
"We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics).",
"The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements.",
"Calculations and simulations will need to keep up with these increased requirements.",
"The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs.",
"These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns.",
"The adaptation of existing CPU-based software to specialized hardware can provide solutions to the experiment-driven challenges, but this requires additional expertise and substantial effort.",
"Artificial intelligence (AI) and machine learning (ML) can also provide solutions to some problems that are more immediately amenable to acceleration on coprocessors.",
"US HEP should, to the extent possible, guide continued development of computing hardware to be maximally useful for solving HEP computing challenges.",
"Important developments include: faster hardware, availability of general-purpose CPUs, universal programming interfaces and portability, automated memory hierarchy, and early access to new hardware.",
"Long-term support of careers for software developers and maintenance for common software tools is essential to the future of HEP calculations and simulations.",
"Efficient utilization of the aforementioned new compute accelerators will require substantial efforts to port existing software to these new paradigms and/or to adopt new AI/ML techniques.",
"Needs in this area include new organizations to ensure software sustainability, permanent software development staff at national labs, joint lab-university appointments for computationally-focused scientists, and increased training across the field.",
"Computing in HEP has long struggled with issues of diversity and inclusion.",
"As the intersection of two highly specialized topics, there is an especially high level of knowledge and skills required to contribute successfully.",
"To get to this level, individuals usually must seek expertise outside of traditional HEP education and training, which typically does not cover computing topics in detail.",
"This poses a discouraging burden, especially for those without connections to such expertise or without the resources to take on such additional work.",
"We hope that substantially expanded support for software and computing training and for software development as a viable career in the HEP ecosystem will alleviate this burden.",
"A larger, healthier HEP computing community can be more welcoming and accessible to participants from all backgrounds."
],
[
"Introduction",
"The Snowmass Theoretical Calculation and Simulation topical working group is cross-cutting and underpins multiple scientific domains.",
"The computing topics discussed overlap with almost all aspects of Snowmass science and almost every Frontier.",
"Increasingly, computation is central to most theoretical and experimental scientific endeavors.",
"The great challenges of science, and in particular physical science supported by the Department of Energy and the National Science Foundation, have a long history of pushing the boundaries of computational science and computing technology.",
"The current Snowmass study indicates this will continue on many fronts.",
"The Theoretical Calculation and Simulation group spans a number of sub-topical themes representing the American Physical Society (APS) Division of Particles and Fields (DPF) Snowmass scientific interests: Cosmic Calculations, Particle Accelerator Modeling, Detector Simulation, Physics (Event) Generators, Perturbative Calculations, and Lattice QCD (quantum chromodynamics).",
"This report first introduces the common challenges and the modern computing landscape in Section , discusses each subtopic in more detail in Sections –, and concludes with recommendations in Section ."
],
[
"Cross-cutting computational challenges",
"The computational landscape of scientific algorithms can broadly be divided into high-performance computing (HPC) and high-throughput computing (HTC) problems.",
"HPC requires the coordinated and cooperative processing of a single large computational problem across multiple nodes with data being exchanged as part of the calculation, often using high-performance interconnects.",
"In contrast, HTC processes a large volume of completely independent computational work units.",
"Depending on the problem type, both HPC and HTC may have a high volume of relatively simple bulk floating point arithmetic that can be accelerated using highly parallel special-purpose computing hardware.",
"Some calculation and simulation tasks, and particularly those that process large data arrays or are machine learning-based, may obtain substantial speed increases from these hardware accelerations.",
"In other cases, algorithms that involve substantial serial performance or many data dependent logical flow decisions will benefit most from hardware that is suited to efficient (and power efficient) serial execution.",
"Detector simulation, event generation, and continuum theoretical calculations are typically HTC tasks, while accelerator modeling, cosmic calculations, and lattice gauge theory are more often HPC tasks.",
"However, exceptions exist, and the use of different algorithms or programming paradigms can sometimes convert HTC to HPC.",
"Memory bandwidth has presented a fundamental rate limit to many calculations.",
"It is common to include computational accelerators that can provide substantially greater floating point throughput and memory bandwidth as coprocessor cards.",
"The most common of these are graphics processor units (GPUs), but Field Programmable Gate Arrays (FPGAs) and other technology options are also common.",
"These often provide distinctly addressable high performance memory to help deliver good performance.",
"Coprocessors are often highly parallel and have limited or fixed function operations, so they are only usable with certain classes of algorithms.",
"Machine learning (ML) algorithms are especially suited to be accelerated on these devices, as described in Section REF .",
"On the other hand, while event reconstruction is easy to parallelize by processing independent events independently on independent serial processor cores of any form, the standard GPU thread model serializes execution when different threads in a group make different data-dependent logical flow decisions (for example, making different event-specific patterns of subroutine calls).",
"From an electronics perspective, it is easier to provide multiple types of memory (small and fast, large and slow) organized in distinct circuit boards to support a high floating point density.",
"The memory technologies are currently split into on-chip caches, in-package DRAM or “high bandwidth memory”, off-package DRAM, and increasingly non-volatile memory.",
"These are listed in order of increasing capacity and decreasing memory bandwidth.",
"The appearance of 2.5D and 3D integration technologies has been transformational and this system-level innovation will likely continue to return greater memory bandwidth over time, giving continued practical gains in high performance computation.",
"Such hierarchical memory systems will likely remain in the future, as they are dictated by physics constraints.",
"The degree of hierarchy is increasing and may grow in the future, independent of the specific processor architecture.",
"In this sense, the trends identified are likely reasonable over a ten-year timeframe.",
"Scientific productivity will be enhanced if data placement is handled automatically and moved infrequently: for example, this should be automatically and efficiently handled by the Unix virtual memory system.",
"This would avoid a requirement that every single scientific programmer laboriously manage the placement and movement of data to specific locations, such as caches.",
"The managing of distinct pointers to distinct “host” and “device” memory spaces is both a recurrent source of programmer error and requires considerable care to ensure that the allocated GPU memory does not exceed the limited capacity, while automatic paging avoids this requirement.",
"Section REF discusses more considerations relevant to scientific programming with modern heterogeneous computing.",
"Interconnects, used for both message passing and filesystem access, are becoming an increasing bottleneck since accelerated computing nodes are becoming rapidly more powerful while interconnect performance gains have not always matched pace.",
"Technologies such as silicon photonics, allowing the integration of optical transceivers in standard silicon chips, are a promising basic technology that could help address interconnect bottlenecks.",
"Similarly, high speed links between GPUs within a computing node are a key enabling technology for many problems.",
"File systems are a key element of any scientific workflow.",
"Data access should be sufficiently fast not to present a bottleneck.",
"Modern computing systems use parallel file systems, such as Lustre, giving up to terabyte/s aggregate access rates to disk.",
"The use of non-volatile memory elements can both cache bursts of data and accelerate metadata access.",
"Data migration to cold or longer term storage can be carried out through either user input or hierarchical storage managers.",
"It is important that these keep pace with growing data volumes and increasing computation speed."
],
[
"Computing hardware acceleration",
"Fixed-function acceleration is possible for problems involving dense matrix multiplication, with tensor cores in GPUs and other hardware specifically designed to handle large matrices.",
"These operations are commonly performed using 64-bit and 32-bit floating point arithmetic, or even on 16-bit floating point arithmetic.",
"Mixed precision functional units are common.",
"Fixed-function dense matrix hardware such as tensor cores do not necessarily accelerate all algorithms.",
"ML is one common and growing source of matrix multiplication operations that can be accelerated in this way.",
"ML can broadly be decomposed into “training” and “inference” phases.",
"Training often involves scientist supervision for parameter tuning, with time to results being critical.",
"Distributed ML training parallelizes the training process.",
"The use of small “mini-batches” in stochastic gradient descent leaves the problem very sensitive to Ahmdahl's law slowdown.",
"In the massively distributed limit [1], the summation across an interconnect of a gradient vector calculated across many randomly selected training samples in a minibatch is a significant performance bottleneck [2].",
"Inference with a previously trained network is typically performed on many independent events of data and is in many cases a trivially parallel (HTC) problem.",
"However, when inference is used in either hard or soft real-time situations (as occurs in HEP experiments), the latency requirements impose challenging performance issues when traditional CPUs are used.",
"Accelerators from Graphcore and Cerebras are good examples of ML-optimized accelerators in addition to more standard GPUs by Nvidia, AMD, and Intel.",
"Reconfigurable computing hardware such as FPGAs or spatial accelerators are powerful options for certain algorithms.",
"They may be accessed via a dataflow model where a hardware circuit is configured in a non-von Neumann approach to stream data from memory, calculate, and store results in memory.",
"They may be particularly effective in eliminating software latency in real-time inference environments [3], [4].",
"Algorithms that are I/O-bound or executed in situations requiring reduced power consumption may also be more efficiently accelerated on FPGAs."
],
[
"Software challenges",
"The computing landscape at this time displays an exciting proliferation of competing computer architectures.",
"It opens vibrant and healthy opportunities for innovation and competition and for the introduction of new ideas.",
"However, it also risks introducing special-purpose or less-general features, compared to previous systems.",
"The memory spaces are often fragmented with different pointer types and data locations.",
"There are multiple instruction-set architectures in use, each with a different programming interface or model.",
"A single program frequently has different segments that target different types of instruction sets within a single computer node.",
"Indeed, some of the computing models available (such as FPGAs or spatial architectures) are not von Neumann computers.",
"Others have dedicated fixed-function acceleration of matrix-matrix multiplication for machine learning.",
"This diversity presents a significant challenge to the scientific programmer with either legacy code or the need to use more than one system.",
"There are a number of different programming models one must consider in the present computing landscape.",
"The following can all in principle be combined with MPI message passing.",
"Present Exascale and Pre-exascale systems support: traditional CPU core, SIMD floating point instructions, and OpenMP threading, CUDA-based GPU programming, HIP-based GPU programming, SYCL-based GPU and FPGA programming, and OpenMP 5.0 offload GPU programming.",
"Future systems may support standards-based C++ parallel STL offload programming; however, multi-system compiler support is currently absent.",
"The challenges of writing high-performance and portable code are threefold: syntactical differences, semantic differences, and data placement.",
"The proliferation of models represents a substantial cost and effort overhead for science exploitation, since multiple implementations of software must be developed and maintained.",
"At this time, due to the diversity of models, abstraction and support for multiple programming models is likely the safest option for portable efficiency on near-term computers.",
"For projects that wish to reuse interfaces to acceleration, several good portability options exist: Alpaka and Department of Energy ASCR projects Kokkos and RAJA.",
"Standards-based programming, such as SyCL, OpenMP, or C++ parallel STL, is to be encouraged.",
"It is important to recognize that not all algorithms are amenable to map to accelerated hardware.",
"This must be considered in the HEP computing landscape to avoid leaving important needs unaddressed.",
"There is much to be said for an element of the computing roadmap using general-purpose processor cores combined with advanced memory technologies such as high bandwidth or hierarchical memory.",
"In this way, the long tail of problems with computational challenges that would be either impossible or prohibitively expensive to port to accelerated hardware may continue to be addressed."
],
[
"Cosmic Calculations",
"In the next decade, a number of powerful observational facilities will start to operate.",
"These next-generation cosmological observations will provide great opportunities to study the early history of universe, the mysteries of dark matter and dark energy, the fundamental particle physics, and possible modifications of Einstein gravity.",
"In order to successfully achieve the observational goals, a state-of-the-art simulation and modeling program is needed [5].",
"Cosmological simulations provide an invaluable tool to optimize the observation design, to interpret observation data, and to help unearth the underlying physics.",
"Next-generation cosmological simulations will increasingly focus on physically rich high-fidelity simulations that can directly connect with observational outputs from multiple different surveys.",
"Several types of cosmological simulations have been employed in the cosmology study.",
"The first-principles N-body simulations that have no free parameters can accurately describe dark matter fluctuations from the largest observable scales down to scales deep into the nonlinear regime.",
"However, the N-body approach does not account for the physics of the baryonic sector.",
"Additional models such as the halo occupation distribution, sub-halo abundance matching, or other schemes are added on top of a simulation to reconstruct galaxies.",
"Hydrodynamical simulations provide a reasonably accurate description of the distribution of baryons, quantify the effects of baryons on various probes of large-scale structure, and provide useful results for the distribution and properties of galaxies and clusters.",
"However, the robustness of the hydrodynamical simulation results depends on the choices of subgrid models and the nature of the cosmic probe in the study.",
"In addition to the aforementioned simulations, other simulations such as beyond $\\Lambda $ CDM simulations that involve the solution of nonlinear Poisson equation in the modified gravity model and radiative transfer simulations that model reionization process have also been used in cosmological studies.",
"Different survey experiments include different observables in the study.",
"Signals from galaxy clustering and lensing are key observables for photometric galaxy surveys.",
"Spectroscopic instruments are used to measure redshifts, radial velocities, gas dynamics and chemical compositions of galaxies.",
"Signals from Baryonic Acoustic Oscillation (BAO) and redshift space distortions (RSD) measurements are used to extract useful cosmological information.",
"Signals from the cosmic microwave background lensing measure the integrated mass between the last scattering surface and us.",
"The Lyman-alpha forest signal in the spectra of distant quasars is used to constrain thermal properties of the intergalactic medium (IGM).",
"The thermal and kinematic Sunyaev Zeldovich (tSZ/kSZ) signals are used to infer the distribution of gas in the universe.",
"The emission from dusty star has been used to delens the cosmic microwave background (CMB).",
"X-ray observations are used to calibrate mass estimates.",
"High-redshift from interferometers can provide useful information about the thermal and ionization state of the intergalactic medium and can be used to learn about dark matter.",
"The diverse observational probes present significant challenges in cosmological simulations.",
"One challenge is the computational cost to include multiple probes with high precision.",
"These simulations need cover a large volume but with sufficient fine resolution for a large number of realizations.",
"In addition, computationally demanding hydrodynamical simulations are required to simulate some observables such as the thermal and kinetic Sunyaev-Zel'dovich effects, tSZ and kSZ, to infer the distribution of gas in the universe, or Lyman-$\\alpha $ , to constrain thermal properties of the intergalactic medium.",
"In some cases, the modeling of certain observables involves a very large dynamical range.",
"Semi-numerical simulations are used when the hydrodynamical simulations cannot cover large enough volume while reaching small enough halos.",
"Calibrations and benchmarks between different approaches are important to interpret new observation data.",
"Another challenge for cosmological simulations is to reproduce observations, for example, correlations between multi-wavelength observables, in consistent galaxy formation models.",
"It is necessary to develop simulation models that can be applied to gravity-only simulations and account for correlations between neutral and ionized gas, stars, and dust in galaxies and galaxy clusters.",
"Also, it is not trivial to simulate massive neutrinos since they make up a non-negligible fraction of the total energy but can be decoupled when relativistic and have a free streaming scale.",
"In the N-body simulations, thermal velocity distributions need to be accounted for in the structure formation calculation on smaller scales.",
"Covering both the large volume required by the weak lensing and maintaining the accuracy at small scales required by strong lensing presents another computational challenge in ray tracing.",
"In the modeling for future analyses of small scale measurements, baryonic feedback effects that can change the local matter density must be included in the cosmological simulations.",
"Besides advancing in terms of increased resolution, larger volumes and better treatments of known physics, additional new physics models need to be included as the reach of surveys expands.",
"These include models of modified gravity with nonlinear Poisson equation for dark energy studies and a variety of dark matter models (warm dark matter, interacting dark matter, self-interacting dark matter, ultralight dark matter, ultraheavy dark matter, multiple component dark matter [6]).",
"Dedicated cosmological simulations are needed for those new models.",
"Next-generation cosmological surveys demand better understanding, mitigation, and control of systematic uncertainties.",
"To attain these goals, it is important to create realistic “virtual universes” from cosmological simulations.",
"Such simulations can be used to solve the inverse problem to infer the physical parameters of different models from the sky survey observation data.",
"However, it is not computationally practical to model all relevant processes from a first-principles approach given the vast dynamic range in cosmology.",
"Fast surrogate models can be used as effective emulators to speed up the solution of this inverse problem.",
"The predictions for cosmological surveys from emulators have been widely used in cosmological studies.",
"It is desirable that these emulators can connect directly to the survey observables to extract cosmological parameters.",
"With the advance of multi-fidelity hydrodynamical simulations, it is also desirable to use emulators to optimize subgrid model parameters and to gain better understanding of the interplay between the subgrid model and cosmology parameters.",
"Cosmological simulations are computationally intensive and demand the use of state-of-the-art computer hardware.",
"With the arrival of exascale supercomputers, cosmological simulations will need to use computer accelerators efficiently and memory access patterns effectively, as well as performance portable programming models and scalable algorithms.",
"Progress has been made to address some of these challenges, for example under the Exascale Computing Project (ECP) led by the Office of Advanced Scientific Computing Research (ASCR) in collaboration with other DOE science program offices [7], [8].",
"Continuous development and support are extremely important to enable full use of these hardware systems.",
"The successful use of the exascale supercomputers will enable gravity-only simulations with unprecedented volume coverage and resolution and hydrodynamics simulations with exceptionally detailed baryonic physics modeling for the cosmological surveys.",
"Scalable analysis approaches are as important as the development of simulation codes since handling and processing of very large data sets generated by the simulation codes require large computing resources in their own right.",
"Besides the development of the simulation codes, the development of analysis tools faces the same challenges as the simulation codes with respect to the efficient usage of the available computer hardware.",
"Co-development of simulation codes and analysis tools is needed for a successful cosmological simulation program.",
"Cosmological simulations are critical for the success of next-generation cosmological observations.",
"The accuracy for these simulations needs to be carefully checked through verification by benchmark among codes and validation with direct comparison against observational data.",
"This requires a concerted effort between different code and analysis development teams and cosmic observation teams.",
"The detailed connection between the simulations and the survey observations has to be established for a successful validation program.",
"In summary, the cosmological simulation community recommended the following advances in order to meet the above challenges and needs [5].",
"Development of scientifically rich and broadly-scoped simulations, which capture the relevant physics and correlations between probes Accurate translation of simulation results into realistic image or spectral data to be directly compared with observations Improved emulators and/or data-driven methods serving as surrogates for expensive simulations, constructed from a finite set of full-physics simulations Detailed and transparent verification and validation programs for both simulations and analysis tools Another area of cosmic calculations is the study of numerical relativity for next-generation gravitational-wave probes of fundamental physics [9].",
"The next-generation gravitational-wave detectors will provide great opportunities to study the nuclear physics of dense matter from the signals of neutron-star or black hole mergers, the nonlinear dynamics of warped spacetime, and the physics beyond general relativity from the signals of binary black holes.",
"To attain these goals requires a new generation of numerical-relativity codes with dramatically improved accuracy that can make effective use of the exascale supercomputers.",
"The higher computational accuracy also results in a significant increase of needs for computational resources in comparison to the present typical weeks to months of runtime on tens to thousands of computer cores.",
"This is due to the higher simulation resolution and the longer simulation to cover the detectors' sensitive frequency spaces.",
"In addition, many simulations will be necessary to span the parameter space of potential signals from black holes and neutron stars.",
"A further topic in cosmic calculations is the modeling of neutrino transport in neutrino-driven core-collapse supernovae [10].",
"A kinetic description of these neutrinos is needed in this model.",
"The solution for the neutrino transport equation is computationally intensive, involving high-dimensional linear algebra.",
"These simulations have been ported to GPU-based supercomputers and will require more computing resources to improve the resolution of the calculations and to explore different physical scenarios."
],
[
"Particle Accelerator Modeling",
"Particle accelerators are regarded as one of the most important inventions of the $20^{\\mathrm {th}}$ century.",
"They provide an invaluable tool in high energy physics study.",
"However, particle accelerators are large and complicated scientific devices that can be thousands of meters long (e.g., the LHC's nearly 27 kilometer circumference) with hundreds of thousands of elements.",
"Their design, construction, and operation are both sophisticated and expensive, and depend critically on computer modeling In the particle accelerator modeling area, there are 26 related letters of intent submitted to this Snowmass process.",
"These letters of intent were summarized in a community-wide white paper that addresses the challenges and needs in modeling particle accelerators [11].",
"Next-generation high energy particle accelerators and colliders will be more complicated and expensive than the existing accelerators.",
"These accelerators may employ both conventional RF cavity and high gradient wakefields from either a plasma channel or a structure waveguide to accelerate charged particle beams to high energy and use strong superconducting magnets and sophisticated control systems to confine these beams.",
"Some potential next-generation particle accelerators include very high energy proton-proton colliders, multi-TeV electron-positron colliders, muon colliders, other very high energy machines such as gamma-gamma and high energy electron-ion colliders, and high intensity accelerators for neutrino physics study.",
"To meet the needs of these next-generation accelerators, it is important for particle accelerator modeling tools to include all types of accelerating structures (e.g., RF-based, plasma-based, structured-based wakefield, plasmonic) and different accelerator components (e.g.",
"superconducting magnets, structured plasmas), and to target the accelerator and beam physics thrust grand challenges on intensity, quality, control, safety, and prediction.",
"For RF-based accelerators, this requires modeling of collective effects with high fidelity over long time scales, improving computational speed to allow for statistical ensembles and design optimization including collective effects, and improved integration with realistic magnet and RF modeling.",
"For plasma-based accelerators, this requires supporting the development of modeling tools and methods that will enable start-to-end simulations that predict the full six-dimension (+ spin) evolution of beams in a plasma-based linear collider, from their creation to their final focusing at the interaction point, and include all the intermediate phases of acceleration, transport, manipulations, and collisions.",
"For structure-based accelerators, this requires the development of efficient and accurate algorithms capable of modeling the beam interaction with its wakefield over long interaction lengths in structures with arbitrary geometries and constitutive parameters.",
"For plasmonic accelerators, this requires the development of new approaches such as a quantum-kinetic method that incorporates the effects of localized petavolts per meter plasmonic fields on the underlying quantum mechanical processes such as the ionic lattice and the energy band structure and to incorporate the multi-physics nature of the unprecedented strongly electrostatic plasmonic modes.",
"For the structured plasma accelerator component, this requires the development and integration of fluid and kinetic codes to meet the modeling demands for a new class of structured plasma devices that couple macroscopic plasma properties with strict requirements on kinetic interactions.",
"The quest for higher accelerating gradients in next-generation particle accelerators puts increasing demands on better understanding the materials of the accelerating structure and relevant phenomena such as RF breakdown.",
"This needs the development of automated scale-bridging methods that can autonomously parameterize higher-level models from large numbers of lower-scale first-principles calculations to enable predictive materials studies over a broad range of materials and conditions.",
"Strong magnets will be highly demanded in the next-generation particle accelerators to reduce the machine size and cost for transverse focusing.",
"Some advanced modeling tools are currently utilized in the US Magnet Development Program.",
"Novel modeling tools with mixed finite element method are needed to improve design time, cost, and performance of future superconducting accelerator magnets.",
"Next-generation particle accelerators will occupy a large land space and consist of many different accelerator segments starting from charged particle sources and ending with final interaction colliders or targets.",
"Different segments of the accelerator are coupled with each other through the charged particle beam transporting through the entire accelerator system.",
"The final particle beam quality does not depends on the performance of a few elements, but on the performance of the entire accelerator system.",
"Without the modeling of the whole accelerator, no project can proceed to the building stage.",
"In order to evaluate the particle accelerator performance and to attain a global optimal design, end-to-end simulations are needed to establish a virtual particle accelerator.",
"The end-to-end simulation denotes a simulation starting with the source that generates charged particles and ending with either the final target or the interaction region.",
"These simulations should include all pertinent physical effects from both first-principles models for high precision accelerator design and fast machine learning-based surrogate models for online particle accelerator tuning.",
"Such a virtual particle accelerator can be used as an emulator of the real physical accelerator in the accelerator operation.",
"One challenge in the end-to-end simulation is to exchange particle distribution data from different components of the particle accelerator.",
"To solve this problem requires supporting efforts to establish standards that would ease the sharing of information and data across codes as well as standards for interfacing codes [12].",
"Another challenge associated with the high fidelity end-to-end accelerator simulation is that it is extremely time consuming (especially with the inclusion of a variety of collective effects), varying from days to weeks and months.",
"The cutting-edge and emerging computing techniques including advanced algorithms, AI/ML methods, and quantum computing may substantially reduce the computational time of accelerator simulation and enable a fast real-time optimization with end-to-end simulations.",
"Advanced algorithms play an important role in the accelerator modeling.",
"High-fidelity self-consistent simulations cannot be done within a reasonable return time without using these algorithms [13], [14].",
"It has also been shown that the advanced algorithms can lead to several orders of magnitude improvement on the computational speed [15].",
"Research and development in this area is needed to explore new algorithms that exhibit better properties, to refine the understanding of the properties and bottlenecks of existing algorithms, and to remove the bottlenecks on cutting-edge heterogeneous computing hardware (e.g., GPU, FPGA), and improve the speed and accuracy of accelerator modeling.",
"Differentiable simulations show great potential to reduce the number of simulations needed for optimization and design of particle accelerators.",
"Using AI/ML surrogate models of the physics simulation can save orders of magnitude computing time [16].",
"A single differentiable simulation can generate the sensitivity information of the final beam quality with respect to thousands of machine parameters.",
"Support is needed for the development of AI/ML modeling techniques and their integration into accelerator modeling and control systems, with an emphasis on fast executing (up to real-time) and differentiable models, continual and adaptive learning for time-varying systems, uncertainty quantification to assess confidence of model predictions, and physics-informed methods to enable broader model generalization to new application conditions.",
"Quantum computing is another area of emerging technology that has the potential to transform some areas particle accelerator simulations by providing exponential improvement of computational efficiency.",
"Support is needed for quantum computing algorithm and code development for accelerator modeling, feasibility study on quantum computing implementation in accelerator modeling.",
"Quantum computing education in the accelerator community is also needed.",
"Particle accelerator modeling is computationally intensive and has utilized supercomputers for many years.",
"It has used both high-performance computing to model collective effects and high-throughput computing to model single particle dynamics.",
"With the arrival of exascale supercomputers and new special purpose computers with hardware accelerators (e.g.",
"GPUs), it is a challenge to achieve the supercomputer peak performance for modeling collective effects in particle accelerators due to deep levels of independent memory caches, on top of global and neighboring communications that are used to exchange information across multiple computer nodes during the simulation.",
"As part of this, the workload needs to be uniformly distributed among many compute units in order to achieve a good parallel efficiency.",
"The current and next-generation supercomputers extensively use the computational accelerators such as GPUs and FPGAs to improve the computational speed of applications.",
"Applications related to single particle dynamics can be effectively simulated on GPUs.",
"It is a challenge for applications including multiple physics models and collective effects to use GPUs effectively.",
"Support is needed to adopt portable, state-of-the-art programming paradigms and frameworks that support both multi-level parallelization and effective dynamic load balancing over all levels of compute parallelism, so that simulations run efficiently on the latest computer architectures with scalable I/O, post-processing and in situ data analysis solutions to support accelerator machine design and operation.",
"Particle accelerator modeling codes were developed with the support of a variety of projects to simulate RF structures, magnets, beam dynamics inside the accelerator structure, laser and beam interaction with plasma, and beam beam and beam wave interactions.",
"These simulation tools are critical for the design and operation of next-generation high energy particle accelerators.",
"Sustainability, reliability, user support and training of these complicated codes are potential challenges in the accelerator community.",
"Further support is needed to maintain these codes with documentation, testing, and benchmark examples.",
"Training classes are needed at the U.S.",
"Particle Accelerator School or other accelerator schools or workshops to help improve the usage of these codes.",
"Another challenge for these accelerator simulation codes is validation with experimental measurements.",
"This is critical for establishing the confidence in the use of these simulation tools for next-generation particle accelerator design applications.",
"The direct comparison between code predictions and accelerator performance and output may be limited by a number of factors, such as not fully characterized machine settings, unknown initial phase space distribution, insufficient range or resolution of diagnostics, and incomplete or missing physics in simulations.",
"Support is needed to connect simulations with direct experimental diagnostics and to carry out detailed benchmark studies with experimental measurements.",
"The particle accelerator modeling codes include a variety of physics models for different accelerator components and involve different research groups.",
"In order to attain the end-to-end simulation for next-generation particle accelerators, different component models need to be integrated into a workflow and ported onto the latest exascale type computer hardware with GPUs.",
"New physical models and capabilities will be added in some of these codes to account for new accelerating and focusing methods.",
"It is desirable to organize the beam and accelerator modeling tools and community through the development of ecosystems of codes, modular libraries, and frameworks that are interoperable via open community data standards for both loosely integrated and tightly integrated workflows.",
"It is desirable to establish open access data repositories for reuse and community-wide AI/ML surrogate model training.",
"It is also desirable to have an organized particle accelerator modeling community (e.g.",
"in the form of centers and consortia) including both academic institutes and industry.",
"This will improve the efficiency of accelerator modeling for next-generation high energy accelerator applications.",
"In summary, to meet the above challenges and needs, the following high-level recommendations are provided from the particle accelerator modeling community [11]: Develop a comprehensive portfolio of particle accelerator and beam physics modeling tools in support of achieving Accelerator and Beam Physics Thrust Grand Challenges on intensity, quality, control, and prediction.",
"Develop software infrastructure to enable end-to-end virtual accelerator modeling and corresponding virtual twins of particle accelerators.",
"Develop advanced algorithms and methods including AI/ML modalities and quantum computing technologies.",
"Develop efficient and scalable software frameworks and associated tools to effectively leverage next-generation high-performance and high-throughput computing hardware.",
"Develop sustainable and reliable code maintenance practices, community benchmarking capabilities, and training opportunities to foster the cooperative application of accelerator software.",
"Organize the beam and accelerator modeling community in open efforts to (a) foster the development of ecosystems of interoperable codes and tools, libraries, and frameworks, based on API and data standards, (b) establish open access and data workflow repositories, and (c) have an organized governance structure."
],
[
"Detector Simulation",
"Accurate detector simulation is crucial for tasks ranging from detector design to data analysis.",
"Geant4 [17], [18], [19] is the primary tool used for detector simulation throughout HEP [20].",
"This C++-based software package was originally released in 1998 as a complete rewrite of the earlier FORTRAN-based Geant3 [21].",
"The Geant4 collaboration and the HEP community engage in continuous research and development to improve the computational performance, refine the models of physical interactions, and incorporate new technical features and models.",
"Geant4 is the center of an ecosystem that includes numerous software packages serving different needs: Geometry/material modeling and navigation: VecGeom, CADMesh, DD4hep, GDML; Physics models: Noble Element Simulation Technique (NEST) for excitation, ionization, etc.",
"; G4CMP for phonons and other condensed matter phenomena; Opticks for optical photon simulation; Other simulation packages (including radiation modeling): MARS, FLUKA, MCNP, SRIM; R&D projects: G4HepEM, AdePT, Celeritas; Fast simulation engines: DELPHES, TrackToy, bespoke experiment-specific solutions, emerging ML approaches.",
"Some of these packages are developed in very close coordination with Geant4, while others are more independent, using Geant4 as a reference or being used as cross-checks for Geant4 results.",
"In addition, Geant4 interfaces with event generators (see Section ) such as pythia 8 for decays of long-lived, metastable, or late-forming particles, and with ROOT and CLHEP for various data analysis and mathematical functions.",
"Custom simulations of electronics responses and other instrumental effects are developed and applied downstream from detector simulation [22].",
"Geant4 is one of the most successful examples of community-supported and -maintained software in HEP.",
"It is even used beyond HEP for medical and astronomical applications.",
"However, HEP needs are the primary drivers of R&D on the software, especially improvements to its computing performance.",
"In particular, the next generation of collider experiments, including the HL-LHC and the various future colliders proposed during the Snowmass process, will pose extreme computing challenges for detector simulation.",
"Compared to present-day experiments, data volumes will increase by at least an order of magnitude, requiring a corresponding increase in the number of simulated events.",
"Detectors will increase in complexity and precision, requiring more detailed geometries and more precise physics models that will use even more computation.",
"At the same time, other aspects of the event processing chain, such as reconstruction, will consume a growing proportion of available computing because of massive increases in simultaneous bunch crossings, or pileup, especially at hadron colliders.",
"Therefore, Geant4 will need to simulate an order of magnitude more events with higher precision while using a smaller proportion of HEP computing.",
"To ground this prediction in reality, Geant4 used 40% of all computing during the LHC Run 2 for CMS and ATLAS [23], and ${\\sim }70\\%$ for LHCb [24], [25].",
"For the former, it is projected that only 14–20% will be available for simulation at the HL-LHC [26], [27].",
"In addition to high energy colliders, experiments of various sizes that search for light or weakly interacting particles rely on Geant4 [28], [29], [30].",
"Such experiments include DUNE, FASER, MicroBOONE, Muon g-2, Mu2e, COHERENT, and direct dark matter detection experiments DAMIC, DarkSide, DARWIN, DEAP, LZ, NEWS-G, PandaX, PICO, SBC, SENSEI, SuperCDMS, and XENON.",
"While many of these experimental collaborations and apparatuses are smaller than those at the CERN LHC, they face similar computational challenges to process petabytes of data, including the corresponding simulation of background and signal processes.",
"The need to meet these challenges with substantially fewer dedicated personnel makes the availability of common software even more important, including both Geant4 and packages like NEST and Opticks that provide specialized physical interaction models.",
"Future neutrino, dark matter, and other similar experiments will obtain even larger data volumes and probe even more precise physics, increasing the computational challenges and the associated need for high-performing and well-supported software.",
"The increasing breadth and precision of next-generation HEP experiments will require corresponding improvements in the physics models used in Geant4 and related software [20].",
"Some areas of interest include liquid argon signal induction, scintillation materials, Cerenkov light propagation, condensed-matter effects, low-energy responses, and rare background processes and interactions.",
"In particular, hadronic interaction models are an area where agreement with data could be improved and access to more measurements, such as hadron-argon interactions, could lead to even further improvement.",
"In general, support for increased effort in these areas is needed to keep up with the demands of new experiments for more precise models, as well as to improve the computational efficiency of the model implementations.",
"The first major evolution in the Geant4 computing model was the introduction of multithreading to reduce memory usage by sharing read-only common data during otherwise embarrassingly parallel event processing [31].",
"This approach has been adopted by the LHC experiments and Mu2e as the experiment software frameworks have similarly evolved to incorporate multithreading in other event processing steps [32].",
"A recently concluded R&D project called GeantV introduced sub-event parallelism using vectorized instructions, primarily targeting many-core CPU architectures, in a new prototype transport engine.",
"The primary findings of this project were that the achievable speedup compared to Geant4 was limited to a factor $2\\pm 0.5$ and that a small percentage of the speedup actually arose from vectorization [33].",
"Nevertheless, the project produced several useful developments, such as VecGeom, and lessons, such as the importance of instruction cache locality.",
"VecGeom provides improved code and vectorization for geometry modeling, enabling a ${\\sim }15\\%$ speedup when used with Geant4.",
"As an illustration of the level of complexity of Geant4, it required 30 FTE-years, spread over a 5-year period, to re-implement its numerous components, even partially or as prototypes, in GeantV: particle transport, geometry modeling and navigation, magnetic field propagation, physical interaction models, and interfaces for experiment software frameworks.",
"The future of detector simulation computing is turning toward the use of GPUs, targeting in particular HPC facilities that are intended to provide a larger fraction of HEP computing for next-generation experiments.",
"Because detector simulation is naturally an HTC problem, nontrivial effort is required to adapt the necessary computations to use computer accelerators effectively.",
"Two ongoing projects, G4HepEM-AdePT and Celeritas, have produced viable prototypes of high-performing GPU-native simulation engines.",
"In particular, the Celeritas effort is led by US national laboratory personnel and was inspired by the earlier porting of the Shift MC radiation transport engine to GPUs [34].",
"Shift is a less general application than Geant4, focusing on neutral particles with few secondary particles produced, limited physical interactions, simple geometries, and no magnetic field propagation.",
"Despite the additional complications encountered in addressing the broad scope of HEP detector simulation, the prototypes have demonstrated promising results, achieving a factor of 40 speedup using 1 Nvidia V100 GPU compared to Geant4 using 7 Power9 CPUs on Summit [35].",
"This can be rephrased as a factor of 200 GPU:CPU equivalence comparing a V100 to a higher-end Xeon CPU, also taking into account power consumption and other relevant factors.",
"Variations in performance from different implementation choices were relatively small, a factor of 2 at most.",
"Scaling this up to the whole Summit supercomputer suggests that its 27 648 GPUs can correspond to 5.5 million CPU cores, an order of magnitude larger than the current worldwide LHC computing grid (WLCG) at 500 000 cores [36].",
"If this performance persists, it has the potential to resolve the major simulation computing challenges.",
"However, we must not underestimate the increased burden of developing and maintaining coprocessor-friendly code, including aspects such as portability to non-Nvidia GPUs and even future non-GPU architectures.",
"It should also be noted that the ongoing projects currently implement only some components of Geant4, including models of electromagnetic interactions, but not yet hadronic interactions, for example.",
"If a complete parallelization of all components is not ultimately possible, the observed speedup will be limited by the remaining serial components, according to Amdahl's law.",
"In terms of personpower, the Celeritas project has so far used 5 FTE-years spread over 2 years, and AdePT is similar.",
"Because this effort builds not just on Geant4, but also on the lessons learned from earlier projects including Shift and GeantV, as well as the products of those efforts such as VecGeom, the total personpower involved is much greater.",
"Another avenue to achieve faster simulation is the use of machine learning to replace or augment existing “classical” (rule-based) simulation engines [37].",
"Full replacement is most commonly pursued using generative models such as generative adversarial networks (GANs), variational autoencoders (VAEs), or normalizing flows (NFs).",
"Augmentation to improve lower-quality results, sometimes called refinement, can be applied to classical fast simulation engines or to generative models, using techniques similar to those above or regression-based approaches.",
"These approaches are largely still in development or just starting to be deployed [38], so their ability to provide an acceptable level of physics fidelity at scale must be assessed carefully.",
"Beyond these ML algorithms, there are proposals to explore the application of differentiable programming directly to simulation engines [37], [39].",
"ML algorithms whose training and inference are performed using industry-standard frameworks are more easily portable to coprocessors such as GPUs, including at HPCs.",
"The possibility also exists to use coprocessors more cost-effectively and efficiently by performing the ML inference as a service, avoiding the need for each CPU to be equipped with a GPU [3].",
"The above solutions, including GPU prototype transport engines and ML algorithms, are promising avenues for faster execution of detector simulation.",
"However, it is unlikely that these will completely supplant Geant4, which provides broad support for myriad use cases.",
"In addition, the results of any new approaches are most often compared to Geant4 to prove their validity.",
"Therefore, some percentage of general purpose von Neumann CPU-based computing will be needed in HEP for the decades to come.",
"The extent to which CPU-based computing can be shifted to less general architectures and devices such as GPUs depends strongly on the availability of experts to develop and maintain the necessary software.",
"There has recently been a severe decrease in funding for permanent positions in HEP to support software developers who work on both these common tools and experiment-specific matters.",
"This threatens the persistence of important knowledge, which must be passed directly from person to person, and the stability and future of these crucial software packages that serve as a foundation for the entire field.",
"There is a strong and vocal consensus throughout the field that this trend must be reversed, with funding for technical positions not just restored, but increased further: “...it is still necessary to support the existing code and make sure that sufficient funding and staffing is provided for maintenance and development of physics algorithms, as well as for adapting the code to any updated CPU hardware, operating systems and new compilers.” [31] “Unfortunately, the Bertini model [of hadronic interactions] has not been actively developed over the last few years due to the lack of personpower.” [6] “A team of highly-skilled physicists and engineers is required to provide the necessary support and developments for Geant4, and the packages extending it, to meet the needs and challenges within the scope of the US HEP experimental program.” [20] “In addition, Geant4's common software, such as physics models, no longer receives any US maintenance funding...",
"This is not sustainable for the long term viability of small experiments.",
"Long term, experiment-agnostic Geant4 support is critical for the success of small experiments...",
"It is therefore essential that there is funding for permanent software and computing experts.” [28] “However, U.S. funding for [the Geant4 toolkit] was discontinued in recent years...",
"Continued support for Geant4 is crucial to the design and construction of future experiments, and for the interpretation of their results...",
"The general challenge of maintaining software and computing talent is exacerbated in the direct detection community by the lack of long term, permanent positions within the experiments...",
"It is therefore essential to provide funding for permanent software and computing experts.” [29] “DM and neutrino collaborations need scientists trained in data acquisition, simulation, and analysis at both the user and developer levels to achieve their science goals.” [30]"
],
[
"Physics Generators",
"The generator software landscape is complex and varied, in order to provide solutions to different aspects: event (matrix element) generation, hadronization and parton shower modeling, underlying event tunes, matching/merging algorithms, multi-parton interactions, particle decays, cross section calculations, parton distribution functions, data formats, and analysis and reinterpretation.",
"There are many software packages that address some of these needs, which often include plugins to perform specific calculations and provide ways for users to plug in their own custom models and computations.",
"A rough and incomplete categorization includes: matrix element generators, which support leading-order (LO) as well as next-to-leading-order (NLO) and sometimes higher order calculations: MadGraph 5_amc@nlo, powheg, sherpa, alpgen, whizard, amegic, comix, helac; parton shower modelers, which also support LO event generation: pythia, herwig, sherpa; neutrino-nucleus interaction generators: genie, neut, NuWro, GiBUU, achilles; plugins for more specific tasks: EvtGen, tauola, photos, dire, vincia; higher-order differential cross section calculation: mcfm, matrix, NNLOJet; one-loop amplitude computation: MadLoop, OpenLoops, recola; utility software: lhapdf, HepMC; analysis and tuning: rivet, mcnntunes, professor, apprentice, nuisance.",
"This wide array of software packages includes a corresponding variety of programming languages and styles—primarily Fortran, C++, and Python—as well as development and maintenance approaches.",
"Unlike detector simulation, in which Geant4 serves as a locus for numerous related efforts as described in Section , most of the software packages listed here are developed largely independently by separate groups.",
"The usage of common data formats and exchange specifications, such as LHEF, UFO, and HepMC, provides some amount of generic interoperability.",
"The percentage of WLCG computing time used for event generation at the CMS and ATLAS experiments is estimated to be 5–12%, with similar or even more substantial requirements expected for ALICE and LHCb [40].",
"However, these summary values hide large variances in tasks and generator software.",
"For example, MadGraph 5_amc@nlo has been found to be substantially faster than sherpa for generation of W+jets events with 0–2 extra partons, but much of the difference depends on parameter choices that may impact physical accuracy in some regions of phase space.",
"In general, for a given task, generator CPU usage scales linearly with the number of events generated.",
"However, moving from LO to NLO and higher orders, or increasing the multiplicity with additional partons [41], for the same physics process typically causes a factorial increase in CPU usage, while different processes may still have very different baseline computational requirements.",
"The memory usage and multithreading abilities of different generator software are also highly variable and can lead to inefficiencies.",
"The range of programming languages and styles can make generators difficult to integrate directly in experiment software frameworks, another potential source of inefficiency.",
"The generator usage at neutrino frontier experiments has not been measured and profiled as systematically, and this subset of the field is generally smaller compared to the energy frontier.",
"Generator CPU usage is projected to increase to 8–20% in the HL-LHC era, when more events and higher precision will frequently be required [26], [27].",
"The increasing precision of upcoming neutrino experiments such as LBNF/DUNE and HyperK should be expected to require similar increases in experiment computing time devoted to event generation.",
"Further future experiments will have different generator needs depending on their initial states.",
"The different categories include high energy hadron colliders, lower (but still relatively high) energy lepton colliders, neutrino experiments, forward physics experiments (such as FASER and SND@LHC), transverse detectors (such as CODEX-b, AL3X, and MATHUSLA), the Electron-Ion Collider (EIC), and proposed muon colliders.",
"The details for the relevant tasks and generators in each category are vast and cannot be summarized here; Ref.",
"[42] presents a comprehensive treatment.",
"As a particular item of interest from the recent Snowmass effort, we highlight new techniques in development for muon collider physics: the use of the effective vector boson approximation for $2 \\rightarrow n$ processes [43] and vector boson fusion as the dominant tool to study the electroweak sector at multiple TeV [44].",
"These processes and approaches have been implemented in new versions of MadGraph 5_amc@nlo, sometimes requiring new features or more involved calculations to ensure stable and reliable results.",
"They illustrate the work that will be needed to model the physics at the next generations of HEP experiments.",
"The interdependence between lattice QCD, nuclear physics, and neutrino event generators is also important to highlight [45].",
"The computational burden of an event generator depends strongly on the precision required, as described above; the calculation techniques used; and any other requirements placed on the output [31], [40], [42].",
"For example, the automated computation of NLO and higher order diagrams, as used in mc@nlo, introduces events with negative weights, which can substantially reduce the statistical power of the final result.",
"There are also inefficiencies in phase space sampling; in slicing, biasing, or filtering to change kinematic features; and in merging to avoid double-counting between matrix element generators and parton shower modelers.",
"Further, experiments' use of generators may involve significant repetition of expensive calculations, especially to assess certain systematic uncertainties that cannot be represented as alternative event weights.",
"Recent efforts coordinated by the HEP Software Foundation (HSF) have identified and resolved several performance bottlenecks, some related to these issues [42].",
"There is ongoing work to continue to address these issues, especially negative weights, which can be reduced, but not yet eliminated, by certain new prescriptions that have not yet achieved wide deployment [40].",
"In general, the development of new theoretical and mathematical tools to perform relevant computations has an outsized impact on event generator computing requirements.",
"At the time of the previous Snowmass reports, generator software was in some ways ahead of other subfields.",
"Machine learning algorithms were employed to derive PDFs [46], and usage of MPI for efficient utilization of existing CPU-based HPC systems was incorporated in sherpa, alpgen, MadGraph 5_amc@nlo, and mcfm, among others.",
"However, more recent initial efforts to port MadGraph 5_amc@nlo to use GPUs did not lead to a production quality result.",
"Revitalizing this task has been identified as a priority, and recent progress is promising.",
"An opportunity comes from the possibility to parallelize the matrix element calculations, which take up more than 90% of the computing time for complex LHC physics processes.",
"Matrix element calculations are a perfect fit for CPU vectorization and hardware acceleration on GPUs.",
"The MadGraph4GPU project, which uses custom CUDA kernels, shows speed improvement factors of ${\\sim }1000$ on an Nvidia V100 GPU, as well as an improvement factor of 3 on CPUs by exploiting vectorization [47].",
"The MadFlow project, which uses a TensorFlow backend, shows a speed improvement of more than an order of magnitude on an Nvidia Titan V GPU and an improvement factor of up to 3 on CPUs [48].",
"A new approach called BlockGen for multi-gluon tree amplitudes shows similar improvements of roughly an order of magnitude on a V100 GPU and 2–6 on CPU, depending on the multiplicity [49].",
"Similar developments for other computationally-intensive generators such as sherpa also need to be prioritized and supported.",
"The use of ML for approximate matrix element calculations, including phase space integration and sampling as well as fully generative models similar to those discussed in Section , is also very promising, but still in relatively early R&D and not yet ready for deployment.",
"Additionally, there are ongoing investigations into ML for parton shower modeling, hadronization, event (un)weighting, and tuning, among other applications [42], [50].",
"Beyond technical developments, other possible improvements that require coordination between different groups have been identified.",
"A common interface, which could also be described as a modular framework, would standardize how generators are used and plug into each other and other software [45], [51], [42].",
"This would be useful not just for theoretical studies, but also for experiment use, especially small experiments with minimal personpower to implement, test, and maintain different interfaces.",
"It could also facilitate more collaboration among generator developers, for example to make faster progress on code modernization and related work.",
"The adoption of common data formats and standards, such as those provided by Les Houches, by more generators and communities could be encouraged as well.",
"While there are clearly a number of potential benefits, both coordination and personpower limitations would need to be overcome in order to realize them.",
"Sharing generator outputs, such as parton-level event records, is another idea to reduce both computation and disk usage [40].",
"At the LHC, ATLAS and CMS often duplicate the work to generate the same physics process at the same precision and to store the result.",
"Avoiding this duplication would require coordination to standardize data formats, host the output, and ensure that the choices of model and generator parameters satisfy different analysis needs.",
"The existing LHC working groups that provide resources for cross sections and other theoretical calculations could serve as a model to organize this kind of coordination.",
"Data preservation organizations could act as a central hub for storage, and an increased emphasis on data preservation and access would also improve the field's capabilities to reinterpret and combine results [42].",
"More broadly, it has been suggested that a diverse and cross-cutting collaboration, inspired to MCNet in Europe with a long-term outlook, could bring together the U.S. event generator community to share resources and ideas [42].",
"The breadth and diversity of generator software brings many advantages and opportunities to HEP.",
"However, it also brings challenges, especially when it comes to keeping up with the increasing demands for both physics precision and computing efficiency.",
"Historically, there has not been a single obvious locus to coordinate common activities, even if MCNet and, more recently, HSF have provided useful forums, especially in the areas of training and computational efficiency, respectively.",
"Towards the future, more emphasis should be placed on establishing and coordinating common activities.",
"and improvements in this area should continue to be emphasized.",
"Projects to reduce the computational burden of event generation, for example by adapting to use GPUs, need a substantial increase in effort in order to be successful.",
"Some of the difficulties that generators face in computing are intrinsically tied to the nature of the calculations being performed, so fundamental research may be required to develop better methods.",
"On the whole, event generators are usually supported by small teams that frequently have little to no dedicated funding and cannot adequately plan for succession.",
"As with detector simulation, it is vital to the entire field that funding is provided for permanent positions to support, improve, and expand these crucial components of HEP software.",
"This is echoed throughout the white papers written during the Snowmass process: “The physics Generators used in the field (eg.",
"Pythia, GENIE, Madgraph, Sherpa) also suffer from lack of stable funding in a similar way.” [28] “The common needs to all direct dark matter detection experiments include: ...",
"Continued support for event generators, including those developed as part of a national security program.” [29] “Participation in generator-related activities is poorly incentivized for both theorists and experimentalists, and opportunities to pursue neutrino generator development as one's primary research activity are rare... A need for greater coordination and prioritization of such activities is widely recognized in the neutrino scattering community.",
"Despite promising initial discussions that have taken place in a series of recent workshops [68, 529–531]; however, neither a clear leadership structure nor significant institutional support to carry out the related work have yet emerged...",
"The cooperation needed for success in neutrino generator development cannot occur to the extent that funding agencies impose rigid boundaries between [high energy and nuclear physics]... [Stitching together multiple models to achieve complete simulations] is ideally done in collaboration with theorists to minimize inconsistencies, but incentives for their direct involvement are currently poor.” [45] “[Theorists]... may not be motivated to work on software optimisations that are not “theoretical” enough to advance their careers.",
"Generator support tasks in the experiments may also not be enough to secure jobs or funding for experimentalists pursuing a career in research.” [31] Ref.",
"[51] specifically surveys the major collider event generators to report the expected level of support through the HL-LHC era: “Funding and career perspectives are a major common issue reported by many generator experts...",
"Generator support tasks in the experiments may also not be enough to secure jobs or funding for experimentalists pursuing a career in research.",
"It is still not clear for people who are doing work on generators or related tools whether it is really possible to have a dedicated career path from now throughout HL-LHC.",
"This is slowly being acknowledged from funding bodies, but the community has low expectations especially because there is no sign that hiring policies will be modified to provide reasonable funding for generator work.",
"The delay caused by the lack of funding for generator work has already started to cause a loss of know-how in some generator packages.” MadGraph 5_amc@nlo: “the collaboration are of the opinion, shared by other groups, that time to work on performance improvements, even by significant factors, is not well recognised by funding agencies...",
"There are no dedicated funds for HL-LHC.” powheg: “there are no funds specifically dedicated to the maintenance of the framework” sherpa: “Several of the Sherpa team are in permanent positions...",
"In the UK, Sherpa posts at IPPP are seen as part of IPPP's core mission and therefore are expected to be treated as core in future funding reviews.” herwig: “Funding is a serious issue... To fund the bread and butter of what the experiments need, one would need different specific funding.” pythia: “the continued development of Pythia8 throughout the HL-LHC era depends on funding agencies and laboratory priorities, though these are not expected to change.",
"The biggest challenge to the collaboration is finding permanent positions for junior people so that they can continue their valuable contributions.” EvtGen: “The core EvtGen team has 4 members who have a fraction of their time funded for EvtGen through an UK STFC grant...",
"There is some funding for a computer engineer to work on code redesign for multithreading.” The developers of the BeAGLE generator, which is used for inelastic scattering in nuclear collisions, state: “The future development of this code is uncertain primarily due to lack of manpower and reliable funding... As discussed above, funding/manpower issues are currently limiting our ability to implement many of these improvements.” [42]"
],
[
"Continuum Field Theory Calculations",
"Continuum field theory calculations include both precision high order perturbative calculations [52] and conformal bootstrap theoretical calculations [53]."
],
[
"Precision perturbative calculation",
"Precision measurements at the LHC and future colliders require theory predictions with uncertainties at the percent level for many observables.",
"Theory uncertainties due to the perturbative truncation are particularly relevant and must be reduced for a large variety of relevant processes.",
"At the high luminosity LHC, even rare processes like Higgs production require theoretical control of cross sections at the 1% level.",
"Reference [52] surveyed the theoretical community and received responses covering 53 scientific publications.",
"This Snowmass whitepaper gave clear motivation for precision as follows: “To fully exploit the physics potential of current and future collider experiments, in particular to unambiguously identify signals of new physics, it is crucial to improve the precision of theoretical predictions.” Since the 2013 Snowmass exercise, the state-of-the-art has moved dramatically and now next-to-next-to-next-to-leading order (N$^3$ LO) QCD calculations for $2\\rightarrow 1$ processes and NNLO QCD calculation for $2\\rightarrow 3$ processes and $2\\rightarrow 2$ processes with internal masses are becoming state-of-the-art and reasonably fast on contemporary computing resources, while automation and more complex final states at NNLO at N$^3$ LO are goals for the Snowmass period.",
"The computational challenge is to perform Feynman loop integrals within perturbative quantum field theory, giving access to the scattering matrix and its analytic structures, often involving non-standard special mathematical functions.",
"The aim is to, eventually, be able to numerically evaluate the expressions such that the error is reliable, the result has sufficient precision, and the computation consumes acceptable resources.",
"Suitable analytical or symbolic preparation of the integrals and amplitudes must be performed and the evaluation of these expressions during Monte Carlo integration over final state phase space, so that they may feed into event generators.",
"Analytical solutions of Feynman integrals in terms of special mathematical functions are preferred from an efficiency perspective, but often require process-specific knowledge or even new mathematical methods.",
"Numerical or semi-analytical methods can allow broader flexibility and better automation.",
"Numerical direct integration approaches are fully established and implemented in codes such as pySecDec and fiesta, but are typically slow in evaluation times.",
"Novel semi-analytical methods based on series expansions seem promising and first public packages such as DiffExp and AMFlow became available recently.",
"Such series expansion methods reduce evaluation times to the order of a minute per phase-space point for typical two-loop integrals.",
"QCD corrections to scattering amplitudes have been fully automated at NLO using libraries like Helac-NLO, MadGraph 5_amc@nlo, nlox, OpenLoops, and recola as well as many other public and private tools.",
"Two and higher loop calculations remain challenging but in the last five years significant progress has been made in many combinations of final state particle multiplicity and number of internal mass scales at two loops, and in more restrictive cases at three and four loops.",
"Further the complete five loop beta function has been calculated.",
"A core computational bottleneck for the calculation of both Feynman amplitudes and loop integrals is in the reduction of linear relations between Feynman integrals (integration-by-parts identities), with fire, reduze, LiteRed and kira being example codes.",
"From a computational perspective the use of modular (finite field) arithmetic in the calculation of integration by parts reduction is an interesting transformation of the computational requirement that may require radically different computing hardware solutions.",
"Reference [52] notes significant difficulties fitting within the constraints of common HPC centers: “These novel techniques have opened the door to perform computer algebraic computations for quantum field theory on HPC systems.",
"These novel methods typically use integer arithmetic rather than floating point arithmetic and, depending on the problem, they may require a significant amount of memory per core.",
"The development of (public) codes for challenging amplitude calculations in a HPC friendly way is an ongoing effort.",
"Currently, in some situations, one may need to resort to implementations that have large memory demands and require run-times that are hard to predict and possibly well beyond available batch limits on a given cluster.” Given the importance of precision theoretical calculation it is imperative that appropriate resources be available to enable them.",
"In 2013, a sample of then state-of-the-art calculations took between 50,000 and 1M core hours.",
"Since then, the per-core CPU performance has improved by about a factor of 5 and the per-node performance by about a factor of 30.",
"At the time, it was hoped to be able to use efficiently either many-core architectures such as Xeon Phi or GPUs.",
"The current state of the art has the additional complexity largely balanced by the growth in per-node performance, reaching up to about 10M core hours with current CPUs, as shown in Fig.",
"REF .",
"Parallelism is typically used within a node, and is indeed in part dictated by growth in core count relative to memory capacity.",
"Multinode parallelism is only exploited as throughput computing making use of trivial parallelism by the majority of codes in the area.",
"[50][h] Computational costs of current state-of-the-art multi-loop calculations in the community, from Ref. [52].",
"Figure: NO_CAPTION Although there is some role for GPUs in numerical integration, they only match a modest part of the scientific requirement [52]: “The use of GPUs is still unclear in our field, since many problems in our field rely on the numerical evaluation or handling of algebraic expressions which are large and/or require irregular memory access patterns.",
"So far, GPUs have found application to cutting-edge problems with the numerical integration of sector decomposed loop integrals.",
"A first step for future applications could include the efficient evaluation of one-loop amplitudes.",
"This would help the huge computational requirements for NLO high-multiplicity evaluations, but also for the real emission integrations for NNLO calculations.",
"Since the efficient use of GPUs is still unclear, future computing for our community will still need to focus on providing CPU resources without attached GPU resources.” Large memory and runtime limitations imposed by typical HPC environments mean that bespoke computing nodes and queues (wherever they are hosted) are required to enable these calculations.",
"Some of the job runtime constraints could be addressed with checkpointing virtual machines but since complex software is involved, some even commercial like Mathematica, a dedicated node environment may be the only practical solution.",
"Machine learning does not at this time play a large role, although some applications have been used in event generation.",
"The licensing of proprietary software like Mathematica or Maple also suggests dedicated computing nodes for this purpose may be cost-effective."
],
[
"Conformal bootstrap",
"The idea of the conformal bootstrap is to constrain and solve conformal field theories (CFTs) using physical consistency conditions like symmetry, unitarity, and causality.",
"By relying on nonperturbative structures, bootstrap methods can work even in strongly-coupled systems where traditional perturbative techniques fail.",
"Over the last few decades, the conformal bootstrap idea has crystallized into two concrete strategies: 1) exploiting exact solvability and 2) deriving bounds using sum rules with positivity constraints and using convex optimization to extract information.",
"When combined with numerical convex optimization, this leads to the numerical conformal bootstrap.",
"The conformal bootstrap is distinguished by being an intrinsically non-perturbative method that yields strict bounds on the theory.",
"Recent successes include precise determinations of critical exponents in physically-relevant theories such as the 3D Ising and O(N) models, constraints on theoretically important theories such as 4D N = 4 supersymmetric Yang-Mills theory and 6D $(2,0)$ SCFTs, and has inspired promising new ideas for bootstrapping S-matrices.",
"The numerical bootstrap is a rapidly-growing field and is anticipated to play a central role in CFTs over the next decade.",
"Targets include 3D scalar models, $O(N)$ vector models in 3D, and in the case of $N=1$ the 3D Ising CFT, while the 3D O(2) model describes the the superfluid transition in liquid helium as well as thin-film superconductors.",
"Here the bootstrap has helped address a controversy, where the results of experiment and the current best Monte Carlo simulations disagreed with each other with 8$\\sigma $ significance, giving results in remarkable agreement with those from Monte Carlo simulations.",
"A second target is 3D Fermionic Models, including 3D CFTs containing N Majorana fermions with Yukawa couplings to one or more scalar fields, often called Gross-Neveu-Yukawa model.",
"The simplest model with $N=1$ Majorana fermion coupled to a real scalar is believed to possess emergent supersymmetry and correspond to the minimal 3D $N=1$ supersymmetric extension of the Ising model, proposed to have a realization on the boundary of topological superconductors.",
"Preliminary studies applying the bootstrap to fermion 4-point functions showed the GNY models to be promising targets for the bootstrap, while further systems can be studied providing a supersymmetric systems including Wess Zumino models.",
"Progress in 3D and 4D gauge theories has been made including including the conformal window of 4D QCD with 12 flavors, establishing a hint of contact with lattice field theory near conformal window results and in $N=4$ supersymmetric Yang Mills theory.",
"From a computational perspective, the whitepaper identifies two key algorithmic challenges: i) to find faster optimization methods, or ways to scale up current tools, and ii) to find efficient methods for exploring high-dimensional spaces of CFT data.",
"High precision arithmetic is required, and often not in native hardware floating point, and used in bespoke semidefinite solver libraries such as SDPB.",
"SDPB can use distributed high precision linear algebra and parallelize over many HPC nodes and hundreds to thousands of cores.",
"The authors are optimistic that further progress will: “enable a new round of fundamental discoveries in theoretical physics, including the numerical classification of CFTs with a small number of relevant operators, definitive answers to longstanding questions about conformal windows of gauge theories in 3 and 4 dimensions, more-or-less complete numerical solutions of the known maximally-supersymmetric CFTs (and their holographic duals), and new robust starting points for Hamiltonian truncation studies of gapped phases.",
"Importantly, the numerical bootstrap can also be used as a discovery tool for finding previously unknown CFTs and as well as for identifying possible analytical solutions of known ones.” The computational requirements so far have involve the use of conventional cluster computers, such as XSEDE and up to 32 nodes and 768 cores.",
"Calculations of different theories range from 200,000 core hours for the 3D Ising model to a few million core hours testing 100 to 1000 points in the CFT parametric space for the 3D $O(3)$ model.",
"Large memory capacity is again significant requirement and is a barrier to efficiency, arising from the high precision (1000 bits) arithmetic and is required for accurately handling large cancellations in the calculation.",
"The authors identify significant software and computing hardware opportunities in future: “further scaling and improving this code, including implementing more sophisticated distributed linear algebra routines, improving memory management to allow scaling past ${\\sim }500$ cores, finding ways to reduce precision requirements, and leveraging new hardware like GPUs and FPGAs.",
"These engineering challenges are tightly coupled to physics: an increase in the scale of solvable semidefinite programs could allow one to explore larger systems of crossing equations and thereby access new CFTs.”"
],
[
"Continuum field theory summary",
"Both multi-loop perturbative calculations and numerical conformal bootstrap have a similar computer hardware requirement.",
"They both have a current software reliance on CPU cores, and a large memory per core requirement.",
"Perturbative calculations have a sizable barrier to the use of GPUs, while the conformal bootstrap approach, after significant software engineering, may be amenable to GPU or FPGA acceleration of high precision arithmetic."
],
[
"Lattice QCD",
"Lattice gauge theory is a systematically improvable theoretical tool for numerical evaluation of the Euclidean Feynman-path integral for Quantum Chromodynamics (QCD).",
"Worldwide lattice gauge theory efforts directly support numerous high-energy physics experiments by calculating properties of hadrons that are vital to interpretation of the experiments.",
"These efforts make significant use of supercomputers and are critically dependent upon continued computing advances.",
"The Flavour Lattice Averaging Group [54] performs a critical review of many important lattice QCD predictions every two to three years.",
"Snowmass Whitepapers have been contributed on the topics relevant to Lattice QCD [55], [45], [56], [57], [58], [59], [60], [61].",
"USQCD also published in 2019 comprehensive set of topical whitepapers and submitted these for consideration by Snowmass [62], [63], [64], [65], [66], [67], [68].",
"The search for new physics requires a joint experimental and theoretical effort.",
"Lattice QCD is an essential tool for obtaining precise model-free theoretical predictions of the hadronic processes underlying many key experimental searches.",
"The role of Lattice QCD in determining the hadronic contributions to the anomalous magnetic moment of the muon was discussed in a Snowmass white paper [59], by the international Muon $g-2$ Theory Initiative white paper [69] and in a white paper by the USQCD collaboration in 2019 [64].",
"These calculations will be critical to the interpretation of results from the Fermilab Muon $g-2$ Experiment [70], which is currently in excellent agreement with results from the earlier Brookhaven experiment [71].",
"Lattice QCD also plays a key role in ab initio prediction of nucleon structure and parton physics [57], [55].",
"Understanding the neutrino-nucleus interaction is critical to the analysis of the Deep Underground Neutrino Experiment.",
"The interaction between an intermediate W or Z boson with with a quark confined inside a neutron or proton within the nucleus is theoretically complex.",
"This was discussed by USQCD in a 2019 white paper [65] and holistically the interplay between lattice QCD, EFTs, nuclear physics, phenomenology, and neutrino event generators was discussed in a contributed Snowmass Whitepaper [45].",
"A comprehensive quark-flavor physics program has been discussed in a USQCD whitepaper [64].",
"Contributed Snowmass white papers include improving the understanding of anomalies in B physics [72], emerging from the the Large Hadron Collider and studied further at Belle II, as well as reconciling CP violation observed in kaon experiments with the standard model and exploring rare kaon decays [56].",
"These phenomena are typically highly suppressed in the standard model and therefore also offer promising avenues for the discovery of new physics.",
"As experimental measurements become more precise over the next decade, lattice QCD will play an increasing role in providing the needed matching theoretical precision.",
"Achieving the needed precision requires simulations with lattices with substantially increased resolution.",
"With finer lattice spacing comes an array of new challenges.",
"They include algorithmic and software-engineering challenges, challenges in computer technology and design, and challenges in maintaining the necessary human resources.",
"The simulations at finer lattice spacing and larger volumes required to realize these goals introduce new challenges [58]: “Meeting them requires new algorithmic research, novel computer hardware design beyond the exascale, improved software engineering, and attention to maintaining human resources.” As a computational problem, Lattice gauge theory is performed on structured Cartesian grids with a high degree of regularity and natural data parallelism.",
"The approach formulates the Feynman path integral for QCD as a statistical mechanical sampling of the related Euclidean space path integral.",
"The sampling is performed by Markov chain Monte Carlo (MCMC) sampling, using forms of the hybrid Monte Carlo (HMC) algorithm [73], [74], [75].",
"Present algorithms for both MCMC sampling and Dirac solvers display growing limitations as substantially greater ranges of energy scales are included in our problem, an algorithmic challenge called critical slowing down.",
"The development of numerical algorithms is a significant intellectual activity that spans physics, mathematics, and computer science.",
"The central repeated operation is the solution of the gauge covariant Dirac equation.",
"The solution is usually performed using iterative Krylov solvers, Newer multigrid [76], [77], [78] solvers have demonstrated order-of-magnitude gains for Wilson fermions by approximately and repeatedly handling degrees of freedom in the low lying eigenspace as a form of preconditioner.",
"The US HEP program is presently focused on the domain-wall and staggered approaches, but corresponding gains for staggered and domain-wall-fermion discretizations are a critical open research activity.",
"A second algorithmic direction is the critical slowing down of MCMC algorithms.",
"These are being studied under Exascale Computing and SciDAC projects and such support is critical to progress in the field.",
"Recent algorithmic directions with different ways of attacking the same problem include applications of machine learning to configuration sampling.",
"This possibility has been discussed in a dedicated Snowmass white paper [79] and will be covered in the report of topical group CompF3 Machine Learning[80].",
"Related Snowmass submissions on tensor networks and quantum simulation will be covered in the report of CompF6 Quantum Computing [81], [82].",
"The Lattice QCD workflow is divided into two phases.",
"First, a MCMC sampling phase generates an ensemble of the most likely gluon field configurations distributed according to the QCD action.",
"The ensemble generation is serially dependent and represents a strong scaling computational problem.",
"Ideally one would be able to use efficiently $O(10^4)$ computing nodes on $O(256^4)$ data points.",
"On the largest scales this becomes a halo-exchange communication problem with a very large interconnect bandwidth requirement since the local data bandwidths vastly exceed those of inter-node communication.",
"In the second phase hadronic observables are calculated on each sampled configuration where many thousands of quark propagators are calculated and assembled into hadronic correlation functions.",
"This both allows more scope for amortizing the setup cost of advanced algorithms like multigrid or deflation, and also has a high degree of trivial parallelism.",
"Scaling Monte Carlo sampling to many computational nodes requires strong scaling and high interconnect bandwidth.",
"Scaling the hadronic observable calculations is not as challenging as multiple configurations can be analyzed at the same time, each job running on a moderate number of nodes.",
"A number of specific simulations, Table REF , have been proposed with estimated costs in a Snowmass white paper [56], and the same methodology can be used to estimate the requirements of the ideal ensemble for flavor physics.",
"The final entry is associated with physics in the B-meson system indicated in Snowmass white paper [72].",
"A $256^3\\times 512$ lattice at a lattice spacing $a = 0.04$ fm ($a^{-1} \\sim 5$ GeV) would allow us to simulate up/down, strange, charm, and bottom quarks at their physical mass in a 10 fm box with $m_\\pi L = 7$ and requires more than a sustained Exaflop year.",
"Several such calculations would be sought by the whole community.",
"These are a modest subset of the many many proposed calculations across the Snowmass contributions, but set the scale of computing sources required in the area of Lattice QCD simulations in support of experiment at beyond-exascale.",
"Between flavor physics, nucleon structure, neutrino scattering, and beyond-the-standard model physics, one could project four or five times this requirement in aggregate across the entire field in the USA alone.",
"On this basis, Ref.",
"[58] estimated that: “These simulation goals therefore clearly demonstrate a need for computers at least 10x more capable than the coming Exaflop computers during the Snowmass period.",
"Since the performance is required to be delivered on a real-code performance basis... more than an order of magnitude improvement, perhaps, from both algorithms and computing are required.” Table: Proposed lattice volumes and cost estimatesin sustained Exaflop hours, scaled from currentsimulations on Cori (NERSC) and Summit (ORNL).Volumes and estimates are proposed in Snowmass white paper , while the finalentry uses the same methodology to estimate the costof the most expensive proposed B-physics capable simulation.The massive vector parallelism of lattice gauge theory is, in principle, amenable to GPU and possibly other acceleration.",
"This imposes a significant additional programmer overhead and the most commonly used packages receive sufficient investment to use the complete range of modern accelerated supercomputers and many of the largest projects use allocations on DOE supercomputer resources.",
"Commonly used Lattice QCD software has been supported by the DOE SciDAC and Exascale Computing Projects.",
"These include Grid [83], [84], [85], MILC [86], [87], CPS [88], Chroma [89] and Quda [90].",
"This has enabled major packages to support the most advanced GPU accelerated HPC computers using software interfaces such as HIP, SYCL, and CUDA APIs in addition to giving good performance on several CPU SIMD architectures.",
"Newer interfaces like OpenMP 5.0 offload and C++17 parallel STL are planned to be adopted as and when appropriate.",
"For smaller projects, especially where rapid development and programmer productivity are at a premium, it is better and more cost effective in terms of human effort to maintain access to a range of CPU resources.",
"USQCD institutional clusters at Brookhaven National Laboratory, Fermilab, and Jefferson Laboratory have been instrumental in supporting the significant number of smaller and experimental projects that would not achieve the return on investment to justify bespoke software development for multiple architectures.",
"Reference [58] notes the heavy reliance on high performance computing places a particularly large dependence on highly tuned bespoke software in this field: “One hopes that this might consolidate the proliferation of programming interfaces as the lack of standardization imposes a very significant software development burden on the science community with duplication of effort for multiple systems.",
"This software development underpins the entire community effort and requires support to make the Snowmass science goals feasible.” “The challenges exist on multiple fronts: intellectual in developing algorithms that evade critical slowing down, software engineering to develop well-performing and portable code on an evolving range of supercomputers and programming models, and technical to remain engaged with the DOE HPC community as systems are planned and developed.” “The community activity is dependent on the existence of bespoke software environments that enable efficient simulation on rapidly evolving supercomputers, which requires significant expertise to develop and sustain.” USQCD has had senior staff, postdoctoral researchers, and post-graduate students funded under the Exascale Computing Project (ECP) and the SciDAC program.",
"It has also funded the development of high performance software portable to Exascale hardware [83], [85].",
"It was noted as important that flexible high-performance software continues to be developed for a diverse range of architectures that tracks the DOE computing program: “the productivity of the community depends on large code bases... which do not have a secure model for development and support which places investments at risk.” Much of the bespoke software development is performed in US National Laboratories.",
"This is required to address HPC architectures as they emerge in the Computational Frontier, developing performance-portable high-level libraries to enable a write once and run anywhere approach that many domain scientists, both in laboratories and universities, can modify effectively.",
"This effort must track the evolution of computing.",
"Finally, it was noted that theoretical particle physics has been one of the last area of physics to recognize the importance of computation in forefront research [58].",
"Continued effort is urgently required to overcome this historical bias and create a vibrant pool of skilled young faculty, and around them their Ph.D. students and research groups.",
"Given the role of lattice gauge theory in confronting hadronic experiment, this field would particularly benefit from a program of joint Lab-University appointments."
],
[
"Conclusions and Recommendations",
"For scientific activities that consume a large amount of computing time and map well to emerging HPC architectures, there is a wonderful opportunity to apply enormous amounts of computation to solve incredibly complex problems.",
"However, common cross-cutting issues have been raised across many different scientific topics.",
"These issues are largely centered on the critical dependency between modern science and computation and software.",
"Computer architectures have necessarily become increasingly complex, diverse, and challenging to program.",
"In order to turn the potential of DOE computing into scientific deliverables, it is critical to develop skills in the community and to fund the required software and algorithmic development.",
"Support for widely used scientific software packages is required to enable the most effective return.",
"Further, accelerated architectures are not fully general purpose and present challenges to algorithms that are strongly serial without any intrinsic vectorization or parallelism potential.",
"Even for those algorithms for which it is possible in principle to use this hardware, the investment to develop software that runs efficiently across many different platforms and programming environments requires a real investment by the domain scientists.",
"This investment of effort may not always be a good value proposition.",
"For these two reasons, there is a continuing need for a mix of computing with both accelerated and general purpose architectures.",
"This mixture should be sized appropriately to accelerate large scale computation where possible but also to effectively support science whose software cannot exploit acceleration.",
"This may occur if it is not possible to port the underlying algorithms to compute accelerators or if a cost-benefit analysis identifies such a port as not being a cost-effective proposition.",
"The latter case likely comprises a “tail” of many smaller computing tasks that run diverse software with limited development effort available.",
"Our recommendations on computing hardware are as follows: Faster computer hardware: For lattice gauge theory applications, there is a need for HPC resources ten times or more faster than planned Exascale computing systems.",
"Support of right-sized CPU clusters: There should be a right-sized (as large as required, as small as possible) provisioning of general purpose computational cores with high performance memory for important algorithms that do not easily map to computational accelerators.",
"National Laboratory institutional (non-HPC) clusters contribute to this role.",
"NERSC is a welcome example of providing both CPU and GPU/accelerator resources to handle a wider variety of problems relevant to HEP.",
"These should accommodate a diverse range of requirements for software, large memory, and long running single node jobs.",
"Universal programming interface: The proliferation of programming interfaces is a barrier to portability and return on investment in software.",
"Simplification and consolidation of the interfaces will improve scientific output.",
"Software portability: The difficulty of programming highly accelerated hardware with performance portability is significant.",
"Appropriate support for software development and maintenance is essential to ensure success of much of the science discussed in this report.",
"Automation of memory hierarchy: Hierarchical memory is a significant burden on scientific programmer productivity.",
"Virtual memory paging systems can alleviate this and can be made efficient by using larger page sizes, if necessary.",
"This may need substantial investment by computer vendors with robust encouragement from the Department of Energy.",
"Early access to new computing hardware: A key element of managing the science program is the early engagement with DOE HPC laboratory sites during the development, and years prior to installation, of major new facilities.",
"The lead time for porting to new architectures lies in the region of multiple years, and early engagement with emerging architectures is required to ensure timely scientific exploitation.",
"The scientific software ecosystem should be nurtured to enable the proposed Snowmass science.",
"There are several steps that could be proactively taken to ensure the maximum scientific return is delivered.",
"Hardware accelerator-friendly, portable common tools: Specific projects to adapt common software packages to run efficiently on GPUs, including those at new HPCs, have shown initial promise with more than order of magnitude speed improvements.",
"These projects should be continued with sufficient effort to deliver complete products and to incorporate portability solutions to support different coprocessor devices and architectures.",
"Adoption of this new software will additionally require expertise devoted to integration and usage in experimental software frameworks, analysis tools, etc., which should be supported as discussed below.",
"The exploration of new machine learning-based methods, as well as technical improvements to existing (CPU-based) software, should also continue.",
"Best practices for common software: Whenever possible, interoperability and common data structures should be encouraged.",
"Existing standard libraries and underlying software technologies should be used when available and feasible.",
"Interoperability, portability, and common, shared software are fundamental pillars of international scientific collaboration and will become increasingly important in the future.",
"Lab-supported software development: Just as the large experiments require talented permanent staff at the Labs to engineer experiments, theoretical calculation and simulation in HEP experiment, in high energy theory, in cosmology, and in large particle accelerator building have large communities often dependent on sophisticated long-term software systems.",
"Career paths must be created to retain some of the most talented experts in software and algorithms.",
"A larger permanent laboratory staff of software development experts could help address HPC architectures as they increase in importance within the Computational Frontier.",
"Organization for long-term common software maintenance: Despite the numerous successes and broad adoption of common software tools across all topical areas in this report, the future of these projects is in jeopardy because of an extreme lack of funding.",
"One priority should be the development, maintenance, and support for common software tools in the areas of accelerator modeling, detector simulation, and physics generators.",
"Because this kind of long-term effort to sustain cross-cutting software does not fit easily into existing modes of funding, new processes and organizations should be established in cooperation with funding agencies.",
"Continuous collaboration with ASCR: A lot of progress has been made in high energy physics applications through collaborations with programs supported by ASCR under the SciDAC and ECP projects.",
"We believe that such collaborations should be encouraged and continuously supported for future computational modeling in high energy physics.",
"The community found the application development focus of the ECP project particularly effective.",
"Joint lab-university tenure track appointments: We believe the DOE should seek to foster the continued development of intellectual leaders in theoretical calculation and simulation.",
"The health of the field requires a similar cohort of individuals at the best universities, reflecting the intellectual vigor and potential of this area to contribute to DOE scientific goals.",
"The creation of such positions can be stimulated by DOE-funded joint, five-year, tenure-track appointments.",
"A good example might be the Jefferson Lab University Relations program of joint and bridged faculty appointments in nuclear physics.",
"Training: Accessible training in both novel architectures and AI/ML with a low barrier to access for graduate students, postdoctoral researchers, and domain scientists is critical to ensuring the skills base exists for productive science in theoretical calculation and simulation.",
"The use of cross-platform performance portable APIs should be encouraged to maximize return on investment in software.",
"Hackathons and code examples are particularly useful."
],
[
"Acknowledgments",
"P. Boyle has been funded by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under the Contract No.",
"DE-SC-0012704 (BNL).",
"K. Pedro is supported by the Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No.",
"DE-AC02-07CH11359 with the U.S. Department of Energy.",
"J. Qiang is supported by the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH11231 (LBNL)."
]
] | 2209.08177 |
[
[
"A Robust and Constrained Multi-Agent Reinforcement Learning Framework\n for Electric Vehicle AMoD Systems"
],
[
"Abstract Electric vehicles (EVs) play critical roles in autonomous mobility-on-demand (AMoD) systems, but their unique charging patterns increase the model uncertainties in AMoD systems (e.g.",
"state transition probability).",
"Since there usually exists a mismatch between the training and test (true) environments, incorporating model uncertainty into system design is of critical importance in real-world applications.",
"However, model uncertainties have not been considered explicitly in EV AMoD system rebalancing by existing literature yet and remain an urgent and challenging task.",
"In this work, we design a robust and constrained multi-agent reinforcement learning (MARL) framework with transition kernel uncertainty for the EV rebalancing and charging problem.",
"We then propose a robust and constrained MARL algorithm (ROCOMA) that trains a robust EV rebalancing policy to balance the supply-demand ratio and the charging utilization rate across the whole city under state transition uncertainty.",
"Experiments show that the ROCOMA can learn an effective and robust rebalancing policy.",
"It outperforms non-robust MARL methods when there are model uncertainties.",
"It increases the system fairness by 19.6% and decreases the rebalancing costs by 75.8%."
],
[
"Introduction",
" Autonomous mobility-on-demand (AMoD) system is one of the most promising energy-efficient transportation solutions as it provides people one-way rides from their origins to destinations , , .",
"Electric vehicles (EVs) are being adopted worldwide for environmental and economical benefits , and AMoD systems embrace this trend without exception.",
"However, the trips sporadically appear, and the origins and destinations are asymmetrically distributed and hard to predict in AMoD systems , .",
"Such spatial-temporal nature of urban mobility increases the management difficulty of a large-scale vehicle fleet and makes the system sensitive to disturbances , , .",
"Moreover, EVs' unique charging patterns (long charging time, high charging frequency, and unpredictable charging behaviors) increase the complexity and uncertainty of the EV AMoD system dynamics , , .",
"In real-world applications, we usually do not have perfectly accurate knowledge of the true system model, e.g., the state transition probability of the AMoD systems, and therefore, there usually exists a model mismatch between the simulator (training environment) and the real-world application (test environment).",
"Thus, existing EV AMoD vehicle coordination methods , , , , , , may have significant performance degradation in the test (true) environment.",
"Despite model-based methods considering prediction errors in mobility demand or vehicle supply , , , , uncertainty in system state transition remains largely unexplored in AMoD systems.",
"More importantly, practical deployment usually needs to satisfy constraints on, e.g.",
"safety and fairness, while maximizing the performance.",
"In this work, we propose a robust and constrained multi-agent reinforcement learning (MARL) framework for EV AMoD systems.",
"The goal is to achieve AMoD system fairness by finding robust rebalancing policies for idle and low-battery EVs with minimal rebalancing cost, under model uncertainty.",
"The advantages of our formulation are two-folds: (i) fairness constraints can still be satisfied even if there is model mismatch; and (ii) the rebalancing cost is still optimized when there is model mismatch.",
"Our Key Contributions are as follows: (1) To the best of our knowledge, this work is the first to formulate EV AMoD system vehicle rebalancing as a robust and constrained multi-agent reinforcement learning problem under model uncertainty.",
"Via a proper design of the state, action, reward, cost constraints, and uncertainty set, we set our goal as minimizing the rebalancing cost while balancing the city's charging utilization and service quality, under model uncertainty.",
"(2) We design a robust and constrained MARL algorithm (ROCOMA) to efficiently train robust policies.",
"The proposed algorithm adopts the centralized training and decentralized execution (CTDE) framework and develops the first robust natural policy gradient (NPG) to improve the efficiency of policy training.",
"(3) We run experiments based on real-world E-taxi system data.",
"We show that our proposed algorithm performs better in terms of reward and fairness, which are increased by 19.6%, and 75.8%, respectively, compared with a non-robust MARL-based method when model uncertainty is present."
],
[
"Related Work",
" AMoD system vehicle rebalancing algorithms re-allocate idle vehicles, sometimes considering charging constraints .",
"Heuristics can lead to sub-optimal rebalancing solutions .",
"Other major categories of AMoD system rebalancing methods include optimization-based algorithms , Model Predictive Control (MPC) and Reinforcement Learning (RL) .",
"Optimization and MPC-based approaches usually formulate the AMoD system vehicle rebalancing problem as a convex optimization or mixed-integer programming problem, where the objective is to improve service quality , or maximize the number of served passengers with fewer vehicles , , .",
"These model-based approaches usually rely on precise knowledge of the probability transition model of the complex dynamics of AMoD systems and, consequently, are sensitive to model uncertainties.",
"Though robust and distributionally robust optimization-based methods have been designed to consider uncertainties caused by mobility demand, supply, or covariates predictions , , , the probability transition error or uncertainty in system dynamics has not been addressed yet.",
"RL-based approaches relax the dependency on system dynamic models.",
"Various RL-based methods, including DQN, A2C and their variants , , , , , , , have been proposed to solve the vehicle rebalancing problem.",
"However, RL suffers from the sim-to-real gap; that is, the gap between the simulator and the real world often leads to unsuccessful implementation if the learned policy is not robust to model uncertainties , .",
"None of the above RL-based rebalancing strategies consider this gap.",
"Given that designing an efficient and robust EV rebalancing method under model uncertainties is still an unsolved challenge, Robust RL aims to find a policy that maximizes the worst-case cumulative reward over an uncertainty set of MDPs , , , .",
"To achieve high system fairness while minimizing rebalancing cost under model uncertainty, we put the fairness constraints in our RL formulation, which is known as Constrained RL that aims to find a policy that maximizes an objective function while satisfying certain cost constraints , .",
"However, applying robust and constrained RL to AMoD rebalancing is still a challenge due to the high-dimensional state and action spaces commonly present in transportation systems.",
"And the problem of robust constrained RL itself is very challenging even in the simple tabular case.",
"Our proposed robust and constrained MARL formulation and algorithm explicitly consider model uncertainties and system fairness to learn robust rebalancing solutions for AMoD systems.",
"And we derive a natural policy gradient for robust and constrained MARL to improve the efficiency of policy training."
],
[
"Problem Statement",
"We consider the problem of managing a large-scale EV fleet to provide fair and robust AMoD service.",
"The goal is to (i) rebalance idle and energy-efficient EVs (we denote them as vacant EVs for notation convenience) among different regions to provide fair mobility service on the passenger's side; (ii) allocate low-battery EVs to charging stations for fair charging service on the EVs' side; (iii) minimize the managing cost of (i) and (ii); (iv) find rebalancing policies robust to model uncertainty, i.e.",
"uncertainty in transition kernel of the MDP.",
"We assume that a city is divided into $N$ regions according to a pre-defined partition method , , .",
"A day is divided into equal-length time intervals.",
"In each time interval $[t, t+1)$ , customers' ride requests and EVs' charging need are aggregated in each region.",
"After the location and status of each EV are observed, a local trip and charging assignment algorithm matches available EVs with passengers and low-battery EVs with charging stations, using existing methods in the literature , , .",
"Then the state information of each region is updated, including the numbers of vacant EVs and available charging spots in each region.",
"Each region then rebalances both vacant and low-battery EVs according to the trained MARL policy.",
"This work focuses on a robust EV rebalancing algorithm design under model uncertainties to maximize the worst-case expected reward of the system while satisfying fairness constraints.",
"For notational convenience, the parameters and variables defined in the following parts of this section omit the time index $t$ when there is no confusion."
],
[
"Preliminary: Multi-Agent Reinforcement Learning",
"We denote a Multi-Agent Reinforcement Learning (MARL) problem by a tuple $G = \\langle \\mathcal {N}, S, A, r,p,\\gamma \\rangle $ , in which $\\mathcal {N}$ is the set of $N$ agents.",
"Each agent is associated with an action $a^i \\in A^i$ and a state $s^i \\in S$ .",
"We use $A = A^1 \\times \\cdots \\times A^N$ to denote the joint action space, and $S = S^1 \\times \\cdots \\times S^N$ the joint state space.",
"At time $t$ , each agent chooses an action $a^i_t$ according to a policy $\\pi ^i: S^i \\rightarrow \\Delta (A^i)$ , where $\\Delta (A^i)$ represents the set of probability distributions over the action set $A^i$ .",
"We use $\\pi = \\prod _i^N \\pi ^i: S \\rightarrow \\Delta (A)$ to denote the joint policy.",
"After executing the joint action, the next state follows the state transition probability which depends on the current state and the joint action, i.e.",
"$p: S \\times A \\rightarrow \\Delta (S)$ .",
"And each agent receives a reward according to the reward function $r^i: S \\times A \\rightarrow \\mathbb {R}$ .",
"Each agent aims to learn a policy $\\pi ^i$ to maximize its expected total discounted reward, i.e.",
"$\\max _{\\pi ^i} v^{\\pi , i}_r(s)$ for all $s \\in S$ , where $v^{\\pi , i}_r(s) = \\mathbb {E}[\\sum _{t = 1}^{\\infty } \\gamma ^{t-1} r_t^i(s_t, a_t) | a_t \\sim \\pi (\\cdot | s_t), s_1 = s]$ which is also known as the state value function for agent $i$ .",
"$\\gamma \\in (0,1)$ is the discounted rate.",
"If these agents belong to a team, the objective of all agents is to collaboratively maximize the average expected total discounted reward over all agents, i.e.",
"$\\max _\\pi v^{\\pi }_{r}(s)$ for all $s \\in S$ , where $v^{\\pi }_{r}(s) = \\mathbb {E}_\\pi [\\sum _{t = 1}^{\\infty } \\gamma ^{t-1} \\sum _{i\\in \\mathcal {N}}r_t^i(s_t, a_t)/N | s_1 = s]$ ."
],
[
"Robust and Constrained Multi-Agent Reinforcement Learning Formulation for EV Rebalancing",
"We formulate the EV rebalancing problem as a robust and constrained MARL problem $G_{rc} = \\langle \\mathcal {N}, S, A, P, r, c, d, \\gamma \\rangle $ , and we define the agent, state, action, probability transition kernel uncertainty set, reward, and cost and fairness constraints as follows."
],
[
"We define for each region a region agent, who determines the rebalancing of vacant and low-battery EVs at every time step.",
"This distributed agent setting is more tractable for large-scale fleet management than a single agent setting because the action space can be prohibitively large if we use a single system-wide agent ."
],
[
"A state $s^{i}$ of a region agent $i$ consists two parts that indicate its spatiotemporal status from both the local view and global view of the city.",
"We define the state $s^i=\\lbrace s_{loc}^i, s_{glo}^i\\rbrace $ , where $s_{loc}^i = (V_i, L_i, D_i, E_i, C_i)$ is the state of region $i$ from the local view, denoting the number/amount of vacant EVs, low-battery EVs, mobility demand, empty charging spots, and total charging spots in region $i$ , respectively.",
"And $s_{glo}^i = (t, pos_i)$ , where $t$ is the time index (which time interval), $pos_i$ is region location information (longitudes, latitudes, region index).",
"The initial state distribution is $\\rho $ ."
],
[
"The rebalancing action for vacant EVs is denoted as $a^i_v = \\lbrace a^i_{v,j} \\rbrace _{j \\in \\text{Nebr}_i}$ , the charging action for low-battery EVs as $a^i_l = \\lbrace a^i_{l,j} \\rbrace _{j \\in \\text{Nebr}_i}$ , where $a^i_{v,j}, a^i_{l,j} \\in [0,1]$ is the percentage of currently vacant EVs and low-battery EVs to be assigned to region $j$ from region $i$ , respectively.",
"And $\\text{Nebr}_i$ is the set consisting of region $i$ and its adjacent regions as defined by the given partition.",
"Therefore $\\sum _{j \\in \\text{Nebr}_i} a^i_{v,j} = 1$ and $\\sum _{j \\in \\text{Nebr}_i} a^i_{l,j} = 1$ for all $i$ .",
"We denote $m^i_{v,j} = h( a^i_{v,j} v^i)$ the actual number of vacant EVs assigned from region $i$ to region $j$ , $m^i_{l,j} = h( a^i_{l,j} l^i)$ the actual number of low-battery EVs in region $i$ assigned to region $j$ .",
"The function $h(\\cdot )$ is used to ensure that the numbers remain as integers and the constraints $\\sum _j m^i_{v,j} = v^i, \\sum _j m^i_{l,j} = l^i$ hold for all $i$ ."
],
[
"We restrict the transition kernel $p$ to a $\\delta $ -contamination uncertainty set $P$ , , , in which the state transition could be arbitrarily perturbed by a small probability $\\delta $ .",
"Specifically, let $\\tilde{p} = \\lbrace \\tilde{p}^a_s \\mid s \\in S, a \\in A \\rbrace $ be the centroid transition kernel, from which training samples are generated.",
"The $\\delta $ -contamination uncertainty set centered at $\\tilde{p}$ is defined as $P := \\bigotimes _{s \\in S, a \\in A}P^a_s$ , where $P^a_s := \\lbrace (1-\\delta )\\tilde{p}^a_s + \\delta q \\mid q \\in \\Delta (S) \\rbrace , s \\in S, a \\in A$ ."
],
[
"Since one of our goals is to minimize the rebalancing cost, we define the shared reward as the negative value of the total rebalancing cost after EVs execute the decisions: $r(s,a) -[ c_v(s,a) + \\bar{\\alpha }c_l(s,a)]$ , where $\\bar{\\alpha }$ is a positive coefficient, and $c_v(s,a), c_l(s,a)$ are moving distances of all vacant and low-battery EVs under the joint state $s$ and action $a$ , respectively.",
"We then define the worst-case value function of a joint policy $\\pi $ as the worst-case expected total discounted reward under joint policy $\\pi $ over $P$ : $v^\\pi _r(s) = \\min _{p \\in P} \\mathbb {E}_\\pi \\left[ \\sum _{t=1}^\\infty \\gamma ^{t-1}{r}_t | s_1 = s \\right]$ .",
"The notation is the same as MARL without considering uncertainty.",
"By maximizing the shared worst-case value function, region agents are cooperating for the same goal."
],
[
"Another goal is to achieve the system-level benefit, i.e., balanced charging utilization and fair service.",
"We define the charging fairness $u_c$ and mobility fairness $u_m$ in Subsection REF .",
"If the values of these fairness metrics are higher than some thresholds by applying a rebalancing policy $\\pi $ , we say the policy $\\pi $ provides fair mobility and charging services among the city.",
"We then augment the MARL problem $G$ with an auxiliary cost function $c$ , and a limit $d$ .",
"The function $c: S \\times A \\rightarrow \\mathbb {R}$ maps transition tuples to cost, like the usual reward.",
"Similarly, we let $v^{\\pi }_c(s)$ denote the worst-case state value function of policy $\\pi $ with respect to cost function $c$ : $v^{\\pi }_c(s) = \\min _{p \\in P}\\mathbb {E}_\\pi [\\sum _{t=1}^{\\infty } \\gamma ^{t-1} c(s_{t}, a_{t}) | s_1 = s]$ .",
"The cost function $c$ is defined as the system fairness (a weighted sum of city's charging fairness $u_c$ and mobility fairness $u_m$ ), i.e., $c(s,a) u_c(s, a) + \\bar{\\beta }u_m(s, a)$ , where $\\bar{\\beta }$ is a positive coefficient.",
"Then the set of feasible joint policies for our robust and constrained MARL EV rebalancing problem is $\\Pi _{C} := \\lbrace \\pi : \\forall s\\in S, v^{\\pi }_c(s) \\ge d \\rbrace $ ."
],
[
"The goal of our robust and constrained MARL EV rebalancing problem is to find an optimal joint policy $\\pi ^*$ that maximizes the worst-case value function subject to constraints on the worst-case expected cost $\\pi ^* = \\operatornamewithlimits{argmax}_{\\pi \\in \\Pi _{C}} v^{\\pi }_r(\\rho )$ , where $v^{\\pi _\\theta }_{\\text{tp}}(\\rho )= \\mathbb {E}_{s\\sim \\rho }[v^{\\pi _\\theta }_{\\text{tp}}(s)]$ , $\\text{tp} \\in \\lbrace r,c\\rbrace $ .",
"We consider policies $\\pi (\\cdot | \\theta )$ parameterized by $\\theta $ and consider the following equivalent max-min problem based on the Lagrangian : $\\max _{\\theta } \\min _{\\lambda \\ge 0} J(\\theta , \\lambda ) := v^{\\pi _\\theta }_r(\\rho ) + \\lambda (v^{\\pi _\\theta }_c(\\rho ) - d),$"
],
[
"Fairness Definition",
"We consider both the mobility supply-demand ratio , , and the charging utilization rate , , , in each region as service quality metrics for the EV AMoD system.",
"However, with limited supply volume in a city, achieving high supply-demand ratios in all regions may not be possible.",
"Keeping the supply-demand ratio of each region at a similar level allows passengers in the city to receive fair service , .",
"Similarly, given a limited number of charging stations and spots, to improve the charging service quality and charging efficiency with limited infrastructure, balancing the charging utilization rate of all regions across the entire city is usually one objective in the scheduling of EV charging , , .",
"The fairness metrics of the charging utilization rate $u_c$ and supply-demand ratio $u_m$ are designed based on the difference between the local and global quantities: $u_c(s,a) = -\\sum ^N_{i = 1} \\left|\\frac{E_i}{C_i}- \\frac{\\sum ^N_{j = 1}E_j}{\\sum ^N_{j = 1}C_j} \\right|$ , $u_m(s,a)= -\\sum ^N_{i = 1} \\left|\\frac{D_i}{V_i^{ava}}- \\frac{\\sum ^N_{j = 1}D_j}{\\sum ^N_{j = 1}V_j^{ava}} \\right|$ , where $V^{ava}_i$ is the number of available EVs in region $i$ .",
"The fairness metrics $u_s(s,a)$ and $u_m(s,a)$ are calculated given the EVs rebalancing action $a$ , and the larger the better.",
"One advantage of the proposed robust and constrained MARL formulation is that the form of the reward/cost function does not need to satisfy the requirements as those of the robust optimization methods , , e.g., the objective/constraints do not need to be convex of the decision variable or concave of the uncertain parameters.",
"In this section, we derive robust natural policy gradient for robust and constrained MARL to efficiently train policies and alleviate overshooting/undershooting and high variance which results in slow convergence .",
"Then we present a robust and constrained MARL algorithm (ROCOMA) to solve the EV rebalancing problem under model uncertainties and fairness constraints."
],
[
"Robust Policy Gradient Descent Ascent",
"The problem (REF ) can be solved by Gradient Descent Ascent (GDA) , which currently is a widely used algorithm for solving minimax optimization problems.",
"At each iteration, GDA simultaneously performs gradient descent update on the variable $\\lambda $ and gradient ascent update on the variable $\\theta $ with step sizes $\\alpha _t$ and $\\beta _{t}$ : $\\theta _{t+1} = \\theta _t + \\alpha _t (\\nabla _\\pi v^{\\pi }_r(\\rho ) + \\lambda _t\\nabla _\\pi v^{\\pi }_c(\\rho )) $ , $\\lambda _{t+1} = \\operatorname{proj}_{\\lambda \\ge 0} \\left[ \\lambda _{t} - \\beta _{t} ( v^{\\pi }_c(\\rho ) - d) \\right]$ .",
"For notation convenience, we omit the subscripts $r$ and $c$ in the value functions when there is no confusion.",
"The robust policy gradient of the value function is given by $\\nabla _\\theta v^\\pi (s_1) = \\sum _{s,a} d^\\pi _{\\gamma ,\\delta ,s_1}(s) \\nabla _\\theta \\pi (a|s)\\phi ^\\pi (\\tau ) + b^\\pi \\propto \\mathbb {E}_{\\pi ,s_1}[\\phi ^\\pi (\\tau )\\nabla \\log \\pi (a|s)+b^\\pi ]$ , where $d^\\pi _{\\gamma , \\delta , s_1} \\propto \\sum _k\\gamma ^k(1-\\delta )^k p^\\pi (s_k=s|s_1)$ is the discounted visitation distribution of $s_{k}=s$ when initial state is $s_1$ and policy $\\pi $ is used; $\\tau $ denotes a trajectory $(s,a,r,c,s^\\prime )$ ; $\\phi ^\\pi (\\tau ) := r+ \\gamma \\delta \\min _sv^\\pi (s)+\\gamma (1-\\delta )v^\\pi (s^\\prime )-v^\\pi (s)$ is the TD residual; $b^\\pi :={\\gamma \\delta }/{(1-\\gamma +\\gamma \\delta )}\\partial _\\theta \\min _{s}v^\\pi (s)$ .",
"Similar to the vanilla policy gradient in the non-robust RL setting, robust policy gradient suffers from overshooting or undershooting and high variance, which results in slow convergence .",
"Hence we propose a robust natural policy gradient (RNPG) method as follows to update the policy along the steepest ascent direction in the policy space, ."
],
[
"Robust Natural Policy Gradient (RNPG)",
"Natural policy gradient (NPG) , , applies a preconditioning matrix to the gradient, and updates the policy along the steepest descent direction in the policy space, .",
"It has been proved that NPG moves toward choosing a greedy optimal action rather than just a better action in the literature .",
"Generally, for a function $L$ defined on a Riemannian manifold $\\Theta $ with a metric $M$ , the steepest descent direction of $L$ at $\\theta $ is given by $-M^{-1}(\\theta )\\nabla L(\\theta )$ , which is called the natural gradient of $L$ .",
"In the policy parameter space $\\left\\lbrace \\pi _\\theta \\right\\rbrace $ , the natural gradient of $L$ at $\\theta $ is given by $\\tilde{\\nabla } L(\\theta )=F(\\theta )^{-1}\\nabla L(\\theta )$ , where $F(\\theta ) := \\mathbb {E}_{s} \\left[ F_s(\\theta ) \\right]$ is the Fisher information matrix at $\\theta $ and $F_s(\\theta ) = \\mathbb {E}_{\\pi (a|s, \\theta )} \\left[ \\frac{\\partial \\log \\pi (a|s, \\theta )}{\\partial \\theta _i} \\frac{\\partial \\log \\pi (a|s, \\theta )}{\\partial \\theta _j} \\right]$ .",
"Although the natural gradient method has been studied in non-robust RL, it is not straightforward to efficiently find the NPG for a robust and constrained MARL problem.",
"We show the natural policy gradient for robust and constrained MARL in the following Theorem REF .",
"We first denote $\\psi ^\\pi (s,a) = \\nabla \\log \\pi (a|s,\\theta )$ and the Fisher information matrix is then given by $F(\\theta ) = \\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}(s)\\pi (a|s)\\psi ^\\pi (s,a)\\psi ^\\pi (s,a)^\\top $ .",
"Theorem 4.1 (Robust Natural Policy Gradient) Let $\\tilde{g}^*$ minimizes the objective $J(\\tilde{g},\\pi _\\theta )$ defined as follows: $\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s)[\\tilde{g}^\\top \\psi ^\\pi (s,a) - \\phi ^\\pi (\\tau ) - b^\\pi ]^2.$ Then $\\tilde{g}^*= F(\\theta )^{-1}\\nabla _\\theta v^\\pi (s_1)$ being the natural policy gradient of the objective function $v^\\pi (s_1)$ .",
"Since $\\tilde{g}^*$ minimizes (REF ), it satisfies the condition $\\partial J/\\partial \\tilde{g}_i = 0$ , which implies: $\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s) \\times \\psi ^\\pi (s,a)[ \\psi ^\\pi (s,a)^\\top \\tilde{g}^* - \\phi ^\\pi (\\tau ) -b^\\pi ] = 0$ or equivalently: $&\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s) \\psi ^\\pi (s,a)\\psi ^\\pi (s,a)^\\top \\tilde{g}^*\\\\= &\\sum _{s,a}d^\\pi _{\\gamma ,\\delta ,s_1}\\pi (a|s) \\psi ^\\pi (s,a)[ \\phi ^\\pi (\\tau ) +b^\\pi ]\\nonumber $ By the definition of Fisher information: $\\text{LHS} = F(\\theta )\\tilde{g}^*$ and $\\text{RHS} = \\nabla _\\theta v^\\pi (s_1)$ , which lead to: $F(\\theta ) \\tilde{g}^* = \\nabla _\\theta v^\\pi (s_1)$ .",
"Solving for $\\tilde{g}^*$ gives $\\tilde{g}^* = F(\\theta )^{-1}\\nabla _\\theta v^\\pi (s_1)$ which follows from the definition of the NPG."
],
[
"Robust and Constrained Multi-Agent Reinforcement Learning Algorithm (ROCOMA)",
"We then propose a robust and constrained MARL (ROCOMA) algorithm to solve the problem (REF ) and train a robust joint policy $\\pi $ using GDA and RNPG.",
"The proposed algorithm is shown in Algorithm REF .",
"Robust and Constrained Multi-Agent Reinforcement Learning Algorithm (ROCOMA) [1] Input $\\zeta ,\\alpha ,\\beta ,\\gamma ,\\delta $ .",
"Initialize $\\theta _0, \\lambda _0$ .",
"$t=0$ to $T$ Estimate $v_r^{\\theta _t}, v_c^{\\theta _t}$ using Algorithm 3 in $j=1$ to $M$ Sample $T_j \\sim \\textit {Geom}(1-\\gamma +\\gamma \\delta )$ , $s_1^j \\sim \\rho $ Sample trajectory from $s^j_1$ : $(s^j_1, a^j_1, \\cdots , s^j_{T_j})$ agent $i=1$ to $N$ $k=1$ to $W$ $\\tilde{g}^j_{t,k+1}(i)= \\arg \\min _{\\tilde{g}} ||\\tilde{g}^j_{t,k}(i)-\\tilde{g}-\\zeta \\nabla _{\\tilde{g}}\\mathcal {L}(\\tilde{g}_{t,k}(i), \\theta _t)||$ , $\\mathcal {L}$ is defined in (REF ) $\\tilde{g}^j_{t,k} = \\sum _{i=1}^N \\tilde{g}^j_{t,k}(i)/N$ $\\tilde{g}_t = \\sum _{j=1}^M\\sum _{k=1}^W \\tilde{g}^j_{t,k}/MW$ $\\theta _{t+1} = \\theta _t + \\alpha _t(\\tilde{g}_{r,t} - \\lambda _t \\tilde{g}_{c,t})$ $\\lambda _{t+1} = \\max \\lbrace \\lambda _t - \\beta _t( \\sum _{j}v_c^{\\theta _t}(s_1^j)/M-d),0\\rbrace $ Output $\\theta _T$ We first randomly initialize the actor neural network parameter $\\theta _0$ and the Lagrange multiplier parameter $\\lambda _0$ .",
"At each iteration $t$ , we first estimate the critic neural networks $v_r^{\\theta _t}, v_c^{\\theta _t}$ under policy $\\pi ^{\\theta _t}$ using Algorithm 3 in .",
"Line REF to line REF in Algorithm REF estimate the RNPG for $v_r^{\\theta _t}$ and $v_c^{\\theta _t}$ .",
"We then sample an initial state $s^j_1$ following the initial distribution $\\rho $ and a time horizon $T_j$ from the geometric distribution $\\textit {Geom}(1-\\gamma + \\gamma \\delta )$ at iteration $j=1, \\cdots , M$ .",
"These samples are used to estimate the RNPG.",
"As shown in Theorem REF , we can get the RNPG of $v^\\pi (s_1)$ by minimizing the objective defined in (REF ).",
"To minimize (REF ) and get the minimizer, we initialize $\\tilde{g}_{t,0}=0$ and use the following stochastic gradient descent (SGD) steps with $l_2$ -projection: $\\tilde{g}_{t,k+1} = \\arg \\min _{\\tilde{g}} ||\\tilde{g}_{t,k}-\\zeta \\nabla _{\\tilde{g}} \\mathcal {L}(\\tilde{g}_{t,k}, \\theta _t) - \\tilde{g}||$ , where $\\zeta $ is the learning rate and $\\mathcal {L}$ is defined in (REF ) as follows.",
"$\\mathcal {L}(\\tilde{g},\\theta ) = \\sum _{\\mathcal {D}(s^j_{T_j})}[\\tilde{g}^\\top \\psi ^{\\theta }(s,a) - \\phi ^\\theta (\\tau ) - b^\\theta ]^2/D,$ where $\\mathcal {D}(s^j_{T_j})$ is a set of trajectories $\\tau $ starting at $s^j_{T_j}$ using policy $\\pi ^{\\theta _t}$ , i.e.",
"$(s^j_{T_j},a,r,c,s^\\prime )$ , $D=|\\mathcal {D}(s^j_{T_j})|$ .",
"After $W$ steps of SDG iterations, the robust natural policy gradient for $v^{\\theta _t}(s^j_1)$ is estimated as $\\sum _{k=1}^W\\tilde{g}_{t,k}^j/W$ .",
"To reduce the computational complexity, we adopt the centralized training and decentralized execution (CTDE) framework in ROCOMA and assume all agents share the same policy $\\pi ^{\\theta ^i}(a^i|s^i)$ , where $\\theta ^1=\\cdots =\\theta ^N=\\theta $ .",
"Then we have $\\nabla \\pi (a|s) = \\sum _{i}^N \\psi ^{\\theta }_i(s,a)$ where $\\psi ^{\\theta }_i(s,a) := \\pi ^{-i}(a^{-i}|s^{-i})\\nabla \\pi ^i(a^i|s^i)$ , $\\pi ^{-i}(a^{-i}|s^{-i}):=\\prod _{j\\ne i}\\pi ^j(a^j|s^j)$ .",
"Therefore, in lines REF to REF , we address the high-dimensional action and state space issue in computing RNPG by using $\\psi ^{\\theta }_i(s,a)$ instead of $\\psi ^{\\theta }(s,a)$ in (REF ).",
"Finally, we update $\\theta _{t+1}$ and $\\lambda _{t+1}$ using GDA."
],
[
"Experiment Setup",
" Three different data sets , including E-taxi GPS data, transaction data and charging station data are used to build an EV AMoD system simulator as the training and testing environment.",
"The simulated map is set as a grid city.",
"The policy networks and critic networks are two-layer fully-connected networks, both with 32 nodes.",
"We use Softplus as activations to ensure the output is positive.",
"The output of policy networks is used to be the concentration parameters of the Dirichlet distribution to satisfy the action constraints (sum to one).",
"We set the maximal training episode number $=20000$ , the maximal policy/critic estimation number $=2000$ , the NRPG SDG iteration number $=500$ , the discount rate $\\gamma = 0.99$ , the perturbed rate $\\delta =0.05$ , the coefficients $\\bar{\\alpha }=\\bar{\\beta }=1$ , the fairness constraint limit $d=-20$ for one simulation step, and use AdamOptimizer with a learning rate of $0.001$ for both policy/critic networks."
],
[
"Experiment Results",
" Our goal of the experiments is to validate the following hypothesis: (1) The proposed ROCOMA can learn effective rebalancing policies; (2) Our proposed method is more robust than a non-robust MARL algorithm through our robust and constrained MARL design.",
"Other than Rebalancing cost: the total moving distance of vacant and low-battery EVs by using rebalancing policy (the lower the better); and System fairness: the weighted sum of mobility and charging fairness (the higher the better); we also monitor Number of expired orders: the total number of canceled orders due to waiting for more than 20 minutes (the lower the better) and Order response rate: the ratio between the number of served demands and the number of total passenger demand (the higher the better).",
"All metrics are calculated in every testing period which consists of 25 simulation steps.",
"We repeat testing for 10 times and show the average values.",
"Table: Conclusion"
]
] | 2209.08230 |
[
[
"Value Summation: A Novel Scoring Function for MPC-based Model-based\n Reinforcement Learning"
],
[
"Abstract This paper proposes a novel scoring function for the planning module of MPC-based model-based reinforcement learning methods to address the inherent bias of using the reward function to score trajectories.",
"The proposed method enhances the learning efficiency of existing MPC-based MBRL methods using the discounted sum of values.",
"The method utilizes optimal trajectories to guide policy learning and updates its state-action value function based on real-world and augmented on-board data.",
"The learning efficiency of the proposed method is evaluated in selected MuJoCo Gym environments as well as in learning locomotion skills for a simulated model of the Cassie robot.",
"The results demonstrate that the proposed method outperforms the current state-of-the-art algorithms in terms of learning efficiency and average reward return."
],
[
"Introduction",
"Reinforcement learning (RL) has recently received great attention from robotics and autonomous systems communities due to its power in handling system non-linearity, uncertainties and complexities.",
"The current RL methods can be classified as Model-free RL (MFRL) and Model-based RL (MBRL).",
"Although MFRL methods have demonstrated proficiency in complex tasks , , they are typically sample-inefficient and suffer from a prolonged training process.",
"In contrast, MBRL methods are usually more sample-efficient, since they learn the dynamic transition model and can use it for planning.",
"In order to learn the model of the environment, methods such as ensemble learning and latent models have been proposed, allowing for fast and robust learning of the environment dynamics.",
"Once the model has been obtained, it can be used together with a planner to achieve a wide variety of tasks and goals.",
"MBRL planner algorithms can be categorized as background planning and discrete-time planning .",
"The former uses the transition data obtained from the model to improve the policy or to learn the value function, while the latter mainly performs planning online for every state that the agent visits.",
"Background planning algorithms include dynamic programming , tabular Dyna , and prioritized sweeping .",
"Examples of discrete-time planning are Monte Carlo Tree Search (MCTS) and Model Predictive Control (MPC) .",
"MCTS algorithms showed superior performance in discrete spaces such as Atari games while MPC algorithms are mainly used for continuous applications such as robotics.",
"POLO is a prominent example of MPC-based MBRL, proposed by Lowrey et al., in which the concepts of value function approximation are combined with MPC to stabilize and accelerate the learning process of the value function .",
"The work on POLO was extended by Morgan et al.",
"through modelling MPC as an actor-critic framework and incorporating the approximation error when learning the dynamics model.",
"Another approach, given by Sikchi et al.",
"integrated the value function learned from model-free off-policy RL into MPC, and proposed actor regularized control to address the issue of actor divergence.",
"Hoeller et al.",
"defined the objective function of MPC in terms of an estimated value function and showed that MPC minimizes an upper bound of the cross-entropy sampling method to the state distribution of the optimal sampling policy.",
"Zong et al.",
"combined the concepts of iLQR control with MFRL, resulting in an algorithm with low computation overhead, high sample efficiency, and robustness against the errors of the model.",
"Hansen et al.",
"used a learned task-oriented dynamics model for MPC and a learned terminal value function, where both the model and the value function are learned jointly by temporal difference learning.",
"The concepts of Model Predictive Path Integral Control (MPPI) and MFRL were combined by Charlesworth et al.",
", resulting in a high-performance algorithm in some challenging simulated manipulation tasks where current RL methods and MPC techniques perform poorly.",
"Typically, MPC methods are based on a random shooting process.",
"This process samples numerous trajectories composed of state-action sequences with a $H$ horizon and evaluates each trajectory with a scoring function .",
"The defined scoring functions in literature can be classified into two types.",
"The first one sums the immediate rewards along the trajectory to calculate the trajectory score $\\sum _{t=0}^{H} \\gamma ^{t}r(s_{t},a_{t})$ , , where $\\gamma \\in [0, 1)$ is the discount factor which balances the emphasis on imminent rewards over distant rewards.",
"The second one sums immediate rewards along the trajectory and adds the estimated value of the terminal state $\\sum _{t=0}^{H-1} \\gamma ^{t}r(s_{t},a_{t}) + \\gamma ^{H}\\hat{V}(s_{H})$ , , , , , , .",
"However, using immediate rewards within the trajectory may not be enough for action selection, since no information about the future is considered.",
"In addition, in environments with sparse reward functions, using immediate rewards within the planning horizon may result in the same score for all trajectories .",
"Moreover, in complex environments, the sampling time is small, and states are close to each other, making the immediate rewards not a good option for differentiating between states.",
"In order to address the aforementioned problems, we propose the use of the discounted summation of estimated state-action values as the scoring function for MPC-based MBRL algorithms, where the score of a trajectory is calculated by summing state-action values rather than rewards.",
"It should be noted that we use state-action values within the planning horizon instead of state values since state-action values pertain to the action taken in the trajectory, making the score accurate within the context of the trajectory.",
"Moreover, summing rewards to score trajectories emphasises actions with greater immediate reward, whereas summing state-action values emphasises actions with greater cumulative rewards, thereby increasing total return over the episode.",
"The results show that the proposed method improves learning efficiency and average return when compared to state-of-the-art baselines.",
"To the best of our knowledge, it is the first time a discounted sum of values has been used in RL.",
"The remainder of the paper is structured as follows: we discuss the related works in Section II while Section III describes preliminaries for key components of MBRL.",
"Section IV introduces the value summation function and its integration in MBRL.",
"The results are presented in Section V. Finally, our contributions and future works are summarized in Section VI.",
"In the context of RL, a problem can be formulated as a finite-horizon Markov Decision Process (MDP) $\\mathcal {M = <S, A, R, P, \\gamma >}$ , where $\\mathcal {S}$ denotes the state space, $\\mathcal {A}$ denotes the action space, $\\mathcal {R : S \\times A} \\rightarrow \\mathbb {R}$ denotes the reward function, $\\mathcal {P : S \\times A \\times S} \\rightarrow \\mathbb {R}$ is the transition model, and $\\gamma \\in [0,1)$ is the discount factor.",
"A policy $\\pi \\in \\Pi : \\mathcal {S \\times A} \\rightarrow \\mathbb {R}$ describes a mapping from states to actions.",
"In RL problems, return is defined as the accumulative discounted reward, given by ${G} = \\sum _{t=0}^{\\infty } {\\gamma ^t r(s_t,\\pi (s_t))}$ .",
"An MDP aims to find the optimal policy $\\pi ^*$ that maximizes the expected return.",
"$V(s)$ denotes the state value, which is defined as the expected return by following the policy $\\pi $ from the state $s$ : $V(s) = \\mathbb {E}_{\\pi } \\left[ \\sum _{t=0}^{\\infty } {\\gamma ^t r(s_t,\\pi (s_t))} \\right], s_0=s$ Similarly, $Q(s,a)$ denotes the state-action value, which is defined as the expected return by taking the action $a$ at the state $s$ and then following the policy $\\pi $ : ${Q(s,a)} = \\mathbb {E}_{\\pi } \\left[ \\sum _{t=0}^{\\infty } {\\gamma ^t r(s_t,\\pi (s_t))} \\right], s_0=s, a_0=a$ In continuous MDPs, the policy and state-action value function are parameterized as $\\pi _{\\theta }$ and $Q_{\\phi }$ , where $\\theta $ and $\\phi $ denote the vector of parameters.",
"The parameters of the state-action value function $Q_{\\phi }$ can be estimated using the following loss function: $J_{\\phi } = \\mathbb {E}_{s,a,r,s^{\\prime } \\sim \\mathcal {D}} \\left[{(r + {\\gamma } {\\max _{a^{\\prime }} {Q_{\\phi }(s^{\\prime },a^{\\prime })}} - Q_{\\phi }(s,a))^2} \\right]$ where $\\mathcal {D}$ is a dataset containing a series of transitions $(s,a,r,s^{\\prime })$ collected by the policy $\\pi _{\\theta }$ .",
"After the state-action value function is approximated, the parameters of the policy are updated using the following function: $\\nabla _\\theta {J_{\\theta }} = \\mathbb {E}_{\\pi _\\theta } \\left[ {\\nabla _\\theta \\log {\\pi _\\theta (s,a)} Q_\\phi (s, a)} \\right]$"
],
[
"MPC-based Model-based Reinforcement Learning",
"MBRL approaches benefit from the model of the environment to plan the best action sequence at each state.",
"In the context of MBRL, the transition model can be known or learned.",
"When the transition model is available, it is used for planning the action sequence at each time step.",
"In this regard, MPC is mostly used to plan the best action sequence.",
"The MPC policy $\\pi _{MPC}$ computes the best action sequence at each state using the transition model in three stages: trajectory sampling, trajectory evaluation, and action selection.",
"In general, an MPC policy is computed as follows: $\\begin{aligned}\\pi _{MPC}(s) = \\arg \\max _{a_{0:H-1}} {\\mathbb {E} \\left[ \\sum _{t=0}^{H-1} {\\gamma ^t r(s_t,a_t) + \\gamma ^H r_f(s_H)} \\right]},\\\\a_t = \\pi _t(s_t), s_0=s\\end{aligned}$ where states evolve according to the on-board model $s_{t+1}=\\hat{f}(s_t,a_t)$ and $\\pi _t$ is a distribution for sampling action at each state.",
"After the optimal action sequence is computed, the first action is executed in the environment and the procedure is repeated at the next time step.",
"It is worth mentioning that, in some cases where MBRL is combined with MFRL, the distribution $\\pi _t$ can be the parameterized policy $\\pi _\\theta $ , called the behavioral policy.",
"Figure: The discounted value summation scoring function assigns greater importance to future rewards than the conventional definition of reward-value functions.",
"The hatched area shows that future rewards are emphasized by the proposed method (γ=0.9,H=100\\gamma =0.9, H=100)."
],
[
"Proposed MBRL Method",
"The proposed Value Summation Model-Based Reinforcement Learning (VS-MBRL) method is presented in this section.",
"The concept of discounted value summation and its boundedness over the infinite horizon is explained.",
"Then, the integration of the proposed scoring function in trajectory evaluation and the corresponding VS-MBRL is discussed."
],
[
"Scoring Function",
"The use of reward-based scoring functions to evaluate trajectories and selecting the best action out of different trajectories can be misleading.",
"The problem with the application of reward-based scoring functions is that the discounted summation of finite-horizon immediate rewards can be significantly biased against future decisions because the impact of future rewards is not considered in the scoring function.",
"Even the selection of greater horizons $H$ cannot alleviate this problem, because they may impose cumulative errors and reduce planning efficiency.",
"Another problem regarding the application of reward-based planning is that in environments with sparse reward feedback, where the total reward is received upon the completion of an episode, the summation of rewards within a planning horizon can be the same across all trajectories .",
"Last but not least, a problem occurs in environments with short sampling times, where the states' difference across each increment of time is too short to have a considerable impact on reward function $r(s_{t},a_{t})$ .",
"For these reasons, incorporating the estimated state-value function in the scoring function can be more insightful.",
"To address these issues, we propose the discounted sum of estimated state-action values to score trajectories as: $S = \\sum _{t=0}^{H}\\gamma ^{t}{Q^{\\pi }}(s_{t},a_{t})$ where the estimated state-action values over a trajectory with the planning horizon $H$ are summed up to determine the trajectory score $S$ .",
"Using the Bellman equation, we can roughly estimate the state-action value of each time-step as follows: $Q^{\\pi }(s_{t},a_{t}) = r(s_{t},a_{t}) + \\gamma Q^{\\pi }(s_{t+1},a_{t+1})$ Using (REF ) iteratively for all time steps within the planning horizon, we can write the scoring function in (REF ) in terms of the immediate rewards as below: $\\begin{aligned}S = & \\sum _{t=0}^{H}\\gamma ^{t}(t+1)r(s_{t}, a_{t}) + \\sum _{t=H+1}^{T}(H+1)\\gamma ^{t}r(s_{t}, a_{t})\\end{aligned}$ See Appendix A for a detailed derivation of (8).",
"The obtained scoring function (REF ) means the rewards within the planning horizon are emphasized by $\\gamma ^{t}(t+1)$ factor, and the rewards outside of the planning horizon are discounted by $(H+1)\\gamma ^{t}$ term.",
"Fig.",
"REF indicates that the value summation method imposes a greater weight on rewards than the reward-value scoring function proposed in .",
"This highlights the potential of the proposed scoring function in better evaluating future rewards and in making future-oriented decisions.",
"It is worth mentioning that over a specific range of horizon $t \\in \\left[2,34 \\right]$ (See Fig.",
"REF ), rewards are weighted by factors greater than 1.",
"We demonstrate that the proposed scoring function (REF ) would not violate the boundedness of summation, even in an infinite-horizon ($H=\\infty $ ) planning case (See Lemma 1).",
"Value Summation MBRL [1] Given $\\hat{f}(s_t,a_t)$ : on-board model, $D$ : replay buffer, $\\pi _{\\theta }(s_{t})$ : actor-network, $Q_{\\phi }(s_{t},a_{t})$ : critic network, $\\gamma $ : discounted factor; ${H}$ : Horizon length; ${N}$ : Number of trajectories; Task Not Completed $state \\leftarrow ReadSensors()$ $n \\leftarrow 0$ to $N$ ${s_{0}} = state$ $S(n) = 0$ $h \\leftarrow 0$ to $H$ $a_{h} = \\pi _{\\theta }(s_{h})$ $s_{h+1} = \\hat{f}(s_{h}, a_{h})$ $q_{h} = Q_{\\phi }(s_{h}, a_{h})$ $S(n) += \\gamma ^{h}q_{h}$ Store $(s_{h}, a_{h})$ in ${D}$ $Train(Q_{\\phi })$ $a_{t} = \\underset{a_{t:t+H}}{\\mathrm {argmax}}\\ S$ $Execute(a_{t})$ $Train(\\pi _{\\theta })$ Figure: The schematic of the Value Summation MBRL algorithm.",
"The al policy generates several trajectories, and the scoring mechanism selects the optimal trajectory with respect to the value summation function.Lemma 1 Consider an infinite horizon action sequence ${A}= \\lbrace a_{0},a_{1},a_{2}, ...\\rbrace $ and its resultant reward sequence ${R} = \\lbrace r_{0}, r_{1}, r_{2}, ... \\rbrace $ .",
"With these assumptions, (REF ) is expressed as below: $S = \\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r(s_{t},a_{t})$ Considering the maximum reward from the infinite reward sequence $R$ as $r_{max} = max \\lbrace r(s_{0},a_{0}), r(s_{1},a_{1}), r(s_{2},a_{2}), ... \\rbrace $ , the scoring function is upper bounded by: $\\begin{aligned}\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r(s_{t},a_{t}) & \\le \\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r_{max}\\\\& = r_{max}\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)\\end{aligned}$ where $\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)$ has an analytical solution as follows (See Appendix B): $\\sum _{t=0}^{\\infty }{\\gamma ^{t}(t+1)} = \\frac{1}{(1-\\gamma )^{2}}$ Substituting (REF ) into (REF ), we have the following relation for the scoring function: $\\sum _{t=0}^{\\infty }\\gamma ^{t}(t+1)r(s_{t},a_{t}) \\le \\frac{1}{(1-\\gamma )^{2}}r_{max}$ Therefore the infinite horizon scoring function has an upper bound.",
"Figure: The performance of value summation model-based reinforcement learning algorithm in comparison to well-known scoring functions.",
"(i) Sum-Value: ∑ t ' =t t+H γ t ' Q π (s t ,a t )\\sum _{t^{\\prime }=t}^{t+H} \\gamma ^{t^{\\prime }}Q^{\\pi }(s_{t},a_{t}) (ii) Sum-Reward: ∑ t ' =t t+H γ t ' r π (s t ,a t )\\sum _{t^{\\prime }=t}^{t+H} \\gamma ^{t^{\\prime }}r^{\\pi }(s_{t},a_{t}) and (iii) Sum-Reward-Value ∑ t ' =t t+H-1 γ t ' r π (s t ,a t )+γ H Q π (s H ,a H )\\sum _{t^{\\prime }=t}^{t+H-1} \\gamma ^{t^{\\prime }}r^{\\pi }(s_{t},a_{t}) + \\gamma ^{H}Q^{\\pi }(s_{H},a_{H})"
],
[
"Value Summation MBRL",
"This subsection presents the VS-MBRL method, which integrates the value summation scoring function into planning (Fig.",
"REF ).",
"This algorithm consists of two sections: gathering trajectories and scoring them.",
"After receiving the current state $s_{t}$ and the instant reward $r_{t}$ from the environment, the on-board model $\\hat{f}(s_{t}, a_{t})$ is used to sample several trajectories with the planning horizon $H$ from the policy $\\pi _{\\theta }$ .",
"The collected trajectories, then, are scored by the value summation function, and the first action from the highest-scored trajectory is executed in the environment.",
"VS-MBRL employs Soft Actor-Critic (SAC) as the underlying actor-critic algorithm to take advantage of the exploration induced by soft policy updates based on the maximum entropy principle.",
"This property makes SAC an excellent candidate for MBRL applications since actions can be sampled directly from the policy $\\pi _{\\theta }$ , and no additional noise is required.",
"However, VS-MBRL can be applied to any developed off-policy actor-critic algorithm.",
"A soft policy evaluation step is conducted in SAC using the soft Bellman backup operator , followed by a soft policy improvement step in which the expected KL divergence is minimized.",
"A description of the VS-MBRL algorithm can be found in Algorithm REF .",
"Using the on-board model $\\hat{f}(s_{t}, a_{t})$ with planning horizon $H$ , the agent shoots a number of trajectories at each time step (line code: $6-17$ ).",
"Eq.",
"(REF ) is applied to evaluate each pair $(s_{t:t+H}, a_{t:t+H})$ (line code: $12-13$ ) such that the corresponding trajectory can be evaluated.",
"The augmented data is collected in the replay buffer ${D}$ , which is then applied to train the state-action value function (line code: 16).",
"The first action from the best-scored trajectory is then executed in the environment (line code: 19).",
"To reduce computation costs, the augmented data is only employed for updating state-action value functions, while the actor-network is updated by $1/{N}^{th}$ that of the critic network, where $N$ is the number of trajectories (line code: 20)."
],
[
"Simulation",
"In this section, we compare the proposed value-summation scoring function with the most recent proposed scoring functions in the literature.",
"The scoring functions used for comparison purposes are: 1) Sum-Reward: Discounted sum of rewards $\\sum _{t=0}^{H} \\gamma ^{t}r(s_{t},a_{t})$ , proposed by .",
"2) Sum-Reward-Value: Discounted sum of rewards added with the estimated value of the terminal state, given by $\\sum _{t=0}^{H-1} \\gamma ^{t}r(s_{t},a_{t}) + \\gamma ^{H}{Q}^{\\pi }(s_{H}, a_{H})$ , proposed by .",
"3) Sum-Value: Our proposed discounted sum of state-action values $\\sum _{t=0}^{H} \\gamma ^{t}Q^{\\pi }(s_{t},a_{t})$ .",
"First, the performance of VS-MBRL is evaluated and compared in standard Gym environments.",
"Then, VS-MBRL is integrated into the process of learning locomotion skills for the Cassie robot and its performance is compared with other baselines.",
"The results indicate that the proposed scoring function outperforms other scoring functions in terms of learning efficiency and average return."
],
[
"Standard Gym Environments",
"We evaluate VS-MBRL in the simulated control tasks of MuJoCo included in the OpenAI-gym library : Ant-v3, Hopper-v3, HalfCheetah-v3, Walker2d-v3, Humanoid-v3, and InvertedDoublePendulum-v2.",
"We compare the average return of VS-MBRL over three trials against other scoring functions.",
"As shown in Fig.",
"REF , the average return of each scoring function is plotted against the time step for each environment.",
"In all environments, except Half-Cheetah-v3 and InvertedDoublePendulum-v2, VS-MBRL outperforms other scoring functions, proving the benefit of using value summation over other methods (See Table REF ).",
"It is speculated that in Half-Cheetah-v3, the estimation of the state-action value function imposes a significant error in planning, especially at the first 200,000 time steps, when the state-action value neural network needs more data.",
"Also, rewards are more useful in simple problems, such as InvertedDoublePendulum-v2, since the state-action value network requires a significant amount of training data.",
"However, these events prove negligible in complex problems, such as Humanoid-v3, Walker2d-v3 and Hopper-v3, where value summation results in fast and stable learning.",
"According to Ant-v3 and HalfCheetah-v3, relying only on reward summation can impede learning for a long time, resulting in poor performance.",
"Table: Performance of Scoring Functions in Different Gym Environments"
],
[
"Cassie's Learning Locomotion Skills",
"Additionally, we show that VS-MBRL is more effective than other scoring functions for the learning of Cassie's locomotion skills.",
"Initially designed and built by Agility Robotics, the Cassie robot is approximately one meter tall and weighs 33 kg.",
"Most of Cassie's mass is centred on the pelvis and each leg has two leaf springs to make it more flexible.",
"Traditional controller design becomes more challenging as a result of underactuation.",
"Xie et al.",
"propose an iterative RL algorithm to increase the stability of locomotion skills through a policy distillation mechanism.",
"In this way, the trained policy is used in the next iteration as an expert $\\pi _{e}$ .",
"The expert's action is then added to that of the behavioral policy $\\pi _{\\theta }$ and fed into the PD controller.",
"Each joint is controlled by a PD control loop, executed at the total rate of 2 kHz, with the targets updated every 30 ms from the policy.",
"We combine VS-MBRL with the controller proposed in , in which VS-MBRL is incorporated into the feedback control loop (Fig.",
"REF ).",
"The controller, therefore, contains an internal loop in which the behavioral policy generates trajectories and scores them.",
"Following this, the optimized action $a_{t}$ is summed up with the reference motion $\\delta a$ and fed into the PD controller.",
"It was decided to set the horizon length $H$ and the number of trajectories $N$ to 3 to reduce computation time.",
"Figure REF compares the performance of VS-MBRL with traditional scoring functions.",
"In the first 400 episodes, there is a negligible difference in performance.",
"Throughout the next 400 episodes, the average return exponentially increases until it reaches a maximum of 250.",
"In comparison to other gym environments, there is a substantial performance gap between scoring functions.",
"This is mainly because, in the simulated model of the Cassie robot for learning locomotion skills, the sampling time is $0.5$ millisecond, which is smaller than that in gym environments (Humanoid's sampling time is 3 millisecond), thus reducing the variation of states and rewards along each trajectory, resulting in difficult reward-based planning.",
"Figure: The application of value summation model-based reinforcement learning algorithm in feedback controller.",
"Reference motions from the expert policy are fed into the algorithm and added to the output.Figure: Value Summation MBRL algorithm (Sum-Value) outperforms other scoring functions in terms of sample efficiency and average return"
],
[
"Conclusion and Discussion",
"This paper presents a model-based reinforcement learning algorithm incorporating discounted value summation into planning.",
"In addition, Soft Actor-Critic was selected as the policy optimization algorithm, in which actions were directly sampled from a behavioral policy.",
"We compared the proposed scoring function with two prevalent planning methods: discounted sum of rewards and discounted sum of rewards added with the estimated value of the terminal state.",
"The performance over some standard baselines indicated the superiority of the proposed method.",
"Moreover, VS-MBRL is applied to learning locomotion skills for a simulated model of the Cassie robot.",
"VS-MBRL proves to be a preferable alternative to current scoring functions, offering considerably greater returns for Cassie's locomotion skills as well as MuJoCo Gym environments."
],
[
"APPENDIX",
"A.",
"Proof of (REF ).",
"Consider the case that we have an MDP with a finite horizon $T$ and a finite planning horizon $H$ .",
"For simplicity, we denote $Q^\\pi (s_t,a_t)$ and $r(s_t,a_t)$ as $Q_t$ and $r_t$ , respectively.",
"Using (REF ) iteratively, we can expand the state-action values of time steps within the planning horizon as follows: $\\begin{aligned}Q_H & = r_H + \\gamma Q_{H+1}\\\\Q_{H-1} & = r_{H-1} + \\gamma r_{H} + \\gamma ^2 Q_{H+1}\\\\& \\vdots \\\\Q_1 & = r_1 + \\gamma r_2 + \\gamma ^2 r_3 + ... + \\gamma ^{H-1} r_{H} + \\gamma ^H Q_{H+1}\\\\Q_0 & = r_0 + \\gamma r_1 + \\gamma ^2 r_2 + ... + \\gamma ^H r_H + \\gamma ^{H+1} Q_{H+1}\\end{aligned}$ The scoring function in (REF ) can be expanded as follows: $S = Q_0 + \\gamma Q_1 + ... + \\gamma ^{H-1} Q_{H-1} + \\gamma ^H Q_H$ Substituting the state-action values from (REF ) in (REF ), we have: $\\begin{aligned}S & = r_0 + \\gamma r_1 + \\gamma ^2 r_2 + ... + \\gamma ^H r_H + \\gamma ^{H+1} Q_{H+1} \\\\& + \\gamma r_1 + \\gamma ^2 r_2 + \\gamma ^3 r_3 + ... + \\gamma ^{H} r_{H} + \\gamma ^{H+1} Q_{H+1} \\\\& \\vdots \\\\& + \\gamma ^{H-1} r_{H-1} + \\gamma ^H r_{H} + \\gamma ^{H+1} Q_{H+1}\\\\& + \\gamma ^H r_H + \\gamma ^{H+1} Q_{H+1}\\\\& = r_0 + 2 \\gamma r_1 + ... + (H+1) \\gamma ^H r_H + (H+1) \\gamma ^{H+1} Q_{H+1}\\\\& = \\sum _{t=0}^{H} \\gamma ^{t}(t+1)r_t + (H+1) \\gamma ^{H+1} Q_{H+1}\\end{aligned}$ Moreover, using the definition of the state-action value, we can write the $Q_{H+1}$ as follows: $Q_{H+1} = \\sum _{t=H+1}^{T} \\gamma ^{t-H-1}r_t$ Finally, after substituting (REF ) in (REF ), we obtain the scoring function in terms of the immediate rewards as follows: $S = \\sum _{t=0}^{H} \\gamma ^{t}(t+1)r_t + \\sum _{t=H+1}^{T} (H+1) \\gamma ^{t}r_t$ B.",
"Proof of the Time Series.",
"From mathematics, we have the following time series with its solution: $\\sum _{t=0}^{\\infty }{\\gamma ^{t}} = \\frac{1}{1-\\gamma }$ The derivative of the time series in (REF ) with respect to $\\gamma $ results in another time series as follows: $\\begin{aligned}\\frac{d}{d\\gamma } \\sum _{t=0}^{\\infty }{\\gamma ^{t}} & = \\frac{d}{dt} (1 + \\gamma + \\gamma ^2 + \\gamma ^3 + ...) \\\\& = 1 + 2 \\gamma + 3 \\gamma ^2 + ... \\\\& = \\sum _{t=0}^{\\infty }{\\gamma ^{t}(t+1)}\\end{aligned}$ Thus, if we take the derivative of both sides of (REF ) with respect to $\\gamma $ , the following relation is obtained: $\\sum _{t=0}^{\\infty }{\\gamma ^{t}(t+1)} = \\frac{1}{(1-\\gamma )^{2}}$ [heading=bibintoc]"
]
] | 2209.08169 |
[
[
"Predicting the Mpemba Effect Using Machine Learning"
],
[
"Abstract The Mpemba Effect -- when a system that is further from equilibrium relaxes faster than a system that is closer -- can be studied with Markovian dynamics in a non-equilibrium thermodynamics framework.",
"The Markovian Mpemba Effect can be observed in a variety of systems including the Ising model.",
"We demonstrate that the Markovian Mpemba Effect can be predicted in the Ising model with several machine learning methods: the decision tree algorithm, neural networks, linear regression, and non-linear regression with the LASSO method.",
"The effectiveness of these methods are compared.",
"Additionally, we find that machine learning methods can be used to accurately extrapolate to data outside the range which they were trained.",
"Neural Networks can even predict the existence of the Mpemba Effect when they are trained only on data in which the Mpemba Effect does not occur.",
"This indicates that information about the effect is contained even in systems where it is not present.",
"All of these results demonstrate that the Mpemba Effect can be predicted in complex, computationally expensive systems, without performing full calculations."
],
[
"Mpemba Effect",
"The Mpemba Effect involves two systems that are not in equilibrium.",
"It occurs when the system that is initially farther from equilibrium becomes closer to equilibrium.",
"The best known example is the claim that hot water can freeze faster than cool water in the same environment.",
"This was named after Erasto Mpemba, who most famously brought it to the attention of the scientific community [1] although it has been debated for thousands of years [2], [3], [4].",
"Several recent theoretical explanations of the effect have been demonstrated.",
"One shows that the Mpemba Effect can be predicted from statistical mechanics [5], [6], [7], [8], [9].",
"Another explanation, Refs.",
"[10], [11] makes use of the framework developed in Ref.",
"[10], which considers the Markovian dynamics as cooling process in the framework of non-equilibrium thermodynamics.",
"We refer to this process as the Markovian Mpemba Effect (MME).",
"The MME is further studied in Refs.",
"[12], [13] Ref.",
"[10] applies the MME to an antiferromagnetic nearest-neighbor interacting Ising spin chain, generalizing the Mpemba Effect to other systems.",
"The one-dimensional Ising spin chain will be the main model used in this paper, enabling us to use machine learning methods to predict the effect, as described in Sec. .",
"The Mpemba Effect traditionally describes systems relaxing towards an equilibrium temperature that is lower than their initial temperature.",
"However, a reverse effect occurs when two systems relax towards equilibrium at a higher temperature.",
"Sometimes the former is referred to as the Mpemba Effect and the latter is referred to as the Inverse Mpemba Effect, but both cases meet the definition of the Mpemba Effect used in this paper, namely that the system that is initially farther from equilibrium becomes closer after a finite time.",
"Unless otherwise specified we refer to the Inverse Markovian Mpemba Effect simply as the Mpemba Effect.",
"This paper is organized as follows.",
"Sec.",
"REF discusses recent works with results that can be related to our results.",
"Sec.",
", describe the formalism (laid out by Ref.",
"[10], [11]) used to determine the Mpemba Effect in the Ising model.",
"Sec.",
", summarizes the various machine learning methods employed.",
"The main results of this work are presented and discussed in Sec. .",
"We conclude in Sec.",
"."
],
[
"Applications and Impact",
"The Mpemba Effect has been observed in a number of systems.",
"In addition to water, [14], [15], it has been predicted in magnetoresistance alloys [16], numerically predicted in colloids [17], predicted analytically and numerically in gasses [18], [19], predicted in particles bounded by anharmonic potentials [20], observed in computational simulations and experiments in gas hydrates [21], and in computational simulations of carbon nanotube resonators [22].",
"It has been demonstrated to have application for faster heating with precooling [23], that the effect happens for temperatures dependent on the maximum work that can be extracted from the system [24], and that the Mpemba Effect can lead to quantum heat engines with greater power output and stability [25].",
"Furthermore, the Mpemba Effect is closely linked to the Kovacs Memory Effect, by which systems out of equilibrium cannot merely be described by macroscopic thermodynamic variables.",
"Knowledge of the Mpemba Effect can be applied when studying memory effects in Refs.",
"[26], [27], [28], [29], [30].",
"These works demonstrate the potential applications of the effect and motivate better statistical understanding.",
"To our knowledge, no work has used machine methods to predict the occurence of the ME in any system.",
"The work presented in this paper uses learning methods to predict the Mpemba Effect in the Ising model.",
"The comparison of methods done in this analysis could be applied to the prediction of the ME in other systems.",
"Machine learning has previously been applied to the Ising model in a number of ways other than the Mpemba Effect: using autoencoding neural networks, [31], using neural networks [32] to predict probability distribution, and comparison of various classification methods such as random forests, decision trees, k nearest neighbors and artificial neural networks [33].",
"These works find several interesting results that can be compared to the work in this paper.",
"Firstly, Ref.",
"[32] found that only the number of nodes in the first layer of the deep neural network could improve the accuracy of the prediction.",
"Secondly, Ref.",
"[33] found that the Decision Tree method was the most accurate predictor.",
"Even though previous results predict different things, comparison with our work could show general trends in machine learning applied to the Ising model.",
"This work uses the formalism developed in Refs.",
"[10], [11] for the Mpemba Effect in the Ising model.",
"We summarize the key equations in the following.",
"The one-dimensional Ising model consists of a chain of $N$ spins, $s_i$ with values of 1 or -1.",
"The energy for a given set of spins is defined to be $E=-J\\sum _{k=0}^N s_k s_{k+1}-h\\sum _{k=1}^N s_k,$ where $J$ and $h$ are parameters respectively giving the field of interaction between neighboring spins and the strength of the external field.",
"A probabilistic distribution $\\vec{p}(t)$ that represents the probability of finding the system in one of the $2^N$ micro-states is defined for the case of a finite number of states, by the equation: $\\frac{d \\vec{p}(t)}{dt}= R(T_b)\\vec{p}(t),$ for $i=1,2,\\cdots , n$ .",
"Matrix multiplication is assumed.",
"Here $R_{ij}(T_b)$ is the transition rate matrix from a state $j$ to state $i$ defined as: $R_{ij} = {\\left\\lbrace \\begin{array}{ll}\\Gamma e^{-\\frac{1}{2}\\beta _b (E_{i} - E_j)},& \\text{if i and j differ by one spin flip} \\\\0,& \\text{if i and j differ by} >1 \\text{ spin flip} \\\\-\\sum _{k\\ne j}R_{{kj}}, & i=j\\\\\\end{array}\\right.",
"}$ At a given temperature, a system will have an equilibrium state, $\\vec{\\pi }(T)=\\frac{e^{-E_i/k_BT}}{\\sum _{i}e^{-E_i/k_BT}}.$ Because the eigenvalues form a complete basis, any state can be expressed as, $\\vec{p}=\\vec{\\pi }(T)+\\sum _{i>1} a_i \\vec{v}_i$ where $v_i$ are the eigenvectors of the rate transition matrix and $a_i$ are coefficients.",
"$\\vec{\\pi }$ is the eigenvector corresponding to an eigenvalue of 0.",
"The general solution to Eq.",
"(REF ) is given by $\\vec{ p}(T;t) = e^{Rt}\\vec{\\pi }(T)= \\vec{\\pi }(T_b) + \\sum _{i > 1} a_i(T) e^{\\lambda _{i}t} \\mathbf {v}_i,$ and $\\lambda _i$ are the eigenvalues of the transition rate matrix.",
"The distance from equilibrium function, $D[\\vec{p}(t); T_b]$ , is given by $D[\\vec{p}(t); T_b]=\\sum _i \\left( \\frac{E_i (p_i-\\pi _i^b)}{T_b}+p_i \\ln {p_i}-\\pi ^b_i \\ln {\\pi _i^b} \\right).$ The Mpemba Effect occurs for any two probability distributions $\\vec{p}_H$ and $\\vec{p}_C$ if $D[\\vec{p}_H(0); T_b]>D[\\vec{p}_C(0); T_b]$ but $D[\\vec{p}_H(t^{\\prime }) T_b]<D[\\vec{p}_C(t^{\\prime }); T_b]$ for some time, $t^{\\prime }$ .",
"Ref.",
"[10] shows that this will occur if $|a_2^H|<|a_2^C|$ ."
],
[
"Machine Learning Methods",
"The data used for the various machine learning methods is generated in the following way.",
"The matrix $R_{ij}$ , given in Eq.",
"(REF ), is calculated for a given $J$ , $h$ , $T$ , and $N$ .",
"Sparse Matrix methods are used to improve the speed of the computation.",
"The Arnoldi method [34] is used to calculate the second greatest (least negative) eigenvalue and the corresponding eigenvector.",
"These correspond to $\\lambda _2$ and $\\mathbf {v}_2$ from Eq.",
"(REF ).",
"These can be used to calculate $a_2$ using the method described of Sec.",
"II in Ref. [11].",
"We use four different machine learning methods to predict the occurrence of the Mpemba Effect in an Ising chain between 5 and 15 spins.",
"We work only with odd-numbered spins because these have different equilibria for the anti-ferromagnetic Ising model.",
"All of these methods have certain free parameters called fit parameters and a loss function which measures the distance of the prediction from the data.",
"The methods use various algorithms to fit the data or match the model to the data."
],
[
"Decision Tree Algorithm (DT)",
"For the decision tree, the classification of data is divided into two classes: for each set of initial conditions, the system can either undergo the IMME or not.",
"We use a data set with in the format of: $\\lbrace N, J, h, T_{c}, T_{h} \\rbrace = {\\left\\lbrace \\begin{array}{ll}1 & \\text{if IMME occurs } \\\\0 & \\text{if IMME does not occur},\\end{array}\\right.",
"}$ The decision tree is trained using the CART (Classification And Regression Tree) algorithm [35].",
"This algorithm uses a flow chart-style tree to classify the input and predict output.",
"Because this method minimizes a loss function different from the other methods we use, we do not record any loss function."
],
[
"Neural Network (NN)",
"Neural networks can be trained to predict the parameters $|a_2|$ given a set of Mpemba parameters.",
"Prediction of $a_2$ is sufficient to determine whether the Mpemba Effect occurs.",
"The network is trained on 4 input parameters, $\\lbrace N, J, h, T \\rbrace = |a_2|$ Note that we multiply each $|a_2|$ by 10000, to make the values a more natural size.",
"This has no impact on the mathematics of prediction for the neural network or any other machine learning method, however, many algorithms are built to select initial values that are more naturally sized.",
"This adjustment prevents the need to change the starting parameters of each algorithm.",
"We find the most accurate neural network consists of 3 hidden layers with 50, 20, and 5 nodes respectively.",
"Each layer as well as the output layer uses the Scaled Exponential Linear Unit (SELU) activation function.",
"The loss function is the Mean Squared Logarithmic Error (MSLE) [36].",
"This is given with the formula $L(y,\\hat{y})=\\frac{1}{n}\\sum _{i=0}^n\\left( \\log (y_i+1)-\\log (\\hat{y}+1) \\right)^2,$ where $y_i$ are the actual values, $n$ is the number of data points, and $\\hat{y}_i$ are the predicted values.",
"The neural network can also be used to directly predict the Mpemba Effect rather than predicting $|a_2|$ .",
"This is done by training the network with data given by Eq.",
"(REF ).",
"The network remains the same except that the output layer has a sigmoid activation function and a binary cross-entropy loss function is used.",
"The case where the Mpemba Effect is predicted directly is referred to as NN2."
],
[
"Linear Regression (LR)",
"Linear regression is used to predict the $|a_2|$ coefficients with the data.",
"We once again minimize the MSLE given in Eq.",
"(REF ).",
"The predicted values, $\\hat{y}_i$ , are given by $&\\hat{y}_i=\\sum _{j=0}^4 c_{j}x_{ij}.",
"\\\\\\nonumber x_{i0}=1; \\hspace{5.69054pt} x_{i1}=n_i; &\\hspace{5.69054pt} x_{i2}=J_i; \\hspace{5.69054pt} x_{i3}=h_i; \\hspace{5.69054pt} x_{i4}=T_i.$ The fit parameters are $c_{j}$ .",
"The MSLE is chosen as the loss function rather than the sum of least squares because it more accurately captures the behavior of data with very different orders of magnitude.",
"Linear regression cannot be used to determine the Mpemba Effect because, for the effect to occur, $|a_2|$ must increase as $T$ increases.",
"If $|a_2|$ is given by a decreasing linear function of $T$ , the $T$ that is farther from equilibrium will always correspond to a larger value of $|a_2|$ .",
"This can be seen in the linear fit in Fig.",
"REF .",
"Nevertheless, linear regression is useful for two reasons.",
"Firstly, the MSLE can be compared to other methods, giving information about their accuracy.",
"Secondly, the values of the fit parameters $c_{j}$ can be used as a crude approximation of the importance of each category of data for predicting $|a_2|$ .",
"The fit parameters are given in Tab.",
"REF .",
"We also quote the correlation coefficents which correspond to an even simpler model.",
"Note that because the linear regression in the fit was performed by minimizing the MSLE rather than the least squares, ordinary relations between correlation coefficients and linear fit parameters do not apply.",
"Table: Upper Row: The linear regression fit parameters.",
"The parameter in column ii corresponds to the fit parameters c i c_i in Eq. ().",
"The first column is labeled c 0 c_0.",
"The other columns are labeled with the parameters that c i c_i is multiplied by.",
"Lower Row: The correlation coefficients between each parameter and a 2 a_2."
],
[
"Nonlinear Regression (NLR) with the LASSO Method",
"The data can be more accurately described with a non-linear model.",
"We use a general expansion of the form $\\hat{y}_i= \\sum _{k_1=0}^N \\ldots \\sum _{k_n=0}^N \\delta _{N, \\sum _i k_i} c_{k_1 \\ldots k_N} \\prod _{j=1}^4P_{k_j}(x_{ij})$ where $P_{k_j}(x_{ij})$ are the Legendre Polynomials mapped to the range of each data set, $N$ is the order up to which we consider, $\\delta _{N, \\sum _i k_i} $ is the Kronecker Delta Function, and $c_{k_1 \\ldots k_N}$ are the fit parameters.",
"In the case when $N=1$ , this expansion reduces to linear regression.",
"The error minimized is once again the MSLE given in Eq.",
"(REF ).",
"For large $N$ , the number of fit parameters lead to a prohibitive computational cost of the minimization.",
"Not all fit parameters at a given order are equally effective at minimization.",
"The LASSO Method [37], [38], [39], [40] is employed in order to determine which fit parameters are most effective and which can be neglected to reduce computational cost.",
"This is done in the following way.",
"We randomly select 1000 data points and use them to calculate a modified loss function given by $L(y,\\hat{y})=\\frac{1}{n}\\sum _{i=0}^n\\left( \\log (y_i+1)-\\log (\\hat{y}+1) \\right)^2 +\\lambda \\sum |c_{ijk\\ell } |.$ The term $\\lambda $ is a penalty term that can be varied.",
"The data points are used to find the fit parameters that lead to local minima for various values of $\\lambda $ .",
"When $\\lambda =0$ , all fit parameters are non-zero.",
"As $\\lambda $ becomes large, all fit parameters approach 0.",
"A number of fits are performed with all fit parameters up to the fourth order with a gradually increasing $\\lambda $ .",
"The fit parameters are ranked according to the value of $\\lambda $ necessary to reach an absolute value less than $10^{-6}$ .",
"This list is ordered in terms of importance for the minimization of the MSLE.",
"A visualization of the ordering can be found in Fig.",
"REF which shows the values of the second order parameters as $\\lambda $ is increased.",
"The lower the value of $\\lambda $ at which the parameters are close to 0, the less important they are.",
"Figure: The best fit values of parameters c 00ij c_{00ij} for ii and jj from 1 to 4 defined in Eq. ().",
"The parameters are obtained by minimizing the loss function given in Eq.",
"() for various λ\\lambda .",
"As λ\\lambda increases, c 00ij c_{00ij} will become small.",
"The order in which the parameters approach 0 gives the order of importance of each parameter in minimizing the MSLE.The parameters are then validated by calculating the MSLE with new data.",
"First the data are validated with a fit involving all parameters.",
"Next, the same validation is performed while excluding the parameter determined to be least important.",
"After that, the least important two parameters are excluded.",
"This process continues until all parameters have been excluded.",
"This process generates a list of validation MSLE with parameters excluded in order of importance from least important to most important.",
"The validation MSLE is plotted in Fig.",
"REF .",
"The plot shows a minimum when 22 parameters are excluded.",
"Thus 22 parameters are excluded for the non-linear regression.",
"This allows for quicker minimization.",
"Figure: The validation MSLE varying the number of excluded parameters.",
"The data were fit to 1000 training points.",
"The parameters are excluded in order of impact as described in the text.",
"The minimum corresponds to a number of excluded parameters that neither overfit nor underfit the 1000 data points.When fitting the data the following procedure is performed.",
"Firstly, we fit the parameters $c_{000i}$ to the data from $i$ from 0 to 4 using the Newton method.",
"The other fit parameters are kept constant at 0.",
"Next we fit the parameters $c_{00ij}$ with $i$ and $j$ from 0 to 4.",
"The initial values of $c_{000i}$ are the best fit values from the previous minimization.",
"The other fit parameters are given initial values 0.",
"All parameters not used in the fit are once again set to 0.",
"The same process is repeated for $c_{0ijk}$ , and finally for $c_{ijk\\ell }$ .",
"This method of finding minimum parameters is employed in order to find a stable local minimum.",
"If all parameters were fitted at once without appropriate starting values, small changes in the fit such as changing the number of data points or increasing the penalty term would result in drastically different local minima.",
"However, it causes the fit to rely most heavily on lower order parameters.",
"The best fit parameters are given in Tab.",
"REF in the Appendix.",
"Perhaps because of the fit procedure, far more higher order parameters than lower order parameters are found by the LASSO to be unnecessary."
],
[
"Comparison of Methods",
"An example of the various methods compared with the data is shown in Fig.",
"REF .",
"Only the methods that predict $|a_2|$ directly are shown.",
"The black dots give the actual computed data.",
"At very low temperatures, they increase slightly with temperature indicating a very weak Mpemba Effect.",
"After that, they decrease smoothly to equilibrium.",
"In order to compare the effectiveness of the various methods, several metrics are used.",
"All of these metrics distinguish between training data, which are used to fix the free parameters or nodes of the model, and validation data, which are not used to determine the free parameters but are used to assess the accuracy.",
"The validation MSLE can be used to compare the accuracy of the Neural Net (NN), Linear Regression (LR) and Non-Linear Regression (NLR).",
"The Decision Tree (DT) and Neural Network trained directly on the Mpemba Effect (NN2) cannot be compared because they minimize different loss function.",
"The positive and negative validation accuracies are also computed.",
"These are calculated by predicting whether the Mpemba Effect occurs for randomly selected $J$ , $h$ , $N$ , and two temperatures.",
"The positive accuracy is the percentage of cases where the method predicts the Mpemba Effect in which the effect actually occurs.",
"The negative accuracy is the percentage of times the method correctly predicts that the effect does not occur.",
"As discussed in Sec.",
"REF , the accuracy is meaningless for LR.",
"For each method, the accuracy increases and the error decreases as the amount of data increases, however it eventually reaches a point at which additional data does not improve the predictions.",
"Tab.",
"REF gives the calculations for the four methods for various numbers of data points.",
"The negative accuracy listed in the table for all four methods is much larger than the positive accuracy.",
"This can be explained by the fact that the Mpemba Effect occurs in approximately $2\\%$ of cases.",
"Methods are more likely to minimize a loss function by predicting that the effect does not occur.",
"For reference, a method that always predicts that the effect does not occur regardless of training data, would have a $0 \\%$ positive accuracy and approximately $98 \\%$ negative accuracy.",
"Figure: An example of |a 2 ||a_2| as a function of temperature.",
"The black dots show the actual calculations compared with the predictions for various methods (colored curves).",
"This occurs for N=7.N=7., J=-6.37J=-6.37, h=0.381h=0.381.",
"The upper row gives a linear axis and the lower row gives a log-axis to show behavior at different orders of magnitude.",
"The NN gives the most accurate prediction followed by NLR and then LR.",
"Note that the models are fit not to this data but to a large set of data including various NN, JJ, hh, and TT.We find that for large data sets, NN2 is most accurate, though NN is almost as accurate and has the added benefit of predicting more information.",
"For very small training sets, NLR is the most effective method at predicting $|a_2|$ , although it fails to reach a high level of accuracy with larger data sets.",
"Table: The error and accuracy for several machine learning methods for varying numbers of training data points.",
"The error is calculated with the MSLE on validation data.",
"The positive/negative accuracy are the fraction of correct results when the method predicts that the effect occurs/does not occur."
],
[
"Extrapolation",
"In the previous section, the parameters for the validation data were different from those of the training data but they were generated in the same range.",
"Machine learning can also be used to extrapolate by validating the model with data in a different range.",
"This is particularly beneficial if the models can make accurate predictions on systems that are more computationally expensive than the systems they are trained on.",
"Out of the parameters $N$ , $J$ , $h$ , and $T$ , only $N$ determines the computational complexity of the calculation.",
"The time to compute $|a_2|$ does not depend on $J$ , $h$ or $T$ .",
"Fig.",
"REF plots the positive accuracy of various methods for predicting the Mpemba Effect in the case when $N=15$ , when it has only been trained on data with $N<15$ .",
"The full results are given in Tab.",
"REF in the Appendix.",
"The lines shown in Fig.",
"REF give a rough approximation for how the accuracy depends on the maximum parameter excluded.",
"Uncertainties are given by the standard deviation of 5 re-samples.",
"When values of $N$ close to 15 are included in the training data, the neural networks are once again the most accurate methods, however, the DT is relatively accurate at making predictions even when only $N=5$ is included.",
"In fact, changing which values of $N$ are included has very little impact on the accuracy of the predictions, indicating that $N$ has very little effect in the prediction with this algorithm.",
"These results agree with the crude prediction from the correlation coefficients, given in Tab.",
"REF .",
"It is reasonable to assume that the methods that are best at long range extrapolation in the ranges we consider will also be most effective at long range extrapolation outside the range we consider or for different systems than the Ising model.",
"However, this has not been demonstrated for certain and merits further investigation.",
"Long range extrapolation could be useful due to the high computational cost for $a_2$ for higher $N$ .",
"The number of states for a given $N$ is $2^N$ , leading to a $2^N\\times 2^N$ transition matrix.",
"The computational cost of the eigenvalues for a $M\\times M$ matrix is $\\mathcal {O}(M^2)$ [41].",
"Thus the computational cost for calculating $a_2$ is $2^{2N}$ .",
"Thus, a calculation with $N=5$ is quicker than a calculation with $N=15$ by a factor of $2^{20}$ .",
"Accurate prediction trained on much simpler systems could save considerable computational cost.",
"Figure: A plot of the positive accuracy vs. maximum value of NN excluded.",
"The full data is given in Tab. .",
"Best fit lines are shown to roughly estimate the relationship between the variablesTable: The predictions for the Mpemba Effect when the neural network is only trained on data where the Mpemba Effect is not present.",
"The neural network is trained on 20000 data points.Notably, the Mpemba Effect can be predicted even if it is trained only with data in which it does not occur.",
"Results for this extrapolation are given in Tab.",
"REF .",
"The first prediction of the Mpemba Effect (ME) is the same as in the previous sections.",
"The second method of prediction, referred to as the Total Mpemba Effect (TME), measures the accuracy of predicting whether or not the Mpemba Effect occurs for any temperature for a given $J$ , $h$ , and $N$ ."
],
[
"The Strong Mpemba Effect",
"In addition to the Mpemba Effect, machine learning methods can be used to predict the Strong Mpemba Effect, defined in Ref. [11].",
"In this situation, the hot system cools exponentially faster than the cold system.",
"The Strong Mpemba effect occurs when $a_2=0$ and $T\\ne T_b$ .",
"The neural network is trained on three input variables, $N$ , $J$ , and $h$ .",
"The output is 1 if the strong Mpemba Effect occurs for any temperature; the output is 0 if the Strong Mpemba Effect does not occur.",
"Results are given in Tab.",
"REF .",
"Table: The validation accuracy for the Strong Mpemba Effect.",
"It is trained with 20000 data points."
],
[
"Conclusion",
"The Mpemba Effect has been shown to occur in many systems, beyond the freezing of water.",
"It has been shown to have applications in quantum heat engines.",
"These applications motivate better predictions of the effect.",
"In order to understand this effect, this work applies statistical methods to the Mpemba Effect in the Ising model.",
"We demonstrate that a number of machine learning methods can be used.",
"Neural networks are the most effective method when a large enough training data set is used.",
"For very small data sets, non-liner regression with the LASSO method may be more effective at predicting $a_2$ .",
"These methods can predict the Mpemba Effect in systems much more complex than those on which they were trained.",
"The decision tree method may be the most effective method at making predictions far outside the range it was trained on.",
"Additionally, the Mpemba Effect can be predicted with neural networks when it is trained only on data where the Mpemba Effect does not occur.",
"This indicates that information about the Mpemba Effect exists in situations in which it does not occur.",
"Finally, we demonstrate that the strong Mpemba Effect can also be predicted using machine learning methods.",
"These predictions were performed for the Mpemba Effect in the Ising model, however, the relative accuracy of each method might hold for the Mpemba Effect in other systems.",
"Determination of the accuracy of various methods in other systems merits further investigation.",
"Acknowledgments — This work was funded by an Ave Maria University undergraduate research grant by Michael and Lisa Schwartz.",
"Thanks to Tomás Licheri for completing crucial tasks necessary for this research."
],
[
"Additional Data",
"In this section we give data for reference that is not used in the main analysis and discussion of the paper.",
"Tab.",
"REF gives the extrapolation accuracy of the methods.",
"Positive accuracies of this table are plotted in Fig.",
"REF and the format is equivalent to Tab.",
"REF .",
"Table: Extrapolation accuracy.",
"The data is trained on 50,000 data points.Tab.",
"REF gives the best fit parameters for non-linear regression.",
"Parameters not included in the table are either mathematically identical to an excluded parameter or shown by the LASSO method to have less impact.",
"The fit was performed with 50000 data points and corresponds to the fifth row of each section of Tab.",
"REF for NLR, however, NLR for any amount of data points greater than 1000 had similar minima.",
"Note that the fit was done to data with $|a_2|$ multiplied by a factor of 10000 as described in the text.",
"To relate these results to data that has not been prepared in this way, one should divide each parameter by 10000.",
"Table: LASSO parameters for a fit with 500000 data points.",
"Fit parameters, c k 1 k 2 k 3 k 4 c_{k_1 k_2 k_3 k_4} are given in Eq. ().",
"All parameters not present in this table are either excluded by the LASSO as described in the main text or unnecessary because they are multiplied by a term identical to that of one an included parameter."
]
] | 2209.08161 |
[
[
"Unusual magnetotransport in twisted bilayer graphene from strain-induced\n open Fermi surfaces"
],
[
"Abstract Anisotropic hopping in a toy Hofstadter model was recently invoked to explain a rich and surprising Landau spectrum measured in twisted bilayer graphene away from the magic angle.",
"Suspecting that such anisotropy could arise from unintended uniaxial strain, we extend the Bistritzer-MacDonald model to include uniaxial heterostrain.",
"We find that such strain strongly influences band structure, shifting the three otherwise-degenerate van Hove points to different energies.",
"Coupled to a Boltzmann magnetotransport calculation, this reproduces previously-unexplained non-saturating $B^2$ magnetoresistance over broad ranges of density near filling $\\nu=\\pm 2$, and predicts subtler features that had not been noticed in the experimental data.",
"In contrast to these distinctive signatures in longitudinal resistivity, the Hall coefficient is barely influenced by strain, to the extent that it still shows a single sign change on each side of the charge neutrality point -- surprisingly, this sign change no longer occurs at a van Hove point.",
"The theory also predicts a marked rotation of the electrical transport principal axes as a function of filling even for fixed strain and for rigid bands.",
"More careful examination of interaction-induced nematic order versus strain effects in twisted bilayer graphene could thus be in order."
],
[
"Introduction",
"The discovery of superconductivity and correlated insulating states in magic-angle twisted bilayer graphene (TBG) [1], [2] placed the material at the forefront of condensed matter physics research [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].",
"The moiré superlattice potential of TBG, resulting from a small relative twist angle $\\theta $ between the graphene layers, can induce nearly flat, topologically non-trivial, isolated bands, consisting of electronic states near the Dirac points of each monolayer of graphene [18].",
"As a result, TBG is an exceptional platform for studying the interplay of electron correlations and band topology [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37].",
"Strain – especially heterostrain consisting of differing lattice distortions in the two layers – is believed to play an important role in the phase diagram of TBG [38], [37], [36].",
"Scanning probe measurements typically find uniaxial heterostrain in the range of $0.1 - 0.7\\%$ in samples fabricated with the tear-and-stack method [3], [9], [11].",
"For heterostrain, as opposed to homostrain, the linear distortion of the moiré unit cell is amplified by a factor of $\\sim 1/\\theta $ relative to the linear distortion of the microscopic atomic lattice.",
"Because we infer twist angle from moiré unit cell area in transport, this effect leads to underestimates of the uncertainty in twist angles presented in transport literature, as noted in Ref.",
"[3].",
"For example, $0.2\\%$ uniaxial heterostrain causes a $\\sim 8\\%$ change in the linear size of the moiré unit cell for a twist angle of $1.38^\\circ $ .",
"However, the effect on the moiré unit cell area is much reduced.",
"In a recent report by some of the authors [Finney et al, Ref.",
"[39]], a TBG sample with a moiré unit cell area of 90 nm$^2$ (corresponding to $\\theta =1.38^\\circ $ , well above the magic angle) displayed several unusual phenomena in magnetotransport.",
"The sample did not exhibit the strong interaction driven effects typically observed in near-magic-angle devices.",
"Rather, over a broad filling range near half filling, the longitudinal magnetoresistivity (MR) exhibited a $B^2$ increase up to $\\approx 5$ T, after which quantum oscillations set in.",
"Such $\\sim 100$ -fold increase in MR was not explained, although the authors conjectured that strain may have played a role based on comparison of a toy Hofstadter model with anisotropy, over a broader range of magnetic field.",
"In this work, we present a systematic theoretical study of the impact of uniaxial heterostrain on the narrow-band dispersion of TBG above the magic angle, analyze its consequences for weak field magnetotransport, and compare it with experimental data from Ref. [39].",
"We base our theory on the Bistritzer-MacDonald (BM) continuum model [18], incorporating heterostrain in the form of a deformation potential, a pseudo-magnetic field [40], [41], and a distortion of the moiré pattern in the interlayer tunneling.",
"Our key theoretical result is that heterostrain lifts the energetic degeneracy of the two Dirac points as well as that of the three van Hove points of a given band.",
"The splitting of the two Dirac points leads to a semimetallic state near the charge neurality point (CNP) with small Fermi pockets.",
"More interestingly, the splitting of the van Hove points leads to open Fermi surfaces (FS) in the filling range bounded by two of the van Hove points.",
"In the weak field semiclassical regime governed by the Boltzmann equation, the open FSs generally lead to a non-saturating $B^2$ MR, explaining the low-field experimental findings of Ref. [39].",
"This theory makes a number of falsifiable predictions.",
"Of note, it predicts a large degree of mixing between longitudinal and transverse MRs within the open FS regime, due to an uncontrolled misalignment of the strain-induced principle axis of transport and the direction of current flow in the Hall bar.",
"It predicts a subtle cusp in resistivity corresponding to the crossing of the lowest-energy van Hove point.",
"Finally, it predicts a Lifshitz transition from two FS pockets to one upon crossing this lower van Hove point.",
"We reanalyze experimental data from [39], and find that these predictions are verified.",
"The theory does not capture the electron-hole asymmetry in the experimental data.",
"The theory also has a few unexpected features.",
"Firstly, the sign change singularity in the Hall number, one on each side of CNP, does not coincide with any of the van Hove points and instead occurs inside the filling range with open FSs.",
"Secondly, the transport principal axis continuously rotates by up to $90^\\circ $ as density is tuned from the CNP to the open FS regime.",
"Such rotation of the transport axes is generally associated with interaction-induced nematic order [14], but here we find that it can arise purely due to strain-induced band structure effects.",
"This work clearly demonstrates that the effects of even miniscule amounts of heterostrain in TBG cannot be neglected.",
"Dramatic and unexpected phenomena occur in strained TBG even in the single-particle regime, without the strong correlation effects that arise near the magic angle.",
"Given the amplifying effect of a small heterostrain on the moiré length scale, it is tantalizing to consider strain engineering of such devices to achieve effects that would be impossible in regular solids due to structural instabilities."
],
[
"Geometric and energetic effects of uniaxial heterostrain on TBG",
"In the limit of small deformations, both the uniaxial heterostrain and a small twist angle are captured via a coordinate transformation: $\\mathbf {r}^{\\prime }_l = \\mathbf {r}+ \\mathbf {u}_l(\\mathbf {r})$ , where $l=t,b$ labels the top (bottom) graphene layers, and $\\mathbf {u}_l(\\mathbf {r}) \\approx \\mathcal {E}_l \\mathbf {r}$ is the local deformation field.",
"The symmetric and antisymmetric part of the $2\\times 2$ tensor $\\mathcal {E}_l$ describes strain and rotation respectively.",
"For twist angle $(\\theta )$ and a uniaxial heterostrain of strength $(\\epsilon )$ and direction $(\\varphi )$ , we parameterize $\\mathcal {E}_{t}=-\\mathcal {E}_{b}\\equiv \\mathcal {E}/2$ , where $\\mathcal {E}\\equiv \\mathcal {T}(\\theta )+ \\mathcal {S}(\\epsilon ,\\varphi )$ , and given by: $\\mathcal {T}(\\theta ) = \\begin{pmatrix} 0 & -\\theta \\\\ \\theta & 0\\end{pmatrix},\\ \\mathcal {S}(\\epsilon ,\\varphi ) = R_{\\varphi }^T \\begin{pmatrix} -\\epsilon & 0\\\\ 0 & \\nu \\epsilon \\end{pmatrix}R_{\\varphi }.$ Here $R_\\varphi $ is the two-dimensional rotation matrix, and $\\nu \\approx 0.16$ is the Poisson ratio [3].",
"Physically, $\\epsilon >0$ corresponds to compressing the top layer while streching the bottom layer along the direction determined by $\\varphi $ , as illustrated in Fig.",
"REF (a).",
"A relative deformation $\\mathcal {E}$ between the graphene bilayers generates a moiré superlattice, with moiré reciprocal lattice vectors $\\mathbf {g}_{i=1,2}=\\mathcal {E}^T\\mathbf {G}_{i=1,2}$ , where $\\mathbf {G}_{i}$ are reciprocal lattice vectors of the undeformed monolayer graphene.",
"The moiré lattice vectors $\\mathbf {L}_{i=1,2}$ are uniquely defined through the relation $\\mathbf {L}_i\\cdot \\mathbf {g}_j=2\\pi \\delta _{ij}$ .",
"It is important to note that only relative deformations generate the moiré superlattice.",
"Homogenous deformations do not play an important role in the narrow band physics, and we neglect it in this work We checked numerically that adding a small homogeneous strain in addition to a heterostrain of similar strength yields almost identical band and transport properties to the case of adding a heterostrain alone..",
"Under rotation $R_\\varphi $ , the strain tensor transforms as a headless vector that remains invariant under $\\varphi \\rightarrow \\varphi +180^\\circ $ .",
"Combined with the $C_{3z}$ symmetry of the undeformed graphene lattice, the strained electronic dispersion within a given graphene valley simply rotates $60^\\circ $ under $\\varphi \\rightarrow \\varphi + 60^\\circ $ .",
"We hereby will only report results for $\\varphi \\in [0^\\circ ,60^\\circ )$ .",
"For concreteness we define the microscopic unit cell vectors $\\mathbf {a}_{i=1,2}$ of undeformed graphene lattice as $ \\mathbf {a}_1 = a(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}),\\ \\mathbf {a}_2 = a(1,0)$ , where $a\\approx 2.46Å$ is the lattice constant.",
"The positions of the sublattice A,B within a unit cell are chosen as $\\vec{\\tau }_A = (0,0)$ and $\\vec{\\tau }_B = \\frac{a}{\\sqrt{3}}(0,1)$ .",
"The reciprocal lattice vectors are $\\mathbf {G}_1 = \\frac{4\\pi }{\\sqrt{3}a}(0,-1)$ and $\\mathbf {G}_2 = \\frac{4\\pi }{\\sqrt{3}a}(\\frac{\\sqrt{3}}{2},\\frac{1}{2})$ .",
"Different conventions lead to different definitions of the Dirac Hamiltonian (see for instance Ref.",
"[38]), but the physics is consistent.",
"Fig.",
"REF (a-b) illustrates the geometric effects of heterostrain for twist angle $\\theta =1.38^\\circ $ .",
"For $\\epsilon =0.2\\%$ , typical in these systems [3], [9], [11], there is a large change in the bond length of the neighboring AA-stacked regions ($\\mathbf {L}_{i=1,2,3}$ ) of the moiré triangular superlattice, which used to form an equilateral triangle at $\\epsilon =0$ .",
"The effect of heterostrain on the moiré unit cell vectors can be estimated to be as large as $\\epsilon /\\theta \\approx 8\\%$ .",
"However, the effect on the moiré unit cell area is much smaller at $\\nu ^2\\epsilon ^2/\\theta ^2$ (see Supplementary Material (SM) Sec.",
"I).",
"Such dramatic amplification of the microscopic strain makes moiré materials ideal for strain engineering not achievable in conventional materials due to structural instability.",
"We proceed to discuss the energetic effects in the context of the continuum BM model [18].",
"We work in the limit where both $\\mathcal {E}_l$ and the wavevector $\\mathbf {k}$ in the moiré Brillouin zone are small, and consider only the leading order terms in both.",
"This would mean, for instance, that terms such as $\\mathcal {E}\\mathbf {k}$ are omitted as higher order terms.",
"This treatment is generally justified away from the magic angle, because higher order terms can play an important role only close to the magic angle where the narrow-band bandwidth is suppressed to a similar energy scale [43], [44].",
"Furthermore, we checked that at $\\theta \\approx 1.38^\\circ $ the effects of such higher order terms are indeed negligibly small.",
"To leading order, the strained BM Hamiltonian for a given valley is given by: $H_{\\eta } = (\\sum _{l=t,b} H_{\\eta ,l}^{intra}) + H_{\\eta }^{inter} ,$ where $\\eta =\\pm 1$ labels $\\mathbf {K}\\ (\\mathbf {K}^{\\prime })$ valleys of monolayer graphene.",
"The interlayer Hamiltonian is given by: $H_{\\eta ,l}^{inter} \\approx \\int \\mathrm {d}^2\\mathbf {r}\\psi ^\\dagger _{\\eta ,t} \\left(\\sum _{j=1,2,3} T_{\\eta ,j} e^{-i\\eta \\mathbf {q}_{j}\\cdot \\mathbf {r}}\\right)\\psi _{\\eta ,b}(\\mathbf {r}) + h.c.,$ where $\\psi _{\\eta ,l}(\\mathbf {r})\\equiv (\\psi _{\\eta ,l,A}(\\mathbf {r}),\\psi _{\\eta ,l,B}(\\mathbf {r}))^T$ is a spinor in the sublattice basis for a given valley and layer.",
"We have suppressed the spin index for simplicity.",
"$\\mathbf {q}_{j=1,2,3}$ are the three nearest neighbor bonds of the reciprocal honeycomb lattice, and $T_{\\eta ,j} = w_0\\sigma _0+w_1\\left(\\cos \\frac{2\\pi (j-1)}{3} \\sigma _x+ \\eta \\sin \\frac{2\\pi (j-1)}{3}\\sigma _y\\right).$ $(\\sigma _0,\\sigma _x,\\sigma _y)$ are Pauli matrices acting on sublattice degrees of freedom.",
"The intra-layer Hamiltonian is given by: $\\begin{split}& H_{\\eta ,l}^{intra} = \\alpha \\sum _{\\mathbf {k}} \\psi ^\\dagger _{\\eta ,l}(\\mathbf {r}) ( [\\mathcal {E}_l] \\sigma _0)\\psi _{\\eta ,l}(\\mathbf {r})\\\\&- \\frac{\\hbar v_F}{a}\\sum _{\\mathbf {k}} \\psi ^\\dagger _{\\eta ,l}(\\mathbf {r}) \\left[ (\\mathbf {k}- \\mathbf {A}_{\\eta ,l})\\cdot (\\eta \\sigma _x,\\sigma _y) \\right]\\psi _{\\eta ,l}(\\mathbf {r}).\\end{split}$ Here the first term is the deformation potential that couples to the electron density.",
"Its value is not precisely known in the literature, with numbers ranging from $-4.1\\ \\mathrm {eV}$ to $30\\ \\mathrm {eV}$ depending on the methodology [45], [46], [47], [48].",
"We use $\\alpha =-4.1\\ \\mathrm {eV}$ in this work based on first principles calculations [48], although the deformation potential does not have an important effect on the band dispersions for heterostrain $\\epsilon \\approx 0.2\\%$ , and only leads to minor quantitative differences.",
"$\\mathbf {A}_{\\eta ,l}$ is the pseudovector potential that comes from changes in the inter-sublattice hopping due to deformations, and changes sign between graphene valleys.",
"It is given as [40], [41]: $\\mathbf {A}_{\\eta ,l} = \\frac{\\sqrt{3}\\beta }{2a}\\eta (\\epsilon _{l,xx}-\\epsilon _{l,yy},-2\\epsilon _{l,xy})$ , where we choose $\\beta \\approx 3.14$ from Refs.",
"[3], [38].",
"We shall further fix $\\hbar v_F/a=2.68\\mathrm {eV}$ , $w_0=88\\mathrm {meV}$ , and $w_1=110\\mathrm {meV}$ in our calculations, and also set $\\hbar =1$ in the remainder of the paper.",
"To leading order approximation, the strained BM Hamiltonian in a given valley (Eq.",
"(REF )) has particle-hole symmetry under $P \\psi _{l}(\\mathbf {r}) = \\sum _{l^{\\prime }}i(\\mu _y)_{ll^{\\prime }}\\psi _{l^{\\prime }}(-\\mathbf {r})$ [49], where $\\mu _y$ is a Pauli matrix acting on the layer degrees of freedom.",
"This means that for every single electron state at energy $E$ and wavevector $\\mathbf {k}$ , there is a state at energy $-E$ and wavevector $-\\mathbf {k}$ .",
"This particle-hole symmetry has been investigated extensively for the unstrained BM model, e.g., Refs.",
"[25], [50], and here it is generalized to the strained case.",
"Since in experiments particle-hole asymmetry is evident for the off-magic-angle device [39], they would come from either higher order gradient terms beyond what's captured in the BM model in Eq.",
"(REF ), or due to interaction effects [51], [52], [53], [54], or their combination.",
"We proceed to discuss the heterostrain effects on the band structure with $\\epsilon =0.2\\%$ and varying direction specified by $\\varphi \\in [0^\\circ ,60^\\circ )$ , depicted in Fig.",
"REF (d-f).",
"For simplicity we only show contour maps of the upper band from valley $\\mathbf {K}$ in the moiré Brillouin zone specified by $\\mathbf {k}=k_1\\mathbf {g}_1+k_2\\mathbf {g}_2$ , where $k_{1,2}\\in [0,1)$ .",
"Heterostrain preserves $C_2T$ and valley $U(1)$ [22] and therefore the lower and upper bands remain connected via two Dirac points.",
"The upper band features six special points — two Dirac points (black stars), three van Hove points (colored dots), and one band maximum (black cross).",
"The six special points of a given band are related to “critical points\" in the context of the Morse theory, which states that $\\sum _{i}(-1)^{\\gamma _i}=\\chi ,$ where $\\gamma _i$ is the index of the $i$ -th critical point, and $\\chi $ is the Euler characteristic of a manifold [55]; $\\chi $ vanishes for the Brillouin zone which is a torus.",
"Although a Dirac point is strictly a point of non-analyticity and is not directly covered by Morse theory, if we imagine adding a tiny gap term it will become a legitimate band extremum and Morse theory applies.",
"Whereas the two band minima (Dirac points) and the band maximum have even $\\gamma $ and so each contributes $+1$ to the sum, every conventional van Hove point (i.e.",
"not a higher order) has an odd $\\gamma $ and contributes $-1$ .",
"Their sum thus vanishes.",
"Therefore, the van Hove points can only be annihilated/created by colliding with local minima/maxima.",
"For a relatively small heterostrain as shown in Fig.",
"REF , the number of special points per band is the same as at $\\epsilon =0$ .",
"However for larger heterostrain (e.g., $\\epsilon =0.5\\%$ , see SM Fig.",
"1), more striking behavior of the special points can occur, such as a change in their total number via afore mentioned collisions and the appearance of tilted type II Dirac cones [56], [57].",
"A key finding of the present work is that the respective energy degeneracies of the two Dirac points and the three van Hove points are lifted by uniaxial heterostrain, and depend sensitively on $\\varphi $ .",
"In the absence of strain [Fig.",
"REF (c)], the three van Hove points are at equal energy, and separate closed contours of constant energy centered around the Dirac points from closed contours centered around band maximum.",
"As illustrated in Fig.",
"REF (d-f), uniaxial heterostrain splits the energy degeneracy of the two Dirac points, leading to a semimetallic state with small Fermi pockets near CNP [38].",
"The three van Hove points also split in energy.",
"The two outermost van Hove points (i.e., closer to the band maximum) bound a filling range of open FSs near $\\nu =2$ , while the innermost van Hove point moves closer to one of the Dirac points.",
"If we continue increasing $\\epsilon $ , a collision of the critical points occurs, the innermost van Hove disappears, the two Dirac points become type-II tilted, and a new ordinary minimum is created.",
"Note that a small mass added to type-II tilted Dirac points won't introduce band extrema and as a consequence type-II tilted Dirac points are not critical points of Morse theory, therefore after the collision Eq.",
"(REF ) still holds.",
"Interestingly, the elongation of the FSs shows a strong filling dependence.",
"Close to the CNP, the bigger Fermi pocket that encloses a Dirac point is stretched along a perpendicular direction to that of the open FSs, see Figs.",
"REF (d-f).",
"As explained later, this leads to a dramatic rotation of the principal transport axis when the filling is tuned from the CNP to the open FS range.",
"The dependence of the energy and filling of the band structure special points on $\\varphi $ at a fixed $\\epsilon $ is shown in Fig.",
"REF (g-h).",
"Of notable interest is the sensitivity of the filling range with open FSs to $\\varphi $ .",
"This filling range must in fact vanish at some $\\varphi $ between $0^\\circ $ and $60^\\circ $ , when the energies of the two outermost van Hove points cross.",
"As seen in Fig.",
"REF (d-f), this also alters the elongation of the open FSs."
],
[
"Boltzmann equation and Magnetoresistivity in TBG",
"Having understood the heterostrain effects on the bandstructure, we proceed to discuss the implications for magnetotransport.",
"We begin by considering the general structure of the two-dimensional resistivity tensor ${\\rho }$ subject to heterostrain.",
"The resistivity tensor is defined via: $\\begin{pmatrix}E_x \\\\ E_y\\end{pmatrix} =\\begin{pmatrix}\\rho _{xx} & \\rho _{xy}\\\\\\rho _{yx} & \\rho _{yy}\\end{pmatrix}\\begin{pmatrix}j_x \\\\ j_y\\end{pmatrix},$ where $\\mathbf {E}=(E_x,E_y)^T$ and $\\mathbf {j}=(j_x,j_y)^T$ are electric field and current vectors respectively.",
"Under rotation by $\\delta \\theta $ , the resistivity tensor transform as: ${\\rho }^{\\prime } = R_{\\delta \\theta }^T {\\rho } R_{\\delta \\theta },\\ R_{\\delta \\theta } = \\begin{pmatrix}\\cos \\delta \\theta & -\\sin \\delta \\theta \\\\\\sin \\delta \\theta & \\cos \\delta \\theta \\end{pmatrix}.$ If the underlying system has a point group symmetry that is higher than $C_{2z}$ (e.g., $C_{3z},C_{6z}$ ), then ${\\rho }=\\rho _0\\mathbb {I} - i \\rho _H \\tau _y$ is the most general form of ${\\rho }$ invariant under such rotations.",
"Here $\\tau _y$ is the Pauli matrix acting in the two-dimensional coordinate basis, $\\rho _0(-B)=\\rho _0(B)$ is the longitudinal resistivity, and $\\rho _H(-B)=-\\rho _H(B)$ is the Hall resistivity.",
"The even/odd parity under time reversal is guaranteed by the Onsager reciprocal relations.",
"Since heterostrain breaks the point group symmetry down to $C_{2z}$ , we generally expect $\\rho _{xx}\\ne \\rho _{yy},\\ \\rho _{xy}\\ne -\\rho _{yx}$ .",
"Nevertheless, it is always possible to define transport principal axes after a suitable rotation $\\delta \\theta $ of the coordinate system, such that: $ {\\rho }_{\\text{principal}}= \\frac{1}{2}(\\rho _1+\\rho _2)\\mathbb {I}+\\frac{1}{2}(\\rho _1-\\rho _2)\\tau _z + \\rho _H i\\tau _y.$ Here $\\rho _{1,2}$ are longitudinal resistivities along the principal transport directions $\\hat{e}_{1,2}$ respectively.",
"The rotation angle $\\delta \\theta $ is determined up to $180^\\circ $ by requiring $\\rho _1< \\rho _2$ .",
"Below we first derive the MR tensor using Boltzmann approach for a general non-interacting electronic system within the relaxation time approximation.",
"Since there is currently insufficient understanding of the scattering mechanisms determining electrical transport in TBG, here we follow Ref.",
"[58] and use relaxation time approximation.",
"We will then present the results for heterostrained TBG, showing that in the open FS region, the low resistivity principal axis ($\\hat{e}_{1}$ ) is nearly perfectly aligned with the shortest moiré bond direction.",
"However there is a dramatic rotation of the principal axis as the filling moves towards the CNP.",
"We further show that the open FSs lead to a $B^2$ non-saturating MR along $\\hat{e}_{2}$ , and a saturating resistivity along $\\hat{e}_{1}$ .",
"For random orientation ($\\theta _0$ ) of the principal axis to the electrical current axis in the Hall bar geometry, e.g., as in Ref.",
"[39], the longitudinal resistivity is given by: $\\rho _{xx} =\\rho _1\\cos ^2 \\theta _0+\\rho _2\\sin ^2\\theta _0$ .",
"It is dominated by the $\\rho _2\\sim B^2$ component, and as a result, the experimental measurements should observe the non-saturating MR component if there is a misalignment with respect to the principal transport axis."
],
[
"Boltzmann equation and method of characteristics",
"We begin with a brief description of the method of characteristics used to solve the Boltzmann equation perturbatively in electric field ${\\bf E}$ but without a restriction on the strength of the perpendicular magnetic field ${\\bf B}=B\\hat{z}$ , as long as the semiclassical regime holds [59].",
"Due to $C_{2z}T$ symmetry of TBG at ${\\bf B}=0$ , there is no Berry curvature contribution to the semiclassical equations of motion.",
"Then, within the relaxation time approximation, the Boltzmann equation for a given energy band becomes $\\frac{\\partial n_\\mathbf {k}}{\\partial t} + (q\\mathbf {E}+q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B}) \\cdot \\frac{\\partial n_\\mathbf {k}}{\\partial \\mathbf {k}} = - \\frac{n_{\\mathbf {k}}-n_{0,\\mathbf {k}}}{\\tau },$ where $q\\mathbf {E}+q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B}$ is the total force on the Bloch electrons, with $\\mathbf {v}_\\mathbf {k}\\equiv \\nabla _{\\mathbf {k}} \\varepsilon _{\\mathbf {k}}$ and charge $q$ ; $n_{0,\\mathbf {k}}$ is the equilibrium Fermi-Dirac distribution and $n_{\\mathbf {k}}$ is the desired non-equilibrium distribution function.",
"We consider a stationary solution to the Boltzmann equation by parameterizing the distribution function as: $n_{\\mathbf {k}} = n_{0,\\mathbf {k}} + n_{1,\\mathbf {k}}.$ As a result, the Boltzmann equation for the deviation of the distribution function from equilibrium is: $(q\\mathbf {E}\\cdot \\mathbf {v}_{\\mathbf {k}}) \\frac{\\partial n_{0,\\mathbf {k}}}{\\partial \\varepsilon _\\mathbf {k}}+(q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B})\\cdot \\frac{\\partial n_{1,\\mathbf {k}}}{\\partial \\mathbf {k}} = - \\frac{n_{1,\\mathbf {k}}}{\\tau }.$ Note that the magnetic field only couples to $n_1$ since $(q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B})\\cdot \\nabla _\\mathbf {k}n_{0,\\mathbf {k}}=(q\\mathbf {v}_\\mathbf {k}\\times \\mathbf {B})\\cdot \\mathbf {v}_{\\mathbf {k}} \\partial _{\\varepsilon _{\\mathbf {k}}} n_{0,\\mathbf {k}}=0$ .",
"In order to solve the above partial differential equation (PDE), we seek a family of curves covering the $\\mathbf {k}$ -space which we parameterize as $\\mathbf {k}(s)$ with $s\\in [0,s_0)$ , such that along these curves the PDE becomes an ordinary differential equation (ODE).",
"If a curve $\\mathbf {k}(s)$ satisfies $ \\frac{d\\mathbf {k}(s)}{d s} = q\\mathbf {v}(s)\\times \\mathbf {B},$ then $n_{1,\\mathbf {k}(s)}\\equiv n_{1}(s)$ satisfies $(q\\mathbf {E}\\cdot \\mathbf {v}_{\\mathbf {k}}) \\frac{\\partial n_{0,\\mathbf {k}}}{\\partial \\varepsilon _\\mathbf {k}}|_{\\mathbf {k}=\\mathbf {k}(s)}+ \\frac{d n_{1}(s)}{d s} = - \\frac{n_{1}(s)}{\\tau }.$ Because $\\frac{d \\varepsilon (s)}{d s} = \\mathbf {v}(s) \\cdot \\frac{d \\mathbf {k}(s)}{d s} = 0,$ the curve $\\mathbf {k}(s)$ must coincide with the contour of constant energy.",
"Thus, the Boltzmann equation becomes: $[q\\mathbf {E}\\cdot \\mathbf {v}(s)] \\frac{\\partial n_{0}(s)}{\\partial \\varepsilon (s)}+ \\frac{d n_{1}(s)}{d s} = - \\frac{n_{1}(s)}{\\tau }.$ The ODE is readily solved with: $n_{1}(s) = \\chi _0 e^{-s/\\tau } - e^{-s/\\tau }\\int _0^{s}\\mathrm {d}s^{\\prime } e^{s^{\\prime }/\\tau } [q\\mathbf {E}\\cdot \\mathbf {v}(s^{\\prime })] \\frac{\\partial n_{0}(s^{\\prime })}{\\partial \\varepsilon (s^{\\prime })}.$ where $\\chi _0$ is a constant determined by the following argument.",
"Since $\\mathbf {k}(s)$ describes a constant energy contour in a two-dimensional Brillouin zone, it is either a closed contour, or several open contours that terminate on boundaries of the Brillouin zone such that they form a closed loop on a torus.",
"In either case, $\\mathbf {k}(s)$ is periodic under $s\\rightarrow s+s_0$ modulo a moiré reciprocal lattice vector, where $s_0$ is the periodicity.",
"The periodicity condition $n_{1}(s_0)=n_{1}(0)$ leads to $\\chi _0 = \\frac{1}{1-e^{s_0/\\tau }}\\int _0^{s_0}\\mathrm {d}s^{\\prime } e^{s^{\\prime }/\\tau }(q\\mathbf {E}\\cdot \\mathbf {v}(s^{\\prime })) \\frac{\\partial n_{0}(s^{\\prime })}{\\partial \\varepsilon (s^{\\prime })},$ which determines the desired $n_1(s)$ .",
"In the low temperature limit, the steady state current from a given energy band is calculated as: $ \\begin{split}j^\\mu & = q\\int \\frac{\\mathrm {d}^2\\mathbf {k}}{(2\\pi )^2} v^{\\mu }_{\\mathbf {k}} n_{1,\\mathbf {k}} \\\\& =\\frac{q^2B}{(2\\pi )^2} \\int \\mathrm {d} {\\varepsilon }\\int _0^{s_0} \\mathrm {d} s v^{\\mu }(s) n_{1}(s)\\\\& = \\frac{q^3B}{(2\\pi )} \\frac{\\tau }{\\omega _c} \\sum _{n=-\\infty }^{\\infty } \\frac{v^{\\mu }_nv^{\\nu }_{-n}}{1+in \\omega _c \\tau }E^{\\nu },\\end{split}$ where $(\\mu ,\\nu )=x,y$ , and we have defined the cyclotron frequency as: $\\omega _c \\equiv 2\\pi /s_0.$ We have also made use of the periodicity of velocity under $s\\rightarrow s+s_0$ to write it in terms of Fourier series, $\\mathbf {v}(s)=\\sum _{n=-\\infty }^{\\infty }\\mathbf {v}_ne^{-in \\omega _c s }$ .",
"To show that the second line of Eq.",
"(REF ) holds, note that at every $\\mathbf {k}$ we can define a local coordinate system $(\\hat{e}_{v},\\hat{e}_{s})$ such that $\\mathbf {v}\\equiv v\\hat{e}_{\\mathbf {v}}$ where $v\\ge 0$ , and $\\hat{e}_{s}=\\hat{e}_{\\mathbf {v}}\\times \\hat{z}$ .",
"The infinitesimal wavevector can be equivalently written as: $\\mathrm {d}\\mathbf {k}=\\mathrm {d}k_x \\hat{e}_x + \\mathrm {d}k_y \\hat{e}_y= \\mathrm {d}k_s \\hat{e}_s + \\mathrm {d}k_v \\hat{e}_v.$ Eq.",
"(REF ) can then be written as ${\\mathrm {d}\\mathbf {k}}/{\\mathrm {d} s} = qvB \\hat{e}_s$ , or equivalently $\\mathrm {d} k_s = qvB\\mathrm {d}s$ .",
"As a result, $\\int \\mathrm {d} k_x \\mathrm {d}k_y = \\int \\mathrm {d} k_s \\mathrm {d}k_v = qB \\int \\mathrm {d}\\varepsilon \\mathrm {d} s.$ The conductivity tensor is therefore given by the following expression: $ \\sigma ^{\\mu \\nu } = \\frac{q^3B}{2\\pi } \\frac{\\tau }{\\omega _c} \\sum _{n=-\\infty }^{\\infty } \\frac{v^{(\\mu )}_nv^{(\\nu )}_{-n}}{1+in \\omega _c\\tau }.$ Eq.",
"(REF ) gives the magnetoconductivity for a given FS contour.",
"In the case of multiple FS contours and multiple bands –as due to spin and valley degeneracy in TBG– conductivities from different FS contours and bands add.",
"Finally, the MR tensor is obtained by inverting the conductivity tensor, i.e., $\\rho = \\left(\\sum _{n,i}\\sigma _{n,i}\\right)^{-1}$ , where $n,i$ are band and contour labels respectively for a given energy level.",
"To better understand Eq.",
"(REF ) consider an example of a parabolic dipsersion with $\\varepsilon _{\\mathbf {k}}=\\frac{1}{2m_0}(k_x^2+k_y^2)$ , where $m_0$ is the bare electron mass.",
"At a fixed energy $\\mu $ the contour is a circle parameterized as: $(k_x,k_y)=\\sqrt{2m_0\\mu }(\\cos \\theta ,\\sin \\theta ),\\ \\theta \\in [0,2\\pi )$ .",
"Using method of characteristics, we get: $\\frac{\\mathrm {d}\\theta }{\\mathrm {d} s} = -\\frac{qB}{m_0}$ , or $\\theta = \\theta _0 - \\omega _0 s$ , where $\\omega _0\\equiv \\frac{qB}{m_0}$ is the cyclotron frequency of bare electrons.",
"This leads to the periodicity in $s$ to be $s_0=2\\pi /\\omega _0$ , where we have chosen the clockwise trajectory such that $s_0>0$ .",
"The Fourier series of the velocity along the constant energy contour is given by: $v_{x}(s)=\\sqrt{\\frac{\\mu }{2m}}\\left(e^{-i \\omega _0 s}+e^{i\\omega _0 s}\\right)$ , and $v_{y}(s)=\\sqrt{\\frac{\\mu }{2m}}\\frac{1}{i}\\left(e^{-i \\omega _0 s}-e^{i\\omega _0 s}\\right)$ .",
"Substituting into Eq.",
"(REF ), we obtain the conductivity tensor: $\\sigma = q^2\\tau \\frac{\\mu }{2\\pi } \\frac{1}{1+\\omega _0^2\\tau ^2}\\begin{pmatrix} 1 & -\\omega _0\\tau \\\\ \\omega _0\\tau & 1 \\end{pmatrix}.$ Note that the total number density of filled electrons is given by $n = \\int \\frac{\\mathrm {d}^2\\mathbf {k}}{(2\\pi )^2} \\Theta (\\mu -\\varepsilon _{\\mathbf {k}}) = \\frac{m_0\\mu }{2\\pi }$ .",
"We therefore reproduce the well known magnetoconductivity tensor: $\\sigma = \\frac{nq^2\\tau }{m_0} \\frac{1}{1+\\omega _0^2\\tau ^2}\\begin{pmatrix} 1 & -\\omega _0\\tau \\\\ \\omega _0\\tau & 1 \\end{pmatrix}.$ In this simple example of a closed FS, the longitudinal resistivity is given by $\\frac{m_0}{nq^2\\tau }$ , independent of the magnetic field.",
"The average of the velocity field, $\\mathbf {v}_{n=0}\\equiv \\frac{1}{s_0}\\int _0^{s_0}\\mathrm {d}s \\mathbf {v}(s)$ , vanishes.",
"However, for an open FS generally $\\mathbf {v}_{n=0} \\ne \\mathbf {0}$ , i.e.. electrons have a finite drift velocity when traversing the contour due to a magnetic field (see SM Fig. 2).",
"The impact of such a finite drift velocity on the magnetotransport can be qualitatively understood using the following example: in the expression for the conductivity tensor (Eq.",
"(REF )), we consider $v^{x}_{n=0}\\ne 0$ but $v^{y}_{n=0}= 0$ .",
"This corresponds to an open FS with a drift velocity along the $x$ direction.",
"In the high field limit ( $\\omega _c\\tau \\propto B \\gg 1)$ , we truncate the Fourier series at the leading order, and as a result, $\\sigma _{\\text{open FS}} \\approx \\frac{q^3B}{2\\pi }\\frac{\\tau }{\\omega _c}\\begin{pmatrix}{(v^{x}_0)^2} & -\\frac{2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau }\\\\ \\frac{2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau } & \\frac{|v^{y}_1|^2}{\\omega _c^2\\tau ^2}\\end{pmatrix},$ where we made use of the equality: $\\mathbf {v}_{-n}=\\mathbf {v}_{n}^*$ .",
"Inverting the matrix, we obtain the MR tensor: $\\begin{split}{\\rho }_{\\text{open FS}} &\\approx \\frac{(2\\pi )\\omega _c}{q^3B \\tau } \\frac{1}{4\\text{Im}(v_{-1}^{x}v_{1}^{y})^2+(v_0^{x})^2|v_{1}^{y}|^2} \\\\& \\times \\begin{pmatrix}|v^{y}_1|^2 & {2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau }\\\\ -{2\\text{Im}(v^{x}_{-1}v^{y}_{1})}{\\omega _c\\tau } & (v^{(x)}_0)^2\\left(\\omega _c\\tau \\right)^2\\end{pmatrix}.\\end{split}$ It is clear that $\\rho _{yy}\\propto B^2$ whereas $\\rho _{xx}\\sim \\mathcal {O}(1)$ .",
"We therefore arrive at the important conclusion that for an open FS, the longitudinal MR has non-saturating $B^2$ behavior along the axis with a zero drift velocity ($\\hat{y}$ in the above example), and saturating behavior along the other axis."
],
[
"Magnetotransport in TBG under heterostrain",
"We proceed to apply the above results to analyze the magnetotransport in TBG.",
"The theory satisfactorily explains the weak-field magnetotransport measurements presented in Ref. [39].",
"We then present two predictions of the theory that we did not anticipate prior to starting this work: the dependence of the principle axis of transport on filling, and the behavior of magnetoresistance and quantum oscillations at densities between the CNP and the onset of quadratic MR.",
"The former is of academic interest, however it cannot be confirmed with our present data sets because of limitations of the Hall bar geometry.",
"The latter can be considered smoking gun evidence for the the presence of the lowest-energy van Hove point and the energetic splitting of the Dirac cones.",
"We do not expect our strained BM model in Eq.",
"(REF ) to yield precise agreement with experiment, so we do not perform fine-tuning of its input parameters.",
"Specifically, the model has particle-hole symmetry, which is absent from experimental measurements.",
"More sophisticated non-interacting model calculations [44], [43] as well as interaction renormalizations [54] are likely necessary to properly account for such details.",
"Although the general phenomena of open FSs and quadratic MR holds for a broad range of heterostrain parameters, we present calculations for $\\theta =1.38^\\circ $ , $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ , parameters chosen to yield reasonable quantitative agreement between the theoretical and experimental results both on the filling range of open FSs, as well as on the frequencies of magnetoresistance oscillations to be presented later.",
"In Fig.",
"REF , we show the computed MR along the principal transport axes (a) and the Hall number (b).",
"For comparison, we plot the experimentally measured longitudinal and transverse resistivities (c) and Hall number (d) for the TBG device studied in Ref. [39].",
"In the filling ranges with open FSs, the calculated $\\rho _2(B)$ exhibits quadratic non-saturating MR, whereas $\\rho _1(B)$ saturates.",
"The filling range for which quadratic MR occurs is bounded by the two outermost van Hove points of the zero-field strained band structure.",
"In experiment, we observe quadratic MR in longitudinal resistivity within a similar range of fillings.",
"More strikingly, we observe quadratic MR in the transverse resistivity as well.",
"In some cases, the symmetric part of the transverse resistivity becomes larger than that of the longitudinal resistivity with field.",
"As discussed earlier, this degree of mixing can be attributed to the misalignment between the strain-induced principal axis of transport and the direction of current flow in the Hall bar geometry.",
"At the first van Hove point ($\\nu \\approx \\pm 0.6$ ), the non-analyticity in the density of states leads to a cusp in the first derivative of the zero-field resistivity with respect to filling (see SM Fig. 6).",
"As shown in Fig.",
"REF (a), at $B\\ne 0$ the longitudinal resistance develops a cusp as a function of filling at the first van Hove point.",
"The cusp becomes more pronounced with increasing $B$ .",
"Experimentally as shown in Fig.",
"REF (c), there is a cusp-like feature developing at $|\\nu |\\sim 0.5-0.8$ depending on the contact pair within the device used, consistent with theoretical predictions.",
"In many contact pairs, this feature presents as a shoulder at $B=0$ , only developing into a cusp at $B \\sim 0.1$ T (see SM Fig. 7).",
"As depicted in Fig.",
"REF (b), the calculated filling dependence of the Hall number shows two singular sign changes inside the open FS regions near $\\nu \\approx \\pm 2$ .",
"The sign changing singularity in the open FS region is $B$ -independent, and is not directly associated with any van Hove point (see SM Fig.",
"5 for a plot of $\\rho _H(B)$ , which crosses zero at the same filling fraction inside the open FS filling range for varying field strength).",
"Moreover, the filling dependence of the Hall number $n_H$ tracks the filling fraction in a broad filling range near the CNP, with the filling range being extended upon increasing $B$ .",
"In Fig.",
"REF (d), we observe the same general shape of the Hall number.",
"Within the open FS filling range, however, the measured Hall number qualitatively deviates from the theoretical curves.",
"We attribute this to a small constant offset in the magnetic field of order 10-20 mT, likely resulting from trapped flux in the superconducting magnet.",
"Here a large quadratic symmetric component of the transverse resistivity is concurrent with a vanishing antisymmetric component.",
"An offset of only a few mT will lead to a small part of the symmetric component mixing into the antisymmetric component, leading to these deviations from theory (See SM Fig. 8).",
"Figure: (a) Rotation of the transport principal axis e ^ 1 \\hat{e}_1 with respect to the global coordinate system for strained BM with ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .",
"The three horizontal dashed lines are the bond directions.",
"In the open FS region, the saturating MR axis is locked to the shortest bond (𝐋 1 \\mathbf {L}_1) direction.",
"However, it rapidly rotates in the closed FS region upon approaching the CNP.",
"(b) Principal transport axes e ^ 1 \\hat{e}_1 (red) and e ^ 2 \\hat{e}_2 (blue) for a few filling fractions.",
"Near the CNP, e ^ 1 \\hat{e}_1 is perpendicular to the shortest moiré bond direction.",
"In the open FS filling range (e.g.",
"ν≈2.13\\nu \\approx 2.13) it is rotated to be parallel to the shortest bond direction.Our calculation finds a dramatic rotation of the principal axis with filling, as illustrated in Fig.",
"REF .",
"In the filling range with open FSs, the principal axis with saturating MR ($\\hat{e}_1$ ) is aligned with direction of the shortest moiré triangular bond, suggesting that the electrons are hopping more efficiently along the shortest bond, which leads to a larger conductivity and therefore a smaller resistivity.",
"Interestingly, when filling is changed from the second van Hove point ($\\nu \\approx \\pm 1.3$ ) to the vicinity of the CNP, $\\hat{e}_1$ rotates dramatically to the perpendicular direction compared to the filling range with open FSs.",
"The rotation of the principal axis is likely due to the opposite elongation of the larger Fermi pocket encircling a Dirac point compared to the open FS contours, see for example Figs.",
"REF (d-f).",
"The rotation of the transport axis with filling purely due to strain-induced bandstructure effects demonstrates that filling dependence of the principal axes orientation need not be associated with interaction induced nematicity [14].",
"Such a filling-dependent rotation of the principal transport axis was not possible to observe in Ref.",
"[39] using the Hall bar geometry, where only $\\rho _{xx}$ and $\\rho _{yx}$ are measured but not $\\rho _{yy}$ .",
"Additional transport measurements are needed, where the filling-dependence of the entire resistivity tensor can be mapped out.",
"Figure: (a) Line cuts of MR near the CNP taken at 26 mK in contact pair 4 - 5 at the indicated field strengths, in Tesla.",
"Vertical dashed lines indicate our estimated location of the lowest-energy van Hove points, based on the cusps in resistivity at low field.",
"Within the region bounded by these points, the quantum oscillations show up before 0.4 T, and their relative strengths do not follow a simple pattern.",
"Outside of this region, the quantum oscillations onset at higher field, and every multiple of 4 quantum Hall filling fraction is observed relatively equally.",
"(b) Fourier transform of the quantum oscillation data with respect to 1/B1/B.",
"It reveals a transition from two pockets to one pocket at the lowest-energy van Hove points.",
"(c) Schematic description of the frequencies observed in panel (b).",
"Red dashed lines are frequencies from the experimental data.",
"Solid black lines are predictions from the theory for ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .",
"The two frequencies f 1 f_1 and f 2 f_2 sum to the one-pocket frequency f 3 f_3 that extends beyond the first van Hove point.",
"They additionally account for the nontrivial relative strengths of the quantum oscillations within the bounds of the first van Hove points.",
"As with other details of this work, the theory predicts electron-hole symmetry, while some asymmetry is observed in experiment.Since this theory predicts a third van Hove point between the CNP and the filling range with open FSs, a direct measurement of this van Hove point is desired.",
"In Fig.",
"REF we reanalyze quantum oscillation measurements of the TBG device discussed in Ref. [39].",
"The effective cyclotron mass $m^*$ is light in the filling range with two small closed Fermi pockets, and dramatically heavier in the filling range with only one closed pocket (Figs.",
"REF (d-f) and SM Fig. 5).",
"The large difference in masses on either side of the innermost van Hove singularity can be used to explain the substantially earlier onset of quantum oscillations with increasing field close to the CNP than away from it, as shown in Fig.",
"REF (a).",
"Fig.",
"REF (b) is a Fourier transform of the quantum oscillation data with respect to $1/B$ .",
"In the filling range of $-0.7\\le \\nu \\le 0.8$ three distinct frequencies $f_{i=1,2,3}$ are clearly observed in the data, with $f_1$ and $f_2$ corresponding to two small Fermi pockets, and $f_3=f_1+f_2$ to the breakdown orbit when the inverse magnetic length is comparable to the momentum space distance between the two small Fermi pockets [60].",
"Outside of the filling range only $f_3$ is observed, showing that there are Lifshitz transitions, one on either side of the CNP, that we ascribe to crossing the lowest-energy van Hove points.",
"Furthermore, these filling fractions also correspond to the cusp-like features in the longitudinal MR data shown in Fig.",
"REF (c), consistent with theoretical predictions for its behavior at van Hove singularities.",
"Therefore, the quantum oscillation data unambiguously demonstrates the existence of a third van Hove singularity at filling fractions between the CNP and the filling range of $B^2$ MR.",
"It is interesting to note again, that the Hall number does not show a sign-changing singularity at this van Hove point, as illustrated in Fig.",
"REF (b) and (d).",
"The frequencies $f_{1,2}$ are a strong constraint on the amount of heterostrain in the TBG sample.",
"Specifically, as illustrated in Fig.",
"REF (c), the frequency $f_2$ is roughly two times $f_1$ , showing that the two small Fermi pockets have an area ratio $\\sim 2:1$ .",
"Theoretically as illustrated by the solid black lines in Fig.",
"REF (c), for a heterostrain strength $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ , the areas $A_{i=1,2}$ of the two small pockets, when converted to frequency via $f_i^{-1}\\equiv (\\Delta \\frac{1}{B})_i=\\frac{2\\pi e}{\\hbar A_i}$ , are in good agreement with experiment.",
"We observe behavior qualitatively similar in all respects to that in Fig.",
"REF (a) in all 3 longitudinal contact pairs for which we have dilution-fridge measurements (see SM Fig. 11).",
"In addition to the quantum oscillation measurements above, we propose an additional experimental procedure for identifying the van Hove points.",
"As usual, at van Hove singularities there are non-analyticities in the electronic density of states.",
"Such non-analyticities will lead to cusps in the first derivative of the zero-field resistivity with respect to filling (see SM Fig. 6).",
"This can be probed via transport measurements, for example, by adding a small ac modulation of the filling or by numerical differentiation of the dc data."
],
[
"Summary and Outlook",
"In summary, we have shown that due to the large size of the moiré unit cell at small twist angles, even a small amount of uniaxial heterostrain on the microscopic scale can lead to dramatic changes in the narrow bands of twisted bilayer graphene.",
"A key feature of the strained bandstructure is the splitting of the respective energetic degeneracies of the two Dirac points and the three van Hove points.",
"The splitting of the two Dirac points leads to a semimetallic state with two small Fermi pockets at the CNP.",
"On the other hand, the two outermost van Hove points bound a broad filling range near $\\nu =\\pm 2$ where the constant energy contours become open.",
"Interestingly, the elongation of the larger Fermi pocket near the CNP is perpendicular to that of the open FSs, the latter being perpendicular to the direction of the shortest moiré triangular bond.",
"We have analyzed the resulting magnetotransport in strained TBG in the framework of the Boltzmann equation using the method of characteristics, treating the magnetic field non-perturbatively.",
"We showed that a non-saturating quadratic longitudinal magnetoresistance in a broad filling range near $\\nu =\\pm 2$ naturally arises due to the heterostrain-induced open Fermi surfaces, therefore explaining the experimental results in the off-magic-angle devices [39].",
"We have also shown that the sign-changing singularities in the Hall number occur in the open FS filling range and are not directly associated with any van Hove singularity as commonly assumed, e.g., in Ref. [61].",
"Furthermore, our results reveal a dramatic rotation of the transport principal axis as the filling is tuned from the charge neutrality point to the filling range of open Fermi surfaces.",
"This is entirely attributed to the strained non-interacting bandstructure effects, and does not require interaction-induced electronic nematicity for explanation.",
"Given the importance of energy-shifted van Hove points in the transport properties of TBG devices, we have analyzed previous quantum oscillation data, which has revealed a Lifshitz transition from two pockets to one pocket at a filling fraction where the innermost van Hove singularity is predicted to occur based on theoretical calculations, therefore offering strong evidence of heterostrain effects on these devices.",
"We have further proposed several additional signatures to look for in future experiments.",
"These include cusps in the derivative of zero field resistivity with respect to filling, a significant difference in cyclotron mass on either side of the innermost van Hove singularity, and a principal transport axis with saturating magnetoresistance in the open Fermi surface filling range.",
"Finally, given the amplifying effect of a small strain at the underlying carbon lattice scale on the moiré lattice scale, the latter of which controls the electronic behavior within the narrow bands, it is tantalizing to consider strain engineering of such devices to achieve effects which would be impossible in regular solids due to structural instabilities."
],
[
"Acknowledgement",
"Funding: X.W.",
"acknowledges financial support from National MagLab through Dirac fellowship, which is funded by the National Science Foundation (Grant No.",
"DMR-1644779) and the state of Florida.",
"O.V.",
"was supported by NSF Grant No.",
"DMR-1916958 and is partially funded by the Gordon and Betty Moore Foundation's EPiQS Initiative Grant GBMF11070, National High Magnetic Field Laboratory through NSF Grant No.",
"DMR-1157490 and the State of Florida.",
"Device measurements and analysis were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-76SF00515.",
"Measurement infrastructure was funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF3429 and grant GBMF9460.",
"D.G.-G. gratefully acknowledges support from the Ross M. Brown Family Foundation.",
"Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.",
"K.W.",
"and T.T.",
"acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).",
"Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822.",
"Author contributions: O.V.",
"conceived the theoretical explanation of the experiments.",
"X.W.",
"and O.V.",
"performed calculations.",
"J.F., L. R., and C.H.",
"fabricated devices.",
"J.F.",
"and X.W.",
"analysed the data.",
"K.W.",
"and T.T.",
"generously supplied the hBN crystals.",
"A.L.S., M.A.K., O.V., and D.G.-G. supervised the experiments and analysis.",
"The manuscript was prepared by X.W.",
"and J.F.",
"with input from all authors.",
"Supplementary Materials for “Unusual magnetoresistance in twisted bilayer graphene from strain induced open Fermi surfaces\" We present additional theoretical results and experimental measurements in support of the main text."
],
[
"Heterostrain effects on the geometry of moiré superlattice",
"Our off-magic-angle twisted bilayer graphene (TBG) devices in Ref.",
"[39] are prepared using the “tear-and-stack\" procedure, and as a result, strain is inevitably introduced.",
"Here we first show that while the moiré unit cell vectors are strongly deformed by even an infinitesimal amount of uniaxial heterostrain in the device, the unit cell area is much less affected.",
"As a result, for the off-magic-angle device studied in Ref.",
"[39], we can have a good estimate of the twist angle ($\\theta $ ) based on the moiré unit cell area alone.",
"In the limit of small deformations, both the uniaxial heterostrain and a small twist angle are captured via a coordinate transformation: $\\mathbf {r}^{\\prime }_l = \\mathbf {r}+ \\mathbf {u}_l(\\mathbf {r})$ , where $l=t,b$ labels the top (bottom) graphene layers, and $\\mathbf {u}_l(\\mathbf {r}) \\approx \\mathcal {E}_l \\mathbf {r}$ is the local deformation field.",
"The symmetric and antisymmetric part of the $2\\times 2$ tensor $\\mathcal {E}_l$ describes strain and rotation respectively.",
"For twist angle $(\\theta )$ and a uniaxial heterostrain of strength $(\\epsilon )$ and direction $(\\varphi )$ , we parameterize $\\mathcal {E}_{t}=-\\mathcal {E}_{b}\\equiv \\mathcal {E}/2$ , where $\\mathcal {E}\\equiv \\mathcal {T}(\\theta )+ \\mathcal {S}(\\epsilon ,\\varphi )$ , and given by: $\\mathcal {T}(\\theta ) = \\begin{pmatrix} 0 & -\\theta \\\\ \\theta & 0\\end{pmatrix},\\ \\mathcal {S}(\\epsilon ,\\varphi ) = R_{\\varphi }^T \\begin{pmatrix} -\\epsilon & 0\\\\ 0 & \\nu \\epsilon \\end{pmatrix}R_{\\varphi }.$ Here $R_\\varphi $ is the two-dimensional rotation matrix, and $\\nu \\approx 0.16$ is the Poisson ratio [3].",
"Physically, $\\epsilon >0$ corresponds to compressing the top layer while streching the bottom layer along the $x$ -axis.",
"A relative deformation $\\mathcal {E}$ between the graphene bilayers generate a moiré superlattice, with moiré reciprocal lattice vectors given by: $\\mathbf {g}_{i=1,2}=\\mathcal {E}^T\\mathbf {G}_{i=1,2},$ where $\\mathbf {G}_{i}$ are reciprocal lattice vectors of the undeformed monolayer graphene.",
"Eq.",
"(REF ) can be used to uniquely determine the three parameters $(\\theta ,\\epsilon ,\\varphi )$ .",
"Additionally it also determines a global angle $\\alpha $ that measures the rotation between the lab and theoretical coordinate systems.",
"Uniaxial heterostrain has a dramatic effect on the distortion of the moiré unit cell vectors, as $|\\delta \\mathbf {g}|/|\\mathbf {g}|\\sim \\mathcal {O}(\\epsilon /\\theta )$ .",
"However, its effect on the moiré unit cell area is much smaller.",
"To show this, note that the area of the moiré Brillouin zone is calculated as: $A_{mBZ} = \\left| (\\mathbf {g}_1 \\times \\mathbf {g}_2 )\\cdot \\hat{z}\\right| = \\left| \\mathbf {g}_1^T (i\\sigma _y) \\mathbf {g}_2 \\right|,$ where on the second equality we have used a vector notation $\\mathbf {g}_i\\equiv (g_{i,x},g_{i,y})^T$ .",
"Following Eq.",
"(REF ), we obtain that the area of the moiré Brillouin zone is independent on $\\varphi $ , and calculated as: $A_{mBZ} = (\\theta ^2-\\nu ^2\\epsilon ^2)A_{BZ},$ where $A_{BZ}\\equiv \\left|(\\mathbf {G}_1\\times \\mathbf {G}_2)\\cdot \\hat{z}\\right|$ is the Brillouin zone area of the undeformed monolayer graphene.",
"The area of the strained moiré unit cell can be calculated in a similar manner, and we get: $A_{m.u.c.}=A_{u.c.",
"}/(\\theta ^2-\\nu ^2\\epsilon ^2)$ , where $A_{u.c.",
"}$ is the unit cell area of undeformed monolayer graphene.",
"Observe that the heterostrain only affects the area of the moiré unit cell by $\\mathcal {O}(\\nu ^2\\epsilon ^2/\\theta ^2)$ which is much smaller than the linear distortion of moiré unit cell vectors.",
"With only a knowledge of the moiré unit cell areas in Ref.",
"[39] (see Table REF ), we estimate the twist angle to be $\\theta \\sim 1.35^\\circ - 1.39^\\circ $ for various contact pairs studied using the Hall bar geometry."
],
[
"Constraining heterostrain from transport measurements",
"For the TBG device studied in Ref.",
"[39], the deformed moiré lattice vectors were not measured.",
"Nevertheless, here we show that magnetotransport measurements, along with theoretical calculations based on the strained Bistrizer-MacDonald (BM) Hamiltonian, offer strong constraints on the heterostrain in the device.",
"We caution, however, that since the strained BM model is an approximate description of the narrow bands of TBG, a precise determination of heterostrain from model calculations is not feasible.",
"First of all, as predicted by theoretical calculations, the van Hove singularities of the band structure lead to non-analytic behavior for the longitudinal magnetoresistance as a function of electron filling.",
"The filling fractions for the six van Hove singularities in the narrow band are listed in Table REF for various contact pairs.",
"Secondly, magnetic oscillations show a Lifshitz transition at the inntermost van Hove singularities ($\\nu _3,\\nu _4$ ), from two small Fermi pockets closer to the charge neutrality point to one Fermi pocket away from it.",
"Furthermore, the areas of the two small Fermi pockets, as revealed by the frequencies of magnetic oscillations, show a $2:1$ or smaller ratio.",
"Both the filling fractions for van Hove singularities and the pocket area size offer strong constraints for the heterostrain.",
"Qualitatively, on the one hand, a broader filling range of open Fermi surfaces can be achieved by increasing the strength of uniaxial heterostrain.",
"On the other hand, to obtain Fermi pocket area sizes near $2:1$ ratio or smaller, a smaller heterostrain is necessary as it leads to a weaker splitting of the two Dirac cones.",
"For theoretical calculations presented in the main text, we find $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ to give reasonably good agreements with both experimental observations described above.",
"A larger heterostrain strength ($\\epsilon =0.3\\%$ ) will lead to a much larger pocket area ratio ( $4:1$ for $\\epsilon =0.3\\%$ and $\\varphi =0^\\circ $ ), inconsistent with magnetic oscillation measurements.",
"On the other hand, a smaller heterostrain strength $\\epsilon =0.1\\%$ decreases the filling range of open Fermi surfaces dramatically, inconsistent with the longitudinal magnetoresistance measurements."
],
[
"Detailed band structure analysis for varying uniaxial heterostrain",
"In the main text we discussed the band structure of the strained TBG for $\\epsilon =0.2\\%$ .",
"The main effect of uniaxial heterostrain is to break the respective energetic degeneracies of the two Dirac points and three van Hove points of a given band, therefore giving rise to a semimetallic state at charge neutrality point, and open Fermi surface regions bounded by the two outermost van Hove points.",
"However for a larger heterostrain, the innermost van Hove point moves closer to one of the Dirac point.",
"As a result, both Dirac cones become type II titled, and the innermost van Hove points of both the upper and lower bands are annihilated.",
"In turn two new band extrema are formed.",
"This is illustrated in Fig.",
"REF .",
"We also explore the possibilities of heterostrain-induced higher order van Hove singularities which is possible for the magic-angle TBG as discussed in Ref. [38].",
"We checked that for $\\theta =1.38^\\circ $ , and up to uniaxial heterostrain strength of $\\epsilon =0.7\\%$ , no higher order van Hove singularities are found.",
"This shows that the band flattening effect at the magic angle may be important for strain engineering of higher order van Hove points.",
"Figure: Velocity field of typical closed and open Fermi surfaces.",
"Whereas for a closed Fermi surface the averaged velocity vanishes, for open Fermi surfaces this is generally violated.In Fig.",
"REF we plot the velocity fields $v_x(s)$ and $v_y(s)$ on typical open and closed Fermi surfaces for the strained TBG, parameterized by $s\\in [0,s_0)$ as defined in the main text.",
"For the closed Fermi surface contours, the averaged velocity, $\\mathbf {v}_{n=0}\\equiv \\frac{1}{s_0}\\int _0^{s_0}\\mathrm {d}s \\mathbf {v}(s)$ , is zero.",
"On the other hand, for a typical open Fermi surface contour, it is finite, and as a result the electron traversing the open Fermi surface contour in the presence of a magnetic field has a finite drift velocity.",
"As discussed in the main text, this is the reason for the non-saturating $B^2$ magnetoresistivity (MR) observed in strained TBG devices."
],
[
"More details on magnetotransport in TBG",
"Here we show that while $B^2$ longitudinal MR generally occurs for strained TBG due to open Fermi surfaces, it does not occur for unstrained devices.",
"In Fig.",
"REF , the longitudinal MRs $\\rho _{xx}$ and $\\rho _{yy}$ as well as the Hall number $n_H$ are plotted for an unstrained BM model calculation.",
"First of all, $\\rho _{xx}=\\rho _{yy}$ since the unstrained TBG has $C_{3z}$ rotational symmetry.",
"Secondly, cusp-like features develop at the triply-degenerate van Hove point at filling fractions $\\nu \\approx \\pm 1.4$ , and are attributed to the non-analyticities in the density of states behavior at the van Hove singularities.",
"Finally, unstrained TBG has saturating MR across all filling range, as illustrated in the inset to Fig.",
"REF (a).",
"Figure: (a) Longitudinal MR ρ xx \\rho _{xx} (dashed) and ρ yy \\rho _{yy} (solid) for strained BM with ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .",
"Different colors represent varying magnetic field strength.",
"Vertical dashed lines are positions of the van Hove points.",
"Shaded areas are open Fermi surface regions.",
"(b) In the closed Fermi surface region, MR saturates at large magnetic fields.",
"(c) In the open Fermi surface region, MR exhibit non-saturating B 2 B^2 dependence along both directions.In Fig.",
"REF we show that for the globally defined coordinate system which is misaligned from the principal transport axis, the $B^2$ behavior generally dominates the MR, and therefore will show up in both $\\rho _{xx}$ and $\\rho _{yy}$ measurements.",
"This remains true for a generic misalignment between the transport axis from experiment and the principal transport axis.",
"Figure: Hall resistivity ρ H \\rho _H at varying magnetic fields.",
"Note that it crosses zero within the filling range of open Fermi surfaces on both sides of the charge neutrality point.",
"These mark the sign-changing singularities in the Hall number depicted in Fig.",
"2(b) of the main text.In Fig.",
"REF we show the Hall resistivity $\\rho _H(B)$ for varying magnetic field strength.",
"Since $\\rho _H = B/n_Hq$ , wherever $\\rho _H(B)$ crosses zero and changes sign, the Hall number displays a sign-changing signularity.",
"Fig.",
"REF clearly shows that $\\rho _H(B)$ crosses zero in the open Fermi surface regions on both sides of the charge neutrality point, and independent on the strength of the $B$ -field.",
"Figure: Filling dependence of the averaged inverse cyclotron mass 1/m * 1/m^*, extracted from the averaged cyclotron frequency ω c ¯=eB m * \\bar{\\omega _c}=\\frac{eB}{m^*}.",
"Here ω ¯ c =1 N∑ n,i ω c,n,i \\bar{\\omega }_c=\\frac{1}{N}\\sum _{n,i}\\omega _{c,n,i}, where nn and ii label the FS (ii) coming from a given band (nn), in units of the bare electron mass.",
"1/m * 1/m^* is larger near the charge neutrality and band edges, explaining the earlier onset of quantum oscillations in these filling regions.In Fig.",
"REF we illustrate the filling-dependent inverse cyclotron mass $1/m^*$ for strained BM model with $\\epsilon =0.2\\%$ and $\\varphi =0^\\circ $ .",
"This is to highlight the dichotomy of light-heavy masses on either side of the innermost van Hove singularities closest to the charge neutrality point.",
"This is consistent with the experimental observation of a much earlier onset field of quantum oscillations in filling range below the innermost van Hove point than above.",
"Figure: (a) Longitudinal resistivitities ρ xx \\rho _{xx} (blue) and ρ yy \\rho _{yy} (orange) at B=0B=0 for strained BM model, with ϵ=0.2%\\epsilon =0.2\\% and ϕ=0 ∘ \\varphi =0^\\circ .",
"(b) The derivative of log resistivity with respect to filling.",
"Gray dotted vertical lines mark positions of the van Hove points, and the yellow shaded area marks the open Fermi surface region.In Fig.",
"REF we also show the filling dependence of the $B=0$ longitudinal resistivities and their derivatives with respect to filling.",
"A key highlight is that the non-analyticities in the density of states at the van Hove points lead to kink-like features in the derivatives, but nearly invisible in the resistitivies themselves."
],
[
"Error analysis of antisymmetrization",
"In Fig.",
"REF we show that the bump-like features in the experimental Hall number plots in filling range of open Fermi surfaces (Fig.",
"2(d) of main text and Fig.",
"REF in the SM) may be attributed to improper antisymmetrization with respect to the $B$ -field, namely, $\\tilde{\\rho }_H(B) = \\frac{\\rho _{yx}(B+\\delta B) - \\rho _{yx}(-B+\\delta B)}{2},$ where $\\delta B$ is a systematic error.",
"The error may be attributed to a small trapped flux of 10 mT in the superconducting magnet, or perhaps an offset in the magnet power supply.",
"Due to the misalignment of transport principal axis with the Hall bar geometry, longitudinal MR also contributes to $\\rho _{yx}(B)$ .",
"In the filling range with open Fermi surfaces, the longitudinal resistance exhibits non-saturating quadratic MR, and will mix into the Hall component which is odd in B.",
"As a result, one expects the improper antisymmetrization error to be largest in this filling range.",
"We investigate this possibility by first fitting a polynomial to the low-field transverse resistivity.",
"This allows us to interpolate the data and add small constant offsets prior to antisymmetrization.",
"Accounting for an offset of roughly 20 mT largely removes the bumps from the data.",
"This offset is larger than what we would expect from trapped flux in a superconducting magnet, however we do not expect the procedure to be accurate to such a fine degree, simply because we do not have fine enough resolution in field to get an accurate polynomial fit."
],
[
"More experimental measurements based on various contact pairs of the Hall bar geometry",
"The device has nine voltage probes on each side.",
"We observe quadratic magnetoresistance regions in roughly half of the device, between the fourth and eighth contacts.",
"We present longitudinal resistivities of these pairs in Fig.",
"REF .",
"In each of these pairs, we observe behavior qualitatively consistent with that presented in the main text.",
"Our Hall measurements (Fig.",
"REF ) are similarly consistent.",
"In Fig.",
"REF , we show quantum oscillations and their Fourier transforms for all three contact pairs for which we have dilution refridgerator data.",
"In all cases, we observe behavior consistent with what we present in the main text: 1) quantum oscillation onset at lower field close to CNP, 2) an irregular pattern of resistivity minima close to CNP, and 3) extra features in the FFT of the quantum oscillations that end at vH1.",
"The density of the first van Hove point is closer to the CNP in the other two contact pairs, and the extra features in the FFT are not as clear."
]
] | 2209.08204 |
[
[
"The trace reconstruction problem for spider graphs"
],
[
"Abstract We study the trace reconstruction problem for spider graphs.",
"Let $n$ be the number of nodes of a spider and $d$ be the length of each leg, and suppose that we are given independent traces of the spider from a deletion channel in which each non-root node is deleted with probability $q$.",
"This is a natural generalization of the string trace reconstruction problem in theoretical computer science, which corresponds to the special case where the spider has one leg.",
"In the regime where $d\\ge \\log_{1/q}(n)$, the problem can be reduced to the vanilla string trace reconstruction problem.",
"We thus study the more interesting regime $d\\le \\log_{1/q}(n)$, in which entire legs of the spider are deleted with non-negligible probability.",
"We describe an algorithm that reconstructs spiders with high probability using $\\exp\\left(\\mathcal{O}\\left(\\frac{(nq^d)^{1/3}}{d^{1/3}}(\\log n)^{2/3}\\right)\\right)$ traces.",
"Our algorithm works for all deletion probabilities $q\\in(0,1)$."
],
[
"Introduction",
"The string trace reconstruction problem, first introduced in 1997 by Levenshtein [17], is concerned with reconstructing an unknown seed string using only noisy samples of the data.",
"The unknown seed string is passed into some noisy channel multiple times, and the resulting error-prone copies are referred to as traces.",
"The goal is to use multiple traces to reconstruct the original seed string with high probability.",
"Levenshtein solved the trace reconstruction problem for a substitution channel, where each symbol of the seed string is mutated independently with constant probability.",
"In 2004, Batu, Kannan, Khanna, and McGregor [2] analyzed the problem for a deletion channel, where symbols of the seed string are each deleted independently with constant probability.",
"The string trace reconstruction problem has applications to computational biology, specifically in the new rapidly-evolving fields of DNA data storage and personalized immunogenics.",
"For example, one might want to reconstruct the correct sequence of nucleotides of a DNA sequence from several traces, each of which has many deletion mutations.",
"It is critical to minimize the number of traces required to reconstruct the seed string with high probability.",
"For example, in the application of DNA data storage, reducing the number of traces results in lower sequencing cost and time [3].",
"However, despite a wealth of recent work and attention on the deletion channel string trace reconstruction problem, for example [5], [10], [11], [12], [13], [14], [18], [20], [21], the current best upper and lower bounds for the number of traces necessary to reconstruct the seed string with high probability remain at $\\exp (\\mathcal {O}(n^{1/5}))$ [4] and $\\tilde{\\Omega }(n^{3/2})$ [5], [12], respectively, where $n$ is the length of the seed string.",
"We remark that a lower bound of $\\exp (\\mathcal {O}(n^{1/3}))$ traces was shown for mean-based algorithms, which are algorithms that only use the empirical means of individual bits in the traces for reconstruction [10], [20].",
"The exponential gap between upper and lower bounds for string trace reconstruction motivates studying variants of the problem for which one may be able to close the gap.",
"Many variants have been recently proposed and studied, for example [1], [6], [7], [9], [16], [19].",
"We focus on a variant known as the tree trace reconstruction problem introduced by Davies, Rácz, and Rashtchian [8].",
"This is a generalization of the vanilla string trace reconstruction problem where the goal is to learn a node-labeled tree, rather than a single string, using traces from a suitably-defined deletion channel.",
"The tree trace reconstruction problem may be directly applicable as well, as research on DNA nanotechnology has demonstrated that DNA molecule structures can be assembled into trees.",
"Recent research has also shown how to distinguish different molecular topologies, such as spiders with three arms from line DNA, using nanopores [15].",
"Davies et al.",
"[8] studied the tree trace reconstruction problem for two special classes of trees: complete $k$ -ary trees and spiders.",
"This paper extends their work on spiders.",
"An $(n,d)$ -spider consists of a single unlabeled root node with paths of $d$ labeled nodes attached to it.",
"In total, there are $n$ labeled nodes.",
"Consider a deletion channel, formally defined in deletion channel, in which every node is independently deleted with probability $q$ .",
"When $d\\ge \\log _{1/q}(n)$ , solving the spider trace reconstruction problem directly reduces to the string trace reconstruction problem [8].",
"This is because in this regime, the legs of the spider are long enough for all of the legs to survive the deletion channel with high probability, so each leg can be considered independently as its own string trace reconstruction problem.",
"Therefore, we assume that $d\\le \\log _{1/q}(n)$ .",
"In this more interesting regime, entire legs are deleted with non-negligible probability.",
"Hence, if one looks at a single trace, it is unclear which of the legs in the seed spider the legs in the trace come from.",
"Davies et al.",
"[8] proved that for deletion probabilities $q<0.7$ , there is some constant $C>0$ that depends only on $q$ such that $\\exp (C\\cdot d(nq^d)^{1/3})$ traces suffice to reconstruct an $(n,d)$ -spider with probability $1-\\mathcal {O}(1/n)$ (we refer to this as with high probability).",
"In this paper, we match this upper bound, up to polylogarithmic factors, but for the full range of deletion probabilities $q\\in (0,1)$ .",
"Furthermore, while Davies et al.",
"[8] used a single variable generating function alongside harmonic analysis, we consider a bivariate generating function, which results in considerably simpler analysis.",
"We use a best-match algorithm coupled with some results about bivariate Littlewood polynomials.",
"We remark that Littlewood polynomials have also been used to analyze a different variant of trace reconstruction known as the matrix reconstruction problem [16].",
"Our main result is the following theorem: Theorem 1.1 Assume that $d\\le \\log _{1/q}(n)$ .",
"For any fixed deletion probability $q<1$ , there exists some constant $C>0$ that depends only on $q$ such that $\\exp \\left(C\\cdot \\frac{(nq^d)^{1/3}}{d^{1/3}}(\\log n)^{2/3}\\right)$ traces suffice to reconstruct an $(n,d)$ -spider with high probability.",
"Note that the upper bound in main result matches the upper bound $\\exp (C\\cdot d(nq^d)^{1/3})$ in [8] up to polylogarithmic factors and works for all deletion probabilities $q\\in (0, 1)$ , not just $q<0.7$ .",
"Furthermore, main result strictly improves upon the upper bound $\\exp (C\\cdot d(nq^d)^{1/3})$ for all $q\\in (0,1)$ when $d = \\omega (\\sqrt{\\log n})$ ."
],
[
"Acknowledgements",
"The authors would like to thank Shyam Narayanan for suggesting the problem."
],
[
"Rooted spiders",
"In this section, we define the objects to be reconstructed: rooted binary-labeled spiders $X$ , as well as an indexing system for their nodes.",
"Definition 2.1 Let $n$ and $d$ be positive integers, and for convenience assume that $d\\mid n$ .",
"An $(n,d)$ -spider $X$ consists of a single unlabeled root node with $\\frac{n}{d}$ paths of $d$ nodes with binary labels from $\\lbrace 0,1\\rbrace $ emanating from it, so there are $n$ labeled nodes in total.",
"We refer to these paths as the legs of the spider.",
"Figure: A binary-labeled (12,4)(12,4) spider, with indexing system drawn in blue to the bottom left of each vertex.An example of a binary-labeled $(12,4)$ -spider is shown in fig:spider.",
"We index each node using two coordinates, where the first coordinate denotes the leg of the spider the node is on, and the second its depth down that leg.",
"This is in contrast to the depth-first-search labelling in [8].",
"In general, we may denote by $a_{i,j}\\in \\lbrace 0,1\\rbrace $ the label of the $(i,j)$ -node and the set of all labels of $X$ as $a=\\lbrace a_{i,j}\\rbrace _{0\\le i<\\frac{n}{d}, 0\\le j<d}$ .",
"For convenience, we define the set $S:=\\lbrace (i,j)\\mid 0\\le i<\\frac{n}{d},0\\le j<d\\rbrace $ , so we can write the labels of $X$ as $a=\\lbrace a_{i,j}\\rbrace _{(i,j)\\in S}$ ."
],
[
"Deletion channel for spiders",
"In the deletion channel for spiders, we start by independently selecting each non-root node for deletion with probability $q$ .",
"Note that we assume the root node is never deleted, as deleting the root node would disconnect the graph.",
"When nodes are deleted, all nodes below it shift upward.",
"If all the nodes in a leg are deleted, the entire leg disappears.",
"If a leg disappears, the remaining legs retain the same left-to-right structure, but it is no longer clear from looking at a trace which leg in the trace corresponds to which leg in the seed.",
"Remark 2.2 For trees that are not spiders, one must be more careful with describing the deletion channel.",
"Davies et al.",
"[8] studied two models, the Tree-Edit-Distance (TED) model and the Left-Propagation Model.",
"However, in the case of spiders, both models equivalent to the deletion channel described above.",
"For convenience in our analysis, after the deletion process we append nodes labeled 0 to the end of each shortened leg until they are of length $d$ again.",
"Also, if any complete legs were deleted, we add a leg of length $d$ with all nodes labeled 0 to the right of the remaining legs.",
"This pads the trace with 0's to form an $(n,d)$ -spider.",
"We refer to the resulting spider as a trace.",
"We remark that this padding process may cause two originally different traces to end up becoming identical.",
"An example of the deletion and padding process is shown in fig:deletion.",
"Figure: An example of the deletion channel applied on a (12,4)(12,4)-spider.",
"The deleted nodes are colored gray.",
"Then we pad nodes labeled 0, colored blue, to form a (12,4)(12,4) spider."
],
[
"Generating function for traces of spiders",
"Though the deletion channel for trees is more complicated than that for strings, it turns out that one can still describe the deletion process explicitly using generating functions.",
"These generating functions will then be used to distinguish between candidate spiders.",
"We begin by defining generating functions which encode the information of the possible traces of a spider: Definition 2.3 Let $a=\\lbrace a_{i,j}\\rbrace _{(i,j)\\in S}$ denote the labels of an $(n,d)$ -spider where $a_{i,j}\\in \\mathbb {R}$ , and let the random variable $b=\\lbrace b_{i^{\\prime },j^{\\prime }}\\rbrace _{(i^{\\prime },j^{\\prime })\\in S}$ denote the labels of its trace from the deletion channel with deletion probability $q$ .",
"Define a generating function $\\sum _{(i^{\\prime },j^{\\prime })\\in S}b_{i^{\\prime },j^{\\prime }}w_1^{i^{\\prime }}w_2^{j^{\\prime }}$ for each possible labeling $b=\\lbrace b_{i^{\\prime },j^{\\prime }}\\rbrace _{(i^{\\prime },j^{\\prime })\\in S}$ of a trace.",
"One of our key observations is that for each $(n,d)$ -spider, we can derive a closed-form formula for the expected value of the generating function of a trace: Lemma 2.4 Let $a=\\lbrace a_{i,j}\\rbrace _{(i,j)\\in S}$ denote the labels of an $(n,d)$ -spider where $a_{i,j}\\in \\mathbb {R}$ , and let the random variable $b=\\lbrace b_{i^{\\prime },j^{\\prime }}\\rbrace _{(i^{\\prime },j^{\\prime })\\in S}$ denote the labels of its trace from the deletion channel with deletion probability $q$ .",
"Define $A_a(w_1,w_2) :=\\mathbb {E}\\left[\\sum _{(i^{\\prime },j^{\\prime })\\in S}b_{i^{\\prime },j^{\\prime }}w_1^{i^{\\prime }}w_2^{j^{\\prime }}\\right]$ to be the expected value of the generating function of a trace, where the expectation is taken over the randomness of the deletion process.",
"Then $A_a(w_1,w_2) = (1-q)\\sum _{(i,j)\\in S}a_{i,j}(q^d+(1-q^d)w_1)^i (q+(1-q)w_2)^j$ for all $w_1,w_2\\in \\mathbb {C}$ .",
"Note that the coordinates of a specific node can only decrease after the deletion process.",
"We compute the probability that the label $b_{i^{\\prime },j^{\\prime }}$ comes from the label $a_{i,j}$ , where $i\\ge i^{\\prime }$ and $j\\ge j^{\\prime }$ .",
"This occurs when: $a_{i,j}$ is preserved, which occurs with probability $1-q$ , Exactly $i^{\\prime }$ of the first $i$ paths are retained, which occurs with probability $\\binom{i}{i^{\\prime }}(1-q^d)^{i^{\\prime }}q^{d(i-i^{\\prime })}.$ Exactly $j^{\\prime }$ of the first $j$ nodes in the path of the node with in $X$ with index $(i,j)$ are retained, which occurs with probability $\\binom{j}{j^{\\prime }}(1-q)^{j^{\\prime }}q^{j-j^{\\prime }}.$ Thus the probability that the label $b_{i^{\\prime },j^{\\prime }}$ comes from the label $a_{i,j}$ is $(1-q) \\binom{i}{i^{\\prime }}(1-q^d)^{i^{\\prime }}q^{d(i-i^{\\prime })} \\binom{j}{j^{\\prime }}(1-q)^{j^{\\prime }}q^{j-j^{\\prime }}.$ We conclude that $\\mathbb {E}&\\left[\\sum _{(i^{\\prime },j^{\\prime })\\in S}b_{i^{\\prime },j^{\\prime }}w_1^{i^{\\prime }}w_2^{j^{\\prime }}\\right]\\\\&=(1-q)\\sum _{(i^{\\prime },j^{\\prime })\\in S}w_1^{i^{\\prime }}w_2^{j^{\\prime }}\\sum _{(i,j)\\in S}a_{i,j}\\binom{i}{i^{\\prime }}(1-q^d)^{i^{\\prime }}q^{d(i-i^{\\prime })}\\binom{j}{j^{\\prime }}(1-q)^{j^{\\prime }}q^{j-j^{\\prime }}\\\\&=(1-q)\\sum _{i=0}^{\\frac{n}{d}-1}\\sum _{j=0}^{d-1}a_{i,j}\\sum _{i^{\\prime }=0}^{i}\\sum _{j^{\\prime }=0}^j\\binom{i}{i^{\\prime }}(1-q^d)^{i^{\\prime }}q^{d(i-i^{\\prime })}w_1^{i^{\\prime }}\\binom{j}{j^{\\prime }}(1-q)^{j^{\\prime }}q^{j-j^{\\prime }}w_2^{j^{\\prime }}\\\\&=(1-q)\\sum _{i=0}^{\\frac{n}{d}-1}\\sum _{j=0}^{d-1}a_{i,j}(q^d+(1-q^d)w_1)^{i}(q+(1-q)w_2)^{j}\\\\&=(1-q)\\sum _{(i,j)\\in S}a_{i,j}(q^d+(1-q^d)w_1)^{i}(q+(1-q)w_2)^{j},$ where we change the order of summation in the second equality and apply the binomial theorem in the third equality.",
"Figure: All possible padded traces for a certain seed (4,2)(4,2)-spider after being passed through the deletion channel, with their associated generating functions.Example 2.5 fig:expectedvalue depicts all $2^4 = 16$ possible deletions that could occur for a specific $(4,2)$ -spider with labels $a_{0,0}=1$ , $a_{1,0}=0$ , $a_{0,1}=1$ , and $a_{1,1}=1$ , shown on the right.",
"The figure also depicts the resulting padded traces and their associated generating functions.",
"Note that $A_a(w_1,w_2)$ , which recall is the expected value of the generating functions of the padded traces, is a weighted average of all the generating functions on the right.",
"For example, if $q=\\frac{1}{2}$ , we can simply average all the values in fig:expectedvalue to get $\\begin{split}A_a(w_1,w_2) &= \\mathbb {E}\\left[\\sum _{0\\le i^{\\prime }<2, 0\\le j^{\\prime }<2}b_{i^{\\prime },j^{\\prime }}w_1^{i^{\\prime }}w_2^{j^{\\prime }}\\right]\\\\&=\\frac{13}{16}+\\frac{3}{16}w_1+\\frac{5}{16}w_2+\\frac{3}{16}w_1w_2\\end{split}$ Note that l1-example equals what we expect from genfunc: $A_a(w_1,w_2) &= (1-q)\\sum _{(i,j)\\in S}a_{i,j}(q^d+(1-q^d)w_1)^i(q+(1-q)w_2)^j\\\\&=\\frac{1}{2}\\cdot \\sum _{0\\le i<2, 0\\le j<2}a_{i,j}\\left(\\frac{1}{4}+\\frac{3}{4}w_1\\right)^i\\left(\\frac{1}{2}+\\frac{1}{2}w_2\\right)^j\\\\&=\\frac{1}{2}\\left[1+\\left(\\frac{1}{2}+\\frac{1}{2}w_2\\right)+\\left(\\frac{1}{4}+\\frac{3}{4}w_1\\right)\\left(\\frac{1}{2}+\\frac{1}{2}w_2\\right)\\right]\\\\&=\\frac{13}{16}+\\frac{3}{16}w_1+\\frac{5}{16}w_2+\\frac{3}{16}w_1w_2.$"
],
[
"Proof of main result",
"In this section we prove main result.",
"Like in previous work on string trace reconstruction, we use a best-match algorithm to reconstruct the spider.",
"As is typical in best-match algorithms, we compare every pair $(X^{(1)}, X^{(2)})$ of candidate spiders to see which spider from the pair is more likely to have produced the observed traces.",
"We repeat this process for each pair of candidates and use the results to select a best possible guess for the original seed spider."
],
[
"Overview of the algorithm",
"We consider all $2^n$ possible candidate spiders $X$ and select a pair of spiders to compare against each other.",
"Suppose we select candidate spiders $X^{(1)}$ and $X^{(2)}$ with labels $a^{(1)}=\\lbrace a_{i,j}^{(1)}\\rbrace _{(i,j)\\in S}$ and $a^{(2)}=\\lbrace a_{i,j}^{(2)}\\rbrace _{(i,j)\\in S}$ , respectively.",
"Now, consider the element-wise difference $a=a^{(1)}-a^{(2)}$ , which is nonzero since $X^{(1)}$ and $X^{(2)}$ are distinct.",
"Let $Y^{(1)}$ and $Y^{(2)}$ denote the random traces with labels $b^{(1)}=\\lbrace b_{i^{\\prime },j^{\\prime }}^{(1)}\\rbrace _{(i^{\\prime },j^{\\prime })\\in S}$ and $b^{(2)}=\\lbrace b_{i^{\\prime },j^{\\prime }}^{(2)}\\rbrace _{(i^{\\prime },j^{\\prime })\\in S}$ , which result from passing $X^{(1)}$ and $X^{(2)}$ respectively through the deletion channel.",
"Now, we compute the difference of the generating functions corresponding to $X^{(1)}$ and $X^{(2)}$ , which is equivalent to plugging $a$ into the expression in genfunc: $\\begin{split}&\\sum _{(i^{\\prime },j^{\\prime })\\in S}(\\mathbb {E}[b_{i^{\\prime },j^{\\prime }}^{(1)}]-\\mathbb {E}[b_{i^{\\prime },j^{\\prime }}^{(2)}])\\cdot w_1^{i^{\\prime }}w_2^{j^{\\prime }}\\\\&=(1-q)\\sum _{(i,j)\\in S}a_{i,j}(q^d+(1-q^d)w_1)^i(q+(1-q)w_2)^j.\\end{split}$ Through a process described in completing proof, we can select some pair of indices $(I,J)\\in S$ , depending on $X^{(1)}$ and $X^{(2)}$ , such that $|\\mathbb {E}[b_{I,J}^{(1)}]-\\mathbb {E}[b_{I,J}^{(2)}]|$ is lower bounded substantially.",
"What this means is that the expected value of some label $b_{I,J}$ in the trace differs significantly depending on whether or not the seed spider was $X^{(1)}$ or $X^{(2)}$ .",
"We can use this information in combination with the empirical expected value $\\mathbb {E}[b_{I,J}]$ among our observed traces to select the better match between $X^{(1)}$ and $X^{(2)}$ , that is, which of $X^{(1)}$ or $X^{(2)}$ is more likely to have produced the empirical expected value $\\mathbb {E}[b_{I,J}]$ .",
"Such a process is known as a mean-based algorithm.",
"We repeat the above comparison for all pairs of spiders and then output the spider $X^*$ that loses against no other spiders, if such a spider exists.",
"If no such spider exists, we can output a uniformly random spider.",
"As the true seed spider is among the $2^n$ candidate spiders, we can use a Chernoff bound to upper bound the probability that it loses against any other candidate spider by $\\mathcal {O}\\left(\\frac{1}{n}\\right)$ .",
"Therefore, so long as we are given enough traces, the true seed spider is outputted by the algorithm with high probability."
],
[
"Littlewood polynomials",
"To analyze the expression in diffspiders, we use bivariate Littlewood polynomials from complex analysis.",
"We begin by defining these polynomials: Definition 3.1 A two-variable polynomial $A(z_1,z_2)$ is called a bivariate Littlewood polynomial if all of its coefficients are in the set $\\lbrace -1,0,1\\rbrace $ .",
"Note that in the right hand side of diffspiders, the coefficients satisfy $a_{i,j}=a_{i,j}^{(1)}-a_{i,j}^{(2)}\\in \\lbrace -1,0,1\\rbrace $ .",
"If we write the right hand side of diffspiders in terms of new variables $z_1=q^d+(1-q^d)w_1$ and $z_2=q+(1-q)w_2$ , then we get $(1-q)\\sum _{(i,j)\\in S}a_{i,j}z_1^iz_2^j,$ which is $(1-q)$ times a nonzero bivariate Littlewood polynomial with $(z_1)$ -degree less than $\\frac{n}{d}$ and $(z_2)$ -degree less than $d$ .",
"In order to lower bound this polynomial for some choice of $z_1$ and $z_2$ , we prove the following lemma concerning bivariate Littlewood polynomials: Lemma 3.2 Let $f(z_1,z_2)$ be a nonzero bivariate Littlewood polynomial with degree $a$ in $z_1$ and degree $b$ in $z_2$ .",
"Then $|f(z_1^*,z_2^*)|\\ge \\exp \\left(-cL_1L_2\\log (ab)\\right)$ for some $z_1^*=\\exp (i\\theta _1)$ and $z_2^*=\\exp (i\\theta _2)$ , where $\\theta _1$ and $\\theta _2$ lie in the ranges $|\\theta _1|\\le \\frac{\\pi }{L_1}$ and $|\\theta _2|\\le \\frac{\\pi }{L_2}$ .",
"Define the 2-variable polynomial $F(z_1,z_2)=\\prod _{\\begin{array}{c}1\\le x\\le L_1 \\\\ 1\\le y\\le L_2\\end{array}}f\\left(z_1e^{2\\pi ix/L_1},z_2e^{2\\pi iy/L_2}\\right).$ Using the maximum modulus principle, which recall says that the modulus $|F|$ of any holomorphic function $F$ achieves its maximum value at the boundary of its domain, we first show that we can find some $z_1^{\\prime }$ and $z_2^{\\prime }$ on the unit circle such that $|F(z_1^{\\prime },z_2^{\\prime })|\\ge 1$ .",
"Note that restricting the domain of a holomorphic function to the unit disk leaves the function holomorphic.",
"Factor $F(z_1,z_2)=z_2^k\\cdot G(z_1,z_2)$ so that $G(z_1,z_2)$ no common factors with $z_2$ .",
"Since $F$ has nonzero coefficients, $G(z_1,0)$ can be viewed as a nonzero polynomial in one variable $z_1$ .",
"We can now factor $G(z_1,0)=z_1^\\ell \\cdot H(z_1)$ so that $H(z_1)$ is nonzero and hence satisfies $|H(0)|= 1$ .",
"By the maximum modulus principle, we can find some $z_1^{\\prime }$ on the unit circle such that $|H(z_1^{\\prime })|\\ge |H(0)|= 1$ .",
"We can apply the maximum modulus principle again to find some $z_2^{\\prime }$ on the unit circle such that $|G(z_1^{\\prime },z_2^{\\prime })|\\ge |G(z_1^{\\prime },0)|$ .",
"Therefore, we can find $z_1^{\\prime }$ and $z_2^{\\prime }$ such that $|F(z_1^{\\prime },z_2^{\\prime })|=|G(z_1^{\\prime },z_2^{\\prime })|\\ge |G(z_1^{\\prime },0)|=|H(z_1^{\\prime })|\\ge |H(0)|= 1.$ Now, applying the definition of $F$ gives $1\\le |F(z_1^{\\prime },z_2^{\\prime })|\\le |f(z_1^{\\prime }e^{2\\pi ix/L_1},z_2^{\\prime }e^{2\\pi iy/L_2})|\\cdot (ab)^{L_1L_2-1}$ for all $1\\le x\\le L_1$ and $1\\le y\\le L_2$ , where we use the fact that $|f(z_1,z_2)|\\le ab$ for $|z_1|=|z_2|=1$ .",
"We can now choose appropriate $x$ and $y$ to rotate $z_1^{\\prime }$ and $z_2^{\\prime }$ along the unit circle in the complex plane so that $z_1^*=z_1^{\\prime }\\cdot e^{2\\pi ix/L_1}=\\exp (i\\theta _1)$ and $z_2^*=z_2^{\\prime }\\cdot e^{2\\pi iy/L_2}=\\exp (i\\theta _2)$ satisfy $|\\theta _1|\\le \\frac{\\pi }{L_1}$ and $|\\theta _2|\\le \\frac{\\pi }{L_2}$ .",
"We conclude that $|f(z_1^*, z_2^*)| \\ge \\frac{1}{(ab)^{L_1 L_2 - 1}} \\ge \\exp (-L_1 L_2 \\log (ab)),$ where $z_1^*=\\exp (i\\theta _1)$ and $z_2^*=\\exp (i\\theta _2)$ satisfy $|\\theta _1|\\le \\frac{\\pi }{L_1}$ and $|\\theta _2|\\le \\frac{\\pi }{L_2}$ .",
"We remark that littlewood is a generalization of [16]."
],
[
"Completing the proof",
"We set the parameters in littlewood to be $L_1=L$ for some constant $L$ to be chosen later, $L_2=1$ , $a \\le \\frac{n}{d}$ , and $b \\le d$ .",
"By littlewood and the triangle inequality, we can lower bound diffspiders as $\\sum _{(i^{\\prime },j^{\\prime })\\in S}|\\mathbb {E}[b_{i^{\\prime },j^{\\prime }}^{(1)}]-\\mathbb {E}[b_{i^{\\prime },j^{\\prime }}^{(2)}]||w_1^*|^{i^{\\prime }}|w_2^*|^{j^{\\prime }}\\ge (1-q)\\exp \\left(-L\\log n\\right)$ for some $z_1^*=\\exp (i\\theta _1)$ and $z_2^*=\\exp (i\\theta _2)$ such that $|\\theta _1|\\le \\pi /L$ and $|\\theta _2|\\le \\pi $ .",
"Recall the change of variables $w_1^*=\\frac{z_1^*-q^d}{1-q^d}\\qquad \\text{and} \\qquad w_2^*=\\frac{z_2^*-q}{1-q}.$ We can upper bound $|w_1^*|$ as $|w_1^*| &=\\frac{|z_1^*-q^d|}{1-q^d}\\\\&\\le \\frac{\\sqrt{\\left(\\cos \\frac{\\pi }{L}-q^d\\right)^2+\\left(\\sin \\frac{\\pi }{L}\\right)^2}}{1-q^d}\\\\&=\\frac{\\sqrt{1-2q^d\\cos \\frac{\\pi }{L}+q^{2d}}}{1-q^d}\\\\&=\\frac{\\sqrt{(1-q^d)^2+2q^d\\left(1-\\cos \\frac{\\pi }{L}\\right)}}{1-q^d}\\\\&=\\left(1+\\frac{2q^d\\left(1-\\cos \\frac{\\pi }{L}\\right)}{(1-q^d)^2}\\right)^{1/2}\\\\&\\le \\exp \\left(\\frac{q^d\\pi ^2}{2L^2(1-q^d)^2}\\right),$ where we use the inequalities $(1+x)^r\\le e^{rx}$ for $r,x\\ge 0$ and $1-\\cos \\frac{\\pi }{L}\\le \\frac{1}{2}\\left(\\frac{\\pi }{L}\\right)^2$ .",
"Therefore, $|w_1^*|^{\\frac{n}{d}}\\le \\exp \\left(\\frac{n}{d}\\cdot \\frac{Cq^d}{L^2(1-q^d)^2}\\right)$ for some constant $C$ .",
"We can also upper bound $|w_2^*|$ as $|w_2^*|=\\frac{|z_2^*-q|}{1-q}\\le \\frac{1+q}{1-q},$ so $|w_2^*|^d\\le \\exp (C^{\\prime }d)$ for some constant $C^{\\prime }$ depending on $q$ .",
"Therefore, by 2 and the fact that $|w_1^*|, |w_2^*| \\ge 1$ , we have $\\exp \\left(\\frac{n}{d}\\cdot \\frac{Cq^d}{L^2(1-q^d)^2}+C^{\\prime }d\\right)\\sum _{(i^{\\prime },j^{\\prime })\\in S}|\\mathbb {E}[b_{i^{\\prime },j^{\\prime }}^{(1)}]-\\mathbb {E}[b_{i^{\\prime },j^{\\prime }}^{(2)}]|\\ge (1-q)\\exp (-L\\log n).$ Thus there exists some pair of indices $(I,J)\\in S$ such that $|\\mathbb {E}[b_{I,J}^{(1)}]-\\mathbb {E}[b_{I,J}^{(2)}]|\\ge \\frac{1-q}{n}\\exp \\left(-\\frac{n}{d}\\cdot \\frac{Cq^d}{L^2(1-q^d)}-C^{\\prime }d-L\\log n\\right) =: \\eta .$ Denote the right hand side of eta by $\\eta $ .",
"Returning now to the best-match algorithm, given two candidate spiders $X^{(1)}$ and $X^{(2)}$ , we define the better match to be $X^{(1)}$ if $\\left|\\frac{1}{T}\\sum _{t=1}^Ts_{I,J}^t-\\mathbb {E}[b_{I,J}^{(1)}]\\right|\\le \\left|\\frac{1}{T}\\sum _{t=1}^Ts_{I,J}^t-\\mathbb {E}[b_{I,J}^{(2)}]\\right|,$ where $s_{I,J}^t\\in \\lbrace 0, 1\\rbrace $ is the value of the node at position $(I, J)$ of the $t$ -th trace.",
"Now, suppose $X^{(1)}=X^*$ is the true seed spider.",
"For all possible seed spiders $X^{(2)}$ , we can use a Chernoff bound to upper bound the failure probability, namely the probability that $X^{(2)}$ is a better match than $X^{(1)}$ , by $\\exp (-T\\eta ^2/2)$ , where $T$ is the total number of traces.",
"Therefore, by a union bound, the probability that $X^*$ loses to at least one other spider is at most $\\mathbb {P}[X^*\\text{ not chosen by algorithm}]&\\le \\sum _{X^{(2)}\\ne X^*}\\mathbb {P}[X^{(2)} \\text{ better match than }X^*]\\\\&\\le 2^n\\cdot \\exp (-T\\eta ^2/2)\\\\&\\le \\exp \\left(n\\log 2-\\frac{T\\eta ^2}{2}\\right).$ For this expression to be at most $\\frac{1}{n}=\\exp (-\\log n)$ , we set $T = \\frac{2}{\\eta ^2}(n\\log 2+\\log n) = \\Theta (\\eta ^{-2} n).$ Plugging in the definition of $\\eta $ from eta yields $T = \\Theta \\left(n^3\\cdot \\exp \\left(\\frac{n}{d}\\cdot \\frac{Cq^d}{L^2(1-q^d)}+C^{\\prime }d+cL\\log n\\right)\\right).$ Note that the $n^3$ term is negligible.",
"The $C^{\\prime }d$ term is also negligible since we are in the regime $d\\le \\log _{1/q}(n)$ .",
"Finally, $1-q^d\\ge 1-q$ depends only on $q$ , so T can be simplified to $T= \\exp \\left(\\Theta \\left(\\frac{nq^d}{dL^2}+L\\log n\\right)\\right).$ To balance these terms, we set $L=\\left(\\frac{nq^d}{d\\log n}\\right)^{1/3}$ to get a final bound of $T = \\exp \\left(C\\cdot \\frac{(nq^d)^{1/3}}{d^{1/3}}(\\log n)^{2/3}\\right),$ where $C$ is a constant that depends only on $q$ .",
"We conclude the proof of main result."
],
[
"Conclusion",
"We presented a mean-based algorithm using Littlewood polynomials that reconstructs $(n,d)$ -spiders with high probability in the regime $d \\le \\log _{1/q}(n)$ , where $q$ is the deletion probability.",
"Our algorithm uses $\\exp \\left(\\mathcal {O}\\left( \\frac{(nq^d)^{1/3}}{d^{1/3}}(\\log n)^{2/3}\\right)\\right)$ traces and works for the full range $q\\in (0,1)$ of deletion probabilities.",
"In light of recent work improving the string trace reconstruction upper bound to $\\exp (\\tilde{\\mathcal {O}}(n^{1/5}))$ using a non-mean-based algorithm [4], it would be interesting to see whether a similar technique could achieve an upper bound of the form $\\exp \\left(\\tilde{\\mathcal {O}}((nq^d)^{1/5})\\right)$ for the spider trace reconstruction problem."
]
] | 2209.08166 |
[
[
"Sobolev orthogonal polynomials on the conic surface"
],
[
"Abstract Orthogonal polynomials with respect to the weight function $w_{\\beta,\\gamma}(t) = t^\\beta (1-t)^\\gamma$, $\\gamma > -1$, on the conic surface $\\{(x,t): \\|x\\| = t, \\, x \\in \\mathbb{R}^d, \\, t \\le 1\\}$ are studied recently, and they are shown to be eigenfunctions of a second order differential operator $\\mathcal{D}_\\gamma$ when $\\beta =-1$.",
"We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th partial derivatives in $t$ variable with respect to $w_{\\beta+s,0}$ over the conic surface plus a sum of integrals over the rim of the cone.",
"Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series.",
"The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\\beta, -s}$ for a positive integer $s$.",
"It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\\mathcal{D}_{\\gamma}$ when $\\gamma = -1$."
],
[
"Introduction",
"Orthogonal polynomials on the conic surface of the revolution were studied recently, which are shown to possess properties parallel to those of spherical harmonics on the unit sphere.",
"Let ${\\mathbb {V}}_0^{d+1}$ be the conic surface ${\\mathbb {V}}_0^{d+1} = \\lbrace (x,t) \\in {\\mathbb {R}}^{d+1}: \\, \\Vert x\\Vert = t, \\, x \\in {\\mathbb {R}}^d, \\, 0 \\le t \\le 1\\rbrace $ in ${\\mathbb {R}}^{d+1}$ .",
"For the weight function ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t) = t^{\\beta }(1-t)^{\\gamma }$ , ${\\beta }> -d$ and ${\\gamma }> -1$ , the orthogonal polynomials with respect to the inner product ${\\langle }f,g{\\rangle }_{{\\beta },{\\gamma }} = b_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\mathsf {w}}_{{\\beta },{\\gamma }}(t) \\mathrm {d}{\\mathsf {m}}(x,t)$ are called the Jacobi polynomials on the cone.",
"These polynomials are studied in [12], [21], [22], [23].",
"It was shown in [21] that these polynomials share many properties of spherical harmonics, including explicit orthogonal basis and an addition formula, which provides essential tools for an extensive study in approximation theory and computational analysis over the cone in [23].",
"Another remarkable property of the Jacobi polynomials on the cone is that they are eigenfunctions of a second-order linear differential operator ${\\mathcal {D}}_{\\gamma }$ when ${\\beta }= -1$ , which is an analog of the Laplace-Beltrami operator on the unit sphere.",
"The purpose of the present paper is to study Sobolev orthogonal polynomials on the conic surface, which are orthogonal with respect to an inner product that contains derivatives.",
"The first case is ${\\langle }f,g{\\rangle }_{{\\beta }, -1} = \\frac{1}{{\\omega }_d}\\int _{{\\mathbb {V}}_0^{d+1}} \\frac{\\partial }{\\partial t} f (x,t)\\frac{\\partial }{\\partial t} g(x,t) t^{{\\beta }+1} \\mathrm {d}{\\mathsf {m}}(x,t)+ \\frac{{\\lambda }}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}}f(\\xi ,1) g(\\xi ,1) \\mathrm {d}\\sigma (\\xi ),$ and we also consider ${\\langle }f,g{\\rangle }_{{\\beta }, -s}$ that involves derivatives up to $s$ order for a positive integer $s$ .",
"There is a reason that we consider only derivatives in the $t$ variable but not the $x$ variable; see the discussion in Section 4.",
"Like in the case of ${\\gamma }> -1$ , our main result provides explicit construction of orthogonal bases and a closed-form formula for the orthogonal projection operator.",
"The study requires an extension of the Jacobi polynomials with a parameter being a negative integer, which needs to satisfy the Sobolev orthogonality of one variable that is inherited from ${\\langle }\\cdot ,\\cdot {\\rangle }_{{\\beta },-s}$ when we restrict the inner product to polynomials depending only on the $t$ variable.",
"Such Sobolev orthogonal polynomials of one variable have been studied by several authors; see, for example, [1], [2], [7], [15], [16], [20] and [11].",
"We shall follow the approach in [20] since it is more convenient for studying orthogonal projection operators and provides a link, in particular, between the Sobolev orthogonal structure and the ordinary orthogonal structure, which is useful for studying the convergence of the Fourier orthogonal series in the Sobolev orthogonal polynomials.",
"In the framework of polynomial approximation theory on the ball and standard or Sobolev orthogonal polynomials, we can refer to [3], [4], [5], [8], [9], [10], [14], [18], among others.",
"In more than one way, our study extends the Jacobi polynomials for ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ on the cone from ${\\gamma }> -1$ to ${\\gamma }= -s$ with $s \\in {\\mathbb {N}}$ .",
"We will show, in particular, that the spectral operator ${\\mathcal {D}}_{-s}$ has the Sobolev orthogonal polynomials as eigenfunctions if $s = 1$ .",
"While the latter fails for $s > 1$ , we do have a clear understanding of what the eigenspaces of ${\\mathcal {D}}_{-s}$ are.",
"For orthogonal polynomials in several variables, our study is also closely related to the Sobolev orthogonal polynomials on the unit ball, which have been extensively studied (see [6] and its references therein).",
"In particular, the description of the eigenspaces of ${\\mathcal {D}}_{\\gamma }$ is similar to the study on the unit ball in [13].",
"The paper is organized as follows.",
"The next section is preliminary, in which we recall two essential ingredients needed for our study, the Jacobi polynomials with negative parameters and spherical harmonics.",
"In Section 3 we review results on ordinary orthogonal polynomials on the conic surface and discuss further properties of the orthogonal projection operators.",
"The Sobolev orthogonal polynomials are defined and studied in Section 4.",
"Finally, the eigenspaces of the operator ${\\mathcal {D}}_{\\gamma }$ are discussed in Section 5."
],
[
"Preliminary",
"The study for orthogonal polynomials on the conic surface follows that of spherical harmonics on the unit sphere.",
"The latter will also be essential for constructing an orthogonal basis on the cone.",
"Another ingredient is the Jacobi polynomial, which we often need an extension to negative parameters in the study of the Sobolev orthogonal polynomials."
],
[
"Jacobi polynomials with negative paramters",
"The Jacobi polynomials $P_n^{({\\alpha },{\\beta })}$ are given explicitly by the hypergeometric function $P_n^{({\\alpha },{\\beta })}(t) = \\frac{({\\alpha }+1)_n}{n!}",
"{}_2F_1 \\left( \\begin{matrix}-n, n+{\\alpha }+{\\beta }+1 \\\\ {\\alpha }+1\\end{matrix}; \\frac{1-x}{2}\\right)$ for ${\\alpha },{\\beta }> -1$ and $n = 0, 1, 2, \\ldots $ .",
"They are orthogonal with respect to the weight function $w_{{\\alpha },{\\beta }}(t) = (1-t)^{\\alpha }(1+t)^{\\beta }$ on $[-1,1]$ with ${\\alpha }, {\\beta }> -1$ , and they satisfy $\\frac{c_{{\\alpha },{\\beta }}}{2^{{\\alpha }+{\\beta }+1}}\\int _{-1}^1 P_n^{({\\alpha },{\\beta })}(t) P_n^{({\\alpha },{\\beta })}(t) w_{{\\alpha },{\\beta }}(t) \\mathrm {d}t = h_n^{({\\alpha },{\\beta })} \\delta _{n,m},$ where $c_{{\\alpha },{\\beta }}$ is the constant so that $h_0^{({\\alpha },{\\beta })} =1$ , $c_{{\\alpha },{\\beta }} = \\frac{\\Gamma ({\\alpha }+{\\beta }+2)}{\\Gamma ({\\alpha }+1)\\Gamma ({\\beta }+1)} \\quad \\hbox{and}\\quad h_n^{({\\alpha },{\\beta })} = \\frac{({\\alpha }+1)_n({\\beta }+1)_n ({\\alpha }+{\\beta }+n+1)}{n!",
"({\\alpha }+{\\beta }+2)_n ({\\alpha }+{\\beta }+2n+1)}.$ For studying the Sobolev orthogonal polynomials, we often need the parameters ${\\alpha }$ or ${\\beta }$ to be negative integers.",
"Such polynomials are discussed already in [17] but they are no longer orthogonal with respect to $w_{{\\alpha },{\\beta }}$ , since the weight function $(1-t)^a(1+t)^{b}$ is no longer integrable if $a$ or $b \\le -1$ .",
"Moreover, $P_n^{({\\alpha },{\\beta })}$ has a degree reduction if $n+{\\alpha }+{\\beta }$ is a negative integer between 1 to $n$ , which causes problems for studying the Sobolev orthogonal polynomials, especially in several variables, since such polynomials are needed for all $n \\in {\\mathbb {N}}_0$ .",
"What we need in this paper are the polynomials $P_n^{({\\alpha },-s)}$ with ${\\alpha }> -1$ and $s \\in {\\mathbb {N}}$ .",
"These polynomials are well defined if $n \\ge s$ and satisfy [17] $P_n^{({\\alpha },-s)} (t) = \\frac{(-{\\alpha }-n)_s}{2^s (-n)_s} (1+t)^s P_{n-s}^{({\\alpha },s)}(t), \\quad n \\ge s,$ which follows from [17] by using $P_n^{({\\alpha },{\\beta })}(-t) = (-1)^n P_n^{({\\beta },{\\alpha })}(t)$ .",
"The definition of $P_n^{({\\alpha },-s)}$ for $n < s$ could be problematic because of the degree reduction.",
"Such polynomials have been studied in the setting of the Sobolev orthogonal polynomials; for example, the Sobolev orthogonality defined via the inner product $ [f,g]_{{\\alpha },{\\beta }}^{-s} : = \\int _{-1}^1 f^{(s)}(t) g^{(s)}(t) w_{{\\alpha },{\\beta }}(t) \\mathrm {d}t + \\sum _{k=0}^{s-1} \\mu _k f^{(k)}(1)g^{(k)}(1),$ where $\\mu _k$ are fixed positive constants and ${\\alpha },{\\beta }> -1$ .",
"The study of such polynomials and their orthogonality has appeared in several papers; see, for example, [1], [7], [2], [15], [16], [19], [20] and [11].",
"There are several ways to define a complete set of orthogonal polynomials for the inner product (REF ).",
"We shall follow the approach given in [20], see also [15], [16], which is more suitable for studying the Fourier orthogonal series.",
"We now recall the necessary result from [20].",
"For convenience, we first define a renormalization of the Jacobi polynomials, $ \\widehat{P}^{(\\alpha ,\\beta )}_{n}(t) = A_n^{({\\alpha },{\\beta })} {P}^{(\\alpha ,\\beta )}_{n}(t) \\quad \\hbox{with}\\quad A_n^{({\\alpha },{\\beta })} = \\frac{2^n}{(n+{\\alpha }+\\beta +1)_{n}}$ for ${\\alpha },{\\beta }> -1$ .",
"This normalization has the advantage that it satisfies, by [17], $ \\frac{d}{dt} \\widehat{P}_n^{({\\alpha },{\\beta })}(t) = \\widehat{P}_{n-1}^{({\\alpha }+1,{\\beta }+1)}(t).$ Now, for ${\\alpha },{\\beta }> -1$ , $s \\in {\\mathbb {N}}$ and $n = 0,1,2,\\ldots ,$ we define a new sequence of polynomials $ \\begin{split}J_n^{({\\alpha }-s,{\\beta }-s)}(t) := {\\left\\lbrace \\begin{array}{ll} \\dfrac{(t+1)^n}{n!",
"}, & 0 \\le n \\le s-1, \\vspace{3.61371pt} \\\\\\displaystyle {\\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}",
"\\widehat{P}_{n-s}^{({\\alpha },{\\beta })}(u) \\mathrm {d}u}, & n \\ge s.\\end{array}\\right.",
"}\\end{split}$ It is easy to see that $J_n^{({\\alpha }-s,{\\beta }-s)}$ is a polynomial of degree $n$ and it satisfies $\\partial ^s J_n^{({\\alpha }-s,{\\beta }-s)}(t) &\\ = \\widehat{P}_{n-s}^{({\\alpha },{\\beta })}(t), \\qquad n \\ge s; \\\\\\partial ^k J_n^{({\\alpha }-s,{\\beta }-s)}(-1) &\\ = {\\left\\lbrace \\begin{array}{ll} \\delta _{k,n}, & n \\le s-1, \\\\0, & n \\ge s,\\end{array}\\right.}",
"\\qquad 0\\le k \\le s-1, $ where $\\partial ^k$ denotes the $k$ -th derivative.",
"These are our polynomials that extend the definition of the Jacobi polynomials to allow negative parameters, which are also orthogonal polynomials with respect to the Sobolev inner product (REF ).",
"More precisely, we have the following [20]: Theorem 2.1 For ${\\alpha },{\\beta }> -1$ and $s \\in {\\mathbb {N}}$ .",
"The polynomial $J_n^{({\\alpha }-s,{\\beta }-s)}$ is orthogonal with respect to the inner product ${\\langle }\\cdot , \\cdot {\\rangle }_{{\\alpha },{\\beta }}^{-s}$ and its norm square is given by $\\left[J_n^{({\\alpha }-s,{\\beta }-s)},J_n^{({\\alpha }-s,{\\beta }-s)}\\right]_{{\\alpha },{\\beta }}^{-s}= {\\left\\lbrace \\begin{array}{ll}\\mu _n & 0 \\le n \\le s-1 \\\\ \\widehat{h}_{n-s}^{({\\alpha },{\\beta })} & n \\ge s \\end{array}\\right.",
"},$ where $\\mu _n$ comes from (REF ), and $\\widehat{h}_n^{({\\alpha },{\\beta })}$ is the norm square of $\\widehat{P}_n^{({\\alpha },{\\beta })}$ , which is given in terms of $h_n^{({\\alpha },{\\beta })}$ by $\\widehat{h}_n^{({\\alpha },{\\beta })} = \\frac{2^{{\\alpha }+{\\beta }+1}}{c_{{\\alpha },{\\beta }}} \\left[A_n^{({\\alpha },{\\beta })}\\right]^2 h_n^{({\\alpha },{\\beta })}.$ Our next proposition shows that, if ${\\alpha }> -1$ and $s\\in {\\mathbb {N}}$ , then the definition $J_n^{({\\alpha },-s)}$ in (REF ) agrees with that of (REF ) when $n\\ge s$ .",
"Proposition 2.2 For ${\\alpha }> -1$ and $s \\in {\\mathbb {N}}$ , $J_n^{({\\alpha },-s)} (t) = \\frac{(n-s)!}{n!}",
"(1+t)^s \\widehat{P}_{n-s}^{({\\alpha },s)}(t), \\qquad n \\ge s.$ In particular, for $n \\ge s$ , $J_n^{({\\alpha },-s)} (t) = \\frac{(-1)^s 2^s}{(-{\\alpha }-n)_s}A_{n-s}^{({\\alpha },s)} P_n^{({\\alpha },-s)}(t).$ We use the hypergeometric expression of the Jacobi polynomials, $P_n^{({\\alpha },{\\beta })}(t) = \\frac{({\\alpha }+1)_n}{n!}",
"{}_2F_1\\left(\\begin{matrix} -n \\,\\,\\, n+{\\alpha }+{\\beta }+1\\\\ {\\alpha }+1 \\end{matrix}; \\frac{1-t}{2} \\right)$ and the fact that $P_n^{({\\alpha },{\\beta })}(t) = (-1)^n P_n^{({\\beta },{\\alpha })}(-t)$ .",
"It follows that $& \\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}",
"P_n^{({\\alpha }+s,0)}(u) \\mathrm {d}u= (-1)^n \\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}",
"P_n^{(0, {\\alpha }+s)}(-u) \\mathrm {d}u \\\\& \\qquad \\quad = (-1)^n \\sum _{k=0}^n \\frac{(-n)_k (n+{\\alpha }+s+1)_k}{k!",
"k!",
"}\\int _{-1}^t \\frac{(t-u)^{s-1}}{(s-1)!}",
"\\left(\\frac{1+u}{2} \\right)^k \\mathrm {d}u \\\\& \\qquad \\quad = \\frac{(-1)^n}{s!}",
"(1+t)^s \\sum _{k=0}^n \\frac{(-n)_k (n+{\\alpha }+s+1)_k}{k!",
"(s+1)_k} \\left(\\frac{1+t}{2} \\right)^k \\\\& \\qquad \\quad = \\frac{(-1)^n}{s!}",
"(1+t)^s \\frac{n!",
"}{(s+1)_n} P_n^{(s,{\\alpha })}(-t)= \\frac{n!}{(n+s)!}",
"(1+t)^sP_n^{({\\alpha },s)}(t).$ Replacing $n$ by $n-s$ and using $A_n^{({\\alpha },s)} = A_n^{({\\alpha }+s,0)}$ , the identity (REF ) then follows from (REF ).",
"The second identity follows from the identity (REF ).",
"It should be pointed out that the integral expression of $J_n^{({\\alpha },-s)}$ for $n \\ge s$ in (REF ) is more convenient for studying the Fourier orthogonal series, as shown in [20] and as we shall see in Section 4 below."
],
[
"Spherical harmonics",
"A homogeneous polynomial $Y$ of $d$ variables is called a solid harmonic if $\\Delta Y =0$ , where $\\Delta $ is the Laplace operator on ${\\mathbb {R}}^d$ .",
"We denote by ${\\mathcal {H}}_m^{d,0}$ the space of homogeneous solid harmonics of degree $m$ in $d$ variables.",
"Thus, if $Y \\in {\\mathcal {H}}_m^{d,0}$ , then $Y(r \\xi ) = r^m Y(\\xi )$ for $\\xi \\in {\\mathbb {S}^{d-1}}$ .",
"Spherical harmonics are restrictions of solid harmonics on the unit ball.",
"We denote the space of spherical harmonics of degree $m$ by ${\\mathcal {H}}_m^d$ .",
"Thus, ${\\mathcal {H}}_m^d = {\\mathcal {H}}_m^{d,0} \\vert _{\\mathbb {S}^{d-1}}$ .",
"It is a common practice to identify ${\\mathcal {H}}_m^{d,0}$ and ${\\mathcal {H}}_m^d$ , we distinguish them to emphasize the dependence on variables for the orthogonal polynomials on the conic surface.",
"It is well known that $\\dim {\\mathcal {H}}_m^d = \\binom{m+d-2}{n}+\\binom{m+d-3}{n-1},\\quad m=1,2,3,\\ldots ,$ and spherical harmonics of different degrees are orthogonal with respect to the surface measure on the unit sphere.",
"Throughout the paper, we denote by $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d\\rbrace $ an orthonormal basis of ${\\mathcal {H}}_m^d$ , so that $\\frac{1}{{\\omega }_d} \\int _{\\mathbb {S}^{d-1}}Y_\\ell ^n(\\xi ) Y_{\\ell ^{\\prime }}^m(\\xi ) \\mathrm {d}\\sigma (\\xi ) = \\delta _{\\ell ,\\ell ^{\\prime }} \\delta _{n,m},$ where $\\mathrm {d}\\sigma $ is the surface measure of ${\\mathbb {S}^{d-1}}$ and ${\\omega }_d$ is the surface area of ${\\mathbb {S}^{d-1}}$ .",
"Let $\\operatorname{proj}_n^{\\mathbb {S}^{d-1}}: L^2({\\mathbb {S}^{d-1}}) \\rightarrow {\\mathcal {H}}_n^d$ be the orthogonal projection operator from $L^2({\\mathbb {S}^{d-1}})$ onto ${\\mathcal {H}}_n^d$ .",
"If $\\lbrace Y_\\ell ^n: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n^{d-1} \\rbrace $ is an orthonormal basis of ${\\mathcal {H}}_n^d$ , then $\\operatorname{proj}_n^{\\mathbb {S}^{d-1}}f (x) = \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_n^d} \\hat{f}_{\\ell }^n Y_\\ell ^n, \\quad \\hat{f}_\\ell ^n = \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} f(\\xi ) Y_\\ell ^n(\\xi ) \\mathrm {d}\\sigma (\\xi ).$ For $f \\in L^2({\\mathbb {S}^{d-1}})$ , its Fourier expansion in spherical harmonics is defined by $f = \\sum _{n=0}^\\infty \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_n^d} \\hat{f}_{\\ell }^n Y_\\ell ^n =\\sum _{n=0}^\\infty \\operatorname{proj}_n^{\\mathbb {S}^{d-1}}f.$ Let $\\Delta _0$ be the Laplace-Beltrami operator, which is the restriction of the Laplacian $\\Delta $ on the unit sphere.",
"Under the spherical polar coordinates $x = r \\xi $ , $r > 0$ , $\\xi \\in {\\mathbb {S}^{d-1}}$ , $\\Delta = \\frac{\\mathrm {d}^2}{\\mathrm {d}r^2} + \\frac{d-1}{r} \\frac{\\mathrm {d}}{\\mathrm {d}r} + \\frac{1}{r^2} \\Delta _0.$ The spherical harmonics are the eigenfunctions of $\\Delta _0$ .",
"More precisely, $\\Delta _0 Y = - n (n+d-2)Y, \\qquad Y \\in {\\mathcal {H}}_n^d, \\quad n=0,1,2,\\ldots .$ The $\\Delta _0$ is a second-order differential operator on the unit sphere.",
"One can also consider spherical gradient $\\nabla _0$ , the first-order differential operators, on the sphere, which is defined by $\\nabla = \\frac{1}{r} \\nabla _0 + \\xi \\frac{\\mathrm {d}}{\\mathrm {d}r}, \\quad x = r \\xi , \\quad \\xi \\in {\\mathbb {S}^{d-1}}.$ The integration by parts formula holds and gives [5], $\\int _{{\\mathbb {S}^{d-1}}} \\nabla _0 f(\\xi ) \\cdot \\nabla _0 g(\\xi ) \\mathrm {d}\\sigma (\\xi ) = - \\int _{{\\mathbb {S}^{d-1}}} \\Delta _0 f(\\xi ) g(\\xi ) \\mathrm {d}\\sigma (\\xi ).$ Together with (REF ), this identity implies that if $\\lbrace Y_\\ell ^n : 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n^d\\rbrace $ is an orthogonal basis of ${\\mathcal {H}}_n^d$ , then $\\int _{{\\mathbb {S}^{d-1}}} \\nabla _0 Y_\\ell ^n(\\xi ) \\cdot \\nabla _0 Y_{\\ell ^{\\prime }}^{n^{\\prime }}(\\xi ) \\mathrm {d}\\sigma (\\xi )={\\lambda }_n \\int _{{\\mathbb {S}^{d-1}}} Y_\\ell ^n(\\xi ) \\cdot Y_{\\ell ^{\\prime }}^{n^{\\prime }}(\\xi ) \\mathrm {d}\\sigma (\\xi ) = {\\lambda }_n \\delta _{\\ell ,\\ell ^{\\prime }} \\delta _{n,n^{\\prime }},$ where ${\\lambda }_n = n(n+d-2)$ , which implies that $\\lbrace Y_\\ell ^n : 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n^d\\rbrace $ is also a family of orthogonal polynomials for the Sobolev inner product defined by, for example, ${\\langle }f,g {\\rangle }_{\\nabla } = \\sum _{k=1}^s \\mu _k\\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\nabla _0^k f(\\xi ) \\cdot \\nabla _0^k g(\\xi ) \\mathrm {d}\\sigma (\\xi ) + \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} f(\\xi ) g(\\xi ) \\mathrm {d}\\sigma (\\xi ),$ where $\\mu _k \\ge 0$ and $s$ is a fixed positive integer, and $\\nabla _0^{2m} := \\Delta _0^m \\quad \\hbox{and} \\quad \\nabla _0^{2m+1} = \\Delta _0^m \\nabla _0.$ In other words, the Sobolev orthogonal polynomials for ${\\langle }\\cdot ,\\cdot {\\rangle }_\\nabla $ on the unit sphere are trivially spherical harmonics themselves."
],
[
"Orthogonal polynomials on the conic surface",
"Orthogonal polynomials on the conic surface ${\\mathbb {V}}_0^{d+1}$ are studied in [21] for the inner product defined by ${\\langle }f, g {\\rangle }_{{\\beta },{\\gamma }} = b_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\mathsf {w}}_{{\\beta },{\\gamma }}(t) \\mathrm {d}{\\mathsf {m}}(x,t),$ where $\\mathrm {d}{\\mathsf {m}}$ is the Lebesgue measure on the conic surface and the weight function ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ is the Jacobi weight function on $[0,1]$ , ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t) = t^{\\beta }(1-t)^{\\gamma }, \\quad {\\beta }> -d, \\quad {\\gamma }> -1$ and $b_{{\\beta },{\\gamma }}$ is the normalization constant so that ${\\langle }1, 1 {\\rangle }_{{\\beta },g} =1$ , which is determined by $\\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) \\mathrm {d}{\\mathsf {m}}(x,t) = \\int _{0}^1 t^{d-1} \\int _{{\\mathbb {S}^{d-1}}} f(t \\xi ,t)\\mathrm {d}\\sigma (\\xi ) \\mathrm {d}t,$ where $\\mathrm {d}\\sigma $ denotes the Lebesgue measure on the unit sphere ${\\mathbb {S}^{d-1}}$ .",
"Then, $b_{\\beta ,\\gamma }= \\frac{1}{\\omega _d} c_{{\\beta }+d-1,{\\gamma }} \\quad \\hbox{with}\\quad c_{{\\beta },{\\gamma }} = \\frac{\\Gamma (\\beta +\\gamma +2)}{\\Gamma (\\beta +1)\\Gamma (\\gamma +1)} \\quad \\hbox{and}\\quad \\omega _{d}=\\frac{2\\pi ^{d/2}}{\\Gamma (d/2)},$ where $\\omega _{d}$ is the surface area of ${\\mathbb {S}^{d-1}}$ .",
"The inner product is well defined for the space ${\\mathbb {R}}[x,t] / {\\langle }t^2 - \\Vert x\\Vert ^2{\\rangle }$ of polynomials modulus the ideal generated by $t^2 -\\Vert x\\Vert ^2$ .",
"For $n \\in {\\mathbb {N}}_0$ , let ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ denote the space of orthogonal polynomials of degree $n$ .",
"Since ${\\mathbb {V}}_0^{d+1}$ is a quadratic surface, so the dimension of the space is the same as that of ${\\mathcal {H}}_n^{d+1}$ .",
"Thus, $\\dim {\\mathcal {V}}_0({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }}) =1$ and $\\dim {\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }}) = \\binom{n+d-1}{n}+\\binom{n+d-2}{n-1},\\quad n=1,2,3,\\ldots .$ An orthogonal basis for ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ is given in [21] in terms of the Jacobi polynomials and spherical harmonics.",
"Let $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d\\rbrace $ be an orthonormal basis of ${\\mathcal {H}}_m^d$ .",
"Define the polynomials, called the Jacobi polynomials on the conic surface, by $ {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })} (x,t) := P_{n-m}^{(2m + {\\beta }+ d-1,{\\gamma })} (1-2t) Y_\\ell ^m (x),$ where, for $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , $Y_\\ell ^m(x)$ is a solid harmonic in ${\\mathcal {H}}_m^{d,0}$ and $Y_\\ell ^m (t\\xi ) = t^m Y_\\ell ^m(\\xi )$ .",
"Then $\\lbrace {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })}: 0 \\le m \\le n, \\,\\, 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m({\\mathbb {S}^{d-1}})\\rbrace $ is an orthogonal basis of ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ , which satisfies $b_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })} (x,t) {\\mathsf {S}}_{m^{\\prime }, \\ell ^{\\prime }}^{n^{\\prime },({\\beta },{\\gamma })} (x,t){\\mathsf {w}}_{{\\beta },{\\gamma }}(t) \\mathrm {d}{\\mathsf {m}}(x,t) = h_{n,m}^{{\\beta },{\\gamma }}\\delta _{n,n^{\\prime }}\\delta _{m,m^{\\prime }}\\delta _{\\ell ,\\ell ^{\\prime }},$ where $h_{n,m}^{{\\beta },{\\gamma }}$ is the square of the $L^2({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }})$ norm of ${\\mathsf {S}}_{m, \\ell }^n$ and $h_{n,m}^{{\\beta },{\\gamma }} = \\frac{({\\beta }+d)_{2m}}{({\\beta }+{\\gamma }+d+1)_{2m}} h_{n-m}^{(2m+{\\beta }+d-1,{\\gamma })}$ with $h_{n-m}^{(2m+{\\beta }+d-1,{\\gamma })}$ defined in (REF ).",
"Of particular interest is the case ${\\beta }= -1$ , for which the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ is an eigenspace of a second order linear differential operator.",
"Parametrizing the space ${\\mathbb {V}}_0^{d+1}$ by $(x,t) = (t\\xi , t)$ and let $\\Delta _0^{(\\xi )}$ be the Laplace-Beltrami operator on the unit sphere in the $\\xi $ variable, which is the restriction of the Laplace operator $\\Delta $ on ${\\mathbb {S}^{d-1}}$ .",
"For ${\\gamma }> -1$ , define $ {\\mathcal {D}}_{\\gamma }= t(1-t)\\frac{\\mathrm {d}^2}{\\mathrm {d}t^2} + \\big ( d-1 - (d+{\\gamma })t \\big ) \\frac{\\mathrm {d}}{\\mathrm {d}t}+ t^{-1} \\Delta _0^{(\\xi )}.$ Theorem 3.1 Let $d\\ge 2$ and ${\\gamma }> -1$ .",
"The orthogonal polynomials in ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ are eigenfunctions of ${\\mathcal {D}}_{{\\gamma }}$ ; more precisely, ${\\mathcal {D}}_{{\\gamma }} u = -n (n+{\\gamma }+d-1) u, \\qquad \\forall u \\in {\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }}).$ The differential operator ${\\mathcal {D}}_{{\\gamma }}$ plays an important role in the study of the Fourier orthogonal series and the best approximation by polynomials on the conic surface [23].",
"For $f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , the Fourier orthogonal series of $f$ is defined by $f = \\sum _{n=0}^\\infty \\operatorname{proj}_n^{{\\beta },{\\gamma }} f,$ where $\\operatorname{proj}_n^{{\\beta },{\\gamma }}: L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }}) \\rightarrow {\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },{\\gamma }})$ is the orthogonal projection operator, which satisfies, using the orthogonal basis given above, $\\operatorname{proj}_n^{{\\beta },{\\gamma }} f = \\sum _{m=0}^n \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_m({\\mathbb {S}^{d-1}})}\\hat{f}_{m,\\ell }^{n, ({\\beta },{\\gamma })} {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },{\\gamma })},\\quad \\hbox{where}\\quad \\hat{f}_{m,\\ell }^{n, ({\\beta },{\\gamma })} := \\frac{{\\langle }f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },{\\gamma })}{\\rangle }_{{\\beta },{\\gamma }}}{h_{m,n}^{{\\beta },{\\gamma }}}.$ In the rest of this section, we consider a property on the derivative of the reproducing kernel.",
"Parametrizing the function $f: {\\mathbb {V}}_0^{d+1} \\rightarrow {\\mathbb {R}}$ by $(x,t) = (t\\xi , t)$ with $\\xi \\in {\\mathbb {S}^{d-1}}$ , it follows that $ \\frac{\\mathrm {d}}{\\mathrm {d}t} f(x,t) = \\frac{\\mathrm {d}}{\\mathrm {d}t} f(t\\xi ,t)= \\xi \\cdot \\nabla _x f(t\\xi ,t) + \\frac{\\partial }{\\partial t} f(t\\xi ,t),$ where $\\frac{\\partial }{\\partial t} = \\partial _{d+1}$ denotes the partial derivative with respect to the $d+1$ variable of $f$ , which should not be confused with the $\\frac{\\mathrm {d}}{\\mathrm {d}t}$ on the left-hand side.",
"Throughout the rest of this paper, we shall adopt the notation $\\partial _t = \\partial _{d+1} = \\frac{\\partial }{\\partial t}$ for functions $f(x,t)$ .",
"We consider the action of $\\partial _t$ on the projection operator.",
"Theorem 3.2 For ${\\beta }> -d$ and ${\\gamma }> -1$ , let $f$ be a differentiable function such that $f\\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ and $\\partial _t f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+1,{\\gamma }+1})$ .",
"Then, for $n =0,1,2,\\ldots $ , $\\frac{\\partial }{\\partial t} \\operatorname{proj}_n^{{\\beta },{\\gamma }} f(x,t) = \\operatorname{proj}_{n-1}^{{\\beta }+1,{\\gamma }+1} \\partial _t f (x,t).$ For $m = n$ , the polynomial ${\\mathsf {S}}_{n,\\ell }^{n,({\\beta },{\\gamma })}(x,t) = Y_\\ell ^n(x)$ , so that $\\frac{\\partial }{\\partial t} {\\mathsf {S}}_{n,\\ell }^{n,({\\beta },{\\gamma })}(x,t) =0$ .",
"For $0 \\le m \\le n-1$ , using the well-known formula for the derivative of the Jacobi polynomials, we obtain $\\frac{\\partial }{\\partial t} {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },{\\gamma })}(x,t) \\, & = \\frac{\\partial }{\\partial t} P_{n-m}^{(2m + {\\beta }+ d-1,{\\gamma })} (1-2t)Y_\\ell ^m (x) \\\\& =\\tau _{n,m} P_{n-m-1}^{(2m + {\\beta }+ d,{\\gamma }+1)} (1-2t) Y_\\ell ^m (x)= \\tau _{n,m} {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}(x,t), $ where $\\tau _{n,m} = - (n+m + {\\beta }+ {\\gamma }+ d)$ .",
"As a consequence of these identities, we see that $\\partial _t \\operatorname{proj}_n^{{\\beta },{\\gamma }} f \\in {\\mathcal {V}}_{n-1}^d({\\mathsf {w}}_{{\\beta }+1,{\\gamma }+1})$ .",
"Consequently, it follows that $& \\left\\langle \\partial _t f, {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}\\right\\rangle _{{\\beta }+1,{\\gamma }+1} =\\left\\langle \\partial _t \\operatorname{proj}_n^{{\\beta },{\\gamma }} f, {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)} \\right\\rangle _{{\\beta }+1,{\\gamma }+1} \\\\& \\qquad \\qquad = \\sum _{k=0}^{n-1} \\sum _{\\nu } \\hat{f}_{k,\\nu }^{n,({\\beta },{\\gamma })} \\tau _{n,k} \\left\\langle {\\mathsf {S}}_{k,\\nu }^{n-1,({\\beta }+1,{\\gamma }+1)},{\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)} \\right\\rangle _{{\\beta }+1,{\\gamma }+1} \\\\& \\qquad \\qquad = \\tau _{n,m} \\hat{f}_{m,\\ell }^{n,({\\beta },{\\gamma })} h_{n-1,m}^{{\\beta }+1,{\\gamma }+1},$ which implies immediately that $\\widehat{\\partial _t f}_{m,\\ell }^{n-1, ({\\beta }+1,{\\gamma }+1)} = \\tau _{n,m} \\hat{f}_{m,\\ell }^{n,({\\beta },{\\gamma })}, \\quad 0 \\le m \\le n-1.$ Consequently, we obtain $\\frac{\\partial }{\\partial t} \\operatorname{proj}_n^{{\\beta },{\\gamma }} f (x,t)\\, & = \\sum _{m=0}^{n-1} \\sum _\\ell \\hat{f}_{m,\\ell }^{n,({\\beta },{\\gamma })}\\tau _{n,m} {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}(x,t) \\\\& = \\sum _{m=0}^{n-1} \\sum _\\ell \\widehat{\\partial _t f}_{m,\\ell }^{n-1, ({\\beta }+1,{\\gamma }+1)} {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta }+1,{\\gamma }+1)}(x,t) \\\\& = \\operatorname{proj}_{n-1}^{{\\beta }+1,{\\gamma }+1} \\partial _t f (x,t),$ where in the first step we used again $\\partial _t {\\mathsf {S}}_{n,\\ell }^n(x,t) = 0$ .",
"For $1 \\le p \\le \\infty $ , let $\\Vert f\\Vert _{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}}$ denote the norm of the space $L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , and we adopt the convention that the space is $C({\\mathbb {V}}_0^{d+1})$ with the norm taken as the uniform norm when $p = \\infty $ .",
"Let $\\Pi _n({\\mathbb {V}}_0^{d+1})$ denote the space of polynomials of degree $n$ restricted on the ${\\mathbb {V}}_0^{d+1}$ .",
"For $f \\in L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , the quantity $E_n(f)_{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}} := \\inf _{P \\in \\Pi _n({\\mathbb {V}}_0^d)} \\Vert f - P \\Vert _{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}}$ is the error of the best approximation by polynomials of degree at most $n$ in the norm of $L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ .",
"We call $\\eta \\in C^\\infty $ an admissible cut-off function if it is supported on $[0,2]$ and satisfies $\\eta (t) = 1$ if $0 \\le t \\le 1$ .",
"Let $\\eta $ be such a function; we define $ Q_{n,\\eta }^{({\\beta },{\\gamma })} f = \\sum _{k=0}^{2n} \\eta \\left(\\frac{k}{n} \\right) \\operatorname{proj}_k^{{\\beta },{\\gamma }} f.$ Then it is known [23] that $Q_{n,\\eta }^{({\\beta },{\\gamma })}$ is a bounded operator in $L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ and it is a polynomial of near best approximation in the sense that $ \\left\\Vert f - Q_{n,\\eta }^{({\\beta },{\\gamma })} f\\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },{\\gamma }}} \\le c E_n (f)_{p,{\\mathsf {w}}_{{\\beta },{\\gamma }}},$ where $c$ is a constant that depends only on $\\eta $ , $p$ , ${\\beta }$ and ${\\gamma }$ .",
"In particular, we obtain the following as a corollary of Theorem REF .",
"Corollary 3.3 Let ${\\beta }> -d$ and ${\\gamma }> -1$ .",
"Let $r$ be a postive integer and let $f \\in C^r({\\mathbb {V}}_0^{d+1})$ such that $\\partial _t^k f \\in L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+k,{\\gamma }+k})$ for $0 \\le k \\le r$ .",
"Then, for $1 \\le p \\le \\infty $ , $\\left\\Vert \\partial _t^k f - \\partial _t^k Q_{n,\\eta }^{({\\beta },{\\gamma })}f\\right\\Vert _{p, {\\mathsf {w}}_{{\\beta }+k,{\\gamma }+k} } \\le c_kE_{n-k} (\\partial _t^k f)_{p, {\\mathsf {w}}_{{\\beta }+k,{\\gamma }+k} }, \\qquad 0 \\le k \\le r.$ This follows immediately from $\\partial _t^k Q_{n,\\eta }^{({\\beta },{\\gamma })} f & = \\sum _{j=0}^{2n} \\eta \\left(\\frac{j}{n} \\right) \\partial _t^j\\operatorname{proj}_j^{{\\beta },{\\gamma }} f= \\sum _{j=k}^{2n} \\eta \\left(\\frac{j}{n} \\right) \\operatorname{proj}_{j-k}^{{\\beta }+k,{\\gamma }+k} f \\\\& = \\sum _{j=0}^{2n-k} \\eta \\left(\\frac{j+k}{n} \\right) \\operatorname{proj}_{j}^{{\\beta }+k,{\\gamma }+k} \\partial _t^k f= {\\mathsf {Q}}_{n,\\eta }^{({\\beta }+k,{\\gamma }+k)} \\partial _t^k f,$ where we define $ {\\mathsf {Q}}_{ n,\\eta }^{({\\beta },{\\gamma })} g = \\sum _{j=0}^{2n-k} \\eta \\left(\\frac{j+k}{n} \\right) \\operatorname{proj}_{j}^{{\\beta },{\\gamma }} g, \\quad 0 \\le k < 2n.$ For fixed $k \\le r$ independent of $n$ , the function $\\eta \\left(\\frac{j+k}{n} \\right)$ plays essentially the same role as $\\eta \\left(\\frac{j}{n} \\right)$ , so that (REF ) holds with ${\\mathsf {Q}}_{n,\\eta }^{({\\beta },{\\gamma })}\\partial _t^k f$ in place of $Q_{n,\\eta }^{({\\beta },{\\gamma })}f$ , form which the proof follows readily."
],
[
"Sobolev orthogonal polynomials on the conic surface",
"As seen in the differential operator (REF ), the derivatives on the conic surface ${\\mathbb {V}}_0^{d+1}$ are partial derivatives with respect to $\\xi $ and $t$ with $x = t \\xi $ for $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ .",
"For the Sobolev inner product, the derivatives in $\\xi \\in {\\mathbb {S}^{d-1}}$ will act on the spherical harmonics and lead to the same orthogonal basis as discussed at the end of Subsection 2.2.",
"Hence, we consider only the derivatives with respect to $\\partial _t$ ; see Remark REF below.",
"For $s \\in {\\mathbb {N}}$ , let us define $W_p^s({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})= \\left\\lbrace f\\in C({\\mathbb {V}}_0^{d+1}): \\partial _t^s f\\in L^p({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta }+s,0})\\right\\rbrace ,$ where $1 \\le p \\le \\infty $ and the space is taken as the $C^s({\\mathbb {V}}_0^{d+1})$ if $p = \\infty $ .",
"Let $s$ be a positive integer and ${\\beta }> -s-d$ .",
"Let ${\\lambda }_1,\\ldots ,{\\lambda }_{r-1}$ be fixed positive numbers.",
"We consider the Sobolev inner product defined by $ {\\langle }f,g{\\rangle }_{{\\beta }, -s} =\\,& \\frac{1}{{\\omega }_d}\\int _{{\\mathbb {V}}_0^{d+1}} \\frac{\\partial ^s}{\\partial t^s} f (x,t)\\frac{\\partial ^s}{\\partial t^s} g(x,t) t^{{\\beta }+s} \\mathrm {d}{\\mathsf {m}}(x,t)\\\\\\ & + \\sum _{k=0}^{s-1} \\frac{{\\lambda }_k}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}}\\frac{\\partial ^k f}{\\partial t^k} (\\xi ,1) \\frac{\\partial ^k g}{\\partial t^k}(\\xi ,1) \\mathrm {d}\\sigma (\\xi ), $ which is evidently an inner product on the space $W_2^s({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ .",
"We denote by ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ the space of orthogonal polynomials of degree $n$ with respect to this inner product.",
"Our first task is to find an orthogonal basis for this space.",
"Recall the modified Jacobi polynomial $J_n^{({\\alpha },-s)}$ defined in (REF ).",
"Let $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d\\rbrace $ be an orthonormal basis of ${\\mathcal {H}}_m^d$ .",
"For $1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d$ , $0\\le m \\le n$ , we define $ {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} (x,t) := J_{n-m}^{(2m+{\\beta }+d-1,-s)}(1-2 t) Y_\\ell ^m(x).$ Theorem 4.1 Let $s \\in {\\mathbb {N}}$ and ${\\beta }> -d-s$ .",
"The polynomials ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}$ , $1 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d$ and $0\\le m \\le n$ consist of a basis of ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ .",
"Moreover, for all $\\ell $ , ${\\mathsf {h}}_{m,n}^{({\\beta },-s)} =\\langle {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}, {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} \\rangle _{{\\beta },-s}$ satisfies ${\\mathsf {h}}_{m,n}^{({\\beta },-s)} = {\\left\\lbrace \\begin{array}{ll}2^{2n-2m} {\\lambda }_{n-m} & 0 \\le n -m \\le s-1, \\\\2^{s - {\\beta }-2m-d}\\, \\widehat{h}_{n-m-s}^{(s+2m+{\\beta }+d-1,0)} & n-m \\ge s.",
"\\end{array}\\right.",
"}$ Let $q_{n-m}(t) = J_{n-m}^{(2m+{\\beta }+d-1, -s)}(t)$ .",
"Changing variable $x = t \\xi $ and using the homogeneity of $Y_\\ell ^m$ , we obtain $\\left\\langle {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}, {\\mathsf {S}}_{m^{\\prime },\\ell ^{\\prime }}^{n^{\\prime },({\\beta },-s)} \\right\\rangle _{{\\beta },-s}= \\, & 2^{ 2s} \\int _0^1 t^{d-1} q_{n-m}^{(s)} (1-2t) q_{n^{\\prime }-m^{\\prime }}^{(s)}(1-2t) t^{{\\beta }+s+m+m^{\\prime }} \\mathrm {d}t \\\\&\\qquad \\times \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}} Y_\\ell ^m (\\xi )Y_{\\ell ^{\\prime }}^{m^{\\prime }} (\\xi ) \\mathrm {d}\\sigma (\\xi ) \\\\& + \\sum _{k=0}^{s-1} {\\lambda }_k 2^{2k} q_{n-m}^{(k)}(-1) q_{n^{\\prime }-m^{\\prime }}^{(k)}(-1) \\\\&\\qquad \\times \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}}^{d-1}} Y_\\ell ^m (\\xi )Y_{\\ell ^{\\prime }}^{m^{\\prime }} (\\xi ) \\mathrm {d}\\sigma (\\xi ).$ Using the orthogonality of $Y_\\ell ^m$ and changing variable $t \\mapsto (1-t)/2$ in the first integral, we see that the expression containing $q_{n-m}$ can be written in terms of the Sobolev inner product $[\\cdot ,\\cdot ]_{{\\alpha },{\\beta }}^{-s}$ defined in (REF ).",
"More precisely, we obtain $\\left\\langle {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}, {\\mathsf {S}}_{m^{\\prime },\\ell ^{\\prime }}^{n^{\\prime },({\\beta },-s)} \\right\\rangle _{{\\beta },-s} =2^{s - {\\alpha }-1} [q_{n-m}, q_{n^{\\prime }-m}]_{{\\alpha },0}^{-s} \\delta _{m,m^{\\prime }} \\delta _{\\ell ,\\ell ^{\\prime }},$ where ${\\alpha }= s+{\\beta }+2m+d-1$ and the inner product $[\\cdot , \\cdot ]_{{\\alpha },0}^{-s}$ is defined as in (REF ) but with $\\mu _k = 2^{2k} {\\lambda }_k/2^{s-{\\alpha }-1}$ .",
"As shown in Theorem REF , the polynomials $J_n^{({\\alpha }-s,-s)}$ are orthogonal with respect to this inner product, regardless of the values of $\\mu _k$ .",
"This proves the orthogonality of ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ .",
"Moreover, the norm ${\\mathsf {h}}_{m,n}^{({\\beta },-s)}$ follows from the norm given in Theorem REF .",
"Remark 4.1 We could also include an additional term in the right-hand side of ${\\langle }\\cdot ,\\cdot {\\rangle }_{{\\beta },-s}$ defined by $\\int _{{\\mathbb {V}}_0^{d+1}} \\nabla _0^s f(x,t) \\cdot \\nabla _0^s g(x,t) \\mathrm {d}{\\mathsf {m}}(x,t),$ where $\\nabla _0$ acts on $\\xi $ .",
"By the discussion for the inner product (REF ) on the unit sphere, the inner product with this additional term has the same orthogonal basis.",
"Recall that the constant $A_n^{({\\alpha },{\\beta })}$ is defined in (REF ).",
"Corollary 4.2 For $0 \\le m \\le n-s$ , $ \\frac{\\partial ^s}{\\partial t^s} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}(x,t) = (-2)^s A_{n-s-m}^{(s+2m+{\\beta }+d-1,0)} S_{m,\\ell }^{n-s, ({\\beta }+s,0)}(x,t),$ and it is equal to zero if $m > n-s$ .",
"Moreover, for $1 \\le k \\le s-1$ and $\\xi \\in {\\mathbb {S}^{d-1}}$ , $ \\frac{\\partial ^k}{\\partial t^k} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}(\\xi ,1) = {\\left\\lbrace \\begin{array}{ll}(-2)^k Y_\\ell ^m(\\xi ) \\delta _{k,n-m}, & m > n-s \\\\0, & m \\le n-s.\\end{array}\\right.",
"}$ If $m \\le n-s$ , by our convention of the derivative $\\partial _t$ over the conic surface and the identity (REF ), $\\frac{\\partial ^s}{\\partial t^s} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}(x,t) \\,& =\\frac{\\partial ^s}{\\partial t^s}\\left[J_{n-m}^{(2m+{\\beta }+d-1,-s)}(1-2t)\\right]Y_\\ell ^m(x) \\\\& = (-2)^{s} \\widehat{P}_{n-m-s}^{(s+2m+{\\beta }+d-1,0)}(1-2t)Y_\\ell ^m(x),$ from which (REF ) follows from (REF ) and the definition of ${\\mathsf {S}}_{\\ell ,m}^{n,({\\beta }+s,0)}$ .",
"Moreover, since $n-m \\ge s$ , for $1 \\le k \\le s-1$ it follows immediately by () that (REF ) holds.",
"If $m > n-s$ , then the derivative in the identity (REF ) is evidently zero since the Jacobi polynomial in ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}$ is of degree $n-m <s$ .",
"Moreover, for $1 \\le k \\le s-1$ , (REF ) follows from ().",
"Our notation for the Sobolev orthogonal polynomials ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ is the same as the one for the ordinary orthogonal polynomials ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },{\\gamma })}$ with ${\\gamma }= -s$ .",
"This is intentional as can be seen in (REF ), which however only works for $n \\ge s$ or ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ for $n-m \\ge s$ .",
"We can, moreover, give an explicit expression of the Sobolev orthogonal polynomials by using the identity (REF ).",
"Corollary 4.3 For $0 \\le \\ell \\le \\dim {\\mathcal {H}}_m^d$ and $0 \\le m \\le n$ , the polynomials $S_{m,\\ell }^{n,({\\beta },-s)}$ satisfy ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}(x,t) = {\\left\\lbrace \\begin{array}{ll} b_{m,n}^s (1- t)^s S_{m,\\ell }^{n-s,({\\beta },s)}(x,t) & n-m \\ge s, \\\\(1-t)^{n-m} Y_\\ell ^m(x), & 0 \\le n-m\\le s-1,\\end{array}\\right.",
"}$ where $b_{m,n}^s = (-1)^s 2^s A_{n-s-m}^{(2m+{\\beta }+d-1,s)}/{(-n-m)_s}$ .",
"In particular, the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ satisfies a decomposition ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s}) = \\bigoplus _{j=0}^{s-1} (1-t)^j {\\mathcal {H}}_{n-j}^{d,0} \\bigoplus (1-t)^s {\\mathcal {V}}_{n-s}^d({\\mathsf {w}}_{{\\beta },s}).$ This follows from the identity (REF ), which shows, together with (REF ), that $J_{n}^{({\\alpha },-s)}(1-2t) = \\frac{(-1)^s 2^s}{(-n)_s} A_{n-s}^{({\\alpha },s)}(1-t)^s P_{n-s}^{({\\alpha },s)}(1-2t)$ for $n\\ge s$ , which implies, with $n$ replaced by $n-m$ , the identity (REF ) when $n - m \\ge s$ .",
"For $n -m \\le s-1$ , (REF ) follows immediately from the definition of $J_n^{({\\alpha },-s)}$ in (REF ).",
"Since $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le m\\rbrace $ is a basis of ${\\mathcal {H}}_n^{d,0}$ , the decomposition (REF ) is an immediate consequence of (REF ).",
"With the expression (REF ) for ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)}$ , we could bypass the integral definition of $J_n^{({\\alpha },-s)}$ in (REF ).",
"However, the integral definition is more convenient for studying the Fourier orthogonal series, as shown below.",
"For $f \\in W_2^s({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ , the Fourier orthogonal series of $f$ is defined by $f = \\sum _{n=0}^\\infty \\operatorname{proj}_n^{{\\beta },-s} f,$ where $\\operatorname{proj}_n^{{\\beta },-s}: W_2^s({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+1,0}) \\rightarrow {\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ is the orthogonal projection operator, which satisfies, using the orthogonal basis given above, $\\operatorname{proj}_n^{{\\beta },-s} f = \\sum _{m=0}^n \\sum _{\\ell }\\hat{f}_{m,\\ell }^{n, ({\\beta },-s)} {\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)},\\quad \\hbox{where}\\quad \\hat{f}_{m,\\ell }^{n, ({\\beta },-s)} := \\frac{\\left\\langle f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}\\right\\rangle _{{\\beta },-s}}{{\\mathsf {h}}_{m,n}^{({\\beta },-s)}}.$ Using the basis in Theorem REF and its corollary, we can derive an integral representation for the projection operator $\\operatorname{proj}_n^{{\\beta },-s} f$ .",
"Theorem 4.4 Let $s$ be a positive integer.",
"For ${\\beta }> -d-s$ , let $f$ be a differentiable function such that $\\partial _t^s f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ .",
"Then, for $n =0,1,2,\\ldots $ and $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , $\\operatorname{proj}_n^{{\\beta },-s} f(x,t) = \\,& \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!}",
"\\operatorname{proj}_{n-m}^{\\mathbb {S}^{d-1}}\\left[\\partial _t^m f(\\cdot ,1)\\right](\\xi ) \\\\& + (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}",
"\\operatorname{proj}_{n-s}^{{\\beta }+s,0} \\left(\\partial _t^s f\\right) (x, v) \\mathrm {d}v. $ Let ${\\alpha }= s+{\\beta }+d-1$ .",
"For $0 \\le m \\le n-s$ , it follows from (REF ) that $\\left\\langle f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}\\right\\rangle _{{\\beta },-s} \\,& =(-2)^s A_{n-m}^{(2m+{\\alpha },0)}\\frac{1}{{\\omega }_d} \\int _{{\\mathbb {V}}_0^{d+1}} \\frac{\\partial ^s}{\\partial t^s} f(x,t) {\\mathsf {S}}_{m,\\ell }^{n-s,({\\beta }+s,0)}(x,t) t^{{\\beta }+s} \\mathrm {d}{\\mathsf {m}}(x,t)\\\\& = (-2)^s A_{n-m}^{(2m+{\\alpha },0)} \\frac{1}{c_{{\\alpha },0}}\\left\\langle \\partial _t^s f, {\\mathsf {S}}_{m, \\ell }^{n-s,({\\beta }+s,0)}\\right\\rangle _{{\\beta }+s,0}$ and the norm of ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ satisfies ${\\mathsf {h}}_{m,n}^{({\\beta },-s)} = 2^{s - {\\beta }-2m-d} \\widehat{h}_{n-m-s}^{({\\alpha }+2m,0)} \\, & =2^{2s} \\frac{1}{c_{2m+{\\alpha },0}} \\left[A_{n-m-s}^{(2m+{\\alpha },0)}\\right]^2 h_{n-s-m}^{(2m+{\\alpha },0)} \\\\& = \\frac{1}{c_{{\\alpha },0}} 2^{2s} \\left[A_{n-m-s}^{(2m+{\\alpha },0)}\\right]^2 \\frac{({\\alpha }+1)_{2m}}{({\\alpha }+2)_{2m}}h_{n-s-m}^{(2m+{\\alpha },0)}\\\\& = \\frac{1}{c_{{\\alpha },0}} 2^{2s} \\left[A_{n-m-s}^{(2m+{\\alpha },0)}\\right]^2 h_{n-s, m}^{{\\beta }+s,0}.$ Consequently, it follows that $\\hat{f}_{m,\\ell }^{n, ({\\beta },-s)} = \\frac{c_{2m+{\\alpha },0}}{c_{{\\beta }+s,0}} \\frac{\\left\\langle \\partial _t^s f, {\\mathsf {S}}_{m, \\ell }^{n-s,({\\beta }+s,0)}\\right\\rangle _{{\\beta }+s,0}}{ (-2)^{s} A_{n-m-s}^{(2m+{\\alpha },0)} h_{n-s, m}^{{\\beta }+s,0}}= \\frac{ \\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)}}{(-2)^{s} A_{n-m-s}^{(2m+{\\alpha },0)}}.$ Now, for $n-s < m \\le n$ , it follows by Corollary REF that $\\left\\langle f, {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}\\right\\rangle _{{\\beta },-s}\\,& = \\sum _{k=0}^{s-1} \\frac{{\\lambda }_k}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\partial _t^k f(\\xi ,1) \\partial _t^k {\\mathsf {S}}_{m, \\ell }^{n,({\\beta },-s)}(\\xi ,1)\\mathrm {d}\\sigma (\\xi ) \\\\& = (-2)^{n-m} \\frac{{\\lambda }_{n-m}}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\partial _t^{n-m} f(\\xi ,1) Y_\\ell ^{m} (\\xi )\\mathrm {d}\\sigma (\\xi ),$ so that, using the norm of ${\\mathsf {h}}_{n,m}^{({\\beta },-s)}$ in Theorem REF , $\\hat{f}_{m,\\ell }^{n, ({\\beta },-s)} = \\frac{1}{(-2)^{n-m}} \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}} \\partial _t^{n-m} f(\\xi ,1) Y_\\ell ^{m} (\\xi )\\mathrm {d}\\sigma (\\xi ),\\quad 0\\le n-m \\le s-1.$ Consequently, by (REF ) and using ${\\alpha }= s+{\\beta }+d-1$ again, it follows that $& \\operatorname{proj}_n^{{\\beta },-s} f(x,t) \\, = \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!",
"}\\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_{n-m}^d} \\frac{1}{{\\omega }_d} \\int _{{\\mathbb {S}^{d-1}}}\\partial _t^{m} f(\\eta ,1) Y_\\ell ^{n-m} (\\eta )\\mathrm {d}\\sigma (\\eta )Y_\\ell ^{n-m} (\\xi ) \\\\& \\qquad \\quad + \\sum _{m = 0}^{n-s} \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_{m}^d}\\frac{ \\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)}}{(-2)^{s} A_{n-m-s}^{(2m+{\\alpha },0)}}\\int _{-1}^{1-2t} \\frac{(1-2t -u)^{s-1}}{(s-1)!}",
"\\widehat{P}_{n-s}^{({\\alpha }+2m,0)}(u) \\mathrm {d}u Y_\\ell ^m(x).$ The inner sum of the first term in the right-hand side is exactly $\\operatorname{proj}_{n-m}^{\\mathbb {S}^{d-1}}[\\partial _t^m f(\\cdot ,1)](\\xi )$ for the function $\\xi \\mapsto \\partial _t^m f(\\xi ,1)$ on the unit sphere.",
"Changing variable $u = 1-2v$ and using (REF ) in the integral, the second term on the right-hand side becomes, $\\sum _{m = 0}^{n-s} \\sum _{\\ell } & \\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)} (-1)^s\\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}",
"P_{n-s}^{({\\alpha }+2m,0)}(1-2v) \\mathrm {d}v Y_\\ell ^m(x) \\\\& = (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}",
"\\sum _{m = 0}^{n-s} \\sum _{\\ell }\\widehat{\\partial _t^s f}_{m,\\ell }^{n-s, ({\\beta }+s,0)} {\\mathsf {S}}_{m,\\ell }^{n-s, ({\\beta }+s,0)} (x,v) \\mathrm {d}v\\\\& = (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}",
"\\operatorname{proj}_{n-s}^{{\\beta }+s,0}\\partial _t^s f (x, v) \\mathrm {d}v.$ Putting the two terms together completes the proof.",
"Corollary 4.5 Let $s$ be a positive integer.",
"For ${\\beta }> -d-s$ , let $f$ be a differentiable function such that $\\partial _t^s f \\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta }+s,0})$ .",
"Then, for $n \\ge s$ , $\\frac{\\partial ^s}{\\partial t^s} \\operatorname{proj}_n^{{\\beta },-s} f(x,t) = \\operatorname{proj}_{n-s}^{{\\beta }+s,0}\\left(\\partial _t^s f\\right) (x,t).$ Taking the derivative $\\partial _t^s$ of (REF ), we see that the first term in the right-hand side is zero, while the second term gives the stated identity.",
"Another consequence of Theorem REF is an expression for the error of approximation.",
"Let $\\eta \\in C^\\infty ({\\mathbb {R}}_+)$ be an admissible cut-off function.",
"We denote by ${\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}$ the near best approximation operator defined in (REF ) but with ${\\gamma }= -s$ .",
"Furthermore, let ${\\mathsf {Q}}_{n-m}^{\\mathbb {S}^{d-1}}f (\\xi )= \\sum _{k=0}^{2(n-m)} \\eta \\left(\\frac{k}{n}\\right) \\operatorname{proj}_k^{\\mathbb {S}^{d-1}}f(\\xi ), \\quad 0 \\le m \\le n,$ be a near best approximation operator of degree $n-m$ on the unit sphere.",
"For $f\\in L^p({\\mathbb {S}^{d-1}})$ , let $E_n^{\\mathbb {S}^{d-1}}(f)_p = \\inf _{Y \\in \\Pi _n({\\mathbb {S}^{d-1}})} \\Vert f - Y\\Vert _{L^p({\\mathbb {S}^{d-1}})}, \\qquad 1 \\le p \\le \\infty $ be the error of best approximation by polynomials on the unit sphere, where the norm is the uniform norm for $p = \\infty $ and $\\Pi _n({\\mathbb {S}^{d-1}})$ denote the space of polynomial of degree at most $n$ restricted on the unit sphere ${\\mathbb {S}^{d-1}}$ .",
"Then (cf.",
"[5]) $\\left\\Vert f - {\\mathsf {Q}}_{n}^{\\mathbb {S}^{d-1}}f \\right\\Vert _p \\le c E_n^{{\\mathbb {S}^{d-1}}}(f)_p, \\qquad 1 \\le p \\le \\infty .$ Theorem 4.6 For $f \\in C^s({\\mathbb {V}}_0^{d+1})$ , $f(x,t) - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f(x,t) \\,& = \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!}",
"\\left[ \\partial _t^m [f(\\cdot ,1)] (\\xi )- {\\mathsf {Q}}_{n-m,\\eta }^{{\\mathbb {S}^{d-1}}} \\left[\\partial _t^m f(\\cdot ,1)\\right](\\xi ) \\right] \\\\& + (-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}",
"\\left[ \\partial _t^s f (x, v)-{\\mathsf {Q}}_{n-s,\\eta }^{{\\beta }+s,0}\\left(\\partial _t^s f\\right) (x, v) \\right] \\mathrm {d}v.$ Multiplying (REF ) by $\\eta \\left( \\frac{k}{n} \\right)$ and summing up over $k$ gives an expression of ${\\mathsf {Q}}_{n,\\eta }^{({\\beta },-s)} f$ .",
"Furthermore, using the Tylor expansion of $t\\rightarrow f(x,t)$ with respect to the $t$ variable at $t =1$ , together with its remainder formula, we can write $f(x,t) = \\sum _{m=0}^{s-1} \\frac{(t-1)^m}{m!}",
"\\partial _t^m [f(\\cdot ,t)] (\\xi ,1) +(-1)^s \\int _{t}^{1} \\frac{(v-t)^{s-1}}{(s-1)!}",
"\\partial _t^s f (x, v) \\mathrm {d}v.$ Together the difference of the two identities gives the error formula for $f -{\\mathsf {Q}}_{n,\\eta }^{({\\beta },-s)} f$ and completes the proof.",
"Taking the $k$ -th order derivative with respect to the $t$ variable for $ 0 \\le t \\le s-1$ , we obtain $\\partial _t^k \\left(f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\right)&(x,t) = \\sum _{m= k}^{s-1} \\frac{(t-1)^{m-k}}{(m-k)!",
"}\\left[ \\partial _t^m [f(\\cdot ,1)] (\\xi )- {\\mathsf {Q}}_{n-m,\\eta }^{{\\mathbb {S}^{d-1}}} \\left[\\partial _t^m f(\\cdot ,1)\\right](\\xi ) \\right] \\\\& + (-1)^{s-k} \\int _{t}^{1} \\frac{(v-t)^{s-k-1}}{(s-k-1)!}",
"\\left[ \\partial _t^s f (x, v)-{\\mathsf {Q}}_{n-s,\\eta }^{({\\beta }+s,0)}\\left(\\partial _t^s f\\right) (x, v) \\right] \\mathrm {d}v,$ moreover, taking the $s$ -th derivative in the $t$ variable, we obtain $ \\partial _t^s \\left(f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\right)(x,t) = {\\mathsf {Q}}_{n-s,\\eta }^{({\\beta }+s,0)}\\left(\\partial _t^s f\\right) (x, t).$ In the above error formulas for $1\\le k \\le s-1$ , the integral is over $v \\in [t,1]$ while the variable is $(x,v) = (t \\xi , v)$ .",
"As it is, we cannot deduce the error estimate for $\\partial _t^k \\big (f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\big )$ immediately from that of $f - {\\mathsf {Q}}_{n-s,\\eta }^{({\\beta }+s,0)}\\big (\\partial _t^s f\\big )$ over ${\\mathbb {V}}_0^{d+1}$ if $1 \\le k \\le s-1$ , although we can if $k =s$ by (REF ).",
"We state the latter case as a corollary.",
"Corollary 4.7 Let $f \\in W_p^s({\\mathsf {w}}_{{\\beta }+s,0})$ .",
"Then, for $1 \\le p \\le \\infty $ , $\\left\\Vert \\partial _t^s f - \\partial _t^s {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)}f \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta }+s,0}}\\le c E_n\\left( \\partial _t^s f \\right)_{p, {\\mathsf {w}}_{{\\beta }+s,0}}.$ We can derive another interesting property of the projection operator for the SOPs, which relies on the property that the Jacobi polynomial $P_n^{({\\alpha },-s)}$ is related to $P_{n-s}^{({\\alpha },s)}$ , as seen in [17].",
"The following lemma shows that our extension $J_n^{{\\alpha },-s}$ satisfies the same property.",
"Theorem 4.8 If $f(x,t) = (1-t)^s g(x,t)$ , then $\\operatorname{proj}_n^{{\\beta },-s} f = (1-t)^s \\operatorname{proj}_{n-s}^{{\\beta },s} g.$ By the definition of the ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-s)}$ , it follows from (REF ) that ${\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} (x,t)= \\sigma _{m,n} (1-t)^s {\\mathsf {S}}_{m,\\ell }^{n-s, ({\\beta },s)}(x,t), \\quad \\sigma _{m,n} := \\frac{(n-s-m)!}{(n-m)!}",
"A_{n-m}^{(2m+{\\beta }+d-1,0)}.$ Expanding $g$ in terms of the Fourier orthogonal series in $L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },1})$ , we obtain $f(x,t) = (1-t)^s g(x,t) \\,& = (1-t)^s \\sum _{n=0}^\\infty \\sum _{m=0}^n \\sum _\\ell \\hat{g}_{m,\\ell }^{n, ({\\beta },s)}{\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },s)}(x,t) \\\\& = \\sum _{n=0}^\\infty \\sum _{m=0}^n \\sum _\\ell \\hat{g}_{m,\\ell }^{n, ({\\beta },s)}\\sigma _{n+s,m}^{-1} {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)}(x,t).$ Applying the orthogonality in the Sobolev inner product, it follows immediately that $\\left\\langle f, {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)} \\right\\rangle _{{\\beta },-s} =\\sigma _{n+1,m}^{-1} \\hat{g}_{m,\\ell }^{n,({\\beta },s)} \\left\\langle {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)}, {\\mathsf {S}}_{m,\\ell }^{n+s, ({\\beta },-s)} \\right\\rangle _{{\\beta },-s}, \\quad 0 \\le m \\le n;$ in particular, with $n$ replaced by $n-s$ , we then obtain $ \\hat{f}_{m,\\ell }^{n,({\\beta },-s)} = \\sigma _{n,m}^{-s} \\hat{g}_{m,\\ell }^{n-s,({\\beta },s)}, \\quad 0 \\le m \\le n-s.$ For $n-s < m \\le n$ , the polynomial $J_{n-m}^{(2m+{\\beta }+d-1,-s)}(t)$ is a polynomial of degree at most $s-1$ , so that $\\partial _t^s {\\mathsf {S}}_{n,\\ell }^{n, ({\\beta },-1)}(x,t) = 0$ ; moreover, $\\partial _t^k f(\\xi ,1) =0$ for $0 \\le k \\le s-1$ , since $f$ contains a factor $(1-t)^s$ .",
"It follows that $\\hat{f}_{n,\\ell }^{n,({\\beta },-s)} = 0$ for $n -s < m \\le n$ by the definition of the Sobolev inner product.",
"Consequently, by (REF ) and (REF ) we obtain $(1-t) \\operatorname{proj}_{n-s}^{{\\beta },s} g\\, & = (1-t) \\sum _{m=0}^{n-s} \\sum _\\ell \\hat{g}_{m,\\ell }^{n-1,({\\beta },s)} {\\mathsf {S}}_{m,\\ell }^{n-s, ({\\beta },s)} \\\\& = \\sum _{m=0}^{n} \\sum _\\ell \\hat{f}_{m,\\ell }^{n,({\\beta },-s)} {\\mathsf {S}}_{m,\\ell }^{n, ({\\beta },-s)} = \\operatorname{proj}_{n}^{{\\beta },-s}f.$ The proof is completed.",
"Corollary 4.9 Let $f \\in L^p({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },0})$ and $f(t) = (1-t) g(t)$ .",
"Then, for $1 \\le p \\le \\infty $ , $\\left\\Vert f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)} f \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },0}}\\le c E_n\\left(g \\right)_{p, {\\mathsf {w}}_{{\\beta },s}} \\le c E_n\\left(g \\right)_{p, {\\mathsf {w}}_{{\\beta },0}}$ Since $f = (1-t)^s g$ , we see that ${\\mathsf {Q}}_{n,\\eta }^{(\\beta , -1)} f = \\sum _{k=0}^{2n} \\eta \\left( \\frac{k}{n}\\right) (1-t)^s \\operatorname{proj}_{k-1}^{{\\beta },s} g= (1-t)^s \\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{\\beta ,s} g,$ where $\\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{(\\beta ,s)}$ is defined as in the proof of Corollary REF .",
"For $p \\ge 1$ , we then obtain $\\left\\Vert f - {\\mathsf {Q}}_{n,\\eta }^{(\\beta , -s)} f \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },0}}& = \\left\\Vert (1-t)^s \\left[ g - \\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{(\\beta , s)} g \\right] \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },0}}\\\\& \\le \\left\\Vert g - \\widetilde{{\\mathsf {Q}}}_{n,\\eta }^{(\\beta , s)} g \\right\\Vert _{p, {\\mathsf {w}}_{{\\beta },s}} \\le c E_n(g)_{p, {\\mathsf {w}}_{{\\beta },s}},$ By the definition of $E_n(g)_{p, {\\mathsf {w}}}$ , it follows immediately $E_n(g)_{p, {\\mathsf {w}}_{{\\beta },s}} \\le E_n(g)_{p, {\\mathsf {w}}_{{\\beta },0}}$ .",
"The proof is completed."
],
[
"Partial differential equation for the Sobolev orthogonal polynomials",
"This section considers partial differential equations satisfied by the Sobolev orthogonal polynomials.",
"As we mentioned in Theorem REF proved in [21], the ordinary orthogonal polynomials in ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ , ${\\gamma }> -1$ , are eigenfunctions of the differential operator ${\\mathcal {D}}_{{\\gamma }}$ , defined in (REF ).",
"For the Sobolev orthogonal polynomials, we need ${\\gamma }$ to be negative integers.",
"We first consider the action of the differential operator when ${\\gamma }$ is a real number."
],
[
"Eigenfunctions of ${\\mathcal {D}}_{{\\gamma }}$ for {{formula:3f962e14-6e6a-454f-99e0-f53f9b58f637}}",
"Recall that the operator ${\\mathcal {D}}_{{\\gamma }}$ is given by ${\\mathcal {D}}_{{\\gamma }} = t(1-t)\\frac{\\mathrm {d}^2}{\\mathrm {d}t^2} + \\big ( d-1 - (d+{\\gamma })t \\big ) \\frac{\\mathrm {d}}{\\mathrm {d}t}+ t^{-1} \\Delta _0^{(\\xi )}.$ We start with a lemma for the action of this operator.",
"Lemma 5.1 Let $p(t)$ be a polynomial in the $t$ variable and $Y(\\xi )$ be a function defined on the unit sphere ${\\mathbb {S}^{d-1}}$ .",
"For $k \\ge 1$ , ${\\mathcal {D}}_{\\gamma } \\left[(1-t)^k\\,p(t) Y(\\xi ) \\right] \\, & = (1-t)^k {\\mathcal {D}}_{\\gamma +2k}\\left[p(t) Y (\\xi )\\right] \\\\& - k\\,(1-t)^{k-1} [d-1-(d+{\\gamma }+k-1)t] p(t)Y(\\xi ).",
"$ For $k=1$ , a quick computation from the definition of ${\\mathcal {D}}_{{\\gamma }}$ shows ${\\mathcal {D}}_{{\\gamma }}[(1-t) p(t) Y(\\xi )]=\\, & \\big ((1-t) \\left[t(1-t)p^{\\prime \\prime }(t) + [d-1-(d+\\gamma +2)t]p^{\\prime }(t)\\right] \\\\& - [d-1-(d+\\gamma )t]p(t)\\big ) Y(\\xi )+ t^{-1}\\,(1-t) p(t) \\, \\Delta _0^{(\\xi )}Y(\\xi ) \\\\= \\, & (1-t){\\mathcal {D}}_{\\gamma +2}[p(t)Y(\\xi )] - [d-1-(d+\\gamma )t]p(t)Y(\\xi ),$ which verifies (REF ) for $k=1$ .",
"Assume that (REF ) holds for $k\\ge 1$ .",
"Then $& {\\mathcal {D}}_{\\gamma }[(1-t)^{k+1} p(t) Y(\\xi )] = {\\mathcal {D}}_{{\\gamma }} [(1-t)^k (1-t) p(t) Y(\\xi )] \\\\& \\quad = (1-t)^k {\\mathcal {D}}_{{\\gamma }+2k}[(1-t)p(t) Y(\\xi )] \\\\&\\qquad - k\\,(1-t)^{k-1} [d-1-(d+\\gamma +k-1)t][(1-t)p(t) Y(\\xi )]\\\\& \\quad = (1-t)^k \\left\\lbrace (1-t){\\mathcal {D}}_{{\\gamma }+2k+2}[p(t) Y(\\xi )] - [d-1-(d+\\gamma +2k)t]p(t) Y(\\xi )\\right\\rbrace \\\\&\\qquad - k\\,(1-t)^{k} [d-1-(d+\\gamma +k-1)t]p(t) Y(\\xi )\\\\& \\quad = (1-t)^{k+1} {\\mathcal {D}}_{{\\gamma }+2k+2}[p(t) Y(\\xi )] - (k+1)(1-t)^k [d-1-(d+\\gamma +k)t]p(t) Y(\\xi ),$ so that (REF ) holds for $k+1$ and the proof is completed by induction.",
"Theorem 5.2 For $0\\le j \\le n$ , let $Z_{j,n}(x,t) = p(t) Y(x)$ with $p$ being a polynomial of degree $j$ in one variable and $Y$ be a solid harmonic polynomial in ${\\mathcal {H}}_{n-j}^{d,0}$ .",
"For $\\gamma \\in {\\mathbb {R}}$ , the only polynomial $p$ for which $Z_{j,n}$ satisfies ${\\mathcal {D}}_{{\\gamma }} Z (x,t) = \\lambda _n^{(\\gamma )} Z(x,t), \\qquad {\\lambda }_n^{({\\gamma })} = -n(n+{\\gamma }+d-1),$ is the Jacobi polynomial $p(t) = c_j\\,P_j^{(2n-2j+d-2, \\gamma )}(1-2t)$ , where $c_j$ is an appropriate constant, if $2n+\\gamma +d-r-1\\ne 0$ for $0\\le r\\le j$ .",
"Without losing generality, we assume $p$ is a monic polynomial, $p(t) = \\sum _{i=0}^j \\,(1-t)^i\\,a_{j,i}, \\quad a_{j,j} = 1.$ Our objective is to determine the coefficients $a_{j,i}$ such that $Z_j$ satisfies (REF ).",
"First, we observe that (REF ) holds if $p(t) =1$ .",
"In this case $Y \\in {\\mathcal {H}}_n^{d,0}$ and $Y(x) = t^n Y(\\xi )$ .",
"Hence, taking the derivative over $t$ and using the relation (REF ), a quick computation shows that $ {\\mathcal {D}}_{{\\gamma }} Y(x)= &\\, {\\mathcal {D}}_{{\\gamma }}[t^n Y(\\xi )] \\\\= &\\, n (n+d-2)Y(\\xi ) - m (m+{\\gamma }+d-1) t^m Y(\\xi ) + t^{m-1} \\Delta _0 Y(\\xi ) \\\\= &\\, -m(m+d+{\\gamma }-1) t^m Y(\\xi ) = -m(m+d+{\\gamma }-1) Y(x).",
"$ For $j \\ge 0$ , we use the identity (REF ) to obtain $& {\\mathcal {D}}_{{\\gamma }} Z_j(x,t) = \\sum _{i=0}^j a_{j,i} {\\mathcal {D}}_{{\\gamma }}\\left[(1-t)^i t^{n-j} Y(\\xi )\\right] \\\\& = \\sum _{i=0}^j a_{j,i} \\left[(1-t)^i {\\mathcal {D}}_{{\\gamma }+ 2i}[Y(x)] - i(1-t)^{i-1}[d-1-(d+\\gamma +i-1)t] Y(x)\\right]$ and then apply (REF ) for $Y\\in {\\mathcal {H}}_{n-j}^{d,0}$ and simplify to obtain ${\\mathcal {D}}_{{\\gamma }} Z_j(x,t) =\\, & \\sum _{i=0}^{j} \\,(1-t)^i \\,a_{j,i} [\\lambda _{n-j}^{(\\gamma +2i)} - i\\,(d+\\gamma +i-1)]\\,Y(x)\\\\&+\\sum _{i=0}^{j-1} (i+1)(1-t)^{i}\\,(\\gamma +i+1) \\,a_{j,i+1}\\,Y(x)\\\\= \\, & (1-t)^j \\,\\lambda _n^{(\\gamma )}\\,Y(x) +\\sum _{i=0}^{j-1} \\,(1-t)^i \\\\& \\times \\lbrace a_{j,i} \\,[\\lambda _{n-j}^{(\\gamma +2i)} - i(d+\\gamma +i-1)] + a_{j,i+1}(i+1)(\\gamma +i+1)\\rbrace Y(x),$ since $a_{j,j} = 1$ .",
"Thus, in order to have $Z_{j,n} (x,t)$ satisfying (REF ), we must have $a_{j,i} \\,[\\lambda _{n-j}^{(\\gamma +2i)} - i(d+\\gamma +i-1)] + a_{j,i+1}(i+1)(\\gamma +i+1) = a_{j,i} \\lambda _n^{(\\gamma )}$ for $0 \\le i \\le j-1$ , which leads to the recurrence relation $a_{j,i} = - \\frac{(i+1)(\\gamma +i+1)}{(j-i)(2n+\\gamma +d-1-j+i)}a_{j,i+1} , \\quad i=0, 1, \\ldots , j-1.$ Solving this recurrence relation and using $a_{j,j} = 1$ , we obtain $a_{j,i} = (-1)^{j-i}\\frac{(i+1)_{j-i}(\\gamma +i+1)_{j-i}}{(j-i)!",
"(2n+\\gamma +d- 1-j+i)_{j-i}}, \\quad 0\\le i \\le j.$ Using the identity $(a+i)_{j-i} = (a)_j/(a)_i$ and $(j-i)!",
"= (-1)^i j!",
"/ (-j)_i$ , we can rewrite it as $a_{j,i} = (-1)^j c_j \\frac{(-j)_i (2n+{\\gamma }+d-j-1)_i}{i!",
"({\\gamma }+1)_i}, \\quad c_j= \\frac{({\\gamma }+1)_j}{(2n+{\\gamma }+d-j-1)_j},$ which is well-defined if $(2n+{\\gamma }+d-j-1)_j \\ne 0$ .",
"Consequently, the polynomial $p$ must be given by $p(x) \\,& = (-1)^j c_j\\phantom{i}_2F_1\\left( \\begin{matrix} -j, 2n+{\\gamma }+d-j-1 \\\\ {\\gamma }+1 \\end{matrix}; 1-t \\right) \\\\& = (-1)^jc_j P_j^{({\\gamma }, 2n-2j+d-2)}(2t-1) =c_j P_j^{(2n-2j+d-2, {\\gamma })}(1-2t),$ which completes the proof.",
"For ${\\gamma }> -1$ , the theorem recovers Theorem REF established in [21].",
"We discuss the case when ${\\gamma }= -s$ and $s \\in {\\mathbb {N}}$ in the next subsection."
],
[
"Eigenfunctions for ${\\mathcal {D}}_{-s}$",
"As a corollary of Theorem REF , we obtain the following result for ${\\mathcal {D}}_{-s}$ .",
"Proposition 5.3 Let $s \\in {\\mathbb {N}}$ and $n \\in {\\mathbb {N}}_0$ .",
"Then the polynomials ${\\mathsf {S}}_{m,\\ell }^{n, (-1,-s)}$ are eigenfunctions of ${\\mathcal {D}}_{-s}$ if and only if $m \\le n-s$ and $m =n$ .",
"By (REF ), the polynomials ${\\mathsf {S}}_{m,-\\ell }^{n,(-1,-s)}$ can be written as ${\\mathsf {S}}_{m,\\ell }^{n,(-1,-s)}(x,y) = c_{n-m} P_{n-m}^{(2m+d-2,-s)}(1-2t) Y_\\ell ^m(x), \\quad n-m \\ge s,$ so that Theorem REF applies for $n - m \\ge s$ , thus ${\\mathsf {S}}_{m,\\ell }^{n, (-1,-s)}$ satisfies (REF ).",
"If $n = m$ , then ${\\mathsf {S}}_{n, \\ell }^{n, (-1,-s)}(x,t) = Y_\\ell ^n(x)$ .",
"As a result, (REF ) holds with $p(z) =1$ .",
"For $1 \\le n-m \\le s-1$ , however, the polynomial $J_{n-m}^{(2m+d-2,-s)}(t) = (1-t)^{n-m}/(n-m)!$ is not the Jacobi polynomial $P_{n-m}^{(2m+d-2,-s)}(t)$ , hence ${\\mathsf {S}}_{n, \\ell }^{n, (-1,-s)}$ is not a solution of (REF ).",
"In the case $s =1$ , the set $\\lbrace m: n -m \\ge s\\rbrace \\cup \\lbrace n\\rbrace = \\lbrace m: 1 \\le m \\le n\\rbrace $ .",
"Hence, we obtain the following corollary.",
"Theorem 5.4 For ${\\gamma }= -1$ , all elements in ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1})$ are eigenfunctions of ${\\mathcal {D}}_{-1}$ ; that is, ${\\mathcal {D}}_{-1} Z = - n (n+ d-2) Z, \\qquad \\forall Z \\in {\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1}).$ Moreover, the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1})$ satisfies a decomposition ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-1}) = {\\mathcal {H}}_n^{d,0} \\cup (1-t) {\\mathcal {V}}_{n-1}^d({\\mathsf {w}}_{-1,1}), \\qquad n =0,1,\\ldots .$ We only need to prove the decomposition.",
"If $m \\le n-1$ , it follows from from (REF ) that ${\\mathsf {S}}_{m,\\ell }^{n,({\\beta },-1)} = c (1-t) {\\mathsf {S}}_{m,\\ell }^{n-1,({\\beta },1)} \\in (1-t) {\\mathcal {V}}_{n-1}^d({\\mathsf {w}}_{-1,1})$ , whereas if $m = n$ , then ${\\mathsf {S}}_{n,\\ell }^{n,({\\beta },-s)} = Y_\\ell ^n \\in {\\mathcal {H}}_n^{d,0}$ .",
"The theorem shows, in particular, that ${\\mathcal {D}}_{{\\gamma }}$ has the complete eigenspaces if ${\\gamma }\\ge -1$ .",
"For $s = 2,3,\\ldots $ , however, not every element of the space ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-s})$ is an eigenfunction of ${\\mathcal {D}}_{ -s}$ by Theorem REF .",
"We can, however, identify the eigenspaces of the operator as follows.",
"Theorem 5.5 For $s = 2,3,\\ldots $ , define ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s}):= {\\mathcal {H}}_{n}^{d,0} \\bigcup _{j=1}^{s-1}P_j^{(2n-2j+d-2,-s)}(1-2t){\\mathcal {H}}_{n-j}^{d,0}\\cup (1-t)^s {\\mathcal {V}}_{n-s}^d ({\\mathsf {w}}_{-1,s}).$ Then ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ is the eigenspace of ${\\mathcal {D}}_{-s}$ ; more precisely, ${\\mathcal {D}}_{-s} Z=\\lambda _n^{(-s)}Z$ for all $Z \\in {\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ .",
"Moreover, the space satisfies $\\dim {\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s}) = \\binom{n+d-1}{d}$ if $-2n-d+j+s+2$ is not a positive integer between 1 and $j$ for $1 \\le j \\le s-1$ and $n \\ge j$ .",
"As in the case of $s = 1$ , using (REF ), the statement on the eigenspace follows readily from Theorem REF .",
"As it is shown for the dimension of ${\\mathcal {V}}_n^d({\\mathsf {w}}_{{\\beta },-s})$ , the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ has the full dimension $\\binom{n+d-1}{d}$ if the Jacobi polynomials $P_j^{(2n-2j+d-2,-s)}$ does not have the degree reduction, which holds if $2n-j+d-2-s +k =0$ for a certain integer $k$ , $1 \\le k \\le j$ , by [17].",
"The theorem shows that the operator ${\\mathcal {D}}_{-s}$ has a complete basis of polynomials as eigenfunctions only if the restriction on $-2n-d+j+s+2$ as stated holds.",
"Example 5.6 If $s = 2$ , then the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-2})$ is given by ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-2}) = {\\mathcal {H}}_n^d \\cup \\left(1+ (2n+d-4)(1-t) \\right) {\\mathcal {H}}_{n-1}^d\\cup (1-t)^s {\\mathcal {V}}_{n-2}^d ({\\mathsf {w}}_{-1,2})$ and it has the full dimension if $-2n-d+5 \\ne 1$ or $2n-d+4 \\ne 0$ for $n \\ge 1$ .",
"Thus, ${\\mathcal {D}}_{-2}$ has a complete basis of polynomials as eigenfunctions if $d$ is odd.",
"In the case that $d$ is even, $ {\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-2})$ is not well-defined for $n = d/2-2$ .",
"We note that the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ does not coincide with ${\\mathcal {V}}_n^d({\\mathsf {w}}_{-1,-s})$ for $s =2,3,\\ldots $ .",
"In the case of $s = -2$ and $d$ is odd, one may ask if there is a Sobolev inner product for which the polynomials in the space ${\\mathcal {U}}_n^d({\\mathsf {w}}_{-1,-s})$ are orthogonal.",
"The authors have no relevant financial or non-financial interests to disclose."
]
] | 2209.08186 |
[
[
"Computing analytic Bayes factors from summary statistics in\n repeated-measures designs"
],
[
"Abstract Bayes factors are an increasingly popular tool for indexing evidence from experiments.",
"For two competing population models, the Bayes factor reflects the relative likelihood of observing some data under one model compared to the other.",
"In general, computing a Bayes factor is difficult, because computing the marginal likelihood of each model requires integrating the product of the likelihood and a prior distribution on the population parameter(s).",
"In this paper, we develop a new analytic formula for computing Bayes factors directly from minimal summary statistics in repeated-measures designs.",
"This work is an improvement on previous methods for computing Bayes factors from summary statistics (e.g., the BIC method), which produce Bayes factors that violate the Sellke upper bound of evidence for smaller sample sizes.",
"The new approach taken in this paper extends requires knowing only the $F$-statistic and degrees of freedom, both of which are commonly reported in most empirical work.",
"In addition to providing computational examples, we report a simulation study that benchmarks the new formula against other methods for computing Bayes factors in repeated-measures designs.",
"Our new method provides an easy way for researchers to compute Bayes factors directly from a minimal set of summary statistics, allowing users to index the evidential value of their own data, as well as data reported in published studies."
],
[
"Introduction",
"In this paper, we develop an analytic formula to compute Bayes factors for repeated-measures designs using only minimal summary statistics from the analysis of variance.",
"Previous attempts to quantify evidence from summary statistics in repeated-measures designs have all relied upon the BIC approximation [21], [4], [6].",
"In contrast, our new method avoids the need for approximation and produces an exact (analytic) Bayes factor directly from the observed $F$ -statistic and associated degrees of freedom.",
"This paper extends the development of the between-subjects Pearson Bayes factor [7] to also consider repeated-measures designs, thus widening the scope and its use for indexing evidential value from summary statistics.",
"Further, our formula is gives researchers a way to compute Bayes factors in repeated-measures designs that does not involve integration [17] or approximation [15], [6].",
"As such, users can easily compute Bayes factors for their own repeated-measures data, and also for any results that are reported in the scientific literature – even null effects.",
"Our method thus further affords researchers the ability to gauge the evidential value of a collection of published results in a straightforward way without the need for raw data."
],
[
"Background",
"We begin with some background on inference in experimental designs with repeated measurements.",
"Let us consider an experiment where $k$ repeated measurements are taken from each of $n$ experimental subjects.",
"We then have a total of $N=nk$ observations.",
"We then place a linear mixed model structure on these observations: $\\mathbf {y} = y_{ij} = \\mu + \\alpha _j + \\pi _i + \\varepsilon _{ij}; \\hspace{8.53581pt} i=1,\\dots ,n;\\hspace{2.84526pt}j=1\\cdots ,k,$ where $\\mu $ represents the grand mean, $\\alpha _j$ represents the treatment effect associated with group $j$ , $\\pi _i$ represents the effect of subject $i$ , and $\\varepsilon _{ij} \\sim \\mathcal {N}(0,\\sigma _{\\varepsilon }^2)$ .",
"Note that the repeated-measures design induces a correlated structure on the data, so the $N=nk$ observations are not independent.",
"As a result, we have $n(k-1)$ independent observations [14].",
"Ultimately, we want to know whether there is a treatment effect (i.e., if there are differences among the treatment groups induced by the repeated measurements).",
"To proceed with hypothesis testing, we define two competing models: $\\mathcal {H}_0:\\alpha _j=0\\text{ for }j=1,\\dots ,k\\\\\\mathcal {H}_1:\\alpha _j\\ne 0\\text{ for some }j$ A classical approach to model selection is the analysis of variance (ANOVA) procedure [9].",
"The analysis of variance procedure works by initially partitioning the total variance in the data $\\mathbf {y}$ into two sources: the variance between the treatment groups, and the residual variance that is left over after accounting for this treatment variability.",
"In repeated-measures designs, we further partition the residual variance by separately breaking out the variance between subjects.",
"Technically, this is done by computing $SS$ (sum of squares) terms, where $SSA = n\\sum _{j=1}^k(\\overline{y}_{\\cdot j}-\\overline{y}_{\\cdot \\cdot })^2, \\hspace{28.45274pt}SSB = k\\sum _{i=1}^n(\\overline{y}_{i\\cdot }-\\overline{y}_{\\cdot \\cdot })^2$ represent the sums of squares corresponding to the treatment effect and the subject effect, respectively; $SST = \\sum _{i=1}^n \\sum _{j=1}^k (y_{ij} - \\overline{y}_{\\cdot \\cdot })^2$ represents the total sum of squares, and $SSR = SST - SSA - SSB$ represents the residual sum of squares left over after accounting for both treatment and subject effects.",
"From here, we compute the $F$ -statistic for the treatment effect in our design as $F=\\frac{SSA}{SSR}\\cdot \\frac{df_{\\text{residual}}}{df_{\\text{treatment}}} = \\frac{SSA}{SSR}\\cdot \\frac{(n-1)(k-1)}{k-1} = \\frac{SSA}{SSR}\\cdot (n-1).$ Conceptually, the $F$ statistic represents the ratio of between-groups variance to the residual variance left over after removing the subject variability.",
"For inference, we compute the likelihood of the observed data $\\mathbf {y}$ under the null hypothesis $\\mathcal {H}_0$ .",
"Specifically, we find the probability of obtaining the observed $F$ statistic (or greater) under $\\mathcal {H}_0$ .",
"If this probability, called the $p$ -value, is small, this indicates that the data $\\mathbf {y}$ are rare under $\\mathcal {H}_0$ , so we can reject $\\mathcal {H}_0$ in favor of the alternative hypothesis $\\mathcal {H}_1$ .",
"Unfortunately, this classical approach to inference is fraught with some issues that undermine its use [21].",
"First, the $p$ -value is not equivalent to the posterior probability $p(\\mathcal {H}_0\\mid \\mathbf {y})$ .",
"Regardless, many researchers believe (incorrectly) that a $p$ -value represents the probability that $\\mathcal {H}_0$ is true [11], and thus take a small $p$ -value to represent evidence for $\\mathcal {H}_1$ .",
"However, [1] demonstrated that $p$ -values classically overestimate this evidence.",
"For example, with a $t$ -test performed on a sample size of 100, a $p$ -value of 0.05 transforms to $p(\\mathcal {H}_0\\mid \\mathbf {y})=0.52$ – rather than reflecting evidence for $\\mathcal {H}_1$ , this small $p$ -value reflects only a slight preference for $\\mathcal {H}_0$ .",
"Second, the evidence provided for $\\mathcal {H}_1$ from this procedure is only indirect – whereas the $p$ -value does measures the predictive adequacy of $\\mathcal {H}_0$ , it makes no such measurement of predictive adequacy for $\\mathcal {H}_1$ .",
"Thus, in this paper we consider a Bayesian approach to model selection.",
"Instead of computing a $p$ -value, we instead compute a Bayes factor [12].",
"The Bayes factor, denoted $\\text{BF}_{01}$ is defined as the ratio of marginal likelihoods for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , respectively.",
"That is, $\\text{BF}_{01} = \\frac{p(\\mathbf {y}\\mid \\mathcal {H}_0)}{p(\\mathbf {y}\\mid \\mathcal {H}_1)}.$ As such, $\\text{BF}_{01}$ indexes the relative likelihood of observing data $\\mathbf {y}$ under $\\mathcal {H}_0$ compared to $\\mathcal {H}_1$ , so $\\text{BF}_{01}>1$ is taken as evidence for $\\mathcal {H}_0$ over $\\mathcal {H}_1$ .",
"Similarly, $\\text{BF}_{01}<1$ is taken as evidence for $\\mathcal {H}_1$ .",
"Note that $\\text{BF}_{01} = 1/\\text{BF}_{10}$ , so if $\\text{BF}_{01}<1$ , we can take the reciprocal and equivalently write $BF_{10} > 1$ .",
"For this reason, it is canonical to report a Bayes factor as a number greater than 1.",
"For example, instead of reporting $\\text{BF}_{01} = 0.25$ , we will take the reciprocal and report $\\text{BF}_{10} = 1/0.25 = 4$ .",
"Both representations imply that the observed data are 4 times more likely under $\\mathcal {H}_1$ than under $\\mathcal {H}_0$ .",
"An equivalent definition of the Bayes factor $\\text{BF}_{01}$ is the extent to which the prior odds for $\\mathcal {H}_0$ over $\\mathcal {H}_1$ are updated after observing data.",
"Thus, the ratio of posterior probabilities for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ can be found by multiplying the ratio of prior probabilities by $\\text{BF}_{01}$ : $\\frac{p(\\mathcal {H}_0\\mid \\mathbf {y})}{p(\\mathcal {H}_1\\mid \\mathbf {y})} = \\text{BF}_{01}\\cdot \\frac{p(\\mathcal {H}_0)}{p(\\mathcal {H}_1)}.$ One immediate consequence of Equation REF is that the Bayes factor can be directly transformed into posterior probability of $\\mathcal {H}_0$ (or if desired, $\\mathcal {H}_1$ ).",
"To see this, we simply solve Equation REF for the posterior probability $p(\\mathcal {H}_0\\mid \\mathbf {y})$ and then apply Bayes' theorem.",
"This gives $p(\\mathcal {H}_0\\mid \\mathbf {y}) &= \\text{BF}_{01} \\cdot \\frac{p(\\mathcal {H}_0)}{p(\\mathcal {H}_1)} \\cdot p(\\mathcal {H}_1\\mid \\mathbf {y})\\\\&= \\frac{\\text{BF}_{01} \\cdot p(\\mathcal {H}_0) \\cdot p(\\mathbf {y}\\mid \\mathcal {H}_1)\\cdot p(\\mathcal {H}_1)}{p(\\mathcal {H}_1)\\cdot p(\\mathbf {y})}\\\\&= \\frac{\\text{BF}_{01}\\cdot p(\\mathcal {H}_0)\\cdot p(\\mathbf {y}\\mid \\mathcal {H}_1)}{p(\\mathbf {y}\\mid \\mathcal {H}_0)\\cdot p(\\mathcal {H}_0) + p(\\mathbf {y}\\mid \\mathcal {H}_1)\\cdot p(\\mathcal {H}_1)}.$ Dividing numerator and denominator by the marginal likelihood $p(\\mathbf {y}\\mid \\mathcal {H}_1)$ yields $p(\\mathcal {H}_0\\mid \\mathbf {y}) = \\frac{\\text{BF}_{01}\\cdot p(\\mathcal {H}_0)}{\\text{BF}_{01}\\cdot p(\\mathcal {H}_0) + p(\\mathcal {H}_1)}.$ By a similar argument, we can show $p(\\mathcal {H}_1\\mid \\mathbf {y}) = \\frac{\\text{BF}_{10}\\cdot p(\\mathcal {H}_1)}{\\text{BF}_{10}\\cdot p(\\mathcal {H}_1) + p(\\mathcal {H}_0)}.$ Commonly, we take a default assumption that both models are equally likely a priori, permitting us to set $p(\\mathcal {H}_0)=p(\\mathcal {H}_1) = 0.5$ .",
"In this case, we get the following simplified formulas: $p(\\mathcal {H}_0\\mid \\mathbf {y}) = \\frac{\\text{BF}_{01}}{\\text{BF}_{01}+1}, \\hspace{28.45274pt} p(\\mathcal {H}_1\\mid \\mathbf {y}) = \\frac{\\text{BF}_{10}}{\\text{BF}_{10}+1}.$ Despite the conceptual simplicity of the Bayes factor, actually computing $\\text{BF}_{01}$ is usually quite difficult.",
"From the conceptual definition, we see that the Bayes factor $\\operatorname{BF}_{01}$ is the ratio of likelihoods for $\\mathbf {y}$ under 0 and 1, respectively.",
"These likelihoods are marginal likelihoods, where each likelihood function $f_i(\\mathbf {y} \\mid \\theta _i)=f(\\mathbf {y} \\mid \\theta _i, i)$ ($i=0,1$ ) is integrated over the prior distribution $\\pi _i(\\theta _i)=\\pi (\\theta _i,i)$ for the vector of model parameters $\\theta _i$ associated with model $i$ .",
"That is, $\\operatorname{BF}_{01} = \\frac{\\int f_0(\\theta _0)\\pi _0(\\theta _0)d\\theta _0}{\\int f_1(\\theta _1)\\pi _1(\\theta _1)d\\theta _1} \\;.$ The marginal likelihoods that form the numerator and denominator of the Bayes factor are in general very difficult to compute.",
"Even if a researcher has sufficient mathematical background to consider this computation, most techniques of integration are difficult to apply.",
"As sample size increases, the likelihood becomes highly peaked at its maximum.",
"But with no prior knowledge of where the maximum occurs, numerical integration methods often fail to converge around this highly peaked area.",
"Also, the parameter vector $\\theta _i$ is often of high dimension.",
"Combined with the peaked integrand, getting numerical methodsto converge is like “finding a needle in a haystack” [20].",
"Thus, our primary goal in this work is to make simplifying assumptions which lead to analytic solutions (i.e., without integral representation) for these marginal likelihoods."
],
[
"The BIC approximation",
"One classic approach to avoid computing marginal likelihoods is to use a calculus trick to approximate them instead.",
"This method is called the BIC approximation [16], [12], [21], [14], and it works by constructing a second-order Taylor approximation of $\\ln m_i(\\mathbf {y}) = \\ln P(\\mathbf {y} \\mid \\theta _i, i)$ centered at the posterior mode for $\\theta _i$ .",
"Since all terms in the Taylor approximation containing the second-derivative or higher are removed, the result only approximates the log marginal likelihood.",
"However, [16] showed that placing a noninformative unit information prior on $\\theta _i$ , the order of the error term in $\\mathcal {O}(1/\\sqrt{N})$ .",
"This results in the approximation $\\ln m_i(\\mathbf {y}) \\approx \\ln P(\\mathbf {y} \\mid \\hat{\\theta }_i,i) - \\frac{k_i}{2}\\ln N$ where $\\hat{\\theta }_i$ is the maximum likelihood estimate for $\\theta _i$ under $i$ , $k_i$ is the number of parameters in $i$ , and $N$ is the total number of independent observations in $\\mathbf {y}$ .",
"Equation REF relates quite well to the Bayesian information criterion (BIC) of [18]: $\\operatorname{BIC}(i)=-2\\ln L_i + k_i\\ln N,$ where $L$ is the maximum likelihood estimate for model $i$ .",
"Combining these two equations yields $\\ln m_i(\\mathbf {y}) \\approx -\\frac{1}{2}\\operatorname{BIC}(i) \\; ,$ or equivalently $m_i(\\mathbf {y}) \\approx \\exp \\Bigl (-\\frac{1}{2}\\operatorname{BIC}(i)\\Bigr ).$ It is then simple to derive the following approximation: $\\operatorname{BF}_{01} = \\frac{m_0(\\mathbf {y})}{m_1(\\mathbf {y})} & \\approx \\frac{\\exp \\Bigl (-\\frac{1}{2}\\operatorname{BIC}(0)\\Bigr )}{\\exp \\Bigl (-\\frac{1}{2}\\operatorname{BIC}(1)\\Bigr )}\\\\ \\nonumber &= \\exp \\Biggl (\\frac{\\operatorname{BIC}(1) - \\operatorname{BIC}(0)}{2}\\Biggr ).$ To use Equation REF for computing Bayes factors, we need only know the BIC values for models 0 and 1.",
"In the context of analysis of variance presented earlier, the BIC can be calculated [16] as $BIC &= N\\ln (1-R^2)+k\\ln N\\\\& = N\\ln \\Biggl (\\frac{SSR}{SST}\\Biggr ) + k\\ln N\\\\ ; .$ One downside to the BIC method is that it requires users to have the “raw” data available in order to compute $SSR$ and $SST$ .",
"One way to improve Equation REF is recast the Bayes factor computation to a form that requires only summary statistics.",
"[4] initially did this for between-subjects designs, and then extended that result to the context of repeated-measures designs [6] by deriving the formula: $\\text{BF}_{01} \\approx \\sqrt{(nk-n)^{k-1}\\cdot \\Biggl (1+\\frac{F}{n-1}\\Biggr )^{n-nk}} \\; .$ The following example nicely illustrates the use of (and a problem with) Equation REF .",
"In a repeated-measures ($k=2$ ) study with $n=18$ participants, [8] observed that mean solution times to subtraction problems were 43 milliseconds faster when the subtraction sign was briefly presented before the actual problem itself, $F(1,17) = 27.17$ , $p<0.001$ .",
"Using Equation REF , we can compute the BIC Bayes factor for these observed data as: $\\text{BF}_{01} & \\approx \\sqrt{(nk-n)^{k-1}\\cdot \\Biggl (1+\\frac{F}{n-1}\\Biggr )^{n-nk}}\\\\& = \\sqrt{(18\\cdot 2-18)^{2-1}\\cdot \\Biggl (1+\\frac{27.17}{18-1}\\Biggr )^{18-18\\cdot 2}}\\\\& = \\sqrt{18^1 \\cdot \\Biggl (1 + \\frac{27.17}{17}\\Biggr )^{-18}}\\\\& = \\sqrt{18\\cdot (2.5982)^{-18}}\\\\& = 0.0007863 \\;.$ By taking the reciprocal and casting this Bayes factor as support for $\\mathcal {H}_1$ , we have $\\text{BF}_{10} \\approx 1/0.0007863 = 1271.79$ .",
"Thus, the BIC Bayes factor tells us that the observed data are approximately 1272 times more likely under $\\mathcal {H}_1$ than under $\\mathcal {H}_0$ .",
"It is important to remember that this Bayes factor is an approximation.",
"This fact is made especially salient by considering the following.",
"[19] showed that under a reasonable class of prior distributions for $p$ -values, an upper bound for the Bayes factor can be computed directly from the $p$ -value as $\\operatorname{BF}_{10} \\le -\\frac{1}{e\\cdot p\\ln (p)}.$ We will now compute this Sellke bound for our previous example.",
"First, we note that for $F(1,17)=27.17$ , the associated $p$ -value is $p=0.0000704$ .",
"This gives us the upper bound $\\operatorname{BF}_{10} & \\le -\\frac{1}{e\\cdot 0.0000704\\cdot \\ln (0.0000704)}\\\\&=546.53.$ From this, the limitation of the BIC approximation is clear.",
"Our computed Bayes factor of 1272 greatly exceeds the Sellke bound of 546.53.",
"In fact, Figure REF shows that this problem persists over a large range of $p$ -values.",
"In the figure, the solid line represents the Sellke bound $B(p)$ for $p$ -values ranging between 0 and 0.02.",
"The dashed line represents the associated repeated-measures BIC Bayes factor of [6], given design parameters equivalent to [8] (i.e., $k=2$ repeated-measures conditions and $n=18$ subjects).",
"Figure: Plot showing that the repeated-measures BIC Bayes factor (dashed line) of is greater than the Sellke bound (solid line) for pp-values ranging between p≈0p\\approx 0 and p=0.02p=0.02."
],
[
"Analytic Bayes factors for repeated-measures designs",
"Against the background of the previous section, we think the need for an analytic Bayes factor for repeated-measures designs is well motivated.",
"Fortunately, recent work by [22] takes an important first step toward this goal.",
"Wang and Sun, motivated by the the work of [10], started with a random effects linear model on the observed data $\\mathbf {y}$ : $\\mathbf {y} = y_{ij}=\\mu + \\alpha _j + \\pi _i + \\varepsilon _{ij},$ where $\\alpha _j \\sim \\mathcal {N}(0,\\sigma ^2_a)$ , $\\pi _i \\sim \\mathcal {N}(0,\\sigma ^2_p)$ and $\\varepsilon _{ij} \\sim \\mathcal {N}(0,\\sigma ^2)$ .",
"Cast in this slightly different context of random effects versus the classical fixed effects model, the analysis of variance procedure amounts to testing whether the random effects term $\\alpha _j$ is identically 0.",
"To capture this constraint, the competing models are defined in a slightly different manner: $0:\\sigma ^2_a=0 \\text{ versus }1:\\sigma _a^2\\ne 0.$ With this setup, [10] placed noninformative priors on $\\mu $ and $\\sigma $ under both 0 and 1 and considered a proper prior on the ratio of variance components $\\tau = \\sigma ^2_a/\\sigma ^2$ under 1.",
"With this prior specification, Garcia-Donato and Sun showed $BF_{10} = \\int _0^{\\infty }(1+\\tau n)^{\\frac{1-k}{2}}\\Biggl (1-\\frac{\\tau n}{1+\\tau n}\\cdot \\frac{SSA}{SST}\\Biggr )^{\\frac{1-N}{2}}\\cdot \\pi (\\tau )d\\tau $ where $\\pi (\\tau )$ is left up to the analyst to choose.",
"[22] used a Pearson Type VI distribution, given by: $\\pi ^{PT}(\\tau ) = \\frac{\\kappa (\\kappa \\tau )^{\\beta }(1+\\kappa \\tau )^{-\\alpha -\\beta -2}}{\\mathcal {B}(\\alpha +1, \\beta +1)}I_{(0,\\infty )}(\\tau )$ where $\\alpha >-1$ and $\\beta >-1$ are shape parameters and $\\kappa >0$ is a scale parameter, and $\\mathcal {B}(x,y) = \\int _0^1t^{x-1}(1-t)^{y-1}dt$ is the standard Beta function.",
"Wang and Sun further reduced the problem of specifying the prior $\\pi ^{PT}$ to choosing one single parameter $\\alpha \\in [-\\frac{1}{2}, 0]$ .",
"They did this by taking $\\kappa =n$ and $\\beta = \\frac{N-k}{2}-\\alpha -2$ .",
"A plot of this prior can be seen in Figure REF ; here, we take the values $n=18$ , $k=2$ , and $N=36$ from our example above, thus setting $\\kappa =18$ and $\\beta =\\frac{36-2}{2}-\\alpha -2$ , where $\\alpha $ ranges among the values $-\\frac{1}{2}, -\\frac{1}{4}, -\\frac{1}{10}, 0$ .",
"As we can see, as $\\alpha $ decreases from to 0 to $-\\frac{1}{2}$ , $\\tau $ becomes more dispersed and less peaked around the mode.",
"This places more prior mass on larger treatment effects than we would see for values of $\\alpha $ closer to 0.",
"Figure: A Pearson Type VI prior for τ\\tau , plotted as a function of shape parameter α\\alpha .Given this choice of prior and simplified parameterization, [22] proved that the Bayes factor derived by Garcia-Donato and Sun (Equation REF ) simplifies to an analytic expression without the need for integral representation: $\\operatorname{BF}_{10} = \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{N-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{N-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{N-k-2}{2}} \\; .$ To apply the Wang and Sun method for computing $\\text{BF}_{10}$ (Equation REF ) in a repeated-measures context, we apply a method of [14] and replace $N$ by $n(k-1)$ [3], [2], [6].",
"This readily gives: $ \\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{n(k-1)-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{n(k-1)-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{n(k-1)-k-2}{2}} \\nonumber \\\\&= \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{nk-n-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{nk-n-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{nk-n-k-2}{2}} \\; .$ We now consider some identities that will greatly simplify Equation REF .",
"First, let $x$ denote the between-groups degrees of freedom, $df_{\\text{treatment}}$ , and let $y$ denote the residual degrees of freedom, $df_{\\text{residual}}$ .",
"As seen earlier, this gives $x=k-1$ and $y=(n-1)(k-1) = nk - n - k +1$ .",
"From here, we can derive the following three identities: $ k=x+1 \\; ; $ $nk-n-k &= (nk-n-k+1)-1\\\\&= y-1 \\; ;$ $nk-n-1 &= (nk-n-k+1)+k-2\\\\&= y+(x+1)-2\\\\&= x + y - 1 \\; .$ We now substitute these identities into Equation REF , giving: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{k}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{nk-n-k}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{nk-n-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{nk-n-k-2}{2}} \\nonumber \\\\&= \\frac{\\Gamma \\Bigl (\\frac{x+1}{2}+\\alpha + \\frac{1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{SSR}{SST}\\Biggr )^{\\alpha -\\frac{(y-1)-2}{2}}\\; .$ By definition, we have $F =\\frac{SSA}{SSR}\\cdot \\frac{df_{\\text{residual}}}{df_{\\text{treatment}}} = \\frac{SSA}{SSR}\\cdot \\frac{y}{x} \\; .$ Also, in a repeated-measures design, subject variability is removed from the total sum of squares term $SST$ .",
"Thus, we can replace $SST$ with $SSA+SSR$ , giving $\\frac{SST}{SSR} &= \\frac{SSA+SSR}{SSR}\\\\&= \\frac{SSA}{SSR}+1\\\\&= \\frac{xF}{y} + 1\\\\&=\\frac{y+xF}{y} \\; .$ Taking the reciprocal of this identity and subsituting back into Equation REF proves the following proposition: Given a repeated-measures analysis of variance summary reported in the form $F(x,y)$ , where $x$ equals the between-treatments degrees of freedom and $y$ equals the residual degrees of freedom, the Bayes factor can be expressed in analytic form as $\\text{BF}_{10} = \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}} \\; ,$ where $\\alpha \\in \\Bigl [-\\frac{1}{2},0\\Bigr ]$ ."
],
[
"Example computations",
"In this section, we provide two examples of computation of the repeated-measures analytic Bayes factor given in Proposition .",
"Both examples come from [8], which was briefly mentioned in the previous section on BIC Bayes factors.",
"In their study, [8] measured adults' response times on a computerized single-digit mental arithmetic task.",
"On some problems, the arithmetic operator (e.g., the addition sign or the multiplication sign) appeared on a computer screen 150 milliseconds before the operands, whereas on other problems, the operator and operands appeared simultaneously.",
"Two critical results appeared and are worth further consideration in our example computations.",
"First, [8] observed that addition problems for which the operator appeared 150 milliseconds before the operands were solved significantly faster than those for which the operator and operands appeared simultaneously, $F(1,17) = 52.36$ , $p<0.001$ .",
"Second, they observed that this pattern did not occur on multiplication problems, $F(1,17) = 1.75$ , $p=0.20$ .",
"On the basis of this claimed null effect, Fayol and Thevenot reasoned that mental processes for addition must be fundamentally different from those involved in multiplication.",
"We can use our repeated-measures analytic Bayes factor formula to index the evidence for $\\mathcal {H}_1$ and $\\mathcal {H}_0$ that come from the addition and multiplication data, respectively.",
"First, let's compute the Bayes factor for the addition result.",
"In addition to the observed $F$ -statistic (52.36) and the relevant degrees of freedom ($x=1$ , $y=17$ ), we must also specify $\\alpha $ , the width of the prior distribution on the variance ratio $\\tau $ .",
"We will employ a “bracketing” approach and compute the Bayes factor at both ends of its consistency range; that is, we will use both $\\alpha =-1/2$ and $\\alpha =0$ .",
"Beginning with $\\alpha =-1/2$ , we substitute the summary statistics into the formula, obtaining the following: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{1}{2}-\\frac{1}{2}+1\\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (-\\frac{1}{2}+1\\Bigr )}\\Biggl (\\frac{17}{17+1\\cdot 52.36}\\Biggr )^{-\\frac{1}{2}-\\Bigl (\\frac{17-3}{2}\\Bigr )}\\\\&=\\frac{\\Gamma \\bigl (1\\bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{1}{2}\\Bigr )}\\Biggl (\\frac{17}{69.36}\\Biggr )^{-\\frac{15}{2}}\\\\&=\\frac{1 \\cdot 5040}{14034.41 \\cdot 1.772454} \\bigl ( 0.245098 \\bigr )^{-7.5}\\\\&=7702.17 \\; .$ Using Equation REF , we can convert this Bayes factor to a posterior probability.",
"Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_1$ is: $p(\\mathcal {H}_1\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{10}}{\\text{BF}_{10}+1}\\\\&= \\frac{7702.17}{7702.17+1}\\\\&= 0.99987 \\;.$ Now, we repeat this calculation, but with $\\alpha =0$ .",
"This gives the following: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{1}{2}+0+1\\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\bigl (0+1\\bigr )}\\Biggl (\\frac{17}{17+1\\cdot 52.36}\\Biggr )^{0-\\Bigl (\\frac{17-3}{2}\\Bigr )}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{3}{2}\\Bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\bigl (1 \\bigr )}\\Biggl (\\frac{17}{69.36}\\Biggr )^{-\\frac{14}{2}}\\\\&=\\frac{0.8862269 \\cdot 5040}{14034.41 \\cdot 1} \\bigl ( 0.245098 \\bigr )^{-7}\\\\&=5989.80 \\; .$ Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_1$ can be computed (using Equation REF ) as: $p(\\mathcal {H}_1\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{10}}{\\text{BF}_{10}+1}\\\\&= \\frac{5989.80}{5989.80+1}\\\\&= 0.99983 \\;.$ Thus, the data observed by [8] are between 5989.80 and 7702.17 times more likely under $\\mathcal {H}_1$ than under $\\mathcal {H}_0$ , with posterior probability for $\\mathcal {H}_1$ between 0.99983 and 0.99987.",
"In all, these data give substantial evidence for an operator priming effect on addition.",
"What can be said about the evidence for $\\mathcal {H}_0$ from their data for multiplication?",
"Recall that they did not find a significant operator priming effect for multiplication, $F(1,17)=1.75$ , $p=0.20$ .",
"We can now perform a similar computation to gauge this evidence exactly.",
"Proceeding as before with $\\alpha =-1/2$ , we obtain: $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&= \\frac{\\Gamma \\Bigl (\\frac{1}{2}-\\frac{1}{2} + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\Bigl (-\\frac{1}{2}+1\\Bigr )} \\cdot \\Biggl (\\frac{17}{17+1\\cdot 1.75}\\Biggr )^{-\\frac{1}{2}-\\frac{17-3}{2}}\\\\&=\\frac{\\Gamma \\bigl (1\\bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{1}{2}\\Bigr )}\\Biggl (\\frac{17}{18.75}\\Biggr )^{-\\frac{15}{2}}\\\\&=\\frac{1 \\cdot 5040}{14034.41 \\cdot 1.772454} \\bigl ( 0.90666667 \\bigr )^{-7.5}\\\\&=0.4225 \\; .$ Since the obtained Bayes factor is less than 1, we take the reciprocal to cast it as evidence for $\\mathcal {H}_0$ : $\\text{BF}_{01} &= \\frac{1}{\\text{BF}_{10}}\\\\&= \\frac{1}{0.4225}\\\\&=2.37 \\;.$ Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_0$ can be computed via Equation REF : $p(\\mathcal {H}_0\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{01}}{\\text{BF}_{01}+1}\\\\&= \\frac{2.37}{2.37+1}\\\\&= 0.70326 \\;.$ Similarly, we can repeat the computation with $\\alpha =0$ : $\\text{BF}_{10} &= \\frac{\\Gamma \\Bigl (\\frac{x}{2}+\\alpha + 1\\Bigr )\\cdot \\Gamma \\Bigl (\\frac{y-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{x+y-1}{2}\\Bigr )\\cdot \\Gamma (\\alpha +1)} \\cdot \\Biggl (\\frac{y}{y+xF}\\Biggr )^{\\alpha -\\frac{y-3}{2}}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{1}{2}+0+1\\Bigr ) \\cdot \\Gamma \\Bigl (\\frac{17-1}{2}\\Bigr )}{\\Gamma \\Bigl (\\frac{1+17-1}{2}\\Bigr )\\cdot \\Gamma \\bigl (0+1\\bigr )}\\Biggl (\\frac{17}{17+1\\cdot 1.75}\\Biggr )^{0-\\Bigl (\\frac{17-3}{2}\\Bigr )}\\\\&=\\frac{\\Gamma \\Bigl (\\frac{3}{2}\\Bigr ) \\cdot \\Gamma \\bigl (8\\bigr )}{\\Gamma \\Bigl (\\frac{17}{2} \\Bigr ) \\cdot \\Gamma \\bigl (1 \\bigr )}\\Biggl (\\frac{17}{18.75}\\Biggr )^{-\\frac{14}{2}}\\\\&=\\frac{0.8862269 \\cdot 5040}{14034.41 \\cdot 1} \\bigl ( 0.90666667 \\bigr )^{-7}\\\\&=0.6319 \\; .$ Taking the reciprocal gives: $\\text{BF}_{01} &= \\frac{1}{\\text{BF}_{10}}\\\\&= \\frac{1}{0.6319}\\\\&=1.58 \\;.$ Assuming equal prior odds for $\\mathcal {H}_0$ and $\\mathcal {H}_1$ , the posterior probability for $\\mathcal {H}_0$ can be computed using Equation REF to be: $p(\\mathcal {H}_0\\mid \\mathbf {y}) &= \\frac{\\text{BF}_{01}}{\\text{BF}_{01}+1}\\\\&= \\frac{1.58}{1.58+1}\\\\&= 0.61240 \\;.$ Thus, data observed by [8] are between 1.58 and 2.37 times more likely under $\\mathcal {H}_0$ than under $\\mathcal {H}_1$ , with posterior probability for $\\mathcal {H}_0$ between 0.61240 and 0.70326.",
"Despite their claim for a null priming effect on multiplication, the evidence for this claim appears to be anecdotal at best."
],
[
"Simulation",
"In this section, we report the results of a simulation study that was designed to benchmark the performance of the analytic Bayes factor in Proposition against two other repeated-measures Bayes factors: the BIC approximation of [6] and the JZS Bayes factor of [17].",
"In this simulation, we used randomly generated datasets that represented several different types of repeated-measures structure.",
"Specifically, our datasets were generated from the linear mixed model $\\mathbf {y} = y_{ij} = \\mu + \\alpha _j + \\pi _i + \\varepsilon _{ij};\\hspace{14.22636pt}i=1,\\dots ,n;\\hspace{8.53581pt}j=1,\\dots ,k,$ where $\\mu $ represents a grand mean, $a_j \\sim \\mathcal {N}(0,\\sigma _{a})$ represent each of the $k$ randomly drawn treatment effects, $\\pi _i \\sim \\mathcal {N}(0,\\sigma ^2_p)$ represent each of the $n$ randomly drawn participant effects, and $\\varepsilon _{ij} \\sim \\mathcal {N}(0,\\sigma _{\\varepsilon }^2)$ represent the normally-distributed error terms.",
"For convenience and brevity of exposition we set $k=3$ , though we saw similar results with other values of $k$ .",
"Also, without loss of generality we set $\\mu =0$ and $\\sigma _{\\varepsilon }=1$ .",
"We then systematically varied the following components of the model: The number of experimental subjects $n$ was set to either $n=10$ , $n=30$ , or $n=80$ ; The intraclass correlation $\\rho $ among the subjects' repeated measurements was set to be either low ($\\rho =0.2$ ) or high ($\\rho =0.8$ ); The size of the treatment effect was manipulated by setting $\\tau = \\sigma _a^2/\\sigma ^2$ to be either $\\tau =0$ , $\\tau =0.5$ , or $\\tau =1$ .",
"For data generated under the condition $\\tau =0$ , the correct model is the null model $0:\\tau =0$ , whereas for data generated under $\\tau =0.5$ and $\\tau =1.0$ the correct model is the alternative model $1:\\tau >0$ .",
"For each combination of number of subjects ($n=10,50,80$ ), treatment effect size ($\\tau =0,0.5,1.0$ ), repeated-measures correlation ($\\rho =0.2,0.8$ ), we generated 1000 simulated datasets.",
"For each of the datasets, we performed a repeated-measures analysis of variance, extracting the $F$ statistic and relevant degrees of freedom ($x$ =between-treatments degrees of freedom and $y$ =residual degrees of freedom).",
"Then we used these values to compute two analytic Bayes factors from Proposition : one using $\\alpha =-\\frac{1}{2}$ and another using $\\alpha =0$ .",
"We also computed the BIC Bayes factor from [6].",
"Finally, we computed the JZS Bayes factor from [17]; note that this Bayes factor can only be computed from the raw data, as the summary statistics alone are not sufficient for its estimation.",
"However, as we wish to show that our analytic Bayes factor outperforms the BIC Bayes factor previously obtained in [6], the JZS Bayes factor is an important benchmark to assess against.",
"All obtained Bayes factors were converted to posterior probabilities via Equation REF , assuming 1-1 prior model odds.",
"To compare the performance of the various computation methods in the simulation, we considered three analyses: we visualized the distribution of posterior probabilities $p(\\mathcal {H}_1\\mid \\mathbf {y})$ ; we calculated the proportion of simulated trials for which the correct model was chosen (i.e., model choice accuracy); we calculated of the proportion of simulated trials for which both methods chose the same model (i.e., model choice consistency).",
"First, let's consider the distribution of posterior probabilities $p(\\mathcal {H}_1\\mid \\mathbf {y})$ (Figure REF ).",
"Here, we constructed boxplots of the posterior probabilities for each of the four Bayes factor methods, split within plots by the number of subjects $n$ , and split across plots to represent all possible combinations of effect size $\\tau $ and repeated-measures correlation $rho$ .",
"We can see that the variability of the posterior probability estimates decreases substantially as the number of subjects $n$ increases.",
"For all simulated datasets in which 0 was the correct model (i.e., $\\tau =0$ , depicted in the first row of Figure REF ), all methods produced posterior probabilities for 1 that were reasonably small.",
"It is striking that in this case, the posterior probabilities derived from our new analytic Bayes factors as well as the BIC approximation of [6] are less than the posterior probabilities derived from the JZS Bayes factor.",
"Note also that setting the prior width to $\\alpha =-1/2$ produces the smallest posterior probabilities.",
"We also note that the separation in performance between the four methods decreases with increasing numbers of subjects $n$ .",
"For datasets in which 1 was the correct model (i.e., $\\tau =0.5$ and $\\tau =1.0$ ; rows 2 and 3 of Figure REF ), a different pattern of results emerged.",
"In these cases, the JZS Bayes factor produced posterior probabilities closer to 1 than did the analytic or BIC Bayes factors.",
"Whereas setting $\\alpha =-1/2$ was preferred in the $\\tau =0$ case, these data reveal that $\\alpha =0$ was the preferred setting when $\\tau > 0$ .",
"Finally, we note that repeated-measures correlation $\\rho $ had little effect on the pattern of posterior probabilities that was observed.",
"Figure: Boxplots depicting the distribution of the posterior probabilities p(ℋ 0 ∣𝐲)p(\\mathcal {H}_0\\mid \\mathbf {y}) for 1000 simulated datasets, split by number of subjects nn (within plots) and effect size τ\\tau and repeated-measures correlation ρ\\rho (across plots).",
"White and light-gray boxes represent the analytic Bayes factor with α=-1 2\\alpha =-\\frac{1}{2} and 0, respectively.",
"Medium gray boxes represent the BIC Bayes factor of .",
"Black boxes represent the JZS Bayes factor of .For the next analysis, we note that even though the distributions of posterior probabilities appear to follow the same pattern across methods, it is unclear whether the analytic Bayes factor proposed in this paper provides the user with correct model choice.",
"Since the data are simulated from target parameters, it is possible for us to gauge this performance exactly.",
"For simulated datasets where $\\tau =0$ , the correct model is $\\mathcal {H}_0$ , whereas when $\\tau =0.5$ or $\\delta =1.0$ , the correct model is $\\mathcal {H}_1$ .",
"Thus, to compare performance of these Bayes factor methods, we calculated model choice accuracy, defined simply as the proportion of the 1000 simulated datasets for which the correct model (0 or 1 was chosen.",
"Model choice was defined by considering $\\mathcal {H}_0$ to be chosen whenever $\\text{BF}_{01}>1$ and $\\mathcal {H}_1$ to be chosen whenever $\\text{BF}_{01}<1$ .",
"The results are displayed in Table REF .",
"Table: Model choice accuracy calculated as the proportion of simulated datasets for which the correct model was chosen.",
"Accuracies are presented as a function of Bayes factor method (analytic with α=-1 2\\alpha =-\\frac{1}{2}, analytic with α=0\\alpha =0, BIC, and JZS), numbers of subjects (n=10,30,80n=10,30,80), effect size (τ=0,0.5,1.0\\tau =0,0.5,1.0), and repeated-measures correlation (ρ=0.2,0.8\\rho =0.2,0.8).All methods were reasonably accurate at choosing the correct model.",
"We observed the highest accuracy for the case where $\\tau =0$ ; here, the analytic methods and the BIC method outperformed the JZS method (mirroring what we saw with the distributions of posterior probabilities in Figure REF ).",
"As before, setting $\\alpha =-1/2$ for our analytic Bayes factor was the most accurate in this case.",
"Accuracy when choosing 1 declined for all methods when $\\tau >0$ .",
"Again, we see that when 1 is the correct model, the JZS Bayes factor is more accurate with respect to model selection.",
"However, we note that setting $\\alpha =0$ increased accuracy relative to $\\alpha =-1/2$ .",
"Finally, we note that accuracy increased with increasing number of subjects $n$ , and there was very little variation between our two repeated-measures correlation settings ($\\rho =0.2$ and $\\rho =0.8$ ).",
"Finally, we calculated model choice consistency, defined as the proportion of simulated datasets for which each of the methods based on summary statistics alone (analytic and BIC) chose the same model as the JZS Bayes factor.",
"This analysis confirms what we observed in the previous two analyses: (1) consistency across methods increases with increasing numbers of subjects $n$ ; and (2) setting $\\alpha =0$ gives Bayes factors that are more consistent with the JZS Bayes factor.",
"Table: Model choice consistency calculated as the proportion of simulated datasets for which each method chose the same model as the JZS Bayes factor.",
"Proportions are presented as a function of Bayes factor method (analytic with α=-1 2\\alpha =-\\frac{1}{2}, analytic with α=0\\alpha =0, and BIC), numbers of subjects (n=10,30,80n=10,30,80), effect size (τ=0,0.5,1.0\\tau =0,0.5,1.0), and repeated-measures correlation (ρ=0.2,0.8\\rho =0.2,0.8)."
],
[
"Conclusion",
"In this paper, we developed an analytic Bayes factor for repeated-measures analysis of variance designs.",
"This Bayes factor gives researchers the ability to obtain Bayes factors directly from a minimal set of summary statistics (namely, the $F$ score and the degrees of freedom).",
"This formula improves upon the repeated-measures BIC Bayes factor formula of [6] in two ways.",
"First, this new analytic formula provides the user with an exact Bayes factor instead of an approximation.",
"Second, our method gives the user the ability to tune the prior used in the computation of the Bayes factor by specifying a hyperparameter $\\alpha $ , which controls the width of the prior distribution of effect sizes $\\tau =\\sigma ^2_a/\\sigma ^2$ .",
"Our simulation study shows that our analytic Bayes factor performs well compared to the JZS Bayes factor of [17] (especially when the prior parameter $\\alpha $ is set to 0).",
"Remarkably, our analytic Bayes factor outperforms the JZS Bayes factor when data are generated under 0.",
"We note that the analytic Bayes factor did not perform quite as well as the JZS Bayes on data generated under 1, but we think this limitation is far outweighed by ease of use of our method.",
"First, our analytic Bayes factor has the unique ability to be computed directly from summary statistics, with no need for raw data or the need to compute a multi-dimensional integral.",
"For this reason, we propose that our analytic Bayes factor will be an invaluable tool for anyone who wishes to assess the evidential value of their own data, as well as data from published studies where raw data is not readily available."
]
] | 2209.08159 |
[
[
"Modeling Quantum Enhanced Sensing on a Quantum Computer"
],
[
"Abstract Quantum computers allow for direct simulation of the quantum interference and entanglement used in modern interferometry experiments with applications ranging from biological sensing to gravitational wave detection.",
"Inspired by recent developments in quantum sensing at the Laser Interferometer Gravitational-wave Observatory (LIGO), here we present two quantum circuit models that demonstrate the role of quantum mechanics and entanglement in modern precision sensors.",
"We implemented these quantum circuits on IBM quantum processors, using a single qubit to represent independent photons traveling through the LIGO interferometer and two entangled qubits to illustrate the improved sensitivity that LIGO has achieved by using non-classical states of light.",
"The one-qubit interferometer illustrates how projection noise in the measurement of independent photons corresponds to phase sensitivity at the standard quantum limit.",
"In the presence of technical noise on a real quantum computer, this interferometer achieves the sensitivity of 11\\% above the standard quantum limit.",
"The two-qubit interferometer demonstrates how entanglement circumvents the limits imposed by the quantum shot noise, achieving the phase sensitivity 17\\% below the standard quantum limit.",
"These experiments illustrate the role that quantum mechanics plays in setting new records for precision measurements on platforms like LIGO.",
"The experiments are broadly accessible, remotely executable activities that are well suited for introducing undergraduate students to quantum computation, error propagation, and quantum sensing on real quantum hardware."
],
[
"Introduction",
"Quantum systems at the forefront of modern research increasingly utilize entanglement for goals ranging from precision sensing to quantum simulation and computation.",
"Prominently, quantum computers employ entanglement to push beyond the capabilities of classical computers, providing a new set of tools that promise to solve complex problems across multiple fields.",
"On the way to more advanced research goals, first generation quantum computers also serve as educational tools.",
"The IBM Quantum Experience, for example, is an online quantum computing platform that provides the opportunity for students to interact with real quantum hardware [1], [2], [3], [4].",
"Here we use the IBM Quantum Experience to teach students about quantum metrology, one of the first areas of quantum science where entanglement has found practical applications.",
"Applications for entanglement in quantum metrology include spin squeezed clocks[5], enhanced biological imaging[6], remote sensing with EPR states [7], and the detection of gravitational waves [8], [9], [10].",
"Our focus for this paper will be on the particularly charismatic example of LIGO [11], which made the first detection of gravitational waves in 2015, realizing a century-old prediction by Einstein [8].",
"At the heart of LIGO is the most sensitive interferometer ever built, whose sensitivity has recently been enhanced by quantum squeezing of light [9].",
"LIGO clearly illustrates the quantum nature of interferometers and the role that entanglement can play in modern metrology.",
"Here, we describe how to use the IBM Quantum Experience to understand the quantum physics underpinning discoveries at LIGO.",
"In Sec.",
", we construct a single-qubit quantum circuit representation of the LIGO interferometer by analyzing the behavior of a photon traveling through the interferometer.",
"In Sec.",
", we present two representations of this quantum circuit - matrices and Bloch spheres - which provide precise mathematical intuition for the behavior of the quantum circuit.",
"We test predictions made by these representations against the results of experiments performed on the IBM Quantum Experience in Sec. .",
"These results allow us to model the nature of quantum noise in the interferometer.",
"We present a calibrated measurement of the sensitivity of the interferometer in Sec. .",
"The tools developed through analysis of the single-qubit interferometer circuit allow us to efficiently assess the enhancement that can be provided by quantum entanglement.",
"In Sec.",
", we present an interferometer circuit with entanglement between two qubits: the simplest possible example of quantum-enhanced metrology.",
"We model the expected behavior of this circuit using matrix mechanics and visualize the expected dynamics using spin Wigner functions [12] before implementing the circuit on a quantum computer.",
"We show in Sec.",
"that entanglement reduces the noise of this interferometer below the standard quantum limit, even in the presence of experimental imperfections.",
"We present our conclusions in Sec.",
"."
],
[
"LIGO as a quantum circuit",
"LIGO utilizes laser interferometry to measure the microscopic spacetime distortions caused by gravitational waves.",
"As shown in Fig.",
"REF , these distortions change the relative length of LIGO's two arms and thus change the interference signal on the detector.",
"The resulting signals are so small that measurements in the frequency bands relevant to the final moments of black hole and neutron star mergers are limited by the inherent quantum shot noise of the laser used in the interferometer [13].",
"Thus there is value in incorporating quantum mechanics into the description of the interferometer, going beyond the classical framework [11] of understanding interferometers with electromagnetic waves.",
"With only one qubit, a quantum circuit model can capture all the salient features of a classical model and demonstrate the effects of quantum projection noise.",
"Repeatedly executing this circuit simulates measurements of the independent photons that make up a coherent laser beam.",
"The quantum circuit model also provides a scalable framework that can incorporate entanglement, which is necessary for quantum enhanced measurement [14].",
"To construct a quantum circuit model of LIGO, we pare the complicated optical system down to a Michelson interferometer, consisting of only a laser, a beam splitter and two mirrors.",
"This geometry, depicted in Fig.",
"REF , captures the essential physics of the system and is often used to illustrate the operation of LIGO [11], [15].",
"We map the two directions that a photon travels in the interferometer to the two states of a single qubit in the quantum circuit.",
"A photon traveling parallel to the laser corresponds to the qubit state $|0\\rangle $ , and a photon traveling in the perpendicular direction corresponds to the state $|1\\rangle $ .",
"We deduce the gates that make up the quantum circuit by analyzing the behavior of each of the optical elements that a photon encounters while traveling through the interferometer.",
"This procedure is similar to that used by Nielsen and Chuang to construct a Mach–Zehnder interferometer in the context of dual-rail quantum computation (Section 7.4.2) [16].",
"A photon in the interferometer is initially emitted by a monochromatic laser source and travels towards the beam splitter.",
"The initial direction of the photon is represented by the quantum state $|0\\rangle $ .",
"At the beam splitter, the photon has 50% chance of reflection.",
"This creates an equal superposition of the photon continuing to travel in the same direction (quantum state $|0\\rangle $ ) and being reflected into the perpendicular direction (quantum state $|1\\rangle $ ), which precisely maps onto the action of a Hadamard gate on a single qubit.",
"After the beam splitter, each state accumulates a phase that is proportional to the distance traveled in the respective arm.",
"A difference in length of the two arms, which can be caused by a gravitational wave, introduces a phase difference $\\phi $ between the two states.",
"This is analogous to the action of an $R_Z(\\phi )$ gate.",
"After retro-reflection at the end of each arm, the photon returns for a second pass through the beam splitter, which is again represented by a Hadamard gate.",
"After the beam splitter, photons in the state corresponding to $|1\\rangle $ are measured on the detector, constituting a measurement in the qubit's computational ($z$ ) basis.",
"Putting together the steps of the photon's journey, the full quantum circuit that represents the LIGO interferometer is $@C=1.0em @R=0.2em @!R {{ {q}_{0} : } & {\\left|0\\right\\rangle } & {\\mathrm {H}} & {\\mathrm {R}_\\mathrm {Z}\\,\\mathrm {(}\\mathrm {\\phi }\\mathrm {)}} & {\\mathrm {H}} & & & \\\\{c:} & {/_{_{1}}} & & & & {_{_{0}}} [-1] & & \\\\}$ This circuit allows us to directly simulate the behavior of a Michelson interferometer on a quantum computer."
],
[
"Expected Behavior of the Single-Qubit Circuit",
"In order to formulate a precise quantum picture of a Michelson interferometer like LIGO, we require a mathematical framework that can predict the behavior of the quantum circuit presented in the prior section.",
"Quantum matrix mechanics provides this framework, but is mathematically removed from a physical or graphical interpretation.",
"We thus present the Bloch sphere representation alongside the matrix representation to help visualize the action of the quantum gates on the qubit.",
"For a single qubit, both complementary representations independently provide a precise accounting of the quantum dynamics.",
"The matrix notation for quantum states represents a qubit as a $2\\times 1$ vector whose entries are complex numbers.",
"Quantum gates are represented by $2\\times 2$ matrices, which act on the state vector through matrix multiplication.",
"Following the convention of Nielsen and Chuang [16], the two basis states $|0\\rangle $ and $|1\\rangle $ are given by $|0\\rangle = \\begin{bmatrix}1 \\\\0\\end{bmatrix} \\text{and }|1\\rangle = \\begin{bmatrix}0 \\\\1\\end{bmatrix}.$ An arbitrary superposition of $|0\\rangle $ and $|1\\rangle $ is given by $|\\psi \\rangle = c_0|0\\rangle + c_1|1\\rangle $ and can be written as the vector $|\\psi \\rangle = \\begin{bmatrix}c_0 \\\\c_1\\end{bmatrix}.$ The squared magnitudes of the coefficients $|c_0|^2$ and $|c_1|^2$ give the probabilities of measuring the qubit in the $|0\\rangle $ and $|1\\rangle $ states, respectively.",
"In the context of the interferometer, only photons in the $|1\\rangle $ state (traveling perpendicularly to the laser output beam) reach the detector whereas photons in the $|0\\rangle $ state (traveling along the laser output beam) return to the laser without being detected.",
"The probabilities $p_1 = |c_1|^2$ and $p_0 = |c_0|^2$ hence respectively correspond to the probabilities of detecting and not detecting each photon.",
"Information about the quantum superposition is encapsulated in the relative phase $\\phi = \\phi _1 - \\phi _0$ between the two complex coefficients, where $\\phi _0$ and $\\phi _1$ are defined by the expression of the complex coefficients in polar form: $c_0 = |c_0|e^{i\\phi _0}$ and $c_1 = |c_1|e^{i\\phi _1}$ .",
"The quantum state $|\\psi \\rangle $ of a qubit can also be represented on the Bloch sphere as seen in Fig.",
"REF .",
"The Bloch sphere is a unit sphere where a vector from the origin to the surface of the sphere represents a single-qubit state.",
"This vector is known as the Bloch vector.",
"The state $|0\\rangle $ is represented by a spin pointing up ($+z$ ) and $|1\\rangle $ is represented by a spin pointing down ($-z$ ).",
"For superpositions of $|0\\rangle $ and $|1\\rangle $ , the $z$ coordinate is given by the relative probability of measuring each of the two states, $p_0 - p_1$ .",
"The angle in the $xy$ -plane is given by the quantum phase $\\phi $ , with $\\phi =0$ corresponding to the $x$ -axis.",
"Quantum gates generate rotations of the state vector on the Bloch sphere.",
"Figure: Visualization on the Bloch sphere of the quantum state of the single-qubit circuit after each gate is applied.",
"The arrow shown on each sphere represents the quantum state |ψ〉=c 0 |0〉+c 1 |1〉.|\\psi \\rangle = c_0|0\\rangle + c_1|1\\rangle .",
"The zz coordinate is the relative probability p 0 -p 1 p_0 - p_1 of measuring |0〉|0\\rangle or |1〉|1\\rangle while the azimuthal angle is the relative phase φ\\phi between the complex coefficients c 0 c_0 and c 1 c_1.",
"The initial state |0〉|0\\rangle is a vector pointing up.",
"The Hadamard gate rotates the state by π\\pi about the zz-axis, then by π/2\\pi /2 about the yy-axis.",
"The R Z (φ)R_Z(\\phi ) gate rotates the state about the zz-axis, with φ=π/3\\phi =\\pi /3 chosen for illustration.",
"The final Hadamard gate maps xx-axis back onto the z-axis, which allows φ\\phi to be measured.During the sequence, the qubit is initialized in the state $|0\\rangle $ , represented by a vector pointing up on the Bloch sphere in Fig.",
"REF a.",
"The first gate applied to this initial state is a Hadamard gate $H$ .",
"The matrix form of $H$ is $H = \\frac{1}{\\sqrt{2}}\\begin{bmatrix}1 & 1 \\\\1 & -1\\end{bmatrix}.$ On the Bloch sphere, the Hadamard gate first rotates the state by $\\pi $ about $z$ and then by $\\pi /2$ about $y$ .",
"This transforms the initial state $|0\\rangle $ into an equal superposition of $|0\\rangle $ and $|1\\rangle $ with a relative phase of 0, $|\\psi \\rangle = \\frac{1}{\\sqrt{2}}\\begin{bmatrix}1 \\\\1\\end{bmatrix}.$ This state vector lies along the $x$ -axis of the Bloch sphere.",
"Following the Hadamard gate, the $R_Z(\\phi )$ gate rotates the state about the $z$ -axis by an angle $\\phi $ .",
"As an example, a rotation by $\\phi = \\pi /3$ is depicted in Fig.",
"REF .",
"In the matrix formulation, the $R_Z(\\phi )$ gate introduces a relative phase of $\\phi $ between the two complex coefficients $c_0$ and $c_1$ via the action of the matrix $R_Z(\\phi ) = \\begin{bmatrix}e^{-i\\phi /2} & 0 \\\\0 & e^{i\\phi /2}\\end{bmatrix}.$ For an interferometer like LIGO, this relative phase $\\phi $ is induced by a path length difference of $\\Delta L = \\phi \\lambda /2\\pi $ where $\\lambda $ is the wavelength of light used in the interferometer.",
"Because the quantum state changes based on the value of $\\phi $ , the detector is sensitive to the effects of gravitational waves.",
"In order to map the phase accumulated due to the $R_Z(\\phi )$ back into the measurement basis ($z$ ), we apply a final Hadamard gate.",
"This gate maps the $x$ -axis back to the $z$ -axis so that phase accumulations in the $xy$ -plane manifest as changes in expectation value for measurements in the standard basis of $|0\\rangle $ and $|1\\rangle $ .",
"Mathematically, we compute the resulting state of the quantum circuit by multiplying together all of the gates so that the final state is $|\\psi \\rangle = HR_Z(\\phi )H|0\\rangle $ , which evaluates to $|\\psi \\rangle =\\frac{1}{2}\\begin{bmatrix}e^{-i\\phi /2} + e^{i\\phi /2}\\\\e^{-i\\phi /2} - e^{i\\phi /2}\\end{bmatrix}=\\begin{bmatrix}\\cos {(\\phi /2)}\\\\-i\\sin {(\\phi /2)}\\end{bmatrix}.$ Here, the probabilities of measuring the final state of the circuit as $|0\\rangle $ and $|1\\rangle $ are given by $p_0 = \\cos ^2(\\phi /2)$ and $p_1 = \\sin ^2(\\phi /2)$ , respectively.",
"We summarize the result of this measurement by defining the polarization operator $\\hat{P}$ such that the polarization is equal to -1 if we measure $|0\\rangle $ and equal to 1 if we measure $|1\\rangle $ .",
"Up to the overall sign, this is the $z$ coordinate of the Bloch vector.",
"In matrix form, the polarization operator is $\\hat{P} = \\begin{bmatrix}-1 & 0 \\\\0 & 1\\end{bmatrix}.$ For the final state $|\\psi \\rangle $ from Eq.",
"(REF ), the expectation value of the polarization operator is $\\langle \\psi | \\hat{P} |\\psi \\rangle = p_1 - p_0 = \\frac{1}{2}\\sin ^2(\\phi /2) - \\frac{1}{2}\\cos ^2(\\phi /2) = -\\cos {\\phi }$ This sinusoidal dependence on $\\phi $ is consistent with the behavior of the $z$ -coordinate of the final state visualized in Fig.",
"REF , where the final state's Bloch vector makes an angle $\\phi $ with respect to the $z$ axis.",
"In the context of LIGO, this dependence on $\\phi $ facilitates sensitivity to signals from gravitational waves.",
"The mapping between the LIGO interferometer and the quantum circuit allows us to simulate the detection of gravitational waves on the quantum computer and better understand the impacts of both fundamental and technical noise sources."
],
[
"Implementing on a quantum computer",
"We implemented the quantum circuit model for LIGO on the IBM Quantum Experience.",
"Quantum circuits were initially constructed manually via the circuit composer interface, which allows quick prototyping without the need for any programming experience.",
"To scale up the number experimental trials and systematically scan through experimental parameters, we used the Qiskit interface, which allows users to program quantum circuits in Python [17].",
"The code for interfacing with the quantum computers was adapted from existing examples utilizing IBM Quantum Experience [1], [3], [17].",
"The code used for all experiments in this paper is available online [18].",
"Experiments were performed on both the quantum computer ibmq_manila and the state vector simulator ibmq_qasm_simulator.",
"The quantum computer ibmq_manila was chosen because, with a quantum volume of 32, it had the largest quantum volume among freely accessible quantum computers [19].",
"The quantum volume is IBM's preferred metric for its quantum computers and is a proxy for overall error rates; it measures the size of the largest quantum circuit that can be executed with acceptable fidelity [20].",
"The simulator ibmq_qasm_simulator explicitly computes the final qubit state and then samples outcomes from the corresponding probability distribution.",
"This provides a reference point for the effects of finite quantum volume and allows us to distinguish the effects of statistical noise from technical noise sources on the quantum computer.",
"Figure REF shows plots of polarization $P$ as a function of $\\phi $ , measured by running the single-qubit circuit from Eq.",
"(REF ) on both the quantum computer ibmq_manila and the simulator ibmq_qasm_simulator.",
"Each data point is obtained by repeatedly executing the quantum circuit and tallying the measurement outcomes 0 and 1 from each shot.",
"A single trial consists of 1024 shots from which we calculate the proportions of shots $p_0$ and $p_1$ with each measurement outcome.",
"Each data point shown is the average polarization $P = p_1 - p_0$ from 5 trials conducted at a fixed value of $\\phi $ .",
"The variability among the 5 trials is smaller than the size of the data points.",
"Figure: Measured polarization P=p 1 -p 0 P = p_1 - p_0 as a function of φ\\phi , computed from the data taken on the quantum computer (purple circle) and the simulator (green square) at 13 angles between 0 and 2π2\\pi , along with the best-fitted curves.",
"Error bars are smaller than the data points.",
"The curves are fitted to the data with amplitude and offset as free parameters.To analyze the relationship between $P$ and $\\phi $ , we fit a cosine curve to the data, with amplitude and vertical offset as free parameters.",
"The fit to the simulator data has an amplitude of 0.999(1) and an offset of -0.0001(7), which are consistent with the expected values of 1 and 0 to within the standard errors reported by the fitting routine (shown in parentheses).",
"This affirms the validity of the fitting routine.",
"The fit to the experimental data from the quantum computer has an amplitude of 0.932(5) and an offset of -0.042(4).",
"The amplitude is reduced from the expected value, mainly due to deviations from theory at positive values of polarization.",
"The majority of these deviations can be explained by readout errors.",
"The readout errors on ibmq_manila vary between qubits and over time, but the overall error rate is typically reported between 2 and 3 percent [19], which corresponds to 4-6% reductions in the amplitude of the polarization.",
"The offset can be explained by unequal rates of the two types of readout error, where the dominant source of error is relaxation from the excited state $|1\\rangle $ to the ground state $|0\\rangle $ during measurement [19].",
"IBM reports an error rates of $2\\times 10^{-4}$ for the main component of Hadamard gate, the $\\sqrt{X}$ gate.",
"This error rate is approximately one hundred times smaller than the readout error rates and thus a negligible contribution to the overall error.",
"Additional errors can come from the $R_z$ gate, whose error rate is not reported by IBM.",
"These results verify that measurements of the single-qubit interferometer circuit depend on the interferometer phase $\\phi $ as expected after incorporating corrections due to dissipation in the real quantum system.",
"Dissipation primarily manifests as readout errors that reduce the amplitude of the polarization curve.",
"The corrections to the curve are important to note as we next seek to invert the relationship and extract the phase of the interferometer from measurement outcomes."
],
[
"Shot noise and the Measurement of Gravitational Waves",
"The sensitivity of an interferometer is defined by the smallest phase shift that can be reliably detected in the measured output of the interferometer.",
"In general, this phase shift is a change from an initial interferometer phase $\\phi _0$ to a new interferometer phase $\\phi = \\phi _0 + \\delta \\phi $ .",
"At LIGO, $\\delta \\phi $ is induced by gravitational waves and $\\phi _0$ corresponds to the initial phase in the absence of gravitational waves.",
"On the quantum computer we can choose $\\delta \\phi $ and $\\phi _0$ arbitrarily to illustrate how shot noise limits the precision with which an interferometer can measure phase shifts.",
"Since $\\delta \\phi $ is small, we can usefully approximate the cosine relationship between polarization and $\\phi $ as locally linear, as depicted in the center panel of Fig.",
"REF .",
"Using this linear relationship, polarization measurements can be used to produce an estimate $\\tilde{\\phi }$ of the phase of the interferometer $\\phi = \\phi _0 + \\delta \\phi $ .",
"Error propagates from measurements of the polarization to the inferred values of $\\tilde{\\phi }$ with a scale factor given by the slope $m$ of the line, where $m=\\sin (\\phi _0)$ for an ideal interferometer.",
"We can express the standard error of the inferred phase $\\tilde{\\phi }$ in terms of the standard deviation of polarization measurement as $\\sigma _{\\tilde{\\phi }} = \\frac{1}{m}\\sigma _{P}.$ Using this expression for the standard deviation of the inferred phase, we can determine theoretically how the precision of the interferometer should scale with number of qubits $N$ that are included in the measurement of average polarization $P$ from a single trial.",
"Here $P = \\frac{1}{N}\\sum _{i=1}^N P_i$ , where $P_i$ represents the measurement of an individual qubit.",
"We can compute $\\sigma _{P} = \\sqrt{\\operatorname{Var}(P)}$ by determining the variance of a polarization measurement.",
"Since each photon is independent and $1/N$ is a constant factor, the variance can be expressed as a sum, $\\operatorname{Var}(P) = \\frac{1}{N^2}\\sum _{i=1}^N \\operatorname{Var}(P_i).$ Using expectation values of the single-qubit polarization operator, we deduce that the variance of a single measurement of an ideal interferometer is $\\operatorname{Var}(P_i) = \\langle P_i^2 \\rangle - \\langle P_i \\rangle ^2 = 1 - \\cos ^2\\phi = \\sin ^2 \\phi .$ Here $\\langle P_i^2 \\rangle = 1$ because both polarization measurement outcomes square to 1, and $\\langle P_i \\rangle = -\\cos {\\phi }$ as derived in Eq.",
"(REF ).",
"Thus, $\\operatorname{Var}(P) = \\sin ^2 \\phi /N$ and $\\sigma _P = \\sin \\phi /\\sqrt{N}$ .",
"Substituting back into Eq.",
"(REF ), the phase sensitivity is $\\sigma _\\phi = \\frac{1}{\\sqrt{N}},$ which is equal to the standard quantum limit and is independent of $\\phi $ .",
"Figure: Polarization measurements at φ=π/2+0.2\\phi = \\pi /2 + 0.2 (right histogram) and the corresponding inferred phases φ ˜\\tilde{\\phi } (top histogram) obtained by inverting the linear approximation (purple solid line) of the best-fitted PP vs φ\\phi cosine curve (black solid curve) at φ 0 =π/2\\phi _0 = \\pi /2.",
"The vertical and horizontal dotted lines represent the original phase value φ 0 =π/2\\phi _0 = \\pi /2 and the corresponding polarization P(φ=π/2)P(\\phi = \\pi /2), respectively.",
"The black curve is the ibmq_manila cosine fit shown earlier in Fig. .",
"The data were taken from 75 sequential polarization measurements, each consisting of N=100N=100 shots.Figure REF shows how shot noise manifests in an interferometer implemented on a quantum computer.",
"The central panel of the figure depicts the experimentally determined relationship between interferometer phase and polarization and a linear approximation of this curve around $\\phi _0 = \\pi /2$ .",
"This initial phase and the corresponding polarization are depicted as dashed lines in the figure.",
"We illustrate the sensitivity of the interferometer to a phase shift $\\delta \\phi = 0.2$ by repeatedly measuring the polarization at the output of a quantum circuit where the interferometer phase was set to $\\phi = \\pi /2 + 0.2$ .",
"From each measurement of the polarization, we obtain an estimate $\\tilde{\\phi }$ using the previously determined relationship between the two variables.",
"Repeatedly measuring polarization illustrates the uncertainty in each measurement due to shot noise and other technical noise sources.",
"Each polarization measurement consists of $N=100$ shots.",
"The histogram shown on the right margin is constructed from 75 independent measurements of probability to illustrate variability.",
"The measured polarization has a standard deviation of 0.109(9), which is within error of theoretical value $\\sigma _P = \\sin {\\phi }/\\sqrt{N}$ at $\\phi = \\pi /2 + 0.2$ .",
"The distribution of polarization maps to a distribution of the inferred values of $\\tilde{\\phi }$ which has a mean of $\\pi /2 + 0.20(1)$ and standard deviation of 0.117(10).",
"The mean is consistent with the initially chosen $\\delta \\phi $ , showing that $\\tilde{\\phi }$ is an unbiased estimator for the phase shifts.",
"The width of the distribution of $\\tilde{\\phi }$ is the sensitivity of the interferometer to phase shifts.",
"The estimated phase sensitivity of 0.117(10) is consistent with shot noise for $N=100$ shots combined with the factor of $1/m = 1.073$ due to the reduced amplitude of the polarization curve as compared to Eq.",
"(REF ).",
"The model we derived in Eq.",
"(REF ) for the shot noise of independent photons predicts that the phase sensitivity scales with the number of shots $N$ as $\\sigma _{\\tilde{\\phi }} \\propto 1/\\sqrt{N}$ .",
"To verify this scaling, we repeat the procedure performed in Fig.",
"REF for values of $N$ between 1 and 1024.",
"We plot the measured sensitivities $\\sigma _{\\tilde{\\phi }}$ in Fig.",
"REF .",
"Shot noise corresponds to a slope of $-1/2$ on the log-log scale depicted in this figure.",
"For all $N$ , the measured sensitivities are slightly above shot noise, but follow the expected scaling, with a slope of -0.506(5) on the log-log plot.",
"On average, the plotted sensitivities are 11(2)% above shot noise, of which 7.3% is due to the reduced amplitude of polarization curve from Fig.",
"REF .",
"The remaining deviation is not statistically significant, but may be due to weak correlations between successive shots, which violate the assumption of independence required for Eq.",
"(REF ).",
"Overall, the results are consistent with the expected scaling for a predominantly shot noise limited device in the presence of some technical noise.",
"The sensitivity of this interferometer is thus primarily limited by the number of qubits queried.",
"This corresponds to noise scaling with the photons or optical power at LIGO.",
"Figure: Standard deviation of inferred φ ˜\\tilde{\\phi } from polarization data taken on ibmq_manila as a function of number of qubits NN.",
"The least-squares best fit line for the log-log plot is shown in solid purple.",
"All data points lie slightly above the standard quantum limit (dashed line) 1/N1/\\sqrt{N}."
],
[
"Modeling Quantum Enhanced Measurement",
"LIGO now utilizes entanglement to improve sensitivity without further increasing optical power.",
"Specifically, LIGO now uses light in a squeezed quantum state where photons are entangled to each other [9].",
"The correlations between the photons in this state improve sensitivity to phase at the expense of amplitude uncertainty.",
"Due to these correlations, the photons are not independent and can circumvent the standard quantum limit.",
"This manipulation of quantum uncertainty is widely used in quantum metrology.",
"Many examples of metrologically useful states can modeled using the language of quantum circuits [14].",
"The simplest form of entanglement involves two qubits.",
"We can create a two-qubit entangled state on a quantum computer by using a combination of $H$ , $X$ , and CNOT gates.",
"Specifically, the controlled not gate (CNOT) flips a target qubit (labeled with $\\bigoplus $ ) based on the state of a control qubit (labeled with $\\bullet $ ).",
"This allows the two qubits to interact, producing an entangled state that can be used as the input to an interferometer circuit in order to achieve improved sensitivity [21].",
"The full circuit is $@C=0.75em @R=0.2em @!R { \\\\{ {q}_{0} : } & { {q}_{0} : } & {\\mathrm {\\left|0\\right\\rangle }} & {\\mathrm {H}} & {1} & {\\mathrm {H}} & {\\mathrm {R_Z}\\,(\\mathrm {\\phi })} & {\\mathrm {H}} & & & & \\\\{ {q}_{1} : } & { {q}_{1} : } & {\\mathrm {\\left|0\\right\\rangle }} & {\\mathrm {X}} & & {\\mathrm {H}} & {\\mathrm {R_Z}\\,(\\mathrm {\\phi })} & {\\mathrm {H}} & & & & \\\\{c:} & {c:} & {/_{_{2}}} & & & & & & {_{_{0}}} [-2] & {_{_{1}}} [-1] & & \\\\}$ To understand this circuit's dynamics, we first construct a mathematical representation of two-qubit states.",
"An arbitrary two-qubit state can be expressed as a superposition of four two-qubit basis states $|\\psi \\rangle = c_0 |00\\rangle + c_1|01\\rangle + c_2|10\\rangle + c_3 |11\\rangle $ .",
"The corresponding matrix notation is the $4 \\times 1$ vector $[c_0, c_1, c_2, c_3]^T$ .",
"Here each basis state is the tensor product of single-qubit basis states for qubits $q_0$ and $q_1$ .",
"For example, $|01\\rangle = |0\\rangle _0 \\otimes |1\\rangle _1$ .",
"In general, states that can be represented as a tensor product $|\\psi \\rangle = |\\varphi _0\\rangle \\otimes |\\varphi _1\\rangle $ are not entangled.",
"If a two-qubit state cannot be factorized as a tensor product of single-qubit states, the two qubits are entangled [16].",
"The first three gates in the two-qubit circuit (Eq.",
"(REF )) transform the initial state $|00\\rangle $ into an entangled state $\\frac{1}{\\sqrt{2}} (|01\\rangle + |10\\rangle )$ , known as a Fock state, to be used for quantum enhanced measurements.",
"The first Hadamard gate and the $X$ gate act independently on the two qubits, preparing the product state $\\frac{1}{\\sqrt{2}}(|0\\rangle _0 + |1\\rangle _0) \\otimes |1\\rangle _1 = \\frac{1}{\\sqrt{2}} (|01\\rangle + |11\\rangle )$ .",
"Next, the CNOT gate produces entanglement by flipping the target bit $q_1$ when the control bit $q_0$ is in the state $|1\\rangle $ .",
"This operation can be represented by the $4 \\times 4$ matrix $\\text{CNOT} = \\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 0 \\\\\\end{bmatrix}.$ Applying this matrix to the state $\\frac{1}{\\sqrt{2}}[0, 1, 0, 1]^T$ produces the Fock state $\\frac{1}{\\sqrt{2}}[0, 1, 1, 0]^T = \\frac{1}{\\sqrt{2}} (|01\\rangle + |10\\rangle )$ .",
"The value of using an entangled state like the Fock state for quantum metrology can be understood visually through the Wigner representation on a spin sphere [12].",
"The Wigner function is a quasi-probability distribution that maps the uncertainty of the quantum state onto the spin sphere.",
"Similar to the Bloch sphere representation, single-qubit gates applied to all qubits rotate the Wigner function without changing the shape of the distribution.",
"This allows for a clean representation of the behavior of each state as it evolves through the interferometer circuit.",
"Figure REF shows the evolution of the two-qubit state after each gate using the Wigner distribution.",
"The single-qubit Bloch spheres from Fig.",
"REF are reproduced in the first row for comparison.",
"The second and third row of Fig.",
"REF compare the evolution of the Wigner distribution when the two qubits start in an untangled state $|00\\rangle $ and when they start in the Fock state $\\frac{1}{\\sqrt{2}}(|01\\rangle + |10\\rangle )$ , respectively.",
"The Wigner function of the unentangled pair is a localized distribution centered about the Bloch vector, with equal uncertainty in both x and y directions.",
"The width of this local distribution (dark blue region) on the unit sphere is $1/\\sqrt{2}$ radians [12], corresponding to the shot noise of two unentangled qubits.",
"For an entangled state like the Fock state, the Wigner distribution is not necessarily symmetric and can have less uncertainty in one direction.",
"The Fock state Wigner function appears as a horizontal ring centered about z=0 in the bottom left sphere in Fig.",
"REF .",
"The thickness of the ring, denoting the uncertainty in the z-direction, is smaller than the width of the unentangled Wigner distribution, at the expense of increased uncertainty in the azimuthal direction.",
"Entanglement also leads to negative Wigner function values at the poles.",
"When properly rotated during the interferometer sequence, this entangled state can exhibit increased sensitivity to the interferometer phase $\\phi $ due to the reduced width and increased spatial structure of Wigner distribution.",
"Figure: Visualization of the quantum state over time in the two-qubit circuit using the spin Wigner distribution W(θ,φ)W(\\theta ,\\phi ).",
"Successive columns depict evolution of the quantum state of the system after each gate in the interferometry sequence is applied.",
"The first row reproduces the single-qubit Bloch spheres from Fig.",
"for comparison.",
"The second row depicts the Wigner distribution for an unentangled initial state |ψ 0 〉=|00〉|\\psi _0\\rangle = |00\\rangle , which is localized about the single-qubit Bloch vector.",
"The third row depicts the Wigner distribution for an entangled pair of qubits initialized in the Fock state |ψ 0 〉=1 2(|01〉+|10〉)|\\psi _0\\rangle = \\frac{1}{\\sqrt{2}}(|01\\rangle + |10\\rangle ), appearing as a horizontal ring.",
"The shape of the Wigner quasi-probability distribution illustrates measurement uncertainty for each quantum state and is rotated under the application of gates in the interferometer sequence in same way as the Bloch vector.To precisely model the behavior of the Fock state in the interferometer requires matrix representations of the three gates used in the interferometer sequence.",
"Since these gates act independently on the two qubits, we can represent each step of the interferometer as the tensor product of two single-qubit gates.",
"The interferometer's beam splitter is represented by a pair of Hadamard gates, $H \\otimes H = \\frac{1}{2}\\begin{bmatrix}1 & 1 & 1 & 1 \\\\1 & -1 & 1 & -1 \\\\1 & 1 & -1 & -1 \\\\1 & -1 & -1 & 1 \\\\\\end{bmatrix}.$ The application of this Hadamard gate transforms the Fock state $\\frac{1}{\\sqrt{2}}(|01\\rangle + |10\\rangle )$ into a cat state $\\frac{1}{\\sqrt{2}}(|00\\rangle - |11\\rangle )$ .",
"Graphically this corresponds to rotating the Wigner distribution by $\\pi /2$ such that the ring is oriented vertically and is sensitive to azimuthal rotations due the interferometer phase.",
"The phase shift portion on the interferometer is given by the product of individual qubit rotations, $R_Z(\\phi ) \\otimes R_Z(\\phi ) = \\begin{bmatrix}e^{-i\\phi } & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & e^{i\\phi } \\\\\\end{bmatrix}.$ The application of this gate to the cat state results in the state $|\\psi \\rangle =e^{-i\\phi }|00\\rangle - e^{i\\phi } |11\\rangle $ , introducing a phase shift of $2\\phi $ between the basis states.",
"The state oscillates as a function of $\\phi $ between the two cat states $\\frac{1}{\\sqrt{2}}(|00\\rangle - |11\\rangle )$ and $\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle )$ with twice the frequency at which states in the single-qubit interferometer oscillate around the Bloch sphere.",
"The final pair of Hadamard gates represents the second pass through a beam splitter and maps the phase shift from the $R_Z$ gates onto the computation basis.",
"The Hadamard gates map one cat state $\\frac{1}{\\sqrt{2}}(|00\\rangle - |11\\rangle )$ back to the initial Fock state, while leaving the other cat state $\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle )$ unchanged.",
"Thus the final state is an oscillation between the Fock state and the cat state, $|\\psi \\rangle = \\frac{1}{2\\sqrt{2}}\\begin{bmatrix}e^{-i\\phi } - e^{i\\phi } \\\\e^{-i\\phi } + e^{i\\phi } \\\\e^{-i\\phi } + e^{i\\phi } \\\\e^{-i\\phi } - e^{i\\phi }\\end{bmatrix}= \\frac{\\cos {\\phi }}{\\sqrt{2}} \\begin{bmatrix}0 \\\\1 \\\\1 \\\\0\\end{bmatrix} - \\frac{i\\sin {\\phi }}{\\sqrt{2}}\\begin{bmatrix}1 \\\\0 \\\\0 \\\\1\\end{bmatrix}.$ The Fock state is detected when the measurements of the two qubits are different, which has a total probability of $p_{01} + p_{10} = \\cos ^2{\\phi }$ .",
"The cat state corresponds to both qubits being in the same state, which has a total probability of $p_{00} + p_{11} = \\sin ^2{\\phi }$ .",
"To encapsulate the oscillation between the cat state and Fock state in a single number, we define a parity operator such that parity is 1 when both qubits are in the same state (e.g., cat state), and parity is -1 when the qubits are in different states (e.g., Fock state).",
"In matrix form, this parity operator is $\\hat{\\Pi } = \\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 \\\\0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}.$ The expectation value of the operator is $\\begin{aligned}\\langle \\Pi \\rangle &= \\langle \\psi |\\hat{\\Pi }| \\psi \\rangle \\\\&= (p_{00} + p_{11}) - (p_{01} + p_{10}) \\\\&= -\\cos {(2\\phi )},\\end{aligned}$ in close analogy to the polarization measurement for a single-qubit interferometer.",
"The key difference between the expectation value of the two-qubit parity measurement (Eq.",
"(REF )) and the expectation value of the average polarization of independent qubits (Eq.",
"(REF )) is the frequency of oscillation when $\\phi $ is varied continuously.",
"As illustrated by Wigner distributions in Fig.",
"REF , an untangled state returns to its original orientation after a rotation of $\\phi = 2\\pi $ whereas the ring-shaped Wigner distribution for an entangled pair of qubits is indistinguishable from its initial state after a rotation by $\\phi =\\pi $ .",
"The increased frequency of oscillation leads to an increased slope $m=2\\sin (2\\phi )$ of the parity vs $\\phi $ curve as compared to the slope $m=\\sin (\\phi )$ of the polarization curve for the single-qubit interferometer.",
"This improves phase sensitivity.",
"In analogy to Eq.",
"(REF ), the phase sensitivity achievable by parity measurement is $\\sigma _{\\tilde{\\phi }} = \\frac{\\sigma _\\Pi }{2\\sin {(2\\phi )}},$ where $\\sigma _{\\Pi }$ is the standard deviation of the average parity.",
"The standard deviation of the average parity in measurement consisting of $n$ shots of the two-qubit interferometer can be computed using the matrix form of the single-shot parity operator $\\hat{\\Pi }_i$ .",
"Assuming all shots are independent, $\\sigma _{\\Pi }^2 = \\frac{\\operatorname{Var}({\\Pi _i})}{n} = \\frac{\\langle \\hat{\\Pi }_i^2 \\rangle - \\langle \\hat{\\Pi }_i \\rangle ^2}{n} = \\frac{\\sin ^2{(2\\phi )}}{n}.$ For a fair comparison to the single-qubit interferometer, we note that $n$ shots involve measurements of $N=2n$ qubits so the properly normalized phase sensitivity is $\\sigma _{\\tilde{\\phi }} = \\frac{1}{\\sqrt{2N}}.$ This is an overall improvement by a factor of $\\sqrt{2}$ as compared to the shot noise in an unentangled system."
],
[
"Quantifying Benefits of Entanglement",
"To demonstrate the improvement in sensitivity introduced by entanglement, we implemented the two-qubit interferometer circuit on the same quantum computer, ibmq_manila, that was used earlier to implement the single-qubit interferometer.",
"We additionally implemented the circuit on the state vector simulator, ibmq_qasm_simulator, to provide a comparison without the effects of technical noise.",
"Figure REF shows the results of parity measurements for each of the two implementations.",
"Each measurement of parity $\\Pi = p_{00} + p_{11} - p_{01} - p_{10}$ is computed from $n = 1024$ shots of the 2-qubit interferometer circuit.",
"The results show the expected sinusoidal dependence on $\\phi $ with a frequency twice that of the single-qubit interferometer.",
"Figure: Measured parity, Π=p 00 +p 11 -p 01 -p 10 \\Pi = p_{00} + p_{11} - p_{01} - p_{10}, as a function of φ\\phi .",
"Purple circles show data from the two-qubit interferometer taken on the quantum computer ibmq_manila at 13 angles between 0 and 2π2\\pi .",
"Green squares show data from the simulator ibmq_qasm_simulator at the same angles.",
"Error bars depict the standard deviation of 5 trials.",
"The curves are fit to the data with amplitude and offset as free parameters.We fit the Manila and simulator data to the expected functional form $\\Pi = -A\\cos {(2\\phi )}+B$ with amplitude and vertical offset as free parameters.",
"The simulator fit has an offset of -0.001(2) and amplitude of 0.997(3), both of which agree well with the expected values of 0 and 1.",
"The Manila fit's offset of -0.007(6) is also consistent with 0, while the amplitude of 0.852(7) corresponds to a 15% reduction from the ideal value.",
"This deviation in amplitude is larger than the corresponding value for the single-qubit interferometer (6.8%) because readout and gate errors in either qubit can flip overall parity, doubling the contribution from these errors.",
"Additionally, the CNOT gate has an error of up to 1% [19], (varying between daily calibrations), which further contributes to the reduction in amplitude.",
"Curves of parity $\\Pi $ vs phase $\\phi $ can be used to infer phase $\\tilde{\\phi }$ from measurements of the two-qubit, in analogy to the procedure illustrated for single-qubit polarization measurements in Fig.",
"REF and Fig.",
"REF .",
"To analyze the phase sensitivity quantitatively, like we did for the one-qubit interferometer, we collected parity data on both the quantum computer, ibmq_manila, and the simulator.",
"Each measurement of parity came from a trial consisting of 1 to 512 sequential shots (corresponding to $N=$ 2 to 1024 qubits), conducted with an interferometer phase $\\phi = \\pi /4$ .",
"From each parity measurement, we inferred a value for the interferometer phase $\\tilde{\\phi }$ using the slope of the cosine fits from Fig.",
"REF .",
"We computed the standard deviation of these phase measurements $\\sigma _{\\tilde{\\phi }}$ using data from 600 trials.",
"The resulting relationship between phase uncertainty and number of qubits contributing to each measurement is plotted in Fig.",
"REF .",
"The measured phase sensitivities for both ibmq_manila and simulator data are consistently below the standard quantum limit $1/\\sqrt{N}$ for unentangled qubits.",
"Fitting the simulator data verifies that the phase sensitivity is $1/\\sqrt{2N}$ , as expected for $N/2$ pairs of entangled qubits.",
"The fit for measurements made on the quantum computer is $(0.83 \\pm 0.02) N^{-0.495(6)}$ , which lies between the standard quantum limit and the limit for entangled pairs of qubits.",
"Overall, the two-qubit interferometer shows an improvement of 17(2)% as compared to the standard quantum limit and 25(3)% when compared to the single-qubit interferometer implemented on the same hardware.",
"This illustrates the benefits that entanglement can provide for quantum measurement, even in the smallest possible system sizes.",
"Figure: Standard deviation of inferred φ ˜\\tilde{\\phi } from polarization data taken on the quantum computer Manila and the state vector simulator as a function of number of qubits NN along with their best-fit lines.",
"All data points lie below the standard quantum limit (dashed line) 1/N1/\\sqrt{N}."
],
[
"Conclusions",
"We have demonstrated how entanglement between two qubits enables measurement with sensitivity below the standard quantum limit.",
"Students working on these experiments gain better comprehension of quantum mechanics, quantum metrology, and entanglement.",
"Running these experiments on a real quantum computer, students practice quantum programming, analyze experimental data, and assess the effects of both statistical uncertainty and technical noise sources.",
"This prepares students with the tools needed to engage in modern research in quantum science and technology.",
"The project presented here requires minimal prerequisites.",
"The current work began as an independent student project through a STEM outreach program that pairs high school student researchers with graduate student mentors [22].",
"This illustrates how the material can be accessible to motivated students for whom this is their first introduction to quantum mechanics.",
"We believe this project would fit well in the undergraduate curriculum, either as a demonstration for an introductory quantum information course or as a project in a modern physics laboratory class.",
"These experiments can be accomplished with minimal resources as the IBM Quantum Experience allows for free remote access.",
"Additionally, students working independently for a class project could extend the techniques presented here to circuits with more than two qubits [4] in order to demonstrate different forms of entanglement and implement a variety of quantum squeezing techniques.",
"We thank Stanford FAST for facilitating our collaboration and providing support for this research.",
"TN acknowledges support from NSF Q-SEnSE and ESC acknowledges the support of the NSF Graduate Research Fellowship.",
"We additionally thank Victoria Borish, Monika Schleier-Smith and Patrick Allamandola for useful discussions."
],
[
"Author contributions",
"All authors contributed equally to this project.",
"CT conducted experiments with guidance from TN and ESC.",
"All authors participated in the analysis of experimental data and the preparation of this manuscript."
]
] | 2209.08187 |
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